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

Abstract

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 ⟨ψ2⟩ and of the Fourier mode ψ̂(k,t) for small wave number k=∣k∣. Upper bounds for ⟨ψ2⟩ and ∣ψ̂(k,t)∣2 are derived. The growth of ⟨ψ2⟩ 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 ∣ψ̂(k,t)∣2≤ck2t2 and ∣ψ̂(k,t)∣2≤c′t2, 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, k5 and k3. The latter is consistent with previous statistical predictions and numerical results.