the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Energyconserving physics for nonhydrostatic dynamics in mass coordinate models
Mark A. Taylor
Peter A. Bosler
Christopher Eldred
Peter H. Lauritzen
Motivated by reducing errors in the energy budget related to enthalpy fluxes within the Energy Exascale Earth System Model (E3SM), we study several physics–dynamics coupling approaches. Using idealized physics, a moist rising bubble test case, and the E3SM's nonhydrostatic dynamical core, we consider unapproximated and approximated thermodynamics applied at constant pressure or constant volume. With the standard dynamics and physics timesplit implementation, we describe how the constantpressure and constantvolume approaches use different mechanisms to transform physics tendencies into dynamical motion and show that only the constantvolume approach is consistent with the underlying equations. Using time step convergence studies, we show that the two approaches both converge but to slightly different solutions. We reproduce the large inconsistencies between the energy flux internal to the model and the energy flux of precipitation when using approximate thermodynamics, which can only be removed by considering variable latent heats, both when computing the latent heating from phase change and when applying this heating to update the temperature. Finally, we show that in the nonhydrostatic case, for physics applied at constant pressure, the general relation that enthalpy is locally conserved no longer holds. In this case, the conserved quantity is enthalpy plus an additional term proportional to the difference between hydrostatic pressure and full pressure.
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The primary motivation of this study is to improve energy treatment in the atmospheric component of the Energy Exascale Earth System Model (E3SM) (Golaz et al., 2019, 2022). This component, called the E3SM Atmosphere Model (EAM) (Rasch et al., 2019), is a close cousin of the Community Atmosphere Model (CAM) (Neale et al., 2012) of the Community Earth System Model (CESM), and the design choices regarding thermodynamics in EAM are inherited from CAM (Williamson et al., 2015).
A recently published comprehensive overview by Lauritzen et al. (2022) of thermodynamics in global atmospheric models describes a few deficiencies in EAM/CAM, including the inconsistent treatment of enthalpy (or energy) fluxes of water forms, both within the atmosphere and across its surface boundaries. In particular, there is a large disagreement between the enthalpy fluxes internal to the model as compared to the enthalpy fluxes implied by the associated mass fluxes (Harrop et al., 2022; Lauritzen et al., 2022). This inconsistency is the result of a few design decisions: (1) there is an absence of water forms, except for the water vapor, in the total mass of moist air; (2) specific heat capacities of dry air are used in place of specific heat capacities of water forms; (3) there is a constant moist pressure assumption, which requires a moist pressure adjustment process; (4) each atmospheric process, including the moist pressure adjustment, is required to conserve energy, which has led to the use of fixers; and (5) the atmosphere and surface components explicitly exchange only mass fluxes of water, not energy fluxes.
Some of the abovementioned issues, numbered (1)–(5), are present in other climate models; for example, nearly all of them, except for the Centre National de Recherches Météorologiques Earth system model (CNRMESM1) (Séférian et al., 2016; Termonia et al., 2018), use specific heat capacities of dry air for water forms. In the E3SM version 1 (E3SMv1) (Golaz et al., 2019), inconsistency (5) is corrected by a socalled internal energy flux (IEFLX), a global energy flux based on the heat capacity of liquid water. However, by design, IEFLX is another global energy fixer, which balances the energy budget of coupled simulations (i.e., simulations with active atmosphere and ocean components) but does not address all deficiencies in the atmospheric models described above.
To eliminate the need for energy fixers and to properly transfer energy of water forms within the atmosphere and at its interface, we first introduce into the model close to theoretical (or unapproximated) specific heat capacities of water forms and include contributions of all water forms into the moist mass. Second, we reconsider how moist physics packages use the first law of thermodynamics to compute temperature tendencies from phase changes. Interpretations of the first law of thermodynamics in physics–dynamics coupling are the primary focus of this study.
For this, we introduce a few assumptions and considerations and derive coupling mechanisms between the moist physics and the adiabatic dynamical core for constantpressure and constantvolume approaches. Our constantpressure approach is an extension of the coupling method that is currently used in EAM/CAM and is applicable to the nonhydrostatic dynamics. For comparison, we also investigate two coupling mechanisms that resemble the coupling in EAM/CAM. One uses the specific heat capacity of the dry air for all species in the moist air, and the other one uses approximated specific heat capacities (in contrast to unapproximated ones mentioned above) of water forms.
Using HOMME, the standalone setup of the dynamical core of EAM, and a simplified moist physics package from Reed and Jablonowski (2012), we provide comparisons of simulations with coupling mechanisms for a test case with a moist rising bubble. This test is commonly used in the literature (Bryan and Fritsch, 2002; Bendall et al., 2020; Liu et al., 2022). Unlike typical largescale tests for climate modeling, the moist rising bubble test is characterized by an unstable initial state with strong vertical velocities, which quickly trigger phase changes crucial for our studies.
In the setup presented here, we show that the constantvolume and constantpressure approaches can be significantly different with large time steps. We also show that using approximations where the specific heat function is held constant when computing the latent heat release (as done in EAM's physics parameterizations) always results in large inconsistencies between the change in the atmosphere's energy and the energy flux of sedimentation. Fixing this inconsistency in EAM requires the use of variable latent heats both when computing the latent heat of phase change and when using this heating term to update the temperature.
Many global atmospheric models use a timesplit or fractionalstep method to separate the dynamics from the physical parameterizations (henceforth referred to as physics), including EAM. In this section, we apply timesplitting to a system of equations, which include phase change and external heating but no external mass fluxes. Splitting the dynamics from the physics, we derive the standard constantvolume and, in the case of hydrostatic dynamics, constantpressure fractional physics steps with nearexact thermodynamics. In the nonhydrostatic case, the constantpressure update is inconsistent with the timesplit equations, and we derive the nonhydrostatic fractional step based on conservation of energy and on a local heating assumption.
We start by considering a system of equations written as
where X(t) represents a state vector of all our prognostic variables at time t, H represents the dynamical terms, and F represents the forcing terms usually computed by the physics. We use a standard timesplit approach and advance the model by one time step Δt via two fractional steps:
with X_{1} denoting the intermediate state after applying the physics fractional step. System (2)–(3) represents a first order in time approximation to Eq. (1). In this example, both fractional steps are being advanced in time with a forward Euler discretization. In practice, for the dynamics time step, more advanced methods are used, such as HEVI–IMEX (Satoh, 2002; Weller et al., 2013; Lock et al., 2014).
HOMME uses a vertically Lagrangian approach (Lin, 2004). Because of this, each dynamics fractional step (3) can advance from a state X_{1} given on arbitrary level positions given by geopotential ϕ and mass coordinate values π, based on hydrostatic pressure. This in turn allows us to consider physics fractional steps that arbitrarily change the state variables, including mass, pressure, and volume, within each model layer. Below we consider physics applied either at constant pressure or at constant volume.
To apply timesplitting to our full set of equations, we start with a generic form of the vertically Lagrangian equations with a terrainfollowing coordinate s and the material derivative $\mathrm{D}/\mathrm{D}t$.
For completeness, we give the equation of state (EOS) and related mass coordinate identities,
In the above equations, u is the 3D velocity; u_{3} is the vertical or radial velocity component; T is temperature; p is the full nonhydrostatic pressure; ρ is density with $\mathit{\alpha}=\mathrm{1}/\mathit{\rho}$; ϕ is the geopotential height; π is the mass coordinate/hydrostatic pressure; $\frac{\partial \mathit{\pi}}{\partial s}$ is the mass coordinate pseudodensity; and q_{i} is a generic total mass mixing ratio that can represent dry air (q_{d}), vapor (q_{v}), liquid water (q_{l}), or ice (q_{f}). For simplicity, we have represented the dynamics terms in the velocity equation by D_{u}. For the HOMME dynamical core used here, the dynamics terms are given in Taylor et al. (2020). We assume the surface elevation, ϕ_{s}, is held fixed in time and impose a pressure boundary condition at the model top, taking p_{top}=π_{top}, set to a global constant based on the desired vertical domain size. We give two different formulations of the thermodynamic equation, Eqs. (5) and (6). The choice of which one to use depends on whether physics is predominantly applied at constant pressure (δp=0) or at constant volume (δϕ=0).
We use the unapproximated thermodynamics (or variable latent heats, as explained further in Sect. 4.3) from Eldred et al. (2022), which is equivalent to the nearexact thermodynamics from Staniforth (2022). The specific heats ${c}_{p}^{*}$ and ${c}_{\mathrm{v}}^{*}$ are given in terms of specific heats of dry air, vapor, liquid, and ice and their mixing ratios; R^{*} is a function given in terms of the gas constants for dry air and vapor and their mixing ratios,
and L_{l} and L_{v} are latent heat constants. The f_{*} terms in the above represent the forcing tendency terms, where f_{u} corresponds to the momentum fluxes and ${f}_{{q}_{i}}$ denotes sources and sinks of water mass from phase changes. The righthand side of the thermodynamic equation includes the energy flux from phase change (${\sum}_{i}{L}_{i}{f}_{{q}_{i}}$), as well as any external heating denoted by f_{T}. The above equations allow for mass flux between species q_{i} via phase changes but have assumed no net mass flux (${\sum}_{i}{f}_{{q}_{i}}=\mathrm{0}$).
In this work we focus on the physics fractional step (2) derived from timesplitting Eqs. (4)–(9). We first expand the material derivative into partial derivatives and dynamics terms and then apply timesplitting, with all the f_{*} physics forcing terms put in the physics fractional step (3) and all remaining terms put in the dynamics fractional step (2). We then assume a forward Euler discretization for the physics fractional step partial derivatives, which we represent by δ (as in $\mathit{\delta}\mathit{u}=\left(\mathit{u}\right(t+\mathrm{\Delta}t)\mathit{u}(t\left)\right)/\mathrm{\Delta}t$). This results in the following physics equations:
where we have again given two formulations of the thermodynamic equation.
2.1 Energy density and column energy
With the unapproximated thermodynamics, the energy density in mass coordinates π can be written as
where we have introduced internal energy (${e}_{i}={c}_{\mathrm{v}}^{*}T+{L}_{\mathrm{v}}{q}_{\mathrm{v}}+{L}_{\mathrm{l}}{q}_{\mathrm{l}}$) and enthalpy ($h={c}_{p}^{*}T+{L}_{\mathrm{v}}{q}_{\mathrm{v}}+{L}_{\mathrm{l}}{q}_{\mathrm{l}}$). The third expression for e given above is useful for certain calculations below and comes from using the identity $(\mathit{\pi}\mathit{\varphi}{)}_{\mathit{\pi}}=\mathit{\varphi}+\mathit{\pi}(\mathit{\varphi}{)}_{\mathit{\pi}}=\mathit{\varphi}\mathit{\pi}\mathit{\alpha}$ and the equation of state. The total column energy is given by
where the additional term represents the potential energy of the atmosphere above the model top (ϕ_{top}). We scale all energy integrals by $\mathrm{1}/g$ and use a normalized horizontal integral so that the global quantities are in units of J m^{−2}. Conserving the total column energy will take into account the work required to change ϕ_{top}.
Our fractional physics step (12–17) conserves the total column energy E in the sense that the change in E will be given by the net external heating, $\mathit{\delta}E=\frac{\mathrm{1}}{g}\int {f}_{T}\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\pi}$.
2.2 Constantvolume and constantpressure updates
From our timesplit physics equations (Eqs. 12–17), we see that the only way to obey the geopotential equation (Eq. 15) is to apply physics at constant volume. In Sect. 2.3 we give the constantvolume procedure to update the prognostic variables following Eqs. (12)–(17), which we refer to as the constantvolume update.
We also consider two additional physics updates that are designed for constant pressure, δp=0. In Sect. 2.4 we show that with the hydrostatic equations, which omit Eqs. (7) and (15), timesplitting naturally leads to a δp=0 update. In the nonhydrostatic equations, it is not possible to impose both δp=0 and δϕ=0, and a constantpressure update cannot be consistent with our physics equations. In Sect. 2.5 we propose an alternative energyconserving δp=0 update. Despite it being inconsistent with our physics equations, numerical results in Sect. 5 demonstrate that it can be practical. Note that modifying the constantvolume update to obey δp=0 as well is impossible because it leads to an overdetermined system which in general will not have energyconserving solutions.
For the derivations of these updates, we simplify the algebra by neglecting momentum tendencies by taking f_{u}=0. We also note that for all updates we have δπ=0, which comes from our assumption of no mass fluxes (8).
2.3 The constantvolume update
The constantvolume update is given by the direct application of the timesplit system (12–13 and 15–17), with constant volume a direct result of Eq. (15). Combined with our assumption of no mass fluxes, we also have constant density (δα=0). The timesplit system reduces to (neglecting the prognostic variables which do not change)
For this update, the model updates q_{i}, ${c}_{\mathrm{v}}^{*}$, and T to obey the above, and then p is updated from the equation of state, holding ϕ constant. This approach represents a firstorder approximation to the original system (Eq. 1); thus it is expected to converge to the correct solution as the time step is decreased. This is the standard physics update used by constant height coordinate models, and it can be straightforwardly used by vertically Lagrangian mass coordinate models. This update conserves energy locally in the sense that the change in energy is given by the external heating (δe=f_{T}), which can be seen directly from Eq. (18).
2.4 The constantpressure update – hydrostatic
Physics parameterizations are often applied at constant pressure (δp=0). An update which holds pressure constant while allowing the volume to change is impossible to derive via timesplitting for the nonhydrostatic equations, since the prognostic equation for layer positions does not have any traditional physics tendency terms; thus any dynamics/physics timesplit approach will lead to δϕ=0 for the update. However, for the hydrostatic equations, which replace Eq. (15) with a diagnostic equation for ϕ (e.g., for CAM/EAM; see Eq. (12.5) in Taylor, 2011), timesplitting the remaining prognostic equations, Eqs. (12), (14), (16), and (17), results in δp=0, and we do have a constantpressure update, given by
The hydrostatic energy is given by
where e_{H} comes from Eq. (20) and makes use of p=π (total pressure equals to the hydrostatic pressure). The hydrostatic timesplit step conserves the hydrostatic column energy in the sense that change in column energy equals the net external heating,
The conservation can be seen by integrating δe_{H} and making use of the fact that the surface elevation ϕ_{surf} remains fixed so that δϕ_{surf}=0.
Changing the volume of a cell does some work on the cells above it, increasing or decreasing their potential energy; thus, any constantpressure update that conserves column energy will not in general satisfy δe=f_{T} within each cell. For this hydrostatic update, the local changes in energy, internal energy, and enthalpy are given by
2.5 The constantpressure update – nonhydrostatic
In the nonhydrostatic case, one cannot derive a δp=0 update consistent with the timesplit equations since the combination δp=0 and δϕ=0 prohibits changes to any state variables. Instead, we look for a δp=0 update that has the same local energy relation as in the hydrostatic equations,
We start from this form since it automatically conserves the correct column energy as can be seen by expanding δe using Eq. (20) and integrating Eq. (27), as in the hydrostatic case.
Expanding δe using Eq. (20) allows us to write (27) as $\mathit{\delta}h+(\mathit{\pi}p)\mathit{\delta}\mathit{\alpha}={f}_{T}$, which can then be written in terms of our prognostic variables as
where the model updates q_{i}, ${c}_{p}^{*}$, and T to obey the above, and then ϕ is updated from the equation of state, holding p constant. For this nonhydrostatic update, the local changes in energy, internal energy, and enthalpy are given by
There are other energyconserving δp=0 updates, but the update in Eqs. (28)–(29) is unique in that it is the only one where phase change or heating localized to a particular layer will induce temperature changes only in that layer and no other layers (shown in Appendix A). We refer to this condition as local heating. We give an example of a nonlocal heating update in Appendix B.
Note that for this fractional step, the update of ϕ from the equation of state could be cast as a forcing term for the geopotential equation.
2.6 Dynamical differences between constant pressure and constant volume
A key difference between the constantpressure and constantvolume approaches is in how latent heat release is translated into physical motion. In the constantvolume approach, during the physics fractional step, each cell's volume and Cartesian position remain fixed, and heating will result in increased pressure. Localized heating will thus result in a pressure gradient, which will induce motion in the dynamics fractional step and in general will result in both vertical and horizontal mass transport. In the constantpressure approach, in the physics fractional step each cell's pressure will remain constant, and the heating will result in vertical expansion of the cell, representing vertical motion. Localized heating will lead to a gradient in the geopotential, which will induce horizontal motion in the dynamics fractional step.
2.7 Enthalpy formulations of the first law of thermodynamics
The first law of thermodynamics states that the energy of the system is conserved up to fluxes. In the hydrostatic case, the first law under constantpressure assumption is given by the thermodynamic equation δh=f_{T}. This form of the first law holds across such a wide range of systems that the first law is often interpreted as local conservation of enthalpy. But here, in the nonhydrostatic case for physics processes that are applied at constant pressure, we must change the above thermodynamic equation in order to conserve energy and to obey the first law. In this case, we derive a new thermodynamic equation. When expressed in terms of enthalpy, this energyconserving thermodynamic equation for nonhydrostatic δp=0 processes can be written as $\mathit{\delta}h+(\mathit{\pi}/p\mathrm{1})\mathit{\delta}\left({R}^{*}T\right)={f}_{T}$, not δh=f_{T}.
Lauritzen et al. (2022) present an overview of challenges and possible solutions regarding modeling sedimentation. The challenges include interactions of the hydrometeors with the surrounding atmosphere and raise questions about representation of different velocities for different species, falling and not falling, and subsequent frictional heating. In the simple moist physics used here, rain falls out of the atmosphere instantaneously with no additional interactions. As with phasechange physics, we consider sedimentation updates appropriate for constantpressure and constantvolume approaches. In both cases we hold the temperature constant. In the constantvolume case, we hold volume constant, update π based on the mass flux, and then update the pressure to be consistent with the EOS. We note that the constantvolume sedimentation procedure can be derived from the timesplit equations if one generalizes Eqs. (13) or (14) to include additional terms induced by the mass fluxes.
For sedimentation used with constantpressure physics, we cannot hold the pressure constant due to the mass flux, so we update π based on the mass flux and hold the nonhydrostatic component of the pressure, p−π, constant. We then update the volume to be consistent with the EOS. The constant p−π sedimentation procedure has the advantage that if the state is in hydrostatic balance, it will remain in hydrostatic balance.
This section covers the setup for our numerical simulations with the physics updates introduced in Sect. 2, applied to unapproximated and approximated thermodynamics in a rising bubble test case with a simplified physics package. It also explains that we evaluate energy fluxes due to sedimentation in each simulation by introducing a baseline, a flux of the internal energy of precipitation.
4.1 Description of simple Reed–Jablonowski moist physics
The simple moist physics package from Reed and Jablonowski (2012) consists of a twostage procedure. First, an amount of condensed water is computed, and a temperature tendency is derived from condensation. For this stage, we compare several different physics updates from Sect. 2. Second, all condensed liquid water is instantaneously removed from the moist air, which can be interpreted as rain falling with infinite speed. For this stage, we use the sedimentation updates given in Sect. 3. In Reed and Jablonowski (2012), for condensation, the temperature tendency given by the conservation of enthalpy is computed using a firstorder Taylor series with respect to temperature for the saturationspecific humidity, ${q}_{\text{sat}}={q}_{\text{sat}}(T,p)$. Here we instead use vapor tendency $\mathrm{\Delta}q=q{q}_{\text{sat}}(T,p)$ explicitly for the current values of T and p.
4.2 Setup of simulations
All runs are performed with the HOMME nonhydrostatic dynamical core with a planar twodimensional domain (Bogenschutz et al., 2023). The initialization procedure for the moist rising bubble is described in Liu et al. (2022). We use the reference state with θ_{0}=300 K, zero background relative humidity, and domain size of $[\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{000},\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{000}]\times [\mathrm{0},\mathrm{20}\phantom{\rule{0.125em}{0ex}}\mathrm{000}]$ m. In notations used by Liu et al. (2022) the initial conditions for the moist warm bubble are given by the potential temperature perturbation maximum Δθ=4.0 K, the relative humidity perturbation maximum Δh=1.0 (not to be confused with enthalpy h from above), and a cosinebell perturbation in the form of an ellipse centered at (0,20 00) m, with 5000 m and 1000 m axis lengths for horizontal and vertical directions, respectively. All simulations use 128 vertical levels and 128 fourthorder spectral elements for the horizontal domain (Taylor, 2011), which corresponds to an approximate horizontal resolution of Δx≃52 m.
In simulations, we vary only time step, $\mathrm{\Delta}t\in \mathit{\{}\mathrm{4.0},\mathrm{2.0},\mathrm{1.0},\mathrm{0.4},\mathrm{0.2},\mathrm{0.1},\mathrm{0.04},\mathrm{0.02},\mathrm{0.002}\mathit{\}}$ s, and the physics updates, which are described below in more detail. We consider physics updates with unapproximated thermodynamics, as well as several approximations. In all cases, for energy diagnostics, we use the definition of energy given by Eq. (21).
4.3 Physics updates
Following considerations presented in Sect. 2 that connect conservation of column energy E in Eq. (21) with local temperature updates in parameterizations, we investigate five updates given in Table 1. The first three updates use unapproximated thermodynamics (or variable latent heats), including a constantvolume approach which makes no approximations other than timesplitting. The remaining two updates introduce specific heatrelated approximations.
Since latent heats of phase changes are defined as a difference of enthalpies of the two phases (Emanuel, 1994), unapproximated thermodynamics that uses specific heat capacities of water species close to theoretical represents variable (with respect to temperature) latent heats of phase changes. That is, whether we impose variable or constant latent heats depends on the choice of specific heat capacities of water forms.
The first update, named CPVLNH, is based on the constantpressure approach and is formulated as
As explained in Sect. 2, this update conserves Eq. (21). Its alias is translated to constant pressure (CP) and variable latent heats (VL), where variable latent heats means that the correct specific heats for each water species are used in enthalpy expressions. In other words, VL in the name of the update indicates that we use the full unapproximated thermodynamics, including the use of ${c}_{p}^{*}$, not ${c}_{p}^{\text{d}}$ (specific heat capacity of the dry air), in Eq. (31). NH in the name of the update stands for nonhydrostatic and denotes that the update conserves nonhydrostatic energy (21). Later, we use HY in the names of updates that conserve only the hydrostatic version of energy (26) and its modifications for updates that use specific heat approximations.
The second update, CVVLNH, is the only one presented here that is based on the constantvolume (CV) approach. It is given by
where, similarly to CPVLNH, we use variable latent heats. As explained in Sect. 2, this update conserves Eq. (21).
The third update, CPVLHY, is a slight modification of CPVLNH,
It is of interest because we suspect that the nonhydrostatic term in Eq. (31) is negligible. The results from both CPVLNH and CPVLHY prove to be indistinguishable for our rising bubble test case.
The other two updates, CPCLHY and CPALHY, are inspired by the EAM design. CPCLHY is given by
and mimics the current EAM. Its name is based on constant latent heats (CL), since the update uses ${c}_{p}^{\mathrm{d}}$, not ${c}_{p}^{*}$.
Since it would require a significant effort to rewrite moist parameterizations in EAM for update (31), one could consider a design where temperature tendencies from a parameterization are computed using ${c}_{p}^{*}$ instead of ${c}_{p}^{\mathrm{d}}$, but ${c}_{p}^{*}$ is held constant during the phasechange physics and only updated after sedimentation. Thus, the CPALHY update is given by
The shorthand AL in its name indicates that only approximate variable latent heats are used.
As for the other updates above, for updates CPCLHY and CPALHY we use the definition of the total column energy (21) when computing energy diagnostics.
4.4 Energy of precipitation and its relation to change of energy in the model
As mentioned in the Introduction, in EAM there is a large disagreement between the change in atmospheric energy as compared to the net flux of energy carried into and out of the model by evaporation and precipitation. This error is significant, and in order to balance the energy budget for simulations with an active ocean component a global fixer must be used (Golaz et al., 2019).
To examine this error in our idealized setup, we look at the total atmospheric energy changes for each of the updates in Table 1 and compare them to the approximate flux F_{P} coming from the internal energy of the precipitation P,
Per discussions in Lauritzen et al. (2022), we define Eq. (36) not with varying temperature but with constant temperature, T_{ref}. Obviously, the true energy flux from precipitation would not be described by Eq. (36). However, Eq. (36) is a very good approximation for the energy of precipitation, as shown in Golaz et al. (2019) and later by our simulations. Using it gives us the same baseline to compare energy of precipitation for simulations with different updates.
Note that definition (36) does not include terms with L_{v} or L_{l} from Eq. (18). The energy of precipitation corresponding to these terms in all updates in Table 1 (as well as in EAM) is treated accurately, unlike the terms that correspond to internal or potential energy. Therefore, in our analysis we focus only on the discrepancy between the globally integrated energy flux without L terms ${\stackrel{\mathrm{\u0303}}{E}}_{t}$, where $\stackrel{\mathrm{\u0303}}{E}=E\frac{\mathrm{1}}{g}\int ({L}_{\mathrm{v}}{q}_{\mathrm{v}}+{L}_{\mathrm{l}}{q}_{\mathrm{l}})\mathrm{d}\mathit{\pi}$ and F_{P} is defined by Eq. (36).
Based on the results from our simulations presented below, we conclude that only updates with full variable latent heats, which correctly treat internal energy of liquid water, accurately represent energy of precipitation.
5.1 Time step convergence
We first examine the error introduced by the physics–dynamics timesplitting. To do this, we consider a fixed spatial discretization so that our system of equations can be represented as a system of ordinary differential equations (ODEs), and we consider the discretization of these ODEs in time. We study the convergence of this ODE system as the time step goes to zero. This approach follows Wan et al. (2015), where CAM physics is shown to converge with respect to time step with fixed horizontal resolution. The time discretization includes the timesplitting error and the truncation error in the forward Euler method used for the physics step and in the Runge–Kutta method used for the dynamics step. These errors are all formally first order or better in Δt; thus, if the ODE has a solution, we expect our discretization to converge to this solution with firstorder accuracy.
For each update in Table 1, we perform simulations with time step Δt varying from 4.0 to 0.002 s, as given in Sect. 4.2. Simulations with 0.002 s are used as reference solutions for convergence studies. Per discussion in Sect. 2, we do not expect the two unapproximated updates (CPVLNH and CVVLNH) to converge to the same unique solution, since only CVVLNH is consistent with the timesplit approach. Nor do we expect the different approximated updates (CPVLHY, CPCLHY, and CPALHY) to agree with any of the other updates. Instead, we study selfconvergence for each update, defining the error with respect to each update's own reference solutions via
where a set of θ_{i} represents a potential temperature field from a simulation with Δt>0.002 s, θ_{ref,i} is the potential temperature field given by a reference solution with Δt=0.002 s, and a_{i} represents the horizontal area weights associated with each nodal value. Sets θ_{i} and θ_{ref,i} are remapped to a uniforminheight vertical grid for the domain [0,15 000] m, and a few horizontal levels at the top and at the bottom of the domain are discarded. The error is computed at t=800 s after the bubble has evolved quite substantially. Plots of θ at t=800 s are shown below in Fig. 4.
In Fig. 1 we present selfconvergence results for all five updates. There CPVLNH and CPVLHY are given by red and black curves, which are practically identical. CVVLNH is represented by the green curve. All of the VL updates show the expected firstorder convergence and have similar errors for small time steps, although CVVLNH has noticeably larger errors for large time steps, which does result in visible differences in θ that are shown below in Fig. 4. The convergence of the nonVL updates, CPALHY and CPCLHY, is given by blue and yellow curves. These updates have similar errors and show the expected firstorder convergence down to about four digits but then fail to continue to converge.
We consider the reference solution for the CVVLNH update the most accurate. For our fixed spatial discretization, the only remaining discretization errors in the CVVLNH update come from time discretization and are driven to zero by time step convergence as shown in Fig. 1. The CPVLNH update has an additional source of error in that it applies a constantpressure assumption, which is not consistent with the timesplit equations. To approximate this error, we measure the difference between the CPVLNH and CVVLNH reference solutions using Eq. (37). At t=800 s, this difference is 0.0026 (shown as the dashed line in Fig. 1). For these two VL updates, the potential temperature fields agree to more than three digits and are visually very close, as shown below in Fig. 4. We consider this difference a crude estimate of the uncertainty introduced by imposing the constantpressure constraint in the physics update.
Figure 2 shows the time evolution of the globally integrated over the domain precipitation rates P for CPVLNH (panel a), CVVLNH (panel b), CPCLHY (panel c), and CPALHY (panel d) plotted against the simulated time for each time step. A plot for CPVLHY is nearly identical to the plot for CPVLNH, and the two reference solutions for these two updates differ by only $\mathrm{3.8}\times {\mathrm{10}}^{\mathrm{5}}$ using Eq. (37), so, CPVLHY is not included in the figure. Precipitation plots can be used to evaluate the sensitivity of each update with respect to different time steps. Qualitatively, all updates have very similar global precipitation rates. We note that the two constantpressure VL updates (panel a) have very little sensitivity, significantly lower than the remaining updates (panels b–d), which we consider to be an advantage of CPVLNH or CPVLHY.
5.2 Energy flux discrepancy
Next we compare energy flux ${\stackrel{\mathrm{\u0303}}{E}}_{t}$ and the approximate precipitation energy flux F_{P} as given by Eq. (36), both globally integrated over the domain, for the reference solutions for each of five updates. Since F_{P} depends on the mass of precipitation, we first confirm that overall globally integrated over the domain mass fluxes from precipitation for all simulations are reasonably close to each other, as shown in Fig. 3a. There, CPVLNH (red), CPVLHY (black), and CVVLNH (green) curves are clustered together, while the nonVL methods, CPALHY (blue) and CPCLHY (yellow), are positioned separately from the other three but next to each other. In Fig. 3b, we first plot in purple F_{P} computed for the precipitation rates of each simulation. Then we plot ${\stackrel{\mathrm{\u0303}}{E}}_{t}\simeq \mathrm{\Delta}\stackrel{\mathrm{\u0303}}{E}/\mathrm{\Delta}t$, where $\mathrm{\Delta}\stackrel{\mathrm{\u0303}}{E}$ is the difference of the total energy of the model (given by Eq. (21) and summed for all columns) for before and after the physics fractional step, adjusted for L terms as discussed in Sect. 4.4. As expected, energy fluxes from the unapproximated VL updates, CPVLNH (red), CPVLHY (black), and CVVLNH (green), are very close to F_{P}. A small difference between F_{P} and ${\stackrel{\mathrm{\u0303}}{E}}_{t}$ for these three updates is due to the temperature variations; the temperature of precipitation in the simulations is slightly smaller than T_{ref}. Another term that possibly could affect the difference between F_{P} and ${\stackrel{\mathrm{\u0303}}{E}}_{t}$ is the potential temperature of precipitation, but it is negligible compared to its internal energy.
For the nonVL updates CPALHY (blue) and CPCLHY (yellow), which have nearly identical values of ${\stackrel{\mathrm{\u0303}}{E}}_{t}$, the difference between ${\stackrel{\mathrm{\u0303}}{E}}_{t}$ and F_{P} is significant,
Since ${c}_{p}^{\mathrm{v}}=\mathrm{1870.0}$ and c_{l}=4188.0 J kg^{−1} K^{−1}, we see that error ${\stackrel{\mathrm{\u0303}}{E}}_{t}{F}_{P}$ is 50 % of the desired flux c_{l}PT_{ref}. This is due to the fact that the originally precipitating water in the model is represented by water vapor, with its enthalpy approximately given by ${c}_{p}^{\mathrm{v}}P{T}_{\text{ref}}$. For both the AL and the CL updates, the phase change physics followed by sedimentation appears to remove vapor from the atmosphere instead of liquid water, which results in ${\stackrel{\mathrm{\u0303}}{E}}_{t}\simeq {c}_{p}^{\mathrm{v}}P{T}_{\text{ref}}$, which in turn leads to (38). In other words, the discrepancy between energies of precipitable water and ${\stackrel{\mathrm{\u0303}}{E}}_{t}$ in CPALHY and CPCLHY is due to the fact that the energy of the liquid water is not properly accounted for throughout the physics update.
5.3 Qualitative comparisons
Figure 4 plots the potential temperature of the bubble at the end of the simulated time, 800 s, for three updates, CPVLNH (top row), CVVLNH (middle row), and CPALHY (bottom row). The results in each column correspond to a different time step size, Δt=4.0, 0.4, 0.04, and 0.002 s, from left to right. Note that simulations with a 0.002 s time step (the rightmost column) are the reference solutions from the convergence plot in Fig. 1. Bubbles from CPVLHY and CPCLHY are not shown: CPVLHY plots are identical to the bubbles from CPVLNH, and CPCLHY bubbles are very similar to the bubbles from CPALHY.
One of the most interesting observations is that while bubbles for CPVLNH and CVVLNH at smaller time steps are very similar, their trajectories towards the reference solutions differ. The shapes in the constantpressure approach become indistinguishable starting at a time step of 0.4 s, while for the constantvolume approach, only the solution for the 0.04 s time step is comparable to the reference solution by eye. In other words, if we consider either reference solution, in panel (d) or panel (h), acceptable, then for this particular test case CPVLNH appears more accurate than CVVLNH when using large time steps.
Separately, for the coarse 4.0 s time steps, the bubble in the constantvolume simulation is more distorted and more turbulent than the bubble from CPVLNH. It is explained in Sect. 2 that in CVVLNH, the energy from condensation is not transferred into moving vertical layers until the dynamics fractional step, while in the case of the constantpressure approach, a part of this energy transfers into vertical motion during the physics fractional step.
Compared to the first two rows, the bubbles from CPALHY in panels (i)–(l) (and CPCLHY, not shown) appear to be warmer and, as a result, moving upward faster, consistent with their approximated temperature update.
The primary focus of this paper is to study different updates in Table 1 at the condensation stage in physics. One could argue that in this case, all simulations should use the same sedimentation routine. To investigate this further, we performed simulations with CVVLNH for condensation and the constantpressure sedimentation. The quantitative results of such simulations and plots are almost identical to those presented. For example, there are no detectable differences in the green convergence curve in Fig. 1. There are only very minor differences for the precipitation and energy flux plots in Figs. 2 and 3. For the bubble plots in the middle row of Fig. 4, switching to the constantpressure sedimentation does not affect the overall structures of the bubbles. It also does not change our conclusions about computational feasibility of the simulations with CVVLNH. Therefore, for simulations with condensation based on CVVLNH, we chose to apply a sedimentation routine based on the constantvolume approach, too, since both obey a timesplit integration concept.
Moist physics packages are designed to conserve energy from phase changes of water forms. A particular form of conservation rule defines how state variables like temperature, pressure, and geopotential are updated during physics–dynamics coupling. We focus on the two most common approaches, constant pressure and constant volume. Considering that the current EAM design is based on a timesplit integration of physics and dynamics, our analysis shows that for the nonhydrostatic model, the constantvolume approach is consistent with the underlying system of equations, while the constantpressure approach is not. The constantpressure approach is attractive for global models since if the initial state is in hydrostatic balance, this balance will be preserved by the physics update, including in the presence of mass fluxes such as sedimentation.
Thermodynamic processes which occur at constant pressure are often shown to locally conserve enthalpy, meaning that changes in enthalpy match the external forcing, and the thermodynamic equation can be written as δh=f_{T}. This relation holds very generally, including in the hydrostatic equations, but it will not conserve energy in the nonhydrostatic equations. For the nonhydrostatic equations we derive the constantpressure energyconserving update $\mathit{\delta}h+(\mathit{\pi}/p\mathrm{1})\mathit{\delta}\left({R}^{*}T\right)={f}_{T}$ and show that this is the unique update which has the additional property that external heating and phase change will only result in local temperature changes. In the rising bubble test case used here, the effect of this correction was negligible.
In order to have the model's energy budget properly account for the energy of the precipitation flux, we study effects of variable, constant, and approximate latent heats during phase transitions in physics. We show that only by using variable latent heats throughout the physics computations can one expect correct accounting of energy fluxes from precipitation. To extend our conclusions for more practical applications like EAM, proper modeling of water energy fluxes would require updating both the moist physics packages and the code which applies the physics tendencies to incorporate unapproximated thermodynamics of water forms.
In this section we give the derivation of the unique nonhydrostatic δp=0 update, Eqs. (28)–(29), which will conserve column energy and ensure a concept of local heating. We consider the model presented in Sect. 2 with both phase changes and external heating f_{T} but no net mass flux or momentum flux. The phase changes are given by $\mathit{\delta}{q}_{i}={f}_{{q}_{i}}$ with associated heating ${\sum}_{i}{L}_{i}{f}_{{q}_{i}}$. We then seek updates for T and ϕ that obey the following constraints:

constant pressure (δp=0);

conservation of column energy ($\mathit{\delta}E=\int {f}_{T}\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\pi}$);

local heating, meaning that latent heat and other sources of external heating will be applied locally, in particular, in regions with ${\sum}_{i}{L}_{i}{f}_{{q}_{i}}=\mathrm{0}$ and f_{T}(π)=0, then δT(π)=0.
In Sect. 2.5, we show that under the δp=0 constraint, the update given by $\mathit{\delta}h+(\mathit{\pi}/p\mathrm{1})\mathit{\delta}\left({R}^{*}T\right)={f}_{T}$ conserves column energy. By inspection one can also see that it obeys the local heating constraint. To show that such an update is unique, we now show the converse: if one assumes δp=0, conservation of the column energy, and local heating, then it must be that $\mathit{\delta}h+(\mathit{\pi}/p\mathrm{1})\mathit{\delta}\left({R}^{*}T\right)={f}_{T}$.
Given the first two requirements, and using Eq. (20), we start with conservation of column energy,
Since we are not considering momentum flux or external mass flux, we have $\mathit{\delta}\mathit{u}=\mathrm{0},\mathit{\delta}\mathit{\pi}=\mathrm{0}$. Combined with δp=0, we derive
where we have introduced Q to denote the sum of the latent and external forcing terms. This integral relation must hold for all possible Q. Combined with our third requirement (δT=0 where ${f}_{{q}_{i}}=\mathrm{0}$ and f_{T}=0) we can show that this integral relation can only hold for all Q if the integrands in Eq. (A2) are equal. To see this in the discrete case, consider heating only in a single arbitrary model layer [π_{1},π_{2}]. Outside that model layer we have no phase change ($\mathit{\delta}{c}_{p}^{*}=\mathit{\delta}{R}^{*}=\mathrm{0}$), and also f_{T}=0; thus δT=0 by our local heating assumption. Thus, outside this region, both integrands are zero, and the integral relation for energy conservation reduces to an integral over the single model layer [π_{1},π_{2}],
The discrete version of this integral is computed as dπ times the layer midpoint values of the integrand; thus these layer midpoint values must be equal, and we have that $\mathit{\delta}\left({c}_{p}^{*}T\right)+(\mathit{\pi}/p\mathrm{1})\mathit{\delta}\left({R}^{*}T\right)=Q$ must hold at every model layer. This equation is equivalent to Eqs. (28)–(29) and the different forms given in Eq. (30). In the continuum, a similar argument can be made by choosing a sequence of Q's which converge to a Dirac delta function at an arbitrary layer π and by examining the convergence of Eq. (A2) with respect to this sequence.
Here we investigate other ways to derive constantpressure updates for the nonhydrostatic model and how the form of the update is connected to the form of the first law of thermodynamics.
In Sect. 2.5 we derive update CPVLNH by considering the energy equation
which leads to the internal energy equation $\mathit{\delta}{e}_{i}=\mathit{\pi}\mathit{\delta}e+{f}_{T}$. In the literature, the first law of thermodynamics is often presented as $\mathit{\delta}{e}_{i}=p\mathit{\delta}\mathit{\alpha}+{f}_{T}$ with quantity pδα defined as the pressure work, but this version of the thermodynamic equation does not conserve column energy (21) for constantpressure processes. It follows that in update CPVLNH the pressure work is given by quantity $\mathit{\pi}\mathit{\delta}\mathit{\alpha}=p\mathit{\delta}\mathit{\alpha}+(\mathit{\pi}p)\mathit{\delta}\mathit{\alpha}$, not pδα.
One can be motivated to explore a constantpressure update with the righthand side given by $\frac{\partial}{\partial \mathit{\pi}}}\left(p\mathit{\delta}\mathit{\varphi}\right)+{f}_{T$ in Eq. (B1). Following Sect. 2, the formulations of this update in terms of e, e_{i}, and h are given by
Therefore, this update has yet another definition of the pressure work, $p\mathit{\delta}\mathit{\alpha}+\left(\mathrm{1}{\displaystyle \frac{\partial p}{\partial \mathit{\pi}}}\right)\mathit{\delta}\mathit{\varphi}$. We attribute inconsistencies in the definition of the pressure work in NH constantpressure updates as compared to quantity pδα to the fact that these updates do not obey a timesplit approach and thus violate our thermodynamic equation (Eq. 13).
As noted in Sect. 2, update (B2) cannot be local. Consider that, in the whole vertical column of the model, there is only one vertical level where a phase change triggers heat release. Then the level expands and the geopotential for the levels above it changes, too. To conserve the total column energy, the h equation from (B2) needs to be satisfied at those levels, which leads to a temperature tendency at levels without a phase change. However, if the locality of the update is not essential for the computational performance, it is to be defined whether update CPVLNH is preferable to the update (B2) or to any other nonlocal update.
The code was developed as a part of the dynamical core (HOMME) code base, which itself is a part of the E3SM repository and is available under a BSD 3clause license. The code, running scripts, and plotting scripts are archived at https://doi.org/10.5281/zenodo.8336379 (Foucar et al., 2023). For instructions, navigate to file components/homme/docstosaveforpaperprovinancejuly22/HOWTOMAKEPAPERPLOTS. Plotting and running scripts use NCL and Python.
All authors contributed to conceptualization and article writing. OG and MAT derived the algorithms. OG implemented the algorithms in HOMME and ran the numerical studies.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
We thank the article reviewers for their helpful and constructive comments.
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC (NTESS), a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration (DOE/NNSA) under contract DENA0003525. This written work is authored by an employee of NTESS. The employee, not NTESS, owns the right, title, and interest in and to the written work and is responsible for its contents. Any subjective views or opinions that might be expressed in the written work do not necessarily represent the views of the U.S. Government. The publisher acknowledges that the U.S. Government retains a nonexclusive, paidup, irrevocable, worldwide license to publish or reproduce the published form of this written work or allow others to do so, for U.S. Government purposes. The DOE will provide public access to results of federally sponsored research in accordance with the DOE Public Access Plan.
This research used a highperformance computing cluster provided by the DOE Office of Science, Office of Biological and Environmental Research Earth Systems Model Development Program area of Earth and Environmental System Modeling and operated by the Laboratory Computing Resource Center at Argonne National Laboratory.
This work was supported by the US Department of Energy (DOE) Office of Science's Advanced Scientific Computing Research (ASCR) and Biological and Environmental Research (BER) Programs under the Scientific Discovery through Advanced Computing (SciDAC 4) ASCR/BER Partnership Program. This research was also supported as part of the Energy Exascale Earth System Model (E3SM) project, funded by BER Earth Systems Model Development Program area of Earth and Environmental System Modeling.
This paper was edited by Yongze Song and reviewed by Thomas Bendall and three anonymous referees.
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 Abstract
 Introduction
 Timesplit physics
 Sedimentation
 Numerical simulations for the moist rising bubble
 Numerical results
 Conclusions
 Appendix A: Uniqueness of the NH update
 Appendix B: Pressure work in NH constantpressure updates
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Timesplit physics
 Sedimentation
 Numerical simulations for the moist rising bubble
 Numerical results
 Conclusions
 Appendix A: Uniqueness of the NH update
 Appendix B: Pressure work in NH constantpressure updates
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References