PROPHET: Overview and Equations

PROPHET (Prognostic Representation Of Physics for Eddy Transport) is ClimaAtmos's representation of turbulence, convection, and the clouds they produce. It is an extended eddy-diffusivity mass-flux (EDMF) scheme in the lineage of [40–44], generalized along the lines of the conditional-filtering framework of [45–47]. The formulation is described in full in the PROPHET paper (Azimi et al., in preparation); this page states the formulation the code implements and maps it onto the source.

The scheme is still called EDMFX throughout the code: the configuration key is turbconv: "prognostic_edmfx", the types are PrognosticEDMFX and EDOnlyEDMFX, and the source files are src/prognostic_equations/edmfx_*.jl. The rename to PROPHET has not been carried into the code.

Three companion pages complete the description: Closures for the parameterized rates and diffusivities, Discretization and Time Stepping for the semi-discrete forms and the implicit solves, and Horizontal Diffusion for the optional horizontal component of the diffusive fluxes. The how-to guide Configuring and Tuning PROPHET covers the configuration options.

What PROPHET adds to a dynamical core

The governing equations include subgrid-scale flux terms: the momentum flux tensor $\boldsymbol{\mathcal{T}}$ and the scalar fluxes $\boldsymbol{\mathcal{F}}_h$, $\boldsymbol{\mathcal{F}}_{q_t}$, $\boldsymbol{\mathcal{F}}_{q_\mu^\sigma}$, and $\boldsymbol{\mathcal{F}}_\chi$. They do not say how those terms are computed. PROPHET closes them.

The distinguishing choice is that it does so with prognostic equations. A conventional convection parameterization diagnoses a mass flux from the instantaneous resolved state, which discards the memory of the convecting population and presumes a separation between the timescale of the parameterized process and the model timestep. PROPHET instead carries subgrid-scale state forward in time: each grid cell holds a small set of subdomains with their own density, vertical velocity, energy, water, and tracer content, advanced by equations that mirror the resolved ones. Convective structures therefore have memory, are advected horizontally by the resolved flow (so mesoscale organization can develop), and conserve mass, water, and energy in the same sense the resolved equations do.

The second consequence of the prognostic formulation is resolution adaptivity. The equations hold whether or not convection is resolved. As the horizontal grid is refined and the resolved flow captures more of the convective variability, the subdomain-to-grid-mean contrasts shrink, the mass-flux part of the subgrid-scale fluxes vanishes smoothly with them, and what remains is a Deardorff-type 1.5-order large-eddy closure. There is no mode switch between the two limits; see The high-resolution limit below.

Subdomain decomposition

Each grid cell is partitioned into $M + 1$ subdomains, indexed $m = 0, \dots, M$. Subdomain $m$ occupies a volume fraction $a^m$ of the cell, and

\[\sum_{m=0}^{M} a^m = 1 .\]

Physically, the subdomains represent coherent structures (updrafts, downdrafts) and the enveloping, more isotropically turbulent remainder; formally, they are the result of a conditional averaging operation on the subgrid flow [45]. Subdomain $m = 0$ is called the environment. It is an ordinary subdomain, distinguished only by having its prognostic equations eliminated. The remaining subdomains $m \ge 1$, collectively the drafts, are the ones the model integrates. The number of drafts is set by updraft_number and is 1 by default; $M = 1$ suffices for shallow convection and stratocumulus [43, 44].

Define the effective density of a subdomain as the product of its volume fraction and its density,

\[\hat{\rho}^m = a^m \rho^m ,\]

which is the model's prognostic mass variable for each draft (the field ρa). Consistency between the subdomains and the grid mean is imposed by three constraints: the effective densities sum to the grid-mean density,

\[\sum_{m=0}^{M} \hat{\rho}^m = \rho ,\]

the effective-density-weighted sum of any specific scalar recovers the grid-mean (Favre-averaged) scalar,

\[\sum_{m=0}^{M} \hat{\rho}^m \psi^m = \rho \psi ,\]

and the subdomain mass fluxes sum to the grid-mean mass flux,

\[\sum_{m=0}^{M} \hat{\rho}^m \boldsymbol{u}^m = \rho \boldsymbol{u} .\]

These constraints are what make the decomposition a decomposition rather than an additional, independent model. They also imply that only $M$ of the $M+1$ sets of subgrid-scale equations are independent of the resolved equations, which is why one set can be dropped.

All subdomains share the resolved pressure $p$, on the grounds that acoustic adjustment is fast compared with the dynamics of interest [42, 45, 47]. This removes the extra acoustic modes the decomposition would otherwise introduce. The effect of subdomain pressure differences is retained as a parameterized pressure (form) drag; see Pressure closure. Subdomain densities are carried separately outside the buoyancy terms: a fully compressible model needs no anelastic approximation within the subgrid, and one would complicate exact energy consistency.

Horizontal velocities are shared across subdomains, and only the vertical velocity differs:

\[\boldsymbol{u}^m = \boldsymbol{u}_h + \boldsymbol{w}^m .\]

This reduces the three-dimensional draft momentum equation to a single equation for the vertical component, and eliminates vertical transport of horizontal momentum by the drafts [41, 42]. In generalized coordinates, $\boldsymbol{u}_h = (u_1, u_2, 0)$ and $\boldsymbol{w}^m = (0, 0, u_3^m)$ in covariant components, so over sloping terrain the shared horizontal component still contributes a nonzero contravariant vertical velocity $u_h^3 = g^{31} u_1 + g^{32} u_2$.

Draft equations

The drafts ($m \ge 1$) are advanced by the following equations. Continuity is in flux form; everything else is in advective form, obtained by combining the flux-form scalar equation with continuity. Advective form is used because the effective densities $\hat{\rho}^m$ can be very small, and dividing by them inside a flux divergence is numerically fragile [47].

Mass continuity:

\[\frac{\partial \hat{\rho}^m}{\partial t} + \nabla \cdot \left[ \hat{\rho}^m (\boldsymbol{u}^m - W_{q_t}^m \hat{\boldsymbol{k}}) \right] = \sum_{n \ne m} \hat{\rho}^m (E^{mn} - \Delta^{mn}) + \hat{\rho}^m \hat{S}_{q_t}^m\]

Momentum. The shared horizontal velocity reduces this to a single equation for $u_3^m$:

\[\frac{\partial \boldsymbol{u}^m}{\partial t} + (2 \boldsymbol{\Omega} + \boldsymbol{\omega}^m) \times \boldsymbol{u}^m = \boldsymbol{b}_{\mathrm{eff}}^m - \nabla \kappa^m + \boldsymbol{S}_u^m - \frac{1}{\rho} \nabla \cdot \left[ \rho (\boldsymbol{\mathcal{T}}^{\mathrm{diff}} + \boldsymbol{\mathcal{H}}_u) \right] + \sum_{n \ne m} E^{mn} (\boldsymbol{u}^n - \boldsymbol{u}^m)\]

Moist static energy:

\[\frac{\partial h_s^m}{\partial t} + \boldsymbol{u}^m \cdot \nabla h_s^m = S_{e,\mathrm{eff}}^m - \boldsymbol{u}^m \cdot \left( - \frac{1}{\rho^m} \nabla p^\dagger + \boldsymbol{b}^m + \boldsymbol{S}_u^m \right) + Q_R^m - \frac{1}{\hat{\rho}^m} \nabla \cdot \left( \hat{\rho}^m \boldsymbol{\mathcal{F}}_{h_s}^{\mathrm{diff},m} \right) - \frac{1}{\rho} \nabla \cdot \left( \rho \boldsymbol{\mathcal{H}}_h \right) + \sum_{n \ne m} E^{mn} (h_s^n - h_s^m) + D_{h_s}^m\]

Total water:

\[\frac{\partial q_t^m}{\partial t} + \boldsymbol{u}^m \cdot \nabla q_t^m = (1 - q_t^m) S_{q_t}^m - \frac{1}{\hat{\rho}^m} \nabla \cdot \left( \hat{\rho}^m \boldsymbol{\mathcal{F}}_{q_t}^{\mathrm{diff},m} \right) - \frac{1}{\rho} \nabla \cdot \left( \rho \boldsymbol{\mathcal{H}}_{q_t} \right) + \sum_{n \ne m} E^{mn} (q_t^n - q_t^m) + D_{q_t}^m\]

Condensate and precipitation, for $\mu \in \{l, i\}$ and $\sigma \in \{\cdot, \mathrm{cl}, \mathrm{pr}\}$ as the microphysics model requires:

\[\frac{\partial q_\mu^{\sigma m}}{\partial t} + \boldsymbol{u}^m \cdot \nabla q_\mu^{\sigma m} = S_{q_\mu^\sigma,\mathrm{eff}}^m - \frac{1}{\hat{\rho}^m} \nabla \cdot \left( \hat{\rho}^m \boldsymbol{\mathcal{F}}_{q_\mu^\sigma}^{\mathrm{diff},m} \right) - \frac{1}{\rho} \nabla \cdot \left( \rho \boldsymbol{\mathcal{H}}_{q_\mu^\sigma} \right) + \sum_{n \ne m} E^{mn} (q_\mu^{\sigma n} - q_\mu^{\sigma m}) + D_{q_\mu^\sigma}^m\]

Tracers:

\[\frac{\partial \chi^m}{\partial t} + \boldsymbol{u}^m \cdot \nabla \chi^m = S_{\chi,\mathrm{eff}}^m - \frac{1}{\hat{\rho}^m} \nabla \cdot \left( \hat{\rho}^m \boldsymbol{\mathcal{F}}_\chi^{\mathrm{diff},m} \right) - \frac{1}{\rho} \nabla \cdot \left( \rho \boldsymbol{\mathcal{H}}_\chi \right) + \sum_{n \ne m} E^{mn} (\chi^n - \chi^m) + D_\chi^m\]

Equation of state: the shared pressure and the subdomain temperature give the subdomain density,

\[\rho^m = \frac{p}{R_m^m T^m} .\]

Note the direction: in the resolved equations, the equation of state returns pressure; here, it returns density, because pressure is shared.

Which water species a draft carries follows the grid-mean microphysics model. In the equilibrium case, only $q_t^m$ is prognostic and the condensate partition comes from saturation adjustment; in the non-equilibrium cases, each draft also prognoses cloud and precipitation species (and, for two-moment microphysics, their number concentrations), each with its own sedimentation flux and sources.

Definitions and assumptions

Perturbation pressure. A hydrostatic reference pressure $p_h$ satisfying $\nabla p_h = -\rho_h \nabla \Phi$ is subtracted to define $p^\dagger = p - p_h$. The reference is the same one the dynamical core uses for accurate evaluation of the pressure-gradient force; the choice affects numerical accuracy, not the dynamics.

Buoyancy. The buoyancy of a subdomain is defined relative to a reference density,

\[\boldsymbol{b}^m = - \frac{\rho^m - \rho_{\mathrm{ref}}}{\rho^m} \nabla \Phi ,\]

and the effective buoyancy driving the subdomain momentum adds the nonhydrostatic pressure gradient and subtracts the pressure drag $\boldsymbol{d}^m$,

\[\boldsymbol{b}_{\mathrm{eff}}^m = - \frac{1}{\rho^m} \nabla p^\dagger + \boldsymbol{b}^m - \boldsymbol{d}^m .\]

Because the effective densities sum to $\rho$, the effective-density-weighted sum of the buoyancies is a grid-mean buoyancy,

\[\sum_{m=0}^{M} \hat{\rho}^m \boldsymbol{b}^m = -(\rho - \rho_{\mathrm{ref}}) \nabla \Phi \equiv \rho \boldsymbol{b} ,\]

so the subdomains exert no net force on the grid mean. Newton's third law imposes the same requirement on the drag, $\sum_m \hat{\rho}^m \boldsymbol{d}^m = 0$, which is why the pressure drag does not appear in the resolved momentum equation.

Effective sources. For a scalar $\psi \ne q_t$, the source that appears in the advective-form equation includes the dilution effect of the mass source,

\[S_{\psi,\mathrm{eff}}^m = S_\psi^m - \psi^m S_{q_t}^m ,\]

and the total-water equation uses the equivalent form $(1 - q_t^m) S_{q_t}^m$. The effective mass source in continuity is

\[\hat{S}_{q_t}^m = S_{q_t}^m - \frac{1}{\hat{\rho}^m} \nabla \cdot \left( \hat{\rho}^m \boldsymbol{\mathcal{F}}_{q_t}^{\mathrm{diff},m} \right) - \frac{1}{\rho} \nabla \cdot \left( \rho \boldsymbol{\mathcal{H}}_{q_t} \right) ,\]

the subdomain counterpart of the grid-mean $\hat{S}_{q_t}$: subgrid-scale moisture transport moves moist air mass, so it appears in the mass budget as well as the water budget.

Sedimentation-divergence term. The terms $D_\psi^m$ arise from converting the flux-form scalar equations to advective form in the presence of sedimentation,

\[D_\psi^m = \frac{1}{\hat{\rho}^m} \left[ \nabla \cdot (\hat{\rho}^m W_\psi^m \hat{\boldsymbol{k}}) - \psi^m \nabla \cdot (\hat{\rho}^m W_{q_t}^m \hat{\boldsymbol{k}}) \right] ,\]

with $W_\psi^m$ the downward mass-weighted sedimentation flux of $\psi$ in subdomain $m$. The physical content beyond ordinary within-draft fallout is geometric: where a draft's volume fraction varies with height, its boundary is not vertical, and sedimenting condensate crosses it. A draft that narrows downward loses falling condensate to its surroundings; one that widens downward gains it. This transfer acts only on condensate species and carries pure condensate rather than a parcel at the source subdomain's composition, which makes it structurally distinct from the dynamical entrainment and detrainment below. For a single draft $m$ exchanging with the environment, updraft_sedimentation! evaluates the first term in two branches according to the sign of $\partial_z a^m$,

\[S_{\psi,\mathrm{sed}}^m = \begin{cases} a^m \partial_z(\rho^m W_\psi^m) , & \partial_z a^m \ge 0 , \\ \partial_z(\hat{\rho}^m W_\psi^m) - \rho^0 W_\psi^0 \partial_z a^m , & \partial_z a^m < 0 , \end{cases}\]

with the environment flux reconstructed as the residual of the grid-mean flux, $\hat{\rho}^0 W_\mu^0 = \rho W_\mu - \hat{\rho}^m W_\mu^m$ and $\rho^0 W_\mu^0 = \max(\hat{\rho}^0 W_\mu^0 / (1 - a^m), 0)$. The sum of the condensate flux convergences $S_{q_t,\mathrm{sed}}^m = \sum_\mu S_{q_\mu,\mathrm{sed}}^m$ is the sedimentation term of the draft mass equation, applied in the analytic $\hat{\rho}^m$ solve, and each intensive draft scalar receives both terms of $D_\psi^m$: its own sedimentation source and the dilution $-\psi^m S_{q_t,\mathrm{sed}}^m / \hat{\rho}^m$ by the sedimented mass. For $h_s^m$, $W_{h_s}^m = \sum_\mu q_\mu^m (h_\mu(T^m) + \Phi) w_\mu^m$ is evaluated inside updraft_sedimentation!, with laterally entrained environment condensate carrying $h_\mu(T^0) + \Phi$.

Subdomain diffusive fluxes. When edmfx_vertical_diffusion is enabled, each draft scalar $\psi^m \in \{h_s^m, q_t^m, q_l^{\mathrm{cl},m}, q_i^{\mathrm{cl},m}, n_l^{\mathrm{cl},m}, n_i^{\mathrm{cl},m}, \chi^m\}$ diffuses down its own intensive vertical gradient $\nabla \psi^m$ [48] (with non-equilibrium microphysics, draft water and moist static energy diffuse on the suspended components $q_{t,\mathrm{eff}}^m = q_t^m - q_{\mathrm{rai}}^m - q_{\mathrm{sno}}^m$ and $h_{s,\mathrm{eff}}^m = h_s^m - (h_l^m - h_d^m) q_{\mathrm{rai}}^m - (h_i^m - h_d^m) q_{\mathrm{sno}}^m$, excluding non-diffusing rain and snow):

\[\left[ \frac{\partial \psi^m}{\partial t} \right]_{\mathrm{diff}} = \frac{1}{\hat{\rho}^m} \nabla \cdot \left( \widehat{\hat{\rho}^m} \, f_m K_h \nabla \psi^m \right) ,\]

where $f_m$ is edmfx_updraft_kh_diffusion_factor (default 1) and $\widehat{\hat{\rho}^m}$ is the harmonic face mean of $\hat{\rho}^m$. The draft mass $\hat{\rho}^m$ evolves only through advection, entrainment, and detrainment, and the moist-mass change from $q_t$ diffusion is absorbed by the environment residual, so dividing the extensive flux divergence by $\hat{\rho}^m$ is the exact intensive tendency. The harmonic mean is the series conductance of the two adjacent cells [49]: because $\widehat{\hat{\rho}^m} / \hat{\rho}^m_c = 2 \hat{\rho}^m_{\mathrm{nb}} / (\hat{\rho}^m_c + \hat{\rho}^m_{\mathrm{nb}}) \le 2$, the division cancels against it, and the intensive tendency is a three-point diffusion of $\psi^m$ with effective face diffusivities between $0$ and $2 f_m K_h$. The draft relaxes toward its vertical neighbors at most twice as fast as grid-mean diffusion at the same $K_h$ for every value of $\hat{\rho}^m$, and the extensive flux across a face adjacent to a draft-free cell vanishes. (An arithmetic face mean amplifies the tendency of a thin draft cell by the area ratio to its neighbors.) Diffusing $\psi^m$ down its own gradient has a fixed point at vertically uniform draft properties and homogenizes internal draft anomalies toward a moist adiabat while keeping the sharp environmental jumps of $\bar{s}_d$ and $\bar{q}_t$ at trade inversions or cloud tops out of the draft tendency. The horizontal counterpart (edmfx_horizontal_diffusion) replaces the harmonic face mean by an element-wise power mean of the draft area, which bounds the relaxation rate in the same way; see PROPHET Horizontal Diffusion.

Energy. The prognostic thermodynamic variable of a draft is the specific moist static energy

\[h_s^m = c_{pm}(q^m)(T^m - T_0) + (q_t^m - q_c^m) L_{v,0} - q_i^m L_{f,0} + \Phi ,\]

not the total energy of the resolved equations. The energy equation neglects the explicit $\partial p^\dagger / \partial t$ but retains the work term $-\boldsymbol{u}^m \cdot (-(\rho^m)^{-1} \nabla p^\dagger + \boldsymbol{b}^m + \boldsymbol{S}_u^m)$, the reversible conversion between kinetic and moist static energy. [50] showed that keeping the perturbation-pressure part of this work matters for updraft temperature and buoyancy in deep convection. The pressure drag is deliberately excluded from the work term: form drag moves macroscopic kinetic energy into turbulence in the surroundings rather than changing the parcel's thermodynamic state, so energy consistency requires it to reappear as a return-to-isotropy source in the turbulence kinetic energy budget.

Because $\hat{\rho}^m$ is held fixed across the diffusive step, water diffusing into a draft displaces an equal mass of dry air and changes the draft's specific enthalpy by $(h_{\mathrm{tot},\mu}^m - h_{\mathrm{tot},d}^m) \, \delta q_\mu^m$. Combining the sensible-heat diffusion $c_p^m \nabla T^m + \nabla \Phi$ with $\sum_\mu (h_\mu^m - h_d^m) \nabla q_\mu^m$ gives the moist-static-energy gradient $\nabla h_s^m$ directly, with $c_p^m$ the heat capacity of the diffusing mixture that the grid-mean flux also uses (see Diffusion). The grid mean receives the same subdomain flux with $(h_\mu^m + \Phi) \nabla q_\mu^m$ in place of $(h_\mu^m - h_d^m) \nabla q_\mu^m$, since diffusing water adds moist mass to the grid mean; the difference, $(h_d^m + \Phi) \nabla q_t^m$, is the enthalpy of the displaced dry air and goes to the environment residual together with the displaced mass (see Coupling to the resolved equations). Under a reference-temperature shift $T_0 \to T_0 + \delta T_0$, $h_s^m$ shifts by $-(1 - q_t^m) c_{pd} \delta T_0 - q_t^m c_{pl} \delta T_0$, so $\delta(\partial_t h_s^m) = (c_{pd} - c_{pl}) \delta T_0 \, \partial_t q_t^m$, which holds identically when $h_s^m$ and $q_t^m$ diffuse down their own gradients with the same operator. For hyperdiffusion, the draft energy equation uses the sensible-heat, geopotential, and water-enthalpy split of Hyperdiffusion.

Coupling to the resolved equations

The subgrid-scale fluxes in the resolved equations are fixed, not chosen: they are whatever makes the sum of the $M+1$ subdomain equations reproduce the resolved equations. Carrying out that sum splits each flux into two parts. For a generic resolved scalar $\psi$,

\[\rho \boldsymbol{\mathcal{F}}_\psi = \underbrace{\sum_{m=0}^{M} \hat{\rho}^m (\boldsymbol{w}^m - \boldsymbol{w})(\psi^m - \psi)}_{\text{mass flux}} + \underbrace{\rho \boldsymbol{\mathcal{F}}_\psi^{\mathrm{diff}}}_{\text{diffusive}} .\]

The first term is the coherent transport by the inter-subdomain circulation, the mass-flux contribution. Only vertical-velocity differences appear, because the horizontal velocity is shared. The second is the transport by intra-subdomain isotropic turbulence, closed by the grid-mean eddy diffusivity of Turbulence kinetic energy and eddy diffusivity. For PrognosticEDMFX, the grid-mean diffusive flux is the sum of the environment and draft diffusive fluxes,

\[\rho \boldsymbol{\mathcal{F}}_\psi^{\mathrm{diff}} = -\hat{\rho}^0 K_h \nabla \psi^0 - \sum_{j=1}^{M} \widehat{\hat{\rho}^j} \, f_j K_h \nabla \psi^j ,\]

for $\psi \in \{q_t^{\mathrm{eff}}, q_l^{\mathrm{cl}}, q_i^{\mathrm{cl}}, n_l^{\mathrm{cl}}, n_i^{\mathrm{cl}}, \chi\}$. Where cloud liquid and ice carry the blended flux of Diffusion, the environment term for $q_l^{\mathrm{cl}}$ and $q_i^{\mathrm{cl}}$ is that flux evaluated with the environment state $(T^0, q_t^0, q_\mu^0)$ and weight $\hat{\rho}^0 K_h$, while the drafts diffuse their cloud species down their own gradients. The grid-mean diffusive enthalpy flux sums the extensive brackets of Diffusion, $c_p^m \nabla T^m + \nabla \Phi + \sum_\mu (h_\mu^m + \Phi) \nabla q_\mu^m$, evaluated with each subdomain's thermodynamic state. For a draft, that bracket exceeds the gradient $\nabla h_{s}^m$ of the moist static energy the draft diffuses by $(h_d^m + \Phi) \nabla q_t^m$, so the energy the environment absorbs from draft diffusion is the enthalpy of the dry air that the diffusing water displaces, accompanying the displaced mass $-\nabla \cdot (\widehat{\hat{\rho}^j} f_j K_h \nabla q_t^j)$ that the environment also absorbs; with face-interpolated coefficients the discrete product rule is exact, so this holds to round-off. Because $\nabla \psi = a^0 \nabla \psi^0 + \sum_j a^j \nabla \psi^j + \sum_j (\psi^j - \psi^0) \nabla a^j$, assembling the grid-mean diffusive flux from the subdomain gradients omits the cross-area term $-\rho K_h \sum_j (\psi^j - \psi^0) \nabla a^j$ (transport associated with $\nabla a^j$ is already carried by the coherent mass flux and detrainment); when edmfx_vertical_diffusion: false, the default, the draft terms are omitted from both the draft and grid-mean equations so the diagnosed environment receives $-\nabla \cdot (\hat{\rho}^0 K_h \nabla \psi^0)$ in either case. The grid-mean diffusive flux is then the environment flux alone, smaller than $-\rho K_h \nabla \bar{\psi}$ by the draft area fraction and without the draft-interior and cross-area transport. Grid-scale species that the drafts do not carry (the P3 ice number and rime quantities) diffuse with $-\rho K_h \nabla \psi$, and so does every scalar for EDOnlyEDMFX.

The consistency constraints make three algebraically equivalent forms of the mass-flux term available,

\[\sum_m \hat{\rho}^m (\boldsymbol{w}^m - \boldsymbol{w})(\psi^m - \psi) = \sum_m \hat{\rho}^m \boldsymbol{w}^m (\psi^m - \psi) = \sum_m \hat{\rho}^m (\boldsymbol{w}^m - \boldsymbol{w}) \psi^m .\]

The code uses the symmetric first form, in which the flux vanishes identically when the subdomains are materially identical, regardless of volume-fraction gradients. That is the discrete counterpart of the property [48] require of the diffusive terms.

The specific fluxes follow. For momentum, the mass-flux part is a tensor product, and because horizontal velocities are shared, it populates only the vertical–vertical entry:

\[\rho \boldsymbol{\mathcal{T}} = \sum_{m=0}^{M} \hat{\rho}^m (\boldsymbol{w}^m - \boldsymbol{w}) \otimes (\boldsymbol{w}^m - \boldsymbol{w}) + \rho \boldsymbol{\mathcal{T}}^{\mathrm{diff}} .\]

For energy, the transported quantity is the specific total enthalpy, since that is what the resolved advective flux carries:

\[\rho \boldsymbol{\mathcal{F}}_h = \sum_{m=0}^{M} \hat{\rho}^m (\boldsymbol{w}^m - \boldsymbol{w}) (h_{\mathrm{tot}}^m - h_{\mathrm{tot}}) + \rho \boldsymbol{\mathcal{F}}_h^{\mathrm{diff}} , \qquad h_{\mathrm{tot}}^m = h_s^m + \kappa^m + \kappa_{\mathrm{iso}} .\]

The mass-flux part carries $h_{\mathrm{tot}}^m$ directly rather than the sensible-heat-plus-water-enthalpy decomposition used for the diffusive part: it represents bodily motion of whole moist parcels, each at its own composition, so the dry-air-diffusion artifact that motivates the decomposition does not arise. Adding the grid-mean $\kappa_{\mathrm{iso}}$ uniformly to every subdomain keeps $\sum_m \hat{\rho}^m h_{\mathrm{tot}}^m = \rho h_{\mathrm{tot}}$; it cancels from the deviation that drives the flux. Water species and tracers take the generic form above.

The scalar mass fluxes are applied by edmfx_sgs_mass_flux_tendency! and the diffusive fluxes by edmfx_sgs_diffusive_flux_tendency!, both in edmfx_sgs_flux.jl, gated on the configuration flags edmfx_sgs_mass_flux and edmfx_sgs_diffusive_flux respectively. Because the total-water flux moves moist air mass, both tendencies also increment the resolved density, consistent with $\hat{S}_{q_t}$ in the continuity equation.

Environment as residual

With the environment equations eliminated, the environment quantities that the flux sums above require are reconstructed from the resolved state and the draft variables:

\[\hat{\rho}^0 = \rho - \sum_{m \ge 1} \hat{\rho}^m , \qquad \psi^0 = \frac{\rho \psi - \sum_{m \ge 1} \hat{\rho}^m \psi^m}{\hat{\rho}^0} , \qquad \hat{\rho}^0 \boldsymbol{w}^0 = \rho \boldsymbol{w} - \sum_{m \ge 1} \hat{\rho}^m \boldsymbol{w}^m .\]

The reconstruction of a specific environment scalar divides by $\hat{\rho}^0$, which is small wherever the drafts nearly fill the cell. specific and the ᶜspecific_env_* helpers in src/utils/variable_manipulations.jl therefore blend the residual toward the grid-mean value below an area threshold a_half; see Environment reconstruction.

The high-resolution limit

The per-subdomain mass-flux term is a product of three factors, $a^m$, $\rho^m (\boldsymbol{w}^m - \boldsymbol{w})$, and $\psi^m - \psi$, each of which responds to refinement. The two contrasts are themselves measures of the subgrid variance among subdomains, and vanish once the resolved flow captures the convective structures. The draft volume fractions $a^m$ can adapt downward as well, through the entrainment/detrainment closures and the surface closure that seeds the drafts, since at fine enough resolution the most buoyant fraction of surface-layer turbulence is resolved rather than represented by a draft. A product vanishes if any factor does, so vanishing contrasts alone are sufficient.

What survives is the diffusive flux,

\[\rho \boldsymbol{\mathcal{F}}_\psi^{\mathrm{diff}} = -\rho K_\psi \nabla \psi , \qquad K_\psi = \frac{K_m}{\mathrm{Pr}_t(\mathrm{Ri})} , \qquad K_m = c_m \, l \sqrt{\kappa_{\mathrm{iso}}} ,\]

together with the deviatoric momentum flux and the prognostic equation for $\kappa_{\mathrm{iso}}$. That is a Deardorff-type 1.5-order large-eddy closure [51, 52]: a prognostic subgrid kinetic energy supplies the velocity scale, the mixing length supplies the length scale, and the trace of the subgrid Reynolds stress is closed by $\kappa_{\mathrm{iso}}$ itself. The mixing length is grid-aware through the resolvability filter scale $l \le \max(\Delta x_h, \Delta z)$, which is inert at global-model horizontal resolutions and reduces to the Deardorff grid-scale bound on isotropic grids.

For this smooth transition to hold discretely, the advection schemes applied to the subdomain variables must not introduce numerical diffusion that mimics a residual subgrid flux; otherwise the scheme approaches the large-eddy limit with a resolution-dependent effective diffusivity. This is one reason the upwinding options for the subdomain variables are configurable; see Discretization and Time Stepping.

Symbols

The remaining closure symbols are defined on the Closures page. Notation and Symbols maps these onto the field names in the code.

Radiation

Radiation is computed once per column, not per subdomain. The formulation partitions the heating rate at the heating-rate level: a clear-sky and an overcast calculation give $Q_R^{\mathrm{clr}}$ and $Q_R^{\mathrm{cld}}$, and each subdomain receives the linear combination weighted by its own cloud fraction,

\[Q_R^m = (1 - f_c^m) Q_R^{\mathrm{clr}} + f_c^m Q_R^{\mathrm{cld}} ,\]

which sums to the grid-mean heating rate because $\sum_m a^m f_c^m = f_c^{\mathrm{tot}}$. A clear subdomain then sees no cloud-top longwave cooling and a fully cloudy one sees all of it.

Where this is implemented