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  • View in gallery
    Fig. 1.

    Time series of the observed large-scale advective tendency for temperature for the GCSS WG4 case 2 dataset, and days 2–8 of the ARM summer 1995 IOP case study.

  • View in gallery
    Fig. 2.

    SCCM ensemble-mean temperature solution and the ensemble standard deviation for the GCSS case study. The ensemble contains 500 members and is based on perturbed initial conditions.

  • View in gallery
    Fig. 3.

    SCCM ensemble-mean temperature solution and the ensemble standard deviation for the ARM case study. The ensemble contains 500 members and is based on perturbed initial conditions.

  • View in gallery
    Fig. 4.

    Examples of solution distribution time series for the GCSS and ARM case studies. The figures show the number of cases exhibiting the solution plotted on the ordinate, as a function of time, for selected model surfaces. The top panels are from the GCSS case study and the bottom panels are from the ARM case study. The solid black line denotes the ensemble mean.

  • View in gallery
    Fig. 5.

    Cases 111 and 175 from the 500-member GCSS ensemble. Panels denote the temperature solution difference from the GCSS ensemble mean.

  • View in gallery
    Fig. 6.

    Cases 191 and 199 from the 500-member ARM ensemble. Panels denote the temperature solution difference from the ARM ensemble mean.

  • View in gallery
    Fig. 7.

    Time series of the ensemble-mean column-integrated condensed water, and 250-mb diabatic heating for the ARM case study. The ensemble standard deviation is shown by the heavy dashed line.

  • View in gallery
    Fig. 8.

    Time series of the observed and ensemble-mean predicted total precipitation rate for the ARM case study, which excludes the CCM3 deep convection scheme. The ensemble standard deviation is given by the dotted line. The upper panel shows the unconstrained solution and the lower panel shows the solution generated with the inclusion of a relaxation term.

  • View in gallery
    Fig. 9.

    Time-averaged ensemble-mean profile of the temperature and specific humidity error for a model configuration that excludes the CCM3 deep convection scheme (see the upper panel in Fig. 8).

  • View in gallery
    Fig. 10.

    Observed and ensemble-mean predicted profiles of dry and moist static energy at hour 240 for the ARM case study (just prior to the large solution bifurcation shown in the lower panels of Fig. 4).

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Assessment of Solution Uncertainties in Single-Column Modeling Frameworks

James J. HackNational Center for Atmospheric Research,* Boulder, Colorado

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John A. PedrettiNational Center for Atmospheric Research,* Boulder, Colorado

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Abstract

Single-column models (SCMs) have been extensively promoted in recent years as an effective means to develop and test physical parameterizations targeted for more complex three-dimensional climate models. Although there are some clear advantages associated with single-column modeling, there are also some significant disadvantages, including the absence of large-scale feedbacks. Basic limitations of an SCM framework can make it difficult to interpret solutions, and at times contribute to rather striking failures to identify even first-order sensitivities as they would be observed in a global climate simulation. This manuscript will focus on one of the basic experimental approaches currently exploited by the single-column modeling community, with an emphasis on establishing the inherent uncertainties in the numerical solutions. The analysis will employ the standard physics package from the NCAR CCM3 and will illustrate the nature of solution uncertainties that arise from nonlinearities in parameterized physics. The results of this study suggest the need to make use of an ensemble methodology when conducting single-column modeling investigations.

Corresponding author address: Dr. James J. Hack, NCAR/CGD, P.O. Box 3000, Boulder, CO 80307-3000.

Email: jhack@ucar.edu

Abstract

Single-column models (SCMs) have been extensively promoted in recent years as an effective means to develop and test physical parameterizations targeted for more complex three-dimensional climate models. Although there are some clear advantages associated with single-column modeling, there are also some significant disadvantages, including the absence of large-scale feedbacks. Basic limitations of an SCM framework can make it difficult to interpret solutions, and at times contribute to rather striking failures to identify even first-order sensitivities as they would be observed in a global climate simulation. This manuscript will focus on one of the basic experimental approaches currently exploited by the single-column modeling community, with an emphasis on establishing the inherent uncertainties in the numerical solutions. The analysis will employ the standard physics package from the NCAR CCM3 and will illustrate the nature of solution uncertainties that arise from nonlinearities in parameterized physics. The results of this study suggest the need to make use of an ensemble methodology when conducting single-column modeling investigations.

Corresponding author address: Dr. James J. Hack, NCAR/CGD, P.O. Box 3000, Boulder, CO 80307-3000.

Email: jhack@ucar.edu

1. Introduction

Numerical modeling of the climate system and assessing its sensitivity to anthropogenic influences is a highly complex scientific problem. Progress toward accurately describing our climate system using global numerical models is primarily paced by uncertainties in the representation of unresolvable physical processes, a topic best known as the physical parameterization problem. A principal example is the treatment of water in the climate system, a constituent that routinely changes phase in the free atmosphere. It is well known that the large-scale water vapor distribution plays a fundamental role in the maintenance of the earth’s climate. Changes in water phase are of comparable importance, where this process represents the major source of diabatic heating in the atmosphere. This diabatic forcing occurs with the release of the latent heat of condensation in clouds, which is most often associated with vertical redistributions of energy and momentum through cloud-scale transport mechanisms driven by the release of latent heat, and the reflection, absorption, and emission of radiation from clouds. All of these diabatic processes occur on scales much smaller than those that can be resolved in global models and must be treated parametrically. Evaluating the many different approaches to parameterizing these types of physical processes in global atmospheric models is both scientifically and computationally demanding.

An alternative approach to testing climate model parameterizations in global atmospheric models is the use of what are called “single-column models” or SCMs. Modeling the atmosphere as a single column is not a new idea (e.g., Silva Dias and Schubert 1977; Ramanathan and Coakley 1978; Lord 1982; Betts and Miller 1986), although the experimental configurations and applications of these tools have grown in sophistication (e.g., see Randall et al. 1996). As the name suggests, the construction of an SCM is analogous to extracting a discrete vertical column from a global model, thus allowing the performance of the parameterized physics for the column to be considered in isolation from the rest of the large-scale model. Various forms of large-scale “forcing” can be used to specify the effects of neighboring columns, as well as to specify the role of physical processes that might otherwise be parameterized (e.g., specification of surface energy exchanges). The SCM approach lacks the more complete feedback mechanisms available to an atmospheric column embedded in a global model, and therefore cannot provide a truly thorough framework for evaluating competing parametric techniques. It can, however, provide an inexpensive first look at some of the characteristics of a particular parameterization approach without having to sort out all the complex nonlinear feedback processes that would occur in a global integration. The simplicity of this framework is a double-edged sword, since experience suggests that the absence of nonlinear large-scale dynamical feedback often makes it difficult to predict the behavior of a physical parameterization in a global model based solely on SCM tests. Thus, the challenge to the single-column modeler is to be able to identify those properties of a new or revised parameterization that will transfer to the more complete global model. To a certain extent this is a signal-to-noise problem where the noise is partly a property of the SCM experimental framework.

The Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment is one of the first examples of an observational field program that provided time-dependent data suitable for driving single-column models (Kuettner 1974), thanks in large part to a very thorough objective analysis conducted by Ooyama (1987). More recently, other field programs have been conducted with an emphasis on collecting measurements of large-scale state variables, which can be used to drive single-column models. Examples include the Atlantic Stratocumulus Transition Experiment (Albrecht et al. 1995), the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE; Webster and Lukas 1992), and the Atmospheric Radiation Measurement (ARM) program (Stokes and Schwartz 1994). At least two single-column modeling intercomparisons based on these observational programs are currently under way, coordinated by the Global Energy and Water Cycle Experiment (GEWEX) Cloud Systems Study (GCSS) working group 4 (e.g., see Moncrieff et al. 1997) and the Department of Energy ARM program (e.g., see Leach et al. 1998). One objective of these intercomparisons is to examine the behavior of different general circulation model physics packages using the identical set of large-scale observations. The hope is that this kind of exercise will help identify the strengths and weaknesses of various parameterization approaches and lead to new scientific insights. The most common experimental framework is to drive the single-column model with a long time series of observed large-scale advection tendencies starting from an observed initial state. The SCM physics packages are then evaluated on the basis of their ability to reproduce the observed time series of quantities like temperature, water vapor, precipitation, cloud liquid water, turbulent heat fluxes, and top-of-atmosphere and surface radiation fluxes.

Like many other model intercomparison initiatives, simulation results submitted to SCM intercomparison case studies generally comprise a single numerical solution that is assumed to be representative. Our investigations with the single-column version of the National Center for Atmospheric Research (NCAR) Community Climate Model have revealed cases of unexpected solution sensitivity to minor changes in the experimental framework. Similar sensitivities have been observed by other single-column modeling groups (Cripe 1998), raising basic questions about the determinism of SCM numerical solutions. In this manuscript we will demonstrate that a single SCM solution is not necessarily representative, and that an ensemble methodology is necessary for a proper characterization of the model physics behavior. We will show that because of nonlinearities in parameterized physics packages, the establishment of solution uncertainties should be a standard component of any single-column modeling investigation.

2. Model description and experimental framework

An SCM is a one-dimensional time-dependent model for which the local time-rate-of-change of the large-scale state variables (e.g., temperature, moisture, momentum, cloud water, etc.) depends on specified horizontal advection tendencies, a specified vertical motion field or a specified vertical advection tendency, and subgrid-scale sources, sinks, and eddy transports. The subgrid-scale contributions are determined by the particular collection of physical parameterizations included in the SCM. Because an SCM lacks the horizontal feedbacks that occur in complete three-dimensional models of the atmosphere, the governing equations are coupled only through the parameterized physics. In a practical sense this means that the thermodynamic and momentum components of the governing equations are generally independent of one another. Although some types of SCM investigations employ a complete system of thermodynamic and momentum equations, typical configurations only treat the thermodynamic budget. This is the approach we will follow, where contributions of the large-scale horizontal momentum field to the parameterized physics will be specified.

A minimal thermodynamic configuration for a single-column model consists of a thermodynamic energy equation and a water vapor mass continuity equation
i1520-0442-13-2-352-e1
where the terms physics, and physics schematically denote the large-scale horizontal advection tendency terms and parameterized physics tendency terms. For the purpose of this discussion, terms other than the physics tendency terms will be referred to as “forcing terms” where the physics tendency terms are viewed as a response to a large-scale forcing.

SCM solutions are obtained by numerically integrating these two budget equations in time starting from an arbitrary initial condition for temperature and moisture. The large-scale terms on the right-hand side of Eqs. (1)–(2) are evaluated numerically, arbitrarily specified, or given by some combination of the two. Minimally, the terms and ω must be specified since these quantities represent a degree of freedom not available to an SCM framework. The vertical advection terms can be numerically evaluated from the specified ω and the temporally evolving temperature and moisture profiles. Alternatively, vertical advection tendencies may be specified, a common configuration for intercomparison studies.

In this study we will use the single-column version of the National Center for Atmospheric Research (NCAR) Community Climate Model, hereafter referred to as the SCCM (see Hack et al. 1998). Solutions will be obtained by specifying a time series for and ω where the large-scale vertical advection terms will be explicitly evaluated using the predicted profiles of temperature and specific humidity. In this case the three-dimensional large-scale advective tendency applied to the physics is not identical to the observed large-scale tendency because the simulated state used to evaluate the vertical component of the large-scale tendency generally differs from the observed state. However, differences in the large-scale temperature and water vapor tendencies turn out to be relatively small, where the two time series are visually indistinguishable from each other for the cases we will examine. We will use the default configuration of the SCCM, which parallels the CCM in its treatment of large-scale vertical advection.Water vapor and other constituents are advected using the CCM semi-Lagrangian procedure (Williamson and Rasch 1994), and temperature is advected with the CCM second-order Eulerian finite-difference scheme (Williamson 1988). The remaining physical parameterization terms will be evaluated using the standard CCM3 physics package (Kiehl et al. 1996).

In what follows, we will explore the SCCM solution sensitivity to initial conditions, as well as to uncertainties in large-scale forcing. Our experimental framework will make use of the GCSS working group 4 (WG4) case 2 forcing dataset obtained from TOGA COARE, and the ARM Summer 1995 intensive observing period (IOP) forcing dataset obtained over the U.S. southern Great Plains. The GCSS WG4 case 2 forcing dataset spans a 7-day period beginning 20 December 1992. This period was chosen by the intercomparison organizers to explore the bulk effects of deep convection. Initial conditions and the time-dependent large-scale forcing terms represent averages over the TOGA COARE intensive flux array. The ARM IOP forcing dataset spans a 17-day period beginning 18 July 1995. As in the case of the GCSS dataset, this initial case study was selected so as to focus on a convectively active period. Initial conditions and the time-dependent large-scale forcing terms are obtained from the variational analysis of Zhang and Lin (1997) using surface flux estimates derived from the ARM Energy Balance Bowen Ratio stations. In the case of the ARM forcing scenario the CCM3 land surface model is initialized with the observed ground temperature and a climatological value for soil water and allowed to internally evaluate surface energy exchanges and the corresponding changes in surface temperature and soil water.

Each of the observational datasets provide time-dependent large-scale observations of temperature, specific humidity, winds, and the corresponding large-scale advection tendencies. The temperature tendency due to large-scale advection for the entire GCSS period and for days 2–8 of the ARM IOP dataset is illustrated in Fig. 1. This large-scale forcing term is dominated by the vertical advection tendency in both datasets. The horizontal advection component plays a considerably larger role in the ARM total advection than it does in the GCSS total advection, but remains a second-order component in the overall thermodynamic energy budget. Inspection of this figure reveals that the ARM large-scale temperature forcing generally consists of stronger events, more tightly focused in time, and with a slight preference for maximizing in the upper troposphere. It is the balance between this large-scale advection term and the unresolved physical process terms that produces the observed large-scale temperature tendency, typically an order of magnitude smaller. The experimental framework explores how well this balance is maintained by initializing the SCCM with the observed temperature and moisture profiles and integrating Eqs. (1) and (2) forward in time subject to the observed time series of and ω. Thus, the time evolution of the solutions depend on how well the parameterized physical processes can reproduce the diagnosed apparent heat source Q1 and apparent moisture sink Q2 (e.g., see Yanai 1973). Since the observations are by definition imperfect, solution errors are the result of errors in the observed large-scale forcing and deficiencies in the parameterized physics. This makes the parameterization evaluation process difficult, but is not the subject of the current investigation. Instead we will focus on the solution uncertainty associated with this kind of experimental framework.

3. Sensitivity to initial condition and forcing uncertainty

Early experimentation with the CCM3 configuration of the SCCM revealed unexpected sensitivity to small changes in the modeling framework. For example, the model would generate slightly different solutions for the same algorithmic form of the physics, but for relatively minor differences in the implementation. In some cases, differences in two solutions could represent a nonnegligible fraction of the error produced by either solution. The sources of solution differences were generally easy to identify, but their mere existence raised serious questions about the representativeness of a single SCCM numerical solution. What made these experiences most surprising was that the CCM physics parameterization package is continually scrutinized for discontinuous behavior, which might affect the growth of small perturbations in the global model’s initial conditions. Indeed, a predictability error growth procedure has been employed for many years to verify the integrity of development versions of the CCM (Rosinski and Williamson 1997), where a well-behaved physics implementation is central to this validation strategy.

The SCCM simulation sensitivity prompted a more systematic investigation of solution uncertainties. Our initial approach was to conduct a large ensemble of single-column model simulations using the GCSS and ARM forcing datasets. Each member of the respective ensemble used an identical large-scale forcing time series, but slightly different initial conditions for temperature and specific humidity. The initial conditions differ for each member of the ensemble by a random perturbation, characterized by a standard deviation of 0.5°C for temperature and a multiplicative modification of the specific humidity field, which gives a maximum standard deviation of 0.5 g kg−1 in the atmospheric boundary layer. Local perturbations to the initial conditions are absolutely bounded by 0.9°C and 6% of the locally observed specific humidity. These perturbations were selected so as to maximize the solution response to different initial conditions within the expected range of measurement, analysis, and experimental setup error (e.g., arbitrary interpolation to model surfaces).

Figures 2 and 3 show the simulated GCSS and ARM ensemble mean temperature errors along with the respective standard deviation of the temperature solution for a 500-member ensemble. The error is defined to be the difference between the ensemble-mean predicted temperature and the observed temperature. A comparison of these two ensembles shows that both solutions exhibit significant departures from the observed temperature time series, where the ARM solution errors become quite severe after about 10 days into the integration period. The magnitude of these errors are typical of other models participating in the ARM SCM intercomparison. Despite the development of nonphysical structures, the ARM solutions are useful for illustrating several points we will make about the utility of this type of experimental framework. In addition to large local errors in the ensemble-mean solutions, each of these examples exhibits relatively large local variability as quantified by the standard deviation. In both cases the largest variability occurs in the lower troposphere and is associated with interactions between convection and the atmospheric boundary layer. The other predicted state variable, specific humidity, shows similar behavior.

Both ensembles contain regions where the standard deviation of the solution is a sizable fraction of the average error. In principle, the local statistical variability could be used to help establish whether solutions from different models, or solution differences from the same model with a revised parameterization package, are statistically or practically significant (e.g., see Livezey and Chen 1983; Tribbia and Baumhefner 1988; Whitaker and Loughe 1998). In other words, the statistical properties of the ensemble could be used to address the question of whether a particular solution difference may simply have occurred by chance. Most statistical techniques require some knowledge about the probability distribution in order to address these issues (e.g., see Ehrendorfer 1994). For example, do the solutions exhibit a normal distribution or do the various solutions tend to cluster? Examination of the solution distributions for the two ensembles reveals a rather interesting behavior, as illustrated in Fig. 4. The information contained in these figures amounts to a probability distribution function time series. Each of the panels show the number of cases exhibiting the solution error plotted on the ordinate (binned by 0.5°C intervals) as a function of time on selected model surfaces. The top panels illustrate a segment of the GCSS solution at approximately 870 mb and 700 mb for days 3–5. Similarly, the bottom panels show the ARM solution properties at 900 mb and 250 mb for days 10–17. All panels show clear solution bifurcations, beyond which the solutions exhibit a multimodal behavior for differing periods of time. The most extreme case is given by the 900-mb ARM solution, which exhibits a variety of preferred states for the last 6 days of the simulation. It is worth noting that in all these cases the ensemble mean (given by the solid black line) has little physical significance since it is highly unlikely to be a realizable state. During these phases of the simulation the ensemble mean is simply the average of two or more largely independent solution states. What is perhaps most intriguing is the tendency for these solution modes to collapse back to a single state, a feature most dramatically illustrated by the 900-mb temperature solutions from the ARM ensemble.

Some additional solution characteristics associated with this clustering can be seen by examining two realizations from each of the solution distributions. Cases 111 and 175 from the GCSS ensemble are compared in Fig. 5, and cases 191 and 199 from the ARM ensemble are compared in Fig. 6. These figures show the single realization solution differences from their respective ensemble mean. The mirror-image-like properties of the two solutions from each ensemble are quite striking and illustrate how the various members of the ensemble can have markedly different characteristics. In the GCSS examples, case 111 is much colder than the ensemble mean in both the upper and lower troposphere during the early phases of the simulation, and considerably warmer for the remaining portion. Case 175 exhibits a response with similar magnitude but with opposite sign. Note that the amplitude of these solution differences, frequently in the range of 2°C, is comparable in magnitude to solution errors characterized by the ensemble mean. Similarly, cases 191 and 199 from the ARM ensemble show solutions on either side of the ensemble mean, where local differences from the ensemble mean can exceed 10°C. As might be expected, these solutions turn out to be members of different model states as characterized by the clustering shown in Fig. 4. Perhaps the most important point is that neither of these solutions considered singularly could be used to properly characterize the behavior of the model physics package, pointing to the need to consider the ensemble properties in any kind of evaluation.

The solution sensitivity to initial conditions is quite remarkable and underscores the potential for other sources of experimental error to contribute to solution uncertainty in this particular SCM framework. One of the obvious sources of error is the large-scale advection tendency time series. For example, the analysis of the divergence field is notoriously prone to error because of the demanding accuracy requirements on observing the horizontal momentum field. Other measurement uncertainties can also play a major role in the derivation of the large-scale forcing terms, such as estimates of surface energy exchange, which can enter into the problem through the objective analysis procedure. An example of this comes from the ARM SCM intercomparison, which includes two variationally analyzed forcing datasets that differ only in their estimates of the surface sensible and latent heat fluxes. These surface flux differences lead to significant differences in the vertical motion field as a consequence of the constrained variational analysis.

We have explored the sensitivity of our SCM solutions to a variety of random perturbations applied to the prescribed vertical motion field. We have examined the effects of random local perturbations (e.g., spatial-temporal noise in a single realization) to the ω time series, and temporally random but vertically coherent perturbations to ω. In both approaches the ensemble-averaged time series of the perturbations mirror their respective vertical motion fields with a standard deviation, which is approximately 10% of the specified ω. The ensemble variability for the GCSS case study including these forcing perturbations is generally much smaller than in our initial condition sensitivity experiments. Nevertheless, the solution distributions exhibit many of the same characteristics, including evidence of solution bifurcation. In the case of the ARM forcing scenario, the variability as measured by the ensemble standard deviation is very similar in structure and magnitude to the initial condition sensitivity tests. The solutions also exhibit the same kind of multiple state separation characteristics as shown earlier in the lower panels of Fig. 4.

As we have demonstrated, the solutions for the prognostic variables show considerable sensitivity to initial condition and forcing perturbation. Examination of other quantities associated with parameterized processes show that they exhibit a comparable, and in some cases even larger uncertainty. Figure 7 shows the ensemble-mean time series for column-integrated cloud condensate, and the total diabatic heating rate at 250 mb as simulated using the ARM IOP forcing scenario. The ensemble standard deviations are also shown in this figure, indicating a substantial variability for these quantities across the members of the ensemble. The standard deviation can often be a large fraction of the ensemble-mean behavior, where examination of individual ensemble members reveals phases of the solution during which geophysical properties like cloud water can vary by a factor of two or more with respect to the ensemble mean. Many of the geophysical variables generated by the various process models in response to the large-scale forcing exhibit multimodal behavior, as was the case with the prognostic variables (Fig. 4). But the more important feature of the solution distributions is that they produce broader excursions from the ensemble mean. This may reflect the fact that the parameterized physics package has many degrees of freedom available to it when responding to a large-scale forcing, and therefore individual processes may exhibit larger variability than what is observed for the large-scale state variables. This highly variable behavior further underscores the need to examine ensembles in order to be able to properly assess differences in parametric approaches.

4. Alternative single-column modeling configurations

An alternative approach to the unconstrained SCM framework discussed in the previous section is to employ some form of relaxation on the solution so that the state variables never drift too far from the observations. In this case an extra term is added to the governing equations so that (1) and (2) become
i1520-0442-13-2-352-e3
where τ is an arbitrary relaxation timescale. For this type of SCM configuration a brute-force ensemble methodology cannot exploit a strategy that perturbs the initial condition since any perturbation is quickly damped by the relaxation procedure. Other methods for generating an ensemble of solutions might be to include perturbations to the adjustment timescale or some form of perturbation to the forcing time series (e.g., with an amplitude reflecting observational uncertainty). Clearly, this is not an ideal framework since it removes or severely damps the temporal degree of freedom available to the SCM, basically leaving the modeler with a diagnostic framework. Nevertheless, it can be a highly illuminating exercise to examine the role of the relaxation term in the context of the overall thermodynamic budget. This term can tell the modeler a great deal about local deficiencies in the solution as well as systematic errors in the parameterized physics response. This framework can also help to expose fortuitously good behavior that might materialize in an unconstrained solution framework, as we will show.

The standard CCM3 physics package treats the process of moist convection using the deep cumulus scheme proposed by Zhang and McFarlane (1995) in conjunction with the scheme developed by Hack (1994) to treat shallow and midlevel convection. This configuration of the model does a relatively poor job of reproducing the observed time series of precipitation, one of the few“observed” geophysical quantities against which an SCM solution can be compared. However, if we disable the deep convection scheme and allow the shallow convection scheme to treat all forms of moist convection (as was done in the CCM2), we find that the unconstrained SCM solution can reproduce the observed precipitation time series with spectacular accuracy, as shown in Fig. 8a. This panel shows the ensemble-mean time series of the total precipitation rate (generated using perturbed initial conditions) along with the ensemble standard deviation for the ARM IOP case study. An interesting feature of this figure is the absence of any significant solution variability except for those phases of the simulation exhibiting anomalous precipitation events. This illustrates one of the strengths of an ensemble approach, where the variability of the solution can be used to estimate the reliability of specific events occurring in any single realization. The overall quality of this solution turns out to be highly misleading since a more thorough examination of the other solution details reveals that the ensemble-mean temperature and water vapor distributions are severely biased, as shown in Fig. 9.

An SCM framework that includes a relaxation forcing is ideally suited to address the question of how the parameterization package would behave if subjected to the same large-scale forcing time series, but for an atmosphere constrained to be close to the observed thermodynamic state. Solutions are shown in Fig. 8b for which a relaxation term has been included in the solution with a 3-h adjustment timescale. An ensemble integration was conducted in which random perturbations were introduced into the vertical motion time series, along the lines of the previously discussed experiments. As in the unconstrained case, the precipitation solution is very stable, with little variability as quantified by the ensemble standard deviation. More importantly, the quality of the ensemble-mean precipitation forecast is markedly degraded, where major events are significantly underestimated or not captured at all. The relaxed solution shows that the precipitation time series from the unconstrained solution is a fortuitous consequence of a systematic bias that is quickly established by this physics configuration when allowed to develop its own state. In this case, the solution drifts toward a state where the lower troposphere is relatively moist and the precipitation time series is simply a reflection of large-scale water vapor convergence. Eliminating the thermodynamic bias provides a more realistic assessment of the physical parameterization response. Furthermore, the solution distribution remains well behaved, where the ensemble-mean quantities have physical significance throughout the entire integration period. Any variability in the solution is attributable to the properties of the physics response when exposed to uncertainties in the large-scale forcing, or to small differences in the large-scale state variables.

5. Concluding remarks

We have explored the solution characteristics for a commonly used single-column modeling experimental framework. This experimental approach obtains solutions to the one-dimensional form of a thermodynamic energy equation and water vapor mass continuity equation by specifying a time series for the large-scale tendency of temperature and water vapor and predicting the evolving state variables based on the parameterized physics response. The underlying assumption for this framework is that a single solution is likely to be representative.

We have employed an ensemble methodology to investigate the sensitivity of single-column modeling solutions to small perturbations in the initial condition. Our experimental approach uses two SCM intercomparison case studies, one from the tropical western Pacific, the other from the U.S. southern Great Plains. Our results reveal a strong solution sensitivity to the initial condition, where the solutions from the various members of the ensemble bifurcate and cluster to form multiple solution states. Random perturbations of the large-scale forcing terms using identical initial conditions lead to similar results.

Our unconstrained solutions can exhibit two or more solution states during various phases of the simulation where in some cases these states oscillate about the ensemble mean solution, as shown in the upper panels of Fig. 4. During these phases of the solution the ensemble mean has little physical significance since it reflects the average of two or more largely independent solution states and does not necessarily represent a realizable solution. This multiple attractor behavior is characteristic of highly nonlinear systems, and illustrates the need for the statistical characterization of single-column model solutions. The most intriguing property of these solutions is the collapse of multiple states back to a single state suggesting the presence of a strong restoring force in the system, which we believe may be associated with the SCM equilibrium state. This behavior deserves additional study, particularly with respect to the other configurations of an SCM that include fewer available degrees of freedom. One example would be to examine the role of vertical advection in a configuration where this term is specified as opposed to being evaluated on the basis of the evolving thermodynamic profile.

We have shown how an unconstrained solution can quickly establish a systematic bias, which can lead to fortuitously good solutions for some of the observed fields. The development of these thermodynamic biases could be of some value if they were representative of the biases that are established in the global model. However, our experience indicates this is not necessarily the case, where the large biases associated with unconstrained SCM solutions appear to be partially a consequence of the lack of large-scale dynamical feedbacks. Thus, errors accumulate as the model is integrated forward in time, where in the later stages of the simulation the physics is responding to a forcing that is no longer consistent with the simulated basic state. If the integration is continued for a long enough period of time, nonphysical structures can develop resulting in large solution bifurcations, as seen in the ARM case study. An example of these structures is shown in Fig. 10, which illustrates the observed and simulated ensemble-mean dry and moist static energy profiles at 240 h into the ARM simulation. This is just prior to the point where the solution begins to bifurcate leading to a handful of extreme simulation states. This figure clearly shows the nonphysical nature of the SCM solution. A feature of highly nonlinear dynamical systems is that the number of attractors tends to increase when they are forced strongly. This seems to be the case for the ARM solution where a complex thermodynamic structure slowly develops, which is then strongly forced, exciting multiple modes of behavior. The GCSS example shows that the nonphysical nature of the solution need not be so extreme, where solution bifurcations materialize after only a few days into the simulation. One must question whether this is a well-posed experimental framework, since the simulations would appear to be of little value once multiple states appear in the solution. At this point in the simulation it is far more instructive to examine the solution distribution properties as opposed to the properties of the ensemble mean.

The incorporation of a relaxation term in the governing equations has a number of desirable consequences. It has the advantage of maintaining a realistic thermodynamic profile for which the parameterized physics response to the large-scale forcing can be evaluated. Sensitivities to the uncertainties in large-scale forcing and the thermodynamic profiles can still be explored with a brute-force ensemble approach. And our tests seem to indicate that the solution distribution remains well behaved, where the ensemble mean has physical significance throughout the entire integration period.

There are many unresolved questions about the most effective experimental framework for conducting single-column modeling studies. One of the foremost questions is whether the detailed reproduction of an observed geophysical time series should be the objective, or whether the reproduction of the statistical properties of that time series is the more realistic goal. One issue that has been documented in this study is that a single-column modeling framework is a complex nonlinear system that should be treated as such. Single solutions to a particular forcing scenario cannot be assumed to be representative. Because of the highly nonlinear character of most physical parameterization packages, the establishment of solution uncertainties should be a standard component of any single-column modeling investigation.

Acknowledgments

The authors would like to acknowledge highly valuable discussions about this work with David Williamson. We sincerely appreciated his interest and suggestions. We also wish to acknowledge Jon Petch’s contributions to the early development of the NCAR single-column model. We have benefited from conversations with David Randall, Tony DelGenio, and Minghua Zhang on the subject of single-column modeling. Ralph Milliff’s comments on an early version of the manuscript were also greatly appreciated. This work was partially supported by the Climate Change Prediction Program (CCPP), which is administered by the Office of Energy Research under the Office of Health and Environmental Research in the Department of Energy Environmental Sciences Division, and by the DOE ARM Program.

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Fig. 1.
Fig. 1.

Time series of the observed large-scale advective tendency for temperature for the GCSS WG4 case 2 dataset, and days 2–8 of the ARM summer 1995 IOP case study.

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 2.
Fig. 2.

SCCM ensemble-mean temperature solution and the ensemble standard deviation for the GCSS case study. The ensemble contains 500 members and is based on perturbed initial conditions.

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 3.
Fig. 3.

SCCM ensemble-mean temperature solution and the ensemble standard deviation for the ARM case study. The ensemble contains 500 members and is based on perturbed initial conditions.

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 4.
Fig. 4.

Examples of solution distribution time series for the GCSS and ARM case studies. The figures show the number of cases exhibiting the solution plotted on the ordinate, as a function of time, for selected model surfaces. The top panels are from the GCSS case study and the bottom panels are from the ARM case study. The solid black line denotes the ensemble mean.

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 5.
Fig. 5.

Cases 111 and 175 from the 500-member GCSS ensemble. Panels denote the temperature solution difference from the GCSS ensemble mean.

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 6.
Fig. 6.

Cases 191 and 199 from the 500-member ARM ensemble. Panels denote the temperature solution difference from the ARM ensemble mean.

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 7.
Fig. 7.

Time series of the ensemble-mean column-integrated condensed water, and 250-mb diabatic heating for the ARM case study. The ensemble standard deviation is shown by the heavy dashed line.

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 8.
Fig. 8.

Time series of the observed and ensemble-mean predicted total precipitation rate for the ARM case study, which excludes the CCM3 deep convection scheme. The ensemble standard deviation is given by the dotted line. The upper panel shows the unconstrained solution and the lower panel shows the solution generated with the inclusion of a relaxation term.

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 9.
Fig. 9.

Time-averaged ensemble-mean profile of the temperature and specific humidity error for a model configuration that excludes the CCM3 deep convection scheme (see the upper panel in Fig. 8).

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

Fig. 10.
Fig. 10.

Observed and ensemble-mean predicted profiles of dry and moist static energy at hour 240 for the ARM case study (just prior to the large solution bifurcation shown in the lower panels of Fig. 4).

Citation: Journal of Climate 13, 2; 10.1175/1520-0442(2000)013<0352:AOSUIS>2.0.CO;2

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

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