Large-scale edge waves generated by hurricane landfall

Authors


Abstract

[1] Direct observations of the storm surge induced by Hurricane Wilma's landfall revealed a formation of a wave pulse propagating alongshore. The height of the wave pulse exceeded 1.5 m in the detided sea level. The duration of this wave pulse was ∼6 h and the propagation speed was of O(10) m s−1. This wave has been identified as an edge wave of large spatial and temporal scales. A set of numerical experiments has been conducted to delineate a generation of edge waves with large spatial and temporal scales by a fast-moving storm system. The model of the coastal ocean was set in a two-dimensional configuration with the continental shelf and slope topography reminiscent of the West Florida shelf. A moving cyclonic system in the gradient wind balance was prescribed analytically. In order to identify a long wave response in the model, a linear boundary problem was solved, yielding dispersion characteristics and a structure of the edge wave modes. A fast-moving storm system crossing the shelf at a right angle produces a nearly symmetrical response of two edge wave trains propagating both downstream (in the direction of a Kelvin wave) and upstream. Typically, zero-mode edge waves dominate the response. As the translation speed of the storm becomes lower, its Eulerian timescale becomes longer and the waves are more affected by the Earth's rotation. In that case, the wave energy propagates predominantly downstream. When the storm trajectory deviates from the normal approach, the edge wave response is not symmetric: most of the energy propagates in the direction of the alongshore component of the storm translation velocity.

1. Introduction

[2] Edge waves are the wave modes trapped in the coastal ocean due to the refraction of long gravity waves over a sloping bottom. Although they were first introduced in the XIX century by Stokes, for a long time edge waves were not recognized as being of any importance for the coastal dynamics [Lamb, 1932]. An analytical solution for a set of edge wave modes over a linear depth profile was obtained by Ursell [1952] and, for a shallow water approximation, by Eckart [1951]. Their amplitude generally decreases offshore and the mode number corresponds to the number of nodal lines of sea surface parallel with the coastline. There is also a zero edge wave mode, sometimes called the Stokes mode [Liu et al., 1998] with no nodal lines and the free surface disturbance exponentially decaying offshore. A bottom slope limits the number of possible edge wave modes while an exponential offshore decay is proportional to the alongshore wave number [Ursell, 1952].

[3] Studies of the edge waves were focused on smaller-scale (wavelength ∼ O(1–10) km), higher-frequency (period ∼ O(1–10) min) waves driven by the incident wind waves or swells [e.g., Guza and Davis, 1974; Foda and Mei, 1981; Buchan and Pritchard, 1995]; while larger-scale edge waves were traditionally related with the effects of a tsunami. In the latter case, part of the incident tsunami wave energy can be transferred to the trapped edge wave modes, which can delay and amplify a sea level response at alongshore locations downstream from the epicenter [i.e., Abe and Ishii, 1987; Shuto and Matsutomi, 1995; Gonzáles et al., 1995; Horillo et al., 2008]. Edge waves were also found to contribute to the generation of meteotsumani [Monserrat et al., 2006]. Meteotsunami are long gravity waves in the tsunami frequency band generated by atmospheric forcing and reaching a high amplitude on the shelf (and especially in the semienclosed basins) due to different resonance mechanisms.

[4] Severe tropical storms are among the most energetic forcing agents for the coastal ocean. Their landfall generates a storm surge which in many cases results in a massive coastal inundation and a loss of human lives, as Hurricane Katrina (August 2005) and Tropical Cyclone Nargis (May 2008) once again proved. The coastal ocean response to the landfall of a tropical cyclone includes a generation of long waves. Some of these waves are trapped on the shelf and propagate along the coastline thus affecting the sea level fluctuations outside the area of a forced storm surge. Most of the previous studies on the generation of trapped wave modes in the coastal ocean by an atmospheric storm system focused on the subinertial coastally trapped waves (CTWs). CTWs are inherently related with the effects of Earth's rotation and propagate in the direction of a Kelvin wave with the coast on their right in the Northern Hemisphere. Hereinafter, this direction is referred to as a downstream. The examples include the works of Martinsen et al. [1979], Carton [1984], Morey et al. [2006], etc. The edge waves received far less attention [e.g., Munk et al., 1956; Greenspan, 1956]; perhaps because they exist at superinertial frequencies and their temporal scales do not match those of the atmospheric systems. In fact, Tang and Grimshaw [1995] included both the subinertial continental shelf waves (barotropic approximation for CTWs) and the edge waves in their storm surge model and found that the first and second shelf wave modes by far dominated the response, while edge waves were relatively insignificant. However, Lighthill [1998] pointed out that the edge wave modes were generated by tropical cyclones in the Bay of Bengal and were observed to propagate along the east coast of India in the upstream direction (opposite to the CTW propagation).

[5] Yankovsky [2008] analyzed an observational data set of the storm surge induced by Hurricane Wilma's landfall on the West Coast of Florida on 24 October 2005. A relatively high temporal and spatial resolution enabled an assessment of the alongshore evolution of this storm surge and, in particular, a detection of the waves traveling downstream (northward) from the hurricane's landfall area. The data revealed a short-lasting wave pulse (∼6 h) clearly belonging to the superinertial domain. This wave was identified as an edge wave of large temporal and spatial scales corresponding to the Eulerian scales of the forcing system.

[6] This paper presents numerical experiments that support an interpretation of the alongshore storm surge evolution induced by Wilma as the generation of a large-scale edge wave. The rest of the paper is organized as follows: section 2 presents an observation of long waves propagating along the Florida coast after Wilma's landfall, section 3 describes a standard case in the set of numerical experiments, section 4 deals with a linear solution for the edge wave modes, as well as with their identification in the standard case, section 5 discusses other numerical experiments, while section 6 summarizes and concludes the paper.

2. Observations

[7] The data collection was requested by FEMA in the anticipation of severe flooding when Wilma started its movement across the Gulf of Mexico toward the West Florida coast. The data set consists of weeklong time series of storm surge (S) and barometric pressure (B) measured by the USGS Florida Integrated Science Center at approximately 30 locations with Hobo water level loggers. More details about the data set can be found in the work of Yankovsky [2008]. The survey area spanned almost 150 km alongshore, although most of the instruments were deployed in the inlets, estuaries (sometimes several km inland), and other semienclosed basins with only a few locations on the exposed coastline. Data from the exposed Gulf of Mexico coastline represent the edge wave propagation best, since the wave signal was delayed in confined basins [Yankovsky, 2008]. Thus presented here are the time series from only five coastal locations, which are shown in Figure 1 and retain the numeration of the original data set. The USGS data are augmented with NOAA measurements at the tide gauge station 8725110 in Naples, FL (Figure 1).

Figure 1.

Map of the West Florida coast in the vicinity of Wilma's landfall showing sites of USGS surge stations (triangles) and the NOAA tide gauge station 8725110 (asterisk). Dashed line is Wilma's track. Isobaths are in meters. P.I., Pine Island.

[8] Hurricane Wilma approached Florida along the straight trajectory at a fairly high translation speed of 20–25 kt (10–13 m s−1 [Pasch et al., 2006]). It made a landfall at approximately 10:30 UTC on 24 October, 2005 at a close-to-normal angle relative to the general orientation of the South Florida coastline (Figure 1). The time of its landfall derived from the barometric pressure measurements at location 27 (closest to the site of a landfall) is depicted in Figure 2. The storm surge loggers were not inundated continuously throughout the deployment; time periods when the instruments were out of the water can be identified as horizontal lines of zero pressure disturbance in Figure 2. The forced storm surge occurred prior to 12 UTC and was stronger in Wilma's southern sector (e.g., location 27, Figure 2) where prevailing winds were onshore, following the cyclonic wind pattern. A few hours after the landfall, at ∼15 UTC, a free wave pulse formed and propagated downstream (northward) along the coastline. At locations 22, 12 and 11 its height exceeded the height of the forced storm surge earlier in time. The sea level measurements were not adjusted to the tidal datum. For this reason, the actual magnitude of the wave pulse is illustrated by a detided sea level record at the NOAA tide gauge station in Naples (Figure 2). The wave height was ∼1.5 m and the wave pulse lasted for approximately 6 h. Its impact on coastal inundation was reduced by a stage of low tide at the time of its propagation [Yankovsky, 2008].

Figure 2.

Time series of storm surge from the USGS array. Light gray lines overlapped with the wave pulse at stations 22 and 12 are low-pass filtered data used to identify the wave pulse arrival time. Time is UTC. Time series are arranged such that their zero pressure disturbances are offset in vertical proportional to the alongshore distance between the stations. Detided sea level time series from the NOAA tide gauge station 8725110 is shown in gray (overlaps surge time series at station 22). Vertical bar indicates Wilma's landfall.

[9] Since the wave clearly belonged to the superinertial frequency band (the inertial period here is 27 h) and was trapped by the coastline, it was identified as an edge wave of large temporal and spatial scales [Yankovsky, 2008]. Its phase speed was assessed as a linear regression coefficient for the wave crest's arrival time at different alongshore locations and was of O(10) m s−1. Two locations (22 and 12) were on the exposed coast 41 km apart, and the delay of the wave crest arrival time gives an alternative estimate for the phase speed, 13.5 m s−1. The time of the wave crest arrival was inferred from the low-pass filtered time series (Figure 2). The wave interacted with complex topography in the proximity of Fort Myers-Sanibel Island (Figure 1), and its amplitude was substantially reduced when it reached location 8 (Figure 2).

3. Numerical Model: Standard Case

3.1. Model Configuration

[10] In order to reproduce a generation of edge waves by a cyclonic storm system crossing the continental shelf, we apply a two-dimensional, vertically integrated, version of the Regional Ocean Modeling System (ROMS) [Song and Haidvogel, 1994; Shchepetkin and McWilliams, 2005]. The model solves nonlinear shallow water equations on an f plane. The momentum advection is approximated with the third-order upstream-biased scheme. Laplacian horizontal viscosity is applied for numerical stability with a viscosity coefficient of 10 m2 s−1.

[11] The model domain is a meridional continental shelf with the origin set in the southwestern corner of the domain (Figure 3). The x coordinate points eastward (toward the coast) and the y coordinate points northward (in the downstream direction). The domain is 400 km wide (x direction) and 800 km long (y direction). The coastal wall is specified at the eastern boundary (x = 400 km), while other boundaries are open (see below). The depth h is uniform in the alongshore direction (that is, all isobaths are parallel with the coastline). The depth is constant near the western boundary (representing the deep ocean) and exponentially decreases onshore in the zone of continental shelf and slope:

equation image

where α = 2 × 10−5 m−1 and x0 = 105 m. The water depth near the coastal wall is ∼5 m. The model grid has a spatial resolution of 2.5 km in both horizontal coordinates. The Coriolis parameter is set to f = 6.39 × 10−5 s−1, which corresponds to the latitude of 26°N.

Figure 3.

Model geometry, plan view. Isobaths are in meters; dashed line is the storm trajectory. Instantaneous position of the storm center (xs, ys) is shown as a circle.

[12] No-slip, no-normal-flow boundary conditions are applied at the coastal boundary. The Orlanski-type radiation boundary conditions [Orlanski, 1976] for both normal and tangential phase speed are applied at the northern and southern boundaries. At the deep ocean (western) boundary, perturbations are radiated with the long gravity wave speed [Chapman, 1985]. A quadratic stress is specified at the bottom with the bottom drag coefficient of 10−3. The model is forced by a variable wind stress field associated with the passage of an atmospheric cyclone.

[13] The center of the storm system has coordinates xs, ys and moves with a translation speed us, vs along x and y coordinates, respectively. In the basic case, the initial position of the cyclone is xs(0) = −100 km, ys(0) = 400 km, and it moves eastward at 10 m s−1 along a straight line perpendicular to the coastline (Figure 3). The translation speed is relatively high and is chosen to be similar to Wilma's translation speed at the time of its landfall. The storm system starts its movement from the outside of the model domain to avoid an impact of the strong wind abruptly applied at the initial moment of integration.

[14] The wind filed is radially symmetric relative to xs, ys and is described by the gradient wind approximation [e.g., Holton, 2004]:

equation image

where Vg is the gradient wind speed (wind blows in the azimuthal direction), R is the distance from the storm center, f is the Coriolis parameter, ρa = 1.2 kg m−3 is the air density, and ∂p/∂n is the radial pressure gradient (positive to the left of the wind direction). The atmospheric pressure disturbance is prescribed as a function of a radial distance R:

equation image

where p0 is the pressure disturbance in the center (xs, ys), and Rm is the radius of the storm system. Equations (2) and (3) yield the analytical expression for the gradient wind:

equation image

In the basic case, Rm = 200 km, β = 15, p0 = 3 hPa and the corresponding radial distribution of a pressure disturbance and a gradient wind are shown in Figure 4. The wind stress is calculated at each time step following Large and Pond [1981] from the circular gradient wind field centered at xs, ys.

Figure 4.

Standard case, radial distribution of (left) pressure perturbation and (right) gradient wind in the cyclonic storm system. Zero radial distance corresponds to the center of the storm (xs, ys).

[15] The model starts from rest and is forced by the wind stress field associated with a propagating atmospheric cyclone. The time step is set to 10 s and the duration of a model run in the standard case is 2.5 days.

3.2. Model Results

[16] The center of the atmospheric storm system crosses the model domain and makes a landfall at 13.9 h at y = 400 km. A free surface displacement near the landfall time (t = 14 h) is shown in Figure 5. At this moment, the displacement is maximal near the coast (where the wind-induced drift is blocked by the coastline) and is confined within 100 km of the landfall site, where the wind is substantial (exceeding 4 m s−1; see Figure 4). The displacement is positive in the southern sector of the landfall area and is negative in the northern sector, following a cyclonic wind circulation. Although a maximum wind is ∼12 m s−1, its brevity and spatial localization leads to a fairly small surge not exceeding 15 cm. The small amplitude of the response warrants the linear dynamics and allows a comparison of model results with the linear wave solution. As the storm systems moves “inland” and the wind subsides, the initial free surface disturbance spreads alongshore, both upstream and downstream, while the maximum disturbance remains trapped by the coastline (Figure 5). Alongshore propagation of the initial free surface disturbance occurs in the form of a wave, as both free surface undulations and a high propagation speed indicate. Wave characteristics can be inferred from the phase diagram (Figure 6) showing a temporal evolution of the free surface disturbance at the coast.

Figure 5.

Standard case, instantaneous free surface elevation (in cm) at 14 (landfall), 18, and 20 h.

Figure 6.

Temporal and alongshore evolution of the free surface disturbance (in cm) at the coast, standard case. The thick solid line represents a phase speed of C = 11.5 m s−1; horizontal bar indicates a time interval of 3.47 h.

[17] Maximum free surface displacements (both positive and negative) occur within the landfall area (300 < y < 500 km) at and soon after the landfall (t = 14–14.7 h). Following the landfall, a wave of positive/negative displacement radiates upstream (southward)/downstream (northward), respectively. Those leading waves have small amplitudes: ∼1 cm for the upstream-propagating and ∼4–5 cm for the downstream-propagating wave. They are followed by waves of opposite displacement with the largest amplitude through the event: ∼6 cm for the wave of positive displacement travelling downstream. This wave is reminiscent of the wave pulse observed after Wilma's landfall. It has a phase speed of 11.5 m s−1; a time interval between the two consecutive zero contours is 3.47 h, giving a wave period of ∼6.92 h. Subsequently, the wave trains quickly decay. Since waves travel in both directions and are trapped by the coastline, we conclude that they are the edge wave modes (further evidence will be presented in the next section). It should be noted that the wave trains propagating upstream and downstream are not entirely symmetric, which might be related with the effect of Earth's rotation.

4. Edge Wave Modes

[18] Dispersion characteristics of edge wave modes are obtained by considering a barotropic inviscid coastal ocean on an f plane. The x coordinate coincides with the coastal wall and the y coordinate points offshore; the orientation of the coordinate system applied here differs from the numerical model described in the previous section but this difference is immaterial for the wave properties. The linear shallow water equations describing free waves in this system are:

equation image
equation image
equation image

where u and v are the alongshore and offshore velocity components, respectively, and η is the free surface perturbation from the equilibrium. Subscripts denote a partial differentiation, where t is the time.

[19] The water depth changes only in the across-shelf direction: h = h(y). We seek a wave-like solution periodic with respect to time and alongshore coordinate, so that the modal structure depends on y only:

equation image

where k is the alongshore wave number and ω is the wave frequency. Solving equations (5) and (6) for u and v and substituting the results in equation (7) yield an equation for the free surface modal structure (where asterisks are omitted for convenience):

equation image

The appropriate boundary conditions for wave modes trapped in the coastal ocean are:

equation image
equation image

The former condition means no flow through the coastal wall, while the latter implies the offshore decay of the wave motion. In numerical calculations, boundary condition (equation (11)) has to be approximated at the finite offshore distance L. Brink [1982] suggested to use the following boundary condition as a proxy for equation (11):

equation image

Equation (8) is nonlinear in ω and k. In order to reduce it to a standard eigenvalue problem, we assume a spectral parameter to be constant equation image = Q and rewrite equations (9), (10), and (12) as:

equation image
equation image
equation image

The boundary problem (equations (13)(15)) is approximated at N grid points with central differences (forward/backward differences at the boundaries y = 0, L, respectively), which yields a matrix eigenvalue problem in the form:

equation image

where A is the N × N tridiagonal matrix, X is the column vector whose elements are η(i) (i = 1,…N), and μ = k2 is the eigenvalue. This eigenvalue problem is solved by using the MATLAB routine EIG. The depth profile h(y) and the spatial discretization are exactly the same as in the numerical model described in the previous section.

[20] The solution corresponds to finding intersection points of the lines of constant phase speed C = ω/k = f/Q with the dispersion curves of trapped wave modes. The positive eigenvalues (real wave numbers) correspond to the wave modes propagating alongshore. The mode number is identified by the number of zero-crossings in the mode structure (eigenvector) η(y). The calculations are repeated for different values of Q, both positive (downstream propagation of the wave mode) and negative (upstream propagation). The results for the three lowest edge wave modes are presented in Figure 7. For a given wave number, the wave frequency and the phase speed increase with the mode number. The trapped wave modes are limited in high frequencies by the dispersion relation for Poincaré waves [Huthnance, 1975]; if ω2gHk2 + f2 then the wave energy radiates offshore in the form of the inertial gravity Poincaré wave. Earth's rotation modifies the edge wave modes: dispersion curves of downstream- and upstream-propagating modes coincide at higher frequencies but diverge at lower frequencies (when the Earth's rotation becomes important). In that, the upstream-propagating modes have higher frequencies (higher phase speed for a specified wave number) than the corresponding downstream-propagating modes; this property was predicted by Huthnance [1975]. The discrepancy is the largest for a zero mode.

Figure 7.

Dispersion diagram of the edge wave modes propagating downstream (solid lines) and upstream (dashed lines). The heavy line outlines a continuum of Poncaré waves. The gray solid line represents a phase speed of C = 11.5 m s−1. Structures of waves marked with triangles are shown in Figure 8.

[21] The downstream-propagating zero mode is in fact a hybrid of two types of waves: an edge wave and a Kelvin wave. While the upstream-propagating zero mode merges with the continuum of Poincaré waves at low wave numbers (Figure 7), a dispersion curve of the dowsnstream-propagating mode at low wave numbers asymptotically approaches a dispersion curve of the Kelvin wave:

equation image

To better illustrate this feature, three zero-mode waves are marked with triangles on the dispersion curve (Figure 7) and their corresponding across-shelf structures are presented in Figure 8 (top). Wave 1 has a structure which extends well beyond the area of a continental shelf and slope (the latter is limited by 300 km) and gradually decays offshore over a deep ocean. It is a Kelvin wave modified by the presence of a variable depth at the coastal wall and decaying offshore on the scale of a barotropic Rossby radius Rd = equation image / f. Its phase speed of 125 m s−1 is very close to 140 m s−1, a phase speed of the “pure” Kelvin wave in equation (17) corresponding to the deep-sea depth of 2000 m. Wave 2 has a lower phase speed of 40 m s−1 and most of its structure concentrates over the area of a variable depth, yet some disturbance extends further beyond, over the deep sea. Finally, wave 3 has a phase speed of only 8.5 m s−1 and represents a “pure” edge wave: its structure is trapped on the shelf and the wave is not affected by the deeper portion of the depth profile. Furthermore, its period is only 2.62 h and the effects of rotation are insignificant.

Figure 8.

Across-shelf structure of the edge wave modes normalized by a free surface elevation at the coast: (top) mode 0, waves 1 (k = 1.072 × 10−6 m−1, ω = 1.34 × 10−4 s−1), 2 (k = 4.171 × 10−6 m−1, ω = 1.668 × 10−4 s−1), and 3 (k = 7.841 × 10−5 m−1, ω = 6.665 × 10−4 s−1) shown on the dispersion diagram in Figure 7 and (bottom) wave modes propagating downstream at C = 11.5 m s−1. Dashed line is the across-shelf free surface displacement from the standard case at t = 21 h, y = 561.25 km.

[22] We now identify the waves generated by the storm system (i.e., Figures 5 and 6) using the solution of a boundary problem (equations (13)(15)). A phase speed of 11.5 m s−1 from Figure 6 is plotted on the dispersion diagram (Figure 7) for the reference, and the wave modes 0, 1 and 2 with the same phase speed are shown in Figure 8 (bottom). These theoretical wave structures are compared with the across-shelf free surface profile from the standard case numerical experiment at t = 21 h and the alongshore coordinate y = 561.25 km. These time and alongshore location correspond to the wave crest passing by and traveling downstream from the landfall site (Figure 6). A free surface disturbance in the numerical model fits almost precisely a zero-mode wave structure. The period of this harmonic wave is 7.08 h (ω = 2.46 × 10−4 s−1), which is close to the wave period of 6.92 h derived from Figure 6. It should be noted here that the wave in the numerical experiment is not a monochromatic harmonic wave (like the one selected from the dispersion curve) but rather is a spectrum produced by the initial disturbance, so the timescale from Figure 6 is used here only as a scale for the comparison purpose. A good agreement between the sea surface perturbation in the numerical model and the theoretical wave mode proves that the long wave response to the storm landfall indeed occurred in the form of edge waves, predominantly of the zero mode, with large spatial and temporal scales matching the scales of the storm system.

5. Additional Numerical Experiments

[23] In this section several other numerical experiments are presented which demonstrate how the long wave response is altered by the storm trajectory, translation speed or wind speed. These calculations are not intended as an exhausting parameter study (the latter is left for the future) but rather as an illustration of a possible importance of the edge waves.

[24] Model case B is the exact repetition of the standard case, except that the translation speed of the atmospheric cyclone is now 5 m s−1 (half of the standard case value). The storm center now makes a landfall at 27.8 h (Figure 9a), compared to 13.9 h in the standard case. The forced storm surge lasts longer than in the standard case, which results in a longer timescale of the trapped wave modes radiated from the landfall area. As the wave period increases, the effects of Earth's rotation become more important (that is, the wave frequency is closer to the inertial frequency). The wave response is more asymmetric than in the standard case, with a larger fraction of the wave energy propagating downstream. Also, the phase speed increases to ∼15 m s−1 for downstream-propagating waves and even faster for the upstream-propagating, following the dispersion characteristics of the edge waves (Figure 7).

Figure 9.

Same as in Figure 6, but for (a) model case B and (b) model case C.

[25] When the storm translation speed is further reduced to 3 m s−1 in model case C (Figure 9b), the wave response becomes subinertial. The landfall in this case occurs at 46.3 h which necessitated a longer model run (72 h). The wave phase speed is reduced to 9.5 m s−1 reflecting an overall slower propagation of the subinertial continental shelf waves (a barotropic limit of CTW). The phase of all wave perturbations travels downstream in this case. However, there is some wave energy spreading upstream of the landfall area (y = 200–300 km) at 47–55 h with the individual sea level contours still tilting in the downstream direction thus indicating downstream phase propagation. This feature is related with the generation of “short” shelf waves with the group velocity and the energy flux in the opposite (upstream) direction relative to the phase speed [e.g., Huthnance, 1975; Wilkin and Chapman, 1987]. Thus the edge waves are replaced by the continental shelf waves when the storm system travels slower.

[26] Time series of the alongshore velocity component and the free surface displacement at ∼150 km downstream from the landfall site (Figure 10) further illustrate the long wave response. The across-shelf wind stress component at y = 351.25 km (where the wind is close to its maximum strength) is shown for the reference. It is obvious that the timescale of the sea level fluctuations increases with the increase of the wind duration or is inversely proportional to the translation speed. The interesting feature is that the alongshore velocity becomes stronger while the free surface displacement becomes smaller as the wave period increases. The increase of the alongshore velocity following the reduction of the translation speed reflects an overall increase of the energy transmitted from the storm system to the ocean. However, as the wave frequency becomes lower, the horizontal divergence of the flow field decreases and in fact the subinertial motions can be treated as nondivergent under the rigid-lid approximation [e.g., Huthnance, 1975]. This tendency depicted in Figure 10, emphasizes the possible danger of large-scale edge waves for coastal flooding if compared to the subinertial coastally trapped waves traditionally associated with storm forcing.

Figure 10.

Time series at the coast (x = 400 km) from model cases: standard (thick black lines), case B (thin black lines), and case C (gray lines) of (top) the across-shelf wind stress component at y = 351.25 km, (center) the free surface disturbance at y = 551.25 km, and (bottom) the alongshelf velocity at y = 550 km. Actual model time is adjusted to plot the time series together.

[27] The edge wave response to the fast-moving storm system traveling at a right angle to the shelf and coastline is almost symmetrical, with wave trains of almost identical amplitude propagating both downstream and upstream. As a result, the wave amplitude is less than a half of the forced surge at the coast (Figure 6). The situation changes dramatically when the storm system coming ashore has an alongshelf component in its translation velocity. In case D (Figure 11a), the storm system travels at the same translation speed of 10 m s−1 as in the standard case, but the storm trajectory is offset by 30° clockwise from the normal angle of approach. That is, the translation velocity has a meridional component of −5 m s−1 (while the zonal component is 8.66 m s−1). The starting point of the storm system's center (xs, ys) is shifted downstream so that the landfall occurs exactly at the same location as in the standard case (y = 400 km), albeit later in time (t = 16.04 h). The edge wave trains generated by the landfall of a cyclone are highly asymmetric in terms of their amplitude: most of the edge wave energy radiates upstream (southward) and the wave amplitude (∼11 cm) is comparable with the magnitude of the forced storm surge at the landfall site. The amplitude of the wave train travelling downstream (northward) is insignificant (∼3 cm). Case E (Figure 11b) is a mirror image of the previous case: the storm system translation velocity now has a downstream component of 5 m s−1, that is, its trajectory is turned by 30° counterclockwise from the normal angle of approach. Most of the edge wave energy in this case radiates downstream, in the direction consistent with the alongshore component of the translation velocity.

Figure 11.

Same as in Figure 6, but for (a) model case D and (b) model case E.

[28] In the last model case F discussed here, the atmospheric pressure anomaly is decreased to −20 hPa, which increases a corresponding value of the maximum gradient wind in the storm system to ∼34 m s−1 (hurricane wind speed). The bottom drag coefficient is also increased to 3 × 10−3, which represents a contribution of surface gravity waves under conditions of stronger wind-forcing. Otherwise the storm structure and translation speed are the same as in the standard case. The resulting phase diagram of the free surface evolution at the coast is shown in Figure 12, now the contours are plotted in meters. Both the nonlinear effects and the bottom friction are potentially more important for the wave propagation in this case. Indeed, the wave amplitude decays much faster than in the standard case. Nevertheless, a characteristic phase speed of the positive-displacement edge wave propagating downstream is close to the standard case (both are ∼11.5 m s−1). Thus the small amplitude solution remains a good proxy for the alongshore propagation of large-scale edge waves under strong wind-forcing.

Figure 12.

Same as in Figure 6, but for model case F. Free surface disturbance is in meters.

6. Discussion and Conclusions

[29] Both the sea level measurements during Wilma's landfall in October 2005 and the idealized numerical experiments demonstrate the generation of large-scale edge waves in the form of a wave pulse or a short wave train. This generation occurs when the storm approaches a coastline along the close-to-normal trajectory and has a high translation speed. This scenario warrants a short Eulerian timescale of the forced surge (in the edge wave frequency domain), which later translates into a freely propagating wave. Unlike CTWs, edge waves travel in both directions relative to the coastline and the wave response is almost symmetric both upstream and downstream of the landfall site if the effect of Earth's rotation is weak (that is, ω > f). The deviations of trajectory from a right angle (or, equivalently, the coastline variations) can be important in making a wave response almost unidirectional relative to the coastline. In particular, most of the energy in such a case will travel in the direction of the alongshore translation velocity component of the storm system. This will increase the amplitude of the edge wave compared to the two-directional wave response.

[30] A few previous observational examples available demonstrate an edge wave generation by tropical storms travelling parallel with the coastline in the upstream direction [e.g., Munk et al., 1956; Beardsley et al., 1977]. In both studies, edge waves with periods of several hours formed a short wave train in the wake of a fast tropical cyclone propagating northeastward along the Middle Atlantic Bight shelf. In both cases the wave amplitude was substantially smaller than in the present case and varied within 50–80 cm. Clearly, this is a different scenario compared to what is proposed in this paper, which motivated additional model experiments with the storm system travelling alongshore. In these model cases the storm system had the same structure as in the standard case and traveled at 10 m s−1 along the coast both downstream and upstream, with the storm center being 100 km offshore. The shelf response includes the forced bulge of a free surface propagating with the storm system, and the long-period free surface oscillations after the storm passage. These slow oscillations have higher amplitudes when the storm travels downstream and are likely related with the subinertial shelf wave modes. In addition, there are higher-frequency long gravity waves radiated offshore, but without a distinctive alongshore phase propagation. Thus there is no visible edge wave generation or their amplitude is so small that their detection requires special analysis techniques. The result is quantitatively similar to that of Tang and Grimshaw [1995] but obviously contradicts the previous observations.

[31] There are two reasons that can explain this contradiction. A storm system traveling alongshore generates wave modes with the phase speed being similar to the translation speed and a wave structure which best fits the forced coastal ocean response. The forced response typically occupies the whole shelf width which would require edge wave modes that are faster than the typical translation speed. The slower coastally trapped waves fit the structure of the moving forcing system much better, unless the translation speed is very high. Indeed, Tang and Grimshaw [1995] considered a storm system travelling at 5.6 m s−1, which is likely too slow to efficiently generate the edge waves. On the other hand, in the observations reported by Munk et al. [1956], the storm travelled alongshore at ∼15 m s−1, faster than in the model cases discussed here. Another possible reason is that both in this study and in the study of Tang and Grimshaw [1995], the coastal ocean was forced by the wind stress, while both Munk et al. [1956] and Greenspan [1956] argued that the edge waves were generated by the atmospheric pressure anomaly associated with the storm system. This notion has an important implication. A cyclonic wind field moving alongshore produces a shelf response reminiscent of the 1st wave mode, that is, surge at the coast and a drop of sea level farther offshore due to the cyclonic wind stress curl. The atmospheric pressure anomaly, on the other hand, produces a sea surface response in the manner of an inverse barometer with a cross-shelf structure that can be approximated by the zero (Stokes) mode. Both Greenspan [1956] and Munk et al. [1956] argued that only a zero edge wave mode is slow enough to be efficiently generated by the traveling storm system. This implies that atmospheric forcing in the form of wind stress only (without corresponding pressure anomaly) can potentially underestimate the role of edge waves in the coastal ocean response to the storm propagating alongshore.

[32] Zero-mode edge waves were generated in our numerical experiments when the storm system made a landfall. They have the simplest possible structure with the free surface exponentially decaying offshore. Interestingly, a zero downstream-propagating wave mode is a “hybrid” wave, as it resembles a Kelvin wave at lower frequencies and an edge wave at higher frequencies. At the intermediate frequencies this wave mode has a close to zero group speed Cg = ∂ω/∂k. This implies that the wave energy won't propagate beyond the area of wave generation. In our case of shelf topography, these waves with a near-zero Cg have periods of ∼10 h but it is possible that in some other areas they might exist at tidal semidiurnal frequencies, which makes the large-scale edge (or hybrid edge-Kelvin) waves potentially important for the tidal dynamics.

[33] Exponential topography applied in our numerical calculations results in a more efficient trapping of long gravity wave energy than over the linear depth profile used for example, in the analytical solution by Ursell [1952]. This is evident in a more rapid offshore decay of a zero mode in the present case (Figure 8, top, waves 2 and 3) compared to the linear depth profile [e.g., Yankovsky, 2008, his Figure 8]. This implies that the large-scale edge waves generated by a fast-moving cyclone can be quite ubiquitous as their generation does not require a very wide shelf. Nevertheless, their detection in observations is a challenge because their fast propagation and a relatively short duration make it difficult to distinguish them from the forced surge. For this reason, numerical experiments offer a reasonable alternative for further insight into their dynamics.

Acknowledgments

[34] This work was supported by the National Science Foundation through grants OCE-0650194 and OCE-0752059. Special thanks are extended to Phil Moore (University of South Carolina) for technical assistance with numerical simulations.