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/

Scale analysis: Time scale

• What is a time scale?
• The characteristic amount of time for something to occur.
• We will use scale analysis, both time and space, to compare things.
• Fast or slow?
• Large or small?
• We always have to compare one thing to another - fast(er) or slow(er).

Time scales

• What is the natural time scale for the Earth’s rotation?

Time scales

• Angular velocity of the Earth
• the Earth’s surface moves through the circumference (at some latitude) in 24 hours.

Two coordinate systems

y

z

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

z’

y’

x’

x

Tangential coordinate system

Ω

R

Earth

Place a coordinate system on the surface.

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x = west to east (longitude)

y = south to north(latitude)

z = local vertical

Φ

a

The component of the angular velocity in the local vertical.

Coriolis parameter is time scale of rotation

f = 2Ωsin(φ)

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at 45°latitude

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f₀ ~ 10^-4 sec^-1

Time scales

• Think about this:
• Given: Rotation of Earth is important to our perception of motion
• Given: We have an object moving through the atmosphere
• Airplane
• Baseball

Time scales

• Back to the object in the atmosphere –
• Airplane
• How far from Denver to Detroit?
• 1000 miles
• How fast does an airplane fly?
• 500 miles per hour
• How long does it take to fly?
• 2 hours
• How do I compare this with the rotation of the Earth?

Time scales

• Back to the object in the atmosphere –
• Airplane
• How do I compare this with the rotation of the Earth?
• If this time scale is approximately the same size as the Earth’s rotation, then the rotation might be important to analyzing the motion.
• What if something moves “fast?” - or “slow?”
• Does the airplane pilot worry about the rotation of the Earth?
• Does the baseball pitcher worry about the rotation of the Earth? (Why does the ball curve?)

Quantitative Comparison

• If something moves “fast,” then U/L is large.

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• If U/L is large relative to f, then rotation is not very important.

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• If something moves “slow,” then U/L is small and rotation might be very important.

Quantitative Comparison

To make a comparison like this form a ratio and ask how that ratio compares to the number one.

Large scale

If the Rossby number is comparable to or smaller than 1, then rotation is important and we say that the flow is “large-scale.”

Typical numbers

In atmosphere a typical velocity above the ground might be 20 m/sec.

The distance from the peak to a trough of a mid-latitude wave might be 1000 km.

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The angular velocity is 7.3 x 10^-5 / sec.

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A representative velocity of the oceanic Gulf Stream might be 100 cm/sec, what is the length scale that would mean that rotation is important?

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A baseball pitcher throws the ball about 90 feet at about 90 miles / hour, is rotation important?

For atmospheric example order 1/7.

For oceanic example order 100/(2*7.3x10-5 L) ~ 1.

• Solve for L in cm

Velocity and distance tell us something about time

distance = rate x time

Estimate time as distance/(some average rate)

L ≡ some characteristic distance

U ≡ some characteristic speed

Characteristic time ≡ L/U

Therefore, for acceleration

D ( )/Dt can be characterized by 1/(L/U)

End: Scale analysis: Time scale

Richard B. Rood

rbrood@umich.edu

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

https://openclimate.org/