Conservation laws, fluxes, and symmetries: lessons from a perturbative approach for self-organized turbulence

Abstract

Some turbulent flows self-organize into large-scale structures, rather than breaking up into ever-smaller scales. Underpinning this phenomenon is the existence of two sign-definite quantities which are conserved by the dynamics. Two-dimensional turbulence is a prime example, where large-scale mean flows, termed condensates, spontaneously emerge. We review a perturbative theoretical framework for the statistical description of such inhomogeneous turbulence, offering new perspectives on the role of the two conserved quantities. We illustrate the universal properties of the theory, comparing results from two-dimensional Navier-Stokes to those from the large-scale-quasi-geostrophic equation. These two models are limiting cases of the shallow water quasi-geostrophic equation, the former exhibiting long-range fluid element interactions, while the latter has local interactions. We then demonstrate these theoretical ideas in two new settings: first, in rotating three-dimensional turbulence, where two-dimensional condensates are known to form. Considering jet-type condensates, we derive the mean-flow profile and discuss a surprising symmetry breaking. Second, we vary the Rossby deformation radius in the shallow water quasi-geostrophic equation. We obtain novel domain-spanning condensates in all tested regimes and show that they follow two-dimensional Navier-Stokes for deformation radii above the forcing scale, and the large-scale quasi-geostrophic equation for those below, demonstrating the power of these asymptotic models.

Figures (9)

• Figure 1: Illustration of the vortex and jet condensate configurations from numerical simulations. Shown are snapshots of the velocity field, with color indicating the magnitude, for (a) Vortex condensate 2DNS with friction $(D =D_\alpha)$, (b) Jet condensate of frictionless 2DNS case ($D=D_\nu$); (c) Vortex condensate in LQG, (d) Jet condensate in LQG. Data from simulations: (a) 2DNS(C), (b) 2DNS(D), (c) LQG(D), and (d) LQG(B), with parameters given in Appendix \ref{['sec:simulation_details']}
• Figure 2: Time-averaged small-scale dissipation $\langle {D_Z} \rangle(x)$ of (a,b) enstrophy in 2DNS and (c,d) kinetic energy in LQG, normalized by the spatial mean $\overline{D_Z}$. Insets show the corresponding mean velocity fields for reference. Simulations shown: (a) 2DNS(C), (b) 2DNS(D), (c) LQG(D), and (d) LQG(B).
• Figure 3: Condensate in frictionless 2DNS ($D = D_\nu$). (a) Mean-flow shear rate $U'=\partial_y U(y)$ ; the dashed line corresponds to the closed solution \ref{['eq:jet_nu']}. (b) Mean enstrophy $U'^2$ (blue), Reynolds stress $\langle {uv} \rangle$ and velocity fluctuations $\langle {v^2} \rangle$ (orange) and $\langle {u^2} \rangle$ (green). For clarity only a single jet is show (corresponding to $-1/4<y<1/4$ in (a). Numerical data correspond to simulation 2DNS(D).
• Figure 4: Simulations of the 3D rotating Navier-Stokes equations (3D-RNS) in a jet geometry. (a,b) Mean-shear profile and (c,d) energy transfer to the condensate for various Rossby numbers. In (b) and (d), the data is rescaled with the theoretical predictions for the rms shear-rate and transfer in gome2025wavesgome2025helicity. (e) Mean momentum flux $J_y$. A constant spatial momentum flux is observed at low rotation, producing an energy flux from anticyclonic to cyclonic regions. Simulations shown: 3D-RNS(A) [Ro=0.23], 3D-RNS(B) [Ro=0.023], 3D-RNS(C) [Ro=0.0046].
• Figure 5: Condensates in (b) 2DNS, (c) SWQG at large deformation radius $L_d \sim L$, (d) LQG and (e) SWQG at small deformation radius $L_d \ll l_f \ll L$. Colors show the vorticity normalized by its rms value. (a) Mean vortex-condensate profile in each case. The white dashed circles in (b-e) indicate the regions over which the mean profiles in (a) are computed. The inner cyan circles delimit the vortex center, shown in (a) by a vertical line. The simulations shown are: (b) 2DNS(C), (c) SWQG(A)$[L_d/L= 0.3$, $l_f/L_d=0.2]$, (d) LQG(D), and (e) SWQG(D)$[L_d/L= 0.04$, $l_f/L_d=3.8]$, with parameters given in Appendix \ref{['sec:simulation_details']}.