Basic equations

We use the linearized equations on an f -plane:

where x = gh' is the geopotential perturbation, and c0 = ^/gH. For the atmosphere, H = RT0/g = P0/pg is the scale height. From Eqn. (4.476), we can derive the vorticity and divergence equations d t i

dD 2

d t whereD is the velocity divergence, and V^ = (d2/dx2) + (d2/dy2). By introducing velocity potential $ and streamfunction f, the horizontal velocity can be decomposed into two components:

Thus

Equation (4.477) is reduced to

is the nondimensional pressure field. Eliminating $ from Eqns. (4.480a) and (4.480c) gives d (V2f - f *) = 0 (4.482)

That means potential vorticity is conserved, i.e.,

Equations (4.480a) and (4.480b) are Laplace equations. Under the boundary conditions of finiteness at infinity, the only possible solutions are df

Eliminating f and n, we can find an equation for $; thus we have a system for the linear barotropic adjustment process:

d dt d 2$ 2 ( d 2$ d 2$ \ 2 $ = c2 + T^T - f $ (4.485b)

dn o

d t where the second equation is a wave equation involved with $ only, so that it can be solved for $ (x, y, t). It is important to remember that the velocity field has been separated into two parts, f (x, y, t) and $ (x, y, t). When $ (x, y, t) is determined, f (x, y, t) and n (x, y, t) can be calculated using Eqn. (4.484).

The final state of geostrophy

From the potential vorticity equation (4.483) we can find out the final state without even solving the time evolution process. Thus

where q(x, y) is the potential vorticity distribution calculated from the initial state. Since the final state is geostrophic, we also have

From these equations we find a single equation for f

where X = co/f is the radius of deformation. This is a Helmholtz equation; its Green function is the modified Bessel function Ko, thus

m j j-w where p = ^- x)2 + (n - y)2. Using polar coordinates, this can be rewritten as

2n Jo Jo fo fo ^2^0 - f no) Ko(X)pdpd0 f f {v2fo - X-2fo) Ko (X) pdpd0 1- j j (x-2fo - f no) *o(x)pdpd0.

Note that the first term is exactly fo; thus fw (x, y) = fo - fo f (x~2fo - f no) Ko(X)pdpd0 (4.491)

Relation between the length scales of initial perturbations.

For simplicity, let us assume that the initial perturbation is confined inside a circle of R = L, and both f o and no are nearly constant. Thus at the origin (o, o)

Using relation d [xK1 (x)] /dx = -xKo (x), we obtain fw = fo - (x 2fo - f no)X2 [-p/Lo ■ Ki (p/Lo)] = fo - (fo - clno/f) [-x ■ Ki (x)], p=L

where x = L/X. When x = L/X ^ 1, xK1 (x) ^ 1, we have fw ™ fo, nw ™ f fo/c2o (4.494)

In this case the streamfunction, and thus the velocity field, basically remain unchanged; however, the pressure field is adjusted toward a state set up by the initial velocity field. When x = L/X > 1, xK1 (x) ^ o, we have fw ^ cono/f, nw ^ no (4.495)

In this case the pressure field remains nearly unchanged; however, the streamfunction, and thus the velocity field, are adjusted toward a state set up by the initial pressure field.

In his seminal paper, Rossby (1938) concluded that the pressure field adjusted toward the velocity field. However, further studies indicated that both the velocity and pressure fields adjust, depending on the initial horizontal scale of the perturbations. For example, Yeh (1957) studied the analytical solutions of the geostrophic adjustment and pointed out that the direction of geostrophic adjustment depends on the initial horizontal scale, in comparison with the radius of deformation.

If the initial perturbation has a small horizontal scale, wave processes can disperse the energy within a time scale shorter than 1/ f. Within such a short time, the vorticity field (or the velocity field) remains basically unchanged; however, the pressure field is altered to be in geostrophic balance with the velocity.

On the other hand, if the initial horizontal scale is large, the adjustment processes take a much longer time, so a new velocity field is established that is in balance with the pressure gradient, and changes in the pressure field stop. As a result, the initial pressure perturbations can mostly remain unchanged.

An interesting application of geostrophic adjustment is the adjustment of free surface elevation and bottom pressure due to surface heating and precipitation (Huang and Jin, 2002b). Surface heating creates a surface elevation anomaly which induces a pressure perturbation in the upper ocean. This perturbation is baroclinic in nature. Since the horizontal scale of surface heating (on the order of a few hundred kilometers or larger) is much larger than the first deformation radius, signals in connection with surface elevation associated with surface heating can survive.

On the other hand, precipitation induces surface elevation and bottom pressure anomalies, which are barotropic perturbation. Precipitation has a horizontal scale on the order of a few hundred kilometers, which is much smaller than the barotropic radius of deformation (on the order of 2,000 km); thus, the initial perturbations in surface elevation and bottom pressure cannot be maintained during geostrophic adjustment. As a result, there is very little signal left behind after the geostrophic adjustment. In fact, signals associated with precipitation cannot be identified from satellite altimetry data.

An example of a vortex

Obukhov (1949) discussed a case in which there was an axi-symmetric velocity field initially, but with no pressure gradient:

where / = R/L0, f = r/L, r = ^x2 + y2. The solution of Eqn. (4.491) is f™ (r) = -A (2 - f2) e-f 2/2, (r) = 2 - f2) e-f2 /2 (4.498)

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