1 of 19

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 19

Vorticity 8: Scaling

3 of 19

Vorticity and Divergence

  • Related to shear �of the velocity field.

∂v/∂x-∂u/∂y

  • Related to stretching �of the velocity field.

∂u/∂x+∂v/∂y

​

4 of 19

Conservation of vertical component of vorticity

5 of 19

Scale Analysis of the Vorticity Equation

  • Changes in relative vorticity are caused by:
      • Divergence
      • Tilting
      • Gradients in density
      • Advection
  • Which of these are most important for large-scale flows?
  • Back to scale analysis…

6 of 19

Scale factors for “large-scale” mid-latitude

7 of 19

Scale of relative vorticity

8 of 19

Compare relative vorticity to planetary vorticity

Usually, planetary vorticity is larger than relative vorticity.

9 of 19

Time rate of change of relative vorticity

10 of 19

Remaining Terms

11 of 19

Scale analysis of the vorticity equation

  • Divergence term largest, followed by horizontal advection and the local time rate of change of relative vorticity
  • Tilting term important where there is large shear and strong horizontal gradient in the vertical velocity (boundary layer, smaller scales)
  • Thermodynamic term important where there are strong density (temperature) gradients that intersect lines of constant pressure (sea breeze, fronts)

12 of 19

Consider divergence term

  • This term scales larger than the other terms.
    • But, remember that the divergence of the horizontal wind is, in actuality, small compared to the change in wind with x,y

13 of 19

Consider divergence term

Divergence scales as the scale of the �ageostrophic wind, so

14 of 19

Consider divergence term

We saw that the relative vorticity is typically about an order of magnitude less than the planetary vorticity. So

15 of 19

So for quasi-nondivergent (i.e., geostrophic) flow

16 of 19

Compare relative vorticity to planetary vorticity and to divergence

Again we see the importance of the rotation of the Earth.

17 of 19

Terms in Vorticity Equation

Time rate of change

Horizontal advection

Vertical advection

Divergence

Tilting

Planetary vorticity advection

Solenoidal term

Time rate of change

Horizontal advection

Divergence

Planetary vorticity advection

18 of 19

Assume balance among terms of 10-10 s-2

19 of 19

End: Vorticity 8: Scaling

Richard B. Rood

rbrood@umich.edu

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

https://openclimate.org/