Integral invariants of two-dimensional and quasigeostrophic shallow-water turbulence

Abstract

We use direct numerical simulations to investigate the low-wavenumber behavior of (i) two-dimensional turbulence and (ii) shallow-water, quasigeostrophic (SWQG) turbulence. For two-dimensional turbulence there are three canonical cases: E(k→0)∼Jk−1, E(k→0)∼Lk, and E(k→0)∼Ik3, where J, L, and I are various integral moments of the two-point vorticity correlation ⟨ωω′⟩. Our simulations confirm that in line with earlier theoretical predictions, J and L are invariants, but that I is time dependent. This, in turn, is consistent with the idea that the triple correlations in two-dimensional turbulence fall off as ⟨ux2ux′⟩∼r−3 at large separation, r, something that we confirm directly in our simulations. We also confirm that a random sea of monopoles leads to E∼Jk−1 turbulence, while a random sea of dipoles yields E∼Lk. Finally, we observe that the integral scale in two-dimensional turbulence grows approximately as ℓ∼t in all three cases, i.e., E∼Jk−1, E∼Lk, and E∼Ik3. The earlier theoretical work is extended to SWQG turbulence. In particular we note that in contrast to two-dimensional turbulence, there are unlikely to be any long-range triple correlations of the form ⟨ux2ux′⟩∼r−3. If this is indeed the case, then Loitsyansky’s integral I is an invariant of SWQG turbulence, as is M, the prefactor in front of k5 in the low-k expansion E∼Ik3+Mk5+O(k7). This suggests that if SWQG turbulence is started with a spectrum steeper than E∼k7, then it will revert to E∼k7, whereas a spectrum shallower than E∼k7 will be invariant at low k. All of these predictions are confirmed by direct numerical simulation.