Chapter 10 - Geostrophic Currents

 Chapter 10 Contents

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. A level surface is a surface of constant gravitational potential, and no work is required to move along a frictionless, level surface (Figure 10.1).

 Figure 10.1 Sketch defining z and r, used for calculating pressure just below the sea surface.

The pressure on the level surface is:

 p = ρ g ( ζ + r ) (10.9)

assuming ρ and g are essentially constant in the upper few meters of the ocean. Substituting this into (10.7a, b), gives the two components (us, vs) of the surface geostrophic current:

 (10.10)

where g is gravity, f is the Coriolis parameter, and ζ is the height of the sea surface above a level surface.

The Oceanic Topography
In §3.4 we define the topography of the sea surface ζ 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 (10.10) the surface geostrophic currents are proportional to the slope of the topography (Figure 10.2), a quantity that can be measured by satellite altimeters if the geoid is known.

 Figure 10.2 The slope of the sea surface relative to the geoid (∂z/∂x) is directly related to surface geostrophic currents vs. The slope of 1 meter per 100 kilometers (10-6rad) is typical of strong currents. Vs is into the paper 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 Φ = 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. This means that 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 ±1 m. Typical slopes are ∂z/x 1-10 microradians for v = 0.1 – 1.0 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 ±1 mm from data collected by the Gravity Recovery and Climate Experiment GRACE satellite mission.

 Figure 10.3 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, which operated from 1992 to 2005, was the first satellite designed to make the much more accurate measurements necessary for observing the permanent (time-averaged) surface circulation of the oceans, tides, and the variability of gyre-scale currents. I twas followed by Jason, launched in 2001, and by the Ocean Surface Topography Mission OSTM (Jason 2) launched in 2008.

Because the geoid was not well known locally before 2004, altimeters were usually flown in orbits that have an exactly repeating ground track. Thus Topex/Poseidon, Jason, and OSTM 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 variability, assuming tides have been removed from the data (Figure 10.4). Mesoscale variability includes eddies with diameters between roughly 20 km and 500 km.

 Figure 10.4 Global map of variance of topography from Topex/Poseidon and ERS satellite altimeter data from 12/92 to 11/98. The variance of topography is an indicator of variability of currents. From AVISO.

The great accuracy and precision of the Topex/Poseidon, Jason and OSTM altimeter systems allow them to measure the oceanic topography over ocean basins with an accuracy of ±2 cm. This allows them to measure:

1. Changes in the mean volume of the ocean (Born et al., 1986, Nerem, 1995);

Global mean sea level measured by accurate altimeters. Click on the image for a zoom.
From Sea-Level Change by Dr. R. Steven Nerem, Colorado Center for Astrodynamics Research, University of Colorado at Boulder.

2. Seasonal heating and cooling of the ocean (Chambers et al., 1998);
3. Tides. See figure 17.13 for a global map of the M2 twice per day lunar tide.(Andersen, Woodworth, and Flather, 1995);
4. The permanent surface geostrophic current system (Figure 10.5);
5. Changes in surface geostrophic currents on all scales (Figure 10.4); and
6. Variations in topography of equatorial current systems such as those associated with El Niño (Figure 10.6).
 Figure 10.5 Global, time-averaged topography of the ocean for the period 1992–2002 from a joint analysis of drifter, satellite altimeter, wind, and the GRACE Gravity Model-01 data by Nikolai Maximenko (IPRC) and Peter Niiler (SIO). Geostrophic currents at the ocean surface are parallel to the contours. Compare with Figure 2.8 calculated from hydrographic data. For more information, see Maximenko (2005) (a 1MB pdf file). From Asia-Pacific Data-Research Center.

 Figure 10.6 Time-longitude plot of sea-level anomalies in the Equatorial Pacific observed by Topex/Poseidon. Warm anomalies are light gray, cold anomalies are dark gray. The anomalies are computed from 10-day deviations from a mean surface computed from 10/3/1992 to 10/8/1995. The data are smoothed with a Gaussian weighted filter with a longitudinal span of 5° and a latitudinal span of 2°. The annotations on the left are cycles of satellite data.

Altimeter Errors (Topex/Poseidon, Jason, and OSTM)
The most accurate observations of the sea-surface topography are from Topex/Poseidon, Jason, and OSTM, and the accuracy has improved each year since the launch of Topex/Poseidon in 1992. Errors for satellite altimeter system are all very small, and they are due to:

1. Instrument noise, ocean waves, water vapor, free electrons in the ionosphere, and mass of the atmosphere. Topex/Poseidon and Jason carry precise altimeter systems able to observed the height of the satellite above the sea surface between ± 66° latitude with a precision of ±1 cm and an accuracy of ±1–2 cm. The systems consist of a two-frequency radar altimeter to measure height above the sea, the influence of the ionosphere, and wave-height. The systems also include a three-frequency microwave radiometer able to measure water vapor in the troposphere.
2. Tracking errors. The satellites carry three tracking systems that enable their position in space, their ephemeris, to be determined with an accuracy of ±1 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 subsatellite track, there is a sampling error. The satellites cannot map the topography between ground tracks, nor can they observe changes with periods less than 2 × 9.9156 d (see §16.3).
4. Geoid error. The permanent topography is not well known over distances shorter than a few hundred kilometers because geoid errors dominate for shorter distances. Maps of the topography smoothed over greater distances are used to study the dominant features of the permanent geostophic currents at the sea surface (Figure 10.5). New satellite systems GRACE and CHAMP are measuring Earth's gravity with enough accuracy that the geoid error will soon be small enough to ignore over distances as short as 100 km.

Taken together, the measurements of height above the sea and the satellite position give sea-surface height in geocentric coordinates with an accuracy of ±1–2 cm. The geoid error adds further errors that depend on the size of the area being measured.

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