Air-sea gas exchange

Air-sea gas transfer is typically parameterised as a function of temperature ($T$) and wind speed ($u_{10}$), and the concentration of the gas in the air ($C_a$) and in the surface water ($C_w$) in the form:

\[F = k(u_{10}, T)(C_w - C_a),\]

where k is the gas transfer velocity.

Our implementation is intended to be generic for any gas, so you can specify air_concentration, water_concentration, and transfer_velocity as any function in GasExchange, but we also provide constructors and default values for carbon dioxide and oxygen.

The wind speed is a top-level property of GasExchange and is passed to the transfer velocity at evaluation time. The default SchmidtScaledTransferVelocity evaluates a $k_{660}(u_{10})$ parameterisation scaled by the Schmidt number. The carbon dioxide and oxygen constructors take wind_speed directly:

CO₂_flux = CarbonDioxideGasExchangeBoundaryCondition(; wind_speed = 5)

wind_speed (like air_concentration) may be a number, a function of (x, y, t), a function of (i, j, grid, clock, model_fields) if discrete_form = true, a Field, or a FieldTimeSeries. If you pass a FieldTimeSeries which lives on a different grid to the model, also pass grid = grid so that it can be wrapped for interpolation onto the model grid.

To setup carbon dioxide and/or oxygen boundary conditions you simply build the condition and then specify it in the model:

using OceanBioME
CO₂_flux = CarbonDioxideGasExchangeBoundaryCondition()
O₂_flux  = OxygenGasExchangeBoundaryCondition()
using Oceananigans

grid = RectilinearGrid(size=(3, 3, 30), extent=(10, 10, 200));

model = NonhydrostaticModel(grid;
                            biogeochemistry = LOBSTER(grid; inorganic_carbon = CarbonateSystem(), oxygen = Oxygen()),
                            boundary_conditions = (DIC = FieldBoundaryConditions(top = CO₂_flux),
                                                    O₂ = FieldBoundaryConditions(top =  O₂_flux)),
                            tracers = (:T, :S))

NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 3×3×30 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme:
│   ├── momentum: Centered(order=2)
│   ├── T: Centered(order=2)
│   ├── S: Centered(order=2)
│   ├── NO₃: Centered(order=2)
│   ├── NH₄: Centered(order=2)
│   ├── P: Centered(order=2)
│   ├── Z: Centered(order=2)
│   ├── DOM: Centered(order=2)
│   ├── sPOM: Centered(order=2)
│   ├── bPOM: Centered(order=2)
│   ├── DIC: Centered(order=2)
│   ├── Alk: Centered(order=2)
│   └── O₂: Centered(order=2)
├── tracers: (T, S, NO₃, NH₄, P, Z, DOM, sPOM, bPOM, DIC, Alk, O₂)
├── closure: Nothing
├── buoyancy: Nothing
└── coriolis: Nothing

Note

All gas exchange models require temperature (T) to be present in the model, and carbon dioxide requires sailinity (S), total inorganic carbon (DIC), and alkalinity (Alk), and optionally can take silicate and phosphate where there names are specified in the keyword argument silicate_and_phosphate_names

Model equations

### Gas transfer velocity

The default gas transfer velocity (ScaledTransferVelocity) returns a velocity in the form:

\[k(u_{10}, T) = cu_{10}^2\left(\frac{Sc(T)}{660}\right)^{-1/2},\]

where $c$ is a coefficient (coeff) which typically is wind product specific with default value $0.266$ cm/hour from Ho et al. (2006), and $Sc$ is gas specific the temperature dependent Schmidt number (the dimensionless ratio of momentum and mass diffusivity) specified as schmidt_number which can be any function of temperature. The default parameterisations is the 4th order polynomial formulation of Wanninkhof (2014).

Currently, the parameters for CO₂ and oxygen are included, but it would be very straightforward to add the parameters given in the original publication for other gases (e.g. inert tracers of other nutrients such as N₂).

Carbon dioxide concentration

For most gasses the water concentration C_w is simply taken directly from the biogeochemical model or another tracer (in which case water_concentration should be set to TracerConcentration(:tracer_name)), but for carbon dioxide it must be derived from the dissolved inorganic carbon (DIC) and Alkalinity by a CarbonChemistry model (please see the docs for CarbonChemistry).

The water concentration is the aqueous carbon dioxide concentration in mmol / m³,

\[C_w = [CO_2(aq)] = DIC\frac{[H^+]^2}{[H^+]^2 + K_1[H^+] + K_1K_2},\]

(CarbonDioxideConcentration), and the air concentration is the dry air mole fraction converted onto the same basis by Dalton's law and a solubility,

\[C_a = x(CO_2)p_{atm}f_f(T, S)\frac{\rho}{10^3}.\]

The solubility $f_f$ is the Weiss and Price (1980) parameterisation (FF), which is the solubility $K_0$ corrected for the water vapour pressure of saturated air and for the non-ideality of the gas phase,

\[f_f = K_0(1 - p_{H_2O})\gamma,\]

and is therefore the right quantity to multiply a dry air mole fraction by. The transfer velocity is a bare piston velocity (its solubility is UnitSolubility), and the atmospheric pressure enters the flux exactly once, on the air side.