Diffusion

Diffusion in ClimaAtmos represents physical subgrid-scale turbulent transport: the vertical mixing of a boundary layer the grid does not resolve, and the horizontal mixing an eddy-viscosity closure supplies. It is distinct from Hyperdiffusion, which is a numerical filter with no physical content, though both smooth gradients and both therefore reduce the spurious oscillations and negative tracer values described in Microphysics.

This page collects what the diffusive terms act on and how they are split between the implicit and explicit halves of the timestep. The closures that supply the diffusivities are documented elsewhere: Large-Eddy Simulation Closures for the resolved-eddy closures, PROPHET: Closures for the TKE-based eddy diffusivity, and the two prescribed profiles below.

Vertical diffusion

Two prescribed-diffusivity closures are available through the vert_diff configuration key. Both are boundary-layer schemes: the diffusivity is a function of height or pressure alone.

Here $\|\boldsymbol{u}_a\|$ and $z_a$ are the wind speed and height of the lowest model level, and $C_E$ is a dimensionless coefficient. Where the pressure falls below 850 hPa, above the boundary layer, the surface-driven diffusivity decays as

\[K_E \exp \left[ - \left( \frac{p_{pbl} - p}{p_{strato}} \right)^2 \right] , \qquad p_{pbl} = 850 \, \mathrm{hPa} , \quad p_{strato} = 100 \, \mathrm{hPa} ,\]

which confines the mixing to the boundary layer and the lower free troposphere. The same value serves as the eddy viscosity for momentum; disable_momentum_vertical_diffusion restricts either closure to the scalars.

Face diffusivities are formed as a harmonic mean of the two neighboring center values rather than an arithmetic one. The flux then collapses at a face separating a turbulent layer from quiescent, strongly stratified air, where an arithmetic mean would leave about $K/2$.

Vertical diffusion can instead come from PROPHET, whose diffusive flux is closed by a prognostic turbulence kinetic energy, or from the LES closures: a vertically acting Smagorinsky–Lilly, or AMD, which always acts on both axes. These paths are alternatives, and a configuration that sets vert_diff alongside turbconv, amd_les, or a vertically acting smagorinsky_lilly is rejected at model construction, as is edmfx_sgs_horizontal_diffusive_flux alongside an LES closure in the horizontal.

What each variable receives

Energy diffuses as a two-piece enthalpy flux rather than as a lumped total enthalpy,

\[\boldsymbol{\mathcal{F}}_E = -\rho K_h \left[ \nabla_v s_d + (h_{\mathrm{eff}} + \Phi) \nabla_v q_t^{\mathrm{eff}} \right] ,\]

with $s_d = h_d + \Phi$ the dry static energy and $h_{\mathrm{eff}}$ the mass-weighted specific enthalpy of the water that actually diffuses. The dry term applies in every configuration, the water term only when $\rho q_t$ is prognostic. Thermodynamics and the Working Fluid explains why the split is needed: diffusing a lumped $h_{tot}$ is not invariant to the reference temperature $T_0$, so it does not conserve energy and water together.

Water diffuses through the effective total water $q_t^{\mathrm{eff}} = q_t - q_{rai} - q_{sno}$. The same flux increments both $\rho q_t$ and $\rho$, so diffusing water carries the corresponding moist-air mass. Cloud liquid and cloud ice have no flux of their own: each takes a share of the aggregate tendency, scaled by the clipped ratio $\min(q_\mu / q_t^{\mathrm{eff}}, 1)$, and the matching number densities scale with it, which preserves the mean particle mass.

Rain, snow, and rain number density are exempt. Diffusing them would smear features such as rain shafts, which evolve through sedimentation and microphysics; see Microphysics.

Passive grid-mean tracers diffuse independently with the full $K_h$.

Momentum receives $\nabla \cdot \boldsymbol{\tau} / \rho$ with the stress $\boldsymbol{\tau} = 2 \rho K_u \boldsymbol{S}$ and $\boldsymbol{S}$ the vertical strain rate, under zero-flux boundary conditions. The surface stress is not applied here; it is added by the separate surface-flux tendency, described in Surface Conditions.

Implicit or explicit

Vertical diffusion is stiff at the timesteps of interest, so implicit_diffusion: true moves the whole tendency into the implicit solve. The model's diff_mode follows that key: under Implicit() the tendency is evaluated from implicit_tendency! and linearized in the Jacobian; under Explicit() it is evaluated from additional_tendency!. The global configurations with a 60 km model top set implicit_diffusion: true, as does every PROPHET configuration except the advection test, and those pair it with approximate_linear_solve_iters: 2. See Implicit Solver.

Horizontal diffusion

Horizontal diffusion comes from an eddy-viscosity closure, never from a prescribed profile: Smagorinsky–Lilly [10], anisotropic minimum dissipation [67], a constant horizontal diffusivity, or the optional PROPHET horizontal diffusive flux. All of them are explicit. See Large-Eddy Simulation Closures and PROPHET Horizontal Diffusion.

The species treatment differs from the vertical closures above. Smagorinsky–Lilly and the constant horizontal diffusion apply a single scalar diffusivity to all grid-scale tracers, precipitation included, while anisotropic minimum dissipation forms a diffusivity per scalar.

Where this is implemented

The keys named here are vert_diff and implicit_diffusion; the horizontal closures are selected with smagorinsky_lilly, amd_les, constant_horizontal_diffusion, and edmfx_sgs_horizontal_diffusive_flux. See Configuration Options.