10.7: Currents from Hydrographic Sections

Lines of hydrographic data along ship tracks are often used to produce contour plots of density in a vertical section along the track. Cross-sections of currents sometimes show sharply dipping density surfaces with a large contrast in density on either side of the current. The baroclinic currents in the section can be estimated using a technique first proposed by Margules (1906) and described by Defant (1961: 453). The technique allows oceanographers to estimate the speed and direction of currents perpendicular to the section by a quick look at the section.

To derive Margules’ equation, consider the slope \(\partial z/\partial x\) of a stationary interface between two water masses with densities \(\rho_{1}\) and \(\rho_{2}\) (see figure \(\PageIndex{1}\)). To calculate the change in velocity across the interface we assume homogeneous layers of density \(\rho_{1} < \rho_{2}\), both of which are in geostrophic equilibrium. Although the ocean does not have an idealized interface as we assumed, and the water masses do not have uniform density, and the interface between the water masses is not sharp, the concept is still useful in practice.

Figure \(\PageIndex{1}\): Slopes \(\beta\) of the sea surface and the slope \(\gamma\) of the interface between two homogeneous, moving layers, with density \(\rho_{1}\) and \(\rho_{2}\) in the Northern Hemisphere. After Neumann and Pierson (1966: 166)

The change in pressure on the interface is: \[\delta p = \frac{\partial p}{\partial x} \ \delta x + \frac{\partial p}{\partial z} \ \delta z \nonumber \]

and the vertical and horizontal pressure gradients are obtained from \((10.2.1)\): \[\frac{\partial p}{\partial z} = -\rho_{1} g + \rho_{1} f v_{1} \nonumber \]

Therefore: \[\begin{align} \delta p_{1} &= -\rho_{1} f v_{1} \ \delta x + rho_{1} g \ \delta z \\ \delta p_{2} &= -\rho_{2} f v_{2} \ \delta x + \rho_{2} g \ \delta \end{align} \nonumber \]

The boundary conditions require \(\delta p_{1} = \delta p_{2}\) on the interface if the interface is not moving. Equating \((\PageIndex{2})\) with \((\PageIndex{3})\), dividing by \(\delta x\), and solving for \(\delta z/\delta x\) gives: \[\frac{\partial z}{\partial x} \equiv \tan \gamma = \frac{f}{g} \left(\frac{\rho_{2} v_{2} - \rho_{1} v_{1}}{\rho_{2} - \rho_{1}}\right) \nonumber \]

Because \(\rho_{1} \approx \rho_{2}\), and for small \(\beta\) and \(\gamma\), \[\begin{align} \tan \gamma &\approx \frac{f}{g} \left(\frac{\rho_{1}}{\rho_{2} - \rho_{1}}\right) \left(v_{2} - v_{1}\right) \\ \tan \beta_{1} &= -\frac{f}{g} v_{1} \\ \tan \beta_{2} &= -\frac{f}{g} v_{2} \end{align} \nonumber \]

where \(\beta\) is the slope of the sea surface, and \(\gamma\) is the slope of the interface between the two water masses. Because the internal differences in density are small, the slope is approximately 1000 times larger than the slope of the constant pressure surfaces.

Consider the application of the technique to the Gulf Stream (figure \(10.5.1\)). From the figure: \(\varphi = 36^{\circ}\), \rho_{1} = 1026.7 \ \text{kg/m}^{3}\), and \(\rho_{2} = 1027.5 \ \text{kg/m}^{3}\) at a depth of 500 decibars. If we use the \(\sigma_{t} = 27.1\) surface to estimate the slope between the two water masses, we see that the surface changes from a depth of 350 m to a depth of 650 m over a distance of 70 km. Therefore, \(\tan \gamma = 4300 \times 10^{-6} = 0.0043\), and \(\Delta v = v_{2} - v_{1} = -0.38 \ \text{m/s}\). Assuming \(v_{2} = 0\), then \(v_{1} = 0.38 \ \text{m/s}\). This rough estimate of the velocity of the Gulf Stream compares well with velocity at a depth of 500 m calculated from hydrographic data (Table \(10.5.3\)) assuming a level of no motion at 2,000 decibars.

The slopes of the constant-density surfaces are clearly seen in figure \(10.5.1\). And plots of constant-density surfaces can be used to quickly estimate current directions and a rough value for the speed. In contrast, the slope of the sea surface is \(8.4 \times 10^{-6}\) or 0.84 m in 100 km if we use data from Table \(10.5.3\).

Note that constant-density surfaces in the Gulf Stream slope downward to the east, and that sea-surface topography slopes upward to the east. Constant pressure and constant density surfaces have opposite slope.

If the sharp interface between two water masses reaches the surface, it is an oceanic front, which has properties that are very similar to atmospheric fronts.

Eddies in the vicinity of the Gulf Stream can have warm or cold cores (figure \(\PageIndex{2}\)). Application of Margules’ method to these mesoscale eddies gives the direction of the flow. Anticyclonic eddies (clockwise rotation in the northern hemisphere) have warm cores (\(\rho_{1}\) is deeper in the center of the eddy than elsewhere) and the constant-pressure surfaces bow upward. In particular, the sea surface is higher at the center of the ring. Cyclonic eddies are the reverse.

Figure \(\PageIndex{2}\): Shape of constant-pressure surfaces \(p_{i}\) and the interface between two water masses of density \(\rho_{1}, \rho_{2}\) if the upper is rotating faster than the lower. Left: Anticyclonic motion, warm-core eddy. Right: Cyclonic, cold-core eddy. Note that the sea surface \(p_{0}\) slopes up toward the center of the warm-core ring, and the constant-density surfaces slope down toward the center. Circle with dot is current toward the reader, circle with cross is current away from the reader. After Defant (1961: 466).