## 1. Introduction

The central thrust in this and an earlier study (Johnson et al. 2000, hereafter Part I) is to develop and apply statistical strategies to assess the accuracy of atmospheric models in the simulation of reversible isentropic processes. The thrust is addressed against the premise that it is relevant to ascertain uncertainties in the simulation of reversibility among different numerical algorithms whether spectral or discrete and also different coordinate systems. The underlying statistical strategy assesses “pure error” in relation to appropriate numerical conservation of dry and moist entropy in accord with the Navier–Stokes equations for the continuum.

The strategy continued in this study assesses numerical accuracy with respect to reversible processes involving the exchanges and transformations of water vapor and cloud water in relation to conservation of equivalent potential temperature as the surrogate for moist entropy. The analysis investigates a model's capability to simulate the future state of a property *f* determined from the set of governing equations in comparison with its proxy *tf* determined from a corresponding simulated transport equation representing the same property. With equality of the initial states by specification and utilizing a proxy continuity equation physically and dynamically consistent with the governing equations in simulating reversibility, the determination of the sum of squared differences of *f* and *tf* over all grid points of the model as a function of time provides a direct assessment of error growth from numerics. This corresponds with the strategy of replications in experimental design and the determination of a sum of squares function within an analysis of variance (Anderson and Bancroft 1952; Box et al. 1978).

Inaccuracies in the simulation of the dry component of entropy will also emerge from the results of this analysis. Of the thermodynamic variables, potential temperature as the dry component of entropy is particularly relevant for assessing the accuracy of hydrostatic models. The potential temperature in combination with the hydrostatic mass specifies the total potential energy, the available potential energy, enthalpy, geopotential energy, dry static energy, and other derived parameters such as the geostrophic wind, static and geostrophic symmetric stability criteria, and the geostrophic Richardson number, etc. The strategy developed permits zonal, meridional, areal, vertical, and temporal assessments and various combinations either for limited domains or the whole atmosphere, and extends to trace constituents including potential vorticity.

The statistical analyses set forth in Part I revealed that truncation errors of numerics possess systematic and random error components and that the optimum strategy is to eliminate bias and minimize random errors (Greenspan 1971). Since random errors cannot be eliminated, the optimum strategy is to achieve a relative frequency distribution of errors equilibrating with time within which even order moments are minimized and odd order moments vanish.

The focus in Part I was on the quasi-horizontal spatial–temporal structure of numerical inaccuracies in simulating reversibility. In this in-depth extension, first attention is given to different presentations of the results from several versions of the National Center for Atmospheric Research Community Climate Model (NCAR CCM) and the University of Wisconsin—Madison (UW) *θ*–*σ* model: zonal–vertical cross sections of inaccuracies, bivariate scatter distributions, relative frequency distributions, and profiles of bias differences, etc. Then a global analysis of variance is formulated in which various components of inaccuracies of models in simulating reversibility are isolated with respect to horizontal and vertical exchange of thermodynamic properties. The variance of the deviation differences quasi-horizontally constitutes the within-group sum of squares. The variance of the quasi-horizontal mean differences about the global mean difference constitutes the among-group sum of squares. Evidence of the vertical growth of systematic error differences within the among-group sum of squares will be presented to illustrate uncertainties in the vertical exchange of entropy. Then utilizing estimates of the random error variance, the uncertainty of whether apparent biases are simply averages from sampling random numerical differences is ascertained. The conditions under which the empirical relative frequency distributions of differences assume the forms of triangular and normal distributions are also appropriately examined. The strategy will also be related to global integral constraints involving entropy and energy exchange to estimate aphysical sources of energy and entropy from numerics.

## 2. Background and experimental strategy

A unique contribution of the prior and current studies lies in setting forth a means to assess nonlinear numerical inaccuracies and inconsistencies in the fully three-dimensional nonlinear baroclinic flow encountered in weather and climate prediction. The differences among the UW hybrid isentropic-sigma (*θ*–*σ*) model (Zapotocny et al. 1994) and the NCAR CCM version 2 (CCM2; Hack et al. 1993) and version 3 (CCM3; Kiehl et al. 1996) were ideal for developing the concept of the pure error sum of squares to assess model numerical accuracies in simulating moist reversible processes and to illustrate the merits of the strategy. There is the difference between models expressed in isentropic and sigma coordinates. There are the differences among Eulerian spectral, semi-Lagrangian, and discrete gridpoint numerics. There is a rich body of literature comparing the climate states of CCM2 and CCM3 and the ensuing improvements made in the development of NCAR's models (Hack et al. 1994; Hack 1998a,b; Kiehl et al. 1998a,b) that enables ready interpretation of the numerical inaccuracies identified in Part I and in this study.

### a. The models

In this study, the NCAR Community Climate Model versions 2 (Hack et al. 1993) and 3 (Kiehl et al. 1996) are run at the standard T42 spectral resolution (approximately 2.8° × 2.8° transform grid) with a vertical resolution of 18 layers. Horizontally the governing equations for mass, vorticity, divergence, and enthalpy are based on an Eulerian spectral transform method with vertical advection being approximated by centered finite differences. Horizontal and vertical transport of water vapor and trace constituent transports are accomplished with shape-preserving semi-Lagrangian transport (SLT; Rasch and Williamson 1990a,b; Williamson and Rasch 1989). The NCAR recommended values of 2.5 × 10^{5} m^{2} s^{−1} and 1.0 × 10^{16} m^{4} s^{−1} for ∇^{2} and ∇^{4} horizontal diffusion coefficients were utilized for divergence, vorticity, and temperature (Hack et al. 1993; Williamson et al. 1995; Acker et al. 1996). See Hack et al. (1993) and Kiehl et al. (1996) for detailed discussions of CCM2 and CCM3.

The UW *θ*–*σ* gridpoint model (Zapotocny et al. 1994, 1997b) has an average of 19.3 layers and consists of a sigma coordinate PBL 150 mb in vertical extent and an isentropic coordinate free atmosphere, which extends up to 1700 K (∼1–2 mb). For this study, the UW *θ*–*σ* model uses 2° lat by 2.5° lon horizontal resolution. The UW model used a 5-min time step while the T42 CCM2 and CCM3 used a 15-min time step. The time differencing, filters, borrowing, etc., in the UW model are described in greater detail in Zapotocny et al. (1997a,b).

### b. Numerical simulation of moist reversible processes

The initial global state of 1200 UTC 4 October 1994 for the 10-day simulations was obtained from the stratospheric version of the Goddard Laboratory for Atmospheres Earth Observing System 4DDA model (GEOS-1; Schubert et al. 1993), commonly referred to as STRATAN. The proxy equivalent potential temperature (*tθ*_{e}) for all grid points at the initial time was simply equated to the equivalent potential temperature (*θ*_{e}) at corresponding points on the information surfaces of each model, being determined from the temperature, pressure, and specific humidity following Bolton (1980). This direct specification avoided interpolation errors between coordinate systems. All simulations were run moist adiabatically except for the effects of skin friction in all models and the CCM diffusion described above. Under the moist-adiabatic constraint of the experiments, the parameterizations involved with radiative transfer, cumulus convection, PBL processes, etc., were suppressed.

In these experiments, continuity equations for water vapor and for cloud water/ice are included explicitly in the governing equations. Although the focus in Part I was on three experiments, the focus in this paper, Part II, will only be on the second experiment since there was little difference in the bivariate distributions and rms differences of the three experiments. From considerations of moist entropy, the second experiment without precipitation was a fully reversible experiment in which cloud formation with diabatic heating from large-scale condensation occurs with RH ≥ 100% while cloud evaporation with diabatic cooling occurs with RH < 100%. Water vapor condenses and creates cloud water/ice, and diabatic heating/cooling from phase changes has been appropriately included as a physical process in the thermodynamics of all models, and in the proxy continuity equation. Thus in all model simulations diabatic vertical transport of all properties and work of expansion and compression occurs.

In this follow-on analysis, five fully reversible experiments are presented. The five experiments include four using various versions of CCM2 and 3 and a fifth using the UW *θ*–*σ* model. One is the default CCM3 in which the enthalpy is simulated by Eulerian spectral numerics and the water vapor, cloud water, and *tθ*_{e} are simulated by semi-Lagrangian numerics. Another is the CCM2/3 mixed experiment in which CCM3 determines *θ*_{e} by its usual simulation of enthalpy by Eulerian spectral numerics and water vapor and cloud water by semi-Lagrangian numerics to be compared with CCM2's simulation of *tθ*_{e} by Eulerian spectral numerics. Another experiment employs CCM2's Eulerian spectral numerics for enthalpy, water vapor, and cloud water to determine *θ*_{e} and also *tθ*_{e}. The final experiment utilizes the mass and momentum distributions from a standard 10-day CCM3 simulation from which CCM3's semi-Lagrangian numerics are then used to simulate the distributions of potential temperature *θ,* water vapor *q*_{υ}, and cloud water *q*_{c} to determine *θ*_{e} and also *tθ*_{e}. The numerics of this experiment differ from Williamson and Olson's (1994) semi-Lagrangian version of CCM3 in that all properties were transported passively as in a chemical transport model.

## 3. Distributions of differences of (*θ*_{e} − *tθ*_{e}) at 10 days

### a. Zonal–vertical and bivariate scatter distributions of differences

Quasi-horizontal distributions of the difference field (*θ*_{e} − *tθ*_{e}) and bivariate scatter distributions were presented for CCM3 and the UW *θ*–*σ* model in Part I. Now zonal-vertical cross sections of the difference fields at day 10 from CCM3, the CCM2/3 mixed, the CCM2 all Eulerian spectral, and CCM3 all semi-Lagrangian simulations for subtropical (24°S) and extratropical latitudes (58°S), are presented in Figures 1a–d and 2a–d, respectively. The panels in the left-hand column of Fig. 3 present the corresponding bivariate scatter distributions at day 10 for the four simulations. Corresponding zonal–vertical cross sections and a bivariate scatter distribution for the UW *θ*–*σ* model are presented in Figs. 4a–c.

The zonal–vertical distributions for CCM3 in Figs. 1a and 2a are by far the most striking with extremely large bias and random error components of differences near the upper and lower boundaries. Corresponding large differences are evident in the scatter portrayed in the bivariate distribution in Fig. 3a1. Since nearly all of the stratospheric values greater than 380 K fall to the right of the equiangular line, the differences of the paired values are positive with *θ*_{e} being larger than *tθ*_{e}. In contrast, tropospheric values less than 360 K tend to fall to the left of the equiangular line with the values of *θ*_{e} being generally less than corresponding values for *tθ*_{e}.

Relative to the scatter portrayed in Fig. 3a1 for CCM3, both the asymmetry and the scatter about the equiangular line are reduced markedly in Figs. 3b1, 3c1, and 3d1 representing the other three simulations, particularly in the upper levels. Thus the results from the spectral Eulerian and semi-Lagrangian simulations were markedly different even though the numerics of all four simulations aim to simulate the same relation between local tendency and transport as discussed earlier. Large bias differences were traced to the limitations of piecewise continuity of the CCM Hermite cubic interpolants and their failure to ensure reversibility in that the estimate of the first derivative of a property at a common arrival level for upward or downward displacements differed (Part I). In effect the representation of Eulerian vertical advection by semi-Lagrangian numerics is double valued for the given arrival level. This condition compromises reversibility, a point studied more extensively by Egger (1999). There was also a source of differences in CCM3 that stemmed from the constrained boundary conditions utilized with the Hermite cubic interpolants (Part I) and smoothing of the temperature distribution across the tropopause region (Ritchie et al. 1995).

Inspection of the free tropospheric portion of the zonal cross sections for the CCM2/3 mixed simulation in Figs. 1 and 2 reveals a noisy but distinct pattern of plus–minus differences extending vertically throughout the troposphere. The CCM3 default and all SLT simulations reveal like patterns extending throughout the troposphere but superimposed on an apparent systematic negative bias. The telltale pattern of plus–minus differences and apparent negative bias stem from coherent truncation errors that develop in simulating vertical advection and reversibility within a stratified baroclinic atmosphere. Here, the existence of strong vertical motion and wind shear associated with baroclinic amplification combined with backing during cold air advection and veering during warm air advection creates strong and differing intensities of horizontal and vertical gradients of temperature, water vapor, and cloud water throughout the baroclinic domain. In this situation, the horizontal and vertical advection of temperature in sigma and isobaric models act opposite to each other and the issue of aspect ratios between horizontal and vertical resolutions enters. Consequently, no unique aspect ratio can be determined in relation to horizontal and vertical resolutions in frontal baroclinic regions, which minimizes truncation errors for a given property everywhere as well as for all properties.

The patterns and vertical extent of the plus–minus differences are somewhat different in the CCM3 all semi-Lagrangian simulation portrayed in Fig. 1d; however, the evidence of systematic patterns of differences within baroclinic waves is still quite pronounced in Fig. 2d. The end result is nonconservation of both dry and moist entropy during the simulation of the slantwise ascent of warm air and expansion coupled with the corresponding descent of cold air and compression. This latter process is intrinsic to reversibility and the maintenance of the kinetic energy of extratropical latitudes.

The pattern of differences in Figs. 1c and 2c for the CCM2 all Eulerian spectral simulation is quite different with more or less uniform negative values everywhere and extremely large differences in the lower troposphere associated with ringing induced by the orography of the Andes. The difficulties of ringing encountered in Eulerian spectral models particularly during the simulation of precipitation over and in the vicinity of orography are long standing (Navarra et al. 1994). The underlying reason for the generality of the negative differences in CCM2's tropospheric region involves an inconsistency between the vertical advection of enthalpy and of potential temperature and the failure to ensure reversibility, which will be discussed later.

### b. Relative frequency and bias distributions

Relative frequency distribution for days 2.5, 5.0, 7.5, and 10.0 and vertical profiles of average differences of *θ*_{e} and *tθ*_{e} for day 10 are presented in the middle and right-hand columns of Fig. 3 for the four simulations utilizing CCM and in Figs. 4d,e for the UW *θ*–*σ* model. Of the four profiles in Fig. 3, the profiles in Figs. 3a3, 3c3, and to a lesser extent 3d3 show vertical distributions with substantial systematic biases, all of which differ from each other. The biases for CCM3 in Fig. 3a3 as determined by mass-weighted layer averages (*δ̂*

In comparing the four relative frequency distributions in Fig. 3, all show a decrease of the peak values and a temporal spreading associated with the growth of differences. Note in Fig. 3c2 at day 2.5, the peak value of 0.79 K^{−1} in the CCM2 all Eulerian spectral simulation with minimal spread is larger than the peak values of the other three distributions, while by day 10 the peak value has decreased to 0.35 K^{−1}. The temporal development of the bias is also evident in Fig. 3c2 in that the interval of maximum frequency shifts systematically toward negative values of (*θ*_{e} − *tθ*_{e}).

The CCM2 all Eulerian spectral profile in Fig. 3c3 shows that the systematic bias noted from the cross sections extends throughout nearly the entire troposphere. There is also the systematic shift of the relative frequency distribution in Fig. 3c2. Since neither the bias nor shifting of the peak occurs in Figs. 3b2, 3b3 for the CCM2/3 mixed simulation, the nature and vertical extent of the bias suggests that the systematic differences developed from an inconsistency in CCM2's numerical simulation of vertical advection of enthalpy relative to the vertical advection of the dry entropy as represented by *θ* within the trace continuity equation. The origin of this inconsistency will be addressed later in view of the changes made in the determination of the substantial derivative of pressure *ω* (Kiehl et al. 1998a).

All in all, the zonal–vertical cross sections provide striking evidence of the long standing difficulties that models encounter in the simulation of transport and reversibility. These difficulties would also be quite pronounced in simulating the transport of ozone, chemical substances, aerosols, and other inert constituents, particularly during stratospheric–tropospheric exchange that is intrinsic to baroclinic development.

## 4. The global sum of squares

The statistical analysis of the pure error sum of squares in Part I and heretofore have focused primarily on the quasi-horizontal, zonal–vertical, and vertical distribution of average differences, bivariate scatter, and relative frequency distributions. Now the focus will turn toward a global perspective by formulating an analysis of variance of the global sum of squares of the pure error difference (*f* − *tf*) partitioned into three components. The three components are the square of the global bias difference, the sum of squares of the deviations of the quasi-horizontal average difference from the global average difference, and the square of the deviations of the field differences from the quasi-horizontal average difference. For details of the classical partitioning of the variance into within- and among-group sum of squares, see chapter 6 of Box et al. (1978).

### a. Definition of quadratic functions, weighted means, and deviations

*w*equals

*ρJ*

_{η}/

*M*and the difference is given by

*δ*

*y*

*z*

*ρ*is the density,

*J*

_{η}is |∂

*z*/∂

*η*|,

*M*is the integrated mass of the region,

*η*is the generalized vertical coordinate, and

*J*

_{η}

*dAdη*is the incremental volume of generalized meteorological coordinates (Johnson 1980). Here

*dA*is the quasi-horizontal incremental surface area

*a*

^{2}cos

*ϕdλdϕ*as projected onto a geopotential surface. Now note that the generalized incremental mass (

*ρJ*

_{η}/

*M*)

*dAdη*being always nonnegative has an equivalence with the probability density function

*p*(

*f*)

*df*. Consider an indefinite integral in three dimensions beginning at a point and sweeping over the entire mass of the atmosphere. In this case, the indefinite integral of the fractional mass increases uniformly from zero to unity and thus corresponds with a cumulative distribution function. The generalized incremental volume

*J*

_{η}

*dAdη*provides for invariant transformation of the global estimates of the mean, quadratic, covariant, and other well-behaved functions including higher-order moments from one coordinate system to another. Thus,

*S*

_{G}(

*δ*) as the integral of the quadratic difference as expressed in (1) and (2) constitutes a mathematically invariant transformation among different coordinate systems for any given region.

*rs*th volume element and of the

*s*th quasi-horizontal layer relative to the total mass

*M*of the global domain are, respectively, defined by

*r*th volume element of the quasi-horizontal layer to the integrated mass

*M*

_{s}of the

*s*th layer given by

*w*

_{r}

_{s}

*ρJ*

_{η}

*dAdη*

_{r}

*M*

_{s}

_{s}

*s*th layer. Here

*M*

_{s}and

*M*are the integrated masses of the

*s*th layer and global domain, respectively, defined by

*R*is the total number of discrete volume elements within each layer and

*S*is the total number of layers.

*f̂*

_{s}), the global average (

*f*

^{*}

_{rs}

*f̂*

^{*}

_{s}

Even though strictly speaking the empirical distributions of the weights *w*_{rs}, *w*_{r}, and *w*_{s} determined from hydrostatic pressure are not simulated without error, the strategy embodied within the pure error sum of squares eliminates this source of error since the empirical calculation of the various weights is always the same for each of the paired values, for example, *w*_{rs}, is the same for *y*_{rs} and *z*_{rs} everywhere.

### b. Definition and expected value of the global sum of squares

*rs*th difference is defined by

*δ*

_{rs}

*y*

_{rs}

*z*

_{rs}

*y*

_{rs}and

*z*

_{rs}expressed as a linear combination of true-values, bias errors (

*λ*), and random errors (ε) are, respectively, given by

*η*

_{rs}equal to

*ζ*

_{rs}everywhere eliminates the true state from consideration and allows the analysis to focus solely on the inaccuracies of the model's numerics. As such, the global sum of squares function reduces to

_{yrs}

_{zrs}

### c. The partitioning of the sum of squares

*S*

_{G}

*δ*

*S*

_{G}

*S*

_{G}

*δ̂*

*S*

_{G}

*δ*

*S*

_{G}(

*S*

_{G}(

*δ̂*

*S*

_{G}(

*δ**). Utilizing (7)–(17) the three component integrals with definitions of corresponding differences are given by

## 5. Analysis of variance for pure error sum of squares

### a. The motivation and formulation

Three purposes of this statistical analysis are to partition variation, relate components of variation with different numerical sources, and assess uncertainty of the sources. As such, the strategy will be to calculate the average difference for each layer, remove the square of this difference from the layer total sum of squares layer by layer, and then utilize the remaining sum of squares, which are utilized to estimate the deviation variation within each layer. The analysis will still include the zonally varying systematic components as the zonal–vertical cross sections indicate. Since there is no means to appropriately proportion the bias and random components of the difference locally, comparisons with estimates of the random error variance among the layers of the given model and with other models are most important.

### b. The global sum of squares and its three components for CCM simulations

For clarity of comparisons, the sum of squares results from the CCM simulations will be discussed fully prior to any discussion of the UW *θ*–*σ* model in order to focus on the relative attributes of the numerics of the four different CCM simulations and not the model coordinate system. There is a fundamental difference in the partitioning of the sum of squares between the results from sigma and hybrid-isentropic coordinates. In sigma models the quasi-horizontal variation of *θ*_{e} within layers involves variation of both *θ* and water substances, while in isentropic layers of the UW *θ*–*σ* model, which do not intersect the planetary boundary layer, the variation involves only water substances superimposed on the uniformity of *θ.*

Tables 1a–d present entries for the global sum of squares *S*_{G}(*δ*) and its three components *S*_{G}(*δ**), *S*_{G}(*δ̂**S*_{G}(_{e}, and the global change Δ_{e}. Tables 2a–d present sums of squares *S*_{s}(*δ*) and its two components *S*_{s}(*δ**) and *S*_{s}(*δ̂**s*th layer at day 10. Tables 2a–d also present corresponding statistics *δ̂**θ̂*_{e}, and Δ*θ̂*_{e} plus a column listing weights to be used later in assessing uncertainty of biases. Taken together, the statistics for the trace *t*_{e} and Δ*t*_{e} can be determined from _{e} in Table 1; likewise the statistics *t**θ̂*_{e} and Δ*t**θ̂*_{e} can be determined from corresponding statistics from Table 2. For the sake of brevity, the comparison of the tabular statistics will largely focus on Tables 1a–d even though the detailed results in Tables 2a–d provide extremely important information on numerical accuracies concerning reversibility level by level for each model.

Consider the situation where the difference *θ̂*_{e} − *t**θ̂*_{e} vanishes within each layer, then both *S*_{G}(*δ̂**S*_{G}(*S*_{G}(*δ**) equals *S*_{G}(*δ*). Two related implications are that the average of a relative frequency distribution of deviations as in Fig. 3 remains centered at zero and the development of systematic differences from vertical advection of *θ̂*_{e} and *t**θ̂*_{e} are precluded. The more general situation is where systematic differences exist with the result that neither *S*_{G}(*δ̂**S*(*S*_{G}(*δ̂**S*_{G}(*δ̂*

The vertical deviation *S*_{G}(*δ̂**S*_{G}(*S*_{G}(*δ*) are considered first for two reasons. There is the need to compare among the different models the magnitude and vertical structure of the mean differences (*θ̂*_{e} − *t**θ̂*_{e}) and (_{e} − *t*_{e}) and the changes Δ*θ̂*_{e} that develop level by level. Even apart from the unrealistic changes in the uppermost layers of CCM3, this need is evidenced by considering the following 10-day changes of *θ̂*_{e} in the lower troposphere where the atmospheric water vapor is maximized. See the second column from the right in Tables 2a–d, which presents the 10-day changes Δ*θ̂*_{e} layer by layer relative to an initial distribution, which was common to all four CCM simulations. In the lowest three layers of the CCM simulations, the largest 10-day decreases of −26.74, −20.64, and −12.84 K occurred in the 992H, 970H, and 929H levels of the CCM2 all Eulerian spectral simulation. Corresponding decreases of −20.12, −15.77, and −8.34 K occurred in the CCM3 and CCM2/3 mixed simulations, while in the CCM3 all SLT the changes were −12.47, −7.07, and −0.21 K. Decreases of *θ̂*_{e} in the lowest layers are expected over the 10-day simulations from upward transport of water vapor and sensible energy in regions where latent heat release has occurred through baroclinic processes. Still, within the lowest layers the outlier of the results among the four CCM simulations is from CCM all SLT, the underlying reason of which may be traced to the constrained boundary conditions utilized with the Hermite interpolants. At the same time, the CCM all SLT biases of −2.93, −1.76, and −1.39 K at the same levels are less than the differences for CCM3 and CCM2, but not for CCM2/3 mixed with values of 1.82, −0.05, and −0.26 K. While the implications of these differences bear closer examination, the key point here from these comparisons is that there is a wide range of results among the CCM simulations that lead to systematic differences in the vertical advection of energy and entropy.

Now compare the global sum of squares *S*_{G}(*δ*) in Tables 1a–d from the four CCM model simulations at days 2.5, 5.0, 7.5, and 10. Inspection of the statistics for day 10 reveals that the sum of squares range from a low of 4.08 K^{2} (2.02 K) for the CCM3 all SLT to a high of 233.24 K^{2} (15.27 K) from CCM3 (values in parenthesis are the rms difference expressed in degrees). The corresponding statistics of 28.00 K^{2} (5.29 K) for CCM2/3 mixed and 27.98 K^{2} (5.29 K) for CCM2 all Eulerian spectral are remarkably similar to each other. The difference between the two is that the simulation of water vapor and cloud water was by SLT numerics in CCM2/3 and by Eulerian spectral numerics in CCM2. At the same time, the sum of squares for *S*_{G}(*δ̂**S*_{G}(^{2} (1.46 K) and 15.03 K^{2} (3.88 K), respectively, from the CCM2 all Eulerian spectral are larger than the corresponding values of 0.09 K^{2} (0.30 K) and 0.03 K^{2} (0.16 K) from the CCM2/3 mixed simulation. Overall, the large differences of globally averaged statistics among the four CCM simulations reflects the uncertainties that still remain in the numerical representation of nonlinear transport processes.

The primary reason for CCM3's large global sum of squares *S*_{G}(*δ*) is *S*_{G}(*δ̂*^{2} (13.99 K), being an order of magnitude larger than corresponding statistics for the other three simulations. This large value for *S*_{G}(*δ̂**θ* determined by the Eulerian spectral advection of enthalpy and the corresponding *tθ* implicitly determined by semi-Lagrangian transport of *tθ*_{e}. Compare statistics for *S*_{s}(*δ**) in Tables 2a–d for the upper five layers 99H through 5H. The result is that the implied trajectories of *θ*_{e} and *tθ*_{e} as, respectively, determined by Eulerian spectral numerics and semi-Lagrangian numerics are quite different. Remember that the condition *θ̇*_{e} equals zero (where the overdot signifies the time rate of change) implies that along the trajectory **r**(*t*) to **r**(*t* + Δ*t*) that both *θ*_{e}[**r**(*t* + Δ*t*)] and *tθ*_{e}[**r**(*t* + Δ*t*)] equals *θ*_{e}[**r**(*t*_{o})]. Thus departures imply different trajectories, and subgrid-scale distributions even though the differences here stem from the inadequacies of numerics.

In contrast, the CCM2/3 mixed simulation in which enthalpy and *tθ*_{e} were simulated by Eulerian spectral numerics and water substances by semi-Lagrangian transport dramatically reduced *S*_{G}(*δ̂*^{2} (0.30 K). Also see *S*_{s}(*δ**) in Table 2b. A comparison of the vertical distributions of *S*_{s}(*δ**) and *δ̂*_{s} layer by layer between CCM3 and CCM2/3 in Tables 2a and 2b shows that this reduction primarily occurred in the upper and lower layers. Here the implied trajectories for *θ*_{e} and *tθ*_{e} at day 10 from CCM2/3 dealing with the systematic component of vertical advection are in relatively close correspondence.

Now note that the magnitudes of *S*_{G}(*δ**) of 37.45 K^{2} (6.12 K) for the CCM3 simulation and 27.88 K^{2} (5.28 K) for the CCM2/3 mixed simulation as global quadratic measures of deviation differences in Tables 1a and 1b are similar. In contrast, note for *S*_{G}(*δ**) in Tables 1c and 1d that lesser values of 10.83 K^{2} (3.29 K) and 3.41 K^{2} (1.85 K) are, respectively, provided by the CCM2 all Eulerian spectral and CCM3 all SLT simulations. The larger values of the squared deviations for CCM3 and CCM2/3 stem from the scatter that develops in the lower troposphere from the differences in the advection of enthalpy by Eulerian spectral methods and of water substances by semi-Lagrangian transport resulting from the failure to ensure reversibility in the low troposphere with its abundance of water vapor. The contrast of these results point out the reduction of squared differences when both *θ*_{e} and *tθ*_{e} are determined by CCM3 semi-Lagrangian numerics and then also CCM2 all Eulerian spectral numerics in that the source of differences stemming from the lack of common numerics for transport is eliminated. Still, in the case of CCM3 all SLT, the reduction of *S*_{G}(*δ̂**θ̂*_{e} and *t**θ̂*_{e} over the 10-day period due to the presence of common biases that develop near the tropopause. See the vertical distribution of Δ*θ̂*_{e} at day 10 in Table 2d with values of −70.43 and −29.21 K that are induced by the same reasons that cause the large positive biased *δ̂**θ̂*_{e} nor *t**θ̂*_{e} evolve in accord with acceptable thermodynamic principles of reversibility.

A comparison of results for *S*_{G}(*δ**) for all four time periods in Tables 1a–d indicates the greater accuracy for the CCM2 all Eulerian spectral except for the CCM3 all SLT at day 10. This greater accuracy of the CCM2 all Eulerian spectral simulation for *S*_{G}(*δ**), however, decreases with time due to the growth of differences by ringing involving orography. See Figs. 1c and 2c.

Now consider *S*_{G}(*δ**) plus *S*_{G}(*δ̂*^{2} (0.63 K), 1.08 K^{2} (1.04 K), 3.19 K^{2} (1.79 K), and 12.95 K^{2} (3.60 K) for days 2.5, 5.0, 7.5, and 10.0, respectively. Here the doubling of the variance at each time period indicates that the growth is exponential over each time interval of 2.5 days. In contrast the sum for the CCM3 all SLT simulation for corresponding times in Table 1d is 3.69 K^{2} (1.92 K), 4.16 K^{2} (2.04 K), 3.93 K^{2} (1.98 K), and 4.05 K^{2} (2.01 K). Except for the last time period, the statistics indicate greater accuracy of the deviation sum of squares for the CCM2 all Eulerian spectral simulation than for the CCM3 all SLT simulation. In the CCM3 all SLT simulation, the sum *S*_{G}(*δ**) and *S*_{G}(*δ̂*

Now contrast the larger global sum of squares *S*_{G}(*S*_{G}(^{2} (0.97 K), 3.73 K^{2} (1.93 K), 8.36 K^{2} (2.89 K), and 15.03 K^{2} (3.88 K), again reveals exponential increase. The values for *S*_{G}(^{2} (0.05 K), 0.00 K^{2} (0.10 K), 0.02 K^{2} (0.14 K), and 0.03 K^{2} (0.16 K), while the values for the CCM3 all SLT simulation in Table 1d are 0.04 K^{2} (0.19 K), 0.04 K^{2} (0.20 K), 0.03 K^{2} (0.17 K), and 0.03 K^{2} (0.16 K), respectively. The uniformity of these values indicates equilibration globally and suggests that the numerics for these two models do not lead to any global mean source or sink of moist entropy above that expected from the random nature of numerical inaccuracies.

### c. The global sum of squares and its components for the UW θ–σ model

Table 3a presents entries for the global sum of squares, its three components, and the global averages _{e}, and Δ_{e} for the four time periods for the UW *θ*–*σ* model. Table 3b presents like statistics level by level for day 10, which correspond with the ones for CCM in Table 2. A comparison of the globally integrated sum of squares in Tables 1a–d and Table 3a reveals *S*_{G}(*δ*) at day 10 is 1.05 K^{2} (1.03 K) in the *θ*–*σ* model as compared with the next closest value of 4.08 K^{2} (2.02 K) from the CCM3 all SLT followed by much larger values of 27.98 K^{2} (5.29 K), 28.00 K^{2} (5.29 K), and 233.24 K^{2} (15.27 K) for the CCM2 all Eulerian spectral, CCM2/3 mixed, and CCM3, respectively. Likewise the UW *θ*–*σ* deviation sum of squares *S*_{G}(*δ**) with a value of 0.70 K^{2} (0.84 K) at day 10 is less than the values of 3.41 K^{2} (1.85 K), 10.83 K^{2} (3.29 K), 27.88 K^{2} (5.28 K), and 37.45 K (6.12 K) from the CCM3 all SLT, CCM2 all Eulerian spectral, CCM2/3 mixed, and CCM3, respectively. However, now note that the value of 0.09 K^{2} (0.30 K) for *S*_{G}(*δ̂*^{2} (0.48 K) for the UW *θ*–*σ* model while all other CCM simulations are larger. At the same time the values for *S*_{G}(^{2} (0.15 K) from the CCM3 default, and 0.03 K^{2} (0.16 K) from the CCM2/3 mixed as well as CCM3 all SLT simulations, respectively, are smaller than the corresponding value of 0.13 K^{2} (0.35 K) for *S*_{G}(*δ̂**θ*–*σ* model. The slightly larger values for the UW *θ*–*σ* model stem from the vertical bias associated with the transport through the *θ*–*σ* interface where grid points emerge and submerge and where interpolation involving noncentered differences is nominally first-order accurate. Here, reversibility is not realized from an inability to properly proportion mass, potential temperature, and water substances in the determination of *θ*_{e} relative to *tθ*_{e} even though conservation of individual properties is satisfied everywhere from the flux formulation for transport. Still the area-average deviation sum of squares *S*_{G}(*δ̂**θ*–*σ* model remains relatively steady and reflects the equilibration of the triangular distribution in differences in Fig. 4d. See the vertical distribution of bias in Fig. 4e.

Preliminary results have been obtained with a generalized UW hybrid *θ*–*η* model with a smooth transition from a coordinate surface (*η*) contiguous with the earth's surface to an isentropic surface that lies above the earth's surface everywhere (Johnson and Yuan 1998), which eliminates the systematic differences that develop from the discrete interface in the UW *θ*–*σ* model. The global values from this model for *S*_{G}(*δ**), *S*_{G}(*δ̂**S*_{G}(*S*_{G}(*δ*) at day 10 are 0.122 K^{2} (0.35 K), 0.009 K^{2} (0.10 K), 0.026 K^{2} (0.16 K), and 0.157 K^{2} (0.40 K). The conclusion is that while the global sum of squares from the UW *θ*–*σ* model are minimal relative to the global sum of squares to all four CCM simulations, the systematic differences from the interface does inflate *S*_{G}(*δ̂**S*_{G}(*θ*–*η* hybrid isentropic model.

## 6. The optimum triangular distribution for the differences (*θ*_{e} − *tθ*_{e})

Within a strategy of assessment of numerical accuracies that includes minimizing the pure error sums of squares, attention will now be directed to several forms of the empirical distributions of pure error differences in search of the optimum form(s) expected from the nature of truncation errors.

First of all there are the considerations of the Central Limit Theorem and the numerous degrees of freedom involved in the computation of the pure error difference in relation to the numerical simulation of *θ*_{e} and *tθ*_{e}. The tendency of the trace *tθ*_{e} is a linear combination of three terms involving nonlinear advection of *θ*_{e}, while the computation of *θ*_{e} is a function of the tendencies of mass, enthalpy (or dry entropy in the UW *θ*–*σ* model), water vapor, and cloud water. Each of the latter four is determined by a linear combination of three terms involving nonlinear advection of the appropriate property. In the tendency equations for water vapor and cloud water, there is also the impact of the Lagrangian sources associated with condensation/evaporation in which the source of one is the sink of the other. The implication of the numerous degrees of freedom involved in the linear combination of terms from the Central Limit Theorem (Cramér 1946) is that a normal distribution of the pure error difference is the most plausible distribution. While each of the terms involving advection (or transport depending on the form of the governing equations) is nonlinear, presumably within each linear term the product of a velocity component and a scalar derivative simply provides for an increase of the randomness needed to ensure a central limit effect.

Inspection of Figs. 3 and 4 suggests three candidate parametric distributions for the pure error differences: a normal distribution for days 5.0, 7.5, and 10.0 for CCM3; a double exponential distribution for CCM2/3; and triangle distributions at days 2.5 and 5.0 for CCM2 all Eulerian spectral and for all four times for the UW *θ*–*σ* model. Clearly the overall comparison reveals that contrary to inference involving the Central Limit Theorem, the distributions for all except CCM3 assume other forms.

While there is no means to directly study the relation of the three different distributions with their parent truncation errors for *θ*_{e} and *tθ*_{e} apart from inspection of the pure differences, three interesting questions emerge. Given the formulation of the set of governing equations for the model, the resolution of the numerics, and the recognition that truncation errors have both bias and random components, why are the relative frequency distributions for all simulations other than by CCM3 more like double exponential and/or triangular than normal? Does the nearness of the empirical distribution to a triangular distribution with its lesser variance provide information concerning optimum accuracy in the simulation of reversibility? Is there a theoretical basis to support a postulate that the triangle distribution is the proper distribution to represent the optimum accuracy attainable in the simulation of reversibility?

Now consider two arbitrary random independent distributions of a property and its trace and then the distribution of their difference in conjunction with the aim to minimize the variance of their differences. From the criterion of minimization of the variance of the difference, one quickly notes that the variance of the difference is minimized if the property and its trace have common distributions, for example, both normal, both double exponential, etc. Common distributions ensure an equal relative frequency for each class interval within the two distributions. As such, in the determination of the pure error difference even though the differences are taken randomly, this commonality maximizes the number of occurrences in which zeros are determined for the pure error difference. The sharp apexes that appear in all the distributions except for CCM3 results from the commonality of the distributions of the truncation errors for a property and its trace. Now contrast the tallness of the peak and its nearer relative frequencies with the length and thickness of tails of the empirical distributions. The former signifies a larger number of small pure error differences with increased accuracy as opposed to the latter, which signifies a larger number of larger differences in either *θ*_{e} and/or *tθ*_{e}. Thus within a given frequency distribution, the more pronounced the tails and associated area, the less pronounced the peak and area under the peak.

Now note for CCM3 and CCM2/3 in Fig. 3 that with time the vertical extent of their peaks decrease and the lateral extent and height of the tails of their distributions increase, indicating amplification of differing systematic truncation errors for *θ*_{e} and *tθ*_{e}. The unbounded nature of these differences is most evident in Figs. 5 and 6 of Part I, where the ordinate is scaled logarithmically to reveal the very large truncation errors in excess of 100 K. In the case of CCM3, the increasing normality of the relative frequency distribution is actually due to the randomness mentioned earlier that is implicitly introduced within different forms of nonlinear advection involved with Eulerian spectral numerics for enthalpy and semi-Lagrangian numerics involving Hermite cubic interpolants for *tθ*_{e}. In essence, the differences in simulations that stem from Eulerian spectral numerics for enthalpy and semi-Lagrangian cubic Hermite interpolants for equivalent potential temperature preclude the commonality of the implicit distributions of truncation errors for *θ*_{e} and for *tθ*_{e}.

The double exponential distribution for CCM2/3 is now attributed to a combination of two effects. With the use of Eulerian spectral numerics for the enthalpy and for the trace *θ*_{e}, common numerics governed by dry-adiabatic processes were realized throughout the upper half of the atmosphere due to the minimal amounts of water substances. This accounts for the sharp apex. In contrast, with the larger amounts of water substances in the lower troposphere being transported by semi-Lagrangian numerics and enthalpy and *tθ*_{e} by Eulerian spectral numerics, the forms of the empirical frequency distributions for *θ*_{e} and *tθ*_{e} diverged. Thus the tails for CCM2/3 develop from large and different truncation errors that amplify with time from the differences of numerics employed.

With these considerations in mind, an important characteristic of the triangle distribution is that the random component of truncation errors for both *θ*_{e} and *tθ*_{e} are bounded within a finite interval as opposed to the normal and double exponential distributions. Thus in the consideration of the three different distributions and implied accuracy, the triangle form of the relative frequency distribution of pure error differences distribution with its sharp peak and lack of tails becomes the candidate that represents greater accuracy.

*a*≤

*x*≤

*a*), the sum of which will generate a triangle distribution (Cramér 1946, 244–246; Weatherburn 1957, p. 44) expressed by

*a*

^{2}/3 is equal to the sum of the variances of the two rectangular distributions, each being equal to

*a*

^{2}/3. The difference of two independent random variates each uniformly distributed within the range (−

*a*≤

*x*≤

*a*) about a common mean true value will be centered at

*x*equal to zero and will likewise generate a corresponding triangular distribution with variance equal to 2/3

*a*

^{2}. The assumption of independence removes any differences between the two distributions as to whether the two random variates are summed or differenced, since the effect of a positive or negative covariance of the random variates on the triangular distribution are removed. See Fig. 5 (after Cramér 1946).

The theoretical development set forth in the appendix employs the uniqueness between a probability density function and its moment generating function in substantiating that the triangle distribution is only realized from either the sum or differences of independent random variates from two uniform distributions. The two distributions must be defined over a common interval with corresponding means and variances. Now given a triangle distribution determined from either the sum or difference of two independent random variates, the development in the appendix also establishes the converse that each of the two independent random variates must be distributed uniformly over a common interval. This conclusion is based on the condition that the moment-generating function of a triangle being equal to [(sinh *at*)/*at*]^{2} is the unique quadratic product of [(sinh *at*)/*at*], which in turn is the unique moment-generating function of a uniform distribution of a random variate. As such, apart from the limitations of the common systematic components of truncation error, theory substantiates the proposition that the nearer the empirical distribution of the pure error differences is to a triangle distribution, the nearer the distributions of truncation errors of *θ*_{e} and of *tθ*_{e} are to corresponding independent distributions of uniform, random variates defined over a common interval. Clearly, if common uniform empirical distributions were actually realized over a common interval, the differences from systematic truncation and their causes within the numerics of the model would have been eliminated. Important considerations for bounding the error distributions of the property and the trace are to ensure proper conservation with respect to transport throughout the global domain and employ numerics that ensure reversibility.

Now consider the results for the UW *θ*–*σ* model in Fig. 4 showing the minimal scatter in the bivariate scatter distribution in Fig. 4c, the form of relative frequency distribution near zero in the region ±0.5 K and the larger spread near the base of the triangle with little change over the 10-day period. As was pointed out in Part I, the relative frequency distribution of differences for the UW *θ*–*σ* model equilibrated with neither the bias nor random components increasing during the 10-day simulation.

Except for the relatively small tails, the profiles of the relative frequency distribution at the four different times in Fig. 4d for the UW *θ*–*σ* model tend to be triangular in form and peak markedly in representing the bulk of approximately 10^{5} gridpoint differences that reside mostly in the isentropic domain. An inspection of the empirical distributions layer by layer reveal that the larger differences stem from the transport within the sigma-coordinate planetary boundary layer and the exchange across the interface between sigma and isentropic domains of the model. As such, the relative frequency distribution of the domain as a whole is actually a profile that is a combination of a triangular distribution for the portion of the model in which isentropic layers do not intersect the sigma domain of the model and normal and double exponential distributions for layers that intersect the sigma domain.

Consider now the range of values for the variances level by level as represented by *S*_{s}(*δ**) in Table 3b. The values for the 324 K and underlying layers are distinctly larger than the values for 332 K and overlying layers, thus delineating variances associated with the pure isentropic portion of the UW *θ*–*σ* model from the portion of the model involving the sigma domain. With appropriate weighting of the squared differences by mass for the free isentropic domain 332 K and above, the estimates of *σ* and *σ*^{2} from Table 3b are 0.20 K and 0.0397 K^{2}, respectively, and the corresponding peak value of a triangle distribution would be 2.05 K^{−1}.

Inspection of (28) and (29) reveals that the peak of the triangular distribution equals (2*a*)^{−1}. Now utilizing the peak value of 2.7 K^{−1} from Fig. 4d and the relation between the peak value and the variance as determined by *a* equal to 0.185, estimates of *σ* and *σ*^{2} are equal to 0.151 K and 0.0228 K^{2}, respectively. The general agreement of the variances estimated statistically from the analysis of variance for the isentropic layers 332 K and higher and the ones based on the peak values of the distribution are noteworthy. As such, the empirical results do not contradict the inference from the appendix that with minimization of the variances of the pure error differences, a model's numerical accuracy in the simulation of reversibility is tending toward its optimum. The overall considerations of the empirical relative frequency distribution layer by layer and globally deserve additional study.

## 7. Global reversibility

### a. Reversibility integral constraints in relation to energy and entropy

The underlying reasons for the systematic differences in CCM2's all Eulerian spectral simulation and also the “hidden” failure of the Hermite interpolants in the CCM3 all SLT simulation to ensure reversibility will now be addressed by considering global constraints in relation to energy and entropy. Nearly all studies of systematic errors in weather and climate models include a focus on either the dry static energy or enthalpy balance. While such assessments of the balance of energies or enthalpy are important, the focus in this study from both theoretical and pragmatic considerations is on potential temperature and/or equivalent potential temperature in relation to reversibility. In actual fact, both the potential and equivalent potential temperatures are closely related in the sense that apart from the very limited regions where phase changes occur, cyclic processes in the free atmosphere for the most part involve the joint conservation of potential and equivalent potential temperature. Note that in atmospheric simulations carried out under dry-adiabatic conditions in the absence of work and net energy fluxes at the upper and lower boundaries of the atmosphere, the global integral of kinetic, internal, and gravitational energies would be conserved, while *θ* is conserved everywhere. Under the fully reversible moist-adiabatic simulations carried out in this study, *θ*_{e} is conserved everywhere while the latent energy of water substances needs to be added to the other three forms of energy in ascertaining global conservation.

### b. Reversibility in relation to CCM2's all Eulerian spectral systematic biased enthalpy

In order to ascertain the thermodynamic source of the bias that developed from Eulerian spectral numerics in CCM2 relative to CCM3, the compounding difficulties of water substances and diabatic heating from condensation/evaporation were eliminated for both CCM2 and CCM3 in follow-on 10-day simulations. Under the dry-adiabatic conditions both global averages *t**t**tθ* was simulated by both Eulerian spectral and SLT numerics using CCM2, but only by SLT numerics using CCM3 since the capability for simulating a trace constituent by Eulerian spectral numerics was not carried forward to CCM3.

Given that the initial global average _{ES} and *t*_{SLT} in CCM3, and no change for *t*_{SLT} in CCM2. In contrast, CCM2's dry Eulerian spectral numerics with its 10-day decrease of −5.02 K for _{ES} fails to ensure integral conservation. Although not presented, the profile of the differences level by level for CCM2 reveals a decrease of *θ̂*_{ES} from day 0 to 10 which corresponds closely with CCM2's negatively biased profile (*θ̂*_{e} − *t**θ̂*_{e}) in Fig. 3c3. This correspondence of differences in combination with the CCM2's global decrease of −5.02 K for _{ES} in Table 4 and −5.08 for _{e} in Table 1c combined with CCM3's near conservation with an increase of only 0.07 K for _{ES} provides firm evidence that the bias develops from CCM2's numerics involving the enthalpy equation itself. Furthermore, the lack of a large systematic difference in CCM3's simulation of the 10-day global change in Δ_{e} of 0.16 K by Eulerian spectral numerics (Table 1a) relative to CCM2's corresponding change of −5.08 K (Table 1c) indicates that the CCM2's systematic bias involving the enthalpy equation has been eliminated in CCM3.

The last row of Table 4 estimates the mean spurious rate of change of dry static energy over the 10 days in W m^{−2} from the relation globally that *c*_{p}*θ* is approximately equal to the enthalpy *c*_{p}*T* plus the geopotential energy *gz* (Berson 1961; Newton 1972). The rate of change of the global integral of the enthalpy *c*_{p}*T* is 0.78 times the tabular values for the rate of change of dry static energy.

In discussing the modification of the enthalpy equation in CCM3 versus CCM2, Kiehl et al. (1996) point to changes made to ensure consistency between the substantial derivative of pressure *ω* as determined by discrete forms of the enthalpy and mass continuity equations. Clearly changes in the determination of *ω* relative to hydrostatic mass continuity impacts vertical advection of enthalpy and also energy transformations as represented by the product of *ω* and specific volume *α.* With the latter being equal to the work of expansion *pdα*/*dt* minus the product of the specific gas constant *R* times *dT*/*dt,* any inaccuracies here impact the transfer of energy by work and thus transformations between internal, geopotential, and kinetic energies; the combination of which precludes appropriate conservation of entropy and the accurate simulation of reversibility that is implicitly associated with adiabatic expansion and contraction.

In the extensive comparison of simulated climate states, various reasons are offered for the improvements achieved by the modifications made in the development of CCM3 from CCM2 (e.g., Gleckler et al. 1995; Hack 1998a,b; Hack et al. 1998; Kiehl et al. 1998a,b). Nearly all of the reasons offered for the improvements involve changes in the parameterizations of subgrid-scale latent and sensible heating and the radiative flux of energy. In view of many changes made among the parameterization of the various components of energy exchange, the isolation of a single component for explaining the improvements did not materialize. However, the tabular results of Table 4, which contrast the 10-day changes of −5.02 and +0.07 K for Δ_{ES} for CCM2 and CCM3, respectively, suggest that a key underlying reason for elimination of flux adjustment at the earth atmospheric interface in CCM3 relative to CCM2 in coupled climate simulations stemmed from modifications in the numerical calculation of *ω* and in turn *ωα.* In this regard, Hack's analyses (1998a,b) pointing toward difficulties with the parameterization of the hydrologic processes with ensuing effects on the radiative fluxes of energy are the most informative.

Now consider the implications of CCM2's dry thermodynamics inducing a cold global tropospheric bias of −5.02 K for ^{−1}. A mean spurious cooling rate of approximately 0.5 K day^{−1} for the entire troposphere estimated by the bias in the profile of entropy is unrealistically large given that the mean infrared atmospheric cooling rate of the global atmosphere is on the order of 2.0 to 2.5 K day^{−1}. This spurious cooling rate when coupled with slowly varying ocean temperatures by virtue of its high heat capacity will induce anomalously intense moist convection.

In terms of a nonconservation of energy, the decrease of 5.02 K for *θ*_{ES} over 10 days corresponds with a spurious global mean enthalpy sink of 45.73 W m^{−2}. Hack's (1998b) globally annual average latent heat fluxes from his Table 1 for the extended 15-yr integration are 104.04 W m^{−2} for CCM2, 89.97 W m^{−2} for CCM3, and 78.0 W m^{−2} from observations. Thus the excesses of latent heat flux within CCM2 compared to CCM3 and observations are 14.07 and 26.04 W m^{−2}, respectively. While the magnitudes of the comparisons differ substantially, the spurious enthalpy sink is in agreement with the sense of the excess flux as emphasized by Hack (1998a,b) and Hack et al. (1998). The spurious enthalpy sink also accounts for the necessity of flux adjustment at the ocean atmosphere interface in CCM2, which was alleviated in the CCM3 coupled model simulations.

### c. CCM3's SLT nonconservation of entropy

At this point, it is important to recognize the limitations of the strategy of using the pure error sum of squares based on the differences of *θ*_{e} and *tθ*_{e} to assess numerical accuracies where bias errors are common in calculating the transport or advection of *θ*_{e} and *tθ*_{e}. This is the case for the CCM3 all SLT where the Hermite cubic functions are used for both the determination of *tθ*_{e} and all the properties that determine *θ*_{e}.

Contrast the CCM3 all SLT extreme 10-day changes as given for Δ*θ̂*_{e} in Table 2d in the upper layers versus the minimal global change as given by Δ*θ̂*_{e} in Table 1d. This situation results from offsetting changes in the lower and upper atmosphere. Although this compensation is indicative of a spurious vertical mixing of the moist entropy, it is the potential temperature as the dry component of entropy, which is in effect being mixed aphysically by the SLT numerics since there is negligible water vapor in the uppermost levels. The mixing of the dry entropy directly impacts the reversible component of total energy, that is, the sum of available potential plus kinetic energy (Lorenz 1955, 1960; Johnson 1989, 2000).

In order to establish the basis of this result, consider that *θ̇*_{e} vanishes everywhere under reversibility of moist thermodynamics. As such, each of the global integrals of the time rate of change of (*θ*_{e}), (*θ*_{e})^{2}, and (_{e})^{2} will be invariant. Now note the invariance of the global integral of (*θ*^{**}_{e}^{2} follows from quadratic expansion of *θ*^{2}_{e}_{e})^{2} and (*θ*^{**}_{e}^{2} where in this subsection *θ*^{**}_{e}*θ*_{e} from the global mean _{e}. Since reversibility of moist processes requires (*θ*_{e})^{2} and (^{2} to be invariant, (*θ*^{**}_{e}^{2} must also be invariant.

Table 5 presents the initial and day 10 normalized sum of squares *S*_{G}(*θ*^{**}_{e}*S*_{G}(_{e}), and *S*_{G}(*θ*_{e}) representing the global integrals of (*θ*^{**}_{e}^{2}, (_{e})^{2}, and (*θ*_{e})^{2} where *S*_{G}(*θ*^{**}_{e}*S*_{G}(_{e}) sum to *S*_{G}(*θ*_{e}). A comparison of the quantities in the parenthesis expressed as rms values of temperature at day 10 with initial values provides a measure of the uncertainty of *θ*_{e} change due to spurious numerics.

Three points from Table 5 are particularly noteworthy. Globally the quadratics of *θ*_{e} and its components from CCM3 and CCM2/3 simulations are conserved to a high degree of accuracy. CCM2 by virtue of its dry thermodynamic bias discussed previously fails to conserve any of the quadratics. The CCM3 all SLT, however, conserves the quadratic of _{e} to a high degree of accuracy, but fails to conserve the global quadratics of *θ*^{**}_{e}*θ*_{e}.

From the perspective of reversible component of total energy, and also the numerics of moist thermodynamics (Johnson 1989, 2000), the failure of CCM3 all SLT to conserve *S*_{G}(*θ*^{**}_{e}*θ*_{e}. Since like analysis of the quadratic components of *θ*^{2} yields similar results for dry-adiabatic simulations, these large decreases in the global integral of (*θ*^{**}_{e}^{2} in effect stem from a systematic spurious vertical mixing of the dry entropy component throughout the atmosphere that increases the entropy of the troposphere while decreasing the entropy of the stratosphere. As a first approximation the percentage decreases of the quadratic energy component is given by one minus the ratio of *S*_{G}(*θ*^{*}_{e}*S*_{G}(*θ*^{*}_{e}

At this point it is important to again emphasize that global values of the pure error sum of squares *S*_{G}(*δ*) by themselves such as the CCM3 all semi-Lagrangian value of 4.08 K^{2} (2.02 K) may be misleading. However, the strategies to assess numerics based on pure error difference in combination with an examination of a model's capability to properly conserve global integrals of *θ* and *θ*_{e} and their quadratics under dry- and moist-adiabatic conditions provide for robust analyses.

## 8. The bias in relation to uncertainty

Inspection of the bivariate scatter distributions, the relative frequency distributions, and the vertical profiles of the mean average differences clearly suggests unquestionable biases; however, at the same time the results for the CCM2/3 mixed experiment and the upper isentropic layers of the UW *θ*–*σ* model indicate minimal differences. Recall that if the differences vanish within each layer, the bias from vertical advection would vanish also. For this reason and also for the sake of simplification and brevity, the focus in this section is limited to considerations of the systematic differences that develop within each layer apart from the related impact of bias on the numerics of vertical advection. The issue then is to seek a means to assess whether the average deviation *δ̂**δ̂*

### a. Underlying distributions of uncertainty

Although a triangular distribution has been set forth as the appropriate distribution for error differences, the appropriate distribution for the mean of the random component of numerical differences as determined by the Central Limit Theorem (Cramér 1946) is the normal distribution. In accord with the Central Limit Theorem, where the degrees of freedom are large and the errors are independent and random all having the same distribution, the distribution of the average of a field tends to the normal distribution given by [*μ,* *σ*^{2}/*N*]. The condition that the linear combination of errors needs to come from the same distribution may be relaxed provided that not any one distribution dominates in the determination of the linear combination of random errors (Cramér 1946).

*t*distribution. Now, in the limit for large

*N,*the

*t*distribution in effect becomes the normal distribution (Keeping 1962). With the number of grid points in excess of 10

^{4}within a model layer extending globally, ample justification is provided for the simplification of utilizing the normal distribution to determine uncertainty. Utilizing this simplification, the underlying condition that the true mean of

*δ̂*

*η̂*

*ζ̂*

*δ*

_{r}are independently randomly distributed with homogenous variance, the distribution of the unit normal deviate

*N**(0,1) appropriate to the distribution of the average

*δ̂*

*T*

*δ̂*

*δ̂*

*σ*

_{δ̂}

*T*(

*δ̂*

*δ̂*

*σ*

_{δ̂}

*α*) confidence limits, the absolute value of the calculated difference

*δ̂*

_{0}employing a two-tailed test must satisfy the following condition:

*δ̂*

_{0}

*T*

_{α/2}

*σ*

_{δ̂}

### b. Unbiased estimators of σ^{2}_{δ̂} and σ^{2}

^{2}

_{δ̂}

*σ*

^{2}

_{δ̂}

*σ*

^{2}from the sums of the differences for a given model layer. In the following notation utilized previously, the pure error sum of squares within the

*s*th layer is defined by

*s*for simplicity,

*S*(

*δ*) is partitioned into a systematic

*S*(

*δ̂*

*S*(

*δ**) sum of squares expressed by

*S*

*δ*

*S*

*δ̂*

*S*

*δ*

*δ*

_{r}as a linear combination of bias (

*λ*

_{r}) and random (ε

_{r}) components

*δ*

_{r}

*λ̂*

_{r}

_{r}

*λ*

^{*}

_{r}

^{*}

_{r}

^{*}

_{r}

*σ*

^{2}

_{m}

*σ*

_{mn}define the variance and covariance in accord with the usual definitions while

*m*and

*n*serve as proxy subscripts. The substitution of (42) into (37) and (39) yields

*S*(

*δ*) reduce to

*S*(

*δ*) and the results of (47) and (48), the unbiased estimators of

*σ̃*

^{2}

_{;af;er}

*σ̃*

^{2}

_{ε}

A key point of the strategy set forth earlier was the need to ensure invariant transformation of the integrals of the mean, quadratic, and higher-order moments among the different coordinate systems. However, there is a complication introduced in the assessment of the uncertainty from the variation of the incremental mass within the model layers of different coordinate systems. Note that if the fractional mass distribution *w*_{r} were all equal, that is, equal to *N*^{−1}, the expectations of (47) and (48), respectively, reduce to *σ*^{2}/*N* and *σ*^{2}[(*N* − 1)/*N*], results that are equivalent with the standard analysis of variance. Given that Σ *w*_{r} equals unity and that Σ *w*^{2}_{r}*N*^{−1}, a comparison of (47) and (48) reveals that the effect of variable mass under the assumption of independence of the random error component and homogeneous variance is to increase the expectation of *S*(*δ̂**S*(*δ**), keeping in mind that the expectation of their sum *S*(*δ*) being *σ*^{2} remains invariant. As such, the effect of variable mass is to increase the range of the confidence limits in a two-tailed test for *δ̂*

The factors Σ *w*^{2}_{r}^{−3} in Tables 2 and 3 for the CCM and UW *θ*–*σ* models, respectively. Since (1 − Σ *w*^{2}_{r}^{−3}, this factor for the most part here is irrelevant in this examination of uncertainty. As such, the major factor impacting uncertainty of *δ̂*_{r} *w*^{2}_{r}*θ*–*σ* model simulations, but not among comparisons of the four CCM simulations.

For completeness, the expected value of the three components and total of the pure error sums of squares are presented in Table 6. With substitution of uniform weighting *w*_{rs} equal to *N*^{−1} everywhere, the last column reveals that partitioning of the total normalized variance *σ*^{2}_{y}*σ*^{2}_{z}*S*_{G}(*δ**), among-group sums of squares *S*_{G}(*δ̂**S*_{G}(*δ̂*

### c. Applications and null hypothesis

*δ̂*

*δ̂*

_{o}is contained within the interval centered at zero and bounded by ±2

*σ.*As such, with the two standard deviation range 95% confidence interval for the unit normal deviate given by 1.96, the bounding interval within which the hypothesis is acceptable is expressed by

For the moment, set aside results from the UW *θ*–*σ* model. Now consider *δ̂**S*_{s}(*δ**) in Tables 2a–d and note apart from the factor 1.96 that the bounding range of *δ̂**S*(*δ**) and the factor (Σ_{r} *w*^{2}_{r}

Utilizing the tabular entry of 1.522 × 10^{−4} for Σ *w*^{2}_{r}^{−2} for multiplication of the values of [*S*(*δ**)]^{1/2} as listed in parenthesis, one determines for all layers other than where zeros are listed that the *δ̂**δ̂*

Now consider the results for *S*_{s}(*δ**), *δ̂**w*^{2}_{r}_{s} from Table 3b for the UW *θ*–*σ* model and note that the random component as estimated by *S*_{s}(*δ**) is substantially less for nearly all layers than like estimates from all the CCM simulations. As a result the bounding interval for which the hypothesis would be accepted is substantially reduced. The consequence of this is that the 260-K layer is the only layer other than ones with zero for *δ̂**S*(*δ**) and Σ *w*^{2}_{r}*δ̂*

The stringent nature of the *t* test in determining the bounding interval for the systematic difference *δ̂**w*^{2}_{r}*N* in the usual determination of the variance of a mean from a random sample, consider that the effective *N* for the factor Σ *w*^{2}_{r}^{−4} from Table 2 corresponds with a sample size of 6570. Utilizing the corresponding values for the UW *θ*–*σ* model from Table 3b, the effective *N* ranges from 281 for the lowest layer to 10 941 for the 445-K layer. Here it is interesting to note that the effective *N* of 10 941 in the upper layers of the hybrid model approaches the actual number of 13 104 for grid points. Given the relatively large variation of the effective *N*, an immediate conclusion is that the variation of mass is a relevant factor in the determination of moments and assessments of uncertainty.

Now note that with an effective *N* approximately equal to 10^{4}, the ratio of the magnitude of the bias |*δ̂*^{−2} of the standard deviation *σ̃**N* under the assumption of independence of the pure error differences *δ*_{m} and *δ*_{n} for all *m* unequal to *n* are extreme. This extreme requirement suggests that the assumption of independence in estimating the variance of the difference *δ̂*

There is at least one factor that implies that the random error variance *σ̃*^{2} may be underestimated. This factor stems from the nature of the atmospheric medium and also numerics in that there will be spatial and temporal correlation of errors induced both vertically and horizontally from the flux form of finite differencing of transport processes and the constraints that enter from ensuring conservation in that the discrete values include bias and random components of properties. Thus in all likelihood the assumption that *σ*_{mn} equals zero in (43) for all *m* and *n* is not satisfied everywhere, even if the differences reduce to the random error component of numerics. Positive correlation of *δ* spatially from numerics simply means that the variance for *δ̂**w*^{2}_{r}

The fact that the empirical relative frequency distributions tend toward the form of a triangle and thus uniformity of the truncation errors for *θ*_{e} and *tθ*_{e} is also indicative of internal constraints from the flux form of differencing. Note that the flux form of differencing entails a linear combination of terms when all components of advection are considered. As such, one would expect the empirical distributions to tend more toward a normal distribution in accord with the Central Limit Theorem. However, given the constraints of local conservation embodied in the flux form of the transport equations, the truncation errors for both the property *θ*_{e} and its trace are included in the numerics that ensure conservation. Thus in this case the Central Limit Theorem is not applicable. In contrast, for example, the form of CCM3's relative frequency distribution tends strongly toward a normal distribution, particularly by day 10. In this simulation with Eulerian spectral numerics for the enthalpy and semi-Lagrangian for *q*_{υ}, *q*_{c}, and *tθ*_{e}, the constraints inherent in the flux formulation of the governing equations are missing.

These considerations, however, are beyond the scope of this study. It is sufficient to state here that the strategy set forth provides meaningful statistics to assess bias and random components of numerical algorithms that are readily compared among different models, keeping in mind that inspection of the differences in the zonal cross sections and also temporal nature of the relative frequency and bivariate scatter distributions are also important.

## 9. Summary, further applications, and implications

Given the relevance of water in its various forms and the difficulties inherent in simulating its exchange, there is urgent need for the critical assessment of the numerical accuracy of weather and climate models in relation to the simulation of hydrological processes and reversibility internal to the atmosphere. Since three-fourths of the atmosphere lies over oceans that are essentially devoid of accurate observations of water substances, coupled with limited capabilities over land, truth concerning the hydrological state of the atmosphere can never be assessed directly. For the most part, the hydrological state of the atmosphere in weather and climate prediction is determined by the model's simulation. Here, the nonlinearity of the governing equations combined with the intrinsic uncertainty of limited observational sampling of water substances in time and space preclude a direct determination of the biases and random errors that develop in prediction models. Ultimately, there is a critical need for the development of global models for weather and climate that accurately simulate all of the processes intrinsic to water in its various forms including precipitation, clouds, reevaporation of clouds, cloud radiative feedback processes, the hygroscopic nature of aerosols, scavenging, etc.

Prime objectives of this study concerning the simulation of reversibility were to set forth a strategy for an in-depth assessment of numerical accuracies, determine the various components of inaccuracies, and ascertain statistical measures of uncertainty. The underlying intrinsic consideration in atmospheric prediction is accuracy. The physical and numerical considerations of entropy and its exchange provide insight into these matters.

Key outcomes ascertained by this analysis of the pure error sum of squares are the following.

The partitioning of the global sum of squares into three components based on the traditional analysis of within- and among-group sum of squares provided unusual insight into different components of error associated with numerics of different models. The structure of the pure error differences associated with horizontal and vertical exchange of

*θ*_{e}and also global nonconservation revealed large differences among the different models.Large differences were associated with numerical inconsistencies in simulating

*θ*_{e}and also its dry component*θ*in model simulations where the enthalpy is predicted by Eulerian spectral numerics and water vapor by semi-Lagrangian numerics. The extreme biases in CCM2 and CCM3 stemmed from the condition that the proxy entropy transport by semi-Lagrangian numerics is inconsistent with the implicit dry entropy transport as determined by Eulerian spectral prediction of enthalpy near the upper and lower boundaries of the models and throughout the stratosphere and the region of the tropopause.Inspection of the zonal distribution of differences in all CCM simulations reveals the plus–minus signature of the truncation errors stemming from the differential advection within amplifying baroclinic waves of extratropical latitudes, except for the CCM2 all Eulerian spectral numerics.

Within the baroclinic waves of extratropical latitudes and particularly in the stratosphere, the magnitude of the pure error differences of the moist entropy that developed between CCM's Eulerian spectral and semi-Lagrangian numerics in dealing with reversible transport processes when divided by the 10-day period of the simulation are in excess of the magnitude of several components of diabatic heating. The source of the truncation errors stems largely from the vertical component of differential advection.

Of all the CCM simulations at day 7.5, the component of the global sum of squares

*S*_{G}(*δ**) from the CCM2 all Eulerian spectral simulation with its minimum value of 1.98 K^{2}(1.41 K) was the most accurate; however, the increase of its pure error difference as well as all three components was exponential with a doubling rate of 2.5 days. By day 10, the global deviation sum of squares exceeded that of the CCM3 all SLT experiment.The statistics for the global sum of squares

*S*_{G}( ) and also the nonconservation globally of moist entropy as determined by Δ*δ̂**θ̂*_{e}readily identified a pervasive numerical bias in CCM2's simulation of*θ*and*θ*_{e}that stemmed from inaccuracies in the determination of*ω*and*ωα.*The ringing from orography was substantial.Temporal monitoring of the peak values and spreading of the relative frequency distributions of pure error differences clearly distinguishes situations in which numerical inaccuracies amplified and/or interacted with the true state. Over the 10 days, the distributions from the CCM3 all SLT and UW

*θ*–*σ*model simulations tended to equilibrate, but not from the CCM3, CCM2/3, and CCM2 all Eulerian spectral simulations.A basis for the application of the triangular distribution as the expected distribution of the differences

*δ*for the relative frequency distribution of the random component of the pure error sum of squares was established theoretically and demonstrated empirically. In the vicinity of small differences near*δ*equal to zero, with the exception of CCM3, the empirical frequency distributions for all simulations reveal a triangular form. Large differences with bias components induced spreading of the tails.From the condition that the moment-generating and probability density functions of a given distribution are uniquely related to each other, the theoretical development of the appendix supports the postulate that the nearer the empirical relative frequency distribution of pure error differences is to the classical triangular distribution, the closer the model's simulation is to the optimum accuracy feasible in ensuring reversibility and appropriate conservation of moist entropy. Implicit in this postulate is the result that the bias components that induce spreading of the tails are suppressed as the variance from the difference between

*θ*_{e}and*tθ*_{e}is minimized.The statistical strategy of the pure errors sum of squares fails to identify numerical inaccuracies in situations where there is a common bias in the simulation of

*θ*_{e}and*tθ*_{e}, such as occurred from the semi-Lagrangian numerics in CCM3 all SLT simulation.In the situation where a common numerical bias occurred, monitoring of the global integrals of the quadratics of potential temperature, its mean, and deviations, identified nonconservation of the quadratic of potential temperature and its deviations of equivalent potential temperature with a 14% global decrease of the latter. The globally integrated 14% decrease of quadratic of the deviations of equivalent potential temperature, which is closely related with the reversible component of total energy as the sum of available potential and kinetic energies, identifies spurious positive definite sources of entropy due to spurious mixing and thus failure to ensure reversibility.

The information gained from application of the statistical strategy in comparisons of the results from the four CCM simulations and the UW

*θ*–*σ*model attests to the robust nature of the statistical assessment developed for isolating inadequacies of models in simulating transport processes and gaining information on their relative accuracies with respect to the appropriate conservation of dry and moist entropy and the simulation of reversibility.The results of this study and also Part I that the simulation of moist and dry reversible processes by models utilizing isentropic coordinates have markedly greater accuracy than models utilizing sigma or other coordinate systems reinforces earlier results (Johnson et al. 1993; Zapotocny et al. 1996, 1997a,b; Reames and Zapotocny 1999a,b).

The findings of this study also support Johnson's (1997) results that the existence of aphysical sources of entropy from numerical mixing of energy constitutes a fundamental physical and logical basis for the “coldness of climate models.”

At this time, it is important to note that the results of the 10-day comparisons do not address the issue of superiority of either isentropic or sigma coordinates in simulating the climate state. The key issue in this study with a focus on hydrologic processes internal to the atmosphere is to ascertain numerical accuracies in simulating reversible processes involving appropriate conservation of dry and moist entropy.

As emphasized overall, the accurate simulation of the atmosphere's hydrologic cycle and reversible isentropic processes will ultimately be crucial to the successful prediction of weather and climate. The inability to conserve appropriately both *θ* and *θ*_{e} in sigma models during long range transport limits accuracies of weather and climate predictions. Models utilizing hybrid isentropic coordinates that minimize the irreversibility of numerics are viable for global modeling. However, from relative considerations, their development is in an early state of infancy. Additional development is needed to realize their expected potential.

## Acknowledgments

This research was supported by NASA under Grants NAG5-4398 and NAG5-9295 and the Department of Energy under Grants DE-FG02-92ER61439 and DE-FG02-01ER63254 and IBM through the donation of an IBM RS/6000 for climate model development and analysis. The authors gratefully acknowledge advice and thoughtful comments of an anonymous reviewer and the assistance of Judy Mohr in preparation of the manuscript. Computations were performed locally on the IBM RS/6000, at the National Energy Research Scientific Computing Center and the NASA Center for Computational Sciences.

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## APPENDIX

### The Inference of Reversibility and the Minimization of Truncation Errors in Relation to Entropy Conservation in Model Simulations

The objective of this appendix is to establish a theoretical basis for the results of this study that an empirical relative frequency distribution of pure error differences will tend more and more toward the classical triangular distribution with decreasing estimates of the variance of the pure error difference. The development will utilize the well-known condition that a given probability density distribution enjoys a unique relation with a given momentum generating function (Weatherburn 1957; Cramér 1946).

*M*(MGF) for the probability density distribution

*p*(

*x*) is defined as the expected value of

*e*

^{tx}given by

*x.*In accord with this definition the MGF of the uniform distribution

*p*(

*x*) equal to (2

*α*)

^{−1}over the interval (−

*a*≤

*x*≤

*a*) and zero elsewhere is given by

*δ*

_{+}of two independent uniformly distributed random variates

*y*plus

*z*over the common interval (−

*a*≤

*y*(or

*z*) ≤

*a*) given by

*δ*

_{+}becomes the quadratic expression for the classical triangle distribution given by

*M*

_{δ+}

*t*

*at*

*at*

^{2}

*δ*

_{+}is expressed by

*δ*

_{−}equal to

*y*minus

*z,*expressed by

*δ*

_{−}becomes

*y*and

*z*with common intervals the MGFs of the difference

*δ*

_{−}and the sum

*δ*

_{+}of two random independent variates are identical.

Furthermore, by virtue of the uniqueness between a MGF and its probability density function and the condition that the MGF of a triangle is the pure simple quadratic of the MGF of a uniform distribution, the conclusion is that each of the underlying distributions of *y* and *z* that determine the pure error difference *δ*_{−} must be uniformly distributed over common intervals {−*a* ≤ [(*y* + *z*) or (*y* − *z*)] ≤ *a*}. Thus in this study, when the correspondence of an empirical relative frequency distribution of pure error differences for moist entropy with a triangle distribution is closer and closer, one may infer that the truncation errors for both *θ*_{e} and *tθ*_{e} are tending more and more to uniform distributions of random variates and thus to the optimum accuracy attainable given the existence of truncation errors from numerics.

Zonal–vertical cross sections of differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] at day 10 for 57.2°S from (a) CCM3, (b) CCM2/3 mixed, (c) CCM2 all Eulerian spectral, and (d) CCM3 all semi-Lagrangian simulations. The contour interval is 2 K

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

Zonal–vertical cross sections of differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] at day 10 for 57.2°S from (a) CCM3, (b) CCM2/3 mixed, (c) CCM2 all Eulerian spectral, and (d) CCM3 all semi-Lagrangian simulations. The contour interval is 2 K

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

Zonal–vertical cross sections of differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] at day 10 for 57.2°S from (a) CCM3, (b) CCM2/3 mixed, (c) CCM2 all Eulerian spectral, and (d) CCM3 all semi-Lagrangian simulations. The contour interval is 2 K

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

(a1)–(d1) The day-10 bivariate distributions of equivalent potential temperature *θ*_{e} (K) vs *tθ*_{e} (K) from CCM3, CCM2/3 mixed, CCM2 all Eulerian spectral, and CCM3 all semi-Lagrangian simulations, respectively. (a2)–(d2) Relative frequency distributions of the pure error difference &lsqb=uivalent potential temperature *θ*_{e}(*K*) minus the trace equivalent potential temperature *tθ*_{e} (K)] at 2.5, 5.0, 7.5, and 10 days, respectively, for the same four simulations. (a3)–(d3) Vertical distribution of the horizontally averaged pure error difference *δ̂*

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

(a1)–(d1) The day-10 bivariate distributions of equivalent potential temperature *θ*_{e} (K) vs *tθ*_{e} (K) from CCM3, CCM2/3 mixed, CCM2 all Eulerian spectral, and CCM3 all semi-Lagrangian simulations, respectively. (a2)–(d2) Relative frequency distributions of the pure error difference &lsqb=uivalent potential temperature *θ*_{e}(*K*) minus the trace equivalent potential temperature *tθ*_{e} (K)] at 2.5, 5.0, 7.5, and 10 days, respectively, for the same four simulations. (a3)–(d3) Vertical distribution of the horizontally averaged pure error difference *δ̂*

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

(a1)–(d1) The day-10 bivariate distributions of equivalent potential temperature *θ*_{e} (K) vs *tθ*_{e} (K) from CCM3, CCM2/3 mixed, CCM2 all Eulerian spectral, and CCM3 all semi-Lagrangian simulations, respectively. (a2)–(d2) Relative frequency distributions of the pure error difference &lsqb=uivalent potential temperature *θ*_{e}(*K*) minus the trace equivalent potential temperature *tθ*_{e} (K)] at 2.5, 5.0, 7.5, and 10 days, respectively, for the same four simulations. (a3)–(d3) Vertical distribution of the horizontally averaged pure error difference *δ̂*

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

(a), (b) The zonal–vertical cross sections of the differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] at day 10 for 24° and 58°S, respectively; (c) the day-10 bivariate distribution of equivalent potential temperature *θ*_{e} (K) vs the trace equivalent potential temperature *tθ*_{e} (K); (d) the relative frequency distribution of differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] for days 2.5, 5.0, 7.5, and 10.0; (e) the day-10 vertical distribution of the horizontally averaged pure error difference, *δ̂**θ*–*σ* model

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

(a), (b) The zonal–vertical cross sections of the differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] at day 10 for 24° and 58°S, respectively; (c) the day-10 bivariate distribution of equivalent potential temperature *θ*_{e} (K) vs the trace equivalent potential temperature *tθ*_{e} (K); (d) the relative frequency distribution of differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] for days 2.5, 5.0, 7.5, and 10.0; (e) the day-10 vertical distribution of the horizontally averaged pure error difference, *δ̂**θ*–*σ* model

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

(a), (b) The zonal–vertical cross sections of the differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] at day 10 for 24° and 58°S, respectively; (c) the day-10 bivariate distribution of equivalent potential temperature *θ*_{e} (K) vs the trace equivalent potential temperature *tθ*_{e} (K); (d) the relative frequency distribution of differences &lsqb=uivalent potential temperature *θ*_{e} (K) minus the trace equivalent potential temperature *tθ*_{e} (K)] for days 2.5, 5.0, 7.5, and 10.0; (e) the day-10 vertical distribution of the horizontally averaged pure error difference, *δ̂**θ*–*σ* model

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

Idealized triangular probability density function over the interval (−2*a* ≤ *x* ≤ 2*a*) as determined from the difference of two rectangular probability density functions (−*a* < *x* < *a*) representing the random rounding error components (after Cramér 1946). See Eqs. (28) and (29)

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

Idealized triangular probability density function over the interval (−2*a* ≤ *x* ≤ 2*a*) as determined from the difference of two rectangular probability density functions (−*a* < *x* < *a*) representing the random rounding error components (after Cramér 1946). See Eqs. (28) and (29)

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

Idealized triangular probability density function over the interval (−2*a* ≤ *x* ≤ 2*a*) as determined from the difference of two rectangular probability density functions (−*a* < *x* < *a*) representing the random rounding error components (after Cramér 1946). See Eqs. (28) and (29)

Citation: Journal of Climate 15, 14; 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

The global sum of squares of pure error differences ^{;t02};t5) and its three components, the horizontal deviation ^{;t02};t5), the vertical average deviation ^{;t02};t5), and the global average difference ^{;t02};t5) for days 2.5, 5.0, 7.5, and 10.0 from (a) CCM3, (b) CMM2/3 mixed, (c) CCM2 all Eulerian spectral, and (d) CCM3 all semi-Lagrangian simulations. Values in parentheses are entries of the square root of the corresponding sums of squares expressed in degrees K. Also presented are the simulated global averages of the pure error difference

The area sum of squares of pure error differences *S _{s}* (δ) (K

^{;t02};t5) and its two components, the horizontal deviation

*S*(δ*) (K

_{s}^{;t02};t5) and the area average difference

*S*

_{s}^{2}) by model layer at day 10 from (a) CCM3, (b) CCM2/3 mixed, (c) CCM2 all Eulerian spectral, and (d) CCM3 all semi-Lagrangian simulations. Values in parenthesis are entries of the square root of the corresponding sums of squares expressed in degrees K. Also presented by model layer for day 10 are the simulated area averages of the pure error difference

_{;t5e}(K), the change of equivalent potential temperature Δ

*(K), and percentage of total atmospheric mass in each layer. Here Σ*

_{e}*w*

^{;t02}

_{;t5r}

(*Continued*)

Table 3a. The global sum of squares of pure error differences *S _{G}* (δ) (K

^{;t02};t5) and its three components, the horizontal deviation

*S*(δ*) (K

_{G}^{;t02};t5), the vertical average deviation

*S*(

_{G}^{;t02};t5), and the global average difference

*S*

_{G}^{2}) for days 2.5, 5.0, 7.5, and 10.0 from the UW θ–σ model simulation. Values in parentheses are entries of the square root of the corresponding sum of squares expressed in degrees K. Also presented are the simulated global averages of the pure error difference

Table 3b. The sum of squares of pure error differences *S _{s}* (

*δ*) (K

^{;t02}) and its two components, the horizontal deviation

*S*(

_{s}*δ**) (K

^{;t02}), and the area average difference

*S*(δ̂) (K

_{s}^{;t02}) by model layer at day 10 from the UW θ–σ model simulation. Values in parentheses are the entries of square root of the corresponding sum of squares expressed in degrees K. Also presented by model layer for day 10 are the simulated area averages of the pure error difference δ̂ (K), the equivalent potential temperature

Global average potential temperatures ^{;ms2};t5) estimates the corresponding spurious global average change of enthalpy. The subscripts ES and SLT denote simulations by Eulerian spectral and semi-Lagrangian numerics, respectively

The global integrals of the quadratic deviation of equivalent potential temperature (^{*}_{;t5e}^{;t02} [*S _{G}* (

^{*}

_{;t5e}

^{;t02};t5)], the quadratic global average equivalent potential temperature

The first two left-hand columns list the three components and total of the normalized sums of squares of the pure error differences in symbolic and mathematical forms as defined by (11)–(25) of section 4, while the definition of the quadratic functions, weighted mean, and deviations are defined by (1)–(10) of section 4. The headings of the columns express the conditions for the determination of the expected values of the sums of squares in conjunction with the simplifications realized from elimination of biases, uniform variance, and uniform weighting. Note that with unbiased pure error differences, and uniform weighting and variance, the components of variance in the right-hand column are equivalent with the standard partitioning where *R* is the number of observations for the groups sum of squares (the number of horizontal incremental grid volumes within a model layer), *S* is the number of groups (the number of model layers), and the product of *R* and *S* is the total number of observations (the total number of incremental grid volumes)