Observations of extreme wave runup events on the US Paciﬁc Northwest coast

. Extreme, tsunami-like wave runup events in the absence of earthquakes or landslides have been attributed to trapped waves over shallow bathymetry, long waves created by atmospheric disturbances, and long waves generated by abrupt breaking. These runup events are associated with inland excursions of hundreds of meters and periods of minutes. While the theory of radiation stress implies that nearshore energy transfer from the carrier waves to the infra-gravity waves can also lead to very large runup, there have not been observations of runup events induced by this process with magnitudes and periods comparable to the other three mechanisms. This work presents observations of several runup events in the US Paciﬁc Northwest that are comparable to extreme runup events related to the other three mechanisms. It also discusses possible generation mechanisms and shows that energy transfer from carrier waves to bound infragravity waves is a plausible

For example, Stockdon et al. (2006) produced a relationship between the 2% exceedance value of runup maxima and the beach slope, wave height, and wave length using data from several natural beaches. Some studies examine the ability of numerical 25 models to simulate runup. For example, Fiedler et al. (2018) show that one-dimensional non-hydrostatic models can predict runup with reasonable accuracy.
Some studies have focused on infrequent runup events with very large magnitudes that are not related to earthquakes or landslides. Aside from being potentially dangerous to beach goers, these runup events are important because they erode and deposit sediments at locations not usually affected by runup (e.g. Dewey and Ryan, 2017) and can potentially damage properties 30 and structures (e.g. Roeber and Bricker, 2015). Observations of such runup events have been so far been attributed to two mechanisms: trapped waves over shallow bathymetry (e.g. Sheremet et al., 2014;Roeber and Bricker, 2015;Montoya and Lynett, 2018) and long waves created by atmospheric disturbances -also known as meteotsunamis (e.g. Monserrat et al., 2006;Olabarrieta et al., 2017). It has also been implied by the theory of radiation stress, that energy transfer from carrier waves to bound infragravity waves can result in infragravity waves of very large heights (e.g. Longuet-Higgins and Stewart, 1962; 35 Battjes et al., 2004), and can potentially lead to very large runup. However, no observed runup with magnitude comparable to those due to the other two mechanisms have been attributed to the energy transfer mechanism.
The primary aim of this work is to show for the first time, through a set of observations, that energy transfer from carrier waves to bound waves is a plausible generation mechanism of runup with magnitudes and periods comparable to those from trapped waves over shallow bathymetry and those from meteotsunamis. The majority of this study is based on a set of obser-40 vations on the PNW coast on January 16, 2016. On this day, at least five different large runup events -some with more than a hundred meters of horizontal excursion, and all at different locations -were captured on video by beach goers. In addition, at least two runup related injury events were documented. The video footage and injuries took place along a 1000 km stretch of coastline within 5 hours of each other. Measurements from a number of instruments at various locations are analyzed. Possible generation mechanisms and comparisons to other similar observations are discussed. Lastly, a method to predict and forecast 45 similar events is presented.

Study site
The wave climate of the PNW coast ( Figure 1) is characterized by large wave heights and long wave periods, especially in the winter. For example, from 2008 to 2018 the median and 95 percentile of significant wave height for the summer month of August are 1.4 m and 2.5 m. For the winter month of January, they are 2.8 m and 5.3 m. For peak wave period they are 8.3 s 50 and 14.8 s for August and 12.9 s and 17.4 s for January (NOAA, 2020a). The reason for this is due to the large fetch and strong winds in the north Pacific storm systems, which are especially effective during the winter as the storm systems move across the ocean basin and achieve landfall (Tillotson and Komar, 1997). The PNW coast is also known for having low-sloping beaches. For example, upper shoreface slope from the central Oregon coast to the central Washington coast ranges between 0.005 to 0.02 (Di Leonardo and Ruggiero, 2015). There have been several 55 studies on runup in this region. For example, Ruggiero et al. (2004) analyzed 1.5-hour water level time-series along several cross-shore transects on Agate Beach (located on central Oregon coast). During this period the offshore wave height and wave period were 2.3 m and 13 s respectively. Runup was found to vary alongshore by a factor of 2 and was found to be proportional to foreshore beach slope. In addition, approximately 96% of the runup energy was contained in low frequencies (less than 0.05 Hz). Fiedler et al. (2015) analyzed runup on a single transect over a 44-day period, also on Agate Beach. During this time the 60 wave height ranged from 0.5 m to 7 m. The top 2% of runup were found to be approximately linearly proportional to the square root of wave height and wave length. In addition, the amplification of low frequency motion was found to decrease dramatically during storms. Holman and Bowen (1984) analyzed wave runup on several locations along the mid-Oregon coast and found that 99.9% of runup variance are attributed to periods of greater than 20 s, and that 83% of runup variance are attributed to periods of greater than 50 s. January 16 16:55, Pacific Beach, Washington: a large runup mobilized several logs and pushed them against a stretch of riprap. A large reflected wave is also seen traveling offshore (YouTube, 2020e).

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Large water-level fluctuations were observed along the same 1000-km stretch of coast during the same time by tide gages with both 6-minute and 1-minute recording intervals. The amplitudes of these water level fluctuations reached as high as about 0.5 m. Further detail is shown in the results sections. Wave runups of similar scale in magnitude have been observed at other times in this region, though typically not with as many video recordings from bystanders across the stretch of coastline as was on January 16, 2016.

Methods
Observations from three sources are presented in this study to provide various data across a range of locations and over different water depths. Table 1 lists the data types, measurement frequency, water depth, and distance from shore for each site. Figure 3 shows the locations of observation sites. Water level, wind speed, and atmospheric pressure from six NOAA CO-OPS stations are used. These gages are located at the coast and span approximately 800 km of coastline between Northern California and 95 Northern Washington. In addition, wave height, wave period, and wave energy density spectra from five NDBC buoys are used. These buoys span a similar range as the NOAA CO-OPS tide gages but are located further offshore on the continental shelf. Also used are water column height from four bottom sensors from the Deep-ocean Assessment and Report of Tsunamis (DART) system. These sensors span approximately 1200 km from north to south and are much further offshore than the NDBC buoys and are in the deep ocean.

Tide gages
The NOAA CO-OPS tide gages (NOAA, 2020b) are located at the coast. Five out of the six tide gages used in this study are located inside estuaries (except for Crescent City which is in a harbor). Figure 4 shows satellite images around each sensor.
Data used in this study at these locations include water level, atmospheric pressure, and wind speed. Water level is measured either from acoustic ranging (at Westport, Garibaldi, South Beach, and Charleston) or microwave radar sensors (at La Push 105 and Crescent City) (NOAA, 2020b). Data is sampled at 1 Hz and averaged over 1 minute to produce 1-minute time series. Frequency analysis on the water level time series are performed via Fast Fourier Transform with 18 degrees of freedom over record lengths between 708 and 715 samples. Atmospheric pressure is measured from pressure sensors mounted between 7.7 m to 11 m above mean sea level (NOAA, 2020b). 21 six-second samples over 2 minutes are averaged and collected every 6 minutes. Wind speed is measured from anenometers mounted between 11 m and 30 m above sea level. 2-minute average of 110 1-Hz samples are collected every 6 minutes.
For the analysis sought in this work, it is important to determine the intensity of water level fluctuations at the tide gages.
One method of representing such intensity is by generating an envelope of the time series. This is often done by using a Hilbert transform. When the time series has signals with high frequencies, however, the resulting Hilbert transform will also contain high frequency signals. In this case, it is necessary then to remove the high frequency signals by a low-pass filter. Another 115 method is to compute the root-mean-square (RMS) of the time series over a specified window. The two methods yield similar results. The RMS method is chosen over the Hilbert transform method for its simpler implementation.

Buoys
NDBC buoys (NOAA, 2020a) are moored buoys located 45 -85 km offshore at water depths of 128-400 m. All stations used in this study are of the 3-meter discus type buoys. Data used at these locations include significant wave height and peak 120 wave period -both recorded at 1-hour intervals. Also used at these locations are wave spectral density, also recorded at 1-hour intervals, across a frequency range of 0.02 Hz to 0.485 Hz. Data acquisition starts at the 20th minute of each hour and continues for 20 minutes. During this time, buoy motions are measured and then transformed from the temporal to the spectral domain.
The energy density spectra derived from wave motions can be useful in characterizing the sea state. A useful parameter that can be computed from ocean wave spectra is the spectral moment: where n, typically an integer, denotes the n-th moment, E(f ) is the energy density, and f is the frequency. In metric units, E(f ) is in m 2 /Hz, f is in Hz, and m n is therefore in m 2 Hz n . Significant wave height is calculated as four times the square root of the zeroth moment of the wave spectra. Peak wave period is calculated as the inverse of the peak frequency.
When n is taken to be negative, wave energy associated with lower frequency is emphasized more than associated with 130 higher frequency. The use of negative moments has been employed by Hwang et al. (2011) to facilitate the separation of swell and wind waves. This is useful in this study as it can serve as an indicator of a swell-dominated sea-state.

Bottom sensors
DART sensors (DART, 2020) are located 280 -560 km offshore at water depths of 2805-4319 m. Water column heights are typically recorded in 15-minute intervals, although 1 min and 15 s intervals are used during special operation modes. In 135 standard operating mode, pressure is measured at 15-second intervals and converted to water column height, but the data is only recorded every 15 minutes and transmitted every hour. When an event is detected by its tsunami detection algorithm, i.e.
when the difference between water column height based on predicted tide and the measured values exceeds a threshold (30 mm in North Pacific), the instrument begins operating in event mode (DART, 2020). During event mode, 15-second values are transmitted in the initial 4 minutes and 15 seconds, followed by four hours of 1-minute averages. Afterward, the system 140 resumes standard operation if no further events are detected.

Observation of environmental conditions
Water level observations at the coast measured by NOAA CO-OPS tide gages in 1-minute increments are shown in Figure 5. A set of water level fluctuations with frequencies higher than the tidal signal and magnitudes as high as 0.5 m can be seen from 145 roughly January 16, 8:00 to January 17, 16:00 (PST, local) across all tide gages used in this study. These magnitudes of water level fluctuations are comparable to those from meteotsunamis and even some tsunamis from earthquakes (Monserrat et al., 2006;Olabarrieta et al., 2017). These fluctuations are more intense in the northern-most (La Push) and the two southern-most tide gages (Charleston and Crescent City) than the three middle tide gages (West Port, Garibaldi, and South Beach). The water level fluctuations first increase, then are sustained around their peak level from January 16 10:00 to January 16 20:00. This Spectral analysis is performed on water level time-series from January 16 10:00 to January 16, 22:00, a period during which intense water level fluctuations persist. An examination of the energy density spectra ( Figure 6) shows the existence of a 155 common peak period at ∼5 minutes across all tide gages (between 4.5 to 5.9 minutes). As described above, a video taken near Charleston during this time showed a lapse of approximately 3 minutes between the trough of a previous large wave runup and the crest of the following large wave runup, suggesting a runup period of approximately 6 minutes. Another spectral peak between approximately 13 to 22 minutes can also be seen for four (Westport, South Beach, Charleston, and Crescent City) of the six stations. These periods corresponds to periods of shelf resonance, and is further discussed in later sections.

La Push
Westport Garibaldi South Beach Charleston Crescent City Atmospheric pressure and wind speed at two tide gage locations are shown in Figure 7. Atmospheric pressure varies over a range of approximately 10 HPa between January 15 12:00 to January 17 12:00. The majority of this variation occurs over two cycles within the two days. 1-hour high-passed time series indicate that the largest high frequency anomaly in atmospheric pressure for this period is about 1 HPa. The largest high frequency wind speed anomaly over this period is about 5 m/s at South Beach and 2.5 m/s at Charleston.

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Significant wave height and peak wave period at NDBC buoys are shown in Figure 8. Wave height is seen to be moderately high for this region (∼4 to 6 m) at the approximate onset of the unusual water level fluctuations reported by the tide gages and throughout the time period during which videos of the large wave runups and injury reports took place. No significant anomaly in wave height was observed. However, large increases of peak wave periods (from ∼12 s to ∼25 s) were observed within the recording interval of 1 hour and very close to the approximate onset of the unusual water level fluctuations at the tide gages.  (right subplot). It can be seen that the significant wave height for the wind component does not vary considerably during the January 16 event. However, significant wave height associated with the swell component across all 5 NDBC buoys increases by approximately 5 m over 12 hours starting close to the onset of the unusual water level fluctuations at the tide gages.
Water column height as measured by DART sensors far offshore (280 -560 km, Figure 3) is shown in Figure 10. Fluctuations 180 of higher-than-usual magnitudes are observed between about 4 hours (sensor 46407) to about 6 hours (sensor 46404) after the approximate onset of large water level fluctuations at the tide gages (i.e. January 16, 8:00). The increase in recording intervals starting near January 16 16:00 was due to transition from 'standard' to 'event' mode, which is triggered by the higher-thanusual magnitude of the water column height fluctuations.

Possible generation mechanisms 185
One possible generation mechanism for large runup involves trapped waves over shallow bathymetry (e.g. Sheremet et al., 2014), such as the continental shelf. On examination of the energy spectra of the onshore water level, the longer period peak (13 to 22 min) is close in magnitude to that due to resonance from the shelf in this region (Allan et al., 2012). For example, Allan et al. (2012) found that the periods of shelf resonance observed after the 2011 Tohoku tsunami were between 17 minutes on January 16, 2016 show that the period of the runup events are closer to 5 minutes than they are to 20 minutes. In addition, large offshore directed waves were detected off the shelf (i.e. at the DART bottom sensors) hours after the initial onset of large runup events, indicating that the returning waves were able to travel past the shelf into deeper water. As such, while shelf resonance may have enhanced the runup events on January 16, they are not likely to be the primary driver.
Extreme runup can also be caused by a phenomenon known as a meteotsunami. In this mechanism, a large atmospheric 195 disturbance travels at the shallow water speed and creates a tsunami-like runup (Monserrat et al., 2006). As described earlier, the largest atmospheric pressure anamoly from January 15 12:00 to January 17 12:00 is about 1 HPa. The wind speed anomaly over this period is about 5 m/s at South Beach and 2.5 m/s at Charleston. In contrast, the atmospheric pressure anomaly and wind speed that led to the meteotsunami analyzed by Olabarrieta et al. (2017) are approximately 5 HPa and 15 m/s, respectively.
Multiple meteotsunamis in the study of Monserrat et al. (2006) are also associated with atmospheric pressure anomaly of 200 around 5 HPa. In addition, the meteotsunami events described by Olabarrieta et al. (2017) and Sheremet et al. (2016) involve one primary large wave (soliton) sometimes followed by large waves with rapidly decaying amplitudes (over a few minutes and with periods of incident waves). In contrast, the January 16, 2016 runup events, as evident by video footage and water level measurements, are recurring events with periods an order of magnitude larger than incident waves and with sustained amplitude over multiple hours. As such, January 16 runup events are not likely meteotsunamis due to lack of significant 205 atmospheric pressure and wind speed anomaly, and the markedly different amplitude-decay and period characteristics than what is discussed in the meteotsunami literature.
A third possible generation mechanism is considered here. As described in the previous section, one of the most striking features of the environmental conditions leading to and during the occurence of the large runup events is the rapid and significant increase in wave energy at very low (<0.06 Hz) frequencies. This observation and the observation of a 5-minute period in water 210 level response at the tide gages suggest a connection between the large runup events and infragravity waves. Specifically, it is known that infragravity waves have periods corresponding to those of wave groups, and a 5-minute period is a plausible period for wave groups when carrier waves have periods of approximately 25 s. For example, 12 waves at 25 s period would make a 5-minute wave group. It is also known that infragravity waves can experience resonant growth under appropriate conditions (Longuet-Higgins and Stewart, 1964;Battjes et al., 2004). Further detail on this mechanism, as well as a method using this 215 mechanism to forecast similar events, are in the discussion section.
Resonant growth of infragravity waves as a driver for large runup events is also supported by the observations of water column heights at the far offshore DART sensors. Specifically, the fluctuations of water column heights at the DART sensors started several hours after the approximate onset of unusual water level fluctuations at the tide gages. This suggests that the heights of the incoming infragravity waves were rather small compared to those of the reflected infragravity waves. This is also 220 consistent with the findings that the most energetic infragravity waves in the deep ocean originate from the nearshore . As a test, it is assumed that the infragravity waves have a period of close to 5 minutes (from the peak in energy density spectrum of the water level at tide gages). Assuming a simple shelf bathymetry, it is estimated that these waves would take approximately 1-1.3 hours to travel from shore to the DART sensors.

Generation of waves with very long periods
The large runup events of January 16, 2016 were associated with a rapid increase of peak period and energy at low frequencies (i.e. Figure 9 and Figure 10). It is thus worth discussing how incident waves of very long periods can be generated. It is known that the height and period of a deep-water wave increase as the wave stays within the wind system, or fetch (Wilson, 1955; 13 https://doi.org/10.5194/nhess-2020- 425 Preprint. Discussion started: 26 January 2021 c Author(s) 2021. CC BY 4.0 License.  Bowyer and MacAfee, 2005). If the fetch moves in the same direction as the waves, the waves would remain in the fetch for a 230 longer period than if the fetch was stationary, and thus grow to be higher and longer.
If, in addition to moving in the wave direction, the fetch also moves at speeds very close to those of the wave groups, waves of even larger heights and periods can be generated due to the longer duration in which the waves stay within the fetch. This has been referred to in the literature as trapped fetch, dynamic fetch, effective fetch, fetch enhancement, and group velocity quasi-resonance (Dysthe and Harbitz, 1987;Bowyer and MacAfee, 2005).

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The likelihood of trapped fetch contributing to the January 16 runup events can be analyzed using storm tracks and wave periods.  Between approximately January 16 17:00 to January 16 22:00, the 46404 station was in higher sampling mode.
are in reasonable alignment with the tracks of the storms. Therefore, trapped fetch was likely to be at least partially responsible for the January 2016 events.
As large as the January 16, 2016 runup events were, they were not the only occurence of extreme runup in this region. On 245 January 18, 2018, video footage (YouTube, 2020f) was taken at a coast of the PNW (Figure 11) that shows recurring extreme runup events with similar characteristics on this day compared to January 16, 2016. Figure 12 ( While it is plausible that a very large and very strong storm could potentially generate waves of very long periods without having a trapped fetch, the trapped fetch provides a mechanism for lesser storms to generate waves of long enough periods to 255 lead to the type of large runup events shown in this study.

Resonant growth of infragravity waves
As shown in Figure 6, a common peak period at approximately 5 minutes is seen in the energy density spectra of the large water level fluctuations at all 6 tide gages along the PNW on January 16, 2016. In addition, a period of approximately 6 minutes can be deduced from one of the videos taken on that day (YouTube, 2020a). A period of approximately 5-6 minutes is also a 260 plausible time-scale for wave groups.
Wave motions with periods in the time-scales of wave groups have long been known to exist (e.g. Munk, 1949;Tucker, 1950). Two mechanisms are known to generate such waves. In the first mechanism, variations of amplitudes within a wave group lead to transfer of momentum in such a way that produces a depression of the mean water level at the location of waves with greater amplitudes and an elevation of mean water level at the location of waves with smaller amplitudes. This so-called 265 'bound infragravity wave' travels at the speed of wave groups towards shore and as a free wave away from shore after reflection ( Longuet-Higgins and Stewart, 1964). In the second mechanism, changes in the cross-shore location of wave breaking releases a free wave towards the shore (Symonds et al., 1982). The relative importance of the second mechanism is known to decrease with decreasing beach slope (List, 1992;Battjes et al., 2004). This suggests that in the case of relatively low beach slope (as is in this study), the first mechanism is important.

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The amplitudes of bound infragravity waves increase shoreward differently compared to free waves. When the carrier wave length is small compared to the water depth, but the group wave length is large compared to the water depth, the infragravity wave amplitude -according to the theory of radiation stress -is (Longuet-Higgins and Stewart, 1964) where a g is the infragravity wave amplitude, a is the carrier wave amplitude, and k is the carrier wavenumber.

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When the carrier waves are in shallow water, i.e. their length is much greater than the water depth, the amplitudes of the bound infragravity waves approach resonance, and the variations of amplitudes with water depth are bound between ∝ h −1/4 and ∝ h −5/2 (e.g. Longuet-Higgins and Stewart, 1962;Battjes et al., 2004). This is in contrast to the shoaling of free waves in shallow water, which varies as ∝ h −1/4 . The resonant growth of bound infragravity waves in shallow water is due to energy transfer from the incident waves to the infragravity waves. When the bottom slope is steep compared to the infragravity wave 280 length, little to no energy transfer exists, and the infragravity wave shoals close to h −1/4 . When the bottom slope is mild compared to the infragravity wave length, a great amount of energy transfer takes place, and the infragravity waves increase in ampltudes close to h −5/2 , i.e. at a much greater rate compared to free waves. Battjes et al. (2004) proposed the following parameter, the normalized slope, as a controlling parameter on the growth of infragravity waves: where h x is a characteristic bed slope, ω is the radial frequency of the infragravity wave, and h here is a characteristic water depth. The results of van Dongeren et al. (2007) confirmed that infragravity wave amplitudes vary close to ∝ h −5/2 when β is small. This growth rate decreases as β increases, and eventually approaches ∝ h −1/4 for large β. It is readily apparent from (3) that shorter infragravity waves are associated with a smaller β. It would seem contradictory, then, that the extreme runup events of January 16, 2016 occured when the infragravity waves were very long.

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To examine the growth of infragravity wave amplitudes from deep to shallow water for different carrier wave periods, a simple model is used. This model assumes a planar bed with constant slope and alongshore uniformity. It employs simple dynamical constraints for the purpose of computing general trends for waves with differing periods. Infragravity wave amplitudes are computed from deep water to the depth at which infragravity waves start to break. Infragravity wave amplitude in deep water follows (2). Carrier wave amplitudes are computed from linear water wave theory. In shallow water, a g ∝ h −α as 295 described earlier, where α depends on β -defined in (3) -and is between -1/4 and -5/2. The characteristic depth h in (3) is taken to be the carrier wave breaking depth, which is determined from the criterion H b = κh b , where H b is the breaking wave height, h b is the breaking depth, and κ is an empirical parameter, taken to be 0.78. The dependence of α on β is obtained from the laboratory and numerical results of van Dongeren et al. (2007, Figure 2). The nearshore slope is taken to be 0.01 (Cohn et al., 2019) and the carrier wave amplitude is taken to be 2.5 m (Figure 8). A carrier wave period of 25 s is chosen to correspond 300 to the peak incident wave period during the large runup events. The results are compared with those from a carrier wave period of 10 s, which is representative of wave periods only hours before the first large runup occurrence.
The result from this simple model shows that 25 s carrier waves enter shallow water at a water depth of 15 m. The corresponding infragravity wave amplitude is 0.87 m. In contrast, for carrier waves of 10 s, the infragravity wave amplitude at this depth is 0.13 m. For carrier waves of 25 s, the infragravity wave resonant growth follows a g ∝ h −1.0 and reaches a maximum Another factor that contributes to the increase in runup magnitude at lower carrier wave frequencies can be attributed to the 310 reduction of energy dissipation. It has been suggested by Battjes et al. (2004) and confirmed by van Dongeren et al. (2007) that the energy dissipation of bound infragravity waves depend on the following parameter: where H is the infragravity wave height near the shoreline. It can be seen that β H is similar to the previously defined β, except for the replacement of h with H. van Dongeren et al. (2007) find that at very low values of β H , the reflection coefficient of 315 the infragravity waves is close to 0, indicating near complete energy dissipation. As β H increases, the reflection coefficients also increase until they approach close to 1, indicating near absence of energy dissipation. The low value of ω observed during the large runup events in this study thus acts to reduce energy dissipation of the bound infragravity wave, leading to larger runup events. To illustrate this effect, the maximum infragravity wave heights from the previous comparison are used in (4), and the resulting β H is related to reflection coefficients from the experiments of van Dongeren et al. (2007, Figure 3). This 320 yields reflection coefficients of 0.50 and 0.40 for carrier wave of 25 s and 10 s, respectively.
In summary, infragravity wave amplitudes near the shore are considerably larger for carrier waves of 25 s than those for carrier waves of 10 s. In addition, infragravity waves associated with 25 s carrier waves also experience less energy dissipation than those associated with 10 s carrier waves. Both of these effects imply enhancements in wave runup.
The plausibility of maintaining a high infragravity wave growth rate and reasonably low energy dissipation is also supported 325 by observations at the DART sensors ( Figure 10). It is seen that the large fluctuations of water column height occur hours after the first occurrence of large water level fluctuations at the shore. This suggests that the incoming infragravity waves, before growth and energy dissipation occurs in the nearshore, are not large enough to cause a significant response at the sensors; whereas the outgoing infragravity waves are able to produce a significant response due to having achieved a reasonable growth and reasonably low energy dissipation before being reflected away from shore. And as stated earlier, this is also consistent with 330 the finding that the most energetic infragravity waves in the deep ocean originate from the nearshore .

A predictor for large runup events due to bound infragravity waves
As shown in the results section, an important observation of the January 16 large runup events is that these events are connected with rapid growth of wave energy in very low frequencies. This connection can be exploited to explore a predictive tool for similar large runup events. The goal of this predictor is to use a metric of ocean waves to predict a metric of water levels at the 335 shore. One approach to this is to use a cutoff frequency to identify the low frequency component of the ocean wave spectra.
However, it was found that the subjectivity of the cutoff frequency makes the predictor less robust. Instead, we explored an approach that uses the negative moments of the ocean wave spectra. In this approach, negative moments (1) at NDBC buoys were correlated to representative measures of the intensity of water level fluctuations at the tide gages. NDBC and tide gage pairs were determined based on proximity. To represent moments of the ocean energy spectra, normalized and non-normalized 340 moments were evaluated. Water levels RMS -hereinafter referred to as η nms -was chosen to represent the magnitude of the water level fluctuations. The best relationship was found between η nms and the non-normalized moments √ m −3 . Although √ m −3 has rather odd units in m/Hz 3/2 , it was found to be a better predictor of η rms than normalized moments, including m −3 f 3 p , m −3 f 3 m , and m −3 f 3 m2 , where f p , f m , and f m2 are peak, mean, and zero-upcrossing frequencies, respectively. It was also found that the use of √ m −3 produced a considerablly improved predictor of η rms compared to √ m 0 (e.g.   La Push-46041 are 35.8 m/Hz 3/2 and 0.035 m, respectively, whereas they are 9.9 m/Hz 3/2 and 0.0028 m, respectively, during the month of August. This suggest that the type of extreme runup events discussed in this work occur much more frequently during winter months than they occur during summer months.

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One reason why large runup events due to bound infragravity waves may occur much more frequently during the winter months than they do in the summer months is related to the fact that the wave periods in this region is much greater in the winter months than they are in the summer months. An increase in the length of the infragravity wave acts to reduce its growth as it approaches shore but also acts to reduce its energy dissipation. This may facilitate a more optimal balance between growth and energy dissipation of the infragravity wave as discussed before.

Conclusions
This work presents an analysis of observations of unusually large runup events that occurred along the PNW coast on January 16, 2016. On this day, video recordings and injury reports document multiple extreme runup events -with horizontal excursions exceeding a hundred meters and periods of minutes -occurring along approximately 1000 km of coastline within 5 hours of each other. Environmental conditions leading up to and during the large runup events are presented.

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The observations show that the large runup events are strongly associated with a rapid increase in wave energy at low frequencies, i.e. the arrival of incident waves with very long periods. In addition, water level measurements at the tide gages show a ∼5 min peak period during this time. The arrival of incident waves with very long periods can be explained by the existence of trapped fetch, which occurs when the fetch moves in the same direction and at speeds close to those of the waves groups. Analysis of storm tracks show that trapped fetch was indeed in effect for this event leading to the large runup events.

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The ∼5 min peak period at the tide gages suggest a link to wave groups and infragravity waves. It is shown using a simple model that a very large carrier wave period results in a very large infragravity wave amplitude at the shore yet maintaining reasonably low energy dissipation. This explanation is supported by far offshore bottom sensors, which did not detect the incoming infragravity waves but did detect the returning ones, even at 3 to 4 km depth.
is developed. The predictive ability is seen to be reasonable for 4 out of the 5 tide gages used in the study, and for two different sets of large runup events. Results from this predictive method suggest that the type of large runup events discussed in this work tend to occur much more frequently in the winter months than they do in the summer months (Tillotson and Komar, 1997, and Figure 1). This is likely owing to the fact that the wave periods are much longer during the winter months (e.g. median = 12.9 s in January) than they are in the summer months (e.g. median = 8.3 s in August). This would result in longer infragravity 395 waves and larger associated runup. The performance demonstrated by this predictive method may be helpful to future efforts in developing forecasting tools for extreme runup events, with the aim of issuing warnings to the public.
Data availability. The data used in this work are publicly available via sources referenced.
Video supplement. The videos referenced in this work are available as cited. Competing interests. The authors declare that they have no conflict of interest.
Acknowledgements. This work was funded by the National Science Foundation under award OCE-1459049. The authors also thank Jeremiah