Exploiting self-organized criticality in strongly stratified turbulence

Abstract

A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal scales, yielding simplified partial differential equations governing the coupled evolution of slow large-scale hydrostatic flows and fast small-scale isotropic instabilities and internal waves. The dynamics captured by the coupled reduced equations is illustrated in the context of two-dimensional strongly stratified Kolmogorov flow. A noteworthy feature of the reduced model is that the fluctuations are constrained to satisfy quasilinear (QL) dynamics about the comparably slowly varying large-scale fields. Crucially, this QL reduction is not invoked as an ad hoc closure approximation, but rather is derived in a physically relevant and mathematically consistent distinguished limit. Further analysis of the resulting slow–fast QL system shows how the amplitude of the fast stratified-shear instabilities is slaved to the slowly evolving mean fields to ensure the marginal stability of the latter. Physically, this marginal stability condition appears to be compatible with recent evidence of self-organized criticality in both observations and simulations of stratified turbulence. Algorithmically, the slaving of the fluctuation fields enables numerical simulations to be time-evolved strictly on the slow time scale of the hydrostatic flow. The reduced equations thus provide a solid mathematical foundation for future studies of three-dimensional strongly stratified turbulence in extreme parameter regimes of geophysical relevance and suggest avenues for new sub-grid-scale parametrizations.

Critical balance and scaling of strongly stratified turbulence at low Prandtl number

“…Many important features of the unity Prandtl-number regime are becoming better understood from a combination of numerical experiments (Waite & Bartello 2004;Brethouwer et al 2007;Riley & Lindborg 2010;Maffioli & Davidson 2016;Lucas, Caulfield & Kerswell 2017;de Bruyn Kops & Riley 2019), observational data Riley & Lindborg 2008;Falder, White & Caulfield 2016;Lefauve & Linden 2022) and theoretical developments (Billant & Chomaz 2001;Lindborg 2006;Chini et al 2022). Strongly stratified turbulence with Pr = O(1) forced at some horizontal length scale leads to emergent vertical scales set by the stratification and exhibits two distinct inertial ranges that transfer energy from the large forcing scales to the small viscous and thermal scales where it is dissipated.…”

On large-scale dynamos with stable stratification and the application to stellar radiative zones

“…We briefly review properties relevant to the dynamo. The turbulent cascade in a stably stratified fluid with energy injection ≈ u 3 rms k f at wavenumber k f and dissipation at viscous wavenumber k ν contains two inertial ranges, one above and one below the Ozmidov wavenumber k O /2π = (N 3 / ) 1/2 where the eddy turnover frequency matches the Brunt-Väisälä frequency (Brethouwer et al 2007;Chini et al 2022). At larger scales with wavenumbers less than k O , the velocity field is highly anisotropic due to the restriction of vertical motions by the stable stratification (Riley & Lelong 2000;Riley & Lindborg 2010).…”

“…The scale separation between k O and k ν is set by the buoyancy Reynolds number k ν = Rb 3/4 k O , where Rb = ReF r 2 . Rb has been found to be the primary control parameter of stably stratified turbulence and needs to be larger than one to avoid the viscosityaffected stratified flow regime where the isotropic inertial range disappears (Billant & Chomaz 2001;Waite & Bartello 2004;Lindborg 2006;Brethouwer et al 2007;Maffioli & Davidson 2016;Chini et al 2022). DNS of strong stably stratified turbulence thus simultaneously requires F r 1 and Rb 1, which is computationally challenging (Bartello & Tobias 2013).…”

“…Such a non-linear dynamo mechanism must also be able to operate in the stably stratificated conditions characteristic of stellar RZs. Stable stratification generally suppresses dynamo action because it imposes a particular form of anisotropy on the velocity field: fluid motions are unrestricted horizontally while vertical motions are rapidly restored (Riley & Lelong 2000;Billant & Chomaz 2001;Lindborg 2006;Brethouwer et al 2007;Chini et al 2022). In the limit of arbitrarily strong stable stratification, the velocity field is approximated by a two component and three dimensional field, which is well known to be unable to support dynamo action (Zeldovich & Ruzmaikin 1980).…”

Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence

“…Chini et al 2014). Recently this asymptotic approach has been successfully extended to the case of a turbulent free shear flows in the presence of a strong stabilizing density stratification (Chini et al 2022); the separation of timescales is ensured by taking the simultaneous limits of small Froude number and large Reynolds number.…”