10.3: Surface Geostrophic Currents from Altimetry

The geostrophic approximation applied at \(z = 0\) leads to a very simple relation: surface geostrophic currents are proportional to surface slope. Consider a level surface slightly below the sea surface, say two meters below the sea surface, at \(z = -r\) (figure \(\PageIndex{1}\)).

Figure \(\PageIndex{1}\): Sketch defining \(\zeta\) and \(r\), used for calculating pressure just below the sea surface.

The pressure on the level surface is: \[p = \rho g (\zeta + r) \nonumber \]

assuming \(\rho\) and \(g\) are essentially constant in the upper few meters of the ocean. Substituting this into \((10.2.2)\) gives the two components \((u_{s}, v_{s})\) of the surface geostrophic current: \[u_{s} = -\frac{g}{f} \frac{\partial \zeta}{\partial y}; \quad\quad v_{s} = \frac{g}{f} \frac{\partial \zeta}{\partial x} \nonumber \]

where \(g\) is gravity, \(f\) is the Coriolis parameter, and \(\zeta\) is the height of the sea surface above a level surface.

The Oceanic Topography

In Section 3.4 we defined the topography of the sea surface \(\zeta\) to be the height of the sea surface relative to a particular level surface, the geoid; and we defined the geoid to be the level surface that coincided with the surface of the ocean at rest. Thus, according to \((\PageIndex{2})\) the surface geostrophic currents are proportional to the slope of the topography (figure \(\PageIndex{2}\)), a quantity that can be measured by satellite altimeters if the geoid is known.

Figure \(\PageIndex{2}\): The slope of the sea surface relative to the geoid \((\partial \zeta/\partial x)\) is directly related to surface geostrophic currents \(v_{s}\). The slope of 1 meter per 100 kilometers \((10 \ \mu \text{rad})\) is typical of strong currents. \(V_{s}\) is into the screen in the northern hemisphere.

Because the geoid is a level surface, it is a surface of constant geopotential. To see this, consider the work done in moving a mass \(m\) by a distance \(h\) perpendicular to a level surface. The work is \(W = mgh\), and the change of potential energy per unit mass is \(gh\). Thus level surfaces are surfaces of constant geopotential, where the geopotential is \(\Phi = gh\).

Topography is due to processes that cause the ocean to move: tides, ocean currents, and the changes in barometric pressure that produce the inverted barometer effect. Because the ocean’s topography is due to dynamical processes, it is usually called dynamic topography. The topography is approximately one-hundredth of the geoid undulations. Thus the shape of the sea surface is dominated by local variations of gravity. The influence of currents is much smaller. Typically, sea-surface topography has amplitude of \(\pm 1 \ \text{m}\) (figure \(\PageIndex{3}\)). Typical slopes are \(\partial \zeta/\partial x \approx 1–10\) microradians for \(v = 0.1-1.0 \ \text{m/s}\) at mid latitude.

The height of the geoid, smoothed over horizontal distances greater than roughly 400 km, is known with an accuracy of \(\pm 1 \ \text{mm}\) from data collected by the Gravity Recovery and Climate Experiment GRACE satellite mission.

Figure \(\PageIndex{3}\): Topex/Poseidon altimeter observations of the Gulf Stream. When the altimeter observations are subtracted from the local geoid, they yield the oceanic topography, which is due primarily to ocean currents in this example. The gravimetric geoid was determined by the Ohio State University from ship and other surveys of gravity in the region. From Center for Space Research, University of Texas.

Satellite Altimetry

Very accurate satellite-altimeter systems are needed for measuring the oceanic topography. The first systems, carried on Seasat, Geosat, ERS–1, and ERS–2 were designed to measure week-to-week variability of currents. Topex/Poseidon, launched in 1992, was the first satellite designed to make the much more accurate measurements necessary for observing the permanent (time-averaged) surface circulation of the ocean, tides, and the variability of gyre-scale currents. It was followed in 2001 by Jason and in 2008 by Jason-2.

Because the geoid was not well known locally before about 2004, altimeters were usually flown in orbits that have an exactly repeating ground track. Thus Topex/Poseidon and Jason fly over the same ground track every 9.9156 days. By subtracting sea-surface height from one traverse of the ground track from height measured on a later traverse, changes in topography can be observed without knowing the geoid. The geoid is constant in time, and the subtraction removes the geoid, revealing changes due to changing currents, such as mesoscale eddies, assuming tides have been removed from the data (figure \(\PageIndex{4}\)). Mesoscale variability includes eddies with diameters between roughly 20 and 500 km.

Figure \(\PageIndex{4}\): Global distribution of standard deviation of topography from Topex/Poseidon altimeter data from 10/3/92 to 10/6/94. The height variance is an indicator of variability of surface geostrophic currents. From Center for Space Research, University of Texas.

The great accuracy and precision of the Topex/Poseidon and Jason altimeter systems allow them to measure the oceanic topography over ocean basins with an accuracy of \(\pm (2–5) \ \text{cm}\) (Chelton et al, 2001). This allows them to measure:

1. Changes in the mean volume of the ocean and sea-level rise with an accuracy of \(\pm 0.4 \ \text{mm/yr}\) since 1993 (Nerem et al, 2006);
2. Seasonal heating and cooling of the ocean (Chambers et al 1998);
3. Open ocean tides with an accuracy of \(\pm (1–2) \ \text{cm}\) (Shum et al, 1997);
4. Tidal dissipation (Egbert and Ray, 1999; Rudnick et al, 2003);
5. The permanent surface geostrophic current system (figure \(\PageIndex{5}\));
6. Changes in surface geostrophic currents on all scales (figure \(\PageIndex{4}\)); and
7. Variations in topography of equatorial current systems such as those associated with El Niño (figure 10.6).

Figure \(\PageIndex{5}\): Global distribution of time-averaged topography of the ocean from Topex/Poseidon altimeter data from 10/3/92 to 10/6/99 relative to the JGM–3 geoid. Geostrophic currents at the ocean surface are parallel to the contours. Compare with figure \(2.3.2\), calculated from hydrographic data. From Center for Space Research, University of Texas.

Figure \(\PageIndex{6}\): Time-longitude plot of sea-level anomalies in the Equatorial Pacific observed by Topex/Poseidon during the 1997–1998 El Niño. Warm anomalies are light gray, cold anomalies are dark gray. The anomalies are computed from 10-day deviations from a three-year mean surface from 3 Oct 1992 to 8 Oct 1995. The data are smoothed with a Gaussian weighted filter with a longitudinal span of 5\(^{\circ}\) and a latitudinal span of 2\(^{\circ}\)◦. The annotations on the left are cycles of satellite data. From Center for Space Research, University of Texas.

Altimeter Errors (Topex/Poseidon and Jason)

The most accurate observations of the sea-surface topography are from Topex/Poseidon and Jason. Errors for these satellite altimeter system are due to (Chelton et al, 2001):

1. Instrument noise, ocean waves, water vapor, free electrons in the ionosphere, and mass of the atmosphere. Both satellites carried a precise altimeter system able to observe the height of the satellite above the sea surface between \(\pm 66^{\circ}\) latitude with a precision of \(\pm(1–2) \ \text{cm}\) and an accuracy of \(\pm (2–5) \ \text{cm}\). The systems consist of a two-frequency radar altimeter to measure height above the sea, the influence of the ionosphere, and wave height, and a three-frequency microwave radiometer able to measure water vapor in the troposphere.
2. Tracking errors. The satellites carried three tracking systems that enable their position in space, the ephemeris, to be determined with an accuracy of \(\pm (1–3.5) \ \text{cm}\).
3. Sampling error. The satellites measure height along a ground track that repeats within ±1 km every 9.9156 days. Each repeat is a cycle. Because currents are measured only along the sub-satellite track, there is a sampling error. The satellite cannot map the topography between ground tracks, nor can they observe changes with periods less than \(2 \times 9.9156 d\) (see Section 16.3).
4. Geoid error. The permanent topography is not well known over distances shorter than a hundred kilometers because geoid errors dominate for short distances. Maps of topography smoothed over greater distances are used to study the dominant features of the permanent geostrophic currents at the sea surface (figure \(\PageIndex{5}\)). New satellite systems grace and champ are measuring earth’s gravity accurately enough that the geoid error is now small enough to ignore over distances greater than 100 km.

Taken together, the measurements of height above the sea and the satellite position give sea-surface height in geocentric coordinates within \(\pm (2–5) \ \text{cm}\). Geoid error adds further errors that depend on the size of the area being measured.