## 1. Introduction

A fundamental problem in atmospheric sciences is the predictability of atmospheric circulations of specific scales. As reviewed by Anthes (1986), the nonlinear nature of atmospheric processes allows cross-scale energy transfer so that the uncertainty or error in any one scale will eventually contaminate all scales; and the existence of atmospheric instabilities will make any error, no matter how small initially, eventually grow and contaminate even a perfect model’s forecast. Understanding the predictability of atmospheric circulations of different scales and identifying the source and nature of errors are both critical steps toward improving weather and climate models.

An important question for numerical weather prediction (NWP) is how much time will pass before a model forecast becomes useless. In reality, there is no unique answer to this general question, since the level of accuracy required for a forecast to be useful depends very much on the purpose for which the forecast is used (Bengtsson and Simmons 1983). Thus, a more specific question should be asked regarding the time limit of the predictability of a specific scale in the multiscale atmosphere.

On one hand, the error growth in a global model for medium-range forecasts on the synoptic scale is assumed to be associated with baroclinic activities. On the other hand, the moist convective instability may limit the forecast skill on the shorter forecast lead time with an increase of the horizontal resolution and the ability to resolve convective systems. In reality, the development of the high-impact weather systems actually results in multiscale interactions, and it is difficult to separate or isolate the effects from different scales. These facts suggest a careful investigation of the scale predictability with a coherent multiscale continuum of atmosphere in mind.

To quantify the practical predictability of a forecast model in terms of a time limit, two aspects should be considered: one is the proper measure used to assess the forecast skill; the other is the criterion used to determine the predictability time limit beyond which a forecast is no longer useful.

In most of the early classical predictability studies, which used either the simplified or ideal models (e.g., Thompson 1957; Lorenz 1963; Lilly 1969; Leith 1971) or the general circulation models (GCMs) (e.g., Charney et al. 1966; Smagorinsky 1969; Jastrow and Halem 1970; Shukla 1981, 1984a), an attempt was made to arrive at a quantitative estimate of the growth rate of an initial error (mostly in terms of the root-mean-square) and to determine the time limit of predictability by means of the error doubling time. This error doubling method was used in a recent study of Hohenegger and Schär (2007) to compare the predictability at synoptic scales versus cloud-resolving scales. However, as shown in Shukla (1985), the error growth rate by itself is not a useful parameter for predictability, partly because it varies significantly for different variables and partly because the ultimate limit of predictability is not only determined by the error growth rate, but also by the saturation value of the error. Neither a larger error, nor a larger error growth rate necessarily means less predictability.

Using a new concept of the ratio of the root-mean-square error to the standard deviation of daily fluctuations, Shukla (1984b, 1985) demonstrated the dependence of predictability on latitude, season, and the variable in question. However, he did not provide an estimate of the predictability in terms of the time limit.

A popular measure of forecast skill for global model evaluations is the anomaly correlation coefficient (ACC; Miyakoda et al. 1972; Hollingsworth et al. 1980; Bengtsson and Simmons 1983; Jolliffe and Stephenson 2012). It calculates the spatial correlation between the forecast and the observed (or analyzed) deviations (i.e., anomalies) from a predefined mean state. The centered version of the ACC also subtracts the spatial mean error. By definition, the ACC is very sensitive to the selected mean state, which can be the 6-h climatology, daily climatology, monthly climatology, seasonal mean, or some running averages. For example, Langland and Maue (2012) calculated the climatology by moving a weighted 21-day-centered mean window at each grid point and for different times. Some empirical thresholds (i.e., 50%, 60%, and 80%) of the ACC are used to quantify the model’s practical predictability time limit (e.g., Hollingsworth et al. 1980). As documented on the ECMWF website, it has been found that the 60% threshold corresponds to the range up to which there is forecast skill for the largest-scale weather patterns; 50% corresponds to forecasts for which the error is the same as that of a forecast based on a climatological average; and 80% would correspond to forecasts for which there is still some skill in large-scale synoptic patterns. Actually, these criteria of the ACC for practical predictability purposes are empirical and can only be applied to larger-scale circulations. As will be shown later, the ACC by itself is not a useful parameter to assess the predictability of smaller scales. If the scale decomposition is applied first, different empirical criteria might be needed to assess the practical predictability of different scales. Furthermore, as argued in Langland and Maue (2012), the ACC is not an optimal metric with which to quantify model forecast skills in all situations, since the ACC can be higher when the large-scale atmospheric flow contains strong anomalies, regardless of whether there is any actual improvement in the model forecast skill. It should be noted that the ACC is strongly modulated by the strength of the anomalies in the forecast and analysis, which may not be related to the model performance.

Another approach to quantify the practical predictability time limit is to compare the forecast error variance with an error criterion derived from the forecasts based on climatology (e.g., Baumhefner 1984; Anthes and Baumhefner 1984; Anthes et al. 1985). As reviewed by Anthes (1986), the limit of atmospheric predictability in global models is considered to be the time required for the difference variance (i.e., the mean square of the difference of some variables) of a pair of solutions that begin with small differences at the initial time to reach the difference variance associated with two randomly chosen atmospheric states. Baumhefner (1984) suggested verifying the forecast by normalizing a conventional measure of skill (such as the root-mean-square error or error variance), as compared with the best case (no error) and/or worst possible case (e.g., the persistence forecast); the limits of predictability for the various scales were estimated by comparing an estimate of the so-called predictability error (i.e., the ensemble average of the root-mean-square difference of the perturbed forecasts) with the root-mean-square climatological forecast error (error from the forecast with climatology) as the error bar. Furthermore, Anthes (1986) suggested estimating the theoretical predictability using twice as much of the climatological forecast error variance as the error variance bar. Obviously, to use this method, scale decomposition is needed to investigate the predictability of different scales.

A fundamental characteristic of the error growth in climate and weather predictions is the upscale cascade of error. Lorenz (1969) first demonstrated the slow inverse cascade of errors from small to large scales using a closure model of two-dimensional flow. It was shown that the time taken for the complete uncertainty at wavenumber

These theoretical estimates of the scale predictability based on the eddy turnover time are not useful for practical application, in part because the error growth rate in a NWP model is also related to the model physics. Actually, the inherent predictability of the atmosphere and the practical predictability of the atmosphere associated with an NWP model are not the same thing. The latter is both state dependent and model dependent and is thus a subject of interest to the model verification community. Hereafter in this paper, the predictability refers to the practical predictability unless otherwise specified.

Dynamical systems in nature, such as atmospheric flows, stock market indices, and heartbeat patterns, etc., exhibit irregular space–time fluctuations on all scales. The fractal or self-similar nature of space–time fluctuations has been identified since the 1970s (Mandelbrot 1975, 1982). Based on a concept in Townsend (1980) that large eddies are the envelopes enclosing small-scale (turbulent) eddy fluctuations, a general systems theory was developed for fractals (Selvam 1990, 1993, 2005, 2007, 2009, 2011). In this theory, the basic concept involves “visualizing the emergence of successively larger scale fluctuations to result from the space–time integration of enclosed smaller scale fluctuations” (Selvam 2009, p. 333); and the hierarchy of self-similar fluctuations generated is manifested as the observed eddy continuum in power spectral analyses of the fractal fluctuations. As shown in Selvam (2011), one primary assumption deduced from this basic concept is that both the radius ratio and the circulation-speed ratio (or eddy-amplitude ratio) of the two successive large and small eddies are equal to the golden ratio

In this paper, we apply Selvam’s general systems theory to forecast error diagnostics and the predictability; we then introduce a new generic method to quantify the scale predictability of the fractal atmosphere following the assumption of a gradual upscale cascade of error. The paper is organized as follows. Section 2 introduces a new generic method, designated as the noise-to-signal ratio (NSR) method, for quantifying the scale predictability of the fractal atmosphere. Section 3 describes the data from the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) forecasts used in this study. Section 4 applies the new NSR method to investigate the scale predictability of the NCEP GFS 500-hPa geopotential height. Section 5 compares the NSR method with the ACC method. Conclusions and a discussion are presented in section 6.

## 2. A new generic method for quantifying the scale predictability: The NSR method

### a. The one-dimensional dataset

In a numerical model, the fractal atmospheric state within a verification domain (any part of the global domain) can be transferred into a one-dimensional dataset. For example, the *t*-hour forecast, the *t*-hour forecast error, and the analysis at the valid time can be expressed by

The value of a variable on each point is associated with the physical location of the point. When the values within a selected domain are transferred into a one-dimensional dataset, the data can be sampled in many different sequential orders. However, no matter which order is selected, the statistics, such as the whole frequency distribution, the mean, and the variance of the dataset, are the same. In other words, when the verification domain in question is verified as a whole (rather than as stratified bins), the actual physical location of each point in this one-dimensional dataset is ignored; from a statistical perspective, each value of the one-dimensional dataset is just one realization of the fractal atmospheric state. The reason to use the Gaussian reduced points is to ensure even sampling over the earth’s surface within the verification domain.

### b. The whole-scale eddies: Perturbations against the instant zonal mean

^{1}with the location index

### c. A new forecast skill score: The *NSR*

*t-*hour forecast against the corresponding analysis) and

From Eq. (4), we can see that the NSR varies from 0 to a value associated with the worst possible forecast, and the smaller the value, the better the forecast skill.

### d. A new method for quantifying the scale predictability

Note that the wavelengths for the same wavenumber vary with latitude (corresponding to a different

*m*th golden square. The total eddy signal

*m*th golden square:

*m*th golden rectangle, the

*m*th golden square, and the largest golden rectangle, respectively, in Fig. 1.

For example, the NSR criteria for the predictability for the scale index

## 3. Data

We now use the 12-hourly NCEP GFS 500-hPa geopotential height 0~192-h forecasts for the Northern Hemisphere initialized four times daily from 1 August to 31 October 2012 to demonstrate the practical applications of the new NSR method for quantifying the scale predictability of the fractal atmosphere in a forecast model. Note that the data in a horizontal resolution of 0.5° is transferred onto a comparable N160 Gaussian reduced grid (https://software.ecmwf.int/wiki/display/EMOS/Reduced+Gaussian+Grids).

As we know, with the continued improvements in model dynamics and physics, data assimilation methods, and observing systems, the quality of global atmospheric analysis has been significantly improved. Although not perfect by any means, the global analysis provides a good estimate of the true atmospheric state for verification purposes where the analysis error can be omitted. For the sake of simplicity, the NCEP GFS analysis is treated as the true atmospheric state for forecast error diagnostics.

## 4. Scale predictability of the NCEP GFS 500-hPa geopotential height derived by the new NSR method

### a. Examples of the one-dimensional dataset

We perform the verification on three different domains: 1) region A [~(15°–35°N and 30°–150°W)]; 2) region B [~(25°–50°N and 30°–150°W)]; and 3) the Northern Hemisphere [~(NH; 0°–90°N and 0°–360°)]. Figure 2 shows the one-dimensional datasets of the analysis eddy amplitude

Although the physical location of each point in each dataset is ignored (as indicated in section 2) when the selected domain is verified as a whole, the indices in Figs. 2 and 3 run from south to north and from west to east starting at the date line or the west edge of each latitude. Therefore, the points in the datasets of Fig. 3a (Fig. 3b) correspond to segments of the points with indices from about 17 000 (28 000) to about 39 000 (52 000) in Fig. 2a (Fig. 2b). In addition, the one-dimensional datasets with point indices running in this order actually carry some structural information on the spatial variations.

As shown by the

For the derivation of the scale predictability based on the NSR method, these variations and diversities are not explicitly considered. The only information derived from the one-dimensional analysis eddy amplitude dataset is a statistical measure: the signal, or the domain-averaged square of the total eddy amplitudes. Therefore, from the statistical perspective, the specific verification domain should have as little diversity as possible in order to derive scale predictability features with better representativeness. For example, the mesoscale circulation systems over the tropics may behave quite differently from those over the polar regions; therefore, the predictability of the mesoscale circulations over the tropics may be quite different from that of the circulation systems with the same horizontal scale over the polar regions. This is a general requirement for any overall spatial verification.

The forecast error amplitude datasets in Figs. 2 and 3 are, in general, analogous to the corresponding analysis eddy amplitude datasets; the larger the analysis eddy amplitude, the larger the forecast error amplitude. As a result, the NSR values for the same lead time are generally of comparable order.

Since the atmosphere is constantly changing, the forecasts valid at a specific time with different lead times have different initial conditions, while the forecasts with different lead times integrated from the same initial conditions should be verified against different evolving atmospheric states. The forecast error amplitudes and the NSR values valid at a specific time do not necessarily increase with the forecast lead time, since they are integrated from different initial conditions. For example, as shown in Fig. 3a, the NSR value of the 168-h forecast is smaller than that of the 144-h forecast. In addition, as shown by the forecast error datasets in Figs. 2 and 3, the error structures for forecasts produced from different initial conditions also differ. However, it is expected that, from the statistical perspective, the NSR values should increase with the forecast lead time until totally saturated. This will be illustrated in the following monthly verification results.

### b. Examples of scale predictability derived by the NSR method

The examples shown in Fig. 3 are used to demonstrate the application of the NSR method to derive the predictability time limit of different wavenumbers. As derived in section 2, the NSR or

Table 1 lists the 500-hPa geopotential height NSR [

Individual examples of 500-hPa geopotential height NSR and

Since the predictability time limit from an individual case perspective, either for an individual valid time or an individual initial time, is not representative, it is desirable to extract meaningful predictability time limits from a statistical perspective. The following two subsections illustrate the time evolution of the

### c. The time evolution of from 1 August to 31 October 2012

Figure 4 shows the time evolution of

We find in Fig. 4 that

We show in Fig. 4 that all of the time series of

Another important feature of the

An interesting feature of the

### d. Mean features of the scale predictability derived by the NSR method

Studied in this paper are three different periods in 2012: August, September, and October. Figure 5 shows the temporal average of the

From Fig. 5, we can see that the

Table 2 summarizes some of the mean features of the scale predictability of the 500-hPa geopotential height in the NCEP GFS in 2012 derived by the NSR method. A coarse linear interpolation or extrapolation is performed if necessary to retrieve the steady growth of the NSR for calculating the time limits of the predictability of the different wavenumbers.

The scale predictability of the 500-hPa geopotential height in the NCEP GFS in 2012 derived by the NSR method (hours).

Figure 6 is designed to effectively present the same information in Table 2, with the derived predictability time limit (a key quantity of interest for a forecaster, as it has a very clear meaning) on the *y* axis as a function of wavenumber (size of meteorological feature) on the upper *x* axis. The wavenumber itself is a function of the scale index (on the *x* axis).

It should be noted that more accurate predictability time limits could be obtained using forecast outputs at higher temporal resolution, especially for the earlier stage forecasts. Users of the NSR should select proper output frequency of the forecasts for their verification purposes with desired accuracy.

The results described above are the features of the practical predictability of the NCEP GFS model for the specific verification variable, time, and space. Different modeling systems should have different practical predictability limits, just as they have different forecast skills. It goes without saying that the NSR method provides a consistent approach for model comparison.

## 5. Comparison of the NSR method with the ACC method

### a. The sensitivity of the *ACC* to the definition of the mean

We tested three different definitions of mean to examine the sensitivity of the ACC to the selection of the mean: the 92-day seasonal mean of the NCEP GFS analysis four times daily from 1 August to 31 October 2012; the 10-day running mean of the NCEP GFS analysis four times daily centered at the valid time; and the instant zonal mean of the NCEP GFS as defined in Eq. (1) of section 2. Figure 7 shows the time evolution of the ACC of the NCEP GFS 500-hPa geopotential height forecasts with lead time of 12, 24, 72, 120, and 168 h from 1 August to 31 October 2012 for region A with the anomalies calculated against these three different definitions of the mean. It should be noted that, because of data availability and also for the sake of simplicity, we do not test against the common climatology set by the WMO or the one operationally used by the NCEP.

As shown in Fig. 7, both the ACC calculated against the seasonal mean and against the instant zonal mean are somehow similar, while their time variation details sometimes differ and the latter is, in general, a little smaller. On the contrary, the ACC calculated against the 10-day running mean is very different, in terms of amplitudes, phases, or peaks; it is much smaller than the ACC calculated against the other two kinds of mean. As expected, the ACC is sensitive to the definition of the mean.

### b. The weak stationarity of the *ACC*

Totally different from the time evolution of the

### c. The mean features of the model predictive skill derived by the *ACC* method

Figure 8 shows the temporal average of the ACC of the NCEP GFS 500-hPa geopotential height with different forecast lead times every 12 h from 12 to 192 h during August, September, and October for region A with the anomalies calculated against different kinds of mean. Consistent with results shown in Fig. 7, the strong sensitivity of the ACC to the different definitions of the mean is quite evident.

Although similar qualitative conclusions can be drawn on the GFS model forecast skills from Fig. 8 and Fig. 4a such that the GFS has better forecast skill in October than in August and September and the model has the worst skill in September for all forecast ranges in region A, it is not possible to derive an objective and quantitative evaluation of the scale predictability from the ACC values in Fig. 8.

In addition, the strong nonlinearity of the ACC’s variation among the lead times is quite evident, especially in shorter-range forecasts. The left side of Fig. 4a and the solid line with crosses in Fig. 5a clearly illustrate that there is a larger rate of increase in the

## 6. Conclusions and discussion

The ACC is one of the standard methods for the evaluation and comparison of the performance of global NWP models. The correlation coefficient of the forecast anomaly field with the analysis anomaly field both valid at the same time is used as a measure of model forecast skill. The calculation of the ACC is known to be sensitive to the definition of the mean state from which the anomaly is extracted. Although the WMO has established a standard definition of the climate mean state for the global NWP community, different operational centers still use different climatologies to calculate the ACC for their own verifications. In addition, the ACC time series have very weak stationarity, which makes it difficult to calculate meaningful long-term mean statistics, and sometimes certain running mean time series have to be derived as smoother replacements but with less representativeness. Since the ACC does not penalize systematic bias and errors in patterns, it is often used together with the bias and RMSE to assess model performance. Another significant shortcoming is that the ACC is strongly modulated by the strength of the anomalies in the forecast and analysis, which may not be related to the model performance. Because the entire anomaly fields are used for the calculation, it is not possible to differentiate the performance of the model on different scales without explicitly performing scale decomposition. For example, the short-range (12–24 h) forecast of a global model should be dominated by errors on the smaller scales. Yet, the ACC is generally very close to one in the short-range (12–24 h) forecasts, because the ACC is dominated by the performance of the model on the larger scale of motions (with large amplitude and high correlation). In other words, the ACC is not sensitive to short-range, small-scale errors. As a result, it is impossible to evaluate the performance of the model on small scales based on the ACC by itself.

In this paper, we propose a new measure of model forecast skill, which we name the NSR. The basic concept is to calculate the ratio of the noise (defined as the average square of the model error amplitudes) to the signal (defined as the average square of total eddy amplitudes) across all scales. On the surface, this is not significantly different from that of the ACC. However, by taking advantage of the intrinsic properties of the self-similarity and the inverse power law of the fractal atmosphere based on the general systems theory of Selvam (1990), and by following the assumption of the upscale cascade of error, we show that the time limit of predictability

There are several other unique advantages of the NSR method. First, the mean (from which the eddies are calculated) is defined by the instant zonal mean. Therefore, by definition, the NSR is adaptive to flow. Second, the time series of the logarithm with base

As a demonstration of the advantages of the NSR method, we evaluate the performance of the NCEP GFS model forecast of the 500-hPa geopotential height from August to October in 2012 and compare that derived with the ACC method. The relative performance of the NCEP GFS in different months is clearly evident in both measures of the NSR and the ACC. However, the strong sensitivity of the ACC to the definition of the mean is shown to be quite substantial. Also, there can be huge variations in the ACC in time for different forecast lead times; the NSR does not have these issues. Moreover, the NSR method provides additional insights on the predictability time limits of different scales, how they differ in different regions dominated by different weather systems, and how they change with time as the season progresses, without the need to explicitly perform the scale decomposition.

With the ability to reveal the predictive skills on different scales as well as the model error growth between scales, the NSR method can provide useful insights on the relative performance of different global models (e.g., the NCEP GFS vs the ECMWF) or the relative performance of the same model at different resolutions (e.g., the NCEP GFS vs the NCEP CFS) on different scales. For example, questions such as “which model performs better on smaller scales (e.g., which may be driven by convection)?” and “which model has a faster error growth rate between scales?” can be addressed. The analysis of the NSR results of a forecast model may provide useful guidance on future model improvement. The NSR method can be applied over subdomains of a global model, which may shed light on model challenges over a specific region. With the aid of a global analysis, the NSR method can also be used to verify a regional model. Exciting developments in NWP are being made in high-resolution convection-permitting models, and in extended-range coupled prediction. It would be interesting to use the NSR method to measure predictability for subdaily forecasts and for monthly forecasts from coupled models. These applications of the NSR method deserve to be pursued in future studies.

There are two important attributes of the new NSR method: 1) to define the NSR as a normalized measure to assess the short-range, small-scale forecast skills and the long-range, large-scale forecast skills in a consistent way; and 2) to identify the golden ratio associated criteria as the thresholds to quantitatively detect the successive scale predictability time limits given the NSR values. The second attribute works only if Selvam’s general systems theory is applicable for the Earth atmosphere. In Selvam’s general systems theory, the atmosphere is assumed fractal, and the fractal dimension is the golden ratio. It is under this theory in addition to the assumption of upscale error cascade that the NSR values could be further interpreted and used to derive the successive scale predictability time limits. We find that Selvam’s theory actually implies a −3 spectral decay law, which has already been demonstrated by many previous theoretical, numerical, or observational studies to be valid for the Earth atmosphere at least down to scales of a few hundreds of kilometers. Many insightful studies (see the appendix) actually suggest that the −3 power law should extend to a much smaller scale than the transition mesoscales identified by the shallower slope of −

In a nutshell, our new NSR method for assessing the short-range, small-scale forecast skills and the long-range, large-scale forecast skills in a consistent way and for quantifying the practical predictability of atmospheric circulations of different scales without the need of explicit scale decomposition could be generic, robust, and useful for model verification.

## Acknowledgments

We thank the reviewers and Christopher A. Davis for their valuable comments, which substantially improved the presentation of this paper. This work is jointly supported by the National Aeronautics and Space Administration and by the National Science Foundation under NSF Award AGS-1033112 and by the NOAA Hurricane Forecast Improvement Project through the Developmental Testbed Center under Award NA08NWS4670046.

## APPENDIX

### Atmospheric Spectrum Structure

In their seminal work, Nastrom and Gage (1985) presented an analysis of the flight observations of wind and temperature, showing a distinct transition from a steep spectral slope of −3 at synoptic scales to a shallower slope of −

Explanations in the literature for the shallower mesoscale spectrum fall into three general categories: 1) an inverse cascade of small-scale energy, produced perhaps by convection; 2) production of gravity waves by divergent flows; or 3) a direct cascade of energy from the large scales. More recent observations and analyses present new facts debating the old explanations of the first two types, and only theories of the third type are left as plausible universal explanations of the Nastrom–Gage spectrum. Therefore, most of the different theories proposed are based on the hypothesis of an energy cascade. Tung and Orlando (2003) proposed a theory that the subdominant −*O*(1 km) scale, independent of energy flux. Tran and Shepherd (2002) also indicated that the choice of forcing mechanisms and dissipation operators has implications for the spectral distribution of energy and enstrophy dissipation and, thus, for the possible existence of energy and enstrophy cascades. Furthermore, the choice of dissipation operators has implications for permissible scaling of the energy spectrum. They remarked that, in choosing forcing mechanisms and dissipation operators for numerical reasons, one should be mindful of these constraints. Tulloch and Smith (2006) proposed a finite-depth surface-quasigeostrophic model that highlights the transition between quasi-two-dimensional barotropic flow and three-dimensional baroclinic flow. They demonstrated how turbulent motions at the synoptic scale can produce a balanced, forward cascade of temperature, resulting in an upper-tropospheric spectrum with a break at a scale that is a function of fundamental background parameters. Later, Tulloch and Smith (2009) showed that a quasigeostrophic model driven by baroclinic instability exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced. The transition wavenumber depends linearly on the ratio of the interior potential vorticity gradient to the surface temperature gradient. Obviously, the transition scales as derived in these numerical models are tunable under different numerical or dynamical assumption details. There are still many limitations and uncertainties in the related numerical studies.

Recently, the spectrum analysis and the structure function analysis are used on high-resolution observations or model simulations to revisit the atmospheric spectrum issue. As reviewed in Augier and Lindborg (2013), while the

Kolmogorov (1941) assumed the presence of anomalous dissipation in developing his power law for 3D isotropic and homogeneous turbulence. His argument for the −

Despite many attempts over the last 30 yr, the atmospheric turbulence spectrum analyzed from the observations has not been fully explained, and the real physical mechanisms that account for the universality of the atmospheric spectrum structure are still unclear. The review presented above raises an interesting and bold question: is the shallower slope of −

We can imagine that the overestimation of the energy at larger scales and/or the underestimation of the energy at smaller scales would distort the energy spectrum to a deeper slope than the real one; on the contrary, the underestimation of the energy in the larger scales and/or the overestimation of the energy in the smaller scales would distort the energy spectrum to a shallower slope. Unfortunately, this usually happens in the practice of spectrum analysis or structure function analysis when the dataset (either from observations or model simulations) is incomplete, not representative, or is overdiffused or overfiltered. Almost all of the observational datasets used to calculate the spectrum structure are either 1D regional segments or 2D regional boxes; that is, they represent only the regional limited sampling of the real atmosphere that actually possesses a full spectrum of scale, ranging from wavenumber 1 to the maximum resolved wavenumber. The structure function analysis should be very sensitive to the length of the data segments; and without the presence of larger scales, the regressed slope for the investigated spectrum band would tend to be shallower. In addition, some techniques, such as detrending, filtering, and combining, etc., introduced in the data preprocessing may distort the energy spectrum results as well. For example, as recognized in Skamarock (2004), the spectrum computed using the full length

It is desirable that a realistic observational atmospheric spectrum is used to check the spectrum behaviors in the numerical model. As shown in Kahn et al. (2011), higher horizontal spatial resolution observations over the entire globe are necessary to observe the global characteristics of small-scale “turbulence,” although this is not an easy undertaking. The limited spatial resolution places a constraint on earlier studies of global atmospheric analyses, restricting the extent to which two-dimensional turbulence theory describes the atmospheric circulation. In addition, the distribution of energy among scales in model simulations may suffer from the deficiencies in model physics and dynamics. With the improvement of meteorological analyses that have much higher spatial resolution and include a well-resolved stratosphere and with the availability of observations that have high horizontal resolution and global coverage, it is expected that the mutual corroboration among theories, observations, and model simulations will help us understand the universality of the atmospheric spectrum structure, determine the intrinsic and practical predictability of atmospheric circulations of different scales, identify the source and nature of errors, and finally improve weather and climate models.

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^{1}

The dataset could also be sampled on the icosahedral or any other grids with fairly uniform spacing, with eddies still defined against the instant zonal mean, which could be easily derived from the interpolated data (at a comparable resolution with the original data) along the corresponding full latitude line.