1 of 18

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 18

QG#: Vorticity Equation Outline

3 of 18

We want an equation for the conservation of vorticity

  • We want an equation that represents the time rate of change of vorticity in terms of sources and sinks of vorticity.

4 of 18

Conservation (continuity) principle

  • dM/dt = Production – Loss

​

Refer to lectures on the conservation principle, derivation of mass conservation equation (continuity equation), derivation of vorticity equation.

5 of 18

Newton’s Law of Motion

Which is the vector form of the momentum equation.

(Conservation of momentum)

​

Where F is the sum of forces acting on a parcel, m mass, v velocity

6 of 18

What are the forces?

  • The total force is the sum of
    • Pressure gradient force
    • Gravitational force
    • Viscous force
    • Apparent forces
  • We derived these forces from first principles

​

7 of 18

We want an equation for the conservation of vorticity

  • We could approach it the same way as momentum, define the sources and sinks of vorticity from first principles.
    • But that is hard to do.
    • What are the first principle sources of vorticity?
  • Or we could use the conservation of momentum, and the definition of vorticity to derive the equation.

8 of 18

Newton’s Law of Motion (components)

9 of 18

Combine the definition of vorticity with the conservation principle

10 of 18

Operate on the momentum equations

11 of 18

Subtract

A time rate of change of vorticity will come from here.

12 of 18

Subtract

The details will depend on d( )/dt.

For an Eulerian fluid d( )/dt = D( )Dt, material derivative

13 of 18

Expand material derivative

14 of 18

Expand material derivative

Dζ/Dt comes from here

15 of 18

Expand material derivative

Other terms (tilting/divergence)

come from here

16 of 18

Expand material derivative

Forcing terms

(solenoid, advection of f,

divergence operating on f)

come from here

17 of 18

Collect terms

  • If you expand the dot products and plug in the Coriolis and pressure gradient force on the right-hand side, you will get the result previously derived in lectures and homework.

18 of 18

QG#: Vorticity Equation Outline

Richard B. Rood

rbrood@umich.edu

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

https://openclimate.org/