1. Introduction
A continental shelf wave (CSW), first observed along the Australian coast by Hamon (1962, 1966), is a type of topographic planetary wave trapped over the continental margin with a frequency less than the local inertial frequency and a wavelength much greater than the depth (LeBlond and Mysak 1978). It is generally accepted that these waves are generated by large-scale weather systems moving across or along the shelf (Robinson 1964; Mysak 1967a,b; Adams and Buchwald 1969; Gill and Schumann 1974; LeBlond and Mysak 1978). In this study, we focus on a storm traveling across the shelf, which might produce a standing wave resulting from resonant motion excited by the forcing, and a CSW freely propagating with the coastline on its right in the North Hemisphere (LeBlond and Mysak 1978; Slørdal et al. 1994; Tang et al. 1998). Wavelet analysis is shown to be able to estimate group speeds by calculating the distance traveled by a wave envelope, which contains waves of a selected frequency interval. For calculating phase speed, cross-wavelet analysis is particularly adapted for studies of nonstationary signals.
Barotropic shelf waves have been observed in many locations; typically, the propagating speeds were within the range of 1–6 m s−1 (Hamon 1966; Isozaki 1968; Mysak and Hamon 1969; Cutchin and Smith 1973; Brooks and Mooers 1977; Enfield and Allen 1983; Camayo and Campos 2006). Higher speeds of barotropic CSW have been found, but were identified as forced waves, whose speed is determined by the speed of the forcing agent (Enfield and Allen 1983; Cuevas et al. 1986; Yankovsky and Garvine 1998). Several numerical simulations of the response of shelf seas to weather forcing have been done in the past and gave results of free-CSW phase and/or group speed (Martinsen et al. 1979; Enfield and Allen 1983; Gordon and Huthnance 1987; Gjevik 1991; Sheng et al. 2006). Nevertheless, to our knowledge, none of them found fast group and phase speeds for CSW by examining the sea level oscillations after a strong storm event, using a wavelet approach.
In this paper we focus on Hurricane Florence, which hit Newfoundland on 13 September 2006 and generated a significant sea level disturbance propagating freely along the eastern Canadian coast. The group and phase speeds of the free-barotropic CSW are estimated using wavelet and cross-wavelet techniques. The speeds were higher than previous observations in many other parts of the world. Wavelet methods are necessarily technical and require detailed explanation. However, the main focuses of this work are the observed high-CSW group and phase speeds. Results are compared with dispersion curves for the first-mode barotropic CSW, using two simplified shelf profiles and with two other storms with different durations (the passage times of the storms over a particular point on the coast) and intensities. Following a summary of the data used (section 2), wavelet and cross-wavelet analyses are presented in section 3, and the results and the conditions for propagation of the CSW generated by the events are discussed in section 4.
2. Data
Three events were studied: Hurricane Florence (12–15 September 2006), Hurricane Isaac (1–3 October 2006), and Tropical Storm Chantal (31 July–2 August 2007). Note that the general term “storm” is used for the three events hereafter. One-minute (min) sea level data records from the Integrated Science Data Management (ISDM), which is a branch of Canada’s Federal Department of Fisheries and Oceans (DFO), were used in this study at six different locations (Fig. 1). Storm tracks and surface atmospheric pressure under the storms (Table 1) were provided by the Canadian Hurricane Centre (CHC), which is a division of the Meteorological Service of Canada. For calculating the mean storm speed across the shelf, only data corresponding to the storm tracks located between latitudes 44° and 49°30′ are considered.
The sea level records were detided using version 1.2 of T_TIDE (Pawlowicz et al. 2002). The long-wave records were extracted by low-pass filtering, with a cutoff frequency of 1.41 cpd. We used an orthogonal quadrature mirror filter type “Symmlet,” with eight vanishing moments (Mallat 2001), to decompose the signal into wavelet coefficients and reconstruct a low-pass-filtered signal by removing the coefficients corresponding to frequencies higher than 1.41 cpd. The result of filtering the detided sea level records is illustrated by Fig. 2 for station Argentia (AR) during the Florence event.
For each station except Lawrence (LW), the 1-h surface atmospheric pressure data used to calculate the expected hydrostatic response of the ocean and the wind speed (Figs. 3 –5) were both provided by the Canadian Daily Climate Data (CDCD), which is operated and maintained by Environment Canada. The times when the storm centers hit the coast were estimated from the storm reports provided by the CHC. The times when the storm’s edges hit the seaboard are graphically approximated from the rise and fall of the hydrostatic ocean response curves (Figs. 3 –5) in AR (the closest station to the storm center), taking into account the longer-term trends. The storm conditions were characterized as the sections above 30% of the maximum hydrostatic sea level elevation under the storm relative to the 2–3-day general trend. This gives an approximation of the storm durations (Table 1), with an arbitrary absolute error of 2 h. The storm width was calculated from the storm duration and mean speed (Table 1).
The three storms had roughly the same path. The intensity of Florence was greater than both Isaac and Chantal (Figs. 3 –5 and Table 1). The great amplitude of the sea level disturbance within the range of 12–45 cm (Table 2), observed at the coast after the passage of Florence (Fig. 3), suggested that the response was barotropic. Furthermore, Clarke and Brink (1985) showed that the response of such a wide, nonequatorial shelf to fluctuating wind forcing should be barotropic.
3. Analysis
a. Wavelet analysis
All the calculations for the normalizations, the wavelet power spectrum (WPS), and the scale-averaged wavelet power (SAWP) followed the methodology described by TC98. To reduce ringing, longer time series (23 days) centered in the event were used for wavelet analysis. Figure 6 shows only a few days before and after the passage of Florence, well within the cone of influence.
The WPSs of Fig. 6a show an equatorward propagation of a 26–30-h period wave along the coast from LW to Yarmouth (YA), after Florence moved onto the shelf. The energy decay, resulting from dissipative effects, is shown by Fig. 6b. Note that the energy was greater in Port aux Basques (PB) than North Sydney (NS), and this might be because station PB is more sheltered (Fig. 1). The maximum energy reached AR a few hours after LW, showing a wave propagation in the opposite direction. The two stations are located at less than half of the storm width (Table 1) from the storm center and may be affected by the forced wave under the storm.
For the other two storms, local responses were observed in the WPSs in AR and LW (not shown) over shorter period bands, including the storm durations (20–24 and 17–21 h for storms Isaac and Chantal, respectively). Nonetheless, there was no significant propagation of sea surface disturbance along the coast.
Table 3 shows the values used to calculate the barotropic CSW group speed along the coast after the passage of Florence. Lag uncertainties among station pairs were estimated by summing the time uncertainty at each station and were used for calculating group-velocity uncertainties (Table 3). Because of high uncertainties, results for station pairs HA–YA have not been used in the calculation. The mean CSW group speed was 11.4 ± 5.9 m s−1.
b. Cross-wavelet analysis
The cross-wavelet transform (XWT) of two signals exposes the regions in time–frequency space with high common power and further reveals information about the phase relationship (Grinsted et al. 2004). The XWTs of the pairs of stations are used to estimate the CSW phase speed. The XWT of two time series Xn and Yn is defined as WXY = WXWY*, where WX is the continuous WT of the time series Xn and the asterisk denotes the complex conjugate. The cross-wavelet power is defined as |WXY|. The complex argument of WXY describes the phase relationship between X and Y in a time–frequency space. As compared to traditional lagged cross-correlation methods, this technique offers the advantage that the time interval is reduced to the duration when the XWT of station pairs has significant power. Thus, the relative phase uncertainty is optimal for each pair and does not include additional errors for the studied event because of noise or other oscillations with similar frequencies, remaining out of the time boundary.
To estimate the confidence interval of the phase difference, we used the circular standard deviation defined as S =
The relative phase relationships can be considered as a measure of the time delay of two periodic signals expressed as a fraction of the wave period. Table 4 shows the values for different station pairs used to calculate the CSW phase velocities along the coast after the passage of Florence. The lags were calculated from the cross-wavelet phase angles (am) as lag = amTm, where Tm is the mean period. The lag errors were estimated as Δlag = lag(S/am + kσ̂T /Tm), where σ̂T is the standard deviation of the period and k is a constant that depends on the confidence level (here, k = 1.96 to satisfy the 95% confidence level).
Results of the XWT (Table 4) between LW and PB show no phase propagation of the CSW. Between LW and NS, the magnitude of the CSW phase speed is not realistic, and the large variability might result from direct influences of the storm. Indeed, it must be kept in mind that records at AR and LW were directly affected by the forced wave under the storm (Fig. 3). We conclude that all station pairs, which include LW, cannot be used in the calculations. The mean phase speed of the CSW calculated from Table 4 is about 16.0 m s−1 with a mean error of 5.1 m s−1.
4. Discussion
The wave observed after the passage of Florence across Newfoundland is expected to be a free CSW (Robinson 1969; Hamon 1966; LeBlond and Mysak 1978). Florence, the slowest storm, was the only one that excited a free CSW within a period band matching the passage of time of the event (28 ± 2 h) over the shelf (Figs. 3 –5). Sheng et al. (2006) observed from subtidal sea level records an equatorward propagation of shelf waves with similar periods (25–28 h) during the first few days after Hurricane Juan moved onto the Scotian shelf. The surface atmospheric pressure at the center of Juan when crossing the shelf was about 976 mb, slightly higher than Florence (964 mb). In terms of wind stress and barometric response of the sea surface, Florence, despite its relative slow speed, was much stronger than the two other storms, Isaac and Chantal. Consequently, one can expect a weaker response to the two faster but less intense storms. Tang et al. (1998)’s model showed that long barotropic shelf waves can easily be excited by a storm with barometric pressures similar to those under Isaac or Chantal. But surprisingly, no significant barotropic sea level disturbance propagation is observed for both Isaac and Chantal within the period ranges fitting the storm durations (Figs. 4 –5). Note that the storms might generate baroclinic disturbances whose surface manifestations are too weak to be observed with this dataset.
The dispersion relation of shelf waves can be determined from a given topographic shelf profile. The 2007 version of the computer program written by Brink and Chapman (1987) was used to compute the dispersion curves of the barotropic shelf wave (Fig. 10a) using the two cross-shelf sections A and B (Fig. 1a). Because of irregularities of the topography, simplified topographies with a cutoff water depth of 4000 m for section A and 3000 m for section B were used in the calculations (Fig. 9).
Low-frequency shelf waves have high group velocities and propagate faster than the maximum possible frequency, which has a zero group velocity. The significant regions on the WPSs (Fig. 6a) become narrower in period as one goes farther from LW, where the response includes periods from 16 to 64 h. The high-frequency waves (16–25 h) have a high wavenumber and a group velocity close or equal to zero (Fig. 10b). Their energy persists for a few hours on the southern–eastern Newfoundland shelf (AR and LW), with no propagation, and dissipates rapidly after the passage of Florence (Fig. 6a). The lower-frequency waves with high group velocities and smaller wavenumbers are weakly dispersive but lose energy resulting from dissipative effects while propagating from LW to YA. Although slowly dissipated, the six WPSs show a persistent period band (26–30 h), including Florence’s duration (28 h or 0.86 cpd), lower than the shelf wave maximum frequency that corresponds to the zero group-velocity regime. The group velocity (cg = dw/dk, where w is the frequency and k is the wavenumber) of a long shelf wave of frequency 0.86 ± 0.08 cpd, estimated from the dispersion curves (Fig. 10a), is within the range of 6.1–15.1 m s−1 and the phase velocity (w/k) is between 12.0 and 18.9 m s−1 (Fig. 10b) for shelf profiles at sections A and B. The mean speeds estimated in this study using wavelet and cross-wavelet analyses (11.4 ± 5.9 and 16 ± 5.1 m s−1 for group and phase speed, respectively) are consistent with these estimations.
Compared to previous observations, the speeds found were high for free-propagating shelf waves, but they are consistent with the computed dispersion curves and numerical models. Gordon and Huthnance (1987) and Tang et al. (1998) used numerical models and obtained dispersion relationships for the first mode of the shelf wave, with group velocities in the nondispersive regime of 13 m s−1 along the Scottish continental shelf and 11.6 m s−1 along the Labrador/Newfoundland shelf, respectively. Sheng et al. (2006) estimated from a nested grid model, a shelf wave phase speed of 15 m s−1 along the Scotian shelf. To our knowledge, speeds of this order of magnitude for a free-barotropic CSW with similar periods have not been detected in the past, using both the wavelet and cross-wavelet techniques.
Gordon and Huthnance (1987) and Tang et al. (1998) pointed out that the ocean response depends strongly on the duration of the atmospheric forcing. After Florence passed over LW and AR, the predominant frequency was slightly lower than the resonant nonpropagating regime for the five stations (Fig. 6a) and similar to the passage time for the storm over the shelf, which is consistent with the continental shelf wave theory. The duration of strong storms like Florence is the key factor that determines the propagation and the frequency band preferentially excited for the free CSW.
For the three storms, the nonpropagating regime is excited only at stations close to the storm track (not shown for storms Isaac and Chantal) in AR and LW (Fig. 6a). It lingers about one day after the storm passes. Section B is steeper than section A (Fig. 9). The former permits higher-frequency shelf waves with a maximum of 1.27 cpd (Fig. 10a), corresponding to the duration of Isaac (1.26 ± 0.08 cpd). Chantal’s duration (1.09 ± 0.08 cpd) was under this maximum but matched the nonpropagating regime in section A. Thus, the durations of Isaac and Chantal were too short to excite any propagating shelf waves.
The study points out the importance of topography and storm duration on the propagation of free CSWs generated by a moving low pressure system (Hamon 1966; Mooers and Smith 1968; Gordon and Huthnance 1987; Tang et al. 1998). The strong amplitude response of the coastal sea level (within the range of 12–45 cm) as a free-barotropic CSW after Florence impacted the Newfoundland shelf was clearly detected in time and frequency using wavelet and cross-wavelet analyses. The CSW period was similar to the storm duration, and the measured fast group and phase speeds were consistent with the dispersion curves for the first-mode CSW computed for the shelf profiles of Newfoundland and Nova Scotia.
Acknowledgments
We thank the DFO, the CHC, and Environment Canada for providing sea level data, storm details, atmospheric pressure, and wind data. The T_TIDE software package was provided by R. Pawlowicz et al. (available online at http://www.eos.ubc.ca/~rich/). The wavelet software was originally written by C. Torrence and G. Compo (available online at http://paos.colorado.edu/research/wavelets/) and the cross-wavelet software was provided by A. Grinsted et al. (available online at http://www.pol.ac.uk/home/research/waveletcoherence/). We are grateful for the advice of Dr. Kenneth Brink, who made the code for calculating the shelf wave dispersion relation (available online at http://www.whoi.edu/cms/files/Fortran_30425.htm). Suggestions provided by Pascal Sirguey were very helpful.
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Locations of sea level stations: AR (47°18′0″N, 53°59′0″W), LW (46°55′0″N, 55°23′24″W), PB (47°34′0″N, 59°08′0″W), NS (46°13′00″N, 60°15′0″W), HA (44°41′0″N, 63°37′0″W), and YA (43°50′0″N, 66°07′0″W) for the Atlantic coast of Canada (Newfoundland and Nova Scotia). Also shown are the tracks of storm centers (circles, crosses, and triangles for Florence, Isaac, and Chantal, respectively), the bathymetry contours of 100, 300, 2000, 3000, 4000, and 5000 m, and the cross-shelf sections A and B.
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
Detided sea level record relative to the mean sea level at AR for Florence before (thin line) and after (thick line) low-pass filtering (with a cutoff frequency of 1.41 cpd).
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
Detided sea level time series (low-pass filtered with a cutoff frequency of 1.41 cpd) relative to the mean sea level for each station (thick solid lines), barometric response of the sea surface (dotted lines) to the atmospheric pressure fluctuations, and wind speed (dashed lines) at stations AR, LW, PB, NS, HA, and YA. Note that neither the barometric response of the sea surface nor the wind speed is shown for LW because of the lack of data. Also shown are the times when Hurricane Florence (front, center, and back) hit the coast of Newfoundland (left, middle, and right thin vertical solid lines, respectively).
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
As in Fig. 3, but for storm Isaac.
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
As in Fig. 3, but for storm Chantal at stations AR, LW, NS, HA and YA. Note that the sea level record for AR is not represented because of lack of data.
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
(a) Morlet wavelet power spectrum of the detided low-pass-filtered sea level records for the six named stations, normalized by 1/(95% confidence level for the corresponding red-noise spectrum) at each scale (power relative to the significant level). Thick vertical lines indicate the time when Florence (front, center, and back of the storm) hit the coast of Newfoundland. (b) The scale-averaged wavelet power time series over the 26–30-h band for AR (Xs), LW (dashed line), PB (dash–dot line), NS (dotted line), HA (solid line), and YA (Os).
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
(a) One-min-interval artificial time series for LW Monte Carlo simulation, containing red noise (α = 0.970) and residual of semidiurnal tide components over the whole time series and a Gaussian-modulated cosine wave of amplitude 19 cm (period randomly chosen within the range of 26–30 h and random phase between 0 − π and from t = 240 h to t = 336 h). (b) Scale-averaged wavelet power time series of the low-pass residual of (a) over the 26–30-h period band. The solid and dashed vertical lines are the times of the maximum-averaged wavelet power and tm = 288 h, respectively.
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
XWT of detided sea level time series at NS and YA after the passage of Florence, normalized by 1/(95% confidence level for the corresponding red-noise spectrum) at each scale (power relative to the significant level). Also shown (as arrows) is the relative phase relationship (with in-phase pointing right, antiphase pointing left, and YA leading NS by 90° pointing straight down).
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
Simplified topography of sections A (solid lines) and B (dotted lines).
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
(a) Dispersion relations for the first mode of the CSW for sections A (solid lines) and B (dotted lines). The thin lines are the durations (represented as frequencies in cpd) of each storm with the estimated error. (b) The phase velocity c (thick solid and dashed line for sections A and B, respectively) and cg (thin solid and dashed line for sections A and B, respectively) are derived from the dispersion curves. The shaded region is the theoretical wavenumber range of the CSW generated by Florence, considering its duration’s error and dispersion relations for both sections A and B.
Citation: Journal of Physical Oceanography 40, 2; 10.1175/2009JPO4204.1
Characteristics of the three storms studied. Values are provided/calculated from the CHC and the CDCD data.
Values of lag-1 autocorrelation parameter (α), noise variance 
Values used for the calculation of the CSW group speed (cg) after the passage of Florence. A positive value of the lag indicates that the second station of the pair leads the first. A positive value of cg indicates that the energy propagates alongshore with the coastline on its right.
Results of the XWT detided sea level records for different Station pairs. A positive value of the lag indicates that the second station of the pair leads the first. A positive value of c indicates that the wave phase propagates alongshore with the coastline on its right.
