Exact instantaneous optimals in the non-geostrophic Eady problem and the detrimental effects of discretization

TL;DR

This paper derives exact formulas for flow configurations that maximize energy growth in the linear Eady problem, analyzes their independence from the Richardson number, and evaluates the impact of different discretization schemes on their accuracy.

Contribution

It provides the first analytical expressions for optimal perturbations in the Eady problem and compares the effects of various discretizations on their growth rates.

Findings

Optimal perturbations are independent of the Richardson number.

Growth rates increase linearly with decreasing horizontal wavelength.

C grid discretizations outperform B grid in accuracy at small scales.

Abstract

We derive exact analytical expressions for flow configurations that optimize the instantaneous growth rate of energy in the linear Eady problem, along with the associated growth rates. These optimal perturbations are relevant linear stability analysis, but, more importantly, they are relevant for understanding the energetics of fully nonlinear baroclinic turbulence. The optimal perturbations and their growth rates are independent of the Richardson number. The growth rates of the optimal perturbations grow linearly as the horizontal wavelength of the perturbation decreases. Perturbation energy growth at large scales is driven by extraction of potential energy from the mean flow, while at small scales it is driven by extraction of kinetic energy from the mean shear. We also analyze the effect of spatial discretization on the optimal perturbations and their growth rates. A second order energy-conserving discretization on the Arakawa B grid generally has too-weak growth rates at small scales and is less accurate than two second order discretizations on the Arakawa C grid. The two C grid discretizations, one that conserves energy and another that conserves both energy and enstrophy, yield very similar optimal perturbation growth rates that are significantly more accurate than the B grid discretization at small scales.