Oceanic flow in which the pressure gradient force is balanced by the Coriolis effect
An example of a geostrophic flow in the Northern Hemisphere.
A northern-hemisphere
gyre
in
geostrophic balance
. Paler water is
less dense
than dark water, but more dense than air; the outwards pressure gradient is balanced by the 90 degrees-right-of-flow
coriolis force
. The structure will eventually dissipate due to friction and mixing of water properties.
A
geostrophic current
is an
oceanic current
in which the
pressure gradient
force is balanced by the
Coriolis effect
. The direction of geostrophic flow is parallel to the
isobars
, with the high pressure to the right of the flow in the
Northern Hemisphere
, and the high pressure to the left in the
Southern Hemisphere
. This concept is familiar from
weather maps
, whose isobars show the direction of
geostrophic winds
. Geostrophic flow may be either
barotropic
or
baroclinic
. A geostrophic current may also be thought of as a rotating shallow water wave with a frequency of zero.
The principle of
geostrophy
or
geostrophic balance
is useful to oceanographers because it allows them to infer
ocean currents
from measurements of the
sea surface height
(by combined
satellite altimetry
and
gravimetry
) or from vertical profiles of
seawater density
taken by ships or autonomous buoys. The major currents of the world's
oceans
, such as the
Gulf Stream
, the
Kuroshio Current
, the
Agulhas Current
, and the
Antarctic Circumpolar Current
, are all approximately in geostrophic balance and are examples of geostrophic currents.
Simple explanation
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Sea water
naturally tends to move from a region of high pressure (or high sea level) to a region of low pressure (or low sea level). The force pushing the water towards the low pressure region is called the pressure gradient force. In a geostrophic flow, instead of water moving from a region of high pressure (or high sea level) to a region of low pressure (or low sea level), it moves along the lines of equal pressure (
isobars
). This occurs because the
Earth
is rotating. The rotation of the earth results in a "force" being felt by the water moving from the high to the low, known as
Coriolis force
. The Coriolis force acts at right angles to the flow, and when it balances the pressure gradient force, the resulting flow is known as geostrophic.
As stated above, the direction of flow is with the high pressure to the right of the flow in the
Northern Hemisphere
, and the high pressure to the left in the
Southern Hemisphere
. The direction of the flow depends on the hemisphere, because the direction of the Coriolis force is opposite in the different hemispheres.
Formulation
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The geostrophic equations are a simplified form of the
Navier?Stokes equations
in a rotating reference frame. In particular, it is assumed that there is no acceleration (steady-state), that there is no viscosity, and that the pressure is
hydrostatic
. The resulting balance is (Gill, 1982):
![{\displaystyle fv={\frac {1}{\rho }}{\frac {\partial p}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/339235aa95c23d59971c35ac49674be8b2e7f2ce)
![{\displaystyle fu=-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdcf096742a347be1155da2986dd6cbd7b61d920)
where
is the
Coriolis parameter
,
is the density,
is the pressure and
are the velocities in the
-directions respectively.
One special property of the geostrophic equations, is that they satisfy the steady-state version of the continuity equation. That is:
![{\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecab62bb3321bc929c0945be2c83dafedae1703f)
Rotating waves of zero frequency
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The equations governing a linear, rotating shallow water wave are:
![{\displaystyle {\frac {\partial u}{\partial t}}-fv=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e78d2153af443f268333dc9d828d3ade0cc012f5)
![{\displaystyle {\frac {\partial v}{\partial t}}+fu=-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/261661bc2d903c2f93fe42c31cf7a8eff4fc9eb8)
The assumption of steady-state made above (no acceleration) is:
![{\displaystyle {\frac {\partial u}{\partial t}}={\frac {\partial v}{\partial t}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/230772a30ce8e502a49362770f52c2c588efc38d)
Alternatively, we can assume a wave-like, periodic, dependence in time:
![{\displaystyle u\propto v\propto e^{i\omega t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cfe22e281a4609ee8cfc32cdbe82e1304ab24c3)
In this case, if we set
, we have reverted to the geostrophic equations above. Thus a geostrophic current
can be thought of as a rotating shallow water wave with a frequency of zero.
See also
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References
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