Limit Cycle Mixing

Abstract

Lagrangian equations for momentum and buoyancy are developed for idealized turbulent fluid elements. The resulting formulation of transport can he viewed as a generalization of mixing length and parcel theories of mixing for application to gridded Eulerian models. This formulation of transport recognizes the mean gradients on the scale of the main transporting eddies and avoids problems with existing methods due to parameterization of fluxes in terms of local gradients between adjacent grid levels. The modeled fluid elements develop relative horizontal motions due to mean vertical shear. Shear-produced horizontal kinetic energy is converted to vertical kinetic energy through modeled pressure adjustments. The fluid element is decelerated through nonlinear pressure drag and small scale diffusion with the ambient fluid while vertical motions are constrained by stable stratification. The linearized version of the equations reproduces classical shear instability governed by a critical Richardson number. With nonlinear pressure drag and small scale diffusion, the element motion adjusts to limit cycle conditions which transport heat and momentum. The limit cycle motion varies from a buoyancy oscillation for large Richardson number to a bimodal limit cycle for small Richardson number. Due to momentum transport by pressure fluctuations, the eddy Prandtl number for stable stratification is generally greater than 1 and increases with stability.