^{1}

^{*}

^{1}

^{1}

Field data and a simplified numerical model were used to investigate the dominant basin-scale internal waves in a tropical Andean reservoir during a dry period. The structure and period of the observed baroclinic oscillations were inferred from spectral analysis of the measured temperature records and the associated isotherm displacements. The dominant oscillation identified from the field data had a vertical structure made of two layers with vertical velocities oscillating out of phase (V2 mode) with a period of 24 h. The theoretical baroclinic vertical modes of the reservoir were estimated by using a simplified eigenfunction model that reproduced the period and vertical structure of the observed dominant mode, indicating also that the horizontal velocity oscillations in every layer were in phase (V2H1 mode) with a period close to 24 h. The wind forcing exhibited a periodic behavior dominated by the 24-h period component and, thus, we concluded that a V2H1 mode was dominant during the survey analyzed and it was excited by resonance with the diurnal wind-forcing. We modeled the reservoir as a linear damped forced mass-spring system to estimate the damping ratio of the baroclinic oscillations and we found that the motions were underdamped with a damping rate similar to the reported in other lakes around the globe. In addition, we estimated the gradient Richardson number due to the dominant V2H1 baroclinic mode and we found that the potential diapycnal mixing by shear was low. Finally, we discuss the vertical structure of the phase of the internal wave oscillations and the potential implications of the identified internal wave field for the ecology of the reservoir. © 2019. Acad. Colomb. Cienc. Ex. Fis. Nat.

A partir de datos de campo y un modelo numérico simple, se investigaron las ondas internas a escala de cuenco en un embalse tropical andino durante un período seco. La estructura y el período de las oscilaciones baroclínicas observadas, se infirieron a partir del análisis espectral de las series de temperatura medidas y de los desplazamientos verticales de las isotermas asociadas. El modo de oscilación dominante identificado a partir de los datos de campo constaba de dos capas que oscilaban con velocidades verticales y fases contrarias (modo V2), y con un período de 24 h. Los modos baroclínicos verticales teóricos del embalse se estimaron a partir de un eigenmodelo simple, el cual predijo el período y la estructura vertical del modo dominante identificado a partir de las observaciones, indicando también que las oscilaciones de la velocidad horizontal en cada capa estaban en fase (modo V2H1) y con un período natural cercano a 24 h. El forzamiento del viento mostró una variabilidad periódica con un período dominante cercano a las 24 h, por lo que concluimos que el modo V2H1 fue el dominante durante el período analizado y sometido a excitación por resonancia con el forzamiento diurno del viento. Se modeló el embalse como un sistema lineal masa-resorte, amortiguado y forzado, con el fin de estimar la relación de amortiguamiento de las oscilaciones baroclínicas, y se obtuvieron oscilaciones subamortiguadas con una tasa de amortiguamiento similar a la reportada en otros lagos alrededor del mundo. También se investigó el potencial del modo V2H1 en la generación vertical de turbulencia debido a inestabilidades cortantes usando el número de Richardson del gradiente, y encontramos que la producción de turbulencia era baja. Por último, se discuten aquí la estructura vertical de la fase en las oscilaciones de las ondas internas y las potenciales implicaciones del campo de ondas internas en la ecología del embalse. © 2019. Acad. Colomb. Cienc. Ex. Fis. Nat.

The existing literature on physical limnology and its linkage with the water quality of temperate inland waters is prolific. However, the investigations in the tropics, and particularly in the tropical Andes, are scarce in spite of the fast growing of human populations in tropical regions, which has created serious environmental problems during the last decades for tropical lakes and reservoirs (

In the particular case of Colombia, numerous reservoirs for hydropower generation and water supply have been constructed during the last few decades to satisfy the growing energy and water demands of the country. Management of these reservoirs requires extensive knowledge and understanding of the physical processes responsible for water circulation, which influence directly water quality, demanding clarification of the predominant processes and time scales over which they evolve (

Stratified lakes and reservoirs can be regarded as three-dimensional, mechanical, viscously damped oscillators characterized by an infinite spectrum of baroclinic (i.e., internal motions in which isopycnal surfaces oscillate) oscillatory natural modes with characteristic periods and spatial structures (

The natural modes are classified depending on the number of nodal points they have in the horizontal

The description of the natural modes has been accomplished in the literature by the implementation of models with simple horizontal geometries (e.g., rectangular, circular, elliptical) and flat bottoms for layered (

Previous studies have dealt with internal waves in Colombian inland waters (Vélez-Castaño & Gómez-Giraldo, 2011;

In this paper, we describe the study site, measurements, data analysis procedure, and the natural eigenfunction model. We describe the evolution of the external forcing and the thermal structure during the survey. Then, we depict the measured internal wave field from spectral analysis of the field data and present the theoretical natural modes by using the eigenmodel. With the aim to show the role of the wind forcing in the excitation of the internal waves, we assimilated the reservoir as a linear damped forced mass-spring system and estimated the damping ratio of the dominant baroclinic mode. Finally, we estimated the potential diapycnal mixing by shear due to the baroclinic modes and summarized the scopes of the numerical model and the relevance of the results for the reservoir ecology.

Where

To obtain the baroclinic modes, we had to find the non-trivial solutions to ^{
iat
}

where

where _{
x
} is the horizontal wave number, _{0} is the phase to be determined from the boundary conditions. Thus, _{
x
}
^{2}

where _{
n
} is the horizontal phase speed and _{
n
} is the vertical structure of the n^{th} vertical mode. This is of the Sturm-Liouville form, which is analogous to the normal modes of oscillations of other systems, e.g., the stretched string. For a continuously stratified lake, there is an infinite set of possible values (eigenvalues):

associated with the corresponding eigenfunctions, the normal modes:

that satisfy

where _{
n
} obeying _{mn} (the eigenfrequencies) associated with the horizontal wavelengths _{
m
} determined by solving eq. 10 that represent the baroclinic modes of the basin. In that way, the natural frequencies and periods _{
mn
} of the baroclinic modes are 27r

To solve the

where _{
n
}
^{2}

where the subscript _{
k
} is the corresponding buoyancy frequency of the

The vertical velocities obtained by the model have arbitrary units because the resultant eigenvectors are not realistic velocity vectors. Thus, the vertical displacement,

Low inflow discharges dominated during the field campaign, with a gradual increase throughout the survey and the occasional arrival of important discharges towards the end of the period. Outflow discharges occurred with a variable operation, with similar magnitudes to those of the inflows when the outtake was in operation. A dominant frequency component in the inflows and outflows time series was not identified, as can be seen from their power spectra (

The evolution of the thermal structure showed a gradual deepening of the isotherms between depths of 10 and 30 meters, which indicates a warming of these waters due to the surface heat fluxes. Towards the end of the survey (from April 6 onwards), there was a change in the deepening rate of the isotherms, with a faster deepening for the isotherms below 20 m deep and an upward trend in the isotherms above 20 m (

Two features of the dynamics of the isotherm displacements stand out. First, there was a dominant periodicity close to 24 h in the vertical displacements that gradually reduced its amplitude towards the end of the survey. Second, the top and bottom isotherms shown in

To identify the nature of the 24-h period oscillations, we calculated the coherence and phase spectra between the temperatures recorded by the thermistors at 6 m and at 20 m and between thermistors at 6 m and at 44 m (

To improve the understanding of the vertical structure of the oscillations, we calculated the coherence and phase for the 24 h period between the temperature at every thermistor depth and the temperature at 6m depth, defined arbitrarily as reference.

Thus, the 24-h peak of the isotherms displacements was associated with a second vertical mode or higher. However, at this point it is not possible to establish with certainty the vertical order of the mode from the field data, as the thermistor chain did not cover the whole depth range of the reservoir and, hence, it could be possible that below 44 m (the location of the deepest thermistor) another change of phase occurs. It should be noted that our data does not strictly exhibit phase differences of 0° or ±180°, which would be expected for pure modes. The discussion of this result is presented below.

The vertical structure of the first three natural modes predicted by the eigenmodel is shown in ^{-1}), 25.2 h (0.11 ms^{-1}), and 35.2 h (0.09 ms^{-1}), respectively.

Previous studies have shown that the periodic operation of the outflow discharge in a reservoir may lead to the excitation of internal waves (

To get a deeper insight into the link of the water column response and the wind-forcing, we assimilated the reservoir to a harmonic, viscously damped wind-forced oscillator. We 24-h bandpass-filtered the observed oscillations of the 25 m depth isotherm and adjusted the output to the oscillation of a linear damped forced mass-spring system (

where _{
lag
} account for the spatial location and _{d} = T(27Π£)^{-1}, the damping ratio _{0},

where _{
si
} is the time of the start of the _{
lag
} were obtained by least-squared fitting

The damping ratio obtained for the V2H1 mode was compared to damping ratios of seiche modes in other lakes around the globe (_{C} = 1), so the oscillations were underdamped, which has also been observed in most stratified systems with few cases (

*Results from the present work

which expresses the ratio of the stability due to the stratification (given by

The vertical displacements predicted by the eigenmodel were scaled by a maximum amplitude of 3 m (

Porce II, the vertical oscillations of up to 4 m at the depth of the submerged gates could be important for inducing essential changes in the quality of the water withdrawn, which in turn may be important for the quality of the river and Porce III reservoir located downstream of Porce II dam. The dominant V2H1 mode during the survey has the peculiarity of generating a periodic contraction and expansion of the middle layers of the reservoir (

River intrusions incorporate nutrients into the upper mixed layer where important primary productivity occurs by the turbulent mixing and advection which are moduled by other physical processes (_{
mix
} ) of substances from the plume up to the surface layer is _{
z
} is the eddy vertical diffusivity and _{
z
} parameterization as a function of _{
z
} at the interior of the water column is of the order of the molecular diffusivity. We assumed a value of _{
z
} = 1x10^{6} m^{2}/s (_{
mic
} 12 years. Thus, we suggest that for the typical dry conditions in Porce II reservoir, the diapycnal mixing induced by the internal waves is negligible and other mechanisms and interactions among processes must be responsible for the major vertical transport and mixing at the lake interior.

According to

More complete models should be used to capture the properties of the wave field in Porce II taking into account both real bathymetry and stratification. The weak and continuous stratification of Porce II, very different from the typically layered-like summer stratification of temperate lakes, as well as the non-simple morphometry of the reservoir, lead us to suggest that future analysis on the baroclinic modes must be addressed by using more elaborated models than the one used in this work.

Internal waves were a prominent feature of the basin-scale hydrodynamics of Porce II reservoir during the dry period studied. An underdamped V2H1 mode was excited in the reservoir by resonance with the diurnal wind forcing and prevailed for several days according to the field observations and the results of a simplified eigenvalue model.

Despite the simplifications involved in the simple model, it has shown to be useful for identifying the period and the spatial structure of the dominant basin-scale oscillations. More detailed models are required to investigate these modes and their effects more thoroughly.

We found that the contribution of the internal waves to the mixing at the interior of the water column was weak, and we suggest that the most important mixing and dissipation region appears to be the turbulent bottom boundary layer driven by seiche currents.

We thank Ana C. Arbeláez, Daniel Largo, Hugo Escorcia, Juan Franco, and Mauricio Toro for their logistic support in the field and to

The present work shows some results of the Master's thesis of Andrés Posada-Bedoya under the supervision of Andrés Gómez-Giraldo and Ricardo Román-Botero. All the authors participated in the analysis and interpretation of the results, as well as in the drafting of the manuscript, the drawing of all the figures, and the preparation of the tables.

The authors declare that there is no conflict of interest of any kind that affects the publication of the results of our research work.

Figure 1S. Schematic view of the structure of various internal wave modes for a rectangular basin of depth H and length L with uniform stratification. The solid lines with arrows indicate the amplitude of the periodic circulation pattern. The dashed line represents the associated amplitude of the isotherms vertical displacement. See the figure 1S in:

Figure 2S. Schematic view of hypothetical mechanisms that could link basin-scale internal waves and water quality in Porce II reservoir. (a1,a2) Difference of the physical and chemical properties of the mass of water extracted by the selective withdrawal during the two states of maximum vertical displacement associated with the V2H1 mode. (b1,b2) Difference in the advance of the river plume front during the evolution of the two states of maximum vertical displacement associated with the V2H1 mode. (c1,c2) Difference on the neutral buoyancy intrusion depth of the river plume for the two states of maximum vertical displacement associated with the V2H1 mode. See the figure 2S in: