Basic dynamics The equation of motion Scale Analysis

Basic dynamics

The equation of motion Scale AnalysisBoussinesq approximation

Geostrophic balance(Reading: Pond and Pickard, Chapters 6-8)

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.

g0=9.80m/s2

1sidereal day =86164s

1solar day = 86400s

Force per unit mass

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 vectorconstant potential energythe 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

Ax=Ay~102-105 m2/sAz ~10-4-10-2 m2/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

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 mH=103 mU= 0.1 m/sT=106 s (~ 10 days)2sin45o=2cos45o≈2x7.3x10-5x0.71=10-4s-1

g≈10 m/s2

≈103 kg/m3

Ax=Ay=105 m2/sAz=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 Equationaccuracy 1 part in 106

Boussinesq approximationDensity 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

Zero order (Geostrophic) balancePressure gradient force = Coriolis force

(accuracy, 1% ~ 1‰)

Re-scaling the vertical momentum equationSince 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‰)