Buoyancy and equation of state

The buoyancy option selects how buoyancy is treated. There are currently three options:

1. No buoyancy (and no gravity).
2. Evolve buoyancy as a tracer.
3. Seawater buoyancy: evolve temperature $T$ and salinity $S$ as tracers with a value for the gravitational acceleration $g$ and an equation of state of your choosing. This is the default setting.

No buoyancy

To turn off buoyancy (and gravity) simply pass buoyancy = nothing to the model constructor.

julia> grid = RegularCartesianGrid(size=(64, 64, 64), extent=(1, 1, 1));

julia> model = IncompressibleModel(grid=grid, buoyancy=nothing)
IncompressibleModel{CPU, Float64}(time = 0 seconds, iteration = 0)
├── grid: RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}(Nx=64, Ny=64, Nz=64)
├── tracers: (:T, :S)
├── closure: IsotropicDiffusivity{Float64,NamedTuple{(:T, :S),Tuple{Float64,Float64}}}
├── buoyancy: Nothing
└── coriolis: Nothing

In this case, you might want to explicitly specify which tracers to evolve. In particular, you may not want to evolve temperature and salinity, which are included by default. To specify no tracers, also passtracers = () to the model constructor.

julia> model = IncompressibleModel(grid=grid, buoyancy=nothing, tracers=())
IncompressibleModel{CPU, Float64}(time = 0 seconds, iteration = 0)
├── grid: RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}(Nx=64, Ny=64, Nz=64)
├── tracers: ()
├── closure: IsotropicDiffusivity{Float64,NamedTuple{(),Tuple{}}}
├── buoyancy: Nothing
└── coriolis: Nothing

Buoyancy as a tracer

To directly evolve buoyancy as a tracer simply pass buoyancy = BuoyancyTracer() to the model constructor. Buoyancy :b must be included as a tracer, for example,

julia> model = IncompressibleModel(grid=grid, buoyancy=BuoyancyTracer(), tracers=(:b))
IncompressibleModel{CPU, Float64}(time = 0 seconds, iteration = 0)
├── grid: RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}(Nx=64, Ny=64, Nz=64)
├── tracers: (:b,)
├── closure: IsotropicDiffusivity{Float64,NamedTuple{(:b,),Tuple{Float64}}}
├── buoyancy: BuoyancyTracer
└── coriolis: Nothing

Seawater buoyancy

To evolve temperature $T$ and salinity $S$ and diagnose the buoyancy, you can pass buoyancy = SeawaterBuoyancy() which is the default.

julia> model = IncompressibleModel(grid=grid, buoyancy=SeawaterBuoyancy())
IncompressibleModel{CPU, Float64}(time = 0 seconds, iteration = 0)
├── grid: RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}(Nx=64, Ny=64, Nz=64)
├── tracers: (:T, :S)
├── closure: IsotropicDiffusivity{Float64,NamedTuple{(:T, :S),Tuple{Float64,Float64}}}
├── buoyancy: SeawaterBuoyancy{Float64,LinearEquationOfState{Float64},Nothing,Nothing}
└── coriolis: Nothing

Without any options specified, a value of $g = 9.80665 \; \text{m/s}^2$ is used for the gravitational acceleration (corresponding to standard gravity) along with a linear equation of state with thermal expansion and haline contraction coefficients suitable for seawater.

If, for example, you wanted to simulate fluids on another planet such as Europa where $g = 1.3 \; \text{m/s}^2$, then use

julia> buoyancy = SeawaterBuoyancy(gravitational_acceleration=1.3)
SeawaterBuoyancy{Float64}: g = 1.3
└── equation of state: LinearEquationOfState{Float64}: α = 1.67e-04, β = 7.80e-04

julia> model = IncompressibleModel(grid=grid, buoyancy=buoyancy)
IncompressibleModel{CPU, Float64}(time = 0 seconds, iteration = 0)
├── grid: RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}(Nx=64, Ny=64, Nz=64)
├── tracers: (:T, :S)
├── closure: IsotropicDiffusivity{Float64,NamedTuple{(:T, :S),Tuple{Float64,Float64}}}
├── buoyancy: SeawaterBuoyancy{Float64,LinearEquationOfState{Float64},Nothing,Nothing}
└── coriolis: Nothing

When using SeawaterBuoyancy temperature :T and salinity :S tracers must be specified. Explicitly this can be accomplished by passing tracers = (:T, :S) to a model constructor.

Linear equation of state

To use non-default thermal expansion and haline contraction coefficients, say $\alpha = 2 \times 10^{-3} \; \text{K}^{-1}$ and $\beta = 5 \times 10^{-4} \text{ppt}^{-1}$ corresponding to some other fluid, then use

julia> buoyancy = SeawaterBuoyancy(equation_of_state=LinearEquationOfState(α=1.67e-4, β=7.80e-4))
SeawaterBuoyancy{Float64}: g = 9.80665
└── equation of state: LinearEquationOfState{Float64}: α = 1.67e-04, β = 7.80e-04

Idealized nonlinear equations of state

Instead of a linear equation of state, five idealized (second-order) nonlinear equation of state as described by Fabien Roquet , Gurvan Madec , Laurent Brodeau , J. Nycander (2015) may be used. These equations of state are provided via the SeawaterPolynomials.jl package.

julia> using SeawaterPolynomials.SecondOrderSeawaterPolynomials

julia> eos = RoquetSeawaterPolynomial(:Freezing)
SecondOrderSeawaterPolynomial{Float64}(0.7718, -0.0491, 0.0, -2.5681e-5, 0.0, -0.005027, 0.0)

julia> buoyancy = SeawaterBuoyancy(equation_of_state=eos)
SeawaterBuoyancy{Float64}: g = 9.80665
└── equation of state: SeawaterPolynomials.SecondOrderSeawaterPolynomials.SecondOrderSeawaterPolynomial{Float64}(0.7718, -0.0491, 0.0, -2.5681e-5, 0.0, -0.005027, 0.0)

TEOS-10 equation of state

A high-accuracy 55-term polynomial approximation to the TEOS-10 equation of state suitable for use in Boussinesq models as described by F. Roquet , G. Madec , Trevor J. McDougall , Paul M. Barker (2015) is implemented in the SeawaterPolynomials.jl package and may be used.

julia> using SeawaterPolynomials.TEOS10

julia> eos = TEOS10EquationOfState()
SeawaterPolynomials.BoussinesqEquationOfState{TEOS10SeawaterPolynomial{Float64},Int64}(TEOS10SeawaterPolynomial{Float64}(), 1020)