The E3SM version 1 Single Column Model

. The single column model (SCM) functionality of the Energy Exascale Earth System Model version 1 (E3SMv1) is described in this paper. The E3SM SCM was adopted from the SCM used in the Community Atmosphere Model (CAM), but has evolved signiﬁcantly since then. We describe changes made to the aerosol speciﬁcation in the SCM, idealizations, and developments made so that the SCM uses the same dynamical core as the full general circulation model (GCM) component. Based on these changes, we describe and demonstrate the seamless capability to “replay" a GCM column using the SCM. We 5 give an overview of the E3SM case library and brieﬂy describe which cases may serve as useful proxies for replicating and investigate some long standing biases in the full GCM runs, while demonstrating that the E3SM SCM is an efﬁcient tool for both model development and evaluation.

only hours in duration. LHC2015 show that CAM5 simulations are very sensitive to the initialization of aerosol for stratiform boundary layer cloud cases, but not for shallow and deep convective cases (because deep and shallow convective microphysics schemes were not tied directly to the aerosol scheme). However, E3SM uses a unified treatment of shallow convection and planetary boundary layer turbulence , thus the shallow convective clouds are tied to the large-scale microphysics scheme, which could lead to more severe impacts and sensivities for the shallow and deep 70 convective cloud regimes if aerosol is not specified adequately.
We have implemented the three options proposed by LHC2015 to initialize aerosol in the E3SM SCM. The first option is to use prescribed aerosol climatology derived from a ten year E3SM present day simulation with climatologically prescribed sea surface temperatures (SSTs). The second option is to specify the droplet and ice concentrations in the microphysics, thus bypassing the aerosol-cloud interaction, and the third option is to use observed aerosol information from the intensive 75 observation period (IOP) forcing file, if it is available. Selecting an aerosol specification option is mandatory for the E3SM SCM and a runtime error will result if a user attempts to run the SCM with no aerosol specification. For the scripts provided in the E3SM SCM case library (see section 3), the most appropriate aerosol specification is already set for each particular case.
Should an E3SM user generate their own forcing and is unsure which option to select, we advise to use the prescribed aerosol specification as a default.

Idealizations
Many published LES comparison studies involving the simulation of boundary layer clouds include "idealizations". As an example, the goal of the LES intercomparison study of the Barbados Ocean and Meteolorological experiment (BOMEX; Holland and Rasmusson 1973) was to investigate the role of turbulence dynamics for the shallow cumulus boundary layer (Siebesma et al. 2002), while avoiding the complications of microphysics and radiation. As such, none of the LESs participating 85 in the study included a microphysical parameterization in their simulation. In addition, the radiative heating tendencies for the LES comparison were included in the large-scale forcing. Should an E3SM SCM user wish to evaluate the turbulence and cloud structure of the BOMEX case against the LES intercomparison study of Siebesma et al. 2002, not only would an applesto-apples comparison would not be possible with an out-of-the-box configuration of the inherited SCM, but it would also be scientifically invalid due to the fact that radiative tendencies would be double counted. While implementing these idealization 90 switches into the model code is rather trivial, it is not an obvious task for the typical SCM user who is not familiar with the code and who may not be aware of the idealizations needed to match LES results. Therefore, with the goal of preventing improper case setups, we have implemented idealization switches into the E3SM SCM code to allow for apples-to-apples comparison with IOP forcings corresponding to the appropriate reference for the particular case (see section 3). The idealization switches added to the E3SM SCM framework includes idealizations related to 95 turning off microphysics and radiation calculations. All relevant switches have been added by default to the run scripts for each particular case, but can be easily switched off by the user if they wish to examine that case using all E3SM physical parameterization schemes. Cases in the E3SM library which include idealizations turned on by default include: ATEX, BOMEX, DYCOMSRF01, DYCOMSRF02, MPACE-B, ARM shallow cumulus, and RICO (see table 1). The remaining cases have no idealizations.

Consistent Dynamical Core
The code required to run the CAM SCM has long been entangled with the Eulerian dynamical core. As a result, the SCM couldn't be run with CAM's current Finite Volume (FV; Lin and Rood 1997) dynamical core or E3SM's Spectral Element (SE; Dennis et al. 2012) dynamical core. This is a problem because while the horizontal advection fields are provided by the IOP forcing files, the dynamical core in the SCM still needs to compute the large scale vertical transport. The Eulerian 105 dynamical core uses a simple Eulerian calculation for the large scale vertical transport, while the SE dynamical core uses a semi-Lagrangian method. Therefore, the inherited SCM was inconsistent with the GCM. In addition, there are stark differences in the numerics between the two dynamical cores; whereas the Eulerian core uses a leapfrog numerical scheme, the SE dynamical core uses a forward in time integration. This results in different coupling between the prescribed and computed dynamical forcing with the physics and results in different dynamics and physics timesteps between the SCM and the GCM run, furthering the 110 inconsistencies between the two configurations.
Ideally, we want the SCM to be as close a proxy to the full GCM run as possible. Thus we upgraded the dynamical core to be the SE core for the E3SM SCM. The major challenge in achieving this goal was the fact that while it was possible to initialize the Eulerian dynamical core with one column, it is not possible to do so with the SE dynamical core which is made up of a series of "elements" on a quadrilateral grid that forms the sphere. Within these elements lie the Gauss-Lobatto-Legendre 115 (GLL) quadrature points, which also correspond to the location of the physics columns. The least invasive way to allow the SCM to work with the SE dycore was to "trick" the model by initializing the dynamical core at a low resolution configuration (we initialize the SCM at ne4 resolution, corresponding to a horizontal grid spacing of approximately 7.5 • at the equator) but initialize only one physics column. Therefore, computation for physics parameterizations are only considered for one column The E3SM SCM library is currently comprised of 25 cases, which range from widely used cases of idealized boundary layer cloud regimes of a few hours in duration to unique cases as that span the duration of years to a decade (i.e. continuous forcing from Atmospheric Radiation Measurement (ARM) Southern Great Plain (SGP) site; Xie et al. 2004). The list of available forcing files and their references can be found on tables 1 and 2. Cases such as DYCOMS, BOMEX, MPACE, RICO, and ATEX are boundary layer cloud cases that are typically used to examine performance of boundary layer, microphysics, and 130 shallow convective parameterizations, while cases such as ARM97, ARM95, TWP, and GATE are cases that can be used to evaluate shallow and deep convective parameterizations. The E3SM SCM library contains IOP files from more recent and modern cases, such as GOAMAZON, RACORO, and DYNAMO; many of these are unique to the E3SM SCM.
The E3SM SCM case library is publicly available on the E3SM SCM Github project wiki (https://github.com/E3SM-Project/scmlib/wiki/E3SM-Single-Column-Model-Case-Library). The SCM user needs only to clone the github repository, 135 which includes the scripts required to run the SCM cases. Note that the code needed to run the E3SM SCM is included with the standard E3SM release code. The user then needs to modify the header of the script for the desired case they wish to perform and then execute the script, which will compile and run the SCM for the desired case. We chose to provide and maintain separate scripts for each particular case, with the unique settings, switches, and idealizations for each case set in the script. An alternative approach is to provide the user with a universal script that can be used to run all cases and to hardcode each case into 140 the E3SM infrastructure as a particular run type (known as a "compset" in the CAM/E3SM parlance). We find that providing unique scripts for each case provides more transparency, while the details of "compsets" tend to remain under-the-hood to most E3SM users. Our approach also provides the user with more flexibility to switch on/off specific idealizations or settings, allowing them to perform sensitivity studies.

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A major advantage of the SCM using the same dynamical core as the full GCM is the ability to easily "replay" a single GCM column with a high degree of accuracy. This is a powerful tool where the user generates IOP forcing from a full E3SM run, with the intention to replicate a column of interest in SCM mode. This can be used to help diagnose model crashes due to unstable physics parameterizations, or to target and address chronic model biases in an efficient manner. It can also help to fill in the gap for a particular regime or location where there is no forcing provided by the E3SM SCM library. The inherited SCM, which 150 Though the E3SM SCM Replay option is accurate, it cannot provide a fully bit-for-bit representation of a GCM column. This is because the GCM and SCM will only give bit-for-bit answers if they do exactly the same calculations. In GCM mode, the end of dynamics state is computed via a series of sub-stepped loops. For the SCM, the net effect of these loops must be encapsulated by the end-of-step values minus the beginning of step values, divided by the timestep. This tendency is then added to the SCM state using forward Euler timestepping. Since the GCM and SCM calculations are not identical, roundoff 160 level differences occur. This issue could in principle be resolved using quadruple precision output but we found the related difficulties associated with this to not be worth a roundoff level gain. Our approximate method has proven suitable for most scientific applications of interest to E3SM users. In section 5.5, we demonstrate an example of using the Replay option.

Applications of the E3SM SCM
In this section we will demonstrate that the SCM can serve as a tool to reproduce and explore climatological biases within the 165 E3SM model. We will also show an example of when the SCM cannot be used as a proxy for the full model. Finally, we will show an example of using the E3SM Replay option. Some major biases in the E3SM model include (but are not limited to) an overestimate of clouds in the Arctic, lack of subtropical maritime stratocumulus, lack of high clouds in the Tropical West Pacific (TWP) warm pool, timing of precipitation in the tropics and mid-latitudes, and a lack of precipitation over the Amazon rainforest (Xie et al. 2018;Zhang et al. 2018;Golaz et al. 2018;Rasch et al. 2019). In this section we will attempt to replicate 170 a select number of these biases with the SCM.
Unless otherwise stated, the SCM results presented in this paper use the short-term hindcast approach (Ma et al., 2015). The SCM is initiated every day at 00Z and run for two days, with prescribed large-scale forcing, surface turbulent fluxes and no nudging. The 24 to 48 hour forecasts in each simulation are then combined as a continuous timeseries. With the hindcast approach, the model is well constrained by the large-scale condition, allowing us to isolate problems related to parameterizations.

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It also avoids the possible impacts of nudging to the clouds and precipitation (Ghan et al. 2000;Randall and Cripe 1999;Zhang et al. 2014). We will, however, explore the differences between nudging and the short-hindcast mode in one example.

Diurnal Cycle of Continental Precipitation
As already stated, SCMs are a useful tool to explore biases due to the model's physical parameterizations. But are there certain conditions and regimes under which the SCM is a better or worse proxy for the full GCM? While it has been demonstrated 180 many times in literature (e.g. Golaz et al. 2002;Bogenschutz and Krueger 2013;Suselj et al. 2013) that boundary layer cloud cases (such as DYCOMS for stratocumulus and BOMEX for shallow cumulus, as an example) can serve as a useful surrogate to explore and improve biases in the global model (due to the important cloud forming processes in these regimes being mostly locally driven), the question of whether precipitation due to deep convective processes can be replicated faithfully in SCMs is less understood. Here we will attempt to replicate E3SM's biases in precipitation, both in the mean state and variability sense, 185 to see when the E3SM SCM may be useful to exploit and investigate these biases.
The diurnal cycle of precipitation, especially over land, is a mode of climate variability that GCMs have long struggled to simulate adequately (Covey et al. 2016;Lee et al. 2007). Over land the late afternoon peak of precipitation is typically associated with the transition of shallow to deep convection while the nocturnal peak is mostly due to elevated convective systems associated with eastward propagating mesoscale convective systems. Many studies have attributed the GCMs inability August of each year. Note that multi-year SCM forcing allows us to perform robust statistical analysis rather than relying on a single case study as typically done in the past with SCM runs.
To see if we can improve this bias in the SCM, we implemented a revised convective triggering function, as described in XIE2019, which has been shown to greatly improve the diurnal cycle of precipitation in E3SM simulations. This new convective triggering is a combination of two methods, known as dynamic Convective Available Potential Energy (dCAPE) 205 and the Unrestricted Launch Level (ULL).
The top row of figure 1 displays the composite of the total precipitation from the periods sampled at the SGP site. While observations show a minimum of precipitation around noon, this is when E3SM SCM shows a maximum precipitation rate. This is representative of the bias found in E3SM simulations for a similar location over the North American Plains subset region in XIE2019, where precipitation was tied a bit too closely to solar insolation and the nocturnal peak of precipitation was 210 not represented. XIE2019 also found that after implementing the revised dCAPE and ULL triggering method the precipitation maximum was shifted to the nocturnal hours (figure 13b of XIE2019). Clearly, not only can the E3SM SCM replicate the original bias found in the global model, but the improved representation due to the new convective triggering is also depicted in our SCM experiments.
Due to the fact that the SCM can replicate the behaviors seen in the global model for this situation, we can further use this 215 SCM case to explore the exact reason for this behavior. The bottom row of figure 1 conditionally samples our dataset for days when the observed precipitation predominately happens in the afternoon and nighttime. We segregate the days with afternoon maximum precipitation by subsetting to days when the observed precipitation has a peak greater than 1 mm/day between 1300 to 2000 LST and when the peak rain rate is 1.5 times greater than any rain rate outside of 1300 to 2000 LST. The nighttime precipitation days are classified as when the rain peak is greater than 1 mm/day with a peak time between 0000 to 0700 LST.

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From this analysis, it is clear that the largest impacts from the improved triggering in terms of precipitation timing occur on days when there is a nocturnal peak of precipitation, which the default E3SM model was missing. The combination of the dCAPE trigger, which prevents the convection scheme from activating too early in the afternoon, and the ULL method which improves the elevated nocturnal convection help to shift the precipitation to the night time hours, on days when it it observed.
Thus, this case makes an example of when the SCM can serve as a good proxy to replicate and improve GCM biases, as well 225 as easily investigating under what scenarios an improved scheme is having the most impact.

Amazon Precipitation Bias
Another major bias in E3SM which is characteristic of most GCMs, is lack of precipitation over the Amazon ( of precipitation. For this location, the radar derived precipitation rate has an annual mean of 6.56 mm/day, the SCM an annual 235 mean of 6.98 mm/day, and the GCM 6.07 mm/day. Therefore, here we see an example where the SCM does not faithfully represent the bias exhibited by the GCM in terms of the climatological rate of precipitation.
The reasoning why the GCM Amazon dry bias cannot be replicated in the SCM is likely because this bias is primarily due to a misrepresentation of the large-scale environmental conditions in the GCM, rather than by parameterized deficiencies. This is important information for E3SM developers and the analysis team. Figure 3 displays the observed composite large-240 scale vertical velocity, relative humidity, and winds at the GOAMAZON location and compares these to the GCM simulated variables. The largest discrepancies between GCM and observations occur during the boreal summer months, which correlates to the period of the most pronounced bias in the climatological rain rate in the GCM. The SCM is driven by observed large-scale forcing, thus is not subject to the errors that drives the GCM biases.
In addition, it is well understood that for deep convection precipitation is usually balanced mainly by advective moisture 245 convergence, which is prescribed in these experiments. Therefore, this is a prime example of a situation where the SCM is not a useful tool to help improve GCM biases but does suggest that efforts should be spent on improving the large-scale circulation, or remote biases, which are probably responsible for the Amazon precipitation bias. The SCM, however, can replicate the bias related to the early onset of precipitation (similar to that seen in fig. 1), thus supporting the idea that the diurnal cycle involves shorter timescales and therefore looks more like the free-running GCM solution than the observed values. To test the impact of this choice, we set the Bergeron efficiency to 1.0. The result is a dramatic decrease in the amount of cloud liquid mixing ratio (third row of fig. 4 and blue curve of bottom row). This example illustrates the ease with which the SCM can be used to explore the impact of parametric assumptions. Note, however, that this quick SCM test may not always capture 265 the sensitivity of the full GCM and our quick test doesn't account for needed retuning to compensate for altered Bergeron efficiency. Though, how E3SM simulations would respond in the climatological sense, and what degree of retuning would be necessary by adjusting this efficiency parameter, is something that the SCM cannot afford to offer.

Hindcast vs. Nudging
As previously mentioned, we chose to perform the majority of experiments in this paper in short-term hindcast mode. However, 270 the E3SM SCM also comes with an option to nudge temperature and moisture to observed values. By default the E3SM SCM uses a nudging timescale of three-hours. It is interesting to note that the solution obtained for the MPACE case is strongly dependent on the technique used to constrain the mean state. The fourth row of fig. 4, which uses nudging, clearly shows a very different solution in terms of the cloud and ice mixing ratio when compared to the hindcast simulation on row two.
The simulation with nudging tends to produce less liquid cloud and virtually no ice. Randall and Cripe (1999) extensively   275 discussed the nudging method for SCMs and conclude that the impact of nudging on SCM simulation depends on the model biases produced without nudging, thus there is no solid theory on what can be expected from a particular model while using nudging. Figure 5 displays the timeseries of the observed temperature profiles for the MPACE period, in addition to the temperature biases for the E3SM SCM runs using hindcast and nudging methods. Obviously, since the nudged run is continually forced 280 towards observations, the temperature bias is near zero for the duration of the run. Conversely, the hindcast run allows the temperature biases to grow over each 48 hr run and are therefore likely to be more representative to the E3SM model bias and therefore provide a more faithful representation of the model. This begs the question; which method should be used for 9 https://doi.org/10.5194/gmd-2020-27 Preprint. Discussion started: 30 April 2020 c Author(s) 2020. CC BY 4.0 License.
E3SM SCM simulations? The answer likely depends on the goal of the particular user. If one simply wants to use the SCM as a proxy for E3SM performance, to replicate GCM biases and provide potential fixes for these biases, then running the SCM in 285 short-term hindcast or free running mode (for short IOP cases) is likely the best option. This will allow the mean state model biases to evolve, but not drift, in a manner similar to the GCM and will likely provide a more faithful representation in terms of cloud representation.
If, however, one is using the SCM for the purposes of parameterization development/implementation and wishes to assess their new parameterization in conditions with little to no mean state bias (e.g. to avoid compensating errors), then the nudging 290 method is likely preferable. For instance, the results seen using nudging vs. hindcast for MPACE clouds may suggest that Arctic clouds simulated in E3SM are an artifact of compensating errors. When the observed temperature and moisture profiles are used, we see the model struggles to produce any ice cloud at all, which is in conflict with observations. This suggests that E3SM developers may need to reevaluate either the parameterizations and/or tuning choices in order to get a desirable solution when the temperature and moisture most resemble observations. In addition, caution should be warranted when using nudging, 295 since constantly nudging to the observed temperature and moisture state inherently breaks the water and energy budget by acting as artificial source. The sequentially splitting techniques that E3SM uses could in theory be obscuring the direct effects of this, which could be leading to the artificial reduction of condensate. Though, this idea needs to be explored more. We wish to replay a column near the location where the bias is most severe. Therefore we choose a location near 5 • N and 140 • E (see yellow star in figure 6). The bias in this location is most prevalent during the boreal summer months, therefore we chose August as the month we will replay in SCM mode. In our experimental setup we simply run the GCM with climatologically prescribed SSTs for a year (starting in January) by configuring the simulation with a single directive ("-e3sm_replay"), 310 which will generate the appropriate forcing to replay a column at every E3SM timestep. To reduce the amount of output generated, we choose to do a regional subset of the forcing (instructions for this provided at the E3SM SCM wiki). We also chose to output initial condition files at the start of every month so that our SCM can start from the same state as the GCM.

Example of Using the E3SM SCM Replay Option
Once the simulation is over we use scripts provided in the E3SM case library to replay our column of choice. The inputs we need to specify are the E3SM generated forcing file, initial condition file, the latitude and longitude we wish to simulate, as 315 well the desired start date and run duration.  Figure 7 clearly shows that E3SM underestimates cloud, not only at the upperlevels but also the lower and mid levels by a substantial amount for this column. While cloud liquid mixing ratio is represented with somewhat reasonable magnitude by the GCM, cloud ice is substantially under predicted. Thus the combination of the low cloud fraction and cloud ice is likely driving the radiation biases seen in the full GCM for this region. From figure 7, it is also clear that the SCM Replay mode is also very capable of reproducing the full GCM, as cloud profiles 325 exhibit nearly identical behavior. As a reminder, fully bit-for-bit results are not expected with the E3SM Replay mode due to the fact that the dynamics tendency calculation is applied differently than in the full GCM. However, we show here that the Replay mode can faithfully represent the behavior of the GCM. While the Replay mode cannot provide information on whether the warm pool cloud bias is due to parameterization deficiencies or discrepancies in the large-scale, we can use the SCM Replay method to perturb parameterization tunable parameters in an efficient way to explore the effect they might have 330 on the high cloud bias.
An example of this, we run the SCM in Replay mode for this column but with the Bergeron efficiency set to 1.0 (blue dashed curve in figure 7), as in section 5.3. In this experiment, while we see noticeable effects in the mid-troposphere in terms of the reduction of cloud liquid, there is little effect towards the increase of cloud fraction or cloud ice mixing ratio. Simultaneously, we also performed several experiments where we perturbed the critical thresholds of the relative humidity for the ice cloud 335 fraction closure (Gettelman et al. 2010), but we saw no noticeable changes in the simulation of the cloud profiles (not shown).
While these experiments were not successful towards improving this bias in E3SM, they allowed us to efficiently rule out potential culprits in the tuning choices while avoiding wasting computational resources of testing the same experiments in long climate integrations.

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This paper describes the E3SM SCM, including modifications made to it since we adopted it from CAM, and how this configuration can be useful for model development and evaluation. A number of important upgrades were made to E3SM SCM since the inherited version, including the ability for the user to specify how the aerosols are treated to avoid unscientific case set ups due to the fact that E3SM initializes all aerosol concentrations to zero. Idealizations have also been implemented and turned on by default, depending on the IOP forcing the user selects, to ensure an apples-to-apples comparison with LES benchmarks that 345 the IOP forcing was meant to replicate. Most importantly, the E3SM SCM is now configured to work with the same dynamical core as the full GCM. This ensures that the SCM runs with the same large-scale vertical advection scheme, time step, and physics-dynamics coupling methods as the full model. It also allows the user to trivially "replay" a column of the full GCM with the SCM without the need to interpolate initial condition files or forcing files from one dynamical core to the next.
(i.e. BOMEX, DYCOMS, RICO) and standard deep convection cases (i.e. ARM97, GATE). We also include IOP forcing files from more recent and modern cases, such as GOAMAZON, RACORO, and DYNAMO; many of these which are unique to the E3SM SCM. For example, the E3SM can simulate conditions at ARM SGP for twelve continuous years. This allows for robust GCM-like statistics to be generated in a computationally efficient manner. Scripts to run each individual case are available at https://github.com/E3SM-Project/scmlib/wiki/E3SM-Single-Column-Model-Case-Library and many have been scientifically 355 validated. The user need only supply paths to relevant output directories if running on E3SM support machines.
We provide some examples of when the E3SM SCM may prove to be a useful proxy for GCM performance. For instance, we are able to successfully replicate the diurnal cycle of precipitation bias in the GCM by using forcing generated at ARM SGP.
This bias is mostly due to deficiencies in the triggering mechanism in the convective parameterization that is unable to properly handle elevated convection. By implementing the trigger improvements documented in XIE2019, we are able to reproduce the 360 same improved diurnal cycle of precipitation in the SCM found in global simulations. However, we were unable to replicate the seasonal cycle of dry Amazon bias with the SCM. We conclude that the root cause of the bias is due to improper representation of the large-scale environment rather than a deficiency with the parameterizations.
Using Arctic clouds as an example, we use the SCM to experiment with tunable parameter changes to evaluate the sensitivity of the high latitude cloud bias. We report positive effects with the tuning of one parameter for this particular regime, but we 365 caution that the SCM cannot inform how a full GCM simulation and radiation balance would be impacted with a modification.
We also compare the SCM in hindcast or free running mode versus a run where the SCM is nudged to observations. By running in hindcast or free-running mode the SCM allows the model biases in temperature and moisture to naturally develop which gives a better proxy with the model behavior in the full GCM and therefore should be used if trying to replicate E3SM behavior.
Nudging the SCM to observations may not provide a proxy with the full GCM and the behavior that could deviate significantly 370 from E3SM global runs. This mode is, however, potentially useful if trying to improve or implement a parameterization while avoiding compensating errors. We caution the user on the potential unintended consequences of adding artificial sources that nudging could introduce.
We also demonstrate that the Replay mode in the E3SM SCM can faithfully replicate a column of the GCM, though bit-forbit replication is not possible in the current implementation. This mode is useful when trying to simulate a particular regime or 375 region that the extensive E3SM case library does not cover. In our example, we replicated the high cloud bias in the Tropical Pacific Warm Pool. While the SCM cannot inform us directly whether biases are caused predominately by deficiencies in the model physics or the large-scale flow; it can provide clues about the culprit. This allows model developers to focus their energies more efficiently towards a solution.
The E3SM SCM is mature and should be a first step in the model physics development and implementation process. With the 380 extensive case library and the ability to simulate many different regimes, users can gain valuable insights on their development efforts and efficiently fix bugs. The SCM is also an important tool for addressing long standing biases in the model; its incredible efficiency makes large sets of perturbed parameter tests easy. In addition, model instabilities that may arise in the full GCM