7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 1
The Effect of Amplitude on the Period of Solitary Waves
M. Preusse1,2*, H. Freistühler1 and F. Peeters2
*Corresponding author, e-mail: martina.preusse@uni-konstanz.de
1 Department of Mathematics and Statistics, University of Konstanz, heinrich.freistuehler@uni-konstanz.de
2 Limnological Institute, University of Konstanz, frank.peeters@uni-konstanz.de
Abstract
The amplitude of solitary waves significantly affects the properties of the waves by changing their degree of nonlinearity. The period of solitary waves e.g. decreases nonlinearly with amplitude. Observations of solitary waves propagating under similar stratification conditions, but with different amplitudes, were compared to identify how the wave period depends on wave amplitude under field conditions. In addition, we employed the Korteweg-de Vries (KdV) equation and the Dubreil-Jacotin-Long (DJL) equation to simulate the dependence of solitary wave period on amplitude. The amplitude dependence of period simulated by the DJL model is in good agreement with the field observations whereas the KdV simulations are only accurate within the lower range of amplitudes of solitary waves occurring in the field.
1. Introduction
Internal solitary waves are often observed, not only in the ocean, but also in long lakes (Horn et al., 2001). They are known as a degeneration mechanism for basin-scale internal waves, where solitary waves are generated by nonlinear steepening of the large-scale wave. Solitons are an important factor in the ecology of lakes, because turbulence generated by breaking high-frequency waves is thought to be the main process driving mixing in the thermocline.
Observations of highly nonlinear or even breaking solitary waves confirm this theory (Moum et al., 2003, Preusse et al., 2010).
Amplitude is one of the main parameters determining the probability of a soliton observed in the field to be accompanied by density inversions, indicating either shear instabilities or trapped cores (Preusse et al., submitted). Moreover, amplitude determines the applicability of models for weakly nonlinear waves. Small-amplitude solitary waves in continuously stratified systems can be simulated by solving the KdV equation (e.g. Ostrovsky and Stepanyants, 2005). However, higher order nonlinearities have to be taken into account for the simulation of large-amplitude solitons. Large-amplitude solitary waves can e.g. be simulated by the stationary DJL equation (Stastna and Lamb, 2002). The effect of amplitude on the solitary wave properties propagation depth, phase velocity, wave length and horizontal and vertical velocities was studied numerically by Vlasenko et al. (2000).
We complement this study by analysing the dependence of period on amplitude numerically.
Moreover, we compare the simulations based on the DJL and KdV equation with periods of solitary waves measured in Lake Constance. The main achievement of our study is the demonstration that the dependence of period on the amplitude of solitary waves observed in the field agrees well with the numerical solutions of the DJL equation.
2. Methods
In order to analyse the dependence of wave period on amplitude for a fixed stratification, the periods of solitary waves propagating in the same wave train were compared. Field data of
Zuerst ersch. in:
Proceedings of VII International Symposium on stratified flows ; Rome, 22nd – 26th August 2011 / ed. by Antonio Cenedese ... (Eds.). - 4 S.. - ISBN 978-88-95814-49-0
Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-204723
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 2 solitary waves were obtained by temperature measurements conducted in the interior of a subbasin of Lake Constance, a freshwater lake, at a water depth of 145 m in 2009 (deepest part of the subbasin) and 100 m in 2010.
Here, we compare solitary wave trains from two observation periods, one from July and the other from October. The background stratifications during the two measuring periods were substantially different with the depth of thermocline defined as the maximum gradient in density being 4 m in July and 10 m in October (Fig. 1).
998 999 1000
5 10 15 20 25 30
0 0.001 0.002 July 2009
Oct. 2010
Depth (m)
Density (kg m )-3 N (1/s )2 2
A) B)
Figure 1. Background stratifications measured shortly before the passage of the solitary wave trains in July (solid line) and October (broken line). Vertical profiles of (A) densities and (B) squared Brunt- Väisälä frequencies.
The properties of the observed solitary waves, such as isothermal displacement η(z,t) at time t and depth z, propagation depth Hp, amplitude a and period T, were automatically extracted from the temperature time series (for details see Preusse et al., submitted). Here, η(z,t) describes the displacement of the isotherm, which is in rest at depth z. For all isotherms the maximum displacement during the passage of a soliton, ηmax(z), was calculated (Fig. 2B). The propagation depth Hp of the soliton is defined as the depth in which ηmax is maximal. The amplitude of the soliton is defined as a = max(ηmax), i.e. the maximum displacement of the isotherm which is undisturbed at Hp. The period T for the individual isotherm displacements was obtained from
∫
= z t dt
z z
T ( , )
) ( 2 ) 1 (
max
η η ,
which is based on the equivalent wave length and the phase velocity (Preusse et al., submitted), and should not be confused with the length of the time interval where the
isothermal displacement is larger than zero. The period of the soliton Tp = T(Hp) is defined as the period of the isothermal displacement η(Hp,t).
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 3 Individual solitons were simulated by the DJL equation (the code was provided by M. Stastna, see Stastna and Lamb, 2002) and the KdV equation (implemented as described in Ostrovsky and Stepanyants, 2005) assuming a zero background current.
3. Results and Discussion
Fig. 2 demonstrates that the vertical and temporal displacement of isotherms by solitary waves measured in the field (July experiment) can successfully be simulated by the DJL and the KdV equation, as long as the amplitudes of the solitons are small. Note that the solution of the KdV equation has a constant period (Fig. 2C).
09:25 09:35 09:45
Period T (min) 7° C
9° C
31. July 2009 A)
10
20
25
Depth (m)
15 5
Max displacement η (m)max
Hp, DJL
B) C)
Data DJL KdV
0 2 4 3 5 7
Tp Hp, DATA
η(Hp,DJL, ) +H. p,DJL
Figure 2. Comparison between observed soliton characteristics of a solitary wave and DJL and KdV simulations: isotherms (A), vertical profiles of maximal displacement (B) and period (C) as measured in Lake Constance (black lines) and as simulated by the DJL equation (grey lines) and KdV equation (broken lines). Propagation depth Hp and period at propagation depth Tp are marked by arrows.
Comparably accurate simulations for large-amplitude solitary waves, applying the complete set of the fully nonlinear Euler equations, were obtained for solitons measured in the
Mediterranean Sea by Vlasenko et al. (2000).
Fig. 3 shows how the simulated period at propagation depth, Tp, for the two different background stratifications (July and October experiment) numerically depends on the amplitude of solitary waves: The period decreases nonlinearly with amplitude, where the decrease in KdV simulations is larger than for DJL simulations and thus the differences between KdV and DJL simulations increase with amplitude.
The measured periods Tp of the observed solitary waves agree well with the periods simulated by the DJL equation, as Fig. 3 demonstrates. For amplitudes larger than 7.5 m, KdV and DJL simulated periods Tp of the solitary waves corresponding to the train observed in October deviate more than 1 min. Since solitary waves with amplitudes larger than 7.5 m are common in Lake Constance (Preusse et al, submitted), nonlinear effects should be considered in lake- wide hydrodynamical models applied to lakes, where solitary waves are known to occur.
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 4 Acknowledgements
We want to thank M. Stastna, who generously provided the numerical code for solving the DJL equation.
References
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4 6 8 10 12 14 16 18 20 2
4 6 8 10 12
Amplitude (m) Period T (min)p
July 2009 DJL KdV Oct. 2010
Figure 3. Influence of amplitude on the period at the propagation depth Hp as measured in Lake Constance for 2 solitary wave trains consisting of several solitary waves, which occurred in summer (empty circles) and autumn (filled circles), and as predicted by the DJL equation (solid lines) and KdV equation (broken lines).
