11.1 Sverdrup's Theory of the Oceanic Circulation
What drives the ocean currents? At first, we might answer, the winds drive the circulation. But if we think more carefully about the question, we might not be so sure. We might notice, for example, that strong currents, such as the North Equatorial Countercurrents in the Atlantic and Pacific Oceans go upwind. Spanish navigators in the 16th century noticed strong northward currents along the Florida coast that seemed to be unrelated to the wind. How can this happen? And, why are strong currents found offshore of east coasts but not offshore of west coasts?
Answers to the questions can be found in a series of three remarkable papers published from 1947 to 1951. In the first, Harald Sverdrup (1947) showed that the circulation in the upper kilometer or so of the ocean is directly related to the curl of the wind stress. Henry Stommel (1948) showed that the circulation in oceanic gyres is asymmetric because the Coriolis force varies with latitude. Finally, Walter Munk (1950) added eddy viscosity and calculated the circulation of the upper layers of the Pacific. Together the three oceanographers laid the foundations for a modern theory of ocean circulation.
11.1 Sverdrup's Theory of the Oceanic Circulation
While Sverdrup was analyzing observations of equatorial currents, he came upon (11.7) below relating the curl of the wind stress to mass transport within the upper ocean. In deriving the relationship, Sverdrup assumed that the flow is stationary, that lateral friction and molecular viscosity are small, and that turbulence near the sea surface can be described using an eddy viscosity. He also assumed that the flow is baroclinic and that the wind-driven circulation vanishes at some depth of no motion. From (8.9 and 8.12) the horizontal components of the momentum equation are:
Sverdrup integrated these equations from the surface to a depth -D equal to or greater than the depth at which the horizontal pressure gradient becomes zero. He defined:
where Mx, My are the mass transports in the wind-driven layer extending down to an assumed depth of no motion.
The horizontal boundary condition at the sea surface is the wind stress, and the boundary at depth -D is zero stress because the currents go to zero:
where Tx and Ty are the components of the wind stress.
Using these definitions and boundary conditions, (11.1) become:
In a similar way, Sverdrup integrated the continuity equation (7.19) over the same vertical depth, assuming the vertical velocity at the surface and at depth -D are zero, to obtain:
Differentiating (11.4a) with respect to y and (11.4b) with respect to x, subtracting, and using (11.5) gives:
where β = ∂f/∂y is the rate of change of Coriolis parameter with latitude, and where curlz(T) is the vertical component of the curl of the wind stress.
This is an important and fundamental result - the northward mass transport of wind driven currents is equal to the curl of the wind stress. Note that Sverdrup allowed f to vary with latitude. We will see later that this is essential.
We calculate β from
where R is Earth's radius and φ is latitude.
Over much of the open ocean, especially in the tropics, the wind is zonal and ∂Ty/∂x is sufficiently small that
Substituting (11.8) into (11.5) , Sverdrup obtained:
Sverdrup integrated this equation from a north-south eastern boundary at x = 0, assuming no flow into the boundary. This requires Mx = 0 at x = 0. Then
where Δx is the distance from the eastern boundary of the ocean basin, and brackets indicate zonal averages of the wind stress (Figure 11.1).
To test his theory, Sverdrup compared transports calculated from known winds in the eastern tropical Pacific with transports calculated from hydrographic data collected by the Carnegie and Bushnell in October and November 1928, 1929, and 1939 between 22°N and 10°S along 80°W, 87°W, 108°W, and 109°W. The hydrographic data were used to compute P by integrating from a depth of D = -1000m. The comparison, Figures 11.2, showed not only that the transports can be accurately calculated from the wind, but also that the theory predicts wind-driven currents going upwind.
Comments on Sverdrup's Solutions
Stream Lines, Path Lines, and the Stream Function
At each instant in time, we can represent the flow field in a fluid by a vector velocity at each point in space. The instantaneous curves that are everywhere tangent to the direction of the vectors are called the stream lines of the flow. If the flow is unsteady, the pattern of stream lines change with time.
The trajectory of a fluid particle, the path followed by a Lagrangean drifter, is called the path line in fluid mechanics. The path line is the same as the stream line for steady flow, and they are different for an unsteady flow.
We can simplify the description of two-dimensional, incompressible flows by using the stream function y defined by:
The stream function is often used because it is a scalar from which the vector velocity field can be calculated. This leads to simpler equations for some flows.
Stream functions are also useful for visualizing the flow. At each instant, the flow is parallel to lines of constant y. Thus if the flow is steady, the lines of constant stream function are the paths followed by water parcels.
The volume rate of flow between any two stream lines of a steady flow is dy, and the volume rate of flow between two stream lines y1 and y2 is equal to y1 - y2. To see this, consider an arbitrary line dx = (dx, dy) between two stream lines (Figure 11.4). The volume rate of flow between the stream lines is:
and the volume rate of flow between the two stream lines is numerically equal to the difference in their values of y.
Now, lets apply the concepts to satellite-altimeter maps of the oceanic topography. In §10.3 we wrote (10.10)
Comparing (11.14) with (11.12) it is clear that
and the sea surface is a stream function scaled by g/f. Turning to Figure 10.6, the lines of constant height are stream lines, and flow is along the lines. The surface geostrophic transport is proportional to the difference in height, independent of distance between the stream lines. The same statements apply to Figure 10.9, except that the transport is relative to transport at the 1000 decibars surface, which is roughly one kilometer deep.
In addition to the stream function, oceanographers use the mass-transport stream function Y defined by:
This is the function shown in Figures 11.2 and 11.3.
|Department of Oceanography, Texas A&M University
Robert H. Stewart, firstname.lastname@example.org
All contents copyright © 2005 Robert H. Stewart,
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Updated on October 18, 2006