On the effect of strain on symmetric instability

In the absence of strain, a strongly baroclinic, geostrophic flow such as a front is unstable to symmetric instability (SI) when its potential vorticity takes the opposite sign of the Coriolis parameter and its Richardson number, Rig, is less than one. SI extracts kinetic energy from vertically sheared geostrophic flows and thus provides a mechanism for the transfer of energy from balanced motions to small-scale turbulence and dissipation.

Such baroclinic flows are often accompanied by a strain field that may be associated with three-dimensional frontal instabilities or with a large-scale confluence. While strain is known to modify baroclinic instability, its effect on SI has yet to be fully explored. A linear fluctuation growth analysis is used to study the energetics and dynamics of SI in the presence of confluence and difluence of varying strengths and for Rig between zero and one. In the analysis the strain rate, geostrophic shear, horizontal density gradient, and stratification are taken to be spatially uniform.

Confluence (difluence) leads to an exponential increase (decrease) of the horizontal density gradient and geostrophic shear with time, driving a thermally direct (indirect) background ageostrophic secondary circulation (ASC) to maintain the thermal-wind balance. A perturbation is added to this ASC taking the form of the fastest growing mode for SI that would develop if no strain had been applied. The analysis reveals that the amplification of the perturbation is suppressed by confluence (while enhanced by difluence) even though the strength of the front increases (decreases). The key to this counterintuitive result is the vertical shear in the background ASC which takes the opposite sign for confluence and difluence. The vertical flux of cross-front momentum associated with the perturbation is upgradient (downgradient) for confluence (difluence) yielding a negative (positive) shear production and a sink (source) of energy for the instability. The suppression of SI by confluence is largest for Rig near one, while the enhancement of SI by difluence is amplified for low Richardson numbers. Both effects are strengthened as the magnitude of the strain increases. The analysis also shows that under difluence the growing instability triggers inertial oscillations and thus provides a mechanism for energy to be transferred from the balanced, geostrophic flow, to unbalanced motions.