# The Azimuth Project Atmospheric and oceanic fluid dynamics (Rev #7, changes)

Showing changes from revision #6 to #7: Added | Removed | Changed

# Contents

## Idea

Atmospheric and oceanic fluid dynamics (AOFD) is about the large scale fluid dynamics of the atmosphere and the oceans and their interactions. With regard to computational fluid dynamics, this discipline concentrates on aspects of solutions of the Navier-Stokes equations that describe large scale effects on a rotating sphere with gravity. This discipline is essential for climate models, especially for general circulation models.

Atmosphere and oceans are strongly influenced by two effects:

• rotation leading to the Coriolis and centrifugal force,

• vertical stratification, that is the formation of vertical layers (strata).

## Details

We will denote three dimensional vectors in $\mathbb{R}^3$ with a “vector arrow” $\vec a$ and their Euclidean length with the same symbol without the arrow $a$. $\vec a \times \vec b$ is the cross product in 3D, and $\vec a \cdot \vec b$ is the Euclidean scalar product.

### Rotating Frame of Reference

It is convenient to describe the atmosphere and ocean from the reference frame that is determined by the surface of the earth, which is not an inertial reference frame due to earth’s rotation. This necessitates a reformulation of the Navier-Stokes equations, which are usually derived and stated with respect to an inertial reference frame.

Our setting consists of an Euclidean three dimensional space $\mathbb{R}^3$ with Cartesian coordinates, which constitutes an inertial reference frame which we denote with $I$, and a sphere $S \subset \mathbb{R}^3$ that is rotating at a constant angular velocity, which is described by the (pseudo)vector $\vec{\omega}$. We denote the rotating reference frame with R.

A vector $\vec{c}$, that is constant in R, is rotating in I, and we have for the rate of change in I:

$\frac{d \vec{c}}{d t} = \vec \omega \times \vec{c}$

We can use this to relate the velocity $\vec{v_I}$ of a particle in the reference frame I to the velocity $\vec{v_R}$ with respect to the rotating reference frame, with $\vec{r}$ denoting the position of the particle:

$(\frac{d \vec{r}}{d t})_I = \vec{v_I} = (\frac{d \vec{r}}{d t})_R + \vec \omega \times \vec{r}$

Differentiating this equation one more time results in the equation relating the accelerations of the two reference frames:

$(\frac{d \vec{v_R}}{d t})_R = \vec{a_R} = (\frac{d \vec{v_I}}{d t})_I - 2 \vec \omega \times \vec{v_R} - \vec \omega \times (\vec \omega \times \vec r)$

The two additional terms on the right describe the Coriolis force and the centrifugal force, (which should be called accelerations instead of forces in this context, but usually aren’t).

#### Centrifugal Force from a Potential

Let $\vec{r}_{\perp}$ denote the perpendicular distance from the axis of rotation, then we have

$\vec \omega \times \vec{r} = \vec \omega \times \vec{r}_{\perp}$

Using the formula…

$\vec \omega \times (\vec \omega \times \vec{r}_{\perp}) = (\vec{\omega} \cdot \vec{r}_{\perp}) \vec{\omega} - (\vec{\omega} \cdot \vec{\omega}) \vec{r}_{\perp}$

…we see that the first term is zero and we get for the centrifugal acceleration

$\vec{a}_{ce} = \omega^2 \vec{r}_{\perp} = - \nabla \phi_{ce}$

with the centrifugal potential

$\phi_{ce} = - \frac{1}{2} \omega^2 \r^2_{\perp}$

This reformulation enables us to incorporate the centrifugal force into any kind of potential that is included in equations in inertial systems. As an example, we will look at the Cauchy momentum equation, which is Newton’s second law for a continuous mass distribution:

$\rho \frac{D v}{d t} = \nabla \cdot {\sigma} + \nabla \phi$

Instead of adding $\vec f$ representing all external forces, we have assumed that all external forces can be derived from a potential $\phi$. Therefore we can formulate this equation in a rotating reference frame by including the Coriolis force and adapting the potential $\phi$ to $\phi'$ including the centrifugal force:

$\rho \frac{D v}{d t} + 2 \vec \omega \times \vec{v} = \nabla \cdot {\sigma} + \nabla \phi'$

#### Spherical Coordinates

We denote spherical coordinates by $(r, \lambda, \theta)$

See

#### The Primitive Equations

The primitive equations are the Navier-Stokes equations on the rotating sphere with three simplifications often used in climate science:

a) the hydrostatic approximation

b) the shallow-fluid approximation

#### Cartesian Approximations

Cartesian Approximations formulate approximate equations of motion of the primitive equations in the tangent plane of a fixed point $(r_0, \lambda_0, \theta_0)$ on the sphere. The simplest one is the f-plane approximation. We use a Cartesian coordinate system $(x, y, z)$ with x being the displacement to the east, y the displacement to the north and z being the height with respect to the surface of the sphere, that is…

$r = r_{sphere} + z$

…with $r_{sphere}$ being the constant radius of the sphere.

We use the following approximation in the tangent plane:

$(x, y, z) \approx (r_{sphere} \lambda \cos{\theta}, r_{sphere} (\theta - \theta_0), z)$

The components of the rotation vector $\vec \omega$ are assumed to be fixed in this approximation:

$\omega_x = 0 \; , \; \omega_y = \omega \cos{(\theta_0)} \; , \; \omega_z = \omega \sin{(\theta_0)}$

We denote the velocities in the tangent plane with $(u, v, w)$. We write

$f_0 = 2 \omega \sin{(\theta_0)}$

The momentum equations of a flow become with this approximation:

$\frac{D u}{D t} - f_0 v = - \frac{1}{\rho} \frac{\partial p}{\partial x}$
$\frac{D v}{D t} + f_0 u = - \frac{1}{\rho} \frac{\partial p}{\partial y}$

and

$\frac{D v}{D t} = - \frac{1}{\rho} \frac{\partial p}{\partial y} - g$

The beta-plane approximation takes further into account, that the Coriolis force may vary from point to point and introduces a linear approximation

$f = f_0 + \beta y$

into the above equations, replacing $f_0$ with $f$.

#### Boussinesq Approximation

The Boussinesq equations? are useful for the approximation of large scale flows in oceans.

## References

• Geoffrey K. Vallis: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation (Cambridge University Press 2006)