The competition between quadratic and integral invariants in inviscid truncated two-dimensional and quasigeostrophic shallow-water turbulence

Abstract

We investigate the behavior of spectrally truncated simulations of inviscid two-dimensional and quasigeostrophic shallow-water (QGSW) turbulence. Under the assumption that the only conserved quantities are the energy and enstrophy (the total energy and the potential enstrophy in the case of QGSW turbulence), it is possible to use arguments from statistical mechanics to predict an absolute equilibrium form for the (total) energy spectrum E(k)=ak/(k2+b), where a and b are constants. However, other robust integral invariants exist in these systems. In strictly two-dimensional turbulence the low-wavenumber spectrum has the expansion E∼Jk−1+Lk+Ik3+O(k5), where J, L, and I are various integral moments of the two-point vorticity correlation function ⟨ωω′⟩. J and L are invariants, while I is generally time dependent. In the case of QGSW turbulence, the equivalent expansion for the total energy spectrum is Et∼LkR2k−1+ÎkR2k+M̂kR2k3+N̂kR2k5+O(k7), where kR is the Rossby deformation wavenumber and Î, M̂, and N̂ are related to further integral moments of the two-point vorticity correlation function. In this system it is found that Î and M̂ are additional invariants. There must, therefore, be competition at the low-wavenumber end of the spectrum between the developing equilibrium spectrum and the region controlled by the integral invariants. We find that while the equilibrium spectrum comes to dominate an increasing range of wavenumbers, it is always blocked from the region k→0 by the presence of the integral invariants. Furthermore, the rate at which the region occupied by the equilibrium spectrum expands is almost independent of the form of both the equilibrium and low-wavenumber spectra. This can be interpreted as the inverse energy cascade, which carries the equilibrium spectrum to low k, being insensitive to the very high and very low k form of the spectrum.