Vertically Sheared Horizontal Flow-Forming Instability in Stratified Turbulence: Linear Stability Analysis using the Analytical Approach to Statistical State Dynamics

Joseph G Fitzgerald, Brian F Farrell

Published 2018-03-02Version 1

Vertically banded zonal jets are frequently observed in weakly or non-rotating stratified turbulence, with the quasi-biennial oscillation in the equatorial stratosphere and the ocean's equatorial deep jets being two examples. Explaining the formation of jets in stratified turbulence is a fundamental problem in geophysical fluid dynamics. Statistical state dynamics (SSD) provides powerful methods for analyzing turbulent systems exhibiting emergent organization, such as banded jets. In SSD, dynamical equations are written directly for the evolution of the turbulent statistics, enabling direct analysis of the statistical interactions between incoherent turbulence and coherent large-scale structure that underlie jet formation. A second-order closure of SSD, known as S3T, has previously been applied to show that meridionally banded jets emerge in barotropic $\beta$-plane turbulence via a statistical instability sometimes called zonostrophic instability (ZI). Two-dimensional (2D) Boussinesq turbulence provides a simple model of non-rotating stratified turbulence analogous to the $\beta$-plane model of planetary turbulence. Jets known as vertically sheared horizontal flows (VSHFs) often emerge in simulations of Boussinesq turbulence, but their dynamics is not yet clearly understood. In this work the S3T analysis of ZI is extended to study VSHF emergence in 2D Boussinesq turbulence using the analytical approach to S3T. VSHFs are shown to form via an instability associated with negative eddy viscosity that is the analog in stratified turbulence of ZI in $\beta$-plane turbulence. This instability is shown to be strikingly similar to ZI, suggesting that jet emergence in both geostrophic and non-rotating stratified turbulence may be conceptualized as instances of the same generic phenomenon.