Forcings and Idealized Cases

A single column or a small box does not simulate its own large-scale environment. The subsidence that caps a stratocumulus deck, the horizontal advection that dries a trade-wind boundary layer, and the geostrophic wind that the surface layer turns against all originate outside the domain. Forcings supply those tendencies as prescribed terms, either from an analytic profile that defines an idealized case or from a data file.

All the forcings described here are applied explicitly, from additional_tendency!, and none has a configuration key of its own. A forcing comes from the setup that initial_condition names, so choosing a case chooses its forcings; the two exceptions are Held–Suarez, selected through the rad key, and the monthly-averaged reanalysis case, selected through external_forcing. See Setups for the hooks a setup implements and Running Single-Column Cases for how to run the cases.

Large-scale subsidence

Subsidence advects scalars vertically with a prescribed velocity $w_{sub}(z)$, negative for descent. The tendency is written in advective rather than flux form,

\[\left. \frac{\partial \chi}{\partial t} \right|_{sub} = -w_{sub} \frac{\partial \chi}{\partial z} ,\]

and discretized as the difference between a flux divergence and $\chi$ times the divergence of the velocity, with first-order upwinding of $\chi$. Writing it this way makes the scheme consistent for a uniform tracer: both terms cancel level by level, so $\chi \equiv 1$ produces no tendency.

The vertical operator carries a zero-flux condition at the top and bottom faces. For the usual case of inflow through the lid, the two terms cancel there as well, which is equivalent to holding $\chi$ at its top-level value in the air descending into the domain.

What subsides depends on which of the two implementations is active. The analytic subsidence of the idealized cases applies to the total enthalpy $h_{tot}$, the total water $q_t$, and, with non-equilibrium microphysics, cloud liquid and cloud ice. The Subsidence() term of the file-driven cases reaches only $h_{tot}$ and $q_t$. Neither touches rain, snow, the hydrometeor number concentrations, the P3 rime variables, or the passive tracers.

Large-scale horizontal advection

Horizontal advection by the large-scale flow enters as prescribed tendencies of temperature and total water, $(\partial T/\partial t)_{adv}$ and $(\partial q_t/\partial t)_{adv}$, supplied as functions of height and, for temperature, of the Exner function. Total water takes the tendency directly, and total energy takes

\[\left. \frac{\partial \rho e_{tot}}{\partial t} \right|_{adv} = \rho \left[ c_{vm} \left. \frac{\partial T}{\partial t} \right|_{adv} + I_v(T) \left. \frac{\partial q_t}{\partial t} \right|_{adv} \right] ,\]

with $c_{vm}$ the isochoric heat capacity of the moist mixture and $I_v$ the internal energy of water vapor. No geopotential term appears: horizontal advection moves specific humidity between columns at the same height.

Column Coriolis and geostrophic forcing

In a single column the horizontal pressure gradient cannot be computed, so it is prescribed through a geostrophic wind $\boldsymbol{u}_g(z)$ and the flow is relaxed toward it by the Coriolis acceleration,

\[\left. \frac{\partial \boldsymbol{u}_h}{\partial t} \right|_{cor} = -f \hat{\boldsymbol{k}} \times (\boldsymbol{u}_h - \boldsymbol{u}_g) .\]

The Coriolis parameter $f$ and both components of $\boldsymbol{u}_g$ come from the setup. A case may set $f = 0$, which switches the term off while leaving the geostrophic profile available to other parts of the setup.

Driving a column from data

Cases driven by GCM output, reanalysis, or a field campaign compose their forcing from a tuple of terms rather than fixing it in the setup. Four terms are available, and each declares the variables it needs from the file:

The default composition applies all four, nudging temperature and humidity as one term and the two wind components as another. The separation matters: VerticalFluctuation supplies a fixed tendency computed from the driving data, whereas Subsidence transports whatever the column currently holds. A dataset that already contains the full vertical advective tendency should use one or the other, not both.

The first three terms accumulate into a single pair of temperature and moisture tendencies, which is then converted to the prognostic variables,

\[\left. \frac{\partial \rho e_{tot}}{\partial t} \right|_{ext} = \rho \left[ c_{vm} \frac{\partial T}{\partial t} + \left( c_{vv} (T - T_0) + L_{v0} - R_v T_0 \right) \frac{\partial q_t}{\partial t} \right] , \qquad \left. \frac{\partial \rho q_t}{\partial t} \right|_{ext} = \rho \frac{\partial q_t}{\partial t} .\]

Nudging

Nudging relaxes a variable toward the driving profile at a rate $\tau^{-1}(z)$,

\[\left. \frac{\partial \psi}{\partial t} \right|_{nudge} = -\frac{\psi - \psi_{data}(z, t)}{\tau(z)} .\]

For scalars, the rate follows a raised cosine in height [81], so that the boundary layer evolves freely while the free troposphere is held near the driving data:

\[\tau^{-1}(z) = \begin{cases} 0 & z < z_i , \\[1ex] \dfrac{1}{2 \tau_r} \left[ 1 - \cos\left( \pi \dfrac{z - z_i}{z_r - z_i} \right) \right] & z_i \le z \le z_r , \\[2ex] \dfrac{1}{\tau_r} & z > z_r . \end{cases}\]

The wind rate is height-independent. The four controlling parameters are

A term may instead be given a constant timescale, a profile, or a height mask.

Where the data comes from

All three sources reach the forcing terms through one canonical, CMIP-named vocabulary, so the terms themselves do not know which source they are reading:

• GCM output at a cfsite, read as time-mean profiles. The subsidence velocity is derived from the pressure velocity, and the vertical-fluctuation tendency is the eddy part of the vertical advection, obtained by subtracting the mean advection from the total on the GCM grid.
• Reanalysis, generated from ERA5 for a given site and date, either as a matched trajectory or as a monthly-averaged repeating day.
• ARM VARANAL field-campaign forcing, converted from pressure levels. Its own vertical advective tendencies are deliberately discarded in favor of the Subsidence() term, so that vertical transport acts on the simulated profiles rather than on the observed ones.

Column Datasets documents the file schema, the per-case defaults, and how to supply your own file; Adding a Column Dataset covers new formats.

Held–Suarez forcing

The Held–Suarez benchmark [79] replaces radiation and boundary-layer physics with Newtonian relaxation toward a prescribed equilibrium temperature and Rayleigh friction on the low-level winds. It is the standard test of a dry dynamical core, and it occupies the radiation slot for the reasons given in Radiation.

Both terms are confined near the surface by the same ramp in the vertical coordinate $\sigma = p/p_s$,

\[f_h(\sigma) = \max \left( 0, \frac{\sigma - \sigma_b}{1 - \sigma_b} \right) ,\]

where the surface pressure is reconstructed hydrostatically from the surface elevation and temperature. The thermal relaxation rate is largest in the tropical boundary layer,

\[k_T(\phi, \sigma) = k_a + (k_s - k_a) \, f_h(\sigma) \cos^4\phi , \qquad k_a = \frac{1}{40 \, \text{day}} , \quad k_s = \frac{1}{4 \, \text{day}} ,\]

and the equilibrium temperature has the familiar equator-to-pole and static stability contrasts, with a stratospheric floor,

\[T_{eq}(\phi, p) = \max \left[ T_{min}, \, \left( T_{eq,0} - \Delta T_y \sin^2\phi - \Delta\theta_z \log\frac{p}{p_0} \cos^2\phi \right) \left( \frac{p}{p_0} \right)^{\kappa_d} \right] .\]

Friction relaxes the horizontal wind toward rest with $k_f = 1/\text{day}$,

\[\left. \frac{\partial \boldsymbol{u}_h}{\partial t} \right|_{HS} = -k_f f_h(\sigma) \, \boldsymbol{u}_h .\]

Two implementation choices are worth knowing. Temperature is diagnosed from the ideal gas law for dry air, so the forcing is independent of moisture even in a moist run. The energy tendency converts the temperature relaxation with the dry isochoric heat capacity $c_{vd}$, consistent with relaxing at constant density. The equator-to-pole contrast and the equatorial equilibrium temperature take different values in dry and moist configurations (ΔT_y_dry/ΔT_y_wet and T_equator_dry/T_equator_wet).

Prescribed radiative forcing

Three cases prescribe a radiative flux profile instead of calling RRTMGP: the DYCOMS cloud-top cooling profile, its ISDAC counterpart, and the TRMM-LBA prescribed heating rate. They are selected through the same rad key as the radiative transfer modes and are described in Radiation.

Prescribed flow

The Shipway and Hill kinematic driver replaces the momentum equation entirely: the vertical velocity is prescribed as a half-sine pulse in time, and the thermodynamic and microphysical variables are advected through it. It is a test of the microphysics and its transport, not of the dynamics, so it requires flat topography and the explicit solver.

What each case forces

Symbols

Where this is implemented