On Turbulent Behavior of the Generalized Surface Quasigeostrophic Equations

TL;DR

This study analyzes turbulence in the two-parameter generalized surface quasi-geostrophic (gSQG) equations on a 2D torus, focusing on energy spectra, enstrophy cascades, and dissipation laws. A Kraichnan-type cascade framework yields a spectrum $\mathcal{E}(\kappa) \sim \eta^{2/3} \kappa_0^{\beta/3} \kappa^{2\beta/3-3}$, supported by high-resolution simulations across $(\alpha,\beta)$, with two key dissipation scales $\kappa_\eta$ and $\kappa_\sigma$ marking inertial-to-dissipation transitions and cascade extent. The authors prove that a direct enstrophy cascade and an upper bound on enstrophy dissipation hold under a turbulence-consistent regime, and they relate $\kappa_\eta$ to the Grashof number $G$ via bounds $G^{1/(3\alpha)} \lesssim \kappa_\eta/\kappa_0 \lesssim G^{2/(3\alpha)}$, while also deriving refined bounds under spectral assumptions. Numerical tests reveal that the predicted power laws hold well away from the fully nonlinear region (near $\beta=\alpha+1$) but break down in the NW quadrant, highlighting nonlinear effects beyond the quasilinear regime. Overall, the work extends 2D turbulence theory to a broader gSQG class, linking spectral behavior, enstrophy cascades, and dissipation with rigorous estimates and detailed numerical verification.

Abstract

Turbulent behavior of the two-parameter family of generalized surface quasigeostrophic equations is examined both rigorously and numerically. We adapt a cascade mechanism argument to derive an energy spectrum that scales as $κ^{2β/3-3}$ where $β$ controls the regularity of the velocity ($β=1$ in the special case of the SQG). Direct numerical simulations indicate that this fits better than $κ^{β/3-3}$ which was derived in earlier work. Guided by earlier work on the 2D Navier-Stokes equations, we prove a certain condition implies a direct cascade of enstrophy, as well as an upper bound on the enstrophy dissipation rate, and sharp bounds on a dissipation wavenumber. The dependence of these rigorous results on the two parameters is demonstrated numerically.

Figures (15)

• Figure 1: Left: Time series plot for energy, enstrophy and $\eta$. Right: Physical space plot at time of maximal $\eta$.
• Figure 2: Left: energy spectra for the NSE. Right: compensated spectra for NSE. The values of $\kappa_\eta$ are indicated by vertical lines, $N=16384$.
• Figure 3: Left: energy spectra for the SQG. Right: compensated spectra for SQG. The values of $\kappa_\eta$ are indicated by vertical lines, $N = 32768$.
• Figure 4: Compensated spectrum plots for $\kappa \in [12,150]$ with different compensated slopes Left: slope $3-\beta$ . Middle: slope $3-\frac{2\beta}{3}$ . Right: $3-\frac{\beta}{3}$ as in Pierrehumbert1994.
• Figure 5: Left: energy spectra for the gSQG. Right: compensated energy spectra. The values of $\kappa_\eta$ are indicated by vertical lines, N=32768.

Theorems & Definitions (21)

• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Proposition 5.1
• proof
• Proposition 5.2
• Theorem 6.1