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Basic concepts of numerical modeling in ocean and atmosphere

Lie-Yauw OeyPrinceton Universitylyo@princeton.edu

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Basic concepts of numerical modeling in ocean and atmosphere

Training Course Session#1

IWMO3, Qingdao, China, Jun/10/2011

L.-Y. Oey

Princeton University

lyo@princeton.edu

Lecture 1: Basic geophysical fluid dynamics

(a) wind waves & flows without rotation;

(b) flows constrained by rotation - Taylor column;

(c) effects of friction: Ekman dynamics;

(d) Sverdrup balance and western boundary current;

(e) instability, waves & eddies.

Lecture 2: Basic numerical techniques

(a) finite-difference grid & approximation;

(b) time and space differencing - the advection equation;

(c) consistency, accuracy & stability;

(d) implicit scheme;

(e) boundary conditions, air-sea coupling.

Lecture 3: Numerical GFD experiments with the Princeton Ocean Model:

Descriptions

(practice to be given in Toni Jordi’s PM session)

(a) 1-D step and block-propagation problems;

(b) Taylor column;

(c) Surface and bottom Ekman layers;

(d) Western Boundary Current – Stommel & Munk;

(e) Estuarine plume debouching onto a continental shelf;

(f) Estuarine plume with tide;

(g) Baroclinic waves and eddies;

(h) Wetting & Drying.

Pls. download codes, runscripts, inputs and outputs etc from:

ftp://aden.princeton.edu/pub/lyo/iwmo3/training-class/

for parallel mpiPOM/sbPOM versions; and from:

ftp://aden.princeton.edu/pub/lyo/pom_gfdex/wmo09training/

anIntroCourseNumOceanExpsUsingPOM/

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for POM08/POM2k versions. Chapter 1: Flows without Rotation

1.1 Waves at the Ocean’s Surface:

Let u = (u, v, w) be the velocity field and z = (x,y,t) be the free surface (Fig.1); and assume ||/H<<1, and also that water density = o = constant:

Fig.1.1-1.

Momentum: o u/t = p’, (1.1-1)

where p’ = ptotal (patmos ogz) (1.1-2)

Then (u)/t = 0, flow is irrotational (1.1-3)

Therefore, u = ; where = velocity potential (1.1-4)

Water is incompressible: .u = 0 2 = 0 (1.1-5)

This is a Laplace eqn.; waves are due entirely to the undulating .

At z = , ptotal = patmos, so that:

(1.1-2) p’ = og, at z = 0 (1.1-6)

Use (1.1-4) & (1.1-6), then:

(1-1) /t = g, at z = 0 (1.1-7)

Also, /t = w = /z, at z = 0, (since D(z)/Dt = 0). (1.1-8)

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(1.1-7) & (1.1-8) 2/t2 = g/z, at z = 0 (1.1-9)

We assume a flat-bottom ocean H(x,y) = h = constant. Then:

w = /z = 0, at z = H, (since D(z+H)/Dt = 0). (1.1-10a)

or ~ 0, z ~ , i.e. for large H (deep water) (1.1-10b)

Eqns.(1.1-5), (1.1-9) and (1.1-10) are the surface-wave equations.

Look for a sinusoidal solution (also simplify to xz-t):

= (z).exp[i(tkx)], = frequency & k = wave number(1.1-11)

(1.1-5)&(1.1-10a) (z) = o cosh[k(z+H)] (1.1-12)

Then the surface condition (1.1-9) gives the dispersion relation:2 = gk.tanh(kH) (dispersion relation) (1.1-13)

Velocity: u = (kgo/). sin(kxt).cosh[k(z+H)]/cosh(kH) (1.1-14a)w = (kgo/).cos(kxt).sinh[k(z+H)]/cosh(kH) (1.1-14b) = o sin(kxt) (1.1-14c)

Shallow-water limit kH << 1 : tanh(kH) ~ kH, c = /k (gH)1/2 (1.1-15) u ~ (kgo/). sin(kxt); w ~ 0

Deep-water limit kH >> 1 : tanh(kH) ~ 1, c = /k (g/k)1/2 (1.1-16) (u,w) ~ (kgo/).exp(kz).{cos(kxt),sin(kxt)}.

Table 1.1-1 Typical values of deep-water waves, H >> /(2) (m) c (m/s) tp (s) Water deeper than H (m) ~ /2

1 1.25 0.8 0.5

10 4 2.5 5

100 12.5 8 50

1000 40 25 500

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1.2 Shallow-Water & Large-Scale Flows of Constant Density ( = o):

When 2H/ << 1, (1.1-15) w ~ 0, and u = (u,v) function(z), so (1.1-7) applies to all z. Then, (1.1-7):

u/t + f k×u = g (1.2-1)

where (because of large scale) a Coriolis acceleration due to the earth rotation 2sin() = f (where = latitude, and = 2/86400 s-1) has been added.

Hydrostatic: ∂p/∂z = −ρg; so that p = pa − ρgz + ρgη (1.2-2)

Mass conservation:

/t + .(Du) = 0, D = H(x,y)+(x,y,t) (1.2-3a)

or, linearized, /t + .(Hu) = 0. (1.2-3b)

Fig.1.2-1. The atmosphere and ocean are thin, so vertical motions are approximately hydrostatic. Large-scale flows are significantly affected by the earth’s rotation.

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An Application of Hydrostacy:(IWMO3.2b)

Fig.1.2-2

Mass × Accel. = Pressure Force

c xyz × u/t = (pA pB) yz

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u/t = p/x

u/t = g/x or g’/x1.3 Step and Block Propagation problems:

Eqns.(1.2-1) & (1.2-3b) in full:

∂∂ t

+ ∂uH∂ x

+ ∂ vH∂ y

=0 (1.3-1a)

( ∂ u∂ t )=+ f v−g

∂∂ x

+(τ ¿¿w x−ru)/H ¿ (1.3-1b)

( ∂ v∂ t )=−fu−g

∂∂ y

+(τ ¿¿w y−rv) /H ¿ (1.3-1c)

For f=0, no wind and no friction, we can derive the wave

equation (c2=gH):

∂2η/∂t2 – (c2∂η/∂x)/x (c2∂η/∂y)/y = 0 (1.3-2)

1-D solution:

Suppose that at t=0, u=0, and = G(x) (1.3-3)

Then the solution to (1.3-2) in 1-D for H=constant is:

= [G(x+ct) + G(x-ct)]/2 (1.3-4)

u = (g/c) [G(x+ct) G(x-ct)]/2 (1.3-5)

where (1.3-5) follows by using (1.3-4) in (1.3-1b).

Step propagation: suppose G(x) = o sgn(x) (1.3-6)

then, = o[sgn(x+ct)+sgn(x-ct)]/2 (1.3-7)

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and u = (og/c)[sgn(x+ct)sgn(x-ct)]/2 (1.3-8)

See figure 1.3-1.

Fig.1.3-1 Step propagation. (IWMO3.2e)

KE/area = u2/2 dz H(og/c)2/2 = go2/2

PE/area = gz dz = g[o2H2]/2

(where is from z=H to z=).

The perturbation PE/area is then go2/2 = KE/area.

Therefore, the initial PE/area due to = o sgn(x)

at t=0 is all converted to KE/area to generate

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current in the direction of “high pressure to low

,” i.e. direction of /x.

Block propagation (fig.1.3-2): (IWMO3.3b)

At t=0, G(x) = o for |x| < L, and =0 for |x|>L. (1.3-9)

In this case, the solution is most easily obtained graphically

(keep in mind equations 1.3-4 and 1.3-5) as shown in fig.1.3-2.

Fig.1.3-2 Block propagation. (IWMO3.3e)

PE/y-length = PE(0) = gz dzdx = go2L

(where is from H to z=, and L to +L).For t > L/c, each block of height o/2 has PE = g(o

2/4)/2×2L = PE(0)/4. The KE of each block = HLu2 = HLo

2g2/4c2 = go2L/4 = PE(0)/4. Total

energy is PE(0)/2 per block, i.e. = PE(0) for 2 blocks.

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Numerical Experiments:The step and block propagation problems can be downloaded from:ftp://aden.princeton.edu/pub/lyo/pom_gfdex/wmo09training/anIntroCourseNumOceanExpsUsingPOM/gill_blok_and_step_prop/

Chapter 2: Flows with Rotation(IWMO3.4b)

In Fig.2-1, outward centrifugal force is balanced by radial pressure force due to gravity associated with sea-surface tilt:

Fig.2-1

V2/r = (p/r)/ = g(h/r) (from 1.2-2) (2-1)

Here, “V” is the azimuthal velocity in a fixed frame of reference. Fluid parcel moves relative to the rotating “table” (the earth) with (u,v) = (radial, azimuthal) velocities as shown. We have:

V = r + v (2-2)

Conservation of Angular Momentum:

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Suppose a parcel is initially at the rim of the rotating table and is at rest relative to the table: (u,v) = (0,0). Imagine that the parcel moves towards the center – then conservation of angular momentum:

V.r = v.r + .r2 = constant = .r12, where r1 = table’s radius

v = [1 (r/r1)2](r12/r) (2-3)

so that the parcel spins faster and faster around the center of the table (in the same direction as , since r/r1 < 1) as it traverses towards the center.GFD3_slow_rotation.mpeg GFD3_fast_rotation.mpeg

Gradient Wind Balance:

Eqn.2-2 into 2-1 gives:v2/r + 2v = g/r, where = h g-12r2/2 (2-4)

where is the height of the free surface relative to a time-independent reference parabolic surface z = g-12r2/2.

Rossby Number Ro:

This is normally defined as Ro = U/(2.L) (2-5)

where U and L are velocity and horizontal length scales respectively. In (2-4) one can define Ro = v/(2r), so that:

Ro2 + Ro = gr-1(/r)/(2)2; where Ro = v/(2r) (2-6)

Cyclostrophic Balance:

For large Ro, i.e. inside a tornado, eqn.(2-6) Ro

2 gr-1(/r)/(2)2 or v2/r g/r (2-8)

Geostrophic Balance:

For small Ro, i.e. large-scale oceanic and atmospheric flows, eqn. (2-6)

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Ro gr-1(/r)/(2)2 or 2v g/r (2-7)

For the remaining of the lecture, we will almost exclusively deal with small Ro.

Taylor Column:For small Ro, the flow is nearly geostrophic, then (c.f. equation 1.2-1)

2 × u = p/ (2-9)

Take the curl of (2-9), and note that

×( × u) = (.u) u(.) + (u.) (.)u = (.)u

[Use ×(a × b) = a(.b) b(.a) + (b.)a (a.)b], and also,

×(p/) = -1×p -2(×p) = -2(×p), we have:

2(.)u = -2(p×) (2-10)

If p and surfaces coincide – fluid is barotropic, then

(.)u = 0 (2-11)

and the u does not change in the direction of the -axis (z), i.e. flow is 2D.

Figure 2-2. Nearly-steady homogeneous (constant-density) flow in an x-periodic channel, 600km by

300km, of depth 200m except at the channel’s center where a cylinder rises 50m above the bottom.

Color is sea-surface height in meters and vectors are velocity at (A) z=0m (i.e. surface), (B) z=90m

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and (C) z=180m. This model calculation was carried out for 100 days when the flow has reached a

nearly steady state. The experiment is meant to illustrate Taylor-Proudman theorem: flow below the

cylinder’s height goes around the cylinder while above it the flow also tends to go around as if the

cylinder extends to the surface. The velocity does not vary with “z” and the vertical velocity (which is

not shown) is nearly zero. GFD7_taylor .mpeg

Thermal Wind Balance – how Gravitational Collapse is arrested by Rotation:If fluid is baroclinic, i.e. p and surfaces do not coincide, then since the pressure field is nearly hydrostatic, i.e. -1p gk (2-12)eqn.(2-10) becomes:

2.u/z = -1gk× (2-13)This is thermal-wind equation – see Fig.2-3.

Fig.2-3 Illustration of thermal wind equation (2-13):1 Vortex tube with background cyclonic is tilted by shear u(z) ..2 producing cyclonic vorticity in x-direction..3 the yz-circulation due to the x-cyclonic vorticity is to tilt the

=constant isopycnal as shown, against 4 the gravitational tendency to collapse it to a level position.

y

zy

z

=const

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Fig.2-4 Note that the u/z produces x-vorticity x in the sense that tends to oppose the gravitational collapse(IWMO3.4e)

Effects of Friction: (IWMO3.5b)∂∂ t

=−∂ U∂ x

−∂V∂ y

(2-14a)

( DuDt )=+ fv−g

∂∂ x

+ ∂∂ z

(K M∂ u∂ z

) (2-14b)

( DvDt )=−fu−g

∂∂ y

+ ∂∂ z

(K M∂ v∂ z

). (2-14c)

Fig.2-5 Modification of geostrophic balance fz×u = p/ by friction.

Dropping the D/Dt and pressure terms from (2-14b,c), and integrating dz form z=E to z=0, we have Ekman transport UE = (UE,VE) produced by wind stress o = (x, y):

fz×UE = o, or fVE = x & +fUE = y (2-15)

.z

xy

x > 0 by +ve uz

x < 0

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similarly for bottom stress b – see Fig.2-5 – note how friction can produce Ekman transport from high to low pressure.

Fig.2-6 Low pressure center produces convergence (left), hence upwelling; and vice versa for high pressure center (right).

Fig.2-7. Nearly-steady homogeneous (constant-density) wind-driven rotational flow in a channel of depth 200m and theoretically unbounded in x, but in practice an x-length = 1000 km is used in the solution. The flow is assumed independent of y. The wind stress is specified in y-direction only: (0, 10-4).(x/500km - 1) m2 s-2. Panel (A) is for the y-directed velocity and panel (B) the x-directed velocity. In (B), profiles of u are also plotted at the four indicated

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x-locations with scales ±0.1 m/s given along the bottom for the first and fourth locations. Detailed plots comparing the profile at x=205 km (i.e. the first x-location) with analytical solution are given in Figure 1-10. The Coriolis parameter is constant, fo = 610-5 s-1, and the vertical eddy viscosity is also constant, K = 510-3 m2 s-1. This numerical solution is at time = 100 days, and differs slightly from the analytical one discussed in the text; instead of the no-slip condition at the bottom, a matching of the velocity to the law-of-the-wall log layer is used [see the POM manual by Mellor, 2004].

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Simple Ideas of Frictional Spindown and Buoyancy Shutdown

Ocean is spun down by friction frictional spindown. But with stratification and

bottom slope, it is possible that bottom mixing (w/stratification+slope) produces

horizontal density gradients, hence thermal-wind shear in the mixed layer so that the

interior flow “sees” a slippery bottom and Ekman pumping & friction are shutdown

buoyancy shutdown.

Rough derivations:

To compute /y due to bottom slope h/y and bottom mixing (downwelling case),

we consider an isopycnal =constant; then = 0 = ¶/¶z h + ¶/¶y y so that

/y = -/z.h/y so that (g/o)/y = N2.hy, where N2 = -g(/z)/o. Or, with b

= -g/o we have: ¶b/¶y = -N2hy where N2 = ¶b/¶z.

By geostrophy within the mixed layer, we have fu = -p/y/o, so that f¶u/¶z = +(g/o)

¶/¶y = -¶b/¶y = N2hy from previous equation. So that integrating across ML, from

z=-h to z=-h+, where = ML thickness:

ui – u = (-h+-z) N2hy assuming that N2 is constant. Therefore: u=ui-(-h+-z) N2hy.

Note:

(a) buoyancy shutdown arises only if bottom has slope h/y0;

(b) if hy is =0, then u = ui extending all the way to bottom, and the interior flow is

then subject to frictional spindown (because then b = -rui operates);

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(c) basically, at buoyancy shutdown, the ui is canceled by thermal-wind part and

b=0.

References:

Garrett, MacCready & Rhines, Ann. Rev. Fluid Mech, 1993, 291-323.

Chapman, JPO, 2002, 336-352.

(IWMO3.5e)

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Convective, Symmetric & Baroclinic Instabilities Made Simple

The atmosphere and ocean are full of waves and eddies.

spatially high-pass satellite SSH

trajectories of all of the 2435 cyclones (blue lines) and 2273 anticyclones (red lines)

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Chapter 3: A Model of Large-Scale Ocean Gyre & Western

Boundary Current

In this section, unless otherwise stated, all variables will be

dimensional. Consider a “box” ocean with sides Lx and Ly on a -plane

(Figure 3-1). A zonal wind stress is applied:

τ w=(−τwo cos ( y /L y) ,0) (3-12)

so that

curl z(τ ¿¿w)=−(τ¿¿wo/Ly)sin( y /L y)¿¿ (3-13)

where wo (which will be set = 10-4 m2s-2 below) is the magnitude of the

(kinematic) wind stress. Initially, the fluid is at rest:

(u, v, w) = (0, 0, 0), and also, free surface = 0 (3-14)

At the four side walls, both the normal and tangential components of the

horizontal velocities are zero:

(u, v) = (0, 0) at x = 0, x = Lx and also at y = 0, y = Ly. (3-15)

At the ocean bottom and at the free surface, equations (1-31) are

satisfied:

w= ∂∂ t

+u∂

∂ x+v

∂∂ y

, at z=( x , y , t ) , (3-16a)

w=−u∂ H∂ x

−v∂ H∂ y

, at z=−H ( x , y ) . (3-16b)

The continuity equation and the x and y momentum equations are then

(c.f. 1-35): (IWMO3.6b)

∂∂ t

=−∂ U∂ x

−∂V∂ y

(3-17a)

( ∂ u∂ t )=+ fv−g

∂∂ x

+ ∂∂ z

(KM∂u∂ z

) (3-17b)

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( ∂ v∂ t )=−fu−g

∂∂ y

+ ∂∂ z

(K M∂ v∂ z

). (3-17c)

where

(U ,V )=∫−H

❑

(u , v ) dz (3-18)

is the depth-integrated transport vector (per unit length). These

equations differ from (1-35) in that the nonlinear advective terms in

momentum equations (3-17b,c) have been dropped. (IWMO3.6e)

Figure 3-1. A “box” ocean circulation driven by eastward (northern half

of basin) and westward (southern half) wind stress distribution.

Depth-integrate (3-17b,c), and again drop nonlinear terms such

as u etc:

( ∂ U∂ t )=+ fV−gH

∂∂ x

+τwx−τb

x(3-19a)

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( ∂ V∂ t )=−fU−gH

∂∂ y

+τwy−τb

y(3-19b)

where w and b are wind and bottom stress vectors respectively. These

equations together with equation (3-17a) constitute there equation for

three unknowns (, U, V). For our purpose, it is more convenient to

write them in terms of the depth-averaged velocity:

(u , v )=∫−H

❑

(u , v )dz /D∫−H

0

(u , v ) dz /H=(U ,V )/H , (3-20)

so that (3-17a) and (3-19) become: (IWMO3.7b)

∂∂ t

+ ∂u H∂ x

+ ∂ v H∂ y

=Q (3-21a)

( ∂ u∂ t )=+ f v−g

∂∂ x

+(τ ¿¿w x−r u) /H ¿ (3-21b)

( ∂ v∂ t )=−f u−g

∂∂ y

+(τ ¿¿w y−r v )/H ¿ (3-21c)

where r is the bottom friction coefficient with dimension m/s, we have

modeled the bottom friction as:

(τ bx , τb

y¿=r (u , v ), (3-22)

and we have also added a source term Q to the right hand side of (3-

21a). This we will use later to model the net effect of precipitation ( P)

minus evaporation (E), i.e.

Q = (P – E)/o (unit is m/s). (3-23)

Let H = constant, a vorticity equation similar to (3-10) may now

be derived from (3-21). Taking the curl of (3-21b,c), we obtain:

∂❑∂t

+❑o v−f∂

∂ t/H=[curl¿¿ z (τ ¿¿w)−Qf −r❑]/H ¿¿ (3-24a)

Apart from the terms involving Q and /t, this equation is equivalent

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to (3-10). The term /t does not appear in (3-10) because it is

neglected when we derived the vertical velocity at the surface (see

equation 1-67). We will explain in the next section when this term may

be deleted; assuming that it can be, then (3-24a) becomes:

∂❑∂t

+❑o v=[curl¿¿ z (τ ¿¿w)−Qf−r❑]/H ¿¿ (3-24b)

The appearance of -Qf together with curlz(w) means that as far as the

large-scale ocean circulation is concerned, effects of wind stress curl

and net precipitation minus evaporation are equivalent. The reason is

that in both cases, the ocean layer beneath the surface boundary layer

(where wind and precipitation etc act) ‘sees’ only vertical mass flux

coming in (or out) of the surface layer. In the case of wind stress curl,

this is “Ekman pumping” (Chapter 1) caused by convergence or

divergence in the surface layer, while in the case of precipitation etc, it

is more direct – due to the flux of water from the atmosphere. We will

model these effects and give a physical interpretation below. Also,

while H is the ocean depth, it may be (and for our purpose will be)

interpreted as the depth of the main thermocline in which the

predominant large-scale currents reside. In that case, the friction r u is

then considered as the friction at the base of the main thermocline.

Sverdrup Relation:

In the open ocean away from side boundaries, the relative

vorticity ❑ is assumed weak, in fact ❑0, and the main vorticity balance

in steady state is, from (3-24b) (omitting Q for simplicity):

❑o v H=❑o V=curl z(τ ¿¿w)¿ (3-25)

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Figure 3-2. A schematic to explain the equatorward Sverdrup transport

V caused by a negative wind stress curl, curlz(w) < 0.

Equation (3-25) is the Sverdrup relation; it shows that in steady motion,

a negative curlz(w) produces southward transport. Figure 3-2 explains

the physics involved. The negative curlz(w) produces convergent flow

in the thin surface (Ekman) layer (Chapter 1) and hence a downward

pumping that squashes the vortex tube in the deeper interior; this is

equivalent to a divergent subsurface flow. Squashing increases the

cross-sectional area of the tube, and if it stays at the same latitude a

negative relative vorticity (i.e. a reduced spin) would be produced, ❑<0

. To keep the same ❑ 0, and assuming that the tube remains

approximately vertical (and it does), the tube must move south along

the curved earth surface where it can stretch and retain its initial zero

spin. The curved surface in this case is the physical equivalence of the

-effect, i.e. the Coriolis parameter f varies with latitude. Gill [1982;

p.465] gives another, equivalent explanation. The divergent subsurface

flow beneath the negative curlz(w) increases the cross-sectional area of

the vortex tube, so the absolute vorticity (i.e. f + ) decreases. But since

f >> , the only way that this decrease can be accomplished is that the

tube moves southward where f is smaller.

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Figure 3-2 illustrates two other features. One is that the effect of

(positive) Q [= (P–E)/o] is similar to (negative) curlz(w): this is

because Q ‘pumps’ water into the subsurface layer and again squashes

vortex tube. The other one is that while the surface is slightly elevated

( > 0) it is not important in the argument given above; in steady state, it

is in fact = 0.

If as shown in Figure 3-1 the Sverdrup transport occurs over

most portion of the ocean basin, then the total southward transport is:

SV ∫0

Lx

curlz (τ ¿¿w )/❑o dx¿ (m3 s-1). (3-26)

Figure 3-3. Annual mean distribution of Ekman pumping, positive =

upwelling, in unit of 10-6 m s-1 (or 0.1 m/day). From Tomczak and

Godfrey.

Figure 3-3 shows a global map of Ekman pumping – note large areas of

downwelling in the subtropical regions, suggesting equatorward

Sverdrup transports there. From (3-26), we have roughly SV curlz(w).Lx/o 2¿ τ w

x ∨¿Lx/(Lyo) (2×10-4/10-11)(Lx/Ly) m3 s-1 20 Sv (1

Sv = 106 m3 s-1) for Lx Ly, ¿ τ wx ∨¿ 10-4 m2 s-2 and o 10-11 m-1s-1. The

corresponding Ekman pumping (downwelling) is from (1-64b) =

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curlz(w)/fo 2¿ τ wx ∨¿/(Lyfo) 1~2×10-6 m s-1 for Ly 2000 km and fo

6×10-5 s-1, roughly equal to the values shown in Figure 3-3. This Ekman

velocity (or Figure 3-3) may be compared with contours of Q [= (P–E

)/o] shown in Figure 3-4, which gives local maxima of roughly 10-7 m

s-1, about 10 times weaker than the Ekman pumping.

Figure 3-4. Annual mean distribution of precipitation minus

evaporation, Q = (P–E)/o, in m yr-1 ( 3×10-8 m s-1); shaded regions are

positive.

Stommel’s Model of the Western Boundary Current (WBC):

In our idealized closed basin (Figure 3-1), the southward

Sverdrup transport must be returned northward. Since this return

cannot be in the mid-ocean (i.e. open ocean) because of the Sverdrup

constraint (as seen above), it must occur near the western and/or eastern

boundaries where the Sverdrup balance is broken. In other words, other

terms in the potential vorticity equation (3-24b), such as friction and/or

nonlinearity (which is neglected in 3-24b), become important near the

boundary. Stommel assumes that (bottom) friction is important so that

(3-24b) becomes, in steady-state:

∂ v∂ x

+( H❑o

r ) v=0. (3-27)

Including curlz(w) adds unnecessary complexity without adding

P a g e | 28

physical insight so it is omitted. Also, near the western or eastern

boundary, ❑∂ v /∂ x. The solution is:

v=C ( y ) exp (−H❑o x /r ) (3-28)

where C(y) is an arbitrary function of y to be determined. Equation (3-

28) shows a northward jet which is maximum at the boundary (for an

eastern boundary, the “x” is replaced by “xLx”) and either decays (if

western boundary) or grow (if eastern boundary) exponentially away

from the boundary. Therefore, the northward return flow must be along

the western boundary and crucially depends on a non-zero o; the width

of this return jet may be estimated as the e-1 x-scale:

xwbc = r/(Ho) (3-29)

This is 50 km for r 10-4 m s-1, H 200 m and o 10-11 m-1 s-1. The

C(y) is determined by requiring that the total northward transport

carried by the WBC is equal to the southward Sverdrup transport at the

same latitude (i.e. same “y”). Thus, ∫0

∞

v H . dx=C ( y )r /❑o=−¿SV ¿, i.e.

C ( y )=−¿o SV /r=∫0

Lx

curl z(τ ¿¿w)dx /r ¿¿ (3-30)

With the aforementioned values of SV 20 Sv, o 10-11 m-1 s-1 and r 10-4 m s-1, C 2 m s-1.

The Stommel WBC solution (3-28) therefore consists of a jet

concentrated against the western boundary. Since this jet has to return

northward all the Sverdrup transport across a narrow width xwbc, the jet’s

velocity is large. The Sverdrup velocity scale is, from (3-25), curlz(w)/(Ho) 2¿ τ w

x ∨¿/(HLyo) (2×10-4/10-11)/(HLy) m s-1 0.1 m s-1

for H = 200 m and Ly = 2000 km; while the Stommel velocity scale is C

2 m s-1. Stommel’s jet is maximum at the coast x = 0 and decays

exponentially offshore to become very weak in the ocean’s interior

where the southward Sverdrup transport prevails. Note that the

Stommel’s jet, being wholly positive, cannot join smoothly to the

Sverdrup flow, nor is it expected to since it is based on a frictional

P a g e | 29

equation that ignores the Ekman pumping by the wind; but this is not

important. Nor is it important that the Stommel’s jet cannot satisfy the

no-slip condition at the coast. The important thing is that: (1) beta (o)

is crucial for the existence of the western-intensified boundary current,

and (2) the WBC jet provides a mechanism that closes the ocean-basin

mass balance. (IWMO3.7e)

Munk’s Model of the Western Boundary Current (WBC):

A slightly more complicated model can be constructed to satisfy

the no-lip condition at x = 0. Instead of the bottom friction we include

horizontal viscosity terms on the RHS of the momentum equations:

∂∂ t

+ ∂u H∂ x

+ ∂ v H∂ y

=Q (3-31a)

( ∂ u∂ t )=+ f v−g

∂∂ x

+ τwx /H+AH ∂2u/∂ x2

(3-31b)

( ∂ v∂ t )=−f u−g

∂∂ y

+τwy /H+AH ∂2 v /∂ y2

(3-31c)

where AH is the horizontal (eddy) viscosity coefficient (unit is m2 s-1),

assumed a constant. If we now take the curl etc as in the derivation of

(3-24b) we obtain:

∂❑∂t

+❑o v=[curl¿¿ z (τ ¿¿w)−Qf ]/H+A H2❑¿¿ (3-32)

Define a stream function (unit m2 s-1) such that:

v=❑x∧u=−¿ y¿ (3-33)

and therefore

¿∂ v /∂ x−∂u /∂ y=¿2 ¿ (3-34)

In the WBC, we now let the beta term balances the horizontal viscosity

term, equation (3-32) in steady state is then (c.f. 3-27):

P a g e | 30

AH ∂4/∂ x4−❑o ∂/∂ x=∂4 /∂❑4−∂/∂=0 (3-35a)

where

= (o/AH)1/3x. (3-35b)

Equation (3-35a) is a fourth-order ordinary differential equation (with

“y” as the ‘floating’ parameter not directly involved), and a solution of

the form exp(m) maybe assumed. Four roots are:

m1 = 0, m2 = 1, m3 = exp(i/3)=(1+i3)/2, m4 = exp(i2/3) =(1i3)/2 (3-36)

The solution then is of the form

= C1 + C2exp(m2) + C3exp(m3) + C4exp(m4) (3-37)

where the C’s are arbitrary function of “y” yet to be determined. The m2

root must be deleted - by choosing C2 = 0, because it leads to a solution

which exponentially grows with (or x). The root m1 leads to C1 –

which is chosen to be C1 = I(y), where the subscript “I” denotes

“interior” (of the ocean away from the boundary) and from the interior

Sverdrup solution (3-25) with vI=∂I /∂ x,

❑I ( x , y )=∫0

x

curl z(¿ τ w)/(H ¿¿o)dx '+❑o( y )¿¿ (3-38)

is the stream function of the interior’s Sverdrup flow; o(y) is an

arbitrary function. This choice is so that ~ I for ~ , since m3 and

m4 give exponentially decaying solutions for large .

Substituting m3 and m4 from (3-36) to (3-37), we obtain then:

= I + e−¿2[C3cos (√32 )+ C4sin(√3

2 )] (3-39)

the C3 and C4 here differ from those in (3-37) but the detail is not

important; they are still arbitrary functions of “y.” The corresponding

velocity components from (3-33) are:

P a g e | 31

u=−∂I

∂ y−e−¿ 2[

∂ C3

∂ ycos(√3

2 )+ ∂ C4

∂ ysin(√3

2 )] (3-40a)

v=e−¿2[(−C3

2+√3C4

2 )cos(√32 )−(√3 C3

2+

C4

2 )sin(√32 )] (3-40b)

The conditions that there can be no normal flow (u=0) and no slip (

v=0) at = 0 give:

C3 = -I(0,y) + K, where K = constant, and (3-41a)

C4=C3/√3. (3-41b)

Therefore, (3-39) gives:

= I(x, y) - I(0,y)e−¿2[cos (√32 )+ 1

√3sin(√3

2 )]+ Ke−¿2[cos (√3

2 )+ 1√3

sin(√32 )] (3-42)

From equation (3-35b), we note that (AH/o)1/3 has the unit of length (m).

Its value is 100 km (for AH = 103 m2 s-1 and o 10 -11 m-1 s-1) which is

thin compared to the width of the basin 2000 km (say); it is the e-

folding decay scale of the western boundary current near x 0. We are

therefore making only a small error if I(0,y) in (3-42) is replaced by

I(x,y). Another way of saying the same thing is that, since I varies

over a large scale >> (AH/o)1/3, I(x, y) I(0,y) near the western

boundary. To determine K, we note that, since our basin is closed, the

stream function around the boundary is zero. Therefore, for to be also

= 0 at x = 0, (3-42) shows that K must also = 0. Therefore, (3-42)

becomes:

= I(x, y) {1 - e−¿2[cos (√32 )+ 1

√3sin(√3

2 )]} (3-43)

The northward velocity is /x:

P a g e | 32

v=(¿o/ AH)13

I (0 , y ) 2

√3sin(√3

2 )e−¿2

¿, = (o/AH)1/3x, (3-44)

in which we have neglected the interior velocity contribution I/x,

which is much smaller. Note that the jet form is again apparent by the

exponential decay term. But in contrast to the Stommel’s solution

equation (3-28), because of the sine function, the northward velocity is

zero at x = 0. Also because of the sine function the velocity becomes

slightly negative (i.e. reversed flow) at the offshore edge of the jet.

Class: estimate the magnitude of v from (3-44).

3-3: The Equivalent Reduced-Gravity Model

In this section non-dimensional variables will be denoted by

subscripts “n.” The above model has one layer and involves the free

surface . We now show that with slight reinterpretations of the

variables, the equations of that model are exactly the same as those

governing the motion of a two-layer ocean in which one of the layers is

infinitely thick and is quiescent. Before we do this, we find out first the

condition when the term involving /t in equation (3-24a) may be

neglected. This is necessary in order to understand the restrictions on

the spatial and temporal scales under which the resulting equation(s)

apply. Instead of (3-24a), we will non-dimensionalize the vorticity

equation that uses the horizontal viscosity instead of the bottom friction,

i.e. we use (3-32) with f(/t)/H retained:

∂❑∂t

+❑o v−f∂

∂ t/H=[curl¿¿ z (τ ¿¿w)−Qf ]/H+AH

2❑¿¿ (3-45)

Define the following non-dimensional variables:

❑=❑n(UL ), o=❑on(U

L2 ), v=vn U ,=❑n( fULg ) , t=tn( L

U ),

w = wno, Q = Qn Qo, AH = AHn AHo. (3-46)

Note that the scale for , i.e. fUL/g, is determined from balancing the

P a g e | 33

pressure gradient and Coriolis terms in equation (3-31b,c). Substitute

(3-46) into (3-45):

∂❑n

∂t n

+❑on vn−( L2

R2 ) ∂❑n

∂t n

=[( Lo

HU 2 )curl¿¿ zn(τ ¿¿wn)−( L2 fQo

HU 2 )Qn]+( AHo

UL )AHn

❑n2❑n ¿¿

(3-47)

where R = (gH)1/2/f is the Rossby radius of deformation. We see that

dropping the term involving n/tn amounts to saying that R >> L, i.e.

the spatial scale of motion is much smaller than the Rossby radius. For

H 1000 m and f 6×10-5 s-1, R 1500 km, so that motions with scales

smaller than about 1000 km will not be much affected by the production

of vorticity due to stretching associated with the motion of the free-

surface. This kind of motion is called a barotropic motion. On the

other hand, as we will now show, R can be quite small for the reduced-

gravity model, in which case the stretching term becomes important.

P a g e | 34

Chapter 4: Two-layer approximation

4.1 2-layer equations

4.2 pressure compensation

4.3 reduced-gravity equivalence

P a g e | 35

Approximate ocean as 2-layer:

Vertical sections of density σθ in the western Atlantic. From Lynn and Reid (1968).

Profile of in situ and potential temperature in the Kermadec Trench in the Pacific on 13 July 1967 at 175.825°E and 28.258°S. Data from Warren (1973)

P a g e | 36

Conservation of volume:∂ hi

∂t+

∂ ui hi

∂ x+

∂ v i hi

∂ y=0, i = 1, 2. (4.1)

Pressure in layer 1, 1 z H1+2,

p1(x,y,z,t) = pa(x,y,t) + g1(1 z) (4.2)

Pressure in layer 2, H1+2 z (H1+H2),

p2(x,y,z,t) = p1(z =H1+2) + g2(H1 + 2 z) (4.3)

Pressure gradients, e.g. /x:p1/x = pa/x + g11/x, in layer 1; (4.4)

p2/x = p1/x + g(1h1)/x, in layer 2, (4.5)

P a g e | 37

where = (2 1), and (4.2) was used for p1(at z

=H1+2), also 2 = H1 + h1 - h1 (see Figure).∂ h1

∂t+

∂ u1 h1

∂ x+

∂ v1 h1

∂ y=0 (4.5a)

( Du1

Dt )=+ f v1−1ρo

∂ pa

∂ x−g

∂❑1

∂ x+τw

x /H (4.5b)

( D v1

Dt )=−f u1−1ρo

∂ pa

∂ y−g

∂❑1

∂ y+τ w

y /H (4.5c)

∂ h2

∂t+

∂ u2 h2

∂ x+

∂ v2 h2

∂ y=0 (4.6a)

( Du2

Dt )=+ f v2−1ρo

∂ pa

∂ x−g

∂❑1

∂ x− g ρ

ρo

∂(¿1−h1)

∂ x¿ (4.6b)

( D v2

Dt )=−f u2−1ρo

∂ pa

∂ y−g

∂❑1

∂ y− g ρ

ρo

∂(¿1−h1)

∂ y¿ (4.6c)

Pressure Compensation

Deep layer-2 currents are often weak, so that (4.6b,c)

can be integrated:

1 + C = (- or 1 = -(

choosing C = (. Where 1 convexes

upwards, 1 > 0, the isopycnal concaves downward,

2 < 0.

Reduced-gravity equation:

∂ h1

∂t+

∂ u1 h1

∂ x+

∂ v1 h1

∂ y=0 (4.7a)

( Du1

Dt )=+ f v1−g ρρo

∂ h1

∂ x+τw

x /H (4.7b)

P a g e | 38

( D v1

Dt )=−f u1−g ρρo

∂ h1

∂ y+τw

y /H (4.7c)

P a g e | 39

Lecture 2: Basic numerical techniques (IWMO3.1beg)

Finite-difference (FD) grid & approximation

u = u(x), 0xL uj = uj(jx), j=1,2, …, J+1 (L2.1)

so that x = L/J.

Infinite-term Fourier expansion:u = ao/2 + 1

(an cos 2n x/L + bn sin 2n x/L), n1. (L2.2)

Note that all wavelengths are represented: L, L/2, L/3,…, L/.

But with J+1 values of uj on the grid, we can at best compute

ao, a1, a2, …, aJ/2, b1, b2, …, bJ/2, (L2.3)

The component with the shortest wavelength has n=J/2:

L/(J/2) = 2L/(L/x) = 2x (L2.4)

Approximating derivatives:

(du/dx)j (uj+1 uj)/x (L2.5)

Accuracy:

Substitute the true solution u(jx) into the RHS of (L2.5), and Taylor-expand:

(uj+1 uj)/x (du/dx)j + (d2u/dx2)jx/2! + (d3u/dx3)j(x)2/3! + … True …..…… Truncation Error ………

= O(x) = (d2u/dx2)jx/2! + (d3u/dx3)j(x)2/3! + … (L2.6)

We then say that the order of accuracy of the finite-difference (FD) approximation (L2.5) is “x,” or O(x).

P a g e | 40

Consistency:

To be consistent, the FD approximation of the derivative (e.g. L2.5) must approach the true derivative, i.e. ~ 0 as x 0.

Clearly, (L2.5) is a consistent approximation to du/dx.

Finite difference schemes

The algebraic equation obtained when derivatives in a differential equation are replaced by FD approximation is called a finite difference approximation to that differential equation, or a finite difference scheme.

Linear advection equation:

u/t + c u/x = 0, u = u(x,t), c = positive constant (L2.7)

General solution is: u = f(x-ct), f = arbitrary function (L2.8)

The “f” is determined by initial condition; for example, if

u(x,0) = F(x), then u = F(xct)is the solution (L2.9)

Fig.L.1 One of the characteristics x-ct=constant of (L2.7).

As time marches forward, the true solution depends on its value “to the left” or “upstream”. This suggests the following FD scheme:

(ujn+1 uj

n)/t + c (ujn uj-1

n)/x = 0 (L2.10)Forward Upstream

The truncation error is then obtained by substituting the true “u”:

P a g e | 41

= [u (jx, (n+1)t) u (jx, nt)]/t + + c [u (jx, nt) u ((j-1)x, nt)]/x (L2.11)

= (2u/t2)t/2 + (3u/t3)t2/3! + … c [(2u/x2)x/2 (3u/x3)x2/3! + …]

= O (x, t).

Consistency of the FD scheme:

As x and t 0, the above FD scheme (L2.10) is said to be consistent since it then becomes closer and closer to the actual differential equation (L2.7).

Convergence:

A FD solution is said to be convergent if for a fixed total time, ujn

u(jx,nt) as x and t 0.

A FD scheme is said to be convergent if it gives a convergent solution for any initial conditions.

A consistent FD scheme does not necessarily mean that its solution approaches the true solution (see Fig.L.2).

Fig.L.2 An example of a consistent FD scheme (L2.10) which does not yield a convergent solution.

In Fig.L.2, solid line is the true-solution characteristic passing through the origin (0,0) and the square grid point where/when the approximate FD solution ( u(0,0)) is desired. The slope of the characteristic is dt/dx = 1/c. On the other hand, using (L2.10), the FD-solution at depends

P a g e | 42

only on those circle grid points. The region defined by the circle grid points is called the domain of dependence of the FD scheme.

As we refine x and t keeping their ratio the same as that shown by the grid rectangles, the FD-solution at may be arbitrarily different from u(0,0). For the FD solution to “know” the u(0,0) value, it is clear that the FD’s domain of dependence must include the origin, i.e. the ratio t/x must be chosen such that it is less than the slope of the solution characteristic “1/c”:

t/x 1/c, or ct/x 1 (L2.12)

In Fig.L2, this can be accomplished by halving the t while keeping the x the same. We note that for the special case when t and x are chosen such that ct/x = 1, the FD-solution at will be exactly equal to the true solution = u(0,0). This is also seen in (L2.10) which then gives ui

n+1 = ui-1

n = ui-2n-1 = … etc.

Condition (L2.12) is a necessary condition for convergence of the FD scheme (L2.10). (Necessary because if we violate the condition, then we know for sure that the solution will not converge. In other words, making sure that the domain of dependence of the FD include the solution characteristic cannot guarantee that the FD-solution converges).

Stability:

What is the behavior of the FD-solution as time-stepping proceed?

Assuming that the true solution is bounded, then a FD-solution ujn is

stable if its error |ujn u(jx,nt)| remains bounded as “n” increases, for

fixed values of t and x.

A FD-scheme is stable if its solution is stable for any initial conditions.

To guarantee that a FD-scheme is stable, we will derive the condition that ensures that the maximum absolute magnitude of the FD solution at a particular time-level “n+1” (say) is less than the corresponding maximum absolute magnitude at the previous time step “n.” This condition will then be a sufficient condition (but obviously may not be necessary).

P a g e | 43

We again take the FD-scheme (L2.10) as an example, and find the sufficient stability condition. Eqn. (L2.10) can be written as:

ujn+1 = (1-) uj

n + uj-1n, = ct/x > 0 (L2.13)

Direct method:

If the coefficients of ujn and uj-1

n are positive, i.e. if

= ct/x 1, and > 0 (L2.14)

then |(1-) ujn + uj-1

n| (1-) |ujn| + |uj-1

n|.

Therefore, taking absolute maximum over all j’s of both sides of (L2.13):

Maxj | ujn+1| (1-) Maxj |uj

n| + Maxj |uj-1n| Maxj |uj

n|

which proves the solution remains bounded with time-stepping. The condition (L2.14) is therefore a sufficient condition for stability of the FD scheme (L2.10). Note that this same condition happens to be also the necessary condition for convergence.

Energy method:

By squaring both sides of (L2.13) and summing over all j’s, the same condition (L2.14), coupled with x-periodic boundary condition, can be shown to also lead to (after some algebra):

j(ujn+1)2 j(uj

n)2

i.e. each and every value of ujn must also be bounded with time-stepping.

von Neumann (Fourier) method:

This method borrows the idea from the analytical method of applying a single Fourier harmonic mode to (L2.7) by the separation of variables:

P a g e | 44

u (x,t) = Re {U(t) eikx } (L2.15)

so that (L2.7) becomes:

dU/dt = ikcU U(t) = U(0) e-ikct and the solution is then:

u (x,t) = Re {U(0) eik(x-ct)} (L2.16)

This shows that each harmonic component (with wavenumber k) is advected at the constant speed “c” with unchanging amplitude.

For the FD-scheme (L2.13), we apply the same idea and assume a solution of the form:

ujn = Re {U(n) eikjx} (L2.17)

where U(n) is the amplitude of the FD-solution at time level “n.” Substitute into (L2.13):

U(n+1)/ U(n) = (1 ) + e-ikx (L2.18)

where || is the amplification factor. Therefore, for the solution to be bounded with time-stepping, we require that || 1. Evaluating ||:

|| = 1 + 2 (1 ) [1 + cos(kx)] (L2.19)

It is clear that for || to be 1:

2 (1 ) [1 + cos(kx)] 0

which can only be satisfied for all k’s if:

= ct/x 1 (L2.14)

which is again the same stability condition (L2.14). This condition is commonly known as the CFL or Courant-Friedich-Levy stability condition, named in honor of the authors who first derived it.

P a g e | 45

An implicit scheme for the diffusion equation

The CFL condition can place an extremely strict (i.e. very small) upper limit for the size of the time step t especially when the spatial grid size (e.g. x) is very small. This is the case in the ocean and atmosphere for the vertical direction because the layer is very thin (~10km) compared to horizontal distances ~ 1000 km. So when finite-differencing in the z-direction, the z can be very small. Implicit scheme removes the restriction on t imposed by the smallness of x.

Consider u/t = 2u/x2 (L2.20)

and we approximate it using the following implicit scheme:

(ujn+1 uj

n)/t = (uj+1n+1 2 uj

n+1 + uj-1n+1)/x2 (L2.21)

Then using the von Neumann analysis, we get:

|imp| = 1/[1 + 2 (1 cos(kx))], where = t/x2

Clearly, |imp| 1 always regardless of the values of t and/or x. For this reason, implicit scheme is used in ocean and atmospheric models especially when approximating the z-direction.

Boundary conditions

Closed (e.g. at the coast) and ocean’s bottom and surface boundary conditions are fairly straight-forward. The trickiest boundaries to treat are “open” boundaries. I have written a set of notes for their treatments, and you may download them from:http://www.aos.princeton.edu/WWWPUBLIC/PROFS/PUBLICATION/OFES20101104.pdf (IWMO3.1end)

A detailed 2-way nesting report can be downloaded from:http://www.aos.princeton.edu/WWWPUBLIC/PROFS/PUBLICATION/OeyAccuracyOfNestedGridOceanModel1996.pdf

P a g e | 46

Lecture 3: The Princeton Ocean Model & GFD

Experiments

http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/

index.html

L3.1 POM08/2k and recent MPI implementations by Toni Jordi

L3.2 Model equations

L3.3 The numerical scheme

L3.4 Idealized experiments

POM User Guide: http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/PubOnLine/POL.html

P a g e | 47

P a g e | 48

L3.4: Idealized Experiments

(a) 1-D step and block-propagation problems;

(b) Taylor column;

(c) Surface and bottom Ekman layers;

(d) Western Boundary Current – Stommel & Munk;

(e) Estuarine plume debouching onto a continental shelf;

(f) Estuarine plume with tide;

(g) Baroclinic waves and eddies;

(h) Wetting & Drying.

Pls. download codes, runscripts, inputs and outputs etc

from:

ftp://aden.princeton.edu/pub/lyo/iwmo3/training-class/

for parallel mpiPOM/sbPOM versions; and from:

ftp://aden.princeton.edu/pub/lyo/pom_gfdex/

wmo09training/anIntroCourseNumOceanExpsUsingPOM/

for POM08/POM2k versions.

2-6: Heuristic derivation of the quasigeostrophic (QG) potential vorticity (PV)

equation for a stratified fluid

P a g e | 49

The inviscid x, y & z-momentum and continuity equations are:

u/t + uu/x + vu/y + wu/z -fv = -r-1(p/x) (2-6.1a)

v/t + uv/x + vv/y + wv/z +fu = -r-1(p/y) (2-6.1b)

p/z = -g (2-6.1c)

u/x + v/y + w/z = 0 (2-6.1d)

Also, a beta-plane is assumed such that:

f = fo + oy (2-6.2)

where fo is the Coriolis parameter at y=0 (around where later we will develop the

instability analysis), o = df/dy, and it is also assumed that |oy| << |fo|. For o 10-11 m-

1 s-1, and |fo| 5×10-5 s-1 or larger (poleward of 20oN/S), we restrict |y| 1000 km or

less.

The symbols have the usual meanings, and r = reference density which is a function

of z only. The basic QG-approximation idea is that the motions are close to being

geostrophic, so we will use the geostrophic velocity assuming constant f fo (which is

‘zeroth-order’) to evaluate the “difficult” terms which in (2-6.1a,b) are the non-linear

and “o” terms.

To the zeroth-order approximation, the motions are nearly geostrophic, so that:

-fovo = -r-1(po/x)

+fouo = -r-1(po/y) (2-6.3a,b)

so that

uo/x + vo/y = -wo/z 0 (2-6.4)

Therefore, wo 0 since it is 0 at the surface. In any case, the zeroth-order or

geostrophic approximation of w = wo is small compared to |uo| or |vo|.1

Substitute the (uo, vo, wo0) into the non-linear and beta parts of (2-6.1), we then can

1 Typically, w = Ekman pumping near the surface ×o/(fo) 0.1(N/m2)/[104(m).10-4(s-1).103(kg/m3)] 10-4 m/s 10 m/day, for a strong wind stress curl of 0.1 N/m2 over a distance of 10 km. Thus comparing to typical |uo| 5×10-2 m/s, the “w” is indeed small. In the open ocean, the “w” is typically smaller than 10 m/day.

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get the following equations for the next-order term with subscripts “1” (u1, v1, p1):

do/dt{uo} – fov1 oyvo = -r-1(p1/x) (2-6.5a)

do/dt{vo} + fou1 + oyuo = -r-1(p1/y) (2-6.5b)

where

do/dt = /t + uo/x + vo/y. (2-6.6)

Taking the (z-component of the) curl of (2-6.5) to eliminate the pressure, and using

the first of (2-6.4) that uo/x + vo/y 0, we get:

do/dt{o} + ovo + fo.u1 = 0, or

do/dt{o + oy} = fow1/z (2-6.7)

after using (2-6.6) and (2-6.1d), i.e. u1/x + v1/y + w1/z = 0.

By vertically integrating (from z=-H to z=0) equation (2-6.7) we recover the QGPV

equation for homogeneous fluid ( = constant), which we studied previously in section

2-1 (see equation (2.10)). For stratified fluid, we need to evaluate w1/z using the

density (or buoyancy b = -g/r) equation [see section 11.1.1 of Marshall and Plumb,

2008] which without heat/salt sources and sinks on the RHS is:

/t + u/x + v/y + w/z = 0 (2-6.8)

The density is given by:

= r(z) + ’(x,y,z,t) (2-6.9)

and the hydrostatic equation is then assumed also for the perturbation density ’ (and

pressure po):

po/z = -g’ (also of course pr/z = -gr) (2-6.10a)

or (po/z)/r = -g’/r = b’ (2-6.10b)

Using the QG-approximation on (2-6.8), we get:

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do/dt{’} + w1(dr/dz) = 0, (2-6.11a)

or multiplying by –g/r, we get do/dt{-g’/r} + w1(-gdr/dz)/r = 0, i.e.

do/dt{b’} + w1N2(z) = 0 (2-6.11b)

where N2(z) = (-gdr/dz)/r is the squared buoyancy (or Brunt-Vaisala) frequency. We

see that (in the absence of sources and sinks) the vertical velocity w1 is related to the

isopycnal movement due to the geostrophic flow:

w1 = do/dt{b’/N2(z)} (2-6.11c)

We can now use w1 in (2-6.7):

do/dt{o + oy} = fow1/z = fodo/dt{[b’/N2(z)]/z}

or upon using the hydrostatic equation (2-6.10b):

do/dt{o + oy} = fodo/dt{[(po/z)/(rN2)]/z}

i.e., do/dt{o + oy + fo[(po/z)/(rN2)]/z} = 0 (2-6.12a)

or, do/dt{(2po)/(for) + oy + fo[(po/z)/(rN2)]/z} = 0 (2-6.12b)

since from (2-6.3a,b), we have for the geostrophic vorticity:

o = vo/xuo/y = 2po/(for) (2-6.13)

For the ocean, r can be assumed to be constant in (2-6.12b), so that equation

becomes:

do/dt{2(po/for) + oy + [(po/for)/z)(fo/N2)]/z} = 0 (2-6.14)

From the geostrophic relation (2-6.3), the po/(for) is equivalent to geostrophic stream

function:

= po/(fo r) (2-6.15)

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so that (2-6.14) is usually expressed in terms of :

do/dt{2 + oy + [(/z)(fo/N2)]/z} = 0 (2-6.16a)

The quantity inside the {..} is the QGPV for a stratified fluid:

Q = 2 + oy + [(/z)(fo/N2)]/z (2-6.16b)

Non-dimensionalization:

We use the following scales:

Table 2-6.1

Scales Variables So that… (primes denote nondimensional variables)

L (x,y) (x,y) = L (x’,y’)

D z z = Dz’

U (u,v) (u,v) = U(u’,v’)

L/U t t = (L/U)t’

fo rUL po po = fo rULpo’, using geostrophic eqn.(2-6.3)

UL = UL’, using eqn. (2-6.15)

Ns N N = NsN’, for some typical Ns value of N

Then the non-dimensionalized form of (2-6.16) is (dropping all primes):

do/dt{2 + y + [S-1(/z)]/z} = 0 (2-6.17a)

and Q = 2 + y + [S-1(/z)]/z (2-6.17b)

where = oL2/U, S = (LD/L)2, and LD = NsD/fo is the baroclinic Rossby radius based on

ocean’s depth D. Note that Ns2D ~ g/r ~ g’, the reduced gravity, so that Ns

2D ~ g’D

~ baroclinic phase speed squared. The nondimensional = o/[(U/L)/L] = o/(o/L) is a

measure of the ratio of gradient of planetary vorticity “o” to the gradient of flow

vorticity “o/L”. By assuming that ~ O(1) in (2-6.17), we are examining an

important GFD case in which the planetary vorticity gradient contributes equally with

the relative vorticity gradient to the overall vorticity balance.

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If the scale of motion is small compared to the Rossby radius, L << LD so that S >> 1,

the effect of vortex-tube stretching to the PV-balance (i.e. the [S-1(/z)]/z term)

is small, and the flow’s relative vorticity (i.e. 2) becomes important. If on the other

hand the scale of motion is large, L >> LD, then S << 1, and the flow’s vorticity

becomes relatively unimportant, i.e. the flow looks horizontally more uniform.

2-7: Baroclinic instability

Fig.2-7.1 suggests that baroclinic instability may occur along special paths “(iii)”

across slanting isopycnals which of course imply the existence of vertical velocity

shear by the thermal-wind equations (from geostrophic eqn.(2-6.3a,b) and hydrostatic

eqn.(2-6.10b)):

k × fo uo/z = b’ (2-7.1a)

or fo uo/z = k × b’ (2-7.1b)

We will now find the condition – what type of vertical shears can produce baroclinic

instability? Or, in general, what type of vertical and horizontal shears can produce

baroclinic and barotropic instabilities? We will use the non-dimensionalized QGPV

equation (2-6.17) so that all variables are from here on non-dimensional.

Dimensional variables will have subscript “*”. Consider an initial or background

state of purely zonal flow Uo(y,z) with stream function (y,z):

Uo(y,z) = /y (note that Vo = /x = 0) (2-7.2)

Introduce perturbation “”, so that the total, time-dependent stream function is:

(x,y,z,t) = (y,z) + (x,y,z,t) (2-7.3)

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Fig.2-7.1. Schematized diagrams showing various scenarios when exchange of fluid

parcels “A” and “B” along the blue path line either leads to increased potential energy

so that the mass center of the system moves up (stable), is unchanged (neutral) or to

decreased potential energy so that the mass center of the system moves down (maybe

unstable). It is easy to see that only for the special path within the “wedge” formed by

the isopycnal (red line) and the horizon (dashed line) in “(iii)” can the exchange

possibly lead to instability (such that after the exchange the two parcels move away

from each other). Such an instability is called baroclinic instability.

The function is perturbation to the initial state ; it represents the structure of the

evolving perturbation field. Substitute (2-7.3) into (2-6.17):

(/t + Uo/x y/x + x/y){q + 2/y2 + y + [S-1/z]/z}

= 0

or (/t + Uo/x)q + J(,q) + x/y = 0 (2-7.4)

where q(x,y,z,t) is the perturbation PV:

q = 2 + [S-1(/z)]/z (2-7.5a)

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and o = 2/y2 + y + [S-1/z]/z (2-7.5b)

is the background or initial state’s PV, and its meridional gradient is (using eqn(2-

7.2)):

o/y = 2Uo/y2 + [S-1Uo/z]/z (2-7.5c)

Notice how the thermal-wind relation leads to the vertical shear Uo/z that then

appears in this last equation for o/y.

How does the structure of Uo(y,z) determine the evolution of the perturbation field ?

That is, given a particular background or initial state Uo(y,z), will the perturbation “injected” on the flow grows or decays? If grow, then the initial state is unstable with

respect to the perturbation . To show that Uo is stable, we must check all possible ’s.

On the other hand, to show instability, we only need to find one perturbation to which

the initial state Uo is unstable.

Linear Stability Analysis:

We assume that || << 1 so that the J(,q) in (2-7.4) is dropped:

(/t + Uo/x)q + x/y = 0 (2-7.6)

The boundary conditions are that vertical velocity w* = wo* + w1* + O(2) is zero at

z=0 (surface) and z=-1 (ocean’s bottom). Here = U/(foL) is the Rossby number

which is assumed to be small, and we already noted previously (see eqn.(2-6.4) and

discussion) that wo* = 0, so that w* w1* The non-dimensional w1 can then be

expressed in terms of using equations (2-6.11c), the hydrostatic relation (2-6.10b),

and (2-6.15) which gives:

b*’ = fo*/z*, hence w1* = do*/dt{fo*/z*/N2} (2-7.7a,b)

and w1 = do/dt{S-1/z} = 0 at z=0 & -1. (2-7.8)

A perturbation energy equation can be derived and it can be shown that:

s (o/y)(<2>/t) dydz = 0 (2-7.9)

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where is the meridional displacement of fluid elements defined by:

/t + Uo/x = /x (2-7.10)

and <.> is a zonal-averaging. From (2-7.9), we see that if there is to be a growth in

the displacement of fluid elements in time, i.e. if <2>/t > 0, then o/y must be

somewhere positive and somewhere else negative in the yz-plane, or o/y can also

be identically zero everywhere. In other words, o/y must vanish on a line in the

yz-plane. Clearly, this condition (i.e. that o/y must vanish somewhere) is not

sufficient.

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