[8] The second problem of Stokes consist in solving the horizontal velocity parallel to a wall, *u*(*z*,*t*), with *z* the vertical coordinate and *t* the time, which is the result of inducing the flow by an oscillatory boundary conditions at the base of *u*(0,*t*) = *u*_{o} cos(*ωt*), where *u*_{o}is the amplitude of the oscillation and *ω* the frequency at which occurs [*Batchelor*, 1967; *Lorke et al.*, 2002]. In this way, shear stresses induces an oscillatory flow in a region characterized by the thickness of the Stokes layer, *d* = 2*π*, above which no horizontal velocities are expected, with *ν* denoting the water viscosity that is considered to be the turbulent eddy viscosity in the benthic boundary layer [*Lorke et al.*, 2002], and is assumed equal to *K*_{ρ} = 2.0 × 10^{−2} m^{2} s^{−1} since the turbulent Prandtl number is close to 1.0 [*Rodi*, 1983; *Pope*, 2000]. Using the velocity profile of the Stokes problem, the work per unit of area of bed, per oscillating period done by the oscillating on the flow is

Then, the net energy per unit of area of bed for a monochromatic oscillation, is calculated as the work per period (*w*, equation (3)) times the number of periods contained on a time *T*_{s}, giving:

In this way, the net work per unit of area of bed done by the earthquake on the reservoir, *W*, can be contrasted with the changes in the potential energy of Rapel reservoir produced by the Maule earthquake (Δ*PE* = 0.8 kJ m^{−2}, Figure 3a), for which it is required knowing *T*_{s}, *ν*, *u*_{o} and *ω*. *T*_{s} is assumed equal to the duration of the earthquake (90 s), and *ν* = *K*_{ρ} as it was previously discussed. The other two variables, *u*_{o} and *ω*, are taken from the accelerometer station located in Melipilla station (see Figure 1a), which is the closest available seismological station of Rapel reservoir (Seismological Service of Universidad de Chile, 2010, http://ssn.dgf.uchile.cl). Land velocities along west–east and south–north axes were computed by integrating in time the accelerations recorded in Melipilla, and long waves were filtered out [*Boore and Bommer*, 2005] by subtracting to the original time series the moving average of the velocity in a time window of 1s. As an example, the resulting time series of ground velocity along the east–west axis is shown in Figure 4a. Furthermore, the energy spectrum of each component of the land velocity was computed, and the specific kinetic energy of the earthquake per unit of frequency, *u*_{o}^{2}(*ω*) used in (4), was computed by adding the energy spectrums of the east–west and south–north components of the horizontal land velocity. Then, the exchanged work per unit of frequency per unit of bed area, *W*(*ω*), was computed based on (4), and the corresponding spectral form is shown in Figure 4b (*ν* = *K*_{ρ} = 2.0 × 10^{−2} m^{2} s^{−1} and *T*_{s} = 90 s). Finally, the total energy per unit of area of bed that entered to the reservoir (Δ*E*) was computed by integrating *W*(*ω*) (Figure 4b) in the range of frequencies of the earthquake (between 10^{−1} and 10^{2} rad s^{−1}, Figure 4b), obtaining Δ*E* = 25.6 kJ m^{−2} that is about 32 times larger than Δ*PE*, which is equivalent to a mixing efficiency of *γ*_{mix} = Δ*PE*(Δ*E*)^{−1} = 3.2%. It worth noticing that if this estimation is done by also considering vertical land velocities Δ*E* = 28.3 kJ m^{−2}, thus showing that in the first approximation the vertical accelerations can be neglected.

[9] The second way of computing the mixing efficiency, *γ*_{mix}, is based on the results of *Shih et al.* [2005] and *Ivey et al.* [2008]. These authors showed that the energetic regime is described by

where *ν*_{m} is the molecular kinematic viscosity (1.3 × 10^{−6} m^{2} s^{−1}), *N* is the buoyancy frequency (rad s^{−1}), and ɛ is the rate of TKE dissipation (m^{2} s^{−3}). Then, using *K*_{ρ} = 2.0 × 10^{−2} m^{2} s^{−1},

and Δ*E* = Δ*PEγ*_{mix}^{−1} = 3000 kJm^{−2} which is equivalent to 24% for a 50 m height water column.