Understanding geostrophic turbulence in a hierarchy of models

Abstract

This thesis is concerned with turbulent motions on the very large scale (on the order of 100 to 1000 km) in the atmosphere and ocean. These large-scale motions are called geostrophic turbulence as they are fundamentally influenced by planetary rotation. However, the full Navier-Stokes equations, which include all aspects of turbulence, are often too complex to yield a meaningful understanding. We examine three idealized models with increasing levels of complexity.

We start with a forced-dissipated, two-dimensional fluid on a $\beta$-plane. The flow is approximated as a single layer and is stirred stochastically, while the spherical geometry is replaced with a tangible plane. Due to the $\beta$-effect, the eddies become anisotropic at a threshold scale called the turbulence-wave crossover scale. The traditional phenomenology assumes the forcing scale to be much smaller than the crossover scale so it has no influence. However, when the forcing scale is comparable to or even larger than the crossover scale, we find the forcing scale strongly influences jet formation. Numerical simulations reveal a new turbulent regime where the forcing scale is larger than the crossover scale, such that eddy/eddy interactions are negligible while eddy/mean-flow interactions dominate the nonlinear energy and enstrophy transfers. We update the current theoretical understanding by proposing a more general formulation of crossover scale and eddy diffusivity.

These results are applied to a simple climate model -- the two-layer, quasi-geostrophic model. We obtain closure theories of eddy heat flux, which crucially depend on two nondimensional numbers: the criticality and the dimensionless friction. We further investigate the role of these two numbers in an idealized dry general circulation model (GCM) that closely resembles the real atmosphere. As the surface friction approaches zero, the dry GCM displays a novel way to equilibrate via an energy-recycling mechanism: eddy kinetic energy is converted back into potential energy by circulations on the largest scale and is then absorbed by radiative damping. When the criticality varies around 1, we explain the change in the horizontal and vertical scales of the eddies with the linear instability theory; and the flow stays weakly nonlinear.