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Basic dynamics
The equation of motion Scale AnalysisBoussinesq approximation
Geostrophic balance(Reading: Pond and Pickard, Chapters 6-8)
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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
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Gravitation and gravity
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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
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Coriolis Force
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In Cartesian Coordinates:
where
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Accounting for the turbulence and averaging within T:
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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.,
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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
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(incompressible)
Reynolds stress has no symmetry:
A more general definition:
if
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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
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Scaling the vertical component of the equation of motion
Hydrostatic Equationaccuracy 1 part in 106
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Boussinesq approximationDensity variations can be neglected for its effect
on mass but not on weight (or buoyancy). Assume that where , we have
neglected
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Geostrophic balance in ocean’s interior
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Scaling of the horizontal components
Zero order (Geostrophic) balancePressure gradient force = Coriolis force
(accuracy, 1% ~ 1‰)
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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
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Taking into the vertical momentum equation, we have
If we scale , and assume
then
and
(accuracy ~ 1‰)