Statistical state dynamics analysis of buoyancy layer formation via the Phillips mechanism in two-dimensional stratified turbulence

Abstract

Horizontal density layers are commonly observed in stratified turbulence. Recent work (e.g. Taylor & Zhou, J. Fluid Mech., vol. 823, 2017, R5) has reinvigorated interest in the Phillips instability (PI), by which density layers form via negative diffusion if the turbulent buoyancy flux weakens as stratification increases. Theoretical understanding of PI is incomplete, in part because it remains unclear whether and by what mechanism the flux-gradient relationship for a given example of turbulence has the required negative-diffusion property. Furthermore, the difficulty of analysing the flux-gradient relation in evolving turbulence obscures the operating mechanism when layering is observed. These considerations motivate the search for an example of PI that can be analysed clearly. Here PI is shown to occur in two-dimensional Boussinesq sheared stratified turbulence maintained by stochastic excitation. PI is analysed using the second-order S3T closure of statistical state dynamics, in which the dynamics is written directly for statistical variables of the turbulence. The predictions of S3T are verified using nonlinear simulations. This analysis provides theoretical underpinning of PI based on the fundamental equations of motion that complements previous analyses based on phenomenological models of turbulence.

Exploiting self-organized criticality in strongly stratified turbulence

“…The advantage of this approach is that there is a clear protocol for discretizing this system in both space and time. In fact, the resulting equations are identical to those that would be obtained via an ad hoc QL approximation of the (2-D) Boussinesq equations, in which flow fields are decomposed into a horizontal (‘streamwise’) average plus a fluctuation about that mean (Fitzgerald & Farrell 2018, 2019). Regarding the spatial discretization, owing to the QL structure, any set of horizontal Fourier modes can be included.…”

Development and long-term evolution of density staircases in stirred stratified turbulence

“…To this end, we first apply an averaging process to equations ( 12)-( 13), and then parameterise any terms involving products of fluctuations in terms of mean quantities. Taking the horizontal average of (13) and applying (16) gives…”

“…The argument advanced above is based on linear stability considerations and hence provides information on the initial evolution to a layered state; when the perturbations attain a sufficiently large amplitude, nonlinear dynamics will come into play. The Phillips effect has been demonstrated to lead to layering in two-dimensional stratified Boussinesq turbulence [16]. In the diffusive regime of double diffusive convection (as found in Polar oceans), the Phillips effect acting on one component of density can lead to layering without the need for double diffusive effects [17].…”

Linear Non-Modal Growth of Planar Perturbations in a Layered Couette Flow

“…The second is that the fluxes in stratified turbulence are expected to amplify any fluctuations in a uniform density gradient [25]. This anti-diffusive mechanism leads to an instability of the statistical state dynamics and forms well-mixed layers separated by thin layers of sharp density gradients [26], a result that is verified by numerical simulations of stratified turbulence [27,28]. As these layers are under shear, the TCI might play a role in the rich phenomenology observed with layer merging/splitting, as well as the re-homogenization of the flow [29].…”