# Predictions for the final equilibrium state of flows on the sphere

## Date

1999

## Authors

Prieto González, Ricardo, author

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## Abstract

Taking as motivation the experimental evidence, both observational and numerical, that un¬ forced high-Reynolds number flows have a tendency to an equilibrium state with dominant coherent structures, two theories that predict the end-state of the flow are extended to the case of a spherical domain in order to apply them to large-scale meteorological problems. The maximum entropy theory is a statistical mechanics approach that abandons the idea of following the precise changes of the fluid, predicting instead a "macroscopic" state, based on the assumption that the most probable macroscopic state is the one that corresponds to the final equilibrium state of the flow; the specification of the actual values of total kinetic energy, angular momentum and circulation of the flow define the structure of the equilibrium state. The minimum enstrophy theory is based on results of low diffusion numerical experiments, where the values of the domain integrated kinetic energy and circulation are approximately constant over time, while the value of the enstrophy (i.e., one-half the domain integrated squared vorticity) has a considerable decay with time. Using the tools of the calculus of variations, the equilibrium state of the flow is predicted by minimizing the enstrophy while keeping constant the initial value of either total kinetic energy or total angular momentum. A nonlinear barotropic non-divergent numerical model on the sphere is used to perform long time integrations of barotropically unstable initial conditions. The type of flows studied are northern hemisphere stratospheric polar vortices, tropical shear layers and alternating zonal jets, the last being integrated both on a rotating and non-rotating sphere. Predictions of the zonally independent equilibrium state are compared with the zonal average of the direct numerical integration after 100 days of evolution for the stratospheric polar vortex experiments. Maximum entropy theory has good predictive skill for the zonal wind and absolute vorticity profiles, as well as for the statistical distribution of traced air parcels. Minimum enstrophy theory has good skill for the zonal wind and absolute vorticity profiles, but just for the cases where mixing is confined to a polar cap, failing in cases where mixing is global or inside a latitude belt; for the latter case a second version of the minimum enstrophy theory, the two-edges problem, shows considerable improvement compared with the one edge solution. For the tropical shear layer experiments, the minimum enstrophy theory with two-edges and constant energy captures a northward displacement of the easterly wind maximum, as well as the flattening of the absolute vorticity profile in tropical regions, behavior which is consistent with the direct numerical integration. Maximum entropy theory qualitatively captures changes of the flow in the southern hemisphere but shows strong sensitivity to small variations of the scale and strength of the initial condition. Predictions from minimum enstrophy theory with two edges and constant angular momentum, with one edge and constant energy, and with one edge and constant angular momentum, show little skill predicting the end-state; the weak decay of absolute enstrophy observed in the direct numerical integration for these cases is a major factor in the predictive skill of the theory. Two-dimensional predictions of equilibrium states were found using maximum entropy theory for an initial condition with alternating zonal jets. For the rotating sphere, a maximum entropy solution was found which contains a number of zonally elongated coherent structures that resemble the direct numerical integration after 150 days of evolution. The non-rotating sphere case reveals the possibility of having more than one equilibrium state, and that the end-state chosen by the nonlinear evolution might be a linear combination of quasi-orthogonal maximum entropy states.

## Description

Fall 1999.

Also issued as author's dissertation (Ph.D.) -- Colorado State University, 2000.

Also issued as author's dissertation (Ph.D.) -- Colorado State University, 2000.

## Rights Access

## Subject

Atmospheric circulation -- Mathematical models

Equilibrium -- Mathematical models