A definition of the background state of the atmosphere using optimal transport

Abstract

The dynamics of atmospheric disturbances are often described in terms of displacements of air parcels relative to their locations in a notional background state. Modified Lagrangian Mean (MLM) states have been proposed by M.E.McIntyre using the Lagrangian conserved variables potential vorticity and potential temperature to label air masses in the instantaneous 3-D atmospheric state, and then rearranging air masses sandwiched between potential vorticity contours in potential temperature layers so that they are zonally symmetric. Methven and Berrisford used conservation of Kelvin's circulation to define the distribution of MLM state mass in zonal angular momentum ($z$) and potential temperature ($\theta$) coordinates given 3-D global data. We prove that for any such mass distribution in $(z,\theta )$ there exists a mapping to a unique mass distribution in geophysical coordinates (latitude and pressure) which minimises energy and completely defines the MLM state in a single hemisphere. Since the state is an energy minimum, it is suitable for quantification of finite amplitude disturbances and examining atmospheric instability. This state is obtained by solving a free surface problem, which we frame as the minimisation of an optimal transport cost over a class of source measures. The solution consists of a source measure, encoding surface pressure, and an optimal transport map, connecting the distribution of mass in geophysical coordinates to the known distribution of mass in $(z,\theta )$. We show that this problem reduces to an optimal transport problem with a known source measure, which has a numerically feasible discretisation, and we present a numerical solution using atmospheric reanalysis data. Additionally, our results hold for a large class of cost functions, and generalise analogous results on free surface variants of the semi-geostrophic equations.