Geostrophic Currents

Full Equations of Motion

continuity equation resulting from assumption of incompressibility.

1 When can acceleration be ignored?

This describes motion that is always being accelerated to the center

The force balance is between the centripetal force and Coriolis force

Where….

R0 = Rossby Number the ratio of centripetal to Coriolis acceleration

When Ro is small, acceleration can be ignored and the force balance is between pressure gradient and Coriolis

You can derive a few more…

2 Momentum Equation for Vertical Motion

This is also known as the hydrostatic equation

Momentum Equation for Horizontal Motion

3 Geostrophic Equations

Expresses the balance between Pressure Gradient and Coriolis force - This is often referred to as Geostrophic Balance

These expression provide a means for determining current velocity from the pressure field which is related to the distribution of density in the ocean. These currents are referred to as Geostrophic Currents

Note: Wherever you find strong pressure gradients in the ocean you will have fast moving geostrophic currents.

Pressure Field in the Ocean

Two ways to express pressure field in the ocean: 1) give the pressure on a series of level surfaces (with respect to the earth’s geopotential surfaces) or 2) give the geopotential heights of equal pressure surfaces. (For more information see Stewart 153…)

D is the Geopotential Height

then…

4 Geopotential height between two equal

pressure surfaces (p1 and p2)

Geopotential height between two equal pressure

surfaces (p1 and p2) at two different hydrographic stations (a and b). α = 1/ρ = specific volume

Difference in geopotential height between two

equal pressure surfaces (p1 and p2) at two different hydrographic stations

slope between two surfaces of constant pressure

Geostrophic at pressure surface a relative to the geostrophic velocity are pressure level b. Note that you need a third hydrographic station to get the slope of D in the north/south direction to compute the u- component of the geostrophic velocity shear.

} p2

Da Db p1

If the lower constant pressure surface (P1) is a constant geopotential surface there is no geostrophic flow at this level and the horizontal pressure gradient in the layer above would have a pressure gradient (and associated geostrophic current) associated with Da-Db.

If the lower constant pressure surface (P1) is not a level geopotential surface the Da-Db will give pressure gradient relative to the lower level and give a geostrophic current in the upper level will be relative relative to the geostrophic current in the lower level.

5 6 7 Ocean Eddies Derived from Topex-Poseidon

Stewart Gives these important steps…

1. Calculate the Da - Db between P1 and P2.

2. Calculate the slope of the upper pressure surface relative to the lower pressure surface

3. Calculate the geostrophic current at the upper surface relative to the lower surface. This is the current shear

4. Integrate the current shear to a depth of known current speed

a) surface current (derive from altimetry) downward

b) depth of no motion upwards

8 i 1 atmospheric density is small ∆Z1

ρ1

i2 ∆Z2

ρ2

Special case of bottom layer velocity = 0

Several Different Variations on Water Column Density Structure given by Knauss…

ρ1

ρ2 

ρ3

ρ1 ρ1 ρ2

ρ3 ρ2 O

i slope of the isopycnal surface flow into page O No flow

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