Current Structure and Measurement

As we have made the linear wave assumption, the vertical velocities induced by these motions are small; however, the horizontal current structure can show significant complexity both in the horizontal and vertical dimension. The complexity in the horizontal direction is due primarily to the presence of boundaries (and hence the horizontal structure of the waves), whereas the vertical complexity is due to both the stratification and the vertical mode.

It is important to tie the vertical position of the measurement location with the likely motion that dominates the flow. The simplest method to determine the likely points of maximum displacement and maximum current is to solve the long linear internal wave problem in a rotating system d2

where wm is the vertical velocity eigenfunction for waves of vertical mode m, z the vertical dimension, N2(z) is the vertical profile of the square of the buoyancy frequency,

where g is the gravitational acceleration, p0 is a reference density (typically the maximum density), p(z) is the vertical profile of density, and cm is the internal wave phase speed for the particular vertical mode in question. This is an eigenvalue problem, such that an infinite number of solutions exist for an infinite number of vertical modes m. For the case of a constant N2, the equation has the sinusoidal solution

where um is the horizontal velocity induced by the wave and «m is a constant. It is quite clear from the above that the position of maximum vertical displacement is offset from the position of maximum horizontal current, which the selection of measurement points needs to take into account. Note that the equivalent depth for each vertical mode in a continuously stratified system can be calculated as

For nonconstant N2(z), eqn. [17] is relatively easily solved numerically as it takes the form of a Sturm-Liouville equation, so from a depth profile of temperature a vertical profile of N2 can be computed and fed into the eigenvalue solver, which will return both the eigenvalues cm and the eigenvectors wm. These eigenvectors will indicate which region of the water column will experience the maximum isotherm oscillations, and the derivate of this eigenvector with respect to z will give the position of the maximum horizontal velocity fluctuations. It is at these locations that thermistors and current meters should be concentrated, respectively. For most vertical modes, concentrating instruments in and around the thermo-cline is sufficient, except for capturing the velocity signal of the first vertical mode in which current measurements are best made either near the surface or bottom.

A key method to link currents with the wave motion described earlier is the use of rotary spectra of currents. By analyzing the different direction of propagation of the currents at different depths and points in space, it is possible to understand not only the predominant frequency of oscillation but also the predominant direction of rotation. Figure 6 shows data from Lake Kinneret in which the basin-scale wave field is dominated by a cyclonic vertical mode one Kelvin wave of period ^24 h, an anticyclonic vertical mode one Poincare wave of period ~12 h, and an anticyclonic vertical mode two Poincare wave of period ^22 h. Analysis of current data collected at this location indicates that the Kelvin wave effect on currents in the thermocline (where the 24 °C

isotherm is located) is weak at this station as the current in the 20-24 h bandwidth is dominated by anticyclonic rotation.

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