Self-organisation through layering of $β$-plane like turbulence in plasmas and geophysical fluids

TL;DR

This work analyzes self-organization and staircase formation in β-plane–like turbulence for plasmas and geophysical fluids using PV-conserving models. It contrasts forcing-driven and instability-driven dynamics across standard (CHM, two-layer QG) and modified (GHM, HW) zonal-flow responses, revealing how PV layering manifests differently: forced systems tend to develop elliptic, merging jets, while instability-driven systems produce straight, stationary jets with fixed width and can even suppress ongoing energy injection. A notable finding is a phase transition in the HW system between 2D turbulence and zonal-flow–dominated states, including a hysteresis that signals bistability. The study highlights the critical role of the zonal-flow response in shaping layering, and shows that saturation can occur without large-scale friction, with implications for understanding PV staircase formation in both plasmas and geophysical flows.

Abstract

Staircase formation and layering is studied in simplified, potential vorticity conserving models of plasmas and geophysical fluids, by investigating turbulent self-organisation and nonlinear saturation with different mechanisms of free energy production -- forcing or linear instability -- and with standard or modified zonal flow responses. To this end, staircase formation in both the standard and modified Charney-Hasegawa-Mima equations with stochastic forcing, along with two different simple instability driven models -- one from a plasma and from a geophysical context -- are studied and compared. In these studies, it is observed that $β$-plane turbulence that does not distinguish between zonal and non-zonal perturbations (i.e., standard zonal response) gradually forms large-scale, elliptic zonal structures that merge progressively, regardless of whether it is driven by forcing (though it should be slow enough to allow wave couplings) or by the baroclinic instability, using for example a two-layer model. Conversely, the plasma system, with its modified zonal response, can rapidly form straight, stationary jets of well-defined size, again regardless of the way it is driven: by stochastic forcing or by the dissipative drift instability. Furthermore, the instability-driven plasma system exhibits a phase transition between a zonal flow dominated state and an eddy dominated state. In both states, saturation is possible without large-scale friction.

Figures (6)

• Figure 1: Summary of the various geometries and conventions used in the geophysical and plasma problems.
• Figure 2: Standard CHM simulation with small-scale forcing. (a) Vorticity snapshot $\zeta(x,y)$ at the final time step and the zonal velocity profile $\overline{v}_{x}$ (grey line). (b) Kinetic energy spectrum $E(k)$ (black), time averaged over $7500 < t< 10000$, decomposed into zonal $E_{Z}(k)$ (red) and non-zonal $E_{turb}(k)$ (green) parts. The forcing around $k=20$ (vertical grey line) is in dotted blue. (c) Kinetic energy spectrum at different timesteps, from early stage (blue) to ZFs (yellow). (d) Hovmöller diagram of the zonal velocity profile $\overline{v}_{x}$.
• Figure 3: Two-Layer QG model results for the top layer. (a) Vorticity snapshot at the final time step and zonal velocity profile (grey line). (b) Kinetic energy spectrum $E(k)$ (black) time averaged over $750<t<1000$, decomposed into zonal $E_{Z}(k)$ (red) and non-zonal $E_{turb}(k)$ (green) parts. (c) Kinetic energy spectrum at different timesteps, from early stage (blue) to ZFs (yellow). (d) Hovmöller diagramm of the zonal velocity profile.
• Figure 4: GHM simulation with small-scale forcing. (a) Vorticity snapshot and zonal velocity profile (grey line) at the final time step. (b) Kinetic energy spectrum $E(k)$ (black) time averaged over $400<t<530$, decomposed into zonal $E_{Z}(k)$ (red) and non-zonal $E_{turb}(k)$ (green) parts. The forcing, which is around $k=10$ (vertical grey line), is in dotted blue. (c) Kinetic energy spectrum at different timesteps, from early stage (blue) to steady ZFs (yellow). (d) Hovmöller diagram of the zonal velocity profile.
• Figure 5: HW simulation with $C=1$, $\kappa=1$. (a) Vorticity snapshot and zonal velocity profile (grey line) at the final time step. (b) Kinetic energy spectrum (black), time averaged over $475<t<950$, decomposed into zonal (red) and non-zonal (green) parts. The positive part of the growth rate, which is maximised at $k_{y0}\approx1.3$ (vertical grey line) is in dotted blue; the growth rate becomes negative at small scales owing to viscous dissipation. (c) Kinetic energy spectrum at different timesteps, from the saturation of the linear instability (blue) to the steady ZFs (yellow). (d) Hovmöller diagram of the zonal velocity profile.