## Abstract

Orthonormal wavelet analysis of the primitive momentum equations enables a new formulation of atmospheric energetics, providing a new description of transfers and fluxes of kinetic energy (KE) between structures that are simultaneously localized in both scale (zonal-wavenumber octave) and location spaces. Unpublished modified formulas for global Fourier energetics (FE) are reviewed that conserve KE for the case of a single latitude-circle and pressure level. The new wavelet energetics (WE) is extended to arbitrary orthogonal analyses of compressible, hydrostatic winds, and to formulating triadic interactions between components. In general, each triadic interaction satisfies a detailed conservation rule. Component “self-interaction” is examined in detail, and found to occur (if other components catalyze) in common analyses except complex Fourier.

Wavelet flux functions are new spatially localized measures of flux across scale, or wavenumber cascade. They are constructed by appropriately constrained partial sums over the scales of wavelet transfer functions. The sum constraints prevent KE “double counting.”

Application to Burgers-shock and Stuart-vortex 1D flow models illustrates appropriate physical interpretations of the new energy budget, compared to purely spatial or wavenumber energetics, and demonstrates methods that deal with asymmetry and lack of translation invariance. Such methods include incorporating all possible periodic translations into the analysis, known as the shift-equivariant wavelet transform. The Burgers shock exhibits in FE a global downscale cascade, whose spatial localization and upscale backscatter near the shock is revealed by WE. The Stuart vortex has zero FE, but its pure translation generates a WE picture that reflects the purely spatial energetics picture.

## 1. Introduction

In recent publications, Fournier (1999a, 2000) demonstrated the utility of orthonormal wavelet analysis (OWA) for efficiently representing atmospheric structures that are simultaneously localized in both scale (i.e., wavenumber band) *and* location. In this paper we develop equations governing atmospheric *energetics* in the wavelet domain, that is, interactions that transfer energy between such structures. We analyze simple examples to establish the appropriate physical interpretation of the new method, and prepare for applications to observations, to appear from Fournier (1999b, submitted to *J. Atmos. Sci.*, hereafter F2).

Associated with the differential equations governing meteorological-field dynamics are conserved quantities, useful for diagnosing the atmospheric state. Generally speaking, conserved quantities are interesting because they constrain the available dynamical-system state space (e.g., Shepherd 1990). For example, dissipative systems such as the atmosphere may create locally organized structures in one quantity by sufficiently increasing the entropy of another (Pandolfo 1993). Different state-space *types,* for example, statistical (mean vs deviation), Fourier, or even “physical” (barotropic vs baroclinic, divergent vs rotational flow, etc.), may be more appropriate than others for any given application. Considerable insight into behavior and predictability of atmospheric phenomena (jet streams, blocking, tropical circulation, etc.) has been gained by studying energetics of such states, as covered by Wiin-Nielsen and Chen (1993). Besides the usual zonal-mean and eddy decomposition, “statistical” state types include EOFs: Wilson and Wyngaard (1996) found, for example, that interscale transfers source kinetic energy (KE) to gravity wave modes and sink KE from entrainment modes.

We investigate here the question, under what conditions the *wavelet* state space is appropriate. Recently, Kishida et al. (1999) found a solenoidal-vector-wavelet state space to be more appropriate than linearly partitioned Fourier space (Zhou et al. 1996), for verifying Kolmogorov's assumption of triadic-interaction scale localness. This was due to the wavelet state space better representing relatively smaller-scale structures, which may also be useful in atmospheric science. The usefulness of wavelet state space is also suggested by the finding of Glendening (2000) that physical-space energetics, partitioned by axial averages along “roll” coherent structures, are sensitive to the local scale, orientation, and quasi-lineality of the rolls. Similarly, Huang (1999) found that optimal energy growth, for small- (but not medium- or large-) scale disturbances constrained in Fourier space, did sensitively depend on zonal variations of the basic state. The joint scale/space energetics that we develop here would be useful in such situations.

The governing PDEs admit several special invariants in absence of forcing, but this study shall be limited to KE. The primitive momentum equations (PME) are our starting point, section 2. In sections 2a–b we sketch the traditional spectral energetics analyses of Saltzman (1957) and subsequent investigators. New wavelet transfer functions (WTFs) and wavelet flux functions (WFFs) of KE are introduced in sections 2c–d. In section 2c the traditional analyses are generalized to *arbitrary* spatially orthogonal projections, extending the Iima and Toh (1995) formulation, and in section 2d a useful manipulation of wavelet-indexed structures is introduced, to derive a measure WFF of spatially localized flux across scales, extending the Meneveau (1991) approach. Construction of triads of transfer between any three “modes” of *any* set of orthogonal spaces is explained in section 2e. Finally, in section 3 we discuss in detail the physical interpretation of wavelet energetics (WE) analyses of two familiar idealized nonlinear flows, those of Burgers (1948) and of Stuart (1967) and Schmid-Burgk (1967). The issue of a particular mode's contribution to its own time evolution is discussed in appendix A. Fournier (1999b) applies WE to the blocking problem using 53 yr of observed global flow data. Table 1 lists all definitions of symbols, notation, etc. not defined elsewhere herein.

## 2. Energetics equations

### a. Zonal-average equations review

Time evolution of horizontal wind (*u, **υ*) is given by the PME [e.g., Holton 1992 (6.1)]. Using the decomposition *f* ≡ *f* + *f*^{★}, Saltzman (1957) rederived the evolution equation for mean eddy KE, *K*, which is

where (letting the length unit be earth's radius)

represents transfer between mean flow and eddies, *C* gives the conversion between specific potential energy and KE, and *D* measures frictional dissipation. The other terms, defined in Table 2, represent horizontal and vertical divergences of *K,* which vanish under integration over a *closed* domain, and so include energy transports across the boundary of any *open* domain considered. Physically, (2.1a) describes KE evolution associated with the collection of zonally localized meteorological phenomena such as storms and quasi-persistent low-pressure systems.

Saltzman (1957) derived similar evolution equations for the zonal-mean-flow KE, 2^{−1}(*u*^{2} + *υ*^{2}), and zonal-mean and eddy forms of the approximate integrand ∝*T*^{★2} of APE. These quantities would complete the atmospheric energetics budget, but they are not investigated in this study. Methods described herein may be applied straightforwardly to generalize those equations.

### b. Fourier-based equations

It is desirable to resolve eddy processes described by (2.1a) into contributions from distinct scales. Traditionally this is done with Fourier series, Table 3. (In section 2c the advantages of applying OWA to this problem will be shown). Saltzman (1957) decomposed *K* = Σ^{∞}_{m=1 }*K*_{m}, where *K*_{m} ≡ *û*_{m}*û*^{*}_{m} + *υ̂*_{m}*υ̂*^{*}_{m}. He introduced the evolution equation

where

(which is the transfer to wavenumber *m* from the zonal mean), *C*_{m} is the conversion from specific potential energy to wavenumber *m, **D*_{m} is the dissipation at wavenumber *m,* and using the Fourier multiplication theorem (FMT) on the TK_{m} expression of Saltzman (1970, Table 1), or (3.14) of Fournier (1998, hereafter F0), yields

(which is the KE triadic transfer to wavenumber *m* from all other nonzero wavenumbers, denoted by *T* in the turbulence literature). Note that every term in (2.2b) reconciles with a respective term in (2.1b), but it is not clear what part of (2.2c) reconciles with the Table-2 terms. [Eqs. (2.1b) and (2.2) correspond to (23) and (47, 48), respectively, of Saltzman (1957).]

Statistical energetics equations (2.1) have been partitioned by wavenumbers, since Σ^{∞}_{m=1} (MK_{m}, *C*_{m}, *D*_{m}) = (MK, *C, **D*) and Σ^{∞}_{m=1 }TK^{S}_{m} = Σ_{a=h,υ} Σ_{b=e,z} BK_{ab}. Thus Saltzman showed that

He later suggested a reformulation for which the (2.3) integrand sum equalled −*K*∇·**V** at every (*ϕ*, *p*), vanishing in the barotropic approximation (Kanamitsu et al. 1972). The physical meaning of such null sums is that TK_{m} describes nonlinear interactions that act to transfer energy between wavenumbers, but create or destroy no net energy. Hansen (1981, p. 31) formulated TK^{S}_{m} = TK_{m} + Σ_{a=h,υ} Σ_{b=e,z} BK_{abm}. Applying FMT to his *C*_{K}(*n*|*m,* ℓ) expression, or (3.26) of F0, yields

TK_{m} is the KE transfer to wavenumber *m* from all other nonzero wavenumbers, *excluding the boundary transports* in Table 4. The advantage of this formulation is that the Σ^{∞}_{m=1} of the Table 4 entries equal the corresponding Table 2 entries, and therefore Σ^{∞}_{m=1} TK_{m} = 0 for *each individual **ϕ* and *p*; so TK_{m} better isolates wave–wave interactions from boundary effects in an open domain.

### c. Equations in the domain of wavelet (or any orthogonal-basis) indexes

#### 1) Formulation

The advantage of wavelet over Fourier energetics formulation is largely in the physical interpretation. As in the Fourier case, nonlinear interactions between particular *scales* can be identified; but in the wavelet case, particular *locations* of the interacting scales are also represented, within a resolution corresponding to the scale. This information is *not available* in Fourier representation.

Applying the wavelet transform [Table 5, reviewed by Fournier (2000, hereafter F1)], to the PME yields

where the ellipses include dissipation effects. To derive the wavelet form, note that

decompose eddy KE into contributions from distinct scales (zonal-wavenumber octaves) and locations, indexed by *j* and *k,* respectively. In his barotropic reformulation mentioned above, Saltzman collected half the trilinear terms to form boundary transports, that are annihilated by integration over a closed domain (Kanamitsu et al. 1972). This was generalized by including pressure-variation effects in the unpublished Fourier energetics of Hansen (1981). In a similar manner, we deduce:

where

is the transfer to scale *j* at location *k* from the zonal mean, *C*_{j,k} ≡ −*p*^{−1}*ω̃*_{j,k}*T̃*_{j,k} is the conversion from APE, *D*_{j,k} is dissipation, the WTF is

[cf. (2.4)], the KE transfer to scale *j* at location *k* from all other *j*′ and *k*′, and other terms, all also at scale *j* and location *k,* are defined in Table 6. Again, there is a one-to-one correspondence between terms (2.5b,c) and in Table 6 on the one hand, and (2.1b) and in Table 2, on the other; *the physical processes have been resolved in both location and scale.* Noncorresponding terms involve ∂_{λ}, for example, (2.5d) and arise from wavelets not being eigenfunctions of the ordinary-derivative operator 𝒟, but are annihilated by Σ_{j,k}.

#### 2) KE conservation by nonlinear eddy interactions

That wave–wave interactions create or destroy no net energy is now expressed by

[In appendix B the numerical method of enforcing (2.6) is explained.] Also note that Σ^{∞}_{j=0 }Σ^{2j−1}_{k=0} (MK_{j,k}, *C*_{j,k}, *D*_{j,k}) = (MK, *C, **D*), the total transfer (2.1b) from mean flow to all eddies, and similarly for other terms mentioned above.

Note that only orthonormality and completeness properties were used to derive (2.5), that is, not any uniquely *wavelet* property such as *two-scale relations* (F1). In the Fourier case, Saltzman used FMT to advantage, a consequence of F_{m}F_{l}F_{n} = *δ*_{0,m+l+n}. There is no similar wavelet identity, although the Parseval theorem still holds, and due to compact support of *W,* the *connection coefficient **W*_{j,k}*W*_{l,m}*W*_{q,r} (e.g., Strang and Nguyen 1995) is an extremely sparse distribution over its indexes. (Such sparseness also implies a form of inequality selection-rule for the triad indexes in section 2e.) Because of this lack, all trilinear terms are calculated by multiplying two factors before transforming. Indeed, the above equations would be obtained for *any* complete, orthogonal basis; the *j, **k* indexes may as well be a single index (see section 2e).

### d. Localized KE flux functions

Adapting the Fourier-based approach of Steinberg et al. (1971), it is useful to construct from the WTF TK_{j,k} a measure FK_{j,k} of total KE *flux* to scale *j* from larger scales *q* < *j, **still localized* at *k,* that is, a WFF. Local downscale (upscale) KE cascades correspond to positive (negative) FK_{j,k}. (In turbulence literature FK is denoted by Π.) The Fourier approach defines

The construction of FK_{j,k} is described in appendix C. An index rearrangement (C.1), or an equivalent energetic bookkeeping, is necessary so that in the Σ^{j}_{q=0} no energy at larger-scale indexes (*q, **r*) is *double counted* for different *k.* Only those elements with *r* accounting (at resolution *q*) for the same location as does *k* (at resolution *j*), will contribute to FK_{j,k}. This makes FK_{j,k} a meaningfully local flux function.

### e. Detailed triadic interactions among wavelet (or any orthogonal basis) indexes

It has been useful in some applications of Fourier-based energetics (section 2b) to decompose the KE transfer (2.4) into detailed transfers among triads (*m, **l, **n*) of wavenumbers. For example, Waleffe (1993) used his proposition that in 3D, two like-helicity wavevector modes (**m**, −**n**) always transfer KE from the opposite-helicity mode **n** − **m**, in order to prove a tendency toward two-dimensionalization of strongly rotating flows. Krishnamurti et al. (1999) showed that individual barotropic-scale triadic interactions contributed importantly to the maintenance of downstream amplification across the Pacific, prior to an intense freeze event over Brazil.

Triads are constructed as follows:

where due to the Σ_{l,n} only the *ln*-symmetric part of “triad-*mln*” [deducible from (2.4)]

need be retained. Underlying the zero-sum rule is the *detailed conservation* rule

(e.g., Lesieur 1990).

Iima and Toh (1995) observed that transfer triads, among *any* orthonormal-basis indexes, may be formulated from the quadratic terms of typical flow equations. It should be mentioned that for their formulation the flow must be incompressible (and the basis must be *t* independent). The WTF function formulation presented here has been extended from that of Iima and Toh (1995) to compressible, hydrostatic flow.

The WTF (2.5e) may be decomposed as

where

The detailed conservation rule for wavelets is

easily proven using the observation that (2.8) is (*jk, **qr*) antisymmetric. Writing (2.8) in antisymmetric form is essential.

Iima and Toh (1995) formulated wavelet-based incompressible triads but made no direct application, and such triads have been employed only recently in the literature (Iima and Toh 1998; Kishida et al. 1999). Fournier (1998) exhibits and discusses triads corresponding to nonlinear interaction of an atmospheric block with two *collections* of cyclone-scale eddies of significant amplitude, scale, and location. This is done by further generalizing (2.8) to

where superscripts *a, **b, **c* denote *any* projections orthogonal with respect to products *uυ* and **U**·**V**. Then detailed conservation takes the form

From the *general triad *(2.9), the Fourier triad (2.7) may be derived by substitutions of the form *u*^{a} → *û*_{m}F_{m}, and the wavelet triad (2.8), by *u*^{a} → *ũ*_{j,k}*W*_{j,k}; that is, individual components constitute 1D orthogonal subspaces. Fournier (1998) uses collections of components to construct specialized multidimensional orthogonal subspaces.

When a coordinate-independent triad is useful, write (2.9) equivalently as

generalizing Eq. (3.4) of Iima and Toh (1995) to compressible hydrostatic flow.

## 3. Wavelet energetics of simple flow models

In this section we discuss energetics examples of simple model flows. Section 3a covers Burgers equation energetics as viewed in Fourier and wavelet analyses. The Burgers equation for a developing *front* (e.g., region of large *strain **U*_{x} < 0) is a common illustration of certain advantages of wavelet representation for nearly singular structures. (In appendix A we discuss how particular components interact *with themselves* for different functional analyses.) In section 3b both types of energetics are applied to Stuart-vortex flow, one of the simplest localized periodic explicit solutions available in nonlinear fluid dynamics. Both these examples will be used to discuss issues regarding the *physical interpretation* of WE, which is less straightforward than in the Fourier case, largely due to the latter's shift invariance. Fourier pays for shift invariance by losing location information; conversely, the cost of location information appears to be a loss of complete shift invariance. However, shift *in*variance may be either *restored* or *replaced* by shift *equi*variance [meaning that a function shift implies the same analysis-shift, as opposed to the *truncated detailed bits* shift reviewed by F1 and discussed in section 3b(2).] The two remedies presented here have been applied for the first time to energetics, by the author.

The first remedy is to regain complete shift invariance by rotating the global coordinate system (parameter *λ*_{ε}) until the most efficient representation is obtained, using *best-shift* criteria reviewed by F1. Thus whatever the original location of the structures, best shift always yields the same coefficients. The best-shift remedy will not be discussed further here, since it is explained by F1, and further applications are presented by F0.

The other remedy, alluded to by F1, is to consider all possible shifts at once. The kind of shift-equivariant WE obtained in this *stationary* or *overcomplete* wavelet analysis will be discussed in sections 3a–b.

### a. Burgers developing shock

Burgers (1948) equation (BE) describes time evolution of a 1D flow ∝*U*(*x, **t*), subject to dissipation (proportional to Re^{−1}) and nonlinear advection. With periodic boundary conditions and initial state *U*(*x,* 0) = −sin*x,* the Cole-Hopf analytic solution, shown in Fig. 1a, is reported, for example, by Platzman (1964). The BE dynamics cause *U* to evolve toward a decaying sawtooth function.

The BE nonlinear term implies that KE dissipation is modified by advection, which in a location-only representation is shown in Fig. 1b. The physical interpretation of this figure is that as the front develops, KE is increased at symmetric locations immediately adjacent, at expense of broad, shallow KE reductions far away. There is no more clue to the scales involved in these transfers than the apparent size of the structures involved.

#### 1) Fourier analysis of Burgers developing shock

The traditional method to describe scale interactions is Fourier series. At small *t,* wavenumber 1 greatly dominates; at large *t, **Û*_{m} approaches the sawtooth-curve result ∼*i*(−1)^{m}*m*^{−1}. For simple 1D flows the wavenumber transfer (2.4) tends to

shown in Fig. 2a. Wavenumber 1 always loses KE, initially to only slightly larger wavenumbers. Eventually the low wavenumbers together give up KE to much larger wavenumbers.

Motion of the KE-receiving wavenumber band toward higher *m* appears more clearly in a plot of the KE flux function *F*_{m}, (Fig. 2b), noting the median-|*F*_{m}| labels. Since *F*_{m} > 0 for all *m,* there is a *uniform* downscale KE cascade in this *global* representation. This figure is similar to figures of Wiin-Nielsen (1994) and Girimaji and Zhou (1995).

#### 2) Wavelet analysis of Burgers developing shock

Figure 3a plots *Ũ*_{j,k} (appendix D) at three times. Generally |*Ũ*_{j,k}| decreases with time for large scales *j* ≤ 1, and grows for small scales *j* > 1, but only at locations *k* ≈ 2^{j−1}, that is, *x* = *λ*_{j,k} ≈ 0. Efficiency of this localization of coefficients (also discussed by F1) appearing in the time evolution is the main motivation behind many attempts to use wavelets for numerical solution (e.g., Bertoluzza et al. 1994; Lazaar et al. 1994; Fournier 1995; Vasilyev et al. 1995; Beylkin and Keiser 1997 and references therein). OWA level *j* corresponds to a difference of smoothings accomplished by low-pass filters differing in scale by a factor of two. Pictographically, the shape of *Ũ*_{1,k} corresponds to *U* > 0 and *U* < 0 lobes (Fig. 1a), and the shape of *Ũ*_{2,k} corresponds to departures of smaller scales as the sawtooth develops. Following figures are similarly laid out, so the arrangement of information here should be made familiar before proceeding.

Figure 4a shows the shift-equivariant wavelet transform (SEWT) *Ũ*_{j,2j−Jℓ} discussed by F1. Small scales are similar in structure to *Ũ*_{j,k} in Fig. 3a, but large scales are different enough to warrant some explanation. Each *Ũ*_{j,2j−Jℓ} is a *correlation* of *U* with a dilated *W,* so large *Ũ*_{j,2j−Jℓ} waves for small *j* indicate strong local correlation between that wavelet and *U.* For instance, *Ũ*_{0,2−Jℓ} at *t* = 0.10 resembles a sine, because *W*_{0,0} ≈ triangle wave ∝−|*x* − *m*_{0}|, whose correlation with negative sine is a sine: |·| sin(·+ 2^{1−J}*π*ℓ) ∝ sin*λ*_{J,ℓ}.

##### (i) Wavelet energetics of Burgers developing shock

Corresponding to *L*_{m }(3.1), in the wavelet representation of this simple system the WTF (2.5e) tends to

shown in Fig. 3b. Physical interpretation of this figure is not too difficult. For *t* = 0.68, *Ũ*_{0,0} (not shown) and *Ũ*_{1,k} lose KE to smaller scales *j* > 1 near the front at *x* = 0 (*k* ≈ 2^{j−1}). Later at *t* = 2.42, this local downscale KE transfer itself has locally shifted downscale, with *Ũ*_{2,k} also losing a little KE, and KE-receiving coefficients (*j* > 2, *k* ≈ 2^{j−1}) becoming increasingly localized. Asymmetry effects are discussed in appendix D.

The WFF (C.2b) is shown in Fig. 3c. Regions of downscale cascade (*F*_{j,k} > 0, loss from larger scales) are clearly away from *x* ≈ 0. Near *x* = 0 there is some localized upscale cascade, or *backscatter,* especially at *t* = 2.4. *This combination of spatial and scale information is not provided by Fourier energetics.*

##### (ii) Stationary wavelet energetics of Burgers developing shock

By regarding the SEWT representation *L*_{j,2j−Jℓ} of the WE (Fig. 4b) one first notes that the joint region in (*j,* ℓ) that receives most KE is moving to higher *j,* closer to the front, as time increases. There are zeros of *L*_{1,21−Jℓ} at |*λ*_{J,ℓ}| ≈ *π*/2. At these points, *Ũ*_{0,0}, instead of *Ũ*_{1,k}, expresses the large-scale KE loss. Let us discuss the particular alternative wavelet-basis location (*λ*_{ε} in Table 5) implied by this observation. Shifted spatial energetics (not shown) are *merely shifted eastward by **π*/2 from Fig. 1b. The well-known phase-shift effect is 〉*U*_{m} = *i*^{−m}*Û*_{m}; thus the shifted Fourier energetics (not shown) are completely unaffected since shift-induced phase factors *i*^{m}*i*^{−m+n}*i*^{−n} = 1 in (3.1). However it so happens that the shifted initial flow 〉*U*(*x,* 0) ≈ 2^{−1/2}*W*_{0,0}(*x*). Comparing the shifted-flow WE (not shown), since 〉*U* so well aligns with *W*_{0,0}, so 〉*U*_{0,0} is the primary KE source for 〉*U,* whereas *Ũ*_{1,k} was the primary source for *U.* This is physically significant in the following way. In order to *compare* Fourier and wavelet energetics, we may remove the latter's location dependence and compare the location sum Σ^{2j−1}_{k=0 }*L*_{j,k} with *L*_{nj}, the Fourier transfer at octave-*j* peak wavenumber. That sum (not shown) wrongly indicates that *n*_{1} = 2 is the primary KE source for *U,* but by the *π*/2 shift the WTF shown in Fig. 2c shows the primary source for 〉*U* at *n*_{0} = 1, and better resembles Fig. 2a. Lastly, Σ^{2J−1}_{ℓ=0 }*L*_{j,2j−Jℓ} (Fig. 2d) also compares reasonably well with *L*_{nj}.

#### 3) Summary of Burgers developing shock analysis

In this section we have seen how origin choice affects WE analysis of the Burgers flow, which evolves a steady discontinuity such as might model a strong atmospheric longitudinal front/strain. Three representations were discussed, besides the purely spatial one. The Fourier basis is shift invariant and so describes nothing about front location, although a global downscale cascade is evident. The OWA basis shows spatial localization of the downscale cascade, but large scales are sensitive to origin shift and wavelet asymmetry. These sensitivities can be corrected as discussed above. SEWT is shift equivariant, but does not contain a single orthonormal basis; rather it contains all possible orthonormal bases arising from shifts. This redundancy slightly complicates the interpretation of large scales. In the next section we examine these issues in regard to a translating pulse of constant shape.

### b. Stuart-vortex flow

Consider a reference frame moving at unit eastward velocity. Total westerly flow ∝*U*(*x, **t*) + 1 resulting from the Stuart-vortex (Schmid-Burgk 1967; Stuart 1967) exact solution to the 2D inviscid vorticity equation is shown in Fig. 1c.

#### 1) Fourier analysis of Stuart-vortex flow

The initial Fourier coefficient

is real (Boyd 2000). Since *U*(*x, **t* + *s*) = *U*(*x* − *s, **t*), time *t* also denotes location; *t* will be referred to implicitly below as a spatial shift. By the phase-shift effect, |*Û*_{m}| is constant and arg*Û*_{m} varies linearly with *t.*

Because ℑ*Û*_{m}(0) = 0, by (3.1), *L*_{m}(*t* = 0) = 0. Since shift phase factors *e*^{−int}*e*^{i(m−n)t}*e*^{int} = 1 in (3.1), so *L*_{m}(*t* > 0) = 0. Thus *there is in the Fourier picture no energetics associated with a purely translating symmetric structure.* But are there energetic transfers in the wavelet picture? To answer this question, first let us regard the purely spatial energetics described by −*U*^{2}*U*_{x} (Fig. 1d). Let us refer to *x* regions where *U* and *U*_{x} have like sign as *stretched* (Holton 1992). Then Fig. 1d shows that in a purely spatial picture of Stuart-vortex dynamics, eastward advection acts to remove KE from stretched regions and give it to *compressed* regions, that is, where *U* and *U*_{x} are opposite.

#### 2) Wavelet analysis of Stuart-vortex flow

As noted elsewhere, OWA is not shift invariant. Figure 3d shows how *Ũ*_{j,k} (phase corrected as in appendix D) depends on vortex location, *t.* At *t* = 3*τ* the vortex has progressed across exactly ¼ of the domain, and each *Ũ*_{j>1,k}(*t*) is purely shifted eastward in *k* by 2^{j−2} from *Ũ*_{j>1,k}(0). However *Ũ*_{j≤1,k}(3*τ*), and all *Ũ*_{j,k}(*t*) for *t* not on the dyadic grid *λ*_{q,r}, are not merely shifted but also *coupled* to other scales. [Similar analysis behavior was demonstrated, for example, by Weng and Lau (1994, Fig. A1).] At *t* = 0, |*Ũ*_{0,0}| is large because *W*_{0,0} well correlates with *U.* As the vortex moves eastward, at *t* = *τ,* more amplitude at smaller scales comes to construct the pulse shape, with *Ũ*_{1,1}. Also recall *W*_{j,k} = 0, so that even though *U* > 0, *U*^{★} has both signs and so other wavelets may arise where *U* is small, because there *U*^{★} < 0 may be large (e.g., *Ũ*_{1,0} < 0). (Since *U* > 0 it might be better to use a *curtailing **R* > 0, reviewed by F1, since *W*^{⊥}_{R,k} > 0 there.) Still smaller wavelets in the east either augment *Ũ*_{1,1}(*t*) [e.g., *Ũ*_{2,2}(*τ*)] or compensate for it [e.g., *Ũ*_{2,3}(2*τ*), *Ũ*_{2,2}(3*τ*)]. Smaller scales *j* > 2 constantly track the location *t.*

##### (i) Wavelet energetics of Stuart-vortex flow

It turns out that, unlike *L*_{m} = 0, *L*_{j,k} ≠ 0 in this case. In fact it displays a complicated structure over both location *k* and scale *j* as shown in Fig. 3e. Pictographically, for fixed *j* the rough structure in the *k* dependence of *L*_{j,k} reflects the −*U*^{★}*U*^{★}_{x} structure inferable from Figs. 1c–d. For example, the pairs (*L*_{5,15}, *L*_{5,16}) at *t* = 0, (*L*_{1,0}, *L*_{1,1}), (*L*_{2,1}, *L*_{2,2}) at *t* = *τ,* (*L*_{1,0}, *L*_{1,1}), (*L*_{3,4}, *L*_{3,5}) at *t* = 2*τ* and others all resemble , and follow location *t* with *k.* Nevertheless the evident *t* dependence of *L*_{j,k} also couples or mixes the scales *j,* so that many exceptions to the pattern appear. Thus the physical local-energetics interpretation of *L*_{j,k} (and hence *F*_{j,k}) ultimately depends on the coordinate-origin choice describing the analyzed flow.

Fortunately the Σ^{j}_{q=0} in (C.2b) partially compensates for scale coupling induced by *t* shift, so that *F*_{j>2,k} structure (Fig. 3f) is qualitatively less *t* dependent, aside from uniform pattern shifts. Sign reversals of *F*_{1,k} after *τ*^{−1}*t* = 1 and 3 reflect the shift of part of the *U*^{★} < 0 segment to the *x*-half-domain formerly occupied by *U*^{★} > 0. This is also the case for *F*_{0,1}(3*τ*). One concludes that large-scale (*j* ≤ 2) WFFs are still sufficiently sensitive to *t* that coordinate-origin choice must be considered part of the overall analysis, which provides motivation for using a best-shift or multiple-shift approach, as well as correcting the phases as in appendix D.

##### (ii) Stationary wavelet energetics of Stuart-vortex flow

Fournier (1998) presents results of applying the *best-shift* coordinate-origin choice to observed atmospheric flows. As an alternative, we discuss here the SEWT WE of Stuart-vortex flow. Figure 4c shows the SEWT, *Ũ*_{j,2j−Jℓ} (phase corrected as in appendix D). For every *j,*

that is, SEWT simply shifts equivariantly with *U.* As in the Burgers case, the resemblance of *Ũ*_{0,2−Jℓ}(0) to a cosine is roughly explained by noting that *U*(*x,* 0) ≈ cos*x* and that |·| cos(·+ 2^{1−J}*π*ℓ) ∝ cos*λ*_{J,ℓ}.

SEWT KE transfer *L*_{j,2j−Jℓ} is shown in Fig. 4d. It is shift equivariant, and so carries *none* of the shift-induced scale coupling discussed in regard to *L*_{j,k}. However *L*_{j,2j−Jℓ} does oscillate as a function of ℓ. Primarily this reflects contributions (from the wavenumber octaves of each *j*) to the curves in Fig. 1d that resemble those in Fig. 4d. The smaller additional oscillations are simply a manifestation of *L*_{j,k} shift sensitivity for any *particular* coordinate origin, expressed by *all* coordinate-origin choices at once in *L*_{j,2j−Jℓ}. Consistent with *L*_{m} = 0 for all *m,* it is observed that Σ^{2J−1}_{ℓ=0 }*L*_{j,2j−Jℓ} = 0 for all *j.*

## 4. Conclusions

Expressions have now been derived for transfer and flux of kinetic energy between modes of an arbitrary orthogonal partitioning of meteorological fields. A particular partition was generated by an orthonormal-wavelet basis, so that transfers and fluxes were quantified, between modes that were simultaneously localized in both scale (zonal-wavenumber octave) and location. Interaction triads were also generally formulated. Results of Meneveau (1991) and Iima and Toh (1995) for incompressible Navier–Stokes type systems have been extended to the compressible hydrostatic case, with the full primitive equation budget now readily at hand.

Applying wavelet energetics to relatively simple model flows showed how to associate wavelet energetics features with structures of the more familiar location and Fourier coefficient pictures. On these models, techniques were tested for remedying lack of shift invariance and removing or explaining artifacts. Further practical and theoretical applications will appear from F2.

The generalization of these results implies applications to and greater understanding of atmospheric energetic phenomena, at least those for which interactions between structures that are simultaneously localized in space and scale are the most significant. Because combining space and scale representations in an orthogonal framework can entail some compromises in unambiguity of interpretation, in cases such as discussed above, care must be taken in applying such combined representations and the associated energetics to too broad a class of phenomena. However such caveats are required for every analysis method; the present paper is offered to provide sufficient details for carrying out wavelet energetics, and to illuminate appropriate precautions.

## Acknowledgments

I thank B. Saltzman, R. B. Smith, K. R. Sreenivasan, R. R. Coifman, A. R. Hansen, P. D. M. Parker, and J. C. van den Berg for comments on an earlier version of this paper, two anonymous reviewers for comments on the revision, M. J. Mohlenkamp for technical consultation and the University of Maryland, College of Computer, Mathematical, and Physical Sciences, Department of Meteorology, Earth System Science Interdisciplinary Center for support while the manuscript was written. This material is based upon work supported by the National Science Foundation under Grant 9420011 and through the National Center for Atmospheric Research Project No. 36211014. The WaveLab© software used in this work is available online (Buckheit et al. 1995).

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### APPENDIX A

#### Self-Interaction in Energetics

In analyzing flow energetics into orthogonal wavelet components, there arises an issue of energy gain from *other* “modes” versus from a mode *to itself*. Consider the simple inviscid (Re → ∞) system: *U*_{t} = −*UU*_{x} and suppose an analysis

where the *analysis* function *ψ*^{n} and *synthesis* function *ψ*_{n} (e.g., Wickerhauser 1994) satisfy

but are otherwise unspecified. In the wavelet case,

and in the complex Fourier case *ψ*^{n}(*x*) = *ψ*^{*}_{n}(*x*) ≡ *F*^{*}_{n}[(2*x* − 1)*π*]. There are also two real Fourier cases

The *self-interaction* question is whether or not a particular mode *n* contributes to its own change. That is, does ∂*U̇*_{n}/∂*U*_{n} = 0? Component *n* self-interaction is determined by

for *U* an *arbitrary* function. (Note that in case *ψ*^{n} = *ψ*_{n} = *F*^{″}_{n} this result is simply the familiar wavenumber *n* interaction with its harmonic 2*n.*) Unless *U* = 0 or *ψ*_{n}𝒟*ψ*^{n} = 0 there must be *some* kind of self-interaction. Even if *U* *ψ*_{n}𝒟*ψ*^{n} = 0, in the general case *U*_{n} ≠ 0 requires *d*_{n}(*x*) ≡ *ψ*_{n}(*x*)𝒟*ψ*^{n}(*x*) be *x* independent. In the complex Fourier case *d*_{n} = −*in* is sufficient, but when *ψ*_{n} = *ψ*^{n} the requirement leads to *ψ*_{n}(*x*) = [*ψ*_{n}(0)^{2} + 2*d*_{n}*x*]^{1/2}, which is not realizable. Hence wavelet and other representations (but not complex Fourier) must include some self-interaction. However, an initial state consisting of a single component *alone* will *not* evolve over *t,* just as the case for a single Fourier mode, because the only interaction is −*U*^{2}_{n}*ψ*^{2}_{n}𝒟*ψ*_{n} = 0.

### APPENDIX B

#### Numerical Considerations

Although in the Fourier basis it is simple to expand nonlinear terms into convolutions, and use *f*_{λ}_{m} = *imf̂*_{m}, in the wavelet basis it is simpler to perform products and ∂_{λ} operations *before* the transform. Error in numerically verifying ∫^{1}_{0 }*u*_{x}*u*^{α }*dx* = 0 was reduced using the scheme (Celia and Gray 1992)

where *γ*_{ℓ} = 5/6, −5/21, 5/84, −5/504, 1/1260. A first-order scheme was used for ∂_{ϕ}.

### APPENDIX C

#### Construction of WFFs

The 2^{J} − 1 elements TK_{j,k} form a pyramidal tableau with 2^{j} elements at level *j. *Fournier (1996) introduced a rectangular *J* × 2^{J−1}-matrix equivalence

The equivalence (C.1) is normalized so that Σ^{2J−1−1}_{ℓ=0} TK_{j}(*λ*_{J,2ℓ}) = Σ^{2j−1}_{k=0} TK_{j,k}, to preserve the spatial-index sum, but make the left-sum limits scale-independent. To illustrate, for *J* → 3 the TK_{j,k} tableau is shown in Table C1. Then by Eq. (C.1) the elements TK_{j}(*λ*_{J,2ℓ}) form Table C2.

Fournier (1996) independently introduced a KE WFF

[It is computationally faster to use (C.1, C.2a) rather than (C.2b); however (C.2b) is comparable to a formulation by Meneveau (1991).] By (2.6) and the normalization choice,

Only those transfer amounts at the correct location *λ*_{J,2ℓ} will contribute to FK_{j}(*λ*_{J,2ℓ}). This makes FK_{j}(*λ*_{J,2ℓ}) a meaningfully local flux function.

### APPENDIX D

#### Wavelet Choice and Symmetry Effects

As suggested by an anonymous reviewer, the wavelet filter dubbed *coiflet* by Daubechies (1992) was used. At length 12, it has minimal nonlinear-phase error (Wickerhauser 1994), and the linear-phase error for *j* > 2 was corrected by shifts *Ũ*_{j,k} ← *Ũ*_{j,k−j[mj]}. SEWT phases were corrected by *Ũ*_{j,2j−Jℓ} ← *Ũ*_{j,2j−J(ℓ−J[2J−jmj])}, for all *j.* The *mode of energy **m*_{j} estimates the cumulative phase-shift of *J* − *j* convolution decimations (see F1) with the filter. {Estimating the more usual *center of energy* 2^{j }∫^{2−1}_{−2−1 }*x**W*_{j,2j−1}(2*πx*)^{2 }*dx* yields the same [*m*_{j}] except for *j* = 1, because of periodicity.} Walden and Contreras Cristan (1998) discuss another approach to phase correction.

Asymmetry in *Ũ*_{j,k} is due to the impossibility of *perfectly* (anti)symmetric real-valued compactly supported orthogonal wavelets, aside from Haar wavelets (Daubechies 1992). The coiflet design trades off some compactness for some symmetry. [Beylkin (1995) gets nearly perfect symmetry by sacrificing one part in *O*(10^{15}) of the reconstruction condition.]

The west–east “dipole” *L*_{1,0} < 0 < *L*_{1,1} at *t* = 0.10 in Fig. 3b implies that KE is being transferred from west to east. That is clearly not the correct physical interpretation. It is an artifact of wavelet asymmetry, which can be seen as follows. Denote (3.2) by *L*_{j,k}(*W, **U*). Then to see *W* asymmetry effects we must compute *L*_{j,k}(*W*^{−}, *U*). Since

## Footnotes

Current affiliation: Department of Meteorology, University of Maryland, College Park, Maryland

*Corresponding author address:* Aimé Fournier, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000. Email: fournier@ucar.edu