Semigeostrophic System Research Articles

On the convexity condition for the Semi-Geostrophic system

We show that conservative distributional solutions to the Semi-Geostrophic systems in a rigid domain are in some well-defined sense critical points of a time-shifted energy functional involving measure-preserving rearrangements of the absolute density and momentum, which arise as one-parameter flow maps of continuously differentiable, compactly supported divergence free vector fields. We also show directly, with no recourse to Monge-Kantorovich theory, that the convexity requirement on the modified pressure potentials arises naturally if these critical points are local minimizers of said energy functional for any admissible vector field. The obligatory connection with the Monge-Kantorovich theory is addressed briefly.

From Euler to the semi-geostrophic system: convergence under uniform convexity

From Euler to the semi-geostrophic system: convergence under uniform convexity

The semi-geostrophic system: weak-strong uniqueness under uniform convexity

We prove a weak-strong uniqueness result for the semi-geostrophic system with constant Coriolis force. The main assumptions on the strong solution are the boundedness of the velocity field as well as the uniform convexity of the Legendre-Fenchel transform of the modified pressure. We give several examples where our results apply, including some classical solutions on the 2-dimensional torus, and the “stationary” solutions for 3DSG (for which the total wind velocity is zero but the pressure may be time-dependent).

Classical Solutions to Semi-geostrophic System with Variable Coriolis Parameter

We prove short time existence and uniqueness of smooth solutions ( in $C^{k+2,\alpha}$ with $k\geq 2$) to the 2-D semi-geostrophic system and semi-geostrophic shallow water system with variable Coriolis parameter $f$ and periodic boundary conditions, under the natural convexity condition on the initial data. The dual space used in analysis of the semi-geostrophic system with constant $f$ does not exist for the variable Coriolis parameter case, and we develop a time-stepping procedure to overcome this difficulty.

Semigeostrophic equations in physical space with free upper boundary

We define Lagrangian solutions in physical space for 3-d incompressible semigeostrophic system with free upper boundary under various conditions for initial data, then prove their existence via the minimization with respect to a geostrophic functional, generalizing the results of Cullen and Feldman (J Math Anal 37(5): 1371–1395, 2006) and Feldman and Tudorascu (Arch Ration Mech Anal 218(1): 527–551 2015) to the situation of free upper boundary. As a byproduct of our proof, we obtain the existence of measure-valued dual space solutions when the initial measure $$\nu _0\in \mathcal {P}_2(\mathbb {R}^3)$$ and is supported on $$\{-\frac{1}{\delta }\le y_3\le -\delta \}$$ .

On the Semi-geostrophic System in Physical Space with General Initial Data

In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic system in physical space is introduced. This is shown to be consistent with previous notions, generalizing them. A weak stability result is obtained first, followed by a general existence result whose proof employs said stability and approximating solutions with regular initial data. The renormalization property ensures the return from physical to dual space; as consequences we get conservation of Hamiltonian energy and some weak time-regularity of solutions.

Solutions of the Fully Compressible Semi-Geostrophic System

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.

On the existence and weak stability of solutions to the compressible semigeostrophic equations

In this article, existence and weak stability of solutions to the compressible semigeostrophic equations are studied both in physical and in dual spaces. We present a proof of the existence of a weak solution to these equations in dual space assuming initial data in L1(R3), extending previous results (Cullen and Maroofi (2003) [8]). An extension to the concept of Lagrangian solutions due to Cullen and Feldman (2006) [6] is presented to the compressible semigeostrophic system, and it is shown the existence of Lagrangian solutions in physical space, with initial potential vorticities in L1, as well a weak stability result with respect to perturbations of the potential vorticity in L1.

On Lagrangian Solutions for the Semi-geostrophic System with Singular Initial Data

We show that weak (Eulerian) solutions for the semi-geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in dual space. However, such solutions are physically relevant to the model. Thus, we discuss a natural generalization of weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We prove existence of such solutions in the case of discrete measures in dual space. We also prove that weak Lagrangian solutions in physical space determine solutions in the dual space. This implies conservation of geostrophic energy along the Lagrangian trajectories in the physical space.

Weak stability of Lagrangian solutions to the semigeostrophic equations

In (Cullen and Feldman 2006 SIAM J. Math. Anal. 37 137–95), Cullen and Feldman proved the existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in Lp, p > 1. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial potential vorticities in L1 converges strongly in Lq, q < ∞, to a Lagrangian solution, in particular extending the existence result of Cullen and Feldman to the case p = 1. We also present a counterexample for Lagrangian solutions corresponding to a sequence of initial potential vorticities converging in . The analytical tools used include techniques from optimal transportation, Ambrosio's results on transport by BV vector fields and Orlicz spaces.

The Radial Solutions of Monge-Ampère Equations and the Semi-geostrophic System

Abstract We first obtain an (optimal) regularity condition for entire radial solutions of Monge- Ampère equations and then study the existence of radial solutions of the Semi-geostrophic system which is coupled by a Monge-Ampère and a transport equation.

Existence of a weak solution for the semigeostrophic equation with integrable initial data

In this article we present a proof of existence of a weak solution for the semigeostrophic system of equations, formulated as an active scalar transport equation with Monge-Ampère coupling, with initial data in . This is an extension of a 1998 result due to Benamou and Brenier, who proved existence with initial data in .

Eady Wave with Thermal Forcing.

A kind of thermal forcing (negative forcing, i.e., cooling) is introduced into Eady waves in the two-dimensional semigeostrophic system. As a result of the forcing, the potential vorticity is Newtonian-damped to a positive constant value. In the presence of the forcing, Eady waves with wave numbers in some range are destabilized (i.e., the growth rate is augmented compared with the adiabatic case) in the far future. However, in the early period up to about twice [t] (the temporal scale of the considered phenomena), the stabilizing effect (i.e., diminishing growth rate) is dominant. The destabilizing effect clearly emerges only after three to four times [t].

Consequences of the Geostrophic Momentum Approximation on Barotropic Instability

Abstract A subsynoptic instability with dominant barotropic signature has been found to be a possible mechanism for the development of frontal waves. Linear semigeostrophic calculations of the growth rate of waves developing along a front exhibiting a low-level potential vorticity band are here compared with the results of similar experiments made with a primitive equation model. The growth rates of the subsynoptic instability are significantly underestimated by the semigeostrophic system, a result which is the opposite of what has been found by previous semigeostrophic–primitive equation comparisons in the case of baroclinic instability calculations. Analytical solutions of the Rayleigh model of pure barotropic instability confirm this behavior of the geostrophic momentum approximation (GMA) in the case of barotropic instability. The inaccuracy of the equations with GMA is very sensitive to the basic wind shear amplitude. Such sensitivity seems to be directly linked with the definition of the semigeostro...

Baroclinic Instability in an Environment of Small Stability to Slantwise Moist Convection. Part I: Two-Dimensional Models

Abstract In the semigeostrophic system, the growth rate of baroclinic waves varies with the inverse square root of the potential vorticity, which acts as the effective static stability. Recent observations in the ascent regions of middle latitude cyclones show that the effective potential vorticity for saturated air is very near zero. In this paper we examine the structure and rate of growth of baroclinic cyclones when the effective potential vorticity is small for upward (saturated) displacements but large in regions of descent. Analytic solutions for two-dimensional disturbances in a two-layer semigeostrophic model and numerical simulations using a multilevel semigeostrophic model show that when the effective potential vorticity is small in regions of upward motion, growth rates are modestly increased and the region of ascent intensifies and collapses onto a thin ascending sheet. In the limit of zero moist potential vorticity the fastest growing wave has a finite growth rate which is about 2.5 times the...

Lee Cyclogenesis. Part I: Analytic Studies

Abstract The growth of synoptic scale cyclones imbedded in a baroclinically unstable zonal flow over a long straight mountain range is investigated. Two different analytical models of the phenomenon are used. The first model uses the linearized quasi-geostrophic equations. It allows a simple superposition of a steady state mountain forced solution and a transient Eady wave. There is no dynamic interaction between the two solutions, but the time evolution of the combined solution reproduces many characteristics of a disturbance passing over the Rocky Mountains. The semigeostrophic equations are used in the second model. These equations allow a linear solution in transform space, but the transformation of the solution to physical space is nonlinear. This allows an interaction between the mountain forced and transient solutions. The minimum pressure developed by the semigeostrophic system is the same as that of the quasi-geostrophic system. However, the shape of the wave is distorted. This effect is caused b...

Semigeostrophic Disturbances in a Stratified Shear Flow over a Finite-Amplitude Ridge

Abstract Steady-state, two-dimensional disturbances forced by flow over a finite-amplitude ridge are considered. The model represents an extension of the one presented by Robinson (1960). This study is based on the semigeostrophic system of equations for uniform potential vorticity flow. The model equations satisfy the Cauchy-Riemann conditions, and solutions for uniform flow over various shaped ridges may be obtained in terms of a complex potential. The novel result is the determination of solutions for disturbances in a zonal current with linear shear. The boundaries are tilled in the cross-stream direction to coincide with basic state potential temperature surfaces. This simplification, which provides isentropic boundaries, permits the solutions for disturbances in a shear flow to be obtained directly from solutions forced by uniform flow over the same ridge. Physical properties of the solutions are presented in terms of three parameters: &epsi/D, r and δ. The amplitude of the ridge is &epsi& and D is ...