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. The main structure of the proof follows the pioneering work of Benamou and Brenier , who analysed the same system but with a rigid boundary condition. However, there are very significant new elements required in our proof of the existence of solutions for the incompressible free boundary problem. The proof uses 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 analogous result for the compressible version of the system.
Free upper boundary value problems for the semi-geostrophic equations
M.J.P. Cullen, D.K. Gilbert, T. Kuna and B. Pelloni
Department of Mathematics and Statistics,
University of Reading,
Reading RG6 6AX, UK
also Met Office, Fitzroy Road,
Exeter EX1 3PB, UK
July 19, 2019
The fully compressible semi-geostrophic system, posed in a domain of the form , with a bounded subset of the physical space, is the following system of equations:
where denotes the lagrangian derivative operator:
The unknowns in the above equations are , , , , ; we assume , and constant, and indeed we will assume in what follows. We also assume . The physical significance of each variable is given in the Appendix.
This system is obtained as an approximation to the laws of thermodynamics and to the compressible Navier-Stokes equations, the fundamental equations that describe the behaviour of the atmosphere, or more precisely the version obtained when viscosity is neglected, known as the Euler equations. The particular approximation made in the derivation of the semi-geostrophic system is valid on scales where the effects of rotation dominate the flow. In this case, the effect of the Coriolis and of the pressure gradient force are balanced, and equation (1.4) is precisely a formulation of hydrostatic and geostrophic balance. The remaining equations formulate other physical properties: (1.1) is the momentum equation; (1.2) represents the adiabatic assumption; (1.3) is the continuity equation and (1.5) is the equation of state which relates the thermodynamic quantities to each other.
The semi-geostrophic system was first introduced by Eliassen  and then rediscovered by Hoskins . It admits more singular behaviour in the solutions than other reductions with a simpler mathematical structure, such as the quasi-geostrophic system, and for this reason this system been used in particular to describe the formation of atmospheric fronts.
For an accurate representation of the behaviour of large-scale atmospheric flow, one should consider the fully compressible semi-geostrophic equations with variable Coriolis parameter and a free upper boundary condition. The complexity of this problem means that so far results have only been obtained after relaxing one or more of these conditions. We give a brief summary of these results.
In , Benamou and Brenier assumed the fluid to be incompressible, the Coriolis parameter constant and the boundaries rigid. The problem they considered, written in dimensionless scalar form, is posed in a fixed domain and given by
The equations are to be solved subject to appropriate initial conditions, and the rigid boundary conditions
where represents the boundary of and is the outward unit normal to .
Using a change of variables, first introduced by Hoskins in , one derives the so-called dual formulation of the system, that elucidates the Hamiltonian structure of the problem. Indeed, in this formulation, the equations are interpreted as a Monge-Ampère equation coupled with a transport problem, and this elegant interpretation yields the proof of the existence of weak solutions of the system in dual space, based on the groundbreaking work of Brenier .
In , Cullen and Gangbo relaxed the assumption of rigid boundaries assuming a more physically appropriate free boundary condition. However, they made the additional assumption of a constant potential temperature, and thus obtained a 2-D system, known as the semi-geostrophic shallow water system, posed on a fixed two-dimensional domain. After passing to dual variables, they showed existence of weak solutions of the resulting dual problem.
The above results were obtained for the dual space formulation of the equations, which is the setting we also consider in the present paper. However, we mention for completeness more recent results regarding the existence of solutions in the original physical variables. The first step in this direction was taken by Cullen and Feldman, who proved in  the existence of Lagrangian solutions in physical variables, a result that was extended in  to the compressible system. Recently, Ambrosio et al have succedeed in proving existence of solutions for the Eulerian formulation, in cases when there are no boundary effects [3, 4].
In this paper, we extend the results above to prove the existence of dual-space solutions for the incompressible system, in three-dimensional space, in a domain with a free upper boundary. This result is stated in Theorem 3.6 , and is a direct but substantial extension of the results of Cullen and Gangbo. The proof differs from the one given in  also in its use of the approach introduced in , namely it exploits the general theory of Hamiltonian ODEs in spaces of probability measures given in . The strategy of the proof is to show that the Hamiltonian of the system, given by the dual energy, satisfies the necessary conditions to invoke the general theory of , and that its superdifferential coincides precisely with the dual velocity of the flow. This, coupled with the existence of the optimal transport map for the given cost function, yields the desired result. We also sketch the extension of this proof to the compressible case. Namely, by writing the equations in pressure coordinates, we extend the result of , who considered the compressible equations but assumed rigid boundary conditions, to the more physically relevant case of free boundary conditions.
We mention that recently Caffarelli and McCann  have developed extensively a general theory of optimal transport in domains with free boundaries. It would be interesting to verify whether these general results can be used to give an alternative proof of the problem considered here.
The paper is organised as follows:
In Section 2, we summarise the results of Benamou and Brenier on the solution of the incompressible 3-D system in dual space, with rigid boundary conditions. The proof of this result sets the strategy for all generalisations, and we highlight how our approach differs from this.
In Section 3, we consider the same problem but assume a more realistic free boundary condition on the top boundary (the surface of the fluid). We first summarise the results for the 2-D case obtained by Cullen and Gangbo, then give the proof for the 3-D case. This is the main result of this paper.
In Section 4, we extend the results to the compressible system. In view of the fact that, in pressure coordinates, the two problems are formally identical, this extension does not introduce any new element.
In the Appendix, we list various definitions and the notation we use throughout, as well as some general results in the theory of optimal transport and Hamiltonian flows that we appeal to in the proof of our results.
2 The incompressible semi-geostrophic system in a fixed domain
We start by describing the strategy common to proving the existence of solutions, in a particular set of coordinates, in all cases we examine. The original approach is due to Benamou and Brenier .
The geostrophic energy, which is conserved by the flow, is given by
An important physical property of the flow described by the semigeostrophic approximation is summarised in the following fundamental principle.
Principle 2.1 (Cullen’s stability principle).
This was expressed in  as the requirement that states corresponding to critical points of (2.1) with respect to such rearrangements of particles in physical space are states in hydrostatic and geostrophic balance. The evolution of states that are critical points of the energy but not minima cannot be described by the semi-geostrophic approximation .
The significance of Brenier’s work is in the elucidation of the precise mathematical meaning of this minimisation principle, and its mathematical formulation in the framework of convex analysis and optimal transport theory. This machinery can be used after a change of variables, introduced by Hoskins  and motivated by physical considerations. In these variables, the problem is formulated mathematically in Hamiltonian form, and the time evolution of the velocity is expressed explicitly.
Formulation in dual variables
The change to dual coordinates is defined by
Note that (1.7) implies
The energy functional (2.1) is formulated in dual variables as
The geostrophic coordinates are related to Cullen’s stability principle through the so-called geopotential , defined as
One can perform a formal variational computation, with respect to variations of particle position satisfying the incompressibility constraint and that conserve absolute momentum so that . This computation indicates that, for the energy in (2.3) to be stationary, it must hold that , and that the condition for the energy to be minimised is that is positive definite, where is the Hessian. Positive definiteness of implies that is convex, see [11, 16, 20, 24]. Hence the stability principle can be formulated as a convexity principle.
Principle 2.2 (Cullen’s convexity principle).
We can now rephrase Cullen’s stability principle as the requirement that which minimises (2.3) is the optimal map in the transport of to with respect to the cost function given by
Brenier’s polar factorization theorem  ensures the existence of a unique such optimal map, and guarantees that this optimal map, for each fixed time , is of the form with a convex function of the space variable .
with initial condition
where the symplectic matrix is defined by
The incompressibility condition and the boundary condition can then be reformulated as
where is the Jacobian matrix of . Hence is a volume preserving mapping of .
Using (2.13), it is possible to derive an evolution equation for in dual space. Namely, for any ,
where the dual velocity is defined (and automatically divergence-free, by its definition) by
with denoting the Legendre transform of :
Equation (2.15) is the weak formulation of the transport equation
The proof of Benamou and Brenier
In order to advect in time using (2.19), the system is discretised in time. Then is used to advect to the next time step, using the transport equation (2.19). Due to the way in which is constructed, we have that and . The measure remains compactly supported within a ball whose radius depends on time.
To solve the transport equation, one must also use a sequence of regularised problems, with Lipschitz continuous velocity field, that approximates . For the approximating problems, the transport equation is uniquely solvable. Then, using the stability property of polar factorisation, one can show that these approximate solutions converge to solutions of the system (2.19)-(2.22).
Let be an open bounded set such that , where is an open ball of radius centred at the origin. Let be a convex bounded function in satisfying
where and is the set of all measurable functions on such that for any and, for any satisfying , we have weakly in (narrowly if );
for any and any . Moreover,
The original result of  makes the assumption in Theorem 2.1. Lopes Filho and Nussenzveig Lopes  extended this result to . Loeper  extended this result further, proving existence and stability of measure valued solutions. In , Faria et al. have extended the results of  for the incompressible equations to the case of an initial potential density in . Faria has recently done the same for the compressible system as well,  .
In view of these results, we will include the case in our main statements below.
The strategy employed to prove Theorem 2.1 can be adapted to prove existence of weak solutions in dual space for the compressible equations [14, 20]. In this paper, we will prove an analogous result for the case of a free boundary condition, using a modification of the original strategy that does not explicitly require the time discretization argument of , but relies instead on the theory of Hamiltonian ODEs of , summarised in the Appendix. This basic structure of proof was already used in .
3 The incompressible free boundary problem
In this section, we study the problem obtained when the rigid boundary condition (1.8) considered in  is replaced by a more physically relevant free boundary condition. To model this situation, the equations (1.7) are to be solved in , where the domain is time-dependent and represents the region occupied by the fluid at time :
Here is a fixed bounded domain with rigid wall boundary conditions, while is unknown and represents the free boundary.
The incompressibility of the flow can be formulated as the requirement that remains constant for all , where denotes the three-dimensional Lebesgue measure. In what follows, we normalise the measure so that
We denote by the probability measure defined on by
We make no a-priori assumption that is a well defined, single valued function, since in principle the free boundary could develop an overhanging profile. Hence our notation in (3.1) is not well defined. However, we will show that the solution indeed corresponds to a well-defined function, so the abuse of notation in our definition of the domain is ultimately justified.
The flat rigid bottom of the domain is defined by .
The boundary conditions we consider are
where is a prescribed constant; for convenience henceforth we take .
In what follows, we first state the results of , obtained by taking advantage of the additional assumption of constant density. This assumption reduces the dimensionality of the problem, so that the governing equations are transformed to the shallow water system.
We then consider variable density and the incompressible three-dimensional problem, and prove our main result.
3.1 Constant density - the 2-D shallow water equations
When the density is assumed constant, the system (1.7) describing the flow of an incompressible fluid reduces to the two-dimensional semi-geostrophic shallow water equations:
Note that the evolution of the free boundary is now explicitly part of the system of governing equations, which are posed in the fixed domain .
The 2-D geostrophic energy associated with the flow is defined by
The dual system in Lagrangian coordinates, obtained after passing to the dual coordinates , , is given by
The main theorem of  is summarised below.
Let be an open connected set. Let be given, . Assume that , are two probability density functions, such that support, where is an open ball of radius centered at the origin. Assume also that the function can be extended to a convex bounded function in and that , satisfy
3.2 Variable density - the incompressible free boundary problem in 3-D
The energy associated with the flow is the geostrophic energy defined by
By a formal but straightforward calculation, it can be shown that, as expected, this energy integral is conserved in time.
Similarly, a formal argument shows that geostrophic and hydrostatic balance can be characterised as a stationary point of the energy in (3.17) with respect to a particular class of variations, supporting the validity of Cullen’s stability principle also in this case.
Remark 3.3 (Support of the density ).
at least if it is assumed that satisfies the bound (3.18), and that the energy is initially bounded.
We also assume that there exists a constant such that for every admissible ,
This assumption will be justified by our solution procedure.
3.2.1 Dual formulation
In what follows, we assume that is an open bounded set. Indeed, we assume there exists such that
The change of variables to the geostrophic coordinates, for each fixed describing the domain, is defined in this case by
This definition of the mapping , and the bound (3.23), imply that, for all , the geostrophic velocity remains bounded.
We will denote the inverse of by (see Theorem 3.9 below);
We show next that, as in the rigid boundary case, the problem can be formulated as an optimal transport problem, whose solution is given by the gradient of a convex function.
The following definition is the analogue of Definition 2.1.
Remark 3.4 (Support of the potential density ).
We show below that the potential density must satisfy the evolution (3.29), Assuming that at time the initial potential density has compact support in , we can deduce that is contained in a bounded open set , depending on the time interval length , such that , for some with . This follows from a standard fixed-point argument; see, for example, [11, 22].
Define the functional
where is defined in (3.2) and the cost function is given by
Principle 3.1 (Cullen’s stability principle).
Namely, given , a stable solution corresponds to the following minimal value for the energy:
where is an appropriate subset of .
3.2.2 Lagrangian formulation and statement of the main theorem
We formulate the semi-geostrophic system in dual variables in Lagrangian form, in a way entirely analogous to the rigid boundary case. This yields
Here, denotes the Legendre transform of the (convex) function and denotes an appropriate minimisation space, which we define in the next section, see (3.38).
At each fixed time , the unknowns in this system are the fluid profile and the geopotential . We can assume that is a well defined function of , an assumption justified by the result of Lemma 3.7 below.
Given and , it is possible to reconstruct . Moreover, we show in Proposition 3.52 below that the pressure is obtained from the solution of the system through the relation
The system is to be solved, in the weak sense of (2.15), given the following initial conditions
satisfying the compatibility condition
We can now state the main theorem. The proof is presented in section 3.4.
Let and let be an initial potential density with support in , where with and is a bounded open set in . Let be given by (3.27).
Then the system of semi-geostrophic equations in dual variables (3.29)-(3.35) with given conditions (3.34), (3.36) satisfying the compatibility condition (3.37), has a stable weak solution such that , where , , and has compact support.
This solution satisfies:
where is a bounded open domain in containing for all .
3.3 The minimisation problem (3.28)
In the rest of this section, we fix the time and often drop the explicit dependence on it from the equations.
Our aim is to prove existence and uniqueness of a minimiser of the functional given by (3.28). We do not follow the strategy employed for the proof of the analogous result for the 2-dimensional problem. Indeed, in our case it does not seem straightforward to prove that the energy functional is strictly convex with respect to . To prove uniqueness of the minimiser, we will consider the Monge-Kantorovich formulation of the problem, following what done in  for the more difficult case of a forced axisymmetric flow.
To be able to prove that the minimisation problem (3.28) admits a solution, we first consider what conditions the problem imposes on the minimisation space .
We start by showing that, for every fixed value of , the minimiser has to correspond to a well defined, single-valued function .
The minimiser of (3.28) is given by a corresponding to with .
Suppose that is multi-valued and define the corresponding domain as . Define . Choose a single valued function such that , and transport map such that
where . The existence of such a map is guaranteed by standard optimal transport results. We choose in such a way that can be expressed as where for all . Let and let denote the optimal map in the transport of to with cost function (3.27). Then, since and is negative, we have