Analysis of Potential Winds

1. Analysis of Potential Winds P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Description of Wind Generated by Geo-Solar Activities…

2. Vortical Nature of Wind • The vertical component of the vorticity at a given latitude f is: • Absolute vorticitywa, • Relative vorticitywr, • Planetary vorticityf. • For the Earth f is positive in NH and negative in SH. The planetary vorticity is typically much bigger than the relative vorticity!?!?!?!?

3. Potential of Absolute Vorticity • Definition • Alternative expression • P is conserved in absence of friction and diabatic heating when following the fluid motion. This model is used as Tracer of Wind…..

4. Diabatic Effects : Column Stretching • As the column stretches the absolute vorticity increases so that the potential vorticity remains constant. • If f is constant then this results in increased relative vorticity (the air parcel spins up).

5. Some examples • You follow the spinning motion of a leaf on the surface of a shallow river. • It is possible to guess the depth of the water ?!?! The leaf will spin up in deeper water.

6. Vorticity Equations • General approach: • Use the momentum equations and take the corresponding partial derivatives to form the curl of the velocity field on the LHS and whatever is the result goes on the RHS. • And … you have a vorticity equation. • Exactly how this is done in Fluid Mechaniscs.

7. Fundamental Force Fields responsible for Wind • Pressure gradient force • Viscous force • Gravitational force • Apparent forces: Centrifugal and Coriolis • Can you think of other classical forces ? • Would they be important in the generation and control of wind? • Electromagnetic force • Total Force field is the vector sum of all of these fields.

8. The Momentum Equation of Real Wind + other forces Pressure gradientforce Viscous force Gravity force = gravitational force + centrifugal force Material derivative of the velocity vector with g = 9.81 m s-2

9. Hydrostatics • A Filed variable Recognized by the Pascal. • Even based on pedagogical principle, to start with simple matters and turn later to the complicated ones, Fluid Mechanics traditionally starts with hydrostatics. These are the usually desired results picturing the connection between pressure p, conservative external force field potential  and density .

10. Effects of the Coriolis force on Wind

11. The Coriolis force • Fictitious force only arising in a rotating frame of reference • It is directed 90º to the right of the wind in the northern hemisphere. • Conversely, it is directed 90º to the left of the wind in the southern hemisphere. • Increases in proportion to the wind speed • Increases with latitude, is zero at the equator. • Degree of influence of Coriolis force on wind field?!?!?

12. Highs and Lows In Northern Hemisphere velocity is deflected to the right by the Coriolis force Motion initiated by pressure gradient & sustained by other forces Opposed by viscosity

13. Momentum Equation with Coriolis Force Material derivative of Pressure gradientforce Viscous force Gravity force Coriolis forces + other forces ???

14. Components of Full Momentum Equation

15. Let’s Think About the Individual Terms These are the curvature terms that arise because the tangential coordinate system curves with the surface of the Earth. The equations are explicitly non-linear.

16. Governing Equations for Wind u, v, w, ρ, p which depend on (x, y, z, t). Can we solve this system of equations?

17. Two dimensional Wind Equations What are the units? Do they check out?

18. Scale Analysis: Let us define:

19. Scales for “large-scale” mid-latitude weather systems

20. The Scale Analysis for Initiation of Winds U*U/L U*U/a U*W/a Uf Wf 10-4 10-5 10-8 10-3 10-3 10-6 10-12 Largest Terms

21. The Largest Terms Note: There is no D( )/Dt term. Hence, no acceleration, no change with time. This is a simple equilibrium Wind. This is the geostrophic balance.