The basic equations, in a reference frame fixed to the earth, are
where
is the mass density (kg/m 
),
the specific volume (1/ 
),
the mass fraction of secondary constituent (kg of S per kg of fluid),
the specific entropy (J/kg 
K),
the relative chemical potential for S (J/kg),
the viscous generation of heat per unit volume and time (W/m ),
the absolute temperature ( 
K),
the pressure (N/m 
),
the fluid velocity vector in earth frame of reference (m/s),
the acceleration due to gravity (m/s ),
the net tide producing force per unit mass (m/s ),
the viscous force per unit volume (N/m ),
the earth's angular velocity (rad/s),
the diffusive flux of S (kg/m s) and
the heat flux by conduction and/or radiation (W/m ).
Auxiliary relations exist for gravitational acceleration
the viscous force per unit volume
the viscous generation of heat per unit volume and time
the viscous stress tensor
the heat flux by conduction and/or radiation
and the diffusive flux of S
where
is the geopotential (J/kg),
the kinematic shear coefficient [L /T],
the bulk viscosity coefficient [L /T],
the diffusivity coefficient for heat [L /T],
the diffusivity coefficient for secondary
constituent [L /T],
the specific heat at constant pressure 
(J/kg K) and
the radiation flux.
Note that for P and T uniform (1.11) reduces to Fick's Law
The second law of thermodynamics requires that
such that the rate of change of total entropy with time in a closed system is never negative.