Physics Department

Seminar Ia

Kelvin-Helmholtz Instability

Author: Aleksandar Vujinovic

Supervisor: Dr. Joze Rakovec

Kranj, May 4, 2015

Abstract

Kelvin-Helmholtz instability (KHI) is an instability at the interface between two parallelstreams with different velocities and densities, with the heavier fluid at the bottom. We willtake a look at two cases. In the first case we have a discontinuous profile of density andvelocity and in the second the profile is continuous. We are interested under what conditionsthe flow instability occurs.

Contents

1 Introduction 1

2 Instability of Discontinuously Stratified Parallel Flows 2

3 Instability of Continuously Stratified Parallel Flows 83.1 Richardson Number Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Conclusion 10

1 Introduction

Kelvin-Helmholtz instability (KHI) was first studied by Hermann von Helmholtz in 1868 andby William Thomson (Lord Kelvin) in 1871. It is a hydrodynamic instability in which incom-pressible and inviscid fluids are in relative and irrotational motion. In KHI as originally studiedby Kelvin and Helmholtz, the velocity and density profiles are uniform in each fluid layer, butthey are discontinuous at the (plane) interface between the two fluids. This discontinuity inthe (tangential) velocity, i.e., the shear flow, induces vorticity at the interface; as a result, theinterface becomes an unstable vortex sheet that rolls up into a spiral.[1] The heavier fluid parcelsfrom lower denser fluid are lifted up and the lighter fluid parcels from lighter upper fluid arepushed down so overall we gain potential energy. The needed energy comes from the kineticenergy of the mean flow.

The name KHI is also used to describe the more general case where the variations of velocityand density are continuous across the interface.

KHI occurs not only in the atmosphere (Fig. 1) and ocean but also in motion of interstellarclouds [2, 3] and clumping in supernova remnants [4] in astrophysics, production of unstableshear layer and vortices in high energy density plasmas [5, 6], and quantized vortices in quantumfluids [7, 8].

Figure 1: Kelvin–Helmholtz instability in atmosphere. [9]

1

2 Instability of Discontinuously Stratified Parallel Flows

Figure 2 shows our two-dimensional model of heavier fluid (ρ2) with velocity u2 on the bottomand lighter fluid (ρ1) with velocity u1 at the top. Vertical and horizontal domains are infiniteto prevent borders to have influence on development of instability. We deal with the problemon sufficiently small scales. The lateral scale must be smaller than effective Rosby radius sowe don’t have to consider Coriolis force exerting influence on the flow. However, it must belarge enough to ignore surface tension, molecular viscosity and density diffusion. By Kelvin’scirculation theorem, the perturbed flow must be irrotational in each layer because the motiondevelops from an irrotational basic flow of uniform velocity in each layer.

It’s a small scale phenomenon which saves us the need to use equations of motion in arotating coordination frame thus simplifying their derivation.

Figure 2: Discontinuous shear across a density interface with ρ2 > ρ1. [10]

I derive the equations as they are derived in [10]. Assuming the velocities we are dealingwith are much lower than sound velocity, we can write the equation for incompressible fluid:

∇ · v = 0. (1)

Because we have potential flow, the velocity is

v = ∇φ. (2)

Substitution into equation (1) gets us Laplace equations:

∇2φ = 0. (3)

Flow is decomposed into a basic state plus perturbations where index j = 1, 2 representsthe variable of first and second layer:

φi = ujx+ φ′j , (4)

where uj represent the basic flow of uniform stream. Using equations (4) and substitutinginto equation (3) gives us

∇2φ′j = 0, (5)

subject to

2

φ′1 → 0 as z →∞,φ′2 → 0 as z → −H.

(6)

For our purpose we can assume H →∞ because depth is much bigger than wave amplitude.At the interface we have kinematic boundary condition which requires that the fluid particle

at the interface must move with the interface. So the vertical velocity just above and belowinterface must be equal. Considering particles just above the interface, this requires

∂φ′j∂z

=dζ

dt=∂ζ

∂t+(uj + u′j

) ∂ζ∂x

at z = ζ. (7)

We expand the term on the left hand side of equation (7) in a Taylor series around z = 0:

∂φ

∂z

∣∣∣∣z=ζ

=∂φ

∂z

∣∣∣∣z=0

+ ζ∂2φ

∂z2+

∣∣∣∣z=0

· · · ≈ ∂φ

∂z

∣∣∣∣z=0

(8)

So for upper and lower layer the kinematic boundary conditions are:

∂φ′j∂z

=∂ζ

∂t+ uj

∂ζ

∂xat z = 0. (9)

Flow dynamics is described by Navier-Stokes equation. For inviscid fluid the equation is

ρ∂v

∂t+ ρ(v · ∇)v = −∇p+ ρg. (10)

Because our flow is irrotational the advection term can be written as

(v · ∇)v =1

2∇v2 − v ×∇× v =

1

2∇v2, (11)

as ∇× v is zero.We rewrite gravity acceleration into a gradient of gravity potential g = −∇(gz) and we get

ρj∂∇φ∂t

+∇(

1

2ρj (∇φ)2 + ρigz

)= −∇pj . (12)

∇ can be written in front of the first term because density is constant and integrating inhorizontal and vertical directions we get unsteady Bernoulli equations

ρj∂φ

∂t+

1

2ρj (∇φ)2 + ρjgz = −pj + Cj , (13)

with Cj as constant. We can calculate them using the initial state of flow before introducingthe perturbation

Cj = pj +1

2ρj u

2j . (14)

The dynamic boundary condition at the interface means that the pressure must be contin-uous across the interface if surface tension is neglected. This requires p1 = p2 at z = 0 so wecan write the equation (14) as

1

2ρ1u

21 − C1 =

1

2ρ2u

22 − C2. (15)

In equation (13) we have a term (∇φ)2 and we substitute equation (4) into it and ignoringhigher order terms we obtain:

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(∇φ)2 =

(u+

∂φ′

∂x

)2

+

(∂φ′

∂z

)2

≈ u2 + 2u∂φ′

∂x. (16)

We use that result in equation (13)

ρj∂φ

∂t+

1

2ρj

(u2j + 2uj

∂φ′j∂x

)+ pj + ρjgζ = Cj , (17)

and requiring (p1 = p2 at z = ζ), we obtain the following condition at the interface:

C1 − ρ1∂φ′1∂t− 1

2ρ1

(u2

1 + 2u1∂φ′1∂x

)− ρ1gζ =

= C2 − ρ2∂φ′2∂t− 1

2ρ2

(u2

2 + 2u2∂φ′2∂x

)− ρ2gζ.

(18)

By summing up equations (15) and (18) we get

ρ1

[∂φ′1∂t

+ u1∂φ′1∂x

+ gζ

]z=0

= ρ2

[∂φ′2∂t

+ u2∂φ′2∂x

+ gζ

]z=0.

(19)

The perturbation therefore satisfy equation (5) and conditions (6), (8), (9), and (19).We solve those equations with

(ζ, φ′1, φ′2) = (ζ, φ1, φ2)eik(x−ct), (20)

where k is real, but c = cr + ici is complex. The flow is unstable if there exists a positive ci.Equation (5) needs solutions of the form

φ1 = Ae−kz,

φ2 = Bekz,

where solutions exponentially increasing from the interface are ignored because of condition(6).Substituting equation (20) into equations (8), (9), and (19) gives us three homogeneous linearalgebraic equations with three unknowns (A,B, ζ):

The Bernoulli equation (19) is

iAkρ1(u1 − c) + iBkρ2(−u2 + c) + gζ(ρ1 − ρ2) = 0. (21)

The kinematic conditions (8) and (9) yield

A+ iζ(u1 − c) = 0 (22)

B + iζ(−u2 + c) = 0. (23)

The three equations (21) to (23) have a non-trivial solution when D (eq. 26) equals zero:D = 0.

Dx = 0 (24)

wherex = [A,B, ζ]T (25)

and

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D =

ikρ1(u1 − c) ikρ2(−u2 + c) g(ρ1 − ρ2)1 0 i(u1 − c)0 1 i(−u2 + c)

(26)

That results in (27):

c2(ρ1 + ρ2)− 2c(u1ρ1 + u2ρ2) +g

k(ρ1 − ρ2) + (u2

1ρ1 + u22ρ2) = 0. (27)

This is a quadratic equation and the solution for c are

c =u1ρ1 + u2ρ2

ρ1 + ρ2±

√(u1ρ1 + u2ρ2)2

(ρ1 + ρ2)2−

gk (ρ1 − ρ2) + u2

1ρ1 + u22ρ2

ρ1 + ρ2. (28)

For instabilities to grow we need c to have imaginary part. The result under square roothas to be negative otherwise the waves are stable.

(u1ρ1 + u2ρ2)2

(ρ1 + ρ2)2<

gk (ρ1 − ρ2) + u2

1ρ1 + u22ρ2

ρ1 + ρ2. (29)

After rearrangement of terms we arrive at

(u1 − u2)2 >g(ρ2

2 − ρ21)

kρ1ρ2. (30)

We can see from equation (28) that for each growing solution there is a correspondingdecaying solution.

Assuming ρ1 ≈ ρ2 ≈ ρ the right side of the equation simplifies in approximate condition(31):

(u1 − u2)2 >2g(ρ2 − ρ1)

kρ. (31)

If u1 6= u2, one can always find short enough wavelengths that satisfies our requirement forinstability. If there is velocity shear we can say that the flow is always unstable to short waves.

The Kelvin–Helmholtz instability is caused by the destabilizing effect of shear, which over-comes the stabilizing effect of stratification. KHI can be generated in the tilting tank laboratoryin which a dense (in Fig. 3 coloured) fluid underlies the lighter fluid. When the tank is tilted thedenser fluid flows down and the lighter fluid flows up and the velocity shear generates instabilityas seen in Fig. 3 and Fig. 4.

Figure 3: Kelvin–Helmholtz instability generated by tilting a horizontal channel containing two liquidsof different densities. The lower layer is dyed. Mean flow in the lower layer is down the plane and thatin the upper layer is up the plane. [11]

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Figure 4: Development of a Kelvin–Helmholtz instability in the laboratory. The upper and faster movinglayer is slightly less dense than the lower layer. At first, waves form and overturn in a two-dimensionalfashion but, eventually, three-dimensional motions appear that lead to turbulence and complete themixing. [12]

At the start we assume flow in each layer is irrotational. However, there exists strongshear vorticity in the interface layer so the vorticity for the whole system does not equal zero.This shear vorticity transforms into circular vorticity which is described in more detail in thefollowing paragraph.

The tendency of vorticity in a parallel shear layer to accumulate into evenly spaced maximais the primary driver of KHI. Given an initial wavelike perturbation, mass conservation requiresaccelerated horizontal flow above the crests and below the troughs (5a) The resulting currentanomalies advect vorticity toward the center of the figure. The vorticity concentration inducesvertical velocity perturbations that amplify the original wave (5b), resulting in positive feedbackand exponential growth of the perturbation. The most-amplified wavelength depends on thedetails of the initial profiles, but typically ranges from 6 to 11 times the initial transition-layerthickness. Stable stratification tends to slow the growth of the billows, but also to acceleratebreaking once the billows reach sufficient amplitude to overturn.[13]

We can use alternative approach to the problem by considering potential and kinetic energy.We take into account only the layer next to interface in which mixing occurs. Because heavier

Figure 5: Schematic representation of the positive feedback that drives shear instability. (a) Vorticityaccumulation due to horizontal advection. (b) Amplification of the initial wave by induced verticalmotions. [13]

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fluid parcels are raised and lighter fluid parcels are lowered the potential energy after mixing israised. For how much is shown in equation (33). The average density is ρ = (ρ1 + ρ2)/2 and wecount the height from where the mixing occurs.

Figure 6: Mixing of a two-layer stratified fluid with velocity shear. Vertical axes are unbounded. Mixingoccurs in the layer close to interface with the height of ∆H. As heavier fluid is raised and lighter fluidlowered so potential energy is increased. That movement is provided by a decrease in kinetic energy ofthe flow. [12]

PE gain = PE final − PE initial =

=

∫ ∆H

0ρgzdz −

[∫ ∆H2

0ρ2gzdz +

∫ ∆H

∆H2

ρ1gzdz

]=

=1

2ρg∆H2 −

[1

2ρ2g

∆H2

4+

1

2ρ1g

3∆H2

4

]=

=1

8(ρ2 − ρ1)g∆H2

(32)

The source for this potential energy increase has to come from somewhere and it comesfrom decrease of kinetic energy during mixing. Momentum is conserved as there are no externalforces and using Bousinessq approximation we see that final velocity is the average of initialvelocities u = (u1 + u2)/2. Kinetic energy loss is

KE loss = KE initial −KE final =

=

∫ ∆H2

0

1

2ρu2

2dz +

∫ ∆H

∆H2

1

2ρu2

1dz −∫ ∆H

0

1

2ρu2dz

=1

2ρu2

2

∆H

2+

1

2ρu2

1

∆H

2− 1

2ρu2∆H

=1

8ρ(u1 − u2)2∆H.

(33)

There has to be bigger kinetic energy loss than potential energy gain for mixing to occur inthe whole layer of ∆H. Necessary condition is thus

g∆H(ρ2 − ρ1)

ρ(u1 − u2)2< 1. (34)

Rearranging the terms we see it is very similar to equation (31)

(u1 − u2)2 >g∆H(ρ2 − ρ1)

ρ. (35)

This is a condition for mixing to occur in the layer ∆H. It is expected that the verticalextent ∆H/2 scales like the wavelength of the longest unstable wave so criterion (31) becomesequality:

7

∆H

2∼ 1

kmin=ρ(u1 − u2)2

2g(ρ2 − ρ1). (36)

3 Instability of Continuously Stratified Parallel Flows

Kelvin and Helmholtz only studied discontinuously stratified flow but the more general case ofcontinuously stratified flow is also commonly called the Kelvin-Helmholtz instability. Such flowwas first studied by Taylor in 1915. He surmised that a gradient Richardson number must be lessthan 1/4 for instability. Prandtl, Goldstein, Richardson, Synge, and Chandrasekhar suggestedvalues from 2 to 1/4 for the critical Richardson number. Miles (1961) was able to prove Taylor’sconjecture and Howard (1961) generalized Miles’ proof which is presented here.[10, 12]

We shall consider a two-dimensional (x, z) inviscid and non-diffusive fluid with horizontaland vertical velocities (u, w), dynamic pressure p, and density anomaly ρ. Two-dimensionaldisturbances are more unstable than three-dimensional which is proved by Squire’s theorem.Kelvin-Helmholtz is basically a two-dimensional phenomenon, only later in its developmentthree-dimensional movements appear which leads to turbulent mixing.

Our basic state consists of a steady, sheared horizontal flow [u = u(z), w = 0] in a verticaldensity stratification [ρ = ρ(z)]. The accompanying pressure field p(z) obeys dp/dz = −gρ(z).We add an infinitesimally small perturbation (u = u+ u′, w = w′, p = p+ p′, ρ = ρ+ ρ′) and afollowing linearization of the momentum equations in x and z direction, continuity and energyequations become:

∂u′

∂t+ u

∂u′

∂x+ w′

∂u

∂z= − 1

ρ0

∂p′

∂x, (37)

∂w′

∂t+ u

∂w′

∂x= − 1

ρ0

∂p′

∂z− ρ′g

ρ0, (38)

∂u′

∂x+∂w′

∂z= 0, (39)

∂ρ′

∂t+ u

∂ρ′

∂x+ w′

∂ρ

∂z= 0. (40)

Defining a stream function through u′ = ∂ψ/∂z, w′ = −∂ψ/∂x the continuity equation canbe satisfied.

Equation (40) can be written as

∂ρ′

∂t+ u

∂ρ′

∂x− ρ0N

2w′

g= 0, (41)

where we have defined N2 = −(g/ρ0)(∂ρ/∂z) as the buoyancy frequency which is also knownas Brunt–Vaisala frequency. It is the angular frequency at which a vertically displaced parcelwill oscillate within a statically stable environment.

Substituting stream functions to equations (37), (38), and (41) gives us:

∂2ψ

∂z∂t− ∂u

∂z

∂ψ

∂x+ u

∂2ψ

∂x∂z= − 1

ρ0

∂p′

∂x,

− ∂2ψ

∂x∂t− u∂

2ψ

∂x2= −gρ

′

ρ0− 1

ρ0

∂p′

∂z,

∂ρ′

∂t+ u

∂ρ′

∂x+ρ0N

2

g

∂ψ

∂x= 0.

(42)

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Equations (42) is solved by assuming normal mode solutions of the form

[ρ′, p′, ψ′] = [ρ, p, ψ]eik(x−ct), (43)

where quantities denoted by ( ) are complex amplitudes. Because the flow is unboundedin x, the wave number k must be real. The eigenvalue c = cr + ici can be complex, and thesolution is unstable if there exists a ci > 0, because then the exponent term is positive andgrows with time.

Substituting normal modes into equations (42) we get

(u− c)∂ψ∂z− ∂u

∂zψ = − 1

ρ0p, (44)

k2(u− c)ψ = −gρρ0− 1

ρ0

∂p

∂z, (45)

(u− c)ρ+ρ0N

2

gψ = 0. (46)

We want to reduce the above system of equations to single equation for ψ. The pressure canbe eliminated by taking the z-derivative of equation (44) and subtracting equation (45). Thedensity can be eliminated by equation (46). Result is

(u− c)(∂2ψ

∂z2− k2ψ

)+

(N2

u− c− ∂2u

∂z2

)ψ = 0, (47)

which is Taylor–Goldstein equation and it governs the behaviour of perturbations in a strat-ified parallel flow.

At the top of the atmosphere and at the ground we impose vertical speed of zero. Thisrequires ∂ψ/∂x = ikψ exp(ikx− ikct) = 0 at the boundaries, which is possible only if

ψ(0) = ψ(H) = 0. (48)

Note that the complex conjugate of the equation is also a valid equation because we can takethe imaginary part of the equation, change the sign, and add to the real part of the equation.Because the Taylor–Goldstein equation does not involve any i, a complex conjugate of theequation shows that if ψ is an eigenfunction with eigenvalue c for some k, then ψ∗ is a possibleeigenfunction with eigenvalue c∗ for the same k. Therefore, to each eigenvalue with a positiveci there is a corresponding eigenvalue with a negative ci. In other words, to each growing modethere is a corresponding decaying mode. A non-zero ci therefore ensures instability.[10]

3.1 Richardson Number Criterion

It’s impossible to solve equations (47) and (48) in the general case for an arbitrary shear flowu(z) so we will derive integral constraints. We replace ψ with

ψ =√u− cφ. (49)

Boundary (47) and and boundary conditions (48) become

∂

∂z

[(u− c)∂φ

∂z

]−

[k2(u− c) +

1

2

∂2u

∂z2+

1

(u− c)

(1

4

(∂u

∂z

)2

−N2

)]φ = 0, (50)

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φ(0) = φ(H) = 0. (51)

Multiplying equation (50) by the complex conjugate φ∗, integrating over the vertical scale,and using conditions (51), we get:

∫ H

0

[N2 − 1

4

(∂u

∂z

)2]|φ|2

(u− c)dz =

∫ H

0(u− c)

(∣∣∣∣∂φ∂z∣∣∣∣2 + k2|φ|2

)dz +

1

2

∫ H

0

∂2u

∂z2dz, (52)

where vertical bars denote the absolute value of complex quantities. The imaginary part ofthis expression is

ci

∫ H

0

[N2 − 1

4

(∂u

∂z

)2]|φ|2

|u− c|2= −ci

∫ H

0

(∣∣∣∣∂φ∂z∣∣∣∣2 + k2|φ|2

)dz, (53)

where ci is the imaginary part of c. If the flow is such that N2 > 14(du/dz)2 everywhere, then

the prior equation states that ci times a positive quantity equals ci times a negative quantity.This is impossible and requires that ci = 0. We define Richardson number as

Ri =N2

(∂u/∂z)2=N2

M2. (54)

Linear stability is guaranteed if the inequality

Ri >1

4(55)

is satisfied everywhere in the flow.This does not mean that flow is unstable if we have Ri <

14 somewhere in the flow. It is a

necessary but not sufficient condition for instability.

4 Conclusion

This paper shows mathematical derivation of equations needed for understanding Kelvin-Helmholtz instability. We see that flow is always unstable for small enough wavelengths (whileignoring surface tension, viscosity and diffusion) if there is velocity shear present in discontin-uously stratified flow.

In continuously stratified flow we define Richardson number and if it is bigger than 1/4 theflow is stable. For flow to be unstable Ri has to be lower than 1/4 but it does not guaranteeinstability.

References

[1] C. Matsuoka, Kelvin-Helmholtz Instability and Roll-up (Scholarpedia, 9(3):11821., 2014).

[2] S. D. Murray, S. D. M. White, J. M. Blondin and D. N. C. Lin, Dynamical instabilitiesin two-phase media and the minimum masses of stellar systems Astrophys. J. 407, (1993),588-596.

[3] M. Vietri et al., The Survival of Interstellar Clouds against Kelvin-Helmholtz InstabilitiesAstrophys. J. 483 (1997), 262-273.

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[4] C-Y. Wang and R. A. Chevalier, Instabilities and Clumping in Type Ia Supernova RemnantsAstrophys. J. 549 (2001), 1119-1134.

[5] E. C. Harding, et al.; Observation of a Kelvin-Helmholtz Instability in a High-Energy-DensityPlasma on the Omega Laser Phys. Rev. Lett. 103 (2009), 045005.

[6] O. A. Hurricane, et al., A high energy density shock driven Kelvin-Helmholtz shear layerexperiment Phys. Plasmas 16 (2009), 056305.

[7] R. Blaauwgeers, et al., Shear Flow and Kelvin-Helmholtz Instability in Superfluids Phys.Rev. Lett. 89 (2002), 155301.

[8] H. Takeuchi, et al., Quantum Kelvin-Helmholtz instability in phase-separated two-componentBose-Einstein condensates Phys. Rev. B 81 (2010), 094517.

[9] http://mentalfloss.com/article/29587/14-gorgeous-kelv-helmz-cloud-formations (30.3.2015).

[10] P. K. Kundu, I. M. Cohen, Fluid Mechanics (Elsevier Academic Press, 3rd edition,2004),476-488.

[11] S. A. Thorpe, Journal of Fluid Mechanics 46: 299–319 (1971).

[12] B. Cushman-Roisin, J. Beckers, Introduction to Geophysical Fluid Dynamics - Physical andNumerical Aspects (Academic Press, 2009), 393-401.

[13] W. D. Smyth and J. N. Moum, Ocean Mixing by Kelvin-Helmholtz Instability Oceanography25 (2012), 140-149.

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