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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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Rotation

We know that many weather systems are characterized by rotating winds.
We know that because of the Earth’s rotation the atmosphere is often observed to rotate around us.

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Circulation of a hurricane

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Wind around a system.

This is the sort of system which is often called a “vortex.”

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Definition of vorticity

Vorticity is the measure of rotation of an air parcel about a vertical axis.

Mathematically, vorticity is the curl of the velocity field.

curl (u)

∇×u

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Forecast: Absolute Vorticity at 500 hPa

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Vorticity: positive and negative

f, the Coriolis parameter represents the vorticity of the Earth, planetary vorticity.

f is positive in northern hemisphere
f is negative in southern hemisphere

Positive vorticity is associated with cyclonic rotation in the northern hemisphere (low pressure systems)
Negative vorticity is associated with cyclonic rotation in the southern hemisphere (low pressure systems)

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Vorticity and angular momentum

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Lets consider a spinning skater

Motion is in the (x,y) plane

Axis of rotation is in the vertical plane

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Formalizing the representation of rotation and divergence

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Suppose we have some flow

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Imagine at the point flow decomposed into two “components”

A “component” that flows into or away from the point.

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Imagine at the point flow decomposed into two “components”

A “component” that flows around the point.

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In general, can imagine a “circulation” around a point.

Circulation based on the component of velocity tangent to some curve summed around the curve.

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Important mathematical and physical operators

We have seen the dot product or divergence.

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Divergence

Related to stretching of the velocity field.

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Important mathematical and physical operators

Which is the definition of vorticity.

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Vorticity

Related to shear of the velocity field.

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Mathematical foundation

Helmholtz decomposition states that any field can be represented as rotational and irrotational pieces.

∇ ∙ u (irrotational)

∇×u (rotational)

Helmholtz decomposition in Wikipedia

https://en.wikipedia.org/wiki/Helmholtz_decomposition