# Deriving the Horizontal Component of Coriolis on a Sphere

On a rotating sphere the Coriolis acceleration in the direction of motion is always zero. It remains a horizontal and vertical component. The vertical component is also called the Eötvös Effect. Often we only need the horizontal component, which always acts to the right on the northern hemisphere and to the left in the southern hemisphere with respect to the direction of motion.

The equation for the horizontal component of the Coriolis acceleration on a sphere is simply:

(1)
with
where'
 $c$ ' =' 'Horizontal component of the Coriolis acceleration, acting perpendicular to the direction of motion. Positive values are acting to the right. $v$ ' =' 'Speed of the moving object with respect to the surface of the sphere $\Omega$ ' =' 'Angular velocity of the sphere $T$ ' =' 'Rotation period $\lambda$ ' =' 'Latitude of the moving object

On this page I derive how this equation follows from the more general equation for the Coriolis acceleration:

(2)
where'
 $\vec c$ ' =' 'Coriolis acceleration in 3D space $\vec v$ ' =' 'Velocity vector in 3D space $\vec \Omega$ ' =' 'Angular rotation vector. The vector is pointing in the direction of the rotation axis and its magnitude is the angular rotation Ω = 2π/T $T$ ' =' 'Rotation period

## Derivation

We can use vector algebra to get the special equation (1) for the surface of a rotating sphere from the general eqaution (2). Here are the steps:

### Local Coordinate Sytem

First we define a local coordinate system for the location of the moving object. Due to rotational symmetry, the longitude does not matter, so for simplicity it is set to 0. We can always make a rotation transformation to get zero longitude.

The local coordinate system consists of 3 unit vectors in sphere centered cartesian coordinates, pointing up, north and east at the location of the moving object. The up-vector is a unit vector pointing from the center of the sphere to the location of the moving object. The east-vector points always in the y-direction of the sphere coordinate system and the north-vector is perpendicular to the up- and east-vectors:

(3)
where'
 $\hat u$ ' =' 'unit vector pointing up at the location of the moving object with respect to the surface of the sphere $\hat e$ ' =' 'unit vector pointing due east $\hat n$ ' =' 'unit vector pointing due north

### Velocity

The velocity-vector depends on the azimuth angle α, which is the angle between the north-vector and the direction of motion along the surface of the sphere. It can be constructed from the local coordinate system. Its magnitude is v. It is expressed in sphere centered cartesian coordinates as follows:

(4)
where'
 $\vec v$ ' =' 'velocity-vector, always parallel to the surface of the sphere at the location of the moving object, i.e. no local up-component $\alpha$ ' =' 'azimuth angle, i.e. direction of motion with respect to the north-vector $\hat n$ in the clockwise direction as viewed in the down direction $-\hat u$ $\lambda$ ' =' 'latitude of the moving object $v$ ' =' 'speed, i.e. magnitude of the velocity vector $\hat u, \hat n, \hat e$ ' =' 'local coordinate system

### Direction of the horizontal Coriolis Acceleration Component

Because the Coriolis acceleration is the vector product of the rotation axis of the sphere and the velocity vector, it acts always perpendicular to this 2 vectors. That means it has no component in the direction of motion. To calculate the local horizontal component we need a vector parallel to the surface pointing to the right with respect to the direction of motion. This right-vector can be constructed using the local coordinate system and the azimuth angle α similarly to the velocity vector:

(5)
where'
 $\hat r$ ' =' 'unit vector pointing parallel to the surface of the sphere pointing to the right with respect to the direction of motion $\vec v$ $\alpha$ ' =' 'azimuth angle, i.e. direction of motion with respect to the north-vector $\hat n$ in the clockwise direction as viewed in the down direction $-\hat u$ $\lambda$ ' =' 'latitude of the moving object $\hat u, \hat n, \hat e$ ' =' 'local coordinate system

### General Coriolis Acceleration Vector

Now that we know the veloctiy vector and the angular rotation vector is given as $\vec \Omega = \Omega \cdot ( 0, 0, 1 )$, we can calculate the Coriolis acceleration in 3D space using the equation (2):

(6)
where'
 $\vec c$ ' =' 'Coriolis acceleration in 3D space in sphere centered cartesian coordinates $\vec v$ ' =' 'velocity vector in sphere centered cartesian coordinaes $\vec \Omega$ ' =' 'angular veloctiy vector in sphere centered cartesian coordinates $v$ ' =' 'speed, i.e. magnitude of the velocity $\Omega$ ' =' 'anular rotation rate, i.e. magnitude of the vector $\vec \Omega$ $\alpha$ ' =' 'azimuth angle, i.e. direction of motion with respect to the north-vector $\hat n$ in the clockwise direction as viewed in the down direction $-\hat u$ $\lambda$ ' =' 'latitude of the moving object

Note: because the Coriolis acceleration is the vector product $\vec v \times \vec \Omega$ and $\vec \Omega$ is pointing in the z-direction of the sphere centered cartesian coordinate system, the Coriolis acceleration z-component is always zero. It's because the vector product of 2 vectors is always perpendicular to the 2 vectors.

### Horizontal Component of the Coriolis Acceleration

To calculate the horizontal component of the Coriolis acceleration we simply have to project the general Corsiolis acceleration from above onto the local right vector $\hat r$. This is achieved by building the scalar product of $\vec c$ with the direction vector $\hat r$:

 (7) (8)

The red part is equal to 1. That means the horizontal component in the right-direction of motion does not depend on the direction of motion. It is simply always to the right (positive c) or to the left (negative c) with respect to the moving object.

So we finally have our simple equation for the horizontal Coriolis component on a rotating sphere:

(9)
where'
 $c$ ' =' 'magnitude of the horizontal Coriolis acceleration acting left/right with respect to the direction of motion parallel to the surface of the rotating sphere $v$ ' =' 'speed of the object along the surface of the rotating sphere $\Omega$ ' =' 'magnitude of the angular veclocity of the rotating sphere $\lambda$ ' =' 'latitude of the moving object

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