Lab 3

1. Lab 3 Estimating the Geostrophic Wind

2. Estimating the Geostrophic Wind • (or “Now I’ve seen it all”!) • Last week, we learned that we could estimate a derivative quantity using finite differencing. Thus, we can express any differential equation as a difference equation.

3. Estimating the Geostrophic Wind • The geostrophic wind  Wind that follows contours or streamfunctions of height or pressure. The strength of the wind is proportional to the pressure gradient

4. Estimating the Geostrophic Wind • Geostrophic balance • Balance condition: A system is said to be in “balance” if one or more “forces” or forcing mechanisms exactly BALANCE one or more forces or forcing mechanisms. • A balanced system: is assumed to be “steady-state” or “conservative” and flow will follow the ‘streamfunction” defined by your balance condition.

5. Estimating the Geostrophic Wind • Examples of “balanced” systems • PGF = Centripetal force  Tornado • PGF = gravity  Hydrostatic balance • PGF = Coriolis force  Geostrophic balance (Earth turning) • PGF = CO + Friction  Ekmann Balance (PBL)

6. Estimating the Geostrophic Wind • PGF = CO +Cent.  Gradient balance • (From Vorticity equation): • vorticity advection = divergence and tilting  • Beltrami or “helical” flow  Mesocyclone

7. Estimating the Geostrophic Wind • Starting with the horizontal Navier-Stokes equation: • We assume frictionless, steady-state flow, thus:

8. Estimating the Geostrophic Wind • Geostrophic Wind

9. Estimating the Geostrophic Wind • where the gradient quantity is our stream (or potential) function and in component form:

10. Estimating the Geostrophic Wind • Then (from Holton):

11. Estimating the Geostrophic Wind • where we can define M as the Montgomery stream function (dry static energy): • (x,y,q) coords • Examine in x,y,p coords:

12. Estimating the Geostrophic Wind • Geostrophic wind Finite difference approximation:

13. Estimating the Geostrophic Wind • Commonly “f” can be approximated as “fo” • or a constant. Thus in the geostrophic wind equation, this was defined as “geostrophy-0” by N. Phillips (1963, Rev. Geophys, 123- 176).

14. Estimating the Geostrophic Wind • Geostrophy - 0

15. Estimating the Geostrophic Wind • If “f” is allowed to vary with latitude, then we call our estimate “geostrophy-1”

16. Estimating the Geostrophic Wind • If we take into account the change in f with latitude, we can define the geostrophic wind as:

17. Estimating the Geostrophic Wind • whose streamfunction is:

18. Estimating the Geostrophic Wind