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CLIMATE/EARTH 401�Geophysical Fluid Dynamics�Dynamical Meteorology

Richard B. Rood

rbrood@umich.edu

https://clasp.engin.umich.edu/people/richard-b-rood/

https://openclimate.org/

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Rotating Coordinate System

We have three types of forces that we consider in dynamical meteorology

Body forces
Surface forces
Apparent forces

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Back to basics: A couple of definitions

Newton’s laws assume we have an “inertial” coordinate system; that is, and absolute frame of reference – fixed, absolutely, in space.
Velocity is the change in position of a particle (or parcel). It is a vector and can vary either by a change in magnitude (speed) or direction.

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In weather and climate

We are not working in an inertial coordinate system.
We are in a rotating coordinate system.

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Apparent forces: A mathematical approach

Non-inertial, non-absolute coordinate system

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One coordinate system related to another by:

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Two coordinate systems

z’ axis is the same as z, and there is rotation of the x’ and y’ axis

z’

y’

x’

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One coordinate system related to another by:

T is time needed to complete rotation.

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Acceleration (force) in rotating coordinate system

The apparent forces that are proportional to rotation and the velocities in the inertial system (x, y, z) are called the Coriolis forces.

The apparent forces that are proportional to the square of the rotation and position are called centrifugal forces.

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Apparent forces

With one coordinate system moving relative to the other, we have the velocity of a particle relative to the coordinate system and the velocity of one coordinate system relative to the other.
This velocity of one coordinate system relative to the other leads to apparent forces. They are real, observable forces to the observer in the moving coordinate system.

The apparent forces that are proportional to rotation and the velocities in the inertial system (x,y,z) are called the Coriolis forces.
The apparent forces that are proportional to the square of the rotation and position are called centrifugal forces.

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The importance of rotation

A number of resources and videos are provided for more insight into the importance of rotation on fluid flow.