in Deep Temperate Lakes
Dissertation
zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.)
- vorgelegt von Martina Preuße - an der
Mathematisch-Naturwissenschaftliche Sektion Fachbereich Biologie
Tag der m¨undlichen Pr¨ufung: 23.07.2012 Referent: Prof. Dr. Frank Peeters Referent: Prof. Dr. Heinrich Freist¨uhler
Referent: Prof. Dr. Andreas Lorke
Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-202319
I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat sud- denly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of Au- gust 1834, was my first chance interview with that singular and beautiful phenomenon...
John Scott Russell [1845]
Zusammenfassung . . . iii
Summary . . . v
Acknowledgments . . . vii
Introduction 1 1 Generation of Turbulence by Internal Waves 9 1.1 Introduction . . . 9
1.2 Methods . . . 11
1.2.1 Measurements . . . 11
1.2.2 Data analysis . . . 13
1.3 Results . . . 16
1.3.1 Degeneration of the Kelvin wave . . . 16
1.3.2 Processes associated with the occurrence of density inversions . . . 18
1.3.3 Generation mechanisms of density inversions . . . 22
1.3.4 Turbulence and mixing . . . 23
1.4 Discussion . . . 25
2 The Effect of Amplitude 29 2.1 Introduction . . . 29
2.2 Methods . . . 30
2.3 Results and Discussion . . . 32
3 Seasonally Varying Wave Properties 35 3.1 Introduction . . . 35
3.2 Methods . . . 38
3.2.1 Experiment . . . 39
3.2.2 Analysis of internal solitary waves . . . 39
3.2.3 Models . . . 44 i
3.3 Results . . . 46
3.3.1 Individual waves . . . 46
3.3.2 Seasonal prototypes . . . 52
3.4 Discussion . . . 56
4 Intrinsic Breaking of Internal Solitary Waves 61 4.1 Introduction . . . 61
4.2 Methods . . . 63
4.3 Results and Discussion . . . 64
4.3.1 Comparison to theory . . . 64
4.3.2 Limiting amplitudes . . . 66
4.3.3 Breaking mechanisms . . . 69
4.4 Conclusions . . . 73
5 Solitary Waves on a Basin-Scale 75 5.1 Introduction . . . 75
5.2 Methods . . . 78
5.2.1 Data analysis . . . 78
5.2.2 Experiments . . . 80
5.2.3 Simulation . . . 81
5.3 Results . . . 82
5.3.1 Origin and propagation of ISWs . . . 82
5.3.2 Fate of ISWs . . . 85
5.3.3 Width of an ISW front . . . 87
5.3.4 Spatial variation of ISW properties . . . 90
5.4 Discussion . . . 92
5.4.1 Conceptual picture of basin-scale propagation of ISWs . . . 92
5.4.2 Implications . . . 93
Discussion 97
References 104
Interne Wellen sind in geschichteten Gew¨assern allgegenw¨artig und erf¨ullen im aquatischen Okosystem eine wichtige Funktion. Im Rahmen einer Energiekaskade ¨¨ uberf¨uhren interne Wellen Energie von großen Skalen in den turbulenten Bereich, in dem Mischung stattfindet.
Interne Solit¨arwellen (ISWn) sind hochfrequente interne Wellen mit hoher Energie, die in der Energiekaskade eine besondere Rolle spielen. Diese Wellen transportieren ihre Energie zu der K¨uste oder in das Littoral, wo ihr Brechen lokal zu Mischung und Resuspension f¨uhrt.
Prim¨ares Ziel dieser Dissertation ist es, die Eigenschaften von ISWn in tiefen Seen der gem¨aßigten Zone zu charakterisieren. Dabei liegt der Fokus auf Eigenschaften, die f¨ur das limnische ¨Okosystem relevant sind, wie die H¨aufigkeit der ISWn und deren Amplitude, Stabilit¨at, Laufmuster und Saisonalit¨at.
Die Experimente zur Bestimmung der Welleneigenschaften wurden im Bodensee durch- gef¨uhrt. Dabei steht der Bodensee beispielhaft f¨ur tiefe Seen der gem¨aßigten Zone, in denen ISWn auftreten, wie Loch Ness, Z¨urichsee oder Babine Lake. Die meisten Welleneigen- schaften wurden aus einem sechsj¨ahriger Temperaturdatensatz extrahiert, der an der tief- sten Stelle eines Nebenbeckens des Bodensees gemessen wurde, dem ¨Uberlingersee. Das beckenweite Laufmuster der ISWn wurde jedoch aus r¨aumlich hochaufgel¨osten Str¨omungs- und Temperaturdaten hergeleitet, in deren Erhebung ¨uber zehn verschiedene Messstellen involviert waren. Um die ¨Ubertragbarkeit der Welleneigenschaften, insbesondere Stabilit¨at und Saisonalit¨at der ISWn, vom Bodensee auf tiefe Seen der gem¨aßigten Zone zu unter- mauern, wurden die gemessenen mit numerisch simulierten Eigenschaften verglichen. Als Grundlage der Simulationen wurden die Korteweg - de Vries Gleichung und die Dubreil- Jacotin-Long (DJL) Gleichung verwendet.
Das Laufmuster der ISWn, das sich aus den Beobachtungen ergeben hat, best¨atigt das typische Bild der Fortpflanzung von ISWn als in etwa parallel zur Hauptachse des Sees dem terminalen Ufer entgegenlaufend, an dem die Wellen brechen. Dar¨uberhinaus weisen die Beobachtungen darauf hin, dass nicht nur am terminalen Ufer, sondern auch an den Ufern parallel zur Laufrichtung Energie dissipiert wird, da ISWn in schmalen Seen
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wie dem ¨Uberlingersee ¨uber die gesamte Breite ausgedehnt sind. Dieses Ph¨anomen der Energiedissipation unterscheidet sich stark von der typischen Situation im Ozean, in der sich die Wellen frei von Ufern ausbreiten, bis sie auf die K¨uste treffen.
Das Hauptergebnis dieser Dissertation besteht in dem erstmals erbrachten Nachweis, dass ISWn auch im Freiwasser regelm¨aßig brechen und dieses Brechen mit der DJL Glei- chung vorhergesagt werden kann. Außerdem liegen in Seen mitunter ¨okologisch relevante Brechungsmechanismen vor, die sich von denen im Ozean unterscheiden. In tiefen Seen der gem¨aßigten Zone k¨onnen sich vor allem w¨ahrend der Erw¨armungsphase in brechenden ISWn oberfl¨achennahe zirkulierende Kerne ausbilden, in denen oberfl¨achennahe Partikel theoretisch ¨uber weite Strecken transportiert werden k¨onnen. Die Messungen weisen je- doch darauf hin, dass die Kerne der Wellen im Feld ¨außerst instabil sind, was die Trans- portf¨ahigkeiten m¨oglicherweise einschr¨ankt. Andererseits bricht ein betr¨achtlicher Anteil, mindestens die H¨alfte, der brechenden ISWn aufgrund von Scherinstabilit¨aten. Eine grobe Quantifizierung der Turbulenz, die von ISWn im Freiwasser generiert wird, legt nahe, dass das Brechen aufgrund von Scherinstabili¨at zur Mischung in der pelagischen Thermokline f¨uhrt.
Insgesamt ergibt sich, dass ISWn nicht nur im Ozean, sondern auch in Seen h¨aufig auftreten k¨onnen. Dar¨uberhinaus besitzen die ISWn Eigenschaften, die f¨ur die Entstehung
¨
okologisch wichtiger, welleninduzierter Prozesse notwendig sind. Solche Prozesse sind beispielsweise Randmischung und Resuspension, Mischung im Freiwasser sowie horizontaler und vertikaler Transport.
Internal waves are ubiquitous in stratified waters. These waves impact on the aquatic ecosystem by transferring energy from large to turbulent scales where mixing takes place.
A special role within this energy cascade is attributed to internal solitary waves (ISWs), i.e. high-frequency internal waves with large energy. These waves are assumed to transport their energy to the coast or the littoral zone, where their breaking locally results in mixing and sediment resuspension.
The central issue of this dissertation is to characterize the properties of ISWs in deep temperate lakes. The focus is on the properties frequency of occurrence, amplitudes, stability, propagation pattern and seasonality, all of which directly related to the relevance of ISWs for the limnological ecosystem.
The experiments carried out to asses the properties of ISWs in lakes were conducted in Lake Constance as a representative of lakes where ISWs occur, e.g. Loch Ness, Lake Zurich or Babine Lake. Most of the wave properties are extracted from a long-term temperature time series measured over six years in the interior of a subbasin of Lake Constance, Lake Uberlingen. However, the basin-scale propagation pattern of ISWs is derived from spatially¨ highly resolved current velocity and temperature measurements involving more than ten different measuring stations. In order to generalize the measuring results and to justify the transfer of wave properties measured in Lake Constance, particularly stability and seasonality, to deep temperate lakes, the measured ISWs were compared with numerical simulations. These simulations are based on the Korteweg - de Vries equation and the Dubreil-Jacotin-Long (DJL) equation.
The propagation pattern of ISWs derived in this study supports the typical under- standing of wave propagation as roughly parallel to the lake’s main axis towards a terminal boundary, where breaking takes place. However, the observations suggest that since ISWs in narrow lakes like Lake ¨Uberlingen are extended over the total width of the lake, energy dissipation has to be expected not only at this terminal boundary of the lake, but also at the boundaries parallel to the propagation direction of the ISWs. This concept of energy
v
dissipation differs strongly from the typical situation in the ocean, where waves propagate unhindered until they shoal upon the shore.
As main result this dissertation demonstrates for the first time that ISWs break reg- ularly not only at the shores but also in the open water and that this breaking can be predicted by the DJL equation. Moreover, ecologically relevant breaking mechanisms exist in lakes which differ from the mechanisms in the ocean. In deep temperate lakes near- surface trapped cores can evolve within breaking waves, particularly during the warming period. Such trapped core waves are theoretically capable of transporting near-surface buoyant particles over large distances in their cores. However, the measurements suggest that the wave cores are highly unstable in the field, which might affect their transport abilities. As indicated by the simulations, a considerable part (at least one half) of the breaking ISWs break due to shear instabilities. A rough quantification of the turbulence generated by ISWs in the open water suggests that shear unstable ISWs induce mixing in the pelagic thermocline.
In summary, the findings of this dissertation suggest that ISWs can occur frequently not only in the ocean, but also in lakes. Moreover, ISWs have the properties required to induce ecologically relevant processes. Among such processes are boundary mixing and resuspension, mixing in the pelagic thermocline as well as horizontal and vertical transport.
I am grateful to my main advisor, Frank Peeters, for the support and guidance he showed me throughout my dissertation writing and particularly for the numerous enlightening discussions. Sincere thanks also go to my supervisor Andreas Lorke, who introduced me into the limnological praxis and theory and to my supervisor Heinrich Freist¨uhler, who left the window to the mathematical point of view open for me.
Besides, I would like to thank my husband, my mother, my family and my friends, who boosted me morally and my colleague Hilmar Hofmann, who was a great information resource. I am truly indebted to the technicians and divers of the limnological institute for their support in the field. Special thanks go to the technicians Beatrix Rosenberg and Joseph Halder for their patience and care during all my experiments. A large part of this work would not have been realized without the radio link to the data buoy, so I also want to thank O. Kotheimer and the School Salem Castle who made this link possible.
It is a great pleasure to thank the limnological community and the members of the limnological institute of the University of Konstanz in particular for welcoming me and my research. I am especially grateful to Marek Stastna, who generously offered me his numerical code for solving the DJL equation.
This study was financially supported by the University of Constance and the German Research Foundation (DFG, SFB 454 / D9 and DFG, PE 701 / 4 - 1).
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Motivation
Solitary waves (SWs) were first named by John Scott Russell, who discovered seemingly unusual waves which propagated as solitary elevations of permanent form on the water surface [Russell, 1845]. Today SWs are known to exist in many fundamental natural phenomena. SWs occur e.g. between different density layers in the atmosphere [Christie, 1992] or as pulse-like signals in neurons [Horn and Opher, 1997], where the stability of the SW-shape allows a transfer of information over large distances. For the same reason, i.e. to enable a long-distance transmission, solitary-like light impulses are used in fiber optics [Maimistov, 2010].
SWs occur also as internal SWs (ISWs) beneath the surface layer in the water column, where the waves propagate between different density layers. Numerous observations of such waves stem from oceanography [Fu and Holt, 1982; Ostrovsky and Stepanyants, 1989; Helf- rich and Melville, 2006; Jackson, 2007]. One source of the generation of ISWs in the ocean are the frequently occurring tides [Ostrovsky and Stepanyants, 1989]. If the amplitude of such a tidal wave is large enough, the wave gradually steepens until it finally decomposes into a train of ISWs [Apel et al., 1985]. This process is described by the Korteweg - de Vries (KDV) equation, where ISWs evolve when the tidal wavefront approaches vertical in order to relax the further steepening process.
Several attempts were made to compare the evolution of ISWs as described by the KDV equation with laboratory experiments. The numerical results agree with laboratory observations even quantitatively [Ostrovsky and Stepanyants, 2005], given the amplitudes of the ISWs are small. SWs, which are exact solutions of the KDV equation are called solitons. SWs, described by
η(x, t) = asech2(x−ct
λ ) (1)
in a two layer system, are an exact solution of the two-layer KDV equation and hence solitons (Fig. 1). Here η is the displacement of the pycnocline at location x and time t,
1
a is the amplitude, c the phase-velocity and λ the wavelength of the wave. By assuming an equipartition of potential and kinetic energy [Moum et al., 2007], the total energy E of solitons given by (1) can be estimated as
E = a24
3∆ρgλ, (2)
where ∆ρ is the density difference between the upper and lower layer and g the gravity constant. Solitons are stationary when using a reference frame moving with the wave, suggesting that SWs observed in the field do not change their properties when propagating undisturbed through the water.
Amplitude a
Displacement η
−500 −300 −100 100 300 500 Wave propagation
x (m)
Figure 1: Schematic of a soliton of depression with wave length λ= 100 mderived via (1). Note that the pycnocline is displaced only in one direction.
Whereas ISWs are assumed to be of stable form in the open water, they are thought to break when they shoal upon sloping topography [Imberger, 1998; Lorke et al., 2006]. Nu- merical investigations [Bourgault et al., 2005] and laboratory experiments [Helfrich, 1992;
Michallet and Ivey, 1999; Boegman et al., 2005b] indicate that these waves can release a large part of their energy in the shoaling process. Hence the ecological consequences of the shoaling of ISWs in the boundary region, i.e. boundary mixing and resuspension, attained special attention. Both processes are of importance for the ecosystem, because they enhance the sediment-water transfer and thus influence the concentration of nutri- ents, dissolved oxygen and suspended particles in the deep littoral [Goudsmit et al., 1997;
MacIntyre et al., 1999; Wuest et al., 2000].
Several studies confirm the important role of ISWs for boundary mixing by associating observations of increased bottom boundary turbulence to the breaking of ISWs [Thorpe et al., 1972; MacIntyre et al., 1999; Michallet and Ivey, 1999; Klymak and Moum, 2003;
Lorke, 2007]. Moreover, field observations demonstrate that shoaling ISWs are able to resuspend sediment [Bogucki et al., 1997; Dickey et al., 1998; Hosegood and van Haren, 2004; Bogucki et al., 2005; Moum et al., 2007] and to transport resuspended sediment along the sloping boundaries [Klymak and Moum, 2003; Scotti and Pineda, 2004]. Hosegood and van Haren [2004] even speculated that ISWs are the dominant mechanism driving on- shelf sediment fluxes. As a result of these observations and speculations the sediment resuspension mechanisms associated with ISWs were investigated numerically and in the laboratory [Stastna and Lamb, 2008; Boegman and Ivey, 2009].
The relevance of ISW-induced boundary turbulence and resuspension for the ecosystem depends strongly on the quantity of such events. This quantity is first of all determined by the quantity of ISWs in the system. The frequency of occurrence of ISWs in lakes depends often on the strength and frequency of wind-forcing [Horn et al., 2001; Boegman et al., 2005b]: a significant proportion of the kinetic energy introduced by the wind forcing at the lake surface is channeled to large-scale internal waves with relatively low frequency [Boegman, 2009]. Just like tidal waves these large internal waves steepen if the wind forcing is sufficiently strong, evolve in propagating surges and finally degenerate into a train of ISWs [Horn et al., 2001]. However, not only the frequency of occurrence of ISWs influences the ecological impact of these waves. A transfer of the ecological importance attributed to ISWs in the ocean to large lakes is thus not complete without a detailed investigation of wave properties and a basin-scale analysis of the propagation pattern of the ISWs.
Whereas the frequency of occurrence of ISWs is a measure of the relevance of ISWs in time, the basin-scale propagation pattern is a measure in space, describing where ISW- related ecological processes might occur. The evolution and fate of ISWs on a two- dimensional spatial scale, i.e. a scale only considering depth and one horizontal direction is well analyzed, both numerically and in the laboratory [Horn et al., 2001; Boegman et al., 2005a; Ostrovsky and Stepanyants, 2005]. Effects of the lake width on ISWs are often neglected for simplicity in laboratory experiments or numerical modeling and were up to now not investigated in detail in the field. Transferring the two-dimensional understanding gained e.g. from laboratory experiments to lakes, ISWs have to be assumed to propagate as extended wave fronts parallel to the shore towards a terminal cross-shore boundary [Thorpe et al., 1972; Imberger, 1998; Boegman et al., 2005b]. However, the width of the basin might affect the propagation behavior of the waves in lakes. Assumptions can e.g. be made about a possible effect of the Coriolis force on the wave properties [Thorpe et al., 1972], which could result in a reduction of the ISW amplitude and thus energy (see formula 2) within the wave front. Considering the lake width as an additional dimension would
also make a deviation of the ISW propagation direction from the shore-parallel direction thinkable and thus a run-up of ISWs on all boundaries of a lake [Lorke et al., 2006]. In this case ISW-induced boundary mixing and resuspension would occur within the entire littoral zone of the lake and not only at the terminal cross-shore boundary.
It is known that the properties of ISWs are mainly determined by stratification and amplitude [Vlasenko et al., 2000]. Since the stratifications in lakes can differ significantly from the stratification in the ocean, the properties of ISWs in lakes can be assumed to be different from ISW properties in the ocean. Such differences might include besides the special shape of ISWs also their breaking mechanism. The breaking mechanism determines the potential of ISWs for either particle transport, e.g. sediment, larvae and plankton, or mixing [Silva et al., 1996; Lamb, 2002; Helfrich and Pineda, 2003] and is hence directly related to different important ecological processes. Even if ISWs are usually assumed to propagate undisturbed through the open water, some observations of ISW breaking in deep ocean water exist [Moum et al., 2003, 2007]. These observations suggest the existence of mechanisms of wave breaking, which are independent of water depth. Such mechanisms recently received increasing attention, both numerically [Lamb, 2002; Stastna and Lamb, 2002; Barad and Fringer, 2010] and in the laboratory [Grue et al., 2000; Carr et al., 2008].
In deep water, waves may start to break either due to shear instabilities or due to convective instabilities [Lamb, 2002; Stastna and Lamb, 2002]. Shear instabilities occur if the ISW-induced current shear is large enough to generate density overturns typically referred to as Kelvin-Helmholtz billows. The shear is considered critical for the formation of billows if it exceeds the Brunt-V¨ais¨al¨a frequency derived from the background stratification by a factor of four [Howard, 1961; Miles, 1961; Thorpe, 2005]. Billowing typically results in the dissipation of energy [Thorpe, 1977]. Convective instabilities occur if the ISW- induced current velocity exceeds the phase velocity of the wave. In this case a (convectively unstable) recirculating wave core is generated. The most important ecologically relevant ability of such convectively unstable ISWs is assumed to be the horizontal transport of buoyant particles in the wave core over long distances throughout the lake [Helfrich and Pineda, 2003; Helfrich and White, 2010].
Lake and ocean wide measurements of turbulence suggest that the majority of the mixing occurs not in the pelagic thermocline, but either at the boundaries or at under- sampled hot spots [Ivey et al., 2008]. Mixing caused by ISW breaking in the deep water is thus considered energetically negligible [Imberger, 1998]. However, also sporadic turbulent fluxes in an otherwise quiet region might have an important ecological impact. The strong stratification in the lake’s interior effectively separates different temperature zones in the
water column, preventing a quick exchange between water packages from adjacent regions.
Hence also dissolved nutrients are prevented from quickly passing separated zones. The breaking of shear-unstable ISWs in the lake’s interior can result in mixing and thus a rapid exchange between adjacent water packages. In this way, nutrients, which are often higher concentrated in the deeper parts of the water column, e.g. due to depletion by phytoplankton in the upper region, would be redistributed. A patch of nutrients provided by a breaking ISW could thus support plankton growth in an otherwise nutrient depleted zone. Although the occurrence of ISW breaking in the pelagic thermocline and the kind of the corresponding breaking mechanism might thus play an important role in the ecosystem of lakes, neither of them have been investigated statistically up to now.
In deep temperate lakes, stratification and thus probably the shape and breaking mech- anism of ISWs depend on season. It can be speculated that the relevance of ISWs for the ecosystem changes significantly with season. This hypothesis is supported by the trivial finding that internal waves are not generated in a well mixed aquatic system. Such kind of stratification dominates during winter in many lakes in the temperate zone. Hence the impact of internal waves in these systems is negligible during winter. However, up to now it was not investigated whether the properties of ISWs change significantly during the stratified period. This lack of investigation is mainly a result of measuring limitations.
Most field studies of ISWs in lakes are short-term studies and carried out during autumn, when stratification closely resembles a two-layer system [Macintyre et al., 2009].
The physical conditions determining the environment for ISWs also influence biological life. Hence many biological constitutionals of the aquatic system undergo annual cycles.
The ecological effect of ISWs on certain constitutionals might thus not only depend on the specific properties of the waves but also on the timing of ISW activity. The spring bloom in the annual cycle of phytoplankton, for example, usually starts shortly after the onset of stratification [Anneville et al., 2005; Peeters et al., 2007]. The number of phytoplankton reduces after the bloom is reached, because of the continuing depletion of nutrients in the upper water column by the phytoplankton itself on the one hand and the increasing predation by zooplankton on the other hand. ISW-induced enhancement of nutrient fluxes would, for example, contribute much more to plankton growth during a time period when the water column is depleted of nutrients, i.e. during or after the spring bloom.
Outline
In this study, the properties of ISWs in Lake ¨Uberlingen, a sub-basin of Lake Constance, are analyzed by a combination of field measurements and numerical modeling. The results of this analysis are used to discuss the ecological relevance of these waves for deep lakes in the temperate zone (discussion).
The aim of chapter 1 is to investigate whether ISW-induced turbulence is negligible in the pelagic thermocline or if it could be of importance for the ecosystem of a lake.
Thus in this chapter the quantity and the timing of internal wave induced turbulence in the pelagic thermocline of Lake ¨Uberlingen are analyzed. Since turbulent events in the pelagic thermocline, if at all existent, have to be expected to occur sporadically and seldom, temperature was measured at a high time resolution, allowing a rough estimation of variation of turbulence over several weeks.
ISW breaking, which might result in the generation of turbulence, and other ISW properties are numerically predictable if background stratification and amplitude of the ISW are known. The applicability of typical equations for the simulation of ISWs, i.e. the KDV and the DJL equation, is tested in chapter 2 for certain ISW amplitudes and stratifications observed in Lake ¨Uberlingen. The purpose of this chapter is identifying the amplitude range in which each model is applicable in order to justify further simulations.
The physical properties of more than 200 ISW trains measured in the center of Lake Uberlingen over a time of 6 years are analyzed in¨ chapter 3. Among these properties are amplitude, number of waves in a wave train, frequency of occurrence of ISWs and ISW breaking, all of them directly related to the relevance of ISWs in Lake ¨Uberlingen. The occurrence of wave breaking in the field is compared with breaking conditions known from laboratory experiments in order to test the laboratory findings under lake conditions and to generalize them for natural stratifications. Moreover, the influence of season on ISW properties is investigated in this chapter. This investigation provides a physical basis for a further discussion whether the importance of ISWs for the ecosystem of a deep lake in the temperate zone can be expected to change with season.
The breaking mechanisms determining the ecological role of the ISWs in the lake’s interior are investigated in chapter 4 via a combination of field measurements and nu- merical modeling. The aim of this chapter is to test the accuracy of numerical predictions of ISW breaking under field conditions. Therefore the occurrence of wave breaking in the field is correlated with numerical simulations of conditions sufficient for wave breaking.
Predictions of the breaking mechanisms are evaluated by a detailed investigation of field measurements of single breaking ISWs.
Chapter 5 investigates the basin-scale propagation of ISWs in order to derive how the wave properties change on their way through Lake Constance and where ISW-induced ecological processes can be expected. Temperature and current time series measured at several different study sites throughout the lake are compared for this purpose. The special focus of this chapter is to analyze the influence of the lake width on the ISW properties and to investigate whether wave breaking can occur over the total width of the lake.
Internal Waves and the Generation of Turbulence in a Large Lake
Martina Preusse, Frank Peeters and Andreas Lorke Limnol. Oceanogr., 55(6), 2010, 2353 – 2365
Abstract
High-resolution thermistor chain data collected between 5 and 20 m in a large stratified fresh- water lake (Lake Constance) at a water depth of 50 m reveal the frequent occurrence of large- amplitude (≥1 m) vertical density inversions that indicate overturns in the pelagic thermocline.
Velocity data collected simultaneously to the temperature measurements suggest that the den- sity inversions are mainly produced by shear instabilities. A comparison between the timing of the passage of the basin-scale internal Kelvin wave and the density inversions demonstrates a pronounced phase relationship, implying that the processes leading to the occurrence of turbu- lence and mixing are connected to the passage of the Kelvin wave. Two different processes are identified during which density inversions were particularly common. An increased number of density inversions and especially high dissipation rates of turbulent kinetic energy were observed when Kelvin wave induced critical shear supported the generation of large Kelvin-Helmholtz bil- lows. A particularly large number of density inversions was also associated with the passage of nonlinear high-frequency waves of large amplitudes. The density inversions typically occurred at the wave troughs, which indicates breaking of these waves. These observations suggest that self-induced shear generated by the basin-scale seiche and by high-frequency internal waves leads to a significant amount of turbulence and mixing in the pelagic thermocline.
1.1 Introduction
Diapycnal exchange in the seasonal thermocline is important for vertical transport of nu- trients, plankton and oxygen [Eckert et al., 2002]. It also affects stratification, which in turn influences the generation of turbulence from large scale processes and thus provides a feedback mechanism on the magnitude of mixing [Lewis et al., 1986; Wuest and Lorke,
9
2009]. Hence, mixing is of great importance both for its ecological implications and its rele- vance to environmental fluid dynamics. The quantification of vertical mass fluxes based on measures of turbulence, however is difficult, since the intensity and occurrence of turbulent mixing is spatially and temporally highly variable [Ivey et al., 2008].
Turbulent mixing in the stratified interior of oceans and lakes has been observed to be too weak to account for turbulent diffusivities estimated from basin-scale tracer budgets [Ledwell et al., 1993; Rudnick and Others, 2003; Wunsch and Ferrari, 2004]. This mismatch is assumed to result from spatial heterogeneity of the mixing. Turbulence and mixing is increased near lateral boundaries, especially near the benthic boundary, in comparison to the interior of the water body Ledwell et al. [1993]; Goudsmit et al. [1997]; Wuest et al.
[2000]. But hotspots of turbulent dissipation are also located in the pelagic zone above the benthic boundary layer, as observed at near-shore slopes [Moum et al., 2007; van Haren, 2009; Shroyer et al., 2010] and ridges or sills [Rudnick and Others, 2003; Lamb, 2004;
Macintyre et al., 2009].
A key idea to explain the observed variability of mixing was that enhanced turbulence may be caused by the ubiquitous internal wave activity. Propagating internal waves with periods between the buoyancy limit and those of basin-scale or inertial waves are common in lakes [Thorpe et al., 1972; Saggio and Imberger, 2001] and in the ocean [Fu and Holt, 1982; Munk and Wunsch, 1998]. At the low-frequency end of the wave spectrum, enhanced turbulence is coupled to the passage of tides in the ocean [Garrett, 2003; Rudnick and Others, 2003] and steep-fronted basin-scale internal seiches in lakes [Macintyre et al., 2009].
The energy cascade from these low-frequency waves to turbulence is assumed to be a major facilitator of mixing [Imberger, 1998; Munk and Wunsch, 1998; Egbert and Ray, 2000].
Considerable effort has been undertaken to understand and quantify the elements of the energy cascade [Imberger, 1998; Horn et al., 2001; Boegman et al., 2005b].
In particular, nonlinear high-frequency internal waves with large amplitudes, gener- ated in the process of degeneration of low-frequency waves, have received special attention [Farmer and Armi, 1999; Boegman et al., 2005a; Helfrich and Melville, 2006]. These nonlin- ear high-frequency waves are ubiquitous in lakes [Thorpe et al., 1996; Boegman et al., 2003;
Lorke et al., 2006] and in the ocean [Orr and Mignerey, 2003; Silva et al., 2009; van Haren, 2009]. They have been observed to propagate several kilometers [Moum et al., 2007] until shoaling on sloping topography [Boegman et al., 2005a; Lorke, 2007; Boegman, 2009] and thereby releasing their energy during the breaking process in the bottom boundary layer.
Wave breaking is an important mixing mechanism in stratified waters [Imberger, 1998;
Garrett, 2003; New et al., 2007]. Instabilities associated with the propagation of large-
amplitude nonlinear internal waves, their superposition, and interaction with background currents also lead to wave breaking in the open water [Moum et al., 2003].
The mixing process associated with internal waves is thought to be caused by the turbulent collapse of density inversions (overturns) [Thorpe, 1977, 2005; Wuest and Lorke, 2009]. Overturns result from physical processes generating shear instabilities [Howard, 1961; Miles, 1961; Fructus et al., 2009], convective instabilities [Orlanski and Bryan, 1969;
Carr et al., 2008] or changes in the effective rate of strain [Alford and Pinkel, 1995]. At present the evolution of these processes, like the occurrence and propagation of nonlinear high-frequency internal waves leading to the generation of overturns, cannot be reproduced in field scale hydrodynamic models ([Boegman et al., 2001] [Horn et al., 2001]. Including internal-wave driven mixing in these coarse-grid models requires a parameterization of the generation of nonlinear high-frequency internal waves from the wave field in lakes and of the mixing efficiency of the various high-frequency processes.
The objectives of this study are to identify the processes leading to large-scale (≥1-m) density inversions in the pelagic thermocline and to assess the relative importance and efficiencies of these processes for the turbulent dissipation of energy in a lake. The follow- ing sections aim to first identify the large-scale generation mechanisms of overturns: large Kelvin-Helmholtz billows generated at the steepened front of a basin-scale Kelvin wave, shoaling of propagating nonlinear internal waves (NLIW) and interaction of high-frequency waves (HFW) with more weakly stratified areas, and second evaluate the relative impor- tance of the mechanisms for the generation of turbulence and mixing in the lake’s interior by estimating the corresponding dissipation rates of turbulent kinetic energy, diapycnal eddy diffusivities, and buoyancy fluxes.
1.2 Methods
1.2.1 Measurements
Measurements of temperature and current velocity were performed in Lake ¨Uberlingen, a sub-basin of Lake Constance (63 km long, 14 km wide), Europe. Lake ¨Uberlingen is a long (length: 20 km), narrow (mean width: 2.3 km) lake with maximum and mean depths of 147 m and 84 m, respectively. The sub-basin is connected to the deep (maximum depth:
252 m) main basin at the Sill of Mainau (Fig. 1.1), where the depth at the thalweg reduces to 100 m [Wessels, 1998]. Three vertical thermistor chains were deployed from 24 June until 09 July 2008 at 60 m depth, 0.4 km offshore, and about 6 km from the western end of the lake. To observe the propagation of high-frequency waves, the chains were arranged
Upper
Lake Constance Lake
Überlingen
15 10 5 0 5 10 10
5 0 5 10 15 20
Distance (m)
Distance (m)
ADCP C1
C2
C3
l1
l2 l3
0 5 10 km φ
Sill of Mainau
-125--225- 175
-25- 25
-75-
Figure 1.1: Lake Constance bathymetry [Wessels, 1998], with the location of the wind station (gray circle); moored thermistor chains C1, C2, and C3; and the ADCP.
in a triangle with side lengths of 13, 21, and 32 m, respectively (Fig. 1.1). The position of the chains was estimated by Global Positioning System (GPS) with an accuracy of
±2.5 m. Each thermistor chain consisted of 14 to 15 individual temperature loggers (TR- 1050, RBR), which were located between 4 and 20 m depth with a vertical spacing of 1 m.
The accuracy of the thermistors is 0.002◦C and the sampling interval was set to 1 s. The internal clocks of the loggers were synchronized at the beginning of the sampling period and showed differences of up to 8 s at the end of the deployment. The corresponding time lags were assumed to depend linearly on time and each temperature series was replaced by a corrected time series using linear interpolation.
A 300-kHz acoustic doppler current profiler (ADCP, RD-Instruments) was deployed at a distance of 10 m from the thermistors at a depth of 70 m. The upward-looking ADCP averaged internally 55 individual profiles taken at 2.72 s intervals providing a sampling interval of 2.5 minutes. The vertical bin size was 0.5 m, resulting in a profiling range spanning 2.7 m above the bottom up to 5 m below the lake surface. The three-dimensional current velocities were recorded in earth coordinates. Velocity estimates with an associated error velocity exceeding 2 m s−1 were disregarded. The error velocity is calculated by the ADCP for each profile and depth bin using redundant data from a fourth acoustic beam.
The resulting mean error velocity was 0.03 m s−1 with a standard deviation of 0.04 m s−1, indicating a rather poor signal to noise ratio in comparison to the observed mean current speeds around 0.05 m s−1.
Wind speed and direction were measured at a land-based meteorological station at the City of Constance by the German Meteorological Service, approximately 1.5 km from the lake and were provided as hourly mean values. The data were assumed to be representative for the wind directions and speeds at Lake ¨Uberlingen [Zenger et al., 1990].
1.2.2 Data analysis
The 10.5◦C isotherm is used as basis for further analysis of the wave motions. Despite the strong temperature fluctuations caused by the internal waves this isotherm could be continuously observed during the entire duration of the experiment. Phase velocity c, direction of phase propagationcdir and wave length λ of HFW are estimated by analyzing the bandpass-filtered (butterworth, lower and upper cut-off frequencies 1/30 min−1 and 1 min−1, respectively) 10.5◦C isotherm displacements observed at the three thermistor chains following Ufford [Ufford, 1947], as follows:
tan(˜cdir) =
hl2 t12
l3 t13 −cos(φ) i
sin−1(φ) (1.1)
λ = T l2 t13
cos(˜cdir) (1.2)
c = λ
T (1.3)
where li, φ are parameters describing the geometry of the triangular mooring (Fig. 1.1) and ˜cdir is the direction of phase propagation relative to the triangle. Wave period T and propagation time tij of a wave from chaini to chain j are determined by auto- and cross- correlations with maximum time lags of 20 and 10 minutes, respectively. For clear signals, such as those from NLIW the direction of propagation can be reliably calculated within an accuracy of±30◦. Error estimates (approximately 65%) of phase velocity and wave length are assessed individually for the observed NLIW using the observed propagation times.
The level of turbulence can be characterized by the number and the magnitude of den- sity inversions in the water column. Since density in Lake Constance is predominantly determined by temperature, the analysis of density inversions is based on thermistor data.
A sorting algorithm is used to estimate vertical displacements (Thorpe displacements) and the Thorpe scale LT, the root mean square of Thorpe displacements including zero values [Thorpe, 1977]. The Thorpe scale is commonly estimated from high-resolution temperature profiles, whereas in the data from the thermistor chains the vertical resolution is limited to 1 m. Because of this poor vertical resolution the estimate ofLT often includes several zero values and can become smaller than the thermistor spacing. To exclude calibration errors
near the accuracy limit of the temperature loggers, displacements are only considered for further analysis if the difference between the measured temperature and the corresponding temperature in the sorted profile exceeds 0.01◦C. Because the mean vertical temperature gradient (dT /dz)(t) at any time t is always larger than 0.03◦C m−1 and 0.49◦C m−1 on average, this threshold is sufficient for resolving the temperature structure in the water column and the accuracy ofLT is rather limited by the vertical resolution of the measure- ments [Stansfield et al., 2001]. Inversions occur likewise frequently and with comparable magnitude at almost all depths at all three chains. On average, 2% of all temperature profiles are unstably stratified, which sums up to 4.8 h of unstable stratification during the measurement period. Maximal displacement lengths reach values up to 6 m. Examples of temperature profiles with inversions are shown in Fig. 1.2.
8 9 10 11
5
10
15
20
19:57 19:58 19:59 inversions
Temperature (°C)
Depth (m)
C1
9 10 11
19:59 20:00 20:01
inversions C2
19:56
19:57 19:58
9 10 11
C3
inversions
a b c
unresolved
Figure 1.2: Temperature profiles with density inversions (black squares) of chains C1, C2, and C3
observed at the indicated times on 08 July 2008. Arrows indicate unresolved inversions with an associated temperature difference of<0.01◦C.
The available potential energy of unstable density fluctuations (APEF) is a measure of the maximum potential energy that can be released to dissipation and mixing by a turbulent overturn [Dillon and Park, 1987]. The APEF can be estimated by comparing observed and sorted density profiles. As recommended by Dillon and Park [Dillon and Park, 1987] for fixed measurement intervals, we use the formula
APEF [J kg−1] ≈ g 2ρ0M
M
X
n=1
dn∆ρn (1.4)
for the estimation of the APEF, where g is the gravitational constant, M is the number of temperature measurements within each profile,ρ0 is the background density, ∆ρn is the Thorpe density fluctuation and dn the Thorpe displacement at depth zn. The calculated values are rough estimates of APEF due to the limited spatial resolution of the temperature measurements.
The rate at which APEF is dissipated in the process of gravitational adjustment depends on stratification, which is described by the buoyancy frequency, N, corresponding to the stable density profile. Under the assumption of isotropic turbulence and a steady-state balance between production and dissipation of potential energy, Dillon and Park [Dillon and Park, 1987] found experimentally that the rate of dissipation of APEF in the seasonal thermocline is proportional to the APEF with the constant of proportionality 4.8 N. This proxy is used to estimate the dissipation rate of turbulent kinetic energy, ,
[W kg−1] ≈ 4.8N APEF. (1.5)
The lower detection limit of this method can be estimated asb ≈10−9 W kg−1 by consid- ering an inverse density profile corresponding to a single 1-m overturn with a temperature inversion of 0.01◦C.
N is estimated by applying 2-m centered differences (and backward or forward differ- ences on the edges) to sorted and 30-s-averaged temperature profiles. The diapycnal eddy diffusivity Kz and the buoyancy flux Jb,
Kz = γmix
N2 (1.6)
Jb = γmix (1.7)
are estimated by using a constant mixing efficiency, γmix, of 0.2 (Osborn 1980). APEF, , Kz and Jb are given as arithmetic averages over the vertical measuring range.
Instabilities in stratified flows typically arise in regions where the Richardson number Ri falls below a critical value of 0.25 [Howard, 1961; Miles, 1961; Thorpe, 2005], where
Ri = N2(z)
(du/dz)2 + (dv/dz)2 (1.8)
with u and v as along-shore and cross-shore velocity, respectively [Howard, 1961; Miles, 1961]. Ri is estimated by using a 2.5-min average of N and order 3 polynomial fits of the vertical profiles of horizontal current velocity.
1.3 Results
1.3.1 Degeneration of the Kelvin wave
An overview of the field data observed during the entire measuring period is given in Fig. 1.3. The wind field (Fig. 1.3a) is dominated by weak to moderate (0.3 to 6.4 m s−1)
2008
28 June 01 July 04 July 07 July
5 10 15 20
Depth (m)
10 14 b 18
W1 W2
Temperature (°C)
P1 P2 P3 P4
1 2 3
20 40 60
-0.08 -0.04 0 0.04 0.08
20 40 60
c
d
Current velocity (m s −1)
Along-shore
Cross-shore Wind
N
5 m s−1
0 2.5
NLIW1 NLIW2
a
Figure 1.3: Overview of field data of the entire sampling period. (a) Vector plot of wind speed (1-h average); reduced wind speeds are colored blue. (b) Five-minute average of temperature at chainC2with a 0.5◦C contour interval. (c) Twenty-minute running average of alongshore current velocity. Arrows mark the occurrence of NLIW and current fronts. (d) Twenty-minute running average of cross-shore current velocity.
westerly winds. The low-frequency oscillations of the depths of isotherms (Fig. 1.3b) are caused by a basin-scale internal seiche with an amplitude exceeding 15 m. The seiche is influenced by Earths rotation and forms a basin-scale internal Kelvin wave with a period generally ranging from 3 to 5 d, depending on stratification [B¨auerle et al., 1998; Boehrer et al., 2000; Appt et al., 2004]. Four full Kelvin wave cycles were recorded (Fig. 1.3b, P1 −P4). After cessation of moderate westerly winds on 04 July and 08 July (Fig. 3a, bars 1 and 2), the Kelvin wave developed steep thermal fronts (Fig. 1.3b, ends of Kelvin wave cycles P3 and P4), indicating the nonlinearity of the basin-scale seiche. The steep thermal fronts were associated with steep alongshore current fronts (Fig. 1.3c, arrows) and NLIW
with amplitudes of up to 15 m and strong current velocities of up to 0.15 m s−1 (Fig. 1.3c, NLIW1 and NLIW2).
10-5 10-4 10-3 100
102 104 106
95 %
10-5 10-4 10-3
10-1 100 101
95 %
along-shore cross-shore
Frequency (Hz)
Power spectral density (m Hz )-1 a 10.5° isotherm
Lmax 3 days
10 min 4 hours
b
2
10 min 4 hours
3 days
Power spectral density (m s Hz ) -12-2
Figure 1.4: Power spectra of (a) 10.5◦C isotherm depth fluctuations and Lmax (chain C1) and (b) alongshore and cross-shore current velocity at 30.8 min depth. Confidence at the 95% level is indicated by dashed lines. Note the different scaling of the vertical axis.
The power spectrum of the fluctuations of isothermal depth has three significant peaks at frequencies corresponding to periods of 3 d, 4 h, and 10 min (Fig. 1.4a). The spectral peak at the lowest frequency is related to the Kelvin wave with a period of 3 d throughout the time of our measurements. The wave with a period of 4 h is a crossshore oscillation, which is strictly limited to Lake ¨Uberlingen [B¨auerle, 1994]. The broad high-frequency peak around 10−3 Hz (5- to 20-min periods) characterizes propagating internal waves.
Current velocity fluctuations were evaluated at 30-m depth below the fluctuating depth of current inversion. All three peaks were observed in the power spectrum of the alongshore velocity fluctuations (Fig. 1.4b), while the only significant peak corresponds to the 4-h period. The power spectrum of the cross-shore velocity does not increase in spectral power toward lower frequencies but also shows small peaks at periods of 4 h and 10 min. The power spectrum of the maximal Thorpe displacements, Lmax, again shows all three peaks, and the high-frequency peak at a period of 16 min is significant. The latter observation indicates periodic occurrence of turbulence correlated with the occurrence of these waves (Fig. 1.4a).
A comparison between the passage of HFW (Fig. 1.5a), the basin-scale internal Kelvin wave (Fig. 1.5b), and the occurrence of density inversions reveals a pronounced phase relationship. The time series of the depth of the 10.5◦C isotherm was low-pass filtered (Butterworth of second order, cutoff frequency: 1/24 h−1) to identify the Kelvin wave in the wave field. The square of the band passfiltered displacement of the 10.5◦C isotherm was used as a measure of the intensity of HFW. The inversions are particularly frequent
−150 −100 −50 0 50 100 150 - 0.1 0 0.1 0.2
Lags (h) Kelvin wave - L Kelvin wave - HFW
T
Correlation coefficient
-15 10
20 15 5
28 June 01 July 04 July 07 July 0.5 1 1.5 2 10.5° isotherm 2.5
Kelvin wave LT
2008
L (m)
Displacement (m)
-5 -6 -7 -8
5 2.5 7.5 HFW
log(ε)
Displacement (m )
log(ε) (W kg )-1 2
a
b
c
T
Figure 1.5: Phase relation between inversions and internal waves (chain C1). (a) Squared band pass- filtered 10.5◦C isotherm displacements (HFW) and dissipation rates. (b) Original and low-pass filtered (cutoff frequency 1/24 h−1) 10.5◦C isotherm displacements (Kelvin wave) and Thorpe scale. (c) Cross- correlation between Thorpe scale and Kelvin wave and between HFW and Kelvin wave. Nearest (relative to 0 lag) local extreme value is a minimum at a phase lag of -15 h.
shortly before the trough of the Kelvin wave passes (Fig. 1.5b) and occur together with the passage of the internal front. Maximal values of Thorpe displacement increase with the amplitude of the basin-scale seiche (Fig. 1.5b) or the intensity of HFW (Fig. 1.5a). The cross-correlation (Fig. 1.5c) between Thorpe scales and Kelvin wave is a periodic function of lag time with a period of about 3 d, the period of the Kelvin wave. The correlation function reaches a local minimum near the −15-h lag, indicating a coupling between the passage of the Kelvin wave front and the occurrence of overturns. Maximal correlation coefficients reach values of almost 0.2 and are highly significant, according to the Students t-test (α < 0.001). Similar magnitudes, periods, and phase lags can be found in the cross-correlation between the Kelvin wave and the intensity of HFW (Fig. 1.5c), indicating a synchronized occurrence of HFW and density inversions at the steepened front of the Kelvin wave, which indicates a causeeffect connection [Alford and Pinkel, 1995].
1.3.2 Processes associated with the occurrence of density inver- sions
In the following analyses, four different conditions under which density inversions occurred are distinguished and the observations are grouped accordingly into four categories of events (Figs. 1.6a, 1.7a): Kelvin wave shear-induced billows (KWB), weak stratification (WStr), nonlinear internal waves (NLIW), and second mode waves (2ndM).
10 15 20
Temperature (°C) f
14:05 14:10 14:15
5 10 15 20
Depth (m)
10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 02:00 04:00 06:00 10
15 20
−0.05 0 0.05 0.1 0.15 m s
Time (hh:mm)
8 10 12 14 16
°C
Time (hh:mm) 10
15
20 g 7°C 10°C
-1
Along-shore velocity
Displacement (m)
KWB WStr1 NLIW1 2 M1
1 2 1 2 1 2
WStr2 nd
b
c
d L , Chain CT 1
L , Chain CT 2
L , Chain C
T 3
a
log(ε) (W kg )-1
Ri < 0.25
Inversion 5 m
20 m
h
i -8 -7 -6 -5
log(ε) e
Figure 1.6: Details of observations during period W1 (cf. Fig. 1.3). (a) Category (b-d) LT at chains C1, C2, andC3. (e) e at chain C1 (f) 1-s temperature (chainC1) from loggers starting at 5 m in depth with a 3-m depth increment. (g) One-second isotherm response (chainC1) with 1◦C temperature intervals (lines), occurrence of temperature inversion (squares), andRi , 0.25 (blue). (h) Twenty-minute average of alongshore current velocity (missing data in gray) and typical current profiles of alongshore (black) and cross-shore velocities (white). One hour on the time scale corresponds to a current velocity of 0.1 m s−1. White arrows indicate KelvinHelmholtz billows (zigzag pattern in the current velocity). (i) Zoom into the area marked by the red arrow in panel g. Contour lines show KelvinHelmholtz billows (white bars) and density inversions (squares).
During the first observation period, W1 (Figs. 1.3b, 1.6), the first two turbulent patches in Fig. 1.6g are associated with oscillations near the buoyancy period 2πN−1 ≥ 4 min, which develop into billow-like structures. The billows occur at the depth of the thermocline (Fig. 1.6i, bars) and the Kelvin waveinduced current shear (Fig. 1.6h, depth of current shear at 14:00). Unstable oscillations, caused by Kelvin waveinduced shear and developing into billows, are grouped into the KWB category. Such instabilities were only observed during the passage of the internal front in period W1.
The following patches of density inversions (Fig. 1.6a: WStr1and WStr2) are associated with a thermocline-wide weakening of the stratification (Fig. 1.6f,g) over the course of 6 h as well as long-lasting billows in the alongshore current velocity at the bottom part of the thermocline (Fig. 1.6h, white arrows). These events are collected into the WStr category.
After the weakening of stratification, NLIW (Fig. 1.6a: NLIW1) with amplitudes of about 5 m (Fig. 1.6g) propagate in the southeast alongshore direction, with a period of 12 min, a phase velocity of 0.28 m s−1, and a wave length of ≈ 200 m. These waves follow an internal current front propagating in an alongshore direction toward the south east (Fig. 1.3c, second arrow), probably the front of the Kelvin wave that was reflected at the western boundary of Lake Constance (compare to [Appt et al., 2004]). NLIW with periods between 5 and 30 min and amplitudes exceeding 3 m are commonly associated with density inversions and represent the NLIW category. Wave packets matching the definition of the NLIW category were found during both periods W1 and W2 (Fig. 1.7a: NLIW2, NLIW3, and NLIW4). The NLIW during W2 both led (NLIW2) and followed (NLIW3 and NLIW4) the entering surge (Fig. 1.3c, third arrow) propagating in an alongshore direction (northwest). Amplitudes and periods of the wave packet depend on their location relative to the fronts: the wave packet NLIW2 propagates with a phase velocity of 0.17 m s−1, a period of 17 min, and a wavelength of 170 m. The estimated direction of the phase velocity in comparison with the direction of current velocities under the wave troughs measured by the ADCP (not shown) indicates that phase velocity and wave orbital velocity under the wave trough have opposite directions, which is in agreement with theoretical models [Vlasenko et al., 2000]. The wave packets NLIW3 (wave lengths ≈ 170 m) and NLIW4 (wave lengths ≈ 75 m) propagate with phase velocities of 0.22 m s−1 and 0.17 m s−1, respectively. They have smaller amplitudes and periods (NLIW3: 13 min; NLIW4: 7 min) than does NLIW2.
The next hot spots of density inversions (Fig. 1.6a: 2ndM1) are surrounded by opening and closing isotherms with an irregular period between 1 and 2 h (Fig. 1.6g), which are accompanied by HFW with amplitudes smaller than 3 m. Such events associated with
Depth (m ) 10
15
20
7 8 9 10 11 12 13
°C
10
50 30
−0. 1
−0.05 0 0.05 0.1 m s-1
Along-shore velocity Occurrence of overturns
C1 C3 C2 10.5°C
b
d
Time (hh:mm)
18:00 20:00 22:00 00:00 02:00 04:00 06:00 08:00 10:00
19:20 20:00 20:40 5
10 15
20 19:45 19:50 19:55 20:00 20:05
7 8 9 10 f
e °C
19:58 20:00 20:02 g
10 9 8 7
°C
Time (hh:mm) 5
2 Mnd 2
NLIW2 NLIW3 NLIW4
a
−8
−7
−6
Depth (m)
c
log(ε) (W kg )Depth (m)-1
Figure 1.7: Details of observations during period W2 (cf. Fig. 3). (a) Category. (b) Occurrence of density inversions (squares) associated with isotherm depth fluctuations (chainC2, 1-s resolution). (c) at chainC2. (d) Five minute and 2-m running average of alongshore current velocity and 10.5◦C isotherm of chainsC1(red),C2 (blue), andC3 (black). (e) One-second temperature (chainC2) with 0.1◦C contour interval (zoom into the square of panel d, showing NLIW2). (f) The trough of a soliton (zoom into the square of panel e). (g) Density inversions are represented by overturns in the troughs (zoom into the square of panel f). Note the different and nonlinear color scaling.
the second vertical mode-type isotherm displacements are grouped into the 2ndM category.
Turbulent patches localized in the center of 2ndM were also found during observation period W2 (Fig. 1.7a: 2ndM2).
1.3.3 Generation mechanisms of density inversions
The median duration of individual density displacements at each thermistor chain was 4 s.
However, maximum durations of inversions reached between 20 and 418 s, depending on the depth in the water column. The magnitude and the time series of the Thorpe scales at the three thermistor chains are highly similar (Fig. 1.6b-d). Maximum values of Thorpe scales and distinct patches of increased numbers of inversions often occurred in all three chains within time periods of 3 min. This similarity supports the conclusion that generation mechanisms acting on spatial scales exceeding the spacing of the chains are responsible for the generation of these density inversions (Fig. 1.7d).
The passage of linear HFW starts synchronously with the passage of the thermal front (Fig. 1.6g). However, density inversions first occur 1 h later together with the KWB (Fig. 1.6c). In contrast to the observations of Alford and Pinkel [Alford and Pinkel, 1995]
in weaker stratification, linear HFW cannot be the only mechanism generating the large- scale density instabilities in these measurements.
The passage of the steepened front of the Kelvin wave at the study site during period W1 is associated with a change in current direction (Fig. 1.3c, first arrow). The alongshore current velocities reach values of up to 0.2 m s−1 in the upper 10 m in depth (Fig. 1.6h) and impose a vertical shear over the water column. This shear is sufficient to decrease the corresponding Ri below the critical value of 0.25 for hours (Fig. 1.6g, blue areas).
Low Ri indicates that the linear HFW and the KelvinHelmholtz billows result from shear instabilities evolving in the strong Kelvin waveinduced background shear. The turbulent collapse of the billows leads to a transfer of energy from the basin-scale wave to turbulent scales.
During the passage of the NLIW1 and the 2ndM1, Ri is above the critical value, except at the edges of the thermocline near 5 and 20 m in depth, where the Kelvin waveinduced current shear is particularly strong (Fig. 1.6h). Overturns corresponding to these events seem to occur without being coupled to lowRi. The interaction of the background current with the current shear from linear HFW and NLIW can induce instabilities, whereas the background shear alone would be too weak to overcome stratification [Boegman, 2009].
Especially in layers with weak stratification, the shear from the linear HFW may be suffi- cient to induce turbulent overturns [Alford and Pinkel, 1995]. Unfortunately, the temporal
