## Abstract

For semigeostrophic (SG) theories derived from the Hamiltonian principles suggested by Salmon it is known that a duality exists between the physical coordinates and geopotential, on the one hand, and isentropic geostrophic momentum coordinates and geostrophic Bernoulli function, on the other hand. The duality is characterized geometrically by a *contact structure.* This enables the idealized balanced dynamics to be represented by horizontal geostrophic motion in the dual coordinates while the mapping back to physical space is determined uniquely by requiring each instantaneous state to be the one of minimum energy with respect to volume-conserving rearrangements within the physical domain.

It is found that the generic contact structure permits the emergence of topological anomalies during the evolution of discontinuous flows. For both theoretical and computational reasons it is desirable to seek special forms of SG dynamics in which the structure of the contact geometry prohibits such anomalies. It is proven in this paper that this desideratum is equivalent to the existence of a mapping of geographical position to a Euclidean domain, combined with some position-dependent additive modification of the geopotential, which results in the SG theory being manifestly Legendre transformable from this alternative representation to its associated dual variables.

Legendre-transformable representations for standard Boussinesq *f*-plane SG theory and for the axisymmetric gradient-balance version used to study the Eliassen vortex are already known and exploited in finite element algorithms. Here, two other potentially useful classes of SG theory discussed in a recent paper by the author are reexamined: (i) the nonaxisymmetric *f*-plane vortex and (ii) hemispheric (variable *f*) SG dynamics. The authors find that the imposition of the natural dynamical and geometrical symmetry requirements together with the requirement of Legendre transformability makes the choice of the *f*-plane vortex theory unique. Moreover, with modifications to accommodate sphericity, this special vortex theory supplies what appears to be the most symmetrical and consistent formulation of variable-*f* SG theory on the hemisphere. The Legendre-transformable representations of these theories appear superficially to violate the original symmetry of rotation about the vortex axis. But, remarkably, this symmetry *is* preserved provided one interprets the metric of the new representation to be a pseudo-Euclidean Minkowski metric. Rotation invariance of the dynamical formulation in physical space is then perceived as a formal Lorentz invariance in its Legendre-transformable representation.

## 1. Introduction

In order to further our understanding of atmospheric and oceanic dynamics it is desirable to possess a set of idealized and simplified equations whose solutions can be obtained with great precision (either numerically or analytically) while realistically treating the dynamical features of interest. In this way the idealized system can provide insights into the *essential* balanced dynamics that a more complete model might often obscure with numerical or gravity wave “noise.” When it comes to the study of discontinuous phenomena, such as atmospheric and oceanic fronts, then the semigeostrophic (SG) systems (Hoskins and Bretherton 1972; Hoskins 1975), which extended the earlier studies of vortex and frontal balance of Eliassen (1951, 1962), seem uniquely suited to the study of the *slow* or *balanced* components of these discontinuities. Being filtered systems of equations, they automatically exclude the obscuring gravity wave components, yet, in their Lagrangian form, they are expressible as a set of parcel-conservation laws requiring no evaluations of spatial derivative. As first pointed out by Cullen (1983), they are therefore able to tolerate contact discontinuities within the fluid and can be integrated using fully Lagrangian finite elements.

Other numerical methods of a Lagrangian character, such as the method of point-vortex advection of Christiansen (1973) or the recently developed method of contour advection of Dritschel and Ambaum (1997), also demonstrate the ability of a Lagrangian style of technique to develop sharp features in an advected tracer. However, SG theory suggests that the production of a frontal interface involves more than an interior rearrangement of preexisting potential vorticity; the vanishing-viscosity limit of SG reveals a process by which an intrusion of a surface of impulsive potential vorticity, originating at the ground, is intruded upward into the interior. It is not obvious that the methods of vortex advection or contour dynamics would be adequately able to handle this process without some refinement or intervention, whereas the SG method is inherently well suited for this type of problem.

The theory of the Lagrangian finite element geometric model form of SG dynamics was put on a firm foundation by Cullen and Purser (1984) and the method was applied to a variety of highly idealized *f*-plane situations by Cullen et al. (1987a,b), by Chynoweth (1987), who constructed the first general geometric model algorithms, and by Shutts (1987), who investigated the ability of SG methods to treat slantwise penetrative convection. Shutts et al. (1988) were also able to demonstrate the applicability of the geometric model, suitably modified, to Eliassen’s axisymmetric balanced vortex in which gradient balance replaces geostrophic balance radially.

Independently, Salmon (1983, 1985), with mainly ocean simulations as his objective, rediscovered and greatly extended SG dynamics by starting with a Hamiltonian (variational) prescription. The filtering assumptions were introduced in a careful way that ensured retention of analogues of the important conservation laws of mass, energy, and potential vorticity. The most significant feature of this alternative derivation was that it admitted a general spatial variation of the Coriolis parameter that hitherto had not been accomplished without violating one or more of the conservation principles. A global SG model based on similar principles was proposed by Shutts (1989). The existence of a contact structure in SG theory was first recognized by Blumen (1981). In an attempt to illuminate the common geometrical features of Salmon’s Hamiltonian model and the conventional SG models, Purser (1993) (henceforth P93) paid particular attention to the contact structure and showed how the SG models could be extended in a way consistent with both the Hamiltonian and geometric model formulations to nonaxisymmetric vortex dynamics on either an *f*-plane or, with proper treatment of the varying Coriolis parameter and sphericity, to the hemisphere. A brief description of contact structure as it relates to Hamiltonian SG dynamics is given in section 2. For further mathematical details, the reader may consult Sewell and Roulstone (1994). Hamiltonian techniques in geophysical fluids are reviewed in Shepherd (1990). A recent interesting development, generalizing the SG models, is provided in McIntyre and Roulstone (1996).

Associated with the contact structure is a duality relation between the primal (or physical) solution, comprising a geopotential function of physical coordinates, and the *dual* solution, comprising a Bernoulli function of isentropic geostrophic momentum coordinates, or a variant of these. The primal solution’s graph (hypersurface in the extended space of physical coordinates **x** augmented by an extra coordinate measuring geopotential *ϕ*) is, in a well-defined sense, the envelope of a continuous family of what we call neutral energy–generating surfaces coexisting in the same extended space. Each neutral energy surface is itself labeled by a dual coordinate **X** and the dual (Bernoulli) potential Φ, and therefore the entire surface can be identified by a single *point* in a dual extended space. The term “duality” recognizes the fact that roles can be reversed. Thus, from the totality of neutral energy surfaces one can consider the subset that passes through a given point **x** and geopotential *ϕ* and notice that the locus of their labels **X** and Φ now constitutes a single new generating surface in extended dual space whose own labels can be taken to be (**x**, *ϕ*). Now, just as the physical solution can be regarded as the envelope “above” (when the sense of increasing *ϕ* is taken as “up”) the neutral energy surfaces labeled by the dual quantities (**X**, Φ), the dual solution can itself be regarded as the envelope “below” those dual generating surfaces labeled by the quantities (**x**, *ϕ*) present in the original physical solution. In this interpretation, we find that the class of finite element solutions are simply those solutions *ϕ*(**x**) whose graphs are each constructed as envelopes of only *finitely many* neutral energy surfaces. Each element is therefore characterized by a physical volume (which it conserves), a single value of **X**, and a single value of Φ.

Cullen and Purser (1984) showed that a trivial transformation of the geopotential of Hoskins’s (1975) constant Coriolis form of SG theory (the “standard theory”) enabled this version to be treated by the methods of Legendre duality. Based on a less obvious mapping to Legendre-transformable form, Shutts et al. (1988) extended the geometrical model to the SG form of the axisymmetric Eliassen vortex model described by Shutts and Thorpe (1978) and by Schubert and Hack (1983). Geometrical implications of Legendre duality were discussed by Purser and Cullen (1987), Chynoweth et al. (1988), and by Chynoweth and Sewell (1989, 1991). Primarily, the practical significance of Legendre transformability is that it leads to the simplest algorithms for the finite element solutions (computations of intersections among curved surfaces is incomparably harder than the equivalent computations for intersecting hyperplanes). However, as we show in section 3, there is another reason for preferring a Legendre-transformable SG theory that is related to the topological structure of the connections between distinct fluid parcels in finite element solutions or in originally smooth solutions during frontal formation. In theories possessing a generic (not Legendre transformable) contact structure it will be shown below that topological anomalies can occur whereby elements may either spontaneously split into disconnected pieces, or else may achieve multiple contact (at disconnected interfaces) with the same neighboring elements. Apart from the obvious computational difficulties implied for finite element algorithms, the possibility of such anomalies makes the uniqueness and regularity of the *weak* (discontinuous) solutions resulting from initially continuous data questionable. This possibility thus undermines the supreme purported virtue of SG dynamics—its ability to accommodate discontinuous solutions. We therefore regard any SG theory *not* possessing a Legendre-transformable representation as structurally deficient.

Two such questionable theories were proposed in P93;one, a nonaxisymmetric generalization of the *f*-plane vortex model; the other, a natural extension of this model to the hemisphere [see Craig (1991) and Magnusdottir and Schubert (1991) for alternative SG treatments not obviously exhibiting the form of contact structure we have described]. In sections 4 and 5, we reexamine the necessary properties of the contact structures of the Hamiltonian SG theories introduced in P93 and propose very minor modifications to the particular formulations suggested there in order to make the modified formulations exactly Legendre transformable. The Legendre-transformable representations of the vortex models appear superficially not to preserve the angular symmetry since the concentric circles of the vortex are mapped to sectors of concentric hyperbolas in each transformed horizontal plane. The symmetry breaking is illusory, however; the new representation *does* preserve the symmetry, provided the space of points **x̂** of the Legendre-transformable representation is regarded as being furnished with the pseudo-Euclidean metric of a (two-dimensional) Minkowski space in place of a true Euclidean metric. The operation of rotation by some angle about the axis of the vortex is then represented by a proportionate Lorentz boost, with cosines and sines of azimuth in the components of the rotation operator being replaced by hyperbolic cosines and sines of the corresponding Lorentz angles. In section 6 we make some remarks concerning the numerical implementation of the proposed new formulations and we conclude in section 7 with some more general suggestions about the potential applications, both in numerical weather prediction and in oceanography, of the type of methods we are advocating.

## 2. Contact structure in SG theories and Hamiltonian dynamics

We shall adopt most of the notational convention of P93. Thus, **x** and **X** are physical and dual coordinates, *μ* and *ν* are physical and dual measures of the respective coordinate volumes, *ϕ* and Φ are physical and dual geopotentials, *η* is the pseudodensity (mass per unit *μ*), and *ρ* is the potential density (mass per unit *ν*). One essential feature of SG theories is that the dynamics is specified by the distribution of geopotential (and boundary constraints) alone. Therefore, the Hamiltonian is expressible in terms only of the geopotential distribution.

Let us define the specific energy of a parcel with physical and dual coordinates **x** and **X** to be ℰ(**x**, **X**). In general, we need not require the domain of **X** to be isometric to the domain of **x**; as we shall see in section 5, it is sometimes more appropriate that the dual space differs from the physical space (e.g., to enable the dual coordinates to be made formally canonical).

We postulate that, at each instant, the collective disposition of the **x** associated with each **X** is such that the energy integral

is minimized with respect to local rearrangements that (i) conserve their **X** on material parcels, (ii) conserve their mass *ρ* on material parcels, (iii) conserve pseudodensity *η,* and (iv) remain inside the bounds defined by the physical domain. The valid solution is then one associated with a scalar function *ϕ* that we identify as the geopotential and that satisfies

where, for each suffix **X**, *ϕ*^{′}_{X}(**x**) denotes the neutral energy function (or its graph, the neutral energy surface):

If the solution *ϕ* follows a neutral energy surface throughout some finite volume, then according to the precepts of SG theory, a prompt **X**-conserving lateral or vertical displacement of any constituent parcel can be achieved with a net change in the total energy of the system. This idea of a neutral energy surface therefore serves to extend the one-dimensional concept of a neutral stratification to the horizontal dimensions also. In the same way that a vertically stable stratification is convex relative to the neutral profiles tangent to it, a symmetrically stable distribution of *ϕ* is (three dimensionally) convex relative to the neutral energy surfaces tangent to *it.*

The dual potentials Φ(**X**) are defined implicitly to be those such that, for each set Σ of **X** space of measure *ν,* the corresponding set *σ* of **x** space of measure *μ* (conserved by potential rearrangements) is obtained as the volume of actual contact:

A more complete discussion of this idea is presented in P93, where it is shown that this prescription provides a definition for the theory’s inherent contact structure and determines the basis for the geometrical duality between the physical solution *ϕ*(**x**) and the dual solution Φ(**X**). Properties of this contact structure are as follows.

The quantity

**X**[and hence ℰ(**x**,**X**)] can be regarded as a unique function jointly of**x**and**∇***ϕ*(by the conditions of the implicit function theorem); the matrix of components, ∂^{2}ℰ/∂*x*_{i}∂*X*_{J}, is therefore not singular.If different solutions

*ϕ*_{1}and*ϕ*_{2}make tangential contact at**x**, then their duals, Φ_{1}and Φ_{2}, make tangential contact at the image,**X**, of**x**.

The quantity ℋ of (2.1) is the Hamiltonian, which directly prescribes the evolution of the flow in **X** space and, indirectly (from the rearrangement result), the flow in physical space also. The variational method for these problems is developed by Salmon (1983, 1985). During the period [*t*_{1}, *t*_{2}] an action integral is extremized:

where the Lagrangian ℒ_{s} is defined as

The horizontal vector field **A** is a time-independent function of the horizontal dual coordinates **X** such that its integral *C,*

in a circuit of constant potential temperature (the vertical dual coordinate *Z*) measures an absolute circulation associated with the effective Coriolis function,

Variations of the action integral with respect to *X* and *Y,* subject to the constraint that parcel values of *Z* and mass remain constant, imply *geostrophic* dynamics in **X** space:

Any circulation integral *C* defined by (2.9) is now a materially conserved quantity. We note that a transformation of dual coordinates, **X** → **X**′, accompanied by a circulation-preserving redefinition of the effective Coriolis function,

leaves the form of (2.9) unchanged. As discussed by Roulstone and Sewell (1996), this enables a choice for *X*′ and *Y*′ to be made such that the new effective Coriolis function *f*′* is constant, whereupon the *X*′ and *Y*′ of each material parcel become canonical coordinates of the Hamiltonian description of the dynamics.

## 3. Legendre duality

### a. Computational advantages of Legendre transformability

As discussed in Schubert (1985) and P93, it is generally possible in SG theory to express the dynamics for *ϕ* or its dual, Φ, in terms of some linear elliptic tendency equation and it is tempting therefore to think that standard computational methods, involving some form of numerical relaxation procedure, will automatically supply a practical way to integrate the time-dependent solutions of interest. However, very frequently the solutions of primary interest in SG studies are of a singular character, such as those describing fronts. Here, the standard gridpoint methods, which rely heavily on the use of spatial differencing, usually become severely compromised by the numerical difficulties associated with evaluating derivatives near the modeled discontinuities or by the spontaneous emergence of (perceived) nonelliptic regions at these places.

As noted in the introduction, an alternative numerical procedure selected specifically to handle these otherwise intractable problems in SG theory is the Lagrangian finite element geometric method proposed by Cullen (1983) and further elaborated by Cullen and Purser (1984), Cullen et al. (1987a,b), Chynoweth (1987), and Chynoweth et al. (1988). The finite elements of this method each conserve their mass and (in adiabatic dynamics, at least) a value of potential temperature that is assumed uniform throughout the element. The horizontal dual coordinates *X* and *Y* are also assumed uniform throughout the element, but subject to change in time according to the dynamics implied by the Hamiltonian, which is evaluated by summing the contributions from each polyhedral element. The total energy associated with each element can be computed as an expression involving the moments of that element and simple functions of its **X**, but at no time is it necessary to evaluate spatial derivatives. As a consequence, the finite element mode of computation stands unsullied by any of the severe numerical problems associated with frontal discontinuities that beset other (e.g., finite difference or spectral) methods of calculation.

In principle, the finite element methods should apply to any SG theory possessing the contact structure described in the previous section. However, as a practical matter, actual implementations of the geometric method have been restricted to the special class of SG dynamics for which a representation (possibly via a nontrivial spatial mapping) exists in which the neutral energy–generating surfaces become hyperplanes in the extended physical space (**x̂**, *ϕ̂*) of this representation. Only in this case do the geometrical calculations involving the surfaces, edges, and vertices of intersections among the various generating surfaces become sufficiently simple to be feasible. The Boussinesq standard *f*-plane SG theories in two and three dimensions have simple Legendre-transformable representations, as exploited by Cullen and Purser (1984) and discussed in detail in Purser and Cullen (1987). Also, Shutts et al. (1988) discovered that the axisymmetric (two-dimensional) variant of SG theory on the *f*-plane (Shutts and Thorpe 1978), in which the radial component is gradient balanced in the sense proposed by Eliassen (Eliassen and Kleinschmidt 1957), possesses a Legendre-transformable representation once the radial coordinate of the vortex has been suitably mapped. This enables the geometric method to be applied, for example, to the investigation of thermally forced solutions in idealized axisymmetric tropical cyclones.

Other potentially useful extensions of SG theory have been formulated, but they appear not to possess Legendre-transformable representations. These include various *f*-plane and hemispheric nonaxisymmetric (three dimensional) generalizations of vortex models (Craig 1991; Magnusdottir and Schubert 1991; P93) and the variable-*f* form of Salmon’s (1985) ℒ_{s} dynamics. While it is obviously desirable, from the computational point of view, to find Legendre-transformable variants of these nonaxisymmetric vortex and variable-*f* theories, we claim that such SG variants are to be preferred also on theoretical grounds. We base this assertion on the following observations concerning the possible forms of finite element solutions (or other singular solutions, such as fronts, which can occur spontaneously from initially smooth data).

### b. Anomalies implied by lack of Legendre transformability

The intersections, in a horizontal surface at some fixed elevation, of the neutral energy surfaces associated with two neighboring dual-space labels, **X**_{1} and **X**_{2}, constitute a family of nonintersecting curves (contours at this horizontal surface of the difference of their respective energy functions ℰ) that cover the area in physical space where both of these particular generating surfaces come into play. If we select one such curve, say *S*_{1,2}, together with some particular point **x** belonging to it, then we can generally find a third neighboring dual-space label, say **X**_{3}, for which the corresponding family of curves formed by all possible intersections, at this same elevation, of neutral energy surfaces *ϕ*^{′}_{2} and *ϕ*^{′}_{3} includes one member, the curve *S*_{2,3}, which is tangent to *S*_{1,2} at **x** but which fails to coincide elsewhere in the immediate neighborhood of **x**. Note, however, that tangency without coincidence of the two curves becomes impossible whenever the contact structure is transformable into one in which the duality takes the special Legendre form, whose intersecting surfaces are always planes. Assuming the label order (1, 2, 3) is monotonic in the sense of the gradients of their respective neutral energy surfaces at **x**, then the two generic possibilities for the general contact structure are as follows.

The curves of intersection

*S*_{1,2}and*S*_{2,3}curl outward at**x**leaving a pinched-off “bow tie”–shaped portion of*ϕ*^{′}_{2}able to form part of the solution surface*ϕ,*but now in two virtually separate pieces (schematically depicted in Fig. 1a).The curves curl inward at

**x**so as to exclude the exposure [under the “sup” operation of (2.2)] of any finite portion of fluid element-2 beyond the single locus of contact,**x**, itself. For slightly perturbed data, the central element reemerges as a “crescent”-shaped, or lenticular, portion of the solution (illustrated in Fig. 1b).

In the former case of the bow tie anomaly, the dynamics potentially permits the spontaneous destruction of the integrity of finite elements. In frontal formation, it would seem to allow the impulsive distribution of potential vorticity associated with the resulting contact discontinuity to be negative, and introduce some undesirable problems associated with guaranteeing uniqueness of solutions with respect to energy minimizing local rearrangements of fluid elements along such a front. In the latter case of the crescent anomaly, the formal difficulties are less severe but involve such configurations as one crescent-shaped finite element being completely surrounded by only two of its neighbors (as illustrated in Fig. 1b). The two outer elements in contact would then share an interface, possibly in many disconnected segments, on which there is also an impulsive distribution of potential vorticity now of positive sign. These theoretical complications are better avoided when the freedom in constructing generalized SG formulations allows one to do so. A result of some help in seeking such anomaly-free formulations is summarized in the following result [which is roughly analogous to the theorem of Darboux in symplectic geometry discussed in Arnold (1980)].

#### 1) Theorem 1

In an arbitrarily differentiable semigeostrophic contact structure, bow tie and crescent anomalies are impossible if and only if the semigeostrophic solutions possess a representation that, in each neighborhood, is Legendre transformable.

#### 2) Remarks

A proof of this result is provided in the appendix. The virtue of this result is that it provides us with a criterion for judiciously modifying the existing nonaxisymmetric vortex and variable-*f* SG theories in order to identify those special forms that are not only known to be anomaly free, but that also promise the possession of Legendre-transformable representations (and hence, computationally feasible dynamics).

### c. Diagnosing the character of potential anomalies

Supposing we wish to determine whether anomalies are potentially present in an SG model endowed with a contact structure and, if so, the character of the anomalies at given primal and dual locations and at different orientations of the intersections. Let these locations be **x**_{0} and **X**_{0} with an orientation defined by **V**, the tangent vector

at **X**_{0} ≡ **X**(0) of a smooth parameterized curve **X**(*s*) where this curve is constrained such that the intersections at **x**_{0} among the neutral energy surfaces *ϕ*^{′}_{X(s)} form a set of curves in **x** that are mutually tangent at **x**_{0}. Without restricting generality, we may adjust the functional form of parameter *s* to make

for some vector **v**_{0} at **x**, **∇** being the gradient operator in **x** space. Differentiating (3.2) with respect to *s* enables us to define *s*-parameterized vector fields **v**(**x**; *s*) generalizing the single vector **v**_{0}:

Adopting the summation convention for repeated indices and the convenient notation

we find that the relation between **V** and **v**_{0} is

A further differentiation provides a formula involving the parametric curvature **V**′ ≡ *d*^{2}**X**(0)/*ds*^{2},

which may be inverted to obtain a direct prescription of this quantity:

in terms of the bilinear operator **G**, which satisfies

Construct any smooth parameterized curve **x**(*r*) passing through **x**_{0} ≡ **x**(0) where it is tangent to all the curves of mutual intersection among the *ϕ*^{′}_{X(s)}. Let **u**(*r*) ≡ *d***x**(*r*)/*dr* be the tangent vector along this new curve and consider the discriminant,

evaluated at *r* = 0 and *s* = 0 (i.e., at **x**_{0} and **X**_{0}). Substituting derivatives of ℰ and henceforth assuming that *ψ* refers to its evaluation at vanishing *r* and *s,* we find that

But, since the first term on the right of (3.10) vanishes, the sign of *ψ* is the same regardless of the particular tangential curve **u**(*r*) that we constructed. Moreover, an examination of the geometry reveals that it is the sign of *ψ* that discriminates between the two kinds of anomalies at this orientation or, if *ψ* = 0, indicates the absence (to first order) of an anomaly here and at this orientation:

Hence, we can substitute for the original vector **u** defining *ψ* any other vector parallel to it. Therefore, let

where the alternating tensor *ε*_{i,j} acts upon a vector to rotate it horizontally by a quarter-turn. Substituting (3.7), we can redefine the discriminant,

where

This provides a straightforward method of analyzing the potential anomalies (if any) of an SG model endowed with a contact structure based on the energy function.

We are now in a position to exploit the ideas introduced in this section and to guide the minor modifications of the *f*-plane vortex and hemispheric SG models of P93 needed to render those theories exactly Legendre transformable.

## 4. The *f*-plane nonaxisymmetric vortex

In this and the following section, we shall simplify the algebraic development by omitting the vertical dimension of the SG theories and therefore omit the associated potential energy contribution to the specific energy function ℰ(**x**, **X**) and to the Hamiltonian. In every case, the potential energy contribution −*Zz* to the total remains unaltered by the various horizontal mappings that we shall be considering.

First, we recall from P93 that, for the axisymmetric *f*-plane vortex at physical radius *r* from the axis and with potential radius *R* (to which the ring of fluid must be expanded or contracted conserving its angular momentum in order to bring it to rest in the rotating framework), the (kinetic component of the) energy function takes the form

If we relate such a vortex to an *unrotated* framework, then the specific energy is just the first term on the right, which is manifestly self-similar with respect to rescaling of either *r* or *R.* In P93 we argued that, in order to accommodate nonaxisymmetric effects consistent with the appropriate (frame relative) definition of *geostrophy* for first-order perturbations about any state of solid-body rotation, then it was necessary for this self-similarity to extend to the form of the energy function generalized in the azimuthal direction, and that the necessary geometrical constrain was that the Hessian of each neutral energy surface evaluated at vanishing relative azimuth (*ξ* − Ξ = 0) should have identical radial and tangential components, where *ξ* and Ξ are the physical and dual azimuth angles about the axis of the vortex. In section 6a of P93 we suggested one particular form of the new energy function, ℰ(*r, ξ*; *R,* Ξ), satisfying this requirement. In the light of theorem 1 and the related discussion of section 3, it is worth reconsidering the exact choice for the azimuthal structure and seeking an alternative functional form for ℰ, equivalent to that proposed in P93 up to second order in relative azimuth, but departing from that form at fourth order in such a way as to avoid the occurrence of bow tie or crescent anomalies in the solution. As at least a necessary condition, we must find that the curves of intersection (at a horizontal level) of the desired neutral energy surfaces will collectively form a biparametric continuous family, just as the lines in a plane form such a family. If we write the energy function in a form that preserves the manifest self-similarity, with respect to radial rescaling, of the first right-hand terms,

for some yet to be determined function *F,* the isotropy of the Hessian at vanishing relative azimuth requires that *F* satisfies the following:

Now, since a limiting case of the energy function obtained as *R* → 0 gives

a necessary condition for obtaining the desired biparametric family of intersections is that *F* has the following property: Given arbitrary constants *R*_{1}, *R*_{2}, Ξ_{1}, and Ξ_{2} then, except for sets in this four-dimensional parameter space of measure zero, a further pair, *R*_{3} and Ξ_{3}, (possibly complex) can be found such that, for all *ξ,*

The only solution of such a problem that also satisfies (4.3) is

If we write *ξ̂* = (8)^{1/2}*ξ* and Ξ̂ = (8)^{1/2}Ξ, then we do indeed confirm that our choice for *F* leads to a Legendre duality,

in the following representation of the physical and dual variables:

While, superficially, it now seems that the angular symmetry in the original description of the dynamics has been destroyed by the intrusion of these cosh and sinh functions of azimuth, in fact, the underlying symmetry remains; in effect, the azimuth angles are subjected to a multiplicative scaling by a constant that happens to be the imaginary number (−8)^{1/2}. Real values are recovered by recognizing the equivalence of such a scaling with a transformation from the horizontal Euclean plane to a two-dimensional Minkowski space, considered either to be a Euclidean space with one coordinate imaginary (the original application of this idea was to special relativity theory), or more conveniently, to be a space of real coordinates but with a pseudo-Euclidean metric,

Then, the cyclic one-parameter group of axial rotations (generating displacements along circles) is now replaced by the one-parameter group of two-dimensional Lorentz boosts (generating displacements along hyperbolas). We note that, in the pseudometric, the radial coordinate *r̂* is recovered from the components *x̂* and *ŷ* by

while scaled Lorentz angles *ξ̂* and Ξ̂ are recovered using

The group of Lorentz boosts is not cyclic. Therefore, transformed back into the original physical domain (*x* and *y*), the neutral energy surfaces formally wind repeatedly around the axis on a Riemann surface, but it is only portions possessing small relative azimuths, |*ξ* − Ξ| ≪ 1, that will ever be of practical significance in constructing a solution.

We see that the latitudinal form of the semigeostrophic transformation is determined by the conservation of angular momentum while, hitherto, there has been considerable ambiguity in the form of the longitudinal part of this transformation. The well-behavedness of the transformation under the extreme conditions in which frontal discontinuities appear dictates a uniquely special form of the function *F* and the associated implied transformation. In this way the former ambiguity is automatically and elegantly disposed of.

It is instructive to see the shapes implied by this construction of the neutral energy and dual generating surfaces. Figure 2a shows some of the curves, passing through a fixed point, formed by intersections of pairs of generating surfaces. Note that some of these curves (cosh type) avoid the axis while others (sinh type) intersect it, according to their orientation relative to radial lines. (This topological distinction is exactly analogous to that in two-dimensional relativity theory between spacelike and timelike lines). The picture corresponding to Fig. 2a for a focus of intersections at some other distance from the axis is essentially no different apart from a trivial change of scale.

Figure 2b shows uniformly spaced contours in physical space (*x, y*) of the speed *u* that one would associate with the kinetic energy function, that is, *u* = (2ℰ)^{1/2}, at the fixed **X** shown by the symbol. Note that near circularity and even spacing of these contours at small amplitudes give way to distorted loops of progressively uneven spacing only when their scale becomes commensurate with the distance to the axis. Figure 2c shows corresponding contours plotted in the dual plane (*X, Y*) when **x** is kept fixed. Radial cross sections corresponding to Figs. 2b and 2c and at vanishing relative azimuth can be seen in Figs. 3a and 3b of P93.

A version of our *f*-plane vortex model for the simplest case of horizontally incompressible barotropic flow, linearized about basic states comprising axisymmetric jets, was shown in P93 to be of comparable accuracy in its treatment of normal mode structures as a nondivergent balance model for the same situations. The interested reader is referred to that paper for illustrative comparisons, which, owing to the linearization, remain equally applicable to our present Legendre-transformable formulation of the vortex SG theory.

## 5. Hemispheric SG theory

We continue to omit from our discussion the vertical components and associated potential energy, but we further simplify the algebra for the hemispheric development by choosing units of time and horizontal distance that make the polar value of the Coriolis parameter and the radius of the earth both unity. Thus, in these units, the Coriolis parameter at latitude *λ* is

where

As discussed in P93, the neutral energy–generating surfaces labeled by the fluid element’s potental radius *R* must possess a vanishing gradient and a horizontally isotropic Hessian of magnitude −*f*^{2} at the location on the earth where *r* = *R* and where physical and dual azimuth (or longitude) angles *ξ* and Ξ are the same. This ensures that, to first order, the SG dynamics reduces to geostrophy. It follows that the radial and tangential components of the Hessian of each equatorially projected neutral energy function are, in our convenient units,

at *r* = *R.* [The angle Λ for which cos(Λ) = *R* is referred to as the potential latitude by Hack et al. (1989).]

In order to obtain a Legendre-transformable representation we seek a mapping of the physical coordinates and geopotential,

such that the corresponding neutral energy surfaces *ϕ̂*′ are linear functions of *x̂* and *ŷ* defined by

and hence, by eliminating the derivatives of Δ*ϕ,* to find that the logarithmic derivative,

satisfies the quadratic equation,

The *β* and *γ* in (5.7) are the following functions of *r*:

The appearance of solutions in pairs corresponds to the fact that the isotropic Hessian condition for generating function *ϕ*′ implies a Hessian condition for the dual generating function Φ′ of identical form except for a sign change. From the formula for the two possible solutions, *p,*

and the definitions (5.8a) and (5.8b), it is apparent that, in order to obtain a real valued logarithmic derivarive *p* over the whole hemispheric range, 0 < *r* < 1, the quantity *α*^{2} cannot be positive. As we have seen, the paradox of imaginary angular scaling is nicely resolved with the aid of a pseudo-Euclidean mapped domain. It is natural, then, to select the *same* angular scaling as used in the *f*-plane vortex so that the hemispheric theory most closely corresponds with the previous development. This judicious choice also appears to lead to the hemispheric theory of greatest formal simplicity and symmetry. Thus, choosing *α*^{2} = −8, our solutions for *p* are

The first solution, *p*^{−}, is the one corresponding to the generalized vortex theory, which gives *r̂* within an arbitrary multiplicative constant and Δ*ϕ* as follows:

As in the *f*-plane vortex, it is convenient to redefine the transformed angular variables so that they become real, that is, replace the definition (5.3b) with *ξ̂* = (+8)^{1/2}*ξ* while simultaneously replacing the circular functions in (5.4a) and (5.4b) by their hyperbolic counterparts. The exact form for the horizontal part of the energy function compatible with this geometry and with a vanishing kinetic energy when (*r, ξ*) = (*R,* Ξ) is

and, when

together with

we reproduce the conditions (4.5) for Legendre duality to hold. Note that (5.13a) confirms that the other solution, *p*^{+}, of (5.10b) provides the dual radial transform. As noted in P93, the representations of the dual radial variables *R̂* in the *f*-plane and hemispheric vortex models are identical if we identify the polar value of Coriolis of the hemispheric case with the *f* used in the *f*-plane vortex model. Both vortex SG theories possess a form of *frame invariance* in the sense that the choice of rotation rate (and hence, Coriolis parameter) used to define the frame of reference can be changed without fundamentally changing the physical content of the theory itself (such a change involves a consistent change in the definition of geopotential and of kinetic energy, of course; the relevant transformations are discussed in P93). This frame invariance appears only to hold in the hemispheric Legendre-transformable case when we adopt our present choice, (−8)^{1/2}, for the *α* in (5.3b).

Figure 3a plots the curves formed by various pairs of intersecting neutral energy surfaces for this hemispheric vortex model. These curves are plotted in transverse Mercator conformal projections in order to minimize the distortion of the shapes of small figures located near the principal meridian. Here we show two foci, since the patterns in physical space are no longer invariant under scaling of *r.* Note that oblique curves never intersect the equator. Figure 3b plots the contours of the energy function, again, for several values of *R. *Figure 3c includes the case of a value *R* slightly exceeding the earth’s radius; since the theory allows values of *R* (but obviously not *r*) larger than the earth’s radius, the implicitly restrictive potential latitude is *not* an appropriate dual coordinate in general applications. This point is further reinforced when we plot the energy contours, for the cases in which *r* is fixed, directly on the projected dual equatorial plane. Figure 3d shows a selection of these contours, together with the projection of lines of latitude and longitude (dotted). From the fact that some contours cross the circle of the projected equator we deduce that this circle represents no intrinsic barrier in the dual domain (recall that dual flow is tangent to these contours of the dual generating surface at the point of its contact with the dual solution).

## 6. Numerical considerations

For a finite element implementation, it is natural to perform almost all the calculations in terms of the Legendre dual coordinates and geopotentials (“hatted” physical and dual variables). As we have seen, the dynamics can be expressed in a convenient (but noncanonical) form when we have both the Hamiltonian and effective Coriolis parameter defined in the system of dual coordinates. In our case, the computation of the Hamiltonian and its first derivative components with respect to Legendre dual space variables requires the intermediate computations of certain moments of each contributing element. A generic element takes its simplest form in these Legendre-transformable physical coordinates (it then has the form of some convex polyhedron) but its internal distribution of effective density *η̂* and of specific energy ℰ are both nonlinear functions of these coordinates. Fortunately, both functions are smooth away from the singularity representing the equator. The element’s total energy and its mass can therefore be approximated to any desired accuracy using an expansion in terms of successive moments of the element combined with the first few Taylor series coefficients, about the same point, of the specific energy density and of the mass density. We shall not pursue these technical issues in great detail, but a brief outline of the basic idea, exemplified by the problem of estimating the mass in an element, is instructive.

Consider the example of the *f*-plane vortex, with uniform effective density *η* in the physical space mapping to a nonuniform counterpart *η̂* in the Legendre-transformable domain via the Jacobian:

The total mass ℐ_{σ}(*η̂*) of a finite element *σ* is therefore the integral

where

We can expand *η̂* about some point **x̂**_{σ} near element *σ* as a Taylor series:

whence

The mass, and more generally mass-weighted moments (of which the parcel-integrated energy is an example), are therefore expressed in terms of the parcel’s ordinary moments in **x̂** space. The latter are relatively straightforward to compute, since each element *σ* comprises a polyhedron.

This technique of employing moment expansions obviously applies to the evaluation of the Hamiltonian as well as to verifying the mass of each element, but it can also be shown that the derivatives of such mass-weighted moments, with respect to variations of the dual extended coordinates, (**X̂**_{τ}, Φ̂_{τ}), for *τ* and *σ* either identical or adjacent, can be evaluated using analogous expansions with moments associated with the interfaces between elements. Such methods should enable the bulk of the computations, associated with both solution generation and the computation of its instantaneous trajectory, to be carried out in the spaces of **x̂** and **X̂**.

Optimal convergence will presumably be obtained when the location of each **x̂**_{σ} at which the partial derivatives are evaluated is close to the center of the corresponding element *σ.* Then a reasonable resolution will ensure that the variations of *η̂,* or of the product of this density with the energy function, will be small enough across the element to make even a short moment expansion an extremely accurate estimate.

## 7. Conclusions

We have shown that the contact structure of a generic form of the Hamiltonian SG theory generalizing the work of Salmon (1985) may imply anomalies in the connectivity of neighboring fluid elements and that these anomalies can only be completely eliminated by ensuring that the contact structure is of the special class that admits a Legendre-transformable representation. We have proceeded to reexamine the SG vortex theories on the *f-*plane and on the hemisphere proposed by this author in a recent paper (P93) and determined the necessary minor modifications required to render them Legendre transformable, and hence, anomaly free. The Legendre-transformable representations exhibit the curious feature of preserving angular symmetry only when we interpret the effective metric as being pseudo-Euclidean; rotational invariance of the original theory takes the form of Lorentz invariance in the new representation and concentric circles in the vortex are mapped to concentric hyperbolas.

The new versions of the SG vortex theories will enable the geometric model techniques of Cullen (1983) and Cullen and Purser (1984) to be extended to fully three-dimensional simulations for which these assumptions of approximately gradient balance are valid. But, unlike the more restrictive model of Shutts et al. (1988), we are now able to handle azimuthally varying components in such solutions, for example, in a simulation of the internal structure of a developing tropical cyclone. A part of one sector of this model can also be adapted to simulations of significantly curved fronts, largely overcoming the defects of standard SG in this area reported by Gent et al. (1994). In the hemispheric case, the distinction between gradient and geostrophic balance of a zonal wind is virtually insignificant (except very close to the pole itself) and so we can legitimately regard the hemispheric vortex model as a minor variant of Salmon’s ℒ_{s} dynamics. We can therefore look forward to future implementations of Salmon’s powerful generalization of SG theory in both the atmosphere and oceans, using the geometric method. In the oceanic case, this will allow simulation of a variety of discontinuous phenomena, such as boundary current separation, outcropping of subsurface layers, and the evolution of unsteady currents where strong thermodynamic and momentum gradients come into play. In the atmospheric case, we shall be able to perform idealized simulations of the entire life cycle of fully nonlinear baroclinic waves with the degree of viscosity or thermal diffusion chosen purely on the basis of physical considerations appropriate to the limiting scale resolved, instead of being mandated, as is typical of most conventional models, by (nonphysical) considerations of computational stability. The SG model simulations thus chosen, while not expected to possess the formal quantitative accuracy expected of primitive equation models at their well-resolved scales, should nevertheless provide the researcher with valuable tools for investigating qualitatively the factors that influence the location and intensities of frontal development under different experimental conditions.

The identified structural deficiency of previous SG models on the hemisphere, which the new formulation avoids, must become manifest in some form in the continuous SG equations. Central to such models is a variable coefficient elliptic equation (for tendency, usually) which must be solved at each instant. The problem with an inconsistent theory appears to be that, as the solution evolves and strong gradients appear, the character of the central equation switches from elliptic to hyperbolic, whereupon there is no longer a definite meaningful solution. Such difficulties are avoided by choosing a Legendre-transformable theory, which guarantees ellipticity indefinitely while the potential vorticity remains positive.

The new hemispheric model is formally self-consistent even in tropical latitudes. It is therefore a candidate for the study of quasi-steady monsoon circulations. As discussed in P93, it should be feasible in practice to combine a pair of hemispheric models of this form with dual coordinates matched at the equator, and allow exchange of mass (conserving this dual coordinate) between the hemispheres, thereby obtaining a fully global model.

Other methods based on generalized “balance” but with consistent analogues of energy conservation and circulation invariants have been proposed recently (e.g., Allen et al. 1990; Allen and Holm 1996; Shapiro and Montgomery 1993) but it is unclear how well these methods are able to accommodate the formation of frontal contact discontinuities. The SG models, while perhaps formally less accurate, *do* possess this ability to handle the formation and evolution of contact discontinuities without difficulty. Because of this, they can be used to generate a number of valuable “benchmark” tests in which such discontinuities are prominent, and against which the more conventional methods of spatial discretization employed by operational forecasting and climate models can be compared and improved. Thus, the methods proposed here could, indirectly, have an impact on the technical development of operational forecast and climate models, now that the typical resolution of such models is beginning to make the proper handling of frontal details a relevant consideration. Finally, it is noted that the semigeostrophic ideas are now beginning to find their way into adaptive methods of data assimilation (Desroziers 1997) and, in this area too, a consistent hemispheric SG theory would be preferable when intense fronts are present.

## Acknowledgments

This work was carried out during the author’s participation in the program “Atmosphere and Ocean Dynamics” at the Isaac Newton Institute (INI), Cambridge, United Kingdom. The author thanks the staff of the INI and the program organizers for their assistance. An earlier draft of this work was improved by the suggestions of Drs. Michael Cullen, Darryl Holm, and Istvan Szunyogh, and Professor Michael Sewell. The author is also grateful to Drs. John Norbury, Ian Roulstone, Rick Salmon, and Professors Tudor Ratiu and Edriss Titi for many stimulating discussions. The author is also grateful for the thoughtful comments of the anonymous reviewers.

## REFERENCES

### APPENDIX

#### Proof of Theorem 1

Write

and use

to define new coordinates **x̂** ≡ (*x̂, ŷ*) where the contact structure is generic, or nondegenerate, in the sense that

By construction, the curvature of the contours of the difference, ℰ_{αβ}, vanishes everywhere in **x̂** space as we take the limit (in any direction), **X** → **X**_{0}. Therefore, prohibition of the bow tie or crescent anomalies described in section 3 requires, for *all* pairs **X**_{α}, **X**_{β}, that the contours of the general difference, ℰ_{αβ}, are also straight relative to the new coordinates **x̂**. Thus we have a generic formula describing the contour:

where the three coefficients of this affine equation are functions jointly of ℰ_{αβ}, **X**_{α}, and **X**_{β}. Consider the first-order variations of these coefficients as we change **X**_{β} in the vicinity of **X**_{0}. Denoting gradient with respect to **X**_{β} at **X**_{0} by **∇**_{β}, and partial derivatives of *b̂* and *ĉ* with respect to ℰ_{α0}, also at **X**_{β} = **X**_{0} by *b̂*_{ℰ} and *ĉ*_{ℰ}, then we find that

Maintaining straight-line contours necessitates that the terms that are quadratic in *x̂* and *ŷ* vanish. Hence,

and, since this is equivalent to saying that the contours of ℰ_{α0} are mutually parallel in *x̂* space, we can therefore alway express each energy difference of the type ℰ_{α0} as a scalar function of a dot product:

for some unit covector **Û**_{α} independent of **x̂**. Finally, we write the general energy difference ℰ_{αβ} ≡ ℰ_{α0} − ℰ_{β0} and substitute the form (A7):

to infer that an infinitesimal displacement *d***x̂** parallel to a contour of this difference must satisfy

In the generic case, **Û**_{α} will not be parallel to **Û**_{β} and so both derivatives *b*^{′}_{α} and *b*^{′}_{β} must vanish in order for (A9) to hold over any finite area. Thus, not only are the contours parallel, but they are also uniformly spaced, implying that *all* the energy differences ℰ_{αβ} are affine functions of the new coordinates **x̂**. By an additive modification of each neutral energy function,

and regarding *ϕ̂*^{′}_{α} as a function of **x̂**, we find now that this modified generating function is expressible in manifestly Legendre-transformable form as asserted:

## Footnotes

*Corresponding author address:* Dr. R. J. Purser, National Centers for Environmental Prediction, W/NP2 WWB Room 207, Washington, DC 20233.

Email: wd23jp@sgi79.wwb.noaa.gov