Understanding Boundary Conditions with Stokes Drift: Eulerian or Lagrangian?

[anpa5279]

Are the velocity BCs in Oceananigans for Eulerian or Lagrangian velocities? In the Langmuir Turbulence Example, the u velocity BC is defined as a wind stress. How are Stokes drift effects accounted for at that boundary?

τx = -3.72e-5 # m² s⁻², surface kinematic momentum flux

u_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(τx))

model = NonhydrostaticModel(; grid, coriolis,
                            advection = WENO(),
                            timestepper = :RungeKutta3,
                            tracers = :b,
                            buoyancy = BuoyancyTracer(),
                            closure = AnisotropicMinimumDissipation(),
                            stokes_drift = UniformStokesDrift(∂z_uˢ=∂z_uˢ),
                            boundary_conditions = (u=u_boundary_conditions, b=b_boundary_conditions))

Since Oceananigans uses the Lagrangian-mean formulation, does stokes_drift = UniformStokesDrift(∂z_uˢ=∂z_uˢ) apply something to the boundary?

[glwagner]

The application of a prescribed wind stress does not depend on whether the prognostic velocity variable is the Eulerian-mean or Lagrangian-mean. You can consider very simple 1D problems to convince yourself of this.

The prognostic variable does matter if the boundary condition is 1) Value or Gradient, or 2) the calculation of the wind stress depends on the prognostic field itself (ie a quadratic drag). In those cases, the variable to which the boundary condition is applied is the Lagrangian-mean velocity. This is also likely what you want...

But neither of those are the case in your example, so you are in the clear.

We've written a preprint that touches on some of these issues if you'd like to take a look: https://glwagner.github.io/assets/pdf/Stokes-drift-ocean-circulation-Wagner-Constantinou-Reichl.pdf