8.2 Thermodynamic and closure of the equations
8.2.2 Thermodynamic state relation
Gibbs’ phase rule and thermodynamic potentials
Gibbs’ phase rule state the the degree of freedomF, i.e. the number of independent intensive variable that completely determines the thermodynamic properties of the system (such as pressure, temperatur and salinity), is equal to
F =C−P + 2, (8.28)
where C is the number of components of the system and P is the number of phases in thermodynamic equilibrium with each other. The ocean consist of two components, salt and freshwater, and consist of one phase, liquid. Therefor, the number of independent variables are three. There are many different independent variables that can be used. These form the basis for the representation of the
thermodynamic relations describing the physical nature of the fluid. The four most common choices of the intensive variables are
ρ−1, s, S
, (p, s, S), ρ−1, T, S
, (p, T, S). (8.29) To represent the thermodynamic state of the fluid, there exist a thermodynamic function that determines the thermodynamic properties of the fluid. This function is often called the thermodynamic potential. It follows from (8.22) and (8.23) that the internal energy per unit massis the thermodynamic potential when (ρ−1, s, S) are chosen as the independent intensive variables. Therefor, it also follows that the equations of state to this potential are given by
T = ∂
∂s
S,ρ
, p=− ∂
∂ρ−1
S,s
, ∆µ= ∂
∂S
s,ρ
. (8.30)
From Euler’s homogeneous function theorem it follows that the internal energy per unit mass can be written as
=T s−pρ−1+µSS+µW (1−S). (8.31) This equation is often called the Euler’s identity. By subtracting the first law of thermodynamics, (8.22), from the total differential of Euler’s identity, (8.31), result in the Gibbs-Durham relation
ρ−1dp−sdT =SdµS+ (1−S)dµW. (8.32) Instead of using (ρ−1, s, S) as independent variables to describe the thermodynamic properties of the fluid, we will choose (p, T, S) as independent variables. The question is now: Which thermodynamic potential is a function of these variables, and what is the equation of state? By adding the differential d(−T s+pρ−1) on both side of (8.22), and define the free enthalpy per unit mass
g =−T s+pρ−1, (8.33)
(8.22) becomes
dg=−sdT +ρ−1dp+ ∆µdS, (8.34)
which states that the free enthalpy per unit mass is the thermodynamic potetial with (p, T, S) as independent variables. From the chain rule it follows that the equations of state is
s=− ∂g
∂T
S,p
, ρ−1 = ∂g
∂p
S,T
, ∆µ= ∂g
∂S
T ,p
. (8.35)
8.2. THERMODYNAMIC AND CLOSURE OF THE EQUATIONS 117 Since the equations of state will be a function of (p, T, S), these equations can be used to transform the entropy equation, (8.26) to prognostic equations for temperature and mass density. The Euler’s identety for the free enthalpy per unit mass is given by substitute the Euler’s identety for the internal energy per unit mass, (8.31) in to the expression for the free enthalpy, (8.33),
g =µSS+µW (1−S). (8.36)
The temperature equation
From the chain rule it follows that the evolution equation for the entropys(p, T, S) is
p,S =cp is the spesific heat capasity. By using the Maxwell relation ∂s
hereβT is the thermal expansion coefficient, (8.26) and (8.8) the entropy equation, (8.37) becomes an evolution equation for the temperature,
ρ cpdT Under adiabitic conditions, the stress tensor reduces to (8.27) and the temperature equation reduces to
This means that the temperature and pressure are related by dT = T βρ cT
pdp. Hence, the temperature is not a conserved quantity under adiabatic conditions, since change in pressure leads to change in temperature. Later we will introduce a potential temperature, which are a conserved quantity under adiabatic conditions.
It will be more practical to deal with potential temperature instead of temperature.
The thermodynamic mass density equation
From the chain rule it follows that the evolution equation for the entropyρ−1(p, T, S) is
where βT, βp and βS are the thermal expansion coefficient, the compresibility coeffisient and the salinity contraction coefficient respectively. By using (8.39) and (8.8), (8.41) becomes an evolution equation for the mass density,
dρ
dt =ρeκdp dt−βT
cp (p∇ ·u−σ :D− ∇ ·Q)− βT
cp ∆µ+T ∂s
∂S
p,T
! +βS
!
∇·JS, (8.42) whereeκ=βp−βTΓ is the adiabatic compressibility coefficient and Γ =βTT /cpρis the adiabatic temperature gradient. The adiabatic compressibility coefficient can also be written as
eκ= 1 ρ
∂ρ
∂p
s,S
= 1 ρ
∂ρ
∂p
T ,S
+ Γ ρ
∂ρ
∂T
p,S
. (8.43)
The relation between the adiabatic compressibility coefficient and the sound ve-locity is given by
c2 = 1
ρeκ (8.44)
It should be notet that (8.42) is not an equation for mass conservation, but an equation for the energy conservation in the fluid.
Heat flux
The total heat fluxQgiven in all the energy equations is often decomposed into one heat flux due to conductivityqcond, and one chemical heat fluxqchem due to change in salt and freshwater consentration. It is well known from thermodynamics that the change in enthalpyh is equal to the amount of heat added to the system due to chemical prosesses, when the pressure is constant. By adding the differential d(pρ−1) on both side of (8.22), and define the enthalpy per unit mass as
h=+pρ−1, (8.45)
(8.22) becomes
dh=T ds+ρ−1dp+ ∆µdS. (8.46)
The enthalpy can be spitt into two parts, one part hS due to the salt, and one parthW due to the fresh water. The total enthalpy is given by the mass-weighted mean
h=hSS+hW(1−S). (8.47)
The chemical heat transport through a surface elementdA is therefor given by qchem·dA= (hsJS+hwJW)·dA. (8.48)
8.2. THERMODYNAMIC AND CLOSURE OF THE EQUATIONS 119 Since the diffusive fluxes satisfies
JS+JW =0, (8.49)
it follows that the chemical heat flux is
qchem = ∆hJS, (8.50)
where ∆h = hS−hW is the partial enthalpy difference. From equation (8.47) it follows that the partial enthalpy difference is given by the thermodynamic state
∆h= ∂h
∂S
p,T
. (8.51)
In order to describe the chemical heat flux with (p, T, S) as independent variables, we need to find the relation between the free enthalpy and enthalpy. By combi-nating (8.33) and (8.45), the relation between the free enthalpy and the enthalpy becomes
g =h−T s, (8.52)
therefor, it follows that (8.51) can be written in terms of free enthalpy and entropy as
∂h
∂S
p,T
= ∂g
∂S
p,T
+ ∂s
∂S
p,T
= ∆µ+T ∂s
∂S
p,T
. (8.53)
The chemical heat flux can then be written as qchem= ∆µ+T
∂s
∂S
p,T
!
JS. (8.54)