6.5 Ocean Modeling

Step 1: Full Navier-Stokes Momentum Equation

Begin with the compressible Navier-Stokes momentum equation where density appears in every term:

$$\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{u}$$

Step 2: Decompose Density into Reference and Perturbation

Write density as a constant reference value plus a small perturbation: $\rho = \rho_0 + \rho'$ where $|\rho'| \ll \rho_0$. Similarly decompose pressure into hydrostatic and dynamic parts:

$$p = p_0(z) + p', \quad \text{where } \frac{dp_0}{dz} = -\rho_0 g$$

Step 3: Substitute and Neglect Small Terms

Substituting into the momentum equation and dividing by $\rho_0$. In the inertial term $\rho D\mathbf{u}/Dt$, replace $\rho$ by $\rho_0$ since $\rho'/\rho_0 \ll 1$. However, in the gravity term, keep $\rho'$ because it multiplies $g$:

$$\rho_0 \frac{D\mathbf{u}}{Dt} = -\nabla p' - \rho' g \hat{z} + \mu \nabla^2 \mathbf{u}$$

Step 4: Resulting Boussinesq Equations

Dividing through by $\rho_0$ yields the Boussinesq momentum equation. The buoyancy $b = -g\rho'/\rho_0$ drives vertical motion while the continuity equation simplifies to incompressible form:

$$\frac{D\mathbf{u}}{Dt} = -\frac{1}{\rho_0}\nabla p' + b\hat{z} + \nu \nabla^2 \mathbf{u}, \quad \nabla \cdot \mathbf{u} = 0$$

Step 5: Physical Justification

In the ocean, $\rho'/\rho_0 \sim 0.01$ (density varies by ~1% from surface to abyss), so neglecting $\rho'$ in inertial terms introduces <1% error. But in the buoyancy term, even small $\rho'$ multiplied by $g \approx 9.81$ m/s$^2$ produces the dominant vertical forcing that drives ocean circulation.

Step 1: Reynolds Decomposition

Decompose each variable into a time-mean (overbar) and turbulent fluctuation (prime): $u = \bar{u} + u'$, $v = \bar{v} + v'$, $p = \bar{p} + p'$. By definition, $\overline{u'} = 0$.

$$u_i = \bar{u}_i + u_i', \quad \overline{u_i'} = 0$$

Step 2: Substitute into the Momentum Equation

Insert the decomposition into the Boussinesq Navier-Stokes equation and consider the nonlinear advection term $u_j \partial u_i / \partial x_j$:

$$(\bar{u}_j + u_j')\frac{\partial (\bar{u}_i + u_i')}{\partial x_j} = \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} + \bar{u}_j \frac{\partial u_i'}{\partial x_j} + u_j' \frac{\partial \bar{u}_i}{\partial x_j} + u_j' \frac{\partial u_i'}{\partial x_j}$$

Step 3: Time-Average and Extract Reynolds Stress

Taking the time average, terms linear in fluctuations vanish ($\overline{u'} = 0$). The quadratic fluctuation term survives as the Reynolds stress tensor. Using the continuity equation $\partial u_j'/\partial x_j = 0$:

$$\overline{u_j' \frac{\partial u_i'}{\partial x_j}} = \frac{\partial}{\partial x_j}\overline{u_i' u_j'} \equiv -\frac{1}{\rho_0}\frac{\partial \tau_{ij}^R}{\partial x_j}$$

Step 4: The RANS Equation

The resulting Reynolds-averaged momentum equation has the same form as the original but with an additional Reynolds stress divergence term requiring parameterisation (closure):

$$\frac{\partial \bar{u}_i}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho_0}\frac{\partial \bar{p}}{\partial x_i} + \bar{b}\delta_{i3} + \nu \nabla^2 \bar{u}_i - \frac{\partial \overline{u_i' u_j'}}{\partial x_j}$$

Step 5: Eddy Viscosity Closure

The closure problem is addressed by parameterising Reynolds stresses using an eddy viscosity $A$ (analogous to molecular viscosity but orders of magnitude larger), yielding the form used in ocean models:

$$-\overline{u_i' u_j'} = A_H \left(\frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i}\right), \quad \text{with } A_H \sim 10^2\text{--}10^4 \;\text{m}^2/\text{s}$$