# Library

Documenting the public user interface.

## Abstract operations

Oceananigans.AbstractOperations.volumeConstant
volume = VolumeMetric()

Instance of VolumeMetric that generates BinaryOperations between AbstractFields and their cell volumes. Summing this BinaryOperation yields an integral of AbstractField over the domain.

Example

julia> using Oceananigans

julia> using Oceananigans.AbstractOperations: volume

julia> grid = RectilinearGrid(size=(2, 2, 2), extent=(1, 2, 3)); c = CenterField(grid);

julia> c .= 1;

julia> c_dV = c * volume
BinaryOperation at (Center, Center, Center)
├── grid: 2×2×2 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
└── tree:
* at (Center, Center, Center)
├── 2×2×2 Field{Center, Center, Center} on RectilinearGrid on CPU
└── Vᶜᶜᶜ at (Center, Center, Center)

julia> c_dV[1, 1, 1]
0.75

julia> sum(c_dV)
6.0
source
Oceananigans.AbstractOperations.ΔzConstant
Δz = ZSpacingMetric()

Instance of ZSpacingMetric that generates BinaryOperations between AbstractFields and the vertical grid spacing evaluated at the same location as the AbstractField.

Δx and Δy play a similar role for horizontal grid spacings.

Example

julia> using Oceananigans

julia> using Oceananigans.AbstractOperations: Δz

julia> grid = RectilinearGrid(size=(1, 1, 1), extent=(1, 2, 3)); c = CenterField(grid);

julia> c_dz = c * Δz # returns BinaryOperation between Field and GridMetricOperation
BinaryOperation at (Center, Center, Center)
├── grid: 1×1×1 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
└── tree:
* at (Center, Center, Center)
├── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
└── Δzᶜᶜᶜ at (Center, Center, Center)

julia> c .= 1;

julia> c_dz[1, 1, 1]
3.0
source
Oceananigans.AbstractOperations.AverageMethod
Average(field::AbstractField; condition = nothing, mask = 0, dims=:)

Return Reduction representing a spatial average of field over dims.

Over regularly-spaced dimensions this is equivalent to a numerical mean!.

Over dimensions of variable spacing, field is multiplied by the appropriate grid length, area or volume, and divided by the total spatial extent of the interval.

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Oceananigans.AbstractOperations.BinaryOperationMethod
BinaryOperation{LX, LY, LZ}(op, a, b, ▶a, ▶b, grid)

Return an abstract representation of the binary operation op(▶a(a), ▶b(b)) on grid, where ▶a and ▶b interpolate a and b to locations (LX, LY, LZ).

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Oceananigans.AbstractOperations.ConditionalOperationMethod
ConditionalOperation(operand::AbstractField;
func = identity,
condition = nothing,
mask = 0)

Return an abstract representation of a masking procedure applied when condition is satisfied on a field described by func(operand).

Positional arguments

• operand: The AbstractField to be masked (it must have a grid property!)

Keyword arguments

• func: A unary transformation applied element-wise to the field operand at locations where condition == true. Default is identity.

• condition: either a function of (i, j, k, grid, operand) returning a Boolean, or a 3-dimensional Boolean AbstractArray. At locations where condition == false, operand will be masked by mask

• mask: the scalar mask

condition_operand is a convenience function used to construct a ConditionalOperation

condition_operand(func::Function, operand::AbstractField, condition, mask) = ConditionalOperation(operand; func, condition, mask)

Example

julia> using Oceananigans

julia> using Oceananigans.Fields: condition_operand

julia> c = CenterField(RectilinearGrid(size=(2, 1, 1), extent=(1, 1, 1)));

julia> f(i, j, k, grid, c) = i < 2; d = condition_operand(cos, c, f, 10)
ConditionalOperation at (Center, Center, Center)
├── operand: 2×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
├── grid: 2×1×1 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── func: cos (generic function with 26 methods)
├── condition: f (generic function with 1 method)

julia> d[1, 1, 1]
1.0

julia> d[2, 1, 1]
10
source
Oceananigans.AbstractOperations.KernelFunctionOperationMethod
KernelFunctionOperation{LX, LY, LZ}(kernel_function, grid;
computed_dependencies=(), parameters=nothing)

Construct a KernelFunctionOperation at location (LX, LY, LZ) on grid an with an optional iterable of computed_dependencies and arbitrary parameters.

With isnothing(parameters) (the default), kernel_function is called with

kernel_function(i, j, k, grid, computed_dependencies...)

Otherwise kernel_function is called with

kernel_function(i, j, k, grid, computed_dependencies..., parameters)

Examples

Construct a kernel function operation that returns random numbers:

random_kernel_function(i, j, k, grid) = rand() # use CUDA.rand on the GPU

kernel_op = KernelFunctionOperation{Center, Center, Center}(random_kernel_function, grid)

Construct a kernel function operation using the vertical vorticity operator valid on curvilinear and cubed sphere grids:

using Oceananigans.Operators: ζ₃ᶠᶠᶜ # called with signature ζ₃ᶠᶠᶜ(i, j, k, grid, u, v)

grid = model.grid
u, v, w = model.velocities

ζ_op = KernelFunctionOperation{Face, Face, Center}(ζ₃ᶠᶠᶜ, grid, computed_dependencies=(u, v))
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Oceananigans.AbstractOperations.UnaryOperationMethod
UnaryOperation{LX, LY, LZ}(op, arg, ▶, grid)

Returns an abstract UnaryOperation representing the action of op on arg, and subsequent interpolation by ▶ on grid.

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Oceananigans.AbstractOperations.∂xMethod
∂x(L::Tuple, arg::AbstractField)

Return an abstract representation of an $x$-derivative acting on field arg followed by interpolation to L, where L is a 3-tuple of Faces and Centers.

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Oceananigans.AbstractOperations.∂yMethod
∂y(L::Tuple, arg::AbstractField)

Return an abstract representation of a $y$-derivative acting on field arg followed by interpolation to L, where L is a 3-tuple of Faces and Centers.

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Oceananigans.AbstractOperations.∂zMethod
∂z(L::Tuple, arg::AbstractField)

Return an abstract representation of a $z$-derivative acting on field arg followed by interpolation to L, where L is a 3-tuple of Faces and Centers.

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Oceananigans.AbstractOperations.@atMacro
@at location abstract_operation

Modify the abstract_operation so that it returns values at location, where location is a 3-tuple of Faces and Centers.

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Oceananigans.AbstractOperations.@binaryMacro
@binary op1 op2 op3...

Turn each binary function in the list (op1, op2, op3...) into a binary operator on Oceananigans.Fields for use in AbstractOperations.

Note: a binary function is a function with two arguments: for example, +(x, y) is a binary function.

Also note: a binary function in Base must be imported to be extended: use import Base: op; @binary op.

Example

julia> using Oceananigans, Oceananigans.AbstractOperations

julia> using Oceananigans.AbstractOperations: BinaryOperation, AbstractGridMetric, choose_location

julia> plus_or_times(x, y) = x < 0 ? x + y : x * y
plus_or_times (generic function with 1 method)

julia> @binary plus_or_times
Set{Any} with 6 elements:
:+
:/
:^
:-
:*
:plus_or_times

julia> c, d = (CenterField(RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1))) for i = 1:2);

julia> plus_or_times(c, d)
BinaryOperation at (Center, Center, Center)
├── grid: 1×1×1 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
└── tree:
plus_or_times at (Center, Center, Center)
├── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
└── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
source
Oceananigans.AbstractOperations.@multiaryMacro
@multiary op1 op2 op3...

Turn each multiary operator in the list (op1, op2, op3...) into a multiary operator on Oceananigans.Fields for use in AbstractOperations.

Note that a multiary operator:

• is a function with two or more arguments: for example, +(x, y, z) is a multiary function;
• must be imported to be extended if part of Base: use import Base: op; @multiary op;
• can only be called on Oceananigans.Fields if the "location" is noted explicitly; see example.

Example

julia> using Oceananigans, Oceananigans.AbstractOperations

julia> harmonic_plus(a, b, c) = 1/3 * (1/a + 1/b + 1/c)
harmonic_plus (generic function with 1 method)

julia> c, d, e = Tuple(CenterField(RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1))) for i = 1:3);

julia> harmonic_plus(c, d, e) # before magic @multiary transformation
BinaryOperation at (Center, Center, Center)
├── grid: 1×1×1 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
└── tree:
* at (Center, Center, Center)
├── 0.3333333333333333
└── + at (Center, Center, Center)
├── / at (Center, Center, Center)
│   ├── 1
│   └── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
├── / at (Center, Center, Center)
│   ├── 1
│   └── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
└── / at (Center, Center, Center)
├── 1
└── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU

julia> @multiary harmonic_plus
Set{Any} with 3 elements:
:+
:harmonic_plus
:*

julia> harmonic_plus(c, d, e)
MultiaryOperation at (Center, Center, Center)
├── grid: 1×1×1 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
└── tree:
harmonic_plus at (Center, Center, Center)
├── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
├── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
└── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
source
Oceananigans.AbstractOperations.@unaryMacro
@unary op1 op2 op3...

Turn each unary function in the list (op1, op2, op3...) into a unary operator on Oceananigans.Fields for use in AbstractOperations.

Note: a unary function is a function with one argument: for example, sin(x) is a unary function.

Also note: a unary function in Base must be imported to be extended: use import Base: op; @unary op.

Example

julia> using Oceananigans, Oceananigans.Grids, Oceananigans.AbstractOperations

julia> square_it(x) = x^2
square_it (generic function with 1 method)

julia> @unary square_it
Set{Any} with 10 elements:
:+
:sqrt
:square_it
:cos
:exp
:interpolate_identity
:-
:tanh
:sin
:abs

julia> c = CenterField(RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1)));

julia> square_it(c)
UnaryOperation at (Center, Center, Center)
├── grid: 1×1×1 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
└── tree:
square_it at (Center, Center, Center) via identity
└── 1×1×1 Field{Center, Center, Center} on RectilinearGrid on CPU
source

Oceananigans.Advection.WENOType
WENO([FT=Float64;]
order = 5,
grid = nothing,
zweno = true,
vector_invariant = nothing,
bounds = nothing)

Construct a weigthed essentially non-oscillatory advection scheme of order order.

Keyword arguments

• order: The order of the WENO advection scheme. Default: 5

• grid: (defaults to nothing)

• vector_invariant: The stencil for which the vector-invariant form of the advection scheme would use. Options VelocityStencil() or VorticityStencil(); defaults to nothing.

• zweno: When true implement a Z-WENO formulation for the WENO weights calculation. (defaults to true)

Examples

julia> using Oceananigans;

julia> WENO()
WENO reconstruction order 5 in Flux form
Smoothness formulation:
└── Z-weno
Boundary scheme:
└── WENO reconstruction order 3 in Flux form
Symmetric scheme:
└── Centered reconstruction order 4
Directions:
├── X regular
├── Y regular
└── Z regular
julia> using Oceananigans;

julia> Nx, Nz = 16, 10;

julia> Lx, Lz = 1e4, 1e3;

julia> chebychev_spaced_z_faces(k) = - Lz/2 - Lz/2 * cos(π * (k - 1) / Nz);

julia> grid = RectilinearGrid(size = (Nx, Nz), halo = (4, 4), topology=(Periodic, Flat, Bounded),
x = (0, Lx), z = chebychev_spaced_z_faces);

julia> WENO(grid; order=7)
WENO reconstruction order 7 in Flux form
Smoothness formulation:
└── Z-weno
Boundary scheme:
└── WENO reconstruction order 5 in Flux form
Symmetric scheme:
└── Centered reconstruction order 6
Directions:
├── X regular
├── Y regular
└── Z stretched
source
Oceananigans.Advection.div_UcMethod
div_uc(i, j, k, grid, advection, U, c)

Calculate the divergence of the flux of a tracer quantity $c$ being advected by a velocity field, $𝛁⋅(𝐯 c)$,

1/V * [δxᶜᵃᵃ(Ax * u * ℑxᶠᵃᵃ(c)) + δyᵃᶜᵃ(Ay * v * ℑyᵃᶠᵃ(c)) + δzᵃᵃᶜ(Az * w * ℑzᵃᵃᶠ(c))]

which ends up at the location ccc.

source
Oceananigans.Advection.div_𝐯uMethod
div_𝐯u(i, j, k, grid, advection, U, u)

Calculate the advection of momentum in the $x$-direction using the conservative form, $𝛁⋅(𝐯 u)$,

1/Vᵘ * [δxᶠᵃᵃ(ℑxᶜᵃᵃ(Ax * u) * ℑxᶜᵃᵃ(u)) + δy_fca(ℑxᶠᵃᵃ(Ay * v) * ℑyᵃᶠᵃ(u)) + δz_fac(ℑxᶠᵃᵃ(Az * w) * ℑzᵃᵃᶠ(u))]

which ends up at the location fcc.

source
Oceananigans.Advection.div_𝐯vMethod
div_𝐯v(i, j, k, grid, advection, U, v)

Calculate the advection of momentum in the $y$-direction using the conservative form, $𝛁⋅(𝐯 v)$,

1/Vʸ * [δx_cfa(ℑyᵃᶠᵃ(Ax * u) * ℑxᶠᵃᵃ(v)) + δyᵃᶠᵃ(ℑyᵃᶜᵃ(Ay * v) * ℑyᵃᶜᵃ(v)) + δz_afc(ℑxᶠᵃᵃ(Az * w) * ℑzᵃᵃᶠ(w))]

which ends up at the location cfc.

source
Oceananigans.Advection.div_𝐯wMethod
div_𝐯w(i, j, k, grid, advection, U, w)

Calculate the advection of momentum in the $z$-direction using the conservative form, $𝛁⋅(𝐯 w)$,

1/Vʷ * [δx_caf(ℑzᵃᵃᶠ(Ax * u) * ℑxᶠᵃᵃ(w)) + δy_acf(ℑzᵃᵃᶠ(Ay * v) * ℑyᵃᶠᵃ(w)) + δzᵃᵃᶠ(ℑzᵃᵃᶜ(Az * w) * ℑzᵃᵃᶜ(w))]

which ends up at the location ccf.

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## Boundary conditions

Oceananigans.BoundaryConditions.BoundaryConditionMethod
BoundaryCondition(Classification::DataType, condition)

Construct a boundary condition of type BC with a number or array as a condition.

Boundary condition types include Periodic, Flux, Value, Gradient, and Open.

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Oceananigans.BoundaryConditions.BoundaryConditionMethod
BoundaryCondition(Classification::DataType, condition::Function;
parameters = nothing,
discrete_form = false,
field_dependencies=())

Construct a boundary condition of type Classification with a function boundary condition.

By default, the function boudnary condition is assumed to have the 'continuous form' condition(ξ, η, t), where t is time and ξ and η vary along the boundary. In particular:

• On x-boundaries, condition(y, z, t).
• On y-boundaries, condition(x, z, t).
• On z-boundaries, condition(x, y, t).

If parameters is not nothing, then function boundary conditions have the form func(ξ, η, t, parameters), where ξ and η are spatial coordinates varying along the boundary as explained above.

If discrete_form = true, the function condition is assumed to have the "discrete form",

condition(i, j, grid, clock, model_fields)

where i, and j are indices that vary along the boundary. If discrete_form = true and parameters is not nothing, the function condition is called with

condition(i, j, grid, clock, model_fields, parameters)
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Oceananigans.BoundaryConditions.FieldBoundaryConditionsType
FieldBoundaryConditions(grid, location, indices=(:, :, :);
west     = default_auxiliary_bc(topology(grid, 1)(), location[1]()),
east     = default_auxiliary_bc(topology(grid, 1)(), location[1]()),
south    = default_auxiliary_bc(topology(grid, 2)(), location[2]()),
north    = default_auxiliary_bc(topology(grid, 2)(), location[2]()),
bottom   = default_auxiliary_bc(topology(grid, 3)(), location[3]()),
top      = default_auxiliary_bc(topology(grid, 3)(), location[3]()),
immersed = NoFluxBoundaryCondition())

Return boundary conditions for auxiliary fields (fields whose values are derived from a model's prognostic fields) on grid and at location.

Keyword arguments

Keyword arguments specify boundary conditions on the 6 possible boundaries:

• west, left end point in the x-direction where i = 1
• east, right end point in the x-direction where i = grid.Nx
• south, left end point in the y-direction where j = 1
• north, right end point in the y-direction where j = grid.Ny
• bottom, right end point in the z-direction where k = 1
• top, right end point in the z-direction where k = grid.Nz
• immersed: boundary between solid and fluid for immersed boundaries

If a boundary condition is unspecified, the default for auxiliary fields and the topology in the boundary-normal direction is used:

• PeriodicBoundaryCondition for Periodic directions
• GradientBoundaryCondition(0) for Bounded directions and Centered-located fields
• nothing for Bounded directions and Face-located fields
• nothing for Flat directions and/or Nothing-located fields
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Oceananigans.BoundaryConditions.FieldBoundaryConditionsType
FieldBoundaryConditions(; kwargs...)

Return a template for boundary conditions on prognostic fields.

Keyword arguments

Keyword arguments specify boundary conditions on the 7 possible boundaries:

• west: left end point in the x-direction where i = 1
• east: right end point in the x-direction where i = grid.Nx
• south: left end point in the y-direction where j = 1
• north: right end point in the y-direction where j = grid.Ny
• bottom: right end point in the z-direction where k = 1
• top: right end point in the z-direction where k = grid.Nz
• immersed: boundary between solid and fluid for immersed boundaries

If a boundary condition is unspecified, the default for prognostic fields and the topology in the boundary-normal direction is used:

• PeriodicBoundaryCondition for Periodic directions
• NoFluxBoundaryCondition for Bounded directions and Centered-located fields
• ImpenetrableBoundaryCondition for Bounded directions and Face-located fields
• nothing for Flat directions and/or Nothing-located fields
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Oceananigans.BoundaryConditions.FluxType
struct Flux <: AbstractBoundaryConditionClassification

A classification specifying a boundary condition on the flux of a field.

The sign convention is such that a positive flux represents the flux of a quantity in the positive direction. For example, a positive vertical flux implies a quantity is fluxed upwards, in the $+z$ direction.

Due to this convention, a positive flux applied to the top boundary specifies that a quantity is fluxed upwards across the top boundary and thus out of the domain. As a result, a positive flux applied to a top boundary leads to a reduction of that quantity in the interior of the domain; for example, a positive, upwards flux of heat at the top of the domain acts to cool the interior of the domain. Conversely, a positive flux applied to the bottom boundary leads to an increase of the quantity in the interior of the domain. The same logic holds for east, west, north, and south boundaries.

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Oceananigans.BoundaryConditions.OpenType
struct Open <: AbstractBoundaryConditionClassification

A classification that specifies the halo regions of a field directly.

For fields located at Faces, Open also specifies field value on the boundary.

Open boundary conditions are used to specify the component of a velocity field normal to a boundary and can also be used to describe nested or linked simulation domains.

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## BuoyancyModels

Oceananigans.BuoyancyModels.BuoyancyMethod
Buoyancy(; model, gravity_unit_vector=ZDirection())

Uses a given buoyancy model to create buoyancy in a model. The optional keyword argument gravity_unit_vector can be used to specify the direction opposite to the gravitational acceleration (which we take here to mean the "vertical" direction).

Example

using Oceananigans

grid = RectilinearGrid(size=(1, 8, 8), extent=(1, 1000, 100))
θ = 45 # degrees
g̃ = (0, sind(θ), cosd(θ))

buoyancy = Buoyancy(model=BuoyancyTracer(), gravity_unit_vector=g̃)

model = NonhydrostaticModel(grid=grid, buoyancy=buoyancy, tracers=:b)
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Oceananigans.BuoyancyModels.LinearEquationOfStateType
LinearEquationOfState([FT=Float64;] thermal_expansion=1.67e-4, haline_contraction=7.80e-4)

Return LinearEquationOfState for SeawaterBuoyancy with thermal_expansion coefficient and haline_contraction coefficient. The buoyancy perturbation $b$ for LinearEquationOfState is

$$$b = g (α T - β S),$$$

where $g$ is gravitational acceleration, $α$ is thermal_expansion, $β$ is haline_contraction, $T$ is temperature, and $S$ is practical salinity units.

Default constants in units inverse Kelvin and practical salinity units for thermal_expansion and haline_contraction, respectively, are taken from Table 1.2 (page 33) of Vallis, "Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation" (2nd ed, 2017).

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Oceananigans.BuoyancyModels.SeawaterBuoyancyType
SeawaterBuoyancy([FT = Float64;]
gravitational_acceleration = g_Earth,
equation_of_state = LinearEquationOfState(FT),
constant_temperature = false,
constant_salinity = false)

Returns parameters for a temperature- and salt-stratified seawater buoyancy model with a gravitational_acceleration constant (typically called $g$), and an equation_of_state that related temperature and salinity (or conservative temperature and absolute salinity) to density anomalies and buoyancy.

constant_temperature indicates that buoyancy depends only on salinity. For a nonlinear equation of state, constant_temperature is used as the temperature of the system. The same logic, with the roles of salinity and temperature reversed, holds when constant_salinity is provided.

For a linear equation of state, the values of constant_temperature or constant_salinity are irrelevant; in this case, constant_temperature=true (and similar for constant_salinity) is valid input.

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Oceananigans.BuoyancyModels.SeawaterBuoyancyType
SeawaterBuoyancy{FT, EOS, T, S} <: AbstractBuoyancyModel{EOS}

BuoyancyModels model for seawater. T and S are either nothing if both temperature and salinity are active, or of type FT if temperature or salinity are constant, respectively.

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Oceananigans.BuoyancyModels.∂x_bMethod
∂x_b(i, j, k, grid, b::SeawaterBuoyancy, C)

Returns the $x$-derivative of buoyancy for temperature and salt-stratified water,

$$$∂_x b = g ( α ∂_x T - β ∂_x S ) ,$$$

where $g$ is gravitational acceleration, $α$ is the thermal expansion coefficient, $β$ is the haline contraction coefficient, $T$ is conservative temperature, and $S$ is absolute salinity.

Note: In Oceananigans, model.tracers.T is conservative temperature and model.tracers.S is absolute salinity.

Note that $∂_x T$ (∂x_T), $∂_x S$ (∂x_S), $α$, and $β$ are all evaluated at cell interfaces in x and cell centers in y and z.

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Oceananigans.BuoyancyModels.∂y_bMethod
∂y_b(i, j, k, grid, b::SeawaterBuoyancy, C)

Returns the $y$-derivative of buoyancy for temperature and salt-stratified water,

$$$∂_y b = g ( α ∂_y T - β ∂_y S ) ,$$$

where $g$ is gravitational acceleration, $α$ is the thermal expansion coefficient, $β$ is the haline contraction coefficient, $T$ is conservative temperature, and $S$ is absolute salinity.

Note: In Oceananigans, model.tracers.T is conservative temperature and model.tracers.S is absolute salinity.

Note that $∂_y T$ (∂y_T), $∂_y S$ (∂y_S), $α$, and $β$ are all evaluated at cell interfaces in y and cell centers in x and z.

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Oceananigans.BuoyancyModels.∂z_bMethod
∂z_b(i, j, k, grid, b::SeawaterBuoyancy, C)

Returns the vertical derivative of buoyancy for temperature and salt-stratified water,

$$$∂_z b = N^2 = g ( α ∂_z T - β ∂_z S ) ,$$$

where $g$ is gravitational acceleration, $α$ is the thermal expansion coefficient, $β$ is the haline contraction coefficient, $T$ is conservative temperature, and $S$ is absolute salinity.

Note: In Oceananigans, model.tracers.T is conservative temperature and model.tracers.S is absolute salinity.

Note that $∂_z T$ (∂z_T), $∂_z S$ (∂z_S), $α$, and $β$ are all evaluated at cell interfaces in z and cell centers in x and y.

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## Coriolis

Oceananigans.Coriolis.BetaPlaneType
BetaPlane([T=Float64;] f₀=nothing, β=nothing,
rotation_rate=Ω_Earth, latitude=nothing, radius=R_Earth)

Return a $β$-plane Coriolis parameter, $f = f₀ + β y$.

The user may specify both f₀ and β, or the three parameters rotation_rate, latitude (in degrees), and radius that specify the rotation rate and radius of a planet, and the central latitude (where $y = 0$) at which the β-plane approximation is to be made.

If f₀ and β are not specified, they are calculated from rotation_rate, latitude, and radius according to the relations f₀ = 2 * rotation_rate * sind(latitude) and β = 2 * rotation_rate * cosd(latitude) / radius.

By default, the rotation_rate and planet radius are assumed to be Earth's.

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Oceananigans.Coriolis.BetaPlaneType
struct BetaPlane{T} <: AbstractRotation

A parameter object for meridionally increasing Coriolis parameter (f = f₀ + β y) that accounts for the variation of the locally vertical component of the rotation vector with latitude.

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Oceananigans.Coriolis.ConstantCartesianCoriolisType
ConstantCartesianCoriolis([FT=Float64;] fx=nothing, fy=nothing, fz=nothing,
f=nothing, rotation_axis=ZDirection(),
rotation_rate=Ω_Earth, latitude=nothing)

Return a parameter object for a constant rotation decomposed into the x, y, and z directions. In oceanography the components x, y, z correspond to the directions east, north, and up. This constant rotation can be specified in three different ways:

• Specifying all components fx, fy and fz directly.
• Specifying the Coriolis parameter f and (optionally) a rotation_axis (which defaults to the z direction if not specified).
• Specifying latitude (in degrees) and (optionally) a rotation_rate in radians per second (which defaults to Earth's rotation rate).
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Oceananigans.Coriolis.FPlaneType
FPlane([FT=Float64;] f=nothing, rotation_rate=Ω_Earth, latitude=nothing)

Return a parameter object for constant rotation at the angular frequency f/2, and therefore with background vorticity f, around a vertical axis. If f is not specified, it is calculated from rotation_rate and latitude (in degrees) according to the relation f = 2 * rotation_rate * sind(latitude).

By default, rotation_rate is assumed to be Earth's.

Also called FPlane, after the "f-plane" approximation for the local effect of a planet's rotation in a planar coordinate system tangent to the planet's surface.

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Oceananigans.Coriolis.HydrostaticSphericalCoriolisMethod
HydrostaticSphericalCoriolis([FT=Float64;]
rotation_rate = Ω_Earth,
scheme = EnergyConservingScheme())

Return a parameter object for Coriolis forces on a sphere rotating at rotation_rate. By default, rotation_rate is assumed to be Earth's.

Keyword arguments

• scheme: Either EnergyConservingScheme() (default), EnstrophyConservingScheme(), or WetCellEnstrophyConservingScheme().
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Oceananigans.Coriolis.NonTraditionalBetaPlaneType
NonTraditionalBetaPlane(FT=Float64;
fz=nothing, fy=nothing, β=nothing, γ=nothing,
rotation_rate=Ω_Earth, latitude=nothing, radius=R_Earth)

The user may directly specify fz, fy, β, γ, and radius or the three parameters rotation_rate, latitude (in degrees), and radius that specify the rotation rate and radius of a planet, and the central latitude (where $y = 0$) at which the non-traditional β-plane approximation is to be made.

If fz, fy, β, and γ are not specified, they are calculated from rotation_rate, latitude, and radius according to the relations fz = 2 * rotation_rate * sind(latitude), fy = 2 * rotation_rate * cosd(latitude), β = 2 * rotation_rate * cosd(latitude) / radius, and γ = - 4 * rotation_rate * sind(latitude) / radius.

By default, the rotation_rate and planet radius is assumed to be Earth's.

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Oceananigans.Coriolis.NonTraditionalBetaPlaneType
struct NonTraditionalBetaPlane{FT} <: AbstractRotation

A Coriolis implementation that accounts for the latitudinal variation of both the locally vertical and the locally horizontal components of the rotation vector. The "traditional" approximation in ocean models accounts for only the locally vertical component of the rotation vector (see BetaPlane).

This implementation is based off of section 5 of Dellar (2011). It conserve energy, angular momentum, and potential vorticity.

References

Dellar, P. (2011). Variations on a beta-plane: Derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere. Journal of Fluid Mechanics, 674, 174-195. doi:10.1017/S0022112010006464

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## Diagnostics

Oceananigans.Diagnostics.StateCheckerMethod
StateChecker(; schedule, fields)

Returns a StateChecker that logs field information (minimum, maximum, mean) for each field in a named tuple of fields when schedule actuates.

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Oceananigans.Diagnostics.AdvectiveCFLMethod
AdvectiveCFL(Δt)

Return an object for computing the Courant-Freidrichs-Lewy (CFL) number associated with time step Δt or TimeStepWizard and the time scale for advection across a cell. The advective CFL is, e.g., $U Δt / Δx$.

Example

julia> using Oceananigans

julia> model = NonhydrostaticModel(grid = RectilinearGrid(size=(16, 16, 16), extent=(8, 8, 8)));

julia> Δt = 1.0;

julia> model.velocities.u .= π;

julia> cfl(model)
6.283185307179586
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Oceananigans.Diagnostics.DiffusiveCFLMethod
DiffusiveCFL(Δt)

Returns an object for computing the diffusive Courant-Freidrichs-Lewy (CFL) number associated with time step Δt or TimeStepWizard and the time scale for diffusion across a cell associated with model.closure. The diffusive CFL, e.g., for viscosity is $ν Δt / Δx²$.

The maximum diffusive CFL number among viscosity and all tracer diffusivities is returned.

Example

julia> using Oceananigans

julia> model = NonhydrostaticModel(grid = RectilinearGrid(size=(16, 16, 16), extent=(1, 1, 1)),
closure = ScalarDiffusivity(; ν = 1e-2));

julia> Δt = 0.1;

julia> dcfl = DiffusiveCFL(Δt);

julia> dcfl(model)
0.256
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## Distributed

Oceananigans.Distributed.DistributedFFTBasedPoissonSolverMethod
DistributedFFTBasedPoissonSolver(global_grid, local_grid)

Return a FFT-based solver for the Poisson equation,

$$$∇²x = b$$$

for MultiArchitectures.

Supported configurations

We support two "modes":

1. Vertical pencil decompositions: two-dimensional decompositions in (x, y)
for three dimensional problems that satisfy either Nz > Rx or Nz > Ry.

2. One-dimensional decompositions in either x or y.

Above, Nz = size(global_grid, 3) and Rx, Ry, Rz = architecture(local_grid).ranks.

Other configurations that are decomposed in (x, y) but have too few Nz, or any configuration decomposed in z, are not supported.

Algorithm for two-dimensional decompositions

For two-dimensional decompositions (useful for three-dimensional problems), there are three forward transforms, three backward transforms, and four transpositions requiring MPI communication. In the schematic below, the first dimension is always the local dimension. In our implementation of the PencilFFTs algorithm, we require either Nz >= Rx, or Nz >= Ry, where Nz is the number of vertical cells, Rx is the number of ranks in x, and Ry is the number of ranks in y. Below, we outline the algorithm for the case Nz >= Rx. If Nz < Rx, but Nz > Ry, a similar algorithm applies with x and y swapped:

1. first(storage) is initialized with layout (z, x, y).
2. Transform along z.

3 Transpose + communicate to storage[2] in layout (x, z, y), which is distributed into (Rx, Ry) processes in (z, y).

1. Transform along x.
2. Transpose + communicate to last(storage) in layout (y, x, z), which is distributed into (Rx, Ry) processes in (x, z).
3. Transform in y.

At this point the three in-place forward transforms are complete, and we solve the Poisson equation by updating last(storage). Then the process is reversed to obtain first(storage) in physical space and with the layout (z, x, y).

Restrictions

The algorithm for two-dimensional decompositions requires that Nz = size(global_grid, 3) is larger than either Rx = ranks[1] or Ry = ranks[2], where ranks are configured when building MultiArch. If Nz does not satisfy this condition, we can only support a one-dimensional decomposition.

Algorithm for one-dimensional decompositions

This algorithm requires a one-dimensional decomposition with either Rx = 1 or Ry = 1, and is important to support two-dimensional transforms.

For one-dimensional decompositions, we place the decomposed direction last. If the number of ranks is Rh = max(Rx, Ry), this algorithm requires that both Nx > Rh and Ny > Rh. The resulting flow of transposes and transforms is similar to the two-dimensional case. It remains somewhat of a mystery why this succeeds (ie, why the last transform is correctly decomposed).

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## Fields

Oceananigans.Fields.AbstractFieldType
AbstractField{LX, LY, LZ, G, T, N}

Abstract supertype for fields located at (LX, LY, LZ) and defined on a grid G with eltype T and N dimensions.

Note: we need the parameter T to subtype AbstractArray.

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Oceananigans.Fields.BackgroundFieldMethod
BackgroundField(func; parameters=nothing)

Returns a BackgroundField to be passed to NonhydrostaticModel for use as a background velocity or tracer field.

If parameters is not provided, func must be callable with the signature

func(x, y, z, t)

If parameters is provided, func must be callable with the signature

func(x, y, z, t, parameters)
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Oceananigans.Fields.FieldMethod
Field{LX, LY, LZ}(grid::AbstractGrid,
T::DataType=eltype(grid); kw...) where {LX, LY, LZ}

Construct a Field on grid with data type T at the location (LX, LY, LZ). Each of (LX, LY, LZ) is either Center or Face and determines the field's location in (x, y, z) respectively.

Keyword arguments

• data :: OffsetArray: An offset array with the fields data. If nothing is provided the field is filled with zeros.
• boundary_conditions: If nothing is provided, then field is created using the default
boundary conditions via FieldBoundaryConditions.
• indices: Used to prescribe where a reduced field lives on. For example, at which k index does a two-dimensional $x$-$y$ field lives on. Default: (:, :, :).

Example

A field at location (Face, Face, Center).

julia> using Oceananigans

julia> grid = RectilinearGrid(size=(2, 3, 4), extent=(1, 1, 1));

julia> ω = Field{Face, Face, Center}(grid)
2×3×4 Field{Face, Face, Center} on RectilinearGrid on CPU
├── grid: 2×3×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── boundary conditions: FieldBoundaryConditions
│   └── west: Periodic, east: Periodic, south: Periodic, north: Periodic, bottom: ZeroFlux, top: ZeroFlux, immersed: ZeroFlux
└── data: 8×9×10 OffsetArray(::Array{Float64, 3}, -2:5, -2:6, -2:7) with eltype Float64 with indices -2:5×-2:6×-2:7
└── max=0.0, min=0.0, mean=0.0

Now, using indices we can create a two dimensional $x$-$y$ field at location (Face, Face, Center) to compute, e.g., the vertical vorticity $∂v/∂x - ∂u/∂y$ at the fluid's surface $z = 0$, which for Center corresponds to k = Nz.

julia> u = XFaceField(grid); v = YFaceField(grid);

julia> ωₛ = Field(∂x(v) - ∂y(u), indices=(:, :, grid.Nz))
2×3×1 Field{Face, Face, Center} on RectilinearGrid on CPU
├── grid: 2×3×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── boundary conditions: FieldBoundaryConditions
│   └── west: Periodic, east: Periodic, south: Periodic, north: Periodic, bottom: Nothing, top: Nothing, immersed: ZeroFlux
├── operand: BinaryOperation at (Face, Face, Center)
├── status: time=0.0
└── data: 8×9×1 OffsetArray(::Array{Float64, 3}, -2:5, -2:6, 4:4) with eltype Float64 with indices -2:5×-2:6×4:4
└── max=0.0, min=0.0, mean=0.0

julia> compute!(ωₛ)
2×3×1 Field{Face, Face, Center} on RectilinearGrid on CPU
├── grid: 2×3×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── boundary conditions: FieldBoundaryConditions
│   └── west: Periodic, east: Periodic, south: Periodic, north: Periodic, bottom: Nothing, top: Nothing, immersed: ZeroFlux
├── operand: BinaryOperation at (Face, Face, Center)
├── status: time=0.0
└── data: 8×9×1 OffsetArray(::Array{Float64, 3}, -2:5, -2:6, 4:4) with eltype Float64 with indices -2:5×-2:6×4:4
└── max=0.0, min=0.0, mean=0.0
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Oceananigans.Fields.ReductionMethod
Reduction(reduce!, operand; dims)

Return a Reduction of operand with reduce!, along dims. Note that Reduction expects reduce! to operate in-place.

Example

using Oceananigans

Nx, Ny, Nz = 3, 3, 3

grid = RectilinearGrid(size=(Nx, Ny, Nz), x=(0, 1), y=(0, 1), z=(0, 1),
topology=(Periodic, Periodic, Periodic))

c = CenterField(grid)

set!(c, (x, y, z) -> x + y + z)

max_c² = Field(Reduction(maximum!, c^2, dims=3))

compute!(max_c²)

max_c²[1:Nx, 1:Ny]

# output
3×3 Matrix{Float64}:
1.36111  2.25     3.36111
2.25     3.36111  4.69444
3.36111  4.69444  6.25
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Oceananigans.Fields.CenterFieldFunction
CenterField(grid; kw...)

Return a Field{Center, Center, Center} on grid. Additional keyword arguments are passed to the Field constructor.

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Oceananigans.Fields.PressureFieldsFunction
PressureFields(grid, bcs::NamedTuple)

Return a NamedTuple with pressure fields pHY′ and pNHS initialized as CenterFields on grid. Boundary conditions bcs may be specified via a named tuple of FieldBoundaryConditions.

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Oceananigans.Fields.PressureFieldsMethod
PressureFields(proposed_pressures::NamedTuple{(:pHY′, :pNHS)}, grid, bcs)

Return a NamedTuple of pressure fields with, overwriting boundary conditions in proposed_tracer_fields with corresponding fields in the NamedTuple bcs.

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Oceananigans.Fields.TendencyFieldsMethod
TendencyFields(grid, tracer_names;
u = XFaceField(grid),
v = YFaceField(grid),
w = ZFaceField(grid),
kwargs...)

Return a NamedTuple with tendencies for all solution fields (velocity fields and tracer fields), initialized on grid. Optional kwargs can be specified to assign data arrays to each tendency field.

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Oceananigans.Fields.TracerFieldsMethod
TracerFields(tracer_names, grid, user_bcs)

Return a NamedTuple with tracer fields specified by tracer_names initialized as CenterFields on grid. Boundary conditions user_bcs may be specified via a named tuple of FieldBoundaryConditions.

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Oceananigans.Fields.TracerFieldsMethod
TracerFields(tracer_names, grid; kwargs...)

Return a NamedTuple with tracer fields specified by tracer_names initialized as CenterFields on grid. Fields may be passed via optional keyword arguments kwargs for each field.

This function is used by OutputWriters.Checkpointer and TendencyFields. 

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Oceananigans.Fields.TracerFieldsMethod
TracerFields(proposed_tracers::NamedTuple, grid, bcs)

Return a NamedTuple of tracers, overwriting boundary conditions in proposed_tracers with corresponding fields in the NamedTuple bcs.

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Oceananigans.Fields.VelocityFieldsFunction
VelocityFields(grid, user_bcs = NamedTuple())

Return a NamedTuple with fields u, v, w initialized on grid. Boundary conditions bcs may be specified via a named tuple of FieldBoundaryConditions.

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Oceananigans.Fields.VelocityFieldsMethod
VelocityFields(proposed_velocities::NamedTuple{(:u, :v, :w)}, grid, bcs)

Return a NamedTuple of velocity fields, overwriting boundary conditions in proposed_velocities with corresponding fields in the NamedTuple bcs.

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Oceananigans.Fields.XFaceFieldFunction
XFaceField(grid; kw...)

Return a Field{Face, Center, Center} on grid. Additional keyword arguments are passed to the Field constructor.

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Oceananigans.Fields.YFaceFieldFunction
YFaceField(grid; kw...)

Return a Field{Center, Face, Center} on grid. Additional keyword arguments are passed to the Field constructor.

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Oceananigans.Fields.ZFaceFieldFunction
ZFaceField(grid; kw...)

Return a Field{Center, Center, Face} on grid. Additional keyword arguments are passed to the Field constructor.

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Oceananigans.Fields.regrid!Method
regrid!(a, b)

Regrid field b onto the grid of field a.

Functionality limitation

Currently regrid! only regrids in the vertical $z$ direction and works only on fields that have data only in $z$ direction.

Example

Generate a tracer field on a vertically stretched grid and regrid it on a regular grid.

using Oceananigans

Nz, Lz = 2, 1.0
topology = (Flat, Flat, Bounded)

input_grid = RectilinearGrid(size=Nz, z = [0, Lz/3, Lz], topology=topology, halo=1)
input_field = CenterField(input_grid)
input_field[1, 1, 1:Nz] = [2, 3]

output_grid = RectilinearGrid(size=Nz, z=(0, Lz), topology=topology, halo=1)
output_field = CenterField(output_grid)

regrid!(output_field, input_field)

output_field[1, 1, :]

# output
4-element OffsetArray(::Vector{Float64}, 0:3) with eltype Float64 with indices 0:3:
0.0
2.333333333333334
3.0
0.0
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## Forcings

Oceananigans.Forcings.AdvectiveForcingType
AdvectiveForcing(scheme=UpwindBiasedFifthOrder(), u=ZeroField(), v=ZeroField(), w=ZeroField())

Build a forcing term representing advection by the velocity field u, v, w with an advection scheme.

Example

Using a tracer field to model sinking particles

using Oceananigans

# Physical parameters
gravitational_acceleration          = 9.81     # m s⁻²
ocean_density                       = 1026     # kg m⁻³
mean_particle_density               = 2000     # kg m⁻³
ocean_molecular_kinematic_viscosity = 1.05e-6  # m² s⁻¹

# Terminal velocity of a sphere in viscous flow
Δb = gravitational_acceleration * (mean_particle_density - ocean_density) / ocean_density
ν = ocean_molecular_kinematic_viscosity

w_Stokes = - 2/9 * Δb / ν * R^2 # m s⁻¹

# output
├── u: ZeroField{Int64}
├── v: ZeroField{Int64}
└── w: ConstantField(-1.97096)
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Oceananigans.Forcings.ContinuousForcingMethod
ContinuousForcing(func; parameters=nothing, field_dependencies=())

Construct a "continuous form" forcing with optional parameters and optional field_dependencies on other fields in a model.

If neither parameters nor field_dependencies are provided, then func must be callable with the signature

func(x, y, z, t)

where x, y, z are the east-west, north-south, and vertical spatial coordinates, and t is time.

If field_dependencies are provided, the signature of func must include them. For example, if field_dependencies=(:u, :S) (and parameters are not provided), then func must be callable with the signature

func(x, y, z, t, u, S)

where u is assumed to be the u-velocity component, and S is a tracer. Note that any field which does not have the name u, v, or w is assumed to be a tracer and must be present in model.tracers.

If parameters are provided, then the last argument to func must be parameters. For example, if func has no field_dependencies but does depend on parameters, then it must be callable with the signature

func(x, y, z, t, parameters)

With field_dependencies=(:u, :v, :w, :c) and parameters, then func must be callable with the signature

func(x, y, z, t, u, v, w, c, parameters)
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Oceananigans.Forcings.DiscreteForcingMethod
DiscreteForcing(func; parameters=nothing)

Construct a "discrete form" forcing function with optional parameters. The forcing function is applied at grid point i, j, k.

When parameters are not specified, func must be callable with the signature

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

where grid is model.grid, clock.time is the current simulation time and clock.iteration is the current model iteration, and model_fields is a NamedTuple with u, v, w and the fields in model.tracers.

Note that the index end does not access the final physical grid point of a model field in any direction. The final grid point must be explicitly specified, as in model_fields.u[i, j, grid.Nz].

When parameters is specified, func must be callable with the signature.

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

Above, parameters is, in principle, arbitrary. Note, however, that GPU compilation can place constraints on typeof(parameters).

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Oceananigans.Forcings.GaussianMaskType
GaussianMask{D}(center, width)

Callable object that returns a Gaussian masking function centered on center, with width, and varying along direction D, i.e.,

exp(-(D - center)^2 / (2 * width^2))

Examples

• Create a Gaussian mask centered on z=0 with width 1 meter.
julia> mask = GaussianMask{:z}(center=0, width=1)
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Oceananigans.Forcings.LinearTargetType
LinearTarget{D}(intercept, gradient)

Callable object that returns a Linear target function with intercept and gradient, and varying along direction D, i.e.,

intercept + D * gradient

Examples

• Create a linear target function varying in z, equal to 0 at z=0 and with gradient 10⁻⁶:

julia> target = LinearTarget{:z}(intercept=0, gradient=1e-6)
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Oceananigans.Forcings.RelaxationType
struct Relaxation{R, M, T}

Callable object for restoring fields to a target at some rate and within a masked region in x, y, z.

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Oceananigans.Forcings.RelaxationMethod
Relaxation(; rate, mask=onefunction, target=zerofunction)

Returns a Forcing that restores a field to target(x, y, z, t) at the specified rate, in the region mask(x, y, z).

The functions onefunction and zerofunction always return 1 and 0, respectively. Thus the default mask leaves the whole domain uncovered, and the default target is zero.

Example

• Restore a field to zero on a timescale of "3600" (equal to one hour if the time units of the simulation are seconds).
using Oceananigans

damping = Relaxation(rate = 1/3600)

# output
Relaxation{Float64, typeof(Oceananigans.Forcings.onefunction), typeof(Oceananigans.Forcings.zerofunction)}
├── rate: 0.0002777777777777778
└── target: 0
• Restore a field to a linear z-gradient within the bottom 1/4 of a domain on a timescale of "60" (equal to one minute if the time units of the simulation are seconds).
dTdz = 0.001 # ⁰C m⁻¹, temperature gradient

T₀ = 20 # ⁰C, surface temperature at z=0

Lz = 100 # m, depth of domain

bottom_sponge_layer = Relaxation(; rate = 1/60,

# output
├── rate: 0.016666666666666666
├── mask: exp(-(z + 100.0)^2 / (2 * 25.0^2))
└── target: 20.0 + 0.001 * z
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Oceananigans.Forcings.ForcingMethod
Forcing(func; parameters=nothing, field_dependencies=(), discrete_form=false)

Returns a forcing function added to the tendency of an Oceananigans model field.

If discrete_form=false (the default), and neither parameters nor field_dependencies are provided, then func must be callable with the signature

func(x, y, z, t)

where x, y, z are the east-west, north-south, and vertical spatial coordinates, and t is time. Note that this form is also default in the constructor for NonhydrostaticModel, so that Forcing is not needed.

If discrete_form=false (the default), and field_dependencies are provided, the signature of func must include them. For example, if field_dependencies=(:u, :S) (and parameters are not provided), then func must be callable with the signature

func(x, y, z, t, u, S)

where u is assumed to be the u-velocity component, and S is a tracer. Note that any field which does not have the name u, v, or w is assumed to be a tracer and must be present in model.tracers.

If discrete_form=false (the default) and parameters are provided, then the last argument to func must be parameters. For example, if func has no field_dependencies but does depend on parameters, then it must be callable with the signature

func(x, y, z, t, parameters)

The object parameters is arbitrary in principle, however GPU compilation can place constraints on typeof(parameters).

With field_dependencies=(:u, :v, :w, :c) and parameters, then func must be callable with the signature

func(x, y, z, t, u, v, w, c, parameters)

If discrete_form=true then func must be callable with the "discrete form"

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

where i, j, k is the grid point at which the forcing is applied, grid is model.grid, clock.time is the current simulation time and clock.iteration is the current model iteration, and model_fields is a NamedTuple with u, v, w, the fields in model.tracers, and the fields in model.diffusivity_fields, each of which is an OffsetArrays (or NamedTuples of OffsetArrays depending on the turbulence closure) of field data.

When discrete_form=true and parameters is specified, func must be callable with the signature

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

Examples

using Oceananigans

# Parameterized forcing
parameterized_func(x, y, z, t, p) = p.μ * exp(z / p.λ) * cos(p.ω * t)

v_forcing = Forcing(parameterized_func, parameters = (μ=42, λ=0.1, ω=π))

# output
ContinuousForcing{NamedTuple{(:μ, :λ, :ω), Tuple{Int64, Float64, Irrational{:π}}}}
├── func: parameterized_func (generic function with 1 method)
├── parameters: (μ = 42, λ = 0.1, ω = π)
└── field dependencies: ()

Note that because forcing locations are regularized within the NonhydrostaticModel constructor:

grid = RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1))
model = NonhydrostaticModel(grid=grid, forcing=(v=v_forcing,))

model.forcing.v

# output
ContinuousForcing{NamedTuple{(:μ, :λ, :ω), Tuple{Int64, Float64, Irrational{:π}}}} at (Center, Face, Center)
├── func: parameterized_func (generic function with 1 method)
├── parameters: (μ = 42, λ = 0.1, ω = π)
└── field dependencies: ()

After passing through the constructor for NonhydrostaticModel, the v-forcing location information is available and set to Center, Face, Center.

# Field-dependent forcing
growth_in_sunlight(x, y, z, t, P) = exp(z) * P

plankton_forcing = Forcing(growth_in_sunlight, field_dependencies=:P)

# output
ContinuousForcing{Nothing}
├── func: growth_in_sunlight (generic function with 1 method)
├── parameters: nothing
└── field dependencies: (:P,)
# Parameterized, field-dependent forcing
tracer_relaxation(x, y, z, t, c, p) = p.μ * exp((z + p.H) / p.λ) * (p.dCdz * z - c)

c_forcing = Forcing(tracer_relaxation,
field_dependencies = :c,
parameters = (μ=1/60, λ=10, H=1000, dCdz=1))

# output
ContinuousForcing{NamedTuple{(:μ, :λ, :H, :dCdz), Tuple{Float64, Int64, Int64, Int64}}}
├── func: tracer_relaxation (generic function with 1 method)
├── parameters: (μ = 0.016666666666666666, λ = 10, H = 1000, dCdz = 1)
└── field dependencies: (:c,)
# Unparameterized discrete-form forcing function
filtered_relaxation(i, j, k, grid, clock, model_fields) =
@inbounds - (model_fields.c[i-1, j, k] + model_fields.c[i, j, k] + model_fields.c[i+1, j, k]) / 3

filtered_forcing = Forcing(filtered_relaxation, discrete_form=true)

# output
DiscreteForcing{Nothing}
├── func: filtered_relaxation (generic function with 1 method)
└── parameters: nothing
# Discrete-form forcing function with parameters
masked_damping(i, j, k, grid, clock, model_fields, parameters) =
@inbounds - parameters.μ * exp(grid.zᵃᵃᶜ[k] / parameters.λ) * model_fields.u[i, j, k]

# output
DiscreteForcing{NamedTuple{(:μ, :λ), Tuple{Int64, Irrational{:π}}}}
├── func: masked_damping (generic function with 1 method)
└── parameters: (μ = 42, λ = π)
source

## Grids

Oceananigans.Grids.LatitudeLongitudeGridType
LatitudeLongitudeGrid([architecture = CPU(), FT = Float64];
size,
longitude,
latitude,
z = nothing,
topology = nothing,
precompute_metrics = true,
halo = nothing)

Creates a LatitudeLongitudeGrid with coordinates (λ, φ, z) denoting longitude, latitude, and vertical coordinate respectively.

Positional arguments

• architecture: Specifies whether arrays of coordinates and spacings are stored on the CPU or GPU. Default: CPU().

• FT : Floating point data type. Default: Float64.

Keyword arguments

• size (required): A 3-tuple prescribing the number of grid points each direction.

• longitude (required), latitude (required), z (default: nothing): Each is either a:

1. 2-tuple that specify the end points of the domain,
2. one-dimensional array specifying the cell interface locations, or
3. a single-argument function that takes an index and returns cell interface location.

Note: the latitude and longitude coordinates extents are expected in degrees.

• radius: The radius of the sphere the grid lives on. By default is equal to the radius of Earth.

• topology: Tuple of topologies (Flat, Bounded, Periodic) for each direction. The vertical topology[3] must be Bounded, while the latitude-longitude topologies can be Bounded, Periodic, or Flat.

• precompute_metrics: Boolean specifying whether to precompute horizontal spacings and areas. Default: true. When false, horizontal spacings and areas are computed on-the-fly during a simulation.

• halo: A 3-tuple of integers specifying the size of the halo region of cells surrounding the physical interior. The default is 3 halo cells in every direction.

Examples

• A default grid with Float64 type:
julia> using Oceananigans

julia> grid = LatitudeLongitudeGrid(size=(36, 34, 25),
longitude = (-180, 180),
latitude = (-85, 85),
z = (-1000, 0))
36×34×25 LatitudeLongitudeGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo and with precomputed metrics
├── longitude: Periodic λ ∈ [-180.0, 180.0) regularly spaced with Δλ=10.0
├── latitude:  Bounded  φ ∈ [-85.0, 85.0]   regularly spaced with Δφ=5.0
└── z:         Bounded  z ∈ [-1000.0, 0.0]  regularly spaced with Δz=40.0
• A bounded spherical sector with cell interfaces stretched hyperbolically near the top:
julia> using Oceananigans

julia> σ = 1.1; # stretching factor

julia> Nz = 24; # vertical resolution

julia> Lz = 1000; # depth (m)

julia> hyperbolically_spaced_faces(k) = - Lz * (1 - tanh(σ * (k - 1) / Nz) / tanh(σ));

julia> grid = LatitudeLongitudeGrid(size=(36, 34, Nz),
longitude = (-180, 180),
latitude = (-20, 20),
z = hyperbolically_spaced_faces,
topology = (Bounded, Bounded, Bounded))
36×34×24 LatitudeLongitudeGrid{Float64, Bounded, Bounded, Bounded} on CPU with 3×3×3 halo and with precomputed metrics
├── longitude: Bounded  λ ∈ [-180.0, 180.0] regularly spaced with Δλ=10.0
├── latitude:  Bounded  φ ∈ [-20.0, 20.0]   regularly spaced with Δφ=1.17647
└── z:         Bounded  z ∈ [-1000.0, -0.0] variably spaced with min(Δz)=21.3342, max(Δz)=57.2159
source
Oceananigans.Grids.RectilinearGridType
RectilinearGrid([architecture = CPU(), FT = Float64];
size,
x = nothing,
y = nothing,
z = nothing,
halo = nothing,
extent = nothing,
topology = (Periodic, Periodic, Bounded))

Creates a RectilinearGrid with size = (Nx, Ny, Nz) grid points.

Positional arguments

• architecture: Specifies whether arrays of coordinates and spacings are stored on the CPU or GPU. Default: architecture = CPU().

• FT : Floating point data type. Default: FT = Float64.

Keyword arguments

• size (required): A tuple prescribing the number of grid points in non-Flat directions. size is a 3-tuple for 3D models, a 2-tuple for 2D models, and either a scalar or 1-tuple for 1D models.

• topology: A 3-tuple (TX, TY, TZ) specifying the topology of the domain. TX, TY, and TZ specify whether the x-, y-, and z directions are Periodic, Bounded, or Flat. The topology Flat indicates that a model does not vary in those directions so that derivatives and interpolation are zero. The default is topology = (Periodic, Periodic, Bounded).

• extent: A tuple prescribing the physical extent of the grid in non-Flat directions. All directions are contructed with regular grid spacing and the domain (in the case that no direction is Flat) is x ∈ (0, Lx), y ∈ (0, Ly), and z ∈ (-Lz, 0), which is most appropriate for oceanic applications with z = 0 usually being the ocean's surface.

• x, y, and z: Each of x, y, z are either (i) 2-tuples that specify the end points of the domain in their respect directions (in which case scalar values may be used in Flat directions), or (ii) arrays or functions of the corresponding indices i, j, or k that specify the locations of cell faces in the x-, y-, or z-direction, respectively. For example, to prescribe the cell faces in z we need to provide a function that takes k as argument and retuns the location of the faces for indices k = 1 through k = Nz + 1, where Nz is the size of the stretched z dimension.

Note: Either extent, or all of x, y, and z must be specified.

• halo: A tuple of integers that specifies the size of the halo region of cells surrounding the physical interior for each non-Flat direction. The default is 3 halo cells in every direction.

The physical extent of the domain can be specified via x, y, and z keyword arguments indicating the left and right endpoints of each dimensions, e.g. x = (-π, π) or via the extent argument, e.g. extent = (Lx, Ly, Lz), which specifies the extent of each dimension in which case 0 ≤ x ≤ Lx, 0 ≤ y ≤ Ly, and -Lz ≤ z ≤ 0.

A grid topology may be specified via a tuple assigning one of Periodic, Bounded, and Flat to each dimension. By default, a horizontally periodic grid topology (Periodic, Periodic, Bounded) is assumed.

Constants are stored using floating point values of type FT. By default this is Float64. Make sure to specify the desired FT if not using Float64.

Grid properties

• (Nx, Ny, Nz) :: Int: Number of physical points in the $(x, y, z)$-direction.

• (Hx, Hy, Hz) :: Int: Number of halo points in the $(x, y, z)$-direction.

• (Lx, Ly, Lz) :: FT: Physical extent of the grid in the $(x, y, z)$-direction.

• (Δxᶜᵃᵃ, Δyᵃᶜᵃ, Δzᵃᵃᶜ): Grid spacing in the $(x, y, z)$-direction between cell centers. Defined at cell centers in $x$, $y$, and $z$.

• (Δxᶠᵃᵃ, Δyᵃᶠᵃ, Δzᵃᵃᶠ): Grid spacing in the $(x, y, z)$-direction between cell faces. Defined at cell faces in $x$, $y$, and $z$.

• (xᶜᵃᵃ, yᵃᶜᵃ, zᵃᵃᶜ): $(x, y, z)$ coordinates of cell centers.

• (xᶠᵃᵃ, yᵃᶠᵃ, zᵃᵃᶠ): $(x, y, z)$ coordinates of cell faces.

Examples

• A default grid with Float64 type:
julia> using Oceananigans

julia> grid = RectilinearGrid(size=(32, 32, 32), extent=(1, 2, 3))
32×32×32 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 1.0)  regularly spaced with Δx=0.03125
├── Periodic y ∈ [0.0, 2.0)  regularly spaced with Δy=0.0625
└── Bounded  z ∈ [-3.0, 0.0] regularly spaced with Δz=0.09375
• A default grid with Float32 type:
julia> using Oceananigans

julia> grid = RectilinearGrid(Float32; size=(32, 32, 16), x=(0, 8), y=(-10, 10), z=(-π, π))
32×32×16 RectilinearGrid{Float32, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 8.0)          regularly spaced with Δx=0.25
├── Periodic y ∈ [-10.0, 10.0)       regularly spaced with Δy=0.625
└── Bounded  z ∈ [-3.14159, 3.14159] regularly spaced with Δz=0.392699
• A two-dimenisional, horizontally-periodic grid:
julia> using Oceananigans

julia> grid = RectilinearGrid(size=(32, 32), extent=(2π, 4π), topology=(Periodic, Periodic, Flat))
32×32×1 RectilinearGrid{Float64, Periodic, Periodic, Flat} on CPU with 3×3×0 halo
├── Periodic x ∈ [3.60072e-17, 6.28319) regularly spaced with Δx=0.19635
├── Periodic y ∈ [7.20145e-17, 12.5664) regularly spaced with Δy=0.392699
└── Flat z
• A one-dimensional "column" grid:
julia> using Oceananigans

julia> grid = RectilinearGrid(size=256, z=(-128, 0), topology=(Flat, Flat, Bounded))
1×1×256 RectilinearGrid{Float64, Flat, Flat, Bounded} on CPU with 0×0×3 halo
├── Flat x
├── Flat y
└── Bounded  z ∈ [-128.0, 0.0]    regularly spaced with Δz=0.5
• A horizontally-periodic regular grid with cell interfaces stretched hyperbolically near the top:
julia> using Oceananigans

julia> σ = 1.1; # stretching factor

julia> Nz = 24; # vertical resolution

julia> Lz = 32; # depth (m)

julia> hyperbolically_spaced_faces(k) = - Lz * (1 - tanh(σ * (k - 1) / Nz) / tanh(σ));

julia> grid = RectilinearGrid(size = (32, 32, Nz), x = (0, 64),
y = (0, 64), z = hyperbolically_spaced_faces)
32×32×24 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 64.0)   regularly spaced with Δx=2.0
├── Periodic y ∈ [0.0, 64.0)   regularly spaced with Δy=2.0
└── Bounded  z ∈ [-32.0, -0.0] variably spaced with min(Δz)=0.682695, max(Δz)=1.83091
• A three-dimensional grid with regular spacing in x, cell interfaces at Chebyshev nodes in y, and cell interfaces stretched in z hyperbolically near the top:
julia> using Oceananigans

julia> Nx, Ny, Nz = 32, 30, 24;

julia> Lx, Ly, Lz = 200, 100, 32; # (m)

julia> chebychev_nodes(j) = - Ly/2 * cos(π * (j - 1) / Ny);

julia> σ = 1.1; # stretching factor

julia> hyperbolically_spaced_faces(k) = - Lz * (1 - tanh(σ * (k - 1) / Nz) / tanh(σ));

julia> grid = RectilinearGrid(size = (Nx, Ny, Nz),
topology=(Periodic, Bounded, Bounded),
x = (0, Lx),
y = chebychev_nodes,
z = hyperbolically_spaced_faces)
32×30×24 RectilinearGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 200.0)  regularly spaced with Δx=6.25
├── Bounded  y ∈ [-50.0, 50.0] variably spaced with min(Δy)=0.273905, max(Δy)=5.22642
└── Bounded  z ∈ [-32.0, -0.0] variably spaced with min(Δz)=0.682695, max(Δz)=1.83091
source
Oceananigans.Grids.new_dataFunction
new_data(FT, grid, loc, indices)

Returns an OffsetArray of zeros of float type FT on architecture, with indices corresponding to a field on a grid of size(grid) and located at loc.

source
Oceananigans.Grids.nodesMethod
nodes(loc, grid; reshape=false)

Return a 3-tuple of views over the interior nodes at the locations in loc in x, y, z.

If reshape=true, the views are reshaped to 3D arrays with non-singleton dimensions 1, 2, 3 for x, y, z, respectively. These reshaped arrays can then be used in broadcast operations with 3D fields or arrays.

source
Oceananigans.Grids.offset_dataFunction
offset_data(underlying_data, grid::AbstractGrid, loc)

Returns an OffsetArray that maps to underlying_data in memory, with offset indices appropriate for the data of a field on a grid of size(grid) and located at loc.

source
Oceananigans.Grids.total_sizeMethod
total_size(loc, grid)

Return the "total" size of a grid at loc. This is a 3-tuple of integers corresponding to the number of grid points along x, y, z.

source
Oceananigans.Grids.xnodesMethod
xnodes(loc, grid, reshape=false)

Return a view over the interior loc=Center or loc=Face nodes on grid in the $x$-direction. For Bounded directions, Face nodes include the boundary points.

Keyword argument

• reshape: With reshape=false (default) the output is a 1D array while with reshape=true the output is a 3D array with size Nx×1×1.

See znodes for examples.

source
Oceananigans.Grids.ynodesMethod
ynodes(loc, grid, reshape=false)

Return a view over the interior loc=Center or loc=Face nodes on grid in the $y$-direction. For Bounded directions, Face nodes include the boundary points.

Keyword argument

• reshape: With reshape=false (default) the output is a 1D array while with reshape=true the output is a 3D array with size 1×Ny×1.

See znodes for examples.

source
Oceananigans.Grids.znodesMethod
znodes(loc, grid, reshape=false)

Return a view over the interior loc=Center or loc=Face nodes on grid in the $z$-direction. For Bounded directions, Face nodes include the boundary points.

Keyword argument

• reshape: With reshape=false (default) the output is a 1D array while with reshape=true the output is a 3D array with size 1×1×Nz.

Examples

julia> using Oceananigans

julia> horz_periodic_grid = RectilinearGrid(size=(3, 3, 3), extent=(2π, 2π, 1), halo=(1, 1, 1),
topology=(Periodic, Periodic, Bounded));

julia> zC = znodes(Center, horz_periodic_grid)
3-element view(OffsetArray(::StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, 0:4), 1:3) with eltype Float64:
-0.8333333333333334
-0.5
-0.16666666666666666
julia> zF = znodes(Face, horz_periodic_grid)
4-element view(OffsetArray(::StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, 0:5), 1:4) with eltype Float64:
-1.0
-0.6666666666666666
-0.3333333333333333
0.0
source

## Lagrangian particle tracking

Oceananigans.LagrangianParticleTracking.LagrangianParticlesMethod
LagrangianParticles(particles::StructArray; restitution=1.0, tracked_fields::NamedTuple=NamedTuple(), dynamics=no_dynamics)

Construct some LagrangianParticles that can be passed to a model. The particles should be a StructArray and can contain custom fields. The coefficient of restitution for particle-wall collisions is specified by restitution.

A number of tracked_fields may be passed in as a NamedTuple of fields. Each particle will track the value of each field. Each tracked field must have a corresponding particle property. So if T is a tracked field, then T must also be a custom particle property.

dynamics is a function of (lagrangian_particles, model, Δt) that is called prior to advecting particles. parameters can be accessed inside the dynamics function.

source
Oceananigans.LagrangianParticleTracking.LagrangianParticlesMethod
LagrangianParticles(; x, y, z, restitution=1.0, dynamics=no_dynamics, parameters=nothing)

Construct some LagrangianParticles that can be passed to a model. The particles will have initial locations x, y, and z. The coefficient of restitution for particle-wall collisions is specified by restitution.

dynamics is a function of (lagrangian_particles, model, Δt) that is called prior to advecting particles. parameters can be accessed inside the dynamics function.

source

## Logger

Oceananigans.Logger.OceananigansLoggerType
OceananigansLogger(stream::IO=stdout, level=Logging.Info; show_info_source=false)

Based on Logging.SimpleLogger, it tries to log all messages in the following format:

[yyyy/mm/dd HH:MM:SS.sss] log_level message [-@-> source_file:line_number]

where the source of the message between the square brackets is included only if show_info_source=true or if the message is not an info level message.

source

## Models

### Non-hydrostatic models

Oceananigans.Models.NonhydrostaticModels.NonhydrostaticModelMethod
NonhydrostaticModel(;     grid,
clock = Clock{eltype(grid)}(0, 0, 1),
buoyancy = nothing,
coriolis = nothing,
stokes_drift = nothing,
forcing::NamedTuple = NamedTuple(),
closure = nothing,
boundary_conditions::NamedTuple = NamedTuple(),
tracers = (),
background_fields::NamedTuple = NamedTuple(),
particles::ParticlesOrNothing = nothing,
velocities = nothing,
pressures = nothing,
diffusivity_fields = nothing,
pressure_solver = nothing,
immersed_boundary = nothing,
auxiliary_fields = NamedTuple(),
)

Construct a model for a non-hydrostatic, incompressible fluid on grid, using the Boussinesq approximation when buoyancy != nothing. By default, all Bounded directions are rigid and impenetrable.

Keyword arguments

• grid: (required) The resolution and discrete geometry on which the model is solved. The architecture (CPU/GPU) that the model is solve is inferred from the architecture of the grid.
• advection: The scheme that advects velocities and tracers. See Oceananigans.Advection.
• buoyancy: The buoyancy model. See Oceananigans.BuoyancyModels.
• coriolis: Parameters for the background rotation rate of the model.
• stokes_drift: Parameters for Stokes drift fields associated with surface waves. Default: nothing.
• forcing: NamedTuple of user-defined forcing functions that contribute to solution tendencies.
• closure: The turbulence closure for model. See Oceananigans.TurbulenceClosures.
• boundary_conditions: NamedTuple containing field boundary conditions.
• tracers: A tuple of symbols defining the names of the modeled tracers, or a NamedTuple of preallocated CenterFields.
• timestepper: A symbol that specifies the time-stepping method. Either :QuasiAdamsBashforth2 or :RungeKutta3.
• background_fields: NamedTuple with background fields (e.g., background flow). Default: nothing.
• particles: Lagrangian particles to be advected with the flow. Default: nothing.
• velocities: The model velocities. Default: nothing.
• pressures: Hydrostatic and non-hydrostatic pressure fields. Default: nothing.
• diffusivity_fields: Diffusivity fields. Default: nothing.
• pressure_solver: Pressure solver to be used in the model. If nothing (default), the model constructor chooses the default based on the grid provide.
• immersed_boundary: The immersed boundary. Default: nothing.
• auxiliary_fields: NamedTuple of auxiliary fields. Default: nothing.
source

### Hydrostatic free-surface models

Oceananigans.Models.HydrostaticFreeSurfaceModels.HydrostaticFreeSurfaceModelMethod
HydrostaticFreeSurfaceModel(; grid,
clock = Clock{eltype(grid)}(0, 0, 1),
buoyancy = SeawaterBuoyancy(eltype(grid)),
coriolis = nothing,
free_surface = ImplicitFreeSurface(gravitational_acceleration=g_Earth),
forcing::NamedTuple = NamedTuple(),
closure = nothing,
boundary_conditions::NamedTuple = NamedTuple(),
tracers = (:T, :S),
particles::Union{Nothing, LagrangianParticles} = nothing,
velocities = nothing,
pressure = nothing,
diffusivity_fields = nothing,
auxiliary_fields = NamedTuple(),
)

Construct a hydrostatic model with a free surface on grid.

Keyword arguments

• grid: (required) The resolution and discrete geometry on which model is solved. The architecture (CPU/GPU) that the model is solve is inferred from the architecture of the grid.
• momentum_advection: The scheme that advects velocities. See Oceananigans.Advection.
• tracer_advection: The scheme that advects tracers. See Oceananigans.Advection.
• buoyancy: The buoyancy model. See Oceananigans.BuoyancyModels.
• coriolis: Parameters for the background rotation rate of the model.
• forcing: NamedTuple of user-defined forcing functions that contribute to solution tendencies.
• free_surface: The free surface model.
• closure: The turbulence closure for model. See Oceananigans.TurbulenceClosures.
• boundary_conditions: NamedTuple containing field boundary conditions.
• tracers: A tuple of symbols defining the names of the modeled tracers, or a NamedTuple of preallocated CenterFields.
• particles: Lagrangian particles to be advected with the flow. Default: nothing.
• velocities: The model velocities. Default: nothing.
• pressure: Hydrostatic pressure field. Default: nothing.
• diffusivity_fields: Diffusivity fields. Default: nothing.
• auxiliary_fields: NamedTuple of auxiliary fields. Default: nothing.
source
Oceananigans.Models.HydrostaticFreeSurfaceModels.ImplicitFreeSurfaceMethod
ImplicitFreeSurface(; solver_method=:Default, gravitational_acceleration=g_Earth, solver_settings...)

Return an implicit free-surface solver. The implicit free-surface equation is

$$$\left [ 𝛁_h ⋅ (H 𝛁_h) - \frac{1}{g Δt^2} \right ] η^{n+1} = \frac{𝛁_h ⋅ 𝐐_⋆}{g Δt} - \frac{η^{n}}{g Δt^2} ,$$$

where $η^n$ is the free-surface elevation at the $n$-th time step, $H$ is depth, $g$ is the gravitational acceleration, $Δt$ is the time step, $𝛁_h$ is the horizontal gradient operator, and $𝐐_⋆$ is the barotropic volume flux associated with the predictor velocity field $𝐮_⋆$, i.e.,

$$$𝐐_⋆ = \int_{-H}^0 𝐮_⋆ \, 𝖽 z ,$$$

where

$$$𝐮_⋆ = 𝐮^n + \int_{t_n}^{t_{n+1}} 𝐆ᵤ \, 𝖽t .$$$

This equation can be solved, in general, using the PreconditionedConjugateGradientSolver but other solvers can be invoked in special cases.

If $H$ is constant, we divide through out to obtain

$$$\left ( ∇^2_h - \frac{1}{g H Δt^2} \right ) η^{n+1} = \frac{1}{g H Δt} \left ( 𝛁_h ⋅ 𝐐_⋆ - \frac{η^{n}}{Δt} \right ) .$$$

Thus, for constant $H$ and on grids with regular spacing in $x$ and $y$ directions, the free surface can be obtained using the FFTBasedPoissonSolver.

solver_method can be either of:

By default, if the grid has regular spacing in the horizontal directions then the :FastFourierTransform is chosen, otherwise the :HeptadiagonalIterativeSolver.

source
Oceananigans.Models.HydrostaticFreeSurfaceModels.PrescribedVelocityFieldsMethod
PrescribedVelocityFields(; u=zerofunc, v=zerofunc, w=zerofunc, parameters=nothing)

Builds PrescribedVelocityFields with prescribed functions u, v, and w.

If isnothing(parameters), then u, v, w are called with the signature

u(x, y, z, t) = # something interesting

If !isnothing(parameters), then u, v, w are called with the signature

u(x, y, z, t, parameters) = # something parameterized and interesting

In the constructor for HydrostaticFreeSurfaceModel, the functions u, v, w are wrapped in FunctionField and associated with the model's grid and clock.

source
Oceananigans.Models.HydrostaticFreeSurfaceModels.SplitExplicitFreeSurfaceType
struct SplitExplicitFreeSurface{𝒩, 𝒮, ℱ, 𝒫 ,ℰ}

The split-explicit free surface solver.

• η::Any

The instantaneous free surface (ReducedField)

• state::Any

The entire state for the split-explicit (SplitExplicitState)

• auxiliary::Any

Parameters for timestepping split-explicit (NamedTuple)

• gravitational_acceleration::Any

Gravitational acceleration

• settings::Any

Settings for the split-explicit scheme (NamedTuple)

source

### Shallow-water models

Oceananigans.Models.ShallowWaterModels.ShallowWaterModelMethod
ShallowWaterModel(; grid,
gravitational_acceleration,
clock = Clock{eltype(grid)}(0, 0, 1),
coriolis = nothing,
forcing::NamedTuple = NamedTuple(),
closure = nothing,
bathymetry = nothing,
tracers = (),
diffusivity_fields = nothing,
boundary_conditions::NamedTuple = NamedTuple(),
timestepper::Symbol = :RungeKutta3,
formulation = ConservativeFormulation())

Construct a shallow water model on grid with gravitational_acceleration constant.

Keyword arguments

• grid: (required) The resolution and discrete geometry on which model is solved. The architecture (CPU/GPU) that the model is solve is inferred from the architecture of the grid.
• gravitational_acceleration: (required) The gravitational acceleration constant.
• clock: The clock for the model.
• momentum_advection: The scheme that advects velocities. See Oceananigans.Advection. Default: UpwindBiasedFifthOrder().
• tracer_advection: The scheme that advects tracers. See Oceananigans.Advection. Default: WENO().
• mass_advection: The scheme that advects the mass equation. See Oceananigans.Advection. Default: WENO().
• coriolis: Parameters for the background rotation rate of the model.
• forcing: NamedTuple of user-defined forcing functions that contribute to solution tendencies.
• closure: The turbulence closure for model. See Oceananigans.TurbulenceClosures.
• bathymetry: The bottom bathymetry.
• tracers: A tuple of symbols defining the names of the modeled tracers, or a NamedTuple of preallocated CenterFields.
• diffusivity_fields: Stores diffusivity fields when the closures require a diffusivity to be calculated at each timestep.
• boundary_conditions: NamedTuple containing field boundary conditions.
• timestepper: A symbol that specifies the time-stepping method. Either :QuasiAdamsBashforth2 or :RungeKutta3 (default).
• formulation: Whether the dynamics are expressed in conservative form (ConservativeFormulation(); default) or in non-conservative form with a vector-invariant formulation for the non-linear terms (VectorInvariantFormulation()).
Formulation-grid compatibility requirements

The ConservativeFormulation() requires RectilinearGrid. Use VectorInvariantFormulation() with LatitudeLongitudeGrid.

source

## Operators

Oceananigans.Operators.div_xyᶜᶜᶜMethod
div_xyᶜᶜᵃ(i, j, k, grid, u, v)

Return the discrete div_xy = ∂x u + ∂y v of velocity field u, v defined as

1 / Azᶜᶜᵃ * [δxᶜᵃᵃ(Δyᵃᶜᵃ * u) + δyᵃᶜᵃ(Δxᶜᵃᵃ * v)]

at i, j, k, where Azᶜᶜᵃ is the area of the cell centered on (Center, Center, Any) –- a tracer cell, Δy is the length of the cell centered on (Face, Center, Any) in y (a u cell), and Δx is the length of the cell centered on (Center, Face, Any) in x (a v cell). div_xyᶜᶜᵃ ends up at the location cca.

source
Oceananigans.Operators.divᶜᶜᶜMethod
divᶜᶜᶜ(i, j, k, grid, u, v, w)

Calculates the divergence $∇·𝐕$ of a vector field $𝐕 = (u, v, w)$,

1/V * [δxᶜᵃᵃ(Ax * u) + δxᵃᶜᵃ(Ay * v) + δzᵃᵃᶜ(Az * w)]

which ends up at the cell centers ccc.

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Oceananigans.Operators.∇²ᶜᶜᶜMethod
∇²ᶜᶜᶜ(i, j, k, grid, c)

Calculate the Laplacian of $c$ via

1/V * [δxᶜᵃᵃ(Ax * ∂xᶠᵃᵃ(c)) + δyᵃᶜᵃ(Ay * ∂yᵃᶠᵃ(c)) + δzᵃᵃᶜ(Az * ∂zᵃᵃᶠ(c))]

which ends up at the location ccc.

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Oceananigans.OutputReaders.FieldDatasetMethod
FieldDataset(filepath;
architecture=CPU(), grid=nothing, backend=InMemory(), metadata_paths=["metadata"])

Returns a Dict containing a FieldTimeSeries for each field in the JLD2 file located at filepath. Note that model output must have been saved with halos.

Keyword arguments

• backend: Either InMemory() (default) or OnDisk(). The InMemory backend will

load the data fully in memory as a 4D multi-dimensional array while the OnDisk() backend will lazily load field time snapshots when the FieldTimeSeries is indexed linearly.

• metadata_paths: A list of JLD2 paths to look for metadata. By default it looks in file["metadata"].

• grid: May be specified to override the grid used in the JLD2 file.

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Oceananigans.OutputReaders.FieldTimeSeriesMethod
FieldTimeSeries(path, name;
backend = InMemory(),
grid = nothing,
iterations = nothing,
times = nothing)

Returns a FieldTimeSeries for the field name describing a field's time history from a JLD2 file located at path.

Keyword arguments

• backend: InMemory() to load data into a 4D array or OnDisk() to lazily load data from disk when indexing into FieldTimeSeries.

• grid: A grid to associated with data, in the case that the native grid was not serialized properly.

• iterations: Iterations to load. Defaults to all iterations found in the file.

• times: Save times to load, as determined through an approximate floating point comparison to recorded save times. Defaults to times associated with iterations. Takes precedence over iterations if times is specified.

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Oceananigans.OutputReaders.FieldTimeSeriesMethod
FieldTimeSeries{LX, LY, LZ}(grid, times, [FT=eltype(grid);]
indices = (:, :, :),
boundary_conditions = nothing)

Return a FieldTimeSeries at location (LX, LY, LZ), on grid, at times.

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## Output writers

Oceananigans.OutputWriters.AveragedTimeIntervalMethod
AveragedTimeInterval(interval; window=interval, stride=1)

Returns a schedule that specifies periodic time-averaging of output. The time window specifies the extent of the time-average, which reoccurs every interval.

output is computed and accumulated into the average every stride iterations during the averaging window. For example, stride=1 computs output every iteration, whereas stride=2 computes output every other iteration. Time-averages with longer strides are faster to compute, but less accurate.

The time-average of $a$ is a left Riemann sum corresponding to

$$$⟨a⟩ = T⁻¹ \int_{tᵢ-T}^{tᵢ} a \mathrm{d} t \, ,$$$

where $⟨a⟩$ is the time-average of $a$, $T$ is the time-window for averaging, and the $tᵢ$ are discrete times separated by the time interval. The $tᵢ$ specify both the end of the averaging window and the time at which output is written.

Example

using Oceananigans.OutputWriters: AveragedTimeInterval
using Oceananigans.Utils: year, years

schedule = AveragedTimeInterval(4years, window=1year)

# output
AveragedTimeInterval(window=1 year, stride=1, interval=4 years)

An AveragedTimeInterval schedule directs an output writer to time-average its outputs before writing them to disk:

using Oceananigans
using Oceananigans.OutputWriters: JLD2OutputWriter
using Oceananigans.Utils: minutes

model = NonhydrostaticModel(grid=RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1)))

simulation = Simulation(model, Δt=10minutes, stop_time=30years)

simulation.output_writers[:velocities] = JLD2OutputWriter(model, model.velocities,
filename= "averaged_velocity_data.jld2",
schedule = AveragedTimeInterval(4years, window=1year, stride=2))

# output
JLD2OutputWriter scheduled on TimeInterval(4 years):
├── filepath: ./averaged_velocity_data.jld2
├── 3 outputs: (u, v, w) averaged on AveragedTimeInterval(window=1 year, stride=2, interval=4 years)
├── array type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
└── max filesize: Inf YiB
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Oceananigans.OutputWriters.CheckpointerMethod
Checkpointer(model; schedule,
dir = ".",
prefix = "checkpoint",
overwrite_existing = false,
cleanup = false,
additional_kwargs...)

Construct a Checkpointer that checkpoints the model to a JLD2 file on schedule. The model.clock.iteration is included in the filename to distinguish between multiple checkpoint files.

To restart or "pickup" a model from a checkpoint, specify pickup=true when calling run!, ensuring that the checkpoint file is the current working directory. See

help> run!

for more details.

Note that extra model properties can be safely specified, but removing crucial properties such as :velocities will make restoring from the checkpoint impossible.

The checkpointer attempts to serialize as much of the model to disk as possible, but functions or objects containing functions cannot be serialized at this time.

Keyword arguments

• schedule (required): Schedule that determines when to checkpoint.

• dir: Directory to save output to. Default: "." (current working directory).

• prefix: Descriptive filename prefixed to all output files. Default: "checkpoint".

• overwrite_existing: Remove existing files if their filenames conflict. Default: false.

• verbose: Log what the output writer is doing with statistics on compute/write times and file sizes. Default: false.

• cleanup: Previous checkpoint files will be deleted once a new checkpoint file is written. Default: false.

• properties: List of model properties to checkpoint. This list must contain [:grid, :architecture, :timestepper, :particles]. Default: [:architecture, :grid, :clock, :coriolis, :buoyancy, :closure, :velocities, :tracers, :timestepper, :particles]

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Oceananigans.OutputWriters.JLD2OutputWriterMethod
JLD2OutputWriter(model, outputs; filename, schedule,
dir = ".",
indices = (:, :, :),
with_halos = false,
array_type = Array{Float32},
max_filesize = Inf,
overwrite_existing = false,
init = noinit,
including = [:grid, :coriolis, :buoyancy, :closure],
verbose = false,
part = 1,
jld2_kw = Dict{Symbol, Any}())

Construct a JLD2OutputWriter for an Oceananigans model that writes label, output pairs in outputs to a JLD2 file.

The argument outputs may be a Dict or NamedTuple. The keys of outputs are symbols or strings that "name" output data. The values of outputs are either AbstractFields, objects that are called with the signature output(model), or WindowedTimeAverages of AbstractFieldss, functions, or callable objects.

Keyword arguments

Filenaming

• filename (required): Descriptive filename. ".jld2" is appended to filename in the file path if filename does not end in ".jld2".

• dir: Directory to save output to. Default: "." (current working directory).

Output frequency and time-averaging

• schedule (required): AbstractSchedule that determines when output is saved.

Slicing and type conversion prior to output

• indices: Specifies the indices to write to disk with a Tuple of Colon, UnitRange, or Int elements. Indices must be Colon, Int, or contiguous UnitRange. Defaults to (:, :, :) or "all indices". If !with_halos, halo regions are removed from indices. For example, indices = (:, :, 1) will save xy-slices of the bottom-most index.

• with_halos (Bool): Whether or not to slice halo regions from fields before writing output. Note, that to postprocess saved output (e.g., compute derivatives, etc) information about the boundary conditions is often crucial. In that case you might need to set with_halos = true.

• array_type: The array type to which output arrays are converted to prior to saving. Default: Array{Float32}.

File management

• max_filesize: The writer will stop writing to the output file once the file size exceeds max_filesize, and write to a new one with a consistent naming scheme ending in part1, part2, etc. Defaults to Inf.

• overwrite_existing: Remove existing files if their filenames conflict. Default: false.

• init: A function of the form init(file, model) that runs when a JLD2 output file is initialized. Default: noinit(args...) = nothing.

• including: List of model properties to save with every file. Default: [:grid, :coriolis, :buoyancy, :closure]

Miscellaneous keywords

• verbose: Log what the output writer is doing with statistics on compute/write times and file sizes. Default: false.

• part: The starting part number used if max_filesize is finite. Default: 1.

• jld2_kw: Dict of kwargs to be passed to jldopen when data is written.

Example

Write out 3D fields for $u$, $v$, $w$, and a tracer $c$, along with a horizontal average:

using Oceananigans
using Oceananigans.Utils: hour, minute

model = NonhydrostaticModel(grid=RectilinearGrid(size=(1, 1, 1), extent=(1, 1, 1)), tracers=(:c,))
simulation = Simulation(model, Δt=12, stop_time=1hour)

file["author"] = "Chim Riggles"
file["parameters/coriolis_parameter"] = 1e-4
file["parameters/density"] = 1027
return nothing
end

c_avg =  Field(Average(model.tracers.c, dims=(1, 2)))

# Note that model.velocities is NamedTuple
simulation.output_writers[:velocities] = JLD2OutputWriter(model, model.velocities,
filename = "some_data.jld2",
schedule = TimeInterval(20minute),

# output
JLD2OutputWriter scheduled on TimeInterval(20 minutes):
├── filepath: ./some_data.jld2
├── 3 outputs: (u, v, w)
├── array type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
└── max filesize: Inf YiB

and a time- and horizontal-average of tracer $c$ every 20 minutes of simulation time to a file called some_averaged_data.jld2

simulation.output_writers[:avg_c] = JLD2OutputWriter(model, (; c=c_avg),
filename = "some_averaged_data.jld2",
schedule = AveragedTimeInterval(20minute, window=5minute))

# output
JLD2OutputWriter scheduled on TimeInterval(20 minutes):
├── filepath: ./some_averaged_data.jld2
├── 1 outputs: c averaged on AveragedTimeInterval(window=5 minutes, stride=1, interval=20 minutes)
├── array type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
└── max filesize: Inf YiB
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Oceananigans.OutputWriters.NetCDFOutputWriterMethod
NetCDFOutputWriter(model, outputs; filename, schedule
dir = ".",
array_type = Array{Float32},
indices = nothing,
with_halos = false,
global_attributes = Dict(),
output_attributes = Dict(),
dimensions = Dict(),
overwrite_existing = false,
compression = 0,
verbose = false)

Construct a NetCDFOutputWriter that writes (label, output) pairs in outputs (which should be a Dict) to a NetCDF file, where label is a string that labels the output and output is either a Field (e.g. model.velocities.u) or a function f(model) that returns something to be written to disk. Custom output requires the spatial dimensions (a Dict) to be manually specified (see examples).

Keyword arguments

• filename (required): Descriptive filename. ".nc" is appended to filename if ".nc" is not detected.

• schedule (required): AbstractSchedule that determines when output is saved.

• dir: Directory to save output to.

• array_type: The array type to which output arrays are converted to prior to saving. Default: Array{Float32}.

• indices: Tuple of indices of the output variables to include. Default is (:, :, :), which includes the full fields.

• with_halos: Boolean defining whether or not to include halos in the outputs. Default: false. Note, that to postprocess saved output (e.g., compute derivatives, etc) information about the boundary conditions is often crucial. In that case you might need to set with_halos = true.

• global_attributes: Dict of model properties to save with every file. Default: Dict().

• output_attributes: Dict of attributes to be saved with each field variable (reasonable defaults are provided for velocities, buoyancy, temperature, and salinity; otherwise output_attributes must be user-provided).

• dimensions: A Dict of dimension tuples to apply to outputs (required for function outputs)

• overwrite_existing: If false, NetCDFOutputWriter will be set to append to filepath. If true, NetCDFOutputWriter will overwrite filepath if it exists or create it if it does not. Default: false. See NCDatasets.jl documentation for more information about its mode option.

• compression: Determines the compression level of data (0-9; default: 0)

Examples

Saving the $u$ velocity field and temperature fields, the full 3D fields and surface 2D slices to separate NetCDF files:

using Oceananigans

grid = RectilinearGrid(size=(16, 16, 16), extent=(1, 1, 1))

model = NonhydrostaticModel(grid=grid, tracers=:c)

simulation = Simulation(model, Δt=12, stop_time=3600)

fields = Dict("u" => model.velocities.u, "c" => model.tracers.c)

simulation.output_writers[:field_writer] =
NetCDFOutputWriter(model, fields, filename="fields.nc", schedule=TimeInterval(60))

# output
NetCDFOutputWriter scheduled on TimeInterval(1 minute):
├── filepath: ./fields.nc
├── dimensions: zC(16), zF(17), xC(16), yF(16), xF(16), yC(16), time(0)
├── 2 outputs: (c, u)
└── array type: Array{Float32}
simulation.output_writers[:surface_slice_writer] =
NetCDFOutputWriter(model, fields, filename="surface_xy_slice.nc",
schedule=TimeInterval(60), indices=(:, :, grid.Nz))

# output
NetCDFOutputWriter scheduled on TimeInterval(1 minute):
├── filepath: ./surface_xy_slice.nc
├── dimensions: zC(1), zF(1), xC(16), yF(16), xF(16), yC(16), time(0)
├── 2 outputs: (c, u)
└── array type: Array{Float32}
simulation.output_writers[:averaged_profile_writer] =
NetCDFOutputWriter(model, fields,
filename = "averaged_z_profile.nc",
schedule = AveragedTimeInterval(60, window=20),
indices = (1, 1, :))

# output
NetCDFOutputWriter scheduled on TimeInterval(1 minute):
├── filepath: ./averaged_z_profile.nc
├── dimensions: zC(16), zF(17), xC(1), yF(1), xF(1), yC(1), time(0)
├── 2 outputs: (c, u) averaged on AveragedTimeInterval(window=20 seconds, stride=1, interval=1 minute)
└── array type: Array{Float32}

NetCDFOutputWriter also accepts output functions that write scalars and arrays to disk, provided that their dimensions are provided:

using Oceananigans

grid = RectilinearGrid(size=(16, 16, 16), extent=(1, 2, 3))

model = NonhydrostaticModel(grid=grid)

simulation = Simulation(model, Δt=1.25, stop_iteration=3)

f(model) = model.clock.time^2; # scalar output

g(model) = model.clock.time .* exp.(znodes(Center, grid)) # vector/profile output

h(model) = model.clock.time .* (   sin.(xnodes(Center, grid, reshape=true)[:, :, 1])
.*     cos.(ynodes(Face, grid, reshape=true)[:, :, 1])) # xy slice output

outputs = Dict("scalar" => f, "profile" => g, "slice" => h)

dims = Dict("scalar" => (), "profile" => ("zC",), "slice" => ("xC", "yC"))

output_attributes = Dict(
"scalar"  => Dict("longname" => "Some scalar", "units" => "bananas"),
"profile" => Dict("longname" => "Some vertical profile", "units" => "watermelons"),
"slice"   => Dict("longname" => "Some slice", "units" => "mushrooms")
);

global_attributes = Dict("location" => "Bay of Fundy", "onions" => 7)

simulation.output_writers[:things] =
NetCDFOutputWriter(model, outputs,
schedule=IterationInterval(1), filename="things.nc", dimensions=dims, verbose=true,
global_attributes=global_attributes, output_attributes=output_attributes)

# output
NetCDFOutputWriter scheduled on IterationInterval(1):
├── filepath: ./things.nc
├── dimensions: zC(16), zF(17), xC(16), yF(16), xF(16), yC(16), time(0)
├── 3 outputs: (profile, slice, scalar)
└── array type: Array{Float32}
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Oceananigans.OutputWriters.WindowedTimeAverageType
WindowedTimeAverage(operand, model=nothing; schedule)

Returns an object for computing running averages of operand over schedule.window and recurring on schedule.interval, where schedule is an AveragedTimeInterval. During the collection period, averages are computed every schedule.stride iteration.

operand may be a Oceananigans.Field or a function that returns an array or scalar.

Calling wta(model) for wta::WindowedTimeAverage object returns wta.result.

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## Simulations

Oceananigans.Simulations.CallbackType
Callback(func, schedule=IterationInterval(1); parameters=nothing)

Return Callback that executes func on schedule with optional parameters. schedule = IterationInterval(1) by default.

If isnothing(parameters), func(sim::Simulation) is called. Otherwise, func is called via func(sim::Simulation, parameters).

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Oceananigans.Simulations.SimulationMethod
Simulation(model; Δt,
stop_iteration = Inf,
stop_time = Inf,
wall_time_limit = Inf)

Construct a Simulation for a model with time step Δt.

Keyword arguments

• Δt: Required keyword argument specifying the simulation time step. Can be a Number for constant time steps or a TimeStepWizard for adaptive time-stepping.

• stop_iteration: Stop the simulation after this many iterations.

• stop_time: Stop the simulation once this much model clock time has passed.

• wall_time_limit: Stop the simulation if it's been running for longer than this many seconds of wall clock time.

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Oceananigans.Simulations.TimeStepWizardType
TimeStepWizard(cfl=0.2, diffusive_cfl=Inf, max_change=1.1, min_change=0.5, max_Δt=Inf, min_Δt=0.0)

Callback for adapting simulation time-steps Δt to maintain the advective Courant-Freidrichs-Lewy (cfl) number, the diffusive_cfl, while maintaining max_Δt, min_Δt, and satisfying max_change and min_change criteria so Δt is not adapted "too quickly".

For more information on the cfl number, see its wikipedia entry.

Example

To use TimeStepWizard, adapt in a Callback and add it to a Simulation:

julia> simulation = Simulation(model, Δt=0.9, stop_iteration=100)

julia> wizard = TimeStepWizard(cfl=0.2)

julia> simulation.callbacks[:wizard] = Callback(wizard, IterationInterval(4))

Then when run!(simulation) is invoked, the time-step simulation.Δt will be updated every 4 iterations. Note that the name :wizard is unimportant.

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Oceananigans.Simulations.run!Method
run!(simulation; pickup=false)

Run a simulation until one of simulation.stop_criteria evaluates true. The simulation will then stop.

Picking simulations up from a checkpoint

Simulations are "picked up" from a checkpoint if pickup is either true, a String, or an Integer greater than 0.

Picking up a simulation sets field and tendency data to the specified checkpoint, leaving all other model properties unchanged.

Possible values for pickup are:

• pickup=true picks a simulation up from the latest checkpoint associated with the Checkpointer in simulation.output_writers.

• pickup=iteration::Int picks a simulation up from the checkpointed file associated with iteration and the Checkpointer in simulation.output_writers.

• pickup=filepath::String picks a simulation up from checkpointer data in filepath.

Note that pickup=true and pickup=iteration fails if simulation.output_writers contains more than one checkpointer.

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## Solvers

Oceananigans.Solvers.BatchedTridiagonalSolverMethod
BatchedTridiagonalSolver(grid; lower_diagonal, diagonal, upper_diagonal, parameters=nothing)

Construct a solver for batched tridiagonal systems on grid of the form

                       bⁱʲ¹ ϕⁱʲ¹ + cⁱʲ¹ ϕⁱʲ²   = fⁱʲ¹,  k = 1
aⁱʲᵏ⁻¹ ϕⁱʲᵏ⁻¹ + bⁱʲᵏ ϕⁱʲᵏ + cⁱʲᵏ ϕⁱʲᵏ⁺¹ = fⁱʲᵏ,  k = 2, ..., N-1
aⁱʲᴺ⁻¹ ϕⁱʲᴺ⁻¹ + bⁱʲᴺ ϕⁱʲᴺ               = fⁱʲᴺ,  k = N

where a is the lower_diagonal, b is the diagonal, and c is the upper_diagonal. ϕ is the solution and f is the right hand side source term passed to solve!(ϕ, tridiagonal_solver, f)

a, b, c, and f can be specified in three ways:

1. A 1D array means that aⁱʲᵏ = a[k].

2. A 3D array means that aⁱʲᵏ = a[i, j, k].

3. Otherwise, a is assumed to be callable:

• If isnothing(parameters) then aⁱʲᵏ = a(i, j, k, grid, args...).
• If !isnothing(parameters) then aⁱʲᵏ = a(i, j, k, grid, parameters, args...).

where args... are Varargs passed to solve_batched_tridiagonal_system!(ϕ, solver, args...).

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Oceananigans.Solvers.FFTBasedPoissonSolverType
FFTBasedPoissonSolver(grid, planner_flag=FFTW.PATIENT)

Return an FFTBasedPoissonSolver that solves the "generalized" Poisson equation,

$$$(∇² + m) ϕ = b,$$$

where $m$ is a number, using a eigenfunction expansion of the discrete Poisson operator on a staggered grid and for periodic or Neumann boundary conditions.

In-place transforms are applied to $b$, which means $b$ must have complex-valued elements (typically the same type as solver.storage).

See solve! for more information about the FFT-based Poisson solver algorithm.

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Oceananigans.Solvers.HeptadiagonalIterativeSolverMethod
HeptadiagonalIterativeSolver(coeffs;
grid,
iterative_solver = cg!,
maximum_iterations = prod(size(grid)),
tolerance = 1e-13,
reduced_dim = (false, false, false),
placeholder_timestep = -1.0,
preconditioner_method = :Default,
preconditioner_settings = nothing,
template = arch_array(architecture(grid), zeros(prod(size(grid)))),
verbose = false)

Return a HeptadiagonalIterativeSolver to solve the problem A * x = b, provided that A is a symmetric matrix.

The solver relies on a sparse version of the matrix A that is stored in matrix_constructors.

In particular, given coefficients Ax, Ay, Az, C, D, the solved problem is

    Axᵢ₊₁ ηᵢ₊₁ + Axᵢ ηᵢ₋₁ + Ayⱼ₊₁ ηⱼ₊₁ + Ayⱼ ηⱼ₋₁ + Azₖ₊₁ ηₖ₊₁ + Azₖ ηₖ₋₁
- 2 ( Axᵢ₊₁ + Axᵢ + Ayⱼ₊₁ + Ayⱼ + Azₖ₊₁ + Azₖ ) ηᵢⱼₖ
+   ( Cᵢⱼₖ + Dᵢⱼₖ/Δt^2 ) ηᵢⱼₖ  = b

To have the equation solved at location {Center, Center, Center}, the coefficients must be specified at:

• Ax -> {Face, Center, Center}
• Ay -> {Center, Face, Center}
• Az -> {Center, Center, Face}
• C -> {Center, Center, Center}
• D -> {Center, Center, Center}

solver.matrix is precomputed with a placeholder timestep value of placeholder_timestep = -1.0.

The sparse matrix A can be constructed with:

• SparseMatrixCSC(constructors...) for CPU
• CuSparseMatrixCSC(constructors...) for GPU

The matrix constructors are calculated based on the pentadiagonal coeffients passed as an input to matrix_from_coefficients function.

To allow for variable time step, the diagonal term - Az / (g * Δt²) is only added later on and it is updated only when the previous time step changes (previous_Δt != Δt).

Preconditioning is done through the various methods implemented in Solvers/sparse_preconditioners.jl.

The iterative_solver used can is to be chosen from the IterativeSolvers.jl package. The default solver is a Conjugate Gradient (cg):

solver = HeptadiagonalIterativeSolver((Ax, Ay, Az, C, D); grid)
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Oceananigans.Solvers.MultigridSolverMethod
MultigridSolver(linear_operation!::Function,
args...;
template_field::AbstractField,
maxiter = prod(size(template_field)),
reltol = sqrt(eps(eltype(template_field.grid))),
abstol = 0,
amg_algorithm = RugeStubenAMG()
)

Returns a MultigridSolver that solves the linear equation $A x = b$ using a multigrid method, where A * x is determined by linear_operation!

linear_operation! is a function with signature linear_operation!(Ax, x, args...) that calculates A * x for given x and stores the result in Ax.

The solver is used by calling

solve!(x, solver::MultigridSolver, b; kwargs...)

for solver, right-hand side b, solution x, and optional keyword arguments kwargs....

Arguments

• template_field: Dummy field that is the same type and size as x and b, which is used to infer the architecture, grid, and to create work arrays that are used internally by the solver.

• maxiter: Maximum number of iterations the solver may perform before exiting.

• reltol, abstol: Relative and absolute tolerance for convergence of the algorithm. The iteration stops when norm(A * x - b) < max(reltol * norm(b), abstol).

• amg_algorithm: Algebraic Multigrid algorithm defining mapping between different grid spacings. Options are RugeStubenAMG() (default) or SmoothedAggregationAMG(). Note: This keyword is relevant only on the CPU.

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Oceananigans.Solvers.PreconditionedConjugateGradientSolverMethod
PreconditionedConjugateGradientSolver(linear_operation;
template_field,
maxiter = size(template_field.grid),
reltol = sqrt(eps(eltype(template_field.grid))),
abstol = 0,
preconditioner = nothing)

Returns a PreconditionedConjugateGradientSolver that solves the linear equation $A x = b$ using a iterative conjugate gradient method with optional preconditioning.

The solver is used by calling

solve!(x, solver::PreconditionedConjugateGradientOperator, b, args...)

for solver, right-hand side b, solution x, and optional arguments args....

Arguments

• linear_operation: Function with signature linear_operation!(p, y, args...) that calculates A * y and stores the result in p for a "candidate solution" y. args... are optional positional arguments passed from solve!(x, solver, b, args...).

• template_field: Dummy field that is the same type and size as x and b, which is used to infer the architecture, grid, and to create work arrays that are used internally by the solver.

• maxiter: Maximum number of iterations the solver may perform before exiting.

• reltol, abstol: Relative and absolute tolerance for convergence of the algorithm. The iteration stops when norm(A * x - b) < tolerance.

• preconditioner: Object for which precondition!(z, preconditioner, r, args...) computes z = P * r, where r is the residual. Typically P is approximately A⁻¹.

See solve! for more information about the preconditioned conjugate-gradient algorithm.

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Oceananigans.Solvers.solve!Function
solve!(ϕ, solver::FFTBasedPoissonSolver, b, m=0)

Solves the "generalized" Poisson equation,

$$$(∇² + m) ϕ = b,$$$

where $m$ is a number, using a eigenfunction expansion of the discrete Poisson operator on a staggered grid and for periodic or Neumann boundary conditions.

In-place transforms are applied to $b$, which means $b$ must have complex-valued elements (typically the same type as solver.storage).

Alternative names for 'generalized' Poisson equation

Equation $(∇² + m) ϕ = b$ is sometimes referred to as the "screened Poisson" equation when $m < 0$, or the Helmholtz equation when $m > 0$.

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Oceananigans.Solvers.solve!Method
solve!(ϕ, solver::BatchedTridiagonalSolver, rhs, args...; dependencies = device_event(solver.architecture))

Solve the batched tridiagonal system of linear equations with right hand side rhs and lower diagonal, diagonal, and upper diagonal coefficients described by the BatchedTridiagonalSolver solver. BatchedTridiagonalSolver uses a modified TriDiagonal Matrix Algorithm (TDMA).

The result is stored in ϕ which must have size (grid.Nx, grid.Ny, grid.Nz).

Reference implementation per Numerical Recipes, Press et. al 1992 (§ 2.4).

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Oceananigans.Solvers.solve!Method
solve!(x, solver::MultigridSolver, b; kwargs...)

Solve A * x = b using a multigrid method, where A is solver.matrix.

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Oceananigans.Solvers.solve!Method
solve!(x, solver::PreconditionedConjugateGradientSolver, b, args...)

Solve A * x = b using an iterative conjugate-gradient method, where A * x is determined by solver.linear_operation

See figure 2.5 in

The Preconditioned Conjugate Gradient Method in "Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods" Barrett et. al, 2nd Edition.

Given:

• Linear Preconditioner operator M!(solution, x, other_args...) that computes M * x = solution
• A matrix operator A as a function A();
• A dot product function norm();
• A right-hand side b;
• An initial guess x; and
• Local vectors: z, r, p, q

This function executes the psuedocode algorithm

β  = 0
r = b - A(x)
iteration  = 0

Loop:
if iteration > maxiter
break
end

ρ = r ⋅ z

z = M(r)
β = ρⁱ⁻¹ / ρ
p = z + β * p
q = A(p)

α = ρ / (p ⋅ q)
x = x + α * p
r = r - α * q

if |r| < tolerance
break
end

iteration += 1
ρⁱ⁻¹ = ρ
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## Time steppers

Oceananigans.TimeSteppers.ClockType
mutable struct Clock{T<:Number}

Keeps track of the current time, iteration number, and time-stepping stage. The stage is updated only for multi-stage time-stepping methods. The time::T is either a number or a DateTime object.

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Oceananigans.TimeSteppers.ClockMethod
Clock(; time, iteration=0, stage=1)

Returns a Clock object. By default, Clock is initialized to the zeroth iteration and first time step stage.

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Oceananigans.TimeSteppers.QuasiAdamsBashforth2TimeStepperMethod
QuasiAdamsBashforth2TimeStepper(grid, tracers,
χ = 0.1;
implicit_solver = nothing,
Gⁿ = TendencyFields(grid, tracers),
G⁻ = TendencyFields(grid, tracers))

Return a 2nd-order quasi Adams-Bashforth (AB2) time stepper (QuasiAdamsBashforth2TimeStepper) on grid, with tracers, and AB2 parameter χ. The tendency fields Gⁿ and G⁻ can be specified via optional kwargs.

The 2nd-order quasi Adams-Bashforth timestepper steps forward the state Uⁿ by Δt via

Uⁿ⁺¹ = Uⁿ + Δt * [(3/2 + χ) * Gⁿ - (1/2 + χ) * Gⁿ⁻¹]

where Uⁿ is the state at the $n$-th timestep, Gⁿ is the tendency at the $n$-th timestep, and Gⁿ⁻¹ is the tendency at the previous timestep (G⁻).

First timestep

For the first timestep, since there are no saved tendencies from the previous timestep, the QuasiAdamsBashforth2TimeStepper performs an Euler timestep:

Uⁿ⁺¹ = Uⁿ + Δt * Gⁿ
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Oceananigans.TimeSteppers.RungeKutta3TimeStepperMethod
RungeKutta3TimeStepper(grid, tracers;
implicit_solver = nothing,
Gⁿ = TendencyFields(grid, tracers),
G⁻ = TendencyFields(grid, tracers))

Return a 3rd-order Runge0Kutta timestepper (RungeKutta3TimeStepper) on grid and with tracers. The tendency fields Gⁿ and G⁻ can be specified via optional kwargs.

The scheme described by Le and Moin (1991) (see H. Le, P. Moin (1991)). In a nutshel, the 3rd-order Runge Kutta timestepper steps forward the state Uⁿ by Δt via 3 substeps. A pressure correction step is applied after at each substep.

The state U after each substep m is

Uᵐ⁺¹ = Uᵐ + Δt * (γᵐ * Gᵐ + ζᵐ * Gᵐ⁻¹)

where Uᵐ is the state at the $m$-th substep, Gᵐ is the tendency at the $m$-th substep, Gᵐ⁻¹ is the tendency at the previous substep, and constants $γ¹ = 8/15$, $γ² = 5/12$, $γ³ = 3/4$, $ζ¹ = 0$, $ζ² = -17/60$, $ζ³ = -5/12$.

The state at the first substep is taken to be the one that corresponds to the $n$-th timestep, U¹ = Uⁿ, and the state after the third substep is then the state at the Uⁿ⁺¹ = U⁴.

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Oceananigans.TimeSteppers.time_step!Method
time_step!(model::AbstractModel{<:QuasiAdamsBashforth2TimeStepper}, Δt; euler=false)

Step forward model one time step Δt with a 2nd-order Adams-Bashforth method and pressure-correction substep. Setting euler=true will take a forward Euler time step.

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Oceananigans.TimeSteppers.time_step!Method
time_step!(model::AbstractModel{<:RungeKutta3TimeStepper}, Δt)

Step forward model one time step Δt with a 3rd-order Runge-Kutta method. The 3rd-order Runge-Kutta method takes three intermediate substep stages to achieve a single timestep. A pressure correction step is applied at each intermediate stage.

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## Turbulence closures

Oceananigans.TurbulenceClosures.AnisotropicMinimumDissipationType
AnisotropicMinimumDissipation{FT} <: AbstractTurbulenceClosure

Parameters for the "anisotropic minimum dissipation" turbulence closure for large eddy simulation proposed originally by Wybe Rozema, Hyun J. Bae, Parviz Moin, Roel Verstappen (2015) and Mahdi Abkar, Hyun J. Bae, Parviz Moin (2016), then modified by Roel Verstappen (2018), and finally described and validated for by Catherine A. Vreugdenhil, John R. Taylor (2018).

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Oceananigans.TurbulenceClosures.AnisotropicMinimumDissipationMethod
AnisotropicMinimumDissipation([time_discretization = ExplicitTimeDiscretization, FT = Float64;]
C = 1/12, Cν = nothing, Cκ = nothing, Cb = nothing)

Return parameters of type FT for the AnisotropicMinimumDissipation turbulence closure.

Keyword arguments

• C: Poincaré constant for both eddy viscosity and eddy diffusivities. C is overridden for eddy viscosity or eddy diffusivity if Cν or Cκ are set, respecitvely.

• Cν: Poincaré constant for momentum eddy viscosity.

• Cκ: Poincaré constant for tracer eddy diffusivities. If one number or function, the same number or function is applied to all tracers. If a NamedTuple, it must possess a field specifying the Poncaré constant for every tracer.

• Cb: Buoyancy modification multiplier (Cb = nothing turns it off, Cb = 1 was used by Mahdi Abkar, Hyun J. Bae, Parviz Moin (2016)). Note: that we do not subtract the horizontally-average component before computing this buoyancy modification term. This implementation differs from Mahdi Abkar, Hyun J. Bae, Parviz Moin (2016)'s proposal and the impact of this approximation has not been tested or validated.

• time_discretization: Either ExplicitTimeDiscretization() or VerticallyImplicitTimeDiscretization(), which integrates the terms involving only z-derivatives in the viscous and diffusive fluxes with an implicit time discretization.

By default: C = Cν = Cκ = 1/12, which is appropriate for a finite-volume method employing a second-order advection scheme, Cb = nothing, which terms off the buoyancy modification term.

Cν or Cκ may be constant numbers, or functions of x, y, z.

Examples

julia> using Oceananigans

julia> pretty_diffusive_closure = AnisotropicMinimumDissipation(C=1/2)
AnisotropicMinimumDissipation{ExplicitTimeDiscretization} turbulence closure with:
Poincaré constant for momentum eddy viscosity Cν: 0.5
Poincaré constant for tracer(s) eddy diffusivit(ies) Cκ: 0.5
Buoyancy modification multiplier Cb: nothing
julia> using Oceananigans

julia> const Δz = 0.5; # grid resolution at surface

julia> surface_enhanced_tracer_C(x, y, z) = 1/12 * (1 + exp((z + Δz/2) / 8Δz));

julia> fancy_closure = AnisotropicMinimumDissipation(Cκ=surface_enhanced_tracer_C)
AnisotropicMinimumDissipation{ExplicitTimeDiscretization} turbulence closure with:
Poincaré constant for momentum eddy viscosity Cν: 0.08333333333333333
Poincaré constant for tracer(s) eddy diffusivit(ies) Cκ: surface_enhanced_tracer_C
Buoyancy modification multiplier Cb: nothing
julia> using Oceananigans

julia> tracer_specific_closure = AnisotropicMinimumDissipation(Cκ=(c₁=1/12, c₂=1/6))
AnisotropicMinimumDissipation{ExplicitTimeDiscretization} turbulence closure with:
Poincaré constant for momentum eddy viscosity Cν: 0.08333333333333333
Poincaré constant for tracer(s) eddy diffusivit(ies) Cκ: (c₁ = 0.08333333333333333, c₂ = 0.16666666666666666)
Buoyancy modification multiplier Cb: nothing

References

Vreugdenhil C., and Taylor J. (2018), "Large-eddy simulations of stratified plane Couette flow using the anisotropic minimum-dissipation model", Physics of Fluids 30, 085104.

Verstappen, R. (2018), "How much eddy dissipation is needed to counterbalance the nonlinear production of small, unresolved scales in a large-eddy simulation of turbulence?", Computers & Fluids 176, pp. 276-284.

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Oceananigans.TurbulenceClosures.ConvectiveAdjustmentVerticalDiffusivityType
ConvectiveAdjustmentVerticalDiffusivity([time_discretization = VerticallyImplicitTimeDiscretization(), FT=Float64;]
convective_κz = 0,
convective_νz = 0,
background_κz = 0,
background_νz = 0)

Return a convective adjustment vertical diffusivity closure that applies different values of diffusivity and/or viscosity depending whether the region is statically stable (positive or zero buoyancy gradient) or statically unstable (negative buoyancy gradient).

Arguments

• time_discretization: Either ExplicitTimeDiscretization() or VerticallyImplicitTimeDiscretization(); default VerticallyImplicitTimeDiscretization().

• FT: Float type; default Float64.

Keyword arguments

• convective_κz: Vertical tracer diffusivity in regions with negative (unstable) buoyancy gradients. Either a single number, function, array, field, or tuple of diffusivities for each tracer.

• background_κz: Vertical tracer diffusivity in regions with zero or positive (stable) buoyancy gradients.

• convective_νz: Vertical viscosity in regions with negative (unstable) buoyancy gradients. Either a number, function, array, or field.

• background_κz: Vertical viscosity in regions with zero or positive (stable) buoyancy gradients.

Example

julia> using Oceananigans

julia> cavd = ConvectiveAdjustmentVerticalDiffusivity(convective_κz = 1)
ConvectiveAdjustmentVerticalDiffusivity{VerticallyImplicitTimeDiscretization}(background_κz=0.0 convective_κz=1 background_νz=0.0 convective_νz=0.0)
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Oceananigans.TurbulenceClosures.IsopycnalSkewSymmetricDiffusivityMethod
IsopycnalSkewSymmetricDiffusivity([time_disc=VerticallyImplicitTimeDiscretization(), FT=Float64;]
κ_skew = 0,
κ_symmetric = 0,
isopycnal_tensor = SmallSlopeIsopycnalTensor(),
slope_limiter = FluxTapering(1e-2))

Return parameters for an isopycnal skew-symmetric tracer diffusivity with skew diffusivity κ_skew and symmetric diffusivity κ_symmetric that uses an isopycnal_tensor model for for calculating the isopycnal slopes, and (optionally) applying a slope_limiter to the calculated isopycnal slope values.

Both κ_skew and κ_symmetric may be constants, arrays, fields, or functions of (x, y, z, t).

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Oceananigans.TurbulenceClosures.RiBasedVerticalDiffusivityType
RiBasedVerticalDiffusivity([time_discretization = VerticallyImplicitTimeDiscretization(),
FT = Float64;]
Ri_dependent_tapering = ExponentialRiDependentTapering(),
ν₀  = 0.30,
κ₀  = 0.42,
κᶜ  = 4.0,
Cᵉ  = 0.57,
Ri₀ = 0.27,
Riᵟ = 0.20,
warning = true)

Return a closure that estimates the vertical viscosity and diffusivit from "convective adjustment" coefficients ν₀ and κ₀ multiplied by a decreasing function of the Richardson number, $Ri$.

Keyword Arguments

• Ri_dependent_tapering: The $Ri$-dependent tapering. Options are: PiecewiseLinearRiDependentTapering(), HyperbolicTangentRiDependentTapering() (default), and ExponentialRiDependentTapering().
• ν₀: Non-convective viscosity.
• κ₀: Non-convective diffusivity for tracers.
• κᶜ: Convective adjustment diffusivity for tracers.
• Cᵉ: Entrainment coefficient for tracers.
• Ri₀: $Ri$ threshold for decreasing viscosity and diffusivity.
• Riᵟ: $Ri$-width over which viscosity and diffusivity decreases to 0.
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Oceananigans.TurbulenceClosures.ScalarBiharmonicDiffusivityType
ScalarBiharmonicDiffusivity([formulation=ThreeDimensionalFormulation(), FT=Float64;]
ν=0, κ=0,
discrete_form = false)

Return a scalar biharmonic diffusivity turbulence closure with viscosity coefficient ν and tracer diffusivities κ for each tracer field in tracers. If a single κ is provided, it is applied to all tracers. Otherwise κ must be a NamedTuple with values for every tracer individually.

Arguments

• formulation:

• HorizontalFormulation() for diffusivity applied in the horizontal direction(s)
• VerticalFormulation() for diffusivity applied in the vertical direction,
• ThreeDimensionalFormulation() (default) for diffusivity applied isotropically to all directions
• FT: the float datatype (default: Float64)

Keyword arguments

• ν: Viscosity. Number, AbstractArray, or Function(x, y, z, t).

• κ: Diffusivity. Number, AbstractArray, or Function(x, y, z, t), or NamedTuple of diffusivities with entries for each tracer.

• discrete_form: Boolean.

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Oceananigans.TurbulenceClosures.ScalarDiffusivityType
ScalarDiffusivity([time_discretization=ExplicitTimeDiscretization(),
formulation=ThreeDimensionalFormulation(), FT=Float64];
ν=0, κ=0,
discrete_form = false)

Return ScalarDiffusivity turbulence closure with viscosity ν and tracer diffusivities κ for each tracer field in tracers. If a single κ is provided, it is applied to all tracers. Otherwise κ must be a NamedTuple with values for every tracer individually.

Arguments

• time_discretization: either ExplicitTimeDiscretization() (default) or VerticallyImplicitTimeDiscretization().

• formulation:

• HorizontalFormulation() for diffusivity applied in the horizontal direction(s)
• VerticalFormulation() for diffusivity applied in the vertical direction,
• ThreeDimensionalFormulation() (default) for diffusivity applied isotropically to all directions
• FT: the float datatype (default: Float64)

Keyword arguments

ν and the fields of κ may be constants (converted to FT), arrays, fields or

• functions of (x, y, z, t) if discrete_form = false
• functions of (i, j, k, grid, LX, LY, LZ) with LX, LY and LZ are either Face() or Center() if discrete_form = true.
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Oceananigans.TurbulenceClosures.SmagorinskyLillyMethod
SmagorinskyLilly(time_discretization = ExplicitTimeDiscretization, [FT=Float64;] C=0.16, Pr=1)

Return a SmagorinskyLilly type associated with the turbulence closure proposed by Lilly (1962) and Smagorinsky (1958, 1963), which has an eddy viscosity of the form

νₑ = (C * Δᶠ)² * √(2Σ²) * √(1 - Cb * N² / Σ²)

and an eddy diffusivity of the form

κₑ = νₑ / Pr

where Δᶠ is the filter width, Σ² = ΣᵢⱼΣᵢⱼ is the double dot product of the strain tensor Σᵢⱼ, Pr is the turbulent Prandtl number, and N² is the total buoyancy gradient, and Cb is a constant the multiplies the Richardson number modification to the eddy viscosity.

Keyword arguments

• C: Smagorinsky constant. Default value is 0.16 as obtained by Lilly (1966).
• Cb: Buoyancy term multipler based on Lilly (1962) (Cb = 0 turns it off, Cb ≠ 0 turns it on. Typically, and according to the original work by Lilly (1962), Cb=1/Pr.)
• Pr: Turbulent Prandtl numbers for each tracer. Either a constant applied to every tracer, or a NamedTuple with fields for each tracer individually.
• time_discretization: Either ExplicitTimeDiscretization() or VerticallyImplicitTimeDiscretization(), which integrates the terms involving only $z$-derivatives in the viscous and diffusive fluxes with an implicit time discretization.

References

Smagorinsky, J. "On the numerical integration of the primitive equations of motion for baroclinic flow in a closed region." Monthly Weather Review (1958)

Lilly, D. K. "On the numerical simulation of buoyant convection." Tellus (1962)

Smagorinsky, J. "General circulation experiments with the primitive equations: I. The basic experiment." Monthly weather review (1963)

Lilly, D. K. "The representation of small-scale turbulence in numerical simulation experiments." NCAR Manuscript No. 281, 0, 1966.

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Oceananigans.TurbulenceClosures.TwoDimensionalLeithType
TwoDimensionalLeith(FT=Float64;
C=0.3, C_Redi=1, C_GM=1,
isopycnal_model=SmallSlopeIsopycnalTensor())

Return a TwoDimensionalLeith type associated with the turbulence closure proposed by Leith (1965) and Fox-Kemper & Menemenlis (2008) which has an eddy viscosity of the form

νₑ = (C * Δᶠ)³ * √(|∇ₕ ζ|² + |∇ₕ ∂w/∂z|²)

and an eddy diffusivity of the form...

where Δᶠ is the filter width, ζ = ∂v/∂x - ∂u/∂y is the vertical vorticity, and C is a model constant.

Keyword arguments

• C: Model constant
• C_Redi: Coefficient for down-gradient tracer diffusivity for each tracer. Either a constant applied to every tracer, or a NamedTuple with fields for each tracer individually.
• C_GM: Coefficient for down-gradient tracer diffusivity for each tracer. Either a constant applied to every tracer, or a NamedTuple with fields for each tracer individually.

References

Leith, C. E. (1968). "Diffusion Approximation for Two‐Dimensional Turbulence", The Physics of Fluids 11, 671. doi: 10.1063/1.1691968

Fox‐Kemper, B., & D. Menemenlis (2008), "Can large eddy simulation techniques improve mesoscale rich ocean models?", in Ocean Modeling in an Eddying Regime, Geophys. Monogr. Ser., vol. 177, pp. 319–337. doi: 10.1029/177GM19

Pearson, B. et al. (2017) , "Evaluation of scale-aware subgrid mesoscale eddy models in a global eddy rich model", Ocean Modelling 115, 42-58. doi: 10.1016/j.ocemod.2017.05.007

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Oceananigans.TurbulenceClosures.VerticallyImplicitTimeDiscretizationType
struct VerticallyImplicitTimeDiscretization <: AbstractTimeDiscretization

A vertically-implicit time-discretization of a TurbulenceClosure.

This implies that a flux divergence such as $𝛁 ⋅ q$ at the n-th timestep is time-discretized as

[∇ ⋅ q]ⁿ = [explicit_flux_divergence]ⁿ + [∂z (κ ∂z c)]ⁿ⁺¹
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## Utilities

Oceananigans.Utils.IterationIntervalMethod
IterationInterval(interval; offset=0)

Return a callable IterationInterval that "actuates" (schedules output or callback execution) whenever the model iteration (modified by offset) is a multiple of interval.

For example,

• IterationInterval(100) actuates at iterations [100, 200, 300, ...].
• IterationInterval(100, offset=-1) actuates at iterations [99, 199, 299, ...].
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Oceananigans.Utils.SpecifiedTimesMethod
SpecifiedTimes(times)

Return a callable TimeInterval that "actuates" (schedules output or callback execution) whenever the model's clock equals the specified values in times. For example,

• SpecifiedTimes([1, 15.3]) actuates when model.clock.time is 1 and 15.3.
Sorting specified times

The specified times need not be ordered as the SpecifiedTimes constructor will check and order them in ascending order if needed.

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Oceananigans.Utils.TimeIntervalType
struct TimeInterval <: AbstractSchedule

Callable TimeInterval schedule for periodic output or diagnostic evaluation according to model.clock.time.

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Oceananigans.Utils.TimeIntervalMethod
TimeInterval(interval)

Return a callable TimeInterval that schedules periodic output or diagnostic evaluation on a interval of simulation time, as kept by model.clock.

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Oceananigans.Utils.WallTimeIntervalMethod
WallTimeInterval(interval; start_time = time_ns() * 1e-9)

Return a callable WallTimeInterval that schedules periodic output or diagnostic evaluation on a interval of "wall time" while a simulation runs, in units of seconds.

The "wall time" is the actual real world time in seconds, as kept by an actual or hypothetical clock hanging on your wall.

The keyword argument start_time can be used to specify a starting wall time other than the moment WallTimeInterval is constructed.

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Oceananigans.Utils.launch!Method
launch!(arch, grid, layout, kernel!, args...; dependencies=nothing, kwargs...)

Launches kernel!, with arguments args and keyword arguments kwargs, over the dims of grid on the architecture arch.

Returns an event token associated with the kernel! launch.

The keyword argument dependencies is an Event or MultiEvent specifying prior kernels that must complete before kernel! is launched.

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Oceananigans.Utils.pretty_filesizeFunction
pretty_filesize(s, suffix="B")

Convert a floating point value s representing a file size to a more human-friendly formatted string with one decimal places with a suffix defaulting to "B". Depending on the value of s the string will be formatted to show s using an SI prefix from bytes, kiB (1024 bytes), MiB (1024² bytes), and so on up to YiB (1024⁸ bytes).

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Oceananigans.Utils.prettytimeFunction
prettytime(t, longform=true)

Convert a floating point value t representing an amount of time in SI units of seconds to a human-friendly string with three decimal places. Depending on the value of t the string will be formatted to show t in nanoseconds (ns), microseconds (μs), milliseconds (ms), seconds, minutes, hours, days, or years.

With longform=false, we use s, m, hrs, d, and yrs in place of seconds, minutes, hours, and years.

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Oceananigans.Utils.with_tracersMethod
with_tracers(tracer_names, initial_tuple, tracer_default)

Create a tuple corresponding to the solution variables u, v, w, and tracer_names. initial_tuple is a NamedTuple that at least has fields u, v, and w, and may have some fields corresponding to the names in tracer_names. tracer_default is a function that produces a default tuple value for each tracer if not included in initial_tuple.

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Oceananigans.Utils.work_layoutMethod
work_layout(grid, dims; include_right_boundaries=false, location=nothing)

Returns the workgroup and worksize for launching a kernel over dims on grid. The workgroup is a tuple specifying the threads per block in each dimension. The worksize specifies the range of the loop in each dimension.

Specifying include_right_boundaries=true will ensure the work layout includes the right face end points along bounded dimensions. This requires the field location to be specified.

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Oceananigans.Utils.@apply_regionallyMacro
@apply_regionally expr

Use @apply_regionally to distribute locally the function calls. Call compute_regionally in case of a returning value and apply_regionally! in case of no return.

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## Units

Oceananigans.Units.GiBConstant
GiB

A Float64 constant equal to 1024MiB. Useful for increasing the clarity of scripts, e.g. max_filesize = 50GiB.

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Oceananigans.Units.KiBConstant
KiB

A Float64 constant equal to 1024.0. Useful for increasing the clarity of scripts, e.g. max_filesize = 250KiB.

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Oceananigans.Units.MiBConstant
MiB

A Float64 constant equal to 1024KiB. Useful for increasing the clarity of scripts, e.g. max_filesize = 100MiB.

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Oceananigans.Units.TiBConstant
TiB

A Float64 constant equal to 1024GiB. Useful for increasing the clarity of scripts, e.g. max_filesize = 2TiB.

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Oceananigans.Units.dayConstant
day

A Float64 constant equal to 24hours. Useful for increasing the clarity of scripts, e.g. stop_time = 1day.

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Oceananigans.Units.daysConstant
days

A Float64 constant equal to 24hours. Useful for increasing the clarity of scripts, e.g. stop_time = 7days.

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Oceananigans.Units.hourConstant
hour

A Float64 constant equal to 60minutes. Useful for increasing the clarity of scripts, e.g. Δt = 1hour.

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Oceananigans.Units.hoursConstant
hours

A Float64 constant equal to 60minutes. Useful for increasing the clarity of scripts, e.g. Δt = 3hours.

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Oceananigans.Units.kilometerConstant
kilometer

A Float64 constant equal to 1000meters. Useful for increasing the clarity of scripts, e.g. Lx = 1kilometer.

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Oceananigans.Units.kilometersConstant
kilometers

A Float64 constant equal to 1000meters. Useful for increasing the clarity of scripts, e.g. Lx = 5000kilometers.

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Oceananigans.Units.minuteConstant
minute

A Float64 constant equal to 60seconds. Useful for increasing the clarity of scripts, e.g. Δt = 1minute.

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Oceananigans.Units.minutesConstant
minutes

A Float64 constant equal to 60seconds. Useful for increasing the clarity of scripts, e.g. Δt = 15minutes.

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Oceananigans.Units.yearConstant
year

A Float64 constant equal to 365days. Useful for increasing the clarity of scripts, e.g. stop_time = 1year.

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Oceananigans.Units.yearsConstant
years

A Float64 constant equal to 365days. Useful for increasing the clarity of scripts, e.g. stop_time = 100years.

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