An empirical parametrization of internal seiche amplitude including secondary effects

Abstract

An internal wave is a propagating disturbance within a stable density-stratified fluid. The internal seiche amplitude is often estimated through theories that describe the amplitude growth based on the Bulk Richardson number (Ri). However, most theoretical formulations neglect secondary effects that may influence the evolution of internal seiches. Since these waves have been pointed out as the most important process of vertical mixing, influencing the biogeochemical fluxes in stratified basins, the wrong estimation may have several impacts on the prediction of the system dynamics. This research paid particular attention to the importance of secondary effects that may play a major role on the basin-scale internal wave amplitude, especially related to the interaction between internal waves and lake boundaries, internal wave depth, and mixing processes due to turbulence. Based on a set of methods, which include auto- and cross-correlations, spectral analysis, and mathematical models, we analyzed the effect of total water depth, wind-resonance, and higher vertical modes on the amplitude growth. We based our analysis on underwater temperature measurements and meteorological data obtained from two small thermally-stratified basins, complemented with numerical simulations. We introduce here a new parametrization which takes into account the total water depth (H), lake length (L), epilimnion thickness ($h_e$), as well as the resonance effect. We observed that the rate of amplitude growth decreases compared to linear theory when $Ri~h_e/L\le 1$. In these cases, we suggests that previous theories overestimate the internal seiche amplitude, neglecting the instabilities generated near the wave crest due to weak stability and wave interactions. However, under shallow thermocline conditions, due to extra pressure in the upper layer, the vertical displacement may be higher than that predicted by the linear theory.

Appendices

Numerical simulations

This appendix contains detailed parameters used to simulate the evolution of basin-scale internal waves in Delft3D. In addition, Tables 1 and 2 presents general data and processed results from all 32 simulated cases, including general parameters of boundary conditions, theoretical results for vertical displacement and degeneration regime.

We used the three-dimensional hydrodynamic model Delft3D-FLOW, which solves the shallow water equations using the hydrostatic assumption [15]. Although Delft3D model may fails to reproduce the internal wave breaking and most of degeneration regimes, it has already been used to describe internal seiche in real lakes and reservoir [16].

The implemented model operated in a horizontal Cartesian grid cells of $90~{\text {m}}\times 90~{\text {m}}$ and 50 fixed layers, with no heat flux at water surface. For the turbulence closure scheme, vertical eddy diffusivity and viscosity was calculated by a $k-\epsilon$ model. The coefficients of background vertical and horizontal viscosity and diffusivity were considered as calibration coefficients and were kept fixed during the simulation. Considering the stability condition, specified by the Courant–Friedrichs number, we used a time step of 1 min to simulate approximately 10 days.

Although Delft3D fails to describe most of the degeneration process of BSIW, Fig. 16 highlights the evolution of the internal bore generated after the degeneration of the basin-scale internal wave detected in simulation 23 (Fig. 11b). The degeneration regime evidenced in the simulation matches with theoretical results [25, 26].

Table 1 Summary of collected data from simulated cases $h_e$ and H indicate the epilimnion and total water depth, respectively

Fig. 16

Internal bore generated as a result of the degeneration process of basin-scale internal wave (Simulation: Run 23)

Table 2 Summary of processed results from simulated cases

Harp lake

For time being this section highlights only one period of basin-scale internal wave activity in Harp Lake. Other periods are summarized in the supplementary material. For Harp Lake, the basin-scale internal waves were excited in summer and spring in the North Hemisphere. The highlighted sub-period is comprised between September 16 and October 20, 2013.

We selected the period from 25th of September to 4th of October since it presented low Ri due to the strong wind events, with mean wind speed $> 3.5~{\text {m/s}}$ and peaks that reached almost $6~{\text {m/s}}$. Homogeneous wind events presented duration of approximately $11~{\text {h}}$ blowing $169~^\circ$ North. The thermal stratification was constant during the period, $\Delta T \approx 15~^{\circ }{\text {C}}$. The strong wind events decreased the Ri from $10^6$ to $10^3$, which lead to the formation of basin-scale internal waves. According to the criteria established by [44], the reduction of Ri value leads to a lake regime of internal seiche dominance. Although it is not a full proof of their existence, this theory gives indication of their generation.

Considering the analyzed period, oscillatory motion was observed mainly in the $8~^{\circ }{\text {C}}$ isotherm, located approximately $1~{\text {m}}$ below the mean thermocline depth ($\approx 7~{\text {m}}$ above the water surface). The first event occurred between September 28 and 30, whilst the second and stronger one was detected in the last five days of the analyzed sub-period, from 1st to 5th of October, 2013. The first one presents mean wind intensity of $2.7~{\text {m/s}}$, whereas the second one a mean wind speed of $3.5~{\text {m/s}}$ (Fig. 17a). Both periods present low Ri and, according to [44], were susceptible to internal seiche formations. Although the Ri does not account the wind direction to analyze the excitation of internal waves, the period presents homogeneous wind events, with mean wind direction to $120^\circ$ and $270^\circ$ North, respectively. Stronger wind events provided higher oscillations, with vertical displacement reaching $1.2~{\text {m}}$, whilst the first period presented maximum vertical displacement of $0.6~{\text {m}}$.

Fig. 17

Thermal and wind speed condition in Harp Lake. a Isotherms time-series in height above the bottom (h.a.b.). b Wind speed time series (Harp Lake; 25/09–04/10)

Figure 18a shows results of the multi-layer hydrostatic linear model with free surface and the power spectral density of isotherms time series. The power spectral density of $8~^{\circ }{\text {C}}$ isotherm shows two prominent peaks (Fig. 18a). Peak (I) and (II) are above the mean red noise spectrum, presenting period of $5~{\text {h}}~30$ and $4~{\text {h}}~30$, respectively. Both periods are close and compatible with the first two vertical modes. Both modes showed similar results since the thermal structure of Harp Lake during this period of analysis is characterized by a thin metalimnion (Fig. 18c).

The $4~{\text {h}}~30$ peak (Fig. 18a; peak I), has lower spectral energy compared to the peak (II), that has periodicity of $5~{\text {h}}~30$. It indicates that the $4~{\text {h}}~30$ fluctuation occurred between September 28 and 30 due to wind event of $2.7~{\text {m/s}}$. The phase analysis indicates that the $8~^{\circ }{\text {C}}$ and $12~^{\circ }{\text {C}}$ isotherms propagates out-of-phase (Fig. 18b), suggesting the occurrence of a higher baroclinic internal wave. The out-of-phase structure may also be observed through Fig. 17b from 1st to 5th of October. However, the detection of the out-of-phase structure is harder for the first period, since the internal seiche amplitude is lower. In both periods the V2H1 mode was more pronounced below the thermocline, in the $8~^{\circ }{\text {C}}$ isotherm, and barely detectable at the $10~^{\circ }{\text {C}}$ isotherm.

Fig. 18

a Power spectral density of the isotherms of $12~^{\circ }{\text {C}}$ and $8~^{\circ }{\text {C}}$. Dashed lines show the mean red noise spectrum for the time series, indicating significant peaks at a $95\%$ confidence level. b spectral phase analyses considering just coherence higher than $80\%$ (Harp Lake; 25/09–05/10), c Vertical temperature profile obtained from Harp Lake in 2nd of October 2013 at 5 a.m., d Zoom-in view of the temperature profile with a second profile $2~{\text {h}}$ late, which is approximately half of the internal seiche period. At $22~{\text {m}}$ above the bottom the water becomes warmer, which leads to a erosion of the $12~^{\circ }{\text {C}}$ isotherm due to internal seiche passage. However, $2~{\text {m}}$ bellow that point, the water becomes colder due to the elevation of the $8~^{\circ }{\text {C}}$ isotherm, that was caused by the oscillation of the second interface, clearly out-of-phase with the $12~^{\circ }{\text {C}}$ isotherm. e a sketch of the V2 mode is presented in (d)

The second vertical mode affects the vertical temperature profile as shown in Fig. 18d, generating vertical displacement in opposite directions. This motion occurs repeatedly, as illustrated schematically in Fig. 18e, until the complete internal seiche dissipation.