1. Introduction
Climate analysis increasingly involves understanding the evolution of relatively weak signals in a high-order chaotic system of strongly interacting components. This hinders climate analysis from observational data alone, because those data records are often too short or too sparse to detect important signals. As a result, climate analysis often relies on numerical simulators of the climate system, such as general circulation models (GCMs), to explore climate sensitivities through experimentation in an Earth-like environment. GCMs also serve as forecast models for a wide range of timescales from short-term weather to long-term climate change that is, for example, associated with increases of concentrations of atmospheric carbon dioxide. Evaluating model error is an important component of research that utilizes GCMs because that process determines how much confidence can be placed in the results. However, evaluating GCMs is a complicated endeavor. GCMs, in order to simulate complex interactions in the actual atmosphere, have attained a corresponding high order of complexity. This makes it difficult to understand the precise correspondence between GCM simulations and reality and, therefore, to diagnose sources of GCM errors (e.g., Randall et al. 2003).
Fortunately, for many applications it is not necessary to consider the full complexity of climate interactions; diagnosis with simpler models can be very useful. Single column models (SCMs) are one such simplification. As their name suggests, SCMs have only one spatial dimension, calculating the time evolution of vertical profiles of temperature and humidity. To obtain this simplification, diabatic tendencies in SCMs are calculated using detailed physics parameterizations while adiabatic tendencies (i.e., advection by the large-scale flow) are prescribed. The combination of detailed physics and an economical framework makes SCMs a valued resource for organized strategies that develop and test physics parameterizations for atmospheric GCMs (e.g., Stokes and Schwartz 1994; Randall et al. 1996; Moncrieff et al. 1997; Randall et al. 2003).
Unfortunately, the simplifications that make SCMs computationally efficient also create sources of error that confuse and compromise results obtained from such models. For example, in the Tropics, temperature fluctuations result from a small imbalance between large diabatic and adiabatic tendencies. In the SCM framework, adiabatic tendencies are prescribed, which decouples them from the diabatic tendencies produced by the model physics. This decoupling can distort cause and effect and can prevent the SCM from maintaining the proper balance between adiabatic and diabatic tendencies. As a result, large errors can occur with damaging consequences for SCM diagnosis (e.g., Ghan et al. 2000; Hack and Pedretti 2000; Xie et al. 2002). For example, the model can develop a grossly unrealistic thermal structure, with disastrous implications for the development of convection and clouds. Diagnosis of these processes in such unrealistic settings is almost meaningless. In addition, errors that arise from the decoupling are largely spurious and can mask sources of error within the model physics (Bergman and Sardeshmukh 2004).
The growth of large errors in SCMs is a familiar problem, and there have been a variety of approaches used to correct them. Some approaches impose corrections directly to the thermal and moisture budgets to maintain realistic atmospheric states for the SCM. The semiprognostic approach specifies all components of atmospheric state except the diabatic tendencies (Lord 1982; Randall et al. 1996). “Nudging” adds relaxation terms to the thermal and moisture equations that guide the SCM's temperature and humidity toward observed profiles (e.g., Ghan et al. 1999; Lohmann et al. 1999; Randall and Cripe 1999). Sobel and Bretherton (2000) prescribe temperature in the free troposphere, but not humidity. In this case, vertical velocity replaces temperature as a prognostic variable. Each of these approaches provides a valuable diagnostic tool in the appropriate context. However, for some applications, we seek to diagnose and improve the impact of model physics on the time evolution of thermal and moisture profiles. For such applications, the corrections imposed by these approaches contaminate results, making a clean diagnosis impossible.
Other approaches circumvent the problem of decoupling by allowing the model physics to equilibrate, in some sense, to the forcing. Emanuel and Živkovíc-Rothman (1999) advocate performing integrations that are longer than the radiative-subsidence timescale (∼30 days) to allow the time evolution of water vapor concentrations to equilibrate to the model physics. Another approach parameterizes the adiabatic tendencies in terms of the time history of the diabatic tendencies (Bergman and Sardeshmukh 2004; Mapes 2003, manuscripts submitted to J. Atmos. Sci.). This allows the SCM to maintain a self-consistent and stable atmospheric state. For both of these approaches, however, the process of equilibration interferes with instantaneous comparisons to observations. In addition, we are left with the nontrivial task of identifying the source of model error, which can be rapidly obscured by interactions among “constituents of the model physics” (i.e., physics parameterizations such as deep convection, radiative transfer, and vertical diffusion) in long model runs (e.g., Klinker and Sardeshmukh 1992).
The methods discussed above have an important common aspect; they correct for the imbalance between adiabatic and diabatic terms in the SCM framework. However, that imbalance, if properly diagnosed, could reveal important sources of model error; by correcting for that imbalance, important diagnostic information is lost. Another alternative, which allows us to utilize that information, is to analyze the short-term behavior of the SCM. Such an analysis is performed before an unrealistic state can develop, eliminating the need to correct the imbalance between adiabatic and diabatic tendencies. Short-term analysis allows us to examine the source of model error before complicated feedbacks within the SCM obscure it. Short-term analysis permits direct comparison to observations and is computationally very inexpensive.
This manuscript presents a diagnosis for SCM physics from the 6-h “state evolution” (i.e., the time evolution of temperature and humidity profiles). Six hours represents the shortest interval over which the SCM can, practically speaking, be directly compared to observations. On the other hand, a diagnosis of the cause of model error is best performed using initial tendencies, which are derived from the state evolution over a single time step. Initial tendencies represent the systematic response of the SCM for a specified initial condition. In fact, initial tendencies, derived from a sufficiently large set of initial conditions, completely characterize the state evolution of a deterministic model. In addition, diagnosing initial tendencies minimizes the feedbacks among constituent parameterizations of the model physics that can obscure “the source of” (i.e., which of the constituent parameterizations contribute to) model error (e.g., Klinker and Sardeshmukh 1992).
The diagnosis presented here has two principal elements: the first relates model errors, which are detected at 6-hourly intervals, to their origin in initial tendencies. To do so, we employ three estimates of the 6-h state evolution in the SCM: the full SCM, a linear version, and an extrapolation of initial tendencies. Intercomparisons between these estimates help us characterize the state evolution of the SCM. If the evolution is highly nonlinear, then the growth of model errors depends sensitively on details of the state evolution over 6-h. In that case, error growth can be too complicated to diagnose in a systematic manner. If the state evolution is “linear” (i.e., is well estimated by the linear SCM), then error growth is determined by initial errors and a single linear operator; we need not concern ourselves with precise details of the state evolution. This makes a systematic diagnosis feasible. However, even for a linear model, feedbacks among the physics constituents can complicate error diagnosis. If the 6-h state evolution of the SCM is merely the extrapolation of initial tendencies, then model error can be diagnosed in the simplest possible terms; we need only analyze the initial tendencies to both detect model errors and identify the sources of those errors. The second element of this work identifies sources of model errors in terms of how individual constituents of the model physics contribute to the origin and amplification of those errors.
The short-term error analysis is performed here for the National Center for Atmospheric Research (NCAR) SCM under tropical conditions. Six-hourly observations from the Tropical Oceans Global Atmosphere Coupled Ocean–Atmosphere Research Experiment (TOGA COARE) provide initial conditions and advective tendencies for the SCM runs. These data also provide observed estimates of 6-h changes from which model errors are detected and characterized. To identify nonlinearities in the state evolution of the NCAR SCM, a linear version of that model is developed. Section 2 describes the NCAR SCM, the linear SCM, and the model experiments. Section 3 analyzes 6-h predictions of temperature and humidity: a comparison of 6-h state changes in the SCM runs to observed values identifies model error. Comparisons of 6-h predictions from the fully nonlinear SCM, from the linear SCM, and extrapolated from initial tendencies provide a detailed characterization of the evolution of model error. In section 4, the contribution by each constituent physics parameterization to model error is analyzed, clearly identifying sources of model error. Concluding remarks are found in section 5.
2. The models and experiments
a. The NCAR SCM
b. The linear SCM
The linear model is constructed empirically from short integrations of the SCM. Such a linear model does not require altering the model equations, which, along with being a significant undertaking, risks introducing unwanted modifications to the model physics. This formulation also allows us to change base states efficiently. Furthermore, initial tendencies for both the linear and fully nonlinear SCMs are identical provided the nonlinear SCM is initialized from the base state. In that case, differences in the subsequent state evolution of the two models result from nonlinearities in the SCM; that is, there is an unambiguous distinction between linear and nonlinear evolutions.
c. The SCM experiments
Observational data for this study were obtained from measurements taken during the TOGA COARE that have been corrected to balance mass, moisture, energy, and momentum budgets (J. L. Lin 2002, personal communication; cf. Zhang and Lin 1997). The TOGA COARE data used here are derived from 21 days of 6-hourly observations. They supply: initial conditions, adiabatic tendencies, base-state conditions for the linear SCM, and an observed estimate of the final state for 84 6-h segments, or cases. For each case, calculations with the NCAR SCM and the linear model were performed. Linear operators were calculated separately for each base state, which allows us to examine the role of nonlinearities separately for each 6-h model run.
In addition to control runs (i.e., with no initial perturbation) of the SCM, 100-member ensembles of the SCM runs were performed for each case. Random initial perturbations for each ensemble member were created level-by-level from uniform distributions of temperature (±1 K) and humidity (±10% of the observed initial value), roughly in accord with ensembles analyzed by Hack and Pedretti (2000). However, since the analysis of these ensembles does not reveal much that is not learned from the analysis of the control runs, results from ensemble runs are not shown in this manuscript. For most cases, 6 h is not sufficient time for the ensemble spread to grow. As a result, the growth of initial errors is small compared to the systematic growth due to initial tendencies and the ensemble mean differs little from the state evolution of the control run.
3. Characterizing error growth
The characterization of error growth in the SCM is an important component of this diagnosis. To do so, we first describe 6-h errors in terms of their amplitude and vertical structure. The amplitude determines the severity of model error. From the vertical structure we can identify which physics constituents have the dominant influence on model error. We then characterize the state evolution of the SCM in terms of a hierarchy of models: an extrapolation of initial tendencies, the linear SCM, and the fully nonlinear SCM. This characterization is important for subsequent efforts that attempt to identify the source of model error. For example, as discussed in section 1, if the 6-h state evolution of the SCM is essentially the extrapolation of initial tendencies, then the source of model error can be diagnosed in the simplest possible terms. If, on the other hand, the 6-h state evolution is highly nonlinear, then a systematic diagnosis is much more difficult.
After a brief description of the analysis technique in section 3a, sec. 3b compares 6-h predictions from the SCM to observations from TOGA COARE to detect and characterize model error. In section 3c, predictions from the SCM and from the linear model are compared to determine the role of nonlinearities for the 6-h state evolution. Finally, section 3d compares extrapolated initial tendencies to 6-h predictions from the SCM to determine the extent to which initial tendencies dictate the 6-h state evolution.
a. Analysis technique
b. Detecting model error
To examine model error, Fig. 1 compares 6-h temperature changes predicted by the SCM (panel a) to observations during TOGA COARE (panel b). Similarly, Fig. 2 compares relative humidity changes. These figures display 6-h changes contoured as functions of pressure and time. The time axis actually represents discrete calculations from the 84 cases in the chronological order that they were observed to occur. There is very little correspondence between the observed and SCM-generated fields, indicating the very unrealistic 6-h state evolution by the NCAR SCM. After only 6-h, the SCM produces temperature changes exceeding 2.0 K with an “overall” (i.e., over all tropospheric levels and all cases) rms value of 1.07 K, whereas observed changes do not exceed 1.0 K (rms value of 0.37 K). Temperature errors are nearly three times as large as the observed variations (overall relative error is 2.8). Six-hour changes of relative humidity predicted by the SCM are also larger than observed (rms values of 0.10 for the SCM compared to 0.05 for observations). Those errors might be even larger if they were not constrained by vapor saturation.
In addition to being large, 6-h errors in the SCM are systematic, indicating the tendency of the model to drift to an unrealistic climate. Temperature predictions in the troposphere are predominantly positive, while relative humidity predictions are typically positive in the upper troposphere, positive near the surface, and negative throughout the midtroposphere.
c. Linear versus nonlinear growth
Nonlinearities are apparently not important for the 6-h evolution of temperature in the SCM. This is demonstrated by the similarity of 6-h predictions from the linear model (Fig. 3) to those of the full SCM (Fig. 1a). The average (i.e., over all 84 cases) bias is only 0.05 K and the average correlation between those two sets of perturbations is 0.78 (on average, the linear prediction explains 64% of the nonlinear prediction). Nonlinearities act to slightly damp perturbations; the average ratio of standard deviations (full SCM divided by the linear SCM) is 0.78.
Nonlinear evolution is, however, important for humidity predictions. The linear model (Fig. 4a) provides a relatively poor reproduction of humidity variations from the SCM (Fig. 2a), particularly in the upper troposphere where 6-h changes predicted by the linear model are several times stronger than those predicted by the full SCM. Much of that discrepancy results from vapor saturation. Water vapor concentrations saturate when the relative humidity reaches 1.0 and subsequent increases of water vapor content leads to condensation in the SCM, which can impact temperature (through latent heat release) but not relative humidity. In the linear model, however, saturation does not occur and humidity is free to increase without bound. Performing additional calculations that limit relative humidity to 100% in the linear model reveals the influence of saturation. Relative humidity perturbations adjusted in this manner (Fig. 4b) provide a much more faithful representation of the SCM; the average correlation between the SCM perturbations and the linear model is enhanced from 0.55 to 0.74 with that simple correction.
d. The influence of initial tendencies
The previous section demonstrated that the 6-h state evolution of the SCM is primarily linear with nonlinear modifications by vapor saturation. The near linearity of the SCM over 6 h makes the diagnosis of model error from 6 h predictions feasible. The diagnosis of model error will be further simplified if 6-h predictions are primarily extrapolations of initial tendencies. In that case, model errors detected at 6-h can be traced to their source in the most direct and unambiguous manner possible. Here, we examine the influence of initial tendencies from comparisons of 6-h SCM predictions (Figs. 1a, 2a) to extrapolated initial tendencies (Fig. 5). These comparisons indicate that initial tendencies dictate, to a large degree, the structure and amplitude of 6-h predictions by the SCM. The patterns of temperature predictions from these two sources (cf. Figs. 5a and 1a) are very similar, particularly below 200 mb. The agreement is not quite as good for relative humidity perturbations, particularly above 300 mb where vapor saturation is known to have a strong influence on humidity predictions (cf. section 3c). Nevertheless, the average correlation between initial tendencies of relative humidity and 6-h predictions is 0.73.
More detailed comparisons between extrapolated initial tendencies and 6-h SCM predictions reveal which aspects of the 6-h predictions are determined directly from initial tendencies and which are influenced more by the subsequent state evolution. These comparisons are shown in Fig. 6 in terms of the bias (Fig. 6a), correlation (Fig. 6b), and ratio of rms deviations from the vertical mean (Fig. 6c) for all 84 cases. The structure and amplitude of deviations from the vertical mean for 6-h temperature predictions are primarily dictated by initial tendencies. For most cases, the correlation between the two predictions is 0.8 or greater (the average correlation is 0.79) and the average ratio of rms deviations is 0.79, indicating that deviations from the vertical mean are typically damped in SCM predictions compared to extrapolated initial tendencies. Initial tendencies, however, do not dictate the vertical mean of 6-h temperature predictions. The average bias for the 84 cases (Fig. 6a) is 0.41 K, which is large compared to the average vertical mean of extrapolated initial tendencies (0.25 K).
The important characteristics of error growth in the NCAR SCM are summarized in Fig. 7, which displays vertical profiles of 6-h changes, averaged over the 84 cases, from: observations (thin solid line), the full SCM (thick solid line), the linear SCM (short-dashed line), and extrapolated initial tendencies (long-dashed line). 1) SCM errors are large. For these cases, the SCM exhibits systematic temperature changes, whereas observed systematic temperature changes are negligible. 2) The evolution of SCM errors is nearly linear. Discrepancies between the linear prediction and the SCM prediction are typically small. Even for levels at which the two predictions disagree most (e.g., 200, 800, and 900 mb), the linear SCM reproduces structural features of the full SCM (e.g., the local minimum at 800 mb; local maxima at 200 and 900 mb). 3) The structure and amplitude of deviations from the vertical mean are primarily determined by initial tendencies. Predictions by the linear SCM and full SCM differ from extrapolated initial tendencies mostly by a vertically uniform temperature increase.
4. Identifying sources of model error
a. Analysis technique
To identify sources of model error, 6-h predictions are performed with different versions of the SCM, each with identical base-state conditions but with altered model physics. These alterations are achieved by switching off different combinations of the seven principal constituents of the model physics adiabatic cooling and dry adiabatic adjustment, deep convection, shallow convection, stable condensation, radiative heating, vertical diffusion, and gravity wave drag. The direct contribution by a physics constituent is calculated by switching off all but that contribution. Since other physics constituents are disabled, the structure of the direct contribution provides a fingerprint that identifies the influence of that physics parameterization in model runs with full physics.
In light of potential interactions between physics constituents, it is important to consider the net contribution of a constituent to the state evolution as well as its direct contribution. The net contribution of a constituent is calculated as the difference of 6-h predictions using the full physics (all constituents enabled) and 6-h prediction with only that constituent switched off. For example, if temperature predictions by the SCM with full physics are nearly identical to predictions by the SCM with only radiative transfer switched off, then we conclude that the net impact of radiative transfer on 6-h predictions is small. In light of the above discussion, the net and direct contributions need not be the same, even if the SCM is linear. In the case that these two contributions differ, it is then necessary to identify interactions among the constituent physics to fully understand the source of error. The discussion hereafter focuses on temperature predictions. Results from the analysis of humidity predictions are, for the most part, redundant.
b. Initial tendency errors
To examine initial tendency errors, we first separate the adiabatic (Fig. 8a) and the diabatic (Fig. 8b) components of the initial temperature tendency. For the SCM to obtain the modest temperature changes that are observed (Fig. 1b), these two components should nearly balance. So, to the extent that vertical temperature advection is reliably obtained from observations during TOGA COARE, the negative of Fig. 8a is a good estimate of what Fig. 8b should look like. There is a slight mismatch of the vertical structures: diabatic heating rates from the SCM tend to concentrate at lower levels than those derived from observations. However, the dominant characteristic of initial tendency error is the poor timing of diabatic tendencies produced by the NCAR SCM: the SCM tends to produce its strongest diabatic heating rates when adiabatic cooling is weakest and its weakest diabatic heating when adiabatic cooling rates are strongest. This problem likely stems from the fact that deep convection is strongest in the NCAR SCM for large values of convective available potential energy (CAPE) of an undiluted parcel (Zhang and McFarlane 1995), while observations show tropical convective activity to be inversely related to undiluted CAPE (e.g., Thompson et al. 1979; Mapes and Houze 1992; McBride and Frank 1999).
The SCM physics modules are now dissected to identify important contributors to initial tendency errors. For this analysis, only diabatic constituents are considered; adiabatic tendencies are disabled. The largest direct contributor to diabatic tendencies is, by far, deep convection (Fig. 9a) as one might expect for a tropical location. The fingerprint of deep convection that is evident in Fig. 9a can be recognized in the diabatic tendencies (Fig. 8b) and even in the 6-h prediction by the full SCM (Fig. 1a). Minor contributors are: vertical diffusion (Fig. 9b), radiative transfer (not shown), and stable condensation (not shown). Direct contributions by the remaining constituents: dry adiabatic adjustment, shallow convection, and gravity wave drag are negligible.
c. 6-h errors
In section 3d, we found that initial tendencies determine, to a large extent, the structure and amplitude of 6-h predictions from the full SCM. However, extrapolated initial tendencies do not precisely reproduce 6-h predictions by the full SCM and, so, the diagnosis of initial tendencies might not reveal all of the important sources of 6-h errors. To verify that the diagnosis of model error from initial tendencies is also valid for 6-h predictions, we examine 6-h predictions for the SCM runs with altered physics. As with the analysis of initial tendencies in the previous section, we examine the direct and net contributions by constituent physics parameterizations to predictions by the full SCM physics. Adiabatic tendencies are disabled for all of these calculations.
For the most part, this analysis confirms the results that were obtained from initial tendencies. Deep convection dominates the direct contributions to 6-h predictions as it did for initial tendencies. This is demonstrated by the small differences between the direct contribution of deep convection (Fig. 11a) and the linear sum of direct contributions from all the constituent physics (Fig. 11b). In addition, the difference between 6-h predictions with all diabatic components enabled (Fig. 12a) and those calculated from the linear sum of individual direct contributions (Fig. 11b) reveals the existence of a strong interaction among physics constituents. That interaction is neutralized by removing vertical diffusion (Fig. 12b)—as with initial tendencies.
The remarkable feature of the 6-h predictions is the strength of the interaction between deep convection and vertical diffusion. Note that the contour intervals in Fig. 12a are three times those in Fig. 12b. Whereas, rms initial tendencies in the troposphere from deep convection are increased by only 24% by interaction, 6-h changes are increased by 340%. Without this interaction, convective heating in the NCAR SCM would be much too weak. Furthermore, this interaction causes strong diabatic heating in all 84 cases, whereas deep convection by itself (Fig. 11a) leads to intermittent diabatic heating. Thus, this interaction plays a large role both for the realistic strength and for the unrealistic timing of convection in the 6-h predictions. The rapid development of convection, regardless of the initial CAPE, might help explain why the NCAR GCM does not produce intermittent convection, as observed, but instead produces persistent convection over warm waters (e.g., Ricciardulli and Garcia 2000).
d. Resolving apparent inconsistencies
Given the strong interaction between vertical diffusion and deep convection, there are two issues that require attention. 1) If 6-h predictions by the SCM are largely extrapolations of initial tendencies, then why is the interaction so much stronger for 6-h predictions than for initial tendencies? 2) If vertical diffusion, which has a very small direct contribution, can have a strong interaction with convection, then adiabatic and diabatic components, which are of comparable strength, might also interact strongly. If that is the case, then disabling adiabatic tendencies to diagnose the source of error can be misleading.
Issue 1 is resolved in two ways. First, the interaction with vertical diffusion amplifies heating by deep convection. Deep convection, when active, provides heating throughout the troposphere and, so, it has a large projection onto the vertical mean temperature change. Thus, the strong impact of this interaction on 6-h predictions is consistent with results in section 3d, where it was shown that the primary difference between 6-h predictions by the full SCM and by extrapolated initial tendencies is a shift of the vertical mean. Second, the amplitude of the direct contribution by deep convection to 6-h predictions by the SCM (Fig. 11a) is much weaker than its direct contribution to extrapolated initial tendencies (Fig. 9a; note the smaller contour intervals in Fig. 11a). On the other hand, using the full physics, the amplitudes of 6-h predictions (Fig. 12a) and extrapolated initial tendencies (Fig. 8b) are comparable. Thus, without vertical diffusion, deep convection in the NCAR SCM diminishes in strength over 6 h due to the depletion of CAPE, which is a nonlinear effect. The role of vertical diffusion, therefore, is to supply the boundary layer with enough moisture to keep convection active over the full 6 h. In other words, rather than amplifying convective heating, the primary role of the interaction between vertical diffusion and deep convection is to counteract a nonlinearity that would otherwise kill off convection. This allows the extrapolated initial tendencies to provide a far representation of 6-h changes.
To resolve issue 2, we examine the interaction between the adiabatic and diabatic (i.e., the full physics) components. To do so, we calculate the sum of the direct contributions by the adiabatic and diabatic components for both initial tendencies (Fig. 13a) and for 6-h predictions (Fig. 13b). The strength of the interaction between diabatic and adiabatic components is determined by comparing these sums to predictions by the fully interactive system. For both initial tendencies (cf. Fig. 13a to Fig. 5a) and 6-h predictions (cf. Fig. 13b to Fig. 1a), this comparison indicates that, indeed, the adiabatic and diabatic components interact. However, this interaction is modest and interferes little with the primary features of either initial tendencies or 6-h predictions. There are two important implications of this result. First, the disabling of adiabatic tendencies that permits us to examine interactions between diabatic constituents is a viable diagnostic technique. Second, since this interaction is much weaker than the interaction between vertical diffusion and deep convection, the advection of remote moisture by the large-scale flow is a much weaker source of CAPE than the vertical diffusion of local moisture from the surface—at least for conditions observed during TOGA COARE.
5. Conclusions
This study utilized short-term predictions to diagnose physics parameterizations in the NCAR atmospheric single column model for tropical conditions observed during TOGA COARE. To circumvent shortcomings of the SCM framework, this procedure focuses on the first 6 h of the SCM state evolution. Short-term analysis diagnoses model error before an unrealistic state develops and before complex interactions in the SCM obscure the source of error while permitting linear analysis. In addition, short-term analysis is computationally economical, which allows the exploration of large volumes of parameter space, yet still permits direct comparisons to observations. The analysis systematically links initial tendencies, from which model error originates, to 6-h predictions, which are compared to observations to detect and characterize model error.
The results from this diagnosis are both practical and illuminating, revealing the utility of short-term predictions. For example: the 6-h state evolution of the NCAR SCM is predominantly linear and the vertical structures of 6-h predictions are primarily extrapolations of initial tendencies. An exception is the nonlinear effect of vapor saturation on humidity predictions. The primary source of model error is not in the strength or vertical structure of diabatic tendencies, but in the timing of those tendencies. The NCAR SCM tends to produce strong diabatic tendencies under meteorological conditions for which observations indicate those tendencies are weak (and vice versa) because closure in the parameterization of deep convection is linked to the CAPE of an undiluted particle while observations indicate that convection is inversely related to CAPE.
The analysis identified a strong interaction between the vertical diffusion of moisture from the surface and deep convection that amplifies the diabatic heating rates from deep convection—even within a single time step. That amplification dominates the 6-h prediction of diabatic heating rates, causing the NCAR SCM to produce strong diabatic heating rates within 6 h, no matter which set of initial conditions is used. In contrast, without that interaction, diabatic heating rates in the NCAR SCM are weaker and more intermittent. This suggests that the misrepresentation of this interaction, in addition to that of the interaction constituents themselves, might be a major contributor to the NCAR GCM's tropical simulation errors.
This analysis also demonstrates the potential ambiguity of incomplete diagnosis. For example, if we were to diagnose the source of model error from the direct contributions, we would conclude that deep convection, as the dominant direct contributor, is the source of model error. On the other hand, if we relied on the net contribution for our diagnosis, we would conclude that vertical diffusion is the source of error. It is only by examining both the direct and net contributions that we identify the importance of the interaction between deep convection and vertical diffusion.
This interaction demonstrates that adjustments made within a time step in the physics module can confuse the diagnosis of model error even if the state evolution from one time step to the next were perfectly linear. It also implies that model errors that seem to arise from one parameterization might actually be caused by a different parameterization. It is, therefore, important to evaluate a physics parameterization in the context of its interactions with all other components of the model. This has important ramifications: changes to GCM physics that are based solely on the improved representation of some physical process, without considering interactions with other components of the model, could actually be detrimental to the performance of the GCM. In fact, a perfect parameterization (e.g., a parameterization of convection that accurately reproduces observed clouds, precipitation, and heating rates from observed large-scale meteorological conditions) is not likely to be the best parameterization (as measured by the realism of climate simulations by the GCM) unless all other components of the GCM are also perfect. Since a perfect model is unattainable, endeavors that seek to develop climate forecast models from first-principle physics alone are fundamentally and irreparably flawed.
This study revealed well-defined shortcomings of the NCAR SCM and identifies their sources. However, to develop a complete diagnostic package for atmospheric physics, there are outstanding issues that need to be addressed: 1) This analysis was performed for only 3 weeks of observed conditions at a specific location in the west Pacific; it has yet to be determined if the errors detected in this study are representative of errors for tropical conditions in general. 2) It has yet to be shown that our analysis can actually guide us toward improving the state evolution of the SCM. 3) We do not know how important errors on such short timescales are for climate simulations that involve much longer timescales. However, it seems inconceivable that short-term errors are not important because processes such as cloud formation and convection, which are fundamental to climate, occur on short timescales. 4) SCMs do not simulate dynamical interactions that are important for GCM simulations; it has yet to be shown that the diagnosis of SCM errors can guide us toward improvements of the full GCM. These are difficult issues to address and are the subject of future research.
Acknowledgments
We wish to thank those who provided important assistance for this work. Discussions with J. Barsugli, H. van den Dool, and other colleagues at NCAR and CDC were helpful for developing the ideas contained in the manuscript. Comments by B. Mapes, A. Sobel, and an anonymous reviewer helped clarify the manuscript. B. Stevens, J. Hack, and J. Pedretti provided important assistance with the NCAR SCM. TOGA COARE data was obtained from M. Zhang (at http://atmgcm.msrc.sunysb.edu/iops.html) with the assistance of Jialin Lin.
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Six-hour changes of temperature from (a) the NCAR SCM and (b) observations from TOGA COARE contoured as functions of pressure and time. The time axis represents 6-h perturbations from 84 discrete cases. Contour intervals are 0.3 K; the 0.0 contour is not shown. Dashed contours represent negative values. Fields have been smoothed by one pass of a 1-2-1 filter in the time domain
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Six-hour changes of relative humidity from (a) the NCAR SCM and (b) observations from TOGA COARE contoured as functions of pressure and time. Contour intervals are 0.03
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Six-hour changes of temperature from the linear SCM contoured as a function of pressure and time. Contour intervals are 0.3 K
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Six-hour changes of relative humidity from (a) the linear SCM and (b) the linear SCM with a saturation correction contoured as functions of pressure and time. Contour intervals are 0.03.
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Extrapolated initial tendencies of (a) temperature and (b) relative humidity contoured as functions of pressure and time. Contour intervals in (a) are 0.3 K and in (b) 0.03
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Case-by-case comparisons of extrapolated initial tendencies of temperature to 6-h predictions. (a) The bias, (b) the correlation, and (c) the ratio of standard deviations (values for 6-h predictions divided by values initial tendencies)
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Six-hour changes of temperature averaged over the 84 cases from: the full SCM (thick solid line), the linear SCM (short-dashed line), extrapolated initial tendencies (long-dashed line), and observations (thin solid line)
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Extrapolated initial tendencies of temperature from the NCAR SCM with altered physics contoured as functions of pressure and time: (a) adiabatic tendencies only and (b) diabatic tendencies only. Contour intervals are 0.3 K
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Extrapolated initial tendencies of temperature from the NCAR SCM with altered physics contoured as functions of pressure and time. Shown are direct contributions by (a) deep convection and (b) vertical diffusion. Contour intervals are 0.3 K
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Extrapolated initial tendencies of temperature from the NCAR SCM with altered physics contoured as functions of pressure and time: (a) the linear sum of calculation from all diabatic constituents and (b) all diabatic constituents active except vertical diffusion (i.e., vertical diffusion and adiabatic tendencies switched off). Contour intervals are 0.3 K
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Six-hour predictions of temperature from the NCAR SCM with altered physics contoured as functions of time and pressure: (a) deep convection alone and (b) the linear sum of the direct contributions by physics constituents. Contour intervals are 0.1 K
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
Six-hour predictions of temperature from the NCAR SCM with altered physics contoured as functions of time and pressure: (a) all diabatic constituents active, and (b) all diabatic constituents except vertical diffusion active. Contour intervals are (a) 0.3 and (b) 0.1 K
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
The linear sum of individual adiabatic and diabatic contributions to temperature contoured as functions of time and pressure. (a) Extrapolated initial tendencies and (b) 6-h predictions by the NCAR SCM. Contour intervals are 0.3 K
Citation: Journal of Climate 16, 22; 10.1175/1520-0442(2003)016<3803:UOSCMD>2.0.CO;2
The amplitudes of unit perturbations are 0.5 K for temperature and 7.5% for relative humidity. The relative amplitude of unit temperature and unit humidity perturbations was chosen based on statistics of 6-h differences in the tropical west Pacific.
