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

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)similar to Jk(-1), E(k -> 0)similar to Lk, and E(k -> 0)similar to Ik(3), where J, L, and I are various integral moments of the two-point vorticity correlation <omega omega(')>. 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 < u(x)(2)u(x)(')>similar to 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 similar to Jk(-1) turbulence, while a random sea of dipoles yields E similar to Lk. Finally, we observe that the integral scale in two-dimensional turbulence grows approximately as l similar to root t in all three cases, i.e., E similar to Jk(-1), E similar to Lk, and E similar to Ik(3). 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 < u(x)(2)u(x)(')>similar to 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 k(5) in the low-k expansion E similar to Ik(3)+Mk(5)+O(k(7)). This suggests that if SWQG turbulence is started with a spectrum steeper than E similar to k(7), then it will revert to E similar to k(7), whereas a spectrum shallower than E similar to k(7) will be invariant at low k. All of these predictions are confirmed by direct numerical simulation. (C) 2008 American Institute of Physics.