  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 turbulence statistics, enabling
direct analysis of the statistical interactions between the incoherent
component of the turbulence and the coherent large-scale structure component
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 referred to as
the zonostrophic instability. Two-dimensional 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 S3T analysis of the
zonostrophic instability is extended to study VSHF emergence in two-dimensional
Boussinesq turbulence using an analytical formulation of S3T amenable to
perturbation stability analysis. VSHFs are shown to form via an instability
that is analogous in stratified turbulence to the zonostrophic instability in
beta-plane turbulence. This instability is shown to be strikingly similar to
the zonostrophic instability, suggesting that jet emergence in both geostrophic
and non-rotating stratified turbulence may be understood as instances of the
same generic phenomenon.
