58
3 Basics of Nonhydrostatic Modelling
diffusion of momentum and density. The onset of free convection can be characterised by a threshold value of the Rayleigh number, introduced by Rayleigh (1916):
Ra =
N
2
d
4
kν
(3.64)
where N is the stability frequency (which is of negative sign for unstable density
stratification), d is the thickness of the unstably stratified fluid layer, k is molecular
diffusivity of density, and ν is kinematic molecular viscosity. Molecular diffusion is
associated with random motions of molecules (Brownian motion). For a turbulent
background fluid, some scientists use turbulent equivalents in Eq. (3.64).
3.14.3 The Flux-Rayleigh Number
The appearance of oceanic convection is the result of a surface buoyancy flux created by air-sea fluxes of heat and freshwater (changing salinity). A linearised equation of state is sufficient to derive surface buoyancy fluxes as a function of these heat
and freshwater fluxes. This equation reads:
ρ(T, S) = ρ o [1 − α(T − T o ) + β(S − S o )]
(3.65)
where the reference density ρ o corresponds to a reference temperature T o and a reference salinity S o . The parameter α is the thermal expansion coefficient and β is an
equivalent coefficient, sometimes called saline contraction coefficient, for salinityrelated variations in density. The surface buoyancy flux B is defined indirectly via
associated density changes in the surface mixed layer of thickness h that follow:
dρ mix
dt
=
ρ o
gh
B
(3.66)
With use of the equation of state (Eq. 3.65) and thermodynamic relations between
surface temperature changes with surface heat fluxes and salinity changes with surface freshwater fluxes, the surface buoyancy flux can be calculated from:
B = g
α
ρ o C P
Q + β(E − P)S mix
where C P is the heat capacity of water, Q is the surface heat flux, S mix is salinity
of the surface mixed layer, and (E − P) is evaporation minus precipitation (units
are m/s). Gill (1982) presents tables of α, β, and C P for seawater as functions of
ambient temperature and salinity.
Free convection develops if a certain threshold value of the flux-Rayleigh number
is exceeded, whereby this number is defined by (Marshall and Schott, 1999):
Ra f =
Bd
4
k 2 ν
3 Basics of Nonhydrostatic Modelling
diffusion of momentum and density. The onset of free convection can be characterised by a threshold value of the Rayleigh number, introduced by Rayleigh (1916):
Ra =
N
2
d
4
kν
(3.64)
where N is the stability frequency (which is of negative sign for unstable density
stratification), d is the thickness of the unstably stratified fluid layer, k is molecular
diffusivity of density, and ν is kinematic molecular viscosity. Molecular diffusion is
associated with random motions of molecules (Brownian motion). For a turbulent
background fluid, some scientists use turbulent equivalents in Eq. (3.64).
3.14.3 The Flux-Rayleigh Number
The appearance of oceanic convection is the result of a surface buoyancy flux created by air-sea fluxes of heat and freshwater (changing salinity). A linearised equation of state is sufficient to derive surface buoyancy fluxes as a function of these heat
and freshwater fluxes. This equation reads:
ρ(T, S) = ρ o [1 − α(T − T o ) + β(S − S o )]
(3.65)
where the reference density ρ o corresponds to a reference temperature T o and a reference salinity S o . The parameter α is the thermal expansion coefficient and β is an
equivalent coefficient, sometimes called saline contraction coefficient, for salinityrelated variations in density. The surface buoyancy flux B is defined indirectly via
associated density changes in the surface mixed layer of thickness h that follow:
dρ mix
dt
=
ρ o
gh
B
(3.66)
With use of the equation of state (Eq. 3.65) and thermodynamic relations between
surface temperature changes with surface heat fluxes and salinity changes with surface freshwater fluxes, the surface buoyancy flux can be calculated from:
B = g
α
ρ o C P
Q + β(E − P)S mix
where C P is the heat capacity of water, Q is the surface heat flux, S mix is salinity
of the surface mixed layer, and (E − P) is evaporation minus precipitation (units
are m/s). Gill (1982) presents tables of α, β, and C P for seawater as functions of
ambient temperature and salinity.
Free convection develops if a certain threshold value of the flux-Rayleigh number
is exceeded, whereby this number is defined by (Marshall and Schott, 1999):
Ra f =
Bd
4
k 2 ν