Diego A. Pantoja, Anatoliy Filonov, Noel Gutiérrez
Departamento de Física
Universidad de Guadalajara, CUCEI
Guadalajara, Mex. 44430
During summer of 2016 an event was register in the volcanic lake of Santa María del Oro, a perturbation that triggers a considerable rise of amplitude on the lake system for at least two days. Due to the scarce observations in the lake, yet it is not definitive what event took place, a storm, a landslide or other. Based on numerical modelling a series of experiments were carry out in order to determine the kind of forcing that created those anomalies. This study is focused in characterizing a submarine landslide and the effects of the wind valley breeze. The results indicate that the lake acts as a dynamical membrane resonating to frequencies at 3.2 min (external seiche) and in a band of the 2-8 hrs (internal seiches) with the 2.64 hrs the most intense. After every forcing the water starts to oscillate uniformly and in a clockwise sense until the energy is dissipated by friction at the internal seiches frequencies.
KeyWords. Internal and external Seiches, Crater Lake, Delft3D model.
|Figure 1: Bathymetry of Santa María del Oro Lake [m] (upper), temperature profile for summer (right) and typical wind valley breeze (bottom).|
According to , a possible landslide took place near the bay at the western section of the lake during august of 2006. The observed perturbations co-oscillated with the background system with a periods between 2 and 6 hrs.
Delft3D is an open-source numerical model developed by WL/Delft Hydraulics and the Delft University of Technology . It includes implementations of several mathematical models for different physical phenomena (currents, transport, wave propagation, morphological developments, etc.). In the present study, it is used to predict the circulation in a lake system. The Delft3D model solves the Navier-Stokes equations for an incompressible fluid, under the shallow water and the Boussinesq assumptions using a finite difference scheme (see ). The model includes the depth-averaged horizontal momentum equations, (shown here in Cartesian coordinates for the sake of clarity):
the depth-averaged continuity equation:
and the vertical momentum equation, which reduces to the hydrostatic pressure relationship via the Boussinesq approximation:
where are the depth-averaged velocity in the and directions, is the Chézy coefficient, is the water depth, is the free surface elevation above the reference plane (at ), is a two dimensional current vector, whose Euclidean norm is , are sinks or sources of water, is Coriolis force, is Reynolds stress, is the gravity, is the horizontal eddy viscosity, is the pressure and the water density. In the horizontal direction, the model uses orthogonal curvilinear grids, which support both cartesian and spherical coordinates. Both coordinate systems permit the management of geometrically complex domains. In the vertical direction, Delft3D uses the or the z coordinate systems. The z coordinate system is made up of layers with predefined depths. In this particular study, the spherical and z-coordinate system are used without boundary conditions. All the simulations were ran for only six days within a mesh of 156 x 126 number of cell in the (x,y)-directions, respectively, m, with 20 z-layer in the vertical distributed according to [10 6 4 3 2] % of the water column, where the first value is assigned to the bottom layer and the last value is assigned to the upper layer (surface). In every cell the number of layers is variable depending on the water depth. The time step used was seg.
|Figure 2: Snapshots of water level six hours later of the simulated landslide. Without considering the first frame, the period expand nearly three hours of simulation. Color represent water level in m.|
|Figure 3: Time series of water level at four observation points. In the upper panel is shown the early development of the water level at the east-west points (left) and at the north-south (right) points, note the x-axis is in minutes. At the lower panel are the corresponding sections through the time evolution during the next 19-hours after three hours the initial perturbation. In this case, the time series is in hours. The thick-line represent the eastern and southern points. Note the change in scales in every plot.|
|Figure 4: Horizontal velocity components at the surface layer (top) and at the sixth layer in the bottom of the lake (bottom). In every row, is shown the time series of the components (left), a representative diagram of the vector velocity (center) and the initial snapshot of the horizontal velocity field (right). The thick line in time series represent the north-south component, the vertical lines the initial time marked as in the first panel whereas the thick vector represents the initial time.|
|Figure 5: Power spectra of the water level at the four points in the lake. The upper lines represent the east-west points, whereas the lower lines the north-south points. The vertical line correspond to the 95% confidence interval. The triangles mark the 2.64 hrs and 3.2 min frequencies.|
|Figure 6: Same as Fig. 5, for Ws (Black lines) and LWs (Blue lines).|
Lake Santa María del Oro can be considered as a circular basin which according to the 2D structure modal configuration, obeys the following standing oscillation from the seiches motion ():
For a basin with radius r, where , and the polar angle. is the Bessel function of order s, and are arbitrary constants and , with the roots of given the normal modes of the system. The period of oscillation of the first external model correspond to , where g is the gravity and H the mean depth. If the lake was considered homogeneous in temperature, the first mode of oscillation (external mode) will remain between 2-3 min, whereas for an stratified lake, the first internal mode remains between 2.1-4.5 hrs ([2,8]). As shown by the combined configuration (Fig. 6), the resonant internal modes can coexist in the same simulation, and in the same band of frequency (2-8 hrs), so there is not a very clear contribution of which forcing caused the sudden perturbation on the observational data. Both simulations, wind and submarine landslide, can cause resonant amplification on the lake. Then, with the information at hand from the numerical outputs we can not confirm that effectively was a landslide that perturb the lake system, there is more work to do in order to support the idea of a topographical perturbation.
This work was supported by the Mexican Secretaría de Educación Publica (SEP) through the Program for Teachers PRODEP of DAP. The AF acknowledge the support of CONACyT.
 A. B. Rabinovich, "Seiches and harbor oscillations", Handbook of coastal and ocean engineering, (2009).
 D. Serrano, A. Filonov and I. Tereshchenko, "Dynamic response to valley breeze circulation in Santa María del Oro, a volcanic lake in México", Geophysical Research Letters, Vol. 29(131649), pp. 27-1-27-4, (2002).
 N. Gutierrez, "In-situ measuments of the coupling between internal waves submarine landslides in a volcanic lake", Tesis de Licenciatura en Física (In spanish), Universidad de Guadalajara, CUCEI, Guadalajara, Mexico, (2017).
 G.R. Lesser, "Development and validation of a three-dimensional morphological model", Coast. Eng. Vol. 51, pp. 883–915, (2004).  Deltares, "Delft3D-FLOW, User Manual", The Netherlands, (2013).
 A. Filonov, I. Tereshchenko, "Thermal lenses and internal solitones in the shallow lake Chapala, Mexico", Chinese Journal of Oceanology and Limnology, Vol. 17, pp. 308-314, (1999).
 A. Filonov, I. Tereshchenko, J. Alcocer, "Dynamic response to mountain breeze circulation in Alchichica, a crater lake in Mexico", Geophysical Research Letters, Vol. 33, L07404, (2006).
 D. Serrano, "Procesos termodinamicos en el lago volcanico de Santa Maria del Oro, Nayarit", Tesis de Doctorado, UNAM, (2004).