Geostrophic flow
Definition of Geostrophic flow:
Flow on a rotating earth along contour lines of equal pressure.
This is the common definition for Geostrophic flow, other definitions can be discussed in the article
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Contents
Coriolis acceleration
In geostrophic flow, the pressure gradient force is balanced by the Coriolis acceleration:
[math]\Large\frac{1}{\rho}\frac{\partial p}{\partial x}\normalsize = f \, v \;,\; \Large\frac{1}{\rho}\frac{\partial p}{\partial y}\normalsize = - f \, u . \qquad (1)[/math]
Meaning of the symbols:
[math]x, \, y\,=[/math] spatial coordinates, along the horizontal Cartesian [math]x[/math] and [math]y[/math] axes
[math]u,\, v\,=[/math] current velocities along the [math]x[/math] and [math]y[/math] axis, respectively
[math]\rho=[/math] seawater density
[math]p=[/math] pressure, satisfying [math]\partial p / \partial z = - \rho g[/math]
[math]z=[/math] coordinate along the vertical axis (upwards positive)
[math]g=[/math] gravitational acceleration
The Coriolis parameter [math]f[/math] is given by [math]f=2 \Omega \sin \phi[/math], where [math]\Omega \approx 7.29211^{−5} \, s^{−1}[/math] is the radial earth rotation frequency and [math]\phi[/math] the elevation radian angle indicating latitude.
Other symbols used in this article:
[math]D=[/math] total water depth
[math]w=[/math] velocity along the vertical [math]z[/math] axis
[math]u_*=[/math] order of magnitude of turbulent velocity fluctuations
Characteristics of geostrophic flow and occurrence
Conditions of geostrophic flow are:
- Frictional effects are small ([math]u_* \lt \lt \sqrt{Df|u|}[/math])
- Flow fluctuations involve time scales much longer than the inertial time scale [math]2 \pi/f[/math]
- Spatial flow variations involve length scales much larger than the inertial scale [math]u/f[/math]
- Absence of density gradients ([math]\rho[/math] independent of [math]x,y,z[/math])
Important characteristics of geostrophic flow are:
- The horizontal current velocity is uniform over the water column
- The flow follows depth contour lines
Conditions for geostrophic flow are seldom met in shallow coastal waters, because of frictional effects and the relatively small scales of temporal and spatial flow fluctuations. In contrast, ocean currents often have characteristics of geostrophic flow. This is the case especially where large-scale bathymetric gradients are important, such as along the continental shelf slope. Flow along depth contour lines limits the seawater exchange across the boundaries between the ocean and shelf seas, as discussed in the article Shelf sea exchange with the ocean.
Water fluxes between ocean and shelf sea require ageostrophic flow (transport across depth contours). Ageostrophic flow is enabled by flow variations on time scales of a day or less and space scales of a few kilometres or less, by depth contours changing direction on similarly short space scales or by enough friction to stop the flow in a day or so. Processes include internal waves and their Stokes drift, tidal pumping, eddies, Ekman transport in the wind-driven surface layer and bottom boundary layer[1].
Mathematical derivations
Uniformity of the horizontal current velocities over the vertical follows from the assumption of homogeneous seawater density, [math]\partial \rho / \partial x = \partial \rho / \partial y =0[/math]. Indeed, derivation of Eq. (1) with respect to the vertical coordinate [math]z[/math] gives [math]\partial u / \partial z = \partial v / \partial z =0[/math].
We consider large spatial scales, but also small enough to assume the Coriolis parameter [math]f[/math] as constant. In that case we have
[math]\Large\frac{\partial u}{\partial x}\normalsize + \Large\frac{\partial v}{\partial y}\normalsize = \Large\frac{1}{\rho f} \Big( \frac{\partial^2 p}{\partial y \partial x}\normalsize - \Large\frac{\partial^2 p}{\partial x \partial y}\normalsize \Big) = 0 . [/math]
From the continuity equation we have [math]\Large\frac{\partial w}{\partial z}\normalsize = -\Bigl[ \Large\frac{\partial u}{\partial x}\normalsize + \Large\frac{\partial v}{\partial y}\normalsize \Bigr] =0 . [/math]
Because [math]\partial w(z) / \partial z =0 [/math] and [math]w=0[/math] at the water surface, the vertical velocity [math]w[/math] vanishes throughout the vertical. As the flow must follow the seabed, we have
[math]w = u \Large\frac{\partial D}{\partial x}\normalsize + v \Large\frac{\partial D}{\partial y}\normalsize =0 . \qquad (2)[/math]
This equation expresses that geostrophic flow is perpendicular to the depth gradient. Geostrophic flow therefore follows contour lines of equal depth.
It can be shown that uniformity of the horizontal current velocities also holds for non-stationary flow driven by earth rotation in the absence of friction and density gradients. The flow equations are in this case
Continuity equation: [math]\Large\frac{\partial D}{\partial t}\normalsize + \vec{\nabla} . (D \vec{u}) =0 . \qquad (3)[/math]
Momentum balance equation: [math]\Large\frac{\partial \vec{u}}{\partial t}\normalsize + (\vec{u}.\vec{\nabla}) \vec{u} + f \, \vec{k} \; \text{x} \; \vec{u} + g \vec{\nabla}D =0 . \qquad (4)[/math]
Here we have used the conventions [math]\quad \vec{\nabla} = (\Large\frac{\partial}{\partial x}\normalsize, \Large\frac{\partial}{\partial y}\normalsize, 0) \quad[/math], [math]\quad \vec{u} = (u, v, 0) \quad [/math], [math]\quad \vec{k} = (0,0,1) . [/math]
We take the curl ([math]\vec{\nabla} \; \text{x}[/math]) of Eq. (4) and use the mathematical equivalences
[math]\vec{\nabla} \; \text{x} \; \vec{k} \; \text{x} \; \vec{u} = \vec{\nabla}.\vec{u} , \quad \vec{\nabla} \; \text{x} \; (\vec{u}.\vec{\nabla}) \vec{u} = (\vec{u}.\vec{\nabla}) \, \omega + \omega \, \vec{\nabla}.\vec{u} ,[/math]
where [math]\omega \equiv \Large\frac{\partial v}{\partial x}\normalsize -\Large\frac{\partial u}{\partial y}\normalsize [/math] is the vorticity, and [math]\vec{\omega}[/math] is the vorticity vector defined by [math]\vec{\omega} = \vec{\nabla} \; \text{x} \; \vec{u} = \omega \, \vec{k} .[/math] The vorticity is a measure of the rotation speed of the flow (twice the angular velocity speed in case of a circular flow pattern).
After some manipulations and using Eq. (3) we find
[math]\Large\frac{d}{dt}\normalsize (\omega + f) = - (\omega+f) \, \vec{\nabla}.\vec{u} ,\qquad (5) [/math]
where [math]\Large\frac{d}{dt}\normalsize = \Large\frac{\partial}{\partial t}\normalsize + \vec{u}.\vec{\nabla}[/math] is the derivative in a frame moving with the flow.
Further manipulation gives
[math]\Large\frac{d}{dt} \Big( \frac{\omega + f}{D} \Big) \normalsize = 0 . \qquad (6)[/math]
This equation expresses conservation of the potential vorticity [math]\omega_{pot} = \Large\frac{\omega + f}{D}\normalsize [/math] along streamlines. The stretching of the water column when the flow is directed towards deeper water results in an increase in vorticity [math]\omega[/math], i.e., a faster rotation of the flow.
Conservation of potential vorticity does not hold in the case of strong friction. Seabed frictional stress [math]\; \tau_b^{(x)} , \; \tau_b^{(y)}[/math] in respectively [math]x[/math] and [math]y[/math] direction changes Eq. (6) into
[math]\Large\frac{d}{dt} \Big( \frac{\omega + f}{D} \Big) \normalsize = \Large \Bigl[ \frac{\partial}{\partial y} \Big( \frac{\tau_b^{(x)}}{\rho D} \Big) - \frac{\partial}{\partial x} \Big( \frac{\tau_b^{(y)}}{\rho D} \Big) \Bigr] \normalsize. \qquad (7)[/math]
Related articles
- Coriolis acceleration
- Shelf sea exchange with the ocean
- Ekman transport
- Coriolis and tidal motion in shelf seas
References
- ↑ Huthnance, J., Hopkins, J., Berx, B., Dale, A., Holt, J., Hosegood, P., Inall, M., Jones, D., Loveday, B.R., Miller, P.I., Polton, J., Porter, M. and Spingys, C. 2022. Ocean shelf exchange, NW European shelf seas: Measurements, estimates and comparisons. Progress in Oceanography 202, 102760
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