Spatial variation of ISW properties - Properties of Internal Solitary Waves in Deep Temperate L

4.4 Conclusions

5.3.4 Spatial variation of ISW properties

A comparison of the amplitude and period of ISWs observed during field campaign two (Fig. 5.5 and Fig. 5.7) and field campaign three (Fig. 5.8) indicates that ISWs propagating in regions with similar water depths and stratification, which are separated by a distance smaller than one half of the Rossby radius, have very similar properties. The statisti-cal comparison of the wave properties of the leading ISWs passing during campaign two (Fig. 5.9) indicates that particularly the amplitudes are similar at station S1 and S2.

This similarity also holds for the observation of temperature inversions associated with the occurrence of ISWs indicating wave breaking [Alford and Pinkel, 1995; Preusse et al., 2010, 2012a]. The breaking of ISWs at the deepest location of Lake ¨Uberlingen (station S1) is accompanied by ISW breaking in the deep near-shore region at station S2 (Fig. 5.7).

In-going Out-going

Figure 5.10: Bending of the ISW front. The phase velocities of the wave fronts were determined from the DJL equation using the amplitudes observed at station S1. A) Statistical comparison of the bending Φ of in- and out-going ISW fronts passing at the two stations S1 and S2 deployed at similar lake depths. The angle Φ is defined in the sketch. Compared to the orthogonal direction to the thalweg, in-going waves are bended backwards towards the southern shore, out-going waves are bended forward. B) Median bending of the in-going (solid line) and out-going (broken line) ISW fronts passing S1 and S2. C) Bending of the ISW front observed at 22 October 2010 passing stations S3, S4 and S5.

Even the fine-scale structure of the breaking process looks very similar at the two different measuring stations (Fig. 5.7). These observations of closely associated temperature inver-sions at stations S1 and S2 suggest that ISW-breaking in the deep water far away from the bottom boundary extends over a large part of the ISW front. Also a simulation of the current velocity and phase velocity of this ISW with the DJL equation indicates that the observed wave amplitude is above the critical limit for wave breaking [for details on the critical limit of wave breaking see Lamb, 2002; Stastna and Lamb, 2002]. According to the simulation, this ISW is breaking due to convective instability, because the maximal waveinduced horizontal current velocity exceeds the phase velocity.

The ISW train observed at the shallow-water boundary station S7 (Fig. 5.8) looks very different from the ISW train observed at the deep-water stations S3 or S4. Individual ISWs passing the shallow-water boundary station S5 are hardly distinguishable, but nevertheless seem to be still existent (Fig. 5.7, S5, arrows). The amplitudes of the ISWs at the near shore station are strongly diminished compared to the amplitudes of the ISWs in the open water far away from the bottom boundary. A possible explanation for the differences between the amplitudes of ISWs along the same ISW front may be a change of lake width in the direction of wave propagation, which might contribute to a redistribution of energy within the ISW front. However, in earlier studies it was demonstrated that the passage of ISWs

at near-shore stations is associated with an increase in turbulence due to ISW breaking [e.g. in Lake Constance by Lorke, 2007; Preusse et al., 2010, for station D and for station H, respectively], suggesting that the boundaries parallel to the propagation direction of the ISWs are locations of enhanced dissipation of ISW energy.

5.4 Discussion

5.4.1 Conceptual picture of basin-scale propagation and struc-ture of ISWs

In summary our observations indicate that

1. The ISW front is extended over the total lake width for a Rossby radius r smaller than the mean lake width L.

2. The propagation direction of the ISW front is roughly parallel to the thalweg.

3. The structure of an ISW train as well as amplitude and period of the leading ISWs of a wave train do not vary strongly within the ISW front at locations with similar deep water depths and separated by a distance smaller than one half of the Rossby radius.

4. ISWs break at the boundaries of the lake, particularly when they reach a terminal boundary orthogonal to the propagation direction where the entire ISW front shoals.

However, as the ISW propagate in along-shore direction, the ISW fronts extend into the boundary region where the interaction with the topography causes breaking and thus energy dissipation of the ISWs along the entire path of the ISW towards the end of the lake.

5. ISW also break in the open water far away from the boundaries. The ISW breaking in the open-water does not occur sporadically in isolated, local events but extends over the entire width of the ISW front.

These observations support the conceptual picture of the propagation and extension of ISWs at a basin scale in Lake Constance summarized in Fig. 5.1C. The front of the ISW is depicted as single line and the arrows indicate the direction of propagation (Fig. 5.1C).

Since the basin-scale internal wave in Lake Constance is a Kelvin-wave, which propagates anti-clockwise around the lake, the ISWs generated at the steepened front of the Kelvin-wave enter Lake ¨Uberlingen from the northern part of the main basin. The ISWs propagate

roughly parallel to the northern and southern boundaries, until they shoal upon the sloping topography of the western boundary. Most of the ISWs break at this terminal cross-shore boundary. However, ISWs are also generated at the reflected internal front of the basin scale wave and propagate from the western end of Lake ¨Uberlingen eastwards. All along their path through the lake, ISWs interact with topography at their outer edges in the shallow waters of near shore regions, probably resulting in localized wave breaking within the boundary region. Furthermore, ISW break ¨Uberlingen, basin-scale waves are not strongly affected by the Earths rotation and therefore typically travel in along-shore direction with little variation in the amplitude of the wave front in cross shore direction. Because ISWs are generated via an energy cascade from the basin-scale low-frequency internal wave to high frequency internal waves, the ISW front also extends over the total lake width and has a similar amplitude along the cross-shore direction. Since wave breaking in the interior of deep lakes mainly depends on the stratification and the size of the amplitude [Lamb, 2002; Preusse et al., 2012b], temperature inversions and wave breaking observed in the interior should occur all along the width of the ISW front. This assumption of ISW front-wide occurrence of ISW breaking in deep waters is supported by our observations of the simultaneous occurrence of wave breaking at the central station S1 and the deep-water boundary station S2.

In the much wider Upper Lake Constance, the ISW front diminishes towards the centre and ISW-breaking is limited to a region closer to shore.

5.4.2 Implications

To our knowledge, lake-wide measurements of ISWs, focusing on the variation of ISW properties in along-shore and cross-shore propagation direction, have not been presented in such detail before. An extensive measuring campaign has been carried out in Lake Kinneret to explain the occurrence of high-frequency waves with substantially smaller amplitudes as observed in this study [Antenucci and Imberger, 2001]. These waves were found to be generated directly by wind-forcing without an energy cascade over the basin-scale seiche. The generation of solitary waves via steepening of a basin-basin-scale seiche as described here is usually coupled to topographical features where the influence of Earth’s rotation diminishes. Thus large-amplitude solitary waves as observed in Lake ¨Uberlingen typically occur in long and narrow lakes such as Loch Ness [Thorpe et al., 1972] and Lake Babine [Farmer, 1978]. However, solitary waves were also reported from long and narrow sub-basins of lakes with main basins affected by Earth’s rotation [Hutter, 2012].

Our observations, supporting the conceptual picture of the propagation and degener-ation of ISWs shown in Fig. 5.1C, provide a basis for a generalizdegener-ation and extrapoldegener-ation of results from studies in which observations of ISW-induced processes were made at only one location in a lake. E.g., the ecological relevance of processes such as ISW-induced vertical advection of plankton [Huber et al., 2011] or of enhanced boundary mixing due to ISWs [Lorke, 2007] that were observed at single locations can be assessed based on the experimentally validated concept of lake wide propagation and degeneration of ISWs.

It is known that the breaking of ISWs leads to increased turbulence [Boegman et al., 2005a; Bourgault et al., 2007; Preusse et al., 2010], and, as suggested by our measurements and former studies [e.g. Lorke, 2007], the breaking takes place at all the boundaries of the lake at which the ISW front passes (Fig. 5.1C, spirals), at the terminal boundary orthogonal to the propagation direction [as in Boegman et al., 2005a; Bourgault et al., 2007], as well as parallel to the propagation direction as in Preusse et al. [2010]. Note, however, that the energy fluxes induced by the breaking of the ISWs at the boundaries differ according to the propagation direction. The shoaling of ISWs at terminal boundaries orthogonal to the propagation direction concerns the entire wave front. The energy dissipated by ISWs breaking at the boundaries parallel to the propagation direction might be replenished by an energy flux horizontally from the wave front to the boundaries.

Lorke [2007], measuring temperature und current fluctuations at a littoral station, observed an increase in turbulence by two orders of magnitude with the occurrence of periodic high frequency temperature fluctuations at a measuring site close to the study site C. The fluctuations in currents and temperatures were interpreted to be associated with breaking high frequency internal waves at this measuring station. Lorke [2007] speculated that similar processes occur along the entire littoral zone, resulting in a diffusivity one order of magnitude larger than a tracer-based estimate of the basin-scale diffusivity in Lake Uberlingen [Maiss et al., 1994]. Since ISWs were also observed in Upper Lake Constance,¨ bottom boundary turbulence can be expected not only at the boundaries all around Lake Uberlingen, but also at the northern shore of Lake Constance (Fig. 5.1C, spirals). Note,¨ however, that the energy flux depends on the energy and number of the ISWs and the latter appears to be significantly larger in Lake ¨Uberlingen than in the rest of the lake due to the ongoing generation process of ISWs in Lake ¨Uberlingen. Hence, one would expect a larger contribution of boundary mixing by breaking ISWs in Lake ¨Uberlingen than in Upper Lake Constance.

In an earlier study analysing more than 200 ISW trains at the central station in Lake Uberlingen [Preusse et al., 2012a], in which more than 15% of the leading ISW were¨

accompanied by temperature inversions. Such inversions are an indication of wave breaking [Alford and Pinkel, 1995; Preusse et al., 2010] and mixing [Thorpe, 1977], whereby mixing is particularly efficient if the breaking occurs due to self-induced shear. ISW breaking in the open water results into mixing distant from the boundaries and leads to a localized flux of nutrients, e.g. phosphate, from the hypolimnion to the epilimnion. In a nutrient depleted epilimnion, e.g. especially after a seasonal plankton bloom, such patches of phosphate might enhance phytoplankton growth and induce plankton patchiness.

Besides mixing, transport due to internal waves is an important ecological feature in stratified lakes. It is known that internal waves vertically displace neutrally buoyant particles over several meters within a few minutes [Huber et al., 2011]. Such vertical motions due to ISWs cause light- and pressure- stress on phytoplankton only on short time scales. However, the effects of the ISWs are amplified, since they typically occur not isolated but in trains consisting of several waves. ISWs have the potential to transport particles horizontally over large distances when the waves form a trapped core [Pineda, 1999]. Trapped cores are formed if the maximal wave induced current velocity exceeds the phase velocity of the ISW, as was determined numerically for the wave showin in Fig. 5.7.

Moreover, dynamical simulation approaches suggest that ISWs generating a trapped core look very similar to the waves shown in Fig. 5.7 [Lamb, 2002] and can occur in stratifications without an upper mixed layer, as is typical for stratifications in deep temperate lakes during the warming period from spring to summer [Preusse et al., 2012b].

Acknowledgments

The field experiments were supported by the technicians Joseph Halder, Beatrix Rosenberg and Alfred Sulger. The divers working with Martin M¨ortl deployed the thermistors in the near shore moorings. This work was financially supported by University of Constance and the German research association (DFG, PE 701 / 4 - 1).

Ecological relevance

ISWs, particularly their breaking, are widely assumed to play an important ecological role in the shore region of the ocean. As will be discussed in the following paragraphs, the present study shows that ISWs in Lake ¨Uberlingen have the physical properties to be of similar importance as in the ocean. The findings suggest that the ecological function of ISWs in Lake ¨Uberlingen, derived from their physical properties, is transferable to other lakes where ISWs have been observed, e.g. Loch Ness, Lake of Zurich, Lake Geneva, Babine Lake or Seneca Lake [Horn et al., 2001, and references therein]. One of the major results presented here, namely the frequent observation of wave breaking in the lake’s interior, indicates that the impact of ISWs on the ecosystem is not restricted to the shore region or the littoral zone. Depending on the breaking mechanism, wave breaking in the open water mainly results in one of two different ecological processes, either mixing or horizontal transport of buoyant particles. Since the breaking mechanisms of the ISWs in Lake ¨Uberlingen change with season, the ecological role of ISWs in the open water of deep temperate lakes can be assumed to change with season and to differ from their role in the ocean.

Frequency of occurrence

ISWs in Lake ¨Uberlingen occur frequently enough to be of similar ecological importance as in the ocean. Two ISW trains are generated on average every second cycle of the basin-scale internal seiche in Lake ¨Uberlingen, as was derived in chapter 3. The first (in-going) train propagates from the main basin into Lake ¨Uberlingen towards the western end of the lake; the second (out-going) train propagates in the opposite direction from the sub-basin into the main basin. Only the in-going train is comparable with the typical tide-generated ISW train in the ocean, since both trains propagate towards sloping topography. However, since also the out-going wave train is ecologically relevant (chapter 5), both propagation directions have to be considered, implying a mean frequency of occurrence of ISW trains every four days from April to October. Similar to ISWs in lakes, which do not occur every cycle of the basin-scale seiche, tide-generated ISWs do not necessarily occur every tidal cycle [e.g. Scotti and Pineda, 2004]. An upper limit for the typical frequency of ISWs in the ocean can thus be derived by assuming that a train of ISWs is

nevertheless generated in the ocean twice a day (semidiurnal tide). Hence ISW trains in the ocean occur at most eight times as often as in Lake ¨Uberlingen. According to chapter 3, a typical wave train in Lake ¨Uberlingen contains on average four large ISWs. The number of ISWs per train in Lake Constance is similar to oceanic observations, e.g. Scotti and Pineda [2004] observed two ISWs at Massachusetts Bay, Klymak and Moum [2003] reported three ISWs at the Oregon shelf and Silva et al. [2009] presented observations of five ISWs per train in the Mozambique Channel.

The frequency of occurrence of ISWs in lakes, if generated by the steepening of a basin-scale seiche, depends on the amplitude of the seiche, topography and stratification [Horn et al., 2001].

Whether ISWs occur, or not, can roughly be estimated for lakes (or sub-basins) not effected by the Earth’s rotation via quantifying topography with the mean lake depth and the stratification with a two-layer approximation. According to Horn et al. [2001], ISWs occur if the mean lake depth is at least 2.25 times as large as the upper layer depthh₁and if the amplitude of the basin-scale seiche aranges between 0.2 times the upper layer depth and the total upper layer depth.

Whereas the first condition is fulfilled in most deep temperate lakes, the second condition, i.e. the large amplitude of the seiche, depends on the wind-forcing. The relation between wind-forcing and seiche amplitude can be estimated with the Wedderburn number W, defined by

W⁻¹ := lu²_∗

where l is the length of the lake, u∗ the surface wind shear velocity depending linearly on the wind-speed and g⁰ is the reduced gravity [Horn et al., 2001; Boegman, 2009]. Note that this equation only holds if the wind stress is applied for sufficient time, i.e. about one quarter of the basin-scale seiche period [Boegman, 2009]. Seiche-generated ISWs are e.g. reported from Loch Ness, Lake of Zurich, Windermere, Babine Lake, Seneca Lake, Kootenay Lake, Balderggersee and Lake Biwa [see Horn et al., 2001, and references therein]. In Lake Constance, ISWs are generated after moderate wind-forcing with wind speeds exceeding 3 m s⁻¹ - 4 m s⁻¹ for about one day.

By choosingLas the length of Lake Constance and U∗ as the critical surface wind shear velocity in Lake Constance, the Wedderburn number suggests that lakes with length land stratifications similar to the stratification in Lake Constance need wind speeds exceeding the critical speed in Lake Constance by the factorp

L/l, since

where u∗ is the critical surface wind shear velocity corresponding to a lake with length l. Small lakes thus need larger speeds for the generation of ISWs than large lakes, however the wind-forcing does not have to last as long. The necessary wind-wind-forcing periodtof a lake with lengthl in comparison to the forcing period T in Lake Constance can be estimated via

4t = 2l

wherec₀ is the phase velocity of the basin-scale internal seiche. Depending on the typical wind-field, ISWs may be generated with similar or even higher frequency of occurrence as in Lake Constance in various other deep temperate lakes. For example, observations of ISWs were re-ported from Lake Babine [Farmer, 1978], which is a narrow lake, located in British Columbia, Canada and is comparable in depth with Lake Constance. Its length exceeds 100 km. An appli-cation of the upper scaling shows that the generation of ISWs in Lake Babine can be expected for relatively small wind-speeds, i.e. for wind-speeds exceeding 2.5 m s⁻¹ - 3 m s⁻¹, which should last about one and a half day. The critical wind-speeds and wind-periods for the generation of ISWs in Lake Geneva, located in Switzerland with a length of 72 km and Seneca Lake, located in New York, USA, with a length of 61 km should be comparable to the critical wind-speeds in Lake Constance. The generation of ISWs in Loch Ness, located in Scotland and Lake Zurich, located in Switzerland, both with a length of about 40 km, requires wind-speeds of 3.5 m s⁻¹ -5 m s⁻¹, however, the wind needs to last only 15 hours. Note that besides the wind-speed itself the wind-direction and timing as well as the basin-scale topography determines whether a surge steep enough to form ISWs is generated. For example, in Lake Biwa, which is broad enough to be influenced by the Earth’s rotation and has a length of approximately 64 km comparable to Lake Constance, ISWs were up to now only observed after the occurrence of a typhoon [Saggio and Imberger, 2001; Boegman et al., 2003]. ISWs seem to appear particularly often in long, narrow lakes, e.g. Loch Ness [Thorpe et al., 1972], Lake Z¨urich and Lake Babine [Farmer, 1978], where

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