1.a. Centrifugal force
Therefore there are more forces: centrifugal and Coriolis. We must acknowledge these because we want to measure motion relative to the rotating earth - this is an "accelerated reference frame" just like the floor of an elevator.
Imagine being on a merry-go-round. Sit still on your horse. You feel thrown radially outward. This is centrifugal force, with
acceleration = (omega)^{2}r,where r is the distance from the center (radius) and omega is the angular speed omega = 2*pi/T where T is the time of one full rotation.
Apply this to the earth: (figure). The radius r = 6000 km, omega = 2*pi/86400 sec, (omega)^{2}r = 3.17 cm/sec^{2}. This would be the centrifugal force at the equator, where it is maximum. At the poles, it is of course 0, since the radius to the axis of rotation is 0 there. (So the actual force is (omega)^{2}r cos(latitude).) Compare this centrifugal force with gravity acceleration of 980 cm/sec^{2}, which pulls downward toward the earth's center. Therefore on the earth, you weigh 0.3% less at the equator than at the pole.
If the solid earth were a true sphere, the motionless ocean would be 20 km deeper at the equator than at the pole. We don't see anything like this. Therefore we conclude: the solid earth itself is spheroidal, with the equatorial radius about 20 km larger than the polar radius.
Thus: the horizontal gravity + horizontal centrifugal force = 0. So we can neglect centrifugal force because the earth itself is spheroidal.
1.b. Coriolis force
Return to the merry-go-round. Sit at the rim, try to throw a pebble towards the center. When there is no rotation - you can hit the center. When the merry-go-round rotation viewed from above is counterclockwise (like earth from Polaris), the pebble hits to the right of center.
(figures)
You'll think, I aimed at the center, but after I'd thrown the pebble, some force deflected it to the right. This is the Coriolis force. Properties of the Coriolis force:
How large is this? see next subtopics
Try this out at the playground, or the Del Mar Fair, or on a lazy susan.
Coriolis Force Acceleration = 2 omega (sin 45)v = 2 x (2 pi/86400)x 0.707 x 100 = 10^{-2} to the east.
They balance! This will be important.
2.b. Coriolis force alone. Start motion impulsively - thereafter if Coriolis is the only force, acceleration is always to the right of the motion. So the motion veers.
Equations for Coriolis force become necessary at this point:
(figure) and period T = (2 pi)/(2 omega sin latitude) = (12 hr)/sin
latitude
We see motion like this very frequently in moored current meter records
- at the local "inertial period" which is 12 hr/sin latitude with a
little spread about this period. What is the inertial period at
different latitudes? (pole, 45 degrees, 30 degrees, equator).
2.c. Geostrophic/ weathermap flow
On a weathermap around a high pressure center:
Meteorologists measure pressure at different locations, and then use the balance PGF = (2 omega sin latitude)speed to get the wind speed. This balance is called geostrophic flow. That is, they can interpret the pressure maps for the winds. This is why oceanographers also want to measure pressure, since the same holds in the ocean.
Weather map for the U.S. for October 8, 2002 from the National Weather Service.
But as we've seen before, oceanographers cannot usually measure pressure differences well enough to obtain the PGF that goes with geostrophic flow (see section 3 below). Even with satellite altimeters that measure the sea surface height pretty well, the error is still several centimeters (with the limitation being the accuracy of the geoid). So oceanographers measure fluid density rho(x,y,z). They then use the hydrostatic balance to figure out the pressure at each depth, based on the mass of fluid above that depth. That is:
PGF_{one_depth} - PGF_{another_depth} = (2 omega sin latitude)(speed_{one_depth} - speed_{another_depth})That is, oceanographers use the mass of fluid to figure out the difference in PGF from one depth to another depth, and therefore get the speed at one depth relative to the speed at another depth.
Oceanographers either draw such maps, say the flow at the surface relative to 1000 dbar for instance, or look at sections to see the difference in speeds.
Atlantic surface height (steric height, something like dynamic
height)
from Reid (1994).
Atlantic 1000 dbar (steric height, something like dynamic
height)
from Reid (1994).
Pacific surface height (steric height, something like dynamic
height)
from Reid (1998).
Pacific 1000 dbar (steric height, something like dynamic
height)
from Reid (1998).
Note the high and low pressure regions. Note how much smaller the
basin-scale pressure gradient force is at 1000 dbar compared with 0
dbar.
(Talley extra: If oceanographers have more information - say, they have measured the flow at one depth using something like a subsurface float deployment, or if they infer the flow at one depth using something like tracer patterns, and perhaps combine these patterns with strong constraints on how much total transport can move through some particular region, they can work out the absolute flow field. This is quite a difficult endeavor. We will look at some examples of such solutions.)
Example of a section across the Gulf Stream:
Figure: Gulf Stream potential density section
Figure: Gulf Stream potential temperature section
Figure: Gulf Stream salinity section
at 26N, Florida Strait, in August, 1981
Direct current measurements - require time and space averaging and/or filtering to remove unwanted signals such as those from tides and internal waves. The remaining signal still contains all of the time scales of the large-scale circulation. It is more or less the case that long time scales go with large horizontal spatial scales. There is no hard and fast rule about how long the time average must be that corresponds with a velocity estimate associated with the pressure gradient. (current meter results, float trajectories and averages.)
Pressure gradients are used to calculate geostrophic flows, as described above. Pressure can be measured in several ways. Altimetry provides the sea surface height to within several centimeters, which is not adequate for the "mean" circulation, but is adequate for determining the variability of the surface currents about an unknown mean. Pressure gauges on the ocean bottom can be calibrated with direct current measurements (say from current meters), and then used in pairs to calculate geostrophic flow. As with direct current measurements, time series signals must be averaged or filtered.
Finally, pressure gradients can be determined from dynamic height (density field), again to within an unknown pressure at each position (as described above, and below). The unknown can be thought of as due to the unknown exact sea surface height relative to the geoid (and which altimetry is mainly not quite adequate to measure).
Velocities associated with the general circulation are then mostly calculated from the pressure gradients.
3.2 Dynamic height. To calculate pressure at a point relative to a deeper or shallower point at the same horizontal location, from the measured density field, using the concepts outlined above.
Use hydrostatic balance, which is the force balance between the vertical pressure gradient and gravity*density.
Dynamic height is an historical quantity, which means that its definition includes some anachronisms owing to the way it had to be calcaulted prior to computers. Define dynamic height D such that
10 delta D = g delta zThus D would be exactly the height z if g = 10 m/sec^{2} and the "10" had units of m/sec^{2}. (However, the "10" is unitless.)
10(D_{2} - D_{1}) = g (integral from 1 to 2) dz
If delta z = 1 meter, delta D = (g delta z)/10 <~ 1 m^{2}/sec^{2} = 1 dynamic meter (definition).
To calculate D from density, use the hydrostatice relation: 0 = -alpha dp - g dz. Then D = -(1/10)integral (from p_{1} to p_{2}) alpha dp', in units of dynamic meters. You see here how the density comes in to work out the pressure change from one depth to another.
For practical purposes, people often use "delta D" instead of "D": alpha = alpha_{35,0,p} + del where del = specific volume anomaly. Then delta D = = -(1/10)integral (from p_{1} to p_{2}) del dp'
The geostrophic velocity is then obtained in terms of dynamic height:
fv = 10 (d/dx)(delta D) and fu = -10 (d/dy)(delta D)
where f is the Coriolis parameter defined before: 2 omega sin(latitude),
x and y are the east and north directions, and u and v are the east and
north velocities.