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Equations of Motion Hydrostatic Approximation

This document revisits the fundamental equations of motion in fluid dynamics, particularly focusing on the hydrostatic approximation. It covers key concepts such as x-momentum, y-momentum, z-momentum, and continuity. The hydrostatic equilibrium emphasizes the balance of forces in a fluid at rest, highlighting the role of pressure gradients and Coriolis acceleration in steady geostrophic flow. With careful analysis of vertical and horizontal moments, we note the dissipation and turbulent mixing that affect fluid dynamics, ensuring a comprehensive understanding of the forces at play.

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Equations of Motion Hydrostatic Approximation

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  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

  5. The momentum equation for vertical velocity is therefore:

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

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

  8. 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

  9. 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

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

  11. Pressure Gradient c c fc Pressure Gradient fc fc fc c c Pressure Gradient Lo Pressure Hi Pressure Pressure Gradient Lo Pressure

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

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

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

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

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