Abstract
We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.
Highlights
The incompressible semi-geostrophic equations (SG) model the large-scale dynamics of rotational atmospheric flows
In [4], those authors proved the existence of global-in-time weak solutions of (1) for initial measures which are absolutely continuous with respect to the Lebesgue measure and have compactly-supported L p density for p > 3
We obtain a solution of (1) for an arbitrary compactly-supported initial measure α by generating a sequenceN∈N of well-prepared discrete measures converging to α in the Wasserstein 2-distance, evolving each of these discrete measures according to the corresponding ODE-initial value problem (IVP) (4), and using compactness in the space of continuous measure-valued maps to pass to the limit as N → ∞
Summary
The incompressible semi-geostrophic equations (SG) model the large-scale dynamics of rotational atmospheric flows. In [4], those authors proved the existence of global-in-time weak solutions of (1) for initial measures which are absolutely continuous with respect to the Lebesgue measure and have compactly-supported L p density for p > 3. The main contribution of this paper is an alternative proof the existence of global-in-time weak solutions of (1) for arbitrary compactly-supported initial measures, which uses recently developed techniques from semi-discrete optimal transport to treat the case where the initial measure is discrete Our application of semi-discrete optimal transport to SG illuminates an explicit and intuitive connection between geostrophic coordinates and corresponding flows in the physical domain Ω It gives a constructive way of determining solutions explicitly, and it forms the basis of an effective numerical scheme, as we illustrate in Sect.
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