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A Unified Theory of AvailablePotenUal Energy’

Theodore G. ShepherdDepartment of PhysicsUniversity of Toronto

Toronto, Ontarjo MSS lA 7

[Original manuscript received 2 June 1992; in revised forîn 16 September 1992]

ABSTRACT Traditional derivations of available potential energy in a variety of contexts,involve combining someform of mass conservation together with energy conservation. Thisraises the questions of why such constructions are required in the first place, and whetherthere is some general method ofderiving the available potential energyfor an arbitraryfiuidsystem. By appealing to the underlying Hamiltonian structure of geophysicalfiuid dynamics,it becomes clear why energy conservation is flot enough, and why other conservation lawssuch as mass conservation needto be incorporated in order to construct an invariant, knownas the pseudoenergy, thaï is a positive-definite functional of disturbance quantities. Theavailable potential energy is just the non-kinetic part of the pseudoenergy, the constructionof which follows a well defined algorithm. Two notable features of the available potentialenergy defined thereby are first, that it is a locally defined quantity, and second, that it isinherently definable at finite amplitude (though one may of course always take the small-amplitude limit if this is appropriate).

The general theory is made concrete by systematic derivations of available potentialenergy in a number of different contexts. Ail the well known expressions are recovered, andsome new expressionsare obtained. The possibility ofgeneralizing the concept of availablepotential energy to dynamically stable basic fiows (as opposed to statically stable basicstates) is also discussed.

RESUMÉ Dans une variété de contextes, l’obtention de l’énergie potentielle disponiblenécessite traditionnellement la combinaison sous une forme ou une autre de la conser-vation de la masse et de celle de l’énergie. Ceci nous amène à nous demanderpourquoi detelles combinaisons sont requises et s’il existe une méthode générale permettant d’obtenirl’énergie potentielle disponible pour un écoulement arbitraire. La structure hamiltoniennede la dynamique des écoulements géophysiques permet de comprendre pourquoi la seuleconservation d’énergie n ‘est pas suffisante, et pourquoi l’incorporation d’autres lois deconservation est essentielle à l’obtention de la pseudoénergie,’ un invariant qui est unefonc-tionnelle définie positive desperturbations. L’énergiepotentielle disponible est simplementla partie non-cinétique de la pseudoénergie dont la construction suit un algorithme biendéfini. Deux caractéristiques essentielles de l’énergie potentielle disponible ainsi obtenuesont:premièrement, qu ‘elle estune quantité localement définie; et deuxièmement, qu’elle est

1A preliminary version of tins paper was presented at the American Meteorological Society’s 8thConference on Atmospheric and Oceanic Waves and Stability (Shepherd, 1991).

ATMOSPHERE-OCEAN 31 (1) 1993, 1—26 0705-5900/93/OOOO-0001$01.25/O© Canadian Meteorological and Oceanographie Society

Z I Theodore G. Shepherd

intrinsèquement définissable à amplitude finie (il est toujours possible toutefois de prendrela limite de faible amplitude si cela est approprié).

La théorie générale est ici concrétisée par des dérivations systématiques de l’énergie po-tentielle disponible pour un ensemble de contextes. Toutes les expressions bien connues sontretrouvées et de nouvelles sont obtenues. On examine également la possibilité de généraliserle concept aux états de base dynamiquement stables (par opposition aux états de base sta-tiquement stables).

i IntroductionThe concept of available potential energy (APE), first promulgated by Margules(1903) and formalized by Lorenz (1955), has had far-reaching applications indynamical meteorology. The initial motivation for the concept came from therealization that most of the potential energy in a stably stratified fluid is trappedin that form, and is flot free to be converted into kinetic energy. Although onemay subtract off the potential energy associated with a suitable resting backgroundflow, the resulting difference energy is typically only linear in the disturbance andthus, being sign-indefinite, is not a satisfactory measure of available potential en-ergy. The solution to this difficulty employed by Lorenz (1955) was to combinethe energy conservation law with another conservation law, namely the conser-vation of mass within each isentropic layer (in fact this represents an infinity ofindependent conservation laws, one for each isentropic layer), with multiplicativecoefficients judiciously chosen so as to yield a new conservation law for a quan-tity, the kinetic plus available potential energy, that is quadratic to leading order indisturbance quantities. The small-amplitude, quadratic form of the APE is the onethat is widely used in diagnostic energy budgets (e.g. Lorenz, 1967).

The same issue arises with the (gravitational) potential energy of incompressibleintemal gravity waves, although it is somewhat disguised in the usual derivations(e.g. Lighthill, 1978, §4.1; Gill, 1982, §6.7). Lt also arises with the (internal) po-tential energy of sound waves in unstratified fluid: in that case energy conservationis combined with conservation of total mass to yield an expression for acousticenergy that is quadratic to leading order in disturbance quantities (e.g. Lighthill,1978, §1.3). All these examples suggest a pattern, namely that when consideringthe potential energy of disturbances to some non-trivial background state one needsto bring in other conservation laws, apart from energy, in order to construct a formof “avaîlable” potential energy that is useful for practical purposes.

The above considerations prompt the following questions. Why does one haveto bring in these other conservation laws? Which ones are needed? Is there anysystematic way in which to construct these forms of available potential energy, ormust one simply be clever?Can one extend the concept of available potential energyto arbitrary fluid systems? And does the concept extend to the case of disturbancesto non-resting basic states (a question explicitly raised by Lorenz (1955, p. 159))?It turfis out that the answers to all of these questions become completely clear andstraightforward once one appeals to the fundamental dynamical structure of thefluid systems, which is concisely expressed within a Hamiltonian formulation. It iS

A Unified Theory of Available Potential Energy I 3

the purpose of this paper to make these answers explicit. In so doing the knownexpressions for available potential energy and acoustic energy referred to above areredenved in a systematic fashion, and some new expressions for available potentialenergy are obtained.

Hamiltonian dynamical structure provides the underpinning for much of theo-retical physics, encompassing such diverse fields as quantum mechanics, statisticalmechanics, relativity, optics and celestialmechanics. This structure provides a com-pact framework within which symmetry properties canbe connected to conservationlaws via Noether’s theorem. Over the last few decades it has become apparent (see,e.g. Morrison, 1982; Marsden, 1984; Amol’d and Khesin, 1992) that the dynamicsof ideal fluids and plasmas also fit within a Hamiltonian framework. One of thereasons why this insight took so long to emerge is that the Eulerian description offluid motion is in virtually every case a non-canonical Hamiltonian system: thatis, it cannot be expressed in terms of canonical (qj,pj) coordinates. A special fea-ture of non-canonical systems is that they possess conservation laws that are notconnected to explicit symmetries of the Hamiltonian, but rather arise from the de-generate nature of the associated Poisson bracket (Littlejohn, 1982). The resultinginvariants are known as Casimir invariants.

It turns out that it is precisely this non-canonical dynamical structure that is re-sponsible for the need to introduce additional conservation laws in the derivationsof available potential energy and acoustic energy described above. These additionalconservation laws are just those involving the Casimir invariants associated withthe non-canonical dynamics. The Haniiltonian structure explains why such a proce-dure is generally required for the construction of an available potential energy fordisturbances to a resting basic state. Perhaps more importantly, it also guaranteesthat such a construction is generically possible for any fiuid system that permitsa Hamiltonian representation, and it provides an explicit algorithm for that con-struction. The resulting invariant is referred to as the pseudoenergy. Although athorough treatment of the Hamiltonian description of fluid dynamics would take usbeyond the scope of this article, the salient points are easily presented in a languageaccessible to the non-specialist, requiring nothing more than variational calculus.Further background may be found in the articles of Benjamin (1984), Holm et ai.(1985), Salmon (1988), and Shepherd (1990, 1992).

The plan of the paper is as follows. First, the derivations of some well knownexpressions for available potential energy are reviewed, in order to fix definitionsand to identify precisely where in those derivations the Casimir invariants are used.The derivation of acoustic energy is in some ways the simplest and is thereforethe logical place to begin (Section 2); the desirability of constructing second-order(quadratic) conservation laws is explained here, following Lighthill (1978, § 1.3).This is followed by the case of incompressible internal gravity waves (Section 3),and Lorenz’s form of available potential energy for compressible, hydrostatic flow(Section 4). In Section 5 the pertinent aspects of non-canonical Hamiltonian dynam-ics are reviewed. This explains why such constructions are necessary, and how tomake them for general Hamiltonian systems. To illustrate the point we return to theforegoing examples to present, in Sections 6—8, systematic derivations of available

4 I Theodore G. Shepherd

potential energy in each case. A notable feature is that the systematically derivedexpressions are inherently finite-amplitude. In Section 9 the power of the methodis demonstrated by an elegant and succinct derivation of Andrews’s (1981) finite-amplitude expression for available potential energy in a non-hydrostatic, stratified,compressible fluid.

The above conservation laws are for disturbances to resting basic states. However,the Hainiltonian structure assures us that the same constructions can be made fordisturbances to any steady solution of the equations, in particular for disturbancesto non-resting basic states.* Indeed, such (Eulerian) pseudoenergy conservationlaws have been derived in a variety of geophysical fluid dynamical contexts (e.g.Andrews, 1983; Mclntyre and Shepherd, 1987; Haynes, 1988; Scinocca and Shep-herd, 1992). Whether they represent a physically meaningful measure of “availableenergy” depends somewhat on the context. For so-called “symmetric flow” theinterpretation tums out to be clear-cut (Cho, et al., 1993). This development issummarized in the present context in Section 10.

The paper concludes with a discussion (Section 11).

2 Energy of acoustic wavesConsider compressible ideal fluid, with no extemal forces. Introduce a restingbasic state p = Po = const., v = 0, where p is the density and v the velocity. Wenow examine the energy of small-amplîtude waves on this background state (cf.Lighthill, 1978, §1.3). The kinetic energy per unit volume Ek is given by

Ek ‘/2p1v12 — 1/2p~~~J~2+O(a3) (2.1)

where a is a measure of the wave amplitude; we assume a « i for the time being.The potential energy per unit volume E~ is given by

~E~= pp&x—~5pp

1(2.2)

poP

where a is the specific volume, p the pressure, and c the speed of sound (i.e.c2 = (dp/dp)( Po))~ whence

i /p~ c2E~=—(p—p

0)—— I —i—— j (p—poV+O(a3) (2.3)

Po 2 xpo Po I

The energy equation takes the form

*Of course this statement transcends the issue of non-canonical dynamics. In contrast to the Eulerian

description, the Lagrangian description of fluid motion is canonical (e.g. Salmon, 1988), and it is weIlknown that tins stmcture permits the general denvasion ofpseudoenergy conservation laws in Lagrangianvariables (Andrews and Mclntyre, 1978a; Ripa, 1981).

A Unified Theory of Available Potential Energy I ~

a(Ek+EP) .VE~+V~(Ekv+pv)=0 (2.4)

There are two difficulties with this situation. First, the energy equation (2.4)is not manifestly in the form of a conservation law. Second, for small-amplitudewaves we have

Ek=0(a2), E~ =0(a) (2.5)

Why is (2.5) unacceptable? In the first place, it seems unnatural to have Ek «Et ifthere are conversions between the two forms of energy. Second,E~ is not of definitesign, 50 that disturbance energy is flot a measure of “size” in any meaningful way.Finally, theoretical analysis is made much more difficuit. In particular, consider aformal solution in ternis of a perturbation expansion in some ordering amplitudeparameter e « 1, viz.

2 2P—PO~Pl~~P2~, V~iVl+C ZF2+” (2.6)

where p~, v1 are calculable from linear theory, P2~ z~=are calculable from second-

order non-linear theory, and so on. In terms of these expansions, we have

Ek = ‘/2PoLz’l12e2+0(&) (2.7)

Ipo i~ po c2 ~ ~2 +0(e3) (2.8)

kVP2~ \~p~ Poil

Now if we look at global energetics, then typically py = O for any sort of phaseaverage since at O(e) the wave has sinusoidal structure in some direction. Thus forself-consistent energetics we need to examine the O(e2) terms, which involve P2.(Generally PT ~ 0.) It follows that disturbance energy cannot be calculated, evento leading order, from linear theory.

Fortunately, there is a remedy for this difficulty (Lighthill, 1978, § 1.3). It turnsout that the terru Po log( P/Po), which is the exact integral of the first term in theexpansion (2.2), satisfies a separate conservation law

(2.9)at

where E0 = po log( P/Pa). and F0 = pov is the rate-of-working by the background

pressure. Note that (2.9) is equivalent to the mass continuity equation

ap(2.10)

at

and is therefore an exact relation. The expansion of E0 in powers of (p — po) is

1P0— Po

(2.11)


which evidently matches E,, to leading order. Subtracting (2.9) from the energyequation (2.4) thus yields

aE1(2.12)

with

[,~ c2, 1 [,~ c2 .,1

E1 = IPoIVV + —(p — poYi = — IPoIVîr + —Plie

2 +0(e3) (2.13)2 L Pc j 2 L Pc J

F1 = (p —pa)v =p1v1e

2 +0(e3) (2.14)

the latter being the rate-of-working by the excess pressure.The following benefits are evident. First, E

1 = 0(a2). Second, E

1 is positivedefinite. Finally, E1 is calculable to leading order from linear theory (i.e. the O(e

2)terms involve only ~‘î, p

1). This form of acoustic energy is therefore preferable tothe form Ek +E,,. As we shaîl see, E1 is actually the small-amplitude approximationto the pseudoenergy density relative to this resting basic state. Note in this respectthat we had to use the mass continuity equation (2.10): for this system, total massis an example of a Casimir invariant (see Section 6).

3 Energy of internai gravity wavesConsider incompressible ideal fluid, stratified under gravity. Introduce a restingbasic state p = po(z), v = 0. We now examine the energy of small-amplitudewaves on this background state (cf. Gill, 1982, §6.7). The kinetic energy per unitvolume Ek is given by

Ek = ‘/2P1v12 — 1/2Pd1.v12 + 0(a3) (3.1)

where a is a measure of the wave amplitude; we assume a « 1 for the time being.The potential energy per unit volume E,, is usually given by pgz, where g is thegravitational acceleration. However, it is convenient to subtract off the backgroundpotential energy pogz, since f pogzdx is just a constant, which leavesE,, = (p — po)gz = [po(z — 1) — po(z)]gz

— —gzp~(z)~ + ‘/2gzp~(z)~2 + 0(a~) (3.2)

where ~ is the vertical particle displacement. The exact energy equation takes theform

a(Ek+E,,) (3.3)

at +V.{(Ek+E,,+pogz+p)v}=0Although (3.3) is manifestly in conservation-law form, we have E,, 0(a) which

is the saine problem (cf. (2.5)) that we had with acoustic energy. Fortunately, asin that case there is a remedy for this difficulty. For incompressible fluids, anyfunction f of p satisfies a separate conservation law

A Unified Theory of Available Potential Energy 17

af(P)Vf~O (3.4)at

which is an exact relation. Note that (3.4) is equivalent to the density evolutionequation

(3.5)

Now, the basic-state stratification pc(z) defines an inverse function Z (O accordingto z Z (pc). If we then choose

f(p) — JgZ(~)di5 (3.6)

note that

f(p)f(pc) gZ(~)d~ =g J[Z(pc)+Z’(pc)(I3~pc)+...]d~

= gz(p — Pc)+ 1/2gZ’(p~)(p — Pc)2 +~“ (3.7)

the first term of which is equal to E,,. From (3.4) together with hydrostatic balancein the basic state (see (9.6)) one obtains the conservation law

~(f(p)—f(pc)) .~-v .V(f(p)—f(pc)) +V~ {(pogz+pc)z’} =0 (3.8)

Finally, subtracting (3.8) from the energy equation (3.3) yields the conservationlaw

aE1

—+S7.Fî=0(a3) (3.9)

at

with

EîZI(PoIzPI2— p~(z) ~‘ — Pc)2) ! (PcIz~iI2 — p~(z) p~) e2 +0(e3) (3.10)

F1 (p —pc)v =plr’ie

2 + O(e3) (3.11)

where p — Pc = epî + C2P2 +..., and similarly for the other variables. The tenu in(3.10) involving (P — Pc)2 is recognized as the usual small-amplitude expression forthe density of available potential energy in an incompressible fiuid (e.g. Lighthill,1978, §4.1; Gill, 1982, §6.7). Since (3.4) is an exact relation, however, it followsthat the conservation law (3.9) can be extended to finite amplitude. In particular,the exact expression for the density of available potential energy is given by

APE=gz(p—pc)— J gZ(is)45=~J g[Zpc+fr—zpc]d~ (3.12)


(Holliday and Mclntyre, 1981, Eq. (2.15)), which evidently reduces to the APE in(3.10) in the small-amplitude limit. When the basic stratification is stable, gZ’( p) <o and (3.12) is positive definite. As pointed out by Holliday and Mclntyre, theexpression (3.12) is notable in that, unlike Lorenz’s available potential energy (seeSection 4), it is a locally defined quantity.

We see the same benefits with (3.10) as with the acoustic energy (2.13): E10(a

2); E1 is positive definite when gp~(z) O and O serves as a vertical coordinate. The overbar denotes a quasi-horizontal average along O surfaces, and p(O) is defined toequal the surface pressure

Pc for O

A Unified Theory of Available Potential Energy I ~

background state so that (p — p) = 0. Using the small-amplitude relation

(4.5)dO

where 0(p) is the average pressure on an isobaric surface, defined to leading orderby p = p(0(p)), the expression (4.4) may be converted to

APE ~ Kc,,g 1~0~< JKî(d~)’O~)2d (4.6)

(Lorenz, 1955, Eq. (8)).As with the available potential energies derived in the previous sections, Lorenz’s

APE is evidently positive definite. As we shaîl see, it too is the non-kinetic part ofthe pseudoenergy relative to the resting background state. Note in this respect thatto introduce (4.2) it was necessary to appeal to conservation of p(O) for each valueof O, which for a hydrostatic fluid is equivalent to conservation of mass withineach isentropic layer: for adiabatic flow the mass within each isentropic layer is aCasimir invariant (see Section 8). Thus Lorenz was implicitly using conservationof mass, in addition to conservation of energy, and was thereby constructing apseudoenergy.

For historical purposes, it is interesting to note that Fj0rtoft (1950) showed thatstably stratified, resting basic states were energy extrema under adiabatic redistri-butions of mass. This variational principle was used by van Mieghem (1956) toderive Lorenz’s small-amplitude fonu of APE (4.6) as the non-kinetic part of theconstrained second variation of energy. This prescient result points the way to thefull non-linear development in Section 8.

5 Hamiltonian structureWhy is there the need for all this manipulation involving invariants other thanenergy? After all, in classical mechanics the equilibrium solutions are usually con-ditional extrema (or critical points) of the potential energy U(q) (e.g. Goldstein,1980), in which case U(q) is locally quadratic in the displacement (q — qc) andthe problems discussed above do not arise.

To understand this, it is useful to consider the symplectic representation of acontinuous Hamiltonian system:

59~1at —j~

1~ (5.1)

where u u(x, t) represents the dynamical variables; 91 is the Hamiltonian (usuallyjust the total energy), which is a functional of the field u(x, t); 8/su is the func-tional or variational derivative; and i0 is the skew-symmetric Poisson tensor (alsosatisfying various algebraic properties). For further discussion of the fonu (5.1)as applied to fluid systems, with numerous examples, one may refer to Morrison(1982), Benjamin (1984), Salmon (1988) or Shepherd (1990).

io I Theodore G. Shepherd

Note that the usual (finite-dimensional) canonical equations

~91 •1~ (5.2)q

1=—, p•——ap, aq,

are a special case of the form (5.1), with

where I is the N x N identity matrix. Darboux’s theorem guarantees that anyHamiltonian representation with a non-singular (i.e. invertible) Poisson tensor maybe locally transfonued into the canonical fonu (5.2) by a suitable change of coordi-nates, 50 that we may consider any representation with non-singular J13 as essentiallycanonical.

However, the representation (5.1) is more general than this, for it allows thepossibility of degenerate (or singular) Poisson tensors (Littlejohn, 1982); in thiscase the Hamiltonian description is said to be non-canonical. A simple exampleof a non-canonical Hamiltonian representation is Euler’s equations describing themotion of a rigid body (see, e.g. Holmes and Marsden, 1983; Holm et al., 1985).A special feature of non-canonical representations is that they possess a class ofinvariants known as Casimir invariants, which are defined to be the solutions of

SC(5.3)

That they are indeed invariants of the dynamics follows from

dC _ f SC ~)u, _ (5cr 591 /591 SC) — I i = —î,J,~— =0 (5.4)dl k Su1’ ~Jt k Su1’ 8u1/ ~5u, Sujj

(where (.,.) is the inner product on the function space {u}), after using (5.1) and(5.3) as well as the skew symmetry of J~,.Now consider disturbances to a steady basic state u = U. If the system is

canonical, in the generalized sense of J~, being invertible, then

591 591

~~iu=’U~~J u=U =0 (5.5)

Hence steady states are conditional extrema of the Hamiltonian, which means that inthe neighbourhood of the equiiibrium U, 91 is quadraîic in disturbance amplitude:

5; — __91(U + Su) = 91(U) + u~, ~ _____ ~u>ou1î+ 0(a3) (5.6)V 2 Su1Su~ j

Thus for a canonical system, none of the difficulties encountered with the potentialenergy in the previous sections arise.

However, those fluid systems are in fact non-canonical (see Sections 6—8), 50

A Unified Theory of Available Potential Energy I i i

that the second implication in (5.5) does not follow since J~i is non-invertible.This means that in the neighbourhood of the equilibrium, 91 is generally linearin disturbance amplitude, and for the reasons described in Section 2 one thereforeseeks to eliminate the 0(a) tenu somehow. Lt turus out that such elimination isalways possible, since

S91 S91 _ SC (5.7)

~ u=U~ u=U u=U

for some Casimir C (the minus sign being by convention). That is, there is alwayssome C such that VC is locally “tangent” to X791 (the “gradients” being taken infunction space). This is because, by the definition (5.3), the “vectors” VC generatedby all the Casimirs span the kemel of the operator J~, (Littlejohn, 1982; Holm etal., 1985).

Thus one may generically construct the functional

~ [ni 91[u] — 91[U] + C [u] — C [U] (5.8)

with C defined by (5.7). .~ is clearly an exact invariant of the non-linear dynamics,since it is made up of exact invariants; in addition, it is of quadratic order indisturbance amplitude for small disturbances, since .~4 [u = U] = O and

~‘ Su) +0(Su2)=(~ u=U u=U Su)+ 0(5u2) = 0(5u2)

These two properties imply that .~4 is what is called a wave activity (Andrews andMclntyre, 1978a; Mclntyre and Shepherd, 1987). In this case, .~4 is a particularfiavour of wave activity known as the pseudoenergy, since it is related to thetemporal symmetry of the dynamics through Noether’s theorem (op. cit.).

Lt is worth emphasizing again that although the detenuination of .~4 is made inthe vicinity of the equilibrium U, the nature of the construction ensures that ~4is anexact, ~finite-amplitudeinvariant. We now show in the following three sections thatthe forms of acoustic energy and available potential energy derived in Sections 2—4follow directly from the pseudoenergy construction (5.8) relative to the relevantresting basic states.

6 Energy of acoustic waves, revisitedThe Eulerian description of three-dimensional fiow of a compressible ideal fluidis Hamiltonian (Morrison, 1982; see, also Shepherd, 1990, §4.5) with dynamicalvariables (z’, i~, p), where Ti is the specific entropy, and Hamiltonian

91 = J{î/2pIz’I2+p~(1 p)}dx (6.1)

with ~E(rj, p) being the internal energy per unit mass. (We consider the case withno external forces.) Using the thermodynamic relationship

h I Theodore G. Shepherd

d~=TdTi+~dp~T~==T ~ _— TE~ -b-- -~ (6.2)

it follows that the functional derivatives of 91 with respect to the dynamical vari-ables are given by

691 _ 691

pz’, =p~E~=pT,

f il 2 ÎII2~EP (6.3)

The Casimir invariants for this system are of the fonu

C =JPC(Ti~ q)dx (6.4)

for arbitrary functions C, where q = [(V x z’). VT]/P is the potential vorticity.Their functional derivatives are given by

6cr 6C= VX(CqVTj), ~ pCryV~(CqVX z’), = C—qCq (6.5)

SpTo construct the acoustic energy we consider disturbances to an isentropic,

homogeneous, resting basic state

z’=0, q=0, rl=Tlc=const., ppcconst. (6.6)

as in Section 2. We then have to choose C in (6.4) so as to satisfy (5.7), or in thiscase

691 _ 6C 691 6C 691 6cr (6.7)STi’ SpSp

when evaluated at the basic state (6.6). Using (6.3) and (6.5) with (6.6), it is clearthat (6.7) is satisfied for

Cq 0, C1 = T, C = —~E(Ti, ~3) p/p (6.8)

For pure acoustic waves we consider an isentropic fluid with p = p(p), 50 it i5readily verified, using (6.2), that the expressions for C and for C1 in (6.8) areconsistent. Since C must be a function of Ti alone, we therefore have

C(1) &E(TI, Pc) —Pc/pc (6.9)

However, the disturbances are isentropic SO TI Tic, and the Casimir invariant tobe used here thus takes the fonu

C = JPC(flc)dX = L~E(flc, Pc) +~c/Pc]JPdx (6.10)


This is just a constant times the total mass, confinuing the statement to that effectmade in Section 2. Lt follows from (6.1) and (6.10) that

91[u] — 91[U] = J{î/2p1z’12 + P~E(flc, P) — pc’E(ric, pc)}dx (6.11)

and

crEn] —cr[U] = —J [~~iic~ Pc)+~] (ppc)dx (6.12)whence the pseudoenergy (5.8) takes the form

= J ~ ‘/~P~z’l2 + P~E(1c, P) — pc’E(Tlc, Pc)d p)] (P—Pc)}dx (6.13)— j—[P~E(I1c~

Noting that

d2[P~E(Tic pi = ~ ~ p)+ = i (6.14)

which when evaluated at p = Pc equals p~’c2, it follows that the small-amplitudeapproximation to (6.13), accurate to 0(a2), is

J 1/~{p0~z’~2+ d

2[~( p)] (P—Pc)2 } dxP=Po

(6.15)

The integrand in (6.15) is just (2.13), as expected. We have thus shown that the wellknown expression for the energy of acoustic waves is systematically derivable asthe small-amplitude approximation to the pseudoenergy relative to the backgroundstate.However, (6.13) is actually an exact, finite-amplitude invariant of the dynamics.

We may use (6.2) to write

in which case the exact expression for the (available) potential energy per unitvolume may be taken from (6.13) as

PLEO1c, p) ~E(T~c,Pc)] —( Pc) P(P) PcPc f— — — 2 ——(P—Pc) (6.17)Pc PcThe expression (6.17) is believed to be new.


7 Energy of internai gravity waves, revisitedThe Euleriandescription of three-dimensional flow of an incompressible ideal fluid,stratified under gravity, is Harniltonian (Benjamin, 1984; Abarbanel et ai., 1986)with dynamical variables (z’, p), and Hamiltonian

91 = J{1/2p1z’12+pgz}dx (7.1)

with691 691

=pz’, = 1/~~z’~2+gz (7.2)

There is a class of Casimir invariants of the fonu

cr =JC(p)dx (7.3)

for arbitrary C, for which

scr scr—=0, —=C(p) (7.4)6z’ Sp

(As in Section 6 the function C in (7.3) can also have a functional dependence onthe pctential vorticity q, bere given by q = (V x z’). Vp, but since the basic statewill have z’ = O we anticipate that this dependence will not be required.)

To construct the available potential energy we consider disturbances to a resting,stably stratified basic state

z’0, PPc(z) (7.5)

as in Section 3. We then have to choose C in (7.3) so as to satisfy (5.7), or in thiscase

691 scr 691 SC6 ~ —-g-— (7.6)

when evaluated at the basic state (7.5). Using (7.2) and (7.4) with (7.5), it is clearthat (7.6) is satisfied for the choice

C’(p) = —gz ~ C(p) = ~JPgZ(j3)d~. (7.7)

Note that the (monotonic) dependence of z on p in the basic state (7.5) defines afunction Z (.) according to z = Z( Pc). The function C( p) provided by (7.7) isjustthe negative of the function f defined in (3.6), as we expect. Lt follows from (7.1)and (7.7) that

91[u]—91[U] = J{1/2p1z’12+(p....pc)gz}dx

zzJ{1/2p1z’12+J gZ(pc)d~ } dx (7.8)

A Unified Theory of Available Potential Energy I ‘s

andf(tP

cr[ti]—crLU]=]~—J gZ( ~)d~ dx (7.9)

whence the pseudoenergy (5.8) takes the form

= J { 1/2p1z’12....J g[Z(pc+P)-..Z(pc)]d~} dx (7.10)The tenu involving the integral over ~ represents the finite-amplitude density ofAPE, recovering (3.12).

Thus we have shown that the known expressions for the available potential energyof a stratified, incompressible fluid are systematically derivable as the non-kineticpart of the pseudoenergy relative to the resting, stably stratified background state.

In the related case of the anelastic equations, the finite-amplitude pseudoenergyanalogous to (7.10), including an expression for APE, has recently been systemat-ically derived by Scinocca and Shepherd (1992, Eq. (3.13b)).

8 Lorenz’s available potential energy, revisitedTo the author’s knowledge, an explicit Hamiltonian fonuulation of three-dimen-sional motion of a compressible, hydrostatic perfect gas has yet to be recorded.fHowever, that need not stop us! Indeed, it may have been noted that in the sys-tematic derivations of pseudoenergy presented in Sections 6 and 7, no explicit useof Hamiltonian structure in the fonu of the J~, operator was required. Providedone knows the relevant invariants of a system, one may just attempt to find a crthat satisfies (5.7) for ah the dynamical variables simultaneously. (This is, for ex-ample, what Haynes (1988) did in deriving pseudomomentum and pseudoenergyconservation laws for this very system of equations: he did not appeal to an explicitHamiltonian fonuulation.) The point is simply that having an underlying Hamil-tonian structure ensures that this strategy will succeed, namely, that a choice ofCasimir cr satisfying the extremal condition (5.7) can be found for an arbitrarysteady solution U.

Although Lorenz (1955) derived bis fundamental expressions in isentropic co-ordinates, for practical applications it is preferable to have results expressed inpressure coordinates. Thus we begin with the Hamiltonian in pressure coordinates,which we know must be given by

91 = J{I/2gîI~~I2 +c,,§10fl(p)}dxhdp (8.1)

where H(p) = (p/p00)K, z’,1 is the horizontal velocity, Xh represents the horizontal

spatial coordinates, and the integral inp is taken from O tope, the surface pressure.There is an issue having to do with how one handles variable pc~ but we ignorethis in what follows. In any case the expression for available potential energy willtum out, in contrast to Lorenz’s expression, to be a locally defined quantity, 50 thatsuch considerations will not be pertinent unless one is interested in global budgets.

lIn tact such a formulation is available in isentropic coordinates (O. Bokhove, pers. commun., 1992).

i6 I Theodore G. Shepherd

The hydrostatic system of equations has three dynamical variables: the two com-ponents of z’,1, plus a thenuodynamic variable that we may conveniently take to beO. In terms of these variables the functional derivatives of the Hamiltonian (8.1)are evidently

691 ~ 6916z’h = g z’,~, = c,,g1fl(p) (8.2)

noting that in these variations p. being an independent variable, is held fixed. Wealso know from the full non-hydrostatic system (cf. (6.4)) that there is a class ofCasimirs of the form

cr = J C(O)dxhdp (8.3)The hydrostatic approximation makes no difference to the invariance of this integral,which is easily verified upon recalling that O is a materially conserved quantity;the absence of a density factor in front of the function C in (8.3) reflects that thefiow is non-divergent in pressure coordinates. From (8.3) we have

=0, ~ =C’(O) (8.4)SZPh 60

Note that in isentropic coordinates the Casimirs (8.3) transfonu to

cr = —g fGC(O)dx,1dO (8.5)where a = p(aO/dzf1 is the isentropic density, after using hydrostatic balance inthe fonu

ap _— —ag (8.6)

From (8.5) it is clear that the mass distribution f adx~ on each isentropic layer isan independent Casimir invariant (taking C(O) to be a delta function localized onthe layer in question), thus confinuing the statement in Section 4 to the effect thatLorenz’s invariants p(O) are Casimir invariants.

To construct the available potential energy we consider disturbances to a resting,stably stratified basic state

z~=0, OOc(p) (8.7)

We then have simply to choose C in (8.3) so as to satisfy (5.7), or in this case

691 scr 691 scr__ = —s--, ~ = (8.8)

when evaluated at the basic state (8.7). Using (8.2) and (8.4) with (8.7), it is clear


that (8.8) is satisfied for the choice

C’(O) = —c,,g1H(p) 4~ C(O) = ~J0c,,g-IH(~(Ô))dÔ (8.9)

Just as with (7.7), the (monotonic) dependence of p on O in the basic state (8.7)defines a function ~P(.) according to p = ~P(Oc). The resulting pseudoenergy isevidently given by

J { 1/2gî~~

2 JB~OOcg~î[n~~p(O+Ô))

— ii( ~P(Oc))]d9}dxhdP (8.10)

which bears a striking resemblance to the incompressible fonu (7.10). The integrandin (8.10), say A, represents (g’ times) the pseudoenergy per unit mass, and isguaranteed to obey a local conservation law of the fonu

~3A—+V.F=0 (8.11)~3t

for some flux F (the details of which will not be worked out here), where V in(8.11) is the gradient operator in (x, y,p)-space. This finite-amplitude fonu of theAPE with O as dependent variable and p as independent variable is believed tobe new. An exact APE in pressure coordinates has been derived by Boer (1989),but differs from the present fonu in that its local value, or density, is not positivedefinite.

The exact, finite-amplitude invariant (8.10) has the small-amplitude limit

.~~J{1/2gîIz~I2+! icc~ ( dOc

Note that the above expressions are more general than those of Lorenz, becausethere is no reference to the domain apart from the integral itself. Rather, they referto departures from any resting basic state (8.7). However, for the special choiceOc(p) = 0(p) it may be seen that the non-kinetic part of (8.12) reduces to the form(4.6) presented earlier.

Indeed, since the expression (8.10) is exact, the choice of the basic state Oc(p)is in some sense arbitrary. Fèr a given isentropic distribution of mass, one maytherefore consider a variety of different basic states (8.7), and seek the one that givesthe minimum available potential energy in (8.10). Such a strategy was explicitlydiscussed by Andrews (1981). Lt seems likely that Lorenz’s choice Oc(p) = O(p)would give the minimum such estimate of APE.

9 Available potential energy of a compressible, stratified fiuidLn this section the power and elegance of the Hamiltonian method is demonstratedby the systematic derivation of a finite-amplitude expression for the available po-


tential energy of a compressible, stratified non-hydrostatic fluid. This result hasprevicusly been derived by Andrews (1981), but using an indirect (albeit inge-nious) manipulation of the goveming equations.

We start with the Hamiltonian structure described in Section 6, but now includinggravitational potential energy. Thus the dynamical variables are (z’, ~ p), and theHamiltonian is

91 = J{1/2p1z’12+pgz+pE(Ti, p)}dx (9.1)

with ~E(Ti,p) being the internal energy per unit mass. The functional derivatives of91 are then (cf. (6.3))

691 591 691 _ p= pz’, -s-— = pT, — î/21z’12+gz

A Unified Theory of Available Potential Energy I i9

sense. However, we can relate z, Pc and Pc to Tic through the (monotonic) depen-dence of ail four variables on z: in particular, we can define the functions Z (~),~R.j.)and ~P(.) according to

z = Z(Tic), Pc = ~&(Tic), Pc = ~P(Tic) (9.9)

Lt follows that the function C may be written as

~P(~C(î) = —gZ(~) — ~E(Tif ~&(Ti))— ~&Mi) (9.10)

We must now check consistency with the middle condition of (9.7). Prom (9.10),

dZ dpc dZ i dpc dZ Pc dPc dZ

di Pcdz dTi p~dz d~ (9.11)

ail tenus in (9.11) having been evaluated at the basic state (9.5). Using (6.2) tosubstitute for ~ and ~ together with hydrostatic balance in the basic state (9.6),it is easily verified from (9.11) that C1 = —T when evaluated at the basic state(9.5), whence the middle condition of (9.7) is satisfied.

We can now deduce, using (9.10), that the available potential energy per unitvolume is given by

APE = (p — pc)gz + P~E(Ti~ p) — Pc~E(Tic, Pc) + pC(Ti) — pcC(Ilc)

= (p—pc)gz+p~E(Ti, P)—Pc~E(Tic, pc)—pgZ(Ti)—p~E(T1, ~&(Ti))

q’ (Ti) + pcgZ(flc) + Pc~ (Tic ~(îc)) + Pc ~P(Tic

The final expression in (9.12) is just p times Andrews’s (1981, Eq. (4.4)) expres-sion for the available potential energy per unit mass, so that we have recoveredAndrews’s result, though far more easily. Andrews shows that the finite-amplitudeexpression is positive definite, and that it reduces in the small-amplitude limit tothe well-known quadratic expression for the available potential energy of acoustic-gravity waves (e.g. Lighthill, 1978, §4.2; Gui, 1982, §6.14). Note also that (9.12)reduces to (6.17> in the special case g 0, Ti Tic~ as it should.

The local pseudoenergy conservation law takes the fonu

.~(1/2PIVI2 +APE>+V{(î/2PIz’I2+APE)V+(p~pc>zJ}z~0 (9.13)

3h

with APE given by (9.12).

10 Available energy for disturbances to non-resting basic statesThe preceding analysis bas demonstrated, by general theory (Section 5) and byexample (Sections 6-9), that finite-amplitude expressions may be generically con-


structed for the available potential energy relative to a stably stratified, resting basicstate for any fluid system, provided the system possesses an underlying Hamiltonianstructure. (This is the case for the full compressible, non-hydrostatic equations, aswell as for most meteorological or oceanographic approximations to them.) Thisavailable potential energy is “available” in the sense of being available for conver-sion into kinetic energy. However, in his pioneering paper Lorenz (1955, p. 159)made the following insightful observation: “There is no assurance inany indivîduaicase that ail the availabie potentiai energy wiii be converted into kinetic energy.For example, if the flow is purely zonai, and the mass and momentum distributionsare in dynamicaliy stable equiiibrium, no kinetic energy at ail can be reaiized. Itmight seem desirable to redefine availabie potentiai energy, so that, in particular,it wiii be zero in the above example.”

This tantalizing prospect was neyer pursued by Lorenz. As we have seen, thesystematic route to an APE is via the construction of a pseudoenergy invariantrelative to a resting basic state. But the general Hamiltonian theory presented inSection 5 requires only that the basic state be a steady solution of the equationsof motion. Lt is therefore obvious that pseudoenergy invariants may be constructed

relative to non-resting as well as resting basic states. (In those cases the functionaldependence of the Casimir invariants on the potential vorticity must be included.)Indeed, such (Eulerian) pseudoenergy conservation laws have been derived in avariety of geophysical fluid dynamical contexts (e.g. Andrews, 1983; Mclntyreand Shepherd, 1987; Haynes, 1988; Scinocca and Shepherd, 1992), though usuallywithin the framework of wave-activily conservation laws generalizing the conceptof wave action (Whitham, 1965; Bretherton and Garrett, 1968). The naturalquestionto ask is whether the pseudoenergy found thereby is a generalization of APE, inthe sense suggested by Lorenz’s observation.

Ln principle, the answer to this question is yes. Consider the situation envisagedby Lorenz, with a dynamically stable zonal flow. If the stability is of the Amol’dtype (see Holm et al., 1985; McLntyre and Shepherd, 1987, §6), in the sense ofbeingassociated with positive-definite pseudoenergy for arbitrary perturbations, then onemay simply choose the basic flow to be the dynamically stable flow: in this case thepseudoenergy will vanish. This makes rigorous the fact that none of the energy inthe stable zonal flow is available for conversion to disturbances. If the initial flow isnot dynamically stable, then one may consider the pseudoenergy of that flow relativeto some stable basic flow as a measure of the available energy in the unstable flow.0f course this measure will depend on the choice of the (dynamically) stable basicflow, just as the expressions for available potential energy depend on the choice ofthe (statically) stable basic state. But one may then consider the available energyin any given initial flow to be the minimum pseudoenergy of that flow over allpossible choices of stable basic flows, which eliminates any reference to a particillarbasic flow. Lndeed, it is precisely this sort of approach that lies behind Shepherd’5(1988) method of deriving saturation bounds on dynamical instabilities: in thatcase pseudomomentum (see Section 11) rather than pseudoenergy conservationwas used, and the saturation bounds were effectively measures of the “availableenstrophy” in the unstable flows. The close similarity between pseudomomentl.lm

A Unifled Theory of Available Potential Energy I 21

and APE was also noted and exploited by Killworth and Mcintyre (1985) to deriveabsorptivity bounds for Rossby-wave critical layers.

One can say, therefore, that the pseudoenergy relative to an Amol’d-stable basicflow is the generalization of available potential energy to non-resting basic states,building in the dynamical stability constraints as suggested by Lorenz. Lt is more-over, like APE itself, an inherently finite-amplitude concept. The difficulty is thatthere is a serious shortage of Amol’d-type stability theorems for geophysically rel-evant flows! For a start, there are no known dynamical stability theorems for fullythree-dimensional flows admitting gravity waves. In the case of quasi-geostrophicbaroclinic flow there is an Arnol’d-type theorem (Holm et al., 1985; Swaters, 1986;Mcîntyre and Shepherd, 1987), and it may well extend to other balanced systems,though this has yet to be established. However, the stable basic flow must haveU/Q~ < O in some frame of reference, where U is the zonal velocity in thatframe and Q~ is the meridional gradient of potential vorticity, and as pointed outby Andrews (1984) this situation is not nonually encountered in the atmosphere.The necessity of choosing a basic flow that is significantly different from the actualatmospheric flow need not in itself preclude obtaining a useful result, as has beendemonstrated by Shepherd’s (1989) pseudomomentum-based saturation bounds forthe Charney problem of baroclinic instability. Nevertheless, the situation would notseem particularly encouraging.

A more satisfactory situation exists for so-called “symmetric flow”, where allvariables are presumed to be independent of one spatial coordinate. This kindof problem has been considered both for axisymmetric planetary circulations and,more recently, for frontal circulations. Lt turus out that the kinetic energy associatedwith the flow along the symmetry axis appears as a fonu of centrifugal potentialenergy for the flow in the plane perpendicular to the symmetry axis, and the La-grangian conservation of absolute angular momentum along the symmetry axis canprovide a significant constraint on the release of the (gravitational) potential energy.These insights led Fj0rtoft (1950) to a variational principle for steady axisymmet-dc flows, which van Mieghem (1956) then used as the basis for constructing asmall-amplitude fonuulation of APE for symmetric disturbances.

More recently, various modified fonus of APE based on Lagrangian parcel dy-namics havebeen proposed (e.g. Emanuel, 1983). One difficulty with all such quan-tities is that they depend strongly on the presumed parcel motion, which of courseis not known a priori. However, the recognition that there exists a finite-amplitudeArnol’d-type stability theorem for symmetric flow (Cho et aI., 1993) — generalizingthe small-amplitude result of Fj0rtoft (1950) — means that the recipe described in thepresent paper for constructing a finite-amplitude APE can be applied inïmediatelyto this problem. This extends van Mieghem’s (1956) small-amplitude analysis tothe non-linear regime.

We briefly summarize titis development for the case considered by Cho et ai.(1993) of non-hydrostatic, adiabatic, Boussinesq flow on anf-plane; further detailsare provided there. We choose y as the symmetry axis. Let i~t be the streamfuncfionfor the flow in the x—z plane, whence oe u~ — = V2~ï is the y-componentof vorticity. If m = v +fr is the y-component of absolute velocity, and O is the


potential temperature, with Occ a constant reference value, then the Hamiltonian isgiven by

91=ff~’1/2IVWI2~mfx~~i (10.1)

Occ J d,dzFrom (10.1) we see that the “kinetic” energy is just that of the motion in the x—zplane, and the potential energy consists of a centrifugal tenu (depending on m) aswell as a gravitational tenu (depending on O). Basic flows that involve no velocitycomponent in the x—z plane are then effectively “resting” basic states within thisframework, and thus the analogy with the previous analysis is very close indeed.

The dynamical variables are (oe, m, O), and the functional derivatives of 91 arethen given by

S91 691 591 _ gz‘~‘ 5m~’ 60 —~ (10.2)

The Casimir invariants are of the fonu

cr = JJC(m~ O)dxdz (10.3)

for arbitrary functions C, with

scr ~o =C acrSoe’ Sm -~-= C8. (10.4)

If we introduce a basic state consisting of a baroclinic flow v = r.tj(x, z) in they-direction, namely

oe = 0, m mc(x, z) = zt~(x, z) +fr, O = Oc(x, z) (10.5)

then the extremal condition (5.7) requires choosing C in (10.3) so as to satisfy

691 _ 5cr 691 _ scr 591 _ acrSoe 6oe’ Sm — Sm’ 50 ~ (10.6)

when evaluated at (10.5). We can see immediately that we must have

Cm(fl10, Oc) =fr, Co(mc, Oc) = ~. (10.7)000

The resulting pseudoenergy is then given by

= JJ{ 1/2 IVi4ïI2 + C(m, O) — C(mc, Oc) — Cm(tfl~j, Oc)(m — mc)— Ce(mc, Oc)(O — Oc)}dxdz (10.8)

The “kinetic” part of (10.8) is evidently positive definite. Whenever the non-kineticpart is positive definite for arbitrary disturbances, it may be regarded as a measure

A Unil’ied Theory of Available Potential Energy I 23

of the available potential energy relative to tite basic flow (10.5). By Taylor’sremainder theorem, titis occurs when

Cmm > 0, Cee > 0, and CmmC88 — (Cm8)

2 > O (10.9)

which afrer using (10.7), together with thenual wind balance in tite basic flow(10.5), can be shown to reduce to Fj0rtoft’s (1950) sufficient conditions for sym-metric stability: viz. inertial stability, static stability and fQ > O where

a(mc, Oc) (10.10)

a(x,z)

is the basic-flow potential vorticity. Note that witen Q is single-signed it followsthat the coordinate transfonuation from (x, z) to (mc, Oc) is non-singular, verifyingthat the expressions (10.7) are indeed well defined.

In particular, if the initial flow is a baroclinic flow V(x, z), O(x, z) that is dynam-ically stable according to the criteria (10.9), then one may simply choose the basicflow (10.5) to be equal to titis initial flow, in whicit case the pseudoenergy (10.8)will vanisit. Titis satisfies Lorenz’s requirement: the APE in such a flow is zero. Onthe other hand, if the initial flow does not satisfy (10.9), titen one may calculate thepseudoenergy of the initial flow relative to different stable basic flows, and choosetite minimum sucit pseudoenergy: titis would provide a measure of the APE in theinitial flow. In principle, titis algorithm is well defined. However, unlike the caseof Lorenz’s APE where one only has to define Oc(p)~ here one must define twofunctions mc(x, z) and Oc(x, z), and it seems that the optimal citoice is far fromobvious. An example of sucit a calculation for a simple case is provided in Cho etal. (1993, §4b).

il DiscussionThe starting point for titis paper was tite observation that in order to construct afonu of available potential energy for disturbances to some non-trivial backgroundstate, one always seems to need to bring in other conservation laws, apart froru thatof energy. Titis is true for acoustic energy (Section 2), for tite available potentialenergy of intemal gravity waves (Section 3), and for Lorenz’ s available potentialenergy (Section 4). In those cases the additional conservation laws involved variousfonus of mass conservation. Titis observation then led to the questions raised inthe titird paragraph of the Introduction.

By treating all these different systems from the common perspective providedby their Hamiltonian structure, clear answers to alI the above-mentioned questionsemerge, as follows. The need to bring in additional conservation laws follows fromthese fluid systems being non-canonical Hamiltonian systems, witicit implies thattheir equilibrium solutions are generally not conditional extrema of tite energy. Titeadditional conservation laws that are required are just those involving the Casimirinvariants associated with tite non-canonical dynamics. The Hamiltonian structureguarantees titat the construction of an available potential energy is generically pos-sible for any fluid system titat permits a Hamiltonian representation; in particular,


the available potential energy is the non-kinetic part of the pseudoenergy defined by(5.8), with cr determined by (5.7), taken relative to some stably stratified, restingbasic state. Two notable features of the available potential energy defined titerebyare first, that it is a locally defined quantity, and second, that it is inherently defin-able at finite amplitude (titougit one may of course always take tite small-amplitudelimit if titis is appropriate). Altitougit titese observations have no doubt been madeby others, they do not seeru to be widely recognized, and in any case have notbeen explicitly recorded hitherto.

Lorenz (1955, p. 159) had raised the question of whetiter the concept of availablepotential energy could somehow be extended to non-resting basic states, in order totake account of dynamic-stability as well as static-stability constraints. The generalHamiltonian theory assures us that titis is indeed possible, at least in principle,provided one can show that the relevant pseudoenergy is of definite sign. Furtiterdiscussion of titis point has been provided in Section 10, including tite case ofsymmetric flow where the situation turus out to be clear-cut.

The Hamiltonian structure also suggests a generalization of available potentialenergy in the following way. Tite conservation laws discussed above are for distur-bances to steady (time-invariant) states, and involve a combination of energy andCasimir invariants titat together is referred to as the pseudoenergy. In a completelyanalogous fasition, Noetiter’s theorem guarantees that second-order disturbance in-variants may be constructed for disturbances to zonally invariant states, and in-volve a combination of momentum and Casimir invariants titat together is referredto as the pseudomomenïum (see, e.g. Shepiterd, 1990, §5). Small-amplitude fonusof pseudomomentum have been widely used in meteorology under the name ofEiiassen-Paim wave activily (Andrews and Mcîntyre, 1976, 1978b; see also Held,1985), and have a pedigree going back to Taylor (1915). Their connection to Hamil-tonian structure was first pointed out by Mcîntyre and Sitepherd (1987, §7). For agiven initial flow, the pseudomomentum provides a way of defining the “availableenstrophy” in that flow that may be converted into eddy enstrophy (cf. Shepherd,1988, 1989).AcknowledgementsThe autitor would like to acknowledge his intellectual indebtedness to Dr M.E.Mcîntyre, whose penetrating insight into fluid-dynamical fundamentals has providedmucit of tite foundation for the syntitesis presented here. Specific comments fromDrs J.C. Bowman, P.H. Haynes and M.E. Mcîntyre have led to improvements inthe manuscript.

Titis research bas been supported by the Natural Sciences and Engineering Re-search Council and by the Atmospheric Environment Service.

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