1 of 12

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/

2 of 12

Thermal Wind: Derivation

3 of 12

Equations of motion in pressure coordinates

Horizontal Momentum (Scaled)

Continuity Equation

Thermodynamic Equation

4 of 12

Hydrostatic Balance

Hydrostatic Equation

(Scaled Vertical Momentum Equation)

5 of 12

Geostrophic wind

6 of 12

What is a tactic for exploring vertical behavior?

  • Or, how would we look at differences in the vertical?

Subtract 🡪 Take a difference 🡪 Take a derivative

7 of 12

Geostrophic wind

Take p, height, derivative

Hydrostatic Equation

(Scaled Vertical Momentum Equation)

8 of 12

Geostrophic wind

Links horizontal temperature gradient

with vertical wind gradient.

Take p, height, derivative

9 of 12

Thermal wind

  • p is an independent variable, a coordinate.
  • Hence, x and y derivatives are taken with p constant.

10 of 12

Schematic of thermal wind.

  • Thickness of layers is related to temperature,
  • which causes a tilt of the pressure surfaces.

11 of 12

Some previous work that has been invoked

  • Physical and intuitive meaning of mathematical manipulations
  • Meaning of independent variables
  • Scaling
  • Geostrophic approximation
  • Thickness

12 of 12

End: Thermal Wind: Derivation

Richard B. Rood

rbrood@umich.edu

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

https://openclimate.org/