Forcing functions

"Forcings" are user-defined terms appended to right-hand side of the momentum or tracer evolution equations. In Oceananigans, momentum and tracer forcings are defined via julia functions. Oceananigans includes an interface for implementing forcing functions that depend on spatial coordinates, time, model velocity and tracer fields, and external parameters.

Forcings are added to Oceananigans models by passing a NamedTuple of functions or forcing objects to the forcing keyword argument in NonhydrostaticModel's constructor. By default, momentum and tracer forcing functions are assumed to be functions of x, y, z, t. A basic example is

u_forcing(x, y, z, t) = exp(z) * cos(x) * sin(t)

grid = RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1))
model = NonhydrostaticModel(grid=grid, forcing=(u=u_forcing,))

model.forcing.u

# output
ContinuousForcing{Nothing} at (Face, Center, Center)
├── func: u_forcing (generic function with 1 method)
├── parameters: nothing
└── field dependencies: ()

More general forcing functions are built via the Forcing constructor described below. Oceananigans also provides two convenience types: * Relaxation for damping terms that restore a field to a target distribution outside of a masked region of space. Relaxation can be used to implement sponge layers near the boundaries of a domain. * AdvectiveForcing for advecting individual quantities by a separate or "slip" velocity relative to both the prognostic model velocity field and any BackgroundField velocity field.

The Forcing constructor

The Forcing constructor provides an interface for specifying forcing functions that

  1. Depend on external parameters; and
  2. Depend on model fields at the x, y, z location that forcing is applied; and/or
  3. Require access to discrete model data.

Forcing functions with external parameters

Most forcings involve external, changeable parameters. Here are two examples of forcing_functions that depend on (i) a single scalar parameter s, and (ii) a NamedTuple of parameters, p:

# Forcing that depends on a scalar parameter `s`
u_forcing_func(x, y, z, t, s) = s * z

u_forcing = Forcing(u_forcing_func, parameters=0.1)

# Forcing that depends on a `NamedTuple` of parameters `p`
T_forcing_func(x, y, z, t, p) = - p.μ * exp(z / p.λ) * cos(p.k * x) * sin(p.ω * t)

T_forcing = Forcing(T_forcing_func, parameters=(μ=1, λ=0.5, k=2π, ω=4π))

grid = RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1))
model = NonhydrostaticModel(grid=grid, forcing=(u=u_forcing, T=T_forcing), buoyancy=SeawaterBuoyancy(), tracers=(:T, :S))

model.forcing.T

# output
ContinuousForcing{NamedTuple{(:μ, :λ, :k, :ω), Tuple{Int64, Float64, Float64, Float64}}} at (Center, Center, Center)
├── func: T_forcing_func (generic function with 1 method)
├── parameters: (μ = 1, λ = 0.5, k = 6.283185307179586, ω = 12.566370614359172)
└── field dependencies: ()
model.forcing.u

# output
ContinuousForcing{Float64} at (Face, Center, Center)
├── func: u_forcing_func (generic function with 1 method)
├── parameters: 0.1
└── field dependencies: ()

In this example, the objects passed to the parameters keyword in the construction of u_forcing and T_forcing –- a floating point number for u_forcing, and a NamedTuple of parameters for T_forcing –- are passed on to u_forcing_func and T_forcing_func when they are called during time-stepping. The object passed to parameters is in principle arbitrary. However, if using the GPU, then typeof(parameters) may be restricted by the requirements of GPU-compiliability.

Forcing functions that depend on model fields

Forcing functions may depend on model fields (velocity, tracers or auxiliary fields) evaluated at the x, y, z where forcing is applied. Here's a somewhat non-sensical example:

# Forcing that depends on the velocity fields `u`, `v`, and `w`
w_forcing_func(x, y, z, t, u, v, w) = - (u^2 + v^2 + w^2) / 2

w_forcing = Forcing(w_forcing_func, field_dependencies=(:u, :v, :w))

# Forcing that depends on salinity `S` and a scalar parameter
S_forcing_func(x, y, z, t, S, μ) = - μ * S

S_forcing = Forcing(S_forcing_func, parameters=0.01, field_dependencies=:S)

grid = RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1))
model = NonhydrostaticModel(grid=grid, forcing=(w=w_forcing, S=S_forcing), buoyancy=SeawaterBuoyancy(), tracers=(:T, :S))

model.forcing.w

# output
ContinuousForcing{Nothing} at (Center, Center, Face)
├── func: w_forcing_func (generic function with 1 method)
├── parameters: nothing
└── field dependencies: (:u, :v, :w)
model.forcing.S

# output
ContinuousForcing{Float64} at (Center, Center, Center)
├── func: S_forcing_func (generic function with 1 method)
├── parameters: 0.01
└── field dependencies: (:S,)

The field_dependencies arguments follow x, y, z, t in the forcing function in the order they are specified in Forcing. If both field_dependencies and parameters are specified, then the field_dependencies arguments follow x, y, z, t, and parameters follow field_dependencies.

Model fields that arise in the arguments of continuous Forcing functions are automatically interpolated to the staggered grid location at which the forcing is applied.

"Discrete form" forcing functions

"Discrete form" forcing functions are either called with the signature

func(i, j, k, grid, clock, model_fields)

or the parameterized form

func(i, j, k, grid, clock, model_fields, parameters)

Discrete form forcing functions can access the entirety of model field data through the argument model_fields. The object model_fields is a NamedTuple whose properties include the velocity fields model_fields.u, model_fields.v, model_fields.w and all fields in model.tracers.

Using discrete forcing functions may require understanding the staggered arrangement of velocity fields and tracers in Oceananigans. Here's a slightly non-sensical example in which the vertical derivative of a buoyancy tracer is used as a time-scale for damping the u-velocity field:

# A damping term that depends on a "local average":
local_average(i, j, k, grid, c) = @inbounds (c[i, j, k] + c[i-1, j, k] + c[i+1, j, k] +
                                                          c[i, j-1, k] + c[i, j+1, k] +
                                                          c[i, j, k-1] + c[i, j, k+1]) / 7

b_forcing_func(i, j, k, grid, clock, model_fields) = - local_average(i, j, k, grid, model_fields.b)

b_forcing = Forcing(b_forcing_func, discrete_form=true)

# A term that damps the local velocity field in the presence of stratification
using Oceananigans.Operators: ∂zᶠᶜᶠ, ℑxzᶠᵃᶜ

function u_forcing_func(i, j, k, grid, clock, model_fields, ε)
    # The vertical derivative of buoyancy, interpolated to the u-velocity location:
    N² = ℑxzᶠᵃᶜ(i, j, k, grid, ∂zᶠᶜᶠ, model_fields.b)

    # Set to zero in unstable stratification where N² < 0:
    N² = max(N², zero(typeof(N²)))

    return @inbounds - ε * sqrt(N²) * model_fields.u[i, j, k]
end

u_forcing = Forcing(u_forcing_func, discrete_form=true, parameters=1e-3)

grid = RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1))
model = NonhydrostaticModel(grid=grid, tracers=:b, buoyancy=BuoyancyTracer(), forcing=(u=u_forcing, b=b_forcing))

model.forcing.b

# output
DiscreteForcing{Nothing}
├── func: b_forcing_func (generic function with 1 method)
└── parameters: nothing
model.forcing.u

# output
DiscreteForcing{Float64}
├── func: u_forcing_func (generic function with 1 method)
└── parameters: 0.001

The annotation @inbounds is crucial for performance when accessing array indices of the fields in model_fields.

Relaxation

Relaxation defines a special forcing function that restores a field at a specified rate to a target distribution, within a region uncovered by a masking function. Relaxation is useful for implementing sponge layers, as shown in the second example.

The following code constructs a model in which all components of the velocity field are damped to zero everywhere on a time-scale of 1000 seconds, or ~17 minutes:

damping = Relaxation(rate = 1/1000)

grid = RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1)) 
model = NonhydrostaticModel(grid=grid, forcing=(u=damping, v=damping, w=damping))

model.forcing.w

# output
ContinuousForcing{Nothing} at (Center, Center, Face)
├── func: Relaxation(rate=0.001, mask=1, target=0)
├── parameters: nothing
└── field dependencies: (:w,)

The constructor for Relaxation accepts the keyword arguments mask, and target, which specify a mask(x, y, z) function that multiplies the forcing, and a target(x, y, z) distribution for the quantity in question. By default, mask uncovered the whole domain and target restores the field in question to 0

We illustrate usage of mask and target by implementing a sponge layer that relaxes velocity fields to zero and restores temperature to a linear gradient in the bottom 1/10th of the domain:

grid = RectilinearGrid(size=(1, 1, 1), x=(0, 1), y=(0, 1), z=(-1, 0))

        damping_rate = 1/100 # relax fields on a 100 second time-scale
temperature_gradient = 0.001 # ⁰C m⁻¹
 surface_temperature = 20    # ⁰C (at z=0)

target_temperature = LinearTarget{:z}(intercept=surface_temperature, gradient=temperature_gradient)
       bottom_mask = GaussianMask{:z}(center=-grid.Lz, width=grid.Lz/10)

uvw_sponge = Relaxation(rate=damping_rate, mask=bottom_mask)
  T_sponge = Relaxation(rate=damping_rate, mask=bottom_mask, target=target_temperature)

model = NonhydrostaticModel(grid=grid, forcing=(u=uvw_sponge, v=uvw_sponge, w=uvw_sponge, T=T_sponge), buoyancy=SeawaterBuoyancy(), tracers=(:T, :S))

model.forcing.u

# output
ContinuousForcing{Nothing} at (Face, Center, Center)
├── func: Relaxation(rate=0.01, mask=exp(-(z + 1.0)^2 / (2 * 0.1^2)), target=0)
├── parameters: nothing
└── field dependencies: (:u,)
model.forcing.T

# output
ContinuousForcing{Nothing} at (Center, Center, Center)
├── func: Relaxation(rate=0.01, mask=exp(-(z + 1.0)^2 / (2 * 0.1^2)), target=20.0 + 0.001 * z)
├── parameters: nothing
└── field dependencies: (:T,)

AdvectiveForcing

AdvectiveForcing defines a forcing function that represents advection by a separate or "slip" velocity relative to the prognostic model velocity field. AdvectiveForcing is implemented with native Oceananigans advection operators, which means that tracers advected by the "flux form" advection term $∇ ⋅ u⃗_slip c$. Caution is advised when $u⃗_slip$ is not divergence free.

As an example, consider a model for sediment settling at a constant rate:

using Oceananigans

r_sediment = 1e-4 # [m] "Fine sand"
ρ_sediment = 1200 # kg m⁻³
ρ_ocean = 1026 # kg m⁻³
Δb = 9.81 * (ρ_ocean - ρ_sediment) / ρ_ocean # m s⁻²
ν_molecular = 1.05e-6 # m² s⁻¹
w_sediment = 2/9 * Δb / ν_molecular * r_sediment^2 # m s⁻¹

sinking = AdvectiveForcing(UpwindBiasedFifthOrder(), w=w_sediment)

# output
AdvectiveForcing with the UpwindBiased scheme:
├── u: ZeroField{Int64}
├── v: ZeroField{Int64}
└── w: ConstantField(-0.00352102)

The first argument to AdvectiveForcing is the advection scheme (here UpwindBiasedFifthOrder()). The three keyword arguments specify the u, v, and w components of the separate slip velocity field. The default for each u, v, w is ZeroField.

Next we consider a dynamically-evolving slip velocity. For this we use ZFaceField with appropriate boundary conditions as our slip velocity:

using Oceananigans
using Oceananigans.BoundaryConditions: ImpenetrableBoundaryCondition

grid = RectilinearGrid(size=(32, 32, 32), x=(-10, 10), y=(-10, 10), z=(-4, 4),
                       topology=(Periodic, Periodic, Bounded))

no_penetration = ImpenetrableBoundaryCondition()
slip_bcs = FieldBoundaryConditions(grid, (Center, Center, Face),
                                   top=no_penetration, bottom=no_penetration)

w_slip = ZFaceField(grid, boundary_conditions=slip_bcs)
sinking = AdvectiveForcing(WENO(; grid), w=w_slip)

# output
AdvectiveForcing with the WENO scheme:
├── u: ZeroField{Int64}
├── v: ZeroField{Int64}
└── w: 32×32×33 Field{Center, Center, Face} on RectilinearGrid on CPU

To compute the slip velocity, we must add a Callbackto simulations.callback that computes w_slip ever iteration:

using Oceananigans.BoundaryConditions: fill_halo_regions!

model = NonhydrostaticModel(; grid, tracers=(:b, :P), forcing=(; P=sinking))
simulation = Simulation(model; Δt=1, stop_iteration=100)

# Build abstract operation for slip velocity
b_particle = - 1e-4 # relative buoyancy depends on reference density and initial buoyancy condition
b = model.tracers.b
R = 1e-3 # [m] mean particle radius
ν = 1.05e-6 # [m² s⁻¹] molecular kinematic viscosity of water
w_slip_op = 2/9 * (b - b_particle) / ν * R^2 # Stokes terminal velocity

function compute_slip_velocity!(sim)
    w_slip .= w_slip_op
    fill_halo_regions!(w_slip)
    return nothing
end

simulation.callbacks[:slip] = Callback(compute_slip_velocity!)

# output
Callback of compute_slip_velocity! on IterationInterval(1)