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 \; , \qquad (1)[/math]
[math] \Large\frac{1}{\rho}\frac{\partial p}{\partial y}\normalsize = - f \, u . \qquad (2)[/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
Characteristics of geostrophic flow
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 Eqs. (1, 2) 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 (3)[/math]
This equation expresses that geostrophic flow is perpendicular to the depth gradient. Geostrophic flow therefore follows contour lines of equal depth.
Vorticity
Vorticity, defined as [math]\omega \equiv \Large\frac{\partial v}{\partial x}\normalsize -\Large\frac{\partial u}{\partial y}\normalsize [/math], is a measure of the rotation speed of the flow (twice the angular velocity speed in case of a circular flow pattern). Vorticity refers here to the large-scale geostrophic gyres occurring in the ocean.
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 (4)[/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 (5)[/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] The vorticity vector [math]\vec{\omega}[/math] is defined as [math]\vec{\omega} = \vec{\nabla} \; \text{x} \; \vec{u} = \omega \, \vec{k} .[/math]
We take the curl ([math]\vec{\nabla} \; \text{x}[/math]) of Eq. (5) 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]
After some manipulations and using Eq. (4) we find
[math]\Large\frac{d}{dt}\normalsize (\omega + f) = - (\omega+f) \, \vec{\nabla}.\vec{u} ,\qquad (6) [/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 (7)[/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 presence of frictional stresses. Seabed frictional stresses [math]\; \tau_b^{(x)} , \; \tau_b^{(y)}[/math] in respectively [math]x[/math] and [math]y[/math] direction or surface wind stresses [math]\; \tau_w^{(x)} , \; \tau_w^{(y)}[/math] change Eq. (7) into
[math]\Large\frac{d}{dt} \Big( \frac{f+\omega}{D} \Big) \normalsize + \Large \frac{1}{D} \Bigl[ \frac{\partial}{\partial y} \Big( \frac{\tau_w^{(x)}}{\rho D} \Big) - \frac{\partial}{\partial x} \Big( \frac{\tau_w^{(y)}}{\rho D} \Big) \Bigr] \normalsize - \Large\frac{1}{D} \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 =0. \qquad (8)[/math]
The change of the potential vorticity in a frame moving with the flow is given by the curl of the seabed stress.
Western boundary currents
Along-shelf geostrophic currents, such as the Gulf Stream, are highly unstable. These boundary currents are strongly meandering and pinch off large eddies[2] (see also Shelf sea exchange with the ocean). It has therefore been suggested that these large horizontal eddies transfer momentum from the boundary current to the shallow shelf and that this process can be described by a horizontal diffusion coefficient [math]K [/math] ('eddy viscosity') [3]. A simple mathematical description can be obtained by assuming that the geostrophic current follows the shelf boundary in [math]S \rightarrow N[/math] direction (Northern Hemisphere), coinciding with the [math]y[/math] axis, and that the mean current velocity [math]v[/math] is stationary and uniform in [math]y[/math] direction. The [math]x[/math] axis is oriented [math]E \rightarrow W[/math] and the mean cross-boundary flow [math]u[/math] is assumed negligible compared to the along-boundary flow. Instead of Eq. (2) we then have the momentum balance in the well-mixed surface layer
[math] \Large\frac{1}{\rho}\frac{\partial p}{\partial y}\normalsize = \Large\frac{\partial}{\partial x}\normalsize \overline{\lt u'v'\gt } = \Large\frac{\partial}{\partial x}\normalsize \Big( K \Large\frac{\partial v}{\partial x}\normalsize \Big) , \qquad (9)[/math]
where [math]u', v'[/math] are temporal and spatial velocity fluctuations related to the eddy motions, and where the square brackets and overline represent the temporal and spatial mean. Cross-differentiating Eqs. (1) and (9) gives
[math]\beta \, v = \Large\frac{\partial^2}{\partial x^2}\normalsize \Big( K \Large\frac{\partial v}{\partial x}\normalsize \Big) , \quad \beta = \Large\frac{df}{dy}\normalsize = 2 (\Omega / R) \cos \phi , \qquad (10)[/math]
where [math]R[/math] is the Earth's radius.
If the eddy viscosity [math]K[/math] is assumed to be an increasing function of the distance [math]x[/math] from the shelf boundary, then the solution of Eq. (10) represents an along-shelf boundary current which is intensified against western shelf boundaries but not against eastern boundaries[4]. This is consistent with the observed pattern of western boundary currents in the Atlantic (Gulf Stream, Brazil Current), Pacific (Kuroshio, East Australian Current) and Indian Oceans (Agulhas Current), see Ocean circulation.
Related articles
- Coriolis acceleration
- Shelf sea exchange with the ocean
- Ekman transport
- Coriolis and tidal motion in shelf seas
- Ocean circulation
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
- ↑ Bower, A. S., H. T. Rossby and J. L. Lillibridge 1985. The Gulf Stream - barrier or blender? J. Phys. Oceanogr. 15: 24–32
- ↑ Munk, W.H. 1950. On the wind-driven ocean circulation. J. Meteorology 7: 79-93
- ↑ Webb, D.J. 1999. An Analytic Model of the Agulhas Current as a Western Boundary Current with Linearly Varying Viscosity. J. Phys. Ocean. 1517-1527
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