Linear theory of roll instabilities in the ocean surface layer

Coherent turbulent roll structures have been modeled and observed at wind-forced ocean frontal zones, yet no comprehensive analytical framework fully explains the characteristics of these features. Using linear stability theory, this study characterizes the parameters of roll-structure formation in stably and unstably stratified Ekman layers, and determines if these structures arise principally through shear or convective instabilities. Results show that perturbations to Ekman shear flow are unstable for Reynolds numbers (R) larger than 35, and that roll formation criteria and orientation for R=500 are similar to those modeled by large-eddy simulations (LES). With the addition of unstable thermal stratification, the shear instability is overtaken by a thermal instability for Rayleigh numbers larger than O(10⁷). Our high-resolution results serve as an update to previous linear stability studies with limited computing resolution. Future study will incorporate a mean lateral thermal gradient and baroclinic instability, and thereby assess all potential mechanisms in concert, as well as compare with LES results for frontal zones.