Evaluation of velocity-related approximations in the nonlinear shallow water equations for the Kuril Islands, 2006 tsunami event at Honolulu, Hawaii


  • Diego Arcas,

    1. NOAA Center for Tsunami Research, Pacific Marine Environmental Laboratory, National Oceanic and Atmospheric Administration, Seattle, Washington, USA
    2. Joint Institute for the Study of the Atmosphere and Ocean, University of Washington, Seattle, Washington, USA
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  • Yong Wei

    1. NOAA Center for Tsunami Research, Pacific Marine Environmental Laboratory, National Oceanic and Atmospheric Administration, Seattle, Washington, USA
    2. Joint Institute for the Study of the Atmosphere and Ocean, University of Washington, Seattle, Washington, USA
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[1] The Nonlinear Shallow Water Equations (NSWE) provide a model for long wave behavior commonly used in tsunami modeling problems in which the scale associated with the surface wavelength is much greater than the ocean depth. This approximation allows for the derivation of the NSWE under the assumptions that the pressure is hydrostatic and the horizontal velocity components uniform along the water column. The present study uses current velocity data acquired by the Kilo Nalu Near-Shore Reef Observatory Acoustic Doppler Current Profiler (ADCP) on the south shore of Oahu (21.288°N, 157.865°W) in the aftermath of the 2006 Kuril tsunami to assess the validity of the NSWE assumptions on the velocity components. ADCP measurements provide information on all three velocity components along the water column, allowing the calculation of terms discarded in the NSWE approximation, containing temporal or vertical derivatives. Comparison of the relative magnitude of terms retained and neglected in the NSWE reveals that in shallow waters (10 m) the size of discarded terms remains smaller, but approaches the order of magnitude of retained terms in the momentum equations. Not all terms present in the equations can be evaluated from the available data, so verification of the assumptions investigated here represents a necessary but not sufficient condition for the validity of the NSWE approximation to model tsunami waves in shallow coastal waters.

1. Introduction

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

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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:

equation image
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equation image
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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.

Figure 1.

(top) Location of the ADCP station relative to the Honolulu coastline and (bottom) profile of the computed maximum depth-averaged current speed along the transect indicated in Figure 1 (top) with a red line. The computed maximum speed values range from (∼0.5 cm/s) in deep water to almost 14 cm/s at the ADCP location.

[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:

equation image

[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 wuz and wvz from the horizontal momentum equations (5) and (6).

[8] In the present study, we make use of the available data to compute estimates of wt and wwz on the LHS of (7) and compare their size to the acceleration of gravity, g, on the RHS. Similarly, we estimate values of wuz and wvz 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, ut and vt.

[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.

2. Method

[10] A sufficiently long data record to capture tidal components with periods longer than those present in the tsunami bandwidth was selected starting approximately 12 hours before tsunami arrival in Honolulu and lasting for approximately 40 hours. All three velocity components at 40 different vertical layers and bottom pressure were recorded. Proximity to the surface of the four upper layers of data points, particularly at low tide resulted in discontinuous time series or bad data measurements. To avoid contamination from these data points, data values from these layers were ignored in our analysis. Next, some isolated missing data points in lower layers were linearly interpolated in time, and both pressure and velocity time series were band-passed filtered between periods of 2 minutes to 2.5 hours using a Butterworth filter. This frequency band captures most of the energy in the tsunami spectrum while discarding higher frequency components due to the effect of ocean swell and lower frequencies associated with tidal oscillations. Finally, the filtered pre-tsunami data were eliminated, resulting in a final filtered data subset starting approximately at the time of tsunami arrival and lasting for more than 24 hours.

[11] Figure 2 represents the recorded wave height and three velocity components as a function of time from a depth of 0.82 m to a depth of 8.57 m from the sea floor, after filtering of non-tsunami frequencies. Only, approximately the first 8 hours of data have been included in Figure 2 for clarity.

Figure 2.

Values of the three velocity components along the water column in cm/s after swell and tidal currents have been filtered out. The top line shows the location of the sea surface for reference during the initial 24-hour period after tsunami arrival.

[12] Next, the temporal and spatial derivatives ut, uz, vt, vz, wt, and wz were computed from the resulting data set by use of a second-order central discretization formula. Use of central discretization approximates the value of the derivative at the center of the spatial and temporal intervals between two neighboring data values, thus defining vertical and temporal derivatives at staggered locations on the grid. The terms wuz, wvz, and wwz, in equations (5)(7) were then constructed for the whole parameter space and their values and those of ut, vt and wt linearly interpolated at the center of each grid cell from their staggered locations. In total, the first and fourth terms on the LHS of equations (5)(7) were computed along the water column as a function of time. Next, three surfaces were created by evaluating the ratios ut/wuz, vt/wvz and g/(wt + wwz) at each cell center. The number of cells where that ratio exceeds 1 and 10 is reflected in Table 1 as a percentage of the total number of cells.

Table 1. Number of Sample Points for Which the Ratios ut/wuz, vt/wvz and g/(wt + wwz) Exceeded 1 and 10 as a Percentage of the Total Number of Sample Points in the Domain
 ut/wuzvt/wvzg/(wt + wwz)

[13] In addition, wave height and depth-averaged values of the two horizontal velocity components as a function of time were evaluated and compared with the corresponding modeled values. Correlation coefficients between observed and simulated data were computed for the first hour of tsunami simulation. Results are shown in Table S1 of the auxiliary material.

3. Results and Discussion

[14] In the NSW approximation dominance and uniformity of the u and v velocity profiles is assumed with the vertical velocity component providing a much smaller (but not negligible) contribution. Deviation from uniformity in the vertical profiles of the horizontal velocity will contribute to discrepancies between observations and numerical solutions of the NSW equations. A reasonable way of quantifying the degree of uniformity in the velocity profiles is to examine the ratio of the squared depth-averaged velocity profile to the depth-averaged velocity profile squared:

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where, for this particular case, the origin of the vertical axis is taken at the sea floor with d1 = 8.57 and d2 = 0.82 indicating the vertical limits of the data above the origin, H is d1-d2, and u represents either one of the horizontal velocity components. When the ratio, ru is close to 1, the velocity profiles exhibit a high degree of uniformity. Small values of this ratio are indicative of vanishing uniformity in the velocity profiles. The value of ru (alt. rv) was computed to be above 0.8 in more than 11% of the data points for the u velocity profiles and in 29% for the v velocity profiles, suggesting a high degree of uniformity along the water column. A plot of the value of this ratio in the intervals between 7 and 9 hours and between 12 and 13 hours after the earthquake is provided in Figures S1 and S2 of the auxiliary material. It is interesting to notice that small values of the ratio are associated with velocity profiles exhibiting small depth-averaged values, as can be observed by examining Animations S1 and S2 of the velocity profiles as a function of time. This is most likely due to the small signal to noise ratio of the data in those realizations.

[15] However, a more direct way to assess the amount of error introduced by the lack of uniformity of u and v is to evaluate the size of the terms eliminated in the NSW approximation where vertical gradients of u and v occur and compare with that of terms retained. It can be seen from the fourth terms in (5) and (6) that almost uniform profiles may contribute significantly to the error in the presence of elevated values of w. Conversely, the effect of significant departure from uniformity can be largely attenuated in the presence of negligible vertical velocity, w. As an example, Figure 3 shows the u, v, and w vertical profiles at the instant when the depth-averaged values of the convective derivatives, wuz and wvz, along the water column showed peak values during the first 8 hours of wave activity (both derivatives peak at approximately the same sampling instant, t = 12.294 hours after earthquake). In this particular case, it becomes evident from the profiles in Figure 3 that peak values of wuz and wvz are caused by the combined effect of steep gradients in the profiles and elevated values in the w velocity.

Figure 3.

Velocity component profiles for (left) u, (middle) v and (right) w along the water column at the time when the depth-integrated values of wuz and wvz reach their maximum value during the first 8 hours of tsunami action. The peak in the values of the convective derivative terms is due to non-uniformities in the profile of the horizontal velocity components and elevated values in the vertical component.

[16] The temporal derivative (first) terms on the LHS of equations (5)(7), ut, vt, and wt are terms conserved in the NSW formulation, while the vertical convective (fourth) terms on the LHS of equations (5)(7), wuz, wvz, and wwz are eliminated; their value, however, can be estimated from the available data. Their magnitude can then be compared to that of the temporal derivatives in the case of equations (5) and (6), and to g in the case of equation (7) to assess the validity of the hydrostatic assumption.

[17] In Table 1, the number of data occurrences with the ratios ut/wuz, vt/wvz and g/(wt + wwz) exceeding 1 and 10 are represented in the first and second rows as a percentage of the total number of data points, indicating how frequently the retained temporal derivative terms ut and vt, are larger than the neglected vertical derivative terms, wuz and wvz and how often the acceleration of gravity exceeded the sum of the temporal and vertical convective derivatives. The results show that in more than 83% of the sampling points for the ratio ut/wuz, and 87% for the ratio vt/wvz the temporal derivatives are of larger magnitude than the vertical convection derivatives. This is indicative of their dominant role in the equation as assumed in the NSW approximation. However, the temporal derivatives are one full order of magnitude larger than the vertical derivatives only 35% and 42% of the time for ut/wuz and vt/wvz, respectively. This is probably not sufficient to completely neglect their contribution to the solution, implying that a certain amount of departure from the physical solution could be attributed to these non-vanishing terms. The combined effects of vertically developing gradients in the u and v velocity profiles, coupled with increasing values of the vertical velocity component w in shallow waters seem to contribute to the growth of wuz and wvz beyond what is expected. Departure from the shallow water approximation is particularly evident at times when the convective derivatives take on unusually elevated values, as is the case in the velocity profiles of Figure 3. Here the size of the convective terms wuz and wvz is only 2 and 1.5 times smaller than ut and vt,, respectively.

[18] It also becomes evident from the third column in Table 1 that the absolute value of the sum of the first and fourth terms, wt and wwz, in the vertical momentum equation (7) is at least an order of magnitude smaller than g everywhere in the domain (Table 1), indicating that hydrostatic conditions will be met in the flow as long as the horizontal convective derivative terms of w (second and third terms, not available from the data) in (7) are also negligibly small compared to the value of g.

[19] Discarded acceleration terms can be expected to play a role in the solution of the Euler equations in shallow water not reflected in the NSW formulation. Figure S3 of the auxiliary material shows a comparison of the observed and modeled wave height and depth-averaged velocity components at the tide gauge location obtained with the NSW tsunami code MOST for approximately the first 6 hours after tsunami arrival. Values of the vertical velocity w are presented in the fourth panel from the top. Table S1 of the auxiliary material shows the correlation coefficients and RMS error values of the modeled and observed data series, indicating higher correlation between model and measurements in the wave height than in the intensity of the velocity components. A number of sources of error from uncertainties in the initial conditions or the bathymetric data to approximations in the numerical algorithm may influence the accuracy of model results [Tang et al., 2009]. We show here by evaluation that the absence of some of the terms in equations (5) and (6) from (2) and (3) may also be contributing to discrepancies between modeled and observed data.

[20] Two additional features become apparent from the data: One is that the vertical velocity component w is, in fact, significantly smaller than either one of the horizontal components as assumed in the NSW formulation, and the other is that wave behavior at the location of the ADCP is essentially linear. This becomes clear if we consider the characteristic formulation of the one-dimensional NSW equations:

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nonlinearity is introduced through the dependence of λ1 and λ2 on u and η. When ∣u∣ + equation imageequation image, as is the case in deep water, wave behavior is essentially linear with nonlinear effects, becoming increasingly important in shallow waters as equation image → 0, but equation image remains of finite size due to the elevated values of u and η. In the specific case under investigation here, (∣u∣ + equation image)/equation image ≈ 1.021 (alt. (∣v∣ + equation image)/equation image ≈ 1.016), where the maximum absolute value of the depth-averaged u and v velocity components during the first 7 hours of tsunami activity have been used to compute an upper bound on the previous expressions, indicating that tsunami waves at the location investigated here are expected to exhibit an essentially linear behavior.

[21] It should also be noted that in the case of larger tsunamis or near-field locations, the linearity conditions in relatively shallow waters presented here for the far-field case will most likely be invalidated by large wave amplitudes and intensity of the currents typically generated in near-field shallow waters.

4. Conclusions

[22] Analysis of current velocities generated by the 2006 Kuril tsunami event south of Honolulu, Hawaii at a depth of approximately 10 m of water shows that the NSW approximation to the Euler equations still seems to be valid in relatively shallow waters, but some of the terms ignored in this formulation are only marginally negligible at this depth, being less than one order of magnitude smaller than non-neglected terms, and can potentially contribute to discrepancies between observed and modeled waveforms.

[23] Since the ADCP sensor only provides data along a single water column, no spatial horizontal derivatives could be computed. Consequently, only those terms present in the Euler equations containing vertical or temporal gradients could be evaluated from the data. Results from this type of analysis can either invalidate the NSW assumption, if a term that is assumed to be negligible is shown to be comparable in magnitude to that of a term that is assumed to be dominant, or provide partial confirmation of the validity, if the size of the neglected term proves to be insignificant compared to that of more dominant retained terms.

[24] The analysis presented here clearly provides partial confirmation of the hydrostatic approximation by showing the magnitude of neglected terms to be at least one order of magnitude smaller than the acceleration of gravity. Partial confirmation of the uniformity of velocity profiles along the water column necessary for full validity of the approximation, however, was unclear, with the magnitude of neglected terms being smaller in size, but comparable to that of retained terms in a significant number of sample points.

[25] For more general conclusions about the validity of the NSW equations for tsunami modeling in shallow coastal waters, more tsunami-generated current data are necessary for different depths, events, and locations. In addition, to evaluate sufficient (as opposed to necessary) conditions for the validity of the approximation, an array of at least three current profilers located along each axis and in close proximity to each other would be needed to allow the reconstruction of terms containing horizontal derivatives, not available from the current data set.


[26] The authors would like to thank Gene Pawlak and Judith Wells from the Kilo Nalu Near-Shore Reef Observatory of the University of Hawaii for sharing their ADCP data. We also thank Liujuan Tang for her valuable comments during the preparation of this manuscript. This publication is partially funded by the Joint Institute for the Study of the Atmosphere and Ocean (JISAO) under NOAA Cooperative Agreement NA17RJ1232, contribution 1868. PMEL contribution 3630.

[27] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.