a. Discrete wavelet analysis

As in Part I, the Meyer wavelet (Meyer 1992) is used. Meyer’s discrete wavelets are basis functions that expand a periodic function in a finite domain (0 ⩽ x ⩽ L). Spatial localization and scale characterizations are represented as follows. The wavelets ϕ_k,j(x) are characterized by two integer parameters (we use a slightly different definition of the subscripts from Part I): the wavenumber k measures the spatial scale, and an index j measures the spatial localization (location) of the wavelet peak. The wavenumber k is a power of two, namely k = 2ⁱ where i = 0, 1, 2, . . . counted in descending direction from the top row in Fig. 4 of Part I. For a given integer wavenumber k, the wavelets represent the modes localized at k different spatial locations, which are constructed by shifting a wavelet form through an equal interval. This spatial shift is indicated by the index j, which increases from j = 1 to j = k as the wavelet localization (identified by its maximum) shifts from left to right.

These wavelets are complete (in the same sense as for a Fourier expansion) and satisfy the orthogonality condition

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where δ_kk′, δ_jj′ are Kronecker’s deltas. By virtue of orthogonality, any periodic function f(x) in the domain 0 ⩽ x ⩽ L can be expanded as

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where the wavelet coefficients are determined by

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and the maximum wavenumber k_max = 2^i_max is equal to half of the total spatial grid points of the data. Equation (3) may be considered as the definition of the wavelet transform (in analogy to the Fourier transform). The last term in Eq. (2) is the domain average f ≡ (∫^L₀ f(x) dx)/L, which can be included as a part of the expansion by introducing a constant wavelet ϕ_0,1(x) = 1 as in Part I.

We apply this wavelet transform sequentially to each horizontal section, (x, y), of the double-periodic simulated fields, that is,

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Here, l_x and l_y are indices for the two horizontal directions, defined by combining the two wavelet parameters as l = k + j and l = 1 for k = 0; N_x and N_y are the number of the data points in the x and y directions, respectively. The wavelet coefficients f̂_lxly(z) are defined as an analytic extension of the one-dimensional case in Eq. (3). Note that the wavelet transform is not applied in the vertical (z) direction because the boundary conditions at top and bottom are not periodic. The two-dimensional wavelet coefficients constitute the wavelet space, in analogy with the Fourier space.

As in Part I, we analyze snapshots of each of three distinctive cloud system regimes—a squall line, a nonsquall cloud cluster, and scattered convection—generated spontaneously in the cloud-resolving model as the large-scale forcing and shear evolve.

b. Decomposition method

The simulated systems are decomposed into the various dynamical components using the invertibility principal of the wavelet coefficients into real (physical) space (Fig. 1). The wavelet transformation [Eq. (3)] is applied to a set of variables consisting of wind vector v, total water condensate t_c, water moisture density q, temperature θ, etc., which can be considered a vector f(x, y, z) = (f₁, f₂, f₃, . . . , f_m), where m is the number of variables. We obtain the wavelet vector f̂(l_x, l_y, z) of the corresponding wavelet coefficients.

It was shown in Part I that the two-scale structure of the mesoscale convective system is realistically depicted in wavelet space, implying that wavelet space, rather than physical space, is the appropriate decomposition medium. In order to implement this idea we define the following general principle. After applying the wavelet transformation, we choose a particular variable (say, f̂_n) which can physically distinguish one dynamical component from another. We classify the wavelet vectors f̂(l_x, l_y, z) into two sets, namely the high component, f̂_H(l_x, l_y, z), and the low component, f̂_L(l_x, l_y, z), depending on whether the absolute value for the wavelet coefficient of this chosen variable, f̂_n, is larger than a specified threshold value f̂_c. Hence, the two wavelet sets are defined by

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for given (l_x, l_y, z), and f̂(l_x, l_y, z) = f̂_H(l_x, l_y, z) + f̂_L(l_x, l_y, z).

The organization of convective cloud systems is largely controlled by wind shear (Moncrieff 1981). Jetlike wind profiles, where shear reverses with height, are conductive to squall lines, a property demonstrated in idealized numerical simulations (Thorpe et al. 1982). In squall lines, shear organizes families of transient cumulonimbus (the convective scale) into a compact structure (the mesoscale convective system) that typically persists for much longer than the constituent cumulonimbus. This multicellular property and the attendant production of mesoscale circulation are clearly evident in the squall line simulations of Lafore and Moncrieff (1989) wherein the basic state was defined from measurements made during a field experiment. Mesoscale convective systems were analytically modeled by Moncrieff (1992) but a rigorous theory has not yet been developed for nonsquall clusters.

Convective systems are fundamentally affected by ambient wind shear, therefore we choose shear as a decomposition criterion. Specifically, we take the wind-speed shear ∂|v̂_H|/∂z, as the criterion (f_n). The vertical derivative is computed by the centered finite differences with a linear extrapolation assumed over the vertical domain boundaries. We plot the vertically averaged maximum pseudospectrum for ∂|v̂_H|/∂z in Fig. 2a along with the same spectrum for the vertical velocity in Fig. 2b, reproduced from Figs. 12–14 of Part I. The mesoscale energy peaks identified for the squall line (top row) and the nonsquall cloud cluster (middle row) cases in Part I have counterparts in the shear spectrum. Hence, we anticipate that the wind speed shear is an appropriate criterion to distinguish the mesoscale from the convective scales: the high-shear component corresponds to the mesoscale mode and the low-shear component to the convective mode. The full wavelet spectra (not shown) indicate that both variables share the same spatial distribution at mesoscales.

Shear has a profound influence on mesoscale organization therefore we use it herein. We take the criterion variable as f̂_n = ∂|v̂_H|/∂z, and the threshold as f̂_c = 〈f̂^′2_n〉^1/2, where 〈 〉 and the prime designate the mean and its deviation over the whole phase domain of (l_x, l_y, z)-space, respectively. This threshold is chosen such that the vertical mass flux is partitioned equally between the high-shear and the low-shear components in the squall line case (see section 3d). We obtain the physical-space structures for two decomposed modes by separately summing the two sets of classified wavelet coefficients by Eq. (4). The completeness and orthogonality of Meyer wavelets allow this decomposition procedure. An identical method was applied by Yamada and Ohkitani (1990) to a boundary-layer vertical velocity time series to distinguish a gusty period from a quiescent period but no attempt was made to correlate it with other variables.

c. Decomposed condensate fields

We apply the wavelet decomposition to the squall line, the nonsquall cloud cluster, and quasi-random scattered convection regimes. The results are shown in Figs. 3, 4, and 5, respectively, as snapshots of condensed water and ice fields. While our decomposition procedure applies to all dependent variables, the condensate field is particularly illustrative. In each figure, the top plate (a) represents the numerically simulated field (detailed in GWMH). The other two plates are the wavelet decomposition: (b) is the low-shear component and (c) the high-shear component. The three-dimensional domain is rotated to give the best perspective for each case. Orientation directions are indicated by N (north), S (south), E (east), and W (west).

Figure 3a is the numerical realization of the squall line, which occurred in an environment characterized by strong lower-tropospheric shear associated with the African easterly jet. The low-shear component of the wavelet decomposition (Fig. 3b) selects an ensemble of cumulonimbus, while the high-shear component of the decomposition successfully identifies the widespread stratiform and mesoscale ascent regions (Fig. 3c). In other words, the wavelet analysis and shear criterion have objectively identified a scale dependence not assumed a priori.

Figure 4a illustrates the simulated nonsquall cloud cluster, which occurs in moderate wind shear and features a less coherent spatial structure than the squall line. The wavelet decomposition nevertheless separates the total field into the two components. Note that the high-shear component (Fig. 4c) is of smaller scale than the corresponding component in the squall line case. The convection is also more intermittent as shown in the decomposed field (Fig. 4b).

Figure 5a represents scattered convection that typically occurs in weak shear conditions. Comparison of the wavelet fields shows that the simulated systems decompose into the convective-scale component (Fig. 5b) but the mesoscale component (Fig. 5c) is absent.

Our wavelet decomposition is based on the hypothesis that a shear criterion is sufficient to identify the convective from the mesoscale components. While the large-scale wind profile controls the convective organization (see introduction), there is no guarantee it would be useful in an objective decomposition of the simulated fields without evoking an explicit scale-dependent criterion.

The analysis of the squall line case (Fig. 3) reproduced features identified in a subjective analysis of such systems and clearly separates the convective-scale and the mesoscale components. The less organized nonsquall cloud clusters are decomposed in Fig. 4, where the mesoscale component has a smaller spatial size, consistent with the dynamics of these systems. The stochastic scattered convection in Fig. 5 is the null test passed by the wavelet procedure; that is, mesoscale organization was not expected and none occurs.

d. Mass fluxes

Our decomposition method can be used to obtain quantities relevant to convective parameterization in objective statistical terms (cf. Donner 1993; Alexander and Cotton 1998). As an example, we computed upward and downward mass fluxes for the total field and for the decomposed fields. We define the upward and downward mass fluxes by

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respectively, where w_↑ = wH(w), w_↓ = wH(−w), and H(x) is the Heaviside step function. The horizontal integral is performed over the whole domain. The corresponding decomposed components, say, w_↑H, w_↓H, w_↑L, and w_↓L, are used for the vertical velocities in order to compute the mass fluxes M_↑H, M_↓H, M_↑L, and M_↓L for decomposed fields, whereas the density ρ and the total water condensate t_c are kept as the total field. The threshold total condensate is t₀ = 0.1 g kg⁻¹. In this convention, the linear sum of the decomposed total mass fluxes M ≡ M_↑ + M_↓ recovers the total mass flux for the total field M = M_H + M_L. The separate sum of the upward and downward components does not exactly recover the total (i.e., M_↑ ≠ M_↑H + M_↑L and M_↓ ≠ M_↓H + M_↓L), because the decomposed upward motion area does not necessarily correspond to an upward motion in the total field. For example, a grid point with mesoscale downdraft may have a positive upward velocity in the total due to a strong embedded convective updraft.²

The results are shown in Fig. 6 for the squall line, the nonsquall cloud cluster, and the scattered convection cases. The solid curves represent the mass fluxes for the total field, the long-dash curves the high-shear component (mesoscale), and the short-dash curves the low-shear component (convective scale). The upward and downward fluxes are represented by positive and negative values, respectively.

With a threshold of the standard deviation of the wind shear, the high-shear and the low-shear components contribute about equally to the mass flux in the squall line case. The high-shear component provides a larger contribution at the lower levels, presumably due to mesoscale downdrafts. The vertical profile of the low-shear mass flux is distributed more homogeneously. The contribution of the high-shear component decreases and its vertical structure is more homogeneous for the nonsquall cloud cluster case. In the scattered convection case, the mass flux is explained mostly by the low-shear (convective) component, and the high-shear component constitutes only a weak up-and-down couplet.

The above decomposed mass-flux analysis can be compared with a more traditional approach based on the columnwise classification in physical space. The precipitation rate is often used as a criterion variable for such columnwise classification (cf. Houze 1977; Tao and Simpson 1989). Note that Churchill and Houze (1984), Tao and Simpson (1989), Steiner et al. (1995), and Xu (1995) take some account of spatial structures. For a simple comparison, we prefer a method strictly based on individual columns as in Houze (1977) that has the same level of complexity as our wavelet decomposition. We therefore classify columns with a precipitation rate higher than 4 mm h⁻¹ as the convective component and all the other columns as the mesoscale component. The result is shown in Fig. 7 in the same format as Fig. 6.

A familiar decomposition of the mass flux into the convective and the mesoscales (cf. Fig. 18 of Tao and Simpson 1989) is qualitatively reproduced in the squall line (not shown). In some respects, this physical space decomposition has a more attractive feature in that the sum of the decomposed upward and downward mass fluxes recover the total upward and downward values, that is, M_↑ = M_↑H + M_↑L and M_↓ = M_↓H + M_↓L, separately. However, the contribution of the convective updrafts and downdrafts is substantially smaller compared to the wavelet-space decomposition, because overlapping convective and mesoscale areas are not taken into account. The unfavorable feature with the physical-space method is that the mass flux is mostly classified as a mesoscale component for the scattered convection case, wherein the mesoscale component should be absent. This occurs because the physical-space rainfall rate criterion does not by itself reflect dynamical aspects. Also lower-level mesoscale downdrafts are not well captured in the squall line case by this method. Rather, convective downdrafts are shallower and confined to the lower level (cf. Fig. 4 of Xu 1995).

The methods of Tao and Simpson (1989) and Xu (1995) were also examined. However, the improvement of the results (not shown) in terms of reduction of the mesoscale component in the scattered convection case was hardly worthwhile in view of the increased complexity. Rather, this demonstrates the strength of our method: by moving to wavelet space, a simple threshold condition can effectively decompose the fields.