Barotropic Radiation Experiment (BARX)

February 14, 2003

The question of how energy flows through the oceans, especially how energy is lost from the currents comprising the general circulation, is receiving much theoretical and modeling attention. This issue is important for understanding not only the structure and transport of the general circulation, but also how water masses and tracers are stirred horizontally and mixed vertically throughout the water column. On theoretical grounds, an important process contributing to the damping of strong currents is radiation of barotropic Rossby waves. The mechanism of Rossby wave excitation by strong meandering currents has been studied extensively by theoreticians in the last two decades. There is some evidence of the existence of this phenomenon between the Gulf Stream and the continental shelf of North America. How much energy this radiation carries away from strong meandering currents, and how fast that energy flux is released to other processes such as diapycnal mixing are issues about which very little is understood. An organized field program specifically aimed at observing radiation toward the open ocean has never been undertaken due to the difficulty of isolating barotropic currents (here meaning simply depth-independent currents).

In addition to providing an introduction to this project, this web page is meant to provide additional details on the properties of tomographic data, and the inversion of that data for current, that do not appear in the text of the proposal to the National Science Foundation. You can read the complete text of the proposal HERE - baro.pdf (1.7 MB).

Principal Investigators: Brian Dushaw (APL-UW, lead institution), Doug Luther (UH), Peter Worcester (SIO), Alan Chave (WHOI), Art Miller (SIO).

The Proposed Array: Tomography, Horizontal Electromagnetic Field Recorders, and Bottom Pressure Recorders

This map shows the proposed array. The stars scattered over the proposed tomography array show nominal locations of 12 pressure and HEF gauges. The contour lines show potential vorticity (f/H) (e.g., Koblinsky, 1990). The heavy blue line to the north is the nominal location of the Gulf Stream.

This antenna for barotropic radiation will combine point (i.e., horizontal electric field recorders - HEFRs) and line integral (i.e., tomography) observations of current and temperature in an array that will be sensitive to phenomena with length scales of 10-1500 km. The HEFR measurements will observe depth-averaged currents produced by phenomena with horizontal scales as short as 10 km (specifically, twice the ocean depth; Chave and Luther, 1990). Single tomography paths (i.e., between two moorings only) will yield currents and temperature averaged along the path, so that the smaller horizontal scale variability is filtered out. The lengths of the acoustic paths range from 500 to 1200 km.

Within southward propagating packets of radiated barotropic Rossby waves, the current variations should be polarized in the northeast-southwest direction, the phase propagation should be northwestward with wavenumbers consistent with the barotropic topographic-Rossby wave dispersion, the group velocity should be southward (likely with a westward component), and there should be oscillatory relative vorticity, concomitant bottom pressure variations, and negligible temperature fluctuations. The array described here has the ability to measure all of these variables.

Tomography, Arrival Patterns, and Resolution

The proposed experiment relies on instruments that are uniquely capable of observing barotropic currents. The figure to the left shows some of the properties of sampling offered by ocean acoustic tomography.

The top panel, (a), shows the predicted acoustic arrival pattern for an 800-km long acoustic path along a diagonal of the northern half of the proposed array. Some 30-40 ray arrivals are expected to be resolved for this area of the ocean. The middle panel, (b), shows the horizontal and vertical sampling of the acoustic rays. Some 40 rays may be resolvable in this case. The next panel, (c). shows a nominal sound speed profile, and panel (d) shows the weighted sampling (resolution) with depth of the rays in panel (b). The weighted sampling as a function of depth is a summation of ds/c^2 over range, where ds is an element of ray arc length and c, is sound speed (the peaks occur at the ray turning depths.)

The panel at lower right, (e), shows the equivalent depth weighting for the current meters of one of the FASINEX experiment moorings; horizontally, the current meter data offer delta function sampling as well.

A Simple Inverse Problem Exercise - Resolution of the Barotropic Mode

The inverse problem involves solving for the current structure using the difference of reciprocal travel times associated with each of the rays shown above. The difference of these travel times is a measure of current because the travel time with the current is different than the travel time against the current.

The inverse requires a model for the current - that is, a set of vertical functions for which the amplitudes are calculated in the inverse. For this exercise, we'll assume that these vertical functions are dynamical modes. The 2001 World Ocean Atlas was used to extract profiles of temperature and salinity for the region of the proposed array. These profiles were then used to calculate sound speed and buoyancy. Buoyancy was then used to calculate a set of vertical modes; these are shown at right, together with a depth-independent mode for barotropic current. For the purposes of this inversion, the modes are normalized so that all of them have a maximum value of 1.

The integrals along the ray paths of the vertical modes were then calculated, taking into account the ray weighting, ds/c^2. In this exercise, range independence is assumed; the result can be readily generalized to include range dependence using a truncated Fourier series. The inverse involves solving the equation d = G*m + e, where d is data (differential travel times), G is the matrix of ray-weighted integrals over the mode functions, and m are mode amplitudes to be solved for. Data noise is e. The inversion assumes a priori weightings (or variances) for mode amplitudes and data noise. The first test of the ability of this inverse problem to resolve the modes is to examine how the a priori uncertainties assumed for the mode amplitudes compares to the formal uncertainties for them - by how much has the assumed variance been reduced?
The a priori variance was assumed to be diagonal:

    0.0400         0         0         0         0         0
         0    0.0400         0         0         0         0
         0         0    0.0400         0         0         0
         0         0         0    0.0400         0         0
         0         0         0         0    0.0400         0
         0         0         0         0         0    0.0400

    0.0000   -0.0001    0.0000   -0.0000   -0.0000   -0.0000
   -0.0001    0.0018   -0.0003    0.0002    0.0022    0.0006
    0.0000   -0.0003    0.0001    0.0003   -0.0006   -0.0001
   -0.0000    0.0002    0.0003    0.0019   -0.0006    0.0005
   -0.0000    0.0022   -0.0006   -0.0006    0.0033    0.0006
   -0.0000    0.0006   -0.0001    0.0005    0.0006    0.0003

By way of a more concrete (somewhat pathological) example, let's assume a surface-intensified current of 0.30 m/s, together with a barotropic (depth-independent) current of 0.05 m/s, and test how the inverse solution compares to the "true" current. The figure at right shows the assumed "true" current (blue) and the inverse solution (red). All modes are weighted equally for this solution. The formal uncertainties in the current profile are given by the dotted red lines. For this example, the differential travel times ranged from 40 to 80 msec (the figure to far right shows the data as a function of ray upper turning depth). For this example, the "true" value for the depth-independent current was 0.05 m/s, and the "true" depth-averaged current was 0.067 m/s. The estimated value for the barotropic mode was 0.065+/-0.002 m/s, and the estimated value for the depth-averaged current was 0.060+/-0.002 m/s.

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