Chapter 10 - Geostrophic Currents
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 Sketch defining z and
r, used for calculating pressure just below the sea surface.
The pressure on the level surface is:
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:
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
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
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
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
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
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.
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
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.
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:
- 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
Change by Dr. R. Steven Nerem, Colorado Center for Astrodynamics Research, University
of Colorado at Boulder.
- Seasonal heating and cooling of the ocean (Chambers et
- Tides. See figure 17.13 for a global map of the M2 twice per day lunar
tide.(Andersen, Woodworth, and Flather, 1995);
- The permanent surface geostrophic current system (Figure 10.5);
- Changes in surface geostrophic currents on all scales (Figure 10.4); and
- 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,
(2005) (a 1MB
pdf file). From Asia-Pacific
|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
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:
- 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,
also include a three-frequency microwave radiometer able to measure water
vapor in the troposphere.
- Tracking errors. The satellites carry three tracking systems that enable
their position in space, their ephemeris, to be determined with an accuracy
- 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).
- 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
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
±1–2 cm. The geoid error adds further errors that depend on the size of
the area being measured.