The Energy Spectrum of Fronts: Time Evolution of Shocks in Burgers‚ Equation

Abstract

Andrews and Hoskins used semigeostrophic theory to argue that the energy spectrum of a front should decay like the ?8/3 power of the wavenumber. They note, however, that their inviscid analysis is restricted to the very moment of breaking; that is, to the instant t = t? when the vorticity first becomes infinite. In this paper, Burgers' equation is used to investigate the postbreaking behavior of fronts. We find that for t > t?, the front rapidly evolves to a jump discontinuity. Combining our analysis with the Eady wave/Burgers? study of Blumen, we find that the energy spectrum is more accurately approximated by the ?8/3 power of the wavenumber, rather than by the k?2 energy spectrum of a discontinuity, for less than two hours after the time of breaking. We also offer two corrections. Cai et al. improve a pseudospectral algorithm by fitting the spectrum of a jump discontinuity. This is not legitimate at t = t? because the front initially forms with a cube root singularity and its spectral coefficients decay at a different rate. Whitham claims that for t > t?, the characteristic equation has two roots. We show by explicit solution that there are actually three.