Smooth Solutions of the Surface Semi-Geostrophic Equations
The semi-geostrophic equations have attracted the attention of the physical and mathematical communities since the work of Hoskins in the 1970s owing to their ability to model the formation of fronts in rotation-dominated flows, and also to their connection with optimal transport theory. In this paper, we study an active scalar equation, whose activity is determined by way of a Neumann-to-Dirichlet map associated to a fully nonlinear second-order Neumann boundary value problem on the infinite strip , that models a semi-geostrophic flow in regime of constant potential vorticity. This system is an expression of an Eulerian semi-geostrophic flow in a co-ordinate system originally due to Hoskins, to which we shall refer as Hoskins’ coordinates. We obtain results on the local-in-time existence and uniqueness of classical solutions of this active scalar equation in Hölder spaces.
Key words and phrases. active scalar equation; surface semi-geostrophic equations; Dirichlet-to-Neumann map; Schauder theory; Monge-Ampère operator.
Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt University, Edinburgh, UK EH14 4AS. Emails: email@example.com, firstname.lastname@example.org
The semi-geostrophic equations (or SG for brevity) constitute a model for the large-scale dynamics of atmospheres and oceans which are dominated by rotational effects. The equations take the form of an active semilinear transport equation in an unknown conservative vector field, and can be considered as a formal vanishing Rossby number limit of the well-known primitive equations; see Section 1.1 for the form of the semi-geostrophic equations in Eulerian coordinates. SG has attracted considerable attention from the mathematical community over the past 20 years as their analysis can be tackled using tools from optimal transport theory and the regularity theory of Alexandrov solutions of the Monge-Ampère equation. In this paper, following Hoskins [Hoskins75], we restrict our attention to incompressible semi-geostrophic flows on an infinite strip in the regime of constant potential vorticity. These are modelled by the following active scalar equation (to which we refer as SSG) on the boundary of the strip in the unknown buoyancy anomaly , namely
where is the -rotated gradient operator on , and is the Neumann-to-Dirichlet map associated to the following time-independent second order fully nonlinear Neumann boundary value problem given by
and is defined pointwise by for all . The reader will notice that the system (1.1) is formally equivalent to the surface quasi-geostrophic model (SQG in short) except that the operator is given by : see [CMT94:SQG] for details.
In this paper, we construct local-in-time smooth solutions of system 1.1 by way of a double fixed point argument in spaces of Hölder continuous functions. The analysis of system (1.2) is tackled, in the regime of small boundary data, by means of classical elliptic theory. Although our main existence result holds only for local-in-time smooth solutions of 1.1, as opposed to global-in-time smooth solutions thereof, it is natural to expect that the dynamics of the system produces discontinuous solutions in finite time. Indeed, in the original widely-cited work of Hoskins and Bretherton [HB72], the authors provide evidence of finite-time singularity generation of the semi-geostrophic equations through a numerical study on the infinite strip .
1.1 Semi-geostrophic dynamics expressed in various coordinate systems
In this and the following section, we provide a brief overview of the state-of-the-art in results on the semi-geostrophic equations. Indeed, the semi-geostrophic equations in Eulerian coordinates, derived in [Eliassen48], in the regime of an incompressible and inviscid flow, comprises the following system
where is the so-called geostrophic velocity field which is that part of the Eulerian velocity field in perfect geostrophic balance, and is the associated ageostrophic velocity field, given by
Moreover, the matrix is given by
The geostrophic velocity and buoyancy anomaly are not independent quantities, but rather are realised as the gradient of a scalar pressure , namely
We refer to (1.3) simply as SG in all that follows.
The construction of any notion of solution (either classical or distributional) of SG expressed in Eulerian coordinates is a difficult problem, and only few results in this direction exist in the literature. By drawing a brief analogy with the theory of water waves, or the free-surface Euler equations (see [lannes:waterwaves]), in the study of SG it is useful to rewrite the governing equations in different coordinate systems, with the hope of constructing some solution of the formally equivalent system therein. Let us now present for the convenience of the reader those versions of SG that have been studied to date and discuss their relationships briefly. Moreover, we provide a mathematical derivation of (1.1) and (1.2), following Hoskins in [Hoskins75], in Section 2.
By introducing the scalar field defined on as
often called the generalised pressure or generalised geopotential, system (1.3) is equivalent to a semilinear transport equation in the unknown conservative vector fields and ,
Here, denotes the identity map on , . Notably, the system (1.4) is not supplemented with an evolution equation for the velocity vector field ; rather, the velocity field must evolve in such a way that the time-dependent vector field remain conservative. One formally equivalent formulation of SG considered in the literature is that in so-called Lagrangian coordinates, namely
where is the Lagrangian flow corresponding to , and the relation between the unknown and is given by
Evidently, the equivalence between the Lagrangian formulation and the previous Eulerian one depends on the regularity of the velocity vector field . One other formally equivalent formulation is expressed in so-called geostrophic coordinates, as studied for instance in the well-known work of Benamou and Brenier [BB98]. Indeed, if we assume that the map is invertible for any , then the scalar field defined as
satisfies a transport equation with a vector field that depends on , namely
This system is also an example of an active scalar equation, where the activity is determined by way of a weak solution of the second boundary-value problem for the Monge-Ampère equation. As is standard, we use the notation to denote the Legendre transform of , while denotes the identity map on . We now discuss the results in the literature regarding the well-posedness of the previous formulations of the system (1.3).
1.2 Brief Review of Existence Results for SG
In [BB98], Benamou and Brenier provided the first result on existence of weak solutions to SG in geostrophic co-ordinates (1.8) in full geostrophic coordinates. The main theorem proved in [BB98] is the following:
The sense in which the quantities above solve the Cauchy problem is defined in Section 5 of [BB98]. The authors perform a time-stepping argument, and solve the Monge-Ampère equation at each time step using the fundamental results of [Brenier91]. Their result is compatible with the independently-derived stability principle of Cullen and Shutts in [CS87]: in this work, the authors affirm that the solutions of SG that are stable (in the sense of inner variation) are those for which the generalised pressure is convex. This result is the first clear connection between the semi-geostrophic equations and the theory of optimal transport. In [CG01], Cullen and Gangbo extend the result by Benamou and Brenier to the shallow water regime, which requires constant potential temperature on a free-surface. In [CM03], Cullen and Maroofi further extend the results from [BB98] and [CG01] to the case of fully compressible semi-geostrophic flow. These results deal with SG in full geostrophic coordinates, as the regularity of the obtained solutions is not sufficient to construct solutions in physical coordinates, either Eulerian or Lagrangian.
In [CF06], Cullen and Feldman make use of the theory by Ambrosio [Ambrosio04:ODE] on the transport equation and ODEs with vector fields of bounded variation, to prove existence of weak solutions to SG in Lagrangian coordinates (1.5), both in domains in and in the regime of shallow water. The main result by Cullen and Feldman is the following.
We invite the reader to consult definition 2.5 in [CF06] for the sense in which is a weak solution of (1.5) and (1.6). It is also worth mentioning the work by Faria, Lopes Filho and Nussenzveig-Lopes [Faria09] in which the authors extend the result in [CF06] to the borderline Lebesgue index case , i.e. the case in which has density in . It is still not clear if one might use these solutions to construct a solution in Eulerian coordinates. Uniqueness of these weak solutions also remains an open question.
The problem of uniqueness of solutions in any context was addressed by Loeper in [Loeper06], in which the author proves existence and stability (in the sense of Shutts and Cullen) of measure-valued solutions to SG in full geostrophic coordinates (1.8) on the torus . In this work, he also studies regularity and uniqueness of smooth solutions, and explores analytical similarities between SG (1.8) and the well-known 2-D incompressible Euler equations in vorticity formulation, namely
The smooth solutions constructed on by Loeper admit the property that is a diffeomorphism, whence smooth solutions in full geostrophic coordinates of (1.8) can be used to construct classical solutions in Eulerian coordinates of (1.4). This is the first result on existence of Eulerian solutions, however the condition of having implies that the vector field cannot lie in for any time . This poses an issue if one is interested in the physical application of the model SG, as such solutions correspond to an unbounded potential temperature, which is given by .
Following the result of De Philippis and Figalli [DF12:reg_MA] on higher regularity of Alexandrov solutions to the Monge-Ampère equation, it became possible to improve the regularity of weak solutions constructed by Benamou and Brenier to the class . Indeed, in [ACDF12:convex] and [ACDF12:periodic], Ambrosio, Colombo, De Philippis and Figalli prove the existence of global weak solutions of SG (1.4) in Eulerian coordinates in the case of a convex 3-D domain in Eulerian coordinates and on the 2-D torus respectively, in the case in which is assumed bounded above and away from zero.
In [FT13], Feldman and Tudorascu demonstrate that weak solutions in Eulerian coordinates have the property that the measure has no atomic part. In order to allow such a case, which is physically pertinent as it corresponds to particular frontal singularities, the authors define a notion of generalised weak solution of SG in Lagrangian coordinates and prove the existence thereof. The authors improve upon this result in [FT15], in which they prove existence of relaxed Lagrangian solutions to SG on a domain in with any general initial data which is convex. Finally, in [FT17], Feldman and Tudorascu address the problem of uniqueness of solutions of SG. In particular, they demonstrate weak-strong uniqueness in Lagrangian coordinates under the assumptions of boundedness of the Eulerian velocity field and uniform convexity of . In [Wilkinson18:SG], the second author proves existence of local-in-time classical solutions of (1.3) in Eulerian coordinates on 3-dimensional smooth bounded simply-connected domains. The technique used relies on the theory of the so-called div-curl systems, and it is consistent with the Stability Principle of Cullen and Shutts.
1.3 Main Result
In this paper, we prove the existence and uniqueness of classical solutions of the initial value problem associated to (1.1) for given smooth initial data. As a minor simplification of the full model originally introduced in Hoskins [Hoskins75], we restrict our interest to the case in which on the lower boundary , so that one need only deal with a single evolution equation (1.1) on the upper boundary , as opposed to two coupled equations on the disconnected set . We comment further on this point in the derivation of the surface semi-geostrophic model from the full semi-geostrophic equations in Section 2 below. We also restrict ourselves to considering SSG on the flat torus , instead of the unbounded plane .
In all that follows, we say that a pair of maps is a local-in-time classical solution of (1.1) associated to initial datum if there exists such that , , with and solving (1.1) pointwise everywhere in , together with . The main result of this paper is the following theorem. We use here the notation to indicate the space of divergence-free functions in .
Theorem 1.1 (Local-in-time existence and uniqueness of classical solutions to SSG).
Suppose . There exists such that given with and , there exists a such that there is an associated local-in-time classical solution of (1.1) on . Moreover, for the classical solution is unique.
As we have presented above, the active vector field depends on the unknown through the Neumann-to-Dirichlet operator associated to the Neumann BVP for the fully-nonlinear equation (1.2). Therefore, an important part of our proof consists in proving that such an operator is well-defined and admits useful analytical properties.
Whilst we believe it is possible to establish the analogue of Theorem 1.1 in the case that is smooth and non-periodic on , we do not do this here.
1.4 Structure of Paper
The paper is structured as follows. In Section 2, we present the derivation of SSG from SG, as originally performed in [Hoskins75]. In Section 3, we introduce a construction of the Neumann-to-Dirichlet operator defined on Hölder spaces, and discuss some of its relevant properties. In Section 4, we prove Theorem 1.1 through an application of Schauder’s fixed point theorem, making use of classical estimates on the solutions of passive transport equations on . In the closing Section 5, we discuss some natural generalisations of our result. Finally, in the Appendix A, for the reader’s convenience, we provide details of the calculations underlying the arguments in the previous sections.
2 Derivation of SSG from SG
The surface semi-geostrophic equations were derived from the semi-geostrophic equations by Hoskins [Hoskins75] in 1975. They arise when one considers the special case of solutions of SG which admit spatially-homogeneous potential vorticity. For the convenience of the reader, we reproduce a derivation of this model here, starting from classical solutions of SG in Eulerian coordinates (which are assumed, but are not known, to exist). It will be helpful in the sequel to consider the system (1.3) expressed in all its components, namely
where is the Coriolis parameter, assumed constant in what follows. Moreover, the Eulerian velocity field is subject to the no-flux constraint on for all times , where denotes the outward unit normal map. As we shall see below, this no-flux assumption is crucial in the derivation of the surface semi-geostrophic equations. We define the vorticity associated to the dynamics of system (2.1) by
and one may readily check that it satisfies the following vorticity equation, namely
pointwise in the classical sense on space-time. If one in turn defines the potential vorticity of the geostrophic flow as
it follows from a calculation that the potential vorticity is conserved along Lagrangian particle trajectories, namely
As such, if one furnishes the system (2.1) with initial data whose associated potential vorticity is spatially inhomogenous, then the corresponding solution also formally has this property. It was discovered by Hoskins that solutions which admit constant potential vorticity on admit a rather beautiful structure in another coordinate system (to which we henceforth refer as Hoskins’ coordinates), details of which we now provide.
2.1 Transformation of Coordinates
Suppose smooth initial data for the dynamics formally generated by (2.1) are given. We now consider the associated 1-parameter family of smooth maps defined by
For each time , the map is assumed to be a -diffeomorphism in what follows. In the sequel, we shall use capital Roman letters, namely , to denote the independent variable for maps defined on considered as the range space of the coordinate transformation for any . Our aim in the sequel is to close a system of equations for a number of ‘natural’ quantities in the coordinate system determined by . In this pursuit, we begin with the following proposition.
For any smooth map , it holds that
where for and each time .
We begin by noticing that for any smooth map , the material derivative of the composition (with respect to the Eulerian velocity field ) in Eulerian coordinates is given by
As such, the corresponding velocity field that advects the flow in Hoskins’ coordinate system is given simply by . If we consider its components, namely
it follows simply that
whence follows the proof of the claim. ∎
It will be helpful in what follows to note that the matrices and are explicitly given by the following expressions:
where we used the notation and denotes the Jacobian of , namely
We shall also employ the notation in what follows. The following observation holds as a consequence of Proposition 2.1 above.
Suppose is a smooth solution of (2.1). It follows that both the maps and defined pointwise as
are smooth solutions of the transport equation
In all that follows, we refer to and as the geostrophic buoyancy anomaly and geostrophic potential vorticity, respectively.
By the definition of vorticity and the form of the matrix in (2.3), one observes that the third row of is given simply by a multiple of the vorticity , namely
For any smooth map , it follows that
with, in particular, it being the case that
by the definition of potential vorticity (2.2). Since the potential temperature and the potential vorticity are Lagrangian invariants with respect to the Eulerian flow , it follows that and are Lagrangian invariants with respect to the flow , namely
which concludes the proof. ∎
The study of and in the coordinate system determined by will, in some sense, ‘replace’ the study of and in Eulerian coordinates. To see how this is the case, we introduce an important streamfunction originally due to Hoskins.
2.2 A New Streamfunction in Geostrophic Coordinates
An important observation of Hoskins was that there exists a time-dependent potential which ‘stores’ all salient features of the dynamics in geostrophic coordinates. Indeed, following [Hoskins75, Section 4], we define a streamfunction in Hoskins’ coordinates by
for and any . A straightforward calculation reveals that
Therefore, by the assumption that be a diffeomorphism of for all times , we have that the following identity between the gradients of the streamfunction and the pressure holds true:
It is this identity which furnishes the link between the semi-geostrophic equations in Eulerian coordinates and the so-called surface semi-geostrophic equations in Hoskins’ coordinates. In order to understand the dynamics of in geostrophic coordinates, we observe the following identities which relate the second derivatives of the pressure to those of the streamfunction :
By taking products and differences in the above, we may identify an equation for the determinant , namely
One can rewrite the determinant in the equation above in terms of the geostrophic potential vorticity and the derivative to obtain
where the geostrophic velocity field is given by
From a structural point of view, system (2.7) constitutes a fully nonlinear PDE (possibly of elliptic type) with non-constant coefficients which is coupled to a pair of transport equations associated to the geostrophic velocity field . Moreover, the boundary condition on both components of is equivalent to on in Hoskins’ geostrophic coordinates. At this moment, we do not know that this system is closed, in the sense that the IBVP associated to (2.7) admits a unique solution in any sense. It is at this point it is prudent to employ the additional assumption that initial data are taken such that the value of given by is constant on . This allows us to eliminate the transport equation for the geostrophic potential vorticity and ensure that the fully nonlinear equation for in (2.7) admit constant coefficients. We outline the structure of the corresponding system below.
2.3 Surface Semi-geostrophic Flow
As intimated above, the equations (2.7) that we obtain in Hoskins’ geostrophic coordinates are not easily studied. We make the additional important assumption of constant potential vorticity at time , namely
for some constant determined in terms of the initial data . Although this assumption might appear to be unnecessarily restrictive at first glance, it is in fact a good approximation when studying real-world atmospheric flows (see [Juckes94:QG]). Mathematically on the other hand, with this simplification and subject to a rescaling, system (2.7) reduces to the following set of coupled equations:
together with the boundary condition on , where the rescaled value of has been simply relabelled as . It turns out that in this setting of spatially inhomogeneous potential vorticity, one need only solve the transport equation on each of the boundary planes , as opposed to on the entire interior of the domain . Indeed, the nomenclature ‘surface semi-geostrophic equations’ comes from the fact that the dynamics in geostrophic coordinates in the bulk is ‘determined’ by the dynamics on the upper and lower boundary planes. Of course, in our work, as the dynamics on the lower boundary portion is trivial, it suffices to understand the evolution of the geostrophic buoyancy on the upper boundary plane alone.
3 Construction and Properties of the Neumann-to-Dirichlet Operator
Our first step in the proof of our main result, namely Theorem 1.1, is to demonstrate that the Neumann-to-Dirichlet map (whose formal definition is given in 3.2 below) associated to the fully nonlinear equation is well defined and admits useful analytical properties. Indeed, this pursuit is captured by the following theorem:
There exists a Fréchet differentiable map defined on a ball such that, for any , is a -classical solution of
on the domain with upper boundary and lower boundary , for any .
In the natural way, we say that is a -classical solution of (3.1) if solves the PDE pointwise everywhere on , and its restriction to the boundary satisfies the given boundary conditions. The zero-mean requirement on is prescribed to guarantee uniqueness of solution to the boundary value problem. Associated to this solution map is the operator defined below, which determines the activity in the surface semi-geostrophic equation (1.1).
The Neumann-to-Dirichlet operator associated to system (1.1) is defined to be
where is the classical restriction operator to the set .
The main idea of the proof of theorem 3.1, and thereby the demonstration that is well defined, is to employ the Banach Fixed Point theorem to guarantee both existence and uniqueness of solutions to (3.1), inspired by the numerical work [BR16:turbulence] by Badin and Ragone on SSG. Indeed, for a given we define the operator as
where the Monge-Ampère-type operator is defined as
and the operator is simply the solution operator associated to the linear Neumann boundary value problem on given by
for any given . It follows that a fixed point of is a classical solution of (3.1) on of class .
The fact that all boundary value problems under study in the sequel are, roughly speaking, periodic in two coordinate directions and non-periodic but of bounded extent in the other makes their analysis slightly awkward. For instance, in the case of the pure Poisson Neumann boundary value problem, one cannot simply apply routine techniques from [GT:ellipticPDEs] to understand their well-posedness. One could, for instance, employ techniques from the monograph [Grubb:book] in order to employ a Method of Reflections-type argument for elliptic equations on the unbounded strip . This is not the approach we adopt in this work, however.
We begin our approach to the proof of Theorem 3.1 by showing well-posedness of the linear BVP (3.4) in Section 3.1. Following this, we look to verify the hypotheses of the Banach Fixed Point Theorem to the operator on a suitable ball in in Section 3.1 thereafter.
3.1 Analysis of the Poisson Problem (3.4)
We begin by recalling the necessary compatibility condition that needs to be satisfied by the inhomogeneity and the boundary datum for the Poisson problem (3.4) be well posed.
Lemma 3.3 (Compatibility condition).
A necessary condition for the problem (3.4) to have a classical -solution on is that and are compatible in the sense that
The well-posedness of the system (3.4), and hence the well-posedness of the operator , is proved by means of Schauder theory. Let us first state the various notions of solution to the Poisson problem with which we work in the sequel.
Given and , a function is said to be:
a weak solution of (3.4) if and