Sea surface
elevation gradients are balanced by geostrophic
surface currents given by
_{}
Satellite altimeter
observations are now precise enough to measure:
·
Changes in mean volume of the ocean (sea level rise and global
warming)
·
Seasonal heating and cooling
·
Tides
·
Mean dynamic topography (on long length scales)
·
Variability in surface geostrophic
currents
·
Variations in the topography of the equatorial current systems
associated with El Nino (but geostrophy can’t be used
at the equator)
Figure
10.4 Global distribution of variance of topography from Topex/
Poseidon altimeter data from 10/3/92 to 10/6/94. The height variance is an indicator
of variability of currents. (From Center for Space Research, University of
Figure
10.5 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
2.8 calculated from hydrographic data. (From Center for Space Research,
University of
From Stewart
Chapter 10
The pressure
gradient associated with the departure of sea surface height from a geopotential is felt throughout the water column, and these
currents are often referred to as barotropic
currents.
They are often said
to be the part of the flow that does not vary with depth, but strictly speaking
barotropic processes are associated with pressure
surfaces that are parallel to density surfaces.
To use geostrophy to infer currents at depth we need to determine
not only the pressure gradient due to the sloping sea surface, but also the
subsurface pressure gradients due to variable density stratification.
_{}
Though we can measure
water pressure with a pressure transducer lowered from a ship, we can’t simply
use this observation because we seldom have an independent way of measuring
depth.
Even if we could
measure depth independently, it would have to be a very precise measurement:
A 10 cm/s current
corresponds to a pressure gradient of
_{} Pa m^{-1 or} 1000 Pa
in 100 km
From the
hydrostatic relation we know that 1000 Pa is equivalent to the pressure change
due to 10 cm of water.
We would need to
know the depth of the pressure gauge to accuracy much better than 10 cm to make
an observation adequate for calculating geostrophic
currents, and we would still need to deal with the issue of the slope of the geoid.
In practice, what
we do in oceanography is to estimate the slope of the geopotential
surface at one depth compared to another, and this
tells us the relative strength of the current at the two depths.
This is a
complimentary approach to that used in satellite altimetry which calculated the
slope of a constant pressure surface (p =
p_{atmosphere}).
Stewart Figure 10.7: Sketch of geometry used for calculating geostrophic current from hydrography.
The steps taken
are:
1. Calculate the
differences in geopotential _{}between two different pressure surfaces 1 and 2
2. Calculate the slope
of the upper surface relative to the lower from observations at two locations A
and B
3. Calculate current
at the upper surface relative to the lower – this is the current shear
4. Integrate
vertically the shear in the current assuming some knowledge of the current at a
reference depth
We use a modified
form of the hydrostatic equation, which for historical reasons is written:
_{}
so that d_{} is the change in potential energy associated with raising 1
kg through a distance of dz. Units are energy per unit mass, J/kg, or m^{2}/s^{2}.
The geostrophic balance is written:
_{}
where _{} is the geopotential along a constant pressure surface.
Now consider how
hydrographic data can be used to evaluate _{}
At station A, the
difference in geopotential between surfaces P_{1} and P_{2} is:
_{}
where the specific volume
anomaly is written as the sum of two parts:
_{}
where _{} is the specific
volume of a standard
The term _{} is the specific volume anomaly, and tables and computer
programs exist for easily calculating this for any observed hydrographic data.
_{}
which is the sum of:
·
the standard geopotential distance
between the pressure surfaces
and in meters would be approximately _{}
- this is what a SCUBA depth gauge measures
·
the geopotential anomaly
_{}
- usually about 0.1% of the geopotential distance
The standard geopotential distance is the same at any horizontal
location in the ocean because there is no variation in the vertical profile of T or S,
so this is not going to enter into the calculation of pressure gradients.
Consider now the geopotential anomaly between P_{1} and P_{2}
at two different stations A and B:
For simplicity,
assume the lower surface is a level
surface i.e. the constant pressure and geopotential
surfaces coincide.
The slope of the
upper surface is:
_{} slope
of constant pressure surface P_{2}
because the standard geopotential distance is the same at stations A and B.
The geostrophic velocity at the upper surface is calculated
from:
_{}
_{}
similar to the way that we
calculated surface velocity from altimetry from the slope of the sea surface
(also a constant pressure surface).
Units: s.(m^{2}/s^{2})/m = m/s (geopotential
anomaly has units of m^{2}/s^{2})
Geopotential anomaly is often
referred to as dynamic height.
Oceanographers also
often scale _{}by 1/g, and call this steric
height, h, with units of
meters.
Steric height measures
variations in the vertical distance between two surfaces of constant pressure,
and should be stated as the steric height of surface p_{1} relative to p_{o},
e.g., the height of the sea surface (p_{1}
= 0) relative to 1000 decibars (approximately
1000 m).
The velocity v is perpendicular to the plane of the
two hydrographic stations and directed into the plane the way the figure is
sketched.
Light on the
right
A useful rule of
thumb is that the flow is such that lighter (less dense, warmer) water is on
the right looking in the downstream direction in the northern hemisphere – light on the right. This only works
if the level surface is below. |
The notions of geopotential anomaly, steric
height, and pressure are somewhat interchangeable. All can be used to visualize
the pressure gradients that give rise to geostrophic
currents.
Tomczak and Godfrey,
Figure 2.7 – Schematic steric height and pressure
section across a cold core eddy:
http://gyre.umeoce.maine.edu/physicalocean/Tomczak/regoc/text/2steric.html
Since the weight of
water above z_{0} must be the same at all locations, the sea surface
must be lower over the region of higher density.
·
pressure surfaces parallel the sea surface, but become flat at
depth
·
some density surfaces outcrop at the surface, and this feature
would have a cold center (possibly visible in satellite imagery)
The distance
between the surface p=0 and reference
level p(z_{0}) is the steric height h(0,z_{0}).
If we plot the
pressure map at constant depth, z = z_{r}, we get:
[See Tomczak and Godfrey, fig. 2.7b]
If we plot the steric height at constant pressure, p = p_{1}, we get:
[See Tomczak and Godfrey, fig. 2.7c]
At any depth level,
contours of steric height coincide with contours of
pressure. Since the shape of the sea surface is so difficult to map accurately,
oceanographers instead use maps of the steric height
relative to a reference level of no motion to map the "dynamic
topography" of the ocean.
This is the
oceanographic equivalent of a meteorologists pressure map, and is an effective
way of visualizing geostrophic currents.
We’ve already seen
in the case of satellite altimetry that we can relatively easily measure the variability
of geostrophic currents, but we are left with
uncertainty in the mean circulation.
Similarly, the
dynamic method used to compute geostrophic velocity
of one depth relative to another leaves us uncertain about the absolute
velocity.
Q: Is it possible
to find a flat surface in the ocean ... one where horizontal pressure
variations vanish?
Yes ... at depth in
most ocean basins the density field is so uniform horizontally that, for
example, steric height of the 1500 m relative to 2000
m varies by only a cm or so.
[Tomczak and Godfrey, fig. 2.8 - dyn
hgt 1500/2000 db and 0/2000 db]
Steric height of the
surface relative to 2000 m shows differences of order 0.5 m in a single basin.
The Southern Ocean
is a marked exception to this – here strong geostrophic
currents extend to the bottom.
Since maps of
dynamic height and pressure are similar, we can sketch the pattern of geostrophic currents on a dynamic height or steric height map.
This method relies
on the assumption that there is little or no flow at, say, 2000 db. Assuming
this, we can compute the dynamic topography at other depths w.r.t.
2000 db (including deeper depths).
This gives ocean
currents at any depth we select, so we get u(z), v(z).
You won’t go too
far wrong in the gyre centers, but there are regions where this won’t work
(notably coastal, and boundary currents).
There are other
observational clues we can use to make a more informed choice of reference
level (such as O_{2} minimum, or breaks in tracer property
distributions).
Oceanographers play
fast and loose with the terms
·
dynamic height/topography
·
steric height
Units are your
friend
·
use them to check how to calculate geostrophic
currents (m/s)
·
f is 10^{-4}
s^{-1}, g is 10 m/s^{2}
·
think about whether you have a sensible answer …
* 0.1 m/s is a moderate to brisk flow
* 1 m/s/ is hauling
* 10 m/s is ballistic
The “light on the
right” rule can be derived another way by reconsidering the geostrophic
balance:
_{}
Multiply
through by r and differentiate w.r.t. z
_{}
Recalling
the hydrostatic relation, we can replace the vertical pressure gradient:
_{}
to get
_{}
We can
simplify this further by approximating the left-hand-side terms.
_{} (the
chain rule)
Consider
the magnitude of the two terms in the expansion.
3 kg/m^{3} / 1000 m * 1 m/s : 1000 kg/m^{3} * 1 m/s / 1000 m
3/1000 : 1
Since _{} varies
so little (3 kg/m^{3}) compared to its mean value, this first term is
negligible, and we can take _{} (and f) outside the vertical derivative,
leaving
_{}
The
slopes of the density field are large (100 to 300 times greater than the
surface) and readily measured from data. So these thermal wind equations give
us a straightforward way to compute the velocity shear. If we have direct
observations of the velocity at some depth, such as from current meters, drifters,
or by assuming a level of no motion, we can compute flow at all other depths.
Rule of thumb: “light on the right” (in the northern hemisphere)
Thermal wind: If in the northern (southern) hemisphere isopycnals slope upward to the left (right) across a current
when looking in the direction of flow, current speed decreases with depth; if
they slope downward, current speed increases with depth.