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Internal waves and the generation of turbulence in the thermocIine of a large lake

Martina Preusse,a,* Frank Peeters,a and Andreas Lorkeb aEnvironmental Physics, Limnological Institute, University of Constance, Con stance, Germany blnstitute for Environmental Sciences, University of Koblenz-Landau, Landau, Germany

Abstract High-resolution thermistor chain data collected between 5 and 20 m in a large stratified freshwater lake (Lake Constance) at a water depth of 60 m reveal the frequent occurrence of large-amplitude (2: I-m) vertical density inversions that indicate overturns in the pelagic thermoc1ine. Velocity data collected simultaneously with the measurements indicate that the density inversions are mainly produced by shear instabilities. A comparison between the timing of the passage of the basin-scale internal and the density inversions demonstrates a pronounced phase relationship, implying that the processes leading to the occurrence of turbulence 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 billows. A particularly large number of density inversions was also associated with the passage of nonlinear high­ waves of large amplitudes. The density inversions typically occurred at the wave troughs, which indicates breaking of these waves. These observations indicate that self-induced shear generated by the basin­ scale and by high-frequency internal waves leads to a significant amount of turbulence and mixing in the pelagic thermoc1ine.

Diapycnal exchange in the seasonal thermoc1ine is scale or inertial waves are common in lakes (Thorpe et al. important for vertical transport of nutrients, plankton, 1972; Saggio and Imberger 2001) and in the (Fu and and (Eckert et al. 2002). It also affects , Holt 1982; Munk and Wunsch 1998). At the Iow-frequency which in turn influences the generation of turbulence from end of the wave spectrum, enhanced turbulence is coupled large-scale processes and thus provides a feedback mech­ with the passage of in the ocean (Garrett 2003; anism on the magnitude of mixing (Lewis et al. 1986; Wuest Rudnick et al. 2003) and steep-fronted basin-scale internal and Lorke 2009). Hence, mixing is of great importance in lakes (Macintyre et al. 2009). The energy cascade both for its ecological implications and for its relevance to from these low-frequency waves to turbulence is assumed environmental . The quantification of to be a major facilitator of mixing (Imberger 1998; Munk vertical mass fluxes based on measures of turbulence, and Wunsch 1998; Egbert and 2000). Considerable however, is difficult, since the intensity and occurrence of effort has been undertaken to understand and quantify the turbulent mixing is spatially and temporally highly variable elements of the energy cascade (Imberger 1998; Horn et al. (Ivey et al. 2008). 2001; Boegman et al. 2005a). Turbulent mixing in the stratified interior of and In particular, non linear high-frequency internal waves lakes has been observed (Ledwell et al. 1993; Rudnick et al. with large amplitUdes, generated in the process of 2003; Wunsch and Ferrari 2004) to be too weak to account degeneration of low-frequency waves, have received special for turbulent diffusivities estimated from basin-scale tracer attention (Farmer and Armi 1999; Boegman et al. 2005b; budgets. This mismatch is assumed to result from spatial Helfrich and Melville 2006). These nonlinear high-frequen­ heterogeneity of the mixing. Turbulence and mixing are cy waves (NLIW) are ubiquitous in lakes (Thorpe et al. increased near lateral boundaries, especially near the 1996; Boegman et al. 2003; Lorke et al. 2006) and in the benthic boundary, in comparison to the interior of the ocean (Orr and Mignerey 2003; Da Silva et al. 2009; Van waterbody (Ledwell et al. 1993; Goudsmit et al. 1997; Haren et al. 2009). They have been observed to propagate Wuest et al. 2000). But hotspots of turbulent dissipation for several kilo meters (Mourn et al. 2007) before they are also located in the above the benthic on sloping topography (Boegman et al. 2005b; Lorke 2007; boundary layer, as observed at nearshore slopes (Mourn et Boegman 2009) and thereby release their energy during the al. 2007; Van Haren 2009; Shroyer et al. 2010) and at ridges breaking process in the bottom boundary layer. Wave or sills (Rudnick et al. 2003; Lamb 2004; Macintyre et al. breaking is an important mixing mechanism in stratified 2009). waters (Imberger 1998; Garrett 2003; New et al. 2007). A key idea used to explain the observed variability of Instabilities associated with the propagation of large­ mixing was that enhanced turbulence may be caused by the amplitude non linear internal waves, their superposition, ubiquitous activity. Propagating internal and interaction with background currents also lead to wave waves with periods between the limit and basin- breaking in the open water (Mourn et al. 2003). The mixing process associated with internal waves is * Corresponding author: [email protected] thought to be caused by the turbulent collapse of density 2353 2354

inversions (overturns) (Thorpe 1977, 2005; Wuest and Lake Lorke 2009). Overturns result from physical processes generating shear instabilities (Howard 1961; Miles 1961; Fructus et aI. 2009), convective instabilities (Orlanski and Bryan 1969; Carr et aI. 2008), or changes in the effective rate of strain (Alford and Pinkel 2000). At present the evolution of these processes, including the occurrence and propagation of NLIW leading to the generation of overturns, cannot easily be reproduced in field-scale hydrodynamic models (Boegman et aI. 2001; Horn et aI. ""' 15 2001). Including internal wave-driven mixing in these 510 Q) coarse-grid models requires a parameterization of the g 5 generation of NLIW from the wave field in lakes and of .s the mixing efficiency of the various high-frequency is 0 processes. 5 5 1'0 km The objectives of this study are to identify the processes ~O leading to large-scale (~ I-m) density inversions in the 1Ol--~~_--":~ pelagic and to assess the relative importance 15 10 5 0 5 10 and efficiencies of these processes for the turbulent Distance (m) dissipation of energy in a lake. The following sections Fig. l. Lake Constance (4T83'99"N, 9°81'89"E) aim to (1) identify the large-scale generation mechanisms of (Wessels 1998), with the location of the wind station (gray circle); overturns (large Kelvin---Helmholtz billows generated at the moored thermistor chains CJ, Cz, and C3; and the ADCP in steepened front of a basin-scale Kelvin wave, shoaling of Lake Uberlingen. propagating NLIW, and interaction of high-frequency waves [HFW] with more weakly stratified areas) and (2) evaluate the relative importance of the mechanisms for the A 300-kHz acoustic Doppler pro filer (ADCP, generation of turbulence and mixing in the lake's interior RD-Instruments) was deployed at a distance of 10 m from by estimating the corresponding dissipation rates of the thermistors at a depth of 70 m. The upward-looking turbulent kinetic energy, diapycnal diffusivities, and ADCP averaged, internally, 55 individual profiles, taken at buoyancy fluxes. 2.72-s intervals, providing a sampling interval of 2.5 min. The vertical bin size was 0.5 m, resulting in a profiling Methods range spanning 2.7 m above the bottom up to 5 m below the lake surface. The three-dimensional current velocities Measurements-Measurements of temperature and cur­ were recorded in earth coordinates. Velocity estimates with rent velocity were performed in Lake UberIingen, a an associated error velocity exceeding 2 m S-I were subbasin of L~~e Constance (63 km long, 14 km wide) in disregarded. The error velocity is calculated by the ADCP Europe. Lake Uberlingen is a long (length, 20 km), narrow for each profile and depth bin using redundant data from a (mean width, 2.3 km) lake with maximum and mean depths fourth acoustic beam. The resulting mean error velocity of 147 m and 84 m, respectively. The sub basin is connected was 0.03 m S-I, with a standard deviation of 0.04 m S-I, to the deep (maximum depth, 252 m) main basin at the Sill indicating a rather poor signal-to-noise ratio in comparison of Mainau (Fig. I), where the depth at the thalweg reduces to the observed mean current speeds of around 0.05 ms-I. to 100 m (Wessels 1998). Wind speed and direction were measured at a land-based Three vertical thermistor chains were deployed from 24 meteorological station at the City of Con stance, approx­ June until 09 July 2008 at 60 m in depth, 0.4 km offshore, imately 1.5 km from the lake, by the German Meteorolog­ and about 6 km from the western end of the lake. To ical Service and were provided as hourly mean values. The observe the propagation of HFW, the chains were arranged data were assumed to be representative for the wind in a triangle with side lengths of 13, 21, and 32 m, directions and speeds at Lake Uberlingen (Zenger et aI. respectively (Fig. 1). The position of the chains was 1990). estimated using a Global Positioning System with an accuracy of ± 2.5 m. Each thermistor chain consisted of 14 Data analysis-The 10SC isotherm is used as a basis for to 15 individual temperature loggers (TR -1050, RBR), further analysis of the wave motions. Despite the strong which were located between 4 and 20 m in depth, with a temperature fluctuations caused by the internal waves, this vertical spacing of I m. The accuracy of the thermistors is isotherm could be continuously observed during the entire 0.002°C, and the sampling interval was set to 1 s. The duration of the experiment. c, direction of internal clocks of the loggers were synchronized at the phase propagation Cdin and wave length I, of HFW are beginning of the sampling period and showed differences of estimated by analyzing the band pass-filtered (Butter­ up to 8 s at the end of the deployment. The corresponding worth, lower and upper cutoff 1/30 min-I and time lags were assumed to depend linearly on time, and I min- I, respectively) ID.5°C isotherm displacements each temperature series was replaced by a corrected time observed at the three thermistor chains, following the series using linear interpolation. method of Ufford (1947), as follows: 2355

a b c 5 Cl

. inversions • inversions • inversions

I unresolved a 10 ~s:: 0 ;S 0- Q) Cl 15

8 9 10 11 9 10 11 9 10 11 Temperature (0C)

Fig. 2. Temperature profiles with density inversions (black squares) of chains Cl. C2, and C3 observed at the indicated times on 08 July 2008. Arrows indicate unresolved inversions with an associated temperature difference of < O.oI T.

12 X tl2 (A.) temperature loggers, displacements are only considered for ---cos 'I' further analysis if the difference between the measured ' ) 13 X tl3 tan (C dir = :c!...... ;...:.;s~i-n-(

o 2.5 5 msl ~I

>-l (l) 3 "0 ..,(l) '"' (l)~ ----0 ---n

.s n ~ fr (l) Cl la

28 June 01 July 04 July 07 July 2008 Fig. 3. Overview of field data of the entire sampling period. (a) Vector plot of wind speed (l-h average); reduced wind speeds are colored blue. (b) Five-minute average of temperature at chain C2 with a OSC contour interval. (c) Twenty-minute running average of alongshore current velocity. Arrows mark the occurrence ofNLIW and current fronts. (d) Twenty-minute running average of cross- current velocity.

The rate at which APEF is dissipated in the process of are estimated by using a constant mixing efficiency, Yl11ix» gravitational adjustment depends on stratification, which is of 0.2 (Osborn 1980). APEF, 1::, K=, and", are given as described by the buoyancy frequency, N, corresponding to arithmetic averages over the vertical measuring range. the stable density profile. Under the assumption of Instabilities in stratified flows typically arise in regions isotropic turbulence and a steady-state balance between where the Richardson number, Ri, falls below a critical production and dissipation of potential energy, Dillon and value of 0.25 (Howard 1961 ; Miles 1961; Thorpe 2005), Park (1987) found experimentally that the rate of where dissipation of APEF in the seasonal thermoc1ine is . N2(z) proportional to the APEF with the constant of propor­ RI = --.--;:----'-'-,----;:- (5) dV tionality of 4.8 N. This proxy is used to estimate the (dUId z)2 + ( ldz)2 dissipation rate of turbulent kinetic energy, e, with U and v as alongshore and cross-shore velocity, e [W kg - I] :::::4.8 x N x APEF (3) respectively (Howard 1961; Miles 1961). Ri is estimated by The lower detection limit of this method can be estimated using a 2.5-min average of N and order 3 polynomial fits of as Bb = 10- 9 W kg- I by considering an inverse density the vertical profiles of horizontal current velocity. profile corresponding to a single I -m overturn with a temperature inversion of 0.01 dc. Results N is estimated by applying 2-m centered differences (and backward or forward differences on the edges) to sorted Degeneration of the Kelvin wave- An overview of the and 30-s- averaged temperature profiles. The diapycnal field data observed during the entire measuring period is eddy diffusivity K= and the buoyancy flux Jb, given in Fig. 3. The wind field (Fig. 3a) is dominated by e weak to moderate (0.3 to 6.4 m S- I) westerly winds. The Kz=Ymix N2 low-frequency oscillations of the depths of isotherms (4) (Fig. 3b) are caused by a basin-scale internal seiche with an amplitude exceeding 15 m. The seiche is influenced by 2357

-0 - along-shore 101 ~ 6 &J 10 -Lmax - cross-shore ~ g "0'" (') .0 '" .U) 104 . 10 min ~ .g 10° ;;: i :::l'" ] q''" ~ 10 2 ~ ...... "' .. _-_...... ~ '.:;;;;;;;;;;;;. 10,1 Ul{, ~ 95%. "-- . ---_. ---- -_.- --..: ::r: p.. 100="-".=.=':':;'=:-_~_~;--__~-;;-_-IL,._~.,.- ___-r-:--,, ___ ~_-' N,:.. 5 3 10,5 10-4 10 10'4 10 Frequency (Hz)

Fig. 4. Power spectra of (a) 1O.5°C isotherm depth fluctuations and Lmax (chain Cl) and (b) alongshore and cross-shore current velocity at 30.8 m in depth. Confidence at the 95% level is indicated by dashed lines. Note the different scaling of the vertical axis.

Earth's rotation and forms a basin-scale internal Kelvin 10-3 Hz (5- to 20-min periods) characterizes propagating wave with a period generally ranging from 3 to 5 d, internal waves. Current velocity fluctuations were evaluat­ depending on stratification (Biiuerle et al. 1998; Boehrer et ed at 30-m depth below the fluctuating depth of current al. 2000; Appt et al. 2004). Four full Kelvin wave cycles inversion. All three peaks were observed in the power were recorded (Fig. 3b, PI-P4). After cessation of moderate spectrum of the alongshore velocity fluctuations (Fig. 4b), westerly winds on 04 July and 08 July (Fig. 3a, bars 1 and while the only significant peak corresponds to the 4-h 2), the Kelvin wave developed steep thermal fronts period. The power spectrum of the cross-shore velocity (Fig. 3b, ends of Kelvin wave cycles P3 and P4), indicating does not increase in spectral power toward lower frequen­ the non linearity of the basin-scale seiche. The steep thermal cies but also shows small peaks at periods of 4 hand 10 min. fronts were associated with steep alongshore current fronts The power spectrum of the maximal Thorpe displacements, (Fig. 3c, arrows) and NLIW with amplitudes of up to 15 m Lmax, again shows all three peaks, and the high-frequency and strong current velocities of up to 0.15 m S-I (Fig. 3c, peak at a period of 16 min is significant. The latter

NLIW1 and NLIW2). observation indicates periodic occurrence of turbulence The power spectrum of the fluctuations of isothermal correlated with the occurrence of these waves (Fig. 4a). depth has three significant peaks at frequencies corre­ A comparison between the passage of HFW (Fig. Sa), sponding to periods of 3 d, 4 h, and 10 min (Fig. 4a). The the basin-scale internal Kelvin wave (Fig. 5b), and the spectral peak at the lowest frequency is related to the occurrence of density inversions reveals a pronounced Kelvin wave with a period of 3 d throughout the time of phase relationship. The time series of the depth of the our measurements. The wave with a period of 4 h is a cross­ lOSC isotherm was low-pass filtered (Butterworth of shore oscillation, which is strictly limited to Lake Uberlin­ second order, cutoff frequency: 1124 h- 1) to identify the gen (Biiuerle 1994). The broad high-frequency peak around Kelvin wave in the wave field. The square of the band pass-

tJ r:c--:---:-----:---:--...,.0.2 7.S~· Cl ~ o 5 (') S 0.1& 2.5 a ~. § s (') 2.5"'::: o o(') Si 2 t""" (') 0' l.5~ a Kelvin - HFW \ I 1'-' wav~ - 0.1 - Kelvin wave - LT . v 0.5 .. -is ~=r--~~~-=~--~~~~ 28 June 01 July 04 July 07 July -150 -100 -50 0 50 100 150 2008 Lags (h) Fig. 5. Phase relation between inversions and internal waves (chain Cl)' (a) Squared band pass-filtered 10SC isotherm displacements (HFW) and dissipation rates. (b) Original and low-pass filtered (cutoff frequency 1124 h- I) 10SC 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. 23S8 filtered displacement of the 10SC isotherm was used as a matching the definition of the NLIW category were found measure of the intensity of HFW. The inversions are during both periods W, and W2 (Fig. 7a: NLIW2• NLIW3, particularly frequent shortly before the trough of the and NLIW4). The NLIW during W2 both led (NLIWz) and Kelvin wave passes (Fig. Sb) and occur together with the followed (NLIW3 and NLIW4) the entering surge (Fig. 3c, passage of the internal front. Maximal values of Thorpe third arrow) propagating in an alongshore direction displacement increase with the amplitude of the basin-scale (north-west). Amplitudes and periods of the seiche (Fig. Sb) or the intensity of HFW (Fig. Sa). The depend on their location relative to the fronts: the wave cross-correlation (Fig. Sc) between Thorpe scales and packet NLIW2 propagates with a phase velocity of Kelvin wave is a periodic function of lag time with a 0.17 m s-', a period of 17 min, and a of period of about 3 d, the period of the Kelvin wave. The 170 m. The estimated direction of the phase velocity in correlation function reaches a local minimum near the comparison with the direction of current velocities under -IS-h lag, indicating a coupling between the passage of the the wave troughs measured by the ADCP (not shown) Kelvin wave front and the occurrence of overturns. indicates that phase velocity and wave orbital velocity Maximal correlation coefficients reach values of almost under the wave trough have opposite directions, which is in 0.2 and are highly significant, according to the Student's agreement with theoretical models (Vlasenko et al. 2000). t-test (a < 0.001). Similar magnitudes, periods, and phase The wave packets NLIW 3 (wave lengths - 170 m) and lags can be found in the cross-correlation between the NLIW4 (wave lengths - 7S m) propagate with phase Kelvin wave and the intensity of HFW (Fig. Sc), indicating velocities of 0.22 m S-1 and 0.17 m S-I, respectively. They a synchronized occurrence of HFW and density inversions have smaller amplitudes and periods (NLIW3: 13 min; at the steepened front of the Kelvin wave, which indicates a NLIW4: 7 min) than does NLIW 2. cause-effect connection (Alford and Pinkel 2000). The next hot spots of density inversions (Fig. 6a: 2ndM,) are surrounded by opening and closing isotherms with an Processes associated with the occurrence of density irregular period between 1 and 2 h (Fig. 6g), which are inversions--In the following analyses, four different condi­ accompanied by HFW with amplitudes smaller than 3 m. tions under which density inversions occurred are distin­ Such events associated with the second vertical mode-type guished and the observations are grouped accordingly into isotherm displacements are grouped into the 2ndM catego­ four categories of events (Figs. 6a, 7a): Kelvin wave shear­ ry. Turbulent patches localized in the center of 2nd M were nd induced billows (KWB), weak stratification (WStr), non­ also found during observation period Wz (Fig. 7a: 2 M2). linear internal waves (NLIW), and second mode waves (2ndM). Generation mechanisms of density inversions-The medi­ During the first observation period, W, (Figs. 3b, 6), the an duration of individual density displacements at each first two turbulent patches in Fig. 6g are associated with thermistor chain was 4 s. However, maximum durations of oscillations near the buoyancy period 2nN-' ;:0: 4 min, inversions reached between 20 and 418 s, depending on the which develop into billow-like structures. The billows occur depth in the water column. The magnitude and the time at the depth of the thermocline (Fig. 6i, bars) and the series of the Thorpe scales at the three thermistor chains are Kelvin wave-induced current shear (Fig. 6h, depth of highly similar (Fig. 6b-d). Maximum values of Thorpe current shear at 14:00). Unstable oscillations, caused by scales and distinct patches of increased numbers of Kelvin wave-induced shear and developing into billows, inversions often occurred in all three chains within time are grouped into the KWB category. Such instabilities were periods of 3 min. This similarity supports the conclusion only observed during the passage of the internal front in that generation mechanisms acting on spatial scales period W,. exceeding the spacing of the chains are responsible for the The following patches of density inversions (Fig. 6a: generation of these density inversions (Fig. 7d). WStr, and WStr2) are associated with a thermocline-wide The passage of linear HFW starts synchronously with weakening of the stratification (Fig. 6f,g) over the course of the passage of the thermal front (Fig. 6g). However, density 6 h as well as long-lasting billows in the alongshore current inversions first occur 1 h later together with the KWB velocity at the bottom part of the thermocline (Fig. 6h, (Fig. 6c). In contrast to the observations of Alford and white arrows). These events are collected into the WStr Pinkel (2000) in weaker stratification, linear HFW cannot category. be the only mechanism generating the large-scale density After the weakening of stratification, NLIW (Fig. 6a: instabilities in these measurements. NLIW,) with amplitudes of about S m (Fig. 6g) propagate The passage of the steepened front of the Kelvin wave at in the south-east alongshore direction, with a period of the study site during period W, is associated with a change 12 min, a phase velocity of 0.28 m s-', and a wave length of in current direction (Fig. 3c, first arrow). The alongshore - 200 m. These waves follow an internal current front current velocities reach values of up to 0.2 m s-, in the propagating in an alongshore direction toward the south­ upper 10 m in depth (Fig. 6h) and impose a vertical shear east (Fig. 3c, second arrow), probably the front of the over the water column. This shear is sufficient to decrease Kelvin wave that was reflected at the western boundary of the corresponding Ri below the critical value of 0.2S for Lake Con stance (compare to Appt et al. [2004]). NLIW hours (Fig. 6g, blue areas). Low Ri indicates that the linear with periods between 30 min and amplitudes HFW and the Kelvin-Helmholtz billows result from shear exceeding 3 m are commonly associated with density instabilities evolving in the strong Kelvin wave-induced inversions and represent the NLIW category. Wave packets background shear. The turbulent collapse of the billows 2359

a

2 b Lr, 1 8 Chain C l • ---E c 0 2 8 0 0', ~ 1 I.• , .., 0.. Chain C2 Cl'" 2 d

10

15

- 0.05 20 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 02:00 04:00 06:00 5

10

15

20 14:10 Time (hh:mm) 14:15

Fig. 6. Details of observations during period W I (cr. Fig. 3). (a) Category (b- d) £Or at chains C l. C 2, and C3. (e) e at chain C l (I) I-s temperature (chain C l) from loggers starting at 5 m in depth with a 3-m depth increment. (g) One-second isotherm response (chain Cl) with 1°C temperature intervals (lines), occurrence of temperature inversion (squares), and Ri < 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 Q.I m S- I . White arrows indicate Kelvin- Helmholtz billows (zigzag pattern in the current velocity). (i) Zoom into the area marked by the red arrow in panel g. Contour lines show Kelvin- Helmholtz billows (white bars) and density inversions (squares). 2360

5

13

12 10 """'E 11 '-' ...c: i5.. 10

10

]: 30 -:El o0.. Cl 50

18:00 20:00 22:00 00:00 02:00 04:00 06:00 08:00 10:00 Time (hh:mm)

10 9 8 7

19:20 20:00 20:40 19 :45 19:50 19:55 20:00 20:05 19:58 20:02 Time (hh:mm)

Fig. 7. Details of observations during period W2 (cf. Fig. 3). (a) Category. (b) Occurrence of density inversions (squares) associated with isotherm depth fluctuations (chain C2, l-s resolution). (c) e at chain C2. (d) Five minute and 2-m running average of alongshore current velocity and IO .5°C isotherm of chains Cl (red), C2 (blue), and C3 (black). (e) One-second temperature (chain C2)with O.l oC contour interval (zoom into the square of panel d, showing NLIW2). (I) The trough of a (zoom into the square of panel e). (g) Density inversions are represented by overturns in the troughs (zoom into the square of panel I). Note the different and nonlinear color scaling. leads to a transfer of energy from the basin-scale wave to wave- induced current shear is particularly strong (Fig. 6h). turbulent scales. Overturns corresponding to these events seem to occur nd During the passage of the NLIWI and the 2 M I , Ri is without being coupled to low Ri. The interaction of the above the critical value, except at the edges of the background current with the current shear from linear thermocline near 5 and 20 m in depth, where the Kelvin HFW and NLIW can induce instabilities, whereas the 2361

Table 1. Occurrence of overturns, total dissipation of potential energy fluctuations (fedt), and logarithmic mean values «8») and variances (0'1) of 8 (neglecting zero values) averaged over the total vertical measuring range for different events occurring at the steepened front of the Kelvin wave.

Occurrence (%) I 8 dt (J kg- I) <8) (W kg-I) 0'; (W kg- l )2 Total time series 100 4.5XIO-3 6.5xI0-s 0.31 KWB 4 9.8X 10-4 3.3XI0-7 0.34 WStr 12 4.2XlO-4 6XlO-s 0.25 NLIW 27 l.l X 10-3 6.9XlO-8 0.27 2ndM 29 l.3XI0- 3 7.9xlO-8 0.26 4 NLIW 2 8 SAx 10- 8.3XI0-s 0.5

background shear alone would be too weak to overcome The measured logarithmic mean = 3.3 X 10-7 W kg-I stratification (Boegman 2009). Especially in layers with of KWB is four times higher than even that for the shoaling weak stratification, the shear from the linear HFW may be NLIW2, indicating that the highest dissipation rates are sufficient to induce turbulent overturns (Alford and Pinkel caused by the large billows. However, during the total 2000). Unfortunately, the temporal resolution of 2.5 min measuring period the different processes, KWB, NLIW, and the spatial resolution of 0.5 m are not sufficient to and 2ndM, rather dissipated a comparable amount of resolve the fine-scale shear induced by the linear HFW and potential energy (Table 1, column 3), between 9.S x 10-4 most NLIW. Nevertheless, the observation of intense and 1.3 x 10-3 J kg-I, because the processes occurred with overturns during WStr and 2"dM supports the hypothesis different frequency (e.g., only a small amount of large that the interaction between HFW-induced shear and billows [Fig. 5g] was generated in the shear induced by the weaker stratifications generates overturns. front of the Kelvin wave during KWB, resulting in a The wave-induced current velocity of the NLIW2 could comparatively low density of overturns of 7.6% [Table 2]). be observed down to the bottom of the water column The arithmetic means, including zero values of dissipa­ (Fig. 7d), indicating that the waves are subject to shoaling. tion, diapycnal eddy diffusivity, and buoyancy flux, for the Most of the energy contained in NLIW is assumed to be various processes and the various passages of the Kelvin released during shoaling (Imberger 1995; Boegman 2009). wave through Lake UberIingen (Kelvin wave cycles P], ... , However, the NLIW energy may not be exclusively P4 in Fig.3b) are shown in Table 2. Period Ps starts dissipated in the bottom boundary layer. This is indicated immediately after period P 4, in which zero dissipation rates by Fig. 7g, in which the overturns occur in the pelagic were added to complete a full Kelvin wave cycle with a thermocline. Density inversions occur predominantly in the period of 3 d (Fig. 4a). Mean values are calculated as wave troughs, where Ri was observed to fall below the averages over the observed 5- to 20-m depth of the water critical value of 0.25 (not shown). Overturns, however, can column and refer to an arbitrary I-kg water packet in the stilI be observed minutes after a trough has passed the thermocline. With the exception of the extremely low value thermistor chain (Fig. 7b). The occurrence of density in cycle P3, the observed mean turbulent diffusivities are at inversions increases during the passage of the wave packet least one order of magnitude larger than the molecular to a mean value of 2S%, whereas the mean displacement diffusivity of heat (1.5 X 10-7 m2 S-I; Pond and Pickard length is only minimally increased, by about 0.2 m. Similar 1975). Dissipation rates (between 2 x 10-I I and I x 10-8 to the echo sounder observations of Mourn et aL (2003; W kg-I), vertical diffusivities (between I x 10-7 and 4 x their figs. 6, 9, 11) of breaking solitary waves and 10-5 m2 S-I), and upward flux of density (between 3 X laboratory experiments (Fructus et aL 2009), the overturns 10- I2 and 2 x 10-9 W kg- I) tend to be highly variable. The indicate self-induced shear instabilities. Subsequent mixing highest values were observed during the passage of the is implied by the slightly lower temperature of 10 0 e to surge. 1O.3°e (green color), instead of 1O.6°e (light orange color), in 11- to 19-m depths and between 19:59 hand 20:00 h in Discussion Fig. 7g and is quantified by the especially high mean diapycnal eddy diffusivity related to wave packet NLIW2 A strong coupling was observed between the occurrence (Table 1). of high-frequency processes and the passage of the steep­ ened front of the Kelvin wave in Lake UberIingen (Fig. 5b). Turbulence and mixing-The APEF varies between 10-4 Energy is known to flow from basin-scale to dissipative­ and 10-7 J kg- I during the entire period of observation scale processes (i.e., from processes with low frequencies (Fig. Sa). The highest values of observed APEF are induced [Kelvin wave, cross-shore oscillation] via high-frequencies 1ld by KWB and NLIW2• Dissipation rates during the total [propagating linear and nonlinear waves, 2 M, KWB] to observation time and during the different processes (for turbulence [Imberger 1995]). The occurrence of high­ instance, KWB and NLIW; Fig. Sb) are approximately frequency processes at the internal front thus indicates a distributed log normally and vary between 10-10 and 10-5 hot spot of the degeneration of the Kelvin wave. Enhanced W kg-I. The observed logarithmic mean and standard dissipation rates, on the other hand, indicate hot spots of deviation of e vary among the different events (Table 1). the degeneration of the higher-frequency processes of 2362

KWB, NLIW, and 2ndM. The distribution of dissipation rates indicates that e is higher within large billows than within shoaling NLIW. But as a result of the Iow density of large billows during KWB both processes are equally important for energy dissipation and buoyancy flux at the front of the Kelvin wave, as the arithmetic mean values indicate. Moreover, the comparison of mean values indicates that both processes induce higher If and :rb than all other observed high-frequency events. Compared to the respective mean values, observed dissipation rates, diffu­ sivities, and buoyancy fluxes are increased by up to four orders of magnitude during the different generation mechanisms of overturns. Overturns are generated via the high-frequency processes, by current shear associated with the Kelvin wave front (Fig. 6g,h), by self-induced shear of NLIW (Fig. 7), and by the suggested interaction of linear HFW and weaker stratification (Fig. 6g). Because the high-frequency processes generating the overturns are coupled with the steep front of the Kelvin wave, high mixing rates can be expected in all areas through which the nonIinear internal surge passes. The estimates of turbulent dissipation rates, vertical mixing, and buoyancy flux are probably valid for the entire nearshore zone of Lake Uberlingen, since the internal surge is known to exist until the second passage of the Mainau Sill on its way back to the main basin (Appt et aI. 2004). Moreover, the surge has also been observed in the interior of Lake UberIingen, which is small compared to the central part of Lake Constance and therefore is not as strongly affected by the (Appt et aI. 2004; Lorke et al. 2006). Of all high-frequency processes generated at the steepened wave front, only the shoaIing NLIW2 observed during the Kelvin wave cycle Ps seemed Fig. 8. Thorpe variables (a) APEF, (b) distributions of to be influenced by the lake-bed topography. While dissipation rate estimates induced by NLIW (black squares) and dissipation rates within the pelagic interior are likely to KWB (open circles). The shaded area shows the distribution of all be somewhat weaker than those observed here, we assume dissipation rates (total). that the estimates associated with the other processes (e.g., Kelvin wave cycles PI-P4) can be applied to the entire interior of Lake UberIingen. To date, no statistics concerning the occurrence of surges and the higher-frequency events generated at the surge have

Table 2. Arithmetic means of 8, Ko, and 1;, and bootstrap 95% confidence limits (10,000 resamples) averaged over different events occurring at the steepened front of the Kelvin wave, individual (Pi, i = 1, ... ,5) or combined (U;i~ I Pi> n = 4, 5) observation periods PI-PS (compare Fig. 3b) and the total vertical measurement range. The first two columns provide the lengths of the observation periods (duration) and the percentage of unstable stratified profiles (overturns).

Duration (h) Overturns (%) s(W kg-I) Ko (m2 S-I) J b (W kg-I) KWB 4 7.6 (6.8±0.8)xI0-S (7.9±0.8)X 10-5 (1A±0.2)x 10-8 Wstr 6.7 13 (1.7±0.2)XIO-S (7A±OA)X 10-5 (3.5±0.3)X 10-9 NLIW 12.1 16.5 (2.5±0.I)X 10-8 (2A±0.I)X 10-4 (5.0±0.3)X 10-9 2nd M 12.2 14.9 (2.3±0.I)XI0-S (1.2±0.I)X 10-4 (4.6±0.3)x 10-9 8 4 8 NLIW2 2 28.1 (7.5±0.7)X 10- (6.6±0.6)X 10- (1.5±0.2)X 10- PI 40 I (1.0±0.I)xl0-9 (5.1 ±0.3)X 10-6 (2.0±0.2)X 10- 10 10 6 P2 68 0.7 (4.5±0.3)X 10- (3.3 ±0.2)X 10- (8.9±0.6)X 10- 11 7 P3 90 0.03 (2.2±0.5)X 10- 11 (l.3±0.3)X 10- (4.3± 1.0) X 10- 12 P4 80 4.8 (1.1 ±O.1)X 10-8 (3.7±0.I)X 10-S (2.3±0.1)X 10-9 Ps 72.8 3.6 (3.8±0.2)X 10-9 (4.2±0.2)X lO-s (7 .6±OA)x 10- 10 9 5 10 U:~l Pi 278 1.7 (3.5±0.2)X 10- (1.2±0.0)X 10- (7.1 ±0.3)X 10- U;~l Pi 350.8 2.1 (3.6±0.I)X 10-9 (1.8±0.I)X 10-5 (7.2±0.3)X 10- 10 2363

been available, other than the general observation (Lorke et Since degeneration of large-scale internal waves via al. 2006) that HFW occur frequently in Lake Constance. HFW is a phenomenon occurring in various lakes (Thorpe Therefore, observed dissipation rates and diffusivities were et al. 1972; Horn et al. 2001; Boegman et al. 2003) and also listed for single Kelvin wave passages in Table 2, which in the ocean in the context of internal tides (Lamb 2004), it allows an estimation of the total averages according to the can be expected that the spatial and temporal highly actual frequency of occurrence of NLIW and KWB. The variable occurrence of turbulence and mixing in the pelagic different duration of the overturn-generating mechanisms thermocline is a common feature (Rudnick et al. 2003; Ivey in the 2-week observation period presented here led to et al. 2008; Macintyre et al. 2009). Mixing in the comparable rates of energy dissipation for the different thermocline of the open water leads to vertical fluxes in processes (Table 1). the lake's interior. These intermittent fluxes of, for The actual values are rough estimates of the APEF, example, nutrients into the epilimnion can locally enhance dissipation rates, and eddy diffusivities associated with the plankton growth and may lead to the generation of overturn-generating processes. Absolute values as well as a plankton patchiness (Mackas et al. 1985). Such patchy comparison of values therefore should be considered a distribution of prey organisms can further affect the tendency. Turbulent patches in the pelagic zone are known intensity and dynamics of predator-prey interactions and to be separated by periods of molecular (Imberger enhance ecosystem productivity (Rovinsky et al. 1997). 1998) and were assumed to have dissipation rates of e = O. In summary, periodic occurrence of turbulence and During the high-frequency processes, the occurrence of mixing was observed in the pelagic thermoc1ine of Lake overturns is enhanced (30% in NLIW2, but only 2% in the Uberlingen. High-frequency processes such as propagating entire observation period). Including an additional number linear HFW and NLIW, 2ndM, and KWB are generated of overturns (the possible unresolved 70% in all high­ within a degeneration process at the steepened front of the frequency processes) with dissipation rates of e = 10-8 W basin-scale Kelvin wave. In this study, turbulence was kg-I (measured 5% percentile of the log-normal distribu­ observed to be produced via these high-frequency process­ tion) above the detection bound eb = 10-9 W kg- 1 would es, by current shear at the current front, and by the self­ add another 4.7 X 10- 10 W kg- 1 to the arithmetic mean. induced shear of internal waves with large amplitudes. Filling all interims with e = 0 W kg-I leads already to an Overturns indicating turbulence were also observed during estimate of e = 3 X 10-9 W kg-I. time periods in which HFW with smaller amplitudes The observed mean diffusivity of 1.2 X 10-5 m2 S-I in propagated in weak stratification. Although for these the pelagic zone is in good agreement with the K= = 3.4 X events the Ri did not fall below the critical value of 0.25 10-5 m2 S-1 estimated with basin-scale tracer measure­ (when an averaging interval of 2.5 min was applied), the ments (Maiss et al. 1994) in the thermocline of Lake shear associated with the HFW, which was not resolved in Constance, indicating that a considerable percentage of the the current velocity measurements, is assumed to be average diffusivities can be attributed to the pelagic zone. sufficient to cause instabilities under weak stratification. The arithmetic mean of nearshore diffusivities of 2 X In addition to shoaling of NLIW, the turbulence-generat­ 10-4 m2 S-I estimated by Lorke (2007) in the thermocline ing mechanisms identified at the internal front of a basin­ of Lake Constance at the benthic boundary is an order of scale Kelvin wave are likely to occur independently of the magnitude higher, indicating enhanced diapycnal mixing at topography, everywhere in the interior of Lake Uberlingen. the boundaries. But at the boundaries, where the stratifi­ The observations in this study indicate that the turbulence cation is weak, the large diffusivities do not result in produced within the stratified interior of the lake leads to similarly large buoyancy fluxes, as observed in the open significant buoyancy flux in the pelagic thermocline, where water. Processes generating overturns in the pelagic zone, the specific contribution depends on the process generating on the other hand, lead to mixing in the strongly stratified the turbulence. Highly variable spatial and temporal thermocline, and, thus, smaller diffusivities are associated occurrence of turbulence and mixing in the pelagic with large buoyancy fluxes. thermocline is likely to be a common feature in many lakes. Maximum estimates of dissipation rates within turbulence hot spots such as KWB exceed 10-5 W kg-I, which is one Acknowledgments order of magnitude higher than the maximum rates We thank E. Biiuerle and H. Hofmann for helpful discussions estimated by Kocsis et al. (1998), also in the pelagic and the technicians J. Halder and A. Sulger for support in the field. thermocline of Lake Constance. The very high temporal We gratefully acknowledge the very helpful comments of two variation in the energy dissipation (Table 2) indicates that anonymous reviewers. This research was funded by the German most of the mixing in the thermocline of the lake's interior Research Foundation (Collaborative Research Centre 454). occurs only during a small time period within a Kelvin wave cycle. Hence, sporadic measurements of energy dissipation References rates or even a high sampling intensity during 3 d can miss ALFORD, H., AND R. PINKEL. 2000. Observations of overturning in the most extreme turbulence events. The arithmetic means of the thermoc1ine: The context of ocean mixing. J. Phys. Ocean the observed energy dissipation rates from the pelagic 30: 805-832, doi: 1O.117511520-0485(2000)030<0805:000ITT> thermocline are in good agreement with the typical 2.0.CO;2 dissipation rates observed in the basin-scale metalimnion ApPT, J., J. IMBERGER, AND H. KOBUS. 2004. Basin-scale motion in oflakes (between 10- 10 and 10-8 W kg-I; Wuest and Lorke stratified upper Lake Constance. Limnol. 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