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Observations of Nonlinear Internal Wave Runup Into the
Surfzone
GREGORY SINNETT1, FALK FEDDERSEN1, ANDREW J. LUCAS2,
GENO PAWLAK1, AND ERIC TERRILL1
1Scripps Institution of Oceanography, La Jolla, California, USA
2Mechanical and Aerospace Engineering, University of California San Diego, La Jolla,
California, USA
JPO, in preparation
October 4, 2017
Corresponding author address:
G. Sinnett, Scripps Institution of Oceanography, 9500 Gilman Drive, La Jolla CA 92093-0209,
gsinnett@ucsd.edu
1
ABSTRACT
Internal wave runup across the nearshore (18 m depth to shore) has not been previously described. Here,
a dense thermistor array on the Scripps Institution of Oceanography pier (to 8 m depth), a mooring in
18 m depth, and an ADCP in 8 m depth are used to study the cross-shore evolution of the rich and
variable nonlinear internal wave (NLIW) field near La Jolla, CA. Isotherm oscillations spanned much of
the water column at a variety of periods from 18 m depth to the shoreline. At times, NLIWs propagated
into the surfzone decreasing temperature by ≈ 1 ◦C in five minutes. When stratification was strong,
temperature variability had a distinct spectral peak near 10 min periods and upslope NLIW propagation
was coherent at semi-diurnal and harmonic periods. When stratification was weak, temperature vari-
ability at all frequencies decreased and was incoherent between 18 m and 6 m depth at semi-diurnal and
harmonic periods, with no distinct 10 min peak. In 8 m depth, onshore coherently propagating NLIW
events (an isolated, rapid and significant temperature drop and recovery lasting 4–6 h) had front velocity
between 1.4 to 7.4 cm s−1, incidence angles of -5◦ to 23◦, and temperature drops of 0.3◦ to 1.7◦. The
front’s position, temperature drop, and two-layer equivalent height of four events were tracked upslope
until propagation terminated at the onshore runup extent. For these events, front position was quadratic
in time, and normalized front temperature drop magnitude and equivalent two-layer height both de-
crease, collapsing as a linearly decaying function of normalized cross-shore distance. The NLIW event
front velocity and deceleration are consistent with two-layer upslope gravity current scalings. During
the NLIW rundown, the gradient Richardson number falls below 0.25 with near-surface cooling and
near-bottom warming in 8 m depth, indicating shear-driven mixing.
2
1. Introduction
Internal waves (internal isopycnal oscillations) are ubiquitous in the coastal ocean. In coastal1
regions, nonlinear internal waves (NLIW) transport and vertically mix sediment, larvae and nu-2
trients (e.g., Leichter et al. 1996; Pineda 1999; Quaresma et al. 2007; Omand et al. 2011). As3
an aggregation mechanism, internal waves can generate patches and fronts of swimming plankters4
(e.g., Lennert-Cody and Franks 1999; Jaffe et al. 2017). In the nearshore (defined here as depths5
h < 20 m) NLIWs can drive nearshore temperature fluctuations of up to 6◦C at tidal and higher6
frequencies (e.g., Winant 1974; Pineda 1991; Walter et al. 2014). The nearshore semi-diurnal7
internal tide can transport nutrients onshore (Lucas et al. 2011), which can drive nearshore phyto-8
plankton blooms (Omand et al. 2012). Nearshore NLIWs were also correlated with the presence of9
phosphate and fecal indicator bacteria near the surfzone (Wong et al. 2012). Although important10
to nearshore ecosystems, the cross-shore transformation of NLIWs in the nearshore, particularly11
to the surfzone, is poorly understood.12
NLIWs that propagate into the nearshore may be either remotely or locally (on the shelf) gen-13
erated (Nash et al. 2012). On the shelf, NLIW generation and propagation depends on on the14
background shelf stratification (e.g., Zhang et al. 2015), and barotropic tides (e.g., Shroyer et al.15
2011) and can be modified by upwelling and regional-scale circulation (Walter et al. 2016). In16
analogy to a surface gravity wave surfzone, as internal waves propagate into shallow water on17
subcritical slopes, they steepen, become highly nonlinear, and dissipate (e.g., Moum et al. 2003;18
MacKinnon and Gregg 2005), creating an “internal surfzone” (e.g., Thorpe 1999; Bourgault et al.19
2008) where mixing is elevated. Highly nonlinear and dissipating NLIW can have both wave and20
bore-like properties when propagating upslope on the shelf from 120 m to 50 m depth (Moum et al.21
2007). NLIWs sometimes form highly nonlinear solitons trailing the leading edge of the dissipat-22
ing internal tidal bore (e.g., Stanton and Ostrovsky 1998; Holloway et al. 1999). Farther onshore,23
internal wave runup occurs as an internal bore in the “internal swashzone”, analogous to surging24
surface gravity wave runup in the swashzone of a beach (e.g., Fiedler et al. 2015).25
In the nearshore, NLIW internal waves have been observed often as internal bores associated26
with the internal tide. In Monterey Bay (h = 15 m), sharp temperature drops in the bottom 10 m27
associated with the M2 (12 hour period) internal tide steepen into a bore front and precede gradual28
cooling over several hours before temperature quickly recovers amid intensified mixing (Walter29
3
et al. 2012). The 12-h evolution of a semi-diurnal non-linear internal bore near Del Mar California30
was tracked between 60 m and 15 m depth (Pineda 1994). In h ≈ 12 m depth, internal tidal bores31
have been related to nutrient and larvae transport (Pineda 1999). Bottom trapped (cold) bores were32
observed near Huntington Beach in the Southern California Bight in depths between 20 m and33
8 m, attributed to breaking semi-diurnal internal waves (Nam and Send 2011). In the nearshore,34
NLIWs can have significant temporal variation (e.g., Suanda and Barth 2015) associated with35
multiple angles of incidence, and can strongly interact with one another (Davis et al. 2017). Thus,36
observational studies of the nearshore internal surfzone and swashzone require dense spatial and37
fast temporal resolution to adequately resolve the onshore evolution of NLIWs.38
The internal surfzone and swashzone have been delineated in laboratory (e.g., Wallace and39
Wilkinson 1988; Helfrich 1992; Sutherland et al. 2013a) and numerical studies (e.g., Arthur40
and Fringer 2014). Laboratory studies of internal bores typically use a layered lock exchange41
(e.g., Shin et al. 2004; Marino et al. 2005) or motor driven paddle to create an internal disturbance42
(e.g., Wallace and Wilkinson 1988; Helfrich 1992), then quantify the speed and shape of the up-43
slope surge of dense water based on layer density differences and total water depth. Analogous44
to surface wave breaking, conditions affecting the internal wave breaking regime and subsequent45
upslope evolution as a bore were found to be a function of an internal Iribarren number Ir (ratio of46
internal wave steepness to bathymetric slope) or offshore wave frequency and amplitude (Suther-47
land et al. 2013a; Moore et al. 2016). The internal Iribarren number also affected the total upslope48
bore dissipation and eventual transport of artificial tracers (Arthur and Fringer 2016). It is not49
clear, however, how well the relationship between idealized laboratory or numerical simulations50
and NLIW runup in the ocean is, since NLIW runup observations across the nearshore are lacking.51
Scripps beach, the Scripps Institution of Oceanography (SIO) Pier (La Jolla, CA), and sur-52
rounding canyons provide a natural laboratory to study NLIWs in shallow environments. Canyon53
currents have been linked to internal waves in this (and other) canyon systems (Shepard et al. 1974;54
Inman et al. 1976). Recent observations in La Jolla canyon show an active internal wave field at55
the semidiurnal frequency. Energy flux is up canyon as internal oscillations transition to higher56
harmonics (M4 and above), indicating onshore propagation of a highly nonlinear and evolving57
internal wave field (Alberty et al. 2017). In 7 m water depth at the end of the Scripps Institution58
of Oceanography (SIO) pier (this study location), bottom temperature can drop rapidly, 5 ◦C over59
minutes (Winant 1974; Pineda 1991). With a 4-element cross-shore array on the Scripps pier, cold60
4
pulses were observed propagating onshore into the surfzone (Sinnett and Feddersen 2014). How-61
ever, details of internal runup until termination, variability and potential impacts to the nearshore62
are not well observed and have primarily been addressed only in laboratory or numerical settings.63
Here, NLIW observations from 18 m depth all the way to the surfzone are described, with64
emphasis on the internal swashzone component of the runup. Experimental details are described65
in section 2, with some of the first time series and spectral observations of NLIW runup in water66
depths as shallow as h = 2 m in section 3. Observations of individual runup events are described in67
section 4 with an emphasis on collapsing relavant physical properties for comparison to laboratory68
and numerical studies. Discussion of these results are in section 5 and concluding remarks are in69
section 6.70
2. Experimental Details
a. Location and Overview
Temperature and current observations at the Scripps Institution of Oceanography (SIO) pier71
(La Jolla California, 32.867N, 117.257W) were made during fall (29 September to 29 October)72
2014 when stratification is strong (Winant and Bratkovich 1981). The SIO pier is 322 m long and73
extends west-north-west (288◦) into water roughly 7.6 m deep. It is ≈ 500 m southeast of Scripps74
Canyon, the northern arm of the La Jolla canyon system (Figure 1a). The shoreline is roughly75
alongshore uniform from 200 m north to 500 m south of the pier, with mean cross-shore slope76
s ≈ 0.027 from the shoreline to h = 18 m depth before a steep canyon break. The reference77
depth (z = 0) is at the mean tide level (MTL), and the cross-shore origin (x = 0) is defined as the78
shoreline at MTL. The x coordinate axis is aligned with the length of the pier (positive onshore)79
making the y axis oriented alongshore (positive toward the north, Figure 1a). The alongshore80
origin (y = 0) is defined at the northern edge of the pier.81
b. Instrumentation
For 30 day experimental period, a vertical temperature chain was deployed at h = 18 m (de-82
noted S18) directly offshore of the pier at x = −657 m, y = 0 m (green star, Figure 1a) with83
5
14 Seabird SBE56 thermistors sampling at 2 Hz spaced 1 m apart extending from 1 m above the84
bed to 3 m below MTL. An additional SBE56 was tethered to a surface float which continually85
sampled near surface temperature at a fixed level relative to the tide. Concurrently, 36 Onset Hobo86
TidBits and 8 Seabird SBE56 thermistors were deployed on the SIO pier pilings (y = 0 m) at87
various cross-shore sites (−273 m < x < −29 m) and vertical locations (−5.9 m < z < 0.1 m)88
(blue and red circles, Figure 1b). These TidBits and SBE56s sampled water temperature at 3 min89
and 15 s intervals respectively, and were calibrated in the SIO Hydraulics Laboratory temperature90
bath, yielding accuracies of 0.01◦C (TidBits) and 0.003◦C (SBE56). The TidBits have a 5-minute91
response time and are capable of resolving oscillations at periods longer than 10 minutes.92
A pier-mounted Seabird SBE 16plus SeaCAT maintained by the Southern California Coastal93
Ocean Observing System (SCCOOS) measured salinity and temperature at x = −246 m and94
z = −5.8 m (roughly 1.2 m above the bed), sampling every 6 min (square, Figure 1b). Salinity95
was linearly related to temperature over the experiment duration at this site, with salinity of 33.5796
±0.05 psu 90% of the time. A pier-end Precision Measurement Engineering (PME) vertical tem-97
perature chain with 1 m vertical resolution maintained by the SIO Coastal Observing Research and98
Development Center (CORDC) provided temperature measurements at 1 Hz sampling rate with99
0.01◦C accuracy (green circles, Figure 1b). This temperature chain was offline from 2 October100
to 6 October, and again from 16 October to 18 October. Four additional SBE56 thermistors were101
mounted 0.3 m above the bed in depth h ≈ 7.6 m at x = −273 m and alongshore locations spaced102
100 m apart (y = −200,−100, 100 and 200 m, red dots, Figure 1a). These instruments were active103
9-30 October and sampled temperature at 1 Hz to capture alongshore variation and incident event104
angle relative to the slope. Temperature data from near-surface pier-mounted thermistors was re-105
moved at times when they were exposed to air (low tide or large waves) following Sinnett and106
Feddersen (2014). For convenience, the pier-mounted instrument site locations near the 8 m, 6 m,107
4 m and 2 m isobaths (x = −273 m, −219 m, −155 m, and −100 m) are referred to as S8, S6, S4108
and S2 (see Figure 1b) throughout the rest of the manuscript.109
Water column velocity was observed by an upward looking Nortek Aquadopp current profiler110
deployed in 7.6 m depth at S8 (black triangle, Figure 1b). It sampled with 1 min averages and111
0.5 m vertical bin size. The ADCP was placed 5 m north of the pier (y = 5) to reduce pier-piling112
flow disturbance, yet still be consistent with pier-mounted thermistors. Velocity data was rotated113
into the x and y coordinate system based on compass headings taken at deployment. Data above114
6
the surface wave trough or in regions with low acoustic return amplitude were removed (≈ 1.5 m115
below the tidal sea surface).116
Meteorological and tide measurements were made by NOAA station 9410230 at S8. Air tem-117
perature and wind speed (two-minute average) were sampled at z ≈ 18 m at six-minute intervals.118
Surface (tidal) elevation η is calculated from an average of 181 one-second samples reported ev-119
ery six minutes. Hourly significant wave height (Hs) and peak period (Tp) were observed by the120
Coastal Data Information Program (CDIP) station 073 (pressure sensor) mounted to a pier piling121
at S8. When observations were not available, (29 September to 21 October) a realtime spectral122
refraction wave model with very high skill initialized from offshore buoys was used (O’Reilly and123
Guza 1991, 1998). Bathymetry was measured from the pier deck using lead-line soundings every124
10 m on 26 September, 10 October, and 24 October. The bathymetry was then interpolated in x125
and the time dependent bathymetry was used when appropriate. The average slope between S8126
and S4 was s = 0.033 with bathymetry variation less than 0.3 m at any location (slope changes127
< 4%) during the experiment. The outer extent of surface wave breaking (surfzone location xsz)128
was estimated by shoaling surface wave conditions observed at S8 over the measured bathymetry129
with the observed tides following Sinnett and Feddersen (2016).130 FIG. 1
c. Background Conditions
The experiment site has a mixed barotropic tide with amplitudes over the 30 day experiment131
period varying between 0.17 m and 1.05 m on a spring-neap cycle, dominated by the lunar semi-132
diurnal (M2) and lunar diurnal (K1) tidal constituents (Figure 2a). Wind conditions were generally133
calm, with a light afternoon sea breeze rarely peaking above 5 m s−1 (Figure 2b). Pier-end (S8)134
significant wave height Hs varied from 0.3 to 1.5 m over the entire experimental period. Surface135
wave events near days 2, 19, 22 and 27 caused significant wave height to peak well above the mean136
Hs ≈ 0.7 m. Air temperature followed a strong diurnal heating and cooling cycle in the first 10137
days of the record, with diurnal variations ≈ 7◦C (Figure 2d, black). The diurnal air temperature138
variation decreased to ≈ 4◦C after day 10, with a subtle cooling trend seen throughout the record.139
Surface water temperature (from the S18 surface thermistor) varied weakly, but contained a diur-140
nal heating and cooling signature (Figure 2d, red). Diurnal air and near-surface (z > −3.5 m)141
water temperature variability was coherent with an ≈ 4 h lag. Diurnal air and water temperature142
7
variability below z = −3.5 m was incoherent.143FIG. 2
3. Month-long Nonlinear Internal Wave observations from 18 m depth to shore
Temperature observations from 18 m depth to near the shoreline at five cross-shore locations144
(Figure 3a-e) highlight the rich and diverse nonlinear internal wave (NLIW) field present during145
the 30 day observational period. The first 10 days were strongly stratified at S18 (x = −657 m,146
h = 18 m) with a large barotropic tide (Figure 3e). During this time, winds were typically calm,147
with a few events where uw > 4 m s−1. Significant wave height averaged 0.8 m during the first four148
days, then decreased to less than 0.5 m and remained small until day 19. An energetic NLIW field149
is present at S18 during the first 10 days, with large vertical isotherm excursions (20◦C isotherm150
displacement is±6 m, Figure 3e). At this time, cross-shore coherent cooling events at semi-diurnal151
and faster time scales are regularly observed in the otherwise warm shallow water and can reduce152
the S4 temperature by 2.25◦C in only 10 min. Clear examples of NLIW cross-shore excursions153
occur near days 1 and 8 (Figure 3). The 10 day period containing strong internal wave activity154
typical of early fall conditions at this site is denoted “period I”.155
The early to late fall transition between period I and the less active remaining 20 days (termed156
“period II”) is characterized by cooling surface water (z > −7 m) and warming at depth (Fig-157
ure 3e). The transition occurs just after day 10, when warm water extended all the way to the158
bottom at S18 with very weak stratification. At this time, surface gravity waves were weak159
(Hs < 0.5 m), sustained winds were moderate (uw < 5 m s−1) with spring barotropic tides (Fig-160
ure 2). At S18, vertical excursions of the 20◦C isotherm were smaller during period II, usually less161
than±3 m. Near surface diurnal temperature oscillations due to solar heating were±0.2◦C at S18,162
increasing to ±0.5◦C at S2, and were coherently observed at all cross-shore locations. Though163
the water was less stratified and isotherm excursions were smaller at S18, NLIW events were still164
observed during period II (notable in Figure 3 near days 12 and 27). Cross-shore coherent NLIW165
events are described in greater detail by zooming in to a time of energetic NLIW activity (identified166
by the black bar, Figure 3e) during period I.167
The 3.5-day energetic NLIW period (Figure 4) had strong stratification and barotropic tides168
but weak winds and surface waves. Temperature variability at all cross-shore locations is strong,169
containing oscillations at periods near M2, the M4 harmonic (6.2 hour period) and higher frequen-170
8
cies. At S18, the T = 20 ◦C isotherm excursions are over 10 m and NLIW events are coherently171
observed all the way to S2. Mid-water column temperature fluctuations are as high as 4.8 ◦C in172
10 min at S18, but decrease onshore with a maximum temperature fluctuation of 2.0 ◦C at S2.173
The first 1.5 days (days 7 - 8.5) are strongly stratified with very warm surface water and a sharp174
thermocline. Near-bottom M4 temperature variability is present at all cross-shore locations. Strat-175
ification is weaker during days 8.5 to 10.5 (Figure 4), yet temperature variability at all locations176
is still observed primarily at M2 periods, although M4 variability is also present particularly at S8177
(Figure 4d).178
High frequency temperature variability (periods shorter than 3 hours) is superimposed on top179
of the M2 and M4 variability. The cross shore evolution of high frequency variability is visible in180
a 9 hour zoom (Figure 5) of the time period indicated by the black bar in Figure 4e. At S18, the181
T = 20 ◦C isotherm gradually rises during the first hour with little high frequency temperature182
variability. Then, at hour 1.5, the 20◦C isotherm plunges roughly 10 m, beginning a series of183
oscillations at ≈ 10 min period that persist over the next six hours (Figure 5e). The first two184
10 m oscillations of the 20 ◦C isotherm near hour 2 are qualitatively similar to a soliton. These185
superimposed high frequency oscillations at S8 are present at S6, but decay in shallower water,186
though some aspects of the high frequency NLIW field are coherent upslope. For example, near187
hour 7 at S18 (the peak of the M4 period event) a pulse of cold water elevates S8 isotherms (lasting188
roughly 10 min). The cold pulse arrives at progressively later times upslope, until it is finally189
observed at S2 just before hour 8 (Figure 5a-d). The pulse propagated onshore at unknown angle190
and affected temperature in water depths as shallow as 2 m, causing temperature there to drop191
0.7 ◦C in five minutes.192
Although occasional pulses of cold water can be tracked coherently upslope, very little high193
frequency energy is coherent between S18 and S8. A further zoom of 1.5 hours shows temperature194
with the 18.1◦C, 19.6◦C and 21.1 ◦C isotherms highlighted to emphasize the lack of cross-shore195
coherence at high frequency (Figure 6). At S18, isotherms are displaced±0.8 m at≈ 10 min period196
(Figure 6e). At S8, isotherm displacements are ±0.4, reduced from S18 (Figure 6d). However,197
isotherm displacements are not coherent between S18 and S8 with near zero correlation for all lags198
during this active 90 minute period. A transition to temperature variability on longer time scales199
and an upslope isotherm tilt is also evident in Figure 6, as the 90 min average 21.1 ◦C isotherm200
depth is approximately 2 m higher at S4 than at S18. At S8, both the 19.6◦C and 21.1◦C isotherms201
9
contain variability at ≈ 10 min periods, particularly in the last 40 min (Figure 6d). Upslope at S6,202
variability at ≈ 10 min period is evident near the bottom (19.6◦C isotherm), but mid-water depths203
(z ≈ −2 m) contain variability at longer time scales (Figure 6c). The resulting variability of the204
21.1 ◦C isotherm at S4 is predominantly at 20 min periods, with less high frequency variability205
than in deeper waters (compare Figures 6b and d).206FIG. 3
FIG. 4
FIG. 5
FIG. 6
The temperature observations (Figures 3–6) with large amplitude isotherm displacements rela-207
tive to water depth, rapid temperature gradients, and M4 harmonics demonstrate the presence of a208
rich NLIW field. Spectral properties of the NLIW field are explored focusing on the mid-water col-209
umn temperature time-series from z = −9 m at S18, and z = −4 m at S6 (Figure 7a,b). During the210
active period I (first 10 days), large (4–5 ◦C) temperature oscillations are present at both locations.211
The less active period II (days 10–30) still has NLIW activity, although the magnitude (1–2 ◦C) is212
much reduced. To contrast these two periods and locations, a seven day time-period is selected to213
represent period I (red and blue, Figure 7a,b) and period II (purple and green, Figure 7a,b). Tem-214
perature spectra of these four time series were calculated with the multi-taper method (Thomson215
1982) using the JLab toolbox (Lilly 2016). The 95% confidence interval (gray shading) is found216
from the χ2k distribution with the 14 degrees of freedom given by the orthogonal Slepian tapers.217
Period I temperature spectra at S18 (red, Figure 7c) has peaks at M2 and M4 frequencies, and218
decays with frequency up to a broad secondary peak at 6–10 cph (7–10 min period), correspond-219
ing to high frequency variability at S18 in Figure 5. Farther upslope, the S6 temperature spectra220
does not have a clearly defined M2 peak and the S6 M2-band variance is 21% that of S18 (blue,221
Figure 7c). However, at S6 a clear and significant M4 peak is present that has essentially the same222
variance as at S18. The M4 peak indicates either M2 to M4 nonlinear energy transfers between223
S18 and S6 or M4 generation in deeper water likely in the offshore canyons (Alberty et al. 2017).224
An additional small S6 spectral peak is evident near M8 (harmonic of M4, 0.33 cph, 3-hour pe-225
riod) with nearly twice as much variance as at S18, suggesting nonlinear energy transfers from226
M4 to M8 between S18 and S6. The S6 spectra falls off similarly to S18, but does not have the227
broad high-frequency spectral peak, consistent with the reduced onshore high frequency variability228
observed in Figure 5. Between S18 and S6, the M2 variability was coherent (≈ 0.78, above the229
99% confidence level of 0.39) with 20 min phase lag, suggesting propagation albeit at an unknown230
angle. Between S18 and S6, M4 variability also was coherent (0.8), as was the M6 and M8 vari-231
ability (at 0.6 and 0.5), albeit more weakly than M2 and M4. However, variability above 1 cph was232
10
not coherent between S18 and S6, similar to the zero lagged correlation of the 19.6 ◦C isotherm233
elevation between S18 and S8 in Figure 6.234
The period II temperature spectra (Figure 7d) were reduced at all frequencies relative to pe-235
riod I. Spectral peaks at M2, M4 and higher harmonics are still present at S18 in period II (purple,236
Figure 7d), but spectral levels are reduced by a factor of 10 at these frequencies. Furthermore,237
the period I S18 elevated variability at > 1 cph is absent during period II. The period II diurnal238
temperature variability at S6 (green, Figure 7d) is slightly elevated relative to S18, consistent with239
increased solar heating and longwave cooling at the shallower S6 depth, and is coherent between240
S6 and S18. At M2 and higher frequencies, the S6 period II spectra has no significant peaks, was241
incoherent with S18, and had a total variance half that of S18.242 FIG. 7
4. Coherent Upslope Evolution of Individual Nonlinear Internal Wave Events
The rich nonlinear internal wave field observed from S18 (18 m water depth) to near-shoreline243
S2 (Figures 3–6) contains cold pulses that propagate coherently upslope (e.g., Figure 5). The244
runup characteristics of these cold pulses ultimately determine the NLIW cross-shore extent and245
impact to the nearshore region, through for example, larval transport (e.g., Pineda 1999). Here, the246
coherent upslope evolution of individual NLIW events is explored in analogy to laboratory studies247
(e.g., Wallace and Wilkinson 1988; Sutherland et al. 2013a). Events are loosely defined as a248
significant and rapid reduction and recovery of temperature near the pier-end over a few hours, and249
are defined quantitatively later. Detailed analysis is restricted to between S8 and just seaward of250
the surfzone at S2 where high thermistor density (Figure 1b) allowed for coherent upslope tracking251
of NLIW events. The time period is narrowed to 9–28 October (experiment days 11 to 30) when252
the S8 (pier-end) alongshore array (red dots, Figure 1a) was concurrently deployed. A single253
NLIW event is examined first to introduce important event parameters (e.g., event front speed cf).254
Analysis is then broadened to multiple events at S8 and farther onshore, leading to scaling the255
upslope NLIW event evolution.256
a. Example NLIW Event Characteristics
An example 5-h long NLIW event occurred on 26 October (red square, experiment day 28257
bottom of Figure 3e) with large surface wave (Hs ≈ 1.2 m) and moderate wind (uw ≈ 3.5 m s−1)258
11
conditions (Figure 2). This event is selected to highlight NLIW runup properties. Prior to the event259
front arrival at hour 1, S8 temperature was essentially constant near 21.2 ◦C and weakly stratified,260
dT/dz < 0.01◦C m−1 (Figure 8a). After the event front arrival, S8 near-bottom temperature fell261
rapidly (≈ 1◦C in 1 min) and the water column stratified (dT/dz > 0.25◦C m−1). Temperature262
fluctuations of O(0.2 ◦C) at 1–30 min time scales are observed throughout the water column.263
Near bottom temperature began to increase after hour 2 (≈ 0.025◦C min−1), while temperature264
in the upper 3 m cooled slightly. The event concluded between hour 2.75 and 4 as the near-bed265
warmed and the near-surface cooled until the water column was again weakly stratified near hour266
4. During this event, the coldest (bottom) S8 temperature was near 19.4◦C (Figure 8a), but the267
coldest (bottom) S18 temperature before the event was near 20.7 ◦C (not shown). Thus, the coldest268
water at S8 during the event originated from a location deeper than 18 m and traveled horizontally269
upslope more than 384 m to reach S8.270
Velocity associated with the upslope NLIW example event was observed by the ADCP at S8.271
Cross-shore (U ) and alongshore (V ) velocities are decomposed into (e.g., for U ) depth-averaged272
(barotropic) velocity U and is the depth-varying (baroclinic) velocity U ′, so that273
U(z, t) = U(t) + U ′(z, t), (1)
and the vertical average of U ′ is zero. The barotropic component is assumed to be irrotational274
in the experiment domain and slowly varying in time. This decomposition is partially aliased275
by the removal of velocity bins near the tidal sea-surface. Prior to the event onset, barotropic276
velocity magnitude was weak (< 0.05 m s−1) as was baroclinic velocity magnitude (almost always277
U ′ < 0.02 m s−1). However, after the abrupt temperature drop at hour 1 signaling the event arrival278
(Figure 8a), baroclinic velocity U ′ increased, with onshore velocity at depth exceeding 0.06 m s−1279
and offshore velocity near the surface (Figure 8b). The baroclinic current was predominantly in the280
cross-shore (U ′) direction, with a weak alongshore (V ′) component (Figure 8c). Near hour 2, the281
direction of U ′ reverses, and thereafter the near-bed flow is offshore and the near-surface flow is282
onshore, coincident with the bottom temperature recovery (Figure 8a and b). During the recovery283
(2.75 h to 4 h), the transition depth between near-surface cooling and near-bed warming is z ≈284
−3 m (Figure 8a), which is also near the U ′ zero crossing depth.285FIG. 8
The near-bottom upslope event temperature evolution (Figure 9) is key to determining event286
12
parameters. Prior to the event start at hour 1, the region from S8 to the shoreline was essentially287
homogeneous in T (Figure 9a). The pier-end near-bottom T was also largely uniform in the along-288
shore (Figure 9b). At each cross-shore and alongshore location, the event arrival is clearly visible289
as a steep drop in T (the event front) that propagates coherently in the alongshore and cross-shore.290
This T drop then slowly reaches a minimum before beginning to recover near hour 2. At S8, the291
overall temperature drop of about 2 ◦C was fairly uniform spanning 400 m in the alongshore (Fig-292
ure 9b). In the cross-shore, the temperature drop is coherent and reduced onshore to x = −137 m293
(black curve in Figure 9a). Onshore of x = −137 m, neither a sharp nor coherent temperature294
drop is observed (dashed curves in Figure 9a). By hour 4 the event is over and temperature has295
largely recovered to the pre-event value, albeit with occasional remnants of colder water upslope296
(e.g., yellow, magenta, and blue curves at hour 4.2 in Figure 9a).297
The example event’s upslope near-bottom temperature evolution (Figure 9) highlights key298
quantifiable event-front characteristics. The sharp temperature drop indicates the event front arrival299
time tf1, defined as when the 3 minute averaged temperature change dT/dt < −0.033 ◦C min−1300
(gray dots Figure 9). Onshore (+x) NLIW event front propagation is evident from the progression301
of tf1 at different cross-shore locations (Figure 9a). Similarly, the alongshore event front arrivals302
(Figure 9b) indicates a south to north (+y) propagation component, consistent with the observed303
baroclinic velocities (positive near-bottom U ′ and weakly positive V ′ at event start, Figure 8b and304
c). At a particular cross-shore location, the event front passes at a time tf2 defined as where the305
3-minute averaged dT/dt > −0.0067 ◦C min−1 (open circles in Figure 9b), corresponding to the306
“nose” of the front (e.g., Arthur and Fringer 2014). Time tf2 does not necessarily correspond to307
the coldest observed event temperature, but rather to when the sharp event front (rapid T drop) has308
passed the sensor. The temperature drop ∆T associated with the event front is then defined as309
∆T = T (tf1)− T (tf2). (2)
At S8, an event is defined to occur when ∆T > 0.3 ◦C over 9 minutes, and is defined to propagate310
farther upslope (onshore) as long as coherent ∆T > 0.15 ◦C. For this example event, S8 ∆T =311
1.26 ◦C, but as the event propagated onshore the magnitude of the coherent event-front decreased312
to ∆T = 0.34 ◦C at x = −137 m (black curve in Figure 9a). As onshore-coherent ∆T > 0.15 ◦C313
was not observed onshore of x = −137 m (dotted lines, Figure 9a), the NLIW event runup cross-314
13
shore extent is defined as xR = −137 m.315FIG. 9
Event front speed cf and angle θ are calculated using the cross-shore and alongshore event316
arrival time and the observed barotropic velocity. The change in event front alongshore arrival317
position versus time dyf/dt at S8 is estimated from the slope of the linear fit of alongshore front318
location yf versus arrival time when ∆T > 0.3 ◦C at three or more alongshore locations. Similarly,319
the S8 cross-shore change in position versus time, dxf/dt, is found from the arrival time difference320
between bottom sensors at S8 (x = −273 m) and x = −246 m. At S8, the event propagation angle321
estimated as322
θ = arctan
(dxf/dt
dyf/dt
), (3)
which is independent of the barotropic current. Although barotropic motions do not affect θ, they323
do affect cf (in this case by approximately 30%). Accounting for barotropic motions, the event324
front speed is,325
cf = dxf/dt cos θ − U cos θ − V sin θ. (4)
For this example event, S8 front speed is cf = 0.06 m s−1 and incidence angle is θ = 11.2◦.326
Observations of ∆T and cf can be related to idealized two layer laboratory and numerical stud-327
ies of NLIW runup with defined layer height (hi) and layer density (ρ) difference ∆ρ (e.g., Suther-328
land et al. 2013a; Arthur and Fringer 2014). Here, the continuously stratified ocean is related to329
an idealized equivalent two-layer fluid with layer density difference ∆ρ = α∆T (where α is the330
coefficient of thermal expansion) and the equivalent two-layer interface height set by equating the331
change in vertically integrated baroclinic potential energy PE associated with the continuously-332
stratified event front to the potential energy change of a two-layer system with ∆ρ and layer depth333
hi. The instantaneous vertically integrated baroclinic potential energy is334
PE(t) =
∫ h
0
(ρ(z′, t)− ρ0)gz′ dz′, (5)
where ρ0 is a constant reference density, g is gravity, the tidally varying water depth is h = h+ η,335
and z′ is a vertical coordinate referenced to the bed. The change in PE associated with the event336
front is337
∆PE = PE(tf2)− PE(tf1). (6)
14
For a two layer system, the equivalent vertically integrated change in potential energy is338
∆PE = ∆ρgz2IW2, (7)
where zIW approximates hi and is the equivalent two-layer interface height above the bed for the339
stratified event. Rearranging (7) gives zIW as a function of ∆PE and ∆ρ,340
zIW =
(2∆PE
∆ρg
) 12
. (8)
The two-layer equivalent interface height is then found from (8) using the change in PE due to the341
event front in the continuously stratified ocean (6) and ∆ρ = α∆T . For the example event, the342
S8 interface height is zIW = 2.48 m, consistent with the large temperature drop in the bottom 2 m343
and weaker drop at shallower depths. Estimation of zIW depends on adequate vertical temperature344
resolution, restricting zIW calculation to cross-shore locations with at least four thermistors in the345
vertical (Figure 1b). Having defined key parameters associated with the NLIW event front (∆T ,346
cf , zIW, and xR), the observed range and upslope (onshore) evolution of individual events are347
investigated next.348
b. Individual NLIW Event Characteristics
Isolated individual NLIW events are defined when ∆T > 0.3 ◦C at S8 and when no other cold349
pulses occur for ±3 h. This second criteria removes overlapping events (discussed later). With350
this criteria, a total of 14 individual NLIW events with 0.3 ◦C < ∆T < 1.7 ◦C were isolated351
at the pier-end (S8) between 9 and 30 October. Two events had ∆T > 1.5 ◦C, six events had352
1.0 ◦C < ∆T < 1.5 ◦C, three events had 0.5 ◦C < ∆T < 1.0 ◦C and three events had 0.3 ◦C <353
∆T < 0.5 ◦C (left column, Figure 10). All fourteen events were observed coherently propagating354
upslope with reduced ∆T so that 54 m farther onshore (at S6) only 10 events were observed, all355
with ∆T > 0.3 ◦C (second column, Figure 10). Despite the onshore reduction in ∆T , six events356
(associated with the largest ∆T at S8) were still observed at S4 (right column, Figure 10). Farther357
upslope ∆T continued to decrease, but at S2 no coherent ∆T > 0.15 ◦C was observed.358 Table 1
FIG. 10
FIG. 11
At S8, the fourteen events propagated upslope with speeds 1.4 cm s−1 < cf < 7.4 cm s−1359
15
(radial magnitude, Figure 11a). These NLIW events also propagated with a range of incidence360
angles (−5◦ < θ < 23◦, Figure 11a) potentially due to the many internal wave generation lo-361
cations nearby. The slight positive mean θ ≈ 5◦, indicates a south to north NLIW propagation362
tendency, suggesting a possible dominant source near the southern La Jolla canyon (Figure 1a)363
through mechanisms described in Alberty et al. (2017). During the example event (Figure 8), the364
inferred large upslope transport of cold water suggests the event is strongly nonlinear. At S8, event365
nonlinearity is quantified with the ratio of near-bed baroclinic velocity magnitude |U ′b| to front366
speed cf , (|U ′b|/cf), where |U ′b| is averaged for 10 minutes between 0.9 m and 1.9 m above the367
bottom after event onset. For linear internal waves |U ′b|/cf � 1. At S8, the example event detailed368
in section 4a has |U ′b|/cf = 0.7 indicating strong nonlinearity. The fourteen isolated NLIW events369
had |U ′b|/cf between 0.3 and 2.0 with a mean value of 0.7.370
For these 14 NLIW events, the observed S8 cf is compared to two-layer gravity current speeds371
(e.g., Sutherland et al. 2013a; Marleau et al. 2014). A flat-bottom two-layer fluid with interface372
height hi in depth h and upper and lower layer densities ρ0 and ρ0 + ∆ρ, respectively has reduced373
gravity g′ = g(∆ρ)/ρ0. The corresponding gravity current Froude number is (Shin et al. 2004)374
F0 =√δ(1− δ), (9)
where δ = hi/h. The speed of the gravity current front is375
cgc = F0(g′h)1/2 = [(1− δ)g′hi]1/2 ≈
[(1− zIW
h
)g′zIW
]1/2, (10)
where, for a NLIW event, zIW is used for the lower layer height, h is the tidally adjusted water376
depth, and ∆ρ is given by α∆T .377
For these 14 NLIW events, the observed S8 upslope event front speed cf is reasonably well378
predicted by the two two-layer gravity current speed cgc (10) with root mean squared (rms) error of379
0.016 m s−1, squared correlation R2 = 0.44 and best-fit slope of 1.15 (Figure 11b). Although cgc380
is biased high relative to cf , this bias could be accounted for by adjusting the F0 definition (9). The381
reasonably good relationship between cf and cgc indicates that these continuously stratified NLIW382
events (e.g., Figure 8) are reasonably well scaled as a two-layer gravity current (e.g., Shin et al.383
16
2004), even though the events propagate at non-zero incidence angles, the bottom slopes weakly,384
and the event may be propagating into inhomogeneous (stratified) water.385
Not all NLIW occurrences are as simple as the example event (Figures 8 and 9) with its clearly386
defined parameters (e.g., tf1, ∆T and zIW). NLIW runup can be complicated, with overlapping387
cold pulses containing differing cf and θ (Figure 12). A near simultaneous initial cold pulse arrival388
at S8 alongshore locations (gray dots, Figure 12b) indicates a NLIW pulse with θ = 2.2◦ which389
propagates onshore (subsequent gray dots, Figure 12a). A second cold pulse is observed roughly390
1.2 h later at S8 cross-shore and alongshore stations (gray crosses, Figure 12a,b) superimposed391
on the first pulse. The second pulse was observed within the surfzone (at this time surfzone wave392
breaking begins at h = 2 m) and propagated south to north at very high angle and with ∆T393
decreasing in the alongshore (∆T = 1.08 ◦C at y = −200 m but ∆T = 0.17 ◦C at y = 200 m).394
Though the onset of the second pulse is cross-shore coherent, the temperature drop was observed395
nearly simultaneously at x = −55 m, y = 0 m and x = −273 m, y = −200 m (gray crosses in396
Figure 12a) before giving a sense of rapid offshore propagation as temperature recovered between397
hours 3 and 4. Although speculative, this pulse may have swept cold water into the surfzone398
at y < 0 m which was then reflected offshore. The second cold pulse propagated through the399
previously conditioned stratification and current. The criteria requiring isolated events removes400
such complicated overlapping cases (Figure 12) where event parameters are difficult to isolate.401 FIG. 12
c. Upslope NLIW Evolution
At S8, 14 isolated NLIW events have ∆T > 0.3 ◦C with no overlapping cold pulses. To402
compare these NLIW events with idealized two-layer laboratory and modeling studies, the event403
propagation angle is restricted to be nearly shore-normal (|θ| < 15◦, eliminating 2 events). A404
further restriction requires the wavefront to be roughly alongshore uniform, where ∆T is within405
0.5 ◦C at four or more alongshore locations (eliminating 8 more events). Background barotropic406
velocity was low during each event (|U | < 1.3 cm s−1). These restrictions result in four remaining407
events (colored markers in Figures 3e) denoted events A–D that are near normally incident and408
propagate into homogeneous conditions. Thus, these representative events are more consistent with409
a two-layer assumption than the total 14 isolated NLIW events at S8. To relate to laboratory two-410
layer internal runup and gravity current studies, these four events are further required to propagate411
17
into homogeneous T at and onshore of an initial cross-shore location x0. Events B and C had412
homogeneous T at and onshore of S8 prior to the event, and thus the initial cross-shore location413
x0 = xS8 = −273 m. Events A and D had some vertical stratification at S8 prior to the event start.414
However, just 27 m onshore T was vertically and onshore homogeneous, so x0 = −246 m for415
events A and D to insure pre-event homogeneous conditions. Note, the example event in Figures 8416
and 9 is event C. The upslope (onshore) evolution of events A–D (colored dots, Figures 3e) are417
explored in detail to highlight NLIW runup characteristics.418
The onshore propagation distance from x0 is ∆x = x− x0 and elapsed time from front arrival419
at x0 is ∆t = t − tf1(x0). Event A–D fronts propagated onshore and slowed down until reaching420
their eventual total runup distance ∆xR = xR− x0 (dots, Figure 13). The upslope transit time was421
between 42–64 min with ∆xR varying between 100-137 m. The two-layer gravity current speed422
(10) can be expressed as dx/dt and the differential equation can be solved for change in cross-423
shore position x assuming a constant Froude number (9) and a constantly sloping bottom. The424
solution is a quadratic relationship between ∆x and ∆t, which has been confirmed in laboratory425
observations of upslope runup of broken internal solitary waves (e.g., Sutherland et al. 2013a). So,426
for each event, the front position ∆x and elapsed time ∆t are fit to a quadratic427
∆x = −df2
(∆t)2 + cf0∆t, (11)
with best-fit NLIW front speed at x0 (cf0) and constant onshore NLIW deceleration df . The fits428
all have high skill (> 0.98, lines Figure 13). The NLIW deceleration df varies between 5.9 ×429
10−6 m s−2 and 2.4 × 10−5 m s−2 (Table 1). Event D had the highest cf0 (7.84 cm s−1), but also430
had the largest deceleration, limiting the runup distance from x0, ∆xR = xR − x0 to 128 m (blue,431
Figure 13). Event A decelerated less than event D, but had a smaller cf0 (5.86 cm s−1) resulting in a432
similar ∆xR. Event C had high cf0 (6.31 cm s−1) and also less deceleration than event D, allowing433
∆xR = 137 m (blue, Figure 13). Event B had the lowest cf0 (4.51 cm s−1) and deceleration, with434
observed ∆xR = 100 m.435
None of these 4 events were observed to propagate coherently into the surfzone (tick marks436
along ordinate axis in Figure 13). During event A, the significant wave height was very small437
(Hs = 0.32 m, Table 1) and the surfzone was narrow. The event runup halted 56 m offshore of the438
estimated surfzone boundary (green tick mark, Figure 13). Event B with Hs = 0.69 m also halted439
18
more than 35 m from the estimated surfzone boundary. Events C and D had Hs > 1 m (Table 1)440
and with the wider surfzone, the total runup distance ∆xR was observed to within 10 m of the441
estimated surfzone boundary. Events C and D both caused thermistors inside the surfzone to cool442
≈ 0.1◦C in six minutes, though this cooling was insufficient to coherently track further onshore as443
an event ∆T .444 FIG. 13
NLIW event front temperature drop ∆T and equivalent two-layer height zIW generally de-445
creases farther upslope (Figure 14a,b). For events A–D, ∆T at x0 (∆T0) varied between 0.96◦C446
and 1.62◦C (Figure 14a), a factor of 1.7. Upslope ∆T decreases differently amongst events, either447
rapidly (event B, black in Figure 14a) or slowly (event A, green). For events B–D, ∆T < 0.41◦C448
at xR. In contrast, the slowly decaying Event A had ∆T = 0.96◦C at xR. Yet, for event A, no449
significant temperature drop was present 20 m onshore of xR. The zIW at x0 (zIW0) varied between450
2.1 m and 2.5 m (Figure 14b), a much smaller range than for ∆T0. Upslope from x0, zIW reduced451
linearly in a relatively similar manner for all events, in contrast to ∆T . At xR, zIW ranges between452
1–1.5 m, still significant compared to zIW0 . For events A–D, the upslope reduction in dimensional453
cf , zIW, and ∆T and the constant deceleration is qualitatively consistent with laboratory observa-454
tions of internal runup of broken internal solitary waves (Wallace and Wilkinson 1988; Helfrich455
1992; Sutherland et al. 2013a).456 FIG. 14
d. Scaling upslope NLIW evolution
The stratified NLIW events A–D have baroclinic velocity structure and temperature structure457
that is qualitatively consistent with an upslope two-layer gravity current (e.g., Figures 8 and 9).458
Events A–D have |U ′b|/cf that is O(1) (Table 1), also consistent with a gravity current. NLIW459
events A–D have constant deceleration (Figure 13) and their density anomaly (∆T ) and height460
(zIW) are reduced onshore consistent with upslope two-layer gravity currents (Marleau et al. 2014).461
Here, the NLIW event parameters (cf0, df , ∆xR, ∆T and zIW) are scaled and compared to gravity462
current scalings.463
The non-dimensional ∆T/∆T0 and zIW/zIW0 dependence upon non-dimensional runup dis-464
tance ∆x/∆xR is examined in analogy with laboratory studies of the upslope propagation of bro-465
ken internal solitary waves (e.g., Wallace and Wilkinson 1988; Helfrich 1992). Upslope event466
front temperature drop ∆T varied substantially (Figure 14a). However, the normalized ∆T/∆T0467
19
largely collapse as a linearly decaying function of ∆x/∆xR (Figure 15a) with best-fit slope −0.61468
and squared correlation R2 = 0.58. For events A–D, the dimensional zIW upslope dependence469
was not as scattered as for ∆T (Figure 14b). Similarly, the non-dimensional zIW/zIW0 collapse470
very well as a linearly decaying function of ∆x/∆xR (Figure 15b) with best fit slope of −0.56471
and R2 = 0.89, again qualitatively consistent with laboratory studies (Wallace and Wilkinson472
1988; Helfrich 1992; Marleau et al. 2014). The collapse of non-dimensional ∆T and zIW sug-473
gests the dynamics of the continuously stratified internal runup into homogeneous water is largely474
self-similar.475FIG. 15
Laboratory two-layer upslope gravity current deceleration is constant and depends upon g′,476
constant bed slope s, and the ratio hi/h where hi represents gravity current height and h is the total477
water depth (Marleau et al. 2014). Adapting this scaling for continuously stratified NLIW event478
deceleration in a continuously stratified ocean results in479
dgc =1
2g′0s
zIW0
h0
(1− zIW0
h0
), (12)
where g′0 zIW0 , h0 are all at evaluated at x0. Here, the averaged bedslope from S8 to S4 is used480
(s = 0.033). The events A–D best-fit front speed at x0 (cf0) and the constant deceleration df (11)481
are compared to the two-layer gravity current scalings for speed cgc (10) and upslope deceleration482
dgc (12). The events A–D cf0 varies from 0.04–0.08 m s−1 and scale well with the two-layer gravity483
current speed cgc estimated at x0 (Figure 16a) with rms error of 0.013 m s−1 and best-fit slope of484
1.17. Events A–D df scales very well with dgc over a large range (factor 2.5) of deceleration485
(Figure 16b) with rms error of 9 × 10−7 m s−2 and best-fit slope of 0.84. The factor 2.5 variation486
in df is largely due to the ∆T0 variations impacting g′0. The small error of the cf0 and df scalings487
indicates that for normally-incident NLIW events propagating upslope into a homogeneous fluid,488
the two-layer gravity current scalings are appropriate.489FIG. 16
Because both non-dimensional ∆T and zIW are largely self-similar with ∆x/∆xR, the upslope490
evolution of an offshore (at x0) observed NLIW runup event can be estimated knowing the total491
runup distance ∆xR. At the onshore runup limit (∆xR), the event front speed cf = dxf/dt = 0.492
20
With the quadratic front evolution, setting the derivative of (11) to zero and substituting yields,493
∆xR =1
2
c2f0df. (13)
The ∆xR estimated from (13) with cf0 and df reproduces the observed ∆xR defined in section494
4a well (Figure 17a), with rms error of 13 m (less than the 18 m cross-shore resolution of the495
thermistor array, Figure 1b) and a best-fit slope of 0.92 that is near-unity. This demonstrates that496
with knowledge of offshore event front parameters (cf0, df , ∆T0, and zIW0) the upslope distribution497
of these parameters can be well estimated.498
However, event front observations from at least three locations along the axis of propagation499
are required to estimate cf0 and df and thus ∆xR via (13). The gravity current scalings for cgc (10)500
and dgc (12) only require vertical temperature coverage at a single location, and can be used to501
estimate502
∆xR =1
2
c2gcdgc
. (14)
The gravity current scaling based ∆xR (14) significantly overpredicts the observed ∆xR (Fig-503
ure 17b), with rms error of 102 m and best-fit slope of 0.55. Relatively small error in cgc and504
dgc (Figure 16) cascade through (14) to generate these large errors. For example, with the best-fit505
slopes for cgc (0.85) and dgc (1.16) and the scaling (14), the predicted best-fit slope is 0.62, which506
is near the observed best-fit slope of 0.55 (Figure 17b). This demonstrates that predictions of total507
runup distance ∆xR are very sensitive to small errors in runup speed and deceleration.508 FIG. 17
5. Discussion
a. Internal runup and comparison to laboratory and numerical studies
For the 14 events at S8, the event front speed is consistent with a internal gravity current509
(Figure 11b) and the ratio |U ′b|/cf is generally O(1), suggesting that these events are internal bores510
(e.g., Pineda 1994; Moum et al. 2007; Walter et al. 2012; Nam and Send 2011). For the four511
isolated (A–D) events, the ratio |U ′b|/cf is also O(1) (Table 1) and the upslope event evolution512
(speed and constant deceleration) is consistent both with upslope gravity currents (Marleau et al.513
21
2014) and internal runup of laboratory broken internal solitary waves (Helfrich 1992; Sutherland514
et al. 2013a). This all indicates that the internal wave breaking begins well offshore of S8 and515
that S8 and onshore locations are located within the internal swashzone where events propagate as516
bores, in analogy with the swashzone of a beach (e.g., Fiedler et al. 2015).517
The evolution of these continuously stratified dense bores propagating upslope into homoge-518
neous fluid are consistent with two-layer upslope gravity current scalings (e.g., Figure 16) using519
near-bed estimated ∆T and an interface height zIW assuming equivalent potential energy (5–8).520
From flat-bottom numerical simulations, a continuously stratified interface between upper and521
lower layers results in a weak decrease, relative to two-layer theory (10), of the gravity current522
speed cgc (White and Helfrich 2014). With stratification similar to that observed in events A-D, the523
continuously stratified model suggests the cgc found from (10) should be reduced by ≈ 5% (White524
and Helfrich 2014). This indicates that applying the two-layer approximation to these continuously525
stratified internal runup events is appropriate and also may account for some of the cgc bias error526
(Figure 11b).527
The constant upslope two-layer gravity current deceleration can be derived by assuming a weak528
slope such that at all locations the front speed follows the Shin et al. (2004) gravity current speed529
(10) with constant Froude number F0 that depends on δ = zIW/h (Sutherland et al. 2013b; Marleau530
et al. 2014). With h(x) = −sx, the quadratic in time dependence of the front position is derived.531
This requires that zIW/h be constant upslope. For the four isolated events A–D, this assumption532
is valid (Figure 18). For all events, zIW/h ≈ 0.3 at ∆x/∆xR = 0 and doesn’t vary by more than533
±0.1 all the way to ∆x/∆xR = 1 (Figure 18). However, unlike two layer systems, here ∆T is not534
constant in the upslope direction resulting in g′ variations.535FIG. 18
The upslope linear reduction in zIW/zIW0 with ∆x/∆xR is qualitatively consistent with labo-536
ratory internal solitary wave runup for all incident wave amplitudes (Wallace and Wilkinson 1988;537
Helfrich 1992), and with laboratory observations of a two-layer upslope gravity current on shallow538
slopes (Marleau et al. 2014). This supports the assumption that such internal bores are self-similar539
(Wallace and Wilkinson 1988). Note, however, that in laboratory internal solitary wave experi-540
ments, the origin (∆x = 0) is the location where solitary wave breaking is initiated (breakpoint),541
which would be offshore of S8.542
The observed linearly-decaying self-similar ∆T/∆T0 decrease with ∆x/∆xR is also qualita-543
tively consistent with two-layer laboratory upslope normalized density decay (Wallace and Wilkin-544
22
son 1988), although again the origin is relative to the internal solitary wave breakpoint. The two-545
layer laboratory density decay also contained significantly more scatter than did normalized height546
(Wallace and Wilkinson 1988), again consistent with these observations (Figure 15). Upslope547
laboratory ∆T/∆T0 decrease with ∆x/∆xR was attributed to mixing and entrainment from the548
surrounding fluid and backflow from previous events. For the continuously stratified events A–D,549
the reduction in ∆T may also be due to mixing at the event front. Prior to the event start, the550
temperature was homogeneous. With the event arrival, the S8 onshore near-bed and offshore near-551
surface flow and the delayed near-surface cooling (Figure 8) also suggests mixing in the event552
front, as the offshore flowing near-surface water would remain warm otherwise. However, the553
observed ∆T reduction may also be because upslope locations are closer to the surface (higher554
z), and S8 temperature drop is reduced at higher z (Figure 8). Some combination of these two555
mechanisms may explain the upslope ∆T decrease.556
b. Potential vertical mixing during the NLIW rundown
Any mixing at the event front cannot be quantified here. However, after internal runup reaches557
xR, dense water then flows back downslope (rundown) during which significant mixing occurs in558
both observations in h ≈ 15 m (Walter et al. 2012) and numerical simulations (Arthur and Fringer559
2014). Here, vertical mixing in the internal swashzone during the rundown of example event560
C (temperature in Figure 8a) is inferred through the evolution of vertically-integrated potential561
energy PE, buoyancy frequency squared N2, shear-squared S2 and gradient Richardson number562
Ri = N2/S2 that indicates when a stratified flow is dynamically unstable (< 0.25).563
The time evolution of PE(t) is estimated with (5) and the potential energy change related to564
the event start is ∆PE(t) = PE(t)−PE(tf1), which evolves due to both reversible (adiabatic advec-565
tion) and irreversible (mixing) density changes. The stratification is given byN2 = (g/ρ0) ∂ρ(z)/∂z,566
where ∂ρ/∂z is found from a least-squares fit over a mid-depth range at S8 (-5.7 m≤ z ≤ -2.7 m).567
Baroclinic velocity shear-squared S2 = (∆U ′/∆z)2 + (∆V ′/∆z)2 (Figure 19c) is found for568
the same mid-depth range. Here, ∆U ′ (and ∆V ′) is the difference between the vertically averaged569
baroclinic velocity near the top of the mid-depth range (-4.2 m ≤ z ≤ -2.7 m) and near the bottom570
of the mid-depth range (-5.7 m ≤ z ≤ -4.2 m). The vertical distance between the centers of the571
two ranges ∆z = 1.5 m.572
23
Before the event arrival, ∆PE, N2 and S2 were consistent and low (dotted lines, Figure 19a-d).573
At the event onset near hour 1, cold water pulsed onshore (Figure 8a and b) elevating ∆PE at S8574
above 100 Jm−2 (Figure 19a). The cold pulse stratified the water column while creating shear at575
mid-depths, leading toN2 and S2 above 4×10−4 s−2 (Figure 19b-c). During the onrush (hour 1 to 2576
when near bottom U ′ was positive, Figure 8b), Ri was near 1, though always above 0.25 indicating577
that local vertical mixing was unlikely (Figure 19d) consistent with model studies on shallow578
slopes (e.g., Moore et al. 2016). Between hours 2 and 3, near-bed U ′ is offshore as cold water579
begins to advect back downslope (Figure 8b). As S8 bottom temperature increases (Figure 8a),580
∆PE and N2 decrease (Figure 19a and b). However, Ri is consistently above the critical value581
(Figure 19d) indicating that local shear-driven mid-water vertical mixing is still unlikely.582
As the rundown intensifies after hour 3 at S8, mid-water S2 at S8 increases again while N2583
is low, causing Ri to drop below the critical value (Figure 19b-d). At this time, shear-driven584
mixing at mid-depths is possible at S8. The timing of this drop in Ri corresponds with a period585
of bottom warming and surface cooling (Figure 8a), with the transition depth between the cooling586
surface and warming bottom near where U ′ changes sign (z ≈ −3.5 m). The direction of U ′ at587
this time (onshore at the surface and offshore at depth) potentially advects recently mixed cooler588
water near the surface offshore of S8 onshore. After the event, the internal swashzone is slightly589
cooler (compare cross-shore bottom temperature before and after the event at locations offshore of590
xR in Figure 9a and vertical temperature structure at S8 in Figure 8a). The difference in mixing591
between internal runup uprush and downrush is consistent with differences in mixing and sediment592
suspension during uprush and downrun in a surface gravity swashzone (Puleo et al. 2000).593FIG. 19
c. Complexity of NLIW runup in the internal swashzone
For the four isolated events, the upslope evolution of event parameters (cf(x), ∆T (x), zIW(x))594
can be predicted (although ∆xR is over-predicted) given water column observations at some off-595
shore location within the internal swashzone. This can provide insight into the onshore transport596
of intertidal settling larvae (e.g., Pineda 1999) and other tracers exchanged with the surfzone.597
However, these four events were relatively simple (isolated, normally-incident, and homogeneous598
pre-event) - analogous to laboratory observations. Even the 14 events at S8 (Figures 10– 11b) were599
relatively simple. These restrictions on event and isolated event definitions eliminated most period600
24
I NLIW cold pulses and several significant events from period II (e.g., Figures 3 and 4).601
In general, the NLIW field is very complex containing large amplitude isotherm oscillations602
over a range of frequencies (M2, its harmonics, as well above 1 cph) which evolve over spring-neap603
conditions. The broad S18 high frequency spectral peak (centered between 6–10 cph) observed604
during period I (red, Figure 7c) is also present in other studies, particularly near topographic fea-605
tures (e.g., Desaubies 1975; D’Asaro et al. 2007). Overlapping cold pulses at variable angles of606
incidence and potential reflection (e.g., Figure 12) are common. This region onshore of a subma-607
rine canyon system (Figure 1a) may also be unusual in terms of the NLIW field. The observed608
complex NLIW conditions demonstrate that the internal swashzone is more complex than that609
described only from isolated events, similar to a surface gravity wave swashzone.610
The wave by wave evolution of internal runup is likely affected by interaction between in-611
dividual runup events. Runup events can interact via bore-bore capture (potentially observed in612
Figure 12), which in a surface gravity swashzone leads to the largest runup events (e.g., Garcıa-613
Medina et al. 2017). Event interactions also can include modification of background conditions614
through which subsequent waves propagate, or interference between the previous event rundown615
and runup of the next event (e.g., Moore et al. 2016; Davis et al. 2017). Laboratory observations616
indicate internal wave breaking and elevated mixing due to interaction with a previous event’s617
downrush (Wallace and Wilkinson 1988; Helfrich 1992). Downslope bottom flow prior to the618
event (e.g., Helfrich 1992; Sutherland et al. 2013a) was occasionally observed (e.g., hour 1.75619
below z = −6 m in Figure 8b). Although not observed here, the downrush may eventually de-620
tach, similar to laboratory observations of gravity current intrusions (Maurer et al. 2010). This621
complexity of the NLIW runup will also impact onshore larval transport and nutrient dispersion.622
Statistics of the offshore surface gravity field can be related to statistics of runup extent for a623
surface gravity swashzone (e.g., Holland and Holman 1993; Raubenheimer and Guza 1996). For624
example offshore significant wave height, peak period, and beach slope can be used to reasonably625
accurately simulate the cross-shore variance of runup excursion up a beach (e.g., Stockdon et al.626
2006; Senechal et al. 2011). In locations less complicated than the submarine canyon system near627
the SIO pier, offshore internal tide and stratification statistics potentially can be combined with628
beach slope information to parameterize internal runup statistics.629
25
6. Summary
For 30 days between 29 September and 29 October 2014, a dense thermistor array sampled630
temperature in depths shallower than 18 m, and a bottom-mounted ADCP in 8 m depth sampled631
1 minute averaged velocity in 0.5 m vertical bins. A rich and variable internal wave field was632
observed from 18 m depth to the shoreline, with isotherm oscillations at a variety of periods, and633
a high frequency spectral peak (near 10 minute periods) when stratification was strong. Isotherm634
excursions were regularly ± 6 m during periods of high stratification; though isotherm excursions635
were less extreme, variability at all frequencies was reduced, and no high frequency spectral peak636
was observed when stratification decreased.637
Cross-shore coherent pulses of cold water at M2 and M4 time scales were regularly observed638
throughout the observational period. NLIW event (rapid temperature drops and recovery) is evident639
from the baroclinic transport of cold water upslope, occasionally causing temperature drops of640
0.7 ◦C in five minutes in water as shallow as 2 m. Fourteen isolated NLIW events were observed in641
8 m depth propagating upslope with speeds (cf) ranging from 1.4 cm s−1 to 7.4 cm s−1, propagation642
angles (θ) from -5◦ to 23◦ and temperature drops (∆T ) between 0.3 ◦C and 1.7 ◦C, decreasing643
upslope. The two-layer equivalent gravity current height (zIW) decreased linearly upslope from644
initial values between 2.1 m and 2.5 m in 8 m depth and was consistent with observations of645
baroclinic velocity. Baroclinic bottom current during the upslope event propagation (|U ′b|) was646
near the event front propagation speed, indicating high non-linearity, with mean |U ′b|/cf = 0.7.647
The upslope evolution of ∆T , zIW, and cf for four representative events most similar to two-648
layer laboratory conditions (alongshore uniform, shore-normal, isolated and propagating into ho-649
mogeneous fluid) are qualitatively consistent with laboratory observations of broken internal wave650
runup. Normalized ∆T and zIW for these events collapse as a linearly decaying function of nor-651
malized runup distance, and upslope gravity current scalings described the front speed cf0 and652
deceleration df well. The associated total runup distance (∆xR) was also well predicted from cf0653
and df , with rms error less than the resolution of the cross-shore thermistor array. However, ∆xR654
prediction with gravity current scalings has significant error due to sensitivity to cf0.655
Depressed temperature remained in the nearshore region for hours, until receding back downs-656
lope. Bottom temperature warmed and surface temperatures cooled during the receding rundown657
of an example event. The gradient Richardson number remained below the critical value (0.25) at658
26
this time, indicating shear driven mixing was occurring consistent with laboratory and modeling659
studies. The four NLIW events selected to compare with laboratory studies are simple cases. In660
general, NLIW runup is more complicated due to superposition (in ways similar to bore-bore cap-661
ture) interaction with previous (receding) events, or as the diverse offshore NLIW field evolves.662
Any understanding of the internal swashzone beyond the most simple cases may require descrip-663
tions of complex interactions or a statistical approach similar to those used to describe the surface664
gravity wave swashzone.665
Acknowledgments. This publication was prepared under NOAA Grant #NA14OAR4170075/ECKMAN,666
California Sea Grant College Program Project #R/HCME-26, through NOAAS National Sea Grant667
College Program, U.S. Dept. of Commerce. The statements, findings, conclusions and recommen-668
dations are those of the authors and do not necessarily reflect the views of California Sea Grant,669
NOAA or the U.S. Dept. of Commerce. Additional support for F. Feddersen was provided by the670
National Science Foundation, and the Office of Naval Research (ONR). ET, AL, were supported by671
***. B. Woodward, B. Boyd, K. Smith, R. Grenzeback, R. Walsh, J. MacKinnon, A. Waterhouse,672
M. Hamann and many volunteer scientific divers assisted with field deployments. Supplemental673
data was supplied by NOAA, CDIP, SCCOOS and CORDC with help from C. Olfe, B. O’Reilly674
and M. Otero. D. Grimes, S. Suanda and others provided feedback that significantly improved the675
manuscript.676
27
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825
826
Generated with ametsocjmk.cls.827
Written by J. M. Klymak828
mailto:jklymak@ucsd.edu829
http://opg1.ucsd.edu/ jklymak/WorkTools.html830
34
Tables
Event Hs (m) uw (m s−1) cf (cm s−1) |U ′b|/cf θ (◦) ∆T0 (◦C) zIW0 (m) df (×10−5 m s−2) ∆xR (m)
A 0.32 1.02 3.8 0.8 2.9 1.29 2.32 1.27 128B 0.69 0.48 3.6 1.2 1.0 0.96 2.10 0.88 100C 1.20 3.36 6.0 0.7 11.2 1.26 2.48 1.32 137D 1.03 2.49 7.4 0.4 3.6 1.62 2.12 2.37 128
Table 1. Summary of example events A–D detailed in section 4. Left to right: Event designator, significant waveheight observed at S8, wind speed, observed event front cross-shore propagation speed, ratio of bottom barocliniccurrent to event propagation speed, propagation angle, event front temperature difference, equivalent two-layer height,deceleration, and total runup distance from x0.
35
Figure Captions
FIG. 1. (a) Google Earth image of the experiment site (La Jolla California) and surroundingnearshore waters. The bathymetry (10 m contour interval, white lines) highlights the La Jolla(southern) and Scripps (northern) Canyon system. The site of a moored temperature chain near the18 m isobath (green star, S18), bottom mounted thermistors (red dots) and the SIO pier (black line)are shown. The x coordinate is chosen to be along the pier (cross-shore). (b) Detail showing thecross-shore (x) instrument deployment locations along the SIO pier (symbols) with reference to themean tide level z = 0 m (blue line), tidal standard deviation (blue dotted) and mean bathymetry(solid black). Three different types of thermistors were deployed, Onset TidBits (blue), SBE56(red) and the Coastal Observing Research and Development Center (CORDC) temperature chain(green). Instrument sites near the 8 m, 6 m, 4 m and 2 m isobaths are indicated (S8, S6, S4, andS2).
FIG. 2. Observed (a) SIO pier tidal elevation η, (b) wind speed uw, (c) significant wave heightHs at the SIO pier-end (S8) and (d) air (black) and surface ocean in 18-m depth (red) temperatureversus time. Wave observations were made by the Coastal Data Information Program (CDIP)station 073. Tidal observations, wind speed and air temperature were observed by NOAA station9410230 located at the pier-end.
FIG. 3. Temperature versus vertical location below the mean tide level (MTL) z and time at cross-shore locations (a) x = −100 m, h ≈ 2 m, denoted S2 (b) x = −155 m, h ≈ 4 m, denoted S4
(c) x = −219 m, h ≈ 6 m, denoted S6 (d) x = −273 m, h ≈ 8 m, denoted S8 (e) x = −657 mh ≈ 18 m, denoted S18. The vertical axes have been scaled to approximate the depth at eachcross-shore location (the vertical scale of plot (e) is compressed to fit on the page). Data in plots(a-d) have been removed (white) when sensors were inoperative or above the water line. Blackdots (right side, all panels) indicate the fixed (relative to MTL) thermistor locations. The blacktriangle in (e) indicates the surface following sensor (surface level η shown as black line). Graysquares at the bottom of (e) indicate the arrival time of isolated events highlighted in section 4,and colored squares indicate the arrival time of events A-D (left to right) which are highlighted indetail in section 4. The black bar on the abscissa indicates the time span highlighted in Figure 4.
FIG. 4. Similar to Figure 3, but with the time axes of all plots zoomed to highlight 3.5 days ofinternal wave activity. The black bar on the abscissa indicates the timespan included in Figure 5.
FIG. 5. Similar to Figure 4, but with the time axes of all plots zoomed to highlight 9 hours begin-
36
ning on 7 October 2014 00:15 PDT. The black bar on the abscissa indicates the timespan includedin Figure 6
FIG. 6. Similar to Figure 5, but with the time axes of all plots zoomed to highlight 1.5 hours be-ginning on 7 October 2014 at 5:49 PDT. The 18.1◦C, 19.6◦C and 21.1◦C isotherms are highlightedwith a black curve in all panels.
FIG. 7. Top panels: Mid-water column temperature at (a) S18 (green star in Figure 1) at z = −9 mand (b) S6 at z = −4 m, versus time. Two 7-day periods containing contrasting internal waveconditions are highlighted at each location. Temperature during days 1.5 - 8.5 in the active period Iis colored red (S18) and blue (S6); and temperature during days 19.2-26.2 in the less active period IIis colored purple (S18) and green (S6). Bottom panels: Temperature spectra vs frequency at bothS18 and S6 for (c) period I and (d) period II. Colors correspond to the highlighted periods in (a)and (b), and frequencies corresponding to the 24-hour (K1), 12-hour (M2), 6-hour (M4), 3-hour(M8) and 10 min periods are highlighted (vertical dotted). The 95% confidence interval for spectraat S18 and S6 are shaded light gray and dark gray, respectively.
FIG. 8. Five-hour time series of a NLIW runup event at S8 beginning at 8:13 am PDT on 26October 2014: (a) S8 1 Hz temperature from near surface to near bed, (b) Cross-shore barocliniccurrent U ′ and (c) alongshore baroclinic current V ′ versus depth and time. In (b) and (c), the bedand mean tidal surface are indicated by the black and blue curves respectively.
FIG. 9. NLIW event example bottom temperature versus time beginning at 8:13 am PDT on 26October 2014 for (a) the cross-shore locations and (b) the pier-end (S8) alongshore locations (reddots in Figure 1a), offset by 0.1◦C in both (a) and (b) for visibility. Gray dots indicate the eventarrival time at all locations and open circles in (a) indicate the temperature change that defines ∆T ,with S8 ∆T = 1.26◦C indicated. The NLIW event ∆T was below the 0.15◦C cutoff thresholdonshore of the runup extent xR = −137 m, indicated by the dotted lines in (a).
FIG. 10. Number (N ) and ∆T distribution (shading) of NLIW events observed between 9 and 30October 2014 at cross-shore locations S8, S6 and S4 (highlighted in Figure 1b). Of the N = 14
events observed at S8, only six were observed at S4, and none with ∆T > 1.5◦C.
FIG. 11. (a) NLIW event front speed cf (radial direction) and incident angle θ at S8 for the 14observed events between 9 and 30 October 2014. Events are plotted as if a viewer were lookingoffshore from the end of the SIO pier. (b) NLIW two-layer gravity current front speed cgc versus
37
event front speed cf (10) for the 14 NLIW events observed at S8. The root-mean-square error is0.016 m s−1, squared correlation R2 = 0.44, and the best-fit slope is 1.15.
FIG. 12. Bottom temperature versus time beginning at 7:37 on 12 October 2014 for (a) cross-shore locations and (b) alongshore locations (as in Figure 9) offset by 0.1◦C for visibility. Graydots indicate the arrival of a NLIW cold pulse propagating with nearly zero angle onshore toxR = −137 m. Gray crosses indicate a second NLIW pulse superimposed on the first, propagatingat high angle from south to north. This second pulse caused significant (≈ 0.8◦C) temperaturereduction in water as shallow as 1 m (x = −55 m).
FIG. 13. NLIW event A-D front propagation distance ∆x versus elapsed time ∆t (colored dots)and quadratic fit (curves) as in (11). There is good agreement (squared correlation> 0.98) betweenthe quadratic fit and observations. For each event, the estimated surfzone boundary location xsz−x0is indicated as a bar on the ordinate.
FIG. 14. NLIW event A-D (colored) (a) ∆T and (b) equivalent two layer interface amplitude zIWvs upslope distance relative to S8 (x − xS8). Note, zIW is only estimated at locations with at leastfour thermistors in the vertical. Events A and D are first estimated at x − xS8 = 27 m as S8 waspre-stratified.
FIG. 15. NLIW event A-D (colored) (a) normalized temperature anomaly ∆T/∆T0 and (b) nor-malized zIW/zIW0 versus normalized upslope propagation distance ∆x/∆xR. The linear fit (dottedblack) to events A-D in (a) has squared correlation R2 = 0.58. The linear fit to all events in (b)has slope -0.56 (dotted black) and R2 = 0.89. Note, zIW is only estimated at locations with fourthermistors in the vertical.
FIG. 16. (a) The best-fit gravity current speed cgc (10) versus NLIW front speed cf0 (11) at x0 withrms error of 0.013 m s−1 and best-fit slope 1.17. (b) Best-fit upslope gravity current decelerationdgc (12) versus NLIW front deceleration df (11) with rms error of 0.2 × 10−5 m s−2 and best-fitslope of 0.84 Both (a) and (b) are for events A-D.
FIG. 17. Predicted NLIW event A-D (colored) propagation distance ∆xR versus observed ∆xRusing (a) best fit front velocity cf0 at x0 and deceleration df , and (b) two layer gravity currentvelocity cgc and deceleration dgc. The rms error is (a) 13 m and (b) 102 m.
FIG. 18. NLIW event A-D (colored) normalized zIW/h versus upslope propagation distance∆x/∆xR.
38
FIG. 19. Five hour time series during event C at S8 of (a) potential energy (5) change from tf1 (b)squared mid-water buoyancy frequency N2, (c) squared mid-water shear S2, and (d) the gradientRichardson number Ri = N2/S2. All values have been lightly smoothed (5-min filter) for visibil-ity. Dotted lines in (a-c) indicate measurements recorded before the arrival of the event front. Thecritical Richardson value (0.25) in (d) indicates the where shear-driven mixing is expected.
39
Figures
250m
10m20m
30m
(a)
(b)
S18
x
y
FIG. 1. (a) Google Earth image of the experiment site (La Jolla California) and surrounding nearshore waters. Thebathymetry (10 m contour interval, white lines) highlights the La Jolla (southern) and Scripps (northern) Canyonsystem. The site of a moored temperature chain near the 18 m isobath (green star, S18), bottom mounted thermistors(red dots) and the SIO pier (black line) are shown. The x coordinate is chosen to be along the pier (cross-shore). (b)Detail showing the cross-shore (x) instrument deployment locations along the SIO pier (symbols) with reference tothe mean tide level z = 0 m (blue line), tidal standard deviation (blue dotted) and mean bathymetry (solid black).Three different types of thermistors were deployed, Onset TidBits (blue), SBE56 (red) and the Coastal ObservingResearch and Development Center (CORDC) temperature chain (green). Instrument sites near the 8 m, 6 m, 4 m and2 m isobaths are indicated (S8, S6, S4, and S2).
40
-101
2(m
)(a)
0
5
uw
(ms!
1) (b)
0
1
Hs(m
) (c)
0 5 10 15 20 25 30Time (Days since 29 Sep 2014)
15
20
25
30
T(/
C) (d) air
water
FIG. 2. Observed (a) SIO pier tidal elevation η, (b) wind speed uw, (c) significant wave height Hs at the SIO pier-end
(S8) and (d) air (black) and surface ocean in 18-m depth (red) temperature versus time. Wave observations were made
by the Coastal Data Information Program (CDIP) station 073. Tidal observations, wind speed and air temperature
were observed by NOAA station 9410230 located at the pier-end.
41
FIG. 3. Temperature versus vertical location below the mean tide level (MTL) z and time at cross-shore locations (a)
x = −100 m, h ≈ 2 m, denoted S2 (b) x = −155 m, h ≈ 4 m, denoted S4 (c) x = −219 m, h ≈ 6 m, denoted
S6 (d) x = −273 m, h ≈ 8 m, denoted S8 (e) x = −657 m h ≈ 18 m, denoted S18. The vertical axes have been
scaled to approximate the depth at each cross-shore location (the vertical scale of plot (e) is compressed to fit on the
page). Data in plots (a-d) have been removed (white) when sensors were inoperative or above the water line. Black
dots (right side, all panels) indicate the fixed (relative to MTL) thermistor locations. The black triangle in (e) indicates
the surface following sensor (surface level η shown as black line). Gray squares at the bottom of (e) indicate the arrival
time of isolated events highlighted in section 4, and colored squares indicate the arrival time of events A-D (left to
right) which are highlighted in detail in section 4. The black bar on the abscissa indicates the time span highlighted in
Figure 4.
42
FIG. 4. Similar to Figure 3, but with the time axes of all plots zoomed to highlight 3.5 days of internal wave activity.
The black bar on the abscissa indicates the timespan included in Figure 5.
43
FIG. 5. Similar to Figure 4, but with the time axes of all plots zoomed to highlight 9 hours beginning on 7 October
2014 00:15 PDT. The black bar on the abscissa indicates the timespan included in Figure 6
44
FIG. 6. Similar to Figure 5, but with the time axes of all plots zoomed to highlight 1.5 hours beginning on 7 October
2014 at 5:49 PDT. The 18.1◦C, 19.6◦C and 21.1◦C isotherms are highlighted with a black curve in all panels.
45
FIG. 7. Top panels: Mid-water column temperature at (a) S18 (green star in Figure 1) at z = −9 m and (b) S6 at
z = −4 m, versus time. Two 7-day periods containing contrasting internal wave conditions are highlighted at each
location. Temperature during days 1.5 - 8.5 in the active period I is colored red (S18) and blue (S6); and temperature
during days 19.2-26.2 in the less active period II is colored purple (S18) and green (S6). Bottom panels: Temperature
spectra vs frequency at both S18 and S6 for (c) period I and (d) period II. Colors correspond to the highlighted periods
in (a) and (b), and frequencies corresponding to the 24-hour (K1), 12-hour (M2), 6-hour (M4), 3-hour (M8) and 10 min
periods are highlighted (vertical dotted). The 95% confidence interval for spectra at S18 and S6 are shaded light gray
and dark gray, respectively.
46
FIG. 8. Five-hour time series of a NLIW runup event at S8 beginning at 8:13 am PDT on 26 October 2014: (a) S8
1 Hz temperature from near surface to near bed, (b) Cross-shore baroclinic current U ′ and (c) alongshore baroclinic
current V ′ versus depth and time. In (b) and (c), the bed and mean tidal surface are indicated by the black and blue
curves respectively.
47
FIG. 9. NLIW event example bottom temperature versus time beginning at 8:13 am PDT on 26 October 2014 for (a)
the cross-shore locations and (b) the pier-end (S8) alongshore locations (red dots in Figure 1a), offset by 0.1◦C in both
(a) and (b) for visibility. Gray dots indicate the event arrival time at all locations and open circles in (a) indicate the
temperature change that defines ∆T , with S8 ∆T = 1.26◦C indicated. The NLIW event ∆T was below the 0.15◦C
cutoff threshold onshore of the runup extent xR = −137 m, indicated by the dotted lines in (a).
48
S8 S6 S4
0
2
4
6
8
10
12
14
N
1:5/C < " T
1:0/C
< "T <
1:5/C
0:5/C
< "T <
1:0/C
0:3/C
< "T <
0:5/C
0:15/C < "T < 0:3/C
No Event
FIG. 10. Number (N ) and ∆T distribution (shading) of NLIW events observed between 9 and 30 October 2014 at
cross-shore locations S8, S6 and S4 (highlighted in Figure 1b). Of the N = 14 events observed at S8, only six were
observed at S4, and none with ∆T > 1.5◦C.
!40/!30/
!20/!10/0/10/
20/
30/
40/
0
0.02
0.04
0.06
0.08
cf (m s!1)
3
(a)
0
0.02
0.04
0.06
0.08
0.1
c gc=
h11!
z IW ~ h
2g
0 zIW
i 1=2
(ms!
1)
0 0.02 0.04 0.06 0.08 0.1cf (m s!1)
(b)
FIG. 11. (a) NLIW event front speed cf (radial direction) and incident angle θ at S8 for the 14 observed events between
9 and 30 October 2014. Events are plotted as if a viewer were looking offshore from the end of the SIO pier. (b) NLIW
two-layer gravity current front speed cgc versus event front speed cf (10) for the 14 NLIW events observed at S8. The
root-mean-square error is 0.016 m s−1, squared correlation R2 = 0.44, and the best-fit slope is 1.15.
49
19.5
20
20.5
21
21.5
22
22.5T
(o C o
ffset
by
0.1)
(a)
x = -55 mx = -73 mx = -100 m (S2)x = -118 mx = -137 mx = -155 m (S4)x = -173 mx = -191 mx = -218 m (S6)x = -246 mx = -273 m (S8)
0 1 2 3 4 5Time (hours from 12 Oct 2014 7:37 PDT)
18
18.5
19
19.5
20
20.5
21
21.5
22
T (o C
offs
et b
y 0.
1)
(b)
y = 200 my = 100 my = 0 my = -100 my = -200 m
FIG. 12. Bottom temperature versus time beginning at 7:37 on 12 October 2014 for (a) cross-shore locations and
(b) alongshore locations (as in Figure 9) offset by 0.1◦C for visibility. Gray dots indicate the arrival of a NLIW
cold pulse propagating with nearly zero angle onshore to xR = −137 m. Gray crosses indicate a second NLIW
pulse superimposed on the first, propagating at high angle from south to north. This second pulse caused significant
(≈ 0.8◦C) temperature reduction in water as shallow as 1 m (x = −55 m).
50
0 20 40 60 80"t (min)
0
50
100
150
"x
(m)
ABCD
FIG. 13. NLIW event A-D front propagation distance ∆x versus elapsed time ∆t (colored dots) and quadratic fit
(curves) as in (11). There is good agreement (squared correlation > 0.98) between the quadratic fit and observations.
For each event, the estimated surfzone boundary location xsz − x0 is indicated as a bar on the ordinate.
0
0.5
1
1.5
2
"T
(/C)
(a)
ABCD
0 50 100 150x! xS8 (m)
0
1
2
3
z IW
(m)
(b)
ABCD
FIG. 14. NLIW event A-D (colored) (a) ∆T and (b) equivalent two layer interface amplitude zIW vs upslope distance
relative to S8 (x− xS8). Note, zIW is only estimated at locations with at least four thermistors in the vertical. Events
A and D are first estimated at x− xS8 = 27 m as S8 was pre-stratified.
51
0
0.2
0.4
0.6
0.8
1"
T="
T0
(a)
ABCD
0 0.2 0.4 0.6 0.8 1"x="xR
0
0.2
0.4
0.6
0.8
1
z IW
=zIW
0
(b)
ABCD
FIG. 15. NLIW event A-D (colored) (a) normalized temperature anomaly ∆T/∆T0 and (b) normalized zIW/zIW0
versus normalized upslope propagation distance ∆x/∆xR. The linear fit (dotted black) to events A-D in (a) has
squared correlation R2 = 0.58. The linear fit to all events in (b) has slope -0.56 (dotted black) and R2 = 0.89. Note,
zIW is only estimated at locations with four thermistors in the vertical.
52
0 0.02 0.04 0.06 0.08 0.1cf0 (m s!1)
0
0.02
0.04
0.06
0.08
0.1c g
c 0=
h11!
z IW
0~ h
0
2g0 0z I
W0
i 1=2
(ms!
1)
(a)
ABCD
0 0.5 1 1.5 2 2.5df (#10!5 m s!2)
0
0.5
1
1.5
2
2.5
dgc
=1 2g0 0s
z IW
0~ h
0
11!
z IW
0~ h
0
2(#
10!
5m
s!2)
(b)
FIG. 16. (a) The best-fit gravity current speed cgc (10) versus NLIW front speed cf0 (11) at x0 with rms error of
0.013 m s−1 and best-fit slope 1.17. (b) Best-fit upslope gravity current deceleration dgc (12) versus NLIW front
deceleration df (11) with rms error of 0.2× 10−5 m s−2 and best-fit slope of 0.84 Both (a) and (b) are for events A-D.
0 50 100 150 200 250Observed "xR (m)
0
50
100
150
200
250
"x
R=
1 2c2 f0
=dob
s(m
)
(a)
ABCD
0 50 100 150 200 250Observed "xR (m)
0
50
100
150
200
250
"x
R=
1 2c2 gc
=dgc
(m)
(b)
FIG. 17. Predicted NLIW event A-D (colored) propagation distance ∆xR versus observed ∆xR using (a) best fit front
velocity cf0 at x0 and deceleration df , and (b) two layer gravity current velocity cgc and deceleration dgc. The rms
error is (a) 13 m and (b) 102 m.
53
0 0.2 0.4 0.6 0.8 1"x="xR
0
0.2
0.4
0.6
0.8
1z I
W(x
)=h(x
)ABCD
FIG. 18. NLIW event A-D (colored) normalized zIW/h versus upslope propagation distance ∆x/∆xR.
0
50
100
"PE
[Jm!
2] (a)
0
2
4
6
8
N2
[s!
2]
#10-4
(b)
0
2
4
6
8
S2
[s!
2]
#10-4
(c)
0 1 2 3 4 5Time (hours from 26 Oct 2014 8:13 PDT)
10-2
10-1
100
101
Ri
(d)
FIG. 19. Five hour time series during event C at S8 of (a) potential energy (5) change from tf1 (b) squared mid-water
buoyancy frequency N2, (c) squared mid-water shear S2, and (d) the gradient Richardson number Ri = N2/S2. All
values have been lightly smoothed (5-min filter) for visibility. Dotted lines in (a-c) indicate measurements recorded
before the arrival of the event front. The critical Richardson value (0.25) in (d) indicates the where shear-driven mixing
is expected.
54
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