Model Top and Sponge Layer

The model top is a rigid lid at a fixed height $z_t$, with insulating and free-slip boundary conditions. A rigid lid reflects upward-propagating waves, so a sponge layer below it absorbs them before they can reflect and contaminate the solution [1].

ClimaAtmos provides two independent sponges: a viscous sponge, which applies horizontal viscous damping to the velocities and most scalars, and Rayleigh damping, which relaxes velocities toward zero. Both are optional and both are confined to the upper part of the domain by the same vertical ramp.

The ramp

Damping has to switch on smoothly, or the onset itself reflects waves. Both sponges use the ramp

\[\beta_{\mathrm{sponge}}(z) = \begin{cases} 0 & z \le z_d, \\[1ex] \sin^2 \left( \dfrac{\pi}{2} \dfrac{z - z_d}{z_t - z_d} \right) & z > z_d, \end{cases}\]

which rises from zero at the onset height $z_d$ to one at the model top [35].

The layer should be roughly 1.5 times the vertical wavelength of the waves it aims to absorb, which is around 2–10 km in the stratosphere [36, 37]. The default onset height is set by the zd_rayleigh and zd_viscous parameters.

Viscous sponge

The viscous sponge applies horizontal viscous damping to the velocities (grid-mean and updraft vertical velocities), the total energy, the water, and the passive grid-scale tracers. As in hyperdiffusion, the precipitating species are exempt: water is damped through the effective total water $q_t^{\mathrm{eff}} = q_t - q_{rai} - q_{sno}$, and the resulting mass tendency is distributed over the suspended cloud species. Turbulence kinetic energy and the PROPHET subdomain scalars are also exempt; they receive only the Rayleigh tracer damping described below. For a scalar $\psi$, the tendency is

\[\rho S_\psi = \dots + \kappa_{\max} \beta_{\mathrm{sponge}} \nabla_h \cdot (\rho \nabla_h \psi),\]

with turbulent diffusivity $\kappa_{\max}$, default $10^6$ in units of m² s⁻¹, set by kappa_2_sponge. For the velocity, the same coefficient multiplies the horizontal vector Laplacian defined in Hyperdiffusion; horizontal density variations are neglected there, as in the hyperdiffusion tendency.

Viscous damping of all variables absorbs waves without applying the zonal-mean torque that Rayleigh damping of horizontal velocities produces [38].

Energy is damped through the same two-piece decomposition that hyperdiffusion and the vertical diffusion use, rather than through $\rho e_{tot}$ directly: a dry-static-energy flux $\rho \nabla_h s_d$, which applies in every configuration, plus a water-enthalpy flux $\rho h_{\mathrm{tot,cl}} \nabla_h q_t^{\mathrm{eff}}$ in moist configurations. Damping the two pieces separately keeps the sponge consistent with the enthalpy-based formulation used elsewhere in the model.

Rayleigh damping

Rayleigh damping relaxes a velocity component linearly toward zero,

\[\boldsymbol{S}_u = \dots - \alpha_{\max} \beta_{\mathrm{sponge}} (\hat{\boldsymbol{k}} \cdot \boldsymbol{u}) \hat{\boldsymbol{k}} ,\]

applied to the vertical velocity with coefficient $\alpha_{\max}$, default 1 s⁻¹, set by alpha_rayleigh_w.

Rayleigh damping is restricted to the vertical velocity for the same reason the viscous sponge handles everything else: relaxing horizontal velocities toward zero applies a zonal-mean torque, which distorts the upper-atmosphere circulation [32, 38].

The implementation also offers damping of horizontal velocity and of tracers, for configurations that need it:

Because the latter two default to zero, the default configuration damps only the vertical velocity, as described above. Some PROPHET configurations enable a small tracer damping. The tracer coefficient acts only on the turbulence kinetic energy and on the PROPHET subdomain scalars, and for the latter the relaxation is toward the grid-mean value rather than toward zero, so only the subgrid-scale departure is damped.

Interaction with timestepping

The Rayleigh damping of the vertical velocity is carried in the vertically implicit part of the tendency split, so its damping timescale does not restrict the timestep; this is why $\alpha_{\max}$ of order 1 s⁻¹ works at timesteps far longer than 1 s. The viscous sponge, which involves horizontal operators, and the optional Rayleigh terms for horizontal velocity and tracers are treated explicitly. See Discretization and Operators for the split and Implicit Solver for the solve.

Where this is implemented

Both sponges are selected with the viscous_sponge and rayleigh_sponge configuration keys; see Configuration Options.