Sea level observations at Cape Bon, Tunisia, and Mazara del Vallo, Sicily, show that large, coherent oscillations exist across the Strait of Sicily with significant energy bands centered at periods of 35.3, 41.6, 50.6, 75.8, and 134.5 min, whose possible structure is confirmed by a numerical approximation to the gravitational barotropic normal modes with realistic topography. It is observed that these oscillations are related to the passage of synoptic weather systems over the region. An investigation on the configuration, phase velocity, and direction of approach of atmospheric disturbances over the region suggests that the oscillations in the Strait of Sicily could be forced by instabilities that develop in large-scale, low pressure fronts that propagate as pressure gravity waves with an approximate phase speed between 24 and 30 m s−1.
The objective of the present paper is to investigate the spatial structure and the mechanisms for the excitation of seiche activity in the region of the Strait of Sicily. Common to several ports on the southern coast of Sicily and the coast of Malta is the presence of sea level oscillations with periods ranging from a few minutes to an hour (Airy 1878; Colucci and Michelato 1976; Drago and Ferraro 1994). In places like the port of Mazara del Vallo, on the southern coast of Sicily, which harbors one of the most important fishing fleets in the Mediterranean, these oscillations sometimes have mischievous consequences. The particular geometry of this port, built around an old estuary, permits the amplification of certain oscillations within the port that can at times turn into a hydraulic jump that propagates up the old estuary producing severe material damage to the local fishing fleet. The phenomenon, known locally as Marrobbio,1 is commonly related to the passage of large-scale atmospheric perturbations through the region. The scales normally associated to large-scale synoptic perturbations (500 km, >1 day) and high-frequency sea level variability (50 km, minutes) are not well matched: a forcing function must either have a time variability close to the modal frequency or at least its spatial structure has to resemble that of the mode. Synoptic atmospheric perturbations, such as atmospheric pressure or wind patterns, generally have characteristic spatial scales of a few hundreds of kilometers and vary in time at periods longer than a few hours. Therefore synoptic atmospheric systems are not likely to excite directly high-frequency motions since, even if realistically they can be considered quite broadband (both in frequency and in wavenumber), they usually do not contain much energy at the scales and periods required for the exitation of seiches. However, synoptic atmospheric systems can, and are known to, develop instabilities that radiate energy in the form of “high frequency” gravity waves (Fujita 1955; Monserrat and Thorpe 1992). The excitation of short period (∼minutes) sea level oscillations near a coast by the passage of atmospheric pressure gravity waves has been observed in other places such as Nagasaki Bay in Japan, where the phenomenon is known as Abiki (Hibiya and Kajiura 1982), the harbor of Ciutadella on the Island of Menorca in the Balearic Islands, where the phenomenon is known as Rissaga (Monserrat et al. 1991). See Rabinovich and Monserrat (1996) for a comprehensive account of observations of “Meteorological Tsunamis” (as this phenomenon is sometimes called). Platzman (1958) in his study of the surge of 26 June 1954 on Lake Michigan showed that resonant coupling between atmospheric forcing and the lake’s response might arise when the propagation speed of the atmospheric disturbance is nearly equal to the speed of free waves on the lake. The case of the Rissaga, in the harbor of Ciutadella, has been the focus of several numerical studies (Tintoré et al. 1988; Gomis et al. 1993). In this harbor the direct excitation of its normal gravity modes by atmospheric waves is not likely due to its relative small size. However, these works investigate the possibility that coastally trapped waves first excited outside the harbor can lead to the excitation of the harbor modes. Another excitation mechanism proposed for the Rissaga are tidally generated internal waves impinging on the coast, which in certain locations, that is, the Caribbean coast of Puerto Rico, have proven to excite short period oscillations on the coast (Giese et al. 1990; Chapman and Giese 1990). However, this type of excitation mechanism is not likely to be applicable to the Marrobbio since the observed Marrobbio events are not correlated with phases of the moon, the M2 semidiurnal tide being the principal tidal component in the region (Lozano and Candela 1995), or with the seasonal variations in stratification.
Our working hypothesis for this investigation is that the excitation of Marrobbio in the Strait of Sicily is related to the passage of atmospheric perturbations over the region, the characteristics of which will be investigated using numerical simulations based on the construction of an approximation to the local barotropic normal modes. This paper is organized as follows: In the next section we describe the characteristics of the bottom pressure observations along with sea level observations in the port of Mazara del Vallo showing the signature of the Marrobbio. The construction of the regional gravity modes is explained, followed by a description of the methodology used to investigate the possible excitation forcing functions using the normal mode representation, which leads to a section describing the different numerical experiments. The paper ends with some concluding remarks and the description of planned future work.
Geographically the Strait of Sicily separates the Western and Eastern Mediterranean basins and has a width of about 140 km and maximum depth of about 400 m (Fig. 1). An interesting characteristic is that on the Sicilian side it has a broad shelf with depths less than 100 m, extending for about 100 km toward Tunisia, with the consequence that about 80% of the cross section has depths less than 200 m. In coordination with the PRIMO 1 studies in the Western Mediterranean, two pressure sensors were installed on the sea bottom across the Strait of Sicily for a period of 1.5 yr (March 1994–October 1995): one at a depth of 7 m near the port of Mazara del Vallo on the southern coast of Sicily and the other at a depth of 6 m off Cape Bon (Ras el Tib), Tunisia. Figure 1 shows the location of both instruments. The original objective of these observations was to study subinertial sea level variability across the strait in relation to the exchange of waters between the Western and Eastern Mediterranean basins. These pressure measurements were done in coordination with moored current observations in the Strait of Sicily by the Istituto per lo Studio Dell’Oceanografia Fisica in La Spezia (Astraldi et al. 1996). However, these latter measurements where concentrated in the central and deep (>50 m) parts of the strait where the two main channels and sills are located. In order to have some estimate of the transport across the Strait at shallow depths, assuming a geostrophic balance in the across strait direction, the two bottom pressure instruments were installed. These instruments [SeaBird Electronics sea gauge pressure, temperature and conductivity (PTC) sensors] were fixed to the seafloor on top of 3-m-long aluminum pipes that were driven into the sandy seafloor with the use of a water pump. This installation procedure provided a stable platform for the instruments, which were serviced every 6 months starting in March 1994 until October 1995, giving three, roughly equal, 6-month-long periods of observations. During the first two observation periods, that is, March 1994–October 1994 and October 1994–April 1995, the instruments were set to sample pressure, temperature, and salinity (conductivity) every 10 minutes, while on the third observation period (April 1995–October 1995) the sampling interval was set to 3 minutes. In this paper we will only discuss the bottom pressure measurements. The temperature and salinity measurements will be discussed elsewhere.
A 50-day sample of raw (Δt = 10 min) pressure measurements is shown in Fig. 2. The plot shows bottom pressure at Mazara del Vallo (upper) and at Cape Bon (lower). On both plots one can observe a tidal signal, about 10 mb in amplitude riding on top a subinertial signal, which is at least of the same amplitude or larger. Also evident in both plots, and particularly more evident in the Mazara record, is the presence of high-frequency (periods <1 h) events that seem to occur simultaneously at both places, although with larger amplitude at Mazara. One of the clearest high-frequency events in the plots is around day 137, which is blown up in Fig. 3 and enhanced by only plotting two days of the high-pass signal, which eliminates anything with periods longer than 1 h. From this figure it becomes evident that the prominent event starting around day 136.8 seems to be occurring at about the same time in both places. The fact that there are such high-frequency events occurring simultaneously at the two sites, which are about 140 km apart, is by itself surprising. More so, when looking at a yearlong time series of high-passed observations at the two sites (Fig. 4), it is evident that these events occur year-round and always simultaneously. Also, when inspecting daily surface atmospheric pressure maps of the Mediterranean (RMS, 1994, 1995) one finds that the presence of a high-frequency event is associated with the passage of a low pressure atmospheric system over the region.
Something still more interesting about the phenomenon is revealed by calculating the cross-spectra between the two sites for different 6-month-long observation periods. In Fig. 5 we show the power spectra, for the first 6 months of bottom pressure observations at Mazara del Vallo and Cape Bon from March to October 1994, along with cross-spectra results (coherence, transfer function amplitude and phase) taking Mazara as input and Cape Bon as output. Marked on the plots are five shaded frequency bands centered at periods of 135, 76, 51, 42, and 36 min, which have significant high coherence across the strait. Note that the high coherent peaks do not necesarily correspond to distinct energy peaks in the power spectrum at each site, indicating the presence of energetic incoherent motions at each side that do not span the strait. The corresponding values of the power spectra, coherence, and transfer function amplitude and phase for these bands are also marked by large dots, which will be kept as reference for subsequent spectral calculation plots. When observing the spectral calculations for the second 6-month observation period, from October 1994 to April 1995 (Fig. 6), it is very striking to find that although the spectra at each site differs in energy levels from the previous 6 months (this is expected since the specific forcing ought to differ between the two observation periods), the coherence and transfer functions remain very similar, that is, there is high coherence for the five mentioned bands, the transfer function amplitude has about the same magnitude for these bands and, what is more interesting, we see the same phase values at these bands. Considering that this is an independent 6-month period of observations, the phase locking observed at these frequency bands implies a very strong coupling of the motions at the two observing sites. In order to examine these events closer, while servicing the instruments on April 1995, we reset the sampling interval to 3 min at the two sites, permitting us to resolve higher frequency motions. Figure 7 shows the power and cross-spectra results for the third period of observations from April to October 1995. Again we see that the coherence and transfer function results confirm the presence of high coherence and similar transfer function amplitude and phase for the five frequency bands mentioned before. It is also interesting to note that at periods shorter than 35 min, the coherence across the strait drops very significantly and there does not seem to be outstanding peaks at higher frequencies than 40 cpd. One possibility for this coupling is if the motions related to these frequency bands are somehow locked to the Strait’s topography and, when excited, oscillate between the two sites maintaining a constant phase relation. That the transfer function phase at all five frequency bands appears to be clustered around zero or 180 degrees is also a clear indication of the dominance of standing waves, spanning the strait, in relation to the phenomenon.
We now link our open sea observations with tide gauge measurements within the port of Mazara del Vallo. In Fig. 8 we show a 5-day record comparing the simultaneous sea level and bottom pressure measurements inside (with the tide gauge) and outside (with the pressure sensor) the port, for a period in October 1995, when a reasonable size Marrobbio occurred. In Fig. 9 we show the corresponding cross-spectra information for the two records indicating high coherence for the bands between 25 and 40 cpd and a slight amplification indicated by the transfer function magnitude. The tide gauge is located well outside the old estuary so the hydraulic jump signature is not evident in the record, although local fishermen did report one in the estuary at the time of the event (612.1 Julian day 1994).
3. Gravitational normal modes
To describe this high-frequency phenomena we disregard rotation and friction, both terms small with respect to the local derivative, and use the linearized, time periodic barotropic equations for momentum and continuity
where the horizontal velocity vector is ueiωt, ω is frequency, t is time, g the acceleration due to gravity, ∇ the horizontal gradient operator, h water depth, and ηeiωt is the sea surface elevation. Combining the two equations in terms of η
and considering solutions in a closed ocean with no flow across the coastal boundaries h∇η · n = 0, where n is the unit vector normal to the boundary, we have an eigenvalue problem that will provide us with a complete basis function set to represent the irrotational motions in the study area.
Initially we constructed normal modes in a coarse 20-km resolution model of the Mediterranean basin. Several gravitational modes with periods close to the ones observed were found that showed activity around the region of the Strait of Sicily (Fig. 10). However, with such coarse resolution it is hard to properly simulate the topographic features of the Strait. It was decided to increase the spatial resolution to 4 km; however, for the whole Mediterranean basin this 4-km resolution is computationally expensive and not necessary as is shown next. An artificial boundary was set to “close” the region of interest for the purpose of the mode calculations. The chosen domain consists of the actual coast and bathymetry of the Strait, resolved to a resolution of 4 km, within a circle of 200-km radius centered roughly at the midpoint of the Strait and contains 10 493 nodes. The effect of the artificial boundaries was investigated by arbitrarily changing the boundaries to allow a larger section of coast, along both Sicily and Tunisia, to be included within the domain. This larger domain, which had 13 085 nodes, allowed additional modes, but it was always possible to identify modes that were compatible between the two different domains, if attention was put on the central region of the strait, say within a 100-km radius circle centered also at the midpoint of the strait. This is because structures generated by the artificial boundaries do not have a very significant effect on the central part of the domain, where the particular topography only allows very specific structures for the barotropic gravitational motions possible.
In Figures 11, 12, and 13 we show examples of some of the gravitational modes resolved that have periods close to the frequency bands the observations indicate as having high coherence across the Strait. These are at periods close to 36, 42, and 51 min, respectively. Besides the period T given by T = 2π/(gλ)1/2, where λ is the eigenvalue and g the acceleration due to gravity, we give the weighted mean depth defined as h = ∫ hη2/∫ η2, where h is the observed depth, η the mode amplitude and the integrals are taken over the part of the domain within a circle of 100-km radius centered at the midpoint of the strait to minimize the effect of the artificial boundary chosen. This depth definition, weighted by the energy of the mode, can be taken as indicative of the mean depth where the mode “lives.” With this depth definition we can calculate a characteristic wavelength L defined by
which is indicated in the figures. Figure 14 shows the values for the period, wavelength, and weighted mean depth for the first 450 modes calculated. Note that although the period and wavelength decrease as a function of mode number, the mean weighted depth tends to remain at around a constant mean value of about 125 m.
To investigate the possible normal mode excitation we now introduce forcing due to atmospheric pressure, wind stress, and friction:
where ηa = −Pa/ρog, the isostatic sea surface elevation due to atmospheric pressure (Pa) and ρo the mean seawater density;
the wind stress in which ρa is air density, CD a drag coefficient and W the horizontal surface wind vector, τb = CBu*u; CB is a bottom drag coefficient and u* a characteristic bottom velocity magnitude. Again there is no normal flow at the boundaries.
To represent the velocity and sea surface elevation within the domain of interest we use the set of functions ϕi defined as the modes in the previous section, that is, as the solutions of the eigenvalue problem defined by ∇ · (h∇ϕi) = −λiϕi at D, with h∇ϕi · n̂ = 0 at ∂D. In terms of this basis, u = ∇ϕ; where ϕ = Σnϕi≥1 Piϕ; and η = Σnϕi≥0 Niηi; where nϕ is the number of functions chosen for the representation and ηi = ciϕi, ci being a suitable constant.
Defining the scalar product [r, g] = ∫D rg dσ, and projecting Eqs. (1) and (2) over the set of functions, that is, [ηi, (1)] and [h∇ϕi, (2)], we arrive at the system of ordinary differential equations
where the state vector
contains the expansion coefficients for η and u, and the vector f is the projection of the forcing fields onto the subset of eigenfunctions chosen.
The local common belief is that the Marrobbio phenomenon is related to a sudden change in atmospheric pressure, as reported in “the notice to mariners” on the chart of the port of Mazara del Vallo. Trying to quantify more specifically if there is a significant correlation of the presence of Marrobbio with the local atmospheric pressure patterns, we analyzed six months of simultaneous meteorologic variables, that is, atmospheric pressure and wind stress fields from (ECMWF) 6-hourly surface data for the region. In Figs. 15 and 16 we show the first EOF modes for the atmospheric pressure and wind stress fields for an area centered around the strait. Calculating the correlation of the absolute value of the high-pass bottom pressure measurements at Mazara del Vallo and the atmospheric pressure and wind stress, first EOF modes show no significant correlation with either of the fields. This is at first surprising, when one actually observes a clear relation of the passage of weather fronts through the region with the occurrence of the phenomenon. We will try to solve this puzzle by performing numerical simulations based on the constructions of the normal gravitational modes for the region.
5. Numerical experiments
Typical average propagation speeds of synoptic weather systems in the Mediterranean are 4–9 m s−1 in summer and 11–18 m s−1 in winter (Garrett 1983). At 18 m s−1 a front takes about 12 hours to traverse the region from west to east. The presence of Marrobbio in Mazara del Vallo was concurrent with the passage of low pressure synoptic weather systems through the region; therefore the first forcing functions tried were synthetic moving, Gaussian-shaped, low pressure systems with a length scale of 500 km × 250 km, a pressure drop of 16 mb, and associated geostrophic wind. It should be noted that it is straightforward to investigate independently the effect of pressure and wind on the excitation of the Marrobbio. Different propagation speeds and angles of approach for the synoptic weather systems were explored, an example of such a synthetic weather system is shown in Fig. 17, where a moving synoptic low pressure perturbation, with a phase velocity of 2 m s−1 and approaching the region from the southwest (i.e., 30° trigonometric), is plotted at four 30-h time intervals. Since the normal modes are orthogonal, once we have defined a forcing function, attention can be given to the set of modes of interest. For these numerical experiments we have decided to use the first 200 functions that cover the frequency range from 0 to 45 cpd (period from ∞ to 30 min). Although this choice will certainly have some effect on the representation, we believe that for the linear model used this is of minor importance since there is no interaction between modes and we are including all the functions necessary to represent all the frequencies that we want to resolve. The mechanical procedure consists of projecting a given forcing function onto the chosen subset of ϕ, and then integrating Eq. (3) over a reasonably long time interval (see the appendix for details in the integration algorithm). For most integrations we neglect the friction term (i.e., τb = 0); therefore if there is excitation of any of the ϕ, it shows in the integration results.
We now look closer to the response of the normal modes near the observation sites at Mazara and Cape Bon. The observations are assumed to correspond to an area of 3 by 3 node points (i.e., 12 km × 12 km area) around the actual observation sites. In this way we reconstruct an approximation to the actual sea level variability caused by the given forcing and also calculate the cross-spectra between the two sites to compare with the observed ones (i.e., Figs. 5, 6, and 7). In Fig. 18 we show an example of the reconstructed cross-spectra from a run done with a synoptic weather system (SWS), which in this case is moving at a phase speed of 10 m s−1, approaching from the southeast (i.e., 120° trigonometric) and we are only looking at the atmospheric pressure effect. The spectral power amplitude for the five bands of interest is about an order of magnitude smaller than the ones observed at both sites; yet the coherence and the transfer function phase maintains a surprisingly similar behavior, as the one in the observations, for these same five bands. Here we see that, although the synoptic weather system is not exciting the sea level at the observed amplitude, the functions used for the representation of the phenomenon are clearly reproducing the observed phase relation between the two sites. Figure 19 gives a synthesis of the results obtained by forcing the model with a SWS, with phase velocities from 2 to 20 m s−1 approaching the region from the southwest (i.e., 60° trigonometric). All other orientations tried, from −180° to 150° at 30° increments, gave comparable results. Both the response due to atmospheric pressure (upper panel) and wind stress (lower panel) are shown. The values indicated for each integration run are the maximum sea level amplitude at the Mazara and Cape Bon sites, as well as the mean standard deviation within a circle of radius 100 km centered in the strait. When comparing with Fig. 4, it is clear that the actual maximum amplitudes observed at the two sites are about an order of magnitude larger than the integration results. Also the atmospheric pressure effect is about an order of magnitude larger than the effect due to wind. These calculations confirm that the synoptic patterns of low pressure systems are not effective to excite the observed motions directly.
Another interesting possibility to explore is related to the fact that SWSs are known to develop instabilities that tend to radiate energy as atmospheric gravity waves, that is, atmospheric pressure jumps that travel as gravity waves close to the bottom of the atmosphere. These have been identified and related to high-frequency sea level variability in other places. A clear example of such a phenomenon was reported in the classical paper by Platzman in 1958 about a seiche excitation by the passage of a squall line over Lake Michigan in 1956. Monserrat et al. (1991) have reported the passage of similar gravity waves over the Balearic Islands and have related them to the excitation of the “Rissaga” in the Port of Ciutadella on the Island of Menorca. That the coupling between an atmospheric pressure jump and a barotropic gravity wave in the ocean is due to a matching of the phase speeds of the two phenomena can be easily seen by doing the following exercise: In a one-dimensional forced problem, for a barotropic homogeneous linear ocean, the momentum equation is
and the continuity equation is
Now if u, η, ηa ∝ ei(wt+kx), co = (gh)1/2 and c = ω/k, then η = ηa/(1 − c2/c2o), which clearly indicates resonance when there is matching between the atmospheric pressure gravity wave phase speed c and the barotropic oceanic gravity wave phase speed co.
To see the possibility of excitation by an atmospheric gravity wave, similar in character to a squall line and possibly related to, and/or produced by, instabilities in SWSs, we have constructed an artificial pressure jump of 4 mb, with a length scale of 20 km, traveling at varying phase speeds and approaching the region from different angles. The spatial and phase speed scales chosen are in accordance with the observations of Platzman (1958) and Monserrat et al. (1991). We have also associated to it a downgradient wind to quantify the wind effect, related to the moving front, in the possible excitation of Marrobbio. Figure 20 shows an example of a pressure jump of 4 mb over a distance of 20 km, traveling at a phase speed of 10 m s−1 and approaching the region from the southwest (i.e., 60° trigonometric). This orientation of 60° trigonometric, which is in a cross-channel direction, turned out to be the one that produced the largest response in the numerical simulations. Figure 21 shows a summary of the results for that orientation but for different phase speeds of the atmospheric front. For a jump with a phase speed between 24 and 30 m s−1, the amplitude of the sea level variability is similar to the observations at both sites (see Figs. 3 and 4). Since the wind-induced excitation is about an order of magnitude smaller than that due to atmospheric pressure, even for the more than realistic wind stress magnitudes used, the wind appears to play a minor role. Figure 22 shows the cross-spectra results for the atmospheric-induced excitation of a front, moving with a phase speed of 27 m s−1 and also approaching the region from the southwest (i.e., 60° trigonometric). The power spectral amplitudes resulting from the integration are higher than the ones observed at the two sites, but one has to remember that this integration is done without friction. However, for the five bands of interest, the cross-spectral results show significant coherence values, and also similar amplitude and phase values for the transfer function estimates between the two sites. Figure 23 shows the cross-spectral results for the same front, approaching the region from the southwest (i.e., 60° trigonometric) and moving at 27 m s−1, but this time the friction term has been turned on with an equivalent e-folding timescale of about 5 days (i.e, CD = 2 × 10−3, u* = 0.1 m s−1 and h ∼ 100 m). For this case the power spectral levels for the five bands of interest are well reproduced at the two sites, while the coherence, transfer function amplitude, and phase remain very close to the observed values.
6. Summary and discussion
The Marrobbio has been characterized as a regional gravitational phenomenon trapped to the local topography. As shown in the observations and also in some of the calculated approximations to the local gravitational modes, a principal oscillation is between the Sicilian and Tunisian coasts, confirming the hypothesis put forward by Airy 120 years ago in relation to the waves causing the seiches observed in Malta (Airy 1878). Our numerical calculations indicate that the Marrobbio can be excited by atmospheric pressure gravity waves traveling through the region at the appropriate phase speed and also with a strong cross-channel component. Maximum response is obtained for atmospheric fronts traveling at 27 m s−1 phase speeds and traversing the region in a cross-channel direction. Atmospheric pressure is the main forcing mechanism for the Marrobbio and produces a response of around an order of magnitude larger than the wind stress forcing. The developed methodology, based on the construction of an approximation to the regional normal modes, is efficient and well adapted to the study of the Marrobbio phenomenon. This work differs from previous seiche excitation studies in its analytical approach based on defining the regional normal modes. However, this type of analysis is common when investigating the response of a shelf region to an incomming tsunami (Van Dorn 1984). The gravitational modes so defined are specific to the local topography and the particular modes coherent between two distinct locations are very specific. In numerical experiments done with our model, we have verified that if the observation locations are moved by a few tens of kilometers the coherent motions vary drastically.
We have advanced an overall sequence of counts conducive to a Marrobbio and established that, indeed, fast atmospheric pressure jumps (squalls) can account for the observations in the strait. However, further observational evidence is required to confirm and better characterize the atmospheric forcing and its causes. An experiment, based on the deployment of a small net of microbarographs over the region along with bottom pressure measurements in and out of the port of Mazara del Vallo, should help to clarify the situation.
We have seen in the course of our calculations that high-frequency gravitational modes are supported in the shelf and slope areas. These modes can be exited by high-frequency fields associated with both atmospheric and oceanic instabilities. Thus we hypothesize that regional normal modes contribute to the variability in most of the world coastal oceans. These modes, which are conspicuous in areas with small tides like the Mediterranean, should be ubiquitous in the world oceans, although usually not easily detectable because of their relative smaller amplitude with respect to the prevailing tide. Even if present, one usually does not pay attention to isolated events, unless a spatial correlation pattern (as the one described in this paper) is observed. The presence of these gravitational modes might contribute to inhomogeneities in mixing patterns along a coast that could help explain the observed variability in biological communities (Pineda 1996; Menge et al. 1997). Furthermore, the cell-like structure of these modes, if sufficiently energetic, should significantly affect the dispersion of tracers and biomass.
Support for the field measurements was provided by Grant OCE-93-13654 from the National Science Foundation and from Contract N00014-94-1-0347 with the Office of Naval Research. Modeling efforts were supported through Contract N00014-95-1-0315 with the Office of Naval Research.
Here we describe an efficient and accurate numerical algorithm to integrate (3). The system of Eq. (3) can be decoupled by a decomposition of the matrix M. Using primes to indicate transposition, we have the following decompositions
where Λ is a diagonal matrix containing the eigenvalues of M and V is the left eigenvector matrix of M, that is, the eigenvectors of M′, and
where the matrix Z contains the right eigenvectors of M.
Transposing Eq. (3) and postmultiplying by matrix V we arrive at a system of decoupled, complex valued, differential equations
where Y′ = y′V and f′V = F′.
Since Z′V = D, where D is a diagonal matrix, the map 𝒵 = D−1Z′, allows us to return to the space of our original coefficients, i.e., Y′𝒵 = y′.
System (A1) permits an analytical integration for piecewise linear forcing functions. Defining tj ⩽ t ⩽ tj + Δt = tj+1, F(t) = Fj + (Fj+1 − Fj)(t − tj), and Y(tj) = Yj, integration of (A1) for a given eigenvalue λ gives
After exact integration the following finite difference scheme is obtained:
where γ = eλΔt; κ = (1/λ)(eλΔt − 1) and κ1 = (1/λ)(κ − Δt).
Corresponding author address: Dr. Julio Candela, Depto. Oc. Fisica, CICESE, P.O. Box 434844, San Diego, CA 92143-4844.
The origin of the word Marrobbio (as spelled in the “notice to mariners” in the chart of the port of Mazara del Vallo) or Marrubbio [as spelled in the paper by Colucci and Michelato (1976)] is not clearly established. Colucci and Michelato claim that Marrubbio translates literally to mad sea, in this case possibly from the fusion of the Italian words mare (sea) and rabbioso (angry, enraged).