This negative buoyancy flux works on mixing at the bottom of the mixed layer, maintaining entrainment velocity w. The governing equations for the physical properties are
dT/dt = -Q/(ρ′CpH) + w(T′-T)/H (1a)
dS/dt = w(S′-S)/H (1b)
dH/dt = w (1c)
w[β(S′-S)-α(T′-T)] = 2∈1E/(Hg) + ∈2αQ/(ρ′Cp)
Here, a prime denotes the value below the mixed layer, ρ is the water density, Cp is the heat capacity of water, and g is the gravitational acceleration. E is the kinetic energy input due to wind stress at the sea surface, and ∈1 and ∈2 are the dissipation coefficients of the kinetic and potential energy inputs, respectively, α=-(∂ρ/∂T)/ρ, and β=(∂ρ/∂S)/ρ.
In addition to these physical properties, CO2 and alkalinity are also considered as model variables. Total carbonate C and alkalinity A increase in fall through winter, as the mixed layer develops and entrain carbonate-rich subsurface water.
dC/dt = -G/H + w(C′-C)/H (2a)
dA/dt = w(A′-A)/H (2b)
The air-sea CO2 flux G is proportional to the piston velocity Vp (proportinal to cubed wind speed) and the difference in partial pressure between the atmosphere Co/γ and the ocean Pco° where γ is the solubility of CO2.
G = Vp(Co-γPco) (3)
The chemical equilibrium is established
Co = K(2C-A)2/(A-C) (4)
where the coefficient K is dependent on water temperature.
