Rip Currents Induced By Small Bathymetric Variation

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Calhoun, Institutional Archive of the Naval Postgraduate School Calhoun: The NPS Institutional Archive Faculty and Researcher Publications Faculty and Researcher Publications 2007 Rip Currents Induced By Small Bathymetric Variation MacMahan, Jamie H. Journal of Geophysical Research http://hdl.handle.net/10945/45735 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/, Rip Currents Induced By Small Bathymetric Variation 1 1 2 Jamie H. MacMahan, Ed B. Thornton, Ad J. H. M. Reniers, Tim P. 1 Stanton, Graham Symonds 1 3 Oceanography Department, Naval Postgraduate School, Monterey, California, USA. 2 Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands and Division of Applied Marine Physics, RSMAS, University of Miami, Miami, FL, USA. 3 CSIRO Marine and Atmospheric Research, Private Bag No. 5, Wembley Way 6913, Australia D R A F T April 3, 2007, 8:32am D R A F T X - 2 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION Abstract. Comprehensive field measurements in the throat of a rip cur- rent have been obtained for the first time as a rip current and its morphology migrated (∼ 4m/day) through a coherent cross- and alongshore array of co-located pressure and velocity sensors. The rip current is associated with a small bathymetric surfzone non-uniformity (1 in 300 alongshore variation). Pulsations of the rip current caused by infragravity (0.004Hz < f < 0.04Hz), very low frequency (< 0.004Hz) motions and tidal variations are 25-50% of the mean current. The presence of the rip current and mild-sloped rip channel do not induce statistically significant alongshore variations in Hrms , but do modify the wave direction and directional spreading. Changes in the directional spreading are correlated with the presence of VLF motions influenced by the presence of the rip current. The mean rip current velocity is maximum shoreward of the mean breaking region, decreases rapidly seaward and is insignificant less than 1.25 surf zone widths from the shoreline. D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X-3 1. Introduction Rip currents are strong, jet-like, seaward-directed flows that originate within the surf zone owing to alongshore gradients in wave-induced radiation stresses and pressure [Bowen, 1969; Dalrymple, 1978; Haller et al., 2002]. Long and Okzan-Haller [2005] showed that refraction of waves over submarine canyons can create variations in wave height alongshore that induce rip currents in the absence of local surfzone bathymetric non-uniformity. However, most field observations of rip currents are coupled to variations in the surfzone alongshore bathymetry, which induce differences in the wave forcing [Shepard et al., 1941; Shepard and Inman, 1950; Sonu, 1972; Aagaard et al., 1997; Brander and Short, 2001, MacMahan et al., 2005]. The degree to which surfzone bathymetric non-uniformities induce a hydrodynamic response, in particular a rip current, is unknown. Within the last decade, more quantitative field and laboratory observations of rip currents have become available [Aagaard et al., 1997; Brander, 1999; Brander and Short, 2000; Brander and Short, 2001; Haller and Dalrymple, 2001; Haas and Svendsen, 2002; Haller et al., 2002; Callaghan et al., 2004; MacMahan et al., 2005]. However, there have been few comprehensive field observations of rip current hydrodynamics owing to the difficulty of deploying instruments in a rip channel, and in part because of the tendency for the rip channels to migrate alongshore. Past field experiments deployed only a few instruments along the axis of the rip channel. During the Nearshore Canyon Experiment (NCEX) in fall 2003 at Torrey Pines, California, a morphologically controlled rip current fortuitously migrated slowly through an extensive cross- and alongshore array of co-located electromagnetic current meters and pressure sensors over 15 days. Compre- D R A F T April 3, 2007, 8:32am D R A F T X - 4 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION hensive measurements of currents in the throat of a rip current were obtained for the first time in the field and are described in the following. 2. Field Site Field observations were obtained during NCEX (October 29 to November 18, 2003; yeardays 302-322). The beach is characterized as near planar with a surfzone slope of 1:70 between 50 ≥ x ≥ −75 and an offshore slope of 1:50 between −75 ≥ x ≥ −200, where the cross-shore distance is positive onshore (see Figure 1). This paper discusses data from two primary arrays deployed: 1) an alongshore array of 6 pressure and bidirectional digital electromagnetic current meters, dems, spanning approximately 500m, and 2) a cross-shore array of 6 pressure and dems ranging from the shoreline to a depth of 3.5 m (Table 1). Incorporated into the cross-shore array were 8 Paroscientific pressure sensors (paros), 6 of which were co-located with the 6 cross-shore dems. Pressure and dem sensors were sampled continuously and synchronously at 15Hz, while the paros were sampled at 1 Hz. The dems were approximately 35cm off the sea bed, except PUV8, which was 1m off the sea bed. Directional wave spectra were measured with an upward-looking, broad-band, acoustic Doppler current profiler located in approximately 6m water depth (x=-246m) in line with the cross-shore array. During the measurement period, the significant wave height (Hmo ) ranged 50 − 125cm, the mean wave period ranged 3.5-12s, and the mean wave direction ranged − 12o −+ 12o (Figure 2). Only a few moderate storm events occurred during the experiment, near the onset and at the end of the measurement period (yeardays 304, 320, and 322). Hourly mean alongshore currents averaged over the alongshore array ranged between − 75cm/s to +25cm/s. Rip current activity occurred between yeardays 303-318, D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X-5 during which time Hmo was approximately 50cm, the wave direction was predominantly shore-normal, and the longshore current was ∼ 0cm/s (Figure 2). 3. Bathymetry Rip channels can be identified in the surfzone bathymetry. Bathymetric surveys were obtained at high tide using a hydrographic surveying system composed of a GPS and sonar mounted onboard a personal water craft. The upper beachface was surveyed at low tide with a GPS mounted on a cart. These surveys were performed 6 times between yeardays 300-316 with the last survey performed 7 days before the end of the experiment (Figure 1). Rip channels are identified as perturbations on the otherwise straight bathymetry contours, either as a depression inside the surfzone cut by the rip current (contours indented shoreward) or as accretion outside the surfzone created by deposition of the rip current (contours bulged seaward). The rip current morphology has ∼ 140 ± 30m channel width based on the bathymetry plots in Figure 1. The rip spacing is ∼ 300-400m based on the rectified video image of yearday 308 (Figure 3). Shepard and Inman [1950] had previously observed rip currents at Torrey Pines with an alongshore spacing of 500m. The depth variation across the rip channel is ∼0.25m in 75m alongshore (Figure 1). It is noted that the rip channel morphology is very broad with the bathymetric variation smaller than the camber (0.5 − 1%) of an athletic playing field (www.drainage.org). 3.1. Migration of Rip Channels The bathymetry for yearday 300 indicates a rip channel located about the 1m contour around y=2800 [Figure 1]. At yearday 303, the rip channel increased in width and mi- D R A F T April 3, 2007, 8:32am D R A F T X - 6 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION grated south to y=2600 in the vicinity of the cross-shore array. The rip channels slowly migrated south during yeardays 308-310. The rip channels are most noticeable at the 1m contour around x=40. The bathymetric contours seaward of the rip channel bend seaward suggesting accretion. The rip channel system continued to migrate south and by yearday 316 was less pronounced. Based on identifying the rip channel from the bathymetry survey and the time between surveys, the average migration of the rip channels was approximately 4m/day to the south 3.2. Non-uniform Bathymetry Non-uniformities of the bathymetry can induce alongshore variations in radiation stresses and pressure gradients to drive circulation [Bowen, 1969; Dalrymple, 1978]. A measure of non-uniformity [Feddersen and Guza, 2003] is the alongshore depth standard deviation, σh (x), defined v u u 1 Z Ly σh (x) = t (h(x, y) − h(x))2 dy, Ly 0 (1) where h(x,y) is the bathymetry between −50 ≤ x ≤ 200 and 2200 ≤ y ≤ 2800, and h(x) is the alongshore mean cross-shore profile. Owing to the fact that the contour lines are not parallel to the y-axis, contour lines are rotated (∼ 2o ), so that 0m contour line is parallel to the shore based on a least-squared fit. Note x’,y’=0,0 in the rotated coordinate frame corresponds to x, y =50, 2500 in Figure 1. h(x0 ) and σh (x0 ) are similar for the 6 surveys, with σh (x0 ) reaching maxima values of 0.24 − 0.37m in 2.6m water depth (Figure 4). The σh (x0 ) values are greatest in the region of breaking waves, where the width of the non-uniformity region is associated with the horizontal excursion (∼ 100m) of the break point over a 2m tide on a 1:50 beach slope (Thornton and Kim, 1993). The non- D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X-7 uniformity of this beach is predominantly associated with the offshore rip morphology (area of accretion) and not with the surfzone rip channels. 4. Rip Currents 4.1. Circulation Hourly mean velocity vectors are used to examine the nearshore circulation during low and high tides for three rip current cases associated with non-uniform bathymetry when the waves were normally incident (yeardays 303, 305, 309) and two alongshore current cases (yearday 320-321) associated with uniform bathymetry and oblique wave incidence (Figure 5). For all wave conditions considered here there is an offshore component of the interior flow due to a rip current or undertow. At the onset of the experiment, alongshore currents were present at high tide, during yearday 303 (Figure 5a). At low tide, the alongshore currents decreased and rip currents dominated at y=2600 (Figure 5b). The nearshore circulation on yearday 305 at high tide shows the strongest offshore flows occurring closest to shore (Figure 5c). The offshore flow is not qualitatively distinguishable between undertow or rip current as the offshore flow component at the alongshore array are all similar. The small bathymetric depression in the vicinity of the cross-shore array has minimal affect at high tide. During low tide, a rip current cell circulation is formed (Figure 5d). The vectors are oriented slightly to the south which is associated with the small oblique wave angle (Figure 2). The video image for this time indicates decreased wave breaking within the rip channel, where the cross-shore array is located (Figure 3). The waves within the rip channel break closer to shore owing to its greater depth. Since the waves break closer to shore in the rip channel, a counter flow circulation develops (Figure 5d). At the shoreward-most PUV sensor, the flow is directed onshore or alongshore D R A F T April 3, 2007, 8:32am D R A F T X - 8 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION in the opposing direction. This is similar to laboratory observations by Haller et al., [2002], but until now has not been observed in the field. The circulation behaves similarly for yearday 309 (compare Figure 5d,f), with the current vectors oriented offshore and northerly. During obliquely incident wave conditions for yearday 320-321, an alongshore current dominates the nearshore (Figure 5g,h). The offshore component of the flow is presumed due to undertow (e.g. Thornton and Guza, 1986). 4.2. Cross-shore Distribution of a Rip Current The cross-shore velocity profile along the axis of a rip current in the field has not been well documented or evaluated (Bowman et al., 1988a,b; Sonu, 1972; Aagaard et al., 1997). Cross-shore velocity profiles are examined at low-mid tide at the cross-shore array for a seaward-directed rip current (yearday 305), shoreward-directed flow associated with the rip current circulation (yearday 311), and an undertow scenario (yearday 317) [Figure 6]. Scenarios (rip current, alongshore current, or undertow) were determined by visual evaluation of the circulation patterns, similar to Figure 5. In each scenario the cross-shore Hrms distribution is similar. The maximum seaward-directed cross-shore velocity occurs inside the surf zone for the rip current, which is similar for undertow, though varying in magnitude. The surf zone is defined as the region inside of teh location where the maximum Hrms occurs (Figure 6). The location of the maximum seaward velocity in the rip current is influenced by the roller of breaking waves, shifting the maximum shoreward [Haller et al., 2002]. The offshore velocity extends barely outside the surf zone. The maximum shoreward-directed flow associated with the cell circulation of the rip current is located closer to the shoreline and no undertow is present. The root mean square IG and VLF variations, or pulsations, about the mean are indicated by vertical bars. The D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X-9 largest pulsations coincide with the maximum mean sea-seaward directed flow (discussed below). The significant flows associated with rip currents occur within the surf zone. The rapid decrease of the seaward velocities is hypothesized to limit the offshore extent of sediment transport, which is supported by the bathymetric observations (Figure 1). Surface traces of sediments and bubbles associated with rip currents are commonly observed outside the surf zone [Shepard et al., 1941]. Rip currents become weaker and surface dominated outside the surf zone based on laboratory observations by Haas and Svendsen [2002] and Dronen et al. [2002]. Their cross-shore velocity measurements at mid-depth in the laboratory were approximately 80% of the value measured at the surface (trough elevation). The accretionary bulge in contours offshore of the rip channels extend between x=-75 and x=-125m in the cross-shore suggesting accretion did not occur beyond x=125m. Aagaard et al. [1997] found that the maximum offshore sediment flux within an alongshore-barred rip current system occurred slightly offshore of the alongshore bar region and quickly decreased, at a location approximately 1/4 the surf zone width offshore. This is consistent with the present findings. 5. Pulsations of Rip Current Pulsations of the rip current are due to infragravity waves, very low frequency motions and tides. The pulsations are of the same magnitude as the mean currents, occasionally doubling the strength of the offshore flow. It is during the pulsations that rip currents are particularly dangerous to swimmers. These contributions are examined separately below. D R A F T April 3, 2007, 8:32am D R A F T X - 10 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION 5.1. Infragravity Waves (0.004 < f < 0.04Hz) MacMahan et al. [2004a] determined that low frequency (infragravity) rip current pulsations were forced by standing infragravity waves for a high-energy, steep-sloping, reflective beach characterized by transverse bars with incised rip channels. This hypothesis is evaluated for this low-energy mild-sloping dissipative beach. Coherence and phase spectra are estimated from the average of four demeaned, quadratically detrended (to remove tides) 15 min segments with 50% overlap between p and u at PUV11 (in the rip channel) and PUV12 (on the shoal) for yearday 309 (Figure 7). The coherence varies between 0 and 0.7 within the infragravity band (0.004 < f < 0.04Hz), whereas the coherency is 1 for the sea-swell band (> 0.04Hz). The phase varies between −90o and +90o within the infragravity band indicative of standing waves (Suhayda, 1974), and is zero for the sea-swell band indicative of progressive waves, consistent with previous field observations at Torrey Pines on a mild sloping beach [Huntley et al. 1981; Guza and Thornton, 1985] and during RIPEX (RIP Current EXperiment in Monterey Bay, CA) for a steep beach [MacMahan et al., 2004a]. The infragravity velocity is smaller within the rip channels owing to the greater depths of the rip channels (Figure 8) [see MacMahan et al., 2004a]. Thus, standing infragravity motions contribute to the forcing of rip current pulsations. The rms infragravity velocity is on the order of 15 cm/s, which is approximately 40% of the measured mean rip current flow velocity. To examine the potential coupling of infragravity waves with the bathymetry the frequency-alongshore wave number (f -ky ) spectra are calculated for u and v using an Iterative Maximum Likelihood Estimator (IMLE) [Pawka, 1983; Oltman-Shay and Guza, 1987]. Spectral estimates with 10 degrees of freedom (frequency resolution of 0.0005Hz) D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 11 were generated from demeaned, quadratically de-trended 3 hour records. The edge waves here have energetic broad peaks centered at f=0.18Hz and ky = ±0.003m−1 (Figure 9) and may be important in rip channel morphodynamics. The maximum energy for mode 2 edge wave occurs at the cross-shore depth corresponding to the location of the alongshore array, which has a wavelength similar to that of the rip current spacing [Eckart, 1951]. This is consistent with Reniers et al. [2006] who found that edge waves were coupled to rip channel morphology at the rip-channel spacing during RIPEX. 5.2. Very Low Frequency Motions (< 0.004Hz) Energy at frequencies < 0.004 Hz and outside the gravity restoring region delineated by the mode zero edge wave curve, is due to very low frequency (VLF) motions (see MacMahan et al., 2004; Reniers et al., 2007; MacMahan et al., 2006). The VLFs have been associated with the presence of rip currents [Smith and Largier, 1995; Haller and Dalrymple, 2001; Brander and Short, 2001; MacMahan et al., 2004b]. The source of rip current VLF motions has been attributed to two mechanisms. The first mechanism was identified as an instability for the case of monochromatic waves and high rip current jet shear conditions in the laboratory [Haller and Dalrymple, 2001] and in a model [Yu and Slinn, 2003]. For these rip current shear instabilities, the VLF energy is predicted to be significantly larger outside the surf zone than inside the surf zone [Yu and Slinn, 2003; Reniers et al., 2007], whereas the laboratory measurements of Haller and Dalrymple [2001] found maximum VLF variance at the rip throat. The second mechanism suggests that VLFs are surf zone eddies generated by wave groups that can potentially be coupled to the rip current morphology [MacMahan et al., 2004b; Reniers et al., 2004, 2007]. Reniers et al. [2007] determined through modeling D R A F T April 3, 2007, 8:32am D R A F T X - 12 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION that rip current related VLFs can be discriminated between rip current shear instabilities and surf zone eddies by the cross-shore distribution of the VLF velocities. For surf zone eddies forced by directionally spread random waves, the VLF velocities have a maximum within the surf zone and decrease shoreward and seaward [Reniers et al., 2007], whereas rip current shear instabilities have a maximum outside the surf zone [Yu and Slinn, 2003]. MacMahan et al. [2004b] found for RIPEX that the VLF velocities associated with surf zone eddies were alongshore uniform within the surf zone and decreased outside the surf zone, supporting the second mechanism. VLFs were observed during NCEX for various wave conditions and alongshore positions of the rip current, and found to be related to the incoming sea-swell energy and modulated by the tide, similar to results by MacMahan et al. [2004b] at RIPEX. In general VLFs propagate alongshore with the phase speed, Cy = f /ky , and their average energy is expected to be homogeneous on an alongshore uniform beach. In contrast, during RIPEX, the VLFs were coupled with the well-established rip channels. The alongshore spatial distributions of VLF velocities (Figure 8, bottom panel) indicate that these motions are not entirely coupled with the rip channel morphology, because their energy occurs along the entire beach. However, there are maxima within the vicinity of the rip currents at the alongshore distance 2600m (Figure 8). During RIPEX, the rip channels were relatively close together (λ = 125m), suggesting rip currents were dependent upon each other, whereas for NCEX the rip currents are much further apart (λ = 300m) and appear independent of each other (Figure 4,8). Shear instabilities of the rip current jet [Haller and Dalrymple, 2001] also could contribute to the maximum VLF velocity within the rip D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 13 current. The rms VLF velocity was on the order 10cm/s, which is approximately 25% of the measured mean rip current flow. 5.3. Tidal Modulation The largest velocity variations within the surf zone are due to tides. In general, the rip current velocity is maximum at low tide [Aagaard et al., 1997; Brander, 1999; Brander and Short, 2000; Brander and Short, 2001; MacMahan et al., 2005]. It is hypothesized that the increase in rip current velocity is due: 1) to the decrease in tidal elevation modifying wave breaking, which affects the corresponding pressure and radiation stress gradients, and 2) to the constriction of the rip current channel owing to continuity. The hourly cross-shore velocity for the alongshore array increases to a maximum slightly below mean sea level (MSL) in the vicinity of the rip channel, and decreases slightly in speed at low tide for yeardays 305-307 (Figure 10). The cross-shore distribution of hourly cross-shore velocities (Figure 10) has a maximum around ebb mid-tide and does not shift seaward during low tide, suggesting that the local bathymetric variability around z=1m is more important in controlling the flow pattern. The hypothesis is that depth-limited wave breaking at MSL induces alongshore variations, but due to the small bathymetric variations, depth-limited breaking occurs at low tide within the rip channel. Note that semi-diurnal tides are slightly larger for yearday 305-307. There is an alongshore southerly shift in rip current maximum during yearday 307 owing to a change in mean wave direction (Figure 2), identified as an increase at PUV 11 and PUV12 relative to PUV5 (Figure 10). For yeardays 308-309, the rip current maximum occurs around low tide. Around MSL on yearday 309, alongshore variations in depth-limited wave breaking are present, inducing alongshore pressure and radiation stress gradients toward the rip channel (Figure D R A F T April 3, 2007, 8:32am D R A F T X - 14 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION 11). At low-tide on yearday 309, owing to the small bathymetric variation, depth-limited breaking occurs within the rip channel and the shoal (Figure 11, top panel). There is minimal alongshore variation in Hrms (Figure 8), suggesting that the alongshore pressure and radiation stress gradients are reduced, but the rip current is still present within the measurements on yearday 309. The offshore extent of the breakerline is shifted seaward around the vicinity of the rip channel (Figure 11, bottom panel). The wave breaking is associated with the small delta that is present in the bathymetric contours on yearday 308 (Figure 1), which was not present on yearday 303. The existence of the small rip current delta is hypothesized to provide a positive feedback on the rip current circulation during low tide. 6. Discussion The observations of the mean velocities at mid-depth of the water column in the throat of the of the rip current found that the maximum occurs inside the surf zone, the velocities have strong pulsation over a range of frequencies owing to infragravity waves, very low frequency motions and tides, and that the velocities surprisingly decay to zero very quickly outside the surf zone (defined by Hrms,max ) within 20% of a surf zone width. Another surprising result is the small size of the alongshore perturbation on the bathymetry comprising the rip channel to support the rip current location. The rip channel spacing was ∼300m with a ∼140m channel width with a depth variation across the rip channel of only ∼0.25m. The measurements show the rip current width (at the 1m contour) was ∼50m, or 30% of the rip channel width. It is hypothesized that the width and strength of the mean rip current jet are a function of positive feed back by perturbations on the wave field by bathymetry and current. To D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 15 examine the positive and negative feed backs that sustain and drive the narrow rip current, the alongshore variations of the wave heights and velocity contributions are examined to determine the strength of the wave-current interaction. Since the cross-shore distributions of the measured rip currents are similar (Figure 6, 10) owing to the fact that the wave forcing on these days was almost the same, only selected days are examined in detail below. Subtle alongshore variations in bathymetry can induce rip currents, which affect the alongshore processes of the mean flows, wave direction, directional spreading, infragravity velocities, and VLF energy. The velocity non-uniformities are related to the offshore bathymetric non-uniformities associated with sediment deposition (z ≥ 1.5m) [Figure 4], which may cause the waves to converge within the vicinity of the rip current owing to refraction. The region of maximum rip current velocity occurs within the surf zone (Figure 6), which is alongshore bathymetrically uniform. Thus, the rip current morphodynamic coupling is dependent upon the sediment delta outside the surf zone and should not be ignored. 6.1. Alongshore Variability of Waves and Velocity Fields The alongshore variations of the cross-shore velocity and wave characteristics at the alongshore array (located at ∼-1m depth contour) were calculated for the mean, sea/swell, infragravity, and very low frequency (VLF) frequency bands for two rip current days comparing high and low tide conditions (Figure 8). The one-hour mean cross-shore currents show well-defined rip currents flowing offshore at low tide and disappearing at high tide. The Hrms statistically does not vary alongshore at the 95% level of significance for either low or high tide. Though statistically insignificant, Hrms values at low tide qualitatively D R A F T April 3, 2007, 8:32am D R A F T X - 16 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION have a slight maximum within the vicinity of the rip current (Figure 8), which may be associated with wave-current interaction owing to its location within a deeper channel, [Haller and Okzan-Haller, 2002], or wave convergence by the offshore contours (z > 1.5m) [Dalrymple and Loranzo, 1978]. 6.2. Wave-Current Interaction Rip currents systems can be classified based on the Froude number [MacMahan et al., 2006], which is similar to quantifying the importance of wave-current interactions [Haller and Okzan-Haller, 2002]. Haller and Okzan-Haller [2002] modeled monochromatic waves in a rip channel using Haller et al. [2002] laboratory measurements and found that not including wave-current interaction leads to incorrect estimates of wave setup in the rip current flow. Yu and Slinn [2003] and Haas et al. [2003] numerically determined that monochromatic wave-current interaction of the rip current limits the offshore extent of the flow. However, Reniers et al. [2006] found that wave-current interaction was less important with random waves. The offshore extent of the rip current and the importance of wave-current interaction have not been measured in the field. Limiting the discussion to normal incident waves on an offshore flowing rip current, U, wave frequency is Doppler shifted, such that (ω − kU ) = σ, where ω is the absolute frequency relative to a stationary observer, k is the wave number, and σ is the intrinsic wave frequency. Differentiating with respect to k, gives Cga − U = Cg , where Cga is the absolute wave group velocity and Cg is the intrinsic wave group velocity [Chawla and Kirby, 2002]. The strongest wave-current interaction occurs at wave blocking when the ratio of |U/Cg | = 1 [Haller and Okzan-Haller, 2002; Chawla and Kirby, 2002]. U/Cg within √ the shallow water limit of linear theory is U/ gh, referred to as the Froude number, where D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 17 h is the local water depth and g is the gravitational acceleration. The Froude number ranges 0 to 1 and is used to evaluate the importance of wave-current interaction for a rip current system. The Froude number was computed for PUV5 and PUV11, separated 25m in the alongshore. The mean Froude number was 0.08 for seaward directed flows only with a maximum value of 0.26, indicating weak wave-current interaction. 6.3. Wave Direction and Directional Spreading The importance of wave-rip current interaction on wave direction and directional spreading is evaluated. Though it was shown that the wave-current interaction had minimal affects on the wave height, wave-current interaction can play an important role in wave direction and directional spreading. The mean wave direction (θmean ) and directional spreading (σθ ) are computed utilizing the low-order Fourier moments of directional-frequency wave spectra [Kuik et al., 1990; Herbers et al., 1999]. The measured directional spreading increases at low tide and within the vicinity of the rip current maximum (Figure 8). When the mean rip current maximum migrates in the alongshore, so does the maximum directional spreading (Figure 8). Though significantly correlated at 95% confidence interval, the correlation between (σθ ) and Umean for the alongshore array for yeardays 305-320 is low, r2 = 0.16 (Figure 12). Herbers et al. [1999] found that directional spreading increases shoreward, in particular, within the surf zone on an alongshore uniform beach. Thus, the directional spreading increases at low tide within the saturated surf zone. The increase in directional spreading within the vicinity of the rip channel is hypothesized to be a result of increased wave refraction by the rip current, bathymetry, and rip current pulsations. Henderson et al. D R A F T April 3, 2007, 8:32am D R A F T X - 18 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION [2006] suggested that the increase in directional spreading within the surf zone, as observed by Herbers et al. [1999], is due to wave refraction owing to the offshore current fluctuations of the shear instabilities. Here, the rip current pulsations provide current fluctuations that increase the variability of the local wave field, which increases the directional spreading. The alongshore directional spreading varies alongshore and is correlated better better with Urms,vlf , (r2 = 0.48), supporting this notion (Figure 8, 12). The Urms,vlf and Umean are also significantly correlated at the 95% confidence interval, suggesting that the presence of the rip current increases the Urms,vlf , which increases (σθ ). However for shore-normal wave conditions (yeardays 305-320), wave-group forced VLF motions (MacMahan et al., 2007) are present in the background increasing (σθ ), similar to the mechanism proposed by Henderson et al. [2006]. The wave direction varies within the rip channel, with a reversal in mean wave direction about shore-normal at low tide on yearday 306 (Figure 8). This suggests wave convergence into the rip current as described by Dalrymple and Loranzo [1978] and Yu and Slinn [2002], but could also be a result of the rip current delta at 3m water depth. 6.4. Narrow Rip Current in a Wide Rip Channel The small alongshore bathymetric variation within the surf zone, aided by the offshore bathymetric variability, create alongshore variations in the wave forcing that induce a rip current. The channel width is relatively wide (140m), yet the offshore direct flow is focused in the center and is narrower than the rip channel. This results in a broad onshore flow and a narrow offshore flow. Arthur [1962] and Bowen [1969] showed that potential vorticity of a rip current is conserved along a streamline, which causes the rip current to D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 19 become narrower as it flows into deeper water. It is surmised that the rip current becomes narrower than the rip channel owing to the conservation of vorticity. In view of this, it is unclear why the rip channel is so wide. One hypothesis is that the antecedent conditions that developed the rip channel induced a wider rip current. However, the dimension of the rip channel was maintained as the rip channel migrated suggesting the antecedent conditions are not important. The migration of the rip current causing the channel to be wider and this is part antecedent morphology. Another hypothesis is that edge waves, VLFs, and rip current shear instabilities, directional wave variability broaden the channel and smooth the bathymetric variations owing to their velocity fluctuations. 6.5. The Dangers of Rip Currents We believe that rip current pulsations are what make rip currents dangerous to swimmers. The VLF and IG pulsations are quasi-Gaussian. Therefore the 3 σ values of pulsations (rms), representing the 1% of occurrence, are larger than 1m/s. The combined mean rip current and pulsations are faster than the average person can swim, even for these relatively mild conditions. Note men’s and women’s Olympic 50m freestyle speeds are approximately 2m/s for ideal conditions, a short-distance, and years of training. The combination of the pulsations, narrow-directed seaward current, and the increased depth of rip channel create a dangerous scenario. The pulsations can create a false-sense of security, because the velocities within the rip channel can be reduced or even shorewarddirected for 15seconds-2.5minutes. When the pulsation reverses to a seaward direction, the pulsation and the rip current quickly move the beach-goer into deeper water, where D R A F T April 3, 2007, 8:32am D R A F T X - 20 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION the beach-goer can not touch the seabed. The magnitude of pulsations and the magnitude of the rip current velocities define the potential danger. 7. Conclusions Comprehensive velocity measurements in the throat of a rip current were obtained for the first time in the field as the rip current migrated through alongshore and cross-shore arrays of pressure and velocity sensors. The maximum mean rip current velocity occurred inside the surf zone. The velocities at mid-depth decayed rapidly outside the surf zone to near zero less than 1.25 surf zone widths from the shoreline. This was surprising given the common observation of surface currents of rip currents identified by sediment plumes or foam extending several surf zone widths offshore. Pulsations of the rip current measured as rms values were the same magnitude as the mean current. The pulsations occurred over a range of frequencies owing to infragravity waves, very low frequency motions and tides. The rip current location was supported by a surprisingly small alongshore perturbation on the bathymetry comprising the rip channel. The rip channel spacing was ∼ 300m with a ∼ 140m channel width with a depth variation across the rip channel of only ∼ 0.25m. The measurements show the rip current width (at the 1m contour) was ∼ 50m, or 30% of the rip channel width. The rip current morphology migrated slowly alongshore at ∼ 4m/day. A sediment bulge was observed offshore of the rip current. The sediment bulge extended ∼ 1.25 surf zone widths from the shoreline at about the same distance offshore where the mean currents decayed to zero. The rip current had a minimal affect on wave heights. The rip current increased the directional wave spreading and VLF energy varied the wave direction. VLF energy was D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 21 significantly correlated at the 95% confidence level to directional spreading, suggesting that the rip currents and background VLF contribute. Infragravity rip current pulsations have minima in the alongshore in the vicinity of the rip current owing to the deeper channels, consistent with shallow water long wave theory. VLFs have maxima in the vicinity of the rip current, and have cross-shore distributions with a maximum in the surf zone for the case of both rip currents and undertow. The rip current had a maximum at mid-tide and decreased slightly at low tide, which is different than previously observed. The maximum at a mid-tide is related to the subtleness of the rip current morphology. Acknowledgments. We thank many folks who assisted in obtaining a great data set: Ron Cowen, Mark Orzech, Jim Stockel, John Woods, Keith Wyckoff, and Rob Wyland. We thank Tom Lippmann and his OSU group and Scripps for performing bathymetric surveys. Cindy Paden, Rob Holman, and Kristen Splinter provided extended rectified Argus images. JM was funded by ONR under contract number N00014-05-1-0154 and N00014-05-1-0352 and the Naval Postgraduate School. EBT and TPS were funded by ONR under contracts n0001405WR20150 and N0001405WR20385. AJHMR was funded by ONR under contract XXXX and the Dutch National Science Foundation (NWO) under contract DCB.5856. GS was a National Research Council Fellow during a sabatical funded in part by the ONR under contract N0001405WR20150. References Aagaard, T., B. Greenwood, and J. Nielsen, Mean currents and sediment transport in a rip channel, Mar. Geol., 140, 24-45, 1997. D R A F T April 3, 2007, 8:32am D R A F T X - 22 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION Arthur, R.S.,A note on the dynamics of rip currents, J. Geophys. Res., 67(7), 2777-2779, 1962. Bowen, A. J., Rip Currents, 1, Theoretical investigations, J. Geophys. Res., 74, 5467-5478, 1969. Bowman, D., D. Arad, D.S. Rosen, E. Kit, R. Golbery, and A. Slavicz, Flow characteristics along the rip current system under low-energy conditions, Mar. Geol., 82, 149-167, 1988a. Bowman, D., D.S. Rosen, E. Kit, D. Arad, and A. Slavicz, Flow characteristics at the rip current neck under low-energy conditions, Mar. Geol., 79, 41-54, 1988b. Brander, R.W., Field observations on the morphodynamic evolution of low wave energy rip current system, Mar. Geol., 157, 199-217, 1999. Brander, R.W., and A.D. Short, Morphodynamics of a large-scale rip current system at Muriwai Beach, New Zealand, Mar. Geol., 165, 27-39, 2000. Brander, R.W., and A.D. Short, Flow kinematics of low-energy rip current systems, J. Coastal Res., 17:2, 468-481, 2001. Callaghan, D.P. T.E. Baldock, P. Nielsen, D.M. Hanes, K. Hass, and J.H. MacMahan, ”Pulsing and circulation in a rip current system”, Proceedings of the 29th International Conference on Coastal Engineering, submitted, Am. Soc. of Civ. Eng., Portugal, 2004. Chawla, A. and J.T. Kirby, Monochromatic and random wave breaking at blocking points, J. Geophys. Res., 107(C7), 3067, doi:10.1029/2001JC001042, 2002. Dalrymple, R.A., Rip Currents and Their Genesis. Summaries, 16th International Conference on Coastal Engineering, Conference Paper No 140, 1978. D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 23 Dalrymple, R.A. and C.J. Lozano, Wave-Current Interation Models for Rip Currents, J. Geophys. Res., 83(C12), 6063-6071, 1978. Dronen, N., H. Karunarathna, J. Fredsoe, B. M. Sumer, and R. Deigaard, An experimental study of rip channel flow, Coastal Eng., 45(3-4), 223-238, 2002. Eckart, C., Surface waves on water of variable depth, Wave Rep. 100, Scripps Institution of Oceanography, University of California, La Jolla, 1951. Feddersen, F., and R. T. Guza, Observations of nearshore circulation: Alongshore uniformity, J. Geophys. Res., 108(C1), 3006, doi:10.1029/2001JC001293, 2003. Guza, R.T. and E.B. Thornton, ”Observations of surf beat,” J. of Geophys. Res., 90, 3161-3171, 1985. Haas, K.A. and I.A. Svendsen, Laboratory measurements of the vertical structure of rip currents, J. Geophys. Res., 107, 2002. Haas, K.A., I.A. Svendsen, M.C. Haller, and G. Zhao, Quasi-three-dimensional modeling of rip current system, J. Geophys. Res., 108, 2003. Haller, M.C., and R.A. Dalrymple,”Rip current instabilities”, J. Fluid Mech., 433, 161192, 2001. Haller, M.C., R.A. Dalrymple, and I.A. Svendsen, ”Experimental study of nearshore dynamics on a barred beach with rip channels,” J. Geophs. Res., 107, 14: 1-21, 2002. Haller, M.C. and T. Ozkan-Haller, Wave breaking and rip current circulation. paper presented at International Conference on Coastal Engineering, Am. Soc. of Civil Eng., Cardiff, pp 705-717, 2002. Henderson, S.M., R. T. Guza, S. Elgar, and T. H. C. Herbers, Refraction of surface gravity waves by shear waves. J. Physical Oceanography, 36 (4), pp 629635, 2006. D R A F T April 3, 2007, 8:32am D R A F T X - 24 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION Herbers, T.H.C., S. Elgar, R. T. Guza, Directional spreading of waves in the nearshore. J. Geophys. Res. 104 (C4), 7683-7693, 1999 Huntley, D.A., R.T. Guza and E.B. Thornton, ”Field observations of surf beat, Part I: Progressive edge waves,” J. of Geophys. Res., 86, 6451-6466, 1981. Kuik, A. J., G. P. van Vledder, and L. H. Holthuijzen, A method for the routine analysis of pitch-and-roll bouy data, J. Phys. Oceanogr., 18, 1020 1034, 1990. Long, J. W., and H. T. Ozkan-Haller , Offshore controls on nearshore rip currents, J. Geophys. Res., 110, C12007, doi:10.1029/2005JC003018, 2006. MacMahan, J. Hydrographic surveying from a personal watercraft. J. Surveying, Engrg. 127 (1), 12-24, 2001. MacMahan, J., A.J.H.M. Reniers, E.B. Thornton and T. Stanton, ”Infragravity rip current pulsations,” J. of Geophys. Res., 109, C01033, doi:10.1029/2003JC002068, 2004a. MacMahan, J. H., A. J. H. M. Reniers, E. B. Thornton, and T. P. Stanton, Surf zone eddies coupled with rip current morphology, J. Geophys. Res., 109, C07004, doi:10.1029/2003JC002083, 2004. MacMahan, E.B. Thornton, T. Stanton, and A.J.H.M. Reniers, ”RIPEX-Rip Currents on a shore-connected shoal beach,” Mar. Geol., 2005. MacMahan, J., E.B. Thornton, and A.J.H.M. Reniers, ”Rip Current Review,” 53(2-3) Coastal Eng., 191-208, 2006. Reniers, A.J.H.M, J.A. Roelvink and E.B. Thornton, ”Morphodynamic modeling of an embayed beach under wave group forcing,” J. of Geophys. Res., 109, C01030,doi:10.1029/2002JC001586, 2004. D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 25 Reniers, A.J.H.M., J. MacMahan, E.B. Thornton and T. Stanton, ”Modeling infragravity motions on a rip-channel beach,” Coast. Eng., 53, 209-222, 2006. Reniers, A.J.H.M., J. MacMahan, E.B. Thornton and T. Stanton, ”Modelling of very low frequency motions during RIPEX,” accepted for publication in J. of Geophys. Res., 2007. Shepard, F.P., K.O. Emery, and E.C. La Fond, Rip currents: a process of geological importance, J. Geol., 49, 337-369, 1941. Shepard, F.P., and D.L. Inman, ”Nearshore water circulation related to bottom topography and refraction,” Trans. Am. Geophys. Union, 31, 196-212, 1950. Smith, J.A. and J.L. Largier, ”Observations of nearshore circulation: rip currents,” J. Geophys. Res., 100, 10967-10975, 1995. Sonu, C.J., ”Field observations of nearshore circulation and meandering currents,” J. Geophys. Res., 77, 3232-3247, 1972. Suhayda, J.N., Standing waves on beaches, J. of Geophys. Res., 79, 3065-3071, 1974. Svendsen, I. A., K. A. Haas, and Q. Zhao, Analysis of rip current systems, in Coastal Engineering, Proceedings of 27th International Conference, edited B. L. Edge, pp. 11271140, Am. Soc. Civ. Eng., New York, 2000. Thornton, E. B., and R. T. Guza, Surf zone longshore currents and random waves: Field data and models, J. Phys. Oceanogr., 16, 11651178, 1986. Thornton, E.B. and C.S. Kim, Longshore Current and Wave Height Modulation at Tidal Frequency Inside the Surf Zone, J. Geophys. Res., 98(C9), 16,509 16,519, 1993. Yu, J. and D.N. Slinn, ”Effects of wave-current interaction on rip currents,” J. Geophys. Res., 108, 3088, doi:10.1029/2001JC001105, 2003 D R A F T April 3, 2007, 8:32am D R A F T X - 26 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION Table 1. NCEX Instrument Locations Cross-shore (m) Alongshore (m) Cross-shore Array PUV3 -5 2610 PUV4 -29 2610 PUV5 -51 2609 PUV6 -79 2609 PUV7 -104 2607 PUV8 -121 2613 PUV9 -54 2728 PUV10 -54 2650 PUV14 -55 2614 PUV5 -51 2609 PUV11 -54 2589 PUV12 -54 2484 PUV13 -55 2289 Alongshore Array D R A F T April 3, 2007, 8:32am D R A F T 7 6 3 2 1 13 0 2400 1 12 0.5 alongshore (m) 8 0.5 13 2600 −0.5 alongshore (m) 1 1 2400 2 10 5 4 11 −1 −1.5 −2 −2.5 −0.5 3 0 −1 6 −2 9 −3 −3.5 7 −1 −1.5 −22.5 − −3 −3.5 −4 −4.5 0.5 0 −4.5 −3 −3.5 13 8 12 1 −1.5 2400 12 2600 −0.5 11 Yearday 308 9 10 5 4 11 X - 27 2800 −4 0.5 0 9 10 2600 −4 alongshore (m) Yearday 303 2800 0.5 Yearday 300 −2.5 2800 −3.5 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION 7 6 12 2400 3 2 1 −0.5 8 13 0.5 −3 −1.5 −1 2600 0 alongshore (m) −3 − 2−.5 2 1 −3.5 −4 −3.5 −4 13 2 1 alongshore (m) 12 2400 3 −1 9 10 5 4 11 −1 −1.5 6 0 alongshore (m) 7 0.5 8 −2 −2.5 2600 −1.5 1 0.5 0 −0.5 −1 −2 −2.5 −3 −4.5 13 2 −0.5 12 2400 3 10 5 4 11 0.5 6 9 −1.5 7 9 10 5 4 11 Yearday 316 2800 −1 8 −3.5 −4 −2 −2.5 −3 5 −3. −4 2600 Yearday 310 2800 −3 −0.5 Yearday 309 2800 −4 −4.5 2 −2 − .5 2200 2200 2200 −150 −100 −50 0 50 −150 −100 −50 0 50 −150 −100 −50 0 50 cross−shore (m) cross−shore (m) cross−shore (m) 2200 2200 2200 −150 −100 −50 0 50 −150 −100 −50 0 50 −150 −100 −50 0 50 cross−shore (m) cross−shore (m) cross−shore (m) Figure 1. Bathymetry referenced to mean sea level (MSL) for yeardays 300, 303, 308, 309, 310, and 316. The circles represent co-located pressure and velocity instruments. Arrows represent rip channel locations, which are referred to in the text. D R A F T April 3, 2007, 8:32am D R A F T X - 28 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION Hmo(m) 1.5 1 0.5 304 306 308 310 312 314 316 318 320 322 304 306 308 310 312 314 316 318 320 322 304 306 308 310 312 314 316 318 320 322 304 306 308 310 312 314 316 318 320 322 304 306 308 310 312 314 yeardays 316 318 320 322 Tmo(s) 0 302 20 10 θomo 0 302 20 0 v(m/s) −20 302 0.5 0 tide(m) −0.5 302 2 0 −2 302 Figure 2. Wave climate measured by directional ADCP in 6m water depth, ap- proximately 250m offshore: significant wave height, Hmo (top), mean wave period (second), peak wave direction (third), alongshore-averaged mean alongshore velocity (fourth), alongshore-averaged mean cross-shore velocity (fifth), and tidal elevation (bottom). D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION X - 29 300 400 NCEX Rectified Argus Image 0 04−Nov−2003 17:00:01 GMT 500 2 600 0 5 Argus x (m) 700 2 0 800 5 900 0 2 10 1000 5 1100 2 5 10 1200 10 1300 −2800 Figure 3. −2600 −2400 −2200 Argus y (m) −2000 −1800 Time-lapse Argus video image at low tide depicting the nearshore breaking wave patterns and the underlying rip current morphology. Bathymetry is in meters. Dots represent the NPS arrays. D R A F T April 3, 2007, 8:32am D R A F T X - 30 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION −1 0 h (m) 1 Survey 300 Survey 303 Survey 308 Survey 309 Survey 310 Survey 316 2 3 4 5 −240 −220 −200 −180 −160 −140 −120 −100 rotated cross−shore distance (m) −80 −60 −40 −240 −220 −200 −180 −160 −140 −120 −100 rotated cross−shore distance (m) −80 −60 −40 0.4 σh (m) 0.3 0.2 0.1 0 Figure 4. The mean alongshore bottom profile, h(x0 ) (top), and alongshore depth variation, σh (x0 ) (bottom), for yeardays 300, 303, 308, 309, 310, 316. D R A F T April 3, 2007, 8:32am D R A F T 0.25 m/s −4 −3 −2 −1 0 1 2600 2800 −8 −7 −6 0.25 m/s −5 −4 −3 −2 −1 −4 −3 −2 −100 −1 0 0 1 2200 −300 −200 2400 c) −100 0 −1 2200 −300 −200 0 1 0 2400 e) −4 −3 −2 −100 −1 0 2200 −300 −200 2600 −8 −7 −6 −5 1 2400 g) −4 −100 −3 −2 0 2200 1 Figure 5. 1 2800 0.25 m/s −1 0 −6 −5 2600 −8 −7 2800 0.25 m/s −4 −3 −2 −1 0 2400 2600 alongshore (m) cross−shore (m) a) −200 −8 −7 −6 −5 cross−shore (m) −8 cross−shore (m) −300 1 2800 cross−shore (m) cross−shore (m) cross−shore (m) cross−shore (m) cross−shore (m) MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION −300 b) −200 −6 −5 −4 −100 0 −200 −3 −2 −1 d) −6 −5 −4 1 2600 −8 −7 −3 −2 0 1 −100 −2 0 0 2400 −100 0 2200 1 −6 0.25 m/s −5 −4 0 2400 2600 −8 −7 −6 −5 −4 −3 −1 f) 2200 −8 −300 −7 −6 h) −200 2800 −1 −1 2200 −300 −7 −200 0.25 m/s 2400 −100 0 −8 −7 0 2200 −300 X - 31 1 2600 −2 −1 −5 −4 −3 0 2400 2600 alongshore (m) 1 2800 0.25 m/s 2800 −8 −7 −6 0.25 m/s 1 2800 Hourly mean velocity vectors for high (left) and low (right) tide during the period when waves were near normally incident and rip currents were present in the cross-shore array for yeardays 303 (a, b), 305 (c,d) and 309 (e,f). Obliquely incident wave conditions for yeardays 320-321 (g,h). D R A F T April 3, 2007, 8:32am D R A F T X - 32 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION (m/s)(m) 0.5 0 −0.5 40 60 80 100 120 140 40 60 80 100 120 140 40 60 80 100 cross−shore (m) 120 140 (m/s)(m) 0.5 0 −0.5 (m/s)(m) 0.5 0 −0.5 Figure 6. Hourly mean cross-shore velocities and Hrms (∇) in the cross-shore for the times when a rip current, shoreward-directed flow associated with the cell circulation, and undertow were present (top -yearday 305, middle -yearday 311, and bottom - yearday 311) at low-mid tide. The inner bars represent VLF pulsations and the outer bars represent IG pulsations. The solid symbol represent PUV11 located 20 meters south of the cross-shore array (Figure 1). D R A F T April 3, 2007, 8:32am D R A F T 0 energy phase 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 f(Hz) 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 f(Hz) 0.07 0.08 0.09 0.1 0.5 0 100 0 −100 0 10 coherence phase X - 33 10 coherence energy MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION 1 0.5 0 100 0 −100 Figure 7. Spectral energy, coherency, and phase for pressure (solid line) and cross-shore velocity (dashed line) estimates for PUV11 (top three sub-panels) and PUV12 (bottom three sub-panels) for yearday 309.27 D R A F T April 3, 2007, 8:32am D R A F T X - 34 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION 0 −0.2 −0.4 umean umean 0 −0.2 −0.4 Hrms 1 Hrms 1 0.5 0.5 θσ 0 30 θσ 0 30 20 20 10 10 θmean θmean 10 10 0 Uig,rms Uig,rms Uvlf,rms −10 0.2 Uvlf,rms −10 0.2 0.1 0.1 0 0.2 0 0.2 Figure 8. 0.1 z(m) z(m) 0.1 0 0 −0.5 −1 −1.5 2200 0 2400 2600 alongshore(m) 2800 0 0 −0.5 −1 −1.5 2200 2400 2600 alongshore(m) 2800 Alongshore variations of hourly umean (top row), Hrms (second row), σmean (third row), θmean (fourth row), Uig,rms (fifth row), Uvlf,rms (sixth row), alongshore bathymetric (bottom row) measurements within the alongshore array for yeardays 306 and 309. Circles represent high tide and squares represent low tide, except for Uig,rms the dashed line represents modelled estimates at low tide. Arrows indicate the rip channel. D R A F T April 3, 2007, 8:32am D R A F T X - 35 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION 50 50 45 45 40 40 0.04 0.04 0.02 25 0.01 20 0.03 30 0.02 25 0.01 20 15 0 −0.02 −0.01 0 ky (1/m) 0.01 15 0 −0.02 0.02 E(m2/s2/Hz/m−1) 30 Hz Hz 0.03 35 E(m2/s2/Hz/m−1) 35 −0.01 10 0 ky (1/m) 0.01 0.02 10 5 5 0 0 Figure 9. F -ky spectra computed using IMLE method for cross-shore (left) and alongshore (right) velocities for the alongshore array for yearday 322.5. Lines represent mode 0, 1, and 2 edge wave dispersion curves for a planar beach with a beach slope of 1:70 [Ekart, 1951]. D R A F T April 3, 2007, 8:32am D R A F T 2700 0.2 2600 0 2500 −0.2 2400 cross−shore(m) 2300 305.5 −0.4 306 306.5 307 307.5 308 308.5 309 309.5 120 0.2 100 tidal elevation(m) 0 80 −0.2 60 −0.4 40 305.5 u(cm/s) MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION u(cm/s) alongshore(m) X - 36 306 306.5 307 306 306.5 307 307.5 308 308.5 309 309.5 307.5 308 time(yearday) 308.5 309 309.5 1 0 −1 305.5 Figure 10. (top) Contour plot of alongshore hourly mean cross-shore velocities as a function of time. (middle) Contour plot of cross-shore hourly mean cross-shore velocities as a function of time. Cross-shore array located in the alongshore at 2610m and the alongshore array located at 55m plotted as dashed lines. (bottom) Tidal elevation. Crossshore velocity colorscale plotted on the right. D R A F T April 3, 2007, 8:32am D R A F T MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION Figure 11. X - 37 (top) Argus variance image around mid-tide for yearday 309. (bottom) Argus variance image around low-tide for yearday 309. D R A F T April 3, 2007, 8:32am D R A F T X - 38 MACMAHAN ET AL.: RIP CURRENTS INDUCED BY SMALL BATHYMETRIC VARIATION 30 0.2 30 20 σθ σθ 20 10 0 0 Urms,vlf (m/s) 0.15 10 rsq=0.16 0.2 0.4 |Umean (m/s)| 0 0 0.1 0.05 rsq=0.48 0.1 Urms,vlf (m/s) 0.2 0 0 rsq=0.28 0.2 0.4 |Umean (m/s)| Figure 12. (left) The correlation between Umean and (σθ ), (middle) and the correlation between Urms,vlf and (σθ ) and (right) the correlation between Umean and Urms,vlf are plotted for the alongshore array for yeardays 305-320. D R A F T April 3, 2007, 8:32am D R A F T