Tidal variations of turbulence at a spring discharging to a tropical estuary
Posgrado en Ciencias del Mar y Limnología, Universidad Nacional Autónoma de México, Del. Coyoacán, México D.F., México
Corresponding author: Gilberto Expósito-Díaz, Posgrado en Ciencias del Mar y Limnología, Universidad Nacional Autónoma de México, Circuito Exterior s/n, Cd. Universitaria, Col. Copilco, Del. Coyoacán, 04510, México D.F., México. (firstname.lastname@example.org)
 Measurements of velocity profiles and near-bottom temperature and pressure were used to determine turbulence properties at a point-source submarine groundwater discharge (brackish) in a tropical estuary. The turbulence properties estimated were Reynolds stress, turbulent kinetic energy production, and vertical eddy viscosity. Results showed a dominance of the zonal Reynolds stress component with maximum of 0.0025 m2 s−2 (2.5 Pa) at low tide. Turbulent kinetic energy production and vertical eddy viscosity values also reached maxima (0.98 W m−3 and ~10−1 m2 s−1, respectively) at low tides. Discharge of brackish water increased at low tides, relative to high tides, as indicated by vertical mean velocity and by mean velocity shear. These maxima were caused by decreasing hydrostatic pressure and likely increasing hydraulic head at the site of discharge. Increased turbulence at low tides was one order of magnitude larger than the turbulence caused elsewhere by tidal flows up to ~2.5 m s−1.
 Estimates of turbulence over the continental shelf, coastal lagoons, and estuaries have been limited by difficulties in obtaining velocity fluctuations throughout the water column. Measurement technologies based on acoustic Doppler current profilers (ADCP) have allowed development of methods that yield vertical profiles of Reynolds stress, turbulent kinetic energy production (TKEP), and vertical eddy viscosity coefficient [Lohrmann et al., 1990; Stacey et al., 1999].
 The purpose of this study is to determine intratidal turbulence variations associated with mixing between direct submarine groundwater discharges, which are brackish, and tropical estuarine waters. The system being studied is located in karst terrain in the Yucatan peninsula in Mexico. This type of terrain is found widespread throughout the world and is characterized by limestone or dolomite bedrock, composed predominantly by calcium carbonate that can be dissolved to form caves and subterranean conduits [Valle-Levinson et al., 2011]. The conduits allow submarine groundwater to discharge directly into the ocean or estuary. The intensity of discharge coming from these systems is determined by differences in pressure in the karst cavity and that exerted by tide at the outflow site [e.g., Kim and Hwang, 2002; Taniguchi, 2002; Valle-Levinson et al., 2011]. These studies have mainly focused on tidal influence on seepage discharges, which increase at low tides.
 Very few studies have been able to identify tidal variations in direct, point source discharges [e.g., Peterson et al., 2009], but no other study, to our knowledge, has investigated the turbulence associated with these discharges or their tidal variations. It is expected that the interaction between the upward discharge from the seabed and the water from the estuary creates turbulence at the transition zone. The turbulence produced, in turn, should depend on discharge intensity and tidal phase. Such is the topic of this investigation.
 In this study, a first attempt is made to quantify turbulence properties in springs discharging brackish water to the ocean. The San Juan spring has a depth of about 4 m, a diameter of 25 m, and is located in the “Capechen-Boca Paila-San Miguel-La Ria” lagoon system, in the Sian Ka'an biosphere reserve in the Mexican Caribbean (Figures 1a, 1b, and 1c). The spring discharges through a cross-section of approximately 15 m2 over the southwestern edge of San Juan conduit (Figure 1d). The entire lagoon system is a shallow basin with an average depth of ~1 m and maximum depths, not including the springs, of ~4 m in the eastern Boca del Río (Figure 1b). Its circulation is estuarine-type owing to buoyancy input in the northern lagoon from direct groundwater discharges and runoff on the western watershed [Chiappa-Carrara et al., 2003].
 The aim of this study is to estimate, with acoustic Doppler current profiler (ADCP) data, variables that can describe turbulence associated with groundwater discharge in a tropical estuary. The variables estimated are Reynolds stress, turbulent kinetic energy production, and vertical eddy viscosity. This assessment seems to be the first of its kind to be disseminated.
2.1 Theoretical Framework
 Turbulence is a condition of the fluid motion in which each velocity component is irregularly distributed in space and time [Trevor et al., 1988]. In geophysical fluids, the Reynolds stress, the turbulent kinetic energy production, and the vertical eddy viscosity coefficient allow descriptions of the turbulence regime [Lohrmann et al., 1990; Stacey et al., 1999].
 The vertical eddy viscosity is a parameter required in analytical and numerical studies of oceanic circulation. It is needed for “turbulence closure” in which turbulent fluctuations of the flow are related to the mean flow. This parameter is a property of the flow rather than a physical constant. It depends on the scale of motion, interaction between motions at different scales, and the degree of stratification of the fluid. In coastal lagoons and estuaries, eddy viscosity values have generally been estimated by traditional methods [e.g., Pacanowski and Philander, 1981]. However, measurement of velocity fluctuations throughout the water column can now be obtained using an ADCP. Thus, this approach has become widely used to obtain a better estimation of the parameters of turbulence in shallow areas [Rippeth et al., 2002; Souza et al., 2004; Osalusi et al., 2009]. The mixing that takes place in estuaries can be related to the shear in the velocity field [Monismith, 2010]. Therefore, it is necessary to determine the velocity shear to derive other turbulence properties.
 Measurements from a moored ADCP were used to analyze the turbulence close to a site of submarine groundwater discharge. A five-minute window from single-ping mode 12 data was used to calculate mean velocities. Reynolds stresses were evaluated following the variance method according to Lohrmann et al.  and Stacey et al. . The instrument's mode-12 operation was used since it produces more accurate results than mode-1 operation [Nidzieko et al., 2006]. The variance method has been used successfully to estimate turbulence parameters associated to different tidal regimes: in tidal channels [Lu and Lueck, 1999; Stacey et al., 1999; Rippeth et al., 2003], in the upper Gulf of California [Souza et al., 2004], and at the bottom boundary layer [Osalusi et al., 2009].
 Reynolds stresses represent the effect of turbulence on the mean motion. Those terms containing the product of the fluctuations of the vertical and each horizontal velocity component that characterize the vertical transport of horizontal momentum by turbulence determine the turbulent kinetic energy production (TKEP), and also the vertical eddy viscosity coefficient (Az). The turbulent kinetic energy production in W m−3 is estimated from the product of the Reynolds stresses and the vertical shear of the horizontal mean velocity components [e.g., Rippeth et al., 2002; Souza et al., 2004; Osalusi et al., 2009]:
where ρ is the water density (1016 kg m−3) and z is the vertical direction. The turbulence generated by spring discharges was estimated with the assumption that Reynolds stresses depend linearly on the spatial derivatives of the large-scale flow velocity [Pond and Pickard, 1978]. The vertical eddy viscosity coefficient was calculated from ADCP data (Figure 1d) with: .
2.2 Field Observations
 Profiles of current velocity were collected from November 23 to 27, 2006, during the dry season. A leveled 1200 kHz ADCP Sentinel (RDI) was deployed near a submarine groundwater discharge, the San Juan spring. The ADCP was installed close to the outflow (Figure 1d) of brackish water (typical outflow salinity is ~20) to collect data in mode 12 using 12 subpings, with a cell size of 20 cm and a blanking distance of 40 cm. The ADCP measured velocity components in 17 cells, plus water temperature and pressure near the seabed. This allowed comparisons of water temperature at bottom and surface by measuring surface temperature with an YSI 556 MPS during the last day of sampling. The spring discharge was estimated from velocity and a nearly uniform distribution throughout the cross section (≈ 15 m2).
3 Results and Discussion
3.1 Mean Velocity and Shear Structure
 Bottom pressure ranged between 4.22 and 4.37 db, indicating a tidal range of only ~0.15 m during the sampling period. Bottom temperature ranged between 20.9 and 28.2 °C with a squared wave-like semi-diurnal variation that featured sudden drops at high tides and rapid, although a bit more gradual, increases at low tides. Thus, an inverse relationship was observed between temperature and pressure at the bottom. Less tidal pressure on the spring favored bottom temperature increases (Figure 2a). At this time of the year, subterranean waters are warmer than estuarine waters. Surface temperature (25.06 °C), measured only on the last sampling day, was 3 °C lower than the groundwater temperature (28 °C) recorded with the ADCP.
 Current velocity time series showed horizontal mean velocity component of up to 0.3 m s−1 (Figures 2b and 2c). The vertical mean velocity component reached positive values of 0.1 m s−1 near the bottom during low tide, a result of increased groundwater outflow (Figure 2d). In general, the strongest velocities were observed during low tide, owing to decreased tidal pressure on the spring, which allowed stronger groundwater outflow relative to high tide. In turn, during high tide, the pressure on the spring increased and the subterranean outflow decreased. This change in discharge strength indicated a clear tidal modulation with values ranging from 1 to 3 m3 s−1 at high and low tide, respectively (Figure 2e).
 During low tide, the u-component of velocity was usually greater than the v-component, inducing an east-northeastward flow. However, near the bottom, the v-component dominated and had an opposite sign compared to the rest of the column (Figure 2c). During the highest tidal elevation, bottom temperature decreased and the horizontal mean flow reversed with respect to that observed during low tide, indicating that the hydrostatic pressure inhibited groundwater discharge. Although no negative values of vertical mean velocity component were observed, seawater inflow into the aquifer could be expected when pressure in the karst cavity becomes lower than the pressure exerted by the tide [Valle-Levinson et al., 2011].
 The u-component of the mean velocity displayed variations throughout the water column during the entire sampling. Figure 3a shows two time series of the u-component, with values of up to 0.30 m s−1 and 0.20 m s−1 at 1.18 and 3.18 m above the bottom (mab), respectively. The greatest velocity differences between those two depths, i.e., the maximum vertical shears at those levels, were observed consistently at low tides. Usually during high tide, the vertical shears were rather small. The temporal means of the shear in four different tidal stages (between low and high tide (flood), high tide, between high and low tide (ebb), low tide) show values ranging between −0.2 and 0.3 s−1 (Figures 3b–e), with a maximum at 1 mab. In some cases, as during the strongest ebb, the shear sign was inverted (Figure 3d). During high tide, at higher pressures than 4.35 db, groundwater outflow was hindered and the shear in the upper layer (2.5 m) was uniform (Figure 3c). Noticeable is the variation in shear profiles among all eight low and ebb tides (Figures 3d and 3e). This is likely a consequence of the varying level of each of the eight low tide periods (black line in Figure 3a), some of them being lower than others. The discharge seems to be quite sensitive to variations in low tide level and therefore the shears are, too.
 The vertical shear time series showed values ranging between −0.6 and 0.9 s−1 (Figure 3f). However, near the bottom the values were around 0.4 s−1, whereas above 1.3 mab the shear was negative. This pattern was inverted during highest tide, with positive shear in the upper layer of 1.8 m thick and negative in the lower layer. These shear patterns were influential to the levels of turbulence observed.
3.2 Analysis of Turbulence
 The turbulence associated to groundwater discharge shows Reynolds stress dominated by the component with maximum values of 2.5 × 10−3 m2 s−2 and minimum of 1.32 × 10−5 m2 s−2 (1.32 × 10−2 Pa) (Figure 4b) during low and high tide, respectively (Figure 4a). The component increases during maximum ebb generating uniform pulses along the water column (Figure 4c). However, during low tides, a layer 1.5 m thick near the bottom shows peaks of with vertical variations generated by the vertical shear of horizontal mean velocity.
 Reynolds stresses respond to variations in vertical shear of the mean horizontal velocity (Figure 3f). The vertical shear is greatest near the bottom, between 0.5 and 1.5 mab, at low tide. Generally, during high tide, the velocities at 1.18 and 3.18 mab are similar and differences between those two levels tend to occur at other tidal stages. During high tide, the increase in hydrostatic pressure hampers outflow of brackish water, which translates into low values of Reynolds stresses.
 The evolution of TKEP is also associated with semidiurnal tidal oscillations, showing an inverse relationship with hydrostatic pressure. Maximum values of TKEP are close to 1 W m−3 at low tide and minima of 4 × 10−3 W m−3 at high tide (Figure 4d). This is a change of at least three orders of magnitude in TKEP within the tidal cycle. The correlation coefficient between the Log10 TKEP and water level (pressure) was ~0.6, which was statistically significant (Figure 4g). However, the variability explained by a linear regression is ~32%, which indicates that there is another factor influencing the TKEP.
 The vertical eddy viscosity coefficient Az ranged between 0.02 and ~1 m2 s−1 with variations related to changes in the water level and vertical shear (Figures 4e and 3f). At low tide, peaks of Az occurred near the bottom, in a lower layer ~1 m thick, while at mid-water and at the surface, the maximum values were about 0.4 m2 s−1. During high tide, this coefficient was practically uniform throughout the water column, with values of order 10−2 m2 s−1. The temporal mean Az profile showed a value of 0.12 m2 s−1 near the bottom, which increased to 0.2 m2 s−1 at 0.5 mab (Figure 4f). Between 1.25 and 2.75 mab, the vertical eddy viscosity coefficient had average values of 0.11 m2 s−1, increasing slightly near the surface to 0.12 m2 s−1. The value of this coefficient seems to have increased slightly near the surface because of static instabilities. Such instabilities were likely related to the jet interacting with denser waters at the limit of vertical excursion for the jet.
 In summary, the turbulence regime produced by spring discharges was obviously related to the intensity of the discharges, which in turn was influenced by the hydrostatic pressure modulated by the tide. It is likely that the tide influenced the piezometric gradient, which was not resolved in these measurements. An increase of the hydrostatic pressure, and concurrent decrease of piezometric gradients, caused a decrease of the groundwater discharge and of turbulence.
 The results presented above are now placed in the context of turbulence properties measured in other environments. Reynolds stresses were of order 10−3 m2 s−2 or one order of magnitude greater than those reported by Souza et al.  in the upper Gulf of California. In that environment, tidal currents (0.6 m s−1) interacting with the bottom produced velocity shears of 0.01 and 0.03 s−1 near the surface and bottom, respectively. In the estuary where the San Juan spring is located, tidal amplitudes are small (< 0.2 m). Although the horizontal mean velocity is markedly smaller than that reported for the upper Gulf of California, the mean velocity shear observed at San Juan spring was one order of magnitude higher. On the other hand, the Reynolds stresses estimated at the spring discharge are comparable to those generated at Menai Strait in Wales (10 m deep), where spring tidal amplitudes are ~3 m and tidal currents are up to 2.5 m s−1 [Rippeth et al., 2002]. Summing-up, Reynolds stresses of order of 10−3 m2 s−2, produced by submarine groundwater discharge during the dry season were comparable to those that occur in tidal channels [e.g., Rippeth et al., 2001], but were one order of magnitude higher than those generated by tidal turbulence in coastal areas [e.g., Souza et al., 2004]. Reynolds stresses at the spring are greater than at typical coastal areas influenced by tidal flows because the spring shows vertical and horizontal velocity components of similar magnitude. In tidal channels, vertical velocities are at least one order of magnitude smaller than horizontal. Although the values of Reynolds stresses may be affected by spatial inhomogeneities of the spring discharge cone, these results are consistent with those obtained by Parra et al. [submitted], with single-point measurements at a spring discharging into a reef lagoon.
 The relationship between water level and TKEP is obvious but with low variance explained by a linear regression. This indicates that there are other factors influencing TKEP such as the hydraulic head, or piezometric gradient between sea level and water table, the porosity (or conductivity) of the karst cavity, and the actual buoyancy of the discharge [Valle-Levinson et al., 2011]. These factors must be taken into account in future studies that should also incorporate hydrological components. A key finding, nonetheless, is that the highest TKEP, the maximum mean (in the turbulence sense) vertical velocity, and the largest spring discharge occur during low tide. When the spring discharge increases so does the vertical shear and the flow becomes unstable. These results were consistent with those reported by Valle-Levinson et al.  at another spring that discharges into the coastal ocean on the northern shore of the Yucatan Peninsula and by Parra et al. [submitted] at a spring discharging into a reef lagoon of Quintana Roo.
 At the spring discharge, TKEP is similar to that reported at Menai Strait in Wales [Rippeth et al., 2002] but the TKEP mechanisms are different for each site. In the spring case, the turbulence generation mechanism is groundwater outflowing from the seabed, with the corresponding increase of Reynolds stress and vertical shear of horizontal mean velocity at low tide. In the tidal channel at Menai Strait, the turbulence generation mechanism is from horizontal currents interacting with bottom stress where maximum values of TKEP are found at peak tidal flows.
 This study was conducted during the dry season of the year, when discharge is relatively low. However, the TKEP values are similar to those found in a tidal channel during spring tide [Rippeth et al., 2002]. Thus, it is expected that springs will increase energy production during the rainy season, when the intensity of groundwater discharge is largest.
 At San Juan spring, Az values were of order 10−1 m2 s−1, two orders of magnitude greater than those reported in the upper Gulf of California by Souza et al.  but of the same order of magnitude as those estimated by Osalusi et al. . In the latter case, estimates were obtained at the bottom boundary layer from Reynolds stresses of order 10−2 m2 s−2, TKEP of order 10−4 W m−3 and vertical shears between 0.1 and −0.05 s−1. The high values of Az obtained at San Juan spring, from ordinary Reynolds stress of order 10−3 m2 s−2 and TKEP between 1 and 10−3 W m−3 were caused by relatively large vertical shears of the mean velocity components (−0.6 to 0.8 s−1).
 In tidal channels and on the continental shelf, the maximum TKEP occurs during maximum ebb and flood and decreases rapidly towards slack tide [e.g., Souza et al., 2004]. On the other hand, in areas of springs discharge, TKEP peaks occur at low tide. At San Juan spring, TKEP shows values of order 1 and 10−3 W m−3 during low and high tide, respectively, showing an inverse relationship with hydrostatic pressure. Moreover, the turbulent kinetic energy production is expected to increase during the rainy season owing to increasing discharge.
 This work was supported by DGAPA-PAPIIT-UNAM, under grant IN115903. We are very grateful to CONACYT, Mexico to sponsor Gilberto Expósito-Díaz during this work. The field assistance of M. Díaz-Flores, D. Salas-Monreal, J. Santamaria, and C. Winant are gratefully appreciated. We thank J. Castro for improving the figures of this article. A.V.L. acknowledges support from NSF project 0825876.