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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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Barotropic Potential Vorticity

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Scaled vorticity equation (terms of 10-10 s-2)

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Potential vorticity

  • Potential Vorticity is a variable that combines the absolute vorticity and some measure of the thickness of a column of air.
  • It links dynamic and thermodynamic properties.
  • In a generalized definition, it is conserved in adiabatic motion.

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Two important definitions

  • barotropic – density depends only on pressure.
    • And by the ideal gas equation, surfaces of constant pressure, are surfaces of constant density, are surfaces of constant temperature.
  • baroclinic – density depends on pressure and temperature.

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Barotropic Potential Vorticity

  • Barotropic potential vorticity is a good framework for thinking about dynamics in the atmosphere.
  • The barotropic vorticity equation is a simple, meaningful predictive model.

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Barotropic Potential Vorticity

Assume constant density

Integrate with height, z1 🡪 z2 over a layer of depth H.

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Thermal Wind in Pressure Coordinates

p is an independent variable - a coordinate.

Hence, x and y derivatives are taken with p constant.

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Thermal Wind for a Barotropic Fluid

  • Barotropic: temperature is constant on a pressure surface
  • This means

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  • Geostrophic wind is constant with height in pressure coordinates in a barotropic fluid

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Barotropic Potential Vorticity

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What happens if the depth (H) is constant?

  • Conservation of potential vorticity becomes conservation of absolute vorticity…

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Barotropic Potential Vorticity

Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.

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What happens if the depth (H) changes?

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Relative vorticity with change of depth

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What happens when the vortex meets the mountain?

Surface with a hill.

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Vorticity and depth

  • There is a relationship between depth and vorticity.
  • As the depth of the vortex changes, the relative vorticity changes in order to conserve the potential vorticity.
  • We have now linked the rotational and irrotational components of the wind.
    • divergence and curl
    • vorticity and divergence
  • Potential vorticity indicates an interplay between relative and planetary vorticity through conservation of absolute angular momentum.

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End: Barotropic Potential Vorticity

Richard B. Rood

rbrood@umich.edu

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

https://openclimate.org/