Equations of Motion Hydrostatic Approximation

1. Equations of MotionHydrostatic Approximation September 22

2. Recall: x-momentum y-momentum z-momentum continuity where:

3. Dissipation/ Turb Mixing Horiz Vert Total Accel Pressure Gradient Coriolis Accel x-momentum y-momentum z-momentum continuity

4. Hydrostatic Equilibrium Typical Scales: with these we can “scale” the equations of motion

6. The momentum equation for vertical velocity is therefore:

7. The momentum equation for vertical velocity is therefore: and the only important balance in the vertical is hydrostatic: Correct to 1:106

8. The momentum equation for horizontal velocity in the x direction is

9. The momentum equation for horizontal velocity in the x direction is The Coriolis balances the pressure gradient, known as the geostrophic balance – similarly for y-direction

10. Steady Geostrophic Flow:Another way to state equations Steady – No acceleration X (eastward) momentum balance Y (northward) momentum balance Z (vertical) momentum balance - Hydrostatic Continuity – Conservation of Mass or Volume

11. Hi P Lo P Lo P cf pgf pgf cf ρ=const u,v=const w/z

12. Pressure Gradient c c fc Pressure Gradient fc fc fc c c Pressure Gradient Lo Pressure Hi Pressure Pressure Gradient Lo Pressure

13. Dissipation/ Turb Mixing Horiz Vert Total Accel Pressure Gradient Coriolis Accel x-momentum y-momentum z-momentum hydrostatic continuity

14. Inertial Motion Total Accel Coriolis Accel balances x-momentum y-momentum z-momentum hydrostatic continuity

15. Dissipation/ Turb Mixing Vertical Coriolis Accel balances x-momentum y-momentum z-momentum hydrostatic continuity Ekman Flow

16. Geostrophic Motion Pressure Gradient Coriolis Accel balances x-momentum y-momentum z-momentum hydrostatic continuity