2.1 Seawater Properties

Step 1: Coulomb Force Between Ions in a Dielectric Medium

In a vacuum, the electrostatic force between two point charges $q_1$ and $q_2$ separated by distance $r$ is given by Coulomb's law. In a dielectric medium with relative permittivity $\epsilon_r$, the medium screens the interaction:

$$F_{\text{Coulomb}} = \frac{q_1 q_2}{4\pi \epsilon_0 \epsilon_r r^2}$$

Step 2: Dielectric Screening in Water

Water has a high dielectric constant $\epsilon_r \approx 80$ at 20 degrees C, arising from the alignment of polar H$_2$O molecules around ions. The interaction potential energy between Na$^+$ and Cl$^-$ in water compared to vacuum is reduced by a factor of 80:

$$U_{\text{ion-ion}}(r) = \frac{z_1 z_2 e^2}{4\pi \epsilon_0 \epsilon_r r} \quad \Rightarrow \quad \frac{U_{\text{water}}}{U_{\text{vacuum}}} = \frac{1}{\epsilon_r} \approx \frac{1}{80}$$

Step 3: Ion-Dipole Interaction Energy

A dissolved ion with charge $q = ze$ interacts with the permanent dipole moment$\mu$ of a water molecule. The interaction depends on the angle $\theta$ between the dipole axis and the ion-dipole separation vector:

$$U_{\text{ion-dipole}} = -\frac{ze\,\mu\cos\theta}{4\pi\epsilon_0 r^2}$$

Step 4: Thermal Average of Ion-Dipole Interaction

At finite temperature, the water dipoles are partially aligned by the ionic field but randomized by thermal fluctuations. Performing the Boltzmann-weighted angular average (Langevin function in the weak-field limit $ze\mu / 4\pi\epsilon_0 r^2 \ll k_BT$):

$$\langle U_{\text{ion-dipole}} \rangle = -\frac{z^2 e^2 \mu^2}{6(4\pi\epsilon_0)^2 k_B T \, r^4}$$

Step 5: Hydration Shell Energy and Ion Dissolution

For an ion to dissolve, the hydration energy (sum of ion-dipole interactions with the first coordination shell of $n$ water molecules at distance $r_0$) must exceed the lattice energy $U_{\text{lattice}}$. The Born solvation energy for transferring an ion of radius $a$ from vacuum to a medium with dielectric constant $\epsilon_r$ is:

$$\Delta G_{\text{solv}} = -\frac{z^2 e^2}{8\pi\epsilon_0 a}\left(1 - \frac{1}{\epsilon_r}\right)$$

Step 6: Debye-Huckel Screening in Electrolyte Solutions

In a solution with many ions (like seawater), each ion is surrounded by an "ionic atmosphere" of counterions that screens its electric field. The Debye-Huckel theory gives the screened potential around an ion as an exponentially decaying Coulomb potential:

$$\phi(r) = \frac{ze}{4\pi\epsilon_0\epsilon_r r}\,e^{-r/\lambda_D} \quad \text{where} \quad \lambda_D = \sqrt{\frac{\epsilon_0 \epsilon_r k_B T}{2 N_A e^2 I}}$$

Step 7: Debye Length in Seawater

The ionic strength of seawater (S = 35) is $I \approx 0.72$ mol/kg. Substituting$\epsilon_r = 80$, $T = 293$ K into the Debye length expression:

$$\lambda_D = \sqrt{\frac{(8.854\times10^{-12})(80)(1.381\times10^{-23})(293)}{2(6.022\times10^{23})(1.602\times10^{-19})^2(720)}} \approx 0.36 \text{ nm}$$

This extremely short Debye length (~3.6 Angstroms, less than one water molecule diameter) means that electrostatic interactions in seawater are screened over sub-nanometer distances, explaining why seawater behaves nearly ideally for many macroscopic thermodynamic properties despite its high ionic concentration.

Step 1: The Equation of State in Differential Form

The density of seawater $\rho$ is a state function of three variables: temperature $T$, salinity $S$, and pressure $p$. The total differential is:

$$d\rho = \left(\frac{\partial\rho}{\partial T}\right)_{S,p} dT + \left(\frac{\partial\rho}{\partial S}\right)_{T,p} dS + \left(\frac{\partial\rho}{\partial p}\right)_{T,S} dp$$

Step 2: Defining the Thermal Expansion Coefficient

The thermal expansion coefficient $\alpha$ quantifies how density changes with temperature at constant salinity and pressure. It is defined as:

$$\alpha = -\frac{1}{\rho}\left(\frac{\partial\rho}{\partial T}\right)_{S,p} \quad \Rightarrow \quad \left(\frac{\partial\rho}{\partial T}\right)_{S,p} = -\rho\,\alpha$$

For seawater, $\alpha$ is typically $1\text{--}3 \times 10^{-4}$ K$^{-1}$. Crucially,$\alpha$ increases with temperature and is near zero (or even negative) at low temperatures, explaining why salinity dominates density near the poles.

Step 3: Defining the Haline Contraction Coefficient

The haline contraction coefficient $\beta$ quantifies how density changes with salinity at constant temperature and pressure:

$$\beta = \frac{1}{\rho}\left(\frac{\partial\rho}{\partial S}\right)_{T,p} \quad \Rightarrow \quad \left(\frac{\partial\rho}{\partial S}\right)_{T,p} = \rho\,\beta$$

For seawater, $\beta \approx 7.5 \times 10^{-4}$ (g/kg)$^{-1}$. Unlike $\alpha$,$\beta$ is relatively constant across the range of oceanic temperatures and salinities.

Step 4: Isothermal Compressibility and Pressure Effects

The isothermal compressibility $\kappa$ captures the pressure dependence of density. It relates to the secant bulk modulus $K$ used in the UNESCO equation:

$$\kappa = \frac{1}{\rho}\left(\frac{\partial\rho}{\partial p}\right)_{T,S} = \frac{1}{K} \quad \Rightarrow \quad \rho(S,T,p) = \frac{\rho(S,T,0)}{1 - p/K(S,T,p)}$$

Step 5: Linearized Equation of State

Combining the definitions of $\alpha$ and $\beta$ with the differential form, and integrating from a reference state $(\rho_0, T_0, S_0)$ at surface pressure:

$$\frac{d\rho}{\rho} = -\alpha\,dT + \beta\,dS + \kappa\,dp$$

$$\rho \approx \rho_0\left[1 - \alpha(T - T_0) + \beta(S - S_0) + \kappa\,p\right]$$

This linearization is valid for small perturbations. For typical oceanic ranges ($\Delta T \sim 30$ K,$\Delta S \sim 5$ g/kg), the nonlinear terms become important, necessitating the full polynomial equation of state.

Step 6: The UNESCO Polynomial Equation of State

The full UNESCO (1983) equation of state is constructed by fitting high-precision laboratory measurements of seawater density. The surface density is a polynomial in $T$ and $S$:

$$\rho(S,T,0) = \rho_w(T) + A(T)\,S + B(T)\,S^{3/2} + C\,S^2$$

where $\rho_w(T)$ is the pure water density (a 5th-order polynomial in $T$), and $A(T)$, $B(T)$ are polynomials capturing the salinity dependence. The$S^{3/2}$ term arises from the Debye-Huckel theory of electrolyte solutions.

Step 7: Secant Bulk Modulus and In-Situ Density

The pressure effect is handled through the secant bulk modulus $K(S,T,p)$, itself a polynomial in all three variables. The in-situ density is then:

$$\rho(S,T,p) = \frac{\rho(S,T,0)}{1 - \frac{p}{K(S,T,p)}} \quad \text{with} \quad K = K_w(T) + K_0(S,T) + K'(S,T)\,p + K''(S,T)\,p^2$$

The full equation involves 41 empirical coefficients, fit to thousands of laboratory measurements with an accuracy of $\pm 0.009$ kg/m$^3$ over the oceanographic range.

Step 8: The Density Ratio and Double Diffusion Criterion

The relative importance of thermal and haline contributions to density stratification is captured by the density ratio $R_\rho$, which determines the susceptibility to double-diffusive instabilities:

$$R_\rho = \frac{\alpha\,\partial T/\partial z}{\beta\,\partial S/\partial z}$$

Salt fingering occurs when $1 < R_\rho < \tau^{-1}$ (warm salty over cool fresh), while diffusive convection occurs when $0 < R_\rho < 1$ (cool fresh over warm salty), where $\tau = \kappa_S/\kappa_T \approx 0.01$ is the diffusivity ratio.

Step 1: Definition of Specific Heat at Constant Pressure

The specific heat capacity at constant pressure is defined as the amount of heat required to raise the temperature of a unit mass of substance by one degree at constant pressure. From the first law of thermodynamics applied to a reversible process:

$$c_p = \left(\frac{\partial h}{\partial T}\right)_{p,S} = T\left(\frac{\partial s}{\partial T}\right)_{p,S}$$

where $h$ is specific enthalpy and $s$ is specific entropy.

Step 2: Relation Between $c_p$ and $c_v$

The difference between heat capacity at constant pressure and constant volume is derived using Maxwell relations. Starting from $h = u + p/\rho$ and using the cyclic relation:

$$c_p - c_v = -\frac{T}{\rho^2}\frac{\left(\frac{\partial\rho}{\partial T}\right)_{p}^2}{\left(\frac{\partial\rho}{\partial p}\right)_{T}} = \frac{T\alpha^2}{\rho\kappa}$$

For seawater, $c_p - c_v \approx 4$ J/(kg K), so $c_v \approx 3989$ J/(kg K) compared to$c_p \approx 3993$ J/(kg K) at S = 35, T = 20 degrees C. The small difference reflects the near-incompressibility of water.

Step 3: Pressure Dependence of $c_p$

How $c_p$ varies with pressure can be derived from a Maxwell relation. Starting from the Gibbs free energy $G(T, p, S)$ and using the equality of mixed partial derivatives:

$$\left(\frac{\partial c_p}{\partial p}\right)_{T,S} = -\frac{T}{\rho}\left[\alpha^2 + \left(\frac{\partial\alpha}{\partial T}\right)_{p,S}\right]$$

This shows that $c_p$ decreases with increasing pressure because the thermal expansion terms are positive. At 4000 dbar, $c_p$ is reduced by approximately 20 J/(kg K) compared to the surface value.

Step 4: Salinity Dependence from Mixing Thermodynamics

The heat capacity of a salt solution differs from pure water because dissolving salt disrupts the hydrogen bonding network. The apparent molar heat capacity $\phi_{C_p}$ of the dissolved salt is defined by:

$$c_p(S,T) = (1 - S/1000)\,c_p^w(T) + \frac{S}{M_{\text{salt}} \cdot 1000}\,\phi_{C_p}(S,T)$$

Since $\phi_{C_p}$ for NaCl is negative (the hydration shells of ions have lower heat capacity than bulk water), $c_p$ decreases with increasing salinity. This yields the approximately linear $c_p \approx 4217 - 3.72S$ dependence seen empirically.

Step 5: Connection to the Adiabatic Lapse Rate

The adiabatic lapse rate $\Gamma$ (temperature change per unit pressure for an isentropic displacement) is directly related to $c_p$ and the thermal expansion coefficient:

$$\Gamma = \left(\frac{\partial T}{\partial p}\right)_{\!s,S} = \frac{\alpha\,T}{\rho\,c_p}$$

This is derived from the Maxwell relation $(\partial T/\partial p)_s = (\partial v/\partial s)_p$combined with $\alpha = -(1/\rho)(\partial\rho/\partial T)_p$ and$c_p = T(\partial s/\partial T)_p$. It shows how $c_p$ controls the compressive heating rate in the deep ocean.

Step 6: The TEOS-10 Gibbs Function Approach

In the modern TEOS-10 framework, all thermodynamic properties including $c_p$ are derived from a single Gibbs potential $g(S_A, T, p)$. The specific heat is obtained as the second temperature derivative:

$$c_p = -T\,\frac{\partial^2 g}{\partial T^2}\bigg|_{p,S_A} \qquad \alpha = \frac{1}{v}\frac{\partial^2 g}{\partial T\,\partial p}\bigg|_{S_A} \qquad \kappa = -\frac{1}{v}\frac{\partial^2 g}{\partial p^2}\bigg|_{T,S_A}$$

This ensures thermodynamic self-consistency: all derived quantities ($\rho, c_p, \alpha, \beta, \kappa, \Gamma$, sound speed, etc.) are mutually consistent because they originate from a single potential function fit to the most accurate laboratory data available.