Chapter 7 - Ocean Circulation

Part 2

Modeling Geostrophic Flow
Estimating Seasurface Slope
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In this part, we go further in our understanding of ocean circulation by explaining how we can indirectly estimate the actual currents in the ocean, by determining the differences in density between two hydrographic stations in the ocean and estimating the slope and the current that results from this slope. This is not covered extensively in the text.


Most of our knowledge of ocean currents comes not from the direct measurement of the circulation, but from an indirect calculation of the slope of the sea surface based on differences in vertical density between two or more points in the ocean separated by 50 to 100 km. The flow of water that results from these ocean slopes are called Geostrophic currents. Geostrophic currents (literally, 'earth-turning') exist when the Coriolis effect, acting perpendicular to the direction of motion, balances the down-slope component of gravity (or the pressure gradient) as shown on the right.

Also note in Fig. 7.8a (and the figure to the right) that this current (over the extent of the North Atlantic Basin) actually follows a curved path called a gyre. Below, in my discussion about estimating a geostrophic current, I focus on the current having a straight path. In reality, we can assume it is straight only over a relatively short distance of a few hundred kilometers.

Note finally, in Fig. 7.8a & b, that the actual height of the sea surface (as determined by satellite data) shows this 'hill' of water on the western side of the North Atlantic Basin. This 'hill' will be discussed in Part 3 of this lesson when we examine why the the western boundary currents in all our ocean basins are intensified. Note also that this 'hill' and the resulting geostrophic currents are created by the surface wind field (Trade Winds and Prevailing Westerlies).

To fully understand this balance, and the examples given below to explain how the balance is achieved, I need to remind you of Newton's first law of motion for a moving mass:

"A mass in motion will remain in motion with a constant velocity (constant speed and direction) so long as the forces acting on it are balanced".

The corollary, therefore, is

"A mass in motion will have a curved path, or will accelerate or decelerate, if the forces acting are it are NOT balanced."

Modeling Geostrophic Flow

Note: I will go into much more depth in this class to show you how to model Geostrophic Flow than is shown in the text, but I think it provides a much better opportunity to tie together multiple concepts to understand this kind of current flow.

First, we will consider the motion of a ball rolling down an inclined frictionless plane and then extend what we learn to the movement of water down a sea surface slope. There are audio and video files at the end of this discussion section that may be very helpful in learning how to model geostrophic flow.

Ball rolling down inclined plane

Consider the motion of a ball rolling down an inclined frictionless plane under two sets of conditions: (1) the plane is fixed to a non-rotating platform; and (2) the plane is fixed to a platform that is rotating counterclockwise. This motion is shown on a cross-sectional view (i.e., looking at it 'from the side") and a plan view (i.e., looking 'down from the top'), as shown below for each of the two sets of conditions.

Non-rotating Platform


Rotating Platform

Fluid flow down a sloping sea surface

It is easy to extend this example to a fluid flowing down a sloping sea surface being moved and deflected by the same two forces described above. At the moment that v becomes constant (with CE and G in balance and vectorally opposite in direction) we have, by definition, a Geostrophic Current.


Note: The slope of the sea surface does not have to be large to support a major Geostrophic Current. The Florida Current, after it has curved around the southern tip of Florida, has a velocity as high as 8 km/hr (about 5 mph) and yet the slope of the sea surface is only about 1 m in 50 km.

Determining the Slope of the Sea Surface - Dynamic Topography

How is the steepness of such a small slope determined?

We must contend with a curved sea surface (the ocean geoid) covered by chaotic ocean waves. We can get quite a good idea of the slope using precision radar satellites, but traditionally the sea surface was estimated indirectly by measuring the internal distribution of vertical density at two or more hydrographic stations (locations on the ocean surface occupied by an oceanographic research vessel) separated horizontally by a few tens to hundreds of kms. The relative slopes of the surface between these stations represents the dynamic topography of the upper mass of the ocean and it is from these estimated slopes that the velocity of the Geostrophic Current can be calculated.

Let me propose a problem that illustrates how the relative slope between two stations is estimated. Only inequalities will be used because, to understand the principles, we do not need to actually calculate densities at the stations -- only to compare them.

Determining the Relative Slopes Between Two Stations

Suppose you had two hydrographic stations (A and B) on the ocean surface separated by 50 km (an arbitrary distance) over a flat bottom, as shown in the cross-section below. I chose a flat bottom so that the average pressures of the water columns at the two stations can be assumed to be constant (Pa = Pb).

Suppose you determine that the average salinities of the water columns at both stations were the same (Sa = Sb), but that the temperature of the water column at Station A (Ta) was greater than the average temperature of the water column at Station B (Tb). Therefore, you would have the following conditions:

Sa = Sb


Ta > Tb

which, because Density (DEN) is prop to 1/T, will result in

DENa < DENb.

This says that the least dense water is at Station A. Therefore, the water column at Station A has a larger volume than the column at Station B (remember that density = mass/unit volume, and we have not changed the mass). If we further assume that the columns of water at both stations have the same diameter, then the increased volume at Station A will result in a greater height of the column at Station A (so Ha > Hb), and the sea surface will slope down from Station A to Station B, as shown below:

When you do this problem on an exam paper, here is what I suggest:

Estimating the Geostrophic Current

Now that we have the slope, we can estimate the direction of the current. Using the same cross-sectional view (and its plan view) of the slope of the sea surface given above (and as shown below), we see that the down-slope component of gravity (G) will point from Station A to Station B. If we assume that the slope was caused by a Geostrophic current, we also know that CE must balance G (and be vectorally opposite to G), because that is what defines a Geostrophic Current and, therefore, that it will point directly up-slope from Station B to Station A.

Note: These force vectoral directions will be true regardless of hemisphere if G and CE are in geostrophic balance (ie., G will always point down slope and CE will always point upslope; opposite to G).
When you do this problem on an exam paper, here is what I suggest:
Once you have the slope, actually draw and label the vectors - G will always point downslope and CE (for a geostrophic current) will by definition point in the opposite direction, or upslope. As you can see in the example, G points downslope from A to B and CE points upslope from B to A, so label them that way on the exam paper (believe me, this will help in the final stage of estimating the geostrophic current).

So, in what direction, relative to this cross-section, will the Geostrophic current actually flow?

Remember that we are assuming that the geostrophic balance has been achieved, so we are NOT looking for a 'curving path'.

To determine the geostrophic current, all we need to know now is the hemisphere.

So, assume we are in the Northern Hemisphere - as you can see in the plan view below on the right, the current will flow parallel to the lines of constant height such that CE will be 90 deg. to the right of the current velocity (v). On the cross-sectional view, this means the current will flow "out of the screen" (shown by the "tip of an arrow") -- think about 'lifting' the plan view up so you are looking at the end. Note that the arrow is coming directly at you, and that v is perpendicular to the line of the cross-sectional plane between Station A and Station B.

In fact, that is the only direction that the current can flow -- either directly at you out of the page or away from you into the page.

Do you see that the current would flow "into the screen" (with the "tail of an arrow" being shown instead) if the stations and slope shown above were in the Southern Hemisphere?

Do you also see that if the slope was up from Station A to Station B (the opposite slope to that shown), the direction of the current for each hemisphere would be just opposite to that discussed above?

I suggest you look at the balance of G and CE (and the resulting geostrophic current) for the following set of conditions for both hemispheres (over and above that used in the illustration above) - these will give you ALL OF THE SETS OF CONDITIONS THAT YOU WILL ENCOUNTER ON THE HOMEWORK AND/OR YOUR EXAM:

Ta < Tb; Sa = Sb

Ta = Tb; Sa < Sb

Ta = Tb; Sa > Sb

For each set of conditions above (including those where the average salinities are not equal), determine the relationship of DENa to DENb and find the slope (and always assume that when DENa < DENb, Station A would have the least dense water; or when DENa>DENb, Station B would have the least dense water).

Though it is not as clear how changes in density due to changes in salinity will result in different heights of the water at the two stations, for this class (and for estimating the slope of the sea surface), you may assume that the column with the lowest salinity (and density) will be higher than the column with the higher average salinity (and density).

Note that the example given only estimates the relative slope between two stations (and, therefore, only the relative Geostrophic current perpendicular to the cross-section between these two station). To find the "true current", we would need to use three or more stations and then vectorally average the relative Geostrophic currents from each of the cross sectional planes between those stations.

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