[60] Each Spray glider deployed in the CCS is equipped with a custom Sontek Argonaut 750 kHz acoustic Doppler profiler (ADP) mounted in the tail. The instrument is oriented such that an upward pitch at the nominal 17° ascent angle and zero roll result in the central axis of the ADP pointing downward so that range bins are depth bins. The three beams of the instrument are aimed 25° off the central axis with one beam looking forward along the long axis of the glider.

[61] The ADP collects a 16 ping ensemble average every 4 m in the vertical during the ascending portion of a dive (Figure B1a). Each ensemble average provides measurements of along-beam speed and return amplitude in five measurement cells for each beam. Each measurement cell extends 4 m in the vertical. For the settings used in the CCS, the first measurement cell is centered 10 m below the glider. The sampling parameters are such that cells from successive ensembles should align as indicated in Figure B1a.

[62] Glider ADP measurements undergo several processing and quality control steps before profiles of ocean velocity are estimated. We calculate the depth of each measurement cell from records of the glider's depth, pitch, and roll during each 16 ping ensemble. Data from ensembles when the glider is pitched or rolled enough to displace measurement cells from their nominal alignment are excluded from further processing. The ADP can be used as an altimeter during the descending portion of a dive; data in cells that are deeper than the altimeter-derived bottom depth are excluded. Along-beam speeds are used to calculate eastward (*u*), northward (*v*), and upward speeds relative to the glider by successive rotations using records of the glider's pitch, roll, and heading during each ensemble average. Velocities relative to the glider that exceed 0.75 m s^{−1} are considered to be erroneous since the speed of the glider through the water is approximately 0.25 m s^{−1} and the range of the ADP is too small (about 20 m) to expect very large relative velocities. We also exclude measurements for which the signal-to-noise ratio is less than 1.0.

[63] Transducer failures have occurred during some deployments. We use the average return strength for each beam during each profile to detect failures of transducers. Any sudden drop in return strength of one beam relative to the other two indicates failure of the respective transducer, and data from that beam are not used in further calculations. The loss of data from one or more beams prohibits calculation of a velocity profile. Transducer failures are the primary cause of the missing ADP velocity profiles shown in Figure 2.

[64] We let *N* be the number of ensemble averages during the ascending portion of a glider dive; the sampling geometry defines a set *N* + 4 estimation bins (Figure B1b). Measurements in the shallowest cell (cell 1) for each ensemble are not used because of ringing of the ADP transducers, so up to four measurements contribute to the estimate in each bin. The exclusion of data from cell 1 results in no data in the uppermost sampling bin, so we only estimate velocity in the *N* + 3 bins with data. The number of measurements contributing to the estimate in a bin is reduced if measurements are excluded during quality control. Because the glider sampling pattern is not perfectly regular, the depth of a given estimation bin is defined to be the mean depth of each good measurement in the bin.

[65] The glider-mounted ADP functions similarly to a lowered acoustic Doppler current profiler (LADCP) deployed from from a research vessel, and our calculation of ocean velocity profiles from the glider-mounted ADP data is based on the LADCP data processing scheme presented by *Visbeck* [2002]. For each valid measurement of horizontal water velocity relative to the glider, (*u*, *v*)_{r}, we have an equation

where (*u*, *v*)_{w} is the ocean velocity at the location of the measurement cell, and (*u*, *v*)_{g} is the velocity of the glider at the same moment. (Note that here the subscript *r* refers to water velocity relative to the glider, which is the opposite of the glider's velocity through the water used in Appendix A.) Both terms on the right hand side of equation (B1) are unknown. There are *N* unknown glider velocities (one for each sampling depth), and *N* + 3 unknown water velocities (one for each estimation bin with data). Excluding data from the shallowest cell for each ensemble, we have at most 4*N* equations of the form of (B1). This system of equations can be written as a matrix equation of the form **Gm** ≅ **d**, where

is the vector of observations of speed relative to the glider in one direction,

is the vector of unknown glider and water velocities in that direction, and

is the matrix of coefficients when all measurements are good. When all measurements are used, **d** has dimensions 4*N* × 1 and **G** has dimensions 4*N* × (*N* + *N* + 3). Loss of measurements reduces only the number of equations in the system, so that, in practice, **d** and **G** have at most 4*N* rows, but **m** always has dimension (*N* + *N* + 3) × 1.

[66] Though the number of equations exceeds the number of unknowns, we still require additional information to solve the system of equations since the ADP data alone can only provide the baroclinic portion of the ocean velocity [*Visbeck*, 2002]. We use the estimate of vertically averaged water velocity during each dive (Appendix A) to reference the ADP shear. This measurement of vertically averaged velocity is valid from the surface to the maximum depth reached by the glider, a range that is offset from the sampling range of the ADP since the ADP samples below the glider. We account for this offset in two ways. First, we exclude ADP velocity estimates in the seven bins (*i*_{bin} = *N* − 2,…, *N* + 4) that are deeper than the glider's maximum depth (Figure B1b) from the constraint. Second, we assume that the near-surface portion of the water column that is not sampled by the ADP has uniform velocity. Under this assumption, we weight the uppermost estimation bin as if it extended to the surface in the vertically averaged velocity constraint. This constraint adds the row

to the matrix **G**. The Δ*z*_{i} are the vertical extents of the velocity bins, which are approximately 4 m, except for Δ*z*_{2} which is larger as discussed above. The corresponding element added to **d** is *U*Σ_{i=2}^{N−3}Δ*z*_{i}, where *U* is the estimated vertically averaged velocity. Since the ADP measures shear only on the ascending portion of each dive and vertically averaged velocity is based on the glider's displacement throughout the entire dive, there is a mismatch in location and time between the shear profile and the barotropic constraint that is unaccounted for. At the 30 km and larger scales considered in this analysis, any errors due to this mismatch should not be significant. The agreement between ADP-derived currents after 30 h filtering and geostrophic currents (e.g., Figure 3) suggests that the induced errors are small.

[67] Ideally, the overdetermined system **Gm** ≅ **d** is now solvable by least squares techniques. However, the loss of equations due to bad measurements can make the system ill conditioned. To further constrain the problem and reduce noise in the solution, we apply the curvature-minimizing smoothness constraint of *Visbeck* [2002] to the horizontal ocean velocities and horizontal glider velocities. These constraints add *N* + 1 and *N* − 2 additional equations to the system, respectively. The additional rows added to **G** are

and

where *w* is a weight that determines the degree of smoothing. We choose *w* = 5 to produce sufficiently smooth velocity profiles. The data vector **d** gains 2*N* − 1 rows of zeros since we seek to minimize curvature in the solution.

[68] We then solve the system for the unknown glider velocities and horizontal ocean velocities using least squares techniques to minimize the *L*_{2} norm of **Gm** − **d**. The solution is