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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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QG1: Scaled Momentum Equation in p Coordinates

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Equations of motion in pressure coordinates�(using Holton’s notation)

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Scale factors for “large-scale” mid-latitude

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Scale of relative vorticity

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From the scale analysis we introduced the non-dimensional Rossby number

A measure of planetary vorticity compared to relative vorticity.

A measure of the importance of rotation.

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Scale analysis of equations in pressure coordinates

  • Start:
    • horizontal flow is approximately geostrophic
    • vertical velocity much smaller than horizontal velocity

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horizontal flow is approximately geostrophic

  • L, length scale, is small compared to the radius of the Earth
  • In the calculation of geostrophic wind, assume f is constant; f = f0

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horizontal flow is approximately geostrophic

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We will scale the material derivative

Ignore // small

This is for use in the advection of temperature and momentum.

ω comes from div(ageostrophic wind)

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Variation of Coriolis parameter

  • L, length scale, is small compared to the radius of the Earth
  • In the calculation of geostrophic wind, assume f is constant; f = f0
  • We cannot assume f is constant in the Coriolis terms…

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Variation of Coriolis parameter

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Variation of Coriolis parameter

Scale of first two terms.

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Wait, wait …

  • A few slides ago didn’t you assume that f was constant when you calculated the geostrophic wind.
  • But then one slide later say that you could not assume f was constant when you calculated the Coriolis term.

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  • Why yes, that’s what I did.

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Forcing terms in momentum equation

approx of coriolis

parameter

Use definition of geostrophic wind �in the pressure gradient force

def’n of the

full wind

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Forcing terms in momentum equation

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Approximate horizontal momentum equation

This equation states that the time rate of change of the geostrophic wind is related to

    • the coriolis force due to the ageostrophic wind and
    • the part of the coriolis force due to the variability of the coriolis force with latitude and the geostrophic wind.

Both of these terms are smaller than the geostrophic wind itself.

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Wait, wait, wait!

  • I learned just a month ago that the geostrophic wind was a diagnostic, there is no acceleration.
    • That for there to be any acceleration of the wind that the wind had to be ageostrophic.

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  • Why yes you did.
    • But, now, we are considering something that is only quasi-geostrophic; that is “almost geostrophic.”

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QG1: Scaled Momentum Equation in p Coordinates

Richard B. Rood

rbrood@umich.edu

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

https://openclimate.org/