Calculating the barotropic pressure contribution to the zonal momentum budget

[scott-conn]

Hi!

I'm interested in looking at terms in the zonal momentum budget for the hydrostatic model. So far, I've been outputting the tendency kernel functions from here:

Oceananigans.jl/src/Models/HydrostaticFreeSurfaceModels/hydrostatic_free_surface_tendency_kernel_functions.jl

Lines 16 to 54 in 11c317d

	"""
	Return the tendency for the horizontal velocity in the ``x``-direction, or the east-west
	direction, ``u``, at grid point `i, j, k` for a `HydrostaticFreeSurfaceModel`.
	The tendency for ``u`` is called ``G_u`` and defined via
	```
	∂_t u = G_u - ∂_x p_n
	```
	where `p_n` is the part of the barotropic kinematic pressure that's treated
	implicitly during time-stepping.
	"""
	@inline function hydrostatic_free_surface_u_velocity_tendency(i, j, k, grid,
	advection,
	coriolis,
	closure,
	u_immersed_bc,
	velocities,
	free_surface,
	tracers,
	buoyancy,
	diffusivities,
	hydrostatic_pressure_anomaly,
	auxiliary_fields,
	forcings,
	clock)
	model_fields = merge(hydrostatic_fields(velocities, free_surface, tracers), auxiliary_fields)
	return ( - U_dot_∇u(i, j, k, grid, advection, velocities)
	- explicit_barotropic_pressure_x_gradient(i, j, k, grid, free_surface)
	- x_f_cross_U(i, j, k, grid, coriolis, velocities)
	- ∂xᶠᶜᶜ(i, j, k, grid, hydrostatic_pressure_anomaly)
	- ∂ⱼ_τ₁ⱼ(i, j, k, grid, closure, diffusivities, clock, model_fields, buoyancy)
	- immersed_∂ⱼ_τ₁ⱼ(i, j, k, grid, velocities, u_immersed_bc, closure, diffusivities, clock, model_fields)
	+ forcings.u(i, j, k, grid, clock, hydrostatic_prognostic_fields(velocities, free_surface, tracers)))
	end

It seems like that will give me everything except the implicit part of the barotropic pressure gradient. If I'm understanding correctly, this means that for a model that has an implicit free surface, I'll be missing the entire contribution of SSH variations to the pressure gradient. Is there some way I can also output this part of the pressure gradient that I'm missing (or at least an approximation to it)?

Thank you!