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Register a new userCullens Stability Principle and Weak Solutions of the Free-surface Semi-geostrophic Equations
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The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimisation argument originally inspired by Cullens Stability Principle, uses optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of Cullens Stability Principle in a rigorous mathematical framework.
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The semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove existence of solutions of the incompressible semi-geostrophic equations in a fully three-dimensional domain with a free upper boundary condition. We show that, using methods similar to those introduced in the pioneering work of Benamou and Brenier, who analysed the same system but with a rigid boundary condition, we can prove the existence of solutions for the incompressible free boundary problem. The proof is based on optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures given by Ambrosio and Gangbo. We also show how these techniques can be modified to yield the same result also for the compressible version of the system.
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.
For the generalized surface quasi-geostrophic equation $$left{ begin{aligned} & partial_t theta+ucdot abla theta=0, quad text{in } mathbb{R}^2 times (0,T), & u= abla^perp psi, quad psi = (-Delta)^{-s}theta quad text{in } mathbb{R}^2 times (0,T) , end{aligned} right. $$ $0<s<1$, we consider for $kge1$ the problem of finding a family of $k$-vortex solutions $theta_varepsilon(x,t)$ such that as $varepsilonto 0$ $$ theta_varepsilon(x,t) rightharpoonup sum_{j=1}^k m_jdelta(x-xi_j(t)) $$ for suitable trajectories for the vortices $x=xi_j(t)$. We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem $$(-Delta)^sW = (W-1)^gamma_+, quad text{in } mathbb{R}^2, quad 1<gamma < frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in cite{chan_uniqueness_2020}.
In this paper, we construct smooth travelling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues of the Lamb dipoles for the two-dimensional incompressible Euler equation. The solutions are obtained by maximization of the energy over some appropriate classes of admissible functions. We establish the uniqueness of maximizers and compactness of maximizing sequences in our variational setting. Using these facts, we further prove the orbital stability of the circular vortex pairs for the gSQG equation.
The fully compressible semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the so-called geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures.
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