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

TL;DR

The authors define an energy-minimising background state for atmospheric MLM labels by formulating the problem as optimal transport between a latitude–pressure source measure and a mass distribution in $(Z,\theta)$ space. They prove existence, uniqueness, and stability of the MLM state under broad cost assumptions, and establish a duality framework linking the primal MLM minimisation to a Kantorovich potential via OT theory. A key contribution is reducing the free-surface MLM problem to a standard OT problem with an extended source/target, and showing a semi-discrete path to numerical computation through Laguerre-cell structures when the target is discrete. This work provides a rigorous, mathematically grounded background state definition suitable for quantifying disturbances and atmospheric instability, with practical implications for numerics and SG-type models.

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 parcels, thus avoiding the need to calculate trajectories explicitly. Methven and Berrisford further defined a zonally symmetric MLM state for global atmospheric flow in terms of mass in zonal angular momentum ($z$) and potential temperature ($θ$) coordinates. We prove that for any snapshot of an atmospheric flow in a single hemisphere, there exists a unique energy-minimising MLM state in geophysical coordinates (latitude and pressure). 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 $(z, θ)$. We show that this problem reduces to an optimal transport problem with a known source measure, which has a numerically feasible discretisation. Additionally, our results hold for a large class of cost functions, and generalise analogous results on free surface variants of the semi-geostrophic equations.

Figures (2)

• Figure 1: Schematic diagram of an MLM state at a snapshot in time. The measure ${\nu}$ represents the distribution of mass in the space of zonal angular momentum $z$ and potential temperature $\theta$. It is assumed to have support contained in a compact set $Y\subset (0,+\infty)^2$. The source space $X=[\varepsilon_0,1-\varepsilon_1]\times[0,+\infty)$ has coordinates $s=\sin(\phi)$, where $\phi$ is latitude, and pressure $p$. The constants $\varepsilon_0,\, \varepsilon_1>0$ exclude the pole and the equator. The surface pressure $\overline{p}$ is a function of $s$ and is an unknown of the problem. It determines the measure ${\mu}_{\bm}$ as the restriction of the Lebesgue measure to the set $X_{\overline{p}}$. Zonal angular momentum $Z$ and potential temperature $\Theta$ are unknown scalar functions of $(s,p)$. The triple $(\bm,Z,\Theta)$ is an MLM state for ${\nu}$ if $(Z,\Theta)$ is an admissible transport map from ${\mu}_{\bm}$ to ${\nu}$.
• Figure 2: Reduction of Problem \ref{['prob:primal']} to the optimal transport problem between the 'extended' source measure ${\mu}_{\mathrm{ext}} = \mathcal{L}^{d} \,\raisebox{-.127ex}{\reflectbox{origin=br]{-90}{$\lnot$}}}\, (B\times [0,P])$ and the 'extended' target measure ${\nu}_{\mathrm{ext}}=\nu + (P-1)\delta_{y_{\mathrm{ext}}}$. Here, ${\mu}_{\mathrm{ext}}=\mathcal{L}^{d} \,\raisebox{-.127ex}{\reflectbox{origin=br]{-90}{$\lnot$}}}\, (B\times [0,P])$, where $B$ is a compact subset of $\mathbb{R}^{d-1}$ and the constant $P>0$ is an upper bound on the minimiser ${\bm}$ of Problem \ref{['prob:primal']}. The measure ${\nu}$ is represented by empty circles. The point $y_{\mathrm{ext}}$ is represented by a filled circle, and is outside the support of ${\nu}$. The cost function is extended to $y_{\mathrm{ext}}$ by zero. The optimal transport map $T_{\mathrm{ext}}$ sends the hatched cell to $y_{\mathrm{ext}}$, the lower boundary of this cell is the graph of $\bm$, and the restriction of $T_{\mathrm{ext}}$ to $X_\bm$ is the optimal transport map from ${\mu}_\bm$ to ${\nu}$.

Theorems & Definitions (37)

• Definition 1.1: MLM state, c.f. Figure \ref{['fig:MLM_schematic']}
• Definition 1.2: Background state cost function
• Remark 1.3
• Definition 2.2: Primal functional
• Remark 2.3
• Lemma 3.1: Weak duality
• proof
• Definition 3.2: Dual functional
• Definition 3.3
• Proposition 3.4