Of great importance in tropical cyclone dynamics is the ocean mixed layer, which provides the energy required for development. The basic processes which maintain the mixed layer are (a) surface solar heating, (b) heat loss from the surface by evaporation and other processes, and (c) stirring by turbulence which entrains water from below which is then homogenized by vertical mixing. (a) decreases the potential energy (PE) of the mixed layer, and (b) and (c) increase the PE. Over the time scale of the development of a TC, it is likely that (b) is the most important of these three processes. Let us consider the energy changes which occur.
The change in potential energy of the mixed layer, dPE = dpwgD (7a)
where dpw = —adT is the change in seawater density, D is the mixed layer depth (assumed to be constant), g is the acceleration of gravity, and a is the coefficient of expansion of seawater. The corresponding change of kinetic energy of the wind, dKE = raudu (7b)
where u is the surface wind speed. On substituting the relations (7a) and (7b), which connect the two variables in the definition of H, into (6), we obtain,
in which A = (a g D )/ (pa u2). Hence H is proportional to the ratio of the change of kinetic energy and potential energy. Furthermore, for typical values of the parameters applicable in the tropics ( a = 0.3 K—1, D = 20m, u = 5 ms and g = 10 ms—2, pa = 1 kgm—3) we find that A « 4 K 1, although there is obviously a considerable variability in this estimate. This result indicates that H is a measure of the efficiency of the conversion between kinetic energy and potential energy. For small values of H the efficiency is low, whereas as H becomes large the efficiency also becomes significant, and for A « 4 K—1, an efficiency of 25% would be obtained for |H|= 1. Two distinct processes are involved, depending on the sign of H.
(a) For H > 0, a decrease in atmospheric KE gives rise to an increase in the PE of the mixed layer. However, due to surface friction (energy dissipation) only a proportion of the KE is available to the mixed layer.
(b) For H < 0, on the other hand, an increase in the PE of the mixed layer gives rise to an increase in atmospheric KE. Here the surface heat transfer from the ocean to the atmosphere causes a convective instability, which is essentially the mechanism through which TCs are initiated.
Phase (b) is the spin-up phase, and phase (a) is the spin-down phase of a loop which characterizes the thermodynamical equilibrium of the coupled air-sea boundary layer, and since the occurrences of the two phases are separated in time and space, also the consequential atmospheric dynamics. Many studies have been reported in the literature of the spin-up phase (b), however, phase (a), has only recently received much attention, stimulated by observations of the reduction in drag coefficient in wind profiles at very high wind speeds (Powell et al. 2003).
As pointed out above, this loop only becomes important as |H| becomes large. This can be established in an alternative manner by considering the rate of change of KE with time, which from (8), is, dKE/dt = —H/AdPE/dt
and eliminating the rate of change of PE with time using (7a) and substituting for A, which yields the development equation, where t = 1/[2HdT/dt] is a time constant for the growth or decay of the atmospheric KE. In phase (b) t > 0, and in phase (a) t < 0. Thus the time constant for development and decay is inversely proportional to H.
For example, in a region where the rate of change of temperature during the growth phase is -0.2°C/day (which on assuming that pw m and the specific heat of water at constant pressure, Cp = 4 103 J kg—1K—1 corresponds with a heat loss per unit area of 200 Wm—2 ) and H = —2.5 K—1 we find that t =1 day indicating an explosive growth process.
In summary, the two phases (a) and (b) are necessary to maintain the statistical equilibrium in the coupled air-sea boundary layer. They are only significant however when |H| is large, which occurs in the tropics. Here, rapid growth rates can occur in regions of positive SST anomaly, where the heat loss from the mixed layer is also large. In the subtropics H « 0, and the efficiency of the energy conversion between the PE of the mixed layer and the atmospheric KE is low, and hence the atmospheric dynamics are mainly controlled by other processes which are essentially independent of the mixed layer, i.e. baroclinic instability.
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