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 temperature 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 Kelvin wave 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- frequency 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 seiche 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 important for vertical transport of nutrients, 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 mech- anism on the magnitude of mixing (Lewis et al. 1986; Wuest and Lorke 2009). Hence, mixing is of great importance both for its ecological implications and for its relevance 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 (Ledwell et al. 1993; Rudnick et al.
2003; Wunsch and Ferrari 2004) to be too weak to account for turbulent diffusivities estimated from basin-scale tracer budgets. This mismatch is assumed to result from spatial heterogeneity of the mixing. Turbulence and mixing are increased near lateral boundaries, especially near the benthic boundary, in comparison to the interior of the waterbody (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 nearshore slopes (Mourn et al. 2007; Van Haren 2009; Shroyer et al. 2010) and at ridges or sills (Rudnick et al. 2003; Lamb 2004; Macintyre et al.
2009).
A key idea used 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 basin-
* Corresponding author: Martina.Preusse@uni-konstanz.de
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 Iow-frequency end of the wave spectrum, enhanced turbulence is coupled with the passage of tides in the ocean (Garrett 2003;
Rudnick et al. 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. 2005a).
In particular, non linear high-frequency internal waves with large amplitUdes, generated in the process of degeneration of low-frequency waves, have received special attention (Farmer and Armi 1999; Boegman et al. 2005b;
Helfrich and Melville 2006). These nonlinear high-frequen- cy waves (NLIW) are ubiquitous in lakes (Thorpe et al.
1996; Boegman et al. 2003; Lorke et al. 2006) and in the ocean (Orr and Mignerey 2003; Da Silva et al. 2009; Van Haren et al. 2009). They have been observed to propagate for several kilo meters (Mourn et al. 2007) before they shoal on sloping topography (Boegman et al. 2005b; Lorke 2007;
Boegman 2009) and thereby release 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 non linear internal waves, their superposition, and interaction with background currents also lead to wave breaking in the open water (Mourn et al. 2003).
The mixing process associated with internal waves is thought to be caused by the turbulent collapse of density 2353
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2354
inversions (overturns) (Thorpe 1977, 2005; Wuest and 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.
2001). Including internal wave-driven mixing in these coarse-grid models requires a parameterization of the generation of NLIW 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 (~ I-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 following sections aim to (1) 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 NLIW, and interaction of high-frequency waves [HFW] with more weakly stratified areas) and (2) evaluate the relative importance 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.
Methods
Measurements-Measurements of temperature and cur- rent velocity were performed in Lake UberIingen, a subbasin of L~~e Constance (63 km long, 14 km wide) in 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. I), 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 in depth, 0.4 km offshore, and about 6 km from the western end of the lake. To observe the propagation of HFW, the chains were arranged in a triangle with side lengths of 13, 21, and 32 m, respectively (Fig. 1). The position of the chains was estimated using a Global Positioning System 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 in depth, with a vertical spacing of I 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.
""' 15
510
Q)g
5is .s
05
Lake
1Ol--~~_--":~
15 10 5 0 5 10 Distance (m)
~O
5 1'0 kmFig. l. Lake Constance (4T83'99"N, 9°81'89"E) bathymetry (Wessels 1998), with the location of the wind station (gray circle);
moored thermistor chains CJ, C z, and C3 ; and the ADCP in Lake Uberlingen.
A 300-kHz acoustic Doppler current pro filer (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 min.
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-I 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-I, with a standard deviation of 0.04 m S-I, indicating a rather poor signal-to-noise ratio in comparison to the observed mean current speeds of around 0.05 ms-I.
Wind speed and direction were measured at a land-based meteorological station at the City of Con stance, approx- imately 1.5 km from the lake, by the German Meteorolog- ical Service 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 aI.
1990).
Data analysis-The 10SC isotherm is used as a 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 propagation Cdin and wave length I, of HFW are estimated by analyzing the band pass-filtered (Butter- worth, lower and upper cutoff frequencies 1/30 min- I and I min- I, respectively) ID.5°C isotherm displacements observed at the three thermistor chains, following the method of Ufford (1947), as follows:
5 a Cl b c
.
inversions • inversions • inversionsI unresolved
10 a
~ 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.) - - - c o s 'I'
( ' ) 13 X tl3
tan C dir = :c!...;...:.;s~i-n-(
<fo-)--
, T x/2 ,
1.= --cos(c dil')
tl3 },
c=T
(1 )
where
h<fo
are parameters describing the geometry of the triangular mooring (Fig. 1) and C'dir is the direction of phase propagation relative to the triangle. Wave period T and propagation time tij of a wave from chain i to chain j are determined by auto- and cross-correlations with maximum time lags of 20 and IQ min, 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 density inversions in the water column. Since density in Lake Con stance 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 of LT 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 (dTldz)(t) at any time t is always larger than O.03°C m-I and 0.49°C m-I, on average, this threshold is sufficient for resolving the temperature structure in the water column, and the accuracy of LT is rather limited by the vertical resolution of the measure- ments (Stansfield et al. 2001). Likewise, inversions occur frequently and with comparable magnitude at almost all depths at all three chains. On average, 2% of all temperature profiles are unstably stratified, which equals 4.8 h of unstable stratification during the measurement period. Maximal displacement lengths reach values of up to 6 m. Examples of temperature profiles with inversions are shown in Fig. 2.
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 (DilIon and Park 1987). The APEF can be estimated by comparing observed and sorted density profiles. As recommended by Dillon and Park (1987) for fixed measurement intervals, we use the formula
-1 g
LM
APEF (J kg 1 ~ -2 - d"I'J.PII
PoM 11=1 (2)
for the estimation of the APEF, where g is the gravitational constant, M is the number of temperature measurements within each profile, Po is the background density, I'J.Pn is the Thorpe density fluctuation, and d" is the Thorpe displace- ment at depth z". The calculated values are rough estimates of APEF as a result of the limited spatial resolution of the temperature measurements.
2356
o
5ms
l~
I2.5
>-l
(l)
"0 3
(l) ..,
'"'
~ (l)
---- n
0---
.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-shore current velocity.
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 (1987) found experimentally that the rate of dissipation of APEF in the seasonal thermoc1ine is proportional to the APEF with the constant of propor- tionality of 4.8 N. This proxy is used to estimate the dissipation rate of turbulent kinetic energy, e,
e [W kg- I] :::::4.8 x N x APEF (3) The lower detection limit of this method can be estimated as Bb
=
10-9 W kg-I by considering an inverse density profile corresponding to a single I -m overturn with a temperature inversion of 0.01d c.
N is estimated by applying 2-m centered differences (and backward or forward differences on the edges) to sorted and 30-s-averaged temperature profiles. The diapycnal eddy diffusivity K= and the buoyancy flux Jb ,
e Kz=Ymix N2
(4)
are estimated by using a constant mixing efficiency, Yl11ix»
of 0.2 (Osborn 1980). APEF, 1::, K=, and", 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
. N2(z)
RI = - - . --;:----'-'-,----;:-
(dUI dz)2
+ (
dVl
dz)2 (5)with U and v as alongshore 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.
Results
Degeneration of the Kelvin wave- An overview of the field data observed during the entire measuring period is given in Fig. 3. The wind field (Fig. 3a) is dominated by weak to moderate (0.3 to 6.4 m S-I) westerly winds. The low-frequency oscillations of the depths of isotherms (Fig. 3b) are caused by a basin-scale internal seiche with an amplitude exceeding 15 m. The seiche is influenced by
&J
106g
.U) .0 104
.g
]
~ 102
~
-Lmax
- along-shore 101 -0 ~ - cross-shore ~
. 10 min
i
'"
"0
'"
(')
~ 10° ;;:
'"
:::l
'"
q'
... "' .. _-_... ~
'.:;;;;;;;;;;;;. 10,1 Ul{,
. ---_. ---- -_.- --.. : ::r:
~
95%. "--p.. 100="-".=.=':':;'=:-_~_~;--_ _ ~-;;-_-IL,._~.,.-_ _ _ -r-:--,, _ _ _ ~_-'
10,5 10-4 105 10'4 103
N,:..
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 wave with a period generally ranging from 3 to 5 d, depending on stratification (Biiuerle et al. 1998; Boehrer et al. 2000; Appt et al. 2004). Four full Kelvin wave cycles were recorded (Fig. 3b, PI-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. 3b, ends of Kelvin wave cycles P3 and P4), indicating the non linearity of the basin-scale seiche. The steep thermal fronts were associated with steep alongshore current fronts (Fig. 3c, arrows) and NLIW with amplitudes of up to 15 m and strong current velocities of up to 0.15 m S-I (Fig. 3c, NLIW1 and NLIW2).
The power spectrum of the fluctuations of isothermal depth has three significant peaks at frequencies corre- sponding to periods of 3 d, 4 h, and 10 min (Fig. 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 cross- shore oscillation, which is strictly limited to Lake Uberlin- gen (Biiuerle 1994). The broad high-frequency peak around
10-3 Hz (5- to 20-min periods) characterizes propagating internal waves. Current velocity fluctuations were evaluat- ed 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. 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 frequen- cies but also shows small peaks at periods of 4 hand 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. 4a).
A comparison between the passage of HFW (Fig. Sa), the basin-scale internal Kelvin wave (Fig. 5b), and the occurrence of density inversions reveals a pronounced phase relationship. The time series of the depth of the lOSC isotherm was low-pass filtered (Butterworth of second order, cutoff frequency: 1124 h-1) to identify the Kelvin wave in the wave field. The square of the band pass- tJ r:
c--:---:---:---:--...,.0.2 7.S~·
5 ~ (') 2.5 S
a
2.5"':::
s
2 t"""
l.5~
1'-' Kelvin wav~ -HFW \ I
Cl o 0.1&
~. §
(') o
o (')
Si
(')0'
a
- 0.1 - Kelvin wave - LT . v
0.5 . .
-is
~=r--~~~-=~--~~~~
28 June 01 July 04 July 07 July 2008
-150 -100 -50 0 50 Lags (h)
100 150
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 measure of the intensity of HFW. The inversions are particularly frequent shortly before the trough of the Kelvin wave passes (Fig. Sb) 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. Sb) or the intensity of HFW (Fig. Sa). The cross-correlation (Fig. Sc) 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 -IS-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 Student's t-test (a < 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. Sc), indicating a synchronized occurrence of HFW and density inversions at the steepened front of the Kelvin wave, which indicates a cause-effect connection (Alford and Pinkel 2000).
Processes associated with the occurrence of density inversions--In the following analyses, four different condi- tions under which density inversions occurred are distin- guished and the observations are grouped accordingly into four categories of events (Figs. 6a, 7a): Kelvin wave shear- induced billows (KWB), weak stratification (WStr), non- linear internal waves (NLIW), and second mode waves (2ndM).
During the first observation period, W, (Figs. 3b, 6), the first two turbulent patches in Fig. 6g are associated with oscillations near the buoyancy period 2nN-' ;:0: 4 min, which develop into billow-like structures. The billows occur at the depth of the thermocline (Fig. 6i, bars) and the Kelvin wave-induced current shear (Fig. 6h, depth of current shear at 14:00). Unstable oscillations, caused by Kelvin wave-induced 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 W,.
The following patches of density inversions (Fig. 6a:
WStr, and WStr2) are associated with a thermocline-wide weakening of the stratification (Fig. 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. 6h, white arrows). These events are collected into the WStr category.
After the weakening of stratification, NLIW (Fig. 6a:
NLIW,) with amplitudes of about S m (Fig. 6g) propagate in the south-east alongshore direction, with a period of 12 min, a phase velocity of 0.28 m s-', and a wave length of - 200 m. These waves follow an internal current front propagating in an alongshore direction toward the south- east (Fig. 3c, second arrow), probably the front of the Kelvin wave that was reflected at the western boundary of Lake Con stance (compare to Appt et al. [2004]). NLIW with periods between Sand 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 W, and W2 (Fig. 7a: NLIW2 • NLIW3, and NLIW4). The NLIW during W2 both led (NLIWz) and followed (NLIW3 and NLIW4) the entering surge (Fig. 3c, third arrow) propagating in an alongshore direction (north-west). Amplitudes and periods of the wave packet depend on their location relative to the fronts: the wave packet NLIW 2 propagates with a phase velocity of 0.17 m s-', 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 NLIW 3 (wave lengths - 170 m) and NLIW4 (wave lengths - 7S m) propagate with phase velocities of 0.22 m S-1 and 0.17 m S-I, respectively. They have smaller amplitudes and periods (NLIW3: 13 min;
NLIW 4: 7 min) than does NLIW 2.
The next hot spots of density inversions (Fig. 6a: 2ndM,) are surrounded by opening and closing isotherms with an irregular period between 1 and 2 h (Fig. 6g), which are accompanied by HFW with amplitudes smaller than 3 m.
Such events associated with the second vertical mode-type isotherm displacements are grouped into the 2ndM catego- ry. Turbulent patches localized in the center of 2nd M were also found during observation period Wz (Fig. 7a: 2ndM2).
Generation mechanisms of density inversions-The medi- an 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. 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. 7d).
The passage of linear HFW starts synchronously with the passage of the thermal front (Fig. 6g). However, density inversions first occur 1 h later together with the KWB (Fig. 6c). In contrast to the observations of Alford and Pinkel (2000) 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 W, is associated with a change in current direction (Fig. 3c, first arrow). The alongshore current velocities reach values of up to 0.2 m s-, in the upper 10 m in depth (Fig. 6h) and impose a vertical shear over the water column. This shear is sufficient to decrease the corresponding Ri below the critical value of 0.2S for hours (Fig. 6g, blue areas). Low Ri indicates that the linear HFW and the Kelvin-Helmholtz billows result from shear instabilities evolving in the strong Kelvin wave-induced background shear. The turbulent collapse of the billows
2
8
1---
E2
0 8
~ 0 1 0..
'"
Cl
210 15
20 5 10 15 20
a b
Lr,
Chain Cl c
0',
Chain C2 d
10:00
•
I .• , . .,
-0.05 12:00 14:00 16:00 18:00 20:00 22:00 00:00 02:00 04:00 06:00
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 Cl. C2, and C3. (e) e at chain Cl (I) I-s temperature (chain Cl) 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
"""' E 10
' - '
...c:
i5..
<I)
Cl 15
10
]: 30 -:El
0.. o Cl
50
13 12 11 10
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 Time (hh:mm)
19:58 20:02
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.loC contour interval (zoom into the square of panel d, showing NLIW2). (I) 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 I). Note the different and nonlinear color scaling.
leads to a transfer of energy from the basin-scale wave to turbulent scales.
During the passage of the NLIWI and the 2ndMI , Ri is above the critical value, except at the edges of the thermocline near 5 and 20 m in depth, where the Kelvin
wave-induced current shear is particularly strong (Fig. 6h).
Overturns corresponding to these events seem to occur without being coupled to low Ri. The interaction of the background current with the current shear from linear HFW and NLIW can induce instabilities, whereas the
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)2Total 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
NLIW2 8 SAx 10-4 8.3XI0-s 0.5
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 sufficient to induce turbulent overturns (Alford and Pinkel 2000). Unfortunately, the temporal resolution of 2.5 min and the spatial resolution of 0.5 m are not sufficient to resolve the fine-scale shear induced by the linear HFW and most NLIW. Nevertheless, the observation of intense overturns during WStr and 2"dM supports the hypothesis that the interaction between HFW-induced shear and weaker stratifications generates overturns.
The wave-induced current velocity of the NLIW 2 could be observed down to the bottom of the water column (Fig. 7d), indicating that the waves are subject to shoaling.
Most of the energy contained in NLIW is assumed to be released during shoaling (Imberger 1995; Boegman 2009).
However, the NLIW energy may not be exclusively dissipated in the bottom boundary layer. This is indicated by Fig. 7g, in which the overturns occur in the pelagic thermocline. Density inversions occur predominantly in the wave troughs, where Ri was observed to fall below the critical value of 0.25 (not shown). Overturns, however, can stilI be observed minutes after a trough has passed the thermistor chain (Fig. 7b). The occurrence of density inversions increases during the passage of the wave packet to a mean value of 2S%, whereas the mean displacement length is only minimally increased, by about 0.2 m. Similar to the echo sounder observations of Mourn et aL (2003;
their figs. 6, 9, 11) of breaking solitary waves and laboratory experiments (Fructus et aL 2009), the overturns indicate self-induced shear instabilities. Subsequent mixing is implied by the slightly lower temperature of 100e to 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 Fig. 7g and is quantified by the especially high mean diapycnal eddy diffusivity related to wave packet NLIW2
(Table 1).
Turbulence and mixing-The APEF varies between 10-4 and 10-7 J kg- I during the entire period of observation (Fig. Sa). The highest values of observed APEF are induced by KWB and NLIW2 • Dissipation rates during the total observation time and during the different processes (for instance, KWB and NLIW; Fig. Sb) are approximately distributed log normally and vary between 10-10 and 10-5 W kg-I. The observed logarithmic mean and standard deviation of e vary among the different events (Table 1).
The measured logarithmic mean
<e>
= 3.3 X 10-7 W kg-Iof KWB is four times higher than even that for the shoaling NLIW2 , indicating that the highest dissipation rates are caused by the large billows. However, during the total measuring period the different processes, KWB, NLIW, and 2ndM, rather dissipated a comparable amount of potential energy (Table 1, column 3), between 9.S
x
10-4 and 1.3 x 10-3 J kg-I, because the processes occurred with different frequency (e.g., only a small amount of large billows [Fig. 5g] was generated in the shear induced by the front of the Kelvin wave during KWB, resulting in a comparatively low density of overturns of 7.6% [Table 2]).The arithmetic means, including zero values of dissipa- tion, diapycnal eddy diffusivity, and buoyancy flux, for the various processes and the various passages of the Kelvin wave through Lake UberIingen (Kelvin wave cycles P], ... , P4 in Fig.3b) are shown in Table 2. Period Ps starts immediately after period P 4, in which zero dissipation rates were added to complete a full Kelvin wave cycle with a period of 3 d (Fig. 4a). Mean values are calculated as averages over the observed 5- to 20-m depth of the water column and refer to an arbitrary I-kg water packet in the thermocline. With the exception of the extremely low value in cycle P3, the observed mean turbulent diffusivities are at least one order of magnitude larger than the molecular diffusivity of heat (1.5 X 10-7 m2 S-I; Pond and Pickard 1975). Dissipation rates (between 2 x 10- I I and I x 10-8 W kg-I), vertical diffusivities (between I x 10-7 and 4 x 10-5 m2 S-I), and upward flux of density (between 3 X
10-I 2 and 2 x 10-9 W kg- I) tend to be highly variable. The highest values were observed during the passage of the surge.
Discussion
A strong coupling was observed between the occurrence of high-frequency processes and the passage of the steep- ened front of the Kelvin wave in Lake UberIingen (Fig. 5b).
Energy is known to flow from basin-scale to dissipative- scale processes (i.e., from processes with low frequencies [Kelvin wave, cross-shore oscillation] via high-frequencies [propagating linear and nonlinear waves, 21ldM, KWB] to turbulence [Imberger 1995]). The occurrence of high- frequency processes at the internal front thus indicates a hot spot of the degeneration of the Kelvin wave. Enhanced dissipation rates, on the other hand, indicate hot spots of the degeneration of the higher-frequency processes of
2362
Fig. 8. Thorpe variables (a) APEF, (b) distributions of dissipation rate estimates induced by NLIW (black squares) and KWB (open circles). The shaded area shows the distribution of all dissipation rates (total).
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
thanall 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 Coriolis force (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 to be influenced by the lake-bed topography. While dissipation rates within the pelagic interior are likely to be somewhat weaker than those observed here, we assume 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) Jb (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
NLIW2 2 28.1 (7.5±0.7)X 10-8 (6.6±0.6)X 10-4 (1.5±0.2)X 10-8
PI 40 I (1.0±0.I)xl0-9 (5.1 ±0.3)X 10-6 (2.0±0.2)X 10-10
P2 68 0.7 (4.5±0.3)X 10-10 (3.3 ±0.2)X 10-6 (8.9±0.6)X 10-11
P3 90 0.03 (2.2±0.5)X 10-11 (l.3±0.3)X 10-7 (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
U:~l Pi 278 1.7 (3.5±0.2)X 10-9 (1.2±0.0)X 10-5 (7.1 ±0.3)X 10-10
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
been available, other than the general observation (Lorke et al. 2006) that HFW occur frequently in Lake Constance.
Therefore, observed dissipation rates and diffusivities were listed for single Kelvin wave passages in Table 2, which allows an estimation of the total averages according to the actual frequency of occurrence of NLIW and KWB. The different duration of the overturn-generating mechanisms in the 2-week observation period presented here led to comparable rates of energy dissipation for the different processes (Table 1).
The actual values are rough estimates of the APEF, dissipation rates, and eddy diffusivities associated with the overturn-generating processes. Absolute values as well as a comparison of values therefore should be considered a tendency. Turbulent patches in the pelagic zone are known to be separated by periods of molecular diffusion (Imberger 1998) and were assumed to have dissipation rates of e = O.
During the high-frequency processes, the occurrence of overturns is enhanced (30% in NLIW2, but only 2% in the entire observation period). Including an additional number of overturns (the possible unresolved 70% in all high- frequency processes) with dissipation rates of e = 10-8 W kg- I (measured 5% percentile of the log-normal distribu- tion) above the detection bound eb = 10-9 W kg-1 would add another 4.7 X 10-10 W kg-1 to the arithmetic mean.
Filling all interims with e = 0 W kg-I leads already to an estimate of
e =
3 X 10-9 W kg-I.The observed mean diffusivity of 1.2 X 10-5 m2 S-I in the pelagic zone is in good agreement with the K=
=
3.4 X10-5 m2 S-1 estimated with basin-scale tracer measure- ments (Maiss et al. 1994) in the thermocline of Lake Constance, indicating that a considerable percentage of the average diffusivities can be attributed to the pelagic zone.
The arithmetic mean of nearshore diffusivities of 2 X 10-4 m2 S-I estimated by Lorke (2007) in the thermocline of Lake Constance at the benthic boundary is an order of magnitude higher, indicating enhanced diapycnal mixing at the boundaries. But at the boundaries, where the stratifi- cation is weak, the large diffusivities do not result in similarly large buoyancy fluxes, as observed in the open water. Processes generating overturns in the pelagic zone, on the other hand, lead to mixing in the strongly stratified thermocline, and, thus, smaller diffusivities are associated with large buoyancy fluxes.
Maximum estimates of dissipation rates within turbulence hot spots such as KWB exceed 10-5 W kg-I, which is one order of magnitude higher than the maximum rates estimated by Kocsis et al. (1998), also in the pelagic thermocline of Lake Constance. The very high temporal variation in the energy dissipation (Table 2) indicates that most of the mixing in the thermocline of the lake's interior occurs only during a small time period within a Kelvin wave cycle. Hence, sporadic measurements of energy dissipation rates or even a high sampling intensity during 3 d can miss the most extreme turbulence events. The arithmetic means of the observed energy dissipation rates from the pelagic thermocline are in good agreement with the typical dissipation rates observed in the basin-scale metalimnion oflakes (between 10-10 and 10-8 W kg-I; Wuest and Lorke 2009), in which the boundaries are still included.
Since degeneration of large-scale internal waves via HFW is a phenomenon occurring in various lakes (Thorpe et al. 1972; Horn et al. 2001; Boegman et al. 2003) and also in the ocean in the context of internal tides (Lamb 2004), it can be expected that the spatial and temporal highly variable occurrence of turbulence and mixing in the pelagic thermocline is a common feature (Rudnick et al. 2003; Ivey et al. 2008; Macintyre et al. 2009). Mixing in the thermocline of the open water leads to vertical fluxes in the lake's interior. These intermittent fluxes of, for example, nutrients into the epilimnion can locally enhance plankton growth and may lead to the generation of plankton patchiness (Mackas et al. 1985). Such patchy distribution of prey organisms can further affect the intensity and dynamics of predator-prey interactions and enhance ecosystem productivity (Rovinsky et al. 1997).
In summary, periodic occurrence of turbulence and mixing was observed in the pelagic thermoc1ine of Lake Uberlingen. High-frequency processes such as propagating linear HFW and NLIW, 2ndM, and KWB are generated within a degeneration process at the steepened front of the basin-scale Kelvin wave. In this study, turbulence was observed to be produced via these high-frequency process- es, by current shear at the current front, and by the self- induced shear of internal waves with large amplitudes.
Overturns indicating turbulence were also observed during time periods in which HFW with smaller amplitudes propagated in weak stratification. Although for these events the Ri did not fall below the critical value of 0.25 (when an averaging interval of 2.5 min was applied), the shear associated with the HFW, which was not resolved in the current velocity measurements, is assumed to be sufficient to cause instabilities under weak stratification.
In addition to shoaling of NLIW, the turbulence-generat- ing mechanisms identified at the internal front of a basin- scale Kelvin wave are likely to occur independently of the topography, everywhere in the interior of Lake Uberlingen.
The observations in this study indicate that the turbulence produced within the stratified interior of the lake leads to significant buoyancy flux in the pelagic thermocline, where the specific contribution depends on the process generating the turbulence. Highly variable spatial and temporal occurrence of turbulence and mixing in the pelagic thermocline is likely to be a common feature in many lakes.
Acknowledgments
We thank E. Biiuerle and H. Hofmann for helpful discussions and the technicians J. Halder and A. Sulger for support in the field.
We gratefully acknowledge the very helpful comments of two anonymous reviewers. This research was funded by the German Research Foundation (Collaborative Research Centre 454).
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