Analysis of Timeseries Data

The various wave, instability and turbulence processes described in this chapter are shown schematically in Figure 17. These processes occur beneath the lake surface and so practicing limnologists do not have the luxury of being able to directly observe them in the field. Some insight regarding their spatial structure is gained from idealized laboratory and computational models but for the most part, limnol-ogists must resort to deciphering timeseries data from thermistor chains, which are the most useful tools in their arsenal. Figure 18 shows many of the processes from Figure 17 as they would appear on thermistor data, which has been contoured to show isotherm displacement timeseries with depth.

During the strong wind event, shear instabilities form at the base of the surface layer (Figure 18(d)). Shortly thereafter, a thermocline jet occurs within a compressed region of the metalimnion and causes a rapid expansion of the strata and localized mixing

Along-slope current

Surface

Along-slope current

Surface

Trapping of waves

Along-slope

Along-slope

Figure 15 Schematic showing wave rays radiating from along-slope flow over bottom topography and propagating toward the littoral zone. (a) The waves are trapped between the mixed surface layer and the lake bed and will eventually break where the bed slope a is critical and the wave frequency o = Nsin a. (b) The mixed fluid created during wave breaking collapses and intrudes into the lake interior, carrying sediment and nutrients from the littoral zone. The local strength of the stratification N' is reduced and the along-slope flow is modified. Adapted From Thorpe SA (1998). Some dynamical effects of internal waves and the sloping sides of lakes. Coastal and estuarine studies: Physical processes in lakes and oceans. American Geophysical Union 54: 441-460.

Figure 15 Schematic showing wave rays radiating from along-slope flow over bottom topography and propagating toward the littoral zone. (a) The waves are trapped between the mixed surface layer and the lake bed and will eventually break where the bed slope a is critical and the wave frequency o = Nsin a. (b) The mixed fluid created during wave breaking collapses and intrudes into the lake interior, carrying sediment and nutrients from the littoral zone. The local strength of the stratification N' is reduced and the along-slope flow is modified. Adapted From Thorpe SA (1998). Some dynamical effects of internal waves and the sloping sides of lakes. Coastal and estuarine studies: Physical processes in lakes and oceans. American Geophysical Union 54: 441-460.

(Figure 18(e)). The basin-scale internal wave energized by the wind event has steepened into an internal surge supporting large amplitude NLIWs of depression and series of step like features resembling internal hydraulic jumps (Figure 18(c)). The surge wave interacts with the lake bed leading to significant mixing over the bottom 20 m of the water column.

The motions described above all result from a single intense wind event. Considering the periodic nature of surface winds, it is not surprising that periodic wave motions occur in lakes over a range of length scales and frequencies. Spectral frequency analysis is used to conveniently analyze timeseries data and determine the relative amount of energy found at each frequency.

Observations from many lakes suggest the existence of a universal frequency spectrum model for lakes. The main features of the internal wave spectrum are shown for several lakes in Figure 19. These lakes range in diameter from 20 km (Lake Biwa) to 10 km (Lake Kinneret) to 1km (Lake Pusiano). Unlike the spectral energy cascade occurring in turbulent flows, internal waves are generated at discrete frequencies throughout the spectrum. Motions are bounded at the low frequency end of the spectrum by H1 seiches that contain the most energy with frequencies between zero and 10~4 Hz. At the high-frequency end of the spectrum, motions are bounded by the high-frequency cut-off N. Shear instabilities cause a sub-N spectral peak with frequency —10~2 Hz and five orders of magnitude less energy than the basin-scale seiches. The middle portion of the spectrum contains freely propagating linear and nonlinear internal waves. The NLIWs are generated under moderate forcing conditions with sech2 or solitary wave profiles and frequencies — 10~3Hz. These waves are short lived because they break upon shoaling topography at the depth of the thermocline. The portion of the spectrum between the basin-scale seiches and NLIWs (i.e., — 10~4Hz) appears to consist of freely propagating gravity waves that have linear or sinusoidal profiles; similar to the broadband background internal wave field associated with the Garrett-Munk spectrum in the ocean. These waves are generated by disturbances within the flow field (radiation from flow over rough topography, wave-wave interactions, mixing regions, intrusions, nonlinear surges and internal hydraulic jumps) where gravity acts as

0.84

0.55

0.25

O Forced H1 seiche □ Nonlinear surge A NLIWs

X H1, H2 and/or H3 seiches

0.67

f/fH

Figure 16 Regime diagram showing the dominant internal wave response under periodic forcing conditions in a long rectangular laboratory tank. Error bars denote the variation in forcing during an experiment (mean ± standard deviation). After Boegman L and Ivey GN (2007) Experiments on internal wave resonance in periodically forced lakes. In Proceedings of the 5th International Symposium on Environmental Hydraulics, 4-7 Dec. 2007, Tempe, Arizona.

Figure 17 Pictorial representation of the various regions of a lake and some of the wave-like physical processes that occur. See also Figure 6.13 in Fischer HB, List, EJ, Koh RCY, Imberger J, and Brooks NH (1979) Mixing in Inland and Coastal Waters. San Diego, CA: Academic Press; Figure 15 in Imberger J (1985) Thermal characteristics of standing waters: An illustration of dynamic processes. Hydrobiologia 125: 7-29; and Figure 7 in Imboden DM and WUest A (1995). Mixing mechanisms in lakes. In Lerman A, Imboden DM, and Gat J (eds.) Physics and Chemistry of Lakes. pp. 83-138. Berlin: Springer.

Figure 17 Pictorial representation of the various regions of a lake and some of the wave-like physical processes that occur. See also Figure 6.13 in Fischer HB, List, EJ, Koh RCY, Imberger J, and Brooks NH (1979) Mixing in Inland and Coastal Waters. San Diego, CA: Academic Press; Figure 15 in Imberger J (1985) Thermal characteristics of standing waters: An illustration of dynamic processes. Hydrobiologia 125: 7-29; and Figure 7 in Imboden DM and WUest A (1995). Mixing mechanisms in lakes. In Lerman A, Imboden DM, and Gat J (eds.) Physics and Chemistry of Lakes. pp. 83-138. Berlin: Springer.

restoring force on fluid parcels displaced from their equilibrium position. The waves in this frequency bandwidth are interacting with one another making it difficult to identify their source. The bandwidth limits on the spectrum depend only upon the stratification and basin size and are independent of the strength of the wind forcing. Stronger winds lead to sharper energy peaks with higher energy content (larger amplitude seiches and more shear instabilities).

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