Away from the ground, the atmosphere adjusts to a geostrophic equilibrium in which the pressure-gradient force balances the Coriolis force associated with a steady flow along the surfaces of constant pressure. If this motion extends to the ground, the effect of turbulent friction is to disrupt this geostrophic balance, thus producing a flow across these surfaces from high to low pressure. Hence, work is being done on the fluid within the surface boundary layer by the pressure-gradient force. This work supplies the necessary energy to maintain this layer against the dissipative forces within it. Accordingly, unless the geostrophic flow is forcibly maintained, it will decay under the action of the bottom friction.
To calculate the typical spin-down time of a geostrophic flow, let us again consider quasi-horizontal motions in a cyclonic vortex. Since the motion is geostrophically balanced away from the ground, the center of the vortex is at low pressure compared to its outer edge. In the surface boundary layer, then, turbulent friction produces a radial flow of matter toward the vortex center. By continuity, this requires upward motion and a compensating outward radial flow above the friction layer. Figure 2.2 presents a
Fig. 2.2. Qualitative sketch of the streamlines of meridional circulation in a mid-latitude cyclonic vortex. The rotation axis is vertical.
qualitative sketch of the streamlines of this secondary flow. Since this slow but inexorable motion approximately conserves angular momentum, high angular velocity fluid is thus progressively replaced by low angular velocity fluid in the atmosphere. As we shall see, this axially symmetric circulation driven by turbulent friction in the surface layers serves to spin down the azimuthal motion of the cyclonic vortex far more rapidly than could turbulent diffusion of momentum. This mechanism, which exchanges mass between the surface boundary layer and the free atmosphere above it, is known as Ekman pumping.
For the sake of simplicity, let us assume that the atmosphere, of height H, is of uniform density. Assume further that in the surface boundary layer, of depth d, the radial inflow of matter is adequately described by Eqs. (2.91) and (2.92). Above the boundary layer, in the free atmosphere, the azimuthal motion of the cyclonic vortex has its relative vorticity - Zg(x, y) - along the z axis (see Eq. [2.37]). By virtue of Eq. (2.79), we thus have dv„ d u„
which is called the geostrophic vorticity. To calculate the upward velocity wE at the top of the boundary layer, let us integrate Eq. (2.73) through the depth of the layer. Because w = 0 at z = 0, it follows that fd(du dv\ we = - — + — dz. (2.95)
Substituting for u and v from Eqs. (2.91) and (2.92), one obtains
This relation merely states that the vertical velocity of the matter that is pumped into the free atmosphere is proportional to the geostrophic vorticity.
For synoptic scale motions, the vorticity equation can be derived from Eqs. (2.74) and (2.75) by cross differentiation with respect to x and y. Neglecting turbulent friction, we thus have
(Compare with Eq. [2.43], which is written in an inertial frame of reference.) If f is regarded as a constant, Eq. (2.97) can be written approximately as dtg dw
d t d z where we have also neglected fg compared to f in the divergence term. Integrating this equation from the top of the boundary layer (z = d) to the top of the atmosphere (z = H), we obtain
d t since d ^ H and w = 0 at z = H. Substituting for wE from Eq. (2.96) and integrating this equation with respect to time, one finds that fg = fg (0)exp[-(fKy/2H2)1/2t ], (2.100)
where fg(0) is the geostrophic vorticity at t = 0. By virtue of Eq. (2.100), the spin-down time of a cyclonic vortex, tsd, is
This result was originally derived by Charney and Eliassen (1949).
To illustrate the problem, we shall let H = 10km, f = 10-4s-1,and KV = 105cm2s-1. By making use of Eq. (2.101), one finds that 'sd & 4 days. In contrast, the characteristic time 'v for turbulent diffusion to penetrate a depth H is of the order H2/KV, which, for the above values of H and KV, gives 'v ^ 100 days, which is much longer than 4 days. We conclude that Ekman pumping is a far more effective mechanism for des'roying a cyclonic vor'ex in 'he Ear'h's a'mosphere 'han is 'urbulen' diffusion of momen'um. Yet, letting Zg = f in Eq. (2.96), which means an intense cyclonic vortex, one finds that the vertical speed wE does not exceed 2.3 cm s-1 at the top of the boundary layer.
As shown in Figure 2.3, it is an analogous meridional circulation that is responsible for the decay of the azimuthal motion created when a cup of tea is stirred. Physically, the spin-down mechanism is essentially that described for the cyclonic vortex, except that in the cup of tea it is the centrifugal force that balances the pressure-gradient force, not the Coriolis force. Visualization of the transient meridional flow is provided by the tea leaves, which are always observed to cluster near the center at the bottom of the spinning fluid. As was shown by Greenspan and Howard (1963), the spin-down time 'sd is, roughly, of the order of (L 2/v^)1/2, where L is a characteristic dimension, parallel to the rotation axis, v is the kinematic viscosity, and ^ is the initial angular velocity. Letting L = 4 cm, v = 10-2 cm2 s-1, and ^ = 2n s-1, one obtains 'sd = 16 s, in agreement with casual observation. One also finds that 'v = L2/v = 1,600 s! Obviously, 'he azimu'hal mo'ion in a cup of 'ea decays much more rapidly 'hrough Ekman pumping 'han by mere viscous diffusion of momen'um.
Fig. 2.3. Qualitative sketch of the streamlines of meridional circulation in a cup of tea. The rotation axis is vertical.
As we shall see in Section 8.4, a similar mechanism may be invoked to explain the high degree of synchronism that is observed in the close binary stars, although the physical cause of Ekman pumping in a nonsynchronous binary component is completely different.
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