Apply boundary conditions

Boundary conditions in ClimaCore attach to operators, not to fields. A finite-difference operator takes one condition per boundary name of its domain; a DG interface flux takes a horizontal boundary condition or a one-sided flux. This page lists the forms and when each applies.

Vertical boundaries (finite-difference operators)

The vertical domain names its two boundaries, (:bottom, :top) by convention, and every C2F operator whose stencil reaches a boundary face takes a keyword of that name.

Condition	Meaning at the boundary face	Accepted by
`SetValue(x₀)`	The operator's argument takes the value `x₀`	`InterpolateC2F`, `DivergenceF2C` (a flux), and, through `DirichletOperator`, `GradientC2F`, `DivergenceC2F`, `CurlC2F`, `UpwindBiasedProductC2F`
`SetGradient(v₀)`	The gradient equals the covariant vector `v₀`	`GradientC2F`
`SetDivergence(d₀)`	The divergence equals `d₀`	`DivergenceC2F`
`SetCurl(c₀)`	The curl equals the contravariant vector `c₀`	`CurlC2F`
`Extrapolate()` / `Outflow()`	Use the nearest interior value (`Outflow(; order)` for higher-order extrapolation)	`InterpolateC2F`, the upwind operators

Steps:

Name the boundaries when constructing the vertical domain, boundary_names = (:bottom, :top).
Give a condition to the outermost operator whose stencil touches the boundary, and only to it; inner operators in a fused expression stay inside the boundary faces. In divf2c(gradc2f(θ)), the divergence takes SetValue fluxes and the gradient takes none:
```
gradc2f = Operators.GradientC2F()
divf2c = Operators.DivergenceF2C(
    bottom = Operators.SetValue(Geometry.WVector(0.0)),
    top = Operators.SetValue(Geometry.WVector(0.0)),
)
@. dydt = divf2c(κ * gradc2f(θ))
```
For a Dirichlet value of the argument itself, use the helper that builds the equivalent gradient stencil: Operators.gradient_c2f_dirichlet(θ; bottom = θ₀, top = Operators.SetGradient(...)) (Solve a column PDE).

A SetGradient prescribes the covariant component ∂x/∂ξ³, the derivative along the third coordinate line. Over terrain, the coordinate surface is tilted, so a zero covariant component is a zero normal derivative only where the surface is flat; the GradientC2F docstring gives the value that prescribes a zero normal derivative on a slope.

Horizontal boundaries

Spectral-element domains are periodic or walled per direction (periodic = true on an IntervalDomain, or boundary_names). On a CG space, the operators impose horizontal conditions through the state, for example by projecting the velocity onto the wall. On a DG space, the interface flux at a boundary face needs the exterior state:

Operators.PeriodicBC(): the domain wraps; no boundary faces exist.
Operators.ReflectingWallBC(): the exterior state mirrors the normal momentum, Operators.ghost_state(bc, normal, argvals⁻), so the wall passes no normal flux.
A one-sided flux function boundary_numflux(normal, argvals⁻) passed to tendency_completion, for any other closure; examples/plane/bickleyjet.jl builds a reflecting wall this way for its noslip case.

With the boundary flux omitted, DG boundary faces contribute a zero flux, which is a closed-boundary closure.

Sponges and damping

Absorbing layers near the model top are tendencies, not boundary conditions; ClimaAtmos's sponge page describes its viscous and Rayleigh sponges, which are written with the operators above.