1. Introduction

The atmospheric conservation (AC) equations are characterized by the presence of nonlinear terms that in the atmosphere describe the energy exchanges between flows characterized by length scales that range from millimeters (Kolmogorov scale) to thousands of kilometers (synoptic scale). The large and continuous variety of scales present in atmospheric flows generates an intrinsic difficulty in the numerical treatment of the AC equations. From scaling considerations, the ratio between the smallest flow scale (η) and a characteristic length scale, L, is approximately proportional to Re^3/4 (Tennekes and Lumley 1972), where Re is the Reynolds number. This means that a three-dimensional numerical description of the planetary boundary layer resolving all scales from η to L would require about 10¹⁵ grid cells. This number is far from being handled nowadays or in the near future by any computing device.

For these reasons, atmospheric numerical modeling is limited to simplified equations that focus on characteristic length scales and filter out all smaller scales. These filtered scales, also known as the subgrid part of the flow, are not explicitly resolved and must be parameterized. Filtering (or averaging) operations represent the first step to a numerical treatment of atmospheric flows as they lead to discretized partial differential equations.

There are two fundamental assumptions that allow the simplification of the AC equations in atmospheric applications, namely,

where f is a generic function of time and space. Equation (1), which is the core of the so-called Reynolds assumptions, relates to the multiple applicability of the averaging operator and guarantees for the cancellation of the average of the fluctuations (f′ = 0); Eq. (2) allows the averaging operator to be commuted with the derivative. Fulfillment of (1) and (2) by a given filter strictly depends on its definition and on the characteristics of the signal to be averaged.

In numerical applications, it is normal practice to avoid defining the averaging operator. Frequent are the examples in which the operator is said to be defined so that it satisfies the Reynolds assumptions but it is never clearly spelled out (e.g., Panofsky 1958; Haltiner and Williams 1980; Holton 1982; Pedlosky 1987). The only indication provided about filter characteristics is that the averaging interval over which the filter is applied is chosen to be “large enough” to average out fast turbulent motions but “short enough” to preserve the trends of the variables. In other words, the flow is supposed to show a well-defined scale separation between turbulent and large-scale parts of the flow. On the other hand, when the averaging operator is explicitly defined, many are the examples in which it is a running mean (Stull 1988; Pielke 1984), although this filter satisfies the Reynolds averaging assumptions only under specific conditions. Equation (1) is satisfied only when a marked scale separation exists in flow motions. If this separation is not marked enough, the use of the running-mean operator generates a subgrid-scale term that not only includes fluctuation covariances (Reynolds terms) but also the so-called Leonard and cross terms (Leonard 1974). A clearer indication about filter definition is normally given in engineering flow studies and for atmospheric flow modeling by the large eddy simulation community (e.g., Deardorff 1970; Orzag 1971; Schumann 1975; Clark et al. 1979; Antonopoulos-Domis 1981; Moin and Kim 1982; Ferziger 1973; Moeng 1984; Moeng and Wyngaard 1988; Anderson 1989; Piomelli et al. 1990; Scotti et al. 1993; Kaimal and Finnigan 1994). Some of those studies have determined a small contribution of the Leonard and cross terms to the subgrid scale but this conclusion refers to engineering flows characterized by a limited range of Reynolds numbers and to relatively small-scale atmospheric flows.

Experimental evidence normally used to support the existence of a scale separation relies on atmospheric power spectra that provide information on the atmospheric energy distribution, for example, the work of van der Hoven (1957). Although this work constitutes an important milestone in atmospheric physics, it has not provided the final evidence that a real scale separation (spectral gap) exists. More recent works indicate that an energy spectrum that concentrates the main part of its energy over a specific range of timescale or length scales is found only for the vertical velocity component whereas horizontal components and scalar quantities spectra (e.g., temperature and moisture) show a spread of energy over all scales ranging from synoptic to microscale motions. Furthermore, these works indicate that horizontal velocity spectra change with both location and time period in which the data are collected (Mantis 1963; Vinnichenko 1970; Young 1987). Courtney and Troen (1990) collected continuous high-frequency data for 1 yr and produced a spectrum in which the so-called spectral gap (SG) contains as much energy as the microscale peak. Nowadays, it is recognized that the SG generally does not exist (e.g., Courtney and Troen 1990;Young 1987). Under these conditions the study of the behavior of the Leonard and Cross terms becomes necessary for a correct estimation of the subgrid-scale (SGS) part of the flow. This is particularly the case for the modeling of flows that encompass parts of the atmospheric spectrum not limited to the neighboring regions of the inertial subrange.

Equation (2) is also strictly connected to the definition of the averaging operator. In atmospheric applications it is common practice to discretize the equations on a nonregular grid (stretched grid, e.g., in the vertical), which gives, on one hand, a finer resolution in the vicinity of the surface and, on the other, reduces the computational effort at higher atmospheric levels where less resolution is required. The use of a stretched grid does not satisfy (2) and correction terms need to be accounted for.

In the present work, Leonard and cross terms have been evaluated for simple energy spectra to illustrate their role in the determination of the SGS term when using running-mean-filtered equations. Furthermore, errors generated by the use of variable grid spacing are estimated in connection to (2). Despite the use of theoretical arguments and predefined signals, the results may be generalized to more complex situations and are important for the definition of the averaging procedure in models and for the treatment of atmospheric data.

2. Multiple application of averaging operators and related assumptions

In this section we summarize the sequence of operation through which the AC equations are customarily filtered. Given u_j, the jth component of the velocity U, and Φ(x, y, z, t), a generic variable, the conservation equation for Φ(x, y, z, t) is given by

where the first and second terms represent the time evolution and transport of Φ, respectively, and where the rhs summarizes production and depletion processes. Variables u_j and Φ can be expressed as follows:

u_ju_ju^′_jΦ(4)

This variable splitting is normally referred to as the Reynolds decomposition. It represents the first step toward a statistical treatment of turbulence. The overbarred quantity represents the resolved part of the flow whereas the primed quantity is the fluctuating component. Both components are space and time dependent but have different characteristic scales. The average indicated above can either be space or time or both, depending on the variability of u_j and Φ (Pielke 1984). Equation (4) can be inserted in (3) to obtain the following equation:

which after a second averaging (filtering) operation reads

In order to eliminate the fluctuating components (but not their correlations) from (6), the averaging operator used in (4) and (6) has to fulfill specific conditions, namely, the well-known Reynolds averaging assumption according to which, given the function f(x, t), the filter satisfies

f(x, t)fx, tf′x, t(7)

and (2). When discretized, (7) is automatically fulfilled for any function, f, whenever the grid and time increments are independent of space and time (Pielke 1984).

Assuming here that the filter and the signal fulfill (7) and (2), (6) can be rewritten as

in which the first and second terms in the lhs represent the total derivative of Φ, the third term represents the divergence of the turbulent flux, and the rhs shows average contributions for production and dissipation of Φ. The second and third terms in the lhs of (8) arise from term II after application of the filter whose properties are mentioned above. In (8), the turbulent part of the flow is now only represented by the fluctuation covariance, which is the only remaining quantity involving the fluctuating components after the filter application. This term is commonly referred to as the subgrid-scale contribution to the flow. Provided the knowledge of sources and sinks, it represents the only unknown in (8) that thus needs to be parameterized. In the next section, we investigate the limit of applicability of assumptions (7) and (2) for different types of filters and signals.

a. Running-mean operator

The purpose of this section is to identify some properties of the running-mean operator. For convenience, the variable Φ(γ) used hereafter is assumed to vary only with γ (generic variable, time or space) since all further considerations easily extend to a dependence to more than one variable. The γ axis is subdivided in intervals of length Δγ_k (numbered with index k) and we concentrate hereafter on the specific interval with boundary indices k and k + 1, having γ₀ as a central point. In this context, the running-mean operator (superscript r) is defined by

The filter defined by (9) does not, in general, satisfy (7). The average of the variables’ product appearing in the second term of (6) may be written as

after Reynolds decomposition and application of successive averaging operations. As (7) is, in general, not fulfilled, the first term in the rhs of (10) does not reduce to a tractable advection term; it is thus common practice to subtract from both sides an advection term as follows:

in which the three terms appearing in the rhs are referred to as Leonard (l), cross (C), and Reynolds terms (ℜ^r), respectively. Many are the interpretations given to these three terms in the literature. Term l was defined by Leonard (1974) as an energy loss of large scales that occurs through the action of nonlinear resolvable scales. Aldama (1990) interpreted the cross term C as a direct interaction between resolved and unresolved scales, which gives rise to backscatter or energy transfer from small to large scales and some energy transfer even to some smaller scales. Instead, ℜ^r is considered as the subgrid term representing the fluctuations’ correlation and often referred to, with abuse of language, as turbulent fluxes. Although l may be explicitly accounted for when γ represents a space variable (Moin and Kim 1982; Piomelli et al. 1990; Leonard 1974; Aldama and Harleman 1991), in some cases l has been neglected and (C + ℜ^r) lumped together to define the SGS term.

Note that the lhs term in (11) represents the complete SGS term. Indeed, evaluated at γ₀, center point of the interval [γ_k, γ_k+1], the term u_jΦ^r(γ₀) − u_j^r(γ₀)Φ^r(γ₀) is the difference between the average of the total product u_jΦ^r and the product of the averages and is thus the fluctuating part of the flow, that is, the SGS term. If Φ is substituted by the wind velocity u_j, u²_j^r(γ₀) − u_j^r2(γ₀) represents the subgrid-scale energy (Nieuwstadt 1990).

It is clear that after applying the Reynolds decomposition, the SGS term is given by the sum of all three terms (l + C + ℜ^r) and not only by (C + ℜ^r) or just ℜ^r, as suggested by several authors. It is interesting to see that although the SGS term expressed as u_jΦ^r(γ₀) − u_j^r(γ₀)Φ^r(γ_O) depends only on information relative to the specific interval [γ_k, γ_k+1], the two successive applications of the running-mean operator makes each term l, C, and ℜ^r dependent on information relative to adjacent intervals (running character of the filter).

b. Local-ensemble average operator

In this section another averaging operator is examined and compared to the running mean. The γ axis is again divided into intervals Δγ_k and we still concentrate on the interval [γ_k, γ_k+1] having γ₀ as central point. Let us now define an averaging operator as

Equation (12) is equivalent to an ensemble average of a continuous function over [γ_k, γ_k+1]. As a matter of fact it is not a proper ensemble mean since it is performed using only Φ values belonging to [γ_k, γ_k+1], that is, over a local ensemble of values. Note that the term ensemble does not refer to an average over independent realizations of the flow. Equation (12) will thus hereafter be referred to as a local-ensemble averaging operator (superscript l). For each γ in the interval [γ_k, γ_k+1], Φ(γ) has thus an identical average value Φ^l(γ₀), which implies that

is automatically fulfilled for all intervals. Furthermore, for the specific point γ₀ we have Φ^r = Φ^l, since the average is performed on the same points. This equalitythough only holds for γ₀. The SGS term is thus given by

For γ₀, we obtain by definition

since (9) and (12) make use of the same set of instantaneous Φ and u_j values for their averages. Note that (14) is evaluated in γ₀ to compare SGS terms that refer to the same interval [γ_k, γ_k+1] obtained with l and r operators. Equation (14) also implies

lγ₀Cγ₀^rγ₀^lγ₀(15)

Equation (15) shows that for a specific point (in this case γ₀), the SGS term obtained via the two averaging methods is identical. It is also important to underline that although ℜ^r and ℜ^l have similar forms (Φ′u_j^′), they represent different physical quantities. While ℜ^l is a pure Reynolds term that represents the total subgrid influence on the evolution of a mean quantity, ℜ^r is only part of it and must be completed by l and C to obtain the same physical quantity.

These two Reynolds terms are identical only if a SG exists. If the averaging interval is located within this gap, the average defined by (9) does not change significantly from one interval to another and we can thus write

Note that the local ensemble averaging operator does not formally require any spectral gap assumption to obtain the complete SGS.

By definition the SGS term expressed as [u_jΦ(γ₀) − u_j(γ₀)Φ(γ₀)], for whatever averaging operator (r or l), only depends upon values of Φ and u_j that belong to [γ_k, γ_k+1], since one single averaging operation is required to obtain this form of the SGS. When Reynolds decomposition is introduced for Φ and u_j, double averaging operations are implied and very different forms of the SGS result for the l and r operators. According to (15) and (16), the absence of a SG and the neglect of l and C may lead to important errors in evaluating the SGS by means of the running-mean operator. Moreover, as the value of ℜ^r is highly dependent on the presence of a spectral gap, inconsistencies may arise whenever ensemble-averaged SGS are parameterized in models based upon running-averaged data treatment or vice versa.

c. Examples

In order to study the behavior of the Leonard and cross terms in relation to the spectral characteristics of a signal, and their role in determining the SGS term, predefined signals with a priori well-defined spectral characteristics are analyzed. For simplicity, only one variable is considered for which the general expression is given by

AαγBβγ(17)

where γ is the independent variable, A and B the amplitudes of the cosine functions, and α and β their frequencies. The parameters defining (17) are varied according to the cases described below:

1. Case 1.S: Signal with a single-spike spectrum. [A = 1, α = 2π/10, B = 0, β = 0]. This case concentrates energy at a single defined period (10γ) with unlimited SG and the amplitude of the signal is one.
2. Case 2.S: Signal with a single-spike spectrum. [A = 1, α = 2π/10, B = 1, β = 2π/500]. The first spike is still at 10γ whereas the second is at 500γ. The two spike modes have equal amplitudes. This case mimics a “small-” and a “large-scale” energy peak separated by an energy gap.

Having defined the datasets with the aforementioned characteristics we now calculate ℜ^l, ℜ^r, l, and C defined for Φ equal to u as

at a fixed position (γ₀) within the dataset and with varying averaging intervals (Δγ) centered in γ₀. Equation (18) represents the variance of Φ obtained with the running and local ensemble filters. Each averaging interval will fall in a specific position of the power spectrum and will capture a certain amount of the available energy to transform it into the SGS part. Note first that the choice of the point γ₀ does not influence the results and second, that all results for ℜ^l, ℜ^r, l, and C presented hereafter are obtained from analytical expressions.

1) Single-spike spectrum (1.S)

In Figure 1, values for ℜ^l, ℜ^r, l, and C as a function of the averaging interval width Δγ are presented (the spectral width is indicated by the horizontal arrow). First, we notice that the equality between SGS^l and SGS^r is fulfilled as can be seen from the comparison between the sum of l, C, and ℜ^r on the one hand and ℜ^l on the other. For averaging intervals smaller than the spike period, ℜ^r and ℜ^l differ by an approximately constant value that corresponds to l + C, but ℜ^r and ℜ^l are equal when the averaging interval coincides with the peak position. This equality is periodically found for values equal to integer multiples of the peak’s period (see appendix A).

When the averaging interval is larger than the spike period, ℜ^l starts oscillating around an asymptotic value (0.5) with a rapidly decreasing amplitude as Δγ increases. Although ℜ^r shows an identical behavior, the amplitude of the oscillations is much larger. While the difference between the two Reynolds terms arises from the fact that l + C still contains part of the SGS after the spike (l + C ≠ 0), the independent behavior of these two terms (l and C) is quite different: after the spike, l suddenly decreases to a negligible value whereas C still contains part of the energy for very large averaging intervals. In fact, for these large averaging intervals, C is finally the only responsible term for the differences between ℜ^r and ℜ^l. The behavior of l after the spike clearly indicates that l is responsible for a transfer of information from the large to the resolvable scales. Indeed, l drops to zero after the spike since no energy from the larger scale is available. Instead, C contains information from both the large (Φ) and small (Φ′) scales. Hence, it remains different from zero even for averaging intervals located well beyond the spike (Φ′ ≠ 0). Indeed, very large averaging intervals are required to reduce the value of C and, thus, the amplitude of the oscillations of ℜ^r. Note that the important parameter in this kind of analysis is more the frequency of the signal rather than its amplitude. While the frequency is strictly linked to the averaging interval width, the amplitude only defines the final value reached by the SGS term (see later). The asymptotic value reached by ℜ^l (0.5) is the true value of the available energy as can easily be determined.

2) Double-spike spectrum

This case includes two spikes located at 10 and 500 γ (Fig. 2) to mimic a large- and a small-scale mode. This example aims at illustrating the behavior of the filters in the presence of a well-defined SG (indicated in Fig. 2) when determining the SGS term. Similar to Fig. 1, Fig. 2 shows the variation of ℜ^r, ℜ^l, l, and C depending on the width of the averaging interval. Three regions may be identified.

1. For averaging intervals located to the left of the 10 γ spike, only part of the small-scale energy is included.
2. For Δγ falling between the two spikes, the small-scale energy is totally included while the large-scale one is only partially accounted for.
3. For Δγ located to the right of 500 γ, all the available energy is captured as subgrid.

As expected, the behavior of the different terms is very similar to the single spike case for Δγ smaller than 10. The terms ℜ^l and ℜ^r grow while approaching the first spike, but when the averaging interval becomes larger than this spike period, ℜ^l rapidly assumes the value 0.5 (corresponding to the available energy) whereas ℜ^r strongly oscillates around this asymptote (Δγ ≥ 100). For averaging intervals located within the SG, the two Reynolds terms increase in a very similar way while l and C tend to zero (Δγ ∼ 500). After the second spike (Δγ = 500), ℜ^l increases and tends to an asymptotic value with rapidly decreasing oscillations. Since the two spikes have an amplitude of 1, the asymptotic value is 1 (0.5 + 0.5). A different behavior is shown by ℜ^r, characterized by larger oscillations, since it only represents part of the SGS term. It grows toward the small-scale spike but sensibly drops in the middle of the scale separation range (Δγ ∼ 60). After this point, it strongly oscillates around ℜ^l with a reduced amplitude as Δγ increases. When the averaging interval is larger than 10γ, that is, within the SG, l already captures information about the presence of the larger-scale spike and shows a negative value. However, it drops suddenly to zero after the 500-γ spike since no energy is captured afterward. Note that since l and C result from a double application of the filter, they include information about the double of the considered interval width: for Δγ between 200 and 300γ, l and C contain information relative to the positions between 100 and 600γ. This explains the large values reached by these two terms around 250γ as they then start capturing information about the 500-γ peak. For this reason, when Δγ is larger than 50, l also decreases gradually to values close to zero since a smaller amount of energy is captured by the filter. On the contrary C shows decreasing values that remain different from zero for all Δγ since the influence of small-scale fluctuations never vanish completely. The amplitude of the oscillations shown by C decreases for larger Δγ since more fluctuations cancel each other.

In order to estimate the role of the SG width, when calculating the SGS contribution with a running mean, we show in the next set of figures similar results to the case (2.S) in which the positions and intensities of the spikes are varied. Let us first consider the case in which the two spikes are located at 10 and 1000γ [A = 1, α = 2π/10, B = 1, β = 2π/1000]. Figure 3 compares the 10% of ℜ^l to the absolute value of (l + C) to quantify the consequence of neglecting these two terms. We see that 0.1ℜ^l grows asymptotically to 0.05 and to 0.10 (dashed line) after the 10- and 1000-γ spikes, respectively. We can identify an averaging interval for which (l + C) is negligible or close to zero and thus for which the SGS calculated with a running or local ensemble mean would give identical results. On the contrary, for averaging intervals close to the peaks, an error much larger than 10% results since (l + C) increases. If the spikes are now chosen at 10 and 100γ (Fig. 4), a proper choice of the averaging interval still can be made (20 < Δγ < 60) for which (l + C) is less than 10% of ℜ^l. Figure 5 shows the case [A = 1, α = 2π/10, B = 1, β = 2π/50] in which the gap separating the spikes has been further reduced. In this case, the choice of an averaging interval that keeps low the value of (l + C) becomes more difficult and restrictive (20 < Δγ < 30). Although the spikes are still separated by a gap, the error made by neglecting the terms l and C is exceeding 10% almost for any averaging interval width, indicating that the mere existence of a scale separation does not guaranty that l and C go to zero.

Let us now investigate the effect of the peak’s amplitude on the resulting range of averaging intervals for which l + C may be neglected. Figure 6 shows a case similar to Fig. 3 in terms of positions of the spikes but where the amplitudes of the signal spikes are changed [A = 1, B = 10]. If in the case of Fig. 3 we could still choose a Δγ as wide as 500γ (100 < Δγ < 600) and make an error smaller than 10%, when the amplitude of the large-scale signal increases, the range of appropriate Δγ is limited to 300. In Fig. 7, the largest amplitude is now set for the small-scale peak [A = 10, B = 1]. In this case the presence of the peak at 1000γ does not influence the choice of the averaging interval width for all values larger than the small-scale spike period.

The cases studied in this section reveal the sensitivity of the running mean operator to the spectral characteristics as well as to the intensity of the signal. In spite of the simplicity of the spectra selected, it clearly appears that the existence of a scale separation does not straightforwardly guarantee to find an averaging interval width such that SGS = ℜ^r. Similarly, the variation of intensity between the large- and small-scale peaks considerably reduces the range of Δγ for which l and C may be confidently neglected. This last example can be generalized to the atmospheric case in which the mesoscale or synoptic spectral peaks contain an amount of energy by far larger than the turbulence one. The existence of an energy gap between the scales thus does not necessarily imply a correct estimate of the SGS with a running-mean operator.

3. Commutation of derivative and average operators and related assumptions

Another important assumption used to simplify the AC equations is

where Φ is the function introduced in section 2. Note that all considerations made hereafter for space-dependent functions hold equally for time-dependent functions. Equation (19), which states that the averaging operator may be commuted with the derivative, allows one to go from (6) to (8) but is only valid for nonvariable grid spacing. If we assume for simplicity that Φ is function only of z, the rhs of (19) may be written as

where h represents the grid spacing that depends on z. The lhs of (19) is directly given by

By substracting (21) from (20) we finally obtain

in which terms I and II represent the derivative of the average and the averaged derivative, respectively, and where term III accounts for the effect of grid stretching. As shown by (22), the variability of h determines the importance of term III compared to I and II, but term III explicitly includes the instantaneous functions Φ(z + h/2) and Φ(z − h/2), which prevents one from obtaining a direct resolution for this term.

Since it is common practice in atmospheric modeling to solve the conservation equations over a nonregular numerical grid to find an optimal compromise between a fine resolution wherever strong gradients may occur (e.g., close to the surface) and a coarser resolution in regions characterized by minor variations of the meteorological variables (e.g., in the upper PBL or in the free troposphere), term III must generally be accounted for.

a. Case study

The aim of this section is to determine the relative importance of term III compared to II in (22) for different grid-stretching factors and types of signal Φ. The function Φ (z) is here defined as the sum of a linear function and sine functions

zAαz,

in which the amplitude A will be varied. The linear trend is here representative of a large-scale motion for which no error is generated due to grid stretching (term III vanishes by definition). By varying A, we will study the relationship between the energy contained within the regularly fluctuating component (sine), the interval width, and the grid-stretching factor.

The discrete numerical grid is defined according to

z_n+1z_nrz_nz_n−1z_n−1(23)

where r acts as the stretching factor. Equation (23) can easily be expressed in a continuous form as (see appendix B)

hzz₁rz,(24)

in which z₁ is the first grid level. Terms II and III in (22) have been calculated analytically for different stretching factors close to those used in mesoscale models for vertical grids.

The ratio of term III to term II can be written as

in which we recognize two contributions. The first is given by the variability of the grid spacing with z (∂h/∂z) and the second (here defined “signal factor”) is only dependent on Φ. Note that the signal factor is closely linked to the intensity of the subgrid-scale part of the flow since {Φ(z) − [Φ(z + h/2) + Φ(z − h/2)]/2} gives the fluctuating component of the signal within interval [z − h/2, z + h/2].

Figures 8 and 9 show term III and the 10% of term II as a function of the grid-cell size h for α = 100 and r = 0.2 and with A = 10 (Fig. 8), and A = 100 (Fig. 9). The first case (not shown in figure) corresponds to A = 0 for which term III and 10% of term II are identically equal to 0 and 0.1 for all h. We see from Figs. 8 and 9 that by increasing the energy contained in the fluctuations, the error (dashed line) increases significantly, especially in vicinity of the peak region where it can exceed 10% (for α = 100). For values of h larger than this peak (h > 100), the error decreases asymptotically to zero as the linear trend starts dominating.

Figures 10 and 11 are similar to Figs. 8 and 9 but for the signal parameters: α = 100 and A = 100 for the stretching factors r = 0.1 (Fig. 10) and r = 0.3 (Fig. 11). In this case, the signal factor remains unchanged and only the grid-stretching factor plays a role. We see that while the error is contained within 10% for r = 0.1 for all h values, this is not the case when r = 0.3 for which the error exceeds this 10% limit for values of h ranging from 50 to 300. Note that the latter values correspond to physical distances [see (23)] ranging from approximately 130 to 950 m, which may represent a significant part of the modeling domain.

In conclusion, it appears that errors generated by grid stretching are significant whenever the energy present in scales smaller or close to the grid spacing becomes important compared to larger scale motions.

4. Conclusions

Atmospheric conservation equations, which describe atmospheric flows, apply to motions ranging from scales of millimeters to thousands of kilometers. Averaging filter operations thus represents the first necessary step toward a numerical treatment of atmospheric flows. In the literature it is common practice to avoid mentioning explicitly the form of the averaging operator and to assume that the filter satisfies the Reynolds assumptions or, in some other cases, the filter is explicitly said to be a running mean. When the nonlinear terms appearing in the AC equations are averaged using a running-mean operator, the SGS term is given by the sum of three components: the familiar Reynolds terms and the so-called Leonard and cross terms. The last two terms can be neglected from the definition of the unresolved part of the flow if and only if 1) the spectral characteristics of the flow show a clear scale separation or SG and 2) the averaging interval is located within this gap. Recent experimental atmospheric spectra clearly point out that, apart from vertical motion, no distinct scale separation appears in the atmospheric frequency distribution of energy. Consequently the neglect of the Leonard and cross terms leads to a miscalculation of the SGS contribution.

In the present work, the relative importance of the Leonard and cross terms has been investigated by means of signals with predefined spectral characteristics and analytical expressions. Another operator, the local-ensemble mean, was defined for which the SGS term is given only by the Reynolds term. The SGS terms obtained through the running and local-ensemble mean were then compared. The role of Leonard and cross terms was investigated as a function of the spectral characteristics of the signal and of the width of the averaging operator. The spectral cases analyzed were a single spike (energy concentrated at one frequency) and a double spike (spikes separated by an energy gap). Results indicated that the sum of the Leonard and cross terms greatly contributes to the SGS value for any position of the averaging interval within the energy spectrum. In the infinite gap case (1.S), the local ensemble and running-mean Reynolds terms converge to the same value for very large averaging intervals. In the proximity of the energy spike, the Leonard and cross terms contain most of the SGS energy. For averaging intervals corresponding to a position located immediately after the spike, the Leonard term collapses to a negligible value. On the other hand, the cross term remains different from zero after the spike as larger averaging intervals still contain energy (fluctuations), but it is destined to reduce to zero as the averaging interval becomes large enough to statistically cancel these fluctuations. Under these latter conditions, Reynolds assumptions are satisfied. The 2.S case indicates that a very large and distinct scale separation is needed in order to neglect the Leonard and cross terms (the running and local-ensemble Reynolds terms coincide). The simple examples show that the error made by omitting the Leonard and cross terms strictly depends on the width of the averaging interval and of the shape of the spectrum. The spectral characteristics define whether the neglect of the Leonard and cross terms would lead to an overestimate or underestimate of the SGS term. The fact that these two terms result from calculations based on information relative to two averaging intervals requires the existence of a scale separation large enough to avoid interferences from larger- or smaller-scale peaks. However, the mere existence of a scale separation does not guarantee that the Leonard and cross terms are automatically negligible. The amount of energy associated with the peaks is also shown to be a relevant parameter in the determination of the role of the Leonard and cross terms: if most of the energy is contained in the large-scale peak, the choice of the optimal averaging interval becomes limited, whereas if most of the energy is contained in the small-scale peak, any averaging interval chosen at frequencies larger that the latter peak will capture all the available energy and will not be affected by the presence of the large-scale peak. If a running mean is used to calculate the SGS of a signal from data, the Leonard and cross terms should be explicitly accounted for.

In the second part of the paper we analyzed the error associated with the assumption that Eq. (2) holds whenever a stretched grid is applied. The considered case is limited to a vertically stretched numerical grid but all considerations can be generalized to a horizontally stretched grid or to variable time step calculations. The analysis of (2) in the case of a stretched grid produces a correction term that accounts for the variability of the grid spacing. The typical grid stretching adopted in mesoscale models in the vertical direction has been considered. The commutation of the averaging and the derivative operators has been investigated using again well-behaving predefined signals and different stretching factors. The results indicate that the error associated to commonly used grid stretching may be significant in representing flows for which a relevant part of the total energy is contained in the subgrid scale or in scales comparable to the grid spacing. On the contrary, for flows characterized by relatively weak subgrid-scale motions (e.g., laminar flows), no error is associated with the commutation of the average and the derivative operators when using a stretched grid.

The natural follow-up of this research will consist in evaluating the Leonard and cross terms as well as the error generated by grid stretching for real atmospheric energy spectra. Considering the recognized variability in space and time of atmospheric spectra, one may also expect to obtain different orders of magnitude of these terms depending on the case considered. Nevertheless, it is clear that the theoretical properties of l, C, and ℜ^r are valid independently of the order of magnitude that they have when evaluated from some real atmospheric spectra.

5. Some arguments for further discussion

The use of a running-mean filter on the AC equations may lead to formal inconsistencies in model applications. In this case, the filter is indeed defined as a running mean but is often used practically as an ensemble average after application of the Reynolds assumptions. Inconsistencies may also arise in model parameterizations of the SGS term when the data and model filters do not coincide. It is important to point out that the calculation of the Leonard and cross terms can easily be obtained from datasets but with the drawback of a reduction in size of the available data due to the implicit need for a double-sized averaging interval. In modeling applications, the main disadvantage resulting from the inclusion of the Leonard and cross terms within the SGS lies in their closure. On the contrary, their use may help to understand different properties and roles of the SGS terms as far as the exchange of the SGS energy is concerned. The fact that the local ensemble leads to a single SGS term (Reynolds term) without the need of an SG could be used for numerical applications. Although one obvious drawback is its noncontinuity, discretized numerical equations are also discontinuous and one could use this operator for the definition of the model filter with more formal consistency. It is important to note that even in the case of the local-ensemble mean, consistency is required between filters used in models and for data treatment.