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  • View in gallery

    Schematic representation of physics–dynamics coupling. (a) Two models: an ocean model and an atmosphere model. Both of these have spatial scales (here indicated by the plane with red lines) and temporal scales (indicated by the blue axis). These are coupled (thick lines); that means one domain in the spatial plane maps into the spatial plane of the other model (thick red line) and similarly in the temporal axis (thick blue line). In the spatial plane, aspects such as grid type, fixed vs variable resolution, one-dimensional vs three-dimensional, and fine vs coarse are shown as some of the aspects of the spatial resolution that can vary between models and do not necessarily have a straightforward mapping. Then, each of these models has its ecosystem of parameterizations (an arbitrary set of processes was chosen here for illustration only), which interact with the model and themselves via coupling. These parameterizations also occupy potentially—or almost certainly—different areas on the spatial plane and temporal axis. All of this exists in front of a background problem of thermodynamics, which ultimately governs them all (or ought to, anyhow). (b) Four-tier scheme of investigation, ranging from (by necessity) abstract analysis via reduced equation sets (with less necessity for abstraction) to simplified physics tests and finally full model runs. The complexity of the analysis increases from one to the other. The manner in which the results and conclusions from the experimentation can inform the production runs ranges from “difficult” (results are expected in the form of guidance or informing a choice that needs to be made in the design phase) to “direct” (a benefit can be demonstrated straightaway by producing an improved forecast).

  • View in gallery

    Global-mean surface temperature change (K) resulting from a doubling of CO2 in simulations conducted with the ECHAM5 atmosphere model (Roeckner et al. 2003, 2006) coupled with a slab ocean. Red and blue markers indicate high- and low-sensitivity models, which differ only in a few uncertain parameters in the physics parameterizations (Klocke et al. 2011). For each time step size listed on the x axis, the global-mean surface temperature change is computed as the difference between a 10-yr present-day simulation and the last 10 years of a 50-yr simulation with doubled CO2. The spatial resolution of the atmosphere model is T31 with 19 layers. Error bars indicate interannual variability of global- and annual-mean surface temperature.

  • View in gallery

    (a) Scatterplots of cloudy mass flux against large-scale mass flux and (b) minus dry mass flux against cloudy updraft mass flux. The mass fluxes have been converted to velocities in units of m s−1 by normalization with density. The data are taken from a height of 3195 m and are averaged in the horizontal to scale of 24 km. Met Office Unified Model.

  • View in gallery

    Convergence to circulation required to maintain Ekman balance of the vertical slice primitive equation simulations (Beare and Cullen 2016) for different time-stepping schemes: implicit, K-update, and Wood et al. (2007). Ro1.7 is shown in gray for reference of the slope (y-axis intercept is arbitrary).

  • View in gallery

    Snapshots of instantaneous (left) 850-hPa vertical pressure velocities and (right) precipitation rates in MITC simulations. (a),(e) CAM-FV; (b),(f) CAM-EUL; and (c),(d),(g),(h) CAM-SE dynamical cores. (c),(g) se_ftype = 1 denotes a physics–dynamics coupling with the long physics time step; (d),(h) se_ftype = 0 couples with a subcycled, short dynamics time step. The physics time steps are 1800 (FV, SE) and 600 s (EUL); the dynamics time steps are 180 (FV), 600 (EUL), and 300 s (SE). In the case of SE with se_ftype=0, the forcing was gradually applied every 300 s. The EUL dynamical core is coupled to the physics in a process split (parallel) way; the SE and FV physics–dynamics coupling is time split.

  • View in gallery

    The 2-yr-mean zonal-mean precipitation rate in four aquaplanet simulations with the CAM5 dynamical cores SE (111 km), FV (111 km), EUL (T85), SLD (T85), and the default CAM5 physics package.

  • View in gallery

    Aquaplanet simulations with the alternative CLUBB PBL, macrophysics, and shallow convection schemes in CAM5. Latitude–pressure cross section of the 1-yr-mean zonal-mean vertical pressure velocity in the tropics for the dynamical cores (a) SE with diffusion (hyperviscosity) coefficient m4 s−1, (b) SE with diffusion coefficient m4 s−1, and (c) SLD without explicit horizontal diffusion. (d)–(f) The 1-yr-mean zonal-mean precipitation rates of the three runs, split into total (red), large-scale (green), and convective (blue) precipitation.

  • View in gallery

    Schematic view of the coupling between the computational domains of the atmosphere model and ocean model , with time advancing to the right. The function represents the parameterization of air–sea fluxes with (), the oceanic (atmospheric) state vector. Term is a given time averaging operator, and , the dynamical time step of the models such that .

  • View in gallery

    Operational ECMWF forecast with a spectral truncation T1279 (a) 16- and (b) 9-km reduced Gaussian grid. Three-day accumulated surface large-scale precipitation for forecasts starting at 0000 UTC 20 May 2015 valid at 0000 UTC 23 May 2015. (c) Study area marked with red square.

  • View in gallery

    Element polynomials in one dimension. The figure shows three elements. The edges of the elements are marked with blue arrows. The red curves are the degree 3 polynomials in each element, and, following the CAM-SE algorithm, the polynomial values from each side of an element boundary are averaged. The filled green circles show the GLL quadrature point values, and the red filled circles are the locations of the GLL quadrature points in each element for . The histogram bar shows the cell-averaged values on an physics grid (each element has been divided into three equal-sized control volumes) obtained by integrating the Lagrange basis functions over the control volumes.

  • View in gallery

    Zonal–time average (top left) surface pressure, (top right) total precipitation rate, (bottom left) total cloud fraction, and (bottom right) albedo as a function of latitude (from the equator to 80°N) for the different configurations of CAM-SE. Temporal averaging over a period of 24 months and mapping to a 1.5° × 1.5° regular latitude–longitude grid was applied for analysis.

  • View in gallery

    Influence of on the resolution sensitivity of the CAM4 physics (precipitation) to QU and VRs using MPAS-A. (a) Sensitivity of equatorial (±2° latitude) precipitation to gridcell size (x axis) in different values of R as represented by three arrows. (b) Fraction of convective precipitation as a function of R (x axis) and gridcell size (240 vs 120 km). (c) Zonal anomaly of precipitation in a VR simulation with . (d) As in (c), but a VR simulation with . (e) Zonal anomaly of velocity potential (shading) and divergent component of wind (arrows) with . (f) As in (e), but for . The solid and dashed circles in (c)–(f) represent the boundaries enclosing the domain with 30-km grid and the transition to 240-km grid domain, respectively.

  • View in gallery

    Illustration of the Ma et al. (2014) and Fowler et al. (2016) approaches for scale-aware convection using the Zhang–McFarlane closure. (a) Term τ from Ma et al. (2014) as a function of grid spacing. (b) The fractional convective cloud cover (σ; red line) and scaling factor for cloud-base mass flux used in Fowler et al. (2016). (c) The cloud-base mass flux (inside y axis) based on the Zhang–McFarlane closure with J kg−1 and J m2 kg−2 and different modifications. Dashed line is the default with s (Default); blue line is τ following Ma et al. (2014); red line is s (Grell and Freitas–Fowler); and green line is combined. The outside y axis in (c) shows the mass increment through the cloud base for s (i.e., multiply each curve by 600). (d) Mass increment through the cloud base is shown for the same cases in (c), using .

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Physics–Dynamics Coupling in Weather, Climate, and Earth System Models: Challenges and Recent Progress

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  • 1 Departamento de Oceanografía Física, Centro de Investigación Científica y Educación Superior de Ensenada, Ensenada, Baja California, México
  • | 2 Pacific Northwest National Laboratory, Richland, Washington
  • | 3 Physical and Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, California
  • | 4 National Center for Atmospheric Research, Boulder, Colorado
  • | 5 Hans Ertel Center for Weather Research, Deutscher Wetterdienst, Offenbach, Germany
  • | 6 Department of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, Michigan
  • | 7 Met Office, Exeter, United Kingdom
  • | 8 CEMPS, Exeter University, Exeter, United Kingdom
  • | 9 INRIA, University of Grenoble–Alpes, LJK, CNRS, Grenoble, France
  • | 10 ECMWF, Shinfield Park, Reading, United Kingdom
  • | 11 Royal Meteorological Institute of Belgium, Brussels, Belgium
  • | 12 Department of Physics and Astronomy, Ghent University, Ghent, Belgium
  • | 13 IAP Kühlungsborn, Leibniz–Institut für Atmosphärenphysik e.V. an der Universität Rostock, Kühlungsborn, Germany
  • | 14 Applied Numerical Algorithms Group, Lawrence Berkeley National Laboratory, Berkeley, California
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Abstract

Numerical weather, climate, or Earth system models involve the coupling of components. At a broad level, these components can be classified as the resolved fluid dynamics, unresolved fluid dynamical aspects (i.e., those represented by physical parameterizations such as subgrid-scale mixing), and nonfluid dynamical aspects such as radiation and microphysical processes. Typically, each component is developed, at least initially, independently. Once development is mature, the components are coupled to deliver a model of the required complexity. The implementation of the coupling can have a significant impact on the model. As the error associated with each component decreases, the errors introduced by the coupling will eventually dominate. Hence, any improvement in one of the components is unlikely to improve the performance of the overall system. The challenges associated with combining the components to create a coherent model are here termed physics–dynamics coupling. The issue goes beyond the coupling between the parameterizations and the resolved fluid dynamics. This paper highlights recent progress and some of the current challenges. It focuses on three objectives: to illustrate the phenomenology of the coupling problem with references to examples in the literature, to show how the problem can be analyzed, and to create awareness of the issue across the disciplines and specializations. The topics addressed are different ways of advancing full models in time, approaches to understanding the role of the coupling and evaluation of approaches, coupling ocean and atmosphere models, thermodynamic compatibility between model components, and emerging issues such as those that arise as model resolutions increase and/or models use variable resolutions.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Markus Gross, mgross@cicese.mx

Abstract

Numerical weather, climate, or Earth system models involve the coupling of components. At a broad level, these components can be classified as the resolved fluid dynamics, unresolved fluid dynamical aspects (i.e., those represented by physical parameterizations such as subgrid-scale mixing), and nonfluid dynamical aspects such as radiation and microphysical processes. Typically, each component is developed, at least initially, independently. Once development is mature, the components are coupled to deliver a model of the required complexity. The implementation of the coupling can have a significant impact on the model. As the error associated with each component decreases, the errors introduced by the coupling will eventually dominate. Hence, any improvement in one of the components is unlikely to improve the performance of the overall system. The challenges associated with combining the components to create a coherent model are here termed physics–dynamics coupling. The issue goes beyond the coupling between the parameterizations and the resolved fluid dynamics. This paper highlights recent progress and some of the current challenges. It focuses on three objectives: to illustrate the phenomenology of the coupling problem with references to examples in the literature, to show how the problem can be analyzed, and to create awareness of the issue across the disciplines and specializations. The topics addressed are different ways of advancing full models in time, approaches to understanding the role of the coupling and evaluation of approaches, coupling ocean and atmosphere models, thermodynamic compatibility between model components, and emerging issues such as those that arise as model resolutions increase and/or models use variable resolutions.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Markus Gross, mgross@cicese.mx

1. Introduction

Weather, climate, and Earth system models approximate the solutions to sets of equations that describe the relevant physics and chemistry. These equations represent, for example, balances of momentum, energy, and mass of the appropriate system. Discrete approximations in space and time to these continuous equations are necessary to solve these equations numerically. Creating a single, coherent, and consistent discretization of an entire system of equations covering the entire range of spatial and temporal scales, even for one component such as the atmosphere, is indeed challenging, if not an impossible task. Even if it is possible, the numerical solution of such a system (spanning all possible scales) is currently beyond the reach of even the most powerful computers. Therefore, the system is separated into components that are discretized mostly independently of each other and then coupled together in some manner. These components can broadly be classified as comprising the resolved fluid dynamical aspects of the atmosphere or the ocean, unresolved fluid dynamical aspects (e.g., those represented by physical parameterizations such as subgrid-scale mixing), and nonfluid dynamical elements such as radiation and microphysical processes.

The challenges associated with bringing together all the various discretized components to create a coherent model will be referred to here as physics–dynamics coupling. The term physics–dynamics coupling has evolved from the fact that the resolved fluid dynamics components are commonly known as the dynamical cores or simply “dynamics,” and the physical parameterizations that represent the unresolved and underresolved processes and the nonfluid dynamical processes are collectively referred to as “physics.” The weather, climate, and Earth system modeling communities have relatively recently started to make focused efforts on addressing physics–dynamics coupling in the broader sense as a topic by itself (Gross et al. 2016a).

Figure 1a schematically shows the variety of model components and the different aspects of discretizing them in both space and time, as well as the coupling between them. For simplicity, Fig. 1a includes only two component models: the atmosphere and the ocean. However, modeling systems often include a large number of other components, such as land, glacier, sea ice, atmospheric chemistry, and ocean biogeochemistry models. These components are inherently coupled to each other through the momentum, mass, and energy exchanges at their interfaces.

Fig. 1.
Fig. 1.

Schematic representation of physics–dynamics coupling. (a) Two models: an ocean model and an atmosphere model. Both of these have spatial scales (here indicated by the plane with red lines) and temporal scales (indicated by the blue axis). These are coupled (thick lines); that means one domain in the spatial plane maps into the spatial plane of the other model (thick red line) and similarly in the temporal axis (thick blue line). In the spatial plane, aspects such as grid type, fixed vs variable resolution, one-dimensional vs three-dimensional, and fine vs coarse are shown as some of the aspects of the spatial resolution that can vary between models and do not necessarily have a straightforward mapping. Then, each of these models has its ecosystem of parameterizations (an arbitrary set of processes was chosen here for illustration only), which interact with the model and themselves via coupling. These parameterizations also occupy potentially—or almost certainly—different areas on the spatial plane and temporal axis. All of this exists in front of a background problem of thermodynamics, which ultimately governs them all (or ought to, anyhow). (b) Four-tier scheme of investigation, ranging from (by necessity) abstract analysis via reduced equation sets (with less necessity for abstraction) to simplified physics tests and finally full model runs. The complexity of the analysis increases from one to the other. The manner in which the results and conclusions from the experimentation can inform the production runs ranges from “difficult” (results are expected in the form of guidance or informing a choice that needs to be made in the design phase) to “direct” (a benefit can be demonstrated straightaway by producing an improved forecast).

Citation: Monthly Weather Review 146, 11; 10.1175/MWR-D-17-0345.1

The parameterizations are typically organized by processes: for example, cumulus convection and cloud microphysics in the atmosphere and lateral and vertical mixing in the ocean. Some of these processes are symbolized in Fig. 1a by clip art icons. Processes reside at different locations in the time–space domain. For example, the characteristic time scales associated with cloud microphysics and planetary-scale advection are vastly different. It can also be shown that the model performance can be improved by grouping specific parameterizations together and using predictors to adjust the input from the dynamics into the parameterizations (Wedi 1999), sampling different times on the time axis.

The wide ranges of spatial and temporal scales that are associated with the different components of weather, climate, and Earth system models have naturally resulted in different focuses in research. The continual increase of resolution means that increasing specialization is needed to address the physical processes that emerge on smaller and smaller scales as the grid size decreases. This specialization inevitably leads to the compartmentalization of the model codes and development teams. This compartmentalization and separation is necessary to understand and gain insights into the complex system and to render the model development manageable and tractable, but they are also in direct conflict with the desirability of unifying processes to allow tighter coupling and to eliminate coupling errors. This conflict is one of the most significant challenges for physics–dynamics coupling.

The compartmentalization leads to what is known as splitting, in which the impact of a process on the evolution of a model state is evaluated in isolation. Splitting assumes that the processes are either evaluated based on the same state and, hence, do not see the impact of other processes on that state, or they are evaluated sequentially (Donahue and Caldwell 2018). Both approaches are inaccurate reflections of reality. While splitting is useful and often unavoidable, it can lead to undesirable features in the numerical solutions. For example, process splitting can impact the model performance when processes compete for limited resources (such as the total water content of a parcel of air). This competition is particularly acute if processes are allowed to operate in isolation for a discrete time that is longer than their appropriate physical time scale. The modeling errors inevitably introduced by splitting are a core theme of the present paper.

Related to weather forecasting models are the examples presented below, such as the negative bias in the 24-h wind forecast noted by Beljaars et al. (2004); the accumulation of convective available potential energy (CAPE), allowing convection to initiate farther from the equator (Williamson and Olson 2003); impacts on the intertropical convergence zone (ITCZ), which may be relevant to forecasting in areas close to the equator; the coupling between the atmosphere and the ocean, which, despite the slowly changing ocean state, has been shown to be vital for forecasting on time scales of hours to weeks (Smith et al. 2018); and examples of the coupling of a weather forecasting model coupled to a regional oceanic model for a realistic simulation of a tropical cyclone. The gray zone topic features examples of gridpoint storms and operational forecasting of downbursts and scale implications on the forecast error growth. The illustrations made using examples of climate models apply directly to both the Earth system models and weather models, though the shorter forecast period may mean that some errors do not manifest themselves directly.

In early coupled climate models, such as the simulations of the global atmospheric circulation coupled to ocean processes presented in the late 1960s by Manabe and Bryan (1969), the much lower spatial resolution and much simpler model formulation were the dominant sources of model error. However, the rapid enhancement of computing capabilities has allowed for a substantial increase in model resolution as well as the incorporation of a much more comprehensive description of subgrid-scale phenomena, such as a more detailed description of microphysical processes. These advances have led to reduced errors in the individual model components. However, the benefits of this reduction in error will not be fully realized if the errors introduced by the coupling between components are not also reduced. Thus, numerical issues in coupling can be a bottleneck in the reduction of overall model error. Therefore, the formulation and implementation of the coupling—ideally, as a minimum—should

  • represent correct asymptotic behavior (see sections 2 and 3);
  • not introduce additional errors between different components, such as atmosphere and ocean (or at least the errors introduced should be smaller than the errors of each of the components; see section 5);
  • respect the physical laws such as conservation of mass, momentum, and energy and the laws of thermodynamics (see section 6);
  • represent accurately the interaction between components that represent a possibly vast range of time and space scales (see section 7);
  • accommodate different types of discretization methods (e.g., spectral transform vs finite difference or finite element methods; see section 8a); and
  • allow the possible use of different resolutions between components including variable and uniform resolutions (see section 8b).
Therefore, as Fig. 1a illustrates, physics–dynamics coupling is not limited only to the interaction between physics and dynamics. A key challenge is the design of time–space integration schemes for the different components that, when combined, reproduce the time–space-averaged behavior of the whole system being modeled.

The remainder of the paper is organized as follows. Section 2 focuses on issues related to process splitting in the time-stepping algorithm. The time–space convergence behavior of current models is also discussed. Section 3 then proceeds to illustrate convergence from the perspective of time–space averaging and the assumption of separation of scales, as well as how to accurately reproduce the asymptotic limits when subgrid transports play a crucial role. Section 4 emphasizes that ideally, there would be a standard test procedure and established benchmark results across a whole range of models, with tests that isolate the components while still reflecting the model complexity and hence maintaining relevance. Section 5 focuses on the coupling between different models, such as atmosphere and ocean. Section 6 highlights the need for thermodynamic compatibility with the laws of thermodynamics. Sections 7 and 8 discuss the complexity of the interaction of parameterizations with increased model resolution, with that increase being either throughout the model domain or through the use of variable resolution within a model domain. Section 8 discusses new and emerging modeling strategies of separating physics and dynamics grids (section 8a) and how time stepping/process splitting (section 2) and scale awareness of deep convection (section 7) can interact and pose a challenge to models using spatially varying horizontal resolution (section 8b). The paper finishes with conclusions and an outlook (section 9).

2. Time-stepping errors introduced by splitting

Models rely on discretizing time and space dimensions to solve their equations numerically. These discrete time steps and grid spacings need to be relatively large to make calculations computationally affordable. Numerical errors arise from both the spatial and temporal discretizations. In this section, the focus is exclusively on time discretization by discussing model behaviors with fixed spatial resolution and different time steps.

a. Impact of time-stepping errors

Time step size can have a substantial impact on the behavior of weather and climate models. For example, one metric of interest for future climate prediction is the change in global-mean surface temperature resulting from a doubling of carbon dioxide (CO2) concentration in the atmosphere. This temperature change was shown to vary by a factor of 2 in one version of the ECHAM5 climate model (Roeckner et al. 2003, 2006) when the model’s time step size was varied between 5 and 40 min (Fig. 2). While solution sensitivity to time step size is not at all surprising from a mathematical perspective, such large discrepancies are undesirable numerical artifacts for model users who assume the models reflect the state-of-the-art understanding of the workings of the real-world system.

Fig. 2.
Fig. 2.

Global-mean surface temperature change (K) resulting from a doubling of CO2 in simulations conducted with the ECHAM5 atmosphere model (Roeckner et al. 2003, 2006) coupled with a slab ocean. Red and blue markers indicate high- and low-sensitivity models, which differ only in a few uncertain parameters in the physics parameterizations (Klocke et al. 2011). For each time step size listed on the x axis, the global-mean surface temperature change is computed as the difference between a 10-yr present-day simulation and the last 10 years of a 50-yr simulation with doubled CO2. The spatial resolution of the atmosphere model is T31 with 19 layers. Error bars indicate interannual variability of global- and annual-mean surface temperature.

Citation: Monthly Weather Review 146, 11; 10.1175/MWR-D-17-0345.1

Sensitivity experiments like the one shown in Fig. 2 are rarely conducted with weather and climate models. Hence, the magnitude of the numerical artifacts is unclear in most models. In practice, model developers often tend to use the longest possible step size and then go through a time-consuming tuning process in which uncertain model parameters are adjusted to match the model output with a chosen set of observations (Hourdin et al. 2017). One can argue that it might be possible to “tune away” the time step sensitivity by using different parameter values for different step sizes; however, there exists the danger that such tuning might result in error compensation that cannot be guaranteed for simulations under different forcing scenarios. Revision of the model and subsequent reduction of the time step sensitivity can provide confidence that results from the numerical models are reasonably accurate solutions of the underlying continuous physics equations, hence improving the credibility of future climate projections.

Strong sensitivities to model time step have been seen in other models as well. Wan et al. (2014) showed that when the physics time step was reduced from the default 30 to 4 min in the Community Atmosphere Model (CAM) version 5, the simulated December–February mean, globally averaged large-scale precipitation rate, liquid water path, and ice water path increased by about 10%, 20%, and 30%, respectively. Zhang et al. (2012) found that the impact of swapping aerosol nucleation parameterizations on sulfuric acid gas and aerosol concentrations was overwhelmed by the effect of changing the time-stepping scheme used for solving the sulfuric acid gas equation in the aerosol–climate model (ECHAM-HAM). For the Integrated Forecast System (IFS), Beljaars et al. (2004) showed that the root-mean-square difference in 10-m wind speed between two 24-h weather forecasts conducted with 10- and 5-min step sizes was 1.39 m s−1. They also showed that this root-mean-square difference could be reduced by about 1/2 when the numerical coupling between the dynamical core and turbulent momentum diffusion was revised to ensure a proper balance between the two processes.

Williamson (2002) mentioned that when the splitting method within the parameterization suite was modified, the National Center for Atmospheric Research (NCAR) Community Climate Model (CCM) version 3 (CCM3) produced a climate equilibrium that was substantially different from the default model in some small contiguous areas. In other areas, the climates were similar, but the balances producing them were different. Most of the studies cited above and the additional examples mentioned below indicate that it is often the combination of coupling between processes and long time steps that cause time-stepping problems in contemporary models. The remainder of this section is focused on coupling issues, though it is acknowledged that long time steps can cause issues within individual processes as well.

b. Splitting in the solution procedure

The process coupling discussed in this section includes the relationship between different parameterizations, the connection between a parameterization and the host model or between different physical phenomena within an individual parameterization. Splitting is employed to evaluate the tendency terms for each process and to combine their effects to advance the discrete solution in time.

The two most popular methods of splitting in operational models are sequential and parallel splitting. In sequential splitting, tendencies of the explicit processes are computed first and are used as input to the subsequent implicit fast process. Sequential splitting is in contrast to parallel splitting, where tendencies of all the parameterized processes are computed independently of each other, using the same fixed state from the beginning of the time step. In other words, in parallel splitting, the individual process can only react to the tendencies from the other processes in the subsequent time step.

Beljaars et al. (2004) advocate sequential splitting with processes ordered from slowest to fastest to allow processes to feed and balance each other within each model step. The benefits of sequential splitting depend on what information from an already-calculated process is used in subsequent process calculations. The IFS uses both state information and tendencies from previous processes in some subsequent process calculations (hereafter referred to as sequential tendency splitting). Therefore, processes see the tendencies of some of the prior processes, but the model state is updated at the end of the time step. CAM physics uses sequential update splitting, where a process operates solely on the model state updated by the immediately preceding process. Since sequential tendency splitting shares more information than sequential update splitting or parallel splitting, it unsurprisingly performs better. More sophisticated coupling has also been shown to be beneficial for specific processes. For example, in the Semi-Lagrangian Averaging of Physical Parameterizations (SLAVEPP) algorithm of Wedi (1999), the tendencies are evaluated at both the departure and arrival points of the semi-Lagrangian trajectory and then averaged.

c. Issues with splitting

Splitting causes an error when interacting processes are considered in isolation. The errors can be large—and the numerical solutions can depend strongly on process ordering—when splitting is used in combination with time steps on the order of, or longer than, the inherent process time scales. Two types of process interactions are commonly seen in the atmosphere: competition and compensation. Competition refers to cases where multiple processes consume the same resources (e.g., cloud water or CAPE), whereas compensation relates to cases where one process is a source for something the other process consumes. A situation for competition arises in the consumption of CAPE, which can be removed by shallow convection, deep convection, or resolved-scale motions. Williamson (2013) provides an example of competition for CAPE in a sequential update split model. Explicit stratiform condensation is considered a fast process in CAM4, and the associated latent heating is applied in a single time step as a hard adjustment, while CAM4’s deep convection parameterization has a fixed time scale of 30 min for CAPE removal. When the model time step is shortened, the ability of these processes to consume convective instability is altered: the fixed time scale process does less, and the hard adjustment does more, resulting in extreme vertical motion and heavy precipitation due to the interaction between the dynamics and the parameterizations. While this might be described as a time step sensitivity, it is instead a sensitivity to the ratio of parameterization time scales, which changes with time step.

Less severe sensitivities have been observed by other investigators in scenarios of competition between processes. Mishra and Sahany (2011) found sensitivity to time step in the average tropical rainfall amount in CAM3 multiyear simulations, noting it was associated with the change in partitioning between convective and large-scale precipitation. Reed et al. (2012) showed sensitivity in the strength of idealized tropical cyclones in high-resolution CAM5 to time step, relating it to the accompanying change to the partitioning between convective and large-scale precipitation. In both studies, the time scale of the convection was not changed, and thus, the ratio of time scales changed. This issue of partitioning is a typical symptom observed in models that use spatial resolutions in the gray zone of cumulus convection (section 7). Although the examples cited above are all from models that use sequential splitting, competition for resources is also a problem for parallel splitting because it can result in unrealistically strong removal of resources. The most egregious cases of this are, for example, negative concentrations of water vapor, hydrometeors, or other tracer species. These are typically resolved by rescaling tendencies to prevent overconsumption. This approach may leave more subtle cases untreated and, where applied, results in transport that does not locally satisfy the transport equations of the model.

Another example of the competition problem was shown by Wan et al. (2013), in which the sulfuric acid condensation and aerosol nucleation acted as two sink processes in the sulfuric acid gas budget in the ECHAM-HAM model. They argued that more accurate simulations of the process rates—and consequently, more accurate near-surface concentrations of aerosol particles and cloud condensation nuclei—can be obtained when a solver handles the competing processes simultaneously without splitting.

The second type of process interaction that can cause a potential splitting problem is the cases of compensation; that is, one process acts as a source for something, whereas the other process acts as a sink. If these processes are coupled by sequential update splitting, the first process might push the quantity of interest to unreasonably high levels, while the second process might pull it to unreasonably low levels. With parallel splitting, the consuming process does not see a state immediately influenced by the source process until the following time step, by which time the excess may have been modified by some other process. An example of such a push/pull problem with sequential update splitting in CAM5 was presented by Gettelman et al. (2015), who note that macrophysics, the interplay of condensation/evaporation and cloud fraction, is the primary source of cloud water, which is subsequently depleted by microphysical processes. By substepping macro- and microphysics together two times during the typical 5–30-min time step, they were able to obtain more realistic model behavior. Wan et al. (2013) describe another push/pull problem related to the sulfuric acid gas budget in ECHAM-HAM. The study compared multiple time-stepping schemes for the coupling of sulfuric acid gas production and condensation. Results show that when the discrete time step is long, compared to the characteristic condensation time scale, sequential splitting between production and condensation leads to a substantial overestimate of the condensation rate, even when the individual processes are represented with accurate solutions of the split equations. When practical to do so, the strongly interacting sources and sinks should be solved simultaneously. A third example is presented by Beljaars et al. (2004) for the IFS. The near-surface wind speed is mainly affected by the pressure gradient force, the Coriolis force, and the turbulent friction. Sensitivity tests showed that if the turbulent diffusion coefficients are computed after the model state variables have been updated by the dynamics-induced tendencies, positive biases in the intermediate wind speeds will lead to overestimation of turbulent friction and thus negative bias in the 24-h wind forecast. These results underline further the relevance of coupling aspects, not only for climate, but also for short- and medium-term weather forecasts.

Splitting would not cause severe problems in the cases of process competition or compensation if the model time step were sufficiently short to resolve the time scales associated with the individual processes and their interactions. In that scenario, the processes—although isolated during a single short time step—could interact indirectly with each other at the next time step via the updated model state. However, many of the parameterized processes are fast, and long model time steps are not uncommon in operational models where the time step correlates with computational cost. Gettelman et al. (2015) note that sequential update splitting with forward Euler time stepping in CAM5 microphysics creates negative cloud water when computed tendencies are multiplied by inappropriately long time steps. This negative cloud water then needs to be removed by schemes that are not physically motivated by the underlying transport equations, such as rescaling, as noted above. Williamson and Olson (2003) found that aquaplanet simulations conducted with the NCAR CCM3 model had a single narrow peak of zonal-mean precipitation at the equator when the Eulerian dynamical core was used, while simulations using the semi-Lagrangian dynamical core had a double ITCZ. A double ITCZ is characterized by a precipitation minimum at the equator and two maxima that are straddling the equator. This sensitivity was attributed to the different time step sizes used for the physics parameterizations in the two model configurations (20 min for Eulerian, 60 min for semi-Lagrangian) rather than the dynamical cores themselves. The explanation the authors provided was that with sequential splitting, longer time steps lead to the accumulation of more CAPE, allowing convection to initiate farther from the equator. The resulting condensational heating and secondary circulation further reinforce convection away from the equator. Similar changes to ITCZ shape in aquaplanet simulations with the CAM3 model have also been reported by Li et al. (2011).

d. Addressing the splitting problem

Tighter coupling between processes is necessary to alleviate the splitting problems noted in sections 2ac. From the perspective of time discretization alone, three strategies have been seen in the literature. The first strategy is the use of shorter time steps to subcycle clusters of strongly interacting processes while keeping the step size of the rest of the model unchanged. Such treatment is applied to large-scale condensation and cloud macrophysics in some versions of CAM5 and its successors (e.g., Gettelman et al. 2015). The second strategy uses sequential tendency splitting to allow faster processes to better react to the effects of slower processes, like the IFS example of dynamics–turbulence coupling in weather forecasts (Beljaars et al. 2004), mentioned earlier in this section. The third strategy is the use of specially designed solvers to handle multiple processes simultaneously, such as the sulfuric acid gas equation example by Wan et al. (2013) discussed in section 2c. Methods of the second and third strategies can be somewhat involved, and their feasibility will depend on the design of the specific parameterizations. Since it can be challenging to formulate a coherent numerical coupling for complex parameterizations that might have been designed with different concepts and use different prognostic variables, attempts to account for process interactions in the continuous or semidiscrete formulation of the equations could also be helpful. For example, thermal instability diagnosed directly from radiative heating profiles is considered in the calculation of entrainment at the top of the cloudy boundary layer in the turbulence schemes by Lock et al. (2000) and Bretherton and Park (2009), which improves the radiation–turbulence coupling from the perspective of time stepping. Some modern parameterizations are designed to handle multiple atmospheric processes in a unified way. Examples include the eddy diffusivity–mass flux (EDMF) scheme of Siebesma et al. (2007) and the Cloud Layers Unified by Binormals (CLUBB) scheme of Golaz et al. (2002a), both of which combine the representations of turbulence and shallow convection. Another example is the parameterization of Park (2014) that represents both shallow and deep convection. Such unified parameterizations provide an opportunity to handle better the interactions between the processes they unify, although those parameterizations can still have strong interactions with other parameterizations, and the time stepping has to be implemented carefully. For instance, CLUBB and cloud microphysics are subcycled together in recent versions of CAM to achieve a tighter coupling.

e. Assessment of time step convergence

Complementary to the design of tighter coupling methods, an assessment of solution behavior in the regime of very short step sizes may provide information to help achieve the ultimate goal of higher accuracy at longer step sizes. In the development of time integration methods for differential equations, convergence analyses that examine whether the numerical error decreases with step size at the expected rate are one of the standard ways for verifying whether the discrete methods and code implementation lead to the intended outcome. Applications of such analysis to the physics parameterizations or full complexity models are rarely seen in the literature. The lack of interest is partly attributable to the concern that physical parameterizations are often designed to work within a particular range of time step sizes, and to use the parameterizations outside of that range may violate physical assumptions, resulting in the model state converging to an unintended or unphysical state. We argue that ideally, the physical assumptions and numerical methods should be clearly separated; the purpose of a time step convergence analysis should be the identification of issues in the numerical methods.

In the absence of analytic solutions, a “proxy ground truth” is needed in a convergence analysis. Recent studies by Teixeira et al. (2007) and Wan et al. (2015) attempted to establish a “proxy ground truth” by running the Navy Operational Global Atmospheric Prediction System (NOGAPS) and CAM5 models with small time step sizes. Wan et al. (2015, p. 216) argued that “convergence toward this proxy [ground truth] is a necessary but insufficient condition for the convergence toward the true solution.” In Teixeira et al. (2007), NOGAPS was found to converge at a first-order rate near the start of the simulations, but the chaotic nature of nonlinear dynamical systems eventually caused simulations with different step sizes to diverge into uncorrelated sequences of weather events, hence loss of convergence. Hodyss et al. (2013) demonstrated with simplified models that when the time stepping scheme does not resolve the parameterized physical processes, the numerical solutions will behave as predicted by the theory of stochastic differential equations. The 1-h simulations that Wan et al. (2015) conducted with the CAM5 model converged at a rate of 0.4 instead of the expected value of 1.0, and the cause was unclear. Given the rare application of such analyses, the understanding of time step convergence in weather, climate, and Earth system models is very limited. Nevertheless, Wan et al. (2015) showed that convergence analysis conducted with individual components of a model could indicate which parts have stronger time step sensitivity and thus require more attention in future development.

With these real-world issues and examples in mind, the paper now proceeds into a more theoretical area, a mathematical analysis approach to the coupling, moving toward the bottom-left of the graph in Fig. 1b.

3. Insights from models with simplified equation sets

In the following two examples, the resolved scale behavior is strongly dependent on the subgrid-scale dynamics. First, the interaction of convection with dynamics is examined, followed by the boundary layer with dynamics. This discussion highlights situations where the combination of resolved and subgrid terms is critical (e.g., in representing the total transport as the sum of resolved and subgrid transport). As the averaging scales such as time step and grid resolution are reduced, the subgrid contribution will diminish and be taken over by the resolved contribution.

a. Interaction of convection with balanced dynamics

In the interaction of convection with balanced dynamics, the spatial averaging scale is assumed sufficiently large, and therefore the semigeostrophic model, which is an accurate approximation to the governing equations on large scales (Cullen 2006), can be used as a proxy for the evolution of the spatially averaged equations. This analysis has the advantage that the “proxy ground truth” (section 2e) is known. The behavior of this model can then be compared with solutions of the exact governing equations with a much finer averaging scale, which consequently resolve convection explicitly. The observed behavior then has implications for the design of models with parameterized convection.

The semigeostrophic model includes the effects of large static stability variations, which are essential in considering interactions with convection. For illustration, the incompressible Boussinesq form of the equations in Cartesian geometry is used. This form uses the ageostrophic wind equation [Eq. (A1)] with the “potential vorticity” matrix [Eq. (A2)] and forcing H [Eq. (A3)]. This forcing includes momentum and thermodynamic forcing terms.

Under semigeostrophic dynamics, the ageostrophic flow is determined diagnostically and includes subgrid as well as resolved fluxes. The ageostrophic motion thus represents a response to the dynamical and physical forcing represented in Eq. (A1). The strength of the response is determined by the eigenvalues of , which represent the inertial and static stability of the atmospheric state. The geostrophic state would be expected to be described by the resolved flow in numerical models. However, the ageostrophic circulation required to maintain geostrophic balance would include subgrid-scale transports as well as resolved ageostrophic transport.

In the presence of moisture, the static stability is reduced by latent heating. This reduction of stability could be expressed, neglecting the condensate loading term in the buoyancy, by replacing the potential temperature θ with the equivalent potential temperature in saturated regions. In the presence of moist instability, would then have a negative eigenvalue. As illustrated by Holt (1989), this will generally result in convective transport rather than continuous vertical motion. The effect is that convective updrafts with any associated convective downdrafts would replace the ascending part of the ageostrophic circulation, while the compensating circulation would be a smooth transport.

The semigeostrophic formulation identifies the convective locations by a negative eigenvalue of the matrix and generates the upward mass transport as modeled by a convection scheme. The downward branch would be determined by mass continuity and the need to maintain balance in the environment. In the tropics, this leads to spreading of the response over a wide area. This process is illustrated using a convection-permitting simulation performed as part of the Earth System Model Bias Reduction and Assessing Abrupt Climate Change project (EMBRACE; http://cordis.europa.eu/project/rcn/99891_en.html). The simulation uses a configuration similar to that used operationally at the Met Office for the United Kingdom–area short-range weather prediction [see Holloway et al. (2012) for details] but with changes made to improve the representation of tropical convection and gravity waves. In this configuration, the model has a horizontal grid spacing of 2.2 km with an 8800 km × 5700 km domain centered on the tropical Indian Ocean and 118 vertical levels with a 78-km lid. Within its domain, the convection-permitting simulation was run freely after being initialized from the operational Met Office global model analysis valid at 0000 UTC 18 August 2011. The lateral boundary conditions were provided every time step by a global model that was reinitialized from Met Office operational analyses every 6 h. The data presented here were taken from 0000 UTC 30 August 2011 after the convection-permitting simulation was fully spun up.

The grid points are classified as cloudy or dry, depending on the presence or not of cloud condensate: the cloudy areas are further subdivided into ascending and descending. The grid points are then aggregated onto a coarser, 24-km grid. This 24-km grid represents a typical resolution at which a convective parameterization is used. Then, for each 24-km grid point, cloudy and dry mass flux, cloudy updrafts and downdrafts, and the total large-scale mass flux are obtained by summing the vertical mass fluxes in the respectively partitioned grid points.

Figure 3 shows that for 24-km grid points that have some cloud, there is a close match between the total large-scale mass flux and the cloudy mass flux, the sum of up- and downdraft; hence, most of the vertical motion happens within the cloudy areas (section 7b). The values of the dry mass flux are unrelated to the cloudy updraft mass flux. Hence, the local compensating subsidence within the 24-km grid box does not match the net upward cloudy mass flux, as is usually assumed in convective parameterizations. The subsidence is instead spread over the whole domain. This spreading is in agreement with the idea that the ascent is represented by convective updrafts, while the subsidence is spread over a much broader region (Bretherton and Smolarkiewicz 1989). This exposition suggests that a radical rethink of convective parameterization strategy is required. An example for convection is the parameterization of Grell and Freitas (2014) or the even more radical approach of Kuell et al. (2007).

Fig. 3.
Fig. 3.

(a) Scatterplots of cloudy mass flux against large-scale mass flux and (b) minus dry mass flux against cloudy updraft mass flux. The mass fluxes have been converted to velocities in units of m s−1 by normalization with density. The data are taken from a height of 3195 m and are averaged in the horizontal to scale of 24 km. Met Office Unified Model.

Citation: Monthly Weather Review 146, 11; 10.1175/MWR-D-17-0345.1

b. Interaction of the boundary layer with balanced dynamics

Shifting the focus from convection to the boundary layer, the effectiveness of different coupling strategies is compared using a simple model as the asymptotic limit of the full equations. A large-scale balance is defined, which should be represented in the resolved numerical solutions, while the circulation required to maintain it will be described by both resolved and subgrid-scale transports. The inclusion of the boundary layer makes a fundamental change to the large-scale balance because of the need to satisfy the no-slip boundary condition. Thus, the balance is defined by the Ekman relations
e1
where are the components of the Ekman velocity, and and , subcomponents of the and introduced above, represent the parameterized friction terms, which will depend on the horizontal momentum as indicated, as well as the thermodynamic structure. These equations can be solved for , given that at the top of the boundary layer and is zero at the ground.

Beare and Cullen (2013) derive equations analogous to Eq. (A1) for the circulation required to maintain Ekman balance in time in the presence of dynamical and physical forcing. The ageostrophic circulation in semigeostrophic theory is an accurate second-order approximation in Rossby number to the velocity in the Euler equations. However, the equivalent circulation in the boundary layer is only first-order accurate, as is the Ekman balance itself.

The effectiveness of schemes to couple the boundary layer with the balanced dynamics is demonstrated by following the method of Cullen (2007). This experiment is described in detail by Beare and Cullen (2016). A vertical slice model is used to construct a sequence of solutions of the boundary layer driven by a baroclinic wave where the Rossby number , with U and L denoting horizontal velocity and length scales, respectively, is progressively reduced. This reduction is achieved by maintaining the same initial structure in the pressure and potential temperature while simultaneously increasing the Coriolis parameter and decreasing the wind speed. The difference between the circulation predicted by the balanced Eq. (1) and the solution of the hydrostatic equations is then calculated. The convergence behavior of the balanced solution to the solution of the hydrostatic primitive equations is as expected. The convergence is of second order outside the boundary layer and first order inside. However, the boundary layer becomes shallower as the Rossby number (Ro) is reduced, giving an overall convergence rate of Ro1.7.

Results are compared using three numerical implementations: standard implicit time stepping, the Wood et al. (2007) scheme, and the K-update scheme. The control simulation uses standard implicit time stepping, but the mixing coefficients and are evaluated only at the beginning of the time step. The Wood et al. (2007) scheme is a stable single step scheme that is unconditionally stable and second-order accurate. This stability and accuracy is achieved by assuming a polynomial dependence of on wind speed. The K-update scheme includes the updated value of the boundary layer mixing coefficient at the new time level in each time step as described by Cullen and Salmond (2003), as well as the more accurate representation of the diffusion process in Wood et al. (2007). This inclusion allows the scheme to represent the balanced solution more accurately.

Figure 4 shows the difference between primitive equation simulations using different boundary layer time-stepping schemes and the balanced model. At smaller Rossby numbers, all primitive equation models follow the ideal Ro1.7 line. However, above Ro = 0.08, the primitive equation model using the implicit scheme starts to deviate significantly above the ideal line and no longer converges at the required rate. The primitive equation model using the K-update scheme deviates slightly above the ideal line at Ro = 0.1. The hydrostatic primitive equation (HPE) model using the Wood et al. (2007) scheme follows the ideal Ro1.7 line for the range of Ro shown. Both the K-update and Wood et al. (2007) schemes account for the variation of the boundary layer diffusion across the time step, giving the improved convergence properties compared to the implicit scheme. The deviation from the Ekman-balanced models thus exposes differences in the numerical methods employed.

Fig. 4.
Fig. 4.