10.4: Geostrophic Currents from Hydrography

The geostrophic equations are widely used in oceanography to calculate currents at depth. The basic idea is to use hydrographic measurements of temperature, salinity or conductivity, and pressure to calculate the density field of the ocean using the equation of state of sea water. Density is used in \((10.2.3)\) to calculate the internal pressure field, from which the geostrophic currents are calculated using \((10.2.4-5)\). Usually, however, the constant of integration in these equations is not known, and only the relative velocity field can be calculated.

At this point, you may ask, why not just measure pressure directly as is done in meteorology, where direct measurements of pressure are used to calculate winds? And, aren’t pressure measurements needed to calculate density from the equation of state? The answer is that very small changes in depth make large changes in pressure because water is so heavy. Errors in pressure caused by errors in determining the depth of a pressure gauge are much larger than the pressure due to currents. For example, using \((10.2.2)\), we calculate that the pressure gradient due to a 10 cm/s current at 30\(^{\circ}\) latitude is \(7.5 \times 10^{-3} \ \text{Pa/m}\), which is 750 Pa in 100 km. From the hydrostatic equation \((10.1.5)\), 750 Pa is equivalent to a change of depth of 7.4 cm. Therefore, for this example, we must know the depth of a pressure gauge with an accuracy of much better than 7.4 cm. This is not possible.

Geopotential Surfaces Within the Ocean

Calculation of pressure gradients within the ocean must be done along surfaces of constant geopotential just as we calculated surface pressure gradients relative to the geoid when we calculated surface geostrophic currents. As long ago as 1910, Vilhelm Bjerknes (Bjerknes and Sandstrom, 1910) realized that such surfaces are not at fixed heights in the atmosphere because \(g\) is not constant, and \((10.1.4)\) must include the variability of gravity in both the horizontal and vertical directions (Saunders and Fofonoff, 1976) when calculating pressure in the ocean.

The geopotential \(\Phi\) is: \[\Phi = \int_{0}^{z} g \ dz \nonumber \]

Because \(\Phi/9.8\) in SI units has almost the same numerical value as height in meters, the meteorological community accepted Bjerknes’ proposal that height be replaced by dynamic meters \(D = \Phi/10\) to obtain a natural vertical coordinate. Later, this was replaced by the geopotential meter (gpm) \(Z = \Phi/9.80\). The geopotential meter is a measure of the work required to lift a unit mass from sea level to a height \(z\) against the force of gravity. Harald Sverdrup, Bjerknes’ student, carried the concept to oceanography, and depths in the ocean are often quoted in geopotential meters. The difference between depths of constant vertical distance and constant potential can be relatively large. For example, the geometric depth of the 1000 dynamic meter surface is 1017.40 m at the north pole and 1022.78 m at the equator, a difference of 5.38 m.

Note that depth in geopotential meters, depth in meters, and pressure in decibars are almost the same numerically. At a depth of 1 meter the pressure is approximately 1.007 decibars and the depth is 1.00 geopotential meters.

Equations for Geostrophic Currents Within the Ocean

To calculate geostrophic currents, we need to calculate the horizontal pressure gradient within the ocean. This can be done using either of two approaches:

1. Calculate the slope of a constant pressure surface relative to a surface of constant geopotential. We used this approach when we used sea-surface slope from altimetry to calculate surface geostrophic currents. The sea surface is a constant-pressure surface. The constant geopotential surface was the geoid.
2. Calculate the change in pressure on a surface of constant geopotential. Such a surface is called a geopotential surface.

Figure \(\PageIndex{1}\): Sketch of geometry used for calculating geostrophic current from hydrography.

Oceanographers usually calculate the slope of constant-pressure surfaces. The important steps are:

1. Calculate differences in geopotential \(\left((\Phi_{A} - \Phi_{B}\right)\) between two constant-pressure surfaces \(\left(P_{1}, P_{2}\right)\) at hydrographic stations A and B (figure \(\PageIndex{1}\)). This is similar to the calculation of \(\zeta\) of the surface layer.
2. Calculate the slope of the upper pressure surface relative to the lower.
3. Calculate the geostrophic current at the upper surface relative to the current at the lower. This is the current shear.
4. Integrate the current shear from some depth where currents are known to obtain currents as a function of depth. For example, from the surface downward, using surface geostrophic currents observed by satellite altimetry, or upward from an assumed level of no motion.

To calculate geostrophic currents oceanographers use a modified form of the hydrostatic equation. The vertical pressure gradient \((10.2.1)\) is written \[\begin{align} \frac{\delta p}{\rho} = \alpha \ \delta p &= -g \ \delta z \\[4pt] \alpha \ \delta p &= \delta \Phi \end{align} \nonumber \]

where \(\alpha = \alpha(S, t, p)\) is the specific volume, and \((\PageIndex{3})\) follows from \((\PageIndex{1})\). Differentiating \((\PageIndex{3})\) with respect to horizontal distance \(x\) allows the geostrophic balance to be written in terms of the slope of the constant-pressure surface using \((10.2.1)\) with \(f = 2\Omega \sin \varphi\): \[\begin{align} \alpha \frac{\partial p}{\partial x} = \frac{1}{\rho} \frac{\partial p}{\partial x} &= -2 \Omega \ v \sin \varphi \\ \frac{\partial \Phi (p = p_{0})}{\partial x} &= -2 \Omega \ v \sin \varphi \end{align} \nonumber \]

where \(\Phi\) is the geopotential at the constant-pressure surface. Now let’s see how hydrographic data are used for evaluating \(\partial \Phi/\partial x\) on a constant-pressure surface. Integrating \((\PageIndex{3})\) between two constant-pressure surfaces \(\left(P_{1}, P_{2}\right)\) in the ocean as shown in figure \(\PageIndex{1}\) gives the geopotential difference between two constant-pressure surfaces. At station A the integration gives: \[\Phi \left(P_{1A}\right) - \Phi \left(P_{2A}\right) = \int_{P_{1A}}^{P_{2A}} \alpha (S,t,p) \ dp \nonumber \]

The specific volume anomaly is written as the sum of two parts: \[\alpha (S,t,p) = \alpha (35,0,p) + \delta \nonumber \]

where \(\alpha(35, 0, p)\) is the specific volume of sea water with salinity of 35, temperature of \(0^{\circ}\text{C}\), and pressure \(p\). The second term \(\delta\) is the specific volume anomaly. Using \((\PageIndex{7})\) in \((\PageIndex{6})\) gives: \[\begin{align*} \Phi \left(P_{1A}\right) - \Phi \left(P_{2A}\right) &= \int_{P_{1A}}^{P_{2A}} \alpha (35,0,p) \ dp + \int_{P_{1A}}^{P_{2A}} \delta \ dp \\[4pt] \Phi \left(P_{1A}\right) - \Phi \left(P_{2A}\right) &= \left(\Phi_{1} - \Phi_{2}\right)_{std} + \Delta \Phi_{A} \end{align*} \]

where \(\left(\Phi_{1} - \Phi_{2}\right)_{std}\) is the standard geopotential distance between two constant-pressure surfaces \(P_{1}\) and \(P_{2}\), and \[\Delta \Phi_{A} = \int_{P_{1A}}^{P_{2A}} \delta \ dp \nonumber \]

is the anomaly of the geopotential distance between the surfaces. It is called the geopotential anomaly. The geometric distance between \(\Phi_{2}\) and \(\Phi_{1}\) is numerically approximately \(\left(\Phi_{2} - \Phi_{1}\right)/g\) where \(g = 9.8 \ \text{m/s}^{2}\) is the approximate value of gravity. The geopotential anomaly is much smaller, being approximately 0.1% of the standard geopotential distance.

Consider now the geopotential anomaly between two pressure surfaces \(P_{1}\) and \(P_{2}\) calculated at two hydrographic stations \(A\) and \(B\) a distance \(L\) meters apart (figure \(\PageIndex{1}\)). For simplicity we assume the lower constant-pressure surface is a level surface. Hence the constant-pressure and geopotential surfaces coincide, and there is no geostrophic velocity at this depth. The slope of the upper surface is \[\frac{\Delta \Phi_{B} - \Delta \Phi_{A}}{L} = \text{slope of constant pressure } P_{2} \nonumber \]

because the standard geopotential distance is the same at stations \(A\) and \(B\). The geostrophic velocity at the upper surface calculated from \((\PageIndex{5})\) is: \[V = \frac{\left(\Delta \Phi_{B} - \Delta \Phi_{A}\right)}{2\Omega \ L \sin \varphi} \nonumber \]

where \(V\) is the velocity at the upper geopotential surface. The velocity \(V\) is perpendicular to the plane of the two hydrographic stations and directed into the plane of figure \(\PageIndex{1}\) if the flow is in the northern hemisphere. A useful rule of thumb is that the flow is such that warmer, lighter water is to the right looking downstream in the northern hemisphere.

Note that I could have calculated the slope of the constant-pressure surfaces using density \(\rho\) instead of specific volume \(\alpha\). I used \(\alpha\) because it is the common practice in oceanography, and tables of specific volume anomalies and computer code to calculate the anomalies are widely available. The common practice follows from numerical methods developed before calculators and computers were available, when all calculations were done by hand or by mechanical calculators with the help of tables and nomograms. Because the computation must be done with an accuracy of a few parts per million, and because all scientific fields tend to be conservative, the common practice has continued to use specific volume anomalies rather than density anomalies.

Barotropic and Baroclinic Flow:

If the ocean were homogeneous with constant density, then constant-pressure surfaces would always be parallel to the sea surface, and the geostrophic velocity would be independent of depth. In this case the relative velocity is zero, and hydrographic data cannot be used to measure the geostrophic current. If density varies with depth, but not with horizontal distance, the constant-pressure surfaces are always parallel to the sea surface and the levels of constant density, the isopycnal surfaces. In this case, the relative flow is also zero. Both cases are examples of barotropic flow.

Barotropic flow occurs when levels of constant pressure in the ocean are always parallel to the surfaces of constant density. Note, some authors call the vertically averaged flow the barotropic component of the flow. Wunsch (1996: 74) points out that “barotropic” is used in so many different ways that the term is meaningless and should not be used.

Baroclinic flow occurs when levels of constant pressure are inclined to surfaces of constant density. In this case, density varies with depth and horizontal position. A good example is seen in figure \(10.5.1\), which shows levels of constant density changing depth by more than 1 km over horizontal distances of 100 km at the Gulf Stream. Baroclinic flow varies with depth, and the relative current can be calculated from hydrographic data. Note, constant-density surfaces cannot be inclined to constant-pressure surfaces for a fluid at rest.

In general, the variation of flow in the vertical can be decomposed into a barotropic component which is independent of depth, and a baroclinic component which varies with depth.