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

View More View Less
  • 1 Department of Atmospheric, Oceanic & Space Science and Laboratory for Scientific Computation, University of Michigan, Ann Arbor, Michigan
© Get Permissions Rent on DeepDyve
Restricted access

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 = tβ 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 > tβ, 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 = tβ because the front initially forms with a cube root singularity and its spectral coefficients decay at a different rate. Whitham claims that for t > tβ, the characteristic equation has two roots. We show by explicit solution that there are actually three.

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 = tβ 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 > tβ, 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 = tβ because the front initially forms with a cube root singularity and its spectral coefficients decay at a different rate. Whitham claims that for t > tβ, the characteristic equation has two roots. We show by explicit solution that there are actually three.

Save