Statistical state dynamics based study of turbulent Eady fronts. Part 2. Finite amplitude equilibria

Streamwise roll circulations commonly observed in frontal regions are primary agents of momentum and tracer transport in the planetary boundary layer (PBL) both in the atmosphere and ocean. Traditionally, the formation of the streamwise roll/streak structure (RSS) has been ascribed to symmetric instability (SI). In part 1, we studied RSS formation in the classical Eady front problem using statistical state dynamics (SSD), which allows incorporating the Reynolds stress (RS) torque instability mechanism together with SI in the dynamics underlying RSS formation. We found using SSD theory that the RS torque mechanism acts synergistically with the SI mechanism in forcing symmetric circulations in fronts when Richardson number Ri < 1, and also that the turbulence-mediated RS torque mechanism supports RSS formation in fronts with Ri > 1 for which the SI mechanism does not operate. Although SI theory provides an explanation for initial roll formation, it leaves open the question of RSS equilibration. An advantage of the SSD formulation of RSS dynamics is that it consistently incorporates the equilibration process. In this paper, we extend perturbation analysis of RSS dynamics in the SSD framework to a nonlinear analysis to understand roll formation, equilibration, and maintenance in the turbulent RSS regime.