2.2 Temperature & Salinity

Step 1: Entropy as a State Variable

For a parcel of seawater undergoing an adiabatic (isentropic) and isohaline displacement, the specific entropy $\eta$ remains constant: $d\eta = 0$. The entropy is a function of temperature, salinity, and pressure:

$$d\eta = \left(\frac{\partial\eta}{\partial T}\right)_{p,S} dT + \left(\frac{\partial\eta}{\partial p}\right)_{T,S} dp = 0$$

Step 2: Identifying the Partial Derivatives

The first partial derivative relates to the heat capacity: $(\partial\eta/\partial T)_{p,S} = c_p/T$. For the second, we use the Maxwell relation derived from the Gibbs free energy$dG = -\eta\,dT + v\,dp + \mu_S\,dS$:

$$\left(\frac{\partial\eta}{\partial p}\right)_{T,S} = -\left(\frac{\partial v}{\partial T}\right)_{p,S} = -\frac{\alpha}{\rho}$$

where $v = 1/\rho$ is specific volume and $\alpha = -(1/\rho)(\partial\rho/\partial T)_{p,S}$ is the thermal expansion coefficient.

Step 3: The Adiabatic Lapse Rate

Substituting into the isentropic condition $d\eta = 0$ and solving for $dT/dp$:

$$\frac{c_p}{T}\,dT - \frac{\alpha}{\rho}\,dp = 0 \quad \Rightarrow \quad \Gamma \equiv \left(\frac{\partial T}{\partial p}\right)_{\!\eta,S} = \frac{\alpha\,T}{\rho\,c_p}$$

This is the adiabatic lapse rate $\Gamma$ -- the rate of temperature increase per unit pressure increase for a parcel displaced adiabatically downward. For typical deep-ocean conditions ($\alpha \approx 1.5 \times 10^{-4}$ K$^{-1}$, $T \approx 275$ K,$\rho \approx 1028$ kg/m$^3$, $c_p \approx 3990$ J/(kg K)):

$$\Gamma \approx \frac{(1.5\times10^{-4})(275)}{(1028)(3990)} \approx 1.0 \times 10^{-8} \text{ K/Pa} \approx 0.10 \text{ °C per 1000 dbar}$$

Step 4: Defining Potential Temperature

The potential temperature $\theta$ is obtained by integrating the lapse rate from the in-situ pressure $p$ down to a reference pressure $p_r$ (usually 0 dbar, the surface). It represents the temperature a parcel would have if moved adiabatically and isohalinally to $p_r$:

$$\theta(S, T, p, p_r) = T - \int_{p_r}^{p} \Gamma(S, T', p')\,dp'$$

Step 5: Numerical Integration (Runge-Kutta Method)

Because $\Gamma$ depends on $T$ and $p$ (and weakly on $S$), the integral must be evaluated numerically. The standard approach uses a 4th-order Runge-Kutta scheme over the pressure interval $[p_r, p]$. Denoting $\Delta p = p - p_r$:

$$k_1 = \Gamma(S, T, p)\,\Delta p$$

$$k_2 = \Gamma(S, T - k_1/2, p - \Delta p/2)\,\Delta p$$

$$k_3 = \Gamma(S, T - k_2/2, p - \Delta p/2)\,\Delta p$$

$$k_4 = \Gamma(S, T - k_3, p_r)\,\Delta p$$

$$\theta = T - \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)$$

Step 6: Physical Interpretation and Magnitude

The difference between in-situ temperature and potential temperature grows with depth. At the ocean floor (~4000 -- 6000 m), this difference is significant:

$$T - \theta \approx \Gamma \cdot p \approx (0.15 \text{ °C}/1000\text{ dbar}) \times 4000\text{ dbar} \approx 0.6\text{ °C}$$

Without this correction, the deep ocean would appear to have a slight temperature increase with depth (due to compressive heating), masking the true stratification. Potential temperature removes this artifact, revealing whether the water column is truly stably stratified.

Step 7: Why $\alpha$ Near Zero Matters at Low Temperature

Since $\Gamma = \alpha T / (\rho c_p)$, the lapse rate approaches zero as $\alpha \to 0$near the temperature of maximum density. For very cold bottom waters ($T < 2$ degrees C),$\alpha$ is small, making $\Gamma$ small and the distinction between $T$ and$\theta$ less important. However, the nonlinear dependence of $\alpha$ on pressure means that at high pressures $\alpha$ is larger even at low temperatures, so the correction remains non-negligible in the deep ocean.

$$\alpha(T, S, p) \approx \alpha(T, S, 0) + \frac{\partial\alpha}{\partial p}\bigg|_{T,S} \cdot p$$

Step 1: Conservation of Heat and Salt During Mixing

When two water masses mix, both heat (proportional to temperature for fixed $c_p$) and salt are conserved. Consider mixing water mass 1 (mass $m_1$, properties $\theta_1, S_1$) with water mass 2 (mass $m_2$, properties $\theta_2, S_2$). Conservation requires:

$$m_1 c_p \theta_1 + m_2 c_p \theta_2 = (m_1 + m_2) c_p \theta_{\text{mix}}$$

$$m_1 S_1 + m_2 S_2 = (m_1 + m_2) S_{\text{mix}}$$

Step 2: The Mixing Fraction Parameterization

Defining the mixing fraction $f = m_1/(m_1 + m_2)$ (the mass fraction of water mass 1), we can write the mixed properties as linear combinations:

$$\theta_{\text{mix}} = f\,\theta_1 + (1-f)\,\theta_2 \qquad S_{\text{mix}} = f\,S_1 + (1-f)\,S_2$$

As $f$ varies from 0 to 1, the point $(\theta_{\text{mix}}, S_{\text{mix}})$ traces a straight line in T-S space connecting $(\theta_1, S_1)$ and $(\theta_2, S_2)$.

Step 3: Eliminating $f$ to Get the Mixing Line Equation

Solving the salinity equation for $f$ and substituting into the temperature equation:

$$f = \frac{S_{\text{mix}} - S_2}{S_1 - S_2} \quad \Rightarrow \quad \theta_{\text{mix}} = \theta_2 + \frac{\theta_1 - \theta_2}{S_1 - S_2}(S_{\text{mix}} - S_2)$$

$$\theta = \theta_2 + \frac{\Delta\theta}{\Delta S}(S - S_2) \quad \text{where} \quad \frac{\Delta\theta}{\Delta S} = \frac{\theta_1 - \theta_2}{S_1 - S_2}$$

This is the equation of a straight line with slope $\Delta\theta/\Delta S$ in T-S space.

Step 4: Three-Component Mixing (Triangular Regions)

For three water masses mixing simultaneously, the mixed point lies within the triangle formed by the three end-members in T-S space. Conservation gives three equations with three unknowns ($f_1, f_2, f_3$):

$$\theta_{\text{mix}} = f_1\theta_1 + f_2\theta_2 + f_3\theta_3$$

$$S_{\text{mix}} = f_1 S_1 + f_2 S_2 + f_3 S_3$$

$$f_1 + f_2 + f_3 = 1$$

This linear system has a unique solution for $(f_1, f_2, f_3)$, allowing quantitative determination of the fraction of each water mass present at any point in the ocean.

Step 5: Cabelling -- The Nonlinear Density Effect

While T-S properties mix linearly, density does not because the equation of state$\rho(\theta, S)$ is nonlinear. The density of the mixture differs from the linear interpolation of the source densities. Expanding $\rho$ to second order around the midpoint:

$$\rho_{\text{mix}} - \bar{\rho} \approx \frac{f(1-f)}{2}\left[\frac{\partial^2\rho}{\partial\theta^2}(\Delta\theta)^2 + 2\frac{\partial^2\rho}{\partial\theta\partial S}\Delta\theta\,\Delta S + \frac{\partial^2\rho}{\partial S^2}(\Delta S)^2\right]$$

where $\bar{\rho} = f\rho_1 + (1-f)\rho_2$ is the linearly interpolated density. Since$\partial^2\rho/\partial\theta^2 > 0$ (density is concave upward in temperature), mixing along isotherms always produces water denser than the linear average -- this is cabelling. The effect is maximized at $f = 0.5$ (equal mixing).

Step 6: The Cabelling Coefficient

The cabelling coefficient $C_b$ quantifies the densification due to mixing. It is defined as:

$$C_b = \frac{\partial^2\rho/\partial\theta^2}{\rho} = \frac{\partial\alpha}{\partial\theta}\bigg|_{p,S}$$

Typical values are $C_b \approx 8 \times 10^{-6}$ K$^{-2}$. For two water masses differing by $\Delta\theta = 10$ degrees C mixing equally ($f = 0.5$), the density increase is approximately:

$$\Delta\rho_{\text{cab}} \approx \frac{\rho\,C_b}{8}(\Delta\theta)^2 \approx \frac{1025 \times 8\times10^{-6}}{8}(10)^2 \approx 0.1 \text{ kg/m}^3$$

Step 7: Thermobaricity -- Pressure-Dependent Mixing

The thermobaric coefficient $T_b$ captures how the thermal expansion coefficient changes with pressure. This means that mixing lines in T-S space produce different density anomalies at different pressures:

$$T_b = \frac{\partial\alpha}{\partial p}\bigg|_{\theta,S} = \frac{\partial^2 v}{\partial\theta\,\partial p}\bigg|_{S}$$

Thermobaricity is important in the Southern Ocean, where cold Antarctic waters become relatively denser at great pressure, potentially triggering deep convection events when perturbations displace parcels to depths where their thermobaric density advantage causes them to sink rapidly.

Step 1: The Problem with Potential Temperature

Potential temperature $\theta$ is defined by integrating the adiabatic lapse rate and is conserved for a single parcel undergoing adiabatic displacement. However, when two parcels with different $(S, \theta)$ properties mix, $\theta$ is not exactly conserved because the specific heat capacity $c_p(S, T, p)$ varies. The heat content is:

$$Q = m \cdot c_p(S, T, p) \cdot T \neq m \cdot c_p^{\text{const}} \cdot \theta$$

Step 2: Quantifying the Non-Conservation Error

When water mass 1 ($\theta_1, S_1$) mixes with water mass 2 ($\theta_2, S_2$), the enthalpy is exactly conserved but $\theta$ is not. The error in treating $\theta$as conservative is proportional to the variation of $c_p$:

$$\delta\theta_{\text{error}} \sim \frac{\Delta c_p}{c_p} \cdot \Delta\theta \sim \frac{c_p(\theta_1, S_1) - c_p(\theta_2, S_2)}{c_p} \cdot (\theta_1 - \theta_2)$$

Since $c_p$ varies by about 5% across the full oceanic range, errors up to 0.05 degrees C can accumulate. While small for individual mixing events, these errors compound along multi-step mixing paths in the global overturning circulation.

Step 3: Specific Enthalpy as the Conserved Quantity

The quantity that is truly conserved during mixing at constant pressure is the specific enthalpy$h$. From the first law, for an adiabatic process at constant pressure and composition:

$$dh = \frac{dp}{\rho} + T\,d\eta + \mu_S\,dS = \frac{dp}{\rho} \quad (\text{for } d\eta = 0, dS = 0)$$

For mixing at constant $p$: $(m_1 + m_2)\,h_{\text{mix}} = m_1\,h_1 + m_2\,h_2$ is exact.

Step 4: Defining Conservative Temperature

Conservative Temperature $\Theta$ is defined so that it is proportional to the potential enthalpy $h^0 = h(S_A, \theta, p_r = 0)$ -- the enthalpy referenced to the sea surface:

$$\Theta = \frac{h^0(S_A, \theta)}{c_p^0} = \frac{h(S_A, \theta, 0)}{c_p^0}$$

where $c_p^0 = 3991.868$ J/(kg K) is a constant reference heat capacity chosen so that$\Theta$ has the same units and approximate magnitude as $\theta$.

Step 5: Why $\Theta$ Is Conservative

Since $c_p^0$ is a constant and enthalpy is conserved during mixing:

$$(m_1 + m_2)\,c_p^0\,\Theta_{\text{mix}} = m_1\,c_p^0\,\Theta_1 + m_2\,c_p^0\,\Theta_2$$

$$\Rightarrow \quad \Theta_{\text{mix}} = f\,\Theta_1 + (1-f)\,\Theta_2 \quad \text{(exactly)}$$

This is exact to the extent that $h^0$ is conserved, which is accurate to within the neglected terms of pressure work during mixing (of order $10^{-12}$ degrees C). By contrast, the non-conservation of $\theta$ during mixing can be up to $10^{-2}$ degrees C.

Step 6: Computing $\Theta$ from $\theta$ and $S_A$

The relationship between $\Theta$ and $\theta$ is obtained by evaluating the Gibbs function $g(S_A, T, p)$ at $p = 0$. The enthalpy at the surface is:

$$h^0 = g - T\frac{\partial g}{\partial T}\bigg|_{p=0} \quad \Rightarrow \quad \Theta = \frac{1}{c_p^0}\left[g(S_A, \theta, 0) - (\theta + 273.15)\frac{\partial g}{\partial T}(S_A, \theta, 0)\right]$$

In practice, this is evaluated using a 75-term polynomial approximation that is accurate to within $\pm 2 \times 10^{-6}$ degrees C. The difference $\Theta - \theta$ reaches a maximum of about 0.05 degrees C for cold, fresh Antarctic waters.

Step 7: The Heat Transport Equation in Terms of $\Theta$

The advantage of Conservative Temperature becomes clear in the ocean heat transport equation. Using $\Theta$, the advection-diffusion equation for heat becomes:

$$\rho_0 c_p^0 \left(\frac{\partial\Theta}{\partial t} + \mathbf{u}\cdot\nabla\Theta\right) = \nabla\cdot(\rho_0 c_p^0 \kappa \nabla\Theta) + \dot{Q}$$

Because $c_p^0$ is a constant, it factors out cleanly. With potential temperature, the variable $c_p(\theta, S)$ would need to remain inside the derivatives, creating spurious sources and sinks of heat in numerical ocean models. The TEOS-10 standard recommends using $\Theta$ as the temperature variable in all ocean models and analyses.