_{1}

^{*}

A series of numerical experiments from a barotropic configuration of the General Curvilinear Ocean Model (GCOM) was conducted to analyze the response to infragravity (IG) waves of the Port of Ensenada, located within Bahia de Todos Santos (BTS), west coast of Mexico. Experiments with forcing frequencies f = 50?1 min?1, f = 30?1 min?1, f = 25?1 min?1 and f = 16.66?1 min?1 showed the expected increase of energy at the corresponding forcing frequency band and also the appearance of secondary peaks of energy at frequency bands f = 8.33?1 min?1 and f = 4.16?1 min?1 which were identified as modes f
_{1 }and f
_{2}; being the band at f = 16.66?1 min?1 the zeroth f
_{0} mode. Maximum peak of spectral energy from the numerical experiments was found at frequency band f
_{0} = 16.66?1 min?1 which agreed with the estimated maximum value of the amplification factor and with the T
_{0} mode of oscillation of the port. Distribution of amplitudes inside PE for modes f
_{0}, f
_{1} and f
_{2} were also presented. Mode f
_{0} represents a quarter-wave oscillation with amplitudes of the same sign; mode f
_{1} has two nodal lines and mode f
_{2} presents and additional one. Corresponding harbor currents were also calculated, they were in the range 20 - 160 cm?s?1. Finally, in order to elucidate the source of the external signals found in the spectral analysis of this study, the natural oscillation modes of the BTS were estimated. Although more studies are needed, BTS oscillation mode T
_{2} = 16.82 min, was identified as the external forcing that excites larger oscillations within the port.

Ports and harbors can be used for commercial, recreational and military purpose. Compared to other kinds of transportation (railroad, airplane, highways), waterborne commerce has two significant advantages: low cost and high capacity. Many nations use waterborne commerce as a major source of transport [

The Port of Ensenada (PE) is located on the west coast of the Baja California Peninsula, approximately 110 km south of the Mexico-US border, within Bahia de Todos Santos (BTS) in the City of Ensenada (^{2} composed of internal areas of navigation and service with a length of berthing of 6014.25 m [

It is well known that harbors with sides on the order of 500 m in length and depths on the order of 10 m are subject to a natural oscillation period on the order of 1 min and longer [

For a simple rectangular basin of length L and uniform depth h with an open end, harbor’s natural oscillation period T (or harbor modes) can be approximately estimated as [

where n is the mode number. Assuming the PE dimensions as L/4 = 2350 m with uniform depth h = 9 m, the fundamental T_{0} mode (or Helmholtz mode) is estimated to be T_{0} = 16.67 min. It describes a quarter-wave oscillation with a single nodal line at the mouth and maximal amplitudes at the head. Since, this is the largest wave period among all the supported standing waves inside PE, it is clear that oscillations of larger period detected at the site would correspond to external signals.

Natural oscillation periods are independent of the external forcing, however, magnitudes of these oscillations highly depend on the external energy source that generates the oscillations [

When IG waves with frequencies close to those of resonating harbor modes arrive into a harbor opening, they can be highly amplified into inner basins resulting in large oscillations of the water surface [

Mexican Navy (SEMAR) maintains a monitoring sea level station inside PE (its location is indicated by the yellow circle in

in ^{−1} min^{−1} is practically the same as the estimated T_{0} mode, it is considered that energy peaks found at frequency bands f = 50^{−1} min^{−1} and 25^{−1} min^{−1} would correspond to signals generated outside the PE. The energy amplification found at bands f < 16.66^{−1} min^{−1} may result from resonance within PE. Registered maximum elevation during that event was A_{n} = 0.6 m [

The response of a harbor to long waves from open sea is well described by a parameter called “Amplification Factor”, H, [

where f is the frequency of the long incoming waves, f_{0} is the resonant frequency of the harbor, and Q is the quality factor, which is a measure of energy damping in the system. In this study, Q was estimated according to [

Harbor’s natural oscillation model, Equation (1), gives natural frequencies of the possible oscillations in an idealized rectangular basin, but it does not provides information of the response to external forcing or the spatial distribution of amplitudes within the basin [

Previous numerical studies on the PE have been focused on wind waves and tides as forcing agents [

In this contribution, a series of numerical experiments are conducted to elucidate the response of the PE to IG wave energy. First, the Helmholtz natural mode is estimated, and then the spatial structure of amplitudes within the harbor under different frequency forcing are obtained. Finally, the spatial distribution of the associated currents is presented. The numerical experiments use a barotropic configuration of the General Curvilinear Ocean Model [

The simulations were carried out using the General Curvilinear Ocean Model [

where u = (u, v, w) is the velocity vector, and i, j and k are the unit vectors in the x, y, and z coordinates respectively, p is the pressure, ρ is the water density and D(= ∇×u) represents the divergence. The dimensionless numbers are the Reynolds number, Re(= UL/ν) and the Froude number F(= U/NL), where U and L are a typical velocity and length respectively, νis the kinematic viscosity and N^{2} = (−g/ρ_{0})∂ρ/∂z is the Brünt-Väisälä frequency. As it is noted, in primitive variables, unknowns are (u, ρ, p) as indicated by Equation (1) and Equation (2), while Equation (3) substitutes the incompressibility condition given bay Ñ×u = 0.

The above set of governing equations is transformed from Cartesian to curvilinear coordinates (i.e. from physical (x, y, z) domain to computational (x, η, ζ) domain and then solved subject to initial and boundary conditions. On solid boundaries u = 0, with no flux of density, ∇ρ∙n = 0 (where n is the unit normal vector), while for p, the boundary condition is determined by substituting u = 0 into the transformed momentum Equation (3) [

The geometry of the study site is well captured by using a three-dimensional boundary-fitted grid. The computational domain consists of 299 × 75 × 11 (ξ × η × ζ) grid points in the x, y and z directions with 10 m horizontal resolution, and 11 vertical layers with 1m minimum distance between points. Grid point distribution at ζ = 11 is shown in

Since, GCOM is based on to pass from a physical space to a computational space through the Jacobian (J) of the transformation, then det(J) ¹ 0 in order to GCOM to perform. Physically, it means that the transformation from one plane to another has to be one-to-one. So, special careful is recommended during the grid elaboration process.

The model is forced by prescribing the velocity field:

for Re = 10^{6}. Since, in this study the model was run in barotropic mode (i.e. uniform density), terms involving F were neglected and Equation (4) represents a passive scalar.

In all numerical experiments the fluid started from rest, and nearly steady solutions, defined by the convergence criterion [

To elucidate the response of the PE basin to long IG waves, first a forcing frequency f = 50^{−1} min^{−1} is considered. The general trend of the resulting spectrum, presented in ^{−1} min^{−1} and f = 16.12^{−1} min^{−1}. In particular, the increase of energy at bands f = 6.28^{−1} min^{−1} and f = 3.63^{−1} min^{−1} agree remarkably well with those presented in ^{−1} min^{−1} and f = 8.33^{−1} min^{−1} that could be related to the first harmonics of the fundamental mode.

Resulting spectra for the numerical experiments with forcing frequency f = 30^{−1} min^{−1}, f = 25^{−1} min^{−1} and f_{0} = 16.66^{−1} min^{−1} are shown in ^{−1} min^{−1} and f = 4.16^{−1} min^{−1 }which can be identified as the resonant frequencies f_{1} and f_{2} respectively. Notice that for f_{0} = 16.66^{−1} min^{−1}, the energy peak is larger than the ones at f = 30^{−1} min^{−1} and f = 25^{−1} min^{−1}. This is due to the fact that f_{0} practically coincides with the estimated resonance Helmholtz mode (T_{0} = 16.65 min) described already and confirms that this is the frequency of the fundamental mode of the PE. Frequency values at bands f_{1} and f_{2} correspond to L/2 = 4700 m and L/4 = 2350 m whose period, T = L(gh)^{−1/2}, is T_{1} = 8.33 min and T_{2} = 4.16 min respectively. Thus, the spatial distribution of amplitudes for f_{0} describes a quarter-wave oscillation with a maximum at the head and amplitudes of the same sign within the basin which means that amplitudes oscillate in phase (

f_{1}, the spatial structure of amplitudes has two nodal lines; one at the head and another some distance away from the entrance ((_{2} the amplitude distribution presents three nodal lines (

The amplification factor, H, as a function of f_{0} obtained for all the numerical experiments is plotted in _{0} = 16.66^{−1} min^{−1} with a Q factor of 19, and decays relatively fast as one moves away from f_{0.} It is worth noting that the amplification factor value at f_{0} predicted by the model agrees very well with the Q-factor, Q = f_{0}/Δf = 21 deduced from the spectrum of ^{−1} from the figure. The similitude of Q values and the good agreement of the results with the theoretical curve, Equation (2), provide robust arguments for the good performance of the model.

Currents associated to oscillations inside a port can generate further resonance and possibly damages to the moored ships [_{0}, f_{1} and f_{2} are shown in _{0}, maxima of velocity (60 and 40 cm∙s^{−1}) are found at the narrow passages inside PE, while at the middle and the head velocities are approximately 20 cm∙s^{−1} (_{1}, contours of magnitude of velocity increase at the narrow passages (80 and 120 cm∙s^{−1}) while at the sites where the basin stretches (I, II and II, ^{−1}. For f_{2}, contours of minimum of magnitude are found at the head (20 cm∙s^{−1}), relative maxima at the middle of the basin (40 - 80 cm∙s^{−1}), and maximum values (160 cm∙s^{−1}) some distance away from the mouth (

Locally, under normal sea conditions, reported velocity of surface currents at the mouth and inside the PE are of the order of 20 - 40 cm∙s^{−1} [^{−1}, 80 cm∙s^{−1} and 100 cm∙s^{−1} for forcing f_{0}, f_{1} and f_{2} respectively. The order of magnitude of these agree very well with the maximum velocities of 62 cm∙s^{−1} estimated with the relation [

using A_{n} = 0.6 m from the observations.

Finally, in order to identify the possible source of the external signals found in the spectral analysis of this study (_{0} = 50.48 min, T_{1} = 25.24 min and T_{2} = 16.82 min. Since,

these values are very close to those found previously in the spectral analysis presented in

Since, the excitation at the entrance of the PE requires the arrival of waves containing energy in the frequency band f_{0} = 16.66^{−1} cpm; BTS oscillation mode T_{2} = 16.82 min could be identified as the excitation source for the oscillations inside PE.

While finishing this contribution, the author noticed the work of Rivera and Peña [_{0} = 51.54 min, T_{1} = 22.88 min and T_{2} = 15.45 min, which agree very well to those in this study. Also their spectral analysis gave energy peaks at frequency bands f_{1} = 45.53^{−1} min^{−1}, f_{2} = 26.79^{−1} min^{−1} and f_{3} = 13.11^{−1} min^{−1}. Although these values are close to those found in this study, they have to be interpreted as no conclusive due to the small length of the time series they used. However, the similitude of their results (obtained from records 30 years apart in time) with those in this study add further robustness to this work.

A series of numerical experiments from a barotropic configuration of the GCOM model was conducted to analyze the response to infragravity (IG) waves of the Port of Ensenada, located within Bahia de Todos Santos (BTS), on the west coast of Mexico.

The fundamental oscillation mode of the port was estimated to be T_{0} = 16.67 min. This value coincided with the third energy peak found at frequency band f_{0} = 16.66^{−1} min^{−1} from spectral estimates of sea level observations inside PE. The remaining two larger period signals were found at frequencies f = 50^{−1} min^{−1} and f = 25^{−1} min^{−1}.

Since the T_{0} = 16.67 min oscillation wave is the largest wave supported by the EP, larger period energy at bands f = 50^{−1} min^{−1} and f = 25^{−1} min^{−1} can be considered as generated outside the port. In order to elucidate their source, a simple exercise to estimate the first oscillation modes for the BTS was performed. It gave T_{0} = 50.48 min, T_{1} = 25.24 min and T_{2} = 16.82 min which agreed very well with the values found in the spectral analysis of this study.

Particularly, the wave energy found in the frequency band f_{0} = 16.66^{−1} min^{−1}, which comes from the T_{2} mode of oscillation of the BTS, coincided with the T_{0} mode of oscillation of the PE; as expected, resonance was found at this frequency (

Numerical experiments with different frequency forcing presented secondary wave energy peaks at bands f = 8.33^{−1} min^{−1} and f = 4.16^{−1} min^{−1} which were identified as modes f_{1} and f_{2}, being f_{0} = 16.66^{−1} min^{−1} the fundamental frequency of the PE. Associated currents (harbor currents) were of the order of 40 - 100 cm∙s^{−1} in agreement to observations.

The results of this study identified first three BTS oscillation modes as the source of the IG waves recorded at PE; however, in order to be conclusive, more studies are needed.

Finally, local Port Authority is planning to extend the main breakwater of the PE to several hundred meters. The good performance of GCOM makes it an ideal tool to obtain and evaluate the desired information for prospects.

Sea level data were kindly provided by Secretaria de Marina (SEMAR, Mexican Navy). Data processing and preliminary spectral analysis were performed by A. Mejia and S. Larios. Computer time was provided by Mexico’s National Oceanographic Data Center, (IIO-UABC).Thanks also go to an anonymous reviewer whose comments helped to improve this paper.

Carlos R. Torres-Navarrete, (2016) Numerical Characterization of Harbor Oscillations in the Port of Ensenada, Mexico. Open Journal of Marine Science,06,395-404. doi: 10.4236/ojms.2016.63033