PROPHET: Horizontal Diffusion

The PROPHET (EDMFX in code) subgrid-scale (SGS) diffusive closure is, by default, vertical only: it parameterizes turbulent transport along the column with an eddy-diffusivity closure and applies the resulting tendency to the grid mean (and, under edmfx_vertical_diffusion, applies the grid-mean tendencies to the prognostic updraft scalars). Two opt-in configuration options add the analogous horizontal down-gradient terms, both defaulting to false:

• edmfx_sgs_horizontal_diffusive_flux adds the horizontal component of the grid-mean environment SGS diffusive flux;
• edmfx_horizontal_diffusion applies the grid-mean horizontal diffusion tendencies to the prognostic EDMFX updrafts, and requires edmfx_sgs_horizontal_diffusive_flux.

Both terms are always explicit: they are applied in the explicit remainder tendency, independently of diff_mode, and never enter the implicit column solve. Each requires a horizontal discretization, so both return immediately on single columns.

Horizontal eddy diffusivity

The horizontal eddy viscosity and diffusivity use the same TKE-based closure as the vertical flux,

\[K_{u,h} = c_m \, l_h \, \sqrt{\kappa_{\mathrm{iso}}}, \qquad K_{h,h} = K_{u,h} \,/\, \mathrm{Pr}_t ,\]

where $l_h$ is the horizontal mixing length (see Mixing length), $\kappa_{\mathrm{iso}}$ is the isotropic subgrid turbulent kinetic energy (Y.c.ρtke, written $e$ in some of the code comments), $c_m$ is the eddy-diffusivity coefficient, and $\mathrm{Pr}_t$ the turbulent Prandtl number. set_horizontal_diffusivities! evaluates $K_{u,h}$ and $K_{h,h}$ at cell centers on each update of the explicit precomputed cache, storing them in ᶜK_u_h and ᶜK_h_h. The horizontal tendencies and the edth/evuh diagnostics read these cached fields.

Unlike the vertical face pipeline (set_face_diffusivities!), the horizontal diffusivities do not include the interfacial entrainment diffusivity $K_e$: that term parameterizes vertical entrainment across an unresolved inversion face ($K_e = \gamma \, w_e \, \Delta z$) and has no horizontal analogue.

Grid-mean flux

The grid-mean tendencies from edmfx_sgs_horizontal_diffusive_flux are, on cell centers,

\[\partial_t (\rho e_\text{tot}) \mathrel{+}= \nabla_h \cdot \left[ \rho \, K_{h,h} \left( \nabla_h s_\text{d} + (h_\text{eff} + \Phi) \, \nabla_h q_\text{tot}^\text{eff} \right) \right],\]

\[\partial_t (\rho q_\text{tot}) \mathrel{+}= \nabla_h \cdot (\rho \, K_{h,h} \, \nabla_h q_\text{tot}^\text{eff}), \qquad \partial_t (\rho \chi) \mathrel{+}= \nabla_h \cdot (\rho \, K_{h,h} \, \nabla_h \chi),\]

where $\nabla_h$ and $\nabla_h\cdot$ are the horizontal gradient and weak divergence, $s_\text{d}$ is the dry static energy, and $\Phi$ is the geopotential. Water diffuses as a single substance, matching the vertical flux: $q_\text{tot}^\text{eff} = q_\text{tot} - q_\text{rai} - q_\text{sno}$ is the water that actually diffuses, and

\[h_\text{eff} = \frac{h_\text{v} q_\text{v} + h_\text{l} q_\text{lcl} + h_\text{i} q_\text{icl}}{\max(q_\text{v} + q_\text{lcl} + q_\text{icl}, \, \epsilon)}\]

is its mass-weighted specific enthalpy, built from clipped inputs. This form avoids the spurious enthalpy flux that diffusing $h_\text{tot}$ directly would produce through the dry-air-mass gradient. $h_\text{eff} + \Phi$ is materialized into a scratch scalar before the weak divergence, because feeding the nested expression directly into $\nabla_h\cdot$ exceeds GPU kernel parameter limits on extruded spaces with warped topography.

The suspended cloud species $q_\text{lcl}$, $q_\text{icl}$ and their number concentrations have no flux of their own. Each takes a share of the aggregate water tendency, scaled by the clipped ratio $\min(q_\mu / q_\text{tot}^\text{eff}, 1)$, and the number densities scale with it too, which preserves the mean particle mass. Rain and snow receive no horizontal transport: the vertical path gives them only the interfacial entrainment diffusivity $K_e$, which has no horizontal analogue. The tracer $\chi$ therefore ranges over the passive tracers alone, diffused with the unscaled $K_{h,h}$.

The aggregate water flux additionally enters the moist-air-mass tendency,

\[\partial_t \rho \mathrel{+}= \nabla_h \cdot (\rho \, K_{h,h} \, \nabla_h q_\text{tot}^\text{eff}),\]

while the species shares do not.

When prognostic TKE is active, the horizontal flux transports TKE and adds the horizontal shear production,

\[\partial_t (\rho \kappa_{\mathrm{iso}}) \mathrel{+}= \nabla_h \cdot (\rho \, K_{u,h} \, \nabla_h \kappa_{\mathrm{iso}}) + 2 \rho \, K_{u,h} \, \mathcal{S}_h : \mathcal{S}_h ,\]

with $\mathcal{S}_h$ the strain rate built from horizontal gradients only (the tensor written $\boldsymbol{\mathcal{E}}$ on the Closures page). The shear production is positive definite; the production from vertical gradients and its stencil are applied by the vertical TKE tendency.

The momentum tendency is the horizontal weak divergence of the SGS stress $\tau = -2 K_{u,h} \mathcal{S}$, with $\mathcal{S}$ the full three-dimensional strain rate of the grid-mean velocity,

\[\partial_t u_h \mathrel{-}= \frac{1}{\rho} \nabla_h \cdot (\rho \, \tau), \qquad \partial_t u_3 \mathrel{-}= \frac{1}{\rho} \nabla_h \cdot (\rho \, \tau),\]

evaluated on cell centers for the horizontal wind and on cell faces for the vertical wind. The vertical wind is included because the horizontal flux of vertical momentum $\overline{u_h' w'}$ is a covariance, which the TKE (the half-trace of the velocity covariance) does not carry, so the down-gradient stress is its only representation. The vertical flux $\overline{w' w'}$ is a variance represented by the TKE, the mass flux, and the pressure closure, so it is not applied as down-gradient diffusion of the vertical wind; the vertical stress divergence on the horizontal wind is handled by the vertical diffusion pathway.

The stress and the shear production are not energetically consistent with each other. The work done by the stress on the resolved flow is $\tau : \nabla_h u$, which includes cross terms such as $\mathcal{S}_{13} \, \partial_x w$, whereas the TKE receives only $2 \rho \, K_{u,h} \, \mathcal{S}_h : \mathcal{S}_h$. The difference is exactly those cross terms. Each form is chosen for a property the other lacks: the stress uses the full strain rate because $\mathcal{S}_{13}$ and $\mathcal{S}_{23}$ are what represent $\overline{u_h' w'}$, and the production uses $\mathcal{S}_h$ because $2 \rho \, K_{u,h} \, \mathcal{S}_h : \mathcal{S}_h$ is positive definite while $\tau : \nabla_h u$ is sign-indefinite and can drive the TKE negative. This is one of the reasons the option is off by default.

Updraft horizontal diffusion

The option edmfx_horizontal_diffusion switches on horizontal diffusion of the prognostic updraft variables. It requires edmfx_sgs_horizontal_diffusive_flux: each subdomain scalar receives the specific tendency of the corresponding grid-mean flux, so every subdomain inherits the same horizontal diffusion as the grid box, matching the uniform vertical treatment of edmfx_vertical_diffusion. For each updraft $j$, the total specific humidity, the cloud species shares, and the SGS tracers receive

\[\partial_t \chi^j \mathrel{+}= \frac{1}{\rho} \nabla_h \cdot (\rho \, K_{h,h} \, \nabla_h \chi),\]

with the same species treatment as the grid mean, and the moist static energy receives the grid-mean total-enthalpy tendency, $\partial_t \mathrm{mse}^j \mathrel{+}= \partial_t(\rho e_\text{tot}) \, / \, \rho$. The area-weighted density $\rho a^j$ is not tendencied, matching edmfx_sgs_diffusive_flux_tendency!. Diffusing each subdomain's own scalars instead would erode the updraft-environment contrasts that the mass-flux decomposition maintains.

Mixing length

The eddy diffusivity depends on direction only through the mixing length: the TKE $\kappa_{\mathrm{iso}}$ is isotropic. The horizontal mixing length is the full physical mixing length $l_\text{phys}$ of the [45] closure (the smooth minimum of the wall, TKE-balance, and static-stability scales, unchanged from the vertical closure), limited from above by a grid scale. It differs from the vertical (master) mixing length only in that grid limiter: the horizontal length uses the spectral-element node scale $\Delta x_h$ (horizontal_filter_scale), whereas the vertical pipeline uses the resolvability filter scale $\Delta_f = \max(\Delta x_h, \Delta z)$ (resolvability_filter_scale),

\[l_h = \min(l_\text{phys}, \, \Delta x_h), \qquad l = \min(l_\text{phys}, \, \Delta_f).\]

Wherever $\Delta x_h \ge \Delta z$ (single columns and GCM resolutions), the two limiters coincide ($\Delta_f = \Delta x_h$), so $l_h = l$ and the horizontal diffusivity equals the one built from the master mixing length. They differ only where $\Delta z > \Delta x_h$ (gray-zone and LES aspect ratios): there $\Delta_f = \Delta z$ while the horizontal length remains limited by $\Delta x_h$, giving the shorter horizontal scale $l_h \le l$ and the ratio $l_h / l = \Delta x_h / \Delta z$ wherever $l_\text{phys}$ exceeds both spacings.

When to enable it

Horizontal EDMF diffusion is intended for configurations whose horizontal node scale $\Delta x_h$ is comparable to or smaller than the physical mixing length, so that the horizontal limiter $\min(l_\text{phys}, \Delta x_h)$ binds. This is the high-resolution and gray-zone regime, where an anisotropic SGS length scale matters.

At coarse horizontal resolution (for example global runs with $\Delta x_h$ of tens of kilometers), the horizontal limiter rarely binds, the horizontal diffusivity reduces to the isotropic environment value, and the term is typically negligible next to resolved horizontal transport. Both options are off by default for that reason.

The Smagorinsky-Lilly and anisotropic-minimum-dissipation closures already supply horizontal SGS diffusion of the same fields, so combining edmfx_sgs_horizontal_diffusive_flux with either is rejected at model construction, as is edmfx_horizontal_diffusion without edmfx_sgs_horizontal_diffusive_flux.

Because the term is explicit, it adds a horizontal diffusive stability limit on the timestep, $\Delta t \lesssim \Delta x_h^2 / (2 K_{h,h})$. At the fine horizontal resolutions where this closure is intended, the timestep is in practice already set by the explicit horizontal acoustic limit $\Delta t \lesssim \Delta x_h / c_s$, which is the more restrictive of the two.

Configuration

edmfx_sgs_horizontal_diffusive_flux: true   # default: false
edmfx_horizontal_diffusion: true            # default: false; requires the flux option

Diagnostics

Three diagnostic variables expose the horizontal closure fields:

• lmixh: the horizontal mixing length $l_h$ [m], recomputed on demand with the grid-scale limit set by the horizontal node spacing;
• edth: the horizontal eddy diffusivity for scalars $K_{h,h}$ [m² s⁻¹], read from the cached ᶜK_h_h;
• evuh: the horizontal eddy viscosity $K_{u,h}$ [m² s⁻¹], read from the cached ᶜK_u_h.

The vertical counterparts are lmix, edt, and evu.