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 Unpublished modified formulas for global Fourier energetics (FE) are reviewed that conserve KE for the case of a single latitude-circle and pressure level.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.

Table 1.

Definitions

Definitions
Definitions

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 ff + f, Saltzman (1957) rederived the evolution equation for mean eddy KE, K, which is

 
formula

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

 
formula

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.

Table 2.

Zonal-average KE boundary-transport terms BKab, where a = h and υ correspond to horizontal and vertical transport due to zonal-average (b = z) or -eddy (b = e) flow

Zonal-average KE boundary-transport terms BKab, where a = h and υ correspond to horizontal and vertical transport due to zonal-average (b = z) or -eddy (b = e) flow
Zonal-average KE boundary-transport terms BKab, where a = h and υ correspond to horizontal and vertical transport due to zonal-average (b = z) or -eddy (b = e) flow

Saltzman (1957) derived similar evolution equations for the zonal-mean-flow KE, 2−1(u2 + υ2), 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 = Σm=1 Km, where Kmûmû*m + υ̂mυ̂*m. He introduced the evolution equation

 
KmtmCmDmSm
(2.2a)

where

 
formula

(which is the transfer to wavenumber m from the zonal mean), Cm is the conversion from specific potential energy to wavenumber m, Dm is the dissipation at wavenumber m, and using the Fourier multiplication theorem (FMT) on the TKm expression of Saltzman (1970, Table 1), or (3.14) of Fournier (1998, hereafter F0), yields

 
formula

(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).]

Table 3.

Fourier series

Fourier series
Fourier series

Statistical energetics equations (2.1) have been partitioned by wavenumbers, since Σm=1 (MKm, Cm, Dm) = (MK, C, D) and Σm=1 TKSm = Σa=h,υ Σb=e,z BKab. Thus Saltzman showed that

 
formula

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 TKm describes nonlinear interactions that act to transfer energy between wavenumbers, but create or destroy no net energy. Hansen (1981, p. 31) formulated TKSm = TKm + Σa=h,υ Σb=e,z BKabm. Applying FMT to his CK(n|m, ℓ) expression, or (3.26) of F0, yields

 
formula

TKm 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 Σm=1 of the Table 4 entries equal the corresponding Table 2 entries, and therefore Σm=1 TKm = 0 for each individual ϕ and p; so TKm better isolates wave–wave interactions from boundary effects in an open domain.

Table 4.

Fourier-spectral boundary-transport terms BKabm to wavenumber m, as in Table 2, except b = e now refers to all other wavenumbers n ≠ 0, m. [BKaem follow from FMT applied to the FNam equation of Hansen (1981, p. 32), or (3.29)–(30) of F0.]

Fourier-spectral boundary-transport terms BKabm to wavenumber m, as in Table 2, except b = e now refers to all other wavenumbers n ≠ 0, m. [BKaem follow from FMT applied to the FNam equation of Hansen (1981, p. 32), or (3.29)–(30) of F0.]
Fourier-spectral boundary-transport terms BKabm to wavenumber m, as in Table 2, except b = e now refers to all other wavenumbers n ≠ 0, m. [BKaem follow from FMT applied to the FNam equation of Hansen (1981, p. 32), or (3.29)–(30) of F0.]

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

 
formula

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

 
formula

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:

 
formula

where

 
formula

is the transfer to scale j at location k from the zonal mean, Cj,k ≡ −p−1ω̃j,kj,k is the conversion from APE, Dj,k is dissipation, the WTF is

 
formula

[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 Σj,k.

Table 5.

Wavelet transform. See F1 and Daubechies (1992, section 9.3)

Wavelet transform. See F1 and Daubechies (1992, section 9.3)
Wavelet transform. See F1 and Daubechies (1992, section 9.3)
Table 6.

Wavelet boundary-transport terms BKabj,k to scale j and location k, as in Table 4, except b &equals a now refers to APE transport

Wavelet boundary-transport terms BKabj,k to scale j and location k, as in Table 4, except b &equals a now refers to APE transport
Wavelet boundary-transport terms BKabj,k to scale j and location k, as in Table 4, except b &equals a now refers to APE transport

2) KE conservation by nonlinear eddy interactions

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

 
formula

[In appendix B the numerical method of enforcing (2.6) is explained.] Also note that Σj=0 Σ2j−1k=0 (MKj,k, Cj,k, Dj,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 FmFlFn = δ0,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 Wj,kWl,mWq,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 TKj,k a measure FKj,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) FKj,k. (In turbulence literature FK is denoted by Π.) The Fourier approach defines

 
formula

The construction of FKj,k is described in appendix C. An index rearrangement (C.1), or an equivalent energetic bookkeeping, is necessary so that in the Σjq=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 FKj,k. This makes FKj,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 nm, 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:

 
formula

where due to the Σl,n only the ln-symmetric part of “triad-mln” [deducible from (2.4)]

 
formula

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

 
formula

(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

 
formula

where

 
formula

The detailed conservation rule for wavelets is

 
formula

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

 
formula

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

 
TKc|(a,b) + TKa|(b,c) + TKb|(c,a) = 0.

From the general triad (2.9), the Fourier triad (2.7) may be derived by substitutions of the form uaûmFm, and the wavelet triad (2.8), by uaũj,kWj,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

 
formula

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 Ux < 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 invariance may be either restored or replaced by shift equivariance [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 λε) 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) = −sinx, 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.

Fig. 1.

(a) U, after Platzman (1964) vs x (211-point grid), shaded by t = 0.58(1, · · · 6) − 0.48. Re = 30. (b) −U2Ux. (c) Stuart-vortex U vs x (27-point grid), shaded by τ−1t = 0, · · · 3; the transverse coordinate y = −0.5. (d) −U2Ux as in (b).

Fig. 1.

(a) U, after Platzman (1964) vs x (211-point grid), shaded by t = 0.58(1, · · · 6) − 0.48. Re = 30. (b) −U2Ux. (c) Stuart-vortex U vs x (27-point grid), shaded by τ−1t = 0, · · · 3; the transverse coordinate y = −0.5. (d) −U2Ux as in (b).

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, Ûm approaches the sawtooth-curve result ∼i(−1)mm−1. For simple 1D flows the wavenumber transfer (2.4) tends to

 
formula

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.

Fig. 2.

(a) Equation (3.1) vs wavenumber n (log scale), shaded by t as in Figs. 1a,b. (b) Fn vs n, with time index labeling median (|Fn|, n). (c) Σ2j−1k=0 Lj,k(〉U) vs wavelet peak-wavenumber nj. (d) Σ2J−1ℓ=0 Lj,2jJ.

Fig. 2.

(a) Equation (3.1) vs wavenumber n (log scale), shaded by t as in Figs. 1a,b. (b) Fn vs n, with time index labeling median (|Fn|, n). (c) Σ2j−1k=0 Lj,k(〉U) vs wavelet peak-wavenumber nj. (d) Σ2J−1ℓ=0 Lj,2jJ.

Motion of the KE-receiving wavenumber band toward higher m appears more clearly in a plot of the KE flux function Fm, (Fig. 2b), noting the median-|Fm| labels. Since Fm > 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 Ũj,k (appendix D) at three times. Generally |Ũj,k| decreases with time for large scales j ≤ 1, and grows for small scales j > 1, but only at locations k ≈ 2j−1, that is, x = λj,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 Ũ1,k corresponds to U > 0 and U < 0 lobes (Fig. 1a), and the shape of Ũ2,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.

Fig. 3.

(a) Wavelet coefficients Ũj,k at three times t = 0.10, 0.68, 2.42 (shading) from Figs. 1a,b. Each row shows bars for a given j vs k. (b) WTF (3.2). (c) WFF Fj,k, as in (a)

Fig. 3.

(a) Wavelet coefficients Ũj,k at three times t = 0.10, 0.68, 2.42 (shading) from Figs. 1a,b. Each row shows bars for a given j vs k. (b) WTF (3.2). (c) WFF Fj,k, as in (a)

Figure 4a shows the shift-equivariant wavelet transform (SEWT) Ũj,2jJ discussed by F1. Small scales are similar in structure to Ũj,k in Fig. 3a, but large scales are different enough to warrant some explanation. Each Ũj,2jJ is a correlation of U with a dilated W, so large Ũj,2jJ waves for small j indicate strong local correlation between that wavelet and U. For instance, Ũ0,2J at t = 0.10 resembles a sine, because W0,0 ≈ triangle wave ∝−|xm0|, whose correlation with negative sine is a sine: |·| sin(·+ 21−Jπℓ) ∝ sinλJ,.

Fig. 4.

(a) SEWT Ũj,2jJm(t) vs [21−J(m − 1) − 1]π. (b) SEWT KE transfer Lj,2jJm

Fig. 4.

(a) SEWT Ũj,2jJm(t) vs [21−J(m − 1) − 1]π. (b) SEWT KE transfer Lj,2jJm

(i) Wavelet energetics of Burgers developing shock

Corresponding to Lm (3.1), in the wavelet representation of this simple system the WTF (2.5e) tends to

 
formula

shown in Fig. 3b. Physical interpretation of this figure is not too difficult. For t = 0.68, Ũ0,0 (not shown) and Ũ1,k lose KE to smaller scales j > 1 near the front at x = 0 (k ≈ 2j−1). Later at t = 2.42, this local downscale KE transfer itself has locally shifted downscale, with Ũ2,k also losing a little KE, and KE-receiving coefficients (j > 2, k ≈ 2j−1) becoming increasingly localized. Asymmetry effects are discussed in appendix D.

The WFF (C.2b) is shown in Fig. 3c. Regions of downscale cascade (Fj,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 Lj,2jJ 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 L1,21−J at |λJ,| ≈ π/2. At these points, Ũ0,0, instead of Ũ1,k, expresses the large-scale KE loss. Let us discuss the particular alternative wavelet-basis location (λε 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 〉Um = imÛm; thus the shifted Fourier energetics (not shown) are completely unaffected since shift-induced phase factors imim+nin = 1 in (3.1). However it so happens that the shifted initial flow 〉U(x, 0) ≈ 2−1/2W0,0(x). Comparing the shifted-flow WE (not shown), since 〉U so well aligns with W0,0, so 〉U0,0 is the primary KE source for 〉U, whereas Ũ1,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 Σ2j−1k=0 Lj,k with Lnj, the Fourier transfer at octave-j peak wavenumber. That sum (not shown) wrongly indicates that n1 = 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 n0 = 1, and better resembles Fig. 2a. Lastly, Σ2J−1ℓ=0 Lj,2jJ (Fig. 2d) also compares reasonably well with Lnj.

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

 
formula

is real (Boyd 2000). Since U(x, t + s) = U(xs, t), time t also denotes location; t will be referred to implicitly below as a spatial shift. By the phase-shift effect, |Ûm| is constant and argÛm varies linearly with t.

Because ℑÛm(0) = 0, by (3.1), Lm(t = 0) = 0. Since shift phase factors eintei(mn)teint = 1 in (3.1), so Lm(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 −U2Ux (Fig. 1d). Let us refer to x regions where U and Ux 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 Ux are opposite.

2) Wavelet analysis of Stuart-vortex flow

As noted elsewhere, OWA is not shift invariant. Figure 3d shows how Ũj,k (phase corrected as in appendix D) depends on vortex location, t. At t = 3τ the vortex has progressed across exactly ¼ of the domain, and each Ũj>1,k(t) is purely shifted eastward in k by 2j−2 from Ũj>1,k(0). However Ũj≤1,k(3τ), and all Ũj,k(t) for t not on the dyadic grid λq,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, |Ũ0,0| is large because W0,0 well correlates with U. As the vortex moves eastward, at t = τ, more amplitude at smaller scales comes to construct the pulse shape, with Ũ1,1. Also recall Wj,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., Ũ1,0 < 0). (Since U > 0 it might be better to use a curtailing R > 0, reviewed by F1, since WR,k > 0 there.) Still smaller wavelets in the east either augment Ũ1,1(t) [e.g., Ũ2,2(τ)] or compensate for it [e.g., Ũ2,3(2τ), Ũ2,2(3τ)]. Smaller scales j > 2 constantly track the location t.

(i) Wavelet energetics of Stuart-vortex flow

It turns out that, unlike Lm = 0, Lj,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 Lj,k reflects the −UUx structure inferable from Figs. 1c–d. For example, the pairs (L5,15, L5,16) at t = 0, (L1,0, L1,1), (L2,1, L2,2) at t = τ, (L1,0, L1,1), (L3,4, L3,5) at t = 2τ and others all resemble , and follow location t with k. Nevertheless the evident t dependence of Lj,k also couples or mixes the scales j, so that many exceptions to the pattern appear. Thus the physical local-energetics interpretation of Lj,k (and hence Fj,k) ultimately depends on the coordinate-origin choice describing the analyzed flow.

Fortunately the Σjq=0 in (C.2b) partially compensates for scale coupling induced by t shift, so that Fj>2,k structure (Fig. 3f) is qualitatively less t dependent, aside from uniform pattern shifts. Sign reversals of F1,k after τ−1t = 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 F0,1(3τ). 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, Ũj,2jJ (phase corrected as in appendix D). For every j,

 
Ũj,2jJ(t = λJ,n) = Ũj,2jJ(ℓ−n+J2J−1)(t = 0);

that is, SEWT simply shifts equivariantly with U. As in the Burgers case, the resemblance of Ũ0,2J(0) to a cosine is roughly explained by noting that U(x, 0) ≈ cosx and that |·| cos(·+ 21−Jπℓ) ∝ cosλJ,.

SEWT KE transfer Lj,2jJ is shown in Fig. 4d. It is shift equivariant, and so carries none of the shift-induced scale coupling discussed in regard to Lj,k. However Lj,2jJ 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 Lj,k shift sensitivity for any particular coordinate origin, expressed by all coordinate-origin choices at once in Lj,2jJ. Consistent with Lm = 0 for all m, it is observed that Σ2J−1ℓ=0 Lj,2jJ = 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.

Fig. 3.

(Continued) (d) Stuart-vortex coefficients Ũj,k as in (a), at four t from Fig. 1c. (e) WTF (3.2) as in (d). (f) WFF (C.2b) as in (d)

Fig. 3.

(Continued) (d) Stuart-vortex coefficients Ũj,k as in (a), at four t from Fig. 1c. (e) WTF (3.2) as in (d). (f) WFF (C.2b) as in (d)

Fig. 4.

(Continued) (c) SEWT Ũj,2jJ as in (a). (d) Lj,2jJm as in (b)

Fig. 4.

(Continued) (c) SEWT Ũj,2jJ as in (a). (d) Lj,2jJm as in (b)

Table C1. TKj,k tableau

Table C1. TKj,k tableau
Table C1. TKj,k tableau

Table C2. TKjJ,2ℓ) tableau

Table C2. TKj (λJ,2ℓ) tableau
Table C2. TKj (λJ,2ℓ) tableau

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: Ut = −UUx and suppose an analysis

 
formula

where the analysis function ψn and synthesis function ψn (e.g., Wickerhauser 1994) satisfy

 
formula

but are otherwise unspecified. In the wavelet case,

 
formula

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

 
formula

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

 
formula

for U an arbitrary function. (Note that in case ψn = ψn = Fn this result is simply the familiar wavenumber n interaction with its harmonic 2n.) Unless U = 0 or ψn𝒟ψn = 0 there must be some kind of self-interaction. Even if Uψn𝒟ψn = 0, in the general case Un ≠ 0 requires dn(x) ≡ ψn(x)𝒟ψn(x) be x independent. In the complex Fourier case dn = −in is sufficient, but when ψn = ψn the requirement leads to ψn(x) = [ψn(0)2 + 2dnx]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 U2nψ2n𝒟ψn = 0.

APPENDIX B

Numerical Considerations

Although in the Fourier basis it is simple to expand nonlinear terms into convolutions, and use fλm = imf̂m, in the wavelet basis it is simpler to perform products and ∂λ operations before the transform. Error in numerically verifying 10 uxuα dx = 0 was reduced using the scheme (Celia and Gray 1992)

 
formula

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

APPENDIX C

Construction of WFFs

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

 
formula

The equivalence (C.1) is normalized so that Σ2J−1−1ℓ=0 TKj(λJ,2ℓ) = Σ2j−1k=0 TKj,k, to preserve the spatial-index sum, but make the left-sum limits scale-independent. To illustrate, for J → 3 the TKj,k tableau is shown in Table C1. Then by Eq. (C.1) the elements TKj(λJ,2ℓ) form Table C2.

Fournier (1996) independently introduced a KE WFF

 
formula

[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,

 
formula

Only those transfer amounts at the correct location λJ,2ℓ will contribute to FKj(λJ,2ℓ). This makes FKj(λJ,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 Ũj,kŨj,kj[mj]. SEWT phases were corrected by Ũj,2jJŨj,2jJ(ℓ−J[2Jjmj]), for all j. The mode of energy mj estimates the cumulative phase-shift of Jj convolution decimations (see F1) with the filter. {Estimating the more usual center of energy 2j 2−1−2−1 xWj,2j−1(2πx)2 dx yields the same [mj] except for j = 1, because of periodicity.} Walden and Contreras Cristan (1998) discuss another approach to phase correction.

Asymmetry in Ũj,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(1015) of the reconstruction condition.]

The west–east “dipole” L1,0 < 0 < L1,1 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 Lj,k(W, U). Then to see W asymmetry effects we must compute Lj,k(W, U). Since

 
formula

so by (3.2), in general Lj,k(W, U) = −Lj,2j+1−k(W, U). Because in this case U = −U and (3.2) is trilinear in U we expect Lj,k(W, U) = Lj,2j+1−k(W, U), which is verified. Removing W asymmetry effects is discussed by F0.

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