Notes from previous years of SIO 210 on Coriolis and geostrophy

These notes are replaced by the readings from the DPO 6th edition text, and backup readings from Stewart online textbook, but you might find them useful in themselves.

Hendershott notes on the effects of rotation (transcript)

We saw in the previous dynamics lecture that the pressure gradient force would produce accelerations in the ocean that are completely unphysical at the large scale, and that frictional forces, even using eddy viscosity, cannot be large enough to balance the PGF, also at this scale. In high-frequency waves (surface waves, higher frequency internal waves), these forces may be enough because there are restoring forces and so velocities never become excessive - gravity, vertical acceleration, horizontal pressure gradient, and horizontal acceleration can balance. As the wave frequencies decrease towards the length of a day though, rotation becomes important even for these gravity waves.

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)2r,
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)2r = 3.17 cm/sec2. 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)2r cos(latitude).) Compare this centrifugal force with gravity acceleration of 980 cm/sec2, 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.


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:

  • Acts only on moving objects
  • Action is proportional to the speed
  • It makes objects go to the right in the northern hemisphere
  • It makes objects go to the left in the southern hemisphere
  • It is biggest at the poles, and goes to zero on the equator, if the motion is parallel to the earth's surface (not vertical).
  • Formally: |acceleration| = 2 omega sin(latitude) x |speed|

    How large is this? see next subtopics

    Try this out at the playground, or the Del Mar Fair, or on a lazy susan.

    2. Large-scale flow with Coriolis force

    2.a. Gulf Stream: PGF acceleration = g deltah/deltax = 103102/(100 x 105) = 10-2 cm/sec2 to the west.

    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:

  • ut = (2 omega sin latitude)v
  • vt = -(2 omega sin latitude)u

    (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:

  • PGF is outward
  • If Coriolis force balances PGF, then CF is inward
  • Therefore the flow is clockwise around the high in the northern hemisphere.

    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:

    PGFone_depth - PGFanother_depth = (2 omega sin latitude)(speedone_depth - speedanother_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:

    1. Measure temperature and salinity and compute the density. Across the Gulf Stream, we see isopycnals plunging hundreds of meters from on to offshore (towards the east if looking at an east-west section).
    2. As one guess, assume there is no flow at some depth, which means that the PGF = 0 at that depth. This was common practice until about 20 years ago, and the depth assumed with no flow was called the "level of no motion". (In reality, there is almost never such a level, but this is a convenient step for teaching how to compute relative speeds.)
    3. Go upward from that level where PGF = 0 on the west side, which is the cold, dense side and calculate how the pressure decreases as you go up. Then go upward from the PGF=0 level on the east side, which is warm and less dense and calculate the pressure decrease. The pressure will decrease faster on the cold, dense side since the water is heavier and more is subtracted at each step.
    4. Therefore at a higher level, the nearshore (cold, dense) P < offshore (warm, light) P. Therefore the PGF is westward at all of these higher levels, above the PGF = 0.
    5. If the flow is geostrophic, which we assert, the Coriolis force is eastward. Therefore the flow is northward.
    6. Since the Pressure gradient persists to the very surface, the surface itself must slope upward from west to east so that the pressure is higher offshore. If in fact we could measure the sea surface height relative to the geoid, this is what we would find (and which is found in various interpretations of satellite altimetry data).
    7. Caveat: we don't really know what the PGF is at a given depth; we just assumed PGF = 0 at some depth since we expect from other observations that the flow is much weaker at depth than at the surface in the Gulf Stream. All that we actually know from measuring the density is the relative flow at one depth compared with another - that is, we know the geostrophic velocity shear and not the absolute geostrophic flow.

    Figure: Gulf Stream potential density section
    Figure: Gulf Stream potential temperature section
    Figure: Gulf Stream salinity section
    at 26N, Florida Strait, in August, 1981

    3. Dynamical quantities for the general circulation. (modified from Talley, 2000 lecture notes)

    3.1 Velocity. Horizontal velocity is measured directly using various current-measuring devices, or calculated from the horizontal pressure gradient. In the latter case, for the large-scale circulation, geostrophy is assumed. Vertical velocity for the large-scale circulation cannot be measured as it is too small, only inferred from the horizontal velocities.

    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 z
    10(D2 - D1) = g (integral from 1 to 2) dz
    Thus D would be exactly the height z if g = 10 m/sec2 and the "10" had units of m/sec2. (However, the "10" is unitless.)

    If delta z = 1 meter, delta D = (g delta z)/10 <~ 1 m2/sec2 = 1 dynamic meter (definition).

    To calculate D from density, use the hydrostatice relation: 0 = -alpha dp - g dz. Then D = -(1/10)integral (from p1 to p2) 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 = alpha35,0,p + del where del = specific volume anomaly. Then delta D = = -(1/10)integral (from p1 to p2) del dp'

  • Atlantic surface height and current names
  • Pacific surface height and current names
  • Indian surface height and current names
    all based on Niiler et al. (2003) surface heights

    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.

    SIO 210 HOME Last modified: Oct. 22, 2014