11.1: Sverdrup's Theory of the Ocean Circulation

While Sverdrup was analyzing observations of equatorial currents, he came upon \((\PageIndex{9})\) below relating the curl of the wind stress to mass transport within the upper ocean. To derive the relationship, Sverdrup assumed that the flow is stationary, that lateral friction and molecular viscosity are small, that non-linear terms such as \(u \ \partial u/\partial x\) are small, and that turbulence near the sea surface can be described using a vertical eddy viscosity. He also assumed that the wind-driven circulation vanishes at some depth of no motion. With these assumptions, the horizontal components of the momentum equation from \(8.3.4-5)\) become:

\[\begin{align} \frac{\partial p}{\partial x} &= f \rho v + \frac{\partial T_{xz}}{\partial z} \\ \frac{\partial p}{\partial y} &= -f \rho u + \frac{\partial T_{yz}}{\partial z} \end{align} \nonumber \]

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: \[\begin{align} \frac{\partial P}{\partial x} &= \int_{-D}^{0} \frac{\partial p}{\partial x} dz, \quad\quad \frac{\partial P}{\partial y} = \int_{-D}^{0} \frac{\partial p}{\partial y} dz, \\ M_{x} &\equiv \int_{-D}^{0} \rho u(z) \ dz, \quad \ M_{y} \equiv \int_{-D}^{0} \rho v(z) \ dz \end{align} \nonumber \]

where \(M_{x}\), \(M_{y}\) 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.

At depth \(-D\) the stress is zero because the currents go to zero: \[\begin{array}{l} T_{xz}(0) = T_{x} \quad\quad T_{xz}(-D) &= 0 \\ T_{yz}(0) = T_{y} \quad\quad T_{yz}(-D) &= 0 \end{array} \nonumber \]

where \(T_{x}\) and \(T_{y}\) are the components of the wind stress. Using these definitions and boundary conditions, \((\PageIndex{1}-2)\) become: \[\begin{align} \frac{\partial p}{\partial x} &= f M_{y} + T_{x} \\ \frac{\partial p}{\partial y} &= -f M_{x} + T_{y} \end{align} \nonumber \]

In a similar way, Sverdrup integrated the continuity equation \((7.7.3)\) over the same vertical depth, assuming the vertical velocity at the surface and at depth \(-D\) are zero, to obtain: \[\frac{\partial M_{x}}{\partial x} + \frac{\partial M_{y}}{\partial y} = 0 \nonumber \]

Differentiating \((\PageIndex{6})\) with respect to \(y\) and \((\PageIndex{7})\) with respect to \(x\), subtracting, and using \((\PageIndex{8})\) gives: \[\begin{align} \beta M_{y} &= \frac{\partial T_{y}}{\partial x} - \frac{\partial T_{x}}{\partial y} \nonumber \\ \beta M_{y} &= \text{curl}_{z} (T) \end{align} \nonumber \]

where \(\beta \equiv \partial f/\partial y\) is the rate of change of Coriolis parameter with latitude, and where \(\text{curl}_{z}(T)\) 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 \(\beta\) from \[\beta \equiv \frac{\partial f}{\partial y} = \frac{2 \Omega \cos \varphi}{R} \nonumber \] where \(R\) is Earth’s radius and \(\varphi\) is latitude.

Over much of the open ocean, especially in the tropics, the wind is zonal and \(\partial T_{y}/\partial x\) is sufficiently small that \[M_{y} \approx -\frac{1}{\beta} \frac{\partial T_{x}}{\partial y} \nonumber \]

Substituting \((\PageIndex{11})\) into \((\PageIndex{8})\), assuming \(\beta\) varies with latitude, Sverdrup obtained: \[\frac{\partial M_{x}}{\partial y} = -\frac{1}{2 \Omega \cos \varphi} \left(\frac{\partial T_{x}}{\partial y} \tan \varphi + \frac{\partial^{2} T_{x}}{\partial y^{2}} R\right) \nonumber \]

Sverdrup integrated this equation from a north-south eastern boundary at \(x = 0\), assuming no flow into the boundary. This requires \(M_{x} = 0\) at \(x = 0\). Then \[M_{x} = -\frac{\Delta x}{2 \Omega \cos \varphi} \left[\Bigl \langle \frac{\partial T_{x}}{\partial y} \Bigr \rangle \tan \varphi + \Bigl \langle \frac{\partial^{2} T_{x}}{\partial y^{2}} \Bigr \rangle R\right] \nonumber \]

where \(\Delta x\) is the distance from the eastern boundary of the ocean basin, and brackets indicate zonal averages of the wind stress (figure \(\PageIndex{1}\)).

Figure \(\PageIndex{1}\): Streamlines of mass transport in the eastern Pacific calculated from Sverdrup’s theory using mean annual wind stress. After Reid (1948).

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 34\(^{\circ}\)N and 10\(^{\circ}\)S and between 80\(^{\circ}\)W and 160\(^{\circ}\)W. The hydrographic data were used to compute \(P\) by integrating from a depth of \(D = -1000 \ \text{m}\). The comparison, figure \(\PageIndex{2}\), showed not only that the transports can be accurately calculated from the wind, but also that the theory predicts wind-driven currents going upwind.

Figure \(\PageIndex{2}\): Mass transport in the eastern Pacific calculated from Sverdrup’s theory using observed winds with \((\PageIndex{11})\) and \((\PageIndex{13})\) (solid lines) and pressure calculated from hydrographic data from ships with \((\PageIndex{6-7})\) (dots). Transport is in tons per second through a section one meter wide extending from the sea surface to a depth of one kilometer. Note the difference in scale between \(M_{y}\) and \(M_{x}\). After Reid (1948).

Comments on Sverdrup’s Solutions

1. Sverdrup assumed i) The internal flow in the ocean is geostrophic; ii) there is a uniform depth of no motion; and iii) Ekman’s transport is correct. I examined Ekman’s theory in Chapter 9, and the geostrophic balance in Chapter 10. We know little about the depth of no motion in the tropical Pacific.
2. The solutions are limited to the east side of the ocean because \(M_{x}\) grows with \(x\). The result comes from neglecting friction, which would eventually balance the wind-driven flow. Nevertheless, Sverdrup solutions have been used for describing the global system of surface currents. The solutions are applied throughout each basin all the way to the western limit of the basin. There, conservation of mass is forced by including north-south currents confined to a thin, horizontal boundary layer (figure \(\PageIndex{3}\)).
3. Only one boundary condition can be satisfied, no flow through the eastern boundary. More complete descriptions of the flow require more complete equations.
4. The solutions do not give the vertical distribution of the current.
5. Results were based on data from two cruises plus average wind data assuming a steady state. Later calculations by Leetmaa, McCreary, and Moore (1981) using more recent wind data produces solutions with seasonal variability that agrees well with observations provided the level of no motion is at 500 m. If another depth were chosen, the results are not as good.
6. Wunsch (1996: §2.2.3), after carefully examining the evidence for a Sverdrup balance in the ocean, concluded we do not have sufficient information to test the theory. He writes:
   The purpose of this extended discussion has not been to disapprove the validity of Sverdrup balance. Rather, it was to emphasize the gap commonly existing in oceanography between a plausible and attractive theoretical idea and the ability to demonstrate its quantitative applicability to actual oceanic flow fields.—Wunsch (1996).
   
   Wunsch, however, notes:
   Sverdrup’s relationship is so central to theories of the ocean circulation that almost all discussions assume it to be valid without any comment at all and proceed to calculate its consequences for higher-order dynamics…it is difficult to overestimate the importance of Sverdrup balance.—Wunsch (1996).
   
   But the gap is shrinking. Measurements of mean stress in the equatorial Pacific (Yu and McPhaden, 1999) show that the flow there is in Sverdrup balance.

Figure \(\PageIndex{3}\): Depth-integrated Sverdrup transport applied globally using the wind stress from Hellerman and Rosenstein (1983). Contour interval is 10 Sverdrups. After Tomczak and Godfrey (1994: 46).

Stream Lines, Path Lines, and the Stream Function

Before discussing more about the ocean’s wind-driven circulation, we need to introduce the concept of stream lines and the stream function (see Kundu, 1990: 51 & 66).

At each instant in time, we can represent a flow field 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 Lagrangian 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 \(\psi\) defined by: \[u \equiv \frac{\partial \psi}{\partial y}, \quad\quad v \equiv -\frac{\partial \psi}{\partial x} \nonumber \]

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 \(\psi\). Thus if the flow is steady, the lines of constant stream function are the paths followed by water parcels.

Figure \(\PageIndex{4}\): Volume transport between stream lines in a two-dimensional, steady flow. After Kundu (1990: 68).

The volume rate of flow between any two stream lines of a steady flow is \(d \psi\), and the volume rate of flow between two stream lines \(\psi_{1}\) and \(\psi_{2}\) is equal to \(\psi_{1} - \psi_{2}\). To see this, consider an arbitrary line \(dx = (dx,dy)\) between two stream lines (figure \(\PageIndex{4}\)). The volume rate of flow between the stream lines is:\[v \ dx + (-u) \ dy = - \frac{\partial \psi}{\partial x} dx - \frac{\partial \psi}{\partial y} dy = - d \psi \nonumber \]

and the volume rate of flow between the two stream lines is numerically equal to the difference in their values of \(\psi\). Now, let’s apply the concepts to satellite-altimeter maps of the oceanic topography. In Section 10.3 I wrote \((\PageIndex{10.3.2})\): \[\begin{align} u_{s} &= -\frac{g}{f} \frac{\partial \zeta}{\partial y} \nonumber \\ v_{s} &= \frac{g}{f} \frac{\partial \zeta}{\partial x} \end{align} \nonumber \]

Comparing \((\PageIndex{16})\) with \((\PageIndex{14})\) it is clear that \[\psi = -\frac{g}{f} \zeta \nonumber \]

and the sea surface is a stream function scaled by \(g/f\). Turning to figure \(10.3.5)\), 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 \(10.6.1\), 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 \(\Psi\) defined by: \[M_{x} \equiv \frac{\partial \Psi}{\partial y}, \quad\quad M_{y} \equiv -\frac{\partial \Psi}{\partial x} \nonumber \]

This is the function shown in figures \(\PageIndex{2}\) and \(\PageIndex{3}\).