We investigate the rapidly changing equilibrium between the momentum sources
and sinks during the passage of a single two-peak storm over the Catalan
inner shelf (NW Mediterranean Sea). Velocity measurements at 24 m water
depth are taken as representative of the inner shelf, and the cross-shelf
variability is explored with measurements at 50 m water depth. During both
wind pulses, the flow accelerated at 24 m until shortly after the wind
maxima, when the bottom stress was able to compensate for the wind stress.
Concurrently, the sea level also responded, with the pressure-gradient force
opposing the wind stress. Before, during and after the second wind pulse,
there were velocity fluctuations with both super- and sub-inertial periods
likely associated
with transient coastal waves. Throughout the storm, the
Coriolis force and wave radiation stresses were relatively unimportant in the
along-shelf momentum balance. The frictional adjustment timescale was around
10 h, consistent with the
The inner shelf, encompassing depths ranging from a few to tens of meters, is dynamically defined as the region that lies between the surf zone (where waves break and the momentum balance is dominated by wave-induced terms) and the mid-shelf (where the along-shelf circulation is usually in geostrophic balance) (Lentz and Fewings, 2012). The circulation over the inner shelf is often investigated through the analysis of the momentum balance in different regions, although most studies have usually focused on those conditions averaged over periods longer than 1 week (Lee et al., 1984; Lentz and Winant, 1986; Lentz et al., 1999, Maza et al., 2006; Fewings and Lentz, 2010; Grifoll et al., 2012, 2013). In contrast, the analysis of the momentum terms at daily and shorter timescales under rapidly changing forcing conditions remains less explored.
Energetic wind events, such as storms, modify the typical pattern of water circulation over the continental shelf. The proximity of the coastline and the relevance of bottom friction prevent the generation of inertial fluctuations, which often prevail in the mid- and outer-shelf following wind pulses (Salat et al., 1992; Shearman, 2005). Additionally, during the passage of storms, the intense wind, and in some cases the associated cooling, affects a large fraction of the water column. The frictional adjustment time, proportional to water depth and inversely proportional to wind stress, is largely reduced nearshore during such energetic events. As a consequence, the magnitude, phase and relative importance of the dominant terms in the momentum balance is modified. For instance, during the passage of the tropical storm Floyd along the US east coast, Kohut et al. (2006) found an increase in both wind stress and the horizontal pressure gradient, with a change in the sign of the terms between the storm and the subsequent relaxation period. During winter wind events on the US east coast, Lee et al. (1984) found evidence of the sea-level slope opposing the wind stress in order to establish a frictional equilibrium that differed from the average conditions. A seasonal study of the Catalan shelf (Grifoll et al., 2013) suggested that the occurrence of one single intense event can dominate the monthly averaged momentum balance, with water piling against the coast as a response to the enhanced wind stress; to balance the surface forcing, the bottom stress term was also increased.
In this paper, we investigate the temporal evolution of the momentum balance terms over relatively short timescales (of order 1 day) during the passage of a storm. The analysis is based on a set of observations from the Catalan inner shelf (offshore the city of Barcelona, NW Mediterranean Sea; Fig. 1). The prevalent momentum terms at two different depths (24 and 50 m) are examined. We take advantage of our setting in a micro-tidal environment to investigate a temporal scale (from hours to a few days) usually not considered in the literature, where the time series are often low-pass filtered to remove the short-term fluctuations (e.g., tidal flow). The goal is to quantify the different momentum terms during the storm and to examine the response timescales of the inner-shelf environment to the principal forcing mechanisms.
Map of the western Mediterranean Sea with the study area
The Catalan shelf is micro-tidal, with tidal amplitudes of the order of 0.1 m. The wind and heat flux regimes exhibit a seasonal cycle associated with the Mediterranean climate and the periodicity of meteorological events in the region. Wind intensity usually has a minimum during summer and is more energetic during fall, winter and spring. During these last seasons, regional storms are predominantly associated with north and northeast winds alternating with northwesterly wind pulses (land winds). Grifoll et al. (2013) analyzed the resulting seasonal circulation pattern over the inner shelf through a combination of numerical and observational techniques. The flow is prevalent in the along-shelf direction year-round, which is consistent with the coastal constraint and the shallowness of the area. The monthly averaged along-shelf momentum balance was between wind stress and pressure gradient, with bottom stress being a second-order term.
In the present study, we focus on a subset of the data analyzed by Grifoll et
al. (2012) that includes an energetic event lasting a few days during March
2011. The bulk of the measurements correspond to a field experiment conducted
over the Catalan inner shelf in the framework of the FIELD_AC project (Grifoll et al., 2012). The data set consisted of velocity time series from three
Acoustic Doppler Current Profiler (ADCP) deployments, each with a pressure
sensor (A1 [AWAC], A2 [AWAC] and A3 [RDI]; Fig. 1). The ADCP bin size was
1 m for A1 and A2, and 2 m for A3 station; the velocity accuracy was
Wave data were recorded through a directional wave buoy [Datawell DWR-G7] moored at A3. A numerical wave model SWAN (Booij et al., 1999) was implemented and calibrated for a region that covers our study area, providing the wave variables with 50 m resolution to estimate the wave-induced momentum terms. The numerical domain is shown in Fig. 1 and the implementation details were presented in Grifoll et al. (2014).
The time series used for estimating the momentum terms have been low-pass
filtered with a 1/12 h
The currents during the entire March–April 2011 field campaign were analyzed in a previous study (Grifoll et al., 2012), highlighting the prevalence of the along-shelf direction and the high correlation between the velocities measured at A1, A2 and A3. For this reason, we focus on the A2 observations, considered to be representative of the dynamics in the inner shelf, and use A1 and A3 to support the momentum-term estimates. The investigation of the cross-shelf variability in the along-shelf momentum equation also uses A3.
We focus on the 12–15 March 2011 period, which includes the passing of a NE
storm with maximum wind velocity of 13 m s
Regional charts of the mean sea-level pressure (HPa) and winds for the sequence 11–14 March 2011. Data source: ERA-Interim global reanalysis from ECMWF.
The along-shelf velocity (Fig. 3b) was characterized by a prevalent southwestward flow in the entire water column, with near-surface velocities typically 4 times larger than the near-bottom flow. The cross-shelf flow (Fig. 3c) was less intense than the along-shelf flow and exhibited a complex vertical structure. As a result, the depth-averaged along-shelf velocities were much larger than the depth-averaged cross-shelf velocities during the two wind peaks (Fig. 3d), reflecting a strong flow polarization associated with the coastal constraint.
During the first day of the storm (12 March), the depth-averaged along-shelf current (Fig. 3d) was toward the southwest with a maximum at 07:00 UTC (2 hours after the wind stress peak). During the calm day (13 March 00:00–22:00 UTC), the wind changed direction slightly toward the northeast (peaking at 15:00 UTC), but the along-shelf currents maintained a similar magnitude and structure than the day before. During the second wind peak the situation repeated itself, but with the along-shelf flow displaying some oscillations. The wind measurements are in good agreement with the values expected from the synoptic charts in Fig. 2.
The cross-shelf currents also displayed a similar time evolution during both wind peaks: the cross-shelf flow intensified with the wind, onshore at the surface and offshore near the bottom; as the wind stress decreased, the flow reversed, turning offshore at the surface and onshore near the bottom (Fig. 3d). During the calm day, the cross-shelf flow was weakly onshore. Following the second wind peak (14 March 16:00 UTC), the surface wind stress decreased gradually from 0.2 Pa to zero (15 March 23:00 UTC). The along-shelf flow remained to the southwest throughout the water column, and the cross-shelf flow was offshore in the sub-surface layers balanced by onshore currents near bottom.
The detided sea level (Fig. 3e) increased during both wind peaks and slowly decreased after the last wind peak. After the storm, the sea level increased as a result of water being piled up against the coast due to the northeasterly wind. The wave conditions measured at A3 were characterized by two significant wave height peaks (Fig. 3f) from the E–SE direction with 8 s period. The peaks of significant wave height followed the peaks of (along-shelf) wind stress, with a delay longer than for the along-shelf currents because of the influence of swell.
Assuming hydrostatic balance, ignoring the sea-level variations as compared
with the total water depth, and neglecting the baroclinic terms (estimated
as small in Grifoll et al., 2012, 2013), the depth-averaged along-shelf
momentum balance equation can be written as
The along-shelf acceleration term at A2 (ACCE. in Eq. 1) is estimated from the observations using finite-centered differences with the velocity recorded at A2 (Fig. 4a). A negative peak is observed in the acceleration time series during the first wind peak. During 13 March, about 1 day after the first wind peak and shortly after the wind weakened (and even reversed), the acceleration term displayed an oscillatory pattern with a repeat interval of about 1 day or less.
Estimates for the along-shelf momentum terms at 25 m. Left-hand-side
terms of Eq. (1):
The non-linear or advective terms are estimated by finite differentiation
between the adjacent ADCP measurements (Kirincich and Barth, 2009; Fig. 4b).
The velocity advection terms (ADVEC. in Eq. 1) were small during the first
peak of the storm but later oscillated in a manner similar to the
acceleration term. There are two additional momentum advection terms related
to changes in the depth of the water column:
The Coriolis term (COR in Eq. 1), computed from the depth-averaged cross-shelf velocities at A2, is small in comparison with the acceleration term during the storm (Fig. 4c). Although the surface and sub-surface cross-shelf flows were relatively important through the water column (Fig. 3c), the depth-averaged cross-shelf flow was much smaller than the along-shelf velocities (Fig. 3d). The size of this term was 4 times smaller than the acceleration term.
The along-shelf wind stress term (W.STR. in Eq. 1; Fig. 4d) was calculated using a neutral drag law (Large and Pond, 1981) from winds measured at the nearby meteorological station (Fig. 1). There was a good correspondence between wind stress and along-shelf velocity (Fig. 3a and d). The maximum (negative) acceleration, however, occurred a few hours before the maximum winds (12 March 05:00 and 12 March 7:00 UTC). After the first wind peak, the acceleration decreased rapidly and changed sign, becoming small but positive during the remaining of the wind pulse. Something similar happened during the second wind peak, but this time characterized by several fluctuations, with acceleration peaks occurring about every 24 h or less (Fig. 4a).
The bottom stress (B.STR. in Eq. 1) is estimated using a linear drag law
(Lentz and Winant, 1986):
The wave-induced mass fluxes (RAD.STR. in Eq. 1) are estimated as follows:
No direct estimate of the along-shelf pressure-gradient force (PRS.GRD. in Eq. 1) can be obtained from the data. The ADCP recorded the pressure in the water column but the distances between ADCPs are not appropriate to capture the along-shelf sea-level variability, as the signal-to-noise ratio is not adequate. Hickey (1984) pointed out that, for spatial scales of the same order of magnitude as the external Rossby radius (about 100 km in the Catalan Sea), the expected sea-level gradient would be only a few centimeters. In our case, the sea-level variations recorded by the pair of pressure sensors in A1 and A2 (separated by only a few kilometers) were of the same order as the accuracy of the devices (order millimeters). As an alternative approximation, an observed pressure-gradient force (PGFO) may be computed using data from a sea-level gauge located in the harbor of Blanes (approximately 64 km to the north; Fig. 1) and the ADCP pressure sensor at A2. This along-shelf pressure gradient remained positive during the entire storm, meaning a downwind accumulation of water, and was reinforced during the two wind peaks (Fig. 4e).
Additionally, we may calculate this along-shelf pressure-gradient force as
the residual from the momentum balance equation (PGFR) as follows:
Both the observed and residual pressure-gradient time series (PGFO and PGFR) reproduce the force direction of the sea-level slope during the wind stress peaks (Fig. 4d). The PGFR includes a contribution by a direct response to the local wind forcing (for instance during 13 March), which is not immediately obvious in PGFO. During the wind peaks, the positive pressure-gradient force partially counterbalanced the wind stress in a manner consistent with other observational studies (Lee et al., 1984; Lentz, 1994; Fewings and Lentz, 2010). Despite its potentially large uncertainty, we use PGFR in the analysis of the momentum balance evolution because it is consistent with the estimates for the other momentum terms.
The cross-shelf variability of the along-shelf momentum is estimated by comparing the inner-shelf results with the momentum terms at 50 m water depth. The acceleration and bottom stress follow a pattern similar to the one observed at 24 m (Fig. 5). During the first wind peak, the acceleration responded to the wind stress, with its maximum occurring before the maximum winds. The maximum bottom stress, as at 24 m, occurred a few hours after the maximum winds. During the second peak, the situation was less clear than at 24 m, with substantial oscillations in the acceleration and bottom stress. The bottom stress reached a minimum toward the end of the negative wind stress pulse.
From the velocities observed at A3, we estimate the surface and bottom
frictional forces, the acceleration and the Coriolis force (Fig. 5). The
surface stress term is estimated with the local wind measured at the CSO,
scaled with the corresponding water depth. The non-linear terms cannot be
estimated due to the lack of additional measurements at 50 m, necessary to
assess the along-shelf gradient. Thus, as the advective and wave-radiation
terms are not available, we estimate the pressure gradient as a residual that
results from combining acceleration, Coriolis and wind and bottom friction
(PGFR
Estimates for the along-shelf momentum terms at 50 m. Left-hand-side
terms of Equation 1: acceleration terms (ACCE.
From the estimated along-shelf momentum terms, we conclude that the primary balance at 24 m (Fig. 4) took place between acceleration, wind stress, bottom friction, momentum advection and pressure force gradient. The Coriolis force and the radiation stress played secondary roles in the momentum balance. In particular, the radiation stresses were 1 order of magnitude smaller than the dominant acceleration, pressure gradient and frictional terms, consistent with other studies that ignored wave forcing outside the surf zone (Lentz et al., 1999; Fewings and Lentz, 2010). In a region 150 km north of our study area, in water depths of 28 m off the Tet River, Michaud et al. (2012) confirmed numerically that the wave effects on the inner-shelf circulation are relatively small even during a storm event.
While our estimates present some uncertainty as a result of instrumental inaccuracies and the intrinsic assumptions in the estimation of the terms, we expect them to be reasonable approximations to the relative size of the dominant terms in the time-varying along-shelf momentum balance. In this section, we combine the wind and velocity observations with the along-shelf momentum estimates to further analyze the inner-shelf response to the changing winds.
During the first peak (12 March), as the wind stress increased, the acceleration term initially became more negative and the bottom stress more positive, as expected from the direction of the flow (Fig. 4a). The peak in the acceleration term occurred 4 h before the wind maximum, as a result of the enhanced frictional dissipation and a rapid change in the residual pressure-gradient force (PGFR), from negative to positive values (Fig. 4a, f and g). The change in the sign of PGFR, indicative of downwind water accumulation, was abrupt, being responsible for switching the acceleration from negative to positive at about 10:00 UTC, hence setting the size and timing of the maximum along-shelf current (Fig. 3d).
During the calm period (13 March), the PGFR once again reversed, likely caused by a relaxation after the first wind peak. The acceleration remained close to zero until 13 March 10:00 UTC and then turned negative, despite the appearance of weak northeast winds. The negative acceleration started at the time when onshore winds were observed (Fig. 3a). The intensification of the southwestward PGFR was locally reflected by a leveling of the sea surface throughout 13 March (Fig. 3e). The sequence of events is consistent with the intensification of a southwestward flow probably in cross-shelf geostrophic balance. During 14 March, when both along- and cross-shelf winds were weak, some nearshore water was progressively released, first through a two-layer baroclinic cross-shelf flow and then by offshore flow in the entire water column (Fig. 3c).
The along-shelf momentum balance during the second wind peak shared some characteristics with the balance during the first wind peak but also displayed important differences. The acceleration term and the PGFR were enhanced following the increase in wind stress, with the along-shelf velocity reaching a maximum at the time of the second wind peak. However, the second wind event also had substantial fluctuations in the acceleration, advective, bottom friction, Coriolis and PGFR terms (Fig. 4). The fluctuations appear as a moderate increase in energy at the 12–16 h band in the wavelet analysis for the depth-averaged currents at station A2 (Fig. 6). These oscillations are consistent with fluctuations in the lowest part of the water column in the cross-shelf velocities (Fig. 3c), and could be explained as a transient coastal current response to the sudden enhanced wind stress in the form of inertio-gravity waves (Kundu et al., 1983; Tintoré et al., 1995). These waves are associated with near-inertial motions resulting from the flow adjustment at the coast. Another plausible explanation of these oscillations is the generation of internal waves dispersed from the surface (wind-mixed layer) to larger depths in the lee of the storm (Gill, 1982; Kundu and Thompson, 1985) or fast coastal Kelvin waves (Csanady, 1982; Gill, 1982). Their proper characterization would require additional observations in the cross- and along-shelf directions jointly with more detailed information of the stratification in the water column before and after the storm.
(Left) wavelet analysis and (right) spectral analysis for the depth-averaged currents at station A2. The solid thin line in the wavelet power spectrum shows the energy values that exceed the significance level; similarly, the dashed line in the global power spectrum shows the significance level. The date indicates 00:00 UTC.
The along-shelf velocity spectral and wavelet analyses also show the
existence of fluctuations with a dominant periodicity of about 2 days,
although only significant for 14 and 15 March (Fig. 6). Sub-inertial
fluctuations in that band of the spectrum may reflect the propagation into
the study area of free coastal waves generated elsewhere during the first
part of the storm. The propagation of topographic waves is briefly considered
in Appendix B, using the Csanady (1982) coastal-strip model to assess the
magnitude of the oscillations for a linear and frictionless coastal band. The
theoretical lowest-mode topographic waves have a period of about 32 h, which
is similar to the dominant period as deduced from the spectral and wavelet
analyses (Fig. 6). Observed sea-level oscillations (Fig. 3e), as large as
0.05 m on timescales of about 1–2 days, gave rise to velocities of about
0.3 m s
The differences between the observed and residual pressure-gradient forces, PGFO and PGFR, are striking (Fig. 3). The PGFR is necessary to balance the flow yet it differs substantially from the pressure gradient as calculated from the A2 and Blanes sea-level gauges (64 km apart). Hence, we may interpret the PGFR as composed of two contributions: (1) a rapid coastal response to the along- and cross-shelf wind forcing, and (2) a relatively slow sea-level adjustment to the propagating storm. Our interpretation is that the PGFO corresponds to the second, relatively smooth, contribution, which would drive the flow in the absence of waves (as it occurs during the first wind event).
Our analysis highlights the importance of the initial conditions (whether the system starts from rest or not) and the complexity of the momentum fluctuations during the development of the storm. A comparison of the temporal evolution of the momentum terms during both wind peaks shows that the role of the acceleration and advective terms is quite different. During the first peak, the advective terms are relatively small as a result of the linear response of the pressure gradient and bottom stress to the wind forcing. After the first peak, however, the acceleration and advective terms display fluctuations that may reflect transient waves. Hence, the along-shelf velocity during the second peak is the cumulative effect of a local response to wind stress combined with the arrival of waves that barely feel the effect of bottom friction.
During the first wind peak, the increase in along-shelf velocity enhanced the
bottom stress, which eventually balanced the joint effect of wind stress and
along-shelf pressure gradient, therefore achieving a complete frictional
adjustment. A measure of the frictional adjustment time can be extracted from
the observations, considering the cross-zero momentum and inflexion points in
the time series. During the first peak (12 March), the flow
started from near rest and the non-linearities were small. After the first peak,
the momentum balance was affected by fluctuations in the acceleration and
advection terms responding to the cross-shelf slope momentum term. Hence, a
maximum value for the frictional adjustment time is estimated as the time
between zero and maximum bottom stress during the first wind pulse, or about
14 h (from 11 March 20:00 UTC to 12 March 10:00 UTC). This frictional adjustment
time doubles the frictional time as computed from the linear drag law of the
bottom stress term (
To provide a framework for the frictional time, we consider the linearized analytical model from Csanady (1982), applied to the first wind peak period. The
along-shelf velocity response to a steady wind stress, considering only
bottom friction, is controlled by the following expression:
It is important to point out that the sea surface adjustment time is similar to the time period between the two wind peaks. The along-shelf momentum generated by the first wind pulse did not have sufficient time to dissipate before the second wind peak arrived. In our case, this is further complicated by the potential arrival of upstream waves. This progressive adjustment may be appreciated in the time evolution of the sea surface, where the sea level approached some near-equilibrium value towards the end of each wind pulse (Fig. 3e). The sea-level time series also illustrates that the adjustment is characterized by at least two separate components: one responding to local wind forcing and the local response to remotely generated waves and another, much slower, associated with the sea-level adjustment at scales of the order of the external Rossby radius (Hickey, 1984), about 100 km in the Catalan shelf.
The short frictional adjustment time is consistent with the study site being
part of the inner shelf during the storm. The inner shelf is defined as the
region where the combined surface and bottom boundary layers occupy the
entire water column (Lentz, 1994). Obviously, the boundaries of the
inner-shelf region vary in time depending on the intensity of the forcing
mechanism. The bottom and surface Ekman depth can be obtained from empirical
formulations such as (Weatherly and Martin, 1978)
Estimates of the surface (continuous line) and bottom (dashed line) Ekman depths (m). The gray patches show the periods where the sum of the surface and bottom Ekman depths exceeded 24 m (the water depth at station A2). The date indicates 00:00 UTC.
The cross-shelf variability of the along-shelf momentum is investigated with
the help of the momentum terms calculated at 50 m water depth. The
frictional adjustment time, as observed through the evolution of the momentum
terms at 50 m, exhibited a 4 h lag when compared with the frictional
adjustment time at 24 m (Sect. 4.1), likely caused by the time delay in the
vertical transfer of momentum. The linear drag coefficient is calculated
following the same approach as for the analysis at 24 m, now adjusting the
PGFR
(Top panel) cross-shelf transect of the frictional adjustment times
for several along-shelf wind stresses
The relative ratio of the fluctuations (standard deviations) in the
acceleration and wind stress terms,
The size of the Coriolis term at 50 m increases in comparison to the size of
this term estimated at 24 m water depth. The standard deviation of the
Coriolis term also increases offshore, from
9.6
The increasing/decreasing importance of the Coriolis/bottom-frictional terms responds to a switch towards the geostrophic balance, typical in the transition from the inner to the mid-shelf. With increasing depth, the frictional effects decrease in the along-shelf momentum balance. Lee et al. (1984) observed that the Coriolis term doubled from 28 to 75 m water depth in the South Atlantic Bight (USA), with progressively smaller frictional terms. At the start of our study period (11 March 2011, Fig. 5), the 50 m water depth Coriolis term was larger than the other terms, suggesting a geostrophic balance during the calm period.
During the storm period, the PGFR
We may finally compare the cross-shelf changes of the frictional and
geostrophic terms in the along-shelf momentum balance (Fig. 8). Their
relative importance can be explored by comparing the bottom frictional time
(Eq. 8) with the inertial time (
We have assessed the effects of the passage of a storm over the inner Catalan shelf (NW Mediterranean Sea) on the along-shelf momentum balance. At 24 m water depth, a primary momentum balance between acceleration, pressure-gradient and frictional forces (surface and bottom) is established. The Coriolis and the wave-induced momentum terms play a second-order role in the momentum balance. Our estimates for the frictional adjustment time and Ekman depth confirm the prevalence of the frictional response of the flow at 24 m. The increasing importance of Coriolis at 50 m corresponds to a shift towards the geostrophic behavior, characterizing the transition from the inner to the mid-shelf.
The storm (12–15 March 2011) had two separate wind peaks that caused
currents with some similarities but also important differences. The main
similarity was the local response to wind forcing, with the along-shelf flow
(towards the southwest) accelerating to a maximum shortly after the wind
peak, at a time when the joint effect of bottom stress and a northeastward
pressure-gradient force (arising from downwind water set up) compensated for
the decreasing wind stress. Such a response was more obvious during the first
peak (13 March), as the system started from a condition of weak along-shelf
flow. During the relatively calm period between both wind peaks (14 March),
the pressure-gradient force turned toward the southwest and the flow remained
in that direction. During this period, the winds were weakly onshore, likely
setting a cross-shelf geostrophic balance. By the end of the calm period, and
lasting through the second wind peak (15 March), the momentum balance was
characterized by the appearance of fluctuations with both super-inertial
(12–16 h) and sub-inertial (1–2 days) periods. During the second wind
peak, the temporal sequence of increased acceleration followed by opposing
bottom stress and pressure gradient reoccurred, but with the along-shelf flow
largely influenced by the sub-inertial (likely topographic) waves, with
velocity amplitudes as large as 0.3 m s
In our analysis, we have focused on the shelf response to a single twin storm, where extensive observational data were available. However, northeasterly energetic wind events are common during spring and fall in the Catalan shelf; therefore, similar events are expected on a yearly basis. The extrapolation of our results to other shelves depends on physical variables such as stratification, river discharge and remote sea-level forcing. In relatively low-energy shelves, such as the Catalan shelf, it is plausible that two-peak storms be commonly characterized by a sequence of a linear response followed by a subsequent non-linear behavior.
Early investigations (e.g., Scott and Csanady, 1976) pointed out that tidally driven fluctuations hinder the analysis of the wind-induced fluctuations over relatively shallow depths, where the frictional adjustment time may be similar or shorter than the tidal period. In these environments, for typical wind events, the wind-forcing, tidal and frictional periods are all comparable; in particular, the semidiurnal and frictional times become similar at depths of about 60 m (Csanady, 1982). In contrast, the micro-tidal nature of Catalan shelf has allowed us to investigate in detail the shelf response at temporal scales shorter than previously investigated for the inner shelf.
A simple model to determine the frictional time adjustment was presented by
Csanady (1981), based on the transport momentum equation in the along-shelf
direction:
Under the assumption that the depth distribution is only a function of the
cross-shelf coordinate, we neglect the along-shelf pressure gradients. Also,
the coastal constraint near the coast implies
The bottom stress was parameterized by Csanady (1981) using a quadratic drag
law equation as a function of the depth-averaged current (
Integrating, the solution for the along-shelf transport follows an
exponential equation
Alternatively, we may consider the bottom frictional term to depend linearly
on the depth-averaged current:
In this case, the solution follows again an exponential solution:
Here, we follow Csanady (1982, Sect. 4.5) to estimate the size of the
coastal propagating anomalies. We look at the non-forced propagation of a
sea surface perturbation, with elevation
The second condition may be simply specified from the requirement of a
finite-size perturbation; from Eq. (B1) this is
equivalent to setting that far enough from the coast the elevation of the
perturbation tends to zero:
Here, we choose
The solution of Eq. (B5) subject to the boundary conditions (B6) and
(B7) is obtained through separation of variables,
Following Csanady (1982), let the water depth be a linear function of the
cross-shelf distance,
From Eq. (B1), the along-shelf velocity is
The periodicity of these perturbations depends on the size of the region
where they are generated, which sets the along-shelf wavenumber
This work was supported by DARDO (ENE2012-38772-C02-02), Rises-AM (GA603396), Plan-Wave (CTM2013-45141-R) and ICoast project (Echo/SUB/2013/661009). We would like to thank Joan Puigdefàbregas, Jordi Cateura and Joaquim Sospedra (LIM-UPC, Barcelona, Spain) for the data acquisition campaign. The authors thank Alexis Beudin (USGS, Woods Hole, USA) and Ken Brink (WHOI, Woods Hole, USA) for a number of useful suggestions. We are also very grateful to our reviewers, Vlado Malačič and one anonymous oceanographer, for their ideas; Vlado Malačič raised the issue of the potential relevance of coastal waves, which meant a substantial reanalysis of our velocity data and a reinterpretation of our results. Finally, we are pleased to acknowledge Gabriel Csanady and Steve Lentz, as their seminal studies on the circulation of the coastal ocean have been a source of inspiration to our work. Edited by: S. Carniel