Basic dynamics ●The equations of motion and continuity Scaling Hydrostatic relation Boussinesq approximation ●Geostrophic balance in ocean’s interior

Newton’s second law in a rotating frame. (Navier-Stokes equation) The Equation of Motion : Acceleration relative to axis fixed to the earth. : Pressure gradient force. : Coriolis force, where : Effective (apparent) gravity. : Friction. molecular kinematic viscosity.

Gravity: Equal Potential Surfaces g changes about 5% 9.78m/s 2 at the equator (centrifugal acceleration 0.034m/s 2, radius 22 km longer) 9.83m/s 2 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 ) The mean ocean surface is not flat and smooth earth is not homogeneous

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 F x is the turbulent (eddy) dissipation If the turbulent flow is incompressible, i.e.,

Eddy Dissipation A x =A y ~ m 2 /s A z ~ m 2 /s >> Reynolds stress tensor and eddy viscosity: Where the turbulent viscosity coefficients are anisotropic., Then

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

Continuity Equation Mass conservation law In Cartesian coordinates, we have or For incompressible fluid, If we define and, the equation becomes

Scaling of the equation of motion Consider mid-latitude (φ≈45 o ) open ocean away from strong current and below sea surface. The basic scales and constants: L=1000 km = 10 6 m H=10 3 m U= 0.1 m/s T=10 6 s (~ 10 days) 2Ωsin45 o =2Ωcos45 o ≈2x7.3x10 -5 x0.71=10 -4 s -1 g≈10 m/s 2 ρ≈10 3 kg/m 3 A x =A y =10 5 m 2 /s A z =10 -1 m 2 /s Derived scale from the continuity equation W=UH/L=10 -4 m/s  

Scaling the vertical component of the equation of motion   Hydrostatic Equation accuracy 1 part in 10 6

Boussinesq approximation Density variations can be neglected for its effect on mass but not on weight (or buoyancy). Assume that where, we have where  Then the equations are where (1) (2) (3) (4) (The term is neglected in (1) for energy consideration.)

Geostrophic balance in ocean’s interior

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

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‰)

Geopotential Geopotential Φ is defined as the amount of work done to move a parcel of unit mass through a vertical distance dz against gravity is (unit of Φ: Joules/kg=m 2 /s 2 ). The geopotential difference between levels z 1 and z 2 (with pressure p 1 and p 2 ) is

Dynamic height Given, we have where is standard geopotential distance (function of p only) is geopotential anomaly. In general, Φ is sometime measured by the unit “dynamic meter” (1dyn m = 10 J/kg). which is also called as “dynamic distance” (D) Note: Though named as a distance, dynamic height (D) is still a measure of energy per unit mass. Units: δ~m 3 /kg, p~Pa, D~ dyn m