Momentum#

This section is aimed at describing the main terms in the momentum equations which are key for numerical modeling. For derivations of these equations, see one of the sources in the references, such as Chapter 7 of Stewart [2008].

Momentum Overview#

Under construction

The Momentum Equations#

The momentum equations describe how momentum change at any given location in the ocean given the forces that act on it. They can be written succintly in vector notation as:

\[\frac{D\textbf{v}}{Dt} = - \frac{1}{\rho}\nabla p - 2\boldsymbol\Omega\times\textbf{v} + \textbf{g} + \textbf{F}_r\]

In this equation, known as the Navier-Stokes equation, the left-hand side is the total derivative of \(\textbf{v}\), and the right-hand side terms are the pressure gradient force (\(-1/\rho \nabla p\)), the Coriolis force (\(2\boldsymbol\Omega\times\textbf{v}\)), gravity (\(\textbf{g}\)), and frictional forces (i.e. viscosity, \(\textbf{F}_r\)). Each of these terms is describe in the following subsections.

The vector notation for these equations can be expanded as

\[\begin{split}\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} &= - \frac{1}{\rho}\frac{\partial p}{\partial x} - 2\Omega v \text{sin}\theta + F_x\\ \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} &= - \frac{1}{\rho}\frac{\partial p}{\partial y} - 2\Omega u \text{sin}\theta + F_y\\ \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} &= - \frac{1}{\rho}\frac{\partial p}{\partial z} + F_z\\\end{split}\]

Note that the small vertical component of the Coriolis has been removed from …

The Pressure Gradient Force#

Under construction

The Coriolis Force#

Under construction

Frictional Forces#

Under construction

The Boussinesq Approximation#

The Boussinesq approximation for momentum is that sea water is, in effect, incompressible. Put into symbols, this means that

\[\frac{1}{\rho} \frac{D \rho}{D t} = 0\]

This Boussinesq approximation is often used when constructing numerical ocean models with fixed grids because the grid cells will not change volume due to changes in density. This is an important consideration because in the real ocean, variations in density do have a measureable effect on ocean volume. In fact, changes in density resulting primarily from increased ocean heat content are a major contributor to contempory sea level rise [Frederikse et al., 2020]. In effect, hen computing density-driven variations in sea level from ocean state estimates, it’s necessary to apply a correction a posteriori [Greatbatch, 1994].

The Continuity Equation#

Applying the conservation equation to momentum (i.e. the conservation of mass in a given volume) results in:

\[\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0\]

This equation is known as the continuity equation.

Under the Boussinesq approximation, this is simplified to:

\[\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0\]

Or, put another way, the flow into a given region but be equal to the flow out.