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Centre for Atmospheric and Oceanic Sciences
INDIAN INSTITUTE OF SCIENCE
BANGALORE 560 012
Lecture Notes on
GEOPHYSICAL FLUID DYNAMICS
Prof. Debasis Sengupta
Workshop on
PHYSICS OF THE ATMOSPHERES & OCEANS
1-12 July 2002
Geophysical Fluid Dynamics (GFD)
What is GFD ?GFD is the theory of large scale, low frequency flow of geophysical fluids (atmo-
sphere, ocean). These flows are strongly influenced by the earth’s rotation. The im-
portance of rotation is estimated from the Rossby number ε, which is the ratio of the
advective time scale L/U and the time scale of the earth’s rotation Ω−1. Large-scale,
low frequency flows have small Rossby number
ε =U
2ΩL≤ 1 (1)
where L and U are horizontal length and velocity scales of the flow and Ω angular
speed of earth rotation. For example an atmospheric flow with L = 1000 km and U = 10
m s−1 has a Rossby number of ε=0.07. An oceanic flow with L = 100 km and U = 1 ms−1
has ε= 0.07. Such flows have time to ”feel” the earth’s rotation, so they come within the
ambit of GFD. Waves on a beach, with L = 100 m and T = 10 sec, do not feel the earth’s
rotation.
The aim of GFD is to develop concepts that help to understand (and predict) geo-
physical flows. These concepts come from the study of idealized ”models” of geophys-
ical flow, which are constructed from the laws of physics with the help of simplifying
assumptions. The development of models is guided by the observed features of geo-
physical flows.
Equations of motionThe equations of motion in an inertial frame of reference are mass conservation (or
continuity) equation
∂ρ
∂t+ ~∇.(ρ~u) = 0 (2)
where ρ is density and ~u velocity, and Newton’s law (momentum conservation) for
a fluid
ρd~u
dt= −~∇p+ ρ~∇φ+ ~F (~u) (3)
1
where p is pressure, ρ ~∇ φ is a conservative body force and ~F (~u) is a nonconservative
fluid friction. In eqs. (2) and (3),
d
dt≡
∂
∂t+ ~u.~∇
If the density is not constant, the momentum and continuity equations do not consti-
tute a closed dynamical system. The first law of thermodynamics plus the equation of
state for the fluid ρ = ρ(T ) must be considered to obtain an equation for ρ , where T is
temperature. 1
In a frame of reference rotating with angular velocity ~Ω,
~uI = ~u+ 2~Ω × ~r
where ~uI is the velocity as seen from an inertial frame of reference. and the equation of
motion (3) becomes
ρ
[d~u
dt+ 2~Ω × ~u
]= −~∇p+ ρ~∇φ+ ~F (4)
The observer in a uniformly rotating coordinate frame sees an extra acceleration, i.e.
the coriolis acceleration 2 ~Ω×~u. Spatial gradients are invariant under the transformation
to a rotating frame; the body force ρ~∇φ has an extra contribution from the centripetal
acceleration; the form of ~F determines if it is invariant or not (if ~F is Newtonian, i.e.
proportional to ∇2~u, it is invariant under transformation to a rotating frame).
The (non-dimensional) Rossby number can now be seen as the ratio between the
coriolis term and the relative acceleration term, i.e.
2~Ω × ~u = O(2ΩU),d~u
dt= O
(U2
L
),
giving
ε =U
2ΩL.
GFD deals with flows for which ε ≤ 1, i.e. the earth’s rotation is important for the
dynamics (the balance of forces). For small -ε flows, it is the coriolis acceleration that
mainly balances any applied force ~Fa (see Fig. 1). This implies that the relative ac-
celeration (or displacement) due to an applied force ~Fa must be at right angles to the1Apart from temperature, salinity or humidity also enter the equation of state for seawater or air.
Separate equations for conservation of salt or moisture are then required to close the system.
2
direction of ~Fa! This ”gyroscopic” property of fluids on a rotating earth is responsible
for the unique character of geophysical flows.
U
Fa
2Ω × U
Ω
Figure 1: Since coriolis acceleration balances the applied force ~Fa, the displacement (in unittime) ~U is to the right of ~Fa.
Scale analysis of equations on a spherical earthEquation (4) takes the following form at a mid-latitude point on the earth’s surface,
with ~u = ui + vj + wk, where i, j, k are unit vectors in the eastward, northward and
vertically upward directions.
du
dt−uv tan θ
Re
+uw
Re
= −1
ρ
∂p
∂x+ 2Ωv sin θ − 2Ωw cos θ + Fx (5a)
dv
dt+u2 tan θ
Re
+vw
Re
= −1
ρ
∂p
∂y− 2Ωu sin θ + Fy (5b)
dw
dt−u2 + v2
Re
= −1
ρ
∂p
∂z− g − 2Ωu cos θ + Fz (5c)
where Re is earth radius, θ is latitude, g is gravitational acceleration and Fx, Fy, Fz, are
the components of fluid friction. For typical scales of atmospheric ”synoptic” flow, U
∼ 10 ms−1, W ∼ 1 cms −1, L ∼ 106 m, D ∼ 104 m (length scale in the vertical),∆P
ρ∼
103 m2 s−2 (horizontal pressure fluctuations),L
U∼ 105 s (advective time scale), the sizes
of other terms in the equations (5a-b) are small compared to the coriolis term and the
pressure gradient term .2, i.e. the dominant balance is
−fv ' −1
ρ
∂p
∂x, fu ' −
1
ρ
∂p
∂y(6)
2The friction terms can be important near boundaries.
3
where f = 2 Ω sin θ. Large-scale flows in the ocean also satisfy (6); such flows are called
”geostrophic”. Note that the local vertical component of coriolis force is the relevant
term in the dynamics. For a given horizontal pressure field, it is possible to calculate
the geostrophic flow ~Vg, which is
~Vg = k ×1
ρf~∇p. (7)
The geostrophic flow is everywhere normal to ~∇p, i.e. the flow is parallel to isobars
(lines of equal p). An atmosphere or ocean with variable density ρo(z) at rest must have
a standard pressure po(z) that depends only on the vertical coordinate z; po and ρo must
be in hydrostatic equilibrium, i.e.
1
ρo
∂po
∂z= −g (8)
In the presence of motion, the pressure and density fields can be written as
p(x, y, z, t) = po(z) + p′(x, y, z, t)
ρ(x, y, z, t) = ρo(z) + ρ′(x, y, z, t); (9)
for an atmosphere or ocean at rest, p′ and ρ′ are zero. Scale analysis of the vertical
momentum equation (5c) shows that ifρ′
ρo
1, then hydrostatic balance holds for the
perturbation pressure and density fields,
∂p′
∂z+ ρ′g = 0, (10)
i.e. vertical accelerations are negligible.
To obtain prediction equations it is necessary to retain the acceleration. The linear
equations are
∂u
∂t− fv = −
1
ρ
∂p
∂x,
∂v
∂t+ fu = −
1
ρ
∂p
∂y
4
VorticityVorticity is an important quantity in GFD. It is defined as
~ω = ~∇ × ~u (11)
From Stokes’ theorem,
∫ ∫A
~ω.ndA =
∮C
~u.d~r, (12)
where C is a contour enclosing the surface A whose normal at each point is the unit
vector n, and d~r is the infinitesimal vector tangent to the curve C at each point. In the
limit of small surface element δA (such that the mean vorticity normal to δA is well-
defined)
ωn =ucl
δA,
where uc is the mean circumferential velocity around the contour C of length L. If δA
is a small circle of radius r,
ωn = uc
2πr
πr2= 2
uc
r(13)
so that locally, the velocity is just twice the average angular velocity. The component of
~ω in the z-direction is
ωz =∂v
∂x−∂u
∂y,
illustrating that a unidirectional flow with shear (e.g. u = u(y) or v = v(x)) has non-
zero vorticity; curved trajectories are not required. Recall that a velocity vector ~u in a
rotating frame of reference appears as ~uI = (~u + 2~Ω × ~r) from an inertial frame, where
~r is the position vector of the fluid element. The absolute vorticity ~ωa (i.e. the vorticity
in an inertial frame) is
~ωa = ~∇ × (~u+ ~Ω × ~r) = ~w + 2~Ω (14)
where ~ω is the vorticity relative to the earth (relative vorticity) and 2~Ω is the planetary
vorticity.
5
At a latitude θ, the component of 2~Ω normal to the earth’s surface is f = 2Ω sin θ.
The size of the vertical component of relative vorticity ωz is O(U
L
). Therefore
ωz
f=
U
fL=
U
2Ω sin θL=
ε
sin θ(15)
Away from the equator sin θ ∼ O(1), so that small Rossby number flows have the prop-
erty that their relative vorticity is small compared to planetary vorticity.
Since vorticity is divergence - free by definition (~∇.~ω = ~∇.(~∇ × ~u) = 0), for an
arbitrary volume V ,
∫ ∫ ∫V
dV (~∇.~ω) = 0.
An equation for the rate of change of relative vorticityd~ω
dtcan be written by taking
the curl of the momentum equation (4). It is
d~ω
dt= (~ωa.~∇)~u− ~ωa(~∇.~u) +
~∇ρ× ~∇pρ2
+ ~∇ ×~F
ρ
The relative vorticity changes as a result of (a) vortex stretching, i.e. change in cross-
sectional area of vortex tubes. 3, due to non zero divergence in the plane normal to the
tube (e.g. if the tube is vertical,∂u
∂x+∂v
∂y6= 0), (b) Vortex tilting due to non-zero shear
u(z) or v(z), (c)non-zero ”baroclinic” vector ~∇ρ× ~∇p, and (d) friction.
A conservation ”law” for vorticityWe shall obtain a conservation law for vorticity in the context of a simple model
described by the ”shallow water equations”. This is not a general law like momentum
conservation (Newton’s second law), but a general principle of great power in GFD.
Consider a sheet of fluid of constant density ρ and mean depth D. We describe the
equations in a Cartesian coordinate system with its origin at latitude θ. We have seen
that the local vertical component of the earth’s rotation is relevant for small Rossby
number flows (eqs. 5 and the subsequent scale analysis). So we assume that the shallow
water system is rotating about the z - axis at Ω sin θ (therefore f = 2Ω sin θ). The surface
3A vortex tube is an imaginary volume whose sides are everywhere parallel to ~ω; it has two end faces
A and B. Since ~ω is divergence free, the flux of vorticity through the end faces is equal, i.e.∫∫
A~ω.nAdA
=∫∫
B~ω.nBdB.
6
of the fluid is at z = h(x, y, t), and the rigid bottom is not flat, but lies at z = hB(x, y). If
D and L are the vertical and horizontal scales of motion, shallow water theory requires
δ =D
L 1. (16)
For constant ρ, the equation of continuity (eq. (2)) reduces to the incompressibility
condition
~∇.~u = 0 (17)
Therefore there is no need to use thermodynamics to calculatedρ
dt; the shallow water
model is purely dynamical. Eq (17) gives
∂u
∂x+∂v
∂y+∂w
∂z= 0 (18)
The first two terms are of size O(U
L
), and it follows that the scale of the vertical
velocity W cannot be larger than O(UD
L
), otherwise eq (18) could not be balanced.
Therefore
W ≤ O(δU) (19)
The shallow water momentum equations are, with p(x, y, z, t) = po(z) + p′(x, y, z, t),
∂u
∂t+
[u∂u
∂x+ v
∂u
∂y+ w
∂u
∂z
]− fv = −
1
ρ
∂p′
∂x(20a)
∂v
∂t+
[u∂v
∂x+ v
∂v
∂y+ w
∂v
∂z
]+ fu = −
1
ρ
∂p′
∂y(20b)
∂w
∂t+ u
∂w
∂x+ v
∂w
∂y+ w
∂w
∂z= −
1
ρ
∂p′
∂z, (20c)
where we have used the hydrostatic relation,
−1
ρ
∂po
∂z+ g = 0, i.e. po(z) = −ρgz
in deriving eq (20c).
Scale analysis of eqs (20) shows that the pressure fluctuation p′ is hydrostatic, i.e. at
depth z,
7
η(x,y,t)
hB(x,y)
H(x,y,t)h(x,y,t)
X
Z D
Figure 2: Geometry of the shallow water model.
p′(x, y, z, t) = ρg(h− z); (21)
Since the horizontal pressure gradient is independent of z, i.e.
∂p
∂x= ρg
∂h
∂x,
∂p
∂y= ρg
∂h
∂y,
the horizontal accelerations are independent of z. So we assume that initially the veloc-
ities u and v are z - independent, implying that they always remain z - independent.
The horizontal momentum equations are
∂u
∂t+ u
∂u
∂x+ v
∂u
∂y− fv = −g
∂h
∂x(22a)
∂v
∂t+ u
∂v
∂x+ v
∂v
∂y+ fu = −g
∂h
∂y(22b)
If u and v are z - independent, eq (18) can be integrated in z to give
w(x, y, z, t) = −z(∂u
∂x+∂v
∂y
)+ w(x, y, z, t) (23)
where w(x, y, z, t) is a constant of integration. At the rigid surface z = hB(x, y), there can
be no normal flow (i.e. the flow must be parallel to the bottom)
w(x, y, hB, t) = u∂hB
∂x+ v
∂hB
∂y,
8
therefore the constant of integration is
w(x, y, t) = u∂hB
∂x+ v
∂hB
∂y+ hB
(∂u
∂x+∂v
∂y
)(24)
So that (23) gives w at any depth
w(x, y, z, t) = (hB − z)
(∂u
∂x+∂v
∂y
)+ hB
(∂u
∂x+∂v
∂y
)(25)
The kinematic boundary condition at the surface is
dh
dt= w =
∂h
∂t+ u
∂h
∂x+ v
∂h
∂yat z = h(x, y, t) (26)
When eqs. (26) and (25) are combined, we get
∂h
∂t+
∂
∂x(h− hB)u +
∂
∂y(h− hB)v = 0 (27)
Since the total depth of the fluid H (x, y, t) = h− hB, the equation of mass conservation
becomes
∂H
∂t+
∂
∂x(uH) +
∂
∂y(vH) = 0 (28)
or
dH
dt+H
(∂u
∂x+∂v
∂y
)= 0 (29)
i.e., horizontal divergence leads to decrease of H ; horizontal convergence leads to in-
crease of H .
Of the three components of vorticity,
wx =∂w
∂y= O
(W
L
)= O
(δU
L
),
wy = −∂w
∂x= O
(W
L
)= O
(δU
L
),
wz =∂v
∂x−∂u
∂y= O
(U
L
),
The vertical component is the dominant part. If h is eliminated from eq (15 a, b) by
differentiating (15a) w.r.t. y and (15b) w.r.t x, we obtain
9
dξ
dt≡∂ξ
∂t+ u
∂ξ
∂x+ v
∂ξ
∂y= −(ξ + f)
(∂u
∂x+∂v
∂y
)(30)
where ξ = wz. The physical content of (30) is that the convergence of absolute vorticity
(i.e. ξ + f ) tubes changes relative vorticity. Using eq. (29), we can write eq. (30) as
dξ
dt=
(ξ + f)
H
dH
dt, (31)
i.e. stretching of vortex tubes(dH
dt> 0
)increases relative vorticity
(dξ
dt> 0
)If f is
constant, (31) may be written
d
dt
(ξ + f
H
)= 0 (32)
The quantity πs =ξ + f
His called ”potential vorticity”, and is conserved following
the flow in shallow water theory.
Thus we have a model where the fluid motion is quasi-two-dimensional (recall that
u and v do not depend on z), and the vorticity is predominantly vertical. As we shall
see, this simple model is very useful indeed!
The Rossby waveThe Swedish meteorologist Carl-Gusstaf Rossby in 1939 introduced homogeneous
(constant ρ) fluid on the surface of a sphere, as the simplest model for the dynamics of
observed large-scale (or planetary) waves in the earth’s atmosphere.
Rossby considered the conservation of potential vorticity
πs =ξ + f
H
on a sphere, where f(= 2Ω sin θ) is a function of latitude. If a fluid parcel moves north
or south through a distance Y , f changes by
∆f =1
Re
∂f
∂θY =
Y
Re
2Ω cos θ (33)
Since ξ = O
(U
L
), ∆f will be of order ξ when
Y
Re
= O(ε tan θ),
10
where
ε =U
fL
Thus in mid-latitudes (tan θ ∼ O(1)), a small north-south displacement gives rise
to a sufficiently large change in f to be dynamically important. Introduce a Cartesian
coordinate system that is tangential to the earth’s surface (Fig. 3), such that f varies as
f = fo + βy, βy fo,
where fo is the coriolis frequency at the central latitude θo,
fo = 2Ω sin θ, β =2Ω
Re
cos θo (34)
If H = h − hB = D + η − hB, where η is the surface displacement due to the motion,
then for small η and hB
πs =fo + βy + ξ
D(1 + η
D− hB
D
)'
(fo + βy + ξ)(1 − η
D+ hB
D
)D
'fo +
(βy + fohB
D
)+ ξ − fo
ηD
D(35)
xy
Figure 3: The beta-plane.
11
We see from eq (35) that the β effect, i.e. the change of f with latitude due to cur-
vature of the earth, is equivalent to bottom relief (through thefohB
Dterm): both affect
the dynamics by changing the ambient potential vorticity. If hB = 0, the conserved
quantity is
πs =1
D
(fo + βy + ξ − fo
η
D
). (36)
The conservation law
(∂
∂t+ u
∂
∂x+ v
∂
∂y
) (fo + βy + ξ − fo
η
D
)= 0,
along with the geostrophic relations
v =g
fo
∂η
∂x, u = −
g
fo
∂η
∂y
and the relation
ξ =∂v
∂x−∂u
∂y=g
fo
(∂2η
∂x2+∂2η
∂y2
)≡
g
fo
∇2η, (37)
leads to the linear evolution equation
∂
∂t
(∇2η −
f2o
gDη
)+ β
∂η
∂x= 0 (38)
where we have dropped all nonlinear terms (involving products of the dependent vari-
able and its derivatives). Notice that the surface displacement η is the stream function
of the geostrophic flow. Since we have used the geostrophic approximation to evaluate
the horizontal velocity, eq (38) represents the time evolution of the geostrophic vorticity.
It is an important equation called the ”quasi-geostrophic” potential vorticity equation.
Recall that the geostrophic equations cannot tell us anything about time evolution of
a flow; (38) is the simplest equation that can be used to study the time evolution of
relative vorticity in geophysical fluid flows.
Equation (38) or its variants (i.e. equations closely related to it) have been very
successful in explaining a number of observed features of large scale flows in the at-
mosphere and ocean. Examples include the evolution of mid latitude weather patterns
(leading to the first successful weather forecasts), and the existence of strong poleward
western boundary currents in the ocean such as the Gulf Stream, the Kuroshio and the
12
Agulhas current. The reason for the success of the quasi-geostrophic potential vorticity
equation is that it contains the essence of vorticity evolution, including the planetary
wave or Rossby wave.
Plane wave solutions of eq. (38)
η ∼ Re exp(i(kx+ ly − σt))
must satisfy the dispersion relation
σ =−βk
k2 + l2 + F, (39)
where F =f 2
o
gD; k and l are wavenumbers in the eastward and northward direction, σ
is frequency. For positive σ, k has to be negative.
The Rossby wave has peculiar properties. The phase speed cx =σ
kis always west-
ward, whereas group velocity Cg =
(∂σ
∂ki,∂σ
∂lj
)can have either eastward or westward
component. Unlike familiar waves such as sound or light waves, the frequency in-
creases with increasing wavelength! Further, as the following calculations show, the
Rossby wave has an upper cutoff in frequency.
The frequency has a maximum (σ is a positive quantity by definition) for a given
value of l, when
∂σ
∂k=
(k2 + l2 + F )β − βk(2k)
(k2 + l2 + F )2= 0,
i.e., at
k = − | (l2 + F )12 |,
So that σmax =β
2(l2 + F )12
. Over all l and k, the Rossby wave cannot have fre-
quency higher than
σ =β
2F12
(40)
i.e. it is a low-frequency wave.
Further interesting properties of the Rossby waves can be studied conveniently with
the help of a dispersion diagram. Equation (39) can be put in the form, with F = 0 for
simplicity,
13
(k −
β
2σ
)2
+ l2 =β2
4σ2,
which is the equation for a circle of radiusβ
2σon the (k, l) plane, with its centre at(
− β
2σ, 0
)(Fig. 4). Wavevectors satisfying the Rossby wave dispersion relation (39)
have their origin at (0, 0) and their tip on the circle. For a given value of l, k can have two
values −k1, and −k2; the wavevector ~K1 with the smaller absolute value of k(= −k1)
represents a wave with longer wavelength than the wavevector ~K2with k = −k2 (see
fig. 4). Waves with wavevectors lying on the right (left) half of the circle are the long
(short) Rossby waves.
k
l
l = loCg1
Cg2
−k1 −k2 (−β/2σ,0)
σ = constant
K1
K2
Figure 4: Dispersion diagram of the Rossby wave.
Consider a Rossby wave with frequency σ + ∆σ (∆σ > 0) on the k, l plane. Its
dispersion curve is a circle with slightly smaller radius, and center at(
−β2(σ + ∆σ)
, 0
).
Now recall that the group velocity of a wave (which is the velocity of energy propaga-
tion) is given by
~cg =
(∂σ
∂ki,∂σ
∂lj
). (41)
This is nothing but the gradient of σ in the (k, l) plane. Since the lines of constant σ are
circles, the gradient must be in the radial direction. As the gradient vector is directed
from low values to high values, the group velocity of a Rossby waves must be directed
towards the center of the circle. Therefore the group velocity of the long Rossby wave
14
(with wavevector ~K1) ~cg1 (see Fig. 4) has a westward component, whereas that of the
short Rossby wave (wavevector ~K2) ~cg2 has an eastward component. It is easy to show
that long Rossby waves carry energy westward much faster than short Rossby waves
carry energy eastward (see the expression for∂σ
∂kon the previous page).
This last property of the Rossby wave provides a simple explanation for the ob-
served ”western intensification” in the ocean. The alongshore (i.e. parallel to the coast-
line) currents near the western boundary of all major ocean basins (both in the northern
and southern hemisphere) are strong compared to currents at eastern boundaries or in
the open ocean away from coasts.
Assume that low frequency fluctuations of surface winds generate a broad spectrum
of Rossby waves (i.e. over a range of frequencies) everywhere in the ocean. At all
frequencies, both long and short Rossby waves will exist. The long Rossby waves will
rapidly carry energy to the west. One would expect that the eastern region of the ocean
will not have energetic flows (i.e. the kinetic energy associated with low frequency
motions will be small). Further, the energy will eventually be carried to the western
boundary region by long Rossby waves. Therefore, the western boundary region will
be a region of energetic flows. Let us do a simple calculation.
Let the governing equation for Rossby waves be
∂
∂t∇2ψ + β
∂ψ
∂x= 0, (42)
which is the same as eq. (38) with F = 0 for simplicity (our results are valid in the
presence of nonzero F as well). 4; ψ is the streamfunction such that
u =−∂ψ∂y
, v =∂ψ
∂x;
ψ is proportional to h or to pressure (see the geostrophic relations). Now consider a
north-south oriented wall (representing a coastline) at x = 0 on the western side of an
ocean basin (Fig. 5). Let a long Rossby wave be incident on this wall (i.e. the incident
wave carries energy towards the wall): It is
ψi = Aiexp i(kix+ liy − σit)4Eq. (38) represents the conservation of relative vorticity due to north-south displacement (the β term)
and due to vortex stretching/shrinking associated with surface displacement (the F term).
15
On encountering the wall, it is reflected as
ψr = Arexp i(krx+ lry − σrt)
The total stream function describing the motion in the ocean 0 < x <∞ is
ψ = ψi + ψr
ψi
ψr
XX=0
Figure 5: Reflection of a long Rossby wave ψi incident on a western boundary.
The condition of no normal flow must be satisfied at the wall, i.e.,
u |x=0=−∂ψ∂y
|x=0= −Ailiexp i(liy − σit)−Arlrexp i(lry − σrt) = 0 (43)
This relation cannot be satisfied at all y and all t unless li = lr and σi = σr. This
implies that the frequency and y - wavenumber of the incident and reflected waves are
identical. Since both waves have to satisfy the dispersion relation for the Rossby wave,
and the reflected wave has to carry energy away from the wall (i.e. ~cg has to have an
eastward x - component), the only possibility is that ψi is the long Rossby wave while ψr
is the short Rossby wave. Therefore the wavelength of the Rossby wave is changed on
reflection. The boundary condition also implies that Ar = −Ai, i.e. there is a 180phase
change on reflection.
The x-wavenumber of the incident wave (the long wave) is
ki =−β2σ
+
√β2
4σ2− l2
while that of the reflected wave (the short wave) is
16
kr =−β2σ
−
√β2
4σ2− l2
The ratio of alongshore speeds (north-south speeds) at the wall associated with the
reflected and incident waves is
| vr || vi |
=|∂ψr
∂x|
|∂ψi
∂x|
∣∣∣∣∣∣∣∣x=0
=kr
ki
> 1,
i.e. the reflected wave has larger v velocity. Since the group speed of the Rossby
wave is
| ~Cg |=β
k2 + l2,
energy in the short wave travels much slower than in the long wave. Any fluid friction
(which we have not considered explicitly) will therefore damp the short wave before it
has moved very far away from the wall. The net result of the reflection process is that
the western boundary region of the ocean has stronger (more energetic) flow than the
open ocean, i.e. ”western intensification”. It can be shown that energy is conserved in
the reflection process, i.e.
< Er >| ~Cgr |=< Ei >| ~Cgi |
across any line parallel to the wall, where < E > is energy density and < E >| ~Cg | is
the energy flux due to the Rossby wave.
At the eastern boundary, the incident short Rossby wave is reflected as a long Rossby
wave that rapidly carries energy to the west. Therefore eastern boundary regions in
mid-latitudes do not have energetic flows. An analysis similar to the one above shows
that there is no wavelength change on reflection from northern or southern boundaries.
In summary, the increase of the coriolis parameter with latitude (i.e. the β-effect)
is responsible for the existence of the Rossby wave. It has some peculiar properties,
including anisotropy, giving rise to westward phase and group propagation of large
scale, low frequency motion in the atmosphere and ocean.
We left out the surface displacement Fη term for simplicity in our analysis of Rossby
wave reflection. However, it can be a very important term for long waves. Consider a
17
very long Rossby wave with purely westward wave vector, such that k2 f 2o
gD. The
dispersion relation (eq. (39)) can be approximated by
σ = −βk
f2o
gD
(44)
the phase speedσ
kfor long Rossby waves goes as f−2
o . In regions relatively close to the
equator, i.e. in the tropics, β varies little with latitude since it goes as cos θ. However,
fo goes as sin θ, which is proportional to θ for moderate values of θ. Equation (44) then
predicts that if a disturbance along the eastern boundary of the ocean generates long
Rossby waves, the phase front of these waves will have a characteristic1
θ2shape : the
wave will travel fast at low latitudes, and its speed will drop off with increasing latitude
on either side of the equator (Fig. 6). Such long Rossby wave phase fronts are often seen
the ocean.
θ = 20οS
θ = 20οN
eastern boundary
east
Figure 6: Phase front of long Rossby waves generated simultaneously all along the easternboundary.
A final point can only be mentioned here (we do not offer a proof): stratification
(i.e. the increase of density ρ(z) with depth z in the ocean, or ρ(p) with pressure p in the
atmosphere) makes the Rossby wave travel much slower than suggested by eqs. (39)
or (43).In the presence of stratification (the real atmosphere and ocean are generally
stably stratified), the F term in the denominator of eq. (38) or (43) isf 2
o
gDe
rather than
18
f 2o
gD; where the ”equivalent depth” De is much smaller than D, the geometric scale in
the vertical direction. In other words, the effective vertical scale of motion is small in
the presence of stratification. Therefore for a given k, the Rossby wave has smaller fre-
quency and propagation speeds than suggested by shallow water theory (with constant
ρ).
In Fig. 6, we did not draw the phase front of the Rossby wave within a few de-
grees north and south of the equator. This is because fo goes to zero at the equator,
and the dynamics of the equatorial region is somewhat different from that of higher
latitudes. We shall not discuss equatorial dynamics here, but mention a few impor-
tant facts. A model similar to the mid-latitude shallow water model can be constructed
for the equatorial region. It can be shown using this equatorial beta plane model that
the equator acts like a waveguide, i.e. there exist several classes of waves which can
propagate along the equator, with amplitude falling sharply away from the equatorial
region. These waves can have simple or complex structure in the y -direction depend-
ing on a meridional mode number. The eastward moving equatorial Kelvin wave is
the simplest of these waves. There is also a family of equatorial Rossby waves, which
are low-frequency waves, just like their off-equatorial counterparts. In addition we
mention the Yanai wave (or mixed Rossby-gravity wave) which can have high or low
frequency, and the high frequency intertia-gravity waves. All these waves can be seen
in the equatorial atmosphere and ocean. Many of them are important players in climate
variability. For example, the oceanic Kelvin wave plays a key role in interannual climate
variability through its influence on El Nino. The theoretical prediction in 1966 of the
existence, structure and propagation characteristics of equatorially trapped waves by
the Japanese scientist Taroh Matsuno is one of the great achievements of Geophysical
Fluid Dynamics. As better observations continue to be made in the tropics from in situ
and space - based sensors, the importance of equatorially trapped waves in climate is
becoming clearer.
Wind stress and forced motionSo far we have discussed free evolution, i.e. the evolution of a geophysical fluid in
the absence of ”external” forcing. An important forcing, both for the atmosphere and
19
ocean, arises from the action of surface stress. Stress arises from the near-surface tur-
bulent transport of momentum in the atmosphere or ocean, or between the atmosphere
and ocean. If we use the momentum equations to study large scale flows, the stress
term in these equations represents the influence of small scale (turbulent) motion on
these flows. We do not discuss the form of the stress in the near-surface atmosphere or
ocean, but assume that a such a stress exists and study its consequences using simple
models.
Since the vertical scale of the boundary layers in the lower atmosphere (1 km) and
upper ocean (10 m-100 m) is small compared to the horizontal scale on which the
stresses vary (100-1000 km), it is the vertical gradient of the horizontal stress that enters
the dynamics. Let the x and y components of the stress ~τ be denoted by τx and τy. Then
the shallow water equations with forcing due to ~τ can be written as
∂u
∂t− fv = −
1
ρ
∂p
∂x+
1
ρ
∂τx
∂z
∂v
∂t+ fu = −
1
ρ
∂p
∂y+
1
ρ
∂τy
∂z(45)
The solution to the linear system can be written as u = up + uE, v = vp + vE , where the
pressure gradient driven flows up and vp satisfy
∂up
∂t− fvp = −
1
ρ
∂p
∂x,
∂vp
∂t+ fup = −
1
ρ
∂p
∂y(46)
and the stress driven flows uE and vE satisfy
∂uE
∂t− fvE =
1
ρ
∂τx
∂z,
∂vE
∂t+ fuE =
1
ρ
∂τy
∂z(47)
In steady state up, vp represent the geostrophic flow. The flow driven by stress uE , vE
is called the Ekman velocity (after the Norwegian explorer-scientist V. Walfrid Ekman).
This flow is confined to the shallow layer of fluid over which the stress acts, called the
Ekman layer. If τx and τy are zero outside the Ekman layer, integration of eq. (47) across
the layer gives
ρ
(∂UE
∂t− fVE
)= −τxs, ρ
(∂VE
∂t+ fUE
)= −τys (48)
if the boundary is below, and
20
ρ
(∂UE
∂t− fVE
)= τxs, ρ
(∂VE
∂t+ fUE
)= τys (49)
if the boundary is above, (see Fig. 7). Here (UE, VE) =∫
(uE, vE)dz is the Ekman volume
transport, and τxs and τys are the values of τx and τy at the surface. The Ekman mass
transport in the atmosphere and ocean add to zero (as can be seen by adding eqs. (48)
and (49)). The volume transport in the atmosphere is about 1000 times that in the ocean,
because the density of air is about 1000 times smaller than water. In steady state, the Ek-
man transport is directed at right angles to the surface stress, illustrating the gyroscopic
property of geophysical fluids. The stress is entirely balanced by coriolis force (in the
absence of pressure gradients) in steady state. For a typical value of surface stress, 0.1
Newton per meter square, the Ekman mass transport in mid-latitudes (f ∼ 10−4s−1) in
the lower atmosphere or upper ocean is about 1000 kgm−1s−1. The Ekman formulas
ρVE =τxs
f, ρUE =
τys
f(50)
are remarkable because transport does not depend on the details of the turbulence that
generates the stress. Experimental verification of eq. (50) is difficult, because it is not
possible to measure accurately the pressure gradient term, which is generally non-zero
in the atmosphere and ocean.
Oceanic Ekman mass transport τs/f
Atmospheric Ekman mass transport τs/f
Surface stress τsZ
Figure 7: The Ekman transport in the upper ocean and lower atmosphere are at right angles tothe surface stress.
The surface stress varies from place to place, and this implies that the (horizontal)
Ekman transport is not constant. If the Ekman transport has non-zero horizontal diver-
21
gence, then fluid must be ”sucked” vertically into or out of the Ekman (or boundary)
layer.
The magnitude of vertical velocity wE just outside the Ekman layer that results from
convergence or divergence of horizontal Ekman transport can be estimated by integrat-
ing the continuity equation. For constant density,
∂u
∂x+∂v
∂y+∂w
∂z= 0; (51)
integrating in z across the Ekman layer, and assuming that w = 0 at the surface gives
ρwE =∂
∂x
(τys
f
)−
∂
∂y
(τxs
f
)(52)
where we have used eqs. (48) and (49). Usually the stress varies over smaller scales
than f , and the ”Ekman pumping” velocity is given by
wE =1
ρ(curl~τs) (53)
where curl~τs =∂τys
∂x−∂τxs
∂y.
Ekman pumping
Surface
Atmosphere boundary layer
Ocean boundary layer
Ekman pumping
Isotherms
Figure 8: Ekman pumping velocity wE is upward in both ocean and atmosphere if curl~τs > 0
22
The sign of wE is the same in the ocean and atmosphere (see Fig. 8), with wE be-
ing 1000 times larger in the atmosphere than in the ocean. Upwelling (upward Ekman
pumping) in the upper ocean below the Ekman layer causes the lines of constant tem-
perature (isotherms) to dome upward; there is upward motion above the atmospheric
boundary layer, which can promote convection. Since f appears in the denominator
of eq. (53), the Ekman pumping velocities are larger near the equator than in mid-
latitudes. At the equator, of course, eqs. (50) or (53) are not valid because f is zero.
Thus not only does surface stress directly drive lateral motion in the Ekman layer, it
also gives rise to three dimensional motion (via continuity) in the region outside the
turbulent boundary layer.
Finally, we obtain a simple relation between the wind field and the ocean transport
including the effect of Ekman pumping. Consider the linear momentum equations for
the ocean on the β plane,
∂u
∂t− fv = −
1
ρ
∂p
∂x,
∂v
∂t+ fu = −
1
ρ
∂p
∂y;
These equations have the same form as equation (45). Note that they are valid (at a
given depth z) for a stratified ocean as well. By cross differentiation, we can form the
vorticity equation
∂ξ
∂t+ f
(∂u
∂x+∂v
∂y
)+ βv = 0. (54)
For an incompressible fluid,
∂ξ
∂t+ βv = f
∂w
∂z(55)
which represents the vorticity balance for a fluid column moving northward in the
planetary vorticity field, in the presence of vortex stretching due to change in w with
depth. For steady flow,
βv = f∂w
∂z. (56)
The interpretation is that if the horizontal velocity field is convergent, the r.h.s is posi-
tive; since vortex tubes are (almost) vertical for small Rossby number flows, stretching
23
of vortex tubes increases the absolute vorticity. Eq. (56) says that this is only possible if
the fluid column moves north. What happens in the presence of turbulent stress?
We have mentioned that stress is important in a shallow (∼ O (100 m)) Ekman layer
in the upper ocean. Assume that at some level −H in the deep ocean (∼ O (1000 m)),
the vertical motion is zero (or negligible). Consider the vertical velocity at the base of
the Ekman layer to act at the ocean surface. Integration of Eq. (56) in z from −H to the
surface then gives
βV =1
ρ
(∂τys
∂x−∂τxs
∂y
)=
1
ρcurl~τs (57)
where V =∫ 0
−Hvdz, and we have used eq. (53) for wE
Equation (57) is called the Sverdrup relation after the Norwegian oceanographer
Harald Sverdrup. It relates the curl of surface wind stress to northward transport in
the entire upper ocean influenced by wind (through Ekman pumping). It is another
form of the steady vorticity balance, and has proved useful for understanding aspects
of large scale wind driven ocean circulation. We note that like the geostrophic relation,
the Ekman formula and the shallow water theory, it is based on drastic assumptions;
its popularity is due to its apparent simplicity.
ReferencesAdrian E. Gill (1982): Atmosphere-ocean Dynamics. Academic Press, London, 662pp.
Joseph Pedlosky (1987): Geophysical Fluid Dynamics, Second Edition. Springer-Verlag,
New York, 710pp.
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