1. Geostrophic adjustment
Question: How does the adjustment to geostrophy take place?
H(1 + ho)
H(1 - ho) x

2. u-momentum
(1)
v-momentum
(2)
continuity
(3)
Take the divergence of (1) and (2) and use (3) to eliminate the
divergence

3. The nonrotating case (f = 0)
the motion remains irrotational (z = 0) then
assume two-dimensional motion
for
for

4. time

5. The rotating case (f nonzero)
The two moving steps will experience a Coriolis force acting
to the right (NH case)
a flow in the -ve y-direction
this flow will experience a Coriolis force in the
-ve x-direction, opposing the flow in the x-direction
The final adjusted state may be calculated without
consideration of the transients

6. The vorticity equation is
The divergence equation is
the linearized form of the
potential vorticity equation
is the perturbation potential vorticity
a parcel retains its original Q’

7. Here, at t = 0, u = v = 0 and h = hosgn(x)
in the steady state
Solution is:
Solution is continuous at
and symmetric about x = 0

8. h x -3 -2 -1 1 2 3 v 1 -3 -2 -1 2 3 x

9. The transient problem
The equation for h is now
Put
Then
Put

10. Can show that at t = 0
Note that
For t > 0
When

12. h h x

13. u u x

14. v v x

15. Energy considerations+ +

16. Total perturbation wave energy per unit length in the y-directionKE PE
is the rate of energy transfer per unit length
in the y-direction at x
Nonrotating situation
at t = 0:
after the passage of the wave
all the perturbation potential energy in a fixed region will be converted into the kinetic energy associated with the steady current that remains after the wave front has passed

17. Rotating situation
Note: the Coriolis forces do not directly appear in the energy
equations
However: the energy changes in the adjustment problem are
profoundly affected by rotation
= one third of the PE released!
The remaining energy is radiated away as inertia-gravity waves in the transient part of the solution.

18. Notes:
Energy is hard to extract from the available potential energy in a rotating
fluid. The geostrophic equilibrium state which is established retains a certain
amount of the initial available potential energy.
The steady solution is not one of rest, but is a geostrophically-balanced flow.
The steady solution is degenerate - any velocity field in geostrophic balance
satisfies the continuity equation exactly. Therefore the steady solution cannot
be found by looking for a solution of the steady-state equations - some other
information is needed.
The required information is supplied by the principle that potential vorticity
is materially conserved (or locally conserved in the linear problem).
The equation for the steady solution contains a length scale, LR = c/|f|, where
c = (gH)1/2 is the wave speed in the absence of rotation.
L << LR, rotation effects are small
for L comparable with or large compared with LR, rotation effects are important

19. Rossby Length
LR is a fundamental length scale in atmosphere-ocean dynamics. It is the horizontal scale at which the gross effects of rotation are of comparable importance with gravitational (or buoyancy) effects
Early in the adjustment stage in the transient problem, the change in level
is confined to a small distance and the horizontal pressure gradient is
comparatively large. Accordingly, gravity dominates the flow behaviour.
Thus at scales small compared with the Rossby length, the adjustment is
approximately the same as in a non-rotating system.
Later, as the change in surface elevation is spread over a distance
comparable with the Rossby length, the Coriolis acceleration becomes just
as important as the pressure gradient term and thus rotation leads to a
response that is radically different from that in the non-rotating case.

20. LR is also an important scale for the geostrophic equilibrium solution as well.
In the problem analyzed, the discontinuity did not spread out indefinitely,
but only over a distance of the order of LR.
For geostrophic, or quasi-geostrophic flow, LR is the scale for which the two
contributing terms to the perturbation potential vorticity Q′ are of the same
order.
For a sinusoidal variation of surface elevation with wavenumber k, the ratio
of the vorticity term to the gravitational term in Q′ is
Therefore, for short waves (1/k << LR) the vorticity term dominates while for
long waves (1/k >> LR) gravitational effects associated with the free surface
perturbation dominate.
For quasi-geostrophic wave motions, the ratio characterizes not only
the partition of perturbation potential vorticity, but also the partition of energy.

21. The perturbation PE integrated over one wavelength is
The perturbation KE integrated over one wavelength is
For quasi-geostrophic waves periodic in x, one can show that (Ex. 9.2)
Thus short-wavelength geostrophic flow contains mainly kinetic energy, whereas long-wavelength geostrophic flow has most of its energy in the potential form.
The situation is different for inertia-gravity waves (see Ex. 9.2).

22. Changes in h are associated with changes in the mass field, whereas changes in z are associated with changes in the velocity field.
For large scales, the potential vorticity perturbation is mainly associated with perturbations in the mass field and energy changes are mainly a result of potential energy changes.
For small scales, the potential vorticity perturbation is mainly associated with perturbations in the velocity field and energy changes are mainly a result of kinetic energy changes.
A distinction can be made between the adjustment processes at different scales.

23. At large scales (1/k >> LR) , it is the mass field that is determined by the initial potential vorticity, and the velocity field is merely that which is in geostrophic equilibrium with the mass field.
In other words, the large-scale velocity field adjusts to be in equilibrium with the large-scale mass field.
In contrast, at small scales (1/k << LR) , it is the velocity field that is determined by the initial potential vorticity, and the mass field is merely that which is in geostrophic equilibrium with the velocity field.
In other words, the mass field adjusts to be in equilibrium with the velocity field.

24. Balanced adjustment

25. Balanced adjustment h
Sdx
h + dh f v u
u + du x y dx

27. Special cases F -a a x
initial state u -a a x
a later state

30. addition
of mass
initial state v v . H u u -a a x

33. Horizontal cross-section through a field of cumulus clouds

34. heating function: δ(y)sin (πz/H)
(a) A heated vertical plate extends between two infinite horizontal parallel plates which are considered free slip and which contain between them a stably-stratified Boussinesq fluid. (b) Linear response to heating of the form $ \delta(y) \sin (\piz/H)$, where $H$ is the distance between the plates and the fluid is inviscid and initially at rest. No Coriolis accelerations are included. (c) Same as (b), but for the case where the fluid diffuses heat and momentum and is rotating, with Coriolis parameter $f$.

35. The response to an idealized slab-symmetric narrow sinusoidal heat source in a uniformly stratified fluid layer. The arrows indicate velocities, and the dashed lines are dye lines, which are equally spaced in the undisturbed flow [from Bretherton(1993)].
The response to an idealized slab-symmetric narrow sinusoidal heat source in a uniformly-stratified fluid layer.

36. the Gaussian heat source
Horizontal section at mid-height of the buoyancy perturbation at three times due to a periodic array of heat sources turned on at $t = 0$. The first panel shows the Gaussian heat source; the times are in units of $\Lambda /c_m$. [From Bretherton (1993)]
Horizontal section at mid-height of the buoyancy perturbation at three times due to a periodic array of heat sources turned on at t = 0.

37. Comparison of horizontal and vertical velocity fields and buoyancy field induced by a heat source for a domain with a rigid lid and a semi-infinite domain. Here the heating aspect ratio is $a/H = 1$ and the nondimensional time is $Nt = 36$. [From Bretherton (1993)].

38. The End