Advection-Diffusion by a Single Traveling Wave

A. H. Sobel and G. R. Flierl

The domain is size pi in y, 2*pi in x. Psi=U=1, k=0.01. Shown is total flux, advective plus diffusive, for various values of c. All calculations done by full spectral code at resolution 64x64 wave numbers. The flux is uniform across the channel, apart from variations whose extremes differ by less than 3 percent of the mean for the low phase speeds, but up to 10 percent of the mean for the highest phase speeds. So this should give some idea of the accuracy of the numerics.

The result at c=0 can't be shown on this semilog plot, but it differs by less than 1 percent from the value for c=0.01; i.e. the curve becomes totally flat for c less than 0.01. Therefore we clearly have a low phase speed regime and a high phase speed regime, separated by a sharp transition in the range of c values 0.05-1.

What I find interesting about this is that when the transition starts to occur, the parameter delta in Flierl and Dewar is already less than O(1). So it seems that the regime delta O(1) or greater is basically the same as delta=infinity, i.e. stationary Rayleigh-Benard convection. Then there is the large c regime, but the transition to that regime occurs when delta is small already, so it can basically be explained by Flierl & Dewar's analysis, i.e. the transition is due to changes in their parameter epsilon, rather than delta. However, the transition occurs between epsilon values of 0.1-1, a regime for which results were not shown in Flierl & Dewar (though if I'm not mistaken their theory is valid there). Of course implicit in all these statements is that the Peclet number remains fixed at some large value, here 100*pi. For smaller Pe things would probably change, but that regime seems less interesting.