10.6: Comments on Geostrophic Currents

Now that we know how to calculate geostrophic currents from hydrographic data, let’s consider some of the limitations of the theory and techniques.

Converting Relative Velocity to Velocity

Hydrographic data give geostrophic currents relative to geostrophic currents at some reference level. How can we convert the relative geostrophic velocities to velocities relative to the Earth?

1. Assume a Level of No Motion: Traditionally, oceanographers assume there is a level of no motion, sometimes called a reference surface, roughly 2,000 m below the surface. This is the assumption used to derive the currents in Table \(10.5.3\). Currents are assumed to be zero at this depth, and relative currents are integrated up to the surface and down to the bottom to obtain current velocity as a function of depth. There is some experimental evidence that such a level exists on average for mean currents (see for example, Defant, 1961: 492).
   Defant recommends choosing a reference level where the current shear in the vertical is smallest. This is usually near 2 km. This leads to useful maps of surface currents because surface currents tend to be faster than deeper currents. Figure \(\PageIndex{1}\) shows the geopotential anomaly and surface currents in the Pacific relative to the 1,000 dbar pressure level.

   

   Figure \(\PageIndex{1}\): Mean geopotential anomaly relative to the 1,000 dbar surface in the Pacific based on 36,356 observations. Height of the anomaly is in geopotential centimeters. If the velocity at 1,000 dbar were zero, the map would be the surface topography of the Pacific. After Wyrtki (1979).

2. Use known currents: The known currents could be measured by current meters or by satellite altimetry. Problems arise if the currents are not measured at the same time as the hydrographic data. For example, the hydrographic data may have been collected over a period of months to decades, while the currents may have been measured over a period of only a few months. Hence, the hydrography may not be consistent with the current measurements. Sometimes currents and hydrographic data are measured at nearly the same time (figure \(\PageIndex{2}\)). In this example, currents were measured continuously by moored current meters (points) in a deep western boundary current and calculated from ctd data taken just after the current meters were deployed and just before they were recovered (smooth curves). The solid line is the current assuming a level of no motion at 2,000 m, the dotted line is the current adjusted using the current meter observations smoothed for various intervals before or after the ctd casts.
3. Use Conservation Equations: Lines of hydrographic stations across a strait or an ocean basin may be used with conservation of mass and salt to calculate currents. This is an example of an inverse problem (Wunsch, 1996 describes the application of inverse methods in oceanography). See Mercier et al. (2003) for a description of how they determined the circulation in the upper layers of the eastern basins of the south Atlantic using hydrographic data from the World Ocean Circulation Experiment and direct measurements of current in a box model constrained by inverse theory.

Figure \(\PageIndex{2}\): Current meter measurements can be used with CTD measurements to determine current as a function of depth avoiding the need for assuming a depth of no motion. Solid line: profile assuming a depth of no motion at 2000 decibars. Dashed line: profile adjusted to agree with currents measured by current meters 1–7 days before the CTD measurements. (Plots from Tom Whitworth, Texas A&M University).

Disadvantage of Calculating Currents from Hydrographic Data

Currents calculated from hydrographic data have been used to make maps of ocean currents since the early 20th century. Nevertheless, it is important to review the limitations of the technique.

1. Hydrographic data can be used to calculate only the current relative to a current at another level.
2. The assumption of a level of no motion may be suitable in the deep ocean, but it is usually not a useful assumption when the water is shallow such as over the continental shelf.
3. Geostrophic currents cannot be calculated from hydrographic stations that are close together. Stations must be tens of kilometers apart.

Limitations of the Geostrophic Equations

I began this section by showing that the geostrophic balance applies with good accuracy to flows that exceed a few tens of kilometers in extent and with periods greater than a few days. The balance cannot, however, be perfect. If it were, the flow in the ocean would never change because the balance ignores any acceleration of the flow. The important limitations of the geostrophic assumption are:

1. Geostrophic currents cannot evolve with time because the balance ignores acceleration of the flow. Acceleration dominates if the horizontal dimensions are less than roughly 50 km and times are less than a few days. Acceleration is negligible, but not zero, over longer times and distances.
2. The geostrophic balance does not apply within about 2\(^{\circ}\) of the equator where the Coriolis force goes to zero because \(\sin \varphi \rightarrow 0\).
3. The geostrophic balance ignores the influence of friction.

Accuracy

Strub et al. (1997) showed that currents calculated from satellite altimeter measurements of sea-surface slope have an accuracy of \(\pm 3-5 \ \text{cm/s}\). Uchida, Imawaki, and Hu (1998) compared currents measured by drifters in the Kuroshio with currents calculated from satellite altimeter data assuming geostrophic balance. Using slopes over distances of 12.5 km, they found the difference between the two measurements was \(\pm 16 \ \text{cm/s}\) for currents up to 150 cm/s, or about 10%.

Johns, Watts, and Rossby (1989) measured the velocity of the Gulf Stream northeast of Cape Hatteras and compared the measurements with velocity calculated from hydrographic data assuming geostrophic balance. They found that the measured velocity in the core of the stream, at depths less than 500 m, was 10–25 cm/s faster than the velocity calculated from the geostrophic equations using measured velocities at a depth of 2000 m. The maximum velocity in the core was greater than 150 cm/s, so the error was \(\approx 10\%\). When they added the influence of the curvature of the Gulf Stream, which adds an acceleration term to the geostrophic equations, the difference in the calculated and observed velocity dropped to less than \(5-10 \ \text{cm/s} \ (\approx 5\%)\).