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.
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).
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.
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. Williams