33 Geostrophic balance

Schematic of geostrophic flow in a horizontal plane in the Northern Hemisphere. The Coriolis force is directed in the direction opposite to the Pressure Gradient Force, implying a current along constant pressure lines.

 

If the Rossby number is low, then rotation is important.  The steady-state force balance between the Coriolis force and the pressure gradient force is called geostrophic balance and is the approximate state of most large-scale flows in the ocean and atmosphere.  For example, the y-component momentum equation in geostrophic balance (see figure at left) would be written:

    \[ 0 = -fu-\frac{1}{\rho_0}\frac{\partial P}{\partial y} \]

Since the current vector is perpendicular to the Coriolis force, it is also perpendicular to the pressure gradient. Flow is thus along lines of constant pressure, called isobars, rather than down the pressure gradient as we would intuit from the smaller scales of our everyday experience.

Key Takeaways

If the pressure field is known, then the geostrophic velocity components can be calculated as:

    \[ u =- \frac{1}{f \rho_0 }\frac{\partial P}{\partial y} \]

    \[ v = \frac{1}{f \rho_0 }\frac{\partial P}{\partial x} \]

Near the surface of the ocean, the shallow pressure is related hydrostatically, P=g \rho_0 \eta, to the elevation of the sea surface, \eta relative to its equilibrium geopotential, z = 0. The near-surface geostrophic flow can be calculated from sea surface height as:

    \[ u_{surface} =- \frac{g}{f }\frac{\partial \eta}{\partial y} \]

    \[ v_{surface} = \frac{g}{f}\frac{\partial \eta}{\partial x} \]

Sea-surface height can be measured with satellite altimeters, although this technology alone is accurate only for detecting how currents change in time because estimates of the time-mean state require knowledge of the gravity field with an accuracy not currently achievable.

In the interior ocean, we cannot measure horizontal pressure gradients directly, but we can use CTD data to measure dynamic height, related to the difference in the pressure gradient between two levels in the vertical. This gives us expressions for differences in geostrophic velocity between two pressure levels:

    \[ u(P)-u(P_{REF}) =- \frac{1}{f }\frac{\partial D}{\partial y} \]

    \[ v(P)-v(P_{REF}) = \frac{1}{f}\frac{\partial D}{\partial x} \]

where in this case

    \[ D=-\int_{P_{REF}}^{P}\frac{1}{\rho}dP \]

A common assumption about the reference level (P_{REF}) velocity is that it is zero at some deep level (i.e., that it is much smaller than the velocity of interest, typically a stronger, shallower flow).  Alternately, the geostrophic velocity can be measured directly at the reference level.

 

The map shows the geostrophic currents field in the Gulf of Mexico. The background image represents the dynamic height.

Maps such as the one here estimate the time-mean geostrophic circulation near the surface of the ocean using an optimal fit to combined data from satellite altimeters, density profiles, and drifting buoys.  This time-mean field is then added to the time-variable field (that can accurately be determined from altimeter data) to produce daily operational maps of ocean currents.

 

 

 

 

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