Abstract

It is shown that there is an exact correspondence between the scalar Rossby–Ertel's potential vorticity (PV) for a field ɛ, and the component of Beltrami's material vorticity along the ɛ-coordinate line (first equivalence theorem). Thus, Rossby–Ertel's PV can be interpreted as a particular case (scalar) of the vectorial Beltrami's material vorticity. The rate of change of Beltrami's vorticity only depends on the curl of the acceleration (or baroclinic–diffusive terms in the rate of change of vorticity) and not on the convective rate of change (involving advection, stretching, and divergence terms). When the motion is circulation preserving (the acceleration is irrotational) Cauchy's vorticity formula states that Beltrami's material vorticity is conserved. However, when the curl of the acceleration is zero only along the direction normal to certain ɛ surfaces, only the ɛ component of Beltrami's material vorticity is conserved. Thus, a second equivalence theorem states that Ertel's PV conservation theorem is equivalent to Cauchy's vorticity formula along the ɛ-coordinate line.

Beltrami's material vorticity is first introduced via Piola's transformations from the spatial vorticity field, but it is shown that a direct definition of Beltrami's material vorticity using the referential velocity and in terms of material variables is also possible. In this approach the current definition of specific PV in terms of the spatial vorticity and the gradient of ɛ becomes instead a relation between both vorticities that can be derived from their respective definitions.

1. Introduction

Sixty years ago Rossby (1940), using the two-dimensional vorticity equation, and the vertically integrated volume conservation equation, for a layer of homogeneous, incompressible fluid, with horizontal velocity uh(xh, t), in a rotating reference frame, found that the ratio of the vertical component of the absolute vorticity (ζak · curlu) to the depth of the fluid layer (D) was materially conserved. Arguing that “the physical meaning of ζa/D is not very clear” Rossby (1940), and also Starr and Neiburger (1940) in a subsequent paper, introduced a virtual (standard) state of the fluid element characterized by ζa0 and D0 (the absolute vorticity and depth, respectively, of the fluid's layer in this reference state), proposed the quantity ζa0 = ζaD0/D as a material invariant, and named it “potential vorticity” (PV). Ertel (1942a, b, c, d; see Schröder 1991), unaware of Rossby's work, discovered that, for a generic field ɛ, when ω · gradε̇ + gradɛ · curla = 0 holds (see list of symbols in Table 1), the conservation law d(αω · gradɛ)/dt = 0 follows, which provided a generalization of the particular case found by Rossby. A detailed account of these items is given in the historical review in Hoskins et al. (1985). The quantity αω · gradɛ is, at the present time, referred to as PV and its conservation law (though, as in Rossby's times, not yet completely understood) is of fundamental importance both in meteorology [see, e.g., Hoskins et al. (1985) and the review by Kurgansky and Tatarskaya (1987)] and oceanography [see the review by Müller (1995)].

Table 1.

List of symbols

List of symbols
List of symbols

This work takes one step forward to better understand the potential vorticity concept. This step consists in identifying PV with an older concept in classical continuum mechanics. It is proved that, when one of the sufficient conditions for the conservation of PV holds (ε̇ = 0), PV density can be identified with one of the components of Beltrami's material description of vorticity (Beltrami's vorticity for short). Since in the very well documented, but more restrictive case of three-dimensional circulation preserving motions, the Cauchy vorticity formula states that Beltrami's vorticity is conserved, the result of the equivalence between Rossby–Ertel's PV and Beltrami's vorticity is of importance because it means that, when the other condition for PV conservation holds (nɛ · curla = 0, where nɛ ≡ gradɛ/|gradɛ|) many of the theorems that apply to circulation preserving motions can be directly applied to special surfaces and directions (identified by the unit vector nɛ) in the fluid. Or, in other words, it means that Rossby–Ertel's PV conservation law is a component of Cauchy's vorticity conservation formula.

The main mathematical tools are briefly introduced in section 2. Section 3 assumes the necessary conditions for PV conservation and provides the main results of this work, namely, the equivalence between Beltrami's material vorticity and PV density, and the equivalence between Cauchy's vorticity formula and the conservation of PV. Some applications of this equivalence theorem are also provided. Finally, concluding remarks are given in section 4.

2. Mathematical preliminaries

This section provides the symbol definitions and the basic kinematical notions necessary for a modern treatment of Beltrami's material vorticity. The reader familiar with these topics may avoid reading this section and pass directly on to section 3.

a. Notation

The motion of a deformable continuum is described by the mapping x = (X, t), where x and X denote the position occupied by a typical particle in the present configuration at time t, and in a fixed reference configuration, respectively. The coordinates x and X are the spatial and material coordinates, respectively. Any function of the spatial variables (x, t) is also a function of the material variables (X, t). Any spatial differentiation may be expressed in terms of material differentiations, and conversely. Differential operations with respect to the spatial variables are denoted ( ),t, grad, curl, and div; while the differential operations with respect to the material variables are denoted d/dt or (˙), Grad, Curl, and Div. The deformation gradient F ≡ ∂/∂X = Gradx = (gradX)−1 and its determinant J = detF > 0.

The tensorial notation basically follows that in Chadwick (1976), Gurtin (1981), and Truesdell (1991). This notation is here preferred instead of the Gibbsian dyadic notation (e.g., Morse and Feshbach 1953; Godske et al. 1957; Tai 1992), more common in fluid dynamics, because it is the one most commonly used in continuum mechanics, especially in the study of deformation. Partial derivatives with respect to Cartesian coordinates {xi}, i = 1, 2, 3 are denoted as f,i ≡ ∂f/∂xi. A tensor S is considered to be a linear map that assigns to each vector b a vector Sb. The tensor product b1b2 of two vectors b1 and b2 is the tensor that assigns to each vector b3 the vector (b2 · b3)b1 [that is, (b1b2)b3 = (b2 · b3)b1]. Given a Cartesian coordinate frame {ei} = {i, j, k} the components Sij of a tensor S are defined by Sijei · Sej, so that S = Sijeiej. In particular, gradb = bi,jeiej. Indices i, j, k, … , ∈ {1, 2, 3}. Summation convection is implicit in repeated indices. The transpose ST of S satisfies Sb1 · b2 = b1 · STb2; in particular (b1b2)T = b2b1. Note that in Gibbsian dyadic notation gradf = fi,jejei, so that expressions of the type (gradb1)b2 in this work correspond to (gradb1)T · b2 = b2 · gradb1 in dyadic notation.

By the chain rule, the well-known formulas = f,t + gradf ·  and = f,t + (gradf) follow for scalar and vector fields, respectively. The expressions Gradf = FT gradf provides a relationship between spatial and material gradients. The particle velocity u = = ∂/∂t. Recall Euler's identity = J divu. The velocity gradient L = ∂u/∂x = gradu, so the advective acceleration is Lu (u · L in dyadic notation). Note that = LF. The deformation changes the line dL, area dS, and volume dV elements into line dl, area ds, and volume dυ, where

 
formula

The vorticity ω = curlu, the acceleration a = = u,t + Lu, and the curl of the acceleration β = curla.

b. Piola's transformation

Let f(x, t), h(x, t) be any vector-valued functions, and g(x, t) be any scalar-valued function. From (1) the following integral expressions follow:

 
formula

Equation (2a) defines the circulation of h along the closed material circuit (L, t) and relates the spatial density h (circulation per unit of line element in the present configuration) to the specific field h° ≡ FTh (circulation per unit of material line element). Equation (2b) defines the flux of f through the area (S, t), and relates the density flux f (flux per unit of area element in the present configuration) to the specific flux f° ≡ J F−1f (flux per unit of material area element). Equation (2c) defines the amount G in a volume (V, t), and relates the spatial density g (amount of G per unit volume in the present configuration) to the specific ĝJ g (amount of G per unit volume in the referential configuration). The Piola transformation π{ · } = J−1F{ · } (see, e.g., Šilhavý 1997, p. 62) provides the transformation from f° to f and vice versa via (2b). It commonly appears in transformations relating the material and spatial descriptions of fields.

Following Casey and Naghdi (1991, hereinafter CN91) we introduce the time-dependent vectors ω° and β° by the Piola transformations ω° = J F−1ω = π−1{ω}, β° = J F−1β = π−1{β}. It seems that the vector ω° was first introduced by Beltrami [1871, see Eqs. (11) and (15) therein] though using a different procedure (see also Truesdell 1954, hereinafter KoV, section 84; and Truesdell and Toupin 1960, section 86). Because of (2b), the flux of ω° through an area element in the referential configuration equals the flux of ω through the corresponding area element in the present configuration. Thus, ω° provides a material description of vorticity, and is referred to as Beltrami's material vorticity.

The velocity and acceleration fields are transformed, accordingly, to (2a), by u° = FTu and a° = FTa. These vectors refer therefore to the circulation of velocity and acceleration, respectively, per unit of material line element. It is important to realize that the above expressions including u°, ω°, a°, and β° may be considered as relations between these terms and the spatial quantities u, ω, a, and β. An alternative definition and interpretation of ω°, in terms of the material velocity in the referential description, is postponed to section 3b, once the relation between Rossby–Ertel's PV and Beltrami's material vorticity has been provided.

c. Vector ω° and the baroclinic–diffusive rate of change of vorticity

It follows, from the results in the previous section, that Curlu° = ω° = π−1{curlu} and Curla° = β° = π−1{curla}. Since for any vector b we have Gradb = ∂b/∂X = ∂b/∂x/∂X = FT gradb, and LTu = ui,juiej = 1/2 gradu2, it follows that ° = F˙Tu = Tu + FT = FTLTu + a° = 1/2 Gradu2 + a°, and taking the Curl we obtain Curl° = Curla°, so that ω̇° = β°. Recall also that, since Divb = J div(J−1Fb), we have Divω° = J divω = 0.

The vector curla may be named the spatial diffusion vector (KoV, section 84). This terminology (see KoV, section 83) distinguishes two types of mechanisms of change of vorticity, the first one depending only upon the present velocity and its spatial derivatives (convective rate) and the second one associated with the acceleration of the particles (diffusive rate). The process of diffusion (KoV, section 85) is different in each special type of model of continuum, while the process of convection (which involves vorticity advection, plus the stretching and divergence terms) is the same for all. Here, to avoid confusion with the terms involving div(gradω) = ∇2ω (called also the diffusive rate of change of vorticity), and to explicitly mention the baroclinic term in the vorticity equation (involving gradα × gradp, where p is the pressure) we shall call the vector curla the baroclinicdiffusive rate of change of vorticity. The vector β° ≡ Curla° may be named the material baroclinic–diffusive vector, in analogy to the spatial baroclinic–diffusive vector curla. Note that, in these terms, the above relation ω̇° = β° means that the rate of change of Beltrami's vorticity is not convective, but only baroclinic–diffusive.

d. Vector ω° and the circulation preserving motion

Consider the circulation along a closed curve (L, t) (a circuit) in the present configuration, and let L be the corresponding circuit (its inverse image) in the reference configuration. Let (S, t) be any surface bounded by (L, t), and S the corresponding surface (inverse image) in the reference configuration. It is well known (e.g., CN91) that the rate of change of the circulation Γ along the material curve (L, t) is

 
formula

A motion is circulation preserving if and only if Γ̇ = 0 for every reducible circuit L. Let the initial value of Beltrami's material vorticity ω°0(X) ≡ ω°(X, t0). From (3) the following conditions for circulation preserving motion are equivalent:

 
formula

The first condition in (4) is the d'Alembert–Euler condition. The second is the Hankel–Appell condition. The last two are alternative ways of expressing the Cauchy's vorticity formula. Any of these conditions is necessary and sufficient for the circulation of every reducible material circuit to remain constant in time, that is, to be a circulation preserving motion.

3. The equivalence theorems

a. Mathematical formulation

Since specific PV can be considered as a class of quantities depending on the particular field ɛ, it is convenient to use the notation [ · ] ≡ J ω · grad( · ) (see appendix A for a brief mathematical theory of general balance equations supporting the use of the terms “specific PV” and “PV density”). We assume now the usual sufficient conditions leading to the conservation of PV. Let the motion be referred to an inertial reference frame; let ε̂ be a conserved quantity (first sufficient condition) (dε̂/dt = 0, e.g., specific entropy θ̂ in an isentropic process). The caret (^) on ε̂ is used, according to (2c), to denote that ɛ is a density, and ε̂ is a specific quantity. Assume (second sufficient condition) that gradε̂ · curla = 0, which is true when the Euler equation holds [a = gradΩ − α gradp, for a potential Ω, and the thermodynamic pressure p = p(α, ε̂)]. Thus, on one hand, the sufficient conditions, Rossby–Ertel's specific potential vorticity, and PV conservation law can be summarized as follows:

 
formula

On the other hand, let us consider a circulation preserving flow; since curla = 0 and dω°/dt = 0 are equivalent necessary and sufficient conditions for circulation preserving motion, we can write, in a way mimicking (5), the following relations for this flow:

 
formula

Comparing the mathematical statements (5) and (6) it is clear that the first sufficient condition for conservation of PV (5a) and the definition of material coordinates (6a) allow us to identify the isosurfaces of the field ε̂ with coordinate surfaces in the referential description (say X = ε̂) and lay down the relation between Rossby–Ertel's PV [X] and the EX component of Beltrami's material vorticity ω°X = EX · ω° as

 
Xω°X
(7)

which states that RossbyErtel's potential vorticity [X] is equal to the X component of Beltrami's material vorticity. Thus, all properties of PV for the field X can be deduced from the properties of Beltrami's material vorticity along the X direction.

The second condition for conservation of PV [(5b) or nɛ · curla = 0] becomes a component (thus a less restrictive condition) of the d'Alembert–Euler condition for circulation preserving motion [(6b), or curla = 0]. The d'Alembert–Euler condition and Cauchy's vorticity formula (dω°/dt = 0, conservation of Beltrami's material vorticity) are equivalent conditions for circulation preserving motion. Using the kinematic identity dω°/dt = β° we can relax the d'Alembert–Euler condition to one fixed direction in the reference configuration, say X, and deduce

 
°Xdtβ°X
(8)

which states, using the Hankel–Appel condition, that Ertel's PV conservation theorem for q̂[X] is equivalent to Cauchy's vorticity formula along the direction EX. This last equivalence theorem is of importance because it states that, when the sufficient conditions for the conservation of specific PV [ɛ] hold, results that hold for circulation preserving motions can be applied to particular directions and surfaces defined by the vector nɛ. Eckart (1960) rederived the conservation ω̇° = 0, that is, Cauchy's formula for circulation preserving motion, but did not notice the relation with Rossby–Ertel's PV conservation theorem. Using a restriction in the choice of material coordinates (namely, that dV = ρdυ) Salmon (1998, p. 202; see also Salmon 1982) noticed that relation with Cauchy's formula, but not with Beltrami's material vorticity.

The above correspondence between Rossby–Ertel's PV and Beltrami's material vorticity is true also when ε̂ is not conserved. In this case Ertel's PV balance equation may be interpreted as a generalized Cauchy's vorticity formula (as noticed in KoV, section 79) where the variable ɛ is not materially conserved (so it does not behave as a proper material variable). The rate of change of the X component of Beltrami's material vorticity can be written in a kinematic way (Viúdez 1999) as

 
°ε̂dtωĖε̂β°ε̂
(9)

where Eε̂ ≡ Gradε̂, ω0ε̂Eε̂ · ω°, and β°ε̂Eε̂ · β°. The vector Ėε̂ may be interpreted as the rate of change of stretching of the ε̂-coordinate lines in the referential configuration (i.e., with respect to material coordinates). It is still possible, therefore, to have dε̂/dt ≠ 0 and PV conservation when ω · gradε̂ = −curla · gradε̂ (strict necessary and sufficient condition for PV conservation). The above circumstance, though probably of reduced physical relevance, can also be interpreted in the present context, by stating that the ε̂ component of Beltrami's material vorticity (ω°ε̂) is conserved if and only if the change of ω°ε̂ due to the stretching of the ε̂ coordinate line in the referential configuration (i.e., with respect to fluid particles) is counterbalanced by the baroclinic–diffusive terms of vorticity along the ε̂-coordinate line.

b. Definition of Beltrami's material vorticity in terms of the referential velocity

It is important to stress that the expression ω° = JF−1ω, based on Piola's transformation [(2c)], may be considered a relation between ω° and ω instead of a definition of ω°. It is a useful relation since it allows one to compute ω° (or any of its components, and hence, specific PV) from J, F, and ω. In fact this is what is done when PV per unit mass, m[ε̂], is computed as αω · gradε̂. However, since ω° = JF−1ω is expressed as a function of the spatial vorticity, it is not a definition of ω° strictly in terms of material (referential) quantities. The same can be said from the relation ω° = Curl(FTu) because it depends on the spatial velocity u. Thus, it seems convenient to provide an alternative definition of ω° dependent only on material variables and on the referential velocity.

In order to do so let us first obtain a relation between u° and the referential velocity v. Let X = (x, t), with the function being the inverse function of for time t constant. The rate of change of fluid particles (or referential positions) that occupy the same place x during the motion is v ≡ ∂(x, t)/∂t. By means of the chain rule applied to the identity function ı with respect to x, that is, ι(x, t) = x,

 
formula

it follows that the relation between the referential velocity v and the fluid velocity u is u = −Fv (see, e.g., Šilhavý 1997, p. 36), and therefore the relation between u° and the referential velocity v is simply

 
uFTFvU2vCv
(11)

where U ≡ (FTF)1/2 is the right stretch tensor (symmetric) for the deformation, and CU2 = FTF is the right Cauchy–Green deformation tensor (symmetric). Thus, the expression

 
ωCv
(12)

provides an alternative definition of Beltrami's material vorticity using the referential velocity in terms of the material variables (X, t). The PV for the field ε̂ = X may therefore be alternatively defined as

 
ωXEXCv
(13)

where EX ≡ GradX. An expression for a° in terms of material variables is not so evident and is included in appendix B.

The components Cii(X, t) (see, e.g., Chadwick 1976, p. 68) are the square of the stretch undergone by a material line element situated at X and aligned with the direction defined by the base vector Ei, while Cij(X, t) (ij) is related to a pair of material line elements situated at X with orthogonal directions specified by Ei and Ej, being the product of the stretches suffered by these material line elements and the cosine of the angle between them in the present configuration. The components of u°(x, t),

 
formula

have the same interpretation in terms of the stretches (per unit time) experienced by the material line element that moves through the spatial point x in the unit of time. Let pi be the principal axes of U (they are termed referential stretch axes, and coincide with the principal axes of C) and λi be the corresponding proper numbers (the proper numbers of C are λ2i). In this orthonormal set of proper vectors the spectral decompositions U = λipipi, and C = λ2ipipi, and therefore u° can be written as u° = −λ2iυipi, where υi = pi · v are the components of v along the referential stretch axes.

c. Examples

Let us consider some examples of the consequences of the equivalence theorems given in section 3a.

1) Vector ω° and the stretch of material lines

As an application of the first theorem (only the first sufficient condition ε̇ = 0 is assumed) let ω = ωm, and ω° = ω°m°, where ω ≡ ‖ω‖, ω° ≡ ‖ω°‖, and m, m° are unit vectors. Recalling that λm = Fm°, λ2 = m° · Cm°, where λ is the stretch of a material line element along the unit vector m°, it follows (CN91) that

 
formula

Application of this result to one particular direction (defined by the unit vector EX) leads to

 
formula

which means that the specific PV can be interpreted as the specific vorticity () divided by the stretch of material elements along the vortex lines times the cosine of the angle ϑ(X, t) = EXm° between the material vortex-line direction and the particular direction EX.

2) Vector ω° and the area stretch

Let N and n be unit vectors normal to the corresponding area elements dS and ds in the reference and present configuration, respectively, and γ = ds/dS the area stretch. Then (CN91), we have

 
ωNγωn
(16)

Applying this result to the particular direction EX leads to

 
ω°XγωnX
(17)

where nX is the unit vector normal to the surface X = constant in the present configuration. The above expression means that the specific PV can be interpreted as the component of the vorticity normal to the X surface times the area stretch experienced by that surface.

3) Vector ω° and the directional circulation preserving motion

Application of the expressions (4) to one particular direction (defined by the unit vector EX) shows that the following results (that may be referred to as directional circulation preserving motion) are equivalent:

 
formula

Equation (18a) is a directional d'Alembert–Euler condition (or second sufficient condition for conservation of PV). Equation (18b) is a directional Hankel–Appell condition. These two state that there is no baroclinic–diffusive change of vorticity along the X-coordinate line. The last two are alternative ways of expressing the directional Cauchy's vorticity formula (conservation of ω°X), (18c) in terms of the X component of Beltrami's material vorticity (or conservation of specific PV), and (18d) in terms of the spatial vorticity. Thus, from (3) it is clear that conditions (18) imply that the flux of vorticity across a deforming area on an X surface (normal to nX at every point) in the present configuration remains constant in time, and therefore it preservers the circulation along the closed curve bounding the area.

The above result is the common interpretation of the conservation of PV in terms of the conservation of circulation on isentropic material circuits in potentiotropic fluids (where density depends only on pressure and entropy) as discussed by Gill (1982, p. 237), or in terms of the conservation of the flux of vorticity across a material surface X = constant whose normal is always perpendicular to the baroclinic vector gradρ × gradp, so that gradX · (gradρ × gradp) = 0 as discussed by Pedlosky (1987, section 2.5).

4) The PV and directional accelerationless motion

Material conservation of a quantity similar to PV, but simpler, results when the flow is accelerationless along one direction (say, along the vertical direction, = 0, which can be justified in hydrostatic flow). Let any quantity Z be materially conserved (Ż = 0) and the Jacobian Jx1,x2[f1, f2] ≡ f1,x1f2,x2f1,x2f2,x1. Since for any two material conserved quantities f1 and f2(1 = 2 = 0) we have d/dt Jx1,x2[f1, f2] = −Jx1,x2[f1, f2]divu, the specific Jacobian Ĵx1,x2[f1, f2] ≡ J Jx1,x2[f1, f2] is also a material invariant, and therefore d/dt{Ĵy,x[w, Z]} = 0. Let ϖ[Z] ≡ Jx,z[υ, Z] + Jz,y[u, Z]. Given the identity curlu · gradZ = Jy,x[w, Z] + Jx,z[υ, Z] + Jz,y[u, Z], the relation between PV density q[Z] and ϖ[Z] is q[Z] = ϖ[Z] + Jy,x[w, Z]. Therefore, when PV is conserved, the specific quantity

 
formula

is also materially conserved. While in (x, y, z) coordinates ϖ[Z] = (υ,xu,y)Z,z + (u,zZ,yυ,zZ,x), in Z coordinates [a nonorthogonal curvilinear coordinate system that considers Z as the vertical coordinate (assuming Z,z ≠ 0) and (x, y) remain unchanged (e.g., isentropic coordinates, Z being specific entropy θ)], ϖ[Z] is expressed in a simpler way as

 
formula

where ũ(x, y, Z, t) = u(x, y, z, t) at z = (x, y, Z, t), etc. Thus ϖ[Z] may be interpreted as the “Z-isosurface spatial vorticity” (υ̃,xũ,y) divided by the vertical line stretch (,Z).

The relation between PV per unit mass m[Z]) and ϖ[Z] per unit mass ϖm[Z] ≡ αϖ[Z]) is m[Z] = ϖm[Z] + αJy,x[w, Z]. For isentropic (θ̇ = 0), mass conserving [d/dt(ρJ) = 0] flow, in isentropic coordinates (θ,z ≠ 0), and assuming now the hydrostatic condition in the chain rule ,θ = p,z,θ = −gρz̃,θ, where g is the acceleration due to gravity, we obtain from (20)

 
formula

where (x, y, p, t) = θ(x, y, z, t) at p = p(x, y, z, t) (in isobaric coordinates p,z ≠ 0). Thus, for hydrostatic flow, ϖ[θ] per unit mass can be interpreted (obviating the constant factor) as the “isentropic spatial vorticity” (υ̃,xũ,y) times the vertical gradient of θ with respect to pressure (,p). Assuming p = p(α, ρ) [which holds, e.g., when an equation of state for the perfect gas = c1T, and the Poisson equation p = c2(T/θ)c3, for constants ci and temperature T, are assumed], we have grad θ · (grad α × grad p) = 0, and both specific PV and ϖm are materially conserved. The conservation of ϖ per unit mass dϖm/dt = 0, with ϖm given by (21), is the conservation given in, for example, Dutton (1976, section 10.2.2), though here it has been obtained in a different way from (20).

5) Vector ω° and Rossby PV

As a final example, consider the very simple case of isochoric and circulation preserving motion (J = 1, dω°/dt = 0) such that the horizontal velocity uh = uh (xh, t) is independent of z. Then xh = h(Xh, t) (a vertical material line element remains vertical along the motion). Since J = 1 we have λγ = 1, where λ = z,Z is the vertical line stretch, and γ = x,Xy,Yy,Xx,Y is the horizontal area stretch. Thus, from (15) and (17), and the condition dω°Z/dt = 0, where ω°Z is the vertical component of ω°, we have

 
ω°Zγωzωzλ
(22)

for every fluid particle. The result ωz/λ = constant is the local expression of Rossby's (1940) integrated equation ζa/D = constant along the motion. The result γωz = constant means that an increase (decrease) in the absolute value of the horizontal area stretch in the fluid element has to be accompanied by a decrease (increase) of the absolute value of the vertical component of the vorticity. These two results are two different ways of expressing the conservation of ω°Z, the simplest nontrivial particular case of Cauchy's material vorticity formula.

4. Concluding remarks

This work has shown that Rossby–Ertel's potential vorticity for the field ε̂ can be put into an exact correspondence with the component of Beltrami's material vorticity along the ε̂-coordinate line. Thus, results that hold for circulation preserving motion (curla = 0) hold also for the less restrictive condition nε̂ · curla = 0, only on ε̂-coordinate surfaces or along ε̂-coordinate lines. Rossby–Ertel's PV conservation theorem for the field ε̂, as a first example, is Cauchy's material vorticity conservation along the ε̂-coordinate line. An immediate consequence of the above is the equivalence between a generalized (vector) Ertel's PV [equal to J (gradϕ)ω, with ˙ϕ = 0] and Beltrami's material vorticity vector, and the equivalence between a generalized PV conservation theorem and a generalized Cauchy's formula.

The main results can be expressed in words in the following way. The term curla may be interpreted as the baroclinic–diffusive rate of change of vorticity, and its vector lines may be called the lines of baroclinic–diffusive change of vorticity, since it is along them that the vorticity is changed by the baroclinic or diffusive terms. In a circulation preserving motion (curla = 0) these lines do not exist. However, the condition curla = 0 can be relaxed to hold just along one particular direction, nε̂ · curla = 0, meaning that there is no baroclinic–diffusive change of vorticity normal to the ε̂ surfaces (baroclinic–diffusive change of vorticity taking place only on ε̂ surfaces). Since the rate of change of material vorticity only depends on the material baroclinic–diffusive vector (the convective terms belong only to the rate of change of spatial vorticity) Beltrami's material vorticity can only change by baroclinic–diffusive processes. If no baroclinic–diffusive change of vorticity along the direction nε̂ takes place (nε̂ · curla = 0) then the component of Beltrami's material vorticity along the ε̂-coordinate lines cannot change and it is therefore conserved. This component of Beltrami's material vorticity is called, in the current geophysical literature, potential vorticity.

The relationship between Rossby–Ertel's PV (and PV conservation) and Beltrami's material vorticity (and Cauchy's vorticity formula) seems to have been hiden in fluid dynamics in general (and in geophysics in particular) for many years. First, as mentioned in CN91, Beltrami in 1871 was the first to provide a formula relating the current velocity field with a time-dependent field ω° (i.e., Jω = Fω°) defined in the reference configuration of the continuum but, apparently, such a formula remained unnoticed ever since. Furthermore, as explained in KoV (section 94), Cauchy's vorticity formula (circulation preserving motion) “lay unnoticed for thirty years after its discovery [1815], and although both Stokes and Kirchhoff appreciated its central importance in classical hydrodynamics, even in recent expositions of the subject it is rarely given the prominence it deserves.” It is therefore quite probable that neither Rossby nor Ertel were fully aware of Beltrami's and Cauchy's contributions on these matters. The work by Casey and Naghdi (1991) seems to be the first one exploiting the Lagrangian description of vorticity based on Beltrami's formula. Other possible difficulties might have been that the concepts of PV density and specific PV (especially that PV is, in general, independent of the mass density), necessary for the understanding of the above results are unclear in the geophysical literature, and that the dyadic notation (preferred in fluid dynamics) differs from the tensorial notation used in modern books in continuum mechanics, especially in the study of deformation. It is hoped that this study, showing the intrinsic relationship between Rossby–Ertel's PV and Beltrami's material vorticity, will help Beltrami's material vorticity gain the recognition that it deserves.

Acknowledgments

I would like to thank D. Dritschel and three anonymous reviewers for their comments. Support for this research has come from the U.K. Natural Environment Research Council (Grant GR3/11899).

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APPENDIX A

The PV Density and Specific PV

For managing balance equations in integral form in a material volume it is useful to employ the kinematic identity

 
formula

For example, the conservation of material volume (identity) is expressed in integral form as

 
formula

where ;tzJ ≡ grad X = J−1, or, in local form, as d;tzJ/dt + ;tzJ div u = 0, or = J div u. Next, we consider some extensive fluid property having a spatial volume density η = η(x, t) obeying a balance equation

 
formula

where hη(x, t) is the true flux vector (in the sense that it does not include the advective term), n is the unit vector normal to the surface ∂υ, and zη(x, t) is the spatial volume density of the rate of supply of the balanced quantity. A balance equation expresses the time derivative of an extensive quantity contained in a volume in terms of its flux through the boundary and the external source of the quantity (see, e.g., Šilhavý 1997, chapter 3). We introduce also the specific value η̂Jη, that is, the amount of the quantity per unit of material volume. The local form of (A3) may therefore be written [using (A1)] in the following equivalent ways:

 
formula

Note that Eqs. (A4b) and (A4c) express the balance of the same quantity though using different fields (the spatial density η and the specific value η̂). This fact is of importance when dealing with the PV balance equation below.

The PV balance for q[ϕ] in the spatial description may be written, in a kinematic way and in the local form, as

 
formula

or, in integral form as

 
formula

where Q is the amount of PV, or simply, the PV (an extensive quantity). Thus, since q[ϕ] = div(ϕω), the amount of PV in a given volume is the flux of ϕω through the volume surface.

Assuming now the usual conditions leading to the conservation of PV (ϕ̇ = 0 and grad ϕ · curl a = 0) PV conservation can be written as

 
formula

The last two equations [as already commented in relation to (A4)] express the conservation of the same quantity (PV), though using different fields, namely, the spatial density q and the specific value q̂. It is therefore consistent to refer to the extensive quantity Q[ϕ] simply as PV for the quantity ϕ, q[ϕ] ≡ ω · grad ϕ as PV density, [ϕ] ≡ Jω · grad ϕ as specific PV (or PV per unit of material volume), and m[ϕ] ≡ αω · grad ϕ as PV per unit of mass (or also specific PV if desired). Note that the relationship between these three quantities is independent of the form of the flux vector hq and source term zq that one may decide to choose in (A5) or (A6).

APPENDIX B

Vector a° in Material Variables

Material time differentiation of (11) leads to a = −ḞvF. Since = Gradu = −Grad (Fv) and

 
formula

we have

 
formula

and therefore

 
formula

The above equation expresses the acceleration a° in terms of the referential velocity v and in material variables. The first term, independent of time derivatives of v and depending upon the relative position of the particles, their material derivatives, and the referential velocity v, may be considered the advective acceleration term (in the reference configuration) of a°. The second term, depending upon the second rate of change of (x, t), may be considered the local acceleration term (in the reference configuration) of a°.

Footnotes

Corresponding author address: Dr. Álvaro Viúdez, School of Mathematics and Statistics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland, United Kingdom. Email: alvarov@mcs.st-and.ac.uk