Balanced dynamics and baroclinic instability: Theory and application to ocean turbulence

The present thesis is concerned with large-scale turbulence in the ocean, and in particular with the problem of what determines the scale and amplitude of mesoscale eddies in the ocean—those eddies with a scale of $O$(100km). While the turbulent roots of atmospheric storms are well understood, an equivalent understanding of the equivalent oceanic problem is forthcoming.; We consider first the associated linear problem of baroclinic instability. Baroclinic instability is the mechanism whereby small perturbations of a putatively steady flow may grow, ultimately to finite amplitude, to form mid-latitude storms in the atmosphere and mesoscale eddies in the ocean. The full Navier-Stokes equations, even with the hydrostatic approximation, are too complex to yield a meaningful physical picture of the instability process, and we examine the properties of four approximate models: the quasi-geostrophic equations, the planetary geostrophic equations, the so-called geostrophic potential vorticity model and a model based on approximations made by way of Hamiltonian dynamics. The first two are appropriate to single parameter regimes, while the latter two span multiple regimes, and in fact include the first two as limits. Numerical results of the linear analysis are compared to those from an un-approximated model, the isopycnal primitive equations. We find that the extended regime models successfully represent the linear properties in all relevant regimes, and that the growth rates on planetary scale occur at much smaller zonal scale than has been previously predicted.; In the second part of the thesis the full oceanic turbulence problem is considered directly through a combination of scaling theory and numerical simulation. The work is motivated in part by satellite observations which indicate that scales of turbulent motion are correlated with the first radius of deformation, a parameter set by the local density stratification, whereas geostrophic turbulence theory predicts a scale which is by eddy energy and the gradient of planetary vorticity (the β effect). A scaling theory which explains the observed correlation is presented. To the extent possible, the theory is tested numerically via simulations made with a fully non-linear, stratified quasi-geostrophic model. The non-uniformity of the oceanic stratification is found to play a key role. As well as explaining the observed scales of oceanic variability, the theory and simulations also predict that the barotropic (vertically averaged) scales of variability will be much larger, and determined by the β effect. This is a measurable hypothesis.