Redalyc.Geostrophic Currents in the Presence of an Internal Waves Field

e-Gnosis E-ISSN: 1665-5745 [email protected] Universidad de Guadalajara México

Plata, Luis; Filonov, Anatoliy; Tereshchenko, Irina; Nelly, Liza; Monzón, César; Avalos, David; Vargas, Carlos Geostrophic currents in the presence of an internal waves field in Bahía de Banderas, México e-Gnosis, núm. 4, 2006, p. 0 Universidad de Guadalajara Guadalajara, México

Available in: http://www.redalyc.org/articulo.oa?id=73000418

How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative © 2006, e-Gnosis [online] Vol. 4, Art. 18 Geostrophic currents in the presense…Plata L. et al.

GEOSTROPHIC CURRENTS IN THE PRESENCE OF AN INTERNAL WAVES FIELD IN BAHÍA DE BANDERAS, MEXICO

CORRIENTES GEOSTRÓFICAS EN PRESENCIA DE UN CAMPO DE ONDAS INTERNAS EN LA BAHÍA DE BANDERAS,MÉXICO

Luis Plata 1, Anatoliy Filonov 2, Irina Tereshchenko 2, Liza Nelly 3, César Monzón 2, David Avalos 4 and Carlos Vargas 2

[email protected] / [email protected] / [email protected] / [email protected] / [email protected] / [email protected] / [email protected]

Recibido: marzo 28, 2006 / Aceptado: diciembre 12, 2006 / Publicado: diciembre 20, 2006

ABSTRACT. The characteristics of internal waves in Bahía de Banderas were determined by means of oscillating CTD casts from a fast oceanographic survey done on April 24 and 25, 2001. Previous studies have shown that the continental shelf of the Mexican Pacific, including Bahía de Banderas, possesses favorable conditions for the generation on internal waves. Fluctuations of the hydrophysical characteristics of the continental shelf caused by the presence and propagation of internal waves are smoothed using a filter whose parameters are determined by the shape of the spatial correlation function of the field pulses relative to the analyzed characteristic. Once internal waves are filtered, temperature, salinity and geostrophic velocity fields are shown for different depths and the general pattern of the geostrophic circulation in the bay is discussed. The strongest currents were present south of Islas Marietas and at the east zone of the bay. It can be stated that the geostrophic circulation in the bay in spring has an important role in the water masses exchange between the zone close to the east coast, inside the bay, and the open ocean.

KEYWORDS. Geostrophic currents, internal waves, temperature and salinity fields, Bahía de Banderas.

RESUMEN. Las características de las ondas internas en la Bahía de Banderas fueron determinadas por medio de lances con un CTD ondulante a partir de un muestreo oceanográfico rápido realizado los días 24 y 25 de abril de 2001. Estudios previos han mostrado que la plataforma continental del Pacífico mexicano, incluyendo la Bahía de Banderas, presenta condiciones favorables para la generación de ondas internas. Las fluctuaciones de las características hidrofísicas en el área de estudio fueron suavizadas mediante un filtro cuyos parámetros están determinados por la forma de la función de correlación espacial de los pulsos del campo de la característica analizada. Una vez filtrado el efecto de las ondas internas, se muestran los campos de temperatura, salinidad y velocidad geostrófica para diferentes profundidades y se discute el comportamiento de la circulación geostrófica en la bahía. Las corrientes más intensas se presentaron en la parte sur de las Islas Marietas y en la zona este de la bahía. Puede establecerse que la circulación geostrófica en primavera en el área de estudio tiene un papel importante en el intercambio de masas entre la zona próxima a la costa este, dentro de la bahía, y el océano abierto.

PALABRAS CLAVE. Corrientes geostróficas, ondas internas, campos de temperatura y salinidad, Bahía de Banderas.

1 Posgrado en Oceanografía Costera, Universidad Autónoma de Baja California, Álvaro Obregón y Julián Carrillo s/n Colonia Nueva, Mexicali, B. C., 21100, México – iio.ens.uabc.mx/

2 Departamento de Física del Centro Universitario de Ciencias Exactas e Ingenierías de la Universidad de Guadalajara. Blvd. Marcelino García Barragán No. 1451, Guadalajara, Jalisco, 44430, México - www.cucei.udg.mx/

3 Posgrado en Desarrollo Sustentable y Turismo, Centro Universitario de La Costa, Universidad de Guadalajara. Av. Universidad de Guadalajara No. 203, Delegación Ixtapa. Puerto Vallarta, Jalisco, 48280, México - www.cuc.udg.mx

4 Posgrado en Ciencias del Mar y Limnologia, Universidad Nacional Autónoma de México. Instituto de Ciencias del Mar y Limnología, Circuito exterior, C. U., Coyoacán, México, 04510 D. F. – www.mar.icmyl.unam.mx

The dynamics of water masses on the continental shelf of the Mexican Pacific coast is affected by barotropic and baroclinic tide. Internal tides are known to cause significant vertical variations in all hydrophysical parameters on the continental shelf. The spatial slopes of the dynamic heights on the continental shelf can be up to one order of magnitude greater than the normal values of the open ocean. Seiwell [1] and later Defant [2] were the first to show that internal tides may cause the temperature and salinity measurements on the continental shelf to provide different results with respect to the geostrophic currents, depending on the tidal phase in which they were taken. Since the publication of Defant’s work [2] on the reality and illusion in oceanographic measurements in regions with intense internal waves, researchers became aware of this intimidating problem and ceased to calculate geostrophic currents even in areas on the continental shelf where geostrophic balance can be maintained.

Filonov et al. [3] proposed a method to conduct a fast oceanographic survey with an oscillating CTD in a grid of numerous and successive stations that would allow them to filter these data and to remove the influence of internal waves. The filtering method proposed by Filonov [4] is based on a smoothing of the fields of temperature and salinity with a filter, whose parameters are determined by the shape of the spatial correlation function of the field’s pulses. The filtering method was successfully tested using data from a fast oceanographic survey near Barra de Navidad, Mexico [4].

Internal waves have important scientific relevance because they play a fundamental role in the vertical and horizontal mixing. The generation of internal wave vary considerably from one place to another as regards the function of the bottom slope ( = dz dx ) and the stratification of the water column (N2). The latter determines the inclination of the upward internal tide energy flux’s path, i.e., the slope of the characteristic ray: =arc tan [(N 2 2)/ (2f 2)]-0.5,where is the barotropic tide wave’s frequency. Energy transmission from barotropic to baroclinic tide is more effective when 1, which is considered a critical value. If < 1 (>1), the energy is transmitted to the coast (out from the coast) [5]. Based on this criterion, previous studies have shown that the continental shelf of the Mexican Pacific, from Manzanillo (Colima) to Bahía de Banderas (Jalisco and Nayarit), possesses favorable conditions for the generation of internal waves of large amplitude [11].

Previous investigations [11-17] have shown that the main characteristics of the internal tides in the continental shelf of the Mexican Pacific coast are very discernible. The internal tide includes a dominant semidiurnal tide, which is represented by a strong signal and a high peak in the spectrum of temperature oscillations. The barotropic tide in the survey area has a mixed character with a dominant semidiurnal component. Coastal areas experienced distorted semidiurnal internal waves traveling coastward. Studies based on the Baines’ model [5] showed that here the energy flow of the barotropic tide to the internal tide can vary significantly at different sections of the slope, depending on the arrival angle of the barotropic wave relative to that of the slope. With a perpendicular arrival, the energy flux has a maximum with the value 763 W/m2, and the extent of the initial internal disturbance is of 4.9 m [11].

Nonlinear disintegration of waves and instability of their sinusoidal shape, as they propagate to the coast, sometimes give rise to the generation of bores or solitons [7, 18, 19]. As a result, the energy from the upward internal tide is completely dispersed on the continental shelf, giving rise to the formation of successively shorter internal waves and, at the same time, provoking changes in the stratification and mixing of the water column.

Study area

Location

Bahía de Banderas (Figure 1), delimited by an imaginary line west of Punta de Mita in the north and Cabo Corrientes in the south, is approximately 1000 km2 (Figure 1). The average width (north-south) of the bay is about 30 km and it has an approximate length (east-west) of 40 km. To the south and the east, the bay is surrounded by a mountain range with maximum heights of 1500 m. To the north, there are ridges with elevations from 500 to 750 m. Among the mountains of the east and the ridges of the north, there is a valley with a width of 15 km. Ameca River flows through this valley, which is the largest one of the region; the mouth of the river is 10 km north of Puerto Vallarta.

Bathymetry

During November-December, 2001 we have taken field measurements of the bay’s depth using an echo sounder L750 Fishfinder (Raytheon Electronics) and GPS (Global Positioning System) GARMIN. The data were then used to draw the bathymetric chart shown in Figure 1. The digital bathymetric matrix, used to perform our calculation of internal waves generation, has a spatial resolution of 50 x 50 m.

The maximum depth of the bay is almost 900 m and the average depth is about 300 m. The bottom slope of the bay at the south edge is steeper than at the north edge. At the north coast, the slope has an average value of 0.012, with a depth of 100 m that is reached 8 km from shore. At the south coast, the average value is 0.080 and the 100 m isobath is 1.5 km from shore. Inside Bahía de Banderas there is a deep canyon, located about 8 km from the south coast. The canyon’s longitudinal axis lies east to west and it extends up to the eastern shoreline of the bay.

Northwest of the bay, there are two little islands called Islas Marietas. From the islands to Punta de Mita, depths are below 25 m and an immersed sandbank has formed, which does not allow the free exchange of water masses between the ocean and the northwestern part of the bay.

Sampling method and data

The oceanographical surveyors covered the bay area (Figure 2) using the research vessel BIP-V, owned by Universidad de Guadalajara, and a towed CTD profiler SBE-19 (Sea-Bird Electronics Inc.), which was placed in a streamlined box. At a vessel speed of 6 knots, the profiler moved near the surface. For measuring purposes, the vessel, while keeping its speed unchanged, made a couple of circles to help the profiler sink to a depth defined by the length of the towing cable of 150 m; then, the vessel moved to the next measurement site [3]. The survey coordinates were recorded with the help of a GPS receiver. The sample rate for depth, temperature, and salinity was twice per second (with a depth step of 0.5 m). The measured data were then smoothed using a cosine filter with a half width of 2 m over depth.

The survey was done from 17:11 h, April 24, 2001 to 21:02 h of the following day, and consisted of 8 transects, perpendicular to the coastline, starting from the 200-300 m isobaths (one station only was at 50 m depth). In any measurement station of our work, internal waves from every part of the bay can arrive from any other place of the bay; after the sum of the interactions of the waves, a random field is generated. Our

ISSN: 1665-5745 -3/43- www.e-gnosis.udg.mx/vol4/art18 © 2006, e-Gnosis [online] Vol. 4, Art. 18 Geostrophic currents in the presense…Plata L. et al. calculations of wavelength (Appendix C) show that the internal waves originated in the central part of the study area have values between 17 and 20 km (Figure 3), of the same order of the basin dimensions; for this reason, they can not cause inclinations of the time series of temperature and salinity. Therefore, the fluctuations in temperature and salinity are generated by short internal waves that are formed by process of disintegration and reflection from the coast. In almost 28 hours of continuous work, the ship covered, at an average velocity of 6 knots (above 11 km/h), more than 102 nautical miles (about 188 km). In all, 71 vertical soundings were made down to a maximum depth of 150 m. Distance between stations in the transects was 2-3 km. Time spent measuring in each cast varied between 2-3 minutes, depending on the depth of each place. During the casts, the CTD sank with a velocity close to 1 m/s, on average.

Salinity data for each vertical cast were treated by the algorithms of Fofonoff and Millard [21] and Trasviña [22], from temperature, conductivity and pressure data. This allowed us to eliminate errors (noise) caused by differences in the response time between conductivity and temperature sensors [23].

Results

The oceanographic survey in Bahía de Banderas was carried out when the semidiurnal tide was approaching its almost maximum height of 1 m corresponding to that day (Figure 3). Figure 4 shows that at the time of measurement the thermocline and the halocline were very close to the sea surface. Maximum temperature gradients were located in the 0-20 m layer, where they reached values of 0.25 °C m-1. From this layer down to 70 m depth, the temperature dropped very slowly, with gradients of 0.1 - 0.05 °C m-1. Salinity shows a vertical change similar to temperature, with a maximum variability from 33.5 up to 34.8 at the thermocline (0-20 m layer). The mean Brunt-Väisälä frequency profile has maximum values between 7.1-14.2 cycles/h at the thermocline (0-20 m layer) and it diminishes to a range of 1.7-2.5 cycles/h to 140 m depth (Figure 5).

Temperature and salinity profiles are displayed in Figure 4. As we can see, the ‘spaghetti’ shape of the profiles showed a very significant variation in the upper layer (from the surface down to 30 m), caused by internal waves influence. Temperature in the thermocline that corresponded to the same layer in different zones of the bay displayed a difference of up to 4 °C. The difference of the position of the thermocline by the vertical line of the same isotherms was of 10-15 m.

Everything mentioned above allows us to affirm that the temperature and salinity profiles belonging to the bay have several levels of ‘noise’ due to internal waves of different amplitudes and periods. The mean level of this ‘noise’ reflects the internal waves intensity for the continental shelf section covering the bay. According to theory, these waves arise as a product of internal tide disintegration on the continental shelf of the bay; due to the reduction of depth, internal waves become nonlinear and disintegrate into groups of short period waves, which propagate in different directions from their places of origin [24].

Temperature and salinity fields after filtering of internal waves

Based on the methodology detailed in Appendices A and B, filtering of spatial fluctuations of temperature and salinity, related with internal waves in Bahía de Banderas was carried out and interpolation of these characteristics was made at the nodes of a regularly spaced 1.5 km mesh.

ISSN: 1665-5745 -4/43- www.e-gnosis.udg.mx/vol4/art18 © 2006, e-Gnosis [online] Vol. 4, Art. 18 Geostrophic currents in the presense…Plata L. et al. As it is proved in a series of theoretical works [25, 26], the optimum shape of the filter to remove random variations of the hydrophysical field caused by internal waves depends on the shape of its spatial correlation function B()x,y . The analyzed oceanographic survey includes P = 71 vertical soundings, corresponding to each horizon from 0 down to 90 m, with a vertical step of 5 m, the spatial correlation function was calculated by means of the following terms: M = P (P 1) / 2 = 71 70 / 2 = 2485,where is the number of possible pair of soundings belonging to the analyzed field. This way, in each spatial interval(,)xy,for our case a square with sides of 1.5 1.5 km, a few hundred sums dropped for small values of i, j, and dozens of sums dropped for maximum values.

In Figure 6 (a, b) we give estimated sections for the normalized spatial correlation function of temperature oscillations using percentages relative to the maximum value along the horizontal axes and the vertical coordinate. The figure shows that maximum values of function B()x,y are reached in the layer corresponding to the thermocline; under this layer, the values drop exponentially. Correlation function has a significant spatial isotropy and a rapid decrease along the x,y axes. This means that the temperature fluctuation field in the bay has been formed by the interaction of internal waves of short length, in this case a few kilometers, which propagate from different generation zones. The correlation function of the salinity field shows the same behavior (not shown). For the calculation of the smoothing weight coefficients, we chose to depict the spatial correlation functions in an analytical way. These were approached by the expression:

() + () = c1 x c2 y () + B x, y e cos 2 d1 x d 2 y .(1)

Coefficients in equation (1) were found with the aid of the minimum squared method. They were calculated for different depth levels at the following limits (for data of temperature oscillations): c1 from 0.0245 to 0.0367; c2 from 0.0225 to 0.0310; d1 from 0.00573 to 0.00619; d2 from 0.00321 to 0.00372. In the case of the mean correlation function for the 0-40 m layer, the one most = ‘contaminated’ by internal waves, the coefficients’ values were: c1 =0.0350; c2 0.0297; = = d1 0.00607; d2 0.00353. Maximum values of mean quadratic errors of smoothing (see formula B3) (1.77 ºC and 1.08 for salinity) were found in the thermocline at the 10 m level. In the remaining levels, the errors were smaller and in the levels below 50 m, they do not go beyond 0.27 ºC and 0.18 for salinity.

In Figure 6 (c, d), mean function B()x,y in the 0-90 m layer is estimated and its approximation B()x,y is calculated by equation (1). As we can see, both figures are practically equal from their maximum value to the 30% value, which we use as the characteristic radius r of the spatial correlation function (see Appendix B).

With regard to the qualitative results of the smoothing and the interpolation for levels in the bay deeper than 30 m, there are few differences among them, so we will look at just a few examples belonging to the upper levels, which were the most contaminated by internal waves. Figure 7 (a to d)andFigure 8 (a to d) show the initial distribution of temperature and salinity respectively, at the 0, 10, 20 and 30 m levels. These figures were obtained with the aid of common interpolation of the measured data at the nodes of the calculation mesh (1.5 1.5 km) through cubic spline. As can be seen, the distribution features of initial temperature and salinity fields are very complex, with many

ISSN: 1665-5745 -5/43- www.e-gnosis.udg.mx/vol4/art18 © 2006, e-Gnosis [online] Vol. 4, Art. 18 Geostrophic currents in the presense…Plata L. et al. crests and valleys of different shapes and ranges caused by the rising and sinking of water masses due to internal waves. Spatial gradients of temperature in some areas of the bay at the 10 m depth reached more than 1 ºC km-1, and, in the case of salinity, they surpassed 1.5 km-1.

Temperature and salinity fields, once filtered of the influence of internal waves (see Figure 7e to h, and Figure 8e to h), show distribution features that change smoothly and in which there are no large spatial gradients. On average, temperature gradient values along the bay are not more than 0.1 ºC/km, and across the bay along the north-south axis they are close to zero. Variations in the smoothed salinity field, compared to the temperature field, are very few and they have maximum values around 0.02 km-1 along the bay. It has to be mentioned that the salinity fields correspond to the dry season (rain season extends from June to September); for this reason, the inflows from rivers like the Ameca are minimum (1 m3 /s or less [27]) and they do not affect the salinity distribution in the bay.

Geostrophic currents

Dynamic heights for shallow regions are generally calculated from the ocean surface down to the bottom [23 28]. Currents in the bay have never been measured, and for this reason we have no recommendations to follow regarding the calculation of geostrophic currents at surface levels or towards the surface of the bay. Geostrophic currents relative to 100 m were calculated using [29]. If we move the reference level to deeper level the qualitative spatial pattern of distribution displayed in Figure 9 do not change considerable.

In Figure 9 there is a noticeable intensification of currents at the NW zone close to Islas Marietas and at the NE zone near the coast. In both cases, velocities are an order of magnitude greater than in the rest of the bay In Figure 9 (a), a qualitative picture of the geostrophic circulation in the bay at the end of April 2001 is shown with large gray arrows. As we can see, the circulation consists of a very weak jet stream of oceanic water; this might be a branch of the Costa Rica Coastal Current which reaches this region of the Mexican Pacific at the end of spring [30]. The jet stream enters the bay close to Cabo Corrientes and then it moves inside the bay in an easterly direction parallel to the south coast. After approximately 10 km, this jet stream divides itself into two branches. The first one, weaker and narrower of about 1-1.5 km in width, flows to the north and after arriving at the center of the bay, it rotates anticlockwise. Near the southern part of Islas Marietas, it then flows out of the bay to the open ocean. The second jet stream branch moves farther from the separation point towards the east, increasing its width and intensity. Later, it flows along the east and north coasts of the bay and then between Punta de Mita and Islas Marietas and finally out of the bay to the open ocean. We can assume that in general terms the character of the geostrophic circulation has not changed with depth, but its values decreased significantly and at levels deeper than 70 m where current values were less than 0.5 cm/s.

Spatial and vertical structure of the geostrophic currents in Bahía de Banderas is shown in Figure 10. These transects complement the circulation sketch and show that the strongest currents were present south of Islas Marietas and at the east zone of the bay, where they appeared down to 50 and 70 m of depth, respectively The flow in a northerly direction towards the center of the bay had its maximum velocity at the surface (5 cm/s) and it reached a depth of just 20 m.

The results obtained have shown that the geostrophic circulation in the bay in the months of spring can play a very important role in the water masses exchange between the regions close to the east coast of the bay which are always more contaminated, and the clean water masses from the open ocean. If we take the main gyre of

ISSN: 1665-5745 -6/43- www.e-gnosis.udg.mx/vol4/art18 © 2006, e-Gnosis [online] Vol. 4, Art. 18 Geostrophic currents in the presense…Plata L. et al. geostrophic currents shown above, which has a mean velocity of 5 cm/s, we find that this flow can follow its entire route in the bay of approximately 70 km during just 16 days, and it can transport to the open ocean a significant volume of surface water which is more contaminated than water at the deeper levels.

Conclusion

Internal tide plays a very important role in the vertical mixing and pollutant dispersal processes. Filtering of internal waves is indispensable for the use of the geostrophic method in currents calculation; without it, values one order of magnitude greater (>20 cm/s) than the expected and measured for more dynamical zones than Bahía de Banderas, like the Gulf of California, are obtained. We should point out that the method used in this work offered the possibility of filtering the random variations of the hydrophysical fields caused by internal waves on the continental shelf. As well, the smoothed fields obtained after filtering were used to calculate the geostrophic currents in Bahía de Banderas. Geostrophic currents are not the dominant currents in the bay. Here, in a partially enclosed place, drift currents caused by marine breeze predominate, buy they play a fundamental roll in the water masses exchange between the bay and the open ocean.

Appendix A: Analytical forms of temperature and salinity fields

Because the oceanographic survey was carried out using the fast technique described by [3], it can be assumed that the temperature and salinity oscillations fields include just three main components: the low frequency component – which depends on its mean gradients in the bay; the random component – caused by pulses originated by disintegration of internal tide waves inside the bay; and the daily behavior component, which depends on the heat exchange at the upper layer of the ocean [4].

To test the homogeneity of the initial data measured during the oceanographic survey in Bahía de Banderas, the vertical casts were divided in two sets of 36 and 35 stations (see Figure 3). Statistical parameters (mean and standard deviation) for each set were calculated for each horizon of analyzed depth (Table I). A comparison of the resultant values corresponding to the salinity and temperature fields, for each horizon, between the three considered sets, show that both of them are significantly homogenous in their fluctuations. The largest differences belong, as it was expected, to the mixing layer (0 to 10 m depth), because numerous dynamical processes (diurnal cooling/heating, wind stress, drift currents) manifest there. However, all the differences are very small (e.g., 0.23°C in temperature, and 0.32 in salinity, at 10 m depth) and they are not significant, therefore, we can consider the temperature and salinity fields as homogeneous.

From this point on we will talk only about the temperature field, because the explanations corresponding to the salinity field are the same. Temperature distribution for a constant level in the bay can be expressed as a function of x, y,andt coordinates, represented by the form:

()= ()+ ( )+ () T x, y,t T x, y T ' x, y,t Tc t (A1) where is the term that describes the low frequency component; T '()x, y,t is the random spatial-temporal caused by short waves generated by the internal tide; is the daily temperature variability, which can be considered as insignificant compared to the random term [4].

ISSN: 1665-5745 -7/43- www.e-gnosis.udg.mx/vol4/art18 © 2006, e-Gnosis [online] Vol. 4, Art. 18 Geostrophic currents in the presense…Plata L. et al. The analytical expression for mean temperature was depicted using the polynomial form:

()= + + + 2 + 2 T x, y a0 a1 x a2 y a3 x a4 y , (A2) where a0, a1, a2, a3, and a4 are coefficients calculated by the minimum squared method with the condition of minimization of the sum:

N [] ()+ + + 2 + 2 2 = Ti a0 a1 xi a2 yi a3 x a4 y min ,(A3) i=1 where N is the total of oceanographical casts.

It was assumed that and this was estimated as: ()()= ()()= 2 2 T ' x, y T x, y T x, y T x, y a0 a1 x a2 y a3 x a4 y . (A4)

Here and in the previous equations, A1-A3, we have the linear dimensions of the Cartesian coordinates axes x, y .

Assuming homogeneity of the temperature deviations field, in each level the estimates of correlated spatially normalized functions are calculated by the equation [4, 31]:

1 PP Bxy(),,= Tlxky() 22 P lk==11 T[]()(l + i x, k + j )y , i = 0, P 1; j = 0, P 1 ,(A5) where i, j are the current number of the values of the correlation function that coincides with the axes of the coordinates x and y;P, the total amount of the values in the temperature field; l,kthe current values in the temperature field; x,y the distance between the values of the temperature field using the axes x and y; 2 = B()0,0 is the dispersion of the temperature field. Because of the central symmetry of the function Bxy(), , negative values of i, j are not included.

Appendix B: Filtering algorithm of internal waves in the bay

Temperature values, ‘cleaned’ of internal waves influence, are calculated for a regular mesh as the sum of the low frequency T()x, y and the random complement, which is found through interpolation of the field deviations using the formula:

(B1)

ISSN: 1665-5745 -8/43- www.e-gnosis.udg.mx/vol4/art18 © 2006, e-Gnosis [online] Vol. 4, Art. 18 Geostrophic currents in the presense…Plata L. et al. where: Ti' is the field deviation value in every M point located at the limits of the region of the characteristic scale of the spatial correlation function; Pi are weight values of the corresponding L = points Pi 1. Weight value of each L point is calculated by means of the normalized correlation i=1 function of the deviation field. This way, the smoothed, interpolated value of the field, is estimated by the expression: MM = TTBB0 ii i,(B2) ii==11 where Bi is the value of the function at the point with a deviation Ti'.

Note that the number of M points is determined by the shape and the characteristic scale r of the spatial correlation function whose approximate value is arrived at using in such a way that it fulfills the relation: B()x,y B()x,y , for every x,y r .Here, =2-0.5 =0.707.In max electronics, based on the above conditions, the width of the directionality diagram can be calculated for the radio antenna through the electric field tension [31]. In the calculations cited above, as a characteristic scale we took the spatial displacement value at the point where the spatial correlation function decreased to 30% of its maximum value.

Because the measurement error of temperature values recorded by the CTD SBE-19 is much less than the errors that arose during the calculation of its smoothed values, the term that deals with the measurement error of the initial values can be suppressed in equation (B1). The mean square error for smoothing and its measure are assessed for each level and node of the mesh through the formulas [26, 32]:

(B3)

Appendix C: Modal analysis for internal waves

Parameters for linear internal waves were found using the numerical solution to the boundary value problem [33] for vertical profiles of the Brunt-Väisälä frequency:

N 22 Wk+=2 t W0 ,(C1) zz h 22 ti where W =0atz=0andz =-H; t is the internal wave frequency; i is the Coriolis parameter; kh is the horizontal wave number; and N2 is the Brunt-Väisälä frequency. Equation (C1) was solved numerically by the Gauss-Seidel method. The eigenfunctions Wn ( , z) and the eigenvalues kn ( ) were used for each fixed mode n and frequency values of internal waves.

References

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ISSN: 1665-5745 -10 / 43- www.e-gnosis.udg.mx/vol4/art18 © 2006, e-Gnosis [online] Vol. 4, Art. 18 Geostrophic currents in the presense…Plata L. et al. 31. Konyaev, K.V. (1990). Spectral Analysis of Physical Oceanographic Data. National Science Foundation, Washington, D.C. 32. Gandin, L.S. (1965). Objective Analysis of Meteorological Fields. Israel Program for Scientific Translations, Jerusalem. 33. Krauss, W. (1966). Interne Wellen. Gebrüder Bornträger, Berlin.

Figure 1. Observation area and bathymetric features of Bahía de Banderas, Mexico.

Figure 2. Ship route and distribution of the profiles made with the oscillating CTD during the fast oceanographic survey of Bahía de Banderas, April 24-25, 2001. The transects are numbered from 1 to 8. (Upper left inset) Sea level variations in Puerto Vallarta during the days of the measurements. (Upper right) The circle shows 30-100% of the spatial correlation function used to smooth the initial temperature and salinity fields. (Bottom right) Position of the polygon and the moorings: the moorings were deployed for ten days with the HOBO autonomous thermographs. (Area within dotted lines) All numerical calculations were done inside the dotted line area. The red line divides the cast in two sets indicated by Roman numbers.

Figure 3. Spatial distribution of the wavelength (in km) of the semidiurnal internal tide in Bahía de Banderas, calculated with the linear model from the data of the oceanographical survey of April 24-25, 2001.

Figure 4. Vertical profiles curves for temperature and salinity from the oceanographical survey of Bahía de Banderas for the period April 24-25, 2001.

Figure 5. Mean vertical profiles of temperature, salinity and Brunt-Väisälä frequency, calculated from 71 vertical casts in Bahía de Banderas for the period of April 24-25, 2001.

Figure 6. Vertical-longitudinal (a) and vertical-latitudinal (b) views of the correlation function of the temperature field, from the oceanographical survey data, April 24-25, 2001. (c, d) Mean function B()x,y in the 0-90 m layer and its approximate value B()x,y calculated by equation (1). Isolines are given in percentages of the maximum value. Shaded area points out the characteristic scale of spatial correlation in the bay.

Figure 7. (a to d) Temperature distribution in Bahía de Banderas for the four upper depths of 0, 10, 20 and 30 m from the oceanographical survey carried out on April 24-25, 2001. (e to h) Same temperature fields after filtering of fluctuations caused by internal waves.

Figure 8. (a to d) Salinity distribution in Bahía de Banderas for the four upper depths of 0, 10, 20 and 30 m from the oceanographical survey carried out on April 24-25, 2001. (e to h) Same salinity fields after filtering of fluctuations caused by internal waves.

Figure 9. Geostrophic velocities (cm/s) in Bahía de Banderas (black arrows), corresponding to the following depths: 0 (a), 10 (b), 20 (c), 30(d), 40(e), 50 (f), 60(g) and 80 m (h). A general pattern of the geostrophic circulation inside the bay is shown with gray arrows in (a).

Figure 10. Vertical-latitudinal (a) and vertical-longitudinal (b) distribution of the geostrophic velocities (in cm/s) in the four transects. The position of the transects can be seen in the upper left inset of the figure. Transect (a) shows the u-velocity component (west-east axis, upper-bottom part of the figure). Transects (b, c, d) show the v-velocity component (north-south axis, right-left part of the figure).

Table I. Statistical parameters for the vertical distribution of temperature (°C) and salinity in Bahía de Banderas, calculated from the data measured in April 24-25, 2001.

Data set Hydrophysical Statistical Depth (m) Parameter Parameter 0 10203040 50 60 70 80 90 100 Mean Temperature 23.63 18.56 16.90 16.15 15.59 15.01 14.57 14.20 13.90 13.65 13.44 35 stations Standard 0.60 0.88 0.58 0.43 0.26 0.23 0.17 0.17 0.20 0.20 0.16 in area I (see Deviation Figure 2) Mean Salinity 32.70 34.17 34.58 34.68 34.63 34.62 34.65 34.67 34.70 34.69 34.71 Standard 1.07 0.59 0.30 0.17 0.15 0.14 0.14 0.12 0.07 0.07 0.07 Deviation Mean Temperature 23.89 18.92 16.90 16.00 15.52 15.00 14.54 14.14 13.73 13.50 13.31 36 stations Standard 0.45 1.23 0.61 0.39 0.39 0.44 0.39 0.26 0.18 0.20 0.22 in area II Deviation Mean Salinity 33.38 34.10 34.62 34.68 34.66 34.64 34.66 34.65 34.73 34.71 34.76 Standard 1.07 0.75 0.34 0.20 0.19 0.17 0.12 0.16 0.10 0.06 0.06 Deviation Temperature Mean 23.76 18.74 16.90 16.08 15.56 15.01 14.56 14.17 13.82 13.58 13.38 71 stations in Standard 0.55 1.08 0.59 0.42 0.33 0.35 0.30 0.22 0.21 0.22 0.20 the study area Deviation Mean Salinity 33.04 34.14 34.60 34.68 34.64 34.63 34.65 34.66 34.72 34.70 34.73 Standard 1.12 0.63 0.32 0.18 0.17 0.16 0.13 0.14 0.08 0.07 0.07 Deviation

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