Basic dynamics The equation of motion Scale Analysis Boussinesq approximation Geostrophic balance (Reading: Pond and Pickard, Chapters 6-8)

The Equation of Motion Newton’s second law in a rotating frame. (Navier-Stokes equation) Force per unit mass : Acceleration relative to axis fixed to the earth. 1sidereal day =86164s 1solar day = 86400s : Pressure gradient force. : Coriolis force, where : Effective (apparent) gravity. g0=9.80m/s2 : Friction. molecular kinematic viscosity.

Gravitation and gravity

Gravity: Equal Potential Surfaces g changes about 5% 9.78m/s2 at the equator (centrifugal acceleration 0.034m/s2, radius 22 km longer) 9.83m/s2 at the poles) • equal potential surface normal to the gravitational vector constant potential energy the largest departure of the mean sea surface from the “level” surface is about 2m (slope 10-5) The mean ocean surface is not flat and smooth earth is not homogeneous

Coriolis Force

In Cartesian Coordinates: where

Accounting for the turbulence and averaging within T:

Given the zonal momentum equation If we assume the turbulent perturbation of density is small i.e., The mean zonal momentum equation is Where Fx is the turbulent (eddy) dissipation If the turbulent flow is incompressible, i.e.,

Eddy Dissipation Then Ax=Ay~102-105 m2/s >> Az ~10-4-10-2 m2/s Reynolds stress tensor and eddy viscosity: , Then Where the turbulent viscosity coefficients are anisotropic. Ax=Ay~102-105 m2/s Az ~10-4-10-2 m2/s >>

Reynolds stress has no symmetry:  A more general definition:   if (incompressible)

Scaling of the equation of motion Consider mid-latitude (≈45o) open ocean away from strong current and below sea surface. The basic scales and constants: L=1000 km = 106 m H=103 m U= 0.1 m/s T=106 s (~ 10 days) 2sin45o=2cos45o≈2x7.3x10-5x0.71=10-4s-1 g≈10 m/s2 ≈103 kg/m3 Ax=Ay=105 m2/s Az=10-1 m2/s Derived scale from the continuity equation W=UH/L=10-4 m/s Incompressible  

Scaling the vertical component of the equation of motion   Hydrostatic Equation accuracy 1 part in 106

Boussinesq approximation Density variations can be neglected for its effect on mass but not on weight (or buoyancy). Assume that where , we have  neglected

Geostrophic balance in ocean’s interior

Scaling of the horizontal components  (accuracy, 1% ~ 1‰) Zero order (Geostrophic) balance Pressure gradient force = Coriolis force

Re-scaling the vertical momentum equation Since the density and pressure perturbation is not negligible in the vertical momentum equation, i.e., , and , The vertical pressure gradient force becomes

Taking into the vertical momentum equation, we have If we scale , and assume then  and (accuracy ~ 1‰)