# The Coriolis Effect in Meterologie

Every rotating pseudo-solid (viscous) body will take a rotationally symmetric oblate shape such that everywhere on the surface the vector sum of the the gravitational force and the "centrifugal force" is orthonal to the surface normal ! This applies also for the Earth and this is illustrated in figure 1

The usual model for the shape of the Earth is that it is an ellipsoid with a semi-major axis of 6378.136 km and a semi-minor axis of 6356.751 km. This is for example illustrated (with some pretty images) here

The latitude of a point on the surface of the Earth is the angle between the outward normal and the equatorial plane. This is so because by astronomic observations the angle between the local normal and the rotational axis can be accurately determined. Note that a plumb line indeed indicates the surface normal as it will align with the vector sum of the gravitational force and the "centrifugal force" as indicated by the green and the black lines of figure 1. As the Earth rotates one complete revolution in a sideral day (23 hours 56 minutes and 4.1 seconds) one can easily for any latitude compute what tangential force component (green line in figure 1) is required to compensate for the tangential component of the "centrifugal force" (black line in figure 1) . The result is displayed in the following figure 2

It is for the middle latitudes that the strongest tangential force of 0.017 Newton per kg is needed while at the poles and on the equator no such force is needed at all!

To understand how the atmosphere moves over the surface of the Earth surface one can use the model of a mass point that moves freely over the surface of such an ellipsoid. The only force that acts on this mass point is this tangential force of up to 0.017 Newton per kg directed towards the closest pole. Note that for this model the motion of the mass point is not affected by the "Earth rotation" at all! It is just about moving over an ellipsoid having the shape of the Earth affected by a tangential force directed towards the closest pole having the magnitude displayed in figure 1.

To investigate the motion of such a mass point relative the surface of the Earth it is useful to transform the equation of motion to an Earth-fixed, rotating coordinate system! This "mathematical appendix" is available for download using URL

https://science-mats.de/coriolis.pdf

Due to the relation illustrated in figure 1 one of the two mathematical transformation term nullifies the physical force which consequently is replace by the other term which is the term "Coriolis Force". This is no real physical force, it is just a mathematical term originating from the transformation to a non-inertial rotating coordinate system. The two "Equations of Motion describe exactly the same motion but have different merits for the analysis and for the propagation with numerical integration! The latter is what is done here to generate the different figures

The relation between the real physical force acting on the mass particle and the virtual "Coriolis Force" is illustrated in figure 4

The physical gravitational force is in direction "North" in the Northern Hemisphere and in direction "South" in the Southern Hemisphere. It is a conservative force only depending on the position of the mass point

The virtual "Coriolis Force" replacing the physical gravitation force in the "Equation of Motion" when a rotating earth fixed coordinate system is used is orthogonal to the direction of motion rotated 90 deg in the clock-wise sense in the Northern Hemisphere and in the anti-clock sense in the Southern Hemisphere. The first interesting observations resulting from the transformation of the "Equation of Motion" to the Earth-fixed rotating system is that the motion of the mass point relative to the Earth surface changes direction but not magnitude. The direction change is always to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. The rate of the direction change is in fact

2 *sin(latitude) times the rotation rate of the Earth

The motions of mass points during the first 5 hours which initially were at latitude 40 deg N and were given the inertial velocities 150 m/s (red) , 357 m/s (black) and 550 m/s (green) towards east are displayed in figur 5. 357 m/s is the velocity of the ground at latitude 40 deg and it is this velocity that results in that the mass point follows the "small circle" corresponding to latitude 40 deg.

If there had been no gravitational force pulling towards north the mass points would all have followed the same "great circle" around the Earth ending its southward motion only at latitude 40 deg S

According to the relation

2 *sin(latitude) times the rotation rate of the Earth

it should take about 32 hours for a mass point to make a full "Coriolis-Type-Whirl" as long as it does not depart too far from latitude 40 deg. Propagating the motion for a mass point having a "surplus velocity" of 100 m/s one gets in the inerial system

It can be seen that the mass point take a somewhat "windling" path from the red point to the blue point, a travel that lasts 32 hours. How this motion appears from the rotating Earth is shown in the next figure 7

The mass point gets almost to latitude 10 deg N before it starts travel back towards north. The rate of "right turn" gets lower when it leaves the "mid latitudes" and approaches the equator where the "Coriolis Effect" is zero. If it had been given a sufficiently high initial velocity it would have crossed the equator and then started to "turn left" instead

## How these trajectories were computed

A grid of 1048 reference points evenly distributed over the Northern Hemisphere of the ellisoid representing the shape of the Earth were defined, see figure 8. The numerical propagation over the surface of the ellipsoid was then replaced with a sequence of standard 2-dimensional orbit propagations in the tangential planes of such grid points using an eigth order corrector/predictor algoritm with Runge/Cutta start up.

The algoritm used was:

From the position vector R and the velocity vector V on the surface of the ellipsoid "mirrors" of these entities were defined in the tangential plane of the closest of these 1048 reference points. These "mirrors" were created by projecting these vectors "up-wards" into this plane.

The "force" assumed for the propagation of these "mirror images" was derived as follows:

1. Find the point on the ellipsoid for which the out-ward normal intersects the plane used for the numerical propagation at the spot were the mass point is situated
2. Use the "force" associated with this point for the propagation of the "mirror image" in the tangential plane of the "grid point"

This then continues until a closer "grid point" is found. When this is the case the position and velocity vectors in the old integration plane are projected to the tangential plane of this new "grid point" and the process continues from the beginning.

This algoritm is equally applicable for the propagation in an inertial system as in a rotating system. The difference is only which "force" is used.

For a non-rotating system this force has magnitude

FORCE=SLATI*RA*OMROT*OMROT

where

• SLATI is sinus for the latitude
• RA is the distance to the rotation axis of the Earth
• OMROT is the rotation rate of the Earth

and is directed north

For a rotating system this force has magnitude

FORCE=2*OMROT*SLATI*V

where

• OMROT is the rotation rate of the Earth
• SLATI is sinus for the latitude
• V is the magnitude of the velocity (relative the rotating frame!)

and is directed orthogonal to the direction of motion (turned 90 deg towards right)