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Strengthening the Linkages Between the Sciences and the Mathematical Sciences
ATen Case Studies of Math/Science Interactions
1Modeling Weather Systems Using Weakly Nonlinear, Unstable Baroclinic Waves
Department of Physical Geography
Woods Hole Oceanographic Institution
The pioneering work of Jule Charney and Eric Eady in the late 1940s showed how the development of large-scale disturbances in atmospheric flow patterns that were associated with emerging weather systems could be explained as a natural instability of the largely westerly, zonal (west to east) winds in Earth's atmosphere. Using a linear, small-amplitude perturbation theory, they were able to explain the basic energy source of the weather wave in terms of the ability of the perturbations to tap into the potential energy distribution in the initial, basic flow on which the disturbances grew and from which they efficiently extracted energy. The linear theory gave reasonably good predictions for the spatial scale of the instabilities and their initial rate of growth. What was lacking in these models was an explanation of what limited the growth of the disturbances as they drained energy from the basic flow, at what amplitude saturation would occur, and what dynamical state would follow that saturation.
Some years ago, I became interested in developing the theory for weakly nonlinear, unstable baroclinic waves as models of weather systems in the atmosphere and eddies in the ocean. The weakly nonlinear theories existing at the time followed the work of Stuart and Watson (for the Orr-Sommerfeld problem) or the Malkus and Veronis approach (for convection). In each of these cases the threshold for instability is determined by a balance of energy release by the unstable mode against internal energy dissipation. That means that the marginally unstable wave already has to have a structure that extracts energy from the mean state. Therefore, a relatively straightforward perturbation expansion around that starting point could give the equation for the evolution of the amplitude of the disturbance in the form suggested by Landau, i.e.
where s is the linear growth rate and N is the fruit of the perturbation calculation (the Landau constant).
However, in the geophysical problem the threshold for instability is determined by overcoming inviscid (adiabatic) constraints associated with what is called potential vorticity conservation. That means the marginal wave (around which a perturbation expansion is started) neither dissipates nor extracts energy. In the absence of an energy-releasing structure in the marginal wave, it is impossible to calculate the effect of the perturbation on the mean and hence to get at the Landau constant.
Fortunately for me, a colleague at Chicago was deep at work on the problem of resonant interactions of capillary waves, and it occurred to me that the method of multiple time scales he was using would allow me to calculate the phase shifts in the wave required for energy extraction