Mathematical modelling of natural phenomena

A. C. Fowler

September 4, 2015

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Waves and inviscid flow 11.0.1 Material derivative . . . . . . . . . . . . . . . . . . . . . . . . 21.0.2 Momentum conservation . . . . . . . . . . . . . . . . . . . . . 31.0.3 Fixed and free surfaces . . . . . . . . . . . . . . . . . . . . . . 6

1.1 Surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Capillary waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2 Interfacial waves . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.3 Rayleigh–Taylor instability . . . . . . . . . . . . . . . . . . . . 15

1.3 Sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Rotating flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.1 Angular velocity . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.2 Poincare waves . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.3 Kelvin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.4 Rossby waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.1 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.2 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . . . . 221.5.3 Asymptotic notation . . . . . . . . . . . . . . . . . . . . . . . 221.5.4 The method of stationary phase . . . . . . . . . . . . . . . . . 22Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Shallow water theory 282.1 The shallow water equations . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.1 A scaling approach . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.1 Characteristics for general systems . . . . . . . . . . . . . . . 332.3 Dam break problem; simple waves . . . . . . . . . . . . . . . . . . . . 34

2.3.1 Similarity solution . . . . . . . . . . . . . . . . . . . . . . . . 362.4 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.1 Generalised simple wave theory . . . . . . . . . . . . . . . . . 372.4.2 The dam break problem, revisited . . . . . . . . . . . . . . . . 402.4.3 The effect of diffusion . . . . . . . . . . . . . . . . . . . . . . . 422.4.4 The issue of boundary conditions . . . . . . . . . . . . . . . . 43

i

2.5 The Korteweg–de Vries equation . . . . . . . . . . . . . . . . . . . . . 442.5.1 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6.1 Classification of partial differential equations . . . . . . . . . . 472.6.2 The Korteweg–de Vries equation . . . . . . . . . . . . . . . . . 48Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Groundwater flow 563.1 The hydrological cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.1 Homogenisation . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.2 Empirical measures . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Groundwater flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 623.3.2 Dupuit approximation . . . . . . . . . . . . . . . . . . . . . . 633.3.3 The seepage face . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Unsaturated soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4.1 The Richards equation . . . . . . . . . . . . . . . . . . . . . . 703.4.2 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . . . . 71

3.5 Immiscible two-phase flows: the Buckley-Leverett equation . . . . . . 723.6 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.6.1 The Taylor–Aris dispersion coefficient . . . . . . . . . . . . . . 763.6.2 Dispersion in practice . . . . . . . . . . . . . . . . . . . . . . . 77

3.7 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.7.1 Taylor dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 78Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Instability 844.0.1 General procedures . . . . . . . . . . . . . . . . . . . . . . . . 844.0.2 Nonlinear diffusion . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1 Saffman–Taylor instability . . . . . . . . . . . . . . . . . . . . . . . . 864.2 Rayleigh–Benard instability . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.1 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3 Double-diffusive convection . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3.1 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.2 Layered convection . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Hydrodynamic stability . . . . . . . . . . . . . . . . . . . . . . . . . . 994.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.5.1 Eigenvalue problems and variational principles . . . . . . . . . 994.5.2 Viscous fingering . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.3 Double-diffusive convection . . . . . . . . . . . . . . . . . . . 994.5.4 Filling box convection . . . . . . . . . . . . . . . . . . . . . . 994.5.5 Layered igneous intrusions . . . . . . . . . . . . . . . . . . . . 1004.5.6 The Orr–Sommerfeld equation . . . . . . . . . . . . . . . . . . 100Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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Appendix 108A.1 Useful vector identities . . . . . . . . . . . . . . . . . . . . . . 108

References 111

iii

Preface

These notes accompany the fourth year course MS4627, Mathematical modelling ofnatural phenomena. My thanks are due to Henry Winstanley and Teresa Kyrke-Smith for their assistance in their development. Revisions may be made during theterm, so you should check on the web for later editions as they become available (seethe date below for the production date of this version). Comments and reports oferrors would be very welcome.

A. C. FowlerSeptember 4, 2015

iv

Chapter 1

Waves and inviscid flow

We begin with the basic kinematics of fluid dynamics, and a terse review of theequations of motion of a fluid.

Continuum mechanics deals with the displacement of matter in three differentforms: solid, liquid and gas. The distinction is easily made, though the edges can beblurred, particularly for some strange materials: silly putty, tomato ketchup, and soon. The displacement of solids is described by the theory of elasticity, and will notconcern us. Subjected to a stress, a solid is distorted (strained) to a new positionof equilibrium: motion is not maintained. In contrast, when a fluid (liquid or gas)is subjected to a stress, it flows, and will continue to do so as long as the stress ispresent: water flows downhill.

The motion of a fluid is described by a vector velocity field u, and this is takento be a function of space x and time t. It is generally taken as an assumption (thecontinuum hypothesis) that u and the other variables describing the motion are C1,meaning they are continuously differentiable functions of their arguments, or smoothfunctions. For the most part, this assumption works well. (And when it does not, wewill find convincing generalisations of our theory.)

The other variables of the motion are the pressure p, and two descriptors of thestate of the fluid, the density ρ and the temperature T . We shall have little further tosay about temperature, and it will mostly be ignored in what follows. Density is themass per unit volume of the fluid, while pressure at a point in a fluid is measured as theforce per unit area exerted on a surface placed in the fluid. A physical interpretationof the pressure thinks of it as the force on the surface resulting from the impact ofthe molecules of the fluid as their momentum is reversed (since Newton’s second lawstates that force is the rate of change of momentum). Pressure is isotropic, that isto say the orientation of the imagined surface in the fluid is irrelevant: the pressureforce acts normally to the surface. The more general case is described by the stress,discussed below.

With these variables to hand, we can start constructing conservation laws, ofwhich the simplest is the conservation of mass equation. We consider a fixed volumeV in the fluid, and compute the rate of change of the mass of fluid within V ; this is

d

dt

∫

V

ρ dV = −∫

∂V

ρu.n dS, (1.1)

1

and follows from the fact that if mass is conserved, the only change of matter insideV can be due to mass flux through its boundary ∂V ; hence the term on the right.Taking the time derivative inside the integral, using the divergence theorem (becausethe variables are C1), and using the fact that V is arbitrary, we are led to the pointform of the equation:

ρt + ∇. (ρu) = 0; (1.2)

here as elsewhere, the subscript t denotes the partial derivative.

1.0.1 Material derivative

A useful idea is that of the material derivative. This is a time derivative of theproperties of a fluid parcel, meaning a fixed blob of fluid. To be more precise, letξ denote the spatial coordinates describing the initial configuration of the fluid att = 0, and let the fluid particle (or element, or parcel) which was at ξ at t = 0 be atposition x at time t > 0. Evidently x = x(ξ, t), and equally evidently

∂x

∂t

∣∣∣∣ξ

= u. (1.3)

The time derivative in (1.3) is called the material or Lagrangian derivative, andwritten as

d

dt=

∂

∂t

∣∣∣∣ξ. (1.4)

It is the time derivative following the fluid.To relate the material derivative to the more useful partial time derivative at the

current time (the Eulerian derivative), we use the chain rule; this leads simply to

d

dt=

∂

∂t+ u .∇, (1.5)

where the time derivative on the right is taken holding x fixed.Now note that the equation of mass conservation can be written in the form

dρ

dt+ ρ∇.u = 0, (1.6)

and in fact this can be directly derived as follows (see also question 1.1). Instead ofbeing fixed, suppose that V (t) is a material volume, always consisting of the samefluid particles. Conservation of mass now takes the form

d

dt

∫

V (t)

ρ dV = 0 =

∫

V (t)

dρ

dtdV + ρ

d

dt(dV ) =

∫

V (t)

(dρ

dt+ ρ∇.u

)dV, (1.7)

since from first principles,d

dt(dV ) = ∇.u dV ; (1.6) then follows from the continuity

of the integrand and the arbitrariness of V (t).

2

Incompressible flow

An incompressible fluid is one which, as the name suggests, maintains its density as

it moves, in other words,dρ

dt= 0; equivalently,

∇.u = 0. (1.8)

Normally, we have ρ = constant, and we shall assume this for an incompressible fluid.

Summation convention

In what follows we stray into the land of vector and tensor calculus, where vectors uhave Cartesian components ui with respect to a normal basis ei, and (second order)tensors σ (a bit like matrices) have components σij. In dealing with these, thesummation convention is widely used (see for example Jeffreys and Jeffreys (1953));this simply states that when a suffix is repeated, summation over the suffix is implied.

Thus, for example, u = uiei, uiui = |u|2, ∇.u =∂ui∂xi

, σii = trσ, and so on.

Two particularly useful tensors are the second order tensor δ, with componentsδij, where

δij =

1, i = j;0, i 6= j.

(1.9)

This is also known as the unit tensor, sometimes written I. The symbol δij is knownas the Kronecker delta. A useful third order tensor is the alternating tensor withcomponents εijk, defined by

εijk =

1, (ijk) = (123), (231), (312),−1, (ijk) = (132), (213), (321),0 otherwise.

(1.10)

It is an exercise to demonstrate the useful identity

εijkεipq = δjpδkq − δjqδkp. (1.11)

1.0.2 Momentum conservation

Two particularly important tensors are the stress tensor σ and the rate-of-straintensor ε, the latter of which is defined by its components:

εij = 12

(∂ui∂xj

+∂uj∂xi

). (1.12)

The stress tensor generalises the idea of pressure. Again, if we draw a surface inthe fluid with normal n, then we define the stress tensor σ by stating that the forceexerted on the surface by the fluid (on the side where the normal points to) is the

3

scalar product σ.n. What this means is that the components of the vector force perunit area are σijnj (summed over j), or equivalently the force per unit area is σijnjei.

1

Conservation of momentum is an integral expression of Newton’s second law, andcan be stated thus:

d

dt

∫

V (t)

ρu dV =

∫

V (t)

ρdu

dtdV =

∫

∂V

σ.n dS +

∫

V (t)

ρf dV, (1.13)

where V (t) is a material volume, and f is a body force per unit mass (usually this isgravity). Using the divergence theorem, assuming smoothness of the variables, andarbitrariness of V (t), we obtain the momentum equation, usually called the Navier–Stokes equation,2

ρdu

dt= ∇.σ + ρf . (1.14)

The Euler equation

Consideration of the form of the stress tensor and its relation to the rate-of-straintensor ε leads to the topic of viscous flow, which we do not pursue. Instead wesuppose that the stresses are purely isotropic, so that the fluid is inviscid,

σij = −p δij, (1.15)

and for an incompressible, inviscid fluid, we have the Euler equations

∇.u = 0,

ρdu

dt= −∇p+ ρf . (1.16)

The vorticity equation

The vorticity of a fluid is defined as

ω = ∇× u; (1.17)

although fluids are (obviously) not rigid bodies, for which an angular velocity iswell defined, the vorticity acts as a kind of measure of local rotational speed. Forexample, the velocity of a rigid body rotation of angular speed Ω about the z axishas u = (−Ωy,Ωx, 0), and thus vorticity 2Ωk, where k is the unit vector in the zdirection; so the vorticity is essentially twice the ‘angular velocity’.

1Tensors are slightly mysterious. A vector is a first order tensor. The important thing about avector is that it has a geometric meaning (location in R3, independent of the choice of coordinatesystem), and the consequence of this is that if we define a change of coordinates e′

i = lijej , whereL = (lij) is an orthogonal matrix, then the corresponding components of a vector u must satisfyu′

i = lijuj , and this defines the tensor property of u. The equivalent property which defines a secondorder tensor is σ′

ij = lipljqσpq.2It is a remarkable fact that the form of this equation can be derived directly from the statistical

mechanics describing the motion of molecules, for which the smoothness assumption does not hold;this derivation is what actually underlies the continuum hypothesis.

4

We can write the Euler equation in the form

ut + ∇(12u2)− u× ω = −1

ρ∇p+ f , (1.18)

and it follows from taking the curl of this and expanding ∇× (u× ω) (see question1.3) that the vorticity satisfies the equation

dω

dt= (ω.∇)u, (1.19)

if the fluid is incompressible. For a compressible fluid, one easily finds the moregeneral

dζ

dt= (ζ.∇)u, (1.20)

providing we assume the equation of state ρ = ρ(p), where

ζ =ω

ρ. (1.21)

Kelvin’s circulation theorem

Let C(t) be a material loop in a fluid, and let S be any (material) surface whoseperimeter is C. The quantity

Γ =

∮

C

u.dr (1.22)

is called the circulation. Computing its derivative, we have

dΓ

dt=

∮

C

[du

dt. dr + u. du

]= −1

ρ

∮

C

∇p. dr = 0. (1.23)

Stokes’s theorem implies ∮

C

u. dr =

∫

S

ω. dS, (1.24)

and thus we can deduce that the vorticity strength is conserved following the flow.3

In particular, if the vorticity is everywhere initially zero, it remains so. A velocityfield for which ω = 0 is called an irrotational flow.

Irrotational, incompressible flow

If ∇× u = 0 everywhere, then there exists a velocity potential φ such that

u = ∇φ. (1.25)

If the fluid is incompressible, then φ satisfies Laplace’s equation

∇2φ = 0. (1.26)

3We omitted the external force field, but the result (1.23) still holds if it is included, providedthe force is derived from a potential, i. e., f = −∇χ, which is for example true for gravity.

5

Another useful relation follows from (1.18), integration of which implies

φt + 12|∇φ|2 +

p

ρ+ gz = f(t), (1.27)

where we take f = −gk to be gravity, directed downwards (k is the unit vector in thez direction). Since φ is only defined up to addition of an arbitrary function of t , wemay choose f(t) as we please; usually it is a convenient constant. (1.27) is one formof the Bernoulli equation.

1.0.3 Fixed and free surfaces

It is not particularly obvious from (1.16) what boundary conditions we might expectto apply, but it becomes clearer when we have incompressible, irrotational flow. Phys-ically, at a solid surface, the normal component of the velocity must vanish, u.n = 0;this is the no-flow-through condition. In addition, we must have force balance; thisis Newton’s third law. In a closed container with solid walls, it is not very obvioushow to do this. Actually, the incompressibility condition in (1.16) allows us to definea vector potential as

u = ∇×ψ, (1.28)

and then taking the curl of (1.16)2 leads to a single equation for ψ:

∇×[d(∇×ψ)

dt− f

]= 0. (1.29)

The point is that if we suppose we can solve this, then the second equation in (1.16)only allows us to specify p at a single point, and the condition of equal pressuresat a boundary actually acts as a boundary condition for the stress in the containingvessel. And actually, since p only appears in (1.16) through its gradient, its value atone point has no effect on the solution.

Effectively, then, the only physically relevant boundary condition is the no flowthrough condition. For an incompressible, irrotational flow, i. e., a potential flow, this

is∂φ

∂n≡ n.∇φ = 0, and we see that this is sufficient to solve Laplace’s equation, up

to addition of an arbitrary function of t, which does not matter since it is only ∇φwhich determines u.

A free surface is one which is able to deform, and typically occurs when theboundary represents the interface between two fluids.The common example whichwill also concern us is an interface between water and air: the surface of a river,lake or ocean, although we could equally be concerned with an oil/water interface,for example. In this case the location of the free boundary is itself a variable, and afurther boundary condition is necessary. This is the kinematic condition, which statesthat a free interface F (r, t) = 0 between two fluids is a material surface; the particlesof each fluid adjoining the interface remain there:

dF

dt= 0. (1.30)

6

Actually, this is a consequence of the continuum hypothesis, which implies that x(ξ, t)is a smooth transformation of the initial configuration ξ. In particular, a materialsurface is mapped into a material surface, and parts of it cannot pop up disconnectedlyin the interior.

The more common way of writing (1.30) is to define the free surface as z =η(x, y, t), so that F = z − η; taking the material derivative leads to the condition

w = ηt + uηx + vηy, (1.31)

where u = (u, v, w). Note that if η is independent of time, this reduces to the no flowthrough condition, since n ∝ (−ηx, ηy, 1).

The force balance condition, or more properly the traction balance condition,takes the form

[σ.n]+− = 0, (1.32)

where the notation represents the jump of the surface stress from one side (−) to theother (+) of the interface. Assuming both fluids are inviscid, this simply implies thatp is continuous. Unlike the case of a fixed boundary, it is important to specify thiscondition all the way along the interface, although as in the earlier discussion we canonly fix p at one point. For the particular case of an atmospheric water/air surface,we may take the atmospheric pressure as a constant, pa, and the Bernoulli conditiontakes the form

φt + 12|∇φ|2 + gη = 0, (1.33)

where we imagine the reference pressure to be at great distance, where there is nomotion and φ = η = 0.

1.1 Surface waves

We are now in a position to study waves on the surface of a liquid. We assume thefluid is incompressible and also irrotational, so that all the above results apply. Weconsider the flow of a liquid with a free surface at z = η and a flat bottom at z = −h,but generally unbounded in the horizontal. The system of equations which describethe flow is then

∇2φ = 0,

φt + 12|∇φ|2 + gη = 0

φz = ηt + φxηx + φyηy

at z = η,

φz = 0 at z = −h. (1.34)

This is a complicated, nonlinear model for the evolution of the two quantities φand η, and it is not easy to solve! Nor does it bear much resemblance to the originalEuler equations, and it is not even clear if it is any easier to analyse; luckily in factit is.

7

The starting point is to describe a basic state. This is an exact solution which isusually a steady state, often trivial to describe. In the present case, the obvious basicstate is the rest state when there is no motion and the top surface is flat; we definethis basic state to be φ = η = 0.

We now ask what happens if we perturb this state slightly, that is to say we nowassume φ and η are small.4 In this case, the quadratic terms in the model will beeven smaller still, and as an approximation we can neglect them. This is the processof linearisation, which leads to an approximate linear model, which is much easier toanalyse.

The linearised version of (1.34) is

∇2φ = 0,

φt + gη = 0

φz = ηt

at z = 0,

φz = 0 at z = −h. (1.35)

Note that because η is small, the values of the variables at z = η can be expandedin Taylor series about z = 0, which is why the surface boundary conditions are takenon z = 0.

This is now a linear set of differential equations with constant coefficients. As aconsequence there are separable solutions, and each of these is an exponential. It isconventional to write

φ = ei(k1x+k2y−ωt)g(z), (1.36)

and these are called normal mode solutions. Conventionally, we take k1 and k2 to bereal, and they are called (longitudinal and transverse) wave numbers. More succinctly,the vector k = (k1, k2) (no longer the vertical unit vector) is called the wave vector,

and the wavelength is then2π

k, where k = |k|.

The reason we take k to be real is that the general solution of the problem may befound by seeking a Fourier transform, and then k is the Fourier transform variable,and thus real; (1.36) then gives the solution at one particular wave vector.

The quantity ω is called the frequency, and is not necessarily real (although itwill be in this problem). Complex values of ω are associated with basic states whichmay be asymptotically stable or unstable, while real values of ω correspond to wavemotions which neither grow nor dissipate.

The function g is also composed of exponentials, but is solved separately becauseit must satisfy boundary conditions. Substituting (1.36) into Laplace’s equation, wetake

g = cosh k(z + h) (1.37)

4Because the quantities are dimensional, ‘small’ must always mean by comparison with somestandard dimensional measure. Later, when we make models dimensionless, ‘small’ will come tomean numerically small, i. e., much less than one.

8

in order to satisfy the basal boundary condition; the amplitude is arbitrary, since theproblem is linear. We can combine the surface conditions to give φtt + gφz = 0 atz = 0, and it then follows from this that

ω2 = gk tanh kh, (1.38)

which determines the frequency in terms of the wavenumber. This relation is calledthe dispersion relation.

Because we can write (1.36) in the form ei(k.r−ωt)g(z), it is clear that these solutionsrepresent sinusoidal travelling waves with wave speed

c =ω

k, c2 =

g

ktanh kh. (1.39)

The waves are called dispersive because c depends on k: waves of different wavelengthtravel at different speeds. Long waves (small k) travel faster than short waves.

The deep water limit of (1.39) is c =

√g

k, while the shallow water limit is c =

√gh,

and in the latter limit the waves are no longer dispersive.

Group velocity

We omitted to mention the form of the water surface, although this is actually whatone observes. Evidently η is also ∝ ei(k.r−ωt), and indeed the most general solutionfor the surface takes the form of a Fourier integral,

η =

∫

R2

A(k)eik.r−ω(k)t dk (1.40)

(dk = dk1dk2 denotes the area element in wave vector space), or in one spatialdimension,

η =

∫ ∞

−∞A(k)eikx−ω(k)t dk. (1.41)

Suppose we drop a stone in a pond; or there is a storm at sea. In either case,an initial local disturbance consisting of a superposition of waves of all wavelengthsspreads out, with the different length waves travelling at different speeds. Naıvely,we might suppose that the disturbance would split up into packets of waves travellingat their phase speed given by (1.39), but this is a little simplistic, as it assumes eachwave pattern forms an infinite train; in reality, waves occur in packets, which onemay conceive of as lying under an overarching envelope; the envelope will generallytravel at a different speed than the individual phase speed of the waves: this speedis called the group velocity.

There are a number of ways of motivating the definition of the group velocity.Suppose an observer travels at speed G so that we write x = Gt; then (1.41) is

η =

∫ ∞

−∞A(k)eitkG−ω(k) dk, (1.42)

9

and we ask what the behaviour of the integral is at large time. The Riemann–Lebesgue

lemma says that η → 0, and integration by parts shows that η = O

(1

t

)providing

kG− ω(k) is monotonic, i. e., ω′(k) 6= G. However, this is only the case if G >√gh,

using (1.38). If G <√gh, then there is a k0 such that ω′(k0) = G, and near this

wavenumber, the exponent in the integral varies more slowly than elsewhere, andtherefore the dominant contribution to the integral comes from its vicinity; this is themethod of stationary phase, and it says that, approximately,

η ≈ A(k0)eitk0G−ω(k0)∫ ∞

−∞e−itω

′′0 (k−k0)2 dk ∼ O

(1√t

)(1.43)

(see question 1.4), which shows that the wave packet of waves with wavenumber k0

travels at a speed G = ω′(k0); this is the group velocity. Since ω is a concave functionof k, the group velocity is less than the phase velocity.

1.2 Capillary waves

1.2.1 Surface tension

Capillary waves are a form of interfacial wave associated with the presence of surfacetension. Surface tension is a property of interfaces, whereby they have an apparentstrength. This is most simply manifested by the ability of small objects which arethemselves heavier than water to float on the interface. The experiment is relativelyeasily done using a paper clip, and certain insects (water striders) have the ability tostay on the surface of a pond.

The simplest way to think about surface tension is mechanically. The interfacebetween two fluids has an associated tension, such that if one draws a line in theinterface of length l, then there is a force of magnitude γl which acts along this line:γ is the surface tension, and is a force per unit length. The presence of a surfacetension causes an imbalance in the normal stress across the interface, as is indicatedin figure 1.1, which also provides a means of calculating it. Taking ds as a short

dθ

dθ dθ

R

/2 /2

γ γ

ds

p

p

+

_

Figure 1.1: The simple mechanical interpretation of surface tension.

10

line segment in an interface subtending an angle dθ at its centre of curvature, a forcebalance normal to the interface leads to the condition

p+ − p− =γ

R, (1.44)

where

R =ds

dθ(1.45)

is the radius of curvature, and its inverse 1/R is the curvature.For a two-dimensional surface, the curvature is described by two principal radii of

curvature R1 and R2, the mean curvature is defined by

κ = 12

(1

R1

+1

R2

), (1.46)

and the pressure jump condition is

p+ − p− = 2γκ = γ

(1

R1

+1

R2

), (1.47)

although this is not much use to us unless we have a way of calculating the curvatureof a surface. This leads us off into the subject of differential geometry, and we do notwant to go there. A better way lies along the following path.

The sceptical reader will in any case wonder what this surface tension actuallyis. It manifests itself as a force, but along a line? And what is its physical origin?The answer to this question veers towards the philosophical. We think we understandforce, after all it pops up in Newton’s second law, but how do we measure it? Pressure,for example, we conceive of as being due to the collision of molecules with a surface,and the measure of the force they exert is due to the momentum exchange at thesurface. We pull on a rope, exerting a force, but the measure of the force is in theextension of the rope via Hooke’s law. Force is apparently something we measure viaits effect on momentum exchange, or on mechanical displacement; we can actuallydefine force through these laws.

The more basic quantity is energy, which has a direct interpretation, whether askinetic energy or internal energy (the vibration of molecules). And in fact Newton’ssecond law for a particle is equivalent to the statement that the rate of change ofenergy is equal to the rate of doing work, and this might be taken as the fundamentallaw.

The meaning of surface tension actually arises through the property of an interface,which has a surface energy γ with units of energy per unit area. The interfacialcondition then arises through the (thermodynamic) statement that in equilibriumthe energy of the system is minimised.

To be specific, consider the situation in figure 1.2, where two fluids at pressuresp− and p+ are separated by an interface with area A. Consider a displacement ofthe interface causing a change of volume dV as shown. Evidently the work done onthe upper fluid is p+ dV , which is thus its change of energy, and correspondingly the

11

dV

p

p_

+

V

A

Figure 1.2: The energetic basis of surface tension.

change for the lower fluid is −p− dV . If the change of interfacial surface area is dA,then the total change of energy5 is

dF = (p+ − p−) dV + γ dA, (1.48)

and at equilibrium this must be zero (since F is minimised). The equilibrium inter-facial boundary condition is therefore

p+ − p− = −γ ∂A∂V

, (1.49)

which, it turns out, is equivalent to (1.47).

Computation of∂A

∂Vcan be done as follows. We consider a displacement of the

interface as shown in figure 1.3. An element of surface A is displaced to A + dA,and we can form a connecting volume dV such that the normal n to the interface isalways parallel to the connecting surface between the end faces A and A + dA. Weneed to distinguish between the normal n to the surface of the connecting volumeand the normal to the interfacial surface. Evidently we have n = n at the end faces,but n.n = 0 on the connecting cylindrical surface.

Applying the divergence theorem, we see that the change in area is

dA =

∫

∂(dV )

n.n dS =

∫

dV

∇.n dV, (1.50)

and thus∂A

∂V= ∇.n. (1.51)

For example, if the interface is represented as z = η(x, y, t), then

∇.n = −∇.

[ ∇η

(1 + |∇η|2)1/2

], (1.52)

5This energy is the Helmholtz free energy.

12

n

n^

A + dA

dV

A

Figure 1.3: Calculation of∂A

∂V.

and for small interfacial displacement, this may be linearised to obtain

2κ = −∂A∂V

= −∇.n = ∇.

[ ∇η

(1 + |∇η|2)1/2

]≈ ∇2η. (1.53)

1.2.2 Interfacial waves

Now let us consider the situation where a fluid of density ρ1 lies on top of a fluid ofdensity ρ2, and there is an interfacial tension of magnitude γ. Each fluid is assumedirrotational and incompressible, and we take the lower fluid to be bounded by a solidwall at z = −h, while the upper fluid extends to z = ∞; we have in mind waterbeneath the atmosphere, but the description of the flow is not limited to these twofluids; for example, we might be interested in the motion of waves at an oil/waterinterface.

Each fluid satisfies∇2φi = 0, (1.54)

where the suffix i = 1, 2 labels the upper and lower fluid respectively. At the interfacez = η between the fluids, we have

φi,z = ηt + φi,xηx + φi,yηy,

φi,t + 12|∇φi|2 +

piρi

+ gη =paρi,

p1 − p2 = −γ∇.n, (1.55)

where the normal n points from fluid 2 to fluid 1; pa is a reference value of the pressurein the undisturbed state, where η = 0 = φi.

For small disturbances η, we linearise the boundary conditions at the interface,whence

φ1,z = φ2,z = ηt,

γ∇2η + ρ1(gη + φ1,t) = ρ2(gη + φ2,t) (1.56)

13

at z = 0. Normal mode solutions satisfying also the no flow through condition atz = −h are

η = eik.r−iωt,

φ1 = A1e−kzeik.r−iωt,

φ2 = A2 coshk(z + h)eik.r−iωt, (1.57)

and application of the interfacial boundary conditions (1.56) leads to the dispersionrelation

ω2 =[g(ρ2 − ρ1) + γk2]k tanh kh

ρ2 + ρ1 tanh kh. (1.58)

This generalises our previous result (1.38) for gravity waves, to which it reduces inthe limit as ρ1, γ → 0.

The Bond number

Fluid flows are characterised by dimensionless numbers, of which the most well knownis the Reynolds number. The inclusion of surface tension is measured by two otherdimensionless numbers, known as the capillary number Ca and the Bond number Bo.The capillary number is defined by

Ca =µU

γ, (1.59)

where µ is the fluid viscosity and U is a typical velocity. Of more interest here is theBond number, defined by

Bo =∆ρgl2

γ, (1.60)

where ∆ρ is the density difference between the fluids, and l is a relevant length scale.If the Bond number is small, then surface tension dominates gravity, and the resultingwaves are called capillary waves.

For water and air, γ = 73 mN m−1, and thus

(γ

∆ρg

)1/2

≈ 2.73 mm. For waves

of length l =2π

k, we then have g∆ρ γk2 if l 17.2 mm, and for these small

wavelength capillary waves, the dispersion relation is approximated by

ω2 ≈ γk3 tanh kh

ρ2

. (1.61)

In deep water (h l

2π), the wave speed is

c2 ≈ γk

ρ2

, (1.62)

and short waves travel fastest, in distinction to gravity waves.

14

1.2.3 Rayleigh–Taylor instability

Another facet of behaviour is illustrated by (1.58) in the case where the upper fluidis heavier than the lower one, i. e., ρ1 > ρ2. In this case the density difference is∆ρ = ρ1 − ρ2, and we have

ω2 ∝ γk2 − g∆ρ. (1.63)

In the absence of surface tension ω is imaginary, and the growth rate of perturbationsσ = −iω can take positive values, indicating that disturbances grow exponentially intime. This behaviour represents linear instability, which will be discussed further inchapter 4, and it represents the obvious fact that when you turn a bucket of waterupside down, the water flows out.

More interestingly, we see that this top-heavy instability can be overcome bysurface tension, specifically if the wavelength l satisfies

l < 2π

(γ

∆ρg

)1/2

. (1.64)

This can be realised by, for example, sucking water into a straw, putting a finger overthe top and holding it vertically; the water will not drain out. The same mechanismexplains the ability of small heavy objects to float on water (for example the insectsknown as water striders, mentioned earlier).

1.3 Sound waves

Noise is caused by the propagation of sound waves in the atmosphere. These waves arecalled by local compression of air, and the propagation of the resulting disturbances.In order to understand sound waves, we must therefore consider the compressibleform of the Euler equations, which is

ρt + ∇. (ρu) = 0,

ρdu

dt= −∇p, (1.65)

in which we ignore the body force term for convenience.

Equation of state

We have not previously flagged it, but evidently the Euler equations provide twoequations for the three unknowns ρ, u and p; another equation is necessary to com-plete the system, and this is known as the equation of state. It is a consequence ofthermodynamics (which itself ultimately relies on statistical mechanics), and providesan algebraic relation between the three quantities ρ, p and temperature T . Actually,we have surreptitiously used the equation of state in discussing incompressible flows,by assuming ρ is constant.

15

The most commonly used form of the equation of state for gases is the perfect gaslaw

ρ =Mp

RT, (1.66)

where M is molecular weight, and R is the universal gas constant. A commonly seenmodification of this is the Van der Waals equation

ρ =M(p+ αρ2)(1− βρ)

RT, (1.67)

which allows a description of phase change between gas and liquid (see question 1.9).Two particular versions of (1.66) are the isothermal equation of state in which

p ∝ ρ, and the adiabatic equation of state in which p ∝ ργ (here γ > 1 is the ratio ofspecific heats, and is not the surface tension); we will generally assume that p = p(ρ)in what follows.

In this case, we can rewrite the Euler equations in the form

ρt + u .∇ρ+ ρ∇.u = 0,

ρut + ρ(u .∇)u + c2∇ρ = 0, (1.68)

where

c =

(dp

dρ

)1/2

(1.69)

is the speed of sound, as we shall see.These equations have a rest state ρ = ρ0, u = 0, in which there is no motion.

For small disturbances to this, we write ρ = ρ0 + ρ1, and then linearise the equations(1.68); this gives

ρ1,t + ρ0∇.u = 0,

ρut + c2∇ρ1 = 0, (1.70)

whence we obtainρ1,tt = c2∇2ρ1, (1.71)

which is the wave equation.Exact solutions are easy to come by, and we will give two examples. The first

describes the production of pitch in musical instruments, and we will consider thecase of a clarinet. Sound is produced at the mouthpiece (x = 0) by blowing past areed which is thus caused to vibrate. The vibration sets up standing waves in thetube of the instrument, which we think of as a one-dimensional cylinder 0 < x < l.Boundary conditions suitable to describe the flow are that u = ρ1,x = 0 at the closed(mouthpiece) end, but at the free (bell) end we prescribe fixed atmospheric pressure,and thus ρ1 = ux = 0. Normal mode solutions are thus

ρ1 = eiωt cosωx

c, (1.72)

16

providing

ω =(2n+ 1)πc

2l, (1.73)

where n is an integer. The fundamental frequency, corresponding to the pitch of thenote, is obtained from taking n = 0, and if written in terms of the Hertz frequencyf = ω/2π, this is

f =c

4l. (1.74)

For a clarinet at room temperature, we have l = 65 cm, c = 340 m s−1, whencef = 131 Hz, corresponding to C below middle C. Actually, for a B[ clarinet, thelowest sounded note is D below middle C.

As a second example, we consider a vibrating source in three dimensions. We cansuppose

ρ1 = Aeiωt at r = a, 6 (1.75)

but what should the other boundary condition be? An obvious answer, at least forthe initial value problem, is to have ρ1 = 0 at ∞, but for the particular solutionsatisfying (1.75), this will not be appropriate.

A clue is afforded by the form of the general solution of the wave equation inconditions of spherical symmetry, which is

ρ1 =1

r[f(r − ct) + g(r + ct)]. (1.76)

Clearly the term in g represents an incoming wave, and Sommerfeld’s radiation con-dition states that this must be suppressed: there is no wave source at infinity. Math-ematically (and in more generality), we prescribe

r(ρ1,t + cρ1,r)→ 0 as r →∞, (1.77)

which forces the solution to represent an outgoing wave. As a consequence the solutionfor our spherically symmetric source is just

ρ1 =Aa

rexp

[iω(t− r

c

)]. (1.78)

1.4 Rotating flows

For large scale motions in the atmosphere or oceans of the Earth, it is sensible towrite the equations of motion with respect to a frame of reference which rotates withthe Earth, since the air or water velocities which one measures are relative to such arotating frame.

6Obviously, ρ1 must be real, but because the problem is linear, it is conventional to allow com-plex exponentials, with the understanding that we are always considering the physical solution tocorrespond to the real part, for example.

17

1.4.1 Angular velocity

In order to do this, we need to understand what is meant by angular velocity. Theangular velocity of a body relative to a fixed point O in the body (which here we willtake as the centre of the Earth) is a vector Ω such that the axis of rotation is parallelto Ω and the angular speed is of magnitude Ω. The velocity of a point with positionvector r (with respect to O) is then Ω× r.

Now suppose ei is a set of Cartesian coordinates fixed in the body. It follows thatei = Ω× ei, and this can be used to define the angular velocity in terms of the rateof change of the frame of reference:

Ω = 12

∑

i

ei × ei. (1.79)

The time derivative of any vector a = aiei (summed) is then

da

dt

∣∣∣∣fix

=da

dt

∣∣∣∣rot

+ Ω× a, (1.80)

and from this it follows (differentiating r twice) that

du

dt

∣∣∣∣fix

=du

dt

∣∣∣∣rot

+ 2Ω× u + Ω× (Ω× r); (1.81)

hence the Euler equations become, in a rotating frame,

ρt + ∇. (ρu) = 0,

ρ

[du

dt+ 2Ω× u

]= −∇p+ ρ∇[1

2|Ω× r|2] + ρf , (1.82)

where we use the fact that

Ω× (Ω× r) = −12∇|Ω× r|2. (1.83)

(See equation (A.13) in the appendix.)

1.4.2 Poincare waves

We now focus on the particular situation on Earth. Both the oceans and the at-mosphere are shallow flows (the horizontal length scales are thousands of kilometres,while the mean depth of the ocean is 4 km and the depth of the troposphere is 10km). As a consequence (further detail of which will appear in chapter 2), the flow isessentially horizontal, and the pressure is essentially hydrostatic. To be specific, wewill assume a layer of variable thickness h, horizontal velocity u = (u, v, 0), constantdensity ρ, and hydrostatic reduced pressure

p− 12ρ|Ω× r|2 ≈ ρg(h− z), (1.84)

18

Ω

θ

λ

x

y z

k

Figure 1.4: Cartesian geometry on the surface of the Earth.

where also we take, as shown in figure 1.4, local Cartesian axes at latitude λ with thez axis pointing vertically upward, x to the east, and y to the north. It follows that

Ω× u = (−Ωv sinλ,Ωu sinλ,−Ωu cosλ), (1.85)

of which the z component is irrelevant since Ωu g.It follows that the equations of motion take the form

ht + (hu)x + (hv)y = 0,

du

dt− fv + ghx = 0,

dv

dt+ fu+ ghy = 0, (1.86)

wheref = 2Ω sinλ. (1.87)

The first of (1.86) is the conservation of mass equation, which can for example beobtained from (1.82)1 by integration vertically from the surface z = 0 to z = h (seequestion 1.10).

The simplest state is the rest state h = h0, u = v = 0, and if we write h = h0 +Hand linearise the system, we have

Ht + h0(ux + vy) = 0,

ut − fv + gHx = 0,

vt + fu+ gHy = 0, (1.88)

with solutions ∝ exp[ik1x+ ik2y − iωt] providing ω = 0 or

ω2 = f 2 + gh0k2. (1.89)

19

This is the dispersion relation for Poincare waves, which represent gravity waves ina rotating frame. The importance of rotation is measured by the Rossby radius ofdeformation, which is defined by

Ld =

√gh0

f. (1.90)

For wavelengths much larger than Ld, i. e., kLd 2π, we have ω ≈ f , correspondingto the pure rotational effect of the Coriolis force. For short wavelengths Ld,

the wave speed isω

k=√gh0, which represents the shallow limit of the ordinary

gravity wave speed. Thus the effects of rotation become significant for length scalescomparable to the radius of deformation.

1.4.3 Kelvin waves

For flow in a channel of width 2L pointing eastward, the normal mode Poincare waves

require k2 =nπ

L, where n is an integer, in order that v = 0 at y = ±L. However, there

is another solution of (1.88) in which v = 0 identically. Substitution of this into (1.88)then implies Htt = gh0Hxx, so that solutions can be taken ∝ exp[my + ik(x ± ct)],providing

m = ± f√gh0

=1

Ld. (1.91)

Conventionally we take solutions which decay away from the channel boundaries, thusfor example

u ∝ exp

(− y

Ld

)eik(x−ct), (1.92)

and the waves propagate eastwards near a southern boundary, or westward near anorthern boundary. These waves are known as Kelvin waves or edge waves.

1.4.4 Rossby waves

What of the other zero of the determinant formed from the normal mode solutions of(1.88), i. e., ω = 0? This solution is associated with a degeneracy of the steady state,and also with solutions of (1.88) which vary slowly in time. To accommodate this,it is necessary in considering (1.86) to retain the nonlinear terms in the acceleration,as we now show. We continue to use the dimensional model, but the issue becomesclearer if the model is first non-dimensionalised, as is done in question 1.11. On thebasis that the length scale of the motion is much less than Ld, we can neglect theterm ht in (1.86)1, and the variation in h is much less than h. We thus put h = h0 inthe mass conservation equation, which we take in the incompressible form

ux + vy = 0. (1.93)

At leading order, we have the geostrophic balance

u = − gfhy, v =

g

fhx, (1.94)

20

which also automatically satisfies mass conservation (if f is constant, though later wewill allow it to vary); again, these arguments can only really be justified by analysingthe problem non-dimensionally.

(1.93) allows us to define a stream function via

u = ψy, v = −ψx, (1.95)

and then the vorticityζ = −∇2ψ. (1.96)

Eliminating h by cross-differentiation in (1.86) yields the exact equation (using (1.93))

dζ

dt+ βv = 0, (1.97)

where

β = fy =2Ω cosλ

R, (1.98)

where R is the radius of the Earth. The beta-plane approximation assumes β isconstant in (1.97).

The quantity ζ is called the potential vorticity, while ζ + f is the absolute vortic-ity, and is conserved following the flow. (1.97) is one form of the quasi-geostrophicpotential vorticity equation, and can be written as a stand alone equation for thestream function ψ:

[∂

∂t+ ψy

∂

∂x− ψx

∂

∂y

]∇2ψ + βψx = 0. (1.99)

As a reference state, we assume a zonal flow ψ = uy (this includes the rest stateas a particular case, but a zonal flow is more realistic atmospherically). We putψ = uy + Ψ and linearise, which leads to the equation

(∂

∂t+ u

∂

∂x

)∇2Ψ + βΨx = 0; (1.100)

solutions are Ψ = exp[ik1x+ ik2y − iωt], and the resulting dispersion relation is

ω

k1

= u− β

k2. (1.101)

The resulting waves drift westward relative to the mean flow.They are called Rossby waves, and they are related to the weather systems which

one sees on television weather maps. These planetary scale waves are caused by aninstability of the basic zonal flow; this is known as baroclinic instability, and arisesthrough the study of a generalisation of (1.99) in which allowance is made for a shearprofile in the zonal flow, and consideration of the energy conservation equation.

21

1.5 Notes and references

1.5.1 Fluid mechanics

The doyen of books on fluid mechanics is probably that of Batchelor (1967); it isfull of wisdom, but is not the most appealing book from which to learn the subject.At a more accessible level is the book by Acheson, which takes as its theme theairfoil, thus leading one through inviscid flow, vortices and circulation to boundarylayer theory and separation. The books by Ockendon and Ockendon (1995, 2004), onviscous and inviscid flow respectively, are succinct and very informative. The latteris a reinvention of the book by Ockendon and Tayler (1983), which itself remains aperfectly valuable discussion of the subject.

Many of the books which discuss wave motion lie in the geophysical arena, andclassics of the subject are the books by Pedlosky (1987) and Gill (1982). A morerecent example is the book by Vallis (2006), which particularly has a lucid discussionof rotating fluid flows.

1.5.2 Non-dimensionalisation

Non-dimensionalisation is one of those tricks of the applied mathematician, appar-ently simple, but actually a technique which can be subtle and which carries a greatdeal of power. It is discussed in many practical books, the now out-dated classicbeing that of Lin and Segel (1974), with more recent treatments being those of Tayler(1986), Fowler (1997) and Howison (2005).

1.5.3 Asymptotic notation

Applied mathematicians use the notation a b, a b, a = O(b), a = o(b), a ∼ b,and so on, almost as a natural part of mathematical language. Behind the usage liesa formal technique, described in many books on asymptotic methods, for examplethose by Bender and Orszag (1979), Murray (1984) or Hinch (1991), and for a rapidsummary see Fowler (1997).

1.5.4 The method of stationary phase

The method of stationary phase is one of a number of asymptotic methods whosediscussion can be found in books on asymptotics such as those cited above. While itspresentation here (see also Ockendon and Tayler (1983) lies at the intuitive level, inreality it is a particular case of the the method of steepest descents, which is a generalmethod used to compute the asymptotic behaviour of Laplace integrals, which are

contour integrals of the form I =

∫

C

g(t)ezt dt, where C is a contour in the complex

t plane, and usually the limiting behaviour of I is sought in the limit z →∞, whereusually the direction of approach is important.

22

Exercises

1.1 Let V (t) be an arbitrary material volume of fluid, and dV a volume element.Explain why

d

dt

∫

V (t)

ρ dV =

∫

V (t)

d

dt(ρ dV ),

whered

dtis the material derivative, and show from first principles that

d

dt(dV ) = ∇.u dV .

Hence deduce the conservation of mass equation in the form

dρ

dt+ ρ∇.u = 0,

explaining in what way the assumptions of smooth ρ and u, and the arbitrarychoice of V (t), are important.

1.2 Derive the form of the Navier–Stokes equation,

ρdu

dt= ∇.σ + ρf ,

by consideration of Newton’s second law for a fluid in a volume V which is fixedin space. (You may assume the form of the conservation of mass equation. It isprobably advisable to treat momentum conservation component by component.)

1.3 Show that the following definitions are correct:

∇φ = ei∂φ

∂xi, ∇.u =

∂ui∂xi

, ∇× u = εijkei∂uk∂xj

.

Show that for an inviscid fluid, ∇.σ = −∇p, and deduce the form of the Eulermomentum equation.

Using the summation convention, show that

(u .∇)u = ∇(12u2)− u× ω,

and that

∇× (u× ω) = u∇.ω − ω∇.u + (ω.∇)u− (u .∇)ω,

where ω = ∇× u is the vorticity, and hence deduce the vorticity equation foran incompressible fluid,

dω

dt= (ω.∇)u.

23

1.4 The method of stationary phase implies that relative to an observer moving ata speed G = ω′(k0), the general solution for the fluid surface elevation η is

η ≈ A(k0)eitk0G−ω(k0)∫ ∞

−∞e−itω

′′0 (k−k0)2 dk.

Use methods of contour integration to calculate

∫ ∞

−∞eiαz

2

dz for real values of

α, and hence show that η ∼ O

(1√t

). (Note: treat carefully the cases α > 0

and α < 0.)

1.5 A stream of depth h flows at constant speed U in the x direction. Write down theequations and boundary conditions for the velocity potential φ, paying particu-lar attention to the choice of the value of the integration constant in Bernoulli’sequation.

Write down the solution of this problem which corresponds to the undisturbedflow of constant speed and depth.

Now consider a small disturbance in the x direction which causes the surface tobe perturbed from z = 0 to z = η(x, t). By linearising about the uniform state,write down a linear set of differential equations and boundary conditions for theperturbed velocity potential Φ and η, and by solving this, derive the dispersionrelation relating wave speed c to wave number k in the form

c = U ±√g

ktanh kh.

Interpret this result physically.

1.6 Hydrostatic pressure in a liquid is given by p = pa−ρgz, where pa is atmosphericpressure, and z = 0 is the liquid surface far from a wall. If the liquid surface isgiven by z = η(x, y), show that η satisfies

∇.

[ ∇η

[1 + |∇η|2]1/2

]= η,

where lengths are made dimensionless with d = (γ/ρg)1/2. If n′ is a unit normalto a contact line at a wall, parallel to the wall and pointing away from the fluid,and n is a unit normal to the contact line which is normal to the liquid airinterface and directed into the air, show that n.n′ = sin θ, where θ is the contactangle (the angle between the liquid surface and the wall, which is considered tobe prescribed). Hence show that at a vertical wall,

(1 + |∇η|2)1/2 = cosec θ,

and thus ∂η/∂n = cot θ, where n is a unit normal to the wall, pointing awayfrom the liquid. Solve this problem for the case of a plane wall at x = 0 with

24

η → 0 as x → ∞, and hence deduce that the capillary rise of the fluid at thewall is

2(γ/ρg)1/2 sin

[π

4− θ

2

].

1.7 A fluid of density ρ1 flows in the x direction at speed U over a fluid of densityρ2 > ρ1 at rest, each fluid being of infinite depth. Write down the equationsgoverning the system, and write down the steady state solution correspondingto the undisturbed state.

By seeking small perturbations in the x direction about this basic state, derivea linearised model for the flow, and hence show that if the perturbed surface(about z = 0) is denoted as η(x, t), there are normal mode solutions in which

η = exp[ikx+ σt],

and derive a quadratic equation for σ in terms of the wave number k.

By solving this, show that if U < Uc, where

Uc =

[(ρ2

2 − ρ21)g

ρ1ρ2k

]1/2

,

σ will be purely imaginary; what does this mean physically?

Show conversely that if U > Uc, then one of the roots has Reσ > 0; what doesthis imply?

Repeat the calculation allowing for a non-zero surface tension γ, and show thatinstability will only occur for a finite band of wave numbers if

U > U∗ = (γg∆ρ)1/4

(2(ρ1 + ρ2)

ρ1ρ2

)1/2

,

where ∆ρ = ρ2 − ρ1. What is this value for air flowing over water?

[This is the Kelvin-Helmholtz instability, and is the explanation for why surfacewaves are generated in the ocean.]

1.8 A source of sound moves along the x axis at speed U < c, so that the boundarycondition for the density perturbation ρ is

ρ = Aeiωt at x = Ut.

By changing coordinates to the moving frame ξ = x − Ut or otherwise, andseeking one-dimensional solutions, show that the solution is

ρ = A exp

[iω

±ct− x±c− U

],

and explain why the + sign applies in x > Ut, and the − sign applies in x < Ut.

25

Deduce that the frequency heard by an observer at x = 0 decreases fromωc

c− Uin t < 0 to

ωc

c+ Uin t > 0. [This explains the Doppler effect, whereby the pitch

of an ambulance siren decreases as it passes you.]

What happens if U > c?

1.9 The Van der Waals equation for a gas takes the form

ρ =M(p+ αρ2)(1− βρ)

RT,

where M , R, α and β are constant, p is pressure, ρ is density and T is temper-ature.

Define the specific volume v =1

ρ, and hence write an expression for p as a

function of v and T . Show that, for fixed values of T , the resulting expression

p(v) is not monotonic if T < Tc =8αM

27βR, and draw the form of the graph of

p(v) for T < Tc and T > Tc.

[The non-monotonicity of the graph represents a mechanism for condensationfrom gas to liquid as the temperature is reduced.]

1.10 An incompressible fluid in 0 < z < h with velocity u = (u, v, w) satisfies the noflow through condition w = 0 at z = 0, and the kinematic condition

w = ht + uhx + vhy at z = h.

By integrating the conservation of mass equation from 0 to h, show that massconservation takes the form

ht + (hu)x + (hv)y = 0.

Show also that this equation can be derived directly from first principles, by

consideration of the rate of change of

∫

A

h dx dy, where A is an arbitrary area

in the (x, y) plane.

1.11 A rotating layer of shallow fluid of depth h and horizontal velocity components(u, v) is described in the rotating frame by the equations

ht + (hu)x + (hv)y = 0,

du

dt− fv + ghx = 0,

dv

dt+ fu+ ghy = 0.

26

Scale the variables by choosing

u, v ∼ U, H =flU

gη, x, y ∼ l, t ∼ l

U,

and show that the dimensionless equations can be written in the form

ε

(l

Ld

)2

ηt + ux + vy = 0,

εdu

dt− v + ηx = 0,

εdv

dt+ u+ ηy = 0,

where

ε =U

fl

is the Rossby number.

Show that if U ∼ 10 m s−1, f ∼ 10−5 s−1, l ∼ 104 km, ε is small. On this basis,show that the motion is approximately geostrophic.

Assuming in addition that l Ld, show how to derive a quasi-geostrophicequation for η by writing u = u0 + εu1 + . . ., etc., and equating terms ofcomparable order in ε. Explain how small you assume the ratio l/Ld to be inyour analysis.

27

Chapter 2

Shallow water theory

Chapter 1 dealt with the equations of inviscid flow, and particularly with their formwhen the flow is incompressible and irrotational, and we showed that small amplitudewater waves on a lake or an ocean are described by a linear theory which describesthe dispersive propagation of waves. In this chapter we turn our attention to thepropagation of waves in a layer of fluid which is shallow; as we shall see, the theorywhich emerges is quite different.

Shallow water theory, as it is called, is applicable in a wide range of circumstances.The flow of a thin layer of liquid is appropriate to the description of streams andrivers, for example, and one might also think of coating and rimming flows: waterdribbling down a windscreen, dripping down a wine glass, or oil lubricating a bearing;these are also shallow flows, but the examples mentioned are better described by theapproximations of lubrication theory, which assumes a shallowness, but also that theReynolds number is small, so that the flows are essentially viscous. For more rapidrivulet or stream flows, this is not the case. Shallow water theory assumes the flow isinviscid, i. e., that the Reynolds number is large, but ignores the consequent fact thatthe flow is turbulent. This, for example, is always the case in rivers. For such flows,the usual method of accommodating turbulence is to allow for a turbulence-inducedfrictional stress on the flow; the resulting equations are then known as the St. Venantequations, but it can be shown that in practice they reduce to the shallow waterequations when the wavelength of the motion is not too large.

2.1 The shallow water equations

There are a number of ways of deriving the shallow water equations, and we will startwith the simplest. We suppose that an incompressible, inviscid fluid flows with itsimpermeable base at z = 0 and its top surface is denoted by z = h(x, y, t). It is aconsequence of conservation of mass (cf. question 1.10) that we have

∂h

∂t+ ∇.

[∫ h

0

u dz

]= 0, (2.1)

where u = (u, v, 0) is the horizontal velocity vector. As an exercise (see question2.1), we find that vertical integration of the momentum equation similarly yields the

28

‘exact’ equation

∂

∂t

∫ h

0

u dz + ∇.

[∫ h

0

(uu) dz

]+ gh∇h = 0, (2.2)

where the only assumption in writing (2.2) is that we have assumed the hydrostaticapproximation

p = pa + ρg(h− z), (2.3)

which is a consequence of the assumption of a shallow layer. In writing (2.2), we takeuu as the tensor with components uiuj, and its divergence is the vector with i-th

component∂(uiuj)

∂xj(summed over j).1

We define the mean horizontal velocity as

u =1

h

∫ h

0

u dz, (2.4)

so that the mass conservation equation (2.1) is just

ht + ∇. (hu) = 0, (2.5)

and we define something known as a profile coefficient as

Dijkl =

∫ h

0

uiuj dz

hukul, (2.6)

whence it follows that the momentum equation takes the form

∂(hu)

∂t+ ∇. [D : (huu)] + gh∇h = 0.2 (2.7)

This fearsome looking equation melts to simplicity with the second basic assump-tion of shallow water theory, which is that the horizontal flow is independent of depth,i. e., u = u. It then follows that (2.7) takes the simpler form

∂(hu)

∂t+ ∇. [(huu)] + gh∇h = 0. (2.8)

Combining this with (2.5), we obtain

ut + (u .∇)u + g∇h = 0, (2.9)

and this, combined with (2.5), gives the shallow water equations.

1Thus ∇. (uu) = (u .∇)u + u(∇.u).2The double dot notation indicates a double scalar product. Since D is a fourth order tensor and

uu is a second order tensor, the double scalar product is a second order tensor, having componentsD : (uu)ij = Dijklukul.

29

The most common application of these equations is for essentially one-dimensionalflows, as appropriate for rivers, for example, and the model takes the simple form,with which we shall largely deal,

ht + (hu)x = 0,

ut + uux + ghx = 0. (2.10)

Note that this model is essentially nonlinear.

2.1.1 A scaling approach

A different and more methodical derivation of the shallow water equations follows ifwe first scale the equations (1.34), which we write in the form

∇2φ = 0,

φz = 0 at z = 0,

φt + 12|∇φ|2 + gh = gh0

φz = ht + ∇φ.∇h

at z = h; (2.11)

here h0 is the depth of the fluid layer when at rest. We non-dimensionalise theequations by choosing scales to balance the principal linear terms in the equations,thus

x, y, z, h ∼ h0, φ ∼ (gh30)1/2, t ∼

(h0

g

)1/2

, (2.12)

and this leads to the dimensionless model

∇2φ = 0,

φz = 0 at z = 0,

φt + 12|∇φ|2 + h = 1

φz = ht + ∇φ.∇h

at z = h, (2.13)

where the variables are now dimensionless; it can be seen that the scaling leads to acanonical formulation in which there are no dimensionless parameters.

The entry of parameters into the problem lies in the choice of initial data. Tobe precise, we will take an initial state of no motion, φ = 0, and an initial surfaceelevation profile having height variation ∆h and wavelength L. These two lengthsintroduce two dimensionless parameters

ε =∆h

h0

, δ =h0

L, (2.14)

and the dimensionless initial surface is then assumed to have the form

h = 1 + εη0(δx) at t = 0, (2.15)

30

where x = (x, y) is the horizontal spatial location, and η0 and its derivatives are O(1).Small amplitude waves, as discussed in chapter 1, follow from consideration of the

limit ε 1, δ = 1. In this case , we rescale the variables as

h = 1 + εη, φ ∼ ε, (2.16)

and this leads to the rescaled model

∇2φ = 0,

φz = 0 at z = 0,

φt + 12ε|∇φ|2 + η = 0

φz = ηt + ε∇φ.∇η

at z = 1 + εη. (2.17)

Expansion in powers of ε now leads to the leading order problem studied in chapter1 (see question 2.2).

In distinction, shallow water theory, or long wave theory, follows from the as-sumptions ε = 1, δ 1. The derivation of the corresponding approximate model issomewhat more tortuous. We rescale the variables as

x =X

δ, φ ∼ t ∼ 1

δ, (2.18)

which leads to the model in the rescaled form

φzz + δ2∇2φ = 0,

φz = 0 at z = 0,

φt + 12

[1

δ2φ2z + |∇φ|2

]+ h = 1

φz = δ2[ht + ∇φ.∇h]

at z = h, (2.19)

where we now take ∇ = (∂X , ∂Y ) to be the horizontal gradient vector (with respectto X).

We write φ as a perturbation expansion

φ = φ0 + δ2φ1 + . . . (2.20)

(a similar expansion for h is unnecessary, at least as far as we take the expansion);substituting this in to the equations and equating powers of δ2, we find

φ0 = φ0(X, t),

φ1,z = −z∇2φ0, (2.21)

and thus the surface boundary conditions become, at leading order,

φ0,t + h+ 12|∇φ0|2 = 1,

φ1,z = ht + ∇φ0.∇h, (2.22)

31

evaluated at z = h. Using (2.21)2, the second of these is just

ht + ∇. (h∇φ0) = 0. (2.23)

(2.23) and (2.22)1 are the shallow water equations. To write them in a more commonform, we define u = ∇φ0 = u(X, t), and we note that therefore ∇ × u = 0 (aswe expect, since the flow is irrotational), and thus ∇(1

2|u|2) = (u .∇)u; taking the

gradient of (2.22)1, we then have

ht + ∇. (hu) = 0,

ut + (u .∇)u + ∇h = 0, (2.24)

which are the shallow water equations. In one spatial dimension, we regain (2.10) (indimensionless form).

2.2 Characteristics

We will now largely deal with the one-dimensional form of the equations, and in thedimensional form (2.10). This is a pair of first order partial differential equations,equivalent to a single second order equation, and its type (hyperbolic, parabolic, ellip-tic) determines the appropriate form of the initial and boundary conditions. Becauseone of the independent variables is time (and we do not expect to have boundaryvalue problems in time), we anticipate that the problem, if well-posed, should not beelliptic. In fact, since we naturally prescribe two initial conditions (for u and h), weexpect the problem to be hyperbolic, and this is indeed the case. In order to showthis, we compute the characteristics of the equations.

A nice way of doing this follows from the following stratagem. Define

c =√gh (2.25)

(note that this is the shallow limit of the linear wave speed for small amplitude waveswhen u = 0). The equations (2.10) then take the form

2ct + cux + 2ucx = 0,

ut + uux + 2ccx = 0, (2.26)

and addition and subtraction of these yields the pair of equations

[∂

∂t+ (u± c) ∂

∂x

](u± 2c) = 0. (2.27)

This shows directly that the characteristics are the curvesdx

dt= u± c, on which

u± 2c are constants: these are known as Riemann invariants. The characteristics arereal and distinct, so the system is hyperbolic, and the characteristic speeds, whichare also the two wave speeds of the motion, are ±c relative to the mean flow speed;

32

as we might expect from linear wave theory. In particular, the wave speeds are non-dispersive.

Because the equations are hyperbolic, appropriate initial data is to prescribe twoquantities, and these are naturally h (or c) and u, and it is also natural to supply twoboundary conditions, but the issue of their location is slightly mysterious.

The Froude number is defined as

F =u√gh. (2.28)

Strictly, this is the local Froude number, as it may vary with position and time inthe flow. More commonly, we define a Froude number for a flow based on typicalscales for u and h. Let us suppose that u > 0, as is the case for flow in a stream,for example. We see that if F > 1 (the flow is supercritical), then both characteristicspeeds are positive, and both waves travel downstream. In this case, it is appropriateto specify two boundary conditions (say for u and h) at an upstream location. Forexample, the supercritical flow in a spillway from a dam at x = 0 has its boundaryconditions applied at x = 0.

On the other hand, if F < 1 (subcritical flow), then one wave moves upstream andone downstream, and it is appropriate to specify boundary conditions at each end ofthe domain. We might then for example specify h at x = l, where (let us say) thestream flows into a reservoir of fixed surface elevation.

Now, this is an odd situation; it says that the ‘right’ boundary conditions toapply depend on the solution itself. And in fact it is easy to imagine a situationwhere, for example, the flow is supercritical at the upstream end, but subcritical atthe downstream end; then we would want to apply three boundary conditions for thesecond order system! Clearly something is amiss.

The resolution of this must await section 2.4, but a clue lies in a simple experiment.My kitchen sink at home is flat-bottomed; the drain lies below the left hand hottap, while the cold tap is to the right. If I switch the cold tap on (the inlet), theflow beneath will be supercritical, but as the flow spreads out towards the drain, itbecomes subcritical. And what is simply observed is that there is a very shallow flowsurrounded by a circular rim where the depth suddenly increases as the flow undergoestransition to a subcritical state. This is a shock wave, and an understanding of thedynamics which cause it to form will provide the resolution of the conundrum of theboundary conditions.

2.2.1 Characteristics for general systems

The calculation of the characteristics for the shallow water equations relied on atrick, but one needs a more general method to establish what they are. So let us nowconsider a system of n first order equations in the form

Aut +Bux = c, (2.29)

where u ∈ Rn, and A and B are matrices.

33

If A and B were scalar, the characteristic speeddx

dt= λ would be λ =

B

A, or more

suggestively, λA − B = 0. The generalisation of this to the n-dimensional case isto find the n values of λ which satisfy det (λA − B) = 0. To see this, we supposethat λ are the eigenvalues of A−1B, so that the matrix P whose columns are thecorresponding eigenvectors diagonalises A−1B as

P−1A−1BP = D = diag (λ). (2.30)

We write u = Pw, and then after some algebra, we find

wt +Dwx = P−1A−1c− (P−1Pt +DP−1Px)w, (2.31)

and this shows that the characteristic equations are indeed

dx

dt= λi,

dwidt

=P−1A−1c− (p−1Pt +DP−1Px)w

i. (2.32)

The system is hyperbolic if all the λi are real, and one would say completely el-liptic if all the eigenvalues are complex; generally such systems will be mixed, andconsequently require a mixture of boundary conditions. While this information isuseful, it does not generally help with the solution, since the right hand sides of thecharacteristic equations will generally involve all the components of w.

2.3 Dam break problem; simple waves

The same is true of the shallow water equations, which despite the presence of Rie-mann invariants, are not generally solvable. An important example where the generalsolution can be given is the dam-break problem, representing as it does the fluid flowdue to the removal of a barrier (the dam) to an upstream lake. In this flow, one of theRiemann invariants is uniformly constant, and the problem is thus reduced in orderand solvable. Such a situation results in what is called a simple wave.

The problem is to solve the shallow water equations in the form (2.27) in t > 0,with initial data being

u = 0, c = c0 for t = 0, x < 0. (2.33)

It seems we should add the condition c = 0 for t = 0, x > 0, but u is not definedthere, and really the extra condition might be phrased as

u = 0, 0 < c < c0 for t = 0, x = 0. (2.34)

The solution proceeds in stages. First, far to the left and near t = 0, we can expectboth the positive and negative characteristics (of respective slopes u ± c ≈ ±c0), tointersect x < 0 at t = 0, in which case u ± 2c = ±2c0 at that point, and thereforeu = 0, c = c0; this region is undisturbed, the characteristics are straight lines x±c0t =constant, and this undisturbed region is therefore bounded by the line x = −c0t, asshown in figure 2.1.

34

0

1

2

3

-3 -2 -1 0 1 2 3

t

x

Figure 2.1: Characteristics for the dam break problem.

Next, the positive characteristics will carry forward into x > −c0t, though theydo not remain as straight lines. But crucially, the Riemann invariant u + 2c = 2c0

is the same constant on all of them. Hence the flow is that of a simple wave. Onthe negative characteristic through a point in x > −c0t, u − 2c is constant (thoughnot = −2c0), and so on the negative characteristic both u and c are constant, andtherefore these characteristics are straight lines. They cannot originate in x < 0, andso must start from the origin, forming a fan of rays called an expansion fan. In thisfan, we therefore have

u+ 2c = 2c0,x

t= u− c, (2.35)

whence the solution is

u = 23

(c0 +

x

t

), c = 1

3

(2c0 −

x

t

), (2.36)

and this provides the solution as long as 0 < c < c0, i. e., for −c0t < x < 2c0t. Thedepth profile is thus given by

h = h0, x < −c0t,

h = 49h0

[1− x

2c0t

]2

, −c0t < x < 2c0t,

h = 0, x > 2c0t, (2.37)

and is illustrated in figure 2.2. Note that the depth at the origin remains constant.One might enquire what happens (mathematically) in the region x > 2c0t. As

regards the solution of the problem as stated, this region is redundant; the positiveand negative characteristics from the initial curve x < 0 span the region t > 0∩x <

35

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

x

h

Figure 2.2: The depth profile given by (2.37); here h0 = c0 = t = 1.

2c0t. On the other hand, we could for example describe the same physical problemby adding extra initial data, for example

u = c = 0 on t = 0, x > 0. (2.38)

(Note that the same physical problem is described by any initial data for u in x > 0.)

In this case the characteristics are degenerate, both collapsing todx

dt= 0, with the

solution being undisturbed: u = c = 0, and the characteristics, both positive andnegative, are vertical. As we discuss further below, this suggests that a shock forms,and indeed u is discontinuous at x = 2c0t; but since h = 0 there, mass and momentumconservation occurs automatically, and the discussion is toothless.

2.3.1 Similarity solution

The attentive reader will note that the statement of the initial and boundary condi-tions for the dam break problem contain no length scale, and this is suggestive of asimilarity solution. Inspection of (2.26) shows that the equations are invariant undera transformation x→ αx, t→ αt, thus suggesting that a suitable similarity variableis x/t, and we could therefore directly seek a solution to the dam break problem inthe form

c = c(ξ), u = u(ξ), ξ =x

t. (2.39)

Substituting these into (2.26), we find

2(u− ξ)c′ = −cu′,(u− ξ)u′ = −2cc′. (2.40)

Dividing the two equations leads todu

dc= ±2, and if we select the positive sign, then

u+ 2c = 2c0, (2.41)

36

where the constant is chosen from the initial condition, which suggests

u = 0, c = c0 at ξ = −∞. (2.42)

Substituting this back into (2.40) leads to

u′ = 0 or u = 23(ξ + c0), (2.43)

which gives the continuous solution as

u = 0, c = c0, ξ < −c0,

u = 23(ξ + c0), c = 1

3(2c0 − ξ), −c0 < ξ > 2c0, (2.44)

which is the same solution obtained earlier.One might wonder why one selects the positive Riemann invariant in (2.41). Se-

lection of the negative sign also leads to a solution satisfying the boundary conditions,which is

u = 0, c = c0, ξ < c0,

u = 2(ξ − c0), c = ξ, ξ > c0, (2.45)

which can be rejected, for example on the basis that it does not satisfy the necessarycondition c→ 0 as ξ →∞.

2.4 Shock waves

One of the distinguishing properties of nonlinear hyperbolic equations is their abilityto form shocks, or shock waves. The term shock wave is familiar from the descriptionof sonic booms as jet aircraft pass overhead, and it refers to a sudden jump in thestate variables. The cause of shock wave formation lies in the fact that neighbouringcharacteristics have the ability to intersect, and it is this which causes the problem.We can illustrate the problem using a generalised simple wave theory for the shallowwater equations.

2.4.1 Generalised simple wave theory

Let us reconsider the shallow water equations, but now in −∞ < x < ∞, and letus suppose that the initial conditions u = u0(x), c = c0(x) satisfy the conditionu0−2c0 = constant, and to be specific we suppose u0 = 2c0. (The reason for choosingthe negative Riemann invariant is that it is more natural to suppose u is large when cis large.) It follows that u = 2c throughout the flow, and therefore the characteristicequations reduce to the single first order equation

vt + vvx = 0, (2.46)

wherev = 3

2u, (2.47)

37

0

1

2

-2 0 2

x

v

Figure 2.3: Distortion of the evolving wave solution of (2.46). The graphs show theevolution of v0 = e−x

2at times t = 0, t = 1, t = 2.

and we take v = v0(x) at t = 0.(2.46) is straightforwardly solved using the method of characteristics. The char-

acteristic equations arex = v, v = 0, (2.48)

where the overdot denotes differentiation with respect to t, and the parameterisedinitial conditions are

v = v0(ξ), x = ξ, at t = 0, (2.49)

and the solution can be written as v = v0(ξ), ξ = x− vt, or implicitly as

v = v0(x− vt), (2.50)

a nonlinear travelling wave. An initial humped function v0(x) will have its higherparts travel more rapidly than the lower parts, and therefore the wave distorts andover-rides itself, as shown in figure 2.3, becoming multi-valued at some point.

For anyone who has been to the beach, this behaviour is familiar as wave-breaking(see question 2.3). For such waves, the surface does indeed fold over as in figure 2.3,although eventually the waves break, but in general, the multi-valuedness cannot besupported. This is true for bores on rivers, or for sonic booms, where certainly thefluid density cannot be multi-valued.

Our escape from this conundrum lies in the realisation that the derivation of thepoint form of the Euler equations assumes smoothness of the variables, but the integralform does not even require continuity. Therefore we can allow discontinuities (shocks)in the solutions, providing we ensure that the integral from of the conservation laws issatisfied across the shock. This leads to the formulation of jump conditions, sometimesreferred to as Rankine-Hugoniot conditions.

The general derivation of a jump condition from a conservation law

φt + qx = 0 (2.51)

38

xf

φ_ q _,

φ q,+ +

xf

xf

+d

Figure 2.4: Derivation of the jump condition.

is as follows. Suppose the integral form of (2.51) is

d

dt

∫ x+

x−

φ dx = −[q]x+x− , (2.52)

where x− and x+ are arbitrary, and suppose the shock position is at xf (t). Weconsider a short time interval dt, and take x− = xf (t), x+ = xf (t) + xf dt, and wesuppose the values of φ and q in front of and behind the shock are φ+, q+ and φ−, q−respectively, as shown in figure 2.4. Evidently, the change in the integral in the timedt is

d

∫ x+

x−

φ dx = (φ− − φ+)xf dt = −dt[q+ − q−], (2.53)

and it follows that the jump condition is simply

xf =[q]+−[φ]+−

. (2.54)

The generalisation to three dimensions is simply made (because a shock is essentiallya one-dimensional object): for the conservation law

φt + ∇.q = 0, (2.55)

the speed of the shock surface in the direction normal to itself is

vn =[q.n]+−[φ]+−

. (2.56)

(For a surface defined by F (x, t) = 0, we have Ft+v.∇F = Ft+vn|∇F | = 0, so that

vn = − Ft|∇F | .)

To apply this to (2.46), we need to know what its proper conservation formis. Evidently there are an infinite number of conserved quantities which lead to(2.46), in fact any function f(v) is conserved (with the corresponding flux being

39

q(v) =

∫ v

0

vf ′(v) dv), and it can be shown (see question 2.4) that the jump con-

ditions corresponding to conservation of mass and momentum in the shallow waterequations correspond in (2.46) to the conservation of 1

2v2 and 1

3v3. We cannot sat-

isfy both of course, and the discrepancy occurs because across a shock, the Riemanninvariant u− 2c is not conserved.

We will nevertheless persevere with (2.46) by way of example. Let us supposethat the initial condition is

v =

1− x2, |x| < 1,0, |x| > 1

at t = 0, (2.57)

and that it is derived from a conservation law for v (with flux 12v2). The solution is

v = 1− (x− vt)2 when v > 0, and solving this yields

v =1− 4xt+ 4t21/2 − 1 + 2xt

2t2. (2.58)

Computing vx, we find that vx → −∞ at t = tc = 12

at x = 1, and this heralds theformation of a shock, which then propagates into x > 1. From the conservation lawvt + (1

2v2)x = 0, we derive the jump condition

xf =[12v2]+−

[v]+−= 1

2v−, (2.59)

since v+ = 0. From (2.58), we therefore have

xf =1− 4txf + 4t21/2 − 1 + 2txf

4t2, xf (

12) = 1, (2.60)

which completes the solution of the problem.At large time, we expect xf →∞, and also that v− → 0, so that also xf → 0 and

xf t. The asymptotic limit of (2.60) is thus

xf ∼xf2t, (2.61)

whence xf ∼√t. In order to find the constant of proportionality, we use conservation

of mass, which says that at large time approximately

∫ xf

0

v dx = 43, which is the initial

value of the integral of v. From (2.58), we have v ≈ x

t, and it follows from this that

xf ∼ 83

√t as t→∞.

2.4.2 The dam break problem, revisited

A proper treatment of shocks in shallow water equations occurs in a modifed versionof the dam break problem, where we suppose a dam in x < 0 of depth hu discharges

40

0

1

2

3

4

5

-10 -5 0 5 10 15 20

t

x

Figure 2.5: Characteristics for the modified dam break problem, with the values ofcd = 1, c− = 2, whence cu = 3.19. The shock is the black line (x = 3.16t).

into a lake of depth hd < hu. The mathematical problem we solve is (2.27), but nowwith initial conditions

u = 0, c = cu, x < 0

u = 0, c = cd, x > 0 (2.62)

at t = 0. The original dam break problem had cd = 0.The principal difference is that there are now two undisturbed regions, u = 0,

c = cu, apparently in x < −cut, and u = 0, c = cd, perhaps in x > −cdt. In−cut < x < cdt we would then have an expansion fan. This would be fine, but thepositive characteristics from x < 0 bend round as do the ones in figure 2.1 and willthus intersect the positive characteristics x − cdt = constant from the downstreamundisturbed region: a shock must form. The precepts of similarity tell us that theshock speed will be a constant V , to be determined.

We are thus led to consider a flow consisting of four regions, as follows, and asshown in figure 2.5. There is an undisturbed region x < −cut, in which u = 0, c = cu.This is region 1 in figure 2.5. There is another undisturbed region downstream inx > V t (V to be determined), where u = 0, c = cd; this is region 4. The shock isat x = V t. Adjoining the upstream undisturbed region 1, there is an expansion fan(region 2) in −cut < x < Ut (U is to be determined), in which u + 2c = 2cu and

u− c =x

t(the negative characteristics go through the origin), whence

u = 23

(cu +

x

t

), c = 1

3

(2cu −

x

t

), (2.63)

and thus the positive characteristics are given by

x = 2cut− 3cuτ2/3t1/3, (2.64)

41

for τ > 0. Finally, between the expansion fan and the shock, Ut < x < V t, there is aplateau region, whose dynamics are determined by the jump conditions at the shock,as described below.

Let us suppose u = u−, c = c− upstream of the shock. The mass and momentumjump conditions

V =[hu]+−[h]+−

=[hu2 + 1

2gh2]+−

[hu]+−(2.65)

imply

u = V − K

c2(2.66)

across the shock (K is constant), and

[K2

c2+ 1

2c2

]+

−= 0. (2.67)

Since u+ = 0 c+ = cd, we have

V =K

c2d

, (2.68)

and since (from upstream) u− + 2c− = 2cu, we have

u− = K

(1

c2d

− 1

c2−

)= 2(cu − c−), (2.69)

which defines K in terms of c−. Using this in the second jump condition (2.67) givesthe equation for c−,

8(cu − c−)2 = (c4− − c4

d)

(1

c2d

− 1

c2−

), (2.70)

which determines c− ∈ (cd, cu) uniquely (by considering the graphs of the left andright hand sides).

We see that c− and therefore u− are constant along the shock, and thus u − 2cis constant throughout the plateau region. Specifically, u + 2c = 2cu and u − 2c =u− − 2c−, and thus u = u− and c = c− are both constant in the plateau, which thushas constant height and speed. Finally, we determine the fan boundary speed U byrequiring that c is continuous there; this leads to

U = 34u− + 1

2cu − 3

2c−. (2.71)

2.4.3 The effect of diffusion

Although we have only shown it in specific examples, we can generally expect non-linear hyperbolic systems to develop shocks; it is in their nature. And while we havecome with a reasonable physical explanation to describe shocks, it is a fact that na-ture avoids such discontinuities where it can. The usual way in which nature avoidsdiscontinuities is by providing a diffusive mechanism, which in the hyperbolic system

42

has been neglected. In the present case, the neglected diffusive mechanism is theviscous transport of stress, which in the case of inviscid (high Reynolds number) fluidflow is largely the transport of Reynolds stresses. A digression on these is thereforenecessary.

Inviscid flows are a consequence of high Reynolds number, but another conse-quence is that such flows, at least where shear is important, are invariably turbulentand thus time-fluctuating. Reynolds himself (1895) introduced the idea of character-ising such flows by writing the velocity components as ui = ui+u′i, where the overbardenotes the (ensenble) mean and the prime the fluctuations. By taking the meanof the Navier-Stokes equations, Reynolds showed that the fluctuating term −ρu′iu′jacted as an effective stress tensor for the flow, dwarfing the effects due to viscosity.

In practice, one needs to constitute the Reynolds stress in terms of the mean flow,and the simplest assumption is to introduce an eddy viscosity ηT by supposing that

−ρu′iu′j = ηT

(∂ui∂xj

+∂uj∂xi

). (2.72)

Typically, the assumed value of ηT is relatively small, however.These turbulent stresses are accommodated in shallow water theory in two ways.

Because the flow is shallow, the principal Reynolds stress is a frictional one, and isusually modelled as an algebraic drag term in the momentum equation: this leadsto the St. Venant equations. However, the longitudinal stresses provide for anotherterm proportional to ux, and when this is accommodated in the vertical integrationof the momentum equation, it modifies the shallow water equations to the form (cf.(2.10)

ht + (hu)x = 0,

ut + uux + ghx =ηTρh

(hux)x . (2.73)

It is an exercise beyond the scope of the present exposition to show that inclusion ofthis term allows for a shock structure, that is to say the shock provides the correctway to describe the limit ηT → 0, but if ηT is non-zero, then there is a thin region ata shock over which h and u change sharply but continuously.

2.4.4 The issue of boundary conditions

The inclusion of an eddy viscosity term provides a resolution for the awkward conceptof discontinuous solutions, but it also provides a resolution of the boundary condi-tion conundrum we discussed earlier: if the Froude number F > 1, we require twoupstream boundary conditions, whereas if F < 1, we need one upstream and onedownstream (and with mixed Froude number, we might require three!).

Now we see that the inclusion of the eddy viscosity term makes the system (2.73)of third order, and this implies that in fact three boundary conditions are necessaryfor the full specification of the problem. We can also see that the system is no longerhyperbolic, and is in fact a mixed system (see question 2.7), with the equation for h

43

being hyperbolic (with wave speed u) and the equation for u being elliptic. Thus thissystem requires two boundary conditions upstream (one each for h and u) and onedownstream (for u).

The limit in which ηT → 0 is a singular approximation, in that it lowers the orderof the system, and therefore only allows us to specify two boundary conditions, butwhich two depends on the sub-characteristics of the system (Kevorkian and Cole 1981)which are the inviscid characteristic speeds u ± √gh. For supercritical flow, whenthese are both positive, we apply the two upstream boundary conditions, and thedownstream one is enabled by a boundary layer there. For subcritical flow, we applyone boundary condition at each end, and the second upstream boundary condition isenabled by a boundary layer at the inlet. And for mixed flow (supercritical upstream,subcritical downstream, for example), we have an internal boundary layer, i. e., ashock, as discussed earlier for the kitchen sink flow; see question 2.8.

2.5 The Korteweg–de Vries equation

Of the two parameters ε and δ measuring the amplitude and length of water waves, wehave seen that the limit ε 1, δ = 1 yields small amplitude linear dispersive waves,while the limit ε = 1, δ 1 yields nonlinear, aggregating waves, which steepen andform shocks. An obvious question is whether a combination of these limits leads to amodel in which the antagonistic effects of nonlinear aggregation and dispersion canlead to a new balance. The answer to this question is yes, and the exercise leads to aformulation of the famous Korteweg–de Vries equation, which has a famous historyin fluid mechanics, and also an extensive set of mathematical offspring.

We begin from the dimensionless water wave model (2.13), but now we chooseε = δ2, and rescale the model as follows:

φ ∼ √ε, t ∼ 1√ε, h = 1 + εη, x = (x, y) ∼ 1

ε, (2.74)

and this leads to the model in the form

φzz + ε∇2φ = 0,

φz = 0 at z = 0,

φt + η + ε[

12|∇φ|2 + φτ

]= 0

φzε

= ηt + ε [∇φ.∇η + ητ ]

at z = 1 + εη, (2.75)

where ∇2 =∂2

∂x2+

∂2

∂y2is the horizontal Laplacian, and we have introduced a second

time scaleτ = εt, (2.76)

on the anticipated basis that the solution will involve a multiple time scale approach

such that∂

∂t−→ ∂

∂t+ ε

∂

∂τ.

44

We define φ(x, 0, t) = Φ(x, t, τ) to be the value of the potential at the base, andthis allows us to derive an approximate solution for φ as a perturbation series (ormore simply as a Taylor series) of the form

φ ∼ Φ− 12εz2∇2Φ + 1

24ε2z4∇4Φ + . . . ; (2.77)

with an obvious notation, this solution is just cos(εz∇)Φ.Substituting this into the surface boundary conditions as written in (2.75), and

expanding to terms of O(ε), leads to

Φt + η = ε[

12∇2Φt − Φτ − 1

2|∇Φ|2

]+ . . . ,

ηt +∇2Φ = ε[−η∇2Φ + 1

6∇4Φ−∇Φ.∇η − ητ

]+ . . . ; (2.78)

elimination of η, using also η ≈ −Φt on the right hand side, yields

Φtt −∇2Φ = ε[

12∇2Φtt − 1

6∇4Φ− Φt∇2Φ− 2∇Φ.∇Φt − 2Φtτ

]+ . . . . (2.79)

As is commonly done, we will proceed with the discussion in one space dimension,so that (2.79) is

Φtt − Φxx = ε[

12Φxxtt − 1

6Φxxxx − ΦtΦxx − 2ΦxΦxt − 2Φtτ

]. (2.80)

Obviously the leading order solution is just

Φ = Φ+(x− t, τ) + Φ−(x+ t, τ). (2.81)

Noting that Φ+t = −Φ+

x and Φ−t = Φ−x , this leads to the approximate form of (2.80)as

Φtt − Φxx = ε[

13

(Φ+xxxx + Φ−xxxx

)+ 2

(Φ+xτ − Φ−xτ

)

−3(Φ−x Φ−xx − Φ+

x Φ+xx

)−(Φ+x Φ−xx − Φ−x Φ+

xx

)]. (2.82)

The last, mixed, term in parenthesis in (2.82) does not produce any secular terms(because the particular solution of uξζ = A′(ξ)B′(ζ) is just u = AB), but the others do(because for example a particular solution of utt−uxx = 2h′(x− t) is u = −th(x− t)),and so they must be suppressed. Defining

u = Φ+x , v = Φ−x , ξ = x− t, ζ = x+ t, (2.83)

this leads to the equation for u in the form

uτ + 32uuξ + 1

6uξξξ = 0, (2.84)

with the same equation for v under the identification u → −v, ξ → −ζ. This is thecelebrated Korteweg–de Vries equation, or KdV equation. As we earlier indicated, itcombines the effects of nonlinearity (the second term) with those of dispersion (thethird).

45

2.5.1 Solitons

The KdV equation (2.84) has travelling wave solutions of the form

u = f(X), X = ξ − cτ, (2.85)

where16f ′′′ + 3

2ff ′ − cf ′ = 0, (2.86)

whencef ′′ + 9

2f 2 − 6cf = 0, (2.87)

and the constant of integration is zero without loss of generality. This has the formof a conservative nonlinear oscillator which has bounded oscillatory solutions of finiteperiod (called cnoidal waves), and solitary waves (waves of infinite period) whichsatisfy f(±∞) = 0, and which are given by

f = 2c sech 2

(√3c

2X

). (2.88)

The cnoidal waves are all unstable. However the solitary waves are stable, andpossess a remarkable particle-like property, which has caused them to be known as‘solitons’: if two of these solitary wave solutions of different amplitudes and thusdifferent speeds approach each other, they undergo a nonlinear interaction, followingwhich they emerge unscathed and proceed on their way, with only a phase shift toshow as a consequence of the collision. Remarkably, exact solutions for these two-soliton solutions can be obtained; indeed, this property extends to N -soliton solutions;and in fact, it can be shown that an arbitrary initial condition for (2.84) satisfyingu → 0 as ξ → ±∞ eventually leads to the formation of a finite number of solitons,together with a dispersive wave train. These remarkable properties are a consequenceof the particular form of the equation, which is one example of a nonlinear evolutionequation which is exactly integrable; other examples are the nonlinear Schrodinger(NLS) equation and the sine–Gordon equation.

Boundary conditions

The KdV equation is third order in space, something which should cause some alarm.It is not particularly obvious what boundary conditions should be applied. If wesuppose the inital condition for the full problem is a local disturbance, then aftera long time the solution will be approximated by two travelling wave packets nearx± t ≈ constant, and thus suggests that the boundary conditions for (2.84) might betaken as u→ 0 as ξ → ±∞, and indeed with these boundary conditions, the model iswell-posed. However, in practice, numerical solutions satisfy the equation on a finitedomain. Commonly, one takes periodic boundary conditions, and this works well.But equally, one might aim to specify values of u and/or uξ at each end; and thisleads to numerical difficulty. Indeed, one would need to apply two conditions at oneend and one at the other, but there seems no physical reason to choose what these

46

are. This is reminiscent of the discussion of the shallow water equations, but withoutthe same resolution.

Because the third space derivative arises partly as a consequence of the identifi-cation of space and time derivatives (see comment after (2.81)), other versions of theKdV equation have been proposed. One of these is the regularised long wave (RLW)equation, also known as the BBM (Benjamin–Bona–Mahony) equation, which is

ut + uux = uxxt, (2.89)

and this has better regularisation properties.To see this, we compare the dispersion for the linearised form (about u = 1, to

keep the advective term) of (2.84) with the corresponding one for (2.89). For thelinearised KdV equation (with coefficients all = 1), ut +ux +uxxx = 0, the dispersionrelation is, for modes exp(ikx− iωt), ω = k− k3; the wave speed is c = 1− k2, whichimplies that changes at high wave number (essentially sharp changes in slope), travelprohibitively fast. It is not as bad as negative diffusion, but it has the same bad

aura. For the BBM equation, the corresponding dispersion relation is ω =k

1 + k2,

and the wave speed goes to zero at large wavenumber. From our discussion above,the resolution of this issue is not clear. What is certain is that the KdV has a verylong-standing pedigree, and has been a subject of intense interest for many years.

2.6 Notes and references

2.6.1 Classification of partial differential equations

Courses on partial differential equations typically teach the classification of types ofsecond order partial differential equation, for example in the form

auxx + 2buxy + cuyy = f(x, y, u, ux, uy), (2.90)

with the different types being associated with the sign of b2−ac: positive, the equationis hyperbolic; negative, it is elliptic; and zero, it is parabolic; and in the three casesthere is a canonical transformation to the respective forms

uξξ − uηη = . . . ,

uξξ + uηη = . . . ,

uηη = . . . , (2.91)

where the right hand sides denote terms of lower order. One can show this by directcalculation (e. g., Carrier and Pearson 1976), but one can also show this by writingthe equation as a second order system. For example, if we define

ψ =

(aux + 2buy

uy

), (2.92)

47

then (2.90) takes the form Aψx + Bψy = . . ., and the roots of det (λA − B) = 0

are λ =b± b2 − ac1/2

a, and thus they are real (corresponding to a hyperbolic

system), complex (an elliptic system) or repeated (generally parabolic) if (b2− ac)1/2

is positive, negative or zero. The important feature of the different systems is in thedifferent forms of boundary conditions which provide a well-posed model, and thisextends to higher order systems, which may be of mixed type. For example, the BBMequation (2.89), when written as a system for u and v = u−uxx appears to be mixedelliptic/hyperbolic, and analysis as a system confirms this, with a single characteristicspeed λ = 0; the other two can be taken as infinite (cf. question 2.7).

2.6.2 The Korteweg–de Vries equation

The KdV equation has a stunningly rich history and lies at the origin of some ofthe most extensive developments of mathematics of recent years; it has been treatedin many books, for example those of Drazin and Johnson (1989) and Newell (1985).The early part of the story is worth re-telling, however. John Scott Russell famouslyobserved what he called a wave of translation on a canal outside Edinburgh in 1834,and reported the observation some ten years later (Russell 1845). In an often-quotedparagraph, he says:

I was observing the motion of a boat which was rapidly drawn along anarrow channel by a pair of horses, when the boat suddenly stopped —not so the mass of water in the channel which it had put in motion; itaccumulated round the prow of the vessel in a state of violent agitation,then suddenly leaving it behind, rolled forward with great velocity, as-suming the form of a large solitary elevation, a rounded, smooth andwell-defined heap of water, which continued its course along the channelapparently without change of form or diminution of speed. I followed iton horseback, and overtook it still rolling on at a rate of some eight ornine miles an hour, preserving its original figure some thirty feet long anda foot to a foot and a half in height. Its height gradually diminished, andafter a chase of one or two miles I lost it in the windings of the channel.Such, in the month of August 1834, was my first chance interview withthat singular and beautiful phenomenon which I have called the Wave ofTranslation. . .

The Korteweg–de Vries equation may describe Scott Russell’s wave, but their pa-per (Korteweg and de Vries 1895) does not reference his specific observation. Togetherwith earlier works by Boussinesq and Rayleigh, they were concerned with the samephenomenon, however. Since the 1960s, there has been an explosion of interest inthis and other such nonlinear evolution equations, and this continues to the presentday.

48

Exercises

2.1 Show that the equation describing conservation of mass of a shallow, incom-pressible, inviscid flow in 0 < z < h is

ht + ∇.

[∫ h

0

u dz

]= 0,

where u = (u, v, 0) is the horizontal velocity vector.

Show further that the horizontal component of momentum conservation,

ut + (u .∇)u + wuz = −1

ρ∇p,

where w is the vertical component of velocity, and ∇ is the horizontal gradientvector, together with a hydrostatic balance

p = pa + ρg(h− z),

lead, when integrated from z = 0 to z = h using the kinematic condition

w = ht + u.∇h at z = h,

to the integrated form

∂

∂t

∫ h

0

u dz + ∇.

[∫ h

0

(uu) dz

]+ gh∇h = 0.

Deduce the (two-dimensional) form of the shallow water equations if it is as-sumed that u is independent of z.

2.2 Write down the equations and boundary conditions suitable to describe themotion of a layer of incompressible, inviscid fluid, initially at rest, and of meandepth h0.. By making the model non-dimensional, show that it can be writtenin the form

∇2φ = 0,

φz = 0 at z = 0,

φt + 12|∇φ|2 + h = 1

φz = ht + ∇φ.∇h

at z = h.

Show that if the initial dimensional surface elevation is given by

h = h0 + ∆hη0(x/L),

49

where η and its derivatives are O(1), then the dimensionless initial elevationintroduces two parameters

ε =∆h

h0

and δ =h0

L.

By assuming ε 1 and δ = 1, show that solutions can be found in the form ofperturbation expansions

h = 1 + εη +O(ε2), φ = εΦ +O(ε2),

and by equating terms of corresponding order in ε, show that Φ and η satisfythe equations of small amplitude wave theory.

2.3 A train of (one-dimensional) ocean waves approaches the shore at x = 0 fromx = +∞ over a sloping base at z = −b(x); the undisturbed sea surface is atz = 0, and the disturbed surface is z = η(x, t), so that the water depth ish = η + b.

Show that the no flow through condition at z = −b takes the form

w = −ub′.

Derive the shallow water equations from first principles, and show that theytake the form

ht + (hu)x = 0,

ut + uux + gηx = 0.

Hence show that ifm = gb′(x) is constant, the Riemann invariants are u±2c−mton x = u± c, where c =

√gh.

Suppose that at t = 0, u = u0(x) and c = c0(x) = K − 12u0(x). Show that

u+ 2c−mt = 2K everywhere, and deduce that on the negative characteristics,

u = u0(ξ) +mt and x = ξ +[

32u0(ξ)−K

]t+ 1

2mt2,

and deduce thatu = mt+ u0[x− 3

2ut+mt2 +Kt].

Hence show that waves will break (i. e., a shock forms) if c′0(ξ) > 0 anywhere.

Do these initial conditions make any physical sense?

2.4 Derive the form of the jump condition at a shock surface for the conservationlaw

φt + ∇.q = 0.

[Hint: take local Cartesian coordinates with one axis normal to the shock sur-face.]

50

Write the one-dimensional shallow water equations in the form corresponding toconservation of mass and momentum density, and hence write down the jumpconditions appropriate for a shock at x = xf (t), both in terms of h and u, andin terms of c =

√gh and u.

Show that for a simple wave in which u = 2c, the shallow water equationsreduce to

vt + vvx = 0,

where v = 32u, and the corresponding jump conditions are

xf = 23

[v3]+−[v2]+−

= 34

[v4]+−[v3]+−

.

Show that both of these are consistent with the equation for v (but cannot bothbe true). [This is because in fact u− 2c is not continuous across the shock.]

2.5 Use the method of characteristics to solve the initial value problem

vt + qx = 0, v(x, 0) = v0(x),

where −∞ < x <∞ and the flux is q(v) = 12v2.

Find an explicit expression for vx in terms of x, t and v, and show that vx →∞as t→ tc, where

tc = minξ:v′0<0

[− 1

v′0(ξ)

].

Sketch a typical evolution of the solution in the cases where (i) v0 is monotoni-cally decreasing; (ii) v0 is monotonically increasing.

Now repeat the calculation for the general flux function q(v). Show that ifq′′(v) > 0, shocks will form in the same way, and derive an expression for tc inthis case.

What are the corresponding results if q′′ < 0?

2.6 A model for traffic flow (see for example the book by Whitham (1974)) on asingle lane road assumes a continuum approximation, and has variables of cardensity ρ and car vehicle speed v. The model takes the form

ρt + (ρv)x = 0,

vt + vvx =V (ρ)− v

τ.

Explain the motivation for this model, and explain the meaning of the time τ .How would you expect the function V (ρ) to behave?

Suppose that τ is small. Write down an approximate solution for v, and deducea first order equation for the density ρ. Can you identify what ‘small’ means inthis context?

51

Consider now this approximate first order equation. Show that for reasonablechoices of V (ρ), the traffic flux q(ρ) = ρV (ρ) is concave (q′′ < 0), and thus thatif the initial density profile increases with distance x, a shock will form in whichthe density increases suddenly. Show that if the traffic is sufficiently congested,the shock will travel backwards. What would an approaching driver experiencein this case?

2.7 By writing the wave equation

δ2utt = uxx

as a pair of first order equations of the form

Aψt +Bψx = 0,

show that the characteristicsdx

dt= λ satisfy λ = ±1

δ. Deduce that the elliptic

system with δ = 0 can be considered as the limit of a hyperbolic system inwhich the characteristic speeds tend to ∞.

The shallow water equations, allowing for a longitudinal viscous term, can bewritten in the form

ht + (hu)x = 0,

ut + uux + ghx =ηTρh

(hux)x ,

where ηT is the eddy viscosity. Suppose that typical depth and velocity scalesare h0 and u0. By scaling the model suitably, show that it can be written indimensionless form as

ht + (hu)x = 0,

F 2(ut + uux) + hx =εTF

2

h(hux)x ,

where the dimensionless parameter εT is defined through

ηT = εTρu0h0,

and has a typical value εT ∼ 0.005. What is the definition of the Froude numberF? Deduce that for F ∼ O(1), the viscous term can be neglected.

Consider now the more general system

ht + (hu)x = 0,

F 2(ut + uux) + hx =εTF

2

hτx,

−δτt + hux = τ.

52

Show that as δ → 0, this model reduces to the viscous shallow water equations.

Show that the characteristic speeds are given by the three roots of the cubic

δλ[F 2(λ− u)2 − h] = εTF2(λ− u),

and that in the limit δ εT , the characteristic speeds are given approximatelyby

λ ≈ u, ±√εTδ,

and deduce that when δ = 0, the system is a mixed hyperbolic-elliptic system.

What happens if εT δ?

2.8 The viscous shallow water equations are written in the dimensionless form

ht + (hu)x = 0,

F 2[(hu)t + (hu2)x] + hhx = ε (hux)x ,

where ε = εTF2, and are subject to the boundary conditions

hu = 1, h = h− at x = 0,

h = h+ at x = L.

Show that steady solutions are determined by the solution of

εhx = g(h) = Kh− F 2 − 12h3,

where the constant K is to be determined, and

h(0) = h−, h(L) = h+.

Assuming ε 1 (more properly ε L) and h± ∼ O(1), show that g must havea positive root, and therefore two positive roots, and that either

(i) g(h+) ≈ 0, in which case the solution has a boundary layer at x = 0,providing h− is not too small and the other root H+ < h+; or,

(ii) g(h−) ≈ 0, and the solution has a boundary layer at x = L, provided h+

is not too large, and the other root H− > h−.

Show that for (i), the condition H+ < h+ is equivalent toF

h3/2+

< 1, and that

for (ii), the condition H− > h− is equivalent toF

h3/2−

> 1.

Deduce that for the inviscid limit ε = 0, the correct boundary conditions tosatisfy are

(i) hu = 1 at x = 0 and h(L) = h+ if Fr+ < 1;

53

(ii) hu = 1 at x = 0 and h(0) = h− if Fr− > 1,

where Fr± represents the local Froude number Fu/√h at the upstream (−) or

downstream (+) end.

What do you think happens if both conditions are satisfied?

2.9 Consider the KdV equation in the form

ut + uux + uxxx = 0.

Show that travelling wave solutions exist of the form u = f(X), X = x − ct,and derive a second order differential equation for f , assuming f(±∞) = 0.

Show that this equation has a first integral

12f ′2 + V (f) = E, †

and give the definition of V (f), assuming V (0) = 0.

Show that the conditions f(±∞) = 0 imply E = 0, and find the explicit solutionfor f in this case.

If, instead, we assume only that f is bounded at ±∞, show that oscillatorysolutions of (†) will occur if −2

3c3 < E < 0, and show that their period will tend

to ∞ as E → 0−. [These correspond to cnoidal waves.]

2.10 [Elliptic functions]

Jacobian elliptic functions are defined in the following way (see Abramowitzand Stegun 1964). First we define

u =

∫ φ

0

dθ

(1−m sin2 θ)1/2.

Then the functions sn (u|m) and cn (u|m) are defined via

sn (u|m) = sinφ, cn (u|m) = cosφ;

where clear, the parameter m can be omitted.

The complete elliptic integral of the first kind is defined by

K(m) =

∫ π/2

0

dθ

(1−m sin2 θ)1/2.

Show that snu and cnu are periodic of period 4K(m).

Show that iff(ξ) = sn ξ, g(ξ) = cn ξ, h(ξ) = cn2ξ,

then

f ′ = (1− f 2)1/2(1−mf 2)1/2,

g′ = −(1− g2)1/2(1−m+mg2)1/2,

h′ = −2h1/2(1− h)1/2(1−m+mh)1/2.

54

(i) Pendulum

Let φ+ω2 sinφ = 0, where the overdots denote differentiation with respectto t. Show that

12φ2 = E + ω2 cosφ,

and show that solutions are periodic if −ω2 < E < ω2.

Show that if f = sin 12φ, then

M

ω2f 2 = (1−Mf 2)(1− f 2),

where

M =2ω2

E + ω2> 1.

Deduce that

f = sn

[ωt√M

∣∣∣∣M],

and use the Jacobi reciprocal relation

sn (u|M) = m1/2sn [M1/2u|m], m =1

M,

to show thatf = m1/2sn (ωt|m).

Where should the integration constant be?

(ii) Cnoidal waves

Suppose f(X) satisfies

f ′2 = R(f) ≡ cf 2 − 13f 3 − k,

where c > 0 and 0 < k < 43c3. Denote 0 < f ∗ < c as the solution of

k = cf ∗2 + 13f ∗3,

and show thatR(f) = 1

3(f + f ∗)(f − f−)(f − f+),

and give formulae for f+ > f− > 0. Show that if

h =f − f−f+ − f−

, ξ =

(f+ + f ∗

3

)1/2X

2and m =

f+ − f−f+ + f ∗

< 1,

then (dh

dξ

)2

= 4h(1− h)(1−m+mh),

and deduce that

f = f− + (f+ − f−)cn2

[(f+ + f ∗

3

)1/2X

2

∣∣∣∣∣m]

;

these are the cnoidal wave solutions of the KdV equation.

55

Chapter 3

Groundwater flow

Groundwater is water in the ground, obviously. But it does not exist in subterraneanlakes, or water pockets. It exists in the interstices of the soil, or rock, which form theupper part of the Earth’s surface. Before we begin, it is worth describing the Earthas it exists below our feet. The radius of the Earth is about 6,370 km, and aroundhalf of that distance at the centre is occupied by the Earth’s core, the outer part ofwhich is (more or less) liquid iron, the motion of which causes the Earth’s magneticfield to exist. Surrounding this is the Earth’s mantle, a composite of crystallinerocks, mostly magnesium and iron silicates, which exists all the way to within tens ofkilometres of the surface. The convective processes in the mantle (which cause andexplain continental drift) cause localised partial melting to occur, as a consequenceof which magma chambers form, and thence volcanoes, and the result of this meltingis that the uppermost part of the Earth consists of a crust which is also rock, butconstructed of the more volatile minerals which melt first; these are generally lighter.Volcanic rocks include basalt and granite for example.

The action of erosive and chemical processes at the Earth’s surface causes otherkinds of rocks to form. These are the processes of geomorphology, which shape thesurface of the Earth. Erosion of rock by water, ice and even wind, breaks rock intosediments, which are washed into streams and thence to the oceans, where they aredeposited in sedimentary sequences. Additionally, micron-sized planktonic speciesdie and sink to the ocean floor, where their hard shells are buried. As these varioussediments are buried, they are compressed and cemented under their own weightto form sedimentary rocks: sand forms sandstone, calcareous plankton form chalkor limestone, clays form shale, and so on. At even greater depth, hydrothermalmineral alteration can occur, forming metamorphic rocks: marble is metamorphosedlimestone, for example. Other types of metamorphic rocks are gneisses and schists.

A further destructive process at the surface is that of weathering, which has asimilar consequence to erosion; indeed, physical weathering, where for example freez-ing of water in cracks causes the break-up of coherent rock, is an initialising processin the formation of particulate rock. Chemical weathering acts in a similar way, butthrough the action of acids in causing chemical reactions which can break down rocks,for example the production of clays from feldspar. Once plants become established,their rotted remains (leaf litter) provides a source of organic material, and in this way

56

soils are formed, through the combination of weathered rock or deposited sedimentand organic matter. Thus at the surface of the Earth, erosion and weathering leads tothe formation of a porous particulate surface layer (soil or sediment), which becomescoarser with depth, and eventually becomes coherent bedrock. The depth of thisunconsolidated porous material is quite variable: zero at mountain tops, or hundredsof metres in sedimentary river basins, for example.

3.1 The hydrological cycle

The porous surface layer consists of loosely aggregated solids, and the space betweenthem is known as the pore space, and its volume fraction is known as the porosity, andusually denoted as φ. Often one thinks of the pore space as consisting of tiny reservoirsconnected by thin channels, though this is not a very realistic representation, unlessthe porosity is small. When rain falls, it infiltrates the pore space of the ground,causing the pore space to be partially occupied by water. The volume fraction of thepore space occupied by water is known as the relative saturation, and denoted S. Thepore water drains down, so that with increasing depth, the saturation increases, untilat some level, it reaches one, and the soil is said to be saturated. This level is knownas the water table, and the unsaturated zone above is known as the vadose zone. Thewater table is a piezometric or phreatic surface, because the water pressure there isequal to the atmospheric pressure.1

On a hillslope, the water table is usually sloped, and as a consequence the ground-water flows down the slope of its own phreatic surface, until this surface reachesground level, which is where groundwater flows into a stream or lake. Thence thecontinental precipitation makes its way back to the ocean, from where evaporation ofwater vapour to the atmosphere leads to cloud formation when condensation occurs.The clouds move over land, and rainfall causes the cycle to begin again; this cycle,one of many on the planet, is the hydrological cycle.

3.2 Darcy’s law

In order to characterise the flow of a liquid in a porous medium, we must relate theflow rate to the pressure gradient. An idealised case is to consider that the poresconsist of uniform cylindrical tubes of radius a; initially we will suppose that theseare all aligned in one direction. If a is small enough that the flow in the tubes islaminar (this will be the case if the associated Reynolds number is <∼ 1000), then

Poiseuille flow in each tube leads to a volume flux in each tube of q =πa4

8µ|∇p|,

where µ is the liquid viscosity, and ∇p is the pressure gradient along the tube. A morerealistic porous medium is isotropic, which is to say that if the pores have this tubularshape, the tubules will be arranged randomly, and form an interconnected network.However, between nodes of this network, Poiseuille flow will still be appropriate, and

1This is not quite accurate, as it neglects the effects of surface tension, discussed further later.

57

an appropriate generalisation is to suppose that the volume flux vector is given by

q ≈ − a4

µX∇p, (3.1)

where the approximation takes account of small interactions at the nodes; the numer-ical tortuosity factor X >∼ 1 takes some account of the arrangement of the pipes.

To relate this to macroscopic variables, and in particular the porosity φ, we observethat φ ∼ a2/d2

p, where dp is a representative particle or grain size so that q/d2p ∼

−(φ2d2

p

µX

)∇p. We define the volume flux per unit area (having units of velocity) as

the discharge u. Darcy’s law then relates this to an applied pressure gradient by therelation

u = −kµ

∇p, (3.2)

where k is an empirically determined parameter called the permeability, having unitsof length squared. In the case where gravity is included, we generalise (3.2) to theform

u = −kµ

[∇p+ ρgk, (3.3)

where k is the unit upwards vector. The discussion above suggests that we can write

k =d2pφ

2

X; (3.4)

the numerical factor X may typically be of the order of 103.To check whether the pore flow is indeed laminar, we calculate the (particle)

Reynolds number for the porous flow. If v is the (average) fluid velocity in the porespace, then

v =u

φ; (3.5)

If a is the pore radius, then we define a particle Reynolds number based on grain sizeas

Rep =2ρva

µ∼ ρ|u|dp

µ√φ, (3.6)

since φ ∼ a2/d2p. Suppose (3.4) gives the permeability, and we use the gravitational

pressure gradient ρg to define (via Darcy’s law) a velocity scale2; then

Rep ∼φ3/2

X

(ρ√gdp dp

µ

)2

∼ 10[dp]3, (3.7)

where dp = [dp] mm, and using φ3/2/X = 10−3, g = 10 m s−2, µ/ρ = 10−6 m2 s−2.Thus the flow is laminar for d < 5 mm, corresponding to a gravel. Only for free flow

2This scale is thus the hydraulic conductivity, defined below in (3.10).

58

through very coarse gravel could the flow become turbulent, but for water percolationin rocks and soils, we invariably have slow, laminar flow.

In other situations, and notably for forced gas stream flow in fluidised beds or inpacked catalyst reactor beds, the flow can be rapid and turbulent. In this case, thePoiseuille flow balance −∇p = µu/k can be replaced by the Ergun equation

−∇p =ρ|u|uk′

; (3.8)

more generally, the right hand side will a sum of the two (laminar and turbulent)interfacial resistances. The Ergun equation reflects the fact that turbulent flow in apipe is resisted by Reynolds stresses, which are generated by the fluctuation of theinertial terms in the momentum equation. Just as for the laminar case, the parameterk′, having units of length, depends both on the grain size dp and on φ. Evidently, wewill have

k′ = dpE(φ), (3.9)

with the numerical factor E → 0 as φ→ 0.

Hydraulic conductivity

Another measure of flow rate in porous soil or rock relates specifically to the passageof water through a porous medium under gravity. For free flow, the pressure gradientdownwards due to gravity is just ρg, where ρ is the density of water and g is thegravitational acceleration; thus the water flux per unit area in this case is just

K =kρg

µ, (3.10)

and this quantity is called the hydraulic conductivity. It has units of velocity. Ahydraulic conductivity of K = 10−5 m s−1 (about 300 m y−1) corresponds to apermeability of k = 10−12 m2, this latter unit also being called the darcy.

3.2.1 Homogenisation

The ‘derivation’ of Darcy’s law can be carried out in a more formal way using themethod of homogenisation. This is essentially an application of the method of multiple(space) scales to problems with microstructure. Usually (for analytic reasons) oneassumes that the microstructure is periodic, although this is probably not strictlynecessary (so long as local averages can be defined).

Consider the Stokes flow equations for a viscous fluid in a medium of macroscopiclength l, subject to a pressure gradient of order ∆p/l. If the microscopic (e. g.,grain size) length scale is dp, and ε = dp/l, then if we scale velocity with d2

p∆p/lµ(appropriate for local Poiseuille-type flow), length with l, and pressure with ∆p, theNavier-Stokes equations can be written in the dimensionless form

∇.u = 0,

0 = −∇p+ ε2∇2u, (3.11)

59

together with the no-slip boundary condition,

u = 0 on S : f(x/ε) = 0, (3.12)

where S is the interfacial surface. Here x is the macroscopic length scale and thedependence of f on x/ε reflects the variation of the interface at the microscale. Themicroscale is explicitly identified by defining x = εξ, and then we seek solutions inthe form

u = u(0)(x, ξ) + εu(1)(x, ξ) . . .

p = p(0)(x, ξ) + εp(1)(x, ξ) . . . . (3.13)

Expanding the equations in powers of ε and equating terms leads to p(0) = p(0)(x),and u(0) satisfies

∇ξ.u(0) = 0,

0 = −∇ξp(1) +∇2

ξu(0) −∇xp

(0), (3.14)

equivalent to Stokes’ equations for u(0) with a forcing term −∇xp(0). If wj is the

velocity field which (uniquely) solves

∇ξ.wj = 0,

0 = −∇ξP +∇2ξw

j + ej, (3.15)

with periodic (in ξ) boundary conditions and wj = 0 on f(ξ) = 0, where ej is theunit-vector in the ξj direction, then (since the equation is linear) we have (using thesummation convention)

u(0) = −∂p(0)

∂xjwj. (3.16)

We define the average flux

〈u〉 =1

V

∫

V

u(0)dV, (3.17)

where V is the volume over which S is periodic.3 Averaging (3.16) then gives

〈u〉 = −k∗.∇p, (3.18)

where the (dimensionless) permeability tensor is defined by

k∗ij = 〈wji 〉. (3.19)

Recollecting the scales for velocity, length and pressure, we find that the dimensionalversion of (3.18) is

〈u〉 = −k

µ.∇p, (3.20)

wherek = k∗d2

p, (3.21)

so that k∗ is the equivalent in homogenisation theory of the quantity φ2/X in (3.4).

3Specifically, we take V to be the soil volume, but the integral is only over the pore space volume,where u is defined. In that case, the average 〈u〉 is in fact the Darcy flux (i. e., volume fluid flux perunit area).

60

3.2.2 Empirical measures

While the validity of Darcy’s law can be motivated theoretically, it ultimately relieson experimental measurements for its accuracy. The permeability k has dimensionsof (length)2, which as we have seen is related to the mean ‘grain size’. If we writek = d2

pC, then the number C depends on the pore configuration. For a tubularnetwork (in three dimensions), one finds C ≈ φ2/72π (as long as φ is relativelysmall). A different and often used relation is that of Carman and Kozeny, whichapplies to pseudo-spherical grains (for example sand grains); this is

C ≈ φ3

180(1− φ)2. (3.22)

The factor (1−φ)2 takes some account of the fact that as φ increases towards one, theresistance to motion becomes negligible. In fact, for media consisting of uncemented(i. e., separate) grains, there is a critical value of φ beyond which the medium as awhole will deform like a fluid. Depending on the grain size distribution, this valueis about 0.5 to 0.6. When the medium deforms in this way, the description of theintergranular fluid flow can still be taken to be given by Darcy’s law, but this nowconstitutes a particular choice of the interactive drag term in a two-phase flow model.At lower porosities, deformation can still occur, but it is elastic not viscous (on shorttime scales), and given by the theory of consolidation or compaction, which we discusslater.

In the case of soils or sediments, empirical power laws of the form

C ∼ φm (3.23)

are often used, with much higher values of the exponent (e.g. m = 8). Such behaviourreflects the (chemically-derived) ability of clay-rich soils to retain a high fraction ofwater, thus making flow difficult. Table 3.1 gives typical values of the permeabilityof several common rock and soil types, ranging from coarse gravel and sand to finersilt and clay.

An explicit formula of Carman-Kozeny type for the turbulent Ergun equation

k (m2) material10−8 gravel10−10 sand10−12 fractured igneous rock10−13 sandstone10−14 silt10−18 clay10−20 granite

Table 3.1: Different grain size materials and their typical permeabilities.

61

expresses the ‘turbulent’ permeability k′, defined in (3.8), as

k′ =φ3dp

175(1− φ). (3.24)

3.3 Groundwater flow

Darcy’s equation is supplemented by an equation for the conservation of the fluidphase (or phases, for example in oil recovery, where these may be oil and water). Fora single phase, this equation is of the simple conservation form

∂

∂t(ρφ) + ∇.(ρu) = 0, (3.25)

supposing there are no sources or sinks within the medium. In this equation, ρ is thematerial density, that is, mass per unit volume of the fluid. A term φ is not presentin the divergence term, since u has already been written as a volume flux (i.e., the φhas already been included in it: cf. (3.5)).

Eliminating u, we have the parabolic equation

∂

∂t(ρφ) = ∇.

[k

µρ∇p

], (3.26)

and we need a further equation of state (or two) to complete the model. The simplestassumption corresponds to incompressible groundwater flowing through a rigid porousmedium. In this case, ρ and φ are constant, and the governing equation reduces (ifalso k is constant) to Laplace’s equation

∇2p = 0. (3.27)

This simple equation forms the basis for the following development. Before pur-suing this, we briefly mention one variant, and that is when there is a compressiblepore fluid (e. g., a gas) in a non-deformable medium. Then φ is constant (so k isconstant), but ρ is determined by pressure and temperature. If we can ignore theeffects of temperature, then we can assume p = p(ρ) with p′(ρ) > 0, and

ρt =k

µφ∇.[ρp′(ρ)∇ρ], (3.28)

which is a nonlinear diffusion equation for ρ, sometimes called the porous mediumequation. If p ∝ ργ, γ > 0, this is degenerate when ρ = 0, and the solutions displaythe typical feature of finite spreading rate of compactly supported initial data.

3.3.1 Boundary conditions

The Laplace equation (3.27) in a domain D requires boundary data to be prescribedon the boundary ∂D of the spatial domain. Typical conditions which apply are a noflow through condition at an impermeable boundary, u.n = 0, whence

∂p

∂n= 0 on ∂D, (3.29)

62

or a permeable surface condition

p = pa on ∂D, (3.30)

where for example pa would be atmospheric pressure at the ground surface. Anotherexample of such a condition would be the prescription of oceanic pressure at theinterface with the oceanic crust.

A more common application of the condition (3.30) is in the consideration of flowin the saturated zone below the water table (which demarcates the upper limit ofthe saturated zone). At the water table, the pressure is in equilibrium with the airin the unsaturated zone, and (3.30) applies. The water table is a free surface, andan extra kinematic condition is prescribed to locate it. This condition says that thephreatic surface is also a material surface for the underlying groundwater flow, sothat its velocity is equal to the average fluid velocity (not the flux): bearing in mind(3.5), we have

∂F

∂t+

u

φ.∇F = 0 on ∂D, (3.31)

if the free surface ∂D is defined by F (x, t) = 0.

3.3.2 Dupuit approximation

One of the principal obvious features of mature topography is that it is relatively flat.A slope of 0.1 is very steep, for example. As a consequence of this, it is typically alsothe case that gradients of the free groundwater (phreatic) surface are also small, and aconsequence of this is that we can make an approximation to the equations of ground-water flow which is analogous to that used in shallow water theory or the lubricationapproximation, i. e., we can take advantage of the large aspect ratio of the flow. Thisapproximation is called the Dupuit, or Dupuit–Forchheimer, approximation.

To be specific, suppose that we have to solve

∇2p = 0 in 0 < z < h(x, y, t), (3.32)

where z is the vertical coordinate, z = h is the phreatic surface, and z = 0 is an im-permeable basement. We let u denote the horizontal (vector) component of the Darcy

flux, and w the vertical component. In addition, we now denote by ∇ =

(∂

∂x,∂

∂y

)

the horizontal component of the gradient vector. The boundary conditions are then

p = 0, φht + u .∇h = w + rp on z = h,

∂p

∂z+ ρg = 0 on z = 0; (3.33)

here we take (gauge) pressure measured relative to atmospheric pressure, and allowfor a recharge rate rp due to precipitation. The condition at z = 0 is that of nonormal flux, allowing for gravity.

63

Let us suppose that a horizontal length scale of relevance is l, and that the corre-sponding variation in h is of order d, thus

ε =d

l(3.34)

is the size of the phreatic gradient, and is small. We non-dimensionalise the variablesby scaling as follows:

x, y ∼ l, z, h ∼ d, p ∼ ρgd,

u ∼ Kd

l, w ∼ Kd2

l2, t ∼ φl2

Kd, (3.35)

where K =kρg

µis the hydraulic conductivity. The choice of scales is motivated by

the same ideas as lubrication theory. The pressure is nearly hydrostatic, and the flowis nearly horizontal.

The dimensionless equations are

u = −∇p, ε2w = −(pz + 1),

∇.u + wz = 0, (3.36)

withpz = −1 on z = 0,

p = 0, ht = w + ∇p.∇h+ r on z = h, (3.37)

where

r =rpl

2

Kd2(3.38)

is the dimensionless precipitation. At leading order as ε → 0, the pressure is hydro-static:

p = h− z +O(ε2). (3.39)

More precisely, if we putp = h− z + ε2p1 + . . . , (3.40)

then (3.36) impliesp1zz = −∇2h, (3.41)

with boundary conditions, from (3.37),

p1z = 0 on z = 0,

p1z = −ht + |∇h|2 + r on z = h. (3.42)

Integrating (3.41) from z = 0 to z = h thus yields the evolution equation for h in theform

ht = ∇. [h∇h] + r, (3.43)

which is a nonlinear diffusion equation of degenerate type when h = 0.

64

This is easily solved numerically, and there are various exact solutions whichare indicated in the exercises. In particular, steady solutions are found by solvingLaplace’s equation for 1

2h2, and there are various kinds of similarity solution. (3.43)

is a second order equation requiring two boundary conditions. A typical situation ina river catchment is where there is drainage from a watershed to a river. A suitableproblem in two dimensions is

ht = (hhx)x + r. (3.44)

At the divide (say, x = 0), we have hx = 0, whereas at the river (say, x = 1), theelevation is prescribed, h = 1 for example. The steady solution is

h =[1 + r − rx2

]1/2, (3.45)

and perturbations to this decay exponentially. If this value of the elevation of thewater table exceeds that of the land surface, then a seepage face occurs, where waterseeps from below and flows over the surface. This can sometimes be seen in steepmountainous terrain, or on beaches, when the tide is going out.

This solution is not uniformly valid at x = 1, where conditions of symmetry at theriver at the base of a valley would imply that u ≡ u.i = 0, and thus px = hx = 0. Inreality we expect all the horizontal flux from the aquifer to find its way to the river,which is here represented as a point at x = 1, z = 1. We suppose, therefore, thatthere is a boundary layer near x = 1, where we rescale the variables by writing

x = 1− εX, w =W

ε, h = 1 + εH, p = 1− z + εP. (3.46)

Substituting these into the two-dimensional version of (3.36) and (3.37), we find

u = PX , W = −Pz, ∇2P = 0 in 0 < z < 1 + εH, 0 < X <∞, (3.47)

with boundary conditions

P = H, εHt + PXHX =W

ε+ r on z = 1 + εH,

PX = 0 on X = 0,

Pz = 0 on z = 0,

P ∼ H ∼ rX as X →∞. (3.48)

At leading order in ε, this is simply

∇2P = 0 in 0 < z < 1, 0 < X <∞,Pz = 0 on z = 0, 1,

PX = 0 on X = 0,

P ∼ rX as X →∞. (3.49)

Evidently, this has no solution unless we allow the incoming groundwater flux rfrom infinity to drain to the river at X = 0, z = 1. We do this by having a singularityin the form of a sink at the river,

P ∼ r

πlnX2 + (1− z)2

near X = 0, z = 1. (3.50)

65

The solution to (3.49) can be obtained by using complex variables and the methodof images, by placing sinks at z = ±(2n+ 1), for integral values of n. Making use ofthe infinite product formula (Jeffrey 2004, p. 72)

∞∏

1

(1 +

ζ2

(2n+ 1)2

)= cosh

πζ

2, (3.51)

where ζ = X + iz, we find the solution to be

P =r

πln

[cosh2 πX

2cos2 πz

2+ sinh2 πX

2sin2 πz

2

]. (3.52)

The complex variable form of the solution is

φ = P + iψ =2r

πln cosh

πζ

2, (3.53)

which is convenient for plotting. The streamlines of the flow are the lines ψ =constant, and these are shown in figure 3.1.

This figure illustrates an important point, which is that although the flow towardsa drainage point may be more or less horizontal, near the river the groundwater seepsupwards from depth. Drainage is not simply a matter of near surface recharge anddrainage. This means that contaminants which enter the deep groundwater mayreside there for a very long time.

A related point concerns the recharge parameter r defined in (3.38). Accordingto table 3.1, a typical permeability for sand is 10−10 m2, corresponding to a hydraulicconductivity of K = 10−3 m s−1, or 3× 104 m y−1. Even for phreatic slopes as low asε = 10−2, the recharge parameter r <∼ O(1), and shallow aquifer drainage is feasible.

0

0.2

0.4

0.6

0.8

1

-3 -2 -1 0 1 2 3

X

z

Figure 3.1: Groundwater flow lines towards a river at X = 0, z = 1.

66

However, finer-grained sediments are less permeable, and the calculation of r fora silt with permeability of 10−14 m2 (K = 10−7 m s−1 = 3 m y−1 suggests thatr ∼ 1/ε2 1, so that if the Dupuit approximation were applicable, the groundwatersurface would lie above the Earth’s surface everywhere. This simply points out theobvious fact that if the groundmass is insufficiently permeable, drainage cannot occurthrough it but water will accumulate at the surface and drain by overland flow. Thefact that usually the water table is below but quite near the surface suggests thatthe long term response of landscape to recharge is to form topographic gradients andsufficiently deep sedimentary basins so that this status quo can be maintained.

3.3.3 The seepage face

A particular case where a seepage face may be appropriate is the situation where waterleaks through a porous wall from a reservoir. If the wall is in 0 < x < 1 (dimen-sionlessly), then the problem is essentially two-dimensional, and suitable boundaryconditions to add to (3.37) for the governing equations (3.36) are

h = 1 at x = 0,

p = 0 at x = 1, z < h(1), (3.54)

and we take r = 0 in (3.44), so that

ht = (hhx)x. (3.55)

A steady solution is possible satisfying (3.54) providing there is no seepage face, andwe choose h = 0 at x = 1; then

h = (1− x)1/2. (3.56)

However, this approximation is not uniformly valid at x = 1, because w = zhxx ∼1

1− x as x→ 1. This motivates us to rescale the variables as

x = 1− ε2X, z = εZ, h = εH, p = εP, w =W

ε2, u =

U

ε, (3.57)

and then the equations (3.36) become

U = PX ,

PZ + 1 = −W,−UX +WZ = 0; (3.58)

the boundary conditions (3.37) and (3.54)2 are

W = 0 at Z = 0,

P = 0, W + PXHX = 0 on Z = H(X),

P = 0 on X = 0, 0 < Z < H(0), (3.59)

67

and the matching condition to the outer solution requires

P ∼ H − Z, H ∼√X as X →∞. (3.60)

It is convenient to defineΠ = P + Z, (3.61)

and to define a stream function ψ such that

U = ψZ = ΠX ,

W = ψX = −ΠZ . (3.62)

Both Π and ψ satisfy Laplace’s equation, indeed Π + iψ is evidently an analyticfunction of X + iZ. The boundary conditions for Π are

Π = H,∂Π

∂n= 0 on Z = H,

Π = Z on X = 0,

ΠZ = 0 on Z = 0,

Π ∼ H ∼√X as X →∞. (3.63)

Alternatively, we can solve Laplace’s equation for ψ, and the corresponding conjugateboundary conditions are

ψ = 12

on Z = H,

ψX = −1 on X = 0,

ψ = 0 on Z = 0,

ψ ∼ Z

2√X

as X →∞. (3.64)

One way of approaching this problem is to use the hodograph transformation. Wecan consider the problem of finding X + iZ as an analytic function of Π + iψ, wherethe domain is mapped as shown in figure 3.2. The boundary conditions then take theform:

Z = Π, X = H−1(Π) on ψ = 12

(AB);

X = 0, Z = Π > 0 on Π = g(ψ) (BC);

Z = 0 on ψ = 0 (CD);

X + iZ ∼ (Π + iψ)2 as Π→∞ (AD). (3.65)

The determination of H uncouples from the problem, but the location of the boundaryBC is determined by the unknown function g(ψ), which is in principle determinedby the extra condition in (3.65)2. There seems to be no simple way to do this.

68

A

B

C

AB

C DD

Z

X Π

ψ

physical space hodograph plane

Figure 3.2: The physical boundary layer space (left) and the corresponding hodographplane (right).

We simplify the problem by writing

X + iZ = (Π + iψ)2 + χ(ζ), (3.66)

where ζ = Π + iψ. This satisfies the conditions on AB, BC and AD provided χ isbounded at ∞, and Imχ = 0 on ψ = 0, 1

2. Since v = Imχ is harmonic, we can write

v = −∞∑

n=1

an sin(2nπψ) exp[−2nπΠ], (3.67)

which derives from the choice

χ =∞∑

n=1

ane−2nπζ . (3.68)

The problem thus reduces to finding a curve BC in the complex ζ space, which startsat ζ = 0 and ends at ζ = Π0 + 1

2i for some Π0 > 0, on which

iΠ = ζ2 +∞∑

n=1

ane−2nπζ . (3.69)

How to do this is not so obvious, however, but one thing is clear: there is no solutionin which BC is vertical in the hodograph plane, corresponding to coalescence of Band C in physical space, and the lack of a seepage face. For then ψ2 =

∑∞1 ane

−2nπiψ,and in particular

∑∞1 an sin 2nπiψ is the Fourier sine series of 0, and thus an ≡ 0,

which is impossible.

3.4 Unsaturated soils

Let us now consider flow in the unsaturated zone. Above the water table, water andair occupy the pore space. If the porosity is φ and the water volume fraction per

69

σas

σaw

σws

airsoilgrain

watersoil

water θ

air

Figure 3.3: Configuration of air and water in pore space. The contact angle θ mea-sured through the water is acute, so that water is the wetting phase. σws, σas andσaw are the surface energies of the three interfaces.

unit volume of soil is W , then the ratio S = W/φ is called the relative saturation.If S = 1, the soil is saturated, and if S < 1 it is unsaturated. The pore space ofan unsaturated soil is configured as shown in figure 3.3. In particular, the air/waterinterface is curved, and in an equilibrium configuration the curvature of this interfacewill be constant throughout the pore space. The value of the curvature depends on theamount of liquid present. The less liquid there is (i. e., the smaller the value of S), thenthe smaller the pores where the liquid is found, and thus the higher the curvature.Associated with the curvature is a suction effect due to surface tension across theair/water interface. The upshot of all this is that the air and water pressures arerelated by a capillary suction characteristic function which expresses the differencebetween the pressures as a function of mean curvature, and hence, directly, S:

pa − pw = f(S). (3.70)

The suction characteristic f(S) is equal to 2σκ, where κ is the mean interfacialcurvature: σ is the surface tension. For air and water in soil, f is positive as water isthe wetting phase, that is, the contact angle at the contact line between air, water andsoil grain is acute, measured through the water (see figure 3.3). The resulting formof f(S) displays hysteresis as indicated in figure 3.4, with different curves dependingon whether drying or wetting is taking place.

3.4.1 The Richards equation

To model the flow, we have the conservation of mass equation in the form

∂(φS)

∂t+ ∇.u = 0, (3.71)

where we take φ as constant. Darcy’s law for an unsaturated flow has the form, withgravitational acceleration included,

u = −k(S)

µ[∇p+ ρgk], (3.72)

70

pa_ pw

S1

drying

wetting

Figure 3.4: Capillary suction characteristic. It displays hysteresis in wetting anddrying.

where k is a unit vector upwards, and the permeability k depends on S. If k(1) = k0

(the saturated permeability), then one commonly writes k = k0krw(S), where krwis the relative permeability. The most obvious assumption would be krw = S, butthis is rarely appropriate, and a better representation is a convex function, such as

krw = S3. An even better representation is a function such as krw =

(S − S0

1− S0

)3

+

,

where S0 is known as the residual saturation. It represents the fact that in fine-grainedsoils, there is usually some minimal water fraction which cannot be removed. It isnaturally associated with a capillary suction characteristic function pa − p = f(S)which tends to infinity as S → S0+, also appropriate for fine-grained soils.

In one dimension, and if we take the vertical coordinate z to point downwards, weobtain the Richards equation

φ∂S

∂t= − ∂

∂z

[k0

µkrw(S)

∂f

∂z+ ρg

]. (3.73)

We are assuming pa = constant (and also that the soil matrix is incompressible).

3.4.2 Non-dimensionalisation

We choose scales for the variables as follows:

f =σ

dpψ, z ∼ σ

ρgdp, t ∼ φµz

ρgk0

, (3.74)

where dp is grain size and σ is the surface tension, assumed constant. The Richardsequation then becomes, in dimensionless variables,

∂S

∂t= − ∂

∂z

[krw

(∂ψ

∂z+ 1

)]. (3.75)

71

To be specific, we consider the case of soil wetting due to surface infiltration: ofrainfall, for example. Suitable boundary conditions for infiltration are

S = 1 at z = 0 (3.76)

if surface water is ponded, or

krw

(∂ψ

∂z+ 1

)= u∗ =

µu0

k0ρwg=µu0

K0

, (3.77)

if there is a prescribed downward flux u0; K0 is the saturated hydraulic conductivity.In a dry soil we would have S → 0 as z → ∞, or if there is a water table at z = zp,S = 1 there.4 For silt with k0 = 10−14 m2, the hydraulic conductivity K0 ∼ 10−7 ms−1 or 3 m y−1, while average rainfall in England, for example, is ≤ 1 m y−1. Thuson average u∗ ≤ 1, but during storms we can expect u∗ 1. For large values of u∗,the desired solution may have S > 1 at z = 0; in this case ponding occurs (as oneobserves), and (3.77) is replaced by (3.76), with the pond depth being determined bythe balance between accumulation, infiltration, and surface run-off.

3.5 Immiscible two-phase flows: the Buckley-Leverett

equation

In some circumstances, the flow of more than one phase in a porous medium isimportant. The type example is the flow of oil and gas, or oil and water (or allthree!) in a sedimentary basin, such as that beneath the North Sea. Suppose thereare two phases; denote the phases by subscripts 1 and 2, with fluid 2 being the wettingfluid, and S is its saturation. Then the capillary suction characteristic is

p1 − p2 = pc(S), (3.78)

with the capillary suction pc being a positive, monotonically decreasing function ofsaturation S; mass conservation takes the form

−φ∂S∂t

+ ∇.u1 = 0,

φ∂S

∂t+ ∇.u2 = 0, (3.79)

where φ is (constant) porosity, and Darcy’s law for each phase is

u1 = −k0

µ1

kr1[∇p1 + ρ1gk],

u2 = −k0

µ2

kr2[∇p2 + ρ2gk], (3.80)

4With constant air pressure, continuity of S follows from continuity of pore water pressure.

72

-0.05

0

0.05

0.2 0.4 0.6 0.8 1

V

S

Figure 3.5: Graph of dimensionless wave speed V (S) as a function of wetting fluidsaturation, indicating the speed and direction of wave motion (V > 0 means wavesmove upwards) if the wetting fluid is more dense. The viscosity ratio µr (see (3.88))is taken to be 30.

with kri being the relative permeability of fluid i.For example, if we consider a one-dimensional flow, with z pointing upwards, then

we can integrate (3.79) to yield the total flux

u1 + u2 = q(t). (3.81)

If we define the mobilities of each fluid as

Mi =k0

µikri, (3.82)

then it is straightforward to derive the equation for S,

φ∂S

∂t= − ∂

∂z

[Meff

q

M1

+∂pc∂z

+ (ρ1 − ρ2)g

], (3.83)

where the effective mobility is determined by

Meff =

(1

M1

+1

M2

)−1

. (3.84)

This is a convective-diffusion equation for S. If suction is very small, we obtainthe Buckley-Leverett equation

φ∂S

∂t+

∂

∂z

[Meff

q

M1

+ (ρ1 − ρ2)g

]= 0, (3.85)

73

which is a nonlinear hyperbolic wave equation. As a typical situation, suppose q = 0,and kr2 = S3, kr1 = (1− S)3. Then

Meff =k0S

3(1− S)3

µ1S3 + µ2(1− S)3, (3.86)

and the wave speed v(S) is given by

v = −(ρ2 − ρ1)gM ′eff(S) = v0V (S), (3.87)

where

v0 =(ρ2 − ρ1)gk0

µ2

, V (S) =χ′(S)

χ(S)2,

χ(S) =µr

(1− S)3+

1

S3, µr =

µ1

µ2

. (3.88)

The variation of V with S is shown in figure 3.5. For ρ2 > ρ1 (as for oil and water,where water is the wetting phase), waves move upwards at low water saturation anddownwards at high saturation.

Shocks will form, but these are smoothed by the diffusion term − ∂

∂z

[Meffp

′c

∂S

∂z

],

in which the diffusion coefficient is

D = −Meff p′c. (3.89)

As a typical example, take

pc =p0(1− S)λ1

Sλ2(3.90)

with λi > 0. Then we find

D = k0p0S2−λ2(1− S)2+λ1

[λ1S + λ2(1− S)

µ1S3 + µ2(1− S)3

], (3.91)

and we see that D is typically degenerate at S = 0. In particular, if λ2 < 2, theninfiltration of the wetting phase into the non-wetting phase proceeds at a finite rate,and this always occurs for infiltration of the non-wetting phase into the wetting phase.

A particular limiting case is when one phase is much less dense than the other,the usual situation being that of gas and liquid. This is exemplified by the problemof snow-melt run-off considered earlier. In that case, water is the wetting phase, thusρ2 − ρ1 = ρw − ρa is positive, and also µw ≈ 10−3 Pa s, µa ≈ 10−5 Pa s, whenceµa µw (µr 1), so that, from (3.86),

Meff ≈k0S

3

µw, (3.92)

at least for saturations not close to unity. Shocks form and propagate downwards(since ρ2 > ρ1). The presence of non-zero flux q < 0 does not affect this statement.

74

Interestingly, the approximation (3.92) will always break down at sufficiently highsaturation. Inspection of V (S) for µr = 0.01 (as for air and water) indicates that(3.92) is an excellent approximation for S <∼ 0.5, but not for S >∼ 0.6; for S >∼ 0.76,V is positive and waves move upwards. As µr → 0, the right hand hump in figure3.5 moves towards S = 1, but does not disappear; indeed the value of the maximumincreases, and is V ∼ µ

−1/3r . Thus the single phase approximation for unsaturated

flow is a singular approximation when µr 1 and 1− S 1.

3.6 Dispersion

Much of the interest in modelling groundwater flow lies in the prediction of solutetransport, in particular in understanding how pollutants will disperse: for example,how do nitrates used for agricultural purposes disperse via the local groundwatersystem? Mostly simply, one would simply add a diffusion term to the advection ofthe solute concentration c:

φct + u.∇c = ∇.[φD∇c]. (3.93)

The diffusive width ∆l of a sharp front travelling at speed u after it has travelled adistance l is of the order of ∆l ∼ (Dl/u)1/2; if we take D ∼ 10−9 m2 s−1, u ∼ 10−6 ms−1 (30 m y−1), l = 103 m, then ∆l ∼ 1 m, and the diffusion zone is relatively narrow.For a more porous sand, the diffusion width is even smaller.

In fact, as velocity increases, the effect of diffusion increases. That this is so isdue to a remarkable phenomenon called Taylor dispersion, described by G. I. Taylorin 1953. Consider the diffusion of a solute in a tube of circular cross section throughwhich a Poiseuille flow passes. If the mean velocity is U and the tube is of radius a,then the velocity is 2U(1− r2/a2), and the concentration satisfies the equation

ct + 2U

(1− r2

a2

)cx = D

(crr +

1

rcr + cxx

), (3.94)

where x is measured along the tube, and r is the radial coordinate. Taylor showed,rather ingenuously, that when the Peclet number Pe = aU/D is large, then the effectof the diffusion term in (3.94) is to disperse the mean solute concentration diffusivelyabout the position of its centre of mass, x = Ut, with a dispersion coefficient ofa2U2/48D. Aris later improved this to

DT =a2U2

48D+D, (3.95)

which is asymptotically valid for x a. The dispersive mechanism is due to the radialvariation of the velocity profile, which can disperse the solute even if the diffusioncoefficient is very small.

75

3.6.1 The Taylor–Aris dispersion coefficient

In order to derive the Taylor–Aris dispersion coefficient (3.95), we firstly scale theequation (3.94) by writing

r ∼ a, x ∼ l, t ∼ l

U, (3.96)

and define

ε =a

l, Pe =

aU

D. (3.97)

The scaled form of (3.94) is then

ct + 2(1− r2)cx =1

εPe

(crr +

1

rcr + ε2cxx

). (3.98)

We assume ε 1, Pe ∼ 1 (or Pe 1), and then write c as an expansion in powersof εPe 1:

c ∼ c(0) + εPe c(1) + . . . . (3.99)

This gives the sequence of equations

c(0)rr +

1

rc(0)r = 0,

c(1)rr +

1

rc(1)r = c

(0)t + 2(1− r2)c(0)

x ,

c(2)rr +

1

rc(2)r = − 1

Pe2c(0)xx + c

(1)t + 2(1− r2)c(1)

x , (3.100)

and so on.Solving these sequentially subject to the boundary conditions that

∂c(i)

∂r= 0 at r = 0, 1, (3.101)

leads first toc(0) = c(0)(x, t), (3.102)

and thenc(1)r = 1

2rc

(0)t +

(r − 1

2r3)c(0)x , (3.103)

and the boundary condition at r = 1 requires

c(0) = C(0)(x− t). (3.104)

Because of this, it is at this point convenient to rewrite the model in terms ofξ = x− t rather than x, and also to introduce a slow time τ = εPe t, in anticipationof the fact that secular terms will appear at O(εPe2). The only difference thismakes is to replace

∂

∂t−→ ∂

∂t− ∂

∂ξ+ (εPe)2 ∂

∂τ, (3.105)

76

and then (3.104) is replaced by

c(0) = C(0)(ξ, τ), (3.106)

and (3.103) can be written as

c(1)r =

(12r − 1

2r3)c

(0)ξ (3.107)

(since C(0)t = 0), with solution

c(1) = C(1)(ξ, τ) + (14r2 − 1

8r4)c

(0)ξ . (3.108)

In terms of ξ and τ , the equation for c(2) is modified to

c(2)rr +

1

rc(2)r = − 1

Pe2c

(0)ξξ + c

(1)t + (1− 2r2)c

(1)ξ + c(0)

τ , (3.109)

and using (3.106) and (3.107), this gives

(rc(2)r )r = r

[C(0)τ −

1

Pe2C

(0)ξξ

]+ (r − 2r3)

[C

(1)ξ + (1

4r2 − 1

8r4)C

(0)ξξ

]. (3.110)

Integration from r = 0 to r = 1 using (3.101) now shows that c(0) = C(0)(ξ, τ) mustsatisfy

c(0)τ =

[1

48+

1

Pe 2

]c

(0)ξξ . (3.111)

Thus the Taylor–Aris dispersion coefficient relative to the mean flow is

a2U2

48D+D. (3.112)

Note thataU

D 1 is not required, only that a l (or equivalently t a/U).

3.6.2 Dispersion in practice

Typically, the dispersion coefficent is generalised for porous media (where we thinkof the pores as being like Taylor’s tube) by writing the dispersion coefficient as

DT = D∗ +D‖, (3.113)

where D∗ represents molecular diffusion and D‖ dispersion in the direction of flow.The tortuosity of the flow paths and the possibility of adsorption on to the solidcauses D∗ to be less than D, and ratios D∗/D between 0.01 and 0.5 are commonlyobserved. In porous media, re-mixing at pore junctions causes the dependence of D‖on the flow velocity to be less than quadratic, and a relation of the form

D‖ = αum, (3.114)

77

where u is the Darcy flux, fits experimental data reasonably well for values 1 <m < 1.2. A common assumption is to take m = 1. Mixing at junctions also causestransverse dispersion to occur, with a coefficient D⊥ which is measured to be lessthan D‖ by a factor of order 102 when Pe 1. Dispersion is thus a tensor property.

If we writeD‖ = α‖|u| (3.115)

for the longitudinal dispersion coefficient, and

D⊥ = α⊥|u| (3.116)

for the lateral dispersion coefficient, then a suitable tensor generalisation is

DTij = α⊥|u|δij + (α‖ − α⊥)

uiuj|u| , (3.117)

where δij is the Kronecker delta. The conservation of solute equation is then

φ∂c

∂t+ u.∇c = ∇.(φDT.∇c] =

∂

∂xi

(φDT

ij

∂c

∂xj

). (3.118)

For a one-dimensional flow in the x direction, c satisfies

∂c

∂t+ v

∂c

∂x=

∂

∂x

[D‖

∂c

∂x

]+

∂

∂y

[D⊥

∂c

∂y

]+

∂

∂z

[D⊥

∂c

∂z

](3.119)

(v = u/φ is the linear velocity) and if the dispersivities are constant, then the solutionfor release of a mass M at the origin at t = 0 is

c =M

8φ(πD‖)1/2D⊥t3/2exp

[−(x− vt)2

4D‖t− r2

4D⊥t

], (3.120)

where r2 = y2 + z2.

3.7 Notes and references

3.7.1 Taylor dispersion

G. I. Taylor was a leading fluid dynamicist at the University of Cambridge, who can beconsidered as the originator of the famous Cambridge school of applied mathematics,which takes as its general philosophy the idea that laboratory experiments are anessential component of the proper practice of applied mathematics, a principle thathas been successfully exported round the world, although the concept still provokesresistance in many mathematics departments, which stubbornly live on in a Cauchy-esque world of abstraction. Taylor followed the lead of Rayleigh, and is justly famousfor many fundamental insights into various aspects of fluid and solid mechanics. Apartfrom Taylor dispersion, we have the Rayleigh–Taylor instability, the Taylor–Greenvortex, the Taylor column, Taylor–Couette flow, etc., etc. Taylor’s (1953) paper

78

introducing his eponymous dispersion coefficient is a typical masterpiece. At thetime, multiple scale methods were barely if at all invented, and Taylor’s treatmentbypasses the method with a bravura page of succinct insight. As is so often the casein these older papers, the scientific intuition of the author is a worthy substitute forthe primitive analytic tools available at the time.

The dispersion coefficient is sometimes called the Taylor–Aris dispersion coeffi-cient after the paper by Aris (1956) which extended Taylor’s analysis using a derivedsequence of moment equations. Again, there is no sign of an asymptotic procedure tobe seen; this, after all, is still several years before the initial application of multiplescale methods to hydrodynamic stability theory (Malkus and Veronis 1958, Stuart1960).

Exercises

3.1 Show that for a porous medium idealised as a cubical network of tubes, thepermeability is given (approximately) by k = d2

pφ2/72π, where dp is the grain

size. How is the result modified if the pore space is taken to consist of pla-nar sheets between identical cubical blocks? (The volume flux per unit widthbetween two parallel plates a distance h apart is −h3p′/12µ, where p′ is thepressure gradient.)

3.2 A sedimentary rock sequence consists of two type of rock with permeabilitiesk1 and k2. Show that in a unit with two horizontal layers of thickness d1 andd2, the effective horizontal permeability (parallel to the bedding plane) is

k‖ = k1f1 + k2f2,

where fi = di/(d1 + d2), whereas the effective vertical permeability is given by

k−1⊥ = f1k

−11 + f2k

−12 .

Show how to generalise this result to a sequence of n layers of thickness d1, . . . , dn.

Hence show that the effective permeabilities of a thick stratigraphic sequencecontaining a distribution of (thin) layers, with the proportion of layers havingpermeabilities in (k, k + dk) being f(k)dk, are given by

k‖ =

∫ ∞

0

kf(k) dk, k−1⊥ =

∫ ∞

0

f(k) dk

k.

3.3 Groundwater flows between an impermeable basement at z = hb(x, y, t) anda phreatic surface at z = zp(x, y, t). Write down the equations governing theflow, and by using the Dupuit approximation, show that the saturated depth hsatisfies

φht =kρg

µ∇.[h∇zp],

79

where ∇ = (∂/∂x, ∂/∂y). Deduce that a suitable time scale for flows in anaquifer of typical depth h0 and extent l is tgw = φµl2/kρgh0.

I live a kilometer from the river, on top of a layer of sediments 100 m thick(below which is impermeable basement). What sort of sediments would thoseneed to be if the river responds to rainfall at my house within a day; within ayear?

3.4 A two-dimensional earth dam with vertical sides at x = 0 and x = l has areservoir on one side (x < 0) where the water depth is h0, and horizontal dryland on the other side, in x > l. The dam is underlain by an impermeablebasement at z = 0.

Write down the equations describing the saturated groundwater flow, and showthat they can be written in the dimensionless form

u = −px, ε2w = −(pz + 1),

pzz + ε2pxx = 0,

and define the parameter ε. Write down suitable boundary conditions on theimpermeable basement, and on the phreatic surface z = h(x, t).

Assuming ε 1, derive the Dupuit-Forchheimer approximation for h,

ht = (hhx)x in 0 < x < 1.

Show that a suitable boundary condition for h at x = 0 (the dam end) is

h = 1 at x = 0.

Now define the quantity

U =

∫ h

0

p dz,

and show that the horizontal flux

q =

∫ h

0

u dz = −∂U∂x

.

Hence show that the conditions of hydrostatic pressure at x = 0 and constant(atmospheric) pressure at x = 1 (the seepage face) imply that

∫ 1

0

q dx = 12.

Deduce that, if the Dupuit approximation for the flux is valid all the way tothe toe of the dam at x = 1, then h = 0 at x = 1, and show that in the steadystate, the (dimensional) discharge at the seepage face is

qD =kρgh2

0

2µl.

80

Supposing the above description of the solution away from the toe to be valid,show that a possible boundary layer structure near x = 1 can be described bywriting

x = 1− ε2X, h = εH, z = εZ, p = εP,

and write down the resulting leading order boundary value problem for P .

3.5 I get my water supply from a well in my garden. The well is of depth h0 (relativeto the height of the water table a large distance away) and radius r0. Show thatthe Dupuit approximation for the water table height h is

φ∂h

∂t=kρg

µ

1

r

∂

∂r

(rh∂h

∂r

).

If my well is supplied from a reservoir at r = l, where h = h0, and I withdrawa constant water flux q0, find a steady solution for h, and deduce that my wellwill run dry if

q0 >πkρgh2

0

µ ln[l/r0].

Use plausible values to estimate the maximum yield (gallons per day) I can useif my well is drilled through sand, silt or clay, respectively.

3.6 A volume V of effluent is released into the ground at a point (r = 0) at time t.Use the Dupuit approximation to motivate the model

φ∂h

∂t=kρg

µ

1

r

∂

∂r

(rh∂h

∂r

),

h = h0 at t = 0, r > 0,∫ ∞

0

r(h− h0)dr = V/2π, t > 0,

where h0 is the initial height of the water table above an impermeable basement.Find suitable similarity solutions in the two cases (i) h0 = 0 (ii) h0 > 0, h−h0 h0, and comment on the differences you find.

3.7 Rain falls steadily at a rate q (volume per unit area per unit time) on a soil ofsaturated hydraulic conductivity K0 (= k0ρwg/µ, where k0 is the saturated per-meability). By plotting the relative permeability krw and suction characteristicσψ/d as functions of S (assuming a residual liquid saturation S0), show that areasonable form to choose for krw(ψ) is krw = e−cψ. If the water table is at depthh, show that, in a steady state, ψ is given as a function of the dimensionlessdepth z∗ = z/zc, where zc = σ/ρwgd (σ is the surface tension, d the grain size)by

h∗ − z∗ = 12ψ − 1

cln

[sinh1

2(ln 1

q∗− cψ)

sinh12

ln 1q∗

],

where h∗ = h/zc, providing q∗ = q/K0 < 1. Deduce that if h zc, thenψ ≈ 1

cln 1

q∗near the surface. What happens if q > K0?

81

3.8 Derive the Richards equation

φ∂S

∂t= − ∂

∂z

[k(S)

µ

∂pc∂z

+ ρg

]

for one-dimensional infiltration of water into a dry soil, explaining the meaningof the terms, and give suitable boundary conditions when the surface flux q isprescribed.

Non-dimensionalise the model, and show that if the surface flux is large com-pared with kρwg/µ, then the Richards equation can be approximated by anonlinear diffusion equation of the form

∂S

∂t=

∂

∂z

[D∂S

∂z

],

and give the definition of D.

Show that, if D = Sm, a similarity solution exists in the form

S = tαF (η), η = z/tβ,

where α =1

m+ 2, β =

m+ 1

m+ 2, and F satisfies

(FmF ′)′ = αF − βηF ′, FmF ′ = −1 at η = 0, F → 0 as η →∞.

Deduce that

FmF ′ = −∫ η0

η

Fdη − βηF,

where η0 (which may be ∞) is where F first reaches zero. Deduce that F ′ < 0,and hence that η0 must be finite, and is determined by

∫ η0

0

F dη = 1.

What happens for t > F (0)−1/α?

3.9 Fluid flows through a porous medium in the x direction at a linear velocity U .At t = 0, a contaminant of concentration c0 is introduced at x = 0. If thelongitudinal dispersivity of the medium is D, write down the equation whichdetermines the concentration c in x > 0, together with suitable initial andboundary conditions. Hence show that c is given by

c

c0

=1

2

[erfc

x− Ut2√Dt

+ exp

(Ux

D

)erfc

x+ Ut

2√Dt

],

where

erfc ξ =2√π

∫ ∞

ξ

e−s2

ds.

82

[Hint: you might try Laplace transforms, or else simply verify the result.]

Show that for large ξ, erfc ξ = e−ξ2

[1√πξ

+ . . .

], and deduce that if x =

Ut+ 2√Dt η, with η = O(1), then

c

c0

≈ 1

2erfc η +O

(1√t

).

Hence show that at a fixed station x = X far downstream, the measured profileis approximately given by

c ≈ c0

[1− 1

2erfc

1

2

(U3

DX

)1/2(t− X

U

)].

This is called the breakthrough curve, and indicates that dispersion causesbreakthrough to occur over a time interval (at large distance) of order ∆tb =(DX/U3)1/2. If D ≈ aU , show that the ratio of ∆tb to tb = X/U is ∆tb/tb ∼(a/X)1/2.

3.10 A blob of contaminant of volume Q is placed at a point in an infinite satu-rated porous medium, and disperses with dispersivity D. Assuming there isno groundwater motion, find the concentration field c(r, t) at later times, as-suming a spherically symmetric solution. [It may be helpful to seek a similaritysolution.]

Plot roughly the evolution with time of the concentration field at a fixed valueof r. Show that if the threshold of detection is a concentration c0, then thecontaminant will remain undetected if

r >

√3

2πe

(Q

c0

)1/3

.

83

Chapter 4

Instability

Stability and instability are fundamental concepts for the applied mathematician. Instudying any mathematical model, we always start at the simplest level, which isthat of the steady state. The first question we can then ask is, is this state stable?The meaning of the question is simply illustrated by standing a pencil on its end.Tap it slightly and it falls over. And since in reality, all physical systems are subjectto perturbation, the practical existence of a mathematical solution representing asteady state only has physical meaning if it is stable, that is to say, when subjected toperturbations, the disturbances die away.1 In this chapter, we will discuss a varietyof fluid mechanical examples in which instability occurs.

4.0.1 General procedures

Suppose we have a nonlinear evolution equation of the form

ut = N (u) in V, B(u) = 0 on ∂V, (4.1)

where N represents a nonlinear partial differential operator and B(u) = 0 representsthe corresponding boundary conditions. Steady states, also called fixed points, equi-libria, critical points, are time-independent functions u0(x) such that N (u0) = 0 andalso satisfy the boundary conditions.

We study the stability of a steady state by putting u = u0 + U, and expand as

N (u) = N (u0) + LU + . . . ,

B(u) = B(u0) +MU + . . . , (4.2)

where the dots indicate quadratic terms. L and M are linear operators, and if weneglect the smaller quadratic terms, we have the linear problem

Ut = L(U) in V, M(U) = 0 on ∂V. (4.3)

1Poincare made the distinction between stability (small disturbances do not grow: the undampedpendulum) and asymptotic stability (they die away: the damped pendulum), but in reality, every-thing is damped, and conservative (undamped) systems are a (useful) mathematical abstraction.

84

This all seems quite vague, but as we shall see in practical examples, concreteresults are easily obtained. Typically, L is a linear partial differential operator. Usu-ally N is autonomous, i. e., independent of time, and in this case (4.3) has separablesolutions U = eσtψ(x).These are called normal modes, and the resulting equation forψ is an eigenvalue problem, of the form

Lψ = σψ in V, Mψ = 0 on ∂V ; (4.4)

ψ is an eigenfunction with corresponding eigenvalue σ, and usually there are aninfinite number of eigenvalues. In a finite domain V , these will form a countable set,and the corresponding eigenfunctions frequently form a complete orthonormal basisfor the general solution of (4.3). In this case the analysis of the eigenvalue problemprovides a complete solution for the linearised problem.

In the case where u0 is independent of x, the linearised system is also autonomousin space, and then the normal mode solutions take the form ψ = weik.x, wherek = (k1, k2, k3) is the wave vector, and the constant vector w is given by a matrixeigenvalue problem

Lw = σw, (4.5)

where, if L = L(∇), then L = L(ik) (for example if L = ∇2, then L = −k2, wherek = |k|). Thus in many examples, the calculation of the eigenvalues σ becomes apurely algebraic problem, and leads to an expression for σ(k), which is actually thesame as the dispersion relation we discussed in chapter 1. Stability is then assessedin the following way: if any of the eigenvalues σ has Reσ > 0, then the steady stateis unstable; but if all the σ have Reσ < 0, the steady state is stable.

4.0.2 Nonlinear diffusion

Let us begin by illustrating this discussion with the model

ut = uxx + V ′(u), u(±1, t) = 0, (4.6)

commonly referred to as a nonlinear diffusion equation, although it is the source whichis nonlinear, not the diffusion. Let us take V (0) = V ′(0) = 0, V ′′(0) > 0, and assumeV is symmetric. As an example, V = 1

2u2 (although actually that choice gives a linear

problem!).Evidently u = 0 is a steady state, and linearisation about this gives the linear

equation ut = uxx + V ′′(0)u, which has normal mode solutions u = eσt+ikx providingσ = V ′′(0)− k2. On an infinite domain, where k can take any value, the steady stateis always unstable. However, on the finite domain, the boundary conditions constrainthe values of k. Specifically, the symmetric modes are proportional to cos kx, and thevalues of k = (n− 1

2)π, where n is an integer (the anti-symmetric modes are sinmπx,

m an integer)). The most unstable mode is that with n = 1, in consequence of whichσ > 0 if V ′′(0) > 1

4π2.

We might ask what happens if V ′′(0) > 14π2? There are other possible steady

states of (4.6), formed from the first integral of the conservative oscillator,

12u2x + V (u) = E. (4.7)

85

With V ′′ > 0, the solutions of this are periodic, and we need to select the value(s) of Esuch that the period is 4 (since evidently the interval (−1, 1) represents a half-period).Supposing we have such a solution u0(x), the linearised equation for perturbations tothis steady state is Ut = Uxx + q(x)U , where q = V ′′[u0(x)], and the normal modesU = eσty(x) are given by the solution of the Sturm–Liouville system

y′′ + [q(x)− σ]y = 0, y(±1) = 0, (4.8)

about which a good deal is known. For the system (4.8), there are a countable numberof real eigenvalues σ1 > σ2 > . . ., with the maximum value being characterised bythe variational principle

σ1 = max

∫ 1

−1

[q(x)y2 − y′2] dx

∫ 1

−1

y2 dx

, (4.9)

and this can be used to provide useful constraints on the stability of the solutions. Inparticular, if V ′′ decreases away from x = 0, the resulting steady state is likely to bestable. More precise information can be gained through a weakly nonlinear stabilityanalysis.

4.1 Saffman–Taylor instability

The first fluid example of instability which we consider is the phenomenon of viscousfingering. It is an easy phenomenon to experimentally realise. On a glass table orsurface (an overhead projector is good for demonstration) squirt a little washing-upliquid. Then gently place a flat glass plate on top of the liquid. If you now pressdown, the liquid will expand, more or less symmetrically: the interface position isstable. now pull the glass plate upwards. Instead of retreating symmetrically, theinterface forms a petal-like structure: it is unstable.

What is happening is an instance of the Saffman–Taylor instability: when a lessviscous fluid displaces a more viscous one, the interface between them is subject to aspatial instability, which leads to a fingering appearance in the interface.

It is very simple to understand this instability, which is commonly done in thecontext of a porous medium, or the mathematically equivalent Hele-Shaw cell (likeour experimental example above). To be specific, we will consider the advance of afluid of viscosity µ− and density ρ− in the vertical direction towards a fluid of viscosityµ+ and density ρ+; the density difference is not necessary for the discussion, in fact.The situation under consideration is shown in figure 4.1.

We label the properties of the fluid above (below) the interface with a + (−). Theequations for incompressible porous flow are actually linear, and are

u = −Π

µ

[∇p+ ρgk

], ∇.u = 0, (4.10)

86

zz = ζ

W

+

W

_

Figure 4.1: Geometric configuration for the Saffman–Taylor instability.

where Π is the permeability, and µ and ρ are the relevant viscosity and density. Wesuppose the fluid flux upwards is W at ±∞, and the conditions at the interface are

w = φζt + uζx + vζy at z = ζ (4.11)

(on both sides) and, if we allow for a surface energy term,

[p]+− = γ∇.

[ ∇ζ

[1 + |∇ζ|2]1/2

], (4.12)

where γ is the surface tension.The basic state is the uniform flow

u = (0, 0,W ), ζ =Wt

φ, (4.13)

whence we have

p± = −[µ±W

Π+ ρ±g

](z − Wt

φ

), (4.14)

which is continuous at the interface z = ζ as required by (4.12).In the perturbed state, we write

p± = −[µ±W

Π+ ρ±g

](z − Wt

φ

)+ P±, z =

Wt

φ+ Z, (4.15)

thus ∇2P± = 0, and the normal mode solutions are

P± = a±eσt+ik.x∓kZ , (4.16)

87

on September 3, 2014rspa.royalsocietypublishing.orgDownloaded from

Figure 4.2: A viscous finger in a channel formed by air advancing into glycerine(Saffman and Taylor 1958).

where k = (k1, k2), x = (x, y), and k = |k|.The interface position is ζ =

Wt

φ+ η, i. e., Z = η, and in the linearised ap-

proximation we take η = aeσt+ik.x. The linearised interface conditions then take theform

φηt = − Π

µ±

∂P±∂Z

, −[µW

Π+ ρg

]+

−η + [P ]+− = γ∇2η at Z = 0, (4.17)

and these lead to

a± = ±µ±φσaΠk

, (4.18)

and thence to the dispersion relation

σ =W∆µ+ ∆ρgΠ− γΠk2k

(µ+ + µ−)φ, (4.19)

where ∆µ = µ+ − µ− and ∆ρ = ρ+ − ρ−.Most simply, if ρ+ = ρ− and γ = 0, the interface is always unstable (σ > 0) if

µ+ > µ−, i. e., the less viscous fluid displaces the more viscous one. The instability isreduced if the upper fluid is less dense, but this is unlikely. Importantly, the growthrate increases indefinitely with k, and this is the sign of an ill-posed model, but theaddition of surface tension causes decay at large wave number, and makes the problemwell-posed.

In practice, the interfacial break-up leads to the formation of long fingers, andhence the phenomenon is called viscous fingering. In a channel, as shown in figure4.2, the fingers form elongate shapes which occupy half the channel width, and forwhich exact solutions of the nonlinear system can be found (using complex variabletheory). In unconfined radial systems, where the less viscous fluid is injected at acentral source, the expanding front breaks up into fingers, which then undergo furthersplitting, eventually forming a quasi-dendritic structure, an example of which is shownin figure 4.3.

88

258 EUROPHYSICS LETTERS

Fig. 1 – Left panel: digitized image of an experimental viscous fingering pattern. Air (black) isinjected into the oil-filled gap (white). The pattern is approximately 22 cm in diameter, cf. [2]. Rightpanel, image courtesy of Ellak Somfai: a di!usion-limited aggregate with 50000000 particles.

a rate proportional to !p, whereas the DLA accretes one particle at a time, changing theLaplacian field after each such growth step. Thus, we refer to viscous fingering and DLA asparallel and serial processes, respectively. In addition, the ultraviolet regularization di!ers; inDLA, p = 0 on the cluster and the regularization is provided by the particle size. In viscousfingering, one solves the problem with the boundary condition p = !", where ! is the surfacetension and " is the local curvature. Finally, viscous fingers are grown in a finite gap and arenot truly 2-dimensional. Accordingly, one can ask whether these two fractal growth problemsare in the same scaling universality class. But to answer this question, one must first definewhat one means by a “scaling universality class.”

Definition of the scaling universality class. having two fractal growth patterns, decidingwhether they belong to the same universality class involves comparing both their geometryand their growth dynamics. The geometric correspondence between the fractal patterns maybe answered by measuring their fractal dimension D0. Denote by Rn the radius of the minimalcircle that contains a fractal pattern. The fractal dimension is defined by how the mass Mn

contained within this circle (number of particles for DLA, area for viscous fingers) scales withRn, Mn ! RD0

n . Measurements of this type indicated a value D0 " 1.71 for both problems,motivating many authors to express the opinion that these two problems are in the sameuniversality class [2,5]. Obviously, the fractal dimension by itself is not su"cient to determinethe growth dynamics, and a more stringent definition of a universality class is necessary.

We propose here that the identity of the scaling properties of the harmonic measure isa su"cient test for two Laplacian growth problems to be in the same universality class.The harmonic measure is the probability for a random walker to hit the boundary of thefractal pattern. It determines the growth in both problems, being proportional to !p at theboundary. Suppose that we know the probability measure µ(s)ds for a random walker tohit an infinitesimal arclength on the fractal boundary. We compute the probability Pi(#) #!

i-th boxµ(s)ds, and then define the generalized dimensions [6] via

Dq # lim!!0

log"N(!)

i=1 P qi (#)

(q $ 1) log #, (1)

Figure 4.3: Digitised image of the formation of a dendritic fingering pattern (Math-iesen et al. 2006, figure 1): air (black) injected centrally into oil (white).

Viscous fingering has become one of those phenomena which has engaged theinterest of a wide variety of sciences (for example the figure 4.3 comes from a studywhich links the phenomenon to that of diffusion-limited aggregation, of interest inthe general area of chaos, fractals and nonlinear science). But it is of wide practicalinterest, particularly because of its importance in the oil industry. Oil extractionoccurs through the drilling of two wells into a sedimentary basin: an extraction welland an injection well. Water is injected into the injection well to drive out the oil,but is thus subject to viscous fingering such as that in figure 4.3. When the waterreaches the extraction well, production is effectively finished, though with much ofthe oil left in the ground.

4.2 Rayleigh–Benard instability

The simplest model of convection is the classical Rayleigh-Benard model in whicha layer of fluid is heated from below, by application of a prescribed temperaturedifference across the layer. Depending on the nature of the boundaries, one may havea no slip condition or a no shear stress condition applied at the bounding surfaces.For the case of mantle convection, one conceives of both the oceans (or atmosphere)and the underlying fluid outer core as exerting no stress on the extremely viscousmantle, so that no stress conditions are appropriate, and in fact it turns out that thisis the simplest case to consider. The geometry of the flow we consider is shown infigure 4.4. It is convenient to assume lateral boundaries, although in a wide layer,these simply represent the convection cell walls, and can be an arbitrary distanceapart.

The equations describing the flow are the Navier-Stokes equations, allied with the

89

wx = 0

Tx = 0

u = 0wx = 0u = 0Tx = 0

z = d

x = a

z = 0x = 0

p = 0, T = T0, w = 0, uz = 0

T = T0 + ∆T , w = 0, uz = 0

Figure 4.4: Geometry of a convection cell.

energy equation and an equation of state, and can be written in the form

ρt + ∇. (ρu) = 0,

ρ[ut + (u .∇)u] = −∇p− ρgk + µ∇2u,

ρcp[Tt + u .∇T ] = k∇2T,

ρ = ρ0[1− α(T − T0)]; (4.20)

in these equations, ρ is the density, u is the velocity, p is the pressure, g is the acceler-ation due to gravity, k is the unit upwards vector, µ is viscosity, cp is the specific heat,T is temperature, k is thermal conductivity, ρ0 is the density at the reference temper-ature T0 at the surface of the fluid layer, and α is the thermal expansion coefficient.The boundary conditions for the flow are indicated in figure 4.4, and correspond toprescribed temperature at top and bottom, no flow through the boundaries, and noshear stress at the boundaries. The lateral boundaries represent stress free ‘walls’,but as mentioned above, these simply indicate the boundaries of the convection cells(across which there is no heat transport, hence the no flux condition for temperature).

To proceed, we non-dimensionalise the variables as follows. We use the convectivetime scale, and a thermally related velocity scale, and use the depth of the box d asthe length scale:

u ∼ κ

d, κ =

k

ρ0cp, t ∼ d2

κ, x ∼ d,

p− [p0 + ρ0g(d− z)] ∼ µκ

d2, T − T0 ∼ ∆T. (4.21)

Here p0 is the (prescribed) pressure at the surface, which we take as constant. Wewould also scale ρ ∼ ρ0, but in the scaled equations below, the density has been

90

algebraically eliminated. The scaled equations take the form

−BTt + ∇. [(1−BT )u] = 0,

1

Pr[1−BT ][ut + (u .∇)u] = −∇p+RaTk +∇2u,

(1−BT )(Tt + u .∇T ) = ∇2T, (4.22)

and the dimensionless parameters are defined as

B = α∆T, Pr =µ

ρ0κ, Ra =

αρ0∆Tgd3

µκ; (4.23)

the parameters Ra and Pr are known as the Rayleigh and Prandtl numbers, respec-tively. The Prandtl number is a property of the fluid; for air it is 0.7, and for waterit is 7. The Rayleigh number is a measure of the strength of the heating. As weshall see, convective motion occurs if the Rayleigh number is large enough, and itbecomes vigorous if the Rayleigh number is large. The parameter B might be termeda Boussinesq number, although this is not common usage.

Suppose we think of values typical for a layer of water in a saucepan. We taked = 0.1 m, µ = 2 × 10−3 Pa s, ∆T = 100 K, α = 3 × 10−5 K−1, ρ0 = 103 kg m−3,κ = 0.3 × 10−6 m2 s−1, g = 9.8 m s−2. Then we have Pr ≈ 7, B ≈ 3 × 10−3, andRa ≈ 5× 107, so that B 1 and Ra 1. This is typically the case. We now makethe Boussinesq approximation, which allows B → 0, and we ignore the terms in Bin (4.22). In words, we assume that the density is constant, except in the buoyancyterm. The mathematical reason for this exception is that, although Ra ∝ B (and soRa → 0 as B → 0), the actual numerical sizes of the two parameters are typicallyvery different. The adoption of the Boussinesq approximation leads to what are calledthe Boussinesq equations of thermal convection:

∇.u = 0,

1

Pr[ut + (u .∇)u] = −∇p+∇2u +RaT k,

Tt + u.∇T = ∇2T, (4.24)

with associated boundary conditions for free slip:

T = 1, u.n = τnt = 0 on z = 0,

T = 0, u.n = τnt = 0 on z = 1, (4.25)

where τnt represents the shear stress.

4.2.1 Linear stability

It is convenient to study the problem of the onset of convection in two dimensions(x, z). In this case we can define a stream function ψ which satisfies

u = −ψz, w = ψx. (4.26)

91

(The sign is opposite to the usual convention; for ψ > 0 this describes a clockwisecirculation.) We eliminate the pressure by taking the curl of the momentum equation(4.24)2, which leads, after some algebra (see also question 4.3), to the pair of equationsfor ψ and T :

1

Pr

[∇2ψt + ψx∇2ψz − ψz∇2ψx

]= RaTx +∇4ψ,

Tt + ψxTz − ψzTx = ∇2T, (4.27)

with the associated boundary conditions

ψ = ∇2ψ = 0 at z = 0, 1,

T = 0 at z = 1,

T = 1 at z = 0. (4.28)

In the absence of motion, u = 0, the steady state temperature profile is linear,

T = 1− z, (4.29)

and the lithostatic pressure is modified by the addition of

p = −Ra2

(1− z)2. (4.30)

(Even if Ra is large, this represents a small correction to the lithostatic pressure, ofrelative size O(B).) The steady state stream function is just

ψ = 0. (4.31)

We define the temperature perturbation θ by

T = 1− z + θ. (4.32)

This yields

1

Pr

[∇2ψt + ψx∇2ψz − ψz∇2ψx

]= ∇4ψ +Ra θx,

θt − ψx + ψxθz − ψzθx = ∇2θ, (4.33)

and the boundary conditions are

ψzz = ψ = θ = 0 on z = 0, 1. (4.34)

In the Earth’s mantle, the Prandtl number is large, and we will now simplify thealgebra by putting Pr =∞. This assumption does not in fact affect the result whichis obtained for the critical Rayleigh number at the onset of convection. The linear

92

stability of the basic state is determined by neglecting the nonlinear advective termsin the heat equation. We then seek normal modes of wave number k in the form

ψ = f(z) eσt+ikx,

θ = g(z) eσt+ikx, (4.35)

whence f and g satisfy (putting Pr =∞)

(D2 − k2)2f + ikRa g = 0,

σg − ikf = (D2 − k2)g, (4.36)

where D = d/dz, andf = f ′′ = g = 0 on z = 0, 1. (4.37)

By inspection, solutions are

f = sinmπz, g = b sinmπz, (4.38)

(n = 1, 2, ...) providing

σ =k2Ra

(m2π2 + k2)2− (m2π2 + k2), (4.39)

which determines the growth rate for the m-th mode of wave number k.Since σ is real, instability is characterised by a positive value of σ. We can see

that σ decreases as m increases; therefore the value m = 1 gives the most unstablevalue of σ. Also, σ is negative for k → 0 or k →∞, and has a single maximum. Sinceσ increases with Ra, we see that σ > 0 (for m = 1) if Ra > Rack, where

Rack =(π2 + k2)3

k2. (4.40)

In turn, this value of the Rayleigh number depends on the selected wave numberk. Since an arbitrary disturbance will excite all wave numbers, it is the minimumvalue of Rack which determines the absolute threshold for stability. The minimum isobtained when

k =π√2, (4.41)

and the resulting critical value of the Rayleigh number is

Rac =27π4

4≈ 657.5; (4.42)

That is, the steady state is linearly unstable if Ra > Rac.For other boundary conditions, the solutions are still exponentials, but the coef-

ficients, and hence also the growth rate, must be found numerically. The resultantcritical value of the Rayleigh number is higher for no slip boundary conditions, forexample, (it is about 1,708), and in general, thermal convection is initiated at valuesof Ra >∼ O(103).

93

4.3 Double-diffusive convection

Double-diffusive convection refers to the motion which is generated by buoyancy, whenthe density depends on two diffusible substances or quantities. The simplest examplesoccur when salt solutions are heated; then the two diffusing quantities are heat andsalt. Double-diffusive processes occur in sea water and in lakes, for example. Othersimple examples occur in multi-component fluids containing more than one dissolvedspecies; convection in magma chambers is one such.

The guiding principle behind double-diffusive convection is still that light fluidrises, and convection occurs in the normal way (the direct mode) when the steady stateis statically unstable (i. e., when the density increases with height), but confoundingfactors arise when, as is normally the case, the two substances diffuse at differentrates. Particularly when we are concerned with temperature and salt, the ratio ofthermal to solutal diffusivity is large, and in this case different modes of convectionoccur near the statically neutral buoyancy state: the cells can take the form of longthin ‘fingers’, or the onset of convection can be oscillatory. In practice, fingers areseen, but oscillations are not.

A further variant on Rayleigh-Benard convection arises in the form of convec-tive layering. This is a long-lived transient form of convection, in which separatelyconvecting layers form, and is associated partly with the high diffusivity ratio, andpartly with the usual occurrence of no flux boundary conditions for diffusing chemicalspecies.

We pose a model for double-diffusive convection based on a density which is relatedlinearly to temperature T and salt composition c in the form

ρ = ρ0[1− α(T − T0) + β(c− c0], (4.43)

where we take α and β to be positive constants; thus the presence of salt makesthe fluid heavier. The equation that then needs to be added to (4.20) is that forconvective diffusion of salt:

ct + u .∇c = D∇2c, (4.44)

where D is the solutal diffusion coefficient, assuming a dilute solution. We adopt thesame scaling of the variables as before, with the extra choice

c− c0 ∼ ∆c, (4.45)

where ∆c is a relevant salinity scale (in our stability analysis, it will be the prescribedexcess salinity at the base of the fluid layer). The Boussinesq form of the scaledequations, based on the assumptions that α∆T 1 and β∆c 1, are then

∇.u = 0,

1

Pr[ut + (u .∇)u] = −∇p+∇2u +RaT k−Rs ck,Tt + u.∇T = ∇2T,

ct + u .∇c =1

Le∇2c. (4.46)

94

The Rayleigh number Ra and the Prandtl number Pr are defined as before, and thesolutal Rayleigh number Rs and the Lewis number Le are defined by

Rs =βρ0∆cgd3

µκ, Le =

κ

D. (4.47)

Note that in the absence of temperature gradients, the quantity −RsLe would bethe effective Rayleigh number determining convection.

4.3.1 Linear stability

Now we study the linear stability of a steady state maintained by prescribed temper-ature and salinity differences ∆T and ∆c across a stress-free fluid layer. In dimen-sionless terms, we pose the boundary conditions

ψ = ∇2ψ = 0 at z = 0, 1,

T = c = 0 at z = 1,

T = c = 1 at z = 0, (4.48)

where as before, we restrict attention to two dimensions, and adopt a stream functionψ. The steady state is

c = 1− z, T = 1− z, ψ = 0, (4.49)

and we perturb it by writing

c = 1− z + C, T = 1− z + θ, (4.50)

and then linearising the equations on the basis that C, θ, ψ 1. This leads to

1

Pr∇2ψt ≈ Ra θx −RsCx +∇4ψ,

θt − ψx ≈ ∇2θ,

Ct − ψx ≈1

Le∇2C, (4.51)

withC = ψ = ψzz = θ = 0 on z = 0, 1. (4.52)

By inspection, solutions satisfying the temperature and salinity equations are

ψ = exp(ikx+ σt) sinmπz,

θ =ik

σ +K2exp(ikx+ σt) sinmπz,

C =ik

σ +K2

Le

exp(ikx+ σt) sinmπz, (4.53)

95

where we have writtenK2 = k2 +m2π2. (4.54)

Substituting these into the momentum equation leads to the dispersion relation de-termining σ in terms of k:

(σ +K2Pr)(σ +K2)

(σ +

K2

Le

)+ k2Pr

[(Rs−Ra)σ

K2+Rs− Ra

Le

]= 0. (4.55)

This is a cubic in σ, which can be written in the form

σ3 + aσ2 + bσ + c = 0, (4.56)

where

a = K2

(Pr + 1 +

1

Le

),

b = K4

(Pr +

1

Le+Pr

Le

)+k2

K2Pr(Rs−Ra),

c =K6

LePr + k2Pr

(Rs− Ra

Le

). (4.57)

Instability occurs if any one of the three roots of (4.56) has positive real part.Since Le and Pr are properties of the fluid, we take them as fixed, and study theeffect of varying Ra and Rs on the stability boundaries where Reσ = 0. Firstly, ifRa < 0 and Rs > 0, then a, b and c are all positive. We can then show (see question4.7) that Reσ < 0 for all three roots providing ab > c, and this is certainly the case ifLe > 1, which is always true for heat and salt diffusion. Thus when both temperatureand salinity fields are stabilising, the state of no motion is linearly stable.

To find regions of instability in the (Rs,Ra) plane, it thus suffices to locate thecurves where Reσ = 0. There are two possibilities. The first is referred to as exchangeof stability, or the direct mode, and occurs when σ = 0. From (4.56), this is when

c = 0, or Rs =Ra

Le− K6

k2Le. This is a single curve (for each k), and since we know that

Reσ < 0 in Ra < 0 and Rs > 0, this immediately tells us that a direct instabilityoccurs if

Ra− LeRs > Rc = mink

K6

k2=

27π4

4. (4.58)

This direct transition is the counterpart of the onset of Rayleigh-Benard convection,and shows that Ra− LeRs is the effective Rayleigh number. This is consistent withthe remark just after (4.47).

The other possibility is that there is a Hopf bifurcation, i. e., a pair of complexconjugate values of σ cross the imaginary axis at ±iΩ, say. The condition for thisis ab = c, which is again a single curve, and one can show (see question 4.8) thatoscillatory instability occurs for

Ra >

(Pr +

1

Le

)Rs

1 + Pr+

(1 +

1

Le

)(Pr +

1

Le

)

PrRc. (4.59)

96

Ra

Rs

U

Z

W

V

os illations

ngers

stati ally stableX

Figure 4.5: Stability diagram for double-diffusive convection.

Direct instability occurs along the line XZ in figure 4.5, while oscillatory insta-bility occurs at the line XW . Between XW and the continuation XU of XZ, thereare two roots with positive real part and one with negative real part. As Ra increasesabove XW , it is possible that the two complex roots coalesce on the real axis, so thatthe oscillatory instability is converted to a direct mode. One can show (see question4.9) that the criterion for this is that b < 0 and

c = 19

[ab+

(a2 − 6b)

3

−a+ (a2 − 3b)1/2

]. (4.60)

For large Rs, this becomes (for k2 =π2

2)

Ra ≈ Rs+3R

1/3c Rs2/3

22/3Pr1/3, (4.61)

and is shown as the line XW in figure 4.5. Thus the onset of convection is oscillatoryonly between the lines XW and XV , and beyond (above) XV it is direct. In practice,oscillations are rarely seen.

97

Fingers

If we return to the cubic in the form (4.55), and consider the behaviour of the rootsin the third quadrant as Ra,Rs→ −∞, it is easy to see that one root is

σ ≈K2

[Ra

Le−Rs

]

Rs−Ra , (4.62)

while the other two are oscillatorily stable (see question 4.10). Thus this growthrate is positive when LeRs < Ra < Rs and grows unboundedly with the wavenumber k (since K2 = k2 + π2 when m = 1). This is an indication of ill-posedness,but in fact we anticipate that σ will become negative at large k. To see when thisoccurs, inspection of (4.55) shows that the neglected terms in the approximation(4.62) become important when k ∼ |Ra|1/4, where σ is maximum (of O|Ra|1/4), andthen σ ∼ −k2 for larger k. Thus in the ‘finger’ regime sector indicated in figure 4.5,the most rapidly growing wavelengths are short, and the resulting waveforms are talland thin. This is what is seen in practice, and the narrow cells are known as fingers.

4.3.2 Layered convection

The linear stability analysis we have given above is only partially relevant to dou-ble diffusive convection. It is helpful in the understanding of the finger regime, butthe oscillatory mode of convection is not particularly relevant. The other principalphenomenon which double diffusive systems exhibit is that of layering. This is atransient, but long-term, phenomenon associated often with the heating of a sta-ble salinity gradient, and arises because in normal circumstances, more appropriateboundary conditions for salt concentration are to suppose that there is no flux at thebounding surfaces.

In pure thermal convection, the heating of an initially stably thermally stratifiedfluid will lead to the formation of a layer of convecting fluid below the stable region.This (single) convecting layer will grow in thickness until it fills the entire layer.This is essentially the ‘filling box’ style of convection. Suppose now we have a stablesalinity gradient which is heated from below. Again a convecting layer forms, whichmixes the temperature and concentration fields to be uniform within the layer. Atthe top of the convecting layer, there will be a step down ∆T in temperature, and astep down ∆c in salinity. It is found experimentally that α∆T = β∆c, that is, theboundary layer is neutrally stable. However, the disparity in diffusivities (typicallyLe 1) means that there is a thicker thermal conductive layer ahead of the interface.In effect, the stable salinity gradient above the convecting layer is heated by the layeritself, and a second, and then a third, layer forms. In this way, the entire fluid depthcan fill up with a sequence of long-lived, separately convecting layers. The layers willeventually merge and form a single convecting layer over a time scale controlled bythe very slow transport of salinity between the convecting layers. Such layers arevery suggestive of some of the fossilised layering seen in magma chambers, but theassociation may be a dangerous one.

98

4.4 Hydrodynamic stability

4.5 Notes and references

4.5.1 Eigenvalue problems and variational principles

The theory of Sturm–Liouville systems of the form (4.8) is expounded in a number oftextbooks, of which a classic is that of Courant and Hilbert (1953). It is a straightfor-ward matter to show that a smooth function which causes (4.9) to have a stationarypoint satisfies the differential equation (4.8), and it is also relatively simple to showthat the stationary value is a local maximum. It is less clear that the maximum of(4.9) over all functions for which the integral exists (actually the Sobolev space H1)is obtained by a smooth function, but it is in fact the case. One way to show thisis to reformulate the differential equation as an integral equation for which a similarmaximisation principle exists. It is a consequence of the Arzela–Ascoli theorem thatthe maximum is obtained by a continuous function, and then direct calculation showsthat this also satisfies the Sturm–Liouville problem. This is one of the few instanceswhere functional analysis has some useful purpose.

4.5.2 Viscous fingering

The phenomenon of viscous fingering, when a less viscous fluid displaces a moreviscous one, was studied by Saffman and Taylor (1958), and the subject has becomesomething of an industry now, with its connections to fractals and dendritic growth,as in figure 4.3 (Mathiesen et al. 2006). A principal practical application is in theoil industry, since the standard method of oil extraction from a borehole is to injectwater down a secondary borehole. The water then displaces the oil in the poroussedimentary rock in the manner suggested by figure 4.3, thus leading to imperfect oilrecovery. A major question is whether the fingering instability can be suppressed.

4.5.3 Double-diffusive convection

The stability analysis and diagram for double-diffusive convection is given for exampleby Baines and Gill (1969), and also in the book by Turner (1979), who also describesa good number of other convective phenomena, including turbulent plume theory andfilling box convection (see below), all in the context of geophysical fluid dynamics.Salt-finger convection in the finite amplitude regime is the subject of a lovely study byHoward and Veronis (1987), which makes use of the large value of the Lewis numberto provide a boundary layer theory for the counter-current convective cells.

4.5.4 Filling box convection

The filling box style of convection alluded to in section 4.3.2 was described by Bainesand Turner (1969); see also Linden (2000). It refers to the situation where a source ofbuoyancy at a boundary drives a convective flow which does not recirculate, because

99

Figure 4.6: Graded layering in the Skaergaard intrusion. Photograph courtesy ofKurt Hollocher.

of the absence of a counteracting buoyancy at the opposing boundary. For example,in a magma chamber (see below), the hot magma is emplaced in the cool crust, andthus is cooled from all sides. The cooling from above induces the sinking of heavyconvective plumes, but since there is no heat source at the base, the cool magmasimply ponds there over time, and in this way the convective flow above slowly buildsup a stably stratified layer at the base: the filling box.

4.5.5 Layered igneous intrusions

The layered igneous intrusions referred to in section 4.3.2 are fossil (that is, solidified)magma chambers which exhibit pronounced layering in their chemistry and miner-alogy. There are many such magma chambers round the world, and many of themare described in the monograph by Wager and Brown (1968). A particularly famousexample is the Skaergaard Intrusion, an illustration of which is shown in figure 4.6.The layering is extensive and consists of alternating bands of different rocks (e. g.,olivine and plagioclase) on a scale of the order of a metre. While the layering isvery suggestive of the layering seen in double-diffusive systems, it appears that theresemblance is merely superficial.

4.5.6 The Orr–Sommerfeld equation

Drazin and Reid (2004)

100

Exercises

4.1 A model for the growth of an advantageous gene in a population which canmigrate is

ut = Duxx + ru(1− u),

with no flux boundary conditions

ux = 0 at x = 0, L.

Here u is the proportion of the population bearing the gene.

Find the steady states of the system, and show that one of them is always stable,but the other is unstable if

r >Dπ2

L2.

4.2 A fluid of viscosity µ1 is injected upwards (along the z axis) at a rate φU intoa porous medium of porosity φ and permeability Π saturated with a fluid ofviscosity µ2. The pressure in each fluid is denoted as p.

Write down the equations and boundary conditions for the flow, assuming theinterface is at z = Ut+ η(x, y, t), and show that the pressure satisfies

∇2p = 0,

p1 = p2,∂p1

∂n= µ

∂p2

∂n

at the interface, where µ = µ1/µ2.

Show also that

φ(U + ηt) = −(1 + |∇η|2)1/2 Π

µi

∂pi∂n

;

at the interface, where i is either 1 or 2, and hence show, by considering smallperturbations away from a uniform state, that the uniform state is unstable ifµ1 < µ2. Comment on whether the model appears to be well-posed.

4.3 A two-dimensional, incompressible fluid flow has velocity u = (u, 0, w), anddepends only on the coordinates x and z. Show that there is a stream functionψ satisfying u = −ψz, w = ψx, and that the vorticity

ω = ∇× u = −∇2ψ j,

and thus thatu× ω = (ψx∇2ψ, 0, ψz∇2ψ),

and hence∇× (u× ω) = (ψx∇2ψz − ψz∇2ψx)j.

Use the vector identity (u .∇)u = ∇(12u2)− u× ω to show that

∇× du

dt=[−∇2ψt − ψx∇2ψz + ψz∇2ψx

]j.

101

Show also that∇× θk = −θxj,

and use the Cartesian identity

∇2 ≡ grad div − curl curl

to show that∇×∇2u = −∇4ψ j.

Hence deduce that the momentum equation for Rayleigh–Benard convectioncan be written in the form

1

Pr

[∇2ψt + ψx∇2ψz − ψz∇2ψx

]= Ra θx +∇4ψ.

4.4 The onset of two-dimensional Rayleigh-Benard convection is studied in the sit-uation where no-slip boundary conditions are applied at the top z = 1 and basez = 0, and leads via the assumptions that the perturbed stream function andtemperature are given by

ψ = f(z)eσt+ikx, Θ = g(z)eσt+ikx,

to the systemσ

Pr(D2 − k2)f = (D2 − k2)2f − ikRa g,

σg + ikf = (D2 − k2)g,

where D =d

dz, and Ra and Pr are the Rayleigh and Prandtl numbers.

What are appropriate boundary conditions for f and g in this case?

Derive a single differential equation for f at the exchange of stability whereσ = 0, and show that it has solutions emz, where m satisfies a polynomialequation which you should derive. Show that the solutions of this polynomialequation can be written, assuming Ra > k4, as

m = ±is, ±p± iq,

where without loss of generality p, q, s > 0. Show that

s = [(k2Ra)1/3 − k2]1/2, q =

√3(k2Ra)1/3

4p,

and give an explicit expression for p. [Hint: one can calculate the square rootp+ iq of α + βeiπ/3 by squaring both sides.]

Show that f is an even function of z − 12, and deduce that

f(z) = B coss(z−12)+C cosh(z−1

2) cos(z−1

2)+G sinh(z−1

2) sin(z−1

2),

and deduce by application of the boundary conditions an implicit equationrelating Ra and k in the form of a determinantal equation.

102

4.5 Write down the equations of motion for thermal convection, assuming theBoussinesq approximation. Make the equations dimensionless, and show thatthe model involves two dimensionless parameters Ra and Pr, the Rayleigh andPrandtl numbers. Write down boundary conditions appropriate to the descrip-tion of a layer of fluid open to the atmosphere above, but resting on a solid (noslip) surface, and heated from below.

Now assume the Prandtl number is infinite. Write the equations in terms ofa suitable stream function ψ, and define a steady state of no motion for thesystem. By linearising the equations, derive a pair of linear equations for thefunctions f and g in the normal modes ψ = f(z) eσt+ikx, θ = g(z) eσt+ikx, whereθ is the temperature perturbation, and hence derive a sixth order equation forthe function f , and give suitable boundary conditions for these at z = 0 andz = 1. Why are the solutions f(z) = sinmπz not appropriate?

4.6 Write down the equations describing the motion and diffusion of heat and saltin a Boussinesq fluid between two parallel plates, and show how they can benon-dimensionalised to obtain the dimensionless model

∇.u = 0,

1

Pr[ut + (u .∇)u] = −∇p+∇2u +RaT k−Rs ck,Tt + u.∇T = ∇2T,

ct + u .∇c =1

Le∇2c,

and define the dimensionless parameters Ra, Rs, Pr and Le.

Assuming a two-dimensional flow, stress-free boundary conditions and pre-scribed basal and surface temperature and concentrations, show that normalmodes can be taken ∝ exp(ikx+σt) sinmπz, and satisfy the dispersion relation

(σ +K2Pr)(σ +K2)

(σ +

K2

Le

)+ k2Pr

[(Rs−Ra)σ

K2+Rs− Ra

Le

]= 0,

where K2 = k2 +m2π2.

If Pr 1 and Le 1, show that instability will occur if

Ra−Rs > K6

k2,

and thus in general if Ra−Rs > 657.5.

4.7 Suppose that σ satisfies

p(σ) ≡ σ3 + aσ2 + bσ + c = 0,

and that a, b and c are positive. Suppose, firstly, that the roots are all real.Show in this case that they are all negative.

103

Now suppose that one root (α) is real and the other two are complex conjugatesβ ± iγ. Show that α < 0. Show also that β < 0 if a > α. Show that a > α ifp(−a) < 0, and hence show that β < 0 if c < ab.

If

a = K2

(Pr + 1 +

1

Le

),

b = K4

(Pr +

1

Le+Pr

Le

)+k2

K2Pr(Rs−Ra),

c =K6

LePr + k2Pr

(Rs− Ra

Le

),

show that a, b, c > 0 if Ra < 0, Rs > 0, and show that if Le > 1, then c < ab.

What does this tell you about the stability of a layer of fluid which is boththermally and salinely stably stratified?

4.8 Suppose that σ satisfies

p(σ) ≡ σ3 + aσ2 + bσ + c = 0,

and that all the roots have negative real part if c < ab. Show that the conditionthat there be two purely imaginary roots ±iΩ is that c = ab, and deduce thatthere are two (complex) roots with positive real part if c > ab. With

a = K2

(Pr + 1 +

1

Le

),

b = K4

(Pr +

1

Le+Pr

Le

)+k2

K2Pr(Rs−Ra),

c =K6

LePr + k2Pr

(Rs− Ra

Le

),

show that this condition reduces to

Ra >

(Pr +

1

Le

)Rs

1 + Pr+

(1 +

1

Le

)(Pr +

1

Le

)

Pr

K6

k2.

Assuming K2 = k2+m2π2, where m is an integer, show that the minimum valueof Ra where this condition is satisfied is when m = 1, and give the correspondingcritical value Raosc.

4.9 On the line XV in figure 4.5, the cubic

p(σ) = σ3 + aσ2 + bσ + c

has two positive real roots β and one negative real root α. Show that thecondition for this to be the case is that

a = α− 2β, b = β2 − 2αβ, c = αβ2,

104

and deduce thataβ2 + 2bβ + 3c = 0. (1)

Show also that at the double root β,

3β2 + 2aβ + b = 0. (2)

Deduce from (1) and (2) that

β =9c− aba2 − 6b

,

and hence, using (2), that

β = 13

[−a+ a2 − 3b1/2

]. (3)

Explain why the positive root is taken in (3), and why we can assume b < 0.

Use the definitions

a = K2

(Pr + 1 +

1

Le

),

b = K4

(Pr +

1

Le+Pr

Le

)+k2

K2Pr(Rs−Ra),

c =K6

LePr + k2Pr

(Rs− Ra

Le

),

to show that if Ra ∼ Rs 1, Ra − Rs 1 and Le 1, then XV isapproximately given by

Ra ≈ Rs+3K2Rs2/3

(4k2Pr)2/3.

4.10 The growth rate σ for finger instabilities is given by

(σ +K2Pr)(σ +K2)

(σ +

K2

Le

)+ k2Pr

[(Rs−Ra)σ

K2+Rs− Ra

Le

]= 0,

and Ra,Rs < 0 with −Ra,−Rs 1; K is defined by K2 = k2 + π2.

Define Rs = Ra r, and consider the behaviour of the roots when Ra → −∞with r fixed. Show that when k is O(1), one root is given by

σ =

(r − 1

Le

)K2

1− r +O

(1

|Ra|

), (∗)

and that this is positive if1

Le< r < 1.

105

Show that the other two roots are of O(|Ra|1/2

), and by putting

σ = |Ra|1/2Σ0 + Σ1 + . . . ,

show that they are given by

σ = ±i kKPr(Ra−Rs)1/2 − 1

2K2

Pr +

1− 1

Le1− r

+O

(1

|Ra|1/2),

and thus represent stable modes.

Show further that when k is large, an appropriate scaling when (∗) breaks downis given by

k = |Ra|1/4α, σ = |Ra|1/4Σ,

and write down the equation satisfied by Σ in this case. Show also that whenα is large, the three roots are all negative, with Σ ∼ −α2S, and S = Pr, 1, or1

Le.

Deduce that the maximal growth rate for finger instability occurs when k ∼|Ra|1/4.

4.11 Sloping convection. A fluid layer of depth d, thermal diffusivity κ and viscos-ity η is inclined at a slope of angle α to the horizontal, and is heated frombelow. Write down suitable dimensional equations for the flow in coordinatesaligned with the layer, assuming it is sufficiently slow that acceleration termsare negligible, and that the Boussinesq approximation is applied.

If the density is ρ0, and the lower surface is a no-slip boundary, while the upperhas zero stress, show that a basic solution has

p = p0(z) = pa + ρ0g(d− z) cosα, u = (u0(z), 0, 0),

where

u0(z) =ρ0g sinα

η(dz − 1

2z2).

Now suppose the density is

ρ = ρ0[1− α(T − T0)],

the base is fixed at a temperature T0+∆T , while the upper surface is maintainedat T0. Non-dimensionalise the model by scaling the variables as

p− p0 =ηκ

d2P, u− u0(z)i =

κ

dU, T − T0 = ∆T θ, x ∼ d, t ∼ d2

κ.

Show that the dimensionless equations become

∇.U = 0,

106

∇P = ∇2U−Ra θ (sinα, 0,− cosα),

θt + U.∇θ +Ra sinα

B(z − 1

2z2)θx = ∇2θ,

and give the definitions of the Rayleigh number Ra and the Boussinesq numberB.

Now seek perturbations of the form

θ = 1− z + Θ(y, z, t),

and show that we can then take

U = (U(y, z, t), ψz,−ψx)

for a suitable transverse stream function ψ, and derive a linearised set of equa-tions for the flow.

If the critical Rayleigh number for a horizontal layer is Rc (≈ 1,108), show

that convective instability occurs for Ra >Rc

cosα, and describe the form of the

resulting motion.

107

Appendix

A.1 Useful vector identities

The following identities are of considerable use (see, for example, Gradshteyn andRyzhik 1980):

∇(a.b) = (a.∇)b + (b.∇)a + a× (∇× b) + b× (∇× a); (A.1)

in particular,(u .∇)u = 1

2∇u2 − u× (∇× u). (A.2)

Another useful result is∇(1

2r2) = r, (A.3)

and more generally

∇f(r) =f ′(r)r

r, (A.4)

so that for example

∇(

1

r

)= − r

r3, (A.5)

which shows that the inverse square law is derived from a potential1

r.

In terms of the alternating tensor, we have

a× b = εijkeiajbk, (A.6)

and similarly

∇× u = εijkei∂uk∂xj

. (A.7)

From (A.6), we have also

a× (b× c) = εijkeiajεkpqbpcq, (A.8)

and using the fact that εijk = εkij, and the vector identity (1.11),

εijkεipq = δjpδkq − δjqδkp, (A.9)

then

a× (b× c) = [δipδjq − δiqδjp]eiajbpcq= eiajbicj − eiajbjci,

= (a.c)b− (a.b)c. (A.10)

In particular,Ω× (Ω× r) = (Ω.r)Ω− Ω2r, (A.11)

108

which is used in the derivation of the rotating frame equations (1.82). Specifically,(1.83) follows firstly from

12|Ω× r|2 = 1

2εijkεipqΩjxkΩpxq

= 12[δjpδkq − δjqδkp]ΩjΩpxkxq

= 12[ΩjΩjxkxk − ΩjΩkxjxk]

= 12[Ω2r2 − (Ω.r)2], (A.12)

and then directly

∇[12|Ω× r|2] = Ω2r− (Ω.r)Ω = −Ω× (Ω× r), (A.13)

using (A.3) and that ∇(Ω.r) = Ω.Another useful formula is

∇× (a× b) = a(∇.b)− b(∇. a) + (b.∇)a− (a.∇)b. (A.14)

in particular, if ∇.u = ∇.ω = 0, then we have

∇× (u× ω) = (ω.∇)u− (u.∇)ω, (A.15)

which is used in deriving the vorticity equation (1.19).In fluid mechanics, we encounter the odd-looking formula

∇2u = ∇ (∇.u)−∇× (∇× u); (A.16)

odd-looking because you can’t take the gradient of a vector, nor therefore the di-vergence of the gradient, i. e., the Laplacian. Slightly more optimistically, we define

∇u to be the tensor with components∂ui∂xj

, and therefore ∇.(∇u) is the vector with

components∂2ui∂xj∂xj

= ∇2ui, and indeed, this is what is meant. However, (A.16) is

a safer definition, because although the components of ∇2u are indeed ∇2ui, this isonly true in cartesian coordinates, and the right hand side is the proper definition inother curvilinear coordinate systems.

To prove the formula, we write the right hand side as

∇ (∇.u)−∇× (∇× u) = ei∂

∂xi

∂uj∂xj− εijkei

∂

∂xjεkpq

∂uq∂xp

= ei

[∂2uj∂xi∂xj

− (δipδjq − δiqδjp)∂2uq∂xj∂xp

]

= ei

[∂2uj∂xi∂xj

− ∂2uj∂xj∂xi

+∂2ui∂xj∂xj

]

= ei∂2ui∂xj∂xj

, (A.17)

109

as assumed.In question 2.1, we deduced the shallow water equations in the form

∂

∂t

∫ h

0

u dz + ∇.

[∫ h

0

(uu) dz

]+ gh∇h = 0. (A.18)

One wonders what to make of the second term. Earlier (in section 1.0.1) we introducedthe idea of second order tensors such as σ, whose components are σij, and we defined

the divergence ∇.σ as the vector ei∂σij∂xj

. In more generality, the tensor ρuu has

components ρuiuj, and so its divergence is

∇. (ρuu) = ei∂(ρuiuj)

∂xj= u∇. (ρu)+ ρ(u .∇)u. (A.19)

This allows a simple derivation of the Navier–Stokes equation, for which the firstprinciples statement of the rate of change of momentum is (for a fixed volume V )

d

dt

∫

V

ρu dV +

∫

∂V

ρuun dS =

∫

∂V

σ.n dS, (A.20)

leading to the point form

∂(ρu)

∂t+ ∇. (ρuu) = ∇.σ, (A.21)

and from (A.19), the left hand side is

u

[∂ρ

∂t+ ∇. (ρu)

]+ ρ

[∂u

∂t+ (u .∇)u

], (A.22)

of which the first term in square brackets vanishes due to mass conservation.

110

References

Abramowitz, M. and I. Stegun 1964 Handbook of mathematical functions. DoverPublications, New York.

Acheson, D. J. 1990 Elementary fluid dynamics. Clarendon Press, Oxford.

Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc.R. Soc. A235, 67–78.

Baines, P. G. and A. E. Gill 1969 On thermohaline convection with linear gradients.J. Fluid Mech. 37, 289–306.

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