a. Sampling theory

The radial velocities (b₁, b₂, b₃, b₄) for a four-beam downward-looking convex head in the Janus configuration (Fig. 1) actually measure the following Cartesian velocities:

View Expanded

where x₀ and y₀ represent the nominal horizontal position of the transducer and z_i is the nominal depth for the ith cell; b_j(x₀, y₀, z_i) represents a beam velocity, positive toward the transducer j = 1–4; u, υ, and w are the Cartesian east, north, and vertical velocities, respectively; θ is the deviation of the axis of the beam from vertical; and s(d_i) = d_i tanθ is half the horizontal beam separation at a depth cell a vertical distance d_i from the transducer face (see Fig. 1). A little algebra isolates the Cartesian velocities in a more convenient form:

View Expanded

If the horizontal dimensions of the feature (e.g., soliton or turbulence patch) are larger than the beam separation, then one can expand (2) in a Taylor series about x₀, y₀, z_i:

View Expanded

The error consists of a term proportional to the distance from the transducer face times the horizontal shear of the perpendicular component at first order and the square of the distance times the second derivative of the normal component velocity at second order. If the horizontal gradients in the velocity field are small, corresponding to large spatial scales, only the zero-order terms are retained, and the transformations normally used to estimate ADCP mean velocities are obtained. Equations (3) represent the spatial convolution inherent in a single ADCP measurement. A temporal averaging may also be performed that further distorts a measurement of the true velocity field.

If the water is incompressible (a good assumption near the surface), then

View Expanded

Substitution into Eq. (3c) yields a first-order correction for w given by

View Expanded

where ŵ is a corrected estimate of w̃. The correction is based on a first-order vertical derivative only, can be obtained readily from a single profile, and is independent of the horizontal orientation (azimuth) of the ADCP.

For turbulence studies, time scales are generally faster than the buoyancy period and the fluctuations are assumed to be advected by the mean flow (e.g., Marsden et al. 1993; McPhee 1994), the “frozen turbulence” hypothesis. Typically, the coordinate frame is rotated along the mean (20 min to hourly averaged) flow and the time derivatives are converted to spatial derivatives. Alternatively, for propagating features, the axes may be rotated along the phase velocity. The downstream spatial derivatives can be converted to time derivatives through

View Expanded

where c and U are the phase speed and mean advective velocities, respectively.

Two methods were considered for the implementation of the corrections. First, a numerical technique was used to invert Eq. (5). This method suffers from an incomplete specification of the initial boundary condition. For the second method, the vertical derivative was approximated from the raw data; that is,

View Expanded

It was found that the two methods, except for the first bin, produced almost identical results and, given its ease of implementation, only (7) will be considered further. Corrections to the horizontal velocity were then made using the corrected vertical velocity as follows:

View Expanded

where the primes indicate the frame rotated along the mean velocity. Note that the correction for the horizontal cross-stream velocity (υ̃′) is assumed to be of second order. The absolute limit of ADCP resolution is, of course, governed by sampling theory. Features must have horizontal length scales greater that twice the bin size and separation to avoid aliasing.

a. Internal waves

The first simulation mimics ensemble-averaging 100 pings in a 2-min ensemble from an ADCP with a 30° transducer angle. Modern pulse-to-pulse coherent ADCPs are quite accurate and for these system characteristics have errors of 0.173 cm s⁻¹ for the vertical and 0.42 cm s⁻¹ for the horizontal velocities, respectively. The rms error in the vertical gradient is given by

σ_d2σ_w̃d²z(9)

and the overall error for the corrected vertical velocity is then

View Expanded

Similarly, the error in the horizontal correction is given by

σ_t2σ_ŵdUct(11)

and the overall error is

View Expanded

Averaging would typically be used for a long-term deployment with a low-frequency ADCP, such as that deployed by Schott and Leaman (1991) to detect vertical convection in the Gulf of Lyons or by Melling et al. (2001) for a long-term survey of the North Water Polynya. Here accuracy, data storage, and battery power considerations may necessitate ensemble averaging. The specific example will be internal solitary waves detected by Marsden et al. (1994b) in the Canadian Arctic. They show that the Benjamin (1966) model gives a crude, but fairly reasonable, estimate of some waves. It is a shallow water, Boussinesq model for which the streamlines are given by

ψzaϕz²λx(13)

with horizontal and vertical velocities

View Expanded

where w and υ are the vertical and horizontal velocities; a and c are the amplitude and phase speed, respectively, and are negative for waves of depression and waves traveling in the negative direction; and ϕ is the eigenvector and ϕ′ its vertical derivative corresponding to the largest eigenvalue (c₀) of the Sturm–Liouville problem:

View Expanded

The phase speed (c) and length scale (λ) of the wave are given by

View Expanded

Profiles of the density, ϕ, and ϕ′ were based on a cast taken at 1201 eastern standard time (EST) 30 April 1992 outside of Resolute, Nunavut, Canada. Details of the cast can be found in Marsden et al. (1994a). Care must be taken in implementing the model. The correction to the vertical velocity field [Eqs. (4) and (5)] is critically dependent on the incompressibility assumption. The model velocities were applied at the streamline depth, the results were differentiated numerically, and the incompressibility condition (critical for this method) was verified.

Equation (14) was then used to calculate a theoretical 15-m solitary wave that was then propagated in the −x direction at the net speed of the wave (c + U) = 40 cm s⁻¹, where c is the intrinsic phase speed from (14) and U is the background tidal flow. Normally, solitons are embedded in the mean flow. The time of the disturbance can be found from the vertical velocity. The background flow is then taken to be an average of the flow immediately preceding and following the flow disturbance. The model was then sampled and averaged at the appropriate depth-dependent separations (i.e., ±d_i tanθ) as outlined in (2) to imitate the spatial convolution inherent in the measurement. These values were compared to an equivalently averaged “direct” wave (i.e., a simulated average of the depth cells at one point directly beneath the transducer face). The model horizontal and vertical velocities prior to convolution and averaging are shown in Figs. 2a and 2b. The maximum υ is about 29 cm s⁻¹ and is highly surface trapped within about 20 m of the ice–water interface, while the model predicts maximum vertical velocities of 7.7 cm s⁻¹ at 63-m depth.

The maximum absolute velocity differences between the direct and convolved wave were calculated to quantify the beam divergence problem. Alternatively, the rms errors could have been used, but these have the disadvantage of presenting an averaged view of the errors. The outer extremities of the disturbance will correspond to low velocities and hence present a conservative estimate. The maximum absolute error for the ŵ (Fig. 3a) peaks at 17-m depth with a difference of about 1.3 cm s⁻¹ (or about 20%) of the peak amplitude. The rms ŵ errors as a function of depth using (9) and (10) are indicated by the solid line. The corrected amplitudes are below instrument noise levels to 40-m depth, below which the correction does not improve the estimate. Figure 3b shows the time series of the direct, or unconvolved (solid line), the convolved (dot–dashed line), and the corrected (dashed line) waves. The large discrepancies at the peaks are evident, as is the efficacy of the velocity correction. Figure 4a shows the expected maximum absolute error of the downstream velocity, u, as a function of depth. Here, a highly idealized correction was calculated using the 0.4 m s⁻¹ phase speed (c + U) in (8). The uncorrected errors are less than 2 cm s⁻¹ to 20 m from the transducer head and increase sharply below to more than 8 cm s⁻¹ at 50-m depth. The solid line indicates expected instrument noise levels as determined from (11) and (12). The corrections are greater than the noise level below 30 m, and indeed errors in beam divergence dominate the signal. The correction reduces the maximum error throughout the water column and is less than the instrument noise level within 40 m of the transducer head. Figure 4b shows the true, convolved, and corrected velocities at 33-m depth. The error is quite profound and can be corrected to nearly match the wave profile. It must be noted, however, that the horizontal phase speed will probably never be known with absolute certainty from a single point measurement, and while the simulations demonstrate the potential gains in correcting the horizontal velocities, these may never be realizable.

b. Turbulence simulations

The previous results demonstrated the effects of beam divergence on internal solitary waves that have length scales that are relatively large compared with beam divergence. Several investigators (e.g., Stacey et al. 1999; Lu and Lueck 1999) have extended the use of the ADCP to turbulence scales. Here, the length scale of the flow would be of the same size as beam divergence, even very close to the transducer head. Marsden (2004, manuscript submitted to J. Phys. Oceanogr.) has investigated turbulence near the ice–water interface in the Canadian Arctic. He used a 614-kHz RDI ADCP set to pulse-to-pulse coherent mode (mode 11), sampling at 1.0 s in 28 depth cells at 0.25-cm increments. The instrument head was positioned 0.54 m below the ice– water interface yielding the first sampling bin at 0.94 m. This mode was very accurate as sampling during quiescent periods indicated rms errors of 0.15 and 1.2 cm s⁻¹ per ping in the vertical and horizontal velocities, respectively. The increased resolution and accuracy came at the cost of penetration, as results could only be obtained within 8 m of the transducer head. The minimum horizontal resolvable length scales, that are not aliased, range from 0.5 m in bin 1, due to the 0.25-m bin size, to 10.2 m at 7 m from the transducer head, due to beam divergence.

Simulations suggest that beam divergence is a problem and can be rectified in the vertical and longitudinal velocities. For turbulence studies, normally (McPhee 1994) the mean flow is determined over, say, a 1-h period, and turbulence parameters are calculated from the deviatory velocities. The form of the streamfunction for modeling turbulent fluctuations was taken to be

View Expanded

The horizontal (u = −∂ψ/∂z) and vertical (w = ∂ψ/ d∂x) velocities were calculated for a number of estimates of b, c, and background flows (U). The values of b and 2.0 × c represent the length scales of the vertical and horizontal velocities, respectively. This form forces the vertical velocity fluctuation to be 0.0 at the wall. The amplitude a was adjusted to give typical velocities found in the data. In all cases, beam divergence was significant for the horizontal velocities and estimates could be improved. Figures 5a and 5b show the vertical and horizontal velocities associated with typical values of b = 2.0, c = 5.5, and U = 40 cm s⁻¹. The vertical velocity field is a doublet with maximum values of 3.0 cm s⁻¹, while the horizontal velocity is surface trapped with a maximum value of about 17.0 cm s⁻¹. The simulations were found to be insensitive to the advection speed U. A large value is preferable, however, to dominate any possible intrinsic phase speed of the eddies and to reduce the correction error of (11). Since the time scale of the fluctuations is greater than the minimum buoyancy period of the water column, the influence of internal waves should be small.

Figure 6a shows that maximum error in the vertical velocity is very small (<1.5 mm s⁻¹), considerably less than the instrument noise level. Figure 6b indicates the direct (solid line), measured (dot–dashed line), and corrected (dashed line) values for the 14th bin, 4.0 m from the transducer face. The errors are small and the correction only marginally improves the results near the transducer head. Figure 7a shows the maximum absolute error of the horizontal velocity as a function of depth. Maximum values of the uncorrected velocities are 2.0 cm s⁻¹, above the instrument noise level, between 1.5- and 6.0-m depth, indicated by the solid line. Figure 7b shows the actual, measured, and corrected values at 4.0-m depth. The extent of the beam divergence error and its subsequent correction are quite striking. The simulations were run for considerable ranges in values for b and c, and the results were similar. Errors in the vertical velocity were negligible, while corrections to the horizontal longitudinal velocities were quite profound and brought maximum errors below the instrument noise level.