Geostrophic flow

Definition of Geostrophic flow: Flow on a rotating Earth for which the horizontal pressure-gradient force and Coriolis acceleration are in approximate balance. The resulting flow is directed approximately along lines of equal pressure. This is the common definition for Geostrophic flow, other definitions can be discussed in the article

Coriolis acceleration

The geostrophic approximation applies to large-scale flows on a rotating Earth for which frictional forces, vertical accelerations and nonlinear inertial effects are small compared to the Coriolis acceleration and horizontal pressure-gradient force. This is generally the case when the Rossby number

[math]Ro = U/(fL)[/math]

is much smaller than 1, where [math]U[/math] and [math]L[/math] are characteristic velocity and length scales (see the article 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 Earth’s angular rotation rate 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

Geostrophic balance can occur both in homogeneous seawater (barotropic flow) and in stratified seawater with density gradients (baroclinic flow).

Conditions of geostrophic flow ([math]Ro \lt 1[/math]) are:

• Frictional effects are small (turbulence-induced frictional velocities [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]

In the special case considered in the following sections, seawater density is assumed homogeneous. Under this assumption:

1. The horizontal current velocity is uniform over the water column
2. 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. Ageostrophic processes include internal waves and their Stokes drift, tidal pumping, eddies, Ekman transport in the wind-driven surface layer and bottom boundary layer.

Barotropic and baroclinic geostrophic flow

If seawater density is horizontally uniform, the horizontal geostrophic velocity does not vary with depth. This situation is called barotropic geostrophic flow.

In the ocean, however, density gradients caused by temperature and salinity differences are often important. In that case the geostrophic velocity varies with depth according to the thermal-wind relation,

[math]f \dfrac{\partial u}{\partial z} = -\dfrac{g}{\rho}\dfrac{\partial \rho}{\partial y}, \qquad f \dfrac{\partial v}{\partial z} = \dfrac{g}{\rho}\dfrac{\partial \rho}{\partial x} . \qquad (3)[/math]

The derivation assumes that the relative vertical change of the density is small compared to the relative change of the velocity, meaning that [math]\partial (\rho v)/dz \approx \rho \partial v/dz[/math] and [math]\partial (\rho u)/dz \approx \rho \partial u/dz[/math]. According to this assumption, the thermal wind equations (3) are obtained by taking the partial [math]z-[/math]derivative of the geostrophic equations (1,2) and substituting [math]\partial p/\partial z \approx – g \rho .[/math]

The thermal wind equations represent baroclinic geostrophic flow.

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 (4)[/math]

This equation expresses that geostrophic flow is perpendicular to the depth gradient – the flow follows contour lines of equal depth. Ocean currents do not cross the continental slope in the absence of friction. Perturbations of the ocean circulation pattern are therefore hardly perceived in the coastal zone.

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. To investigate departures from steady geostrophic balance and the influence of Earth rotation on large-scale ocean circulation, we now consider the depth-averaged shallow-water equations. The flow equations are in this case

Continuity equation: [math]\Large\frac{\partial D}{\partial t}\normalsize + \vec{\nabla} . (D \vec{u}) =0 . \qquad (5)[/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 (6)[/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. (6) 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. (5) we find

[math]\Large\frac{d}{dt}\normalsize (\omega + f) = - (\omega+f) \, \vec{\nabla}.\vec{u} ,\qquad (7) [/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.

Potential-vorticity conservation is one of the fundamental constraints governing large-scale ocean circulation. It explains why ocean currents tend to follow bathymetric contours and why changes in water-column thickness are associated with changes in flow vorticity.

Further manipulation gives

[math]\Large\frac{d}{dt} \Big( \frac{\omega + f}{D} \Big) \normalsize = 0 . \qquad (8)[/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. Equation (7) also shows that flow along bathymetric depth contours ([math]dD/dt = 0 [/math]) implies conservation of the total vorticity [math]\omega+f[/math].

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. (8) 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 (9)[/math]

The change of the potential vorticity in a frame moving with the flow is given by the curl of the seabed stress. Equation (9) shows that wind stress and bottom friction modify the potential vorticity of the flow through the curl of the applied stresses. Frictional effects therefore enable transport across depth contours and exchange between shelf seas and the open ocean.

Geostrophic adjustment

Large-scale ocean flows tend toward geostrophic balance through a process known as geostrophic adjustment. When pressure gradients are initially unbalanced, gravity waves and inertial oscillations redistribute mass and momentum until an approximate balance between pressure-gradient force and Coriolis acceleration is established.

Western boundary currents

The influence of Earth rotation on large-scale ocean circulation becomes especially important in western boundary currents such as the Gulf Stream and Kuroshio. These currents are highly unstable; they are strongly meandering and pinch off large eddies (see also Shelf sea exchange with the ocean). It has therefore been suggested that these large horizontal eddies can transfer momentum from the boundary current to the shallow shelf due to frictional processes and that this process can be described by a horizontal diffusion coefficient [math]K [/math] ('eddy viscosity') .

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 (10)[/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 (10) 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 (11)[/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, for example [math]\, K=Ax, \; A \approx 0.1 m/s[/math], then the solution of Eq. (10) with boundary conditions [math]v=0[/math] at [math]x=0[/math] and [math]x \rightarrow \infty[/math], represents a current along the western ocean boundary with width given by [math]l \approx \sqrt{A/\beta}[/math] . The order of magnitude of the modeled width is 100 km, which is consistent with the observed width of western boundary currents in the Atlantic (Gulf Stream, Brazil Current), Pacific (Kuroshio, East Australian Current) and Indian Oceans (Agulhas Current), see Ocean circulation.

In summary

Geostrophic balance is one of the central organizing principles of large-scale ocean circulation. Although exact geostrophic conditions are rarely met in nature, many ocean currents can be understood as small departures from geostrophic flow caused by friction, density gradients, wind forcing and time-dependent motions.

Related articles

Coriolis acceleration

Shelf sea exchange with the ocean

Ekman transport

Coriolis and tidal motion in shelf seas

Ocean circulation

References

The main author of this article is Job Dronkers Please note that others may also have edited the contents of this article. Citation: Job Dronkers (2026): Geostrophic flow. Available from http://www.coastalwiki.org/wiki/Geostrophic_flow [accessed on 7-05-2026] For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Dronkers J