1 Introduction
Abstract

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 [7], 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 [5]. 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

1 Introduction

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:

(1.1)
(1.2)
(1.3)
(1.4)
(1.5)

where denotes the lagrangian derivative operator:

(1.6)

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 [17] and then rediscovered by Hoskins [21]. 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 [7], 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

(1.7)

The equations are to be solved subject to appropriate initial conditions, and the rigid boundary conditions

(1.8)

where represents the boundary of and is the outward unit normal to .

Using a change of variables, first introduced by Hoskins in [21], 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 [8].

This result was generalised in [15] to prove existence of weak solutions for the 3-dimensional compressible system (1.1)-(1.5), still assuming a fixed boundary and a rigid boundary condition.

In [13], 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 [12] the existence of Lagrangian solutions in physical variables, a result that was extended in [14] 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 [13] also in its use of the approach introduced in [14], namely it exploits the general theory of Hamiltonian ODEs in spaces of probability measures given in [5]. 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 [5], 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 [15], 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 [9] 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 [7].

Let be a fixed bounded domain, and a fixed constant. Consider the system of equations (1.7), with suitable prescribed initial conditions and the rigid boundary conditions given by (1.8).

The geostrophic energy, which is conserved by the flow, is given by

(2.1)

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).

Stable solutions of (1.7)-(1.8) correspond to solutions that, at each fixed time , minimise the energy given by (2.1) with respect to the rearrangements of particles, in physical space, that conserve the absolute momentum and the density .

This was expressed in [24] 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 [11].

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 [21] 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

(2.2)

Note that (1.7) implies

The energy functional (2.1) is formulated in dual variables as

(2.3)

The geostrophic coordinates are related to Cullen’s stability principle through the so-called geopotential , defined as

(2.4)

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).

Minima of the energy (2.1), with respect to variations as in Principle 2.1, correspond to a geopotential , as given by (2.4), which is a convex function of .

We can now express the dual formulation in the language of optimal transport theory, [6, 25].

Definition 2.1.

The potential density associated to the system (1.7) is the push forward of the Lebesgue measure of the domain through the map given by (2.2):

(2.5)

This means that the measure is defined by

and satisfies the change of variable formula

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

(2.6)

Brenier’s polar factorization theorem [8] 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 .

Hence defining as in (2.2) and as in (2.4), we can use the fact that , to rewrite (1.7)-(1.8) as the following system of equations for , :

(2.7)
(2.8)
(2.9)
(2.10)

with initial condition

(2.11)

where the symplectic matrix is defined by

(2.12)

We now write (2.7)-(2.11) in Lagrangian form. We define the Lagrangian flow map corresponding to the velocity , i.e.

and can then rewrite (2.7), (2.9), as first done in [12], in the form

(2.13)

The incompressibility condition and the boundary condition can then be reformulated as

(2.14)

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 ,

(2.15)

where the dual velocity is defined (and automatically divergence-free, by its definition) by

(2.16)

with denoting the Legendre transform of :

(2.17)

Equation (2.15) is the weak formulation of the transport equation

(2.18)

Combining (2.18), (2.16) and the weak formulation of the Monge-Ampère equation (2.14) yields the semi-geostrophic equations in dual variables

(2.19)
(2.20)
(2.21)

where is defined by (2.12) and by (2.17); is the unique optimal transport map of to .

Equation (2.21) expresses the energy minimisation requirement, hence it is a precise mathematical formulation of Cullen’s principle. Equations (2.19)-(2.21) are supplemented with the initial condition

(2.22)

Note that we require that is a given measure with compact support contained in some ball .

The proof of Benamou and Brenier

To prove the existence of weak solutions of the system (2.19)-(2.22), the following strategy was introduced in [7]:

  • Given the compactly supported, absolutely continuous measure at a given fixed time , compute the velocity field from (2.21) and (2.20).

  • 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).

This strategy gives a proof of the main result [7, Theorem 5.1]; our slightly more general statement is taken from [12, Theorem 2.3]:

Theorem 2.1.

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

(2.23)

for some . Then, for , there exist functions on , on such that satisfy (2.19)-(2.21)and the initial condition (2.22) in the weak sense. In addition,

  1. , satisfy

    where and is the set of all measurable functions on such that for any and, for any satisfying , we have weakly in (narrowly if );

  2. where ;

  3. satisfies

    for any and any . Moreover,

  4. , .

Remark 2.2.

The original result of [7] makes the assumption in Theorem 2.1. Lopes Filho and Nussenzveig Lopes [23] extended this result to . Loeper [22] extended this result further, proving existence and stability of measure valued solutions. In [19], Faria et al. have extended the results of [12] 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, [18] .

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 [7], but relies instead on the theory of Hamiltonian ODEs of [5], summarised in the Appendix. This basic structure of proof was already used in [14].

3 The incompressible free boundary problem

In this section, we study the problem obtained when the rigid boundary condition (1.8) considered in [7] 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 :

(3.1)

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

(3.2)

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

(3.3)
(3.4)

where is a prescribed constant; for convenience henceforth we take .

In what follows, we first state the results of [13], 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:

(3.5)
(3.6)
(3.7)
(3.8)

where , , and all equations are to be solved for . The system (3.5)-(3.8) is to be considered with the prescribed initial and boundary conditions

(3.9)

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

(3.10)

The dual system in Lagrangian coordinates, obtained after passing to the dual coordinates , , is given by

(3.11)
(3.12)
(3.13)
(3.14)
(3.15)

The main theorem of [13] is summarised below.

Theorem 3.1.

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

(3.16)

Then, for , there exist functions on , on such that satisfy (3.11)-(3.15) and the initial condition (3.15) in the weak sense. In addition , satisfy the regularity stated in (i)-(iv) of Theorem 2.1.

3.2 Variable density - the incompressible free boundary problem in 3-D

We now consider the incompressible semi-geostrophic system (1.7) in the region given by (3.1), with boundary conditions (3.3)-(3.4).

The energy associated with the flow is the geostrophic energy defined by

(3.17)

By a formal but straightforward calculation, it can be shown that, as expected, this energy integral is conserved in time.

Proposition 3.2.

The system (1.7)-(3.34) conserves the energy integral in (3.17).

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 ).

We can assume that there exists such that the density satisfies

(3.18)

This follows from assuming the bound at time and employing the third of equations (1.7). The full arguments are presented in [20].

Note that the incompressibility condition as expressed by (3.2) and the conservation of energy (3.17) imply that any sufficiently regular which is a solution of the system has to satisfy

(3.19)

at least if it is assumed that satisfies the bound (3.18), and that the energy is initially bounded.

Indeed,

(3.20)

and

(3.21)

We also assume that there exists a constant such that for every admissible ,

(3.22)

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

(3.23)

This bound follows from the bound (3.18) on , and from the fact that can be assumed to remain bounded. The latter is guaranteed by condition (H1), see section 3.4.

The change of variables to the geostrophic coordinates, for each fixed describing the domain, is defined in this case by

where

(3.24)

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.

We use (3.24) to rewrite the energy in (3.17), at fixed time , as the following functional in dual space:

(3.25)

The following definition is the analogue of Definition 2.1.

Definition 3.1.

Given as in (3.2), define the potential density associated with the flow described by (1.7)-(3.4) as the push forward of the measure under the map given by (3.24).

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

(3.26)

where is defined in (3.2) and the cost function is given by

(3.27)
Principle 3.1 (Cullen’s stability principle).

At each fixed time , the pair corresponding to a solution of (1.7) with boundary conditions (3.4) minimises the energy (3.25) amongst all pairs where is given by (3.2) and .

Namely, given , a stable solution corresponds to the following minimal value for the energy:

(3.28)

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

(3.29)
(3.30)
(3.31)
(3.32)

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

(3.33)

The system is to be solved, in the weak sense of (2.15), given the following initial conditions

(3.34)
(3.35)
(3.36)

satisfying the compatibility condition

(3.37)

It is not difficult to show that, formally, (3.29)-(3.35) yields a stable solution of (1.7), see [20]:

Lemma 3.5.

A sufficiently regular solution of (3.29)-(3.35) yields a solution of (1.7) with initial condition (3.34) and boundary conditions (3.3)-(3.4).

We can now state the main theorem. The proof is presented in section 3.4.

Theorem 3.6.

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 [10] 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 .

Lemma 3.7.

The minimiser of (3.28) is given by a corresponding to with .

Proof.

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