Coriolis acceleration
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Formal derivation of the Coriolis acceleration
Coriolis acceleration arises when motion is described in a rotating reference frame attached to the Earth. It is therefore a purely kinematic effect associated with the use of rotating coordinates.
Time differentiation in a rotating frame introduces an additional term related to the rotation of the coordinate axes wikipedia:
[math](d \vec r / dt)_{fixed frame} = (d \vec r / dt)_{rot} + \vec \Omega \times \vec r = \vec u + \vec \Omega \times \vec r \, , \qquad (1)[/math]
where [math]\vec r[/math] is the position vector of a fluid parcel ([math]\vec r =0[/math] at the Earth’s centre), [math]\vec u[/math] is the velocity measured in the rotating Earth frame, and [math]\vec \Omega[/math] is the Earth’s angular velocity vector directed along the Earth’s rotation axis. The Earth’s angular rotation rate is approximately [math]\Omega \approx 7.2921 \times 10^{-5} \; s^{-1} .[/math]
Differentiating Eq. (1) once more gives
[math](d^2 \vec r /dt^2)_{fixed frame} = d \vec u /dt + 2 \vec \Omega \times \vec u + \vec \Omega \times (\vec \Omega \times \vec r) \, . \qquad (2)[/math]
The three terms on the right-hand side represent respectively: the acceleration measured in the rotating Earth frame, the Coriolis acceleration, and the centrifugal acceleration.
The centrifugal acceleration is largely balanced by gravity and by adjustment of the equilibrium sea surface.
If no external or frictional forces act on the fluid parcel, then
[math](d^2 \vec r /dt^2)_{fixed frame} =0 \, .[/math]
To derive the commonly used local expressions for the Coriolis acceleration, we use the coordinate system shown in Figure 1. Here [math]\theta[/math] is longitude, [math]\phi[/math] latitude and [math]R[/math] the Earth radius. In the local east-north-up coordinate system the vector components are
[math]\vec u = (u = R \, \cos \phi \, \dfrac{d\theta}{dt} , \; v = R \, \dfrac{d\phi}{dt} , \; 0) ,[/math]
[math]\vec \Omega = (0, \; \Omega \, \cos \phi, \; \Omega \, \sin \phi) .[/math]
Substitution into Eq. (2) gives the horizontal components of the Coriolis acceleration:
[math]du/dt = f v, \qquad dv/dt = - fu, \qquad f=2 \Omega \sin \phi \, . \qquad (3)[/math]
where [math]f[/math] is the Coriolis parameter. These equations use the traditional approximation, in which only the vertical component of the Earth’s rotation vector is retained.
The Coriolis acceleration varies with latitude; the Coriolis acceleration vanishes at the equator and it is greatest at the poles.
Heuristic derivation of the Coriolis acceleration
The Coriolis acceleration can also be understood more intuitively when fluid velocities ([math]u,v[/math]) are much smaller than the Earth’s surface rotation velocity [math]\; U = R \, \Omega \, \cos \phi .[/math]
Due to the Earth’s rotation, a centrifugal acceleration acts perpendicular to the Earth’s rotation axis. At the Earth’s surface, part of this acceleration is balanced by gravity and part by the equilibrium sea-surface slope.
If seawater is initially at rest, these forces are in equilibrium. When water moves eastward with velocity [math]u[/math], the centrifugal acceleration changes slightly, producing a small acceleration toward the equator:
[math] dv/dt = - \sin \phi [(U+u)^2 - U^2]/R \cos\phi \approx - fu [/math] with [math]f = 2 \Omega \sin{\phi} .[/math]
The approximation is valid because in natural situations [math]|u| \lt \lt |U|[/math].
Although physically correct, this argument should not be interpreted as the fundamental origin of the Coriolis acceleration. The Coriolis effect does not depend on gravity and arises fundamentally from describing motion in a rotating reference frame, as shown by Eq. (1).
A second heuristic interpretation follows from angular momentum conservation. When fluid is travelling northward along the earth's surface with velocity [math]v = R \, d\phi/dt ,[/math] it experiences a decrease of surface rotation velocity [math]U[/math]. The principle of angular momentum conservation implies that the fluid will accelerate in order to conserve its angular momentum. This can be expressed as
[math]\dfrac{d}{dt} [(U + u) R \cos \phi ] = \dfrac{d}{dt} [ \Omega R^2 \cos^2 \phi + u R \cos \phi] =0 .[/math]
From this equation we find [math]du/dt = fv[/math], assuming again that [math]|u| \lt \lt |U| .[/math]
Consequences of Coriolis acceleration
Fluid parcels moving freely on the rotating Earth experience a deflection:
- to the right in the Northern Hemisphere ([math]\phi\gt 0[/math]),
- to the left in the Southern Hemisphere ([math]\phi\lt 0[/math]).
If friction and external forces are absent, a fluid parcel with initial horizontal velocity [math]V[/math] undergoes inertial motion, according to Eq. (3):
[math]u = V \sin ft , \qquad v = V \cos ft \, . \qquad (4)[/math]
The parcel follows an inertial circle with radius
[math]R_i = V/f .[/math]
Such motions are sometimes observed when strong river outflows enter wide coastal basins.
The importance of Coriolis acceleration depends on the ratio between inertial and rotational effects. A useful dimensionless measure is the Rossby number,
[math]Ro = V/(fL) ,[/math]
where [math]V[/math] is a characteristic flow velocity and [math]L[/math] a characteristic horizontal length scale.
For [math]Ro \gg 1[/math], Earth's rotation has little influence on the motion. This is typically the case for rapidly varying motions such as surface-wave orbital velocities.
When [math]Ro[/math] is similar or less than 1, Coriolis acceleration strongly influences the flow. This occurs for tidal motion in shelf seas and for large-scale wind-driven circulation, and ocean currents.
The influence of Coriolis acceleration on tidal motion in shelf seas is described in the articles Coriolis and tidal motion in shelf seas and Ocean and shelf tides.
On the ocean basin scale, the latitudinal variation of the Coriolis acceleration (Eq. 3) has to be taken into account. The influence on large scale ocean circulation patterns is discussed in the articles Geostrophic flow and Ekman transport.
Related articles
- Ekman transport
- Geostrophic flow
- Ocean and shelf tides
- Tidal motion in shelf seas
- Ocean circulation
- Vorticity
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