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Feb 15, 1999 - div(c1 c2) curlc1 · c2 curlc2 · c1. (4). We introduce also the kinematic equations of vorticity and the Beltrami equation in the spatial description.
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On Ertel’s Potential Vorticity Theorem. On the Impermeability Theorem for Potential Vorticity ´ LVARO VIU´DEZ* A Department of Meteorology, Naval Postgraduate School, Monterey, California (Manuscript received 8 November 1996, in final form 6 April 1998) ABSTRACT Potential vorticity (PV) is usually defined as av · gradf, where a is the specific volume, v is vorticity, and f is any quantity, usually a conserved one. The most common derivation of the PV theorem therefore uses the component of the vorticity equation normal to the f surfaces. Since PV can also be expressed as a div(u 3 gradf ) and a div(vf ), alternative derivations of the PV conservation law are introduced. In these derivations the PV conservation theorem is considered as the divergence of the projection (weighted by |grad f |) of the equation of motion onto the direction of gradf, or, alternately, as the divergence of a f -weighted vorticity equation. The first of these interpretations is closely related to the procedure of considering every f surface as a surface of constraint for the infinitesimal virtual displacements used in variational methods, and therefore it is closely related to a Hamiltonian derivation of the PV theorem. The different expressions are presented using the spatial as well as the material description of the fields. The kinematical foundations of the PV theorem in the material description are especially simple because they only involve derivative commutations with respect to the material variables. It is also provided a precise mathematical expression for the so-called impermeability theorem, clarifying the sense in which such a theorem can be understood. In order to do so it is necessary to introduce a suitable transformation of the fluid velocity. An immediate consequence of such a transformation is that the quantity f and the quantity v · gradf (also called potential vorticity substance per unit volume) behave as a label of the particles and as the ‘‘density,’’ respectively, of the transformed fluid. The impermeability theorem is then an expression of the conservation of the ‘‘mass’’ and of the conservation of the identity of the particles in the transformed fluid.

1. Introduction Since the discovery of the potential vorticity (PV) theorem by Ertel (1942a,b,c,d), PV has been extensively applied in the field of geophysical fluid dynamics (see, e.g., the review of Hoskins et al. 1985). This use of PV in the analysis of geophysical fluid motion has become so widespread that any step toward a better understanding of Ertel’s PV theorem (hereafter just PV theorem) represents, undoubtedly, an important step. Since a very important contribution to the understanding of a theorem resides in its derivation in addition to its simple exposition, those theorems based on a large and complicated proof appear obscure, whereas those based on a short and simple proof appear clear. Specifically one desires to make the derivation in a simple way and in

* Current affiliation: Departament de Fı´sica, Universitat de les Illes Balears, Palma de Mallorca, Spain. ´ lvaro Viu´dez, Departament de Corresponding author address: A Fı´sica, Universitat de les Illes Balears, 07071-Palma de Mallorca, Spain. E-mail: [email protected]

q 1999 American Meteorological Society

a general framework where a comparison with other developments in physics is possible. The PV conservation theorem has been derived in several different ways since Ertel’s original work in 1942. The derivation that appears in most text books (e.g., Dutton 1976, 344; Pedlosky 1987, 38) follows Ertel’s original derivation and is based on the appropriate factor multiplication and combination of the vorticity equation, the continuity equation, and the law for the rate of change of a general fluid property f, with all the above equations expressed in the spatial (or Eulerian) description. Some years after Ertel’s work, Truesdell (1951) introduced, from the Beltrami vorticity equation, the kinematic foundation of Ertel’s theorem also in the spatial description. From Truesdell’s kinematic identity, the PV conservation theorem follows directly by assuming certain dynamic and thermodynamic properties of the fluid. More recently Ripa (1981) and Salmon (1982, 1983, 1988) derived the PV theorem from Hamilton’s principle in the material (or Lagrangian) description. In this variational approach the conservation of PV corresponds, via Noether’s theorem, to the symmetry of the Lagrangian under particle-label variations that leave the density and entropy unchanged.

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Some years later, Haynes and McIntyre (1987, hereafter HM87) derived several conclusions about the evolution of PV substance (PV substance per unit volume is defined as v · gradf ) in the presence of diabatic heating and frictional forces. Their conclusions stimulated abundant further work (not always absent of certain controversy) in meteorology (Danielsen 1990; Haynes and McIntyre 1990; McIntyre and Norton 1990; Bretherton and Scha¨r 1993; Scha¨r 1993; Koshyk and McFarlane 1996) as well as in physical oceanography (Marshall and Nurser 1992; Csanady and Vittal 1996). The purpose of this work is to introduce other possible derivations of the PV theorem based on the principle of conservation of momentum, or derived from kinematic identities, and to show how new and already established results are related. We also provide some clarifying results concerning the so-called impermeability theorem developed by Haynes and McIntyre (1987, 1990). Most of the formulas are presented in both material and spatial descriptions thanks to an extensive use of the material description of vorticity and other kinematic theorems given by Casey and Naghdi (1991).

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TABLE 1. List of symbols. Symbol

Description

F 5 Gradx J 5 detF u 5 x˙ L 5 gradu v 5 curlu a 5 u˙ b 5 curla v8 5 JF 2T · v b8 5 JF 2Tb u8 5 F · u a8 5 F · a F* 5 JF 2T f F 5 gradf r a 5 r21 f 5 gradV p s S 5 grads J (ƒ1, ƒ 2, ƒ 3 ) P 5 av · S S8 5 Grads h 5 v·S

Deformation gradient Determinant of F Fluid velocity Velocity gradient Vorticity Acceleration Curl of the acceleration Material vorticity Material curl of a Transformed velocity Transformed acceleration Adjugate of F Generic scalar field Gradient of f Mass density Specific volume Extraneous specific force Thermodynamic pressure Specific entropy Gradient of s Jacobian of functions ƒ i Potential vorticity (PV) Material gradient of s ‘‘PV substance’’ per unit volume

2. Kinematical preliminaries In this section we introduce very briefly the symbol definitions and the basic kinematical notions. Basically we follow the notation in Truesdell (1954, hereafter KoV), Truesdell and Toupin (1960, hereafter CFT), Serrin (1959), and Casey and Naghdi (1991). The motion of a deformable continuum is described by the mapping x 5 x(X, t), where X and x denote, respectively, the position occupied by a typical particle in a fixed reference configuration and in the present configuration at time t. The coordinates X are the material coordinates, and the variables X, t are the material variables. The coordinates x are the spatial coordinates, and the variables x, t are the spatial variables. 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 or ( ),t , grad, curl, and div; whereas the differential operations with respect to the material variables are denoted d/dt or (˙), Grad, Curl, and Div. The deformation gradient (a second rank tensor) and its determinant are F 5 ]x/]X 5 Gradx and J 5 detF . 0, respectively (see Table 1 for a list of symbols). For any function f the expression f ,t and grad f shall denote the partial time derivative and the gradient of the function g(x, t) at a given t such that f 5 g(x, t). The expression ˙f and Grad f shall denote the partial time derivative and the gradient of the function G(X, t) at a given t such that f 5 G(X, t) 5 g(x21 (x, t), t); that is, ˙f 5 ]G(X, t)/]t (see, e.g., Truesdell 1991, 103). By the chain rule, the well-known formulas ˙f 5 f ,t 1 x˙ · grad f and f˙ 5 f,t 1 x˙ · gradf follow for scalar and vector fields, respectively. The expression Grad f 5

F · grad f provides a relationship between spatial and material gradients. The notations divf and Divf stand for the traces of gradf and Gradf, respectively. The particle velocity u 5 x˙ 5 ]x/]t. The velocity gradient L 5 ]u/]x 5 gradu. Note that F˙ 5 F · L. The vorticity, the acceleration, and the curl of the acceleration are v 5 curlu, a 5 u˙ 5 ]u(X, t)/]t, and b 5 curla, respectively. Following Casey and Naghdi (1991) we introduce the time-dependent vectors v8 and b8 by the Piola transformations v8 5 JF2T · v 5 p21{v}, b8 5 p21{b}. The vector v8, first introduced by Beltrami (see also KoV, §84, CFT, §86); provides a material description of vorticity and it will be referred to for short as the material vorticity. Recall that Divv8 5 J divv 5 0. The velocity and acceleration fields are transformed by u8 5 F · u and a8 5 F · a. Then it follows that Curlu8 5 v8 5 p21{curlu} and Curla8 5 b8 5 p21{curla}. The symbol 8 is •always attached to a single letter; thus u˙ 5 a but u˙8 [ u8 ± a8 (the overbar covers the quantity to which the dot operation is applied). Note also that v ˙ 8 5 b8 and Curlu˙8 5 Curla8. This last vector may be named the material diffusion vector, in analogy to the spatial diffusion vector curla (KoV, §84). This terminology has sense when the fluid acceleration is included in a specific dynamical equation (see next section). In the case of a homogenous fluid, the curl of the fluid acceleration depends only on the diffusive terms in the dynamical equations. We will make use of the relations (F · c1 ) 3 (F · c 2 ) 5 F* · (c1 3 c 2 ), div( ) 5 J 21 Div[F* · ( )], where the adjugate F* [ JF . 2T

(1)

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If the referential configuration corresponds to the fluid spatial configuration at some time t 5 t 0 , let v8(X) 5 0 v8(X, t 0 ), then

1

v (X, t) 5 J 21 v 08 1

E

t

t0

2

b 8 dt · F,

(2)

which may be called the basic vorticity formula since from it all properties of the vorticity are directly and easily derived (KoV, §85). The first term of the rhs of (2) represents the change in vorticity by convection, whereas the second term represents the change in vorticity by diffusion. For a function f we define F 5 gradf. We shall use very frequently the vorticity divergence identities

v · F 5 div(u 3 F) 5 div(vf ).

(3)

Relations (3)1–2 and (3)1–3 represent equivalently the most general ‘‘pure’’ vorticity divergence formulas in the sense defined by Howard (1958). [Subindices attached to an equation’s number refer to the different quantities in that equation separated by the symbol 5; e.g., relation (3)1–2 refers to the relation between terms (3)1 and (3) 2 .] Relation (3)1–2 is a particular case of div(c1 3 c 2 ) 5 curlc1 · c 2 2 curlc 2 · c1 .

(4)

We introduce also the kinematic equations of vorticity and the Beltrami equation in the spatial description

v ˙ 5 v · L 2 v divu 1 b, •

Jv 5 Jv · L 1 Jb.

(5) (6)

The first two and the last quantities on the rhs of (5) may be called the convective and the diffusive rate of change of vorticity, respectively (KoV, §83). Equation (6) is deduced from (5) by using divu 5 J˙ /J. Having introduced the kinematical vorticity equations (5) and (6), we need to introduce also the kinematical vorticity equation in the material description. This is simply the already mentioned relation

v ˙ 8 5 b8.

(7)

It states that the rate of change of material vorticity v8 equals the material diffusive rate of change of vorticity. Equation (5) may be obtained from (7) by replacing v8 and b8 in function of v and b. There are no convective rates of change in the material description as they exist only in the spatial description. This is because the advective derivative and local time derivative terms belong only to the temporal derivative of a function expressed in spatial variables. The basic vorticity formula (2) is, in fact, a direct integration of the material Beltrami vorticity equation. The absence of a convective rate of change in the material description of vorticity is very convenient since, quoting KoV (p. 155) ‘‘Particular models of continua . . . are defined by giving a particular functional form to the dependence of the acceleration x¨ upon other variables. Thus the process of diffusion is

509

different in each special type of continuum, while that of convection is the same for all. . . . The foregoing analysis justifies the statement that the dynamical specification of a medium consists in the statement of what conditions give rise to diffusion of vorticity and how great the quantity of diffusion then results.’’ The following conditions are equivalent

b 5 0,

b 8 5 0,

v 8 5 v 08,

Jv 5 v 08 · F. (8)

The first (8)1 is the d’Alembert–Euler condition. [When the same equation number refers to different equations separated by colons, the subindex i refers to the ith equation.] The second (8) 2 is the Hankel–Appell condition. The last two are alternative ways of expressing the Cauchy 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. Derivations of the PV theorem Since the PV theorem is a consequence of several physical assumptions, the different PV derivations depend basically on (i) the relative order in which these physical assumptions and algebraic operations are introduced, (ii) the way in which the motion is described (basically material or spatial descriptions), and (iii) how the physical information is presented (from the conservation of linear momentum, from a Hamiltonian principle, from the Lagrangian equations, etc.). From the physical point of view the importance of the different derivations resides in a large part in how the different sufficient physical conditions for the conservation of PV are introduced. Since the strictly sufficient and necessary conditions do not have to correspond to easily realizable physical assumptions, we shall focus on common sufficient conditions that are easy to grasp from the physical point of view. The sufficient and necessary conditions, which can be deduced from the kinematic approach (to be introduced below), do not have an easy physical interpretation. The necessary conditions that we shall require are that the fluid be perfect and that the flow conserves entropy. a. The PV theorem from the momentum equation 1) SPATIAL

DESCRIPTION

The most frequent way of deriving the PV theorem follows Ertel’s derivation and basically presents the PV conservation law from the combination of three equations in the spatial configuration. The first one is the Cauchy equation of motion for a perfect fluid, or Euler equation, a 5 f 2 a gradp, where a [ r

21

(9)

is the specific volume, f 5 gradV is the

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extraneous force per unit mass derivable from a potential V, and p is the thermodynamic pressure. [Here a is the fluid acceleration in an inertial frame.] The second is the continuity equation

a˙ 5 a divu,

(10)

and the third is the conservation of specific entropy s s˙ 5 0.

(11)

Although s in (11) can, in general, be considered as any tensorial quantity f (Ertel 1942d; Truesdell 1951) it will be considered here as the specific entropy field s. Considering f as a general tensorial quantity is useful in order to obtain the strictly necessary conditions for the conservation of the quantity av · gradf but, since we are more interested in the physical interpretation, we shall attach from the beginning a specific physical meaning to f by considering it to be the entropy s. In this way (11) may be considered the energy equation (the system is isentropic), which, together with the equation of state in the form p 5 p(a, s),

(12)

makes the system (9)–(12) have as many equations as unknowns (u, p, a, and s). Note that by virtue of (12) the Jacobian J ( p, a, s) is null J ( p, a, s) 5 (gradp 3 grada) · grads 5 0.

(13)

Let S [ grads. The appropriate manipulation of these equations leads to

a˙ v · S 5 av · S divu, av ˙ · S 5 2av · S divu 1 av · L · S, ˙ 5 2av · L · S. av · S

(14)

The first (14)1 is the continuity equation (10) multiplied by the v projection of S. The second (14) 2 is the aS component of the vorticity equation [the curl of (9)]. Here we have made use of (13). The third (14) 3 represents the av projection of the gradient of the rate of change of the conserved entropy s. The addition of (14)1,2,3 provides the conservation theorem for the potential vorticity P [ av · S ˙ 5 0. P (15) It is clear from the above (14)1,2,3 that this derivation of the PV theorem is based on the expression for PV (3)1 . This widely used derivation, based on the vorticity equation, might, however, have inhibited the consideration of other possible derivations. Another equally valid derivation is that based on the expression for PV (3) 2 . This alternative way consists first in eliminating from the analysis the component of the momentum equation (9) in the direction of S. This is done by taking the cross product of (9) with S a 3 S 5 f 3 S 2 a gradp 3 S.

(16)

This equation represents the tangential projection

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(weighted by S) of the Euler equation (9) onto the direction of S. The term tangential projection is used here, up to a minus sign, in the sense described in KoV (§4) or Ericksen (1960, 315), and it is not to be confused with the projection of a vector onto the plane normal to S (Truesdell 1991, 315). Thus, the components of the extraneous force f and the pressure gradient (and hence of the acceleration) in the direction of S are eliminated from the analysis and only the isentropic component of these terms remain. The next step is to eliminate from the analysis both terms on the rhs of (16). This is done by taking the divergence of (16) because of the identity (4) and the fact that div(curlc) 5 0. Thus we obtain div(a 3 S) 5 J( p, a, s) 5 0,

(17)

with the last result following from (13). Taking into account the relation •

a div(a 3 S) 5 a div(u 3 S)

(18)

[which only assumes s˙ 5 0 and the continuity equation (10), see the appendix], we obtain the conservation theorem (15) for P 5 a div(u 3 S). Note that in this derivation no use of the vorticity equation is made (we have emphasized this fact by writing P in the divergence form, which does not include explicitly the vorticity vector). This derivation, specifically the term (18)1 , shows that the PV conservation equation can be alternatively considered as the divergence of the projection (weighted by S) of the Euler equation onto the direction of S. Therefore, since the S component of the Euler equation is, in fact, not included in the analysis, the PV conservation theorem may be conveniently handled by the introduction of isentropic coordinates (coordinates for the s surfaces or, in the most general case, coordinates for the f surfaces). This derivation, and in particular (17), shows that any extra term appearing on the rhs of the momentum equation (9), for example a viscous term, will appear as an extra divergence term in the equation for the rate of change of P. This fact clearly shows that the local time derivative of v · S can always be written as a divergence term, a result obtained in a different way by Haynes and McIntyre (1987). 2) MATERIAL

DESCRIPTION

As already mentioned, all the above derivations may be formulated in the material description. This may be carried out with the use of the vectors u8, a8, v8, introducing the material description of S by S8 [ Grads 5 F · grads 5 F · S, and with the help of the relations (2). For example, the Euler equation (9) in the material description is a8 5 GradV 2 a Gradp,

(19)

(see, e.g., Serrin 1959, 136; Hollmann 1964). The material equation of continuity (20) can be written in the following ways

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r J 5 0,

r J 5 ro,

(20)

(e.g., Serrin 1959, 133; CFT, §156). The scalar product of the Curl of (19) with S8 [ Grads is

b 8 · S8 5 (Gradp 3 Grada) · Grads 5 JJ ( p, a, s). (21) The potential vorticity in the material description is v8 · S8 because Jv · S 5 J(J 21 F T · v o ) · (F21 · S o ) 5 F 2T · (F T · v o ) · S o 5 v o · S o .

(22)

b. The kinematic approach Although the PV conservation is not a kinematic identity, we refer to this type of derivation as the kinematical approach since the application of the dynamic assumptions is the last step in this method. These derivations lead to what may be interpreted as the kinematical equivalent of the Ertel PV theorem. The kinematic identities of the Ertel’s PV theorem in the spatial description may be written in correspondence to identities (3) in the following equivalent ways

• ˙ o 1 bo · Fo, vo · Fo 5 vo · F •

˙ o ) 1 Div(a o 3 F o ), Div(u o 3 F o ) 5 Div(u o 3 F Div(v o f) 5 Div(v of ˙ ) 1 Div( b o f).

(24)

•

˙ [Grad In the above we have used Gradf ˙ 5 Gradf 5 F8 and (˙) commute], the rule for the derivative of a scalar product, and the identities v ˙ 8 5 b8, Curlu˙8 5 Curla8, and CurlF8 5 0. (Recall that u˙8 5 a8 1 Gradu · u 5 a8 1 F · L · u 5 a8 1 F˙ · u.) The terms in each column in these equations are related through identities of the type (3) but applied in the material description. Note that one of the advantages of working in the material description with expressions involving material time derivatives is that differentiations with respect to the material variables (X, t) commute, in the same way that differentiations with respect to the spatial variables (x, t) do. From (24)1 it is evident that the identity (23)1 in the material description is simply the rule of differentiation of the scalar product v8 · F8 and the use of (7). ˙ 5 By applying the condition f ˙ 5 0 (which implies F8 0), the terms in the second column in (24) are zero. Using the F8 component of the Curl of the Euler equation (21) and the thermodynamic condition J ( p, a, s) 5 0, the terms in the third column are also zero, from which the conservation of PV in the material description follows. The two formulas (23) 2 and (24) 2 play the same role in the derivation introduced in the previous subsection as (23)1 and (24)1 in the derivation of the PV theorem from the vorticity equation. c. Relation to the derivation from Hamilton’s principle

•

Jv · F 5 Jv · gradf ˙ 1 Jb · F, •

J div(u 3 F) 5 J div(u 3 gradf ˙ ) 1 J div (a 3 F), •

third column are also zero. Then using the material equation of continuity (20) in the first column, we obtain the PV conservation theorem (15). The material description of (23) is

•

A result that may be also obtained using (2) in a more convoluted way vo · So 5 Div(u 3 So) 5 Div[(F · u) 3 (F · S)] 5 Div[F* · (u 3 S)] 5 J div(u 3 S) 5 Jv · S [see also the derivation in Salmon (1982)]. Therefore the conservation of PV can be written • ˙ o 5 b o · So 1 v o · S ˙o 5 0 vo · So 5 v ˙ o · So 1 vo · S by virtue of (21), (13), (11), (7). Having presented the above instructive examples we will not continue with this method of using first the dynamical/thermodynamical equations and then their algebraic manipulations. In order to present the results in a more ordered and concise way the dynamical/thermodynamical laws may be used as the last step of the derivation process. This is done in the next subsection.

J div(v f ) 5 J div(vf ˙ ) 1 J div( bf ),

511

(23)

or in other possible combinations resulting from choosing one term in every column of these equations. Equation (23)1 was first obtained by Truesdell (1951; KoV, §79), who chose as the starting point the vorticity diffusion equation of Beltrami in the spatial configuration (6). The different terms in every column of (23) are related through identities of the type (3). A direct derivation of (23) 2 is included in the appendix. Although the above identities (23)1,2,3 contain no dynamic information, they have been derived so as to easily incorporate the physical assumptions leading to the PV conservation theorem. If f is a conserved property of the fluid (f ˙ 5 0), the terms in the second column of (23) are zero. Using (13) together with (9), the terms on the

While the kinematical approach is independent (up to the last step) of any dynamical equation, the other derivations presented in subsection 3a started from the momentum equation (9). However, since classical mechanics can also be founded in different principles not including the principle of conservation of momentum (e.g., variational principles of d’Alembert, the Lagrangian equations, Hamilton principle, etc.), the PV conservation theorem can also be derived in alternative ways from these principles. [A discussion about the relevance of variational principles in continuum mechanics may be found for instance in CFT (section 231) and Salmon (1988).] The objective of this subsection is to briefly relate the derivations of the PV theorem in the previous subsections with those obtained from Hamilton’s principle (Ripa 1981; Salmon 1982, 1983, 1988). Here we follow the review in Salmon (1988), who distinguished between homentropic and nonhomentropic

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fluids. For a homentropic fluid (the entropy field is homogeneous) he derived from a Hamiltonian principle the equation

v ˙ 8 5 0.

(25)

˙ 5 0. The If f is a conserved quantity, f ˙ 5 0, then F8 PV theorem then follows in the material description • ˙ 5 v8 · F8 5 0. The relation with the derivations in P the previous subsections is clear if we note that for a homentropic fluid gradp 3 grada 5 0 (i.e., the flow is barotropic). Therefore from the Euler equation (9) b 5 curla 5 0, then the d’Alembert–Euler condition holds, and especially from (8) 3 follows (25), which is equivalent to the Cauchy vorticity formula (8) 4 . This derivation, which is a derivation of (25) rather than a derivation of the PV theorem, is therefore related to the derivation in sections 3a and 3b based on (3)1 because it is based on the vorticity equation, although in this case it is applied to the more restrictive case of circulation-preserving motion (8). Palmer (1988) also considered barotropic flow. Using his notation for v9 [ Jv and for the Lie derivative LUv9 5 v9 ,t 1 u · gradv9 2 v 9 · L, from the Beltrami formula (6) and the d’Alembert–Euler condition (b 5 0) follows that LUv9 5 0, which may be considered a way of expressing (25) in the spatial description. In the derivation for nonhomentropic flow, since s˙ 5 0 (and it is assumed everywhere that Grads ± 0), a material coordinate of X (or particle label) can be identified with s (say a rectangular Cartesian material coordinate Z 5 s). Then the PV theorem (15) as derived in Salmon (1988) follows in the material description from the variational Hamilton’s principle. This conservation theorem corresponds to the Hamiltonian symmetry associated with the virtual displacements in the referential space (or particle label variations) that satisfy the constraint of not modifying the density and the constraint of remaining within surfaces of constant entropy. This second derivation corresponds to the second derivation presented in subsection 3a [i.e., based on (3) 2 ] because only the component of the momentum equation tangent to the s surfaces is taken into account from the beginning (particle label variations implying entropy changes are excluded). In the d’Alembert’s differential variational principle (e.g., Vujanovic and Jones 1989, 7) this approach corresponds to considering only the virtual work due to infinitesimal virtual displacements dr (or variations for a fixed time) compatible with the constraint s˙ 5 0. This can be done by taking the cross product of the equation of motion with virtual displacements of the form SS21 dr, in an analogous way to (16). Note that s˙ 5 0 may be interpreted as a constraint to the more general flow described by (9) or (19). This is an integrable constraint and therefore it can be reduced in the material description to an equality depending only on position and time, becoming equivalent to a conventional holonomic coordinate constraint in the form

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f 9 i (x, t) 5 f (x, t) 2 f i 5 0, which is the equation of the f i surface (see, e.g., Santilli 1978, 220). 4. The impermeability theorem (IT) for PV In the above section I have adopted the usual sufficient conditions for the conservation of PV. These sufficient conditions, expressed in Eqs. (9)–(12), include the conservation of mass, the conservation of entropy s, and the absence of viscous terms in the conservation of momentum (9). It is also clear that, in the general case, when s˙ ± 0 and viscous terms are included in the ˙ ± 0). balance of momentum, PV is not conserved (P Whether the rate of change of the nonconserved PV is due to a flux or to a source appears to be a matter of definition. This is because of the well-known tautologic character of the general balance or general conservation law, which derives in the equivalence of surface and volume sources (e.g., CFT; §157; Truesdell 1991, §iii.5). Let us define h [ v · F. ‘‘PV substance’’ (PVS) per unit volume is defined as v · S. (Defining PVS per unit volume as rP, though obviously equal to v · S, is not appropriate since it makes explicit a dependence with the density that does not exist; i.e., r is not needed for such a definition.) The impermeability theorem (IT) states that, in some sense, (i) ‘‘there can be no net transport of PVS across any isentropic surface’’ and that (ii) ‘‘PVS can neither be created nor destroyed within a layer bounded by two isentropic surfaces’’ (HM87). The objective of this section is to clarify the sense of the IT. We will see next that the ‘‘proofs’’ that there is no flux of PVS across isentropic surfaces (HM87) are simply a consequence of assigning, to fictitious particles of PVS, a velocity field whose component normal to the f surface coincides with the speed of displacement of the f surface. In order to clarify this point, it is convenient to introduce the rate of change of a field function f measured by an observer moving with an arbitrary velocity v, dv f/dt 5 f ,t 1 v · grad f 5 ˙f 1 (v 2 u) · grad f. (26) Note that v, being arbitrary, does not have to be equal to the fluid velocity u. From this definition and the vorticity equation (5), we can obtain the kinematic identities d vh /dt 1 h divv 5 h˙ 1 h divu 1 div[(v 2 u) h] 5 div[(v 2 u)h 1 vf ˙ 1 a 3 F] 5 div{[(v 2 u) ` v] · F 1 v d vf /dt 1 a 3 F} 5 div{v d vf /dt 1 [v 3 (v 2 u) 1 a] 3 F}, (27) where c1 ` c 2 5 c1c 2 2 c 2c1 is the outer product of the vectors c1 and c 2 (e.g., Hestenes 1993). Since c · (c1 ` c 2 ) · c 5 0 for any vectors c, c1 , and c 2 , we observe that only the vectorial term (v dv f /dt) inside the divergence of (27) 4 has a nonzero component in the direction of F, that is, in the direction normal to the f

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surfaces. This can be also observed by noting that (c1 ` c 2 ) · c 5 (c 2 3 c1 ) 3 c, so [(v 2 u) ` v] · F 5 [v 3 (v 2 u)] 3 F as it is written in (27) 5 . Relation (27)1–3 , or (27)1–4 , is the kinematic equivalent, in differential form, of the integral equation (4.5) in HM87. We integrate (27) 1–4 over a volume V (t), bounded by a surface ]V (t), formed by fictitious particles whose velocity is v. We need to apply the formula dv dt

E

f dV 5

V (t)

E

(d v f /dt 1 f divv) dV

(28)

V (t)

(CFT, §81), where here dv /dt indicates that the volume of integration is material with respect to the velocity v [formula (26)1–2 cannot be applied directly here because the integral is not a function of x]. The result of that integration is the equation (4.5) in HM87, which properly written is dv dt

E R

h dV

V

2

{v d vf /dt 1 [v 3 (v 2 u) 1 a] 3 F} · ds 5 0.

]V

(29)

In the above integration we have assumed that v is an arbitrary well-behaved flow. Thus it satisfies a one-toone mapping of the form x 5 x9(X9, t), and therefore fictitious h particles are never created nor destroyed. For any partial surface belonging to ]V (t) coincident with a f -surface ]V f , we have ds 5 F21Fds, and therefore the surface integral (29) on a ]V f surface can be written If 5 2

E

F h d vf /dt ds. 21

(30)

]Vf

If v is defined as to fulfill dv f /dt 5 0,

(31)

then I f 5 0 and the integral (29) need only to be computed on non-]V f surfaces. Note that the condition (31) can be used to define only the component of v in the direction of F, that is, normal to the f surface. This component is also called the normal velocity or the speed of displacement of the f surface (e.g., Serrin 1959, 137; CFT, §177; Truesdell 1991, 106). This result can be stated as follows: if fictitious h particles are defined as those having a velocity v such that dv f /dt 5 0, then the rate of change of h of a moving volume of h particles bounded by a surface ]V depends only on the values of the fields in the non-]V f surfaces of ]V. Note that this result is purely kinematic, and that, in fact, it could be used alternatively as a definition of the component of v in the direction of F. A large part of the controversy associated with this and similar statements (Danielsen 1990; Haynes and McIntyre 1990, hereafter HM90) is, in my opinion, a consequence that

513

(i) Eq. (29) was not originally formulated in a kinematic way (HM87 started this derivation from the momentum equation), (ii) the differential form (27)1–3 was not provided, and (iii) the symbol d/dt was used in the first term of the integral equation (29) instead of dv /dt. Note that we can employ neither d/dt (i.e., dv /dt for v 5 u) nor ]/]t (i.e., dv /dt for v 5 0) in the rhs of (29) because the control volume chosen in HM87 was neither material nor spatial. This is because it is assumed that the f surfaces do not have to be either material or steady. Evidently the above result can be expressed in a simple and general way that clearly manifests its tautologic character: fictitious particles of XXX having a velocity v such that dv f /dt 5 0 never cross f surfaces. If XXX is PVS and f is entropy, we obtain IT(i). This statement is, in fact, equivalent to the well-known feature that the normal velocity of a f surface is y n 5 2|F|21 f ,t . As already mentioned, the condition (31) only defines the component of v normal to the f surfaces. The component of v tangent to the f surfaces was implicitly defined in further works (HM90; McIntyre and Norton 1990) in order to make the component of the vector in the divergence of (27) 3,4 that is tangent to the f surface also zero. Then, since both normal and tangent components have to be zero (and it is assumed that F ± 0 everywhere), this vector in the divergence of (27) 3,4 has to be the vector zero. Thus, the vector field v that satisfies this condition is simply v 5 u 2 vf ˙ h21 2 a 3 Fh21

(32)

(it is assumed that h ± 0). Then for this defined velocity field v we have from (27)1–3 the equivalent expressions d vh /dt 1 h divv 5 0,

h ,t 1 div(hv) 5 0.

(33)

These equations state that h behaves as the ‘‘density’’ of the fictitious flow v. Or, in other words, the role that h plays in the flow v is equivalent to the role that r plays in the real flow u. The above statement may be considered the basis of the analogy between the fictitious fluid and flow (h, v) and the real one (r, u). Note that h is a pseudoscalar and that it is indefinite (not positive definite); however, since (32) assumes h ± 0, it is also assumed that the density of an h particle never changes sign. Integration of (10) and (33) leads to the integral expressions d dt dv dt

E E

r dV 5

V

V

h dV 5

d dt dv dt

E E

dM 5 0,

B

dM9 5 0.

(34)

B9

Equation (34)1 expresses the conservation of mass of a body B of fluid. The second (34) 2 also establishes the conservation of ‘‘mass’’ dM9 5 h dV of a body B9 of the fictitious fluid of PVS. Recall that the volume in (34) 2 refers to a volume that is material with respect to v. Equation (34) 2 includes IT(ii) since the f surfaces

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are material with respect to the velocity v. It is in the sense of (34) 2 that the rate of change of ∫ h dV [Eq. (5.1) in HM90] does not change. This rate of change is different from the rate of change in (34)1 . To provide the kinematic meaning of (31) we recall that the flow u satisfies also the conservation of the identity of the particles ˙ 5 0, X (35) (e.g., Serrin 1959, 149). Equation (35) is not very often used because most applications only require solving for the velocity field, however (35) is necessary in order to obtain the trajectories of the fluid particles. Equation (31) establishes that f is a label of the fluid particle whose flow is v. What is the integral form of this conservation equation (31)? Differential and integral forms of conservation laws are related through the continuity equation. However, v does not satisfy a continuity equation with density r (10) but with density h (33)1 . Then from (28) and (33)1 the integral form of (31) is simply dv dt

E

V

hf dV 5

dv dt

E

f dM9 5 0.

(36)

B9

From the above results we conclude that the IT is basically the conservation of mass of a fluid having a particle label f, density h, and velocity v. Finally, the IT is sometimes enunciated by saying that a vector j exists such that the conservation of h can be written in the form

h,t 1 divj 5 0.

(37)

Note that, since the local and the advective derivative of h appear in (37) in different terms, the terms in this expression are not Galilean invariant. On the other hand, Ertel (1960b) demonstrated that the material derivative of any function f can be expressed as a spatial divergence times the specific volume of the fluid. His derivation starts from expressing f as the Jacobian of three suitably chosen functions f i , i •5 1, 2, 3, times the specific volume; that is, ˙f 5 aJ (f i ) 5 a divb, where b is a function of f i . In the case of PV, this may correspond to the choice f 3 5 s, and the representation of the vorticity v 5 gradf 1 3 gradf 2 , where f 1 and f 2 are the Monge’s potentials (e.g., Ericksen 1960, §35; in general this representation enjoys only a local validity). In the case of h, and in local form, this result is simply [J (f i )],t 1 div[uJ (f i ) 2 b] 5 0, which has the form (37). The basis of Ertel’s procedure is that any function f can be written as the Jacobian of three suitably chosen functions f i , and that the rate of change of the Jacobian J (f i ) can be written as J (f i ) times the divergence of a vector function b that depends on f i . In the case of barotropic flow the Euler equation (9) implies (8) and then it is always possible to choose f 1 and f 2 such that f˙ 1 5 f˙ 2 5 0 (see Clebsh transformation, KoV, §101; CFT, §136). Then the conservation of PV can be easily derived since d/dt[(aJ (f 1 , f 2 , s)] 5 aJ 21 d/dt[(Gradf 1 3 Gradf 2 ) · Grads] 5 0, because s˙ 5 0, commutation

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of Grad and d/dt operators and the continuity equation (see Ertel 1960a; Hollmann 1964). Let us write the different kinematical identities involving h,t . These are (v · F) ,t 5 v · F ,t 1 v ,t · F, [div(u 3 F)] ,t 5 div(u 3 F ,t ) 1 div(u ,t 3 F), [div(vf )] ,t 5 div(vf ,t ) 1 div(v ,t f ),

(38)

or other possible combinations resulting from choosing one term in every column of these equations. Equation (38) 3 was introduced by Bretherton and Scha¨r (1993). The identity presented by HM87 [their Eq. (5.5)] is a combination of the third term in (38) 2 and the second term in (38) 3 . Now, if these kinematic identities, which involve only the local change of h and commutation of derivatives with respect to the spatial variables (x, t), are compared with the equations for the rate of change of P in the material description (24), we observe that they have. the same algebraic structure. Recall that the symbol ( ) is also a partial derivative of a function expressed in material variables (X, t). This correspondence is a consequence of the ‘‘principle of duality’’ (CFT, §14). When minuscules and majuscules are used to denote the components of vectors and tensors with respect to basis vectors in the spatial and material description, respectively, as well as derivatives with respect to spatial and material variables (e.g., gradf 5 f ,ie i and Gradf 5 f ,AE A ), the principle of duality establishes that in any given equation, majuscules and minuscules may be interchanged. This means that when we derive (38) and (24) we follow the same algebraic operations, although in different spaces (material and spatial). Therefore the identities for h involving local time derivatives in the spatial description, are the duals of the identities for the rate of change of P in the material description. This principle is purely formal, the meaning of the two sets of equations being different. Sufficient conditions for the PV conservation can be easily derived from (24) but not from (38). 5. Discussion and conclusions In this work I have introduced, in the spatial and material description, alternative derivations of the PV theorem. The most common derivation is based on (3)1 and uses the component of the vorticity equation normal to isentropic surfaces. In an alternative derivation, based on (3) 2 , the PV conservation law is basically considered as the divergence of the projection (weighted by S) of the momentum equation onto the direction of S. This interpretation is closely related to the procedure of considering every f surface as a surface of constraint for the infinitesimal virtual displacements used in variational methods, and therefore it is closely related to a Hamiltonian derivation of the PV theorem. Another divergence-based derivation is related to (3) 3 . It considers

15 FEBRUARY 1999

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the PV conservation law as the divergence of a f weighted vorticity equation. Since there are no convective rates of change in the material description, one could think that the rate of change of vorticity due to stretching and rotation of vortex lines disappears. Strictly speaking the vorticity equation states that curlu˙ 5 curlf g and Curla8 5 curlf g8 (in the spatial and material description, respectively). Here f g and f g8 stands for the specific total force. Since in the material description the operator d/dt and spatial derivatives do not commute, one may express curlu˙ as v ˙ 1 v divu 2 v · L (5). The case curlf g 5 0, is of special importance because then curlu˙ 5 0 [the motion is circulation-preserving, see (8)] and then the balance v ˙ 1 v divu 2 v · L 5 0 occurs. If furthermore the fluid is incompressible (divu 5 0), this balance is simply reduced to v ˙ 2 v · L 5 0. This is a useful relation because if it is known, at one spatial point x and at time t, that v · L ± 0 then we can deduce that the vorticity of the fluid particle being in that point x at time t is changing. Conversely, if it is observed that the vorticity of the fluid particle being at point x at time t is changing, we can deduce that v · L ± 0 at that point x at time t. But, it is probably not correct to assign a cause–effect relation to this balance saying that, for example, v ˙ is due to v · L (the opposite relation, that v · L is due to v ˙ , could also be equally taken). Thus, though observation of v · L is a useful way to deduce v ˙ (in large part because the conditions curlf g 5 0 and divu 5 0 are approximately found in nature), v · L is not the physical cause of the rate of change of v. The balance v ˙ 2 v · L 5 0 belongs only to the material description. It is, in part, a consequence of having adopted the spatial description because it is derived from the noncommutation of d/dt and spatial derivatives. In the material description such a balance has no sense, but this does not mean that any real physical effect is missing. In the material description the circulating-preserving condition is instead visualized [see (2) and (8)1,2 ] by noting that v8 5 v80 , so the material vorticity vector of every particle (i.e., at every point of the X space) does not change with time. The sufficient conditions for the conservation of PV are the common ones, expressed in Eqs. (9)–(12), and include the conservation of mass, the conservation of entropy s, and the absence of viscous terms in the conservation of momentum (9). The sufficient and necessary condition can be obtained by equating to zero the rhs of identities of the type (23) and (24) after replacing a and f ˙ by their corresponding expressions in the physical laws for these quantities, but in general this condition does not have an easy physical interpretation. The kinematical foundations of the PV theorem in the material description are especially simple because they involve only derivative commutation and application of the identities v ˙ 8 5 b8 and Curlu˙8 5 Curla8. This simplicity is due to the absence of ‘‘advective

515

terms’’ in the expressions for the rate of change of f and vorticity. A final comment about the PV theorem concerns its interpretation. The conservation of PV under certain conditions provides a useful tool for obtaining material changes in one of the quantities a, v, or grads in terms of the values and material changes of the others. In the description of oceanographic or meteorological processes, the PV conservation is sometimes used to explain or infer the cause of motion, for example, saying that changes in v are a consequence of the changes in a grads. There is however nothing in the derivation of PV that supports such an interpretation. A precise mathematical formulation that clarifies the sense in which the IT can be understood has been also provided in this article. The mathematical formulation of this theorem is Eqs. (31) and (33)1 in the differential form, and Eqs. (36) and (34) 2 in the integral form. In order to provide the mathematical expression of this theorem it is necessary to introduce a fictitious flow v, given by (32). As a consequence of this definition the quantity v · F behaves as the density and f behaves as a particle label of the transformed fluid. Although this fictitious flow v was in different ways mentioned in some previous works (Haynes and McIntyre 1990; Xu 1992; Bretherton and Scha¨r 1993), it was not considered a basic requirement for the IT. It is for this reason that this theorem was, to my knowledge, never explicitly expressed in mathematical terms. Had the IT ever been expressed in mathematical terms, the field v would have explicitly appeared in the formulation. I emphasize therefore that the introduction of the flow v is not an optional interpretation, or a form of envisioning, imaging, or picturing the IT, but that this theorem is a direct consequence of the introduction of such a flow. The definition of v by (32) is, in fact, a particular transformation of the velocity field u. The IT belongs to this transformed flow, not to the original flow u. An immediate consequence of this transformation is that the quantity v · gradf (or PVS when f is specific entropy) behaves as the transformed density, and f behaves as a particle label of the transformed fluid. Any other transformation of the form v 5 u 2 vf ˙ h21 2 (u˙ 2 f gradl ) 3 Fh21

(39)

with l an arbitrary scalar field, will satisfy also the IT because the field f gradl 3 F is tangent to the f surfaces and because it is nondivergent. Equation (39) defines therefore the class of transformations of the velocity field that satisfies (31) and (33)1 . Thus the choice of v is not unique. A specific choice of l determines the gauge. Since no special conditions have been invoked in the derivation of the above results, it is clear that the IT is strictly a kinematic theorem [as was already noted in HM87 and Bretherton and Scha¨r (1993)]. Any reference in its derivation to forces, friction, heating, or thermodynamic quantities is therefore superfluous if not confusing. Transformation (39) can now be applied to

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the system of equations (9)–(12) with f ˙ equal to s˙ 5 0. By choosing the gauge f gradl 5 a [this is an allowed gauge because, due to (13), a satisfies F · curl(f gradl ) 5 0, i.e., S · curla 5 0], these equations are sufficient for the flow v to be equal to the fluid flow u. Therefore the sufficient conditions for the conservation of PV, included in (9)–(12), are also sufficient for v to be the fluid flow u. Since the IT basically states that the ‘‘mass’’ of the transformed fluid is conserved and that f is a particle label of the transformed fluid (PVS), the usefulness and advantages of working with the transformed field v instead of with u still have to be demonstrated. Acknowledgments. I am grateful to Prof. Haney (NPS) for his comments and corrections that have considerably improved this work. Partial support for this study was obtained through a postdoctoral grant from the Ministerio Espan˜ol de Educacio´n y Ciencia and through the Office of Naval Research. APPENDIX Direct Derivation of (23) 2 •

In order to relate div(a 3 F) with div(u 3 F) directly from the acceleration it is easier to start from the Lagrange’s formula because it avoids the use of the dyadic L: a 5 u,t 1 v 3 u 1 gradu 2 /2. Taking the component of this equation normal to F and computing its divergence we obtain div(a 3 F) 5 div(u ,t 3 F) 1 div[(v 3 u) 3 F] 5 [div(u 3 F)] ,t 2 v · F ,t 1 v · L · F 1 u · gradv · F 1 v · F divu 5 [div(u 3 F)] ,t 2 div(u 3 gradf )1 u · gradF · v 1 u · gradv · F 1 v · F divu •

5 div(u 3 F) 2 div(u 3 gradf ˙) 1 div(u 3 F) divu, where we have used u · gradF · v 1 u · gradv · F 5 u · grad(v · F) 5 u · grad[div(u 3 F)]. Using divu 5 J˙ /J we obtain the kinematic identity (23) 2 . REFERENCES Bretherton, C. S., and C. Scha¨r, 1993: Flux of potential vorticity substance: A simple derivation and a uniqueness property. J. Atmos. Sci., 50, 1834–1836. Casey, J., and P. M. Naghdi, 1991: On the Lagrangian description of vorticity. Arch. Ration. Mech. Anal., 115, 1–14. Csanady, G. T., and G. Vittal, 1996: Vorticity balance of outcropping isopycnals. J. Phys. Oceanogr., 26, 1952–1956.

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Danielsen, E. F., 1990: In defense of Ertel’s potential vorticity and its general applicability as a meteorological tracer. J. Atmos. Sci., 47, 2013–2020. Dutton, J. A., 1976: The Ceaseless Wind. An Introduction to the Theory of Atmospheric Motion. McGraw-Hill, 579 pp. Ericksen, J. L., 1960: Tensor fields. Handbuch der Physik III/1, S. Flu¨gge, Ed., Springer-Verlag, 902 pp. Ertel, H., 1942a: Ein neuer hydrodynamischer Erhaltungssatz. Naturwiss., 30, 543–544. , 1942b: Ein neuer hydrodynamischer Wirbelsatz. Meteor. Z., 59, 277–281. ¨ ber das Verha¨ltnis des neuen hydrodynamischen Wir, 1942c: U belsatzes zum Zirkulationssatz von V. Bjerknes. Meteor. Z., 59, 385–387. ¨ ber hydrodynamische Wirbelsa¨tze. Phys. Z., 43, 526– , 1942d: U 529. , 1960a: Teorema sobre invariantes sustanciales de la Hidrodina´mica. Gerl. Beitr. Geophys., 69, 290–293. , 1960b: Relacio´n entre la derivada individual y una cierta divergencia espacial en Hidrodina´mica. Gerl. Beitr. Geophys., 69, 357–361. Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional and other forces. J. Atmos. Sci., 44, 828–841. , and , 1990: On the conservation and impermeability theorems for potential vorticity. J. Atmos. Sci., 47, 2021–2031. Hestenes, D., 1993: New Foundations for Classical Mechanics. Kluwer Academic, 644 pp. Hollmann, G., 1964: Ein vollsta¨ndiges System hydrodynamischer Erhaltungssa¨tze. Arch. Meteor. Geophys. Bioklim., A14, 1–13. Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877–946. Howard, L. N., 1958: Divergence formulas involving vorticity. Arch. Ration. Mech. Anal., 1, 113–123. Koshyk, J. N., and N. A. McFarlane, 1996: The potential vorticity budget of an atmospheric general circulation model. J. Atmos. Sci., 53, 550–563. Marshall, J. C., and A. J. G. Nurser, 1992: Fluid dynamics of oceanic thermocline ventilation. J. Phys. Oceanogr., 22, 583–595. McIntyre, M. E., and W. A. Norton, 1990: Dissipative wave–mean interactions and the transport of vorticity or potential vorticity. J. Fluid Mech., 212, 403–435. Palmer, T. N., 1988: Analogues of potential vorticity in electrically conducting fluids. Geophys. Astrophys. Fluid Dyn., 40, 133–145. Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2d ed. SpringerVerlag, 710 pp. Ripa, P., 1981: Symmetries and conservation laws for internal gravity waves. Amer. Inst. Phys. Conf. Proc., 76, 281–306. Salmon, R., 1982: Hamilton’s principle and Ertel’s theorem. Amer. Inst. Phys. Conf. Proc., 88, 127–135. , 1983: Practical use of Hamilton’s principle. J. Fluid Mech., 132, 431–444. , 1988: Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech., 20, 225–256. Santilli, R. M., 1978: Foundations of Theoretical Mechanics I. Springer-Verlag, 266 pp. Scha¨r, C., 1993: A generalization of Bernoulli’s theorem. J. Atmos. Sci., 50, 1437–1443. Serrin, J., 1959: Mathematical principles of classical fluid mechanics. Handbuch der Physik VIII/1. S. Flu¨gge, Ed., Springer-Verlag, 471 pp. Truesdell, C. A. T., 1951: On Ertel’s vorticity theorem. Z. Angew. Math. Phys., 2, 109–114. , 1954: The Kinematics of Vorticity. Indiana University Press, 232 pp. , 1991: A First Course in Rational Continuum Mechanics. Vol. 1. 2d ed. Academic Press, 391 pp. , and R. Toupin, 1960: The classical field theories. Handbuch der Physik III/1. S. Flu¨gge, Ed., Springer-Verlag, 902 pp. Vujanovic, B. D., and S. E. Jones, 1989: Variational Methods in Nonconservative Phenomena. Academic Press, 370 pp. Xu, Q., 1992: Formation and evolution of frontal rainbands and geostrophic potential vorticity anomalies. J. Atmos. Sci., 49, 629– 648.