1. Lab 3 Estimating the Geostrophic Wind

2.  (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,  ) 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