## 1. Introduction

[2] In the context of long wave theory, the nonlinear shallow water equations (NSWE)

where *u and v* are the two horizontal depth-integrated velocity components, *d* is the still ocean depth, *g* is the acceleration of gravity, and *h* is the total water depth as sum of *d* and the disturbed water surface *η*, can be derived from the incompressible continuity and Euler equations:

where *u, v*, and *w* are the three velocity components and *p* is the pressure, under the assumptions that accelerations in the vertical direction are negligible (i.e., the pressure in the fluid is essentially hydrostatic), the two horizontal velocity components, *u* and *v*, are uniform along the water column (depth-integrated), and the vertical velocity *w* is negligibly smaller than *u* and *v* [*Johnson*, 1997]. The nonlinear analytical [*Synolakis*, 1986, 1987] and numerical [*Liu et al.*, 1995; *Titov and Synolakis*, 1997] and even linear analytical [*Synolakis*, 1986, 1987; *Kânoğlu and Synolakis*, 1998] solutions of shallow water models have repeatedly shown good agreement in wave height when compared with experimental results [*Synolakis*, 1987; *Liu et al.*, 1995; *Titov and Synolakis*, 1997]. The NSW approximation has also shown remarkable agreement with observed tsunami wave height during real tsunami events in both deep and shallow waters [*Wei et al.*, 2008; *Arcas and Titov*, 2006; *Borrero et al.*, 2009; *Tang et al.*, 2009], however, current velocity comparisons with observations during an event are rare, with *Fritz et al.* [2006] providing one of the few examples of surface velocity estimates during the inundation stage of the 2004 Indian Ocean event based on the application of a particle image velocimetry analysis to the motion of Lagrangian markers in the flow, and *Synolakis* [1986] and *Kânoğlu* [2004] providing experimental and analytical results of the wave front progression of a solitary wave from deep to shallow water and during run-up. *Fritz et al.* [2006] observed peak speeds of up to 5 m/s during the on land flow of the 2004 Indian Ocean event in Banda Aceh, and *Synolakis* [1986] reported an abrupt deceleration of the wave front speed of a solitary wave as it reaches the shoreline, followed by a mild acceleration, before decelerating again to complete the run-up process. It is also well known that before the wave nears the shoreline, current speeds are expected to increase from their deep ocean values to the more elevated shallow water values presented here, as reflected in the velocity profile of Figure 1 (bottom), where the computed depth-averaged speeds increase from approximately 0.5 cm/s in deep water to almost 14 cm/s at the ADCP location. Three-dimensional tsunami current speed data with high spatial resolution and sampling frequency are, however, nonexistent, to the knowledge of these authors with this paper providing a first investigation of the three-dimensional vertical and temporal structure of the flow.

[3] On 15 November 2006 at 11:14 UTC an earthquake with moment magnitude 8.3 was recorded (46.607°N, 153.230°W) 445 km east-northeast of Kuril'sk, Kuril Islands, approximately 30.3 km deep (USGS). A transpacific tsunami was generated, taking approximately 6.5 hours to reach Honolulu, Hawaii, with wave heights exceeding 1 m in some coastal areas. An RDI 1200 kHz Sentinel Acoustic Doppler Current Profiler (ADCP) sensor originally installed to study the nearshore coral reef environment [*Herbert et al.*, 2007; *Pawlak et al.*, 2009] was located approximately 430 m offshore of Honolulu at a depth of 10 m below mean sea level. Figure 1 (top) shows the location of the sensor relative to the coastline of Honolulu. The sensor sampling frequency was 1 Hz, with an accuracy in the pressure measurement of 1 cm of water and better than 0.4 cm/s in the depth-averaged velocity measurements [*Bricker et al.*, 2007]. Measurements of the bottom pressure and three velocity components were made at a maximum of 40 bins spaced 0.25 m apart along the water column from a depth of 0.82 m to 8.57 m, above the seafloor with *u* (zonal) and *v* (meridian) velocities defined positive toward the east and north, respectively, and *w* velocities positive upward. The purpose of the present study is to use these tsunami current speed data to investigate some of the necessary (but not sufficient) conditions for the application of the NSWE approximation to the Euler equations in tsunami modeling.

[4] The temporal and spatial resolution of the data recorded allows for the evaluation of accurate estimates of both temporal and vertical derivatives at a single location and for the construction of terms present in the Euler equations containing such derivatives.

[5] It is interesting to note that no evidence of boundary layer was observed in the velocity profiles, suggesting that either the location of the deepest bin of recorded values at 0.82 m above the seafloor lies above the thickness of the boundary layer, or that the tsunami frequency is too high to allow the formation of a fully developed boundary layer.

[6] The hydrostatic assumption in the NSWE implies that vertical accelerations in the problem are negligibly small compared to the acceleration of gravity and vertical pressure gradient, effectively eliminating all terms on the left-hand side (LHS) of equation (7) to obtain:

[7] The assumption of depth-averaged horizontal velocity components effectively eliminates from the equations any vertical derivatives of *u* and *v* and is, therefore, equivalent to considering uniformity of velocity profiles along the water column, resulting in the elimination of the convective terms *wu*_{z} and *wv*_{z} from the horizontal momentum equations (5) and (6).

[8] In the present study, we make use of the available data to compute estimates of *w*_{t} and *ww*_{z} on the LHS of (7) and compare their size to the acceleration of gravity, *g*, on the RHS. Similarly, we estimate values of *wu*_{z} and *wv*_{z} and compare those to the size of the only terms retained on the LHS of (5) and (6), which can be evaluated from the available data, namely, *u*_{t} and *v*_{t}.

[9] Lastly, we present a comparison of modeled and observed wave heights and depth-averaged velocity components at the station location using the tsunami modeling code MOST (Method of Splitting Tsunamis) [*Titov and González*, 1997] which has been validated and verified extensively as specified by *Synolakis et al.* [2008], and examine their correlation and RMS values. We conclude that the marginal violation of vertical uniformity of horizontal velocity profiles can, at least in part, be responsible for some of the discrepancies observed between modeled and measured waveforms.