Large-scale dynamics in two-dimensional Euler and surface quasigeostrophic flows

The large-scale dynamics in classical two-dimensional Euler and surface quasigeostrophic flows are studied by examining the evolution of the mean-square stream function <psi(2)> and of the Fourier mode psi(k,t) for small wave number k=parallel to k parallel to. Upper bounds for <psi(2)> and parallel to psi(k,t)parallel to(2) are derived. The growth of <psi(2)> is at most quadratic in time t and nearly quadratic in time for surface quasigeostrophic and Euler flows, respectively. At the modal level, it is found that parallel to psi(k,t)parallel to(2)<= ck(2)t(2) and parallel to psi(k,t)parallel to(2)<= c(')t(2), where c and c(') are constant, for the surface quasigeostrophic and Euler cases, respectively. These bounds imply a steep energy spectrum at small k respectively, k(5) and k(3). The latter is consistent with previous statistical predictions and numerical results. (c) 2006 American Institute of Physics.