Lecture Notes on Introduction to Geophysical Fluid Dynamics · Lecture Notes on Introduction to...

Lecture Notes on Introduction to GeophysicalFluid Dynamics

Xiaoming WangDepartment of Mathematics and Geophysical Fluid Dynamics Institute

Florida State UniversityTallahassee, FL32306, USA

July 27, 2006

Contents

1 Introduction 31.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Rotation and stratification . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Stratification . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Rotating frame and Coriolis force . . . . . . . . . . . . . . . . 71.4 Hydrostatic approximation . . . . . . . . . . . . . . . . . . . . 101.5 Shallow water equation . . . . . . . . . . . . . . . . . . . . . . 131.6 Elementary exact solutions . . . . . . . . . . . . . . . . . . . . 16

2 Rotation Effect 212.1 Geostrophic balance . . . . . . . . . . . . . . . . . . . . . . . 212.2 Linear waves in rotating shallow water equation . . . . . . . . 222.3 Formal zero Rossby number limit . . . . . . . . . . . . . . . . 282.4 Derivation of QG via two scale expansion . . . . . . . . . . . . 312.5 Ekman Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5.1 Bottom Ekman layer . . . . . . . . . . . . . . . . . . . 432.5.2 Surface Ekman Layer . . . . . . . . . . . . . . . . . . . 46

2.6 Svedrup transport and westward intensification . . . . . . . . 49

3 Stratification Effect 553.1 Formal zero Froude number limit . . . . . . . . . . . . . . . . 55

3.1.1 Formal zero Froude number limit . . . . . . . . . . . . 553.1.2 Vertical collapse . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Internal (linear) waves . . . . . . . . . . . . . . . . . . . . . . 64

1

2 CONTENTS

4 Combined Rotation and Stratification Effects 674.1 Thermal wind . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Derivation of the continuously stratified QG . . . . . . . . . . 684.3 Two layer model, and One and One-half layer model . . . . . 73

5 References 77

Chapter 1

Introduction

1.1 Basic equations

We start with the following compressible Navier-Stokes system coupled withheat conduction and under the influence of buoyancy forcing. The systemconsists of the balance of momentum, conservation of mass, and energy con-servation.Compressible Navier-Stokes system

momentum ρD

v

Dt+∇p = µ

∇ ·(∇ v +

(∇ v)tr)

+ (λ+ µ)∇(∇·

v)− ρg

k︸︷︷︸buoyancy

mass∂ρ

∂t+∇ ·

(ρ

v)

= 0

energyDT

Dt= κ∆T +

µ

∫Ω

∣∣∣∇ v∣∣∣2 + 2λ

∫Ω

∣∣∣∇· v∣∣∣2

where ~v is the velocity of a fluid particle at spatial location ~x and time t,ρ is the density, p is the pressure, µ > 0 is the 1st viscosity of the fluid,λ + 2

3µ is the 2nd viscosity coefficient, g is the gravitational constant, T is

the temperature, κ is the thermal diffusive coefficient of the fluid, DDt

is the

3

4 CHAPTER 1. INTRODUCTION

total derivative, or material derivative, i.e.,

D

Dt=

∂

∂t+ ~v · ∇.

The compressible NSE is extremely complicated (see Lions 1996 and thereferences therein). The first step in our study is to make the followingBoussinesq approximation:

ρ = ρb everywhere except in buoyancy term

ρ = ρb + ρ

ρ = −α (T − Tb) in buoyancy term

where Tb is a reference background temperature.We then arrive at the following

Boussinesq system

Dv

Dt+∇p = ν∆

v −gρ

ρb

k

divv= 0

Dρ

Dt= κ∆ρ

Other physical, chemical or biological processes can be incorporated aswell. For instance salinity equation would take the form

DS

Dt= Qsource −Qsink.

It is worthwhile to point out that even with the Boussinesq approxima-tion, the problem is still extremely complicated. For instance, the wellposed-ness (existence and uniqueness of solutions with given initial/boundary value,continuous dependence of solution on given data) of the Boussinesq systemis a major open problem which embodies the wellposedness problem of thethree dimensional Navier-Stokes equations for homogeneous incompressiblefluids (one of the seven million dollar millennium problems posed by the ClayInstitute www.claymath.org/Millennium Prize Problems/).

We also notice that the fluid motion is incompressible under the Boussi-nesq approximation. Therefore tools for incompressible fluids can be applied(Majda and Bertozzi 2001, Temam 2000).

The Boussinesq system can be also derived via kinetic theory.

1.2. ROTATION AND STRATIFICATION 5

1.2 Rotation and stratification

The main concern of geophysical fluid dynamics is fluid motion at large scales.Due to the large scale of motion, the rotation effect of the earth as well asthe stratification of the fluids become import as we shall illustrate below.

1.2.1 Rotation

In order to see the importance of the earth’s rotation we recall the followingfacts on a few geophysical fluid phenomena.

horizontal scale L ≈ 103 km (meandering jet stream), gulf-stream, hur-ricane 102 km, great red spot 104 km.

typical velocity U ≈ 10 m/s.

eddy turn over time T = LU≥ 105 s

1 day = 24× 60× 60 = 86400s

From this we see that the eddy turn over time is usually longer or comparableto a day. Thus the motion will ”feel” the earth’s rotation. Hence the rotationof the earth has to be taken into consideration.

The nondimensional quantity that characterizes the relative importanceof rotation is the Rossby number which is defined as

Ro =2π/Ω

L/U=

time of one revolution

eddy turn over time

Ω =2π

24h+

2π

1y∼= 7.29× 10−5

where Ω is the angular velocity (of the earth).Usually

Ro<∼ 1

implies the importance of rotation.


1.2.2 Stratification

Next, we illustrate the circumstances where stratification is relatively impor-tant.

Suppose we have two parcels of fluids with densities ρ(z + h) and ρ(z)respectively with z being the vertical coordinate. Suppose each parcel hasvolume V . Newton’s law leads to

g (ρ (z + h)− ρ (z))V︸︷︷︸buoyancy force

= ρ (z)Vd2h

dt2.

Now suppose we assume the density satisfies the following relationship

ρ = ρb + ρ(z) + ρ′︸︷︷︸ρ

where ρ(z) is a linear function.We apply leading order approximation for ρ on both sides of the equation,

i.e., neglecting the higher order term ρ′ on the left hand side and approximateρ by ρb on the right hand side, we deduce

d2h

dt2− g

ρb

dρ

dzh = 0

i.e., the distance between the two parcels satisfies a linear 2nd order ODE.The motion would be periodic (oscillatory) provided

dρ

dz< 0

or lighter parcel sits on top of a heavier parcel. The frequency of the mo-tion is apparently of great importance and it it is call the Brunt-Vaisalafrequency which is defined as

N2 = − g

ρb

dρ

dz.

The relative importance of stratification is then defined in terms of theFroude number which is given by

Fr =1/N

H/U=

U

H√− g

ρb

dρdz

=buoyancy time

eddy turn over time

1.3. ROTATING FRAME AND CORIOLIS FORCE 7

where H is the typical length in the vertical direction and U is the typicalvelocity in the vertical direction.

We see thatFr

<∼ 1

implies relative importance of stratification.An alternative way of interpreting the Froude number is the following

identity1

2Fr2 =

12ρbU

2

(H(−dρdz

))gH=

kinetic energy

potential energy

i.e., one-half of the square of the Froude number is the ratio of kinetic energyand potential energy.Remark: For atmosphere and oceans at large scale, both rotation and strat-ifications are important. See Gill 1982, Pedlosky 1987 for for details. Seealso Chapter 4.

1.3 Rotating frame and Coriolis force

We now consider the equation of motion in a reference frame which is itselfrotating with an angular velocity Ω with respect to some inertial referenceframe.

Assume the axis of rotation is

k , the angular velocity is Ω, the inertial

reference frame is determined by

i ,

j ,

k and the rotating reference frame

determined by

i′

,

j′

,

k′

.Any vector ~r can be represented in both the inertial reference frame and

the rotating reference frame as

r= x

i +y

j +z~k = x′

i′

+y′

j′

+z~k.

Recall that x′

= x cos Ωt+ y sin Ωt,

y′

= −x sin Ωt+ y cos Ωt.


Thus dx

′

dt=

dx

dtcos Ωt+

dy

dtsin Ωt− Ωx sin Ωt+ Ωy cos Ωt,

dy′

dt= −dx

dtsin Ωt+

dy

dtcos Ωt− Ωx cos Ωt− Ωy sin Ωt.

We also recall that

d

i′

dt= Ω

k ×

i′

,

d

j′

dt= Ω

k ×

j′

.

Therefore

dr

dt=dx

′

dt

i′

+dy

′

dt

j′

+dz

dt

k +Ω

k ×(x′

i′

+y′

j′)

i.e.,

dr

dt=d

r

dt′+ Ω

k ×r

where ddt′

is the rate of change in rotating framework.This further implies,

d2 r

dt2=

(d

dt′+ Ω

k ×)2

r=

d2 r

dt′2+ 2Ω

k ×d

r

dt′+ Ω2~k ×

(

k ×~r).

Consequently, the momentum equations in rotating framework take the form

Dv

Dt+ 2Ω

k ×v︸︷︷︸

Coriolis

+Ω2

k ×(

k ×r)

+∇p = ν∆v −gρ

ρb

k .

We notice that the centrifugal term Ω2

k ×(

k ×r)

is not important since

k ×(

k ×r)

=

k ×(−y

i +x

j)

= −y

j −x

i= −1

2∇(x2 + y2

)and thus can be absorbed into the pressure term.

1.3. ROTATING FRAME AND CORIOLIS FORCE 9

We then define the geopotential

Φ = φ+ φC

= −GMr− 1

2

(x2 + y2

)Ω2.

We now have the followingRotating Boussinesq equations

Dv

Dt+ 2Ω

k ×v +∇p = ν∆

v·− gρ

ρα

k,

divv= 0,

Dρ

Dt= κ∆ρ.

(1.1)

Next we consider rotating coordinates on the sphere so that

k 6= rotation-axis. In this case we have,

Dv

Dt+ 2

Ω ×v +

Ω ×(

Ω ×r)

+∇p = ν∆v −gρ

ρb

k,

Ω ×(

Ω ×r)

= ∇φC , φC = −1

2r2⊥Ω2,

2

Ω ×v= 2

∣∣∣∣∣∣∣

i

j

k0 Ωh Ωv

u v w

∣∣∣∣∣∣∣ .In terms of the components of the velocity field ~v = (u, v, w), we have

Du

Dt− 2Ωvv + 2Ωhw = −∂

≈p

∂x+ ν∆u,

Dv

Dt+ 2Ωvu = −∂

≈p

∂y+ ν∆v,

Dw

Dt− 2Ωhu = −∂

≈p

∂z+ ν∆w − gρ

ρb

.


We customary set Ωh = 0. However, no general justification for such an ap-proximation is available except in some special cases (see Gill 1982, Pedlosky1987, Cushman-Roisin 1994).

The next approximation is in terms of the Coriolis parameter

f = 2Ωv.

We approximate f by a linear function, i.e., we take the beta plane ap-proximation

f = f0 + βy (f0 = Ω sin θ0, β = Ω cos θ0)

and we arrive at the followingβ-plane equations

Du

Dt− (f0 + βy) v = −∂

≈p

∂x+ ν∆u,

Dv

Dt+ (f0 + βy)u = −∂

≈p

∂y+ ν∆v,

Dw

Dt= −∂

≈p

∂z− gρ

ρb

+ ν∆w,

∂u

∂x+∂v

∂y+∂w

∂z= 0,

Dρ

Dt= κ∆ρ.

In the case of β = 0 we recover the rotating Boussinesq system. This is alsocalled f -plane approximation.

1.4 Hydrostatic approximation

Another important feature of geophysical flows is the relative shallowness ofthe fluids.

1.4. HYDROSTATIC APPROXIMATION 11

We consider here domain with small aspect ratio, i.e.,

vertical scale

horizontal scale=H

L 1.

The smallness of aspect ratio usually implies the smallness of the verticalacceleration, i.e.

Dw

Dt 1.

Therefore, we deduce from the 3rd momentum balance with the β planeequations, the following hydrostatic balance neglecting viscosity

∂≈p

∂z= −gp

ρb

.

Consequently we have the followingPrimitive equations (for the oceans)

Du

Dt− fv = −∂

≈p

∂x+ ν∆u,

Dv

Dt+ fu = −∂

≈p

∂y+ ν∆v,

0 = −∂≈p

∂z− gρ

ρb

,

divv = 0,

Dρ

Dt= κ∆ρ.

(1.2)

We now present a heuristic argument for hydrostatic approximation basedon scaling. Since

H

L= ε 1,

we introduced the re-scaled variables (should all be order one)

V (x, y, z) =

(u (x, y, εz) , v (x, y, εz) ,

1

εw (x, y, εz)

)


we then have

div

V =∂u

∂x+∂v

∂y+∂w

∂Z= 0,

W∂h

∂z= w

∂h

∂Z,

DW

Dt=

1

ε

Dw

Dt,

where

Z = εz, h = h (x, y, Z) ,W (x, y, z) =1

εw (x, y, εz) .

Therefore, DwDt

is a small quantity. Utilizing this in the 3rd momentum equa-

tion we deduce the hydrostatic balance∂p

∂z= − ρ

ρb

g neglecting dissipation.

A slightly more careful dimensional analysis is the following (neglecting dis-sipation):

Typical horizontal velocity U

horizontal length scale L

vertical velocity W

vertical length scale H

The incompressibility implies

U

L≈ W

H

thus

W ≈ H

LU ≈ εU U

The dimensional analysis of the horizontal momentum takes the followingform (neglecting viscosity)

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z= fv − ∂p

∂x

U

T,

U2

L,

U2

L,

WU

H, ΩU,

P

L.

1.5. SHALLOW WATER EQUATION 13

All terms are all order 1The dimensional analysis of the vertical momentum equation takes the

form

∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ w

∂w

∂z= −∂p

∂z− ρ

ρb

g

W

T,

UW

L,

UW

L,

W 2

H,

P

H,

g∆ρ

ρp

εU

T, εU2

L, εU2

L, εU2

L, 1

εPL, g∆ρ

ρp

Notice thatg∆ρ

ρp

≈ 10−2 UW

L

(102

103· 10−2 ≡ 10−3

)for geophysical applications (see Pedlosky 1987, Cushman-Roisin 1994). Thusthe leading order balance is the hydrostatic balance.

Remark. viscosity term can be included, ν∂2

v

∂z2survives.

Remark. Mathematical justification is still open.

1.5 Shallow water equation

In this section, we derived the shallow water equations.We start with the β-plane equations (neglecting viscosity and stratifica-

tion) with hydrostatic approximation which takes the form

Du

Dt− fv = − 1

ρb

∂p

∂x,

Dv

Dt+ fu = − 1

ρb

∂p

∂y,

g = − 1

ρb

∂p

∂z,

∂u

∂x+∂v

∂y+∂w

∂z= 0.


Now let hb(x, y) be the bottom topography from a reference height z = 0and let h(x, y, t) be the height of the water surface from the same referenceheight z = 0 (see figure).

We consider flows with initially depth (z) independent horizontal velocity;i.e.,

∂u0

∂z=∂v0

∂z= 0.

This implies∂2w0

∂z2= 0

by the incompressibility condition.It is easy to check that this depth independence is preserved under the

dynamics

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z− fv = − 1

ρp

∂(p+ ρbgz)

∂x,

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z+ fu = − 1

ρp

∂(p+ ρbgz)

∂y,

∂u

∂x+∂v

∂y+∂w

∂z= 0,

0 =∂(p+ ρbgz)

∂z.

Our goal now is to derive the equations satisfied by the horizontal velocity(u, v) and surface water height h.

Integrating the incompressibility condition in z, we have(∂u

∂x+∂v

∂y

)∫ h

hb

dz + w∣∣hhb

= 0.

Obviously, water can not penetrate the bottom topography, i.e., there is nonormal flow at z = hb, or ~v · ~n = 0. Therefore,

(u, v, w) ·(∂hb

∂x,∂hb

∂y,−1

)= 0 at z = hb

which implies

w (x, y, hb) = u∂hb

∂x+ v

∂hb

∂y.

1.5. SHALLOW WATER EQUATION 15

Next, at the surface of the water, particles remain on top surface. Therefore

∂h

∂t= ~v · ~n sec θ

where θ is the angle formed by the z axis and the normal to the surface (seefigure).

This implies,

w (x, y, h) =∂h

∂t+ u

∂h

∂x+ v

∂h

∂yat z = h.

Combining these equations, we deduce(∂u

∂x+∂v

∂y

)(h− hb) +

∂h

∂t+ u

∂h

∂x+ v

∂h

∂y− u

∂

∂xhb − v

∂

∂yhb = 0

or equivalently

∂h

∂t+

∂

∂x((h− hb)u) +

∂

∂y((h− hb)v) = 0.

On the other hand, the hydrostatic balance dictates

p(x, y, z) = ρbg(h− z) + p0

where p0 is the pressure at the surface of the water. Assume the surfacepressure to be a constant, we then have derived the followingrotating shallow water equations

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y− fv = −g∂h

∂x,

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ fu = −g∂h

∂x,

∂h

∂t+

∂

∂x((h− hb)u) +

∂

∂y((h− hb)v) = 0.

(1.3)

The mathematical investigation of the shallow water equation is very chal-lenging. It is easy to see that it is equivalent to 2D isentropic compressibleEuler equations with γ = 2 when f and hb are zero. Thus singularity couldform among others. The interested reader is referred to Majda 1984 for moreresults on compressible fluid systems.


1.6 Elementary exact solutions

Here we collect a few exact solutions to the primitive equations. These exactsolutions will be useful in our understanding of basic flow phenomena.

We first consider two dimensional homogenous flows, i.e., we postulate

the Ansatzv=

(vH , 0

), ρ = 0. The primitive equations then reduce to the

followingBarotropic equations

DH vH

Dt+ f

v⊥H +∇Hp = ν∆H

vH

divHvH= 0

where

DH

Dt=

∂

∂t+(

vH ·∇H

)∇H =

(∂

∂x,∂

∂y

)

∆H =∂2

∂x2+

∂2

∂y2

We recover the 2D Navier-Stokes equations for homogeneous incompressibleNewtonian fluids in the case of f = 0.

The barotropic equation can be written in terms of vorticity and streamfunctions. Let ψ be a stream function, i.e.,

vH= ∇⊥ψ =

(−∂ψ∂y,∂ψ

∂x

)and the associated (relative)vorticity ω is given by

ω = curlv= −∂u

∂y+∂v

∂x= ∆ψ.

Taking the curl of the barotropic equations, we haveBarotropic equations in vorticity-stream function form

∂ω

∂t+ J (ψ, ω) +

∂f

∂yψx = ν∆ω,

ω = ∆ψ.(1.4)

1.6. ELEMENTARY EXACT SOLUTIONS 17

Next we utilize the β-plane approximation f = f0 + βy and introduce thepotential vorticity

q = ω + f(y) = ∆ψ + f(y)

we then deduce an alternative form ofbarotropic equation in potential vorticity

∂q

∂t+ J(ψ, q) = ν∆q, (1.5)

here the Jacobian is given by

J(ψ, ϕ) = ∇⊥ψ · ∇ϕ.

An easy way to find exact solution is to assume that there is a functionalrelationship between the stream function ψ and the potential vorticity q, i.e.

q = F (ψ)

so that the nonlinear term vanishes. One of the simplest relationship is alinear relationship

q = µψ.

We then recover solutions for 2D flows (see Majda and Bertozzi, 2001, Majdaand Wang 2006).

Another way to get exact solution to the barotropic equation is to imposelinear relationship between the relative vorticity ω and the stream functionψ, i.e.

ω = µψ, (∆ψ = µψ) .

This leads to an eigenvalue-eigenfunction problem for ψ at each time instance.The time evolution of the stream function is then governed by

∂∆ψ

∂t+ β

∂ψ

∂x= 0,

∂ψ

∂t+β

µ

∂ψ

∂x= 0.

The solutions are the Rossby waves

ψ(x, y, t) = ψ0

(x− β

µt, y

)


which are dispersive waves in the sense that different waves (µ) have differentpropagation speed.

In the periodic case (with period 2π in all directions), the Rossby wavescan be calculated explicitly:

∆ψ = µψ ⇐⇒ µ = −∣∣∣k ∣∣∣2 = Λ, ψ =

∑|k |2=Λ

ψke

ik·x

ψ(

x, t)

=∑

∣∣∣k ∣∣∣2=Λ

Ake

i

k ·

x+

βk1t

|k |2

+ C.C.

ψ0

(x)

=∑

∣∣∣k ∣∣∣=Λ

Akei

k ·

x + C.C.

Thus the zonal phase speed of the wave is

ω(

k)

k1

=− βk1

|k |2

k1

= − β∣∣∣k ∣∣∣2 .Hence the waves are not only dispersive, but the higher frequency part moveslower! This is one of the distinctive features of waves in geophysical fluiddynamics.

Next, we consider a one dimensional ] exact solution to Boussinesq systemand discuss its relationship to the Brunt-Vaisala frequency. In this case, wepropose the following Ansatz

vH ≡ 0, p ≡ 0, w = w(t)

ρ = ρ(z) =dρ

dz· z (linear stratification)

The Boussinesq system then reduces to∂w

∂t= − ρ

ρb

g,

∂ρ

∂t+ w

∂ρ

∂z= 0.

1.6. ELEMENTARY EXACT SOLUTIONS 19

Thus∂2w

∂t2+

(− g

ρb

∂ρ

∂z

)w = 0,

∂2ρ

∂t2+

(− g

ρb

∂ρ

∂z

)ρ = 0.

The fluid has a stably stratified background if

dρ

dz< 0.

We then obtain the frequency of oscillation, is again the Brunt-Vaisala fre-quency

N2 =

(− g

ρb

dρ

dz

).

We see that initial perturbation leads to persistent oscillation with BV fre-quency.

On the other hand, in the unstably stratified case, i.e.

dρ

dz> 0

Small perturbation leads to unbounded solution.The interested reader is referred to Majda and Wang 2006 and the ref-

erences therein for more exact solutions illustrating different facets of thephysics.

Chapter 2

Rotation Effect

The main purpose of this chapter is to study the effect of rotation. Wewill see that the dynamics can be simplified under the assumption of strongrotation (small Rossby number).

2.1 Geostrophic balance

We first recall the rotating Boussinesq equations (1.1)

Dv

Dt+

f × v +

1

ρb

∇p = ν∆v − ρ

ρb

g

k,

divv= 0,

Dρ

Dt= κ∆ρ.

(Same thing works with the Primitive equations or shallow water equation.)We consider here the small Rossby number Ro limit without stratification,

i.e., ρ ≡ 0 and

Ro =1ΩLU

1.

21

22 CHAPTER 2. ROTATION EFFECT

We also assume a small Ekman number (which characterizes the relativeimportance of viscosity)

Ekman number = Ek =rotation time

viscous time=

1Ω

H2

ν

=ν

ΩH2 1.

As Ro → 0, the leading order balance (see non-dimensionalization insection 4.2) is the well-knowngeostrophic balance (Taylor Proudman)

~f × ~v +1

ρ0

∇p = 0. (2.1)

This leads to, under the f -plane approximation,

−f0v +1

ρb

∂p

∂x= 0,

f0u+1

ρb

∂p

∂y= 0.

In another word,

u = − ∂

∂y(

1

f0ρb

p),

v =∂

∂x(

1

f0ρb

p).

We notice that the horizontal velocity field is incompressible with theleading order pressure field serving as scaled stream function. Thus the levelsets of the pressure field are stream lines. Hence geophysical flows do notflow from high to low but flow in transversal direction. This also gives us ahint that two dimensional incompressible flows/models are possibly relevantfor geophysical fluid dynamics.

2.2 Linear waves in rotating shallow water

equation

Here we consider linearized shallow water equation assuming the velocitiesare small and the height is a small perturbation of a constant (average) heightH. We also assume zero bottom topography for simplicity, i.e., hb = 0.

2.2. LINEAR WAVES IN ROTATING SHALLOW WATER EQUATION23

Let h = H + h′, the linearized rotating shallow water equations(LRSWE) takes the form, after dropping the prime,

∂u

∂t− fv = −g∂h

∂x,

∂v

∂t+ fu = −g∂h

∂y,

∂h

∂t+H

(∂u

∂x+∂v

∂y

)= 0.

(2.2)

In the case without lateral boundary, it is easy to derive two types of exactsolutions to LRSWE.Solution 1. Geostrophic balance (steady solution)

fv⊥

= −g∇h,

divv = 0,

v =

g

f∇⊥h.

Solution 2. General plane wave solution to LRSWE. We look for solutionsof the form

v= (u, v, h)tr = e

i(

k ·x−w

(k

)t)

r

where ~k represents a wave vector here.The plane wave ansatz in LRSWE leads to the following eigenvalue-

eigenvector problem

A(

k)

r=

(−ω

(

k)I + A1k1 + A2k2 − iB

)r= 0


where

A1 =

0 0 g0 0 0H 0 0

A2 =

0 0 00 0 g0 H 0

B =

0 −f 0f 0 00 0 0

A(

k)

=

−ω(

k)

+if gk1

−if −ω(

k)

gk2

Hk1 Hk2 −ω(

k)

.

In order to ensure the existence of non-trivial solutions, we must have thefollowing solvability condition

0 = detA(

k)

= −ω3(

k)

+ ω(

k)(

Hg∣∣∣k ∣∣∣2 + f 2

)= −ω

(

k)(

ω2(

k)2

−(gH∣∣∣k ∣∣∣2 + f 2

)).

There are two possible scenarios.

Case 1 ω(

k ) = 0 (steady waves)

In this case A(

k)

can be reduced to

0 0 0

−igk1

f0 1

igk2

f1 0

and the solutions are

h = fH cos(

k ·x),

v = −gH

∣∣∣k ∣∣∣ sin(

k ·x)

k⊥∣∣∣k ∣∣∣ .


These are shear flows in the direction of

k .

Case 2 ω(

k)

= ±√gH∣∣∣k ∣∣∣2 + f 2 (Poincare waves).

In this case the matrix A leads to

0 0 0

−ωk1 + ifk2

H∣∣∣k ∣∣∣2 1 0

−ωk2 + ifk1

H∣∣∣k ∣∣∣2 0 1

.

These correspond to inertial-gravity waves taking the form

h = h0ei(

k ·x−ω

(k

)t),

v =

ωh

H∣∣∣k ∣∣∣2

k −ifh

H∣∣∣k ∣∣∣2

k⊥.

The real part of the inertial gravity waves takes the form

v =

ω(

k)h0

H∣∣∣k ∣∣∣2 cos

(

k ·x −ω

(

k)t)

k∣∣∣k ∣∣∣ +fh0

H∣∣∣k ∣∣∣ sin

(

k ·x −ω

(

k)t)

k⊥∣∣∣k ∣∣∣ ,

h = h0 cos(

k ·x −ω

(

k)t).

When the rotation is relatively weak in the sense that

f = 0 or∣∣∣k ∣∣∣2 f2

gH

we see that rotation is not important and

ω(

k)∼ ±

√gH∣∣∣k ∣∣∣ .


This corresponds to classical gravity waves

∂2h

∂t2− gH∆h = 0.

On the other hand, for ∣∣∣k ∣∣∣2 f 2

gH

we haveω(

k)∼ ±f.

The solutions look like circular inertial oscillation (thus inertial).

We can imagine the behavior of the solutions for intermediate wavenumbers must be a mixture of the two.

The next exact solution involves a lateral boundary.Kelvin waves: We now consider waves governed by the linearized rotatingshallow water equation in half space x > 0. This kind of geometry occurs inoceanography among others.

We impose the following kind of ansatzu = 0v = v(x, y, t)h = h(x, y, t)

which is consistent with the existence of a impermeable lateral boundary.It is then easy to check that the non-zero velocity v and the perturbation

height h must satisfy the linear wave equation

∂2v

∂t2= c2

∂2v

∂y2,

∂2h

∂t2= c2

∂2h

∂y2

where c =√gH is the speed of gravity waves.

This implies, according to D’Alembert’s formula,

v = V1 (x, y + ct) + V2 (x, y − ct)


which further implies, after utilizing the 2nd equation in (2.2),

h = −

√H

gV1 (x, y + ct) +

√H

gV2 (x, y − ct) + r(x, t).

Notice that the remainder term r must satisfy

∂r

∂t= 0

thanks to the 3rd equation in LRSWE. This implies that r is a constantfunction in time and thus not a wave. Therefore, we can set r to zero.

Next, we utilize the 1st equation in (2.2), and we deduce, without loss ofgenerality,

∂V1

∂x= − f√

gHV1, ⇒ V1 = V10 (y + ct) e−

xR ,

∂V2

∂x=

f√gH

V2, ⇒ V2 = V20 (y − ct) exR ,

where

R =

√gH

f=c

f

is the Rossby deformation radius.Notice the solution V2 grows without bound in x and thus is not physical

and hence must vanish identically. Therefore we have onlyu = 0

v =√gH F (y + ct) e−

xR

h = H F (y + ct) e−xR .

(2.3)

Notice that these waves, the so-called Kelvin waves, are trapped waves. TheRossby deformation radius is approximately the distance travelled by the

waves in one rotation period

(2π

f

). It is also the reciprocal of the cut-off

wave number where rotation dominates.


2.3 Formal zero Rossby number limit

In this section, we study the formal zero Rossby number limit of the rotatingshallow water equation (1.3). As for any physical problems, we start theprocess with proper non-dimensionalization of the rotating shallow waterequations (1.3).

We first introduce the typical/reference scales

typical horizontal length scale : L

vertical length scale : H

velocity : U

time (eddy turn over) scale : T = LU

vertical perturbation scale : h0

The non-dimensional rotating shallow water equations takes the formD

v

Dt+ 1

Ro

v⊥

+ ΘFr2∇h = 0,

DhDt− 1

Θ

v ·∇

(hb

H

)+ 1

Θ

(1 + Θh− hb

H

)div

v= 0,

(2.4)

where the non-dimensional parameters are defined as

Ro =U

Lf︸︷︷︸rotation time

eddy turn over

, Fr =U√gH︸︷︷︸

typical velocitygravity wave speed

, Θ =h0

H︸︷︷︸characteristic height variation

mean height

.

We now impose the following assumption on the non-dimensional param-eters

Ro = ε 1,

Θ = FRo = Fε 1,

Fr =√

ΘRo = F12 Ro = F

12 ε 1,

and we also introduce the notation

hb = h0hb

2.3. FORMAL ZERO ROSSBY NUMBER LIMIT 29

Remark: The assumptions on the parameters are consistent with large scaleatmosphere and ocean dynamics. The assumption on the Rossby number isclear since we are interested in the regime of fast rotation. It is then clearthat Θ

Fr2 must be of the order of 1ε

in order to have a balanced non-trivialleading order dynamics. If we impose balanced dynamics in the h equationas well, i.e., the leading order in h equation of the order of 1

ε, the assumption

on Θ follows, which further implies the assumption on Fr.Under these assumptions we may re-write the non-dimensional rotating

shallow water equation asD

v

Dt+

1

ε

v⊥

+1

ε∇h = 0,

Dh

Dt− v ·∇hb +

1

Fε

(1 + εFh− εF hb

)div

v= 0.

(2.5)

Our goal now is to derive an effective equation satisfied by the limitvorticity or its variation. This is of course inspired by our experience withtwo dimensional incompressible inviscid fluids.

Recall that the (local) vorticity is defined as

ω = curlv= −∂u

∂y+∂v

∂x.

Taking the curl of the velocity equations in the rotating shallow water equa-tions (1.3), we deduce

∂ω

∂t+

v ·∇ω + (ω + f) div

v +

Df

Dt= 0

which is the same as

D(ω + f)

Dt+ (ω + f) div

v= 0.

Recall that the h now is the perturbation from the average height H. There-fore the height equation in (1.3) can be written as

D (h+H − hb)

Dt+ (h+H − hb) div

v= 0.

Combining the two identities we deduce

D (ω + f)

Dt− (ω + f)

(h+H − hb)

D (h+H − hb)

Dt= 0


which is the same asD

Dt

(ω + f

h+H − hb

)= 0. (2.6)

The physical interpretation is that the vertically averaged total vorticity ispreserved along stream line.

It is easy to see that the non-dimensional version of this conservation ofvertically averaged total vorticity takes the form

D

Dt

(Ro ω + 1

1 + Θh− hb

H

)= 0,

or equivalently, under the assumptions on the parameters,

D

Dt

(1 + εω

1 + εFh− εF hb

)= 0 (2.7)

since Ro= ε, Θ = εF , hb = h0hb .We are now in position to postulate the following formal series for the

velocity field, height and vorticity

v(

x, t)

=v

(0)(

x, t)

+ εv

(1)(

x, t)

+ ε2 v

(2)(

x, t)

+ · · · ,

h(

x, t)

= h(0)(

x, t)

+ εh(1)(

x, t)

+ ε2h(2)(

x, t)

+ · · · ,

ω(

x, t)

= ω(0)(

x, t)

+ εω(1)(

x, t)

+ ε2ω(2)(

x, t)

+ · · · .

Inserting these formal series into the non-dimensional rotating shallow waterequation, we have, to the leading order (1/ε),

v

(0)⊥= −∇h(0).

Thus

divv

(0)= 0,

v

(0)= ∇⊥h(0).

Inserting these formal series into the conservation of vertically averaged totalvorticity equation we deduce, to the leading order (order ε),

D

Dt

(ω(0) − Fh(0) + Fhb

)= 0.

2.4. DERIVATION OF QG VIA TWO SCALE EXPANSION 31

Thus we have derived the followingQuasi-geostrophic equations

∂

∂tq + J (ψ, q) = 0,

q = ∆ψ − Fψ + Fhb,

(2.8)

where ψ = h(0).When F = 0, we recover the 2D Euler system in vorticity-stream function

formulation.

2.4 Derivation of QG via two scale expansion

The derivation of the quasi-geostrophic equation in the last section reliedheavily on the conservation of vertically averaged total vorticity, and wedo not have information on what happenes to various kinds of waves (forinstance, inertial gravity waves). In this section, we approach the zero Rossbynumber limit from a different perspective. We recognize that small Rossbynumber indicates that there are two apparently different time scales: thefast time scale of rotation, and the slow time scale of vortical motion. Thissuggests that we should try a two time scale expansion of the problem.

Recall that we can write (2.5) in the form (separating the fast rotationand the rest) under the same assumption on the parameters as in the previoussection,

∂

V

∂t+

1

εL

V +B(

V ,

V)

= 0

where ε is the Rossby number, and zero bottom topography (hb = 0) isassumed for simplicity, and

V=

( v

F12h

), L

(

V)

=

(v⊥

+F− 12∇(F

12h)

F− 12 div

v

)

B(

V ,

V)

=

( v ·∇

v

div(F

12h

v))

The choice of F 1/2h is to ensure the anti-symmetry of L as we shall see later.


For convenience we define the new perturbative surface height as

h = hnew = F12hold.

The non-dimensional rapid rotating shallow water equation takes the form

∂

V

∂t+

1

εL

V +B(

V ,

V)

= 0 (2.9)

with ~V , L and B given above and with F12h replaced by h.

Apparently the asymptotic behavior of the system at vanishing Rossbynumber (ε) depends on properties of L.

It is easy to see that L is anti-symmetric (skew-symmetric). Indeed,(L

V ,

W)

= −(

V ,L

W),

L(

V)

=

(v⊥

+F− 12∇h

F− 12 div

v .

)The eigenvalues/eigenvectors of L can be calculated explicitly in the pe-

riodic case. Indeed, we haveeigenvalue/eigenfunction of L (periodic case)

V = eik ·

x r ,

L(

V)

= eik ·

xL(i

k)

r ,

L(i

k)

=

0 −1 iF− 1

2k1

1 0 iF− 12k2

iF− 12k1 iF− 1

2k2 0

.

Remark: We notice that L(i

k ) is skew-symmetric which corresponds to theskew-symmetry of L .

Now the eigenvalue/eigenfunction problem for the operator L reduces to

the eigenvalue/eigenvector problem for the matrix L(i

k ).


Eigenvalue problem:

L(i

k)

r= iω

r⇔

(L(i

k)− iω

)r= 0

Characteristic equation:

det(L(i

k)− iω

)= iω

(ω2 −

(1 + F−1

∣∣∣k ∣∣∣2))eigenvalues:

gravity mode ω(−1)k

= −ω(

k)

= −(

1 + F−1∣∣∣k ∣∣∣2) 1

2

,

vortical mode ω(0)(

k) = 0,

gravity mode ω(1)k

= ω(

k)

=

(1 + F−1

∣∣∣k ∣∣∣2) 12

.


The corresponding (right) eigenvectors in the case of

k 6= 0 are

gravity moder

(1)k

=

−ω(

k)k1 + ik2

√2ω(

k) ∣∣∣k ∣∣∣

−ω(

k)k2 − ik1

√2ω(

k) ∣∣∣k ∣∣∣

F−1/2∣∣∣k ∣∣∣

√2ω(

k)

,

r

(−1)k

=

ω(

k)k1 + ik2

√2ω(

k) ∣∣∣k ∣∣∣

ω(

k)k2 − ik1

√2ω(

k) ∣∣∣k ∣∣∣

F− 12 |

k |√

2ω(

k)

,

vortical moder

(0)k

=

−iF− 1

2k2

iF12k1

1

.


The eigenvectors in the case of

k= 0 are

r

(−1)k

=

− i√

2

1√2

0

,r

(0)k

=

001

,r

(1)k

=

i√2

1√2

0

.

We then observe the followingSymmetry relations for the eigenvectors(

r

(α)k

)∗=

r

(−α)

−k

for α = ±1, 0

where ∗ denotes complex conjugation.Next, we apply the general idea of two scale expansion to RSWE type

problem

∂

V

∂t+

1

εL

V↑

linear

+B(

V ,

V)

↑bilinear

= 0.

We introduce the fast rotation time scale τ =t

εand notice that ∂

∂t=

∂∂t

+ 1ε

∂∂τ

. We further postulate the following formal asymptotic expansion

for ~V

V=

V0

(t, τ)

∣∣∣∣τ= t

ε

+ ε

V1

(t, τ)

∣∣∣∣τ= t

ε

+ h.o.t

together with the sublinear growth condition

limτ→∞

V1

(t, τ)

τ= 0.

We then deduce the leading order expansion

∂

V0

∂τ+ L

V0

= 0.

Therefore,

V0

(t, τ) = e−LτV (t).


However, we do not have the dynamics of V on the slow time scale. Thedynamics of V will be derived utilizing the sublinear growth condition.

For this purpose, we consider the next order expansion

∂

V1

∂τ+ L

V1

= −

∂

V0

∂t+B

(

V0

,

V0) .

Duhammel’s principle implies that

V1

(t, τ) = e−Lτ

V1

(t, τ)

∣∣∣∣t=0

−e−Lττ∂V

∂t(t)−e−Lτ

∫ τ

0

eLsB(e−LsV , e−LsV

)ds.

Sublinear growth condition dictates the following effective equation∂V

∂t+ lim

τ→∞

1

τ

∫ τ

0

eLsB(eLsV · e−LsV

)ds = 0,

V (t)∣∣t=0

=

V 0 .

Original solution then takes the form

V=

V0

(t, τ) + o (1) = e−L(tε)V

(x, t)

+ o (1) .

In order to derive the effective equation explicitly, one needs to evaluate

1

τ

∫ τ

0

esLB(e−sLV , e−sLV

)ds.

For this purpose, we consider eigenmode expansion for V

V(

x, t)

=∑

k∈ZZ2

1∑α=−1

eik ·

xa

(α)k

(t)r

(α)k.

The reality condition implies(a

(α)k

(t))∗

= a(−α)

−k

(t) .


Since the eik ·

x r

(α)k

are eigenfunctions of L with corresponding eigenvalue

iω(α)k

, we have

e−sLV(

x, t)

=∑

k∈ZZ2

1∑α=−1

a(α)ke

i

(k ·

x−ω

(α)k

s

)r

(α)k.

Thus we may write down explicit form for the bilinear term B(

v ,

v)

=((v ·∇)

v

div (hv )

)in terms of the eigenmodes as

B(e−sL, V , e−sLV

)=

∑k∈ZZ2

1∑α=−1

∑k 1+

k 2=

k

1∑α1,α2=−1

a(α1)k 1

· a(α2)k 2

C(α1,α2,α)

(k 1,

k 2,

k )e

i

(k ·

x−

(ω

(α1)k 1

+ω(α2)k 2

)s

) r

(α)k

where

Cα1,α2,αk 1,

k 2,

k

=i

2

(v

(α1)k 1

·

k 2

)⟨r

(α2)k 2

,r

(α)k

⟩+

(v

(α2)k 2

·

k 1

)⟨r

(α1)k 1

,r

(α)k

⟩+

(v

(α1)k 1

·

k 1

)h

(α2)k 2

h(α)k

+

(v

(α2)k 2

·

k 2

)h

(α1)k 1

h(α)k

.

It then follows

esLB(e−sLV , e−sLV

)=

∑k∈ZZ2

1∑α=−1

∑k 1+

k 2=

k

1∑α1,α2=−1

ei

(k ·

x−

(ω

(α1)k 1

+ω(α2)k 2

−ω(α)k

)s

)C

(α1,α2,α)k 1,

k 2,

ka

(α1)k 1

(t)a(α2)k 2

(t)

r

(α)k.

In the case of

k 1 +

k 2=

k but

ω(α1)k 1

+ ω(α2)k 2

− ω(α)k6= 0

we have an oscillatory function and thus

limτ→∞

1

τ

∫ τ

0

ei

(k ·

x−

(ω

(α1)k 1

+ω(α2)k 2

−ω(α)k

)s

)C

(α1,α2,α)k 1,

k 2,

ka

(α1)k 1

(t)a(α2)k 2

(t)ds = 0.


Therefore nontrivial contribution comes only if the following resonance con-ditions are satisfied

k 1 +

k 2=

k,

ω(α1)k 1

+ ω(α2)k 2

= ω(α)k.

Hence we can write down the modewise equation for effective dynamics

da(α)k

dt+

∑(

k 1,k 2,α1,α2

)∈S

α,k

C(α1,α2,α)k 1,

k 2,

ka

(α1)k 1

(t)a(α2)k 2

(t) = 0

where the resonance set is given by

Sα,

k

=(

k 1,

k 2, α1, α2

) ∣∣∣

k 1 +

k 2=

k, ω(α1)k 1

+ ω(α2)k 2

= ω(α)k

.

Resonance happens if(

k 1,

k 2, α1, α2

)∈ S

α·k

It is easy to see that there are three types of possible resonances:

1. resonance among vortical modes,

2. resonance among gravity modes,

3. resonance between vortical and gravity modes.

We now study the three types of possible resonances.Case 1: Resonance among vortical modes.

Since ω(α)k

= 0 for all vortical modes, we have resonance among vortical

modes if

k 1 +

k 2=

k .Case 2: Resonance among gravity modes.

We will show that there is NO 3-wave interaction.Indeed, if there were resonance among three gravity waves, we must have,

according to the resonance condition,

k 1 +

k 2=

k,(1 + F−1

∣∣∣k 1

∣∣∣2) 12

±(

1 + F−1∣∣∣k 2

∣∣∣2) 12

=

(1 + F−1

∣∣∣k ∣∣∣2) 12,


(the 2nd equation is ω(

k 1

)+ ω

(

k 2

)= ω

(

k)

taking into consideration of

two gravity modes for each

k )On the other hand, the elementary inequality(

1 + (x+ y)2) 12 <

(1 + x2

) 12 +

(1 + y2

) 12

tells us

ω(

k 1

)+ ω

(

k 2

)=

(1 + F−1

∣∣∣k 1

∣∣∣2) 12

+

(1 + F−1

∣∣∣k 2

∣∣∣2) 12

>

(1 + F−1

(∣∣∣k 1

∣∣∣+ ∣∣∣k 2

∣∣∣)2) 1

2

>

(1 + F−1

(∣∣∣k 1 +

k 2

∣∣∣)2) 1

2

= ω(

k)

which is a contradiction.Case 3: gravity/vortical mode interaction.

subcase 1: 2 vortical and 1 gravity. This is impossible since ω(0)k

= 0 for

all vortical modes.subcase 2: 2 gravity and 1 vortical. We must have, in the case,

k 1 +

k 2=

k,

ω(

k 1

)+ ω

(

k 2

)= ω

(

k) .

Without loss of generality we may assume

ω(

k)

= 0.

This implies ∣∣∣k 1

∣∣∣ =∣∣∣k 2

∣∣∣ .It is easy to check that the C vanishes in this case.


Next we decompose V into vortical and gravity components

V = V I + V II ,

V I =∑

k∈ZZ2

a(0)kei

k ·

x r

(0)k

(vortical),

V II =∑

k∈ZZ2

∑α=±1

a(α)kei

k ·

x r

(α)k

(gravity).

The effective equation for the vortical component in terms of the modes takesthe form

da(0)k

dt+

∑k 1+

k 2=

k

C(0,0,0)k 1,

k 2,

k, a

(0)k 1

a(0)k 2

+∑

k 1+

k 2=

k∣∣∣k 1

∣∣∣=∣∣∣k 2

∣∣∣

∑α1+α2=0

a1=±

C(α1,α2,0)k 1,

k 2,

ka

(α1)k 1

a(α2)k 2

= 0.

Direct calculation yields

C(1,−1,0)k 1,

k 2,

k 1+

k 2

= C(−1,1,0)k 1,

k 2,

k 1+

k 2

= 0 for

k 1 +

k 2=

k∣∣∣k 1

∣∣∣ =∣∣∣k 2

∣∣∣ .Therefore, we have the evolution equation for the vortical modes

da(0)k

dt+

∑k 1+

k 2=

k

C(0,0,0)k 1,

k 2,

ka

(0)k 1

a(0)k 2

= 0,

where

C(0,0,0)k 1,

k 2,

k

=

F− 32

(

k 1 ·

k⊥2

)(∣∣∣k 2

∣∣∣2 − ∣∣∣k 1

∣∣∣2)2ω(

k 1

)ω(

k 2

)ω(

k 1 +

k 2

) .

Let

V I =

( v I

hI

)=

uI

vI

hI

,


we then havev I= F− 1

2∇⊥hI .

Next, we define the potential vorticity

qI = ωI − F12hI

= F12 ∆hI − F

12hI

=∑k∈ZZ2

a(0)kei

k ·

x

(−(∣∣∣k ∣∣∣2 F 1

2 + F12

))h

(0)k

= −∑k∈ZZ2

a(0)kei

k ·

xF

12ω(

k)h

(0)k.

It is easy to check that the potential vorticity satisfies the quasi-geostrophicequations (QG)

DqIDt

=∂qI∂t

+v I ·∇qI = 0.

We now go back to the velocity field and recall

V = e−τLV (t) + h.o.t

= e−τLV I(t) + e−τLV II(t) + h.o.t

= V I(t) + e−τLV II(t)︸︷︷︸↓0

+ h.o.t.

This tells us that V I , the vortical part of the velocity, is the weak limit of ~V ,or the result of filtering ~V (filtering out the fast gravity waves modes in V II).We see that we have obtained additional (versus the approach in section 2.3)information on the dynamics by going through the two time scale expansion.Moreover, the vortical modes are not influenced by the gravity modes as isapparent from the quasi-geostrophic equations. However, the gravity modeswill be influenced by the vortical modes as we shall see below.

Next we consider dynamics of the gravity waves V II


For gravity modes a(α)k

, α = ±1, three wave resonances occur only if

α1 = α, α2 = 0,

k 1 +

k 2=

k,∣∣∣k 1

∣∣∣ =∣∣∣k ∣∣∣ ,

or

α2 = α, α1 = 0,

k 1 +

k 2=

k,∣∣∣k 2

∣∣∣ =∣∣∣k ∣∣∣ .

The 2nd case is the same as the 1st with the role of

k 1,

k 2 switched.In terms of the eigenmode, we have

da(α)k

dt= −

∑∣∣∣k 1

∣∣∣=∣∣∣k ∣∣∣2C

(α,0,α)k 1,

k−

k 1,

ka

(0)k−

k 1

a(α)k 1

where the interaction coefficients C(α,0,α)k 1,

k 2,

k,∣∣∣α∣∣∣ = 1 is given by (use the

explicit eigenfunctions)

C(1,0,1)k 1,

k−

k 1,

k

=

(C

(−1,0,−1)k 1,

k−

k 1,

k

)∗= − F− 1

2

4ω(

k 2

)ω(

k)2 ∣∣∣k ∣∣∣2

(∣∣∣k ∣∣∣2(

k 1 ·

k⊥)−(

k 1 ·

k)(

k 1 ·

k⊥))

+iω(

k)(

2∣∣∣k ∣∣∣4 − ∣∣∣k ∣∣∣2 (

k 1 ·

k)−(

k 1 ·

k⊥)2)

.

Remark: Dynamics on the shell |

k |2 = Λ are independent since

da(α)k

dt=

∑∣∣∣k 1

∣∣∣2=Λ

C(α)k 1,

ka

(α)k 1

,∣∣∣k ∣∣∣2 = Λ

C(1)k 1,

k

(V I

)=

(C(−1)

k ,

k 1

(V I

))∗= −C(1,0,1)

k 1,

k−

k 1,

ka

(0)k−

k 1

.

2.5. EKMAN LAYERS 43

The interested reader may consult Majda 2002 and the references therein forrigorous justification of this asymptotic expansion.

2.5 Ekman Layers

In this section we introduce bottom/surface boundary layer due to the com-bined effect of rotation and viscosity. These boundary layers are relativelystable and are named after Ekman who first discovered these phenomena.

2.5.1 Bottom Ekman layer

We start with the primitive equations neglecting stratification

DvH

Dt+ f

v⊥H= ∇Hp+ ν∆

vH ,

0 =∂p

∂z,

divHvH +

∂w

∂z= 0.

Suppose that the fluids occupy half space z > 0 and boundary conditionsare

~vH = 0, w = 0, at z = 0( bottom)

together with periodic boundary conditions in the horizontal directions orfree-slip/no-penetration horizontal boundary conditions in the case of hori-zontally bounded domains.

We are particularly interested in the case of fast rotation, i.e.,

Ro =L

Uf 1.

Thus, to the leading order, we have geostrophic balance

fv⊥H= −∇H p.


It is then obvious that the geostrophic balanced velocity field ~vH cannotsatisfy the vertical boundary conditions. This implies that we need to keep

the vertical viscosity term ν∂2

vH

∂z2effective in a thin (boundary) layer at

least. The thickness of the layer is about

√ν

f= d as we shall demonstrate

below.We now have, with the vertical viscosity term, to the leading order, in

the boundary layer,

−fv = −∂p∂x

+ ν∂2u

∂z2,

fu = −∂p∂y

+ ν∂2v

∂z2,

0 = −∂p∂z,

∂u

∂x+∂u

∂y+∂w

∂z= 0.

On the other hand, in the interior (outside of the boundary layer), we havethe geostrophic balance

−fv = −∂p∂x,

fu = −∂p∂y.

Since∂p

∂z= 0 everywhere (in and out of the boundary) by the hydrostatic

equation, we havep = p.

Therefore we deduce−f (v − v) = ν

∂2u

∂z2

(= ν

∂2 (u− u)

∂z2

),

f (u− u) = ν∂2v

∂z2

(= ν

∂2 (v − v)

∂z2

).


This set of linear ODEs can be solved explicitly with the characteristic equa-tion and its roots given by

λ4 +1

d4= 0,

1

de±

π4i,

1

de(π±

π4)i

where

d =

√ν

f.

Notice1

de±

π4i lead to exponential growth and has to be eliminated. We then

have, to the leading orderu = u

(1− e

− z√2d cos

z√2d

)− ve

− z√2d sin

z√2d,

v = ue− z√

2d sinz√2d

+ v

(1− e

− z√2d cos

z√2d

).

Remark: Boundary layer thickness is proportional to ∼√

2d =2ν

f. It is

called the Ekman depth.Remark: the velocity field in the boundary layer is transversal to the interiorvelocity field (u, v). Indeed,

~vH =

1− e

− z√2d cos

z√2d

−e−z√2d sin

z√2d

e−√

z√

2d sinz√2d

1− e− z√

2 cosz√2d

vH

≈

z√2d

− z√2d

z√2d

z√2d

vH

=z

d

1√2− 1√

2

1√2

1√2

vH


i.e.,vH is a rotation of vH plus a scaling of a factor of

z

dfor small z. The

rotation is of an angle ofπ

4counter-clockwise.

Remark: Another interesting phenomena is Ekman pumping. Utilizing theleading order approximations above we have,

∂u

∂x+∂v

∂y=

(∂u

∂x+∂v

∂y

)(1− e

− z√2d cos

z√2d

)+

(∂u

∂y− ∂v

∂x

)e− z√

2d sinz√2d.

Therefore

w(x, y, z) =

∫ z

0

∂w

∂z(z′) dz′

= −∫ z

0

(∂u

∂x+∂v

∂y

)(z′) dz′

= ω(x, y)

∫ z

0

e− z′√

2d sinz′√2ddz′

=1

f∆p(x, y)

∫ z

0

e− z′√

2d sinz′√2ddz′

where

ω =d√2f

∆p(x, y).

Thus we see that the vertical velocity w at the end of the boundary layer(thus the same as in the interior of the domain) is non-trivial although it isweak. We have up-welling (positive w) provided that ∆p > 0, i.e., ω > 0and we have down-welling (negative w) provided that ω < 0.

2.5.2 Surface Ekman Layer

We now consider wind stress (τx, τy) driven flows. In this case we will see asurface Ekman layer.

Again we assume the fluid velocity in the interior, (u, v), satisfies geostrophicbalance

−fv = −∂p∂x

fu = −∂p∂y

.


The leading order dynamics in the boundary layer takes the form

fv = −∂p∂x

+ ν∂2u

∂z2,

fu = −∂p∂y

+ ν∂2v

∂z2,

0 =∂p

∂z,

∂u

∂x+∂v

∂y+∂x

∂z= 0.

This leads to

−f (v − v) = ν∂2u

∂z2

(= ν

∂2(u− u)

∂z2

),

f (u− u) = ν∂2v

∂z2

(= ν

∂2(v − v)

∂z2

),

∂u

∂x+∂v

∂y+∂w

∂z= 0.

The boundary conditions are, at z = 0

ρ0ν∂u

∂z= τx,

ρν∂v

∂z= τy,

w|z=0 = 0,

and at infinity,u→ u, v → v, as z → −∞.


Again the solutions can be calculated explicitly

(d =

√ν

f

)

u = u+1

ρ0fde

z√2d

τx cos

(z√2d− π

4

)− τy sin

(z√2d− π

4

),

v = v +1

ρ0fde

z√2d

τx sin

(z√2d− π

4

)− τy cos

(z√2d− π

4

).

Remark: The departure of the flow in the boundary layer from the interiorflow is exclusively due to wind stress (independent of interior flow), and could

be large

(∼ 1

d=

√f

ν

), and the flow in the boundary layer always has an

45 angle to the wind stress.Remark: Wind stress could also produce Ekman pumping. In deed,

w = limz→−∞

w = −∫ 0

−∞

∂w

∂zdq =

∫ 0

−∞

(∂u

∂x+∂v

∂y

)dz

=

∫ 0

−∞

(∂(u− u)

∂x+∂(v − v)

∂y

)dz

=

∫ 0

−∞

(ν

f

∂

∂x

(∂2v

∂z2

)− ν

f

∂

∂y

(∂2u

∂z2

))dz

=ν

f

∂2v

∂x∂z

∣∣∣∣z=0

− ν

f

∂2u

∂y∂z

∣∣∣∣z=0

=1

ρ0f

(∂τy∂x

− ∂τx∂y

).

Therefore we have up-welling if f > 0 (northern hemisphere) and curl τ =∂τy∂x

− ∂τx∂y

> 0, i.e., counter-clockwise wind stress. Similar statement can be

made for down-welling.An unresolved issue here is the interior flow in the horizontal directions.

The horizontal velocity is not determined in the bottom or surface Ekmanlayer calculations. However, it can be determined by utilizing both the topand the bottom Ekman layers.

2.6. SVEDRUP TRANSPORT AND WESTWARD INTENSIFICATION49

According to our calculation of wind stress driven surface Ekman layer

w = limz→−∞

w =1

ρ0f

(∂τy∂x

− ∂τx∂y

)On the other hand, according to our calculation of the bottom Ekman layer,

w = limz→∞

w =

√νf

√2f

∆p(x, y)

Thus, matching the vertical velocity in the interior we deduce

∆p(x, y) =

√2f

ρ0

√ν

(∂τy∂x

− ∂τx∂y

).

We may then recover the horizontal velocities in the interior using geostrophicbalance

−v = −∂p∂x, fu = −∂p

∂y.

Some mathematical justifications of the Ekman layers are available.

2.6 Svedrup transport and westward intensi-

fication

In this section we consider wind stress driven ocean flows in a basin withlateral boundary at x = 0 and x = L. Instead of using the f plane approxi-mation as in the previous sections, we utilize the β-plane approximation

f = f0 + β0y |β0y| f0.

Let the wind stress on the surface of the ocean be (τx, τy)Just as in the calculation for Ekman layers, leading order dynamics at


fast rotation takes the form

−fv = −∂p∂x

+ ν∂2u

∂z2,

fu = −∂p∂y

+ ν∂v

∂z2,

0 = −∂p∂z,

∂u

∂x+∂v

∂y+∂w

∂z= 0.

Here we have replaced p/ρ0 by p in order to simplify notations.Neglecting viscosity, we deduce

− (f0 + β0y) v = −∂p∂x,

(f0 + β0y)u = −∂p∂y,

0 = −∂p∂z,

∂u

∂x+∂v

∂y+∂w

∂z= 0.

Cross differentiation (in order to eliminate p), we have,

−β0v − f∂v

∂y= f

∂u

∂x

or

β0v = −f(∂u

∂x+∂v

∂y

)= f

∂w

∂z.

Thus we have, approximately, since |β0y| f0,

β0v = f0∂w

∂z.


This tells us vertical gradient in the vertical velocity produces non-trivialpoleward flows.

An interesting question then is how to estimate∂w

∂z. A simple (and crude)

estimate of the gradient is via finite difference

∂w

∂z≈ wtop − wbottom

H

≈ 1

ρ0f0H

(∂τy∂x

− ∂τx∂y

)− d√

2f0H∆p

where we have used our previous calculations on Ekman pumping for thebottom and surface Ekman layers.

Thus

β0

f0

∂p

∂x≈

(1

ρ0f0H

(∂τy∂x

− ∂τx∂y

)− d√

2f0H∆p

)f0,

β0∂p

∂x≈

(1

ρ0H

(∂τy∂x

− ∂τx∂y

)− d√

2H∆p

)f0.

Neglecting viscosity (since it is small), we have, approximately,

β0∂p

∂x=

f0

ρ0H

(∂τy∂x

− ∂τx∂y

).

Therefore, non-trivial wind stress and β plane could produce southward/northwardflows. This flows is called Svedrup flow. In the case of trade wind (westwardat low latitude) and westerlies (at mid latitude) wind stress, the Svedrup flowis southward.

One deficiency of the calculation above is that the physical no-flow bound-ary condition at lateral boundary can not be satisfied in general. Indeed,

since u = − 1

f0

∂p

∂y= 0 at x = 0, L, it is not consistent with the equation

without viscosity. Thus Svedrup flow is the first order approximation. To fixthis, we retain the viscosity term

d√2H

∆p+β0

f0

∂p

∂x=

1

ρ0H

(∂τy∂x

− ∂τx∂y

)


where d =√

νf0

as before. This is equivalent to

−√ν√

2f0H∆p− β0

f0

∂p

∂x= − 1

ρ0H

(∂τy∂x

− ∂τx∂y

).

We recover the Svedrup relation at vanishing viscosity of course.Notice that the characteristics go from x = L to x = 0. Therefore we

may have a boundary layer at the west end of the domain (x = 0). Utilizingstretched variable

x =x√ν,

∂

∂x=

1√ν

∂

∂x,

we see that the leading order ( 1√ν) dynamics is governed by

− 1√2f0H

∂2pint

∂x2− β0

f0

∂pint

∂x= 0

where pint is the interior (of the boundary layer, and scaled) stream function.It is easy to see then

pint(x, y) = p1(y) + p2(y)e−

√2β0H√f0√

νx.

Notice that the 2nd term on the right hand side is of boundary layer type andit is located at the west end of the domain. Thus it is also called westwardintensification. It is related to Gulf streams. (see Cushman-Roisin 1994 formore details). We can now present an approximate solution in the wholedomain as

p = p1(y) + p2(y)e−√

2β0H√f0ν

x − f0

ρ0β0H

∫ L

x

(∂τy∂y

− ∂τx∂y

)dx.

We consider the following special case in order to further understand theproblem. We consider northern hemisphere and assume the following windstress which corresponds to trade wind near equator and westerly winds atmid-latitude

τy = 0, τx = τx(y),dτx(y)

dy> 0.

We then deduce

p = p1(y) + p2(y)e−√

2β0H√f0ν

x+

f0

ρ0β0H

∂τx∂y

(L− x).


The lateral no-flow boundary conditions∂p

∂y= 0 at x = 0, L lead to

p′1(y) + p′2(y)e

−√

2β0H√f0ν Lx = 0,

p′1(y) + p′2(y) +f0

ρ0β0H

d2τxdy2

Lx = 0.

Thereforep1(y) = p10 (constant),

p2(y) = − f0

ρ0β0H

dτxdy

Lx + β0.

It is worthwhile to point out that the boundary layer flow northward,since

v =1

f0

∂p

∂x∼

√2

ρ0

√νf0

dτxdy

Lxe−√

2β0H√f0ν

x.

Thus the Gulf streams bring warm water near equator northward and warmcertain regions.

Chapter 3

Stratification Effect

In this chapter, we focus on the effect of stratification. In particular, we willconsider the situation where stratification effect is much stronger than rota-tion effect. This is physically relevant for meso-scale atmosphere et cetera.

3.1 Formal zero Froude number limit

3.1.1 Formal zero Froude number limit

We first recall the Boussinesq equations without rotation

D~v

Dt= − 1

ρb

∇p− gρ

ρb

~k,

Dρ

Dt= −wdρ

dz,

∇· v = 0.

55

56 CHAPTER 3. STRATIFICATION EFFECT

Here and below in this section we use the following notation

vH= (u, v),

v= (

vH , w),

ρ = ρb − bz + ρ = ρ+ ρ,

DH

Dt=

∂

∂t+

vH ·∇H ,

∇H =

(∂

∂x,∂

∂y

),

divHvH=

∂u

∂x+∂v

∂y.

We also introduce the following typical scales

L: length scale

U : velocity

Te :=L

U: eddy turnover time

ρb: background density

p: mean pressure

TN = N−1: buoyancy time

where N =(− g

ρb

dρdz

) 12

is the Brunt-Vaisala frequency.

We now non-dimensionalize the Boussinesq system

x′=

x

L, t′ =

t

Te

,v′=

v

U, ρ′ =

ρ

ρbB, p′ =

p

p

3.1. FORMAL ZERO FROUDE NUMBER LIMIT 57

where B is a numerical factor to be determined, and we have

Dv′

Dt′= −

(p

ρbU2

)∇p′ −

(BgL

U2

)ρ′

k,

Dρ′

Dt′=

(− L

Bρb

dρ

dz

)w′,

divv′

= 0.

The non-dimensional parameters (numbers) in the equations are

Froude Fr =TN

Te

=U

LN: relative importance of stratification versus eddy turnover,

Euler P =p

ρbU2:pressure

inertial,

Γ =NU

gB=

U(gB)

1N.

Thus the non-dimensional Boussinesq equations take the form, after wedrop the primes,

Dv

Dt= −P∇p− (Γ)−1 (Fr)−1 ρ

k,

Dρ

Dt= Γ (Fr)−1w,

divv= 0.

We introduce the following change of variable in order to further simply theappearance of the system

Pp −→ p,

ρ

Γ−→ ρ.

,

Further more, we assume relative importance of stratification, i.e.,

Fr = ε 1,


we have

Dv

Dt= −∇p− 1

ερ

k,

Dρ

Dt=

1

εw,

divv= 0.

Next, we propose the following formal series in ε

v =

v

(0)(

x, t)

+ εv

(1)(

v , t)

+ o(ε2),

ρ = ρ(0)(

x, t)

+ ερ(1)(

v , t)

+ o(ε2),

p = ε−1p(−1)(

x, t)

+ p(0)(~x, t) + o (ε) .

Notice that the form of p is different from the form ofv . The difference is

needed as we shall see below.Collecting the leading order ε−1 terms, we have

∇Hp(−1) = 0,

∂p(−1)

∂z= −ρ(0),

w(0) = 0.

The first equation above implies

p(−1) = p(−1)(z, t)

The last equation in the leading order expansion and the divergence free

condition divv

(0)= 0 implies horizontal incompressibility

divHv

(0)

H = 0.


Next, we move onto the next order ε0 = 1, and we have

DH v

(0)

H

Dt= −∇Hp

(0),

0 = −∂p(0)

∂z− ρ(1),

divHv

(0)

H = 0.

Thus the overall leading order dynamics isDH

v(0)

H

Dt= −∇Hp

(0),

divHv

(0)

H = 0,

and

w(0) = 0,

∇Hp(−1)

(x, t)

= 0,

∂p(−1)

∂z

(x, t)

= −ρ(0)(

x, t).

We observe that the dynamics at each level (z value) is simply the 2D in-compressible Euler system. The dynamics at different levels are completelydecoupled, and the leading order vertical velocity vanishes identically.

3.1.2 Vertical collapse

An interesting phenomena observed in strongly stratified decaying flows isthe so called vertical collapsing where initially arbitrary vorticity field tendto collapse into horizontal ones (see Majda 2002 and the references therein).In order to understand this phenomena, we first generalize the formal asymp-


totic to rotating Boussinesq with dissipation

Dv

Dt+ f

k ×v= −∇p+ ν∆

v − g

ρb

ρ

k,

Dρ

Dt= −wdρ

dz(ρ = ρ− bz) ,

divv= 0.

The addition of eddy/Newtonian viscosity ν leads to two additional non-dimensional numbers

Reynolds number Re =Tν

Te

=L2

νLU

=LU

ν=

viscous time

eddy turn over time,

Rossby number Ro =U

Lf=

1f

LU

=rotation time

eddy turn over time=Tf

Te

.

Thus the non-dimensional Boussinesq system takes the form

Dv

Dt+

1

Ro

k ×v= −P∇p+

1

Re∆

v − 1

Γ

1

Frρ

v ,

Dρ

Dt=

Γ

Frw,

divv= 0.

Introducing again the change of variables

Pp→ p,ρ

Γ→ ρ, Fr = ε 1,

we have

Dv

Dt+

1

Ro

k ×v= −∇p+

1

Re∆

v −1

ερ

k,

Dρ

Dt=

1

εw,

divv= 0.


Utilizing a formal series in ε as in the inviscid case and collecting leadingorder terms we have

DH v

(0)

H

Dt+

1

Ro

v

(0)⊥H = −∇Hp

(0) +1

Re∆H

v

(0)

H +1

Re

∂v

(0)

H

∂z2,

divHv

(0)

H = 0,w(0) = 0,

∂p(−1)

∂z(z, t) = −ρ(0) (z, t) ,

∇Hp(−1) = 0.

We observe that when ν → 0; i.e., Re → ∞, we recover the layered 2D

incompressible Euler. However, with viscosity, layers “talk” to each othervia dissipation in the vertical direction

1

Re

∂2 v

(0)

H

∂z2

and this may lead to vertical collapse as we shall see below. It is convenientto introduce the vorticity-stream function formulation.

For this purpose we first decompose the horizontal velocity into its meanand pure fluctuation part (which can be represented using stream function)

v

(0)

H =

V H +∇⊥Hψ,

V H =1

LxLy

∫[0,Lx]×[0,Ly ]

v

(0)

H (x, y, z) dx dy.

We then observe that

ω = curl

~VH +∇⊥Hψ

0

=

∂

V⊥H

∂z

0

+

ω′H

ω

,

ω = ∆Hψ,ω′H=

∂

∂z∇Hψ.


Therefore, we have the vorticity stream function formulation as

∂ω

∂t+

V H ·∇Hω + JH (ψ, ω) =1

Re∆Hω +

1

Re

∂2ω

∂z2,

∆Hψ = ω,

JH (ψ, ω) = ∇⊥Hψ · ∇Hω,

∂

V H

∂t+

1

Ro

V⊥H=

1

Re

∂2

V H

∂z2.

We now give an explicit example exhibiting vertical collapse. We considersolution on a single energy shell, say the ground energy shell

ψ (x, y, z, t) =∑

∣∣∣∣k j

H

∣∣∣∣2=1

Aj (z, t) eik

j

H ·x He−

tRe ,

V H (z, t) =∑

k

R−1(t)eikze−k2tRe

V H0 (k),

where

kH= (k1, k2) ,xH= (x, y)

and

R(t) =

cos

t

Rosin t

Ro

− sint

Rocos t

Ro

.

The introduction of R(t) is useful since

∂

∂t

(R(t)

V H

)=

1

Re

∂2

∂z2

(R (t)

V H

).

We now introduce the special initial data for the mean horizontal velocity

V H (z, 0) =

(Vmax sin z

0

)=

Vmaxeiz

2i0

− Vmax

e−iz

2i0


and we have

V H (z, t) = R−1 (t) e−t

Re

(Vmax

2ieiz − Vmax

2ie−iz

0

)

= R−1 (t) e−t

ReVmax

(sin z

0

).

Therefore, the vortcity equation leads to

∂Aj

∂t(z, t) + i

kj

H ·

V H (z, t)Aj (z, t) =1

Re

∂2Aj

∂z2.

We also introduce the following initial data for the stream function for thepure fluctuation part of the horizontal velocity

ψ (x, y, z, 0) = sinx+ sin y

=1

2ieix − 1

2ie−ix +

1

2ieiy − 1

2ie−iy.

For large t,

V H∼ 0, and thus the Aj equation is approximately

∂Aj

∂t∼ 1

Re

∂2Aj

∂z2.

HenceAj ∼ e−

tRe

which further impliesψ ∼ e−

2tRe

while

V H∼ e−t

Re .

Therefore, the long time dynamics is dominated by vertical shear

V H andthus

ω = curl

(

V H +∇⊥ψ0

)∼

(∂

∂z

V⊥H

0

).

Hence we observe vertical collapse in vorticity since the vertical componentis basically zero. The interested reader is referred to Majda (2002) and thereferences therein for more details on zero Froude number limit.


3.2 Internal (linear) waves

In this section, we introduce another kind of wave, the so-called internal waveinduced by stratification of fluids.

These waves travel internally. This is different from surface waves likewater waves that we are all familiar with.

We postulate the following hypothesis:

• no rotation f ≡ 0

• no dissipation ν = 0

• small amplitude so that linearization is justified.

Under these hypothesis, the appropriate model now is the following lin-earized inviscid non-rotating Boussinesq

∂v

∂t= − 1

ρb

∇p− 1

ρb

gρ

k,

∂ρ

∂t= −wdρ

dz,

divv= 0.

We look for plane wave solutions, i.e., we use the nsatz:

v = ~vei(

k ·

x−ωt),

ρ = ρei(k ·

x−ωt).

It is then easy to check that we have non-trivial solution (wave) if the fol-lowing dispersion relation is satisfied

ω2 = N2

∣∣∣kH

∣∣∣2∣∣∣k ∣∣∣2where

kH= (k1, k2) and N2 = − gρb

dρdz

.

3.2. INTERNAL (LINEAR) WAVES 65

As a specific example, we have

u = U sin (k1x+ k3z − ωt) ,

w = −k1

k3

U sin (k1x+ k3z − ωt) ,

ρ = −A cos (k1x+ k3z − ωt) ,

A =ρbN

g

√k2

1 + k23

k3

U.

It is easily calculated that the phase speed is

cphase =ω√

k21 + k2

3

and it is in the (k1, k3) direction.The group velocity is calculated as

Cgx =∂ω

∂k1

=2ωk2

3

k1 (k21 + k2

3),

Cgz =∂ω

∂k3

= − 2ωk3

k21 + k2

3

.

Therefore(Cgx.Cgz) ⊥ (k1, k3)

i.e., energy propagates in a direction perpendicular to the phase.The interested reader is referred to Gill (1982) or Pedlosky (1987) for

more details on internal waves.

Chapter 4

Combined Rotation and

Stratification Effects

We studied the effect of rotation and stratification separately in the previ-ous two chapters. However, in many applications in ocean and atmospheredynamics, both rotation and stratification are important. Thus we need tostudy combined rotation and stratification effects. We will start with the ro-tating Boussinesq equations (1.1) or the primitive equations (1.2). Our goalis again to identify key phenomena and derive simplified effective dynamics.

4.1 Thermal wind

The first phenomena associated with combined rotation and stratificationeffect is the so-called thermal wind. Here we consider linearized inviscidstationary primitive equations

−fv = − 1

ρb

∂p

∂x,

fu = − 1

ρb

∂p

∂y,

∂p

∂z= −ρg.

67

68CHAPTER 4. COMBINED ROTATION AND STRATIFICATION EFFECTS

An easy consequence is, after eliminating the pressure field, we have thefollowing thermal wind equations

∂v

∂z= − g

ρbf

∂ρ

∂x,

∂u

∂z=

g

ρbf

∂ρ

∂y.

(4.1)

We see that horizontal variation of the density leads to vertical variation ofthe horizontal velocities. It is called thermal wind since the density is relatedto the temperature (high temperature implies low density and vice versus).

The interested reader is referred to Pedlosky (1987) and Gill (1982) formore on thermal winds.

4.2 Derivation of the continuously stratified

QG

Here we formally derive the effective equations for the rotating Boussinesqsystem at small Rossby and small Froude number. Recall that the rotatingBoussinesq equations (1.1) can be written as

Dv

Dt+ f

k ×v = − 1

ρb

∇p− g

ρb

ρ

k,

divv = 0,

Dρ

Dt= −wdρ

dz,

where ρ is the deviation of the density from the background averaged linearstratification, i.e.

ρ = ρ+ ρ = ρb − bz + ρ.

We non-dimensionalize the system as before, and we introduce the fol-lowing typical/reference quantities

4.2. DERIVATION OF THE CONTINUOUSLY STRATIFIED QG 69

L: horizontal length scale

U : horizontal velocity scale

Te =L

U: eddy turnover time

TR = f−1: rotation time

TN = N−1: buoyancy time

ρb: background density

p: mean pressure

N : =

(− g

ρb

dρ

dz

) 12

Brunt-Vaisala frequency

The non-dimensionalization leads to the following non-dimensional pa-rameters: the Rossby number Ro, the Froude number Fr, the Euler numberP and Γ,

Ro =TR

Te

=U

Lf, Fr =

TN

Te

=U

LN, P =

p

ρbU2, Γ =

NU

gB.

The non-dimensional form of the rotating Boussinesq system is (afterdropping the primes)

Dv

Dt+

1

Ro

k ×v = −P∇p− 1

Γ

1

Fr

ρ

k,

divv = 0,

Dρ

Dt= Γ (Fr)

−1w.

We now postulate the following standing


Assumptions.

Ro = ε 1 (fast rotation)

Fr = FRo = Fε 1 (strong rotation)

P = (Ro)−1 = 1ε

(geostrophic balance)

Γ = order 1

Remark: Without loss of generality we can set Γ = 1 with the change of

variableρ

Γ→ ρ .

Remark: Likewise, P need not be exactly1

ε. We can rescale p as long as

P ∼ 1ε

.Under the standing assumption, the rotating Boussinesq equation can be

re-casted as

Dv

Dt= −1

ε

(

k ×v +∇p+

1

Fρ

k

),

divv= 0,

Dρ

Dt=

1

ε

1

Fw.

We then propose a formal series expansion in ε

v(

x, t)

=v

(0)(

x, t)

+ εv

(1)(

x, t)

+ o(ε2),

ρ(

x, t)

= ρ(0)(

x, t)

+ ερ(1)(

x, t)

+ o(ε2),

p(

x, t)

= p(0)(

x, t)

+ εp(1)(

x, t)

+ o(ε2).

4.2. DERIVATION OF THE CONTINUOUSLY STRATIFIED QG 71

It is then easy to see that the leading order dynamics (1

ε) is formally

v

(0)

H = ∇⊥Hp

(0), (geographic)

∂p(0)

∂z= − 1

Fρ(0), (hydrostatic)

w(0) = 0 ⇒ divH∇⊥Hp

(0) = 0. (incompressible)

However, we do not have dynamics for the velocity field. This is similarto the case of zero Rossby number limit for shallow water equations that wediscussed in chapter 2. Just as in that case, we have two ways of derivingthe effective dynamics:

• derivation using conservation of vorticity (Ertel’s theorem). This isrelatively easier but less revealing.

• Derivation via two time scale expansion. This is more involved butmore revealing.

We leave the detail of the two time scale approach to the interested reader(see Majda 2002 for details).

Here we only present the derivation utilizing Ertel’s Theorem. We firstrecall Ertel’s theorem. We then derive effective dynamics utilizing Ertel’stheorem. Ertel’s theorem will be verified afterwards.

Theorem (Ertel).D

Dt

((w +f

k)· ∇ρ

)= 0.

Remark:ω +f

k=ωa is called the total or absolute vorticity where

ω=

curlv= ∇×

v .Since ρ = ρ+ ρ = ρb − bz + ρ, an equivalent form of Ertel’s Theorem is

D

Dt

((ω3 + f)

dρ

dz+

ωH ·∇H ρ+ (ω3 + f)

∂ρ

∂z

)= 0

and its non-dimensional form is

D

Dt

ω3 −

1

Ro· Fr

Γ

∂ρ

∂z− Fr

Γ

(ωH ·∇H ρ+ ω3

∂ρ

∂z

)= 0

which is equivalent to, with our standing hypothesis,

D

Dt

ω3 − F

∂ρ

∂z− εF

(ωH ·∇Hρ+ ω3

∂ρ

∂z

)= 0.


Utilizing Ertel’s theorem, we deduce that the leading order dynamics is

w(0) = 0,

∂ρ(0)

∂z= −F ∂

2p(0)

∂z2,

DH

Dt

ω

(0)3 + F 2∂

2p(0)

∂z2

= 0.

Thus we have derived the followingQuasi-geostrophic equations for stably stratified flow

DH

Dt

(∆Hψ + F 2 ∂ψ

∂z2

)= 0, (4.2)

vH = ∇⊥ψ,

w = 0,

ρ = −F ∂ψ∂z.

Remark: We notice that ψ = p(0) is the stream function, i.e., the leadingorder pressure field is the stream function.

A slightly more general form can be derived in the same fashion

DH

Dt

(∆Hψ + F 2 ∂ψ

∂z2+ βy

)= D3ψ + F

where D3ψ is some 3D dissipation operator.We now verify Ertel’s theorem.

4.3. TWO LAYER MODEL, AND ONE AND ONE-HALF LAYER MODEL73

Proof of Ertel’s Theorem assuming f is a constant.

D

Dt

((ω +f

k)· ∇ρ

)=

(ω +f

k)· DDt∇ρ+

Dω

Dt· ∇ρ

= − ωa ·

(∇ v

tr)∇ρ+

Dω

Dt· ∇ρ

=

((∇ v)

ω − (~v · ∇) f

k +(∇ v)f

k +g

ρb

(∇⊥

Hρ

0

))·∇ρ−

ωa ·(∇ v

tr)∇ρ

= − ωa ·

(∇ v

tr)∇ρ+

(∇ v)

ωa ·∇ρ

= 0.

The continuous stratified quasi-geostrophic equations is the starting pointof many important theoretical studies of geophysical problems (see Pedlosky1987, Majda 2002, Majda and Wang 2006 among others). The interestedreader is referred to Majda 2002 and the references therein for rigorous jus-tification of the derivation above, including a two time scale derivation.

4.3 Two layer model, and One and One-half

layer model

Here we derive the two layer model and the one and one-half layer model fromthe continuously stratified model that we derived in the previous section.

We first derived the two layer model with equal depth via truncating thecontinuously stratified quasi-geostrophic (4.2) in the first two vertical modes.

Assuming aspect ratio 1, we look for stream function of the form (trun-cated to the first two vertical modes)

ψ (x, y, z, t) = ψb(x, y, t) + ψt(x, y, t)√

2 sin z


where the first part is the barotropic part and the second part is baroclinic.The Jacobian then takes the form

J

(ψ,∆ψ + F 2∂

2ψ

∂z2

)= J (ψb,∆ψb) +

√2 sin z J

(ψb,∆ψt − F 2ψt

)+√

2 sin z J (ψt,∆ψb) + 2 sin2 z J(ψt,∆ψt − F 2ψt

).

We then project the equation (4.2) to the bases 1,√

2 sin z and we have

∂

∂t∆ψb + J (ψb,∆ψb) + J

(ψt,∆ψt − F 2ψt

)+ β

∂ψb

∂x= 0,

∂

∂t

(∆ψt − F 2ψt

)+ J (ψt,∆ψb) + J

(ψb,∆ψt − F 2ψt

)+ β

∂ψt

∂x= 0.

Introducing the new variables

ψ1 = ψb + ψt

ψ2 = ψb − ψt

we have the followingTwo Layer Equations

∂

∂t(∆ψ1 − F ′ (ψ1 − ψ2)) + J (ψ1,∆ψ1 − F ′ (ψ1 − ψ2)) + β

∂ψ1

∂x= 0,

∂

∂t(∆ψ2 − F ′ (ψ2 − ψ1)) + J (ψ2,∆ψ2 − F ′ (ψ2 − ψ1)) + β

∂ψ2

∂x= 0,

(4.3)

where F ′ =F 2

2.

Next, we derive the one and one-half layer model from the two layer modelas the depth of the bottom layer approaches infinity.

We first recall the general two layer model with possibly different depths(Pedlosky 1987 or Gill 1982)

∂

∂t(∆ψ1 − F1 (ψ1 − ψ2)) + J (ψ1,∆ψ1 − F1 (ψ1 − ψ2)) + β

∂ψ1

∂x= 0,

∂

∂t(∆ψ2 − F2 (ψ2 − ψ1)) + J (ψ2,∆ψ2 − F2 (ψ2 − ψ1)) + β

∂ψ2

∂x= 0,

where the Fjs are inverse proportional to the depth Hj of the jth layer.

4.3. TWO LAYER MODEL, AND ONE AND ONE-HALF LAYER MODEL75

Now let the depth of the top layer, H1, be fixed, and let the depth of the

bottom layer (H2) approaches infinity, i.e.,H1

H2

= ε → 0. Such a situation

arises in ocean where a thin mixing layer sitting on a much thicker inertiallayer.

The assumption on the depths is equivalent to

F1 fixed,

F2 = εF1.

Utilizing this assumption in the two layer model and formally setting εto zero, we have the followingOne and One-Half Layer Model

∂

∂t(∆ϕ1 − F1ϕ1 + F1ϕ2) + J (ϕ1,∆ϕ1 − F1ϕ1 + F1ϕ2 + βy) = 0,

∂

∂t(∆ϕ2) + J (ϕ2,∆ϕ2 + βy) = 0.

(4.4)

We see that the bottom layer satisfies the usual barotropic quasi-geostrophicequation while F1ϕ2 serve as an effective topography (could be time depen-dent) for the top layer.

The one layer model is easy to derive from the two layer or the continu-ously stratified model by truncation to the barotropic mode

∂

∂t(∆Ψb + βy) + J (Ψb,∆Ψb + βu) = 0. (4.5)

The derivation above can be generalized to produce the following generalone layer equation with topography h, beta plane effect, dissipation andexternal forcingOne Layer Model

∂

∂t

(∆ψ − F 2ψ + h+ βy

)+ J

(ψ,∆ψ − F 2ψ + h+ βy

)= Dψ + F . (4.6)

One final remark is that our knowledge on geophysical flows is very lim-ited, even for the seemingly simple one layer model. The interested reader isreferred to Majda and Wang 2006 for more on the dynamics and statisticaltheories of one layer geophysical flows.

Chapter 5

References

1. Batchelor, G.K., 1967, An Introduction to Fluid Dynamics, CambridgeUniv. Press.

2. Cushman-Roisin, B., 1994, Introduction to Geophysical Fluid Dynam-ics, Prentice Hall.

3. Gill, A.E., 1982, Atmosphere-Ocean Dynamics, New York: AcademicPress.

4. Lions, P-L., 1996, Mathematical topics in fluid mechanics , OxfordUniversity Press, New York.

5. Majda, A., 1984, Compressible fluid flow and systems of conservationlaws in several space variables, Springer-Verlag, New York.

6. Majda, A., 2002, Introduction to PDEs and Waves for the Atmosphereand Ocean, AMS.

7. Majda, A. and Bertozzi, A., 2001, Vorticity and Incompressible Flow,Cambridge Univ. Press.

8. Majda, A. and Wang, X., 2006, Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows, Cambridge Univ. Press.

9. Pedlosky, J., 1987, Geophysical Fluid Dynamics, Springer-Verlag, NewYork.

77

78 CHAPTER 5. REFERENCES

10. Temam, R., 2000, Navier-Stokes equations, AMS-Chelsa.

Lecture Notes on Introduction to Geophysical Fluid Dynamics · Lecture Notes on Introduction to...

Documents

Transcript of Lecture Notes on Introduction to Geophysical Fluid Dynamics · Lecture Notes on Introduction to...