1. Capillary wave turbulence at the charged surface of liquid hydrogen M.Yu. Brazhnikov, A.A. Levchenko, L.P. Mezhov-Deglin, Institute of Solid State Physics RAS, Russia G.V. Kolmakov ISSP RAS & Lancaster University A.N. Silchenko, P.V.E. McClintock Lancaster University, UK

2. Short content Introduction Liquid hydrogen as a test fluid for studies of nonlinear wave phenomena. Linear and nonlinear waves at the charged surface of liquid hydrogen. Steady state turbulence of capillary waves. Relaxation of the turbulent cascade after switching off the outer driving force. Suppression of turbulence by an additional low- frequency driving. Conclusions.

3. Properties of liquid hydrogen, helium and water Liquid Hydrogen, T=15 K Liquid Helium, T=4.2 K Water, T=300 K Density , g/cm 3 0.0760.1451.0 Surface tension , dyn/cm 2.70.1277 Capillary length, cm 1.180.181.74 Nonlinearity coefficient (  /  3 ) 1/4, cm 9/4 /g 1/2 s 1/2 8.92.53.0 Kinematical viscosity, cm 2 s 0.00260.000020.01 Dielectric permittivity  1.261.04781

4. Usage of liquid hydrogen as a test liquid for studies of capillary turbulence High nonlinearity coefficient, low viscosity: The inertial range of frequencies is an order wider in “conventional fluids” (water). Possibility to create quasi-2D charged layer below the liquid surface. The dispersion management by application of external electric field. Small density, excitation of surface oscillations by weak oscillating electrical field. Driving force acts directly on the surface. Very pure conditions of excitations (e.g., with respect to the vibrating platform techniques). Spectral characteristic and angle dependence of the driving force can be varied in a controllable way in vide limits.

5. Experiments with the charged surface of liquid hydrogen LIQUID HYDROGEN K.R. Atkins, Phys. Rev. 116, 1339 (1959) P e = p 0 e 2 / (2 v H r 4 ); R 4 + = p 0 e 2 / (2 v H P s ) p 0 – polarizability, 1·10 -24 см 3 v H – effective volume  4.4 10 -23 см 3 Estimated R+ ≈ 45 Å. Experimental R+ ≈ 15 Å E Radioactive plate (source of charges) d R.A. Ferrel, Phys. Rev. 108, 167 (1937) W=(  h 2 /2mR 2 ) + 4  R 2 + 4  R 3 P /3 m – mass of electron  - surface tension. At zero pressure R 4 = (  h 2 / 8m  ). R-  10Å. Positive charge Negative charge U

6. Linear waves on the neutral surface of liquid Dispersion law for linear capillary-gravity waves (waves of small amplitude) In accordance with the previous table, for water at T=20 C one has: k c =3.6 cm -1, for liquid hydrogen at T=15 K one has: k c =5.32 cm -1. In case of long (gravity) waves, k<k c, the dispersion relation reads For short (gravity) waves, k>kc, the dispersion relation is the following g is the free fall acceleration  is the surface tension  Is the density of liquid

7. Linear waves on the charged surface of liquid hydrogen What is the influence of charges localised below the surface? Here we consider the case where the quasi-two dimensional charged layer screens totally the external electric field (E≠0 in gas phase, only) – that corresponds to condition of experiments with the positively charged surface of liquid hydrogen. E d U Mean distance between charged clusters in the 2D layer is equal to r ~ n -1/2 ~10 -5 cm In experiments with charged surface the two-dimensional density of positive charges stabilised below the surface is equal to n~10 10 charges/cm 2 Three types of charges in liquid H 2 : Negative charges – electron bubbles (cannot be stabilised below the surface due to relatively high probability of quantum tunnelling through the surface); and negatively charged clusters (snowballs) H 2 – complexes surrounded by a solidified layer of hydrogen. Positive charges – positively charged clusters (used in our experiments with nonlinear capillary waves). (Mezhov-Deglin and Levchenko, 1997) r The sigh of charges localised below the surface is defined by polarity of the voltage U applied between the surface and the upper electrode.

8. It is of importance that in the case of waves on the charged surface of liquid there is an additional parameter with the dimensionality of length – the distance d between the charged surface and the upper flat electrode. And this distance should be compared with the capillary wavelength c. (Note that we consider the waves on the surface of deep layer of liquid, <<h, where h is the depth of liquid, so th(kh)  1.) So, in case of waves with the wave length >r~10 -5 cm we may consider the distribution of charges quasi-continuously. E d U The dispersion law for capillary waves at the charged surface of liquid in “quasi- continuous” approximation has been calculated by D. Chernikova in 1976.

9. 1. Large d (d>> c ) Evolution of the spectrum of waves on the charged surface of liquid with increasing the voltage U (the case of large d). This type of spectrum has been studies by P. Leidered in 1979, in experiments with positively charged interface in a phase-separated 4 He- 3 He mixture. In this case the dispersion relation reads: The term in the dispersion relation, which appears due to electrical forces, is underlined.

10. The dispersion relation: At high electric fields a flat surface becomes unstable. The instability (in case of fixed total number of charges) leads to formation of multi-charged dimples with characteristic sizes ~ k c -1, or to formation of dimple crystal, in dependence of total number of charges – the reconstruction of the surface takes place. Dimple crystal formed on the free surface of superfluid helium at high electric fields (Leiderer, 1979). Spectrum of waves on the flat charged surface of liquid at high electric field E>E c.

11. In the case where the distance d is relatively small, d<a the dispersion law acquires the following simple form So, electrical field renormalizes the free fall acceleration g: At G<0 a flat surface becomes unstable, but the instability is developed at small k << k c, first (long-wave instability). E<E c E>E c 2. Small d (d<< c ) where the effective free fall acceleration is give be the following equation The effective free fall acceleration is decreased with increasing the electric field. Unstable region.

12. Reconstruction of the charged surface of liquid hydrogen at high electric fields At U>U c the surface is reconstructed and a stationary, solitary-like wave of the surface deformation is formed, where Solitary wave of deformation formed on the charged surface of liquid hydrogen at U>U c A.A. Levchenko et al, 1997. Dependence of the maximal angle of reflection of a laser beam from the surface and of its derivative on the voltage applied. Z, mm r, mm

13. U, V T, K At high voltages, U>U c2, the geyser-like discharge of the surface take place. Snapshot of the surface at the moment of the geyser formation. Phase diagram of soliton stability at the charged surface of liquid hydrogen. The first critical voltage is equal to

14. Spectrum of linear oscillations of the charged liquid hydrogen surface Scheme of measurements Intensity Measured intensity of the laser beam measured by a photodetector. The dispersion curve for the waves on the charged surface of liquid hydrogen. The depth of the liquid layer was 2.7 mm. The DC voltage was U = 283 V (circles), 954 V (squares), and 1080 (diamonds). Log k (cm -1 ) Log  (Hz) At G  0 the surface waves have a purely capillary dispersion law for all wave length. So, we can manage the dispersion of waves by changing the applied voltage U

15. Log  Log E  PUMPING DAMPING For capillary wave turbulence (V.E. Zakharov, A.N. Filonenko, 1967): In the inertial range: Energy flux P >0 (E is the only integral of motion, no inverse cascade) INERTIAL RANGE E  ~  -3/2 Nonlinear capillary waves on the surface of liquid hydrogen Multiple frequency generation and cascade-like transfer of energy to high frequency due to nonlinearity The dispersion relation for capillary waves is of decay type  ~ k 3/2 k=k1+k2=1+2k=k1+k2=1+2 E  ~  -3/2

16. The amplitude of three-wave interaction for capillary waves The kinetic equation for capillary waves The theory of weak turbulence presents a basis for description of dynamics of capillary waves. (Zakharov, Filonenko, 1967) -- “number of waves” with the wave vector k. -- the damping coefficient for capillary waves. a k (t) – normalised (canonical) amplitude of the wave. where

17. Steady-state spectrum of capillary turbulence (Zakharov, Filonenko, 1967). Where k p is the characteristic scale of pumping, k b is the viscous scale (short-wave edge of the inertial range).

18. The observed spectra of capillary waves on the liquid surface (experiments with water in a cell placed on a vibrating platform). E. Henry, P. Alstrom and M.T. Levinsen, Europhys. Lett, 52, 27 (2000). 1.W. Wright, R. Hiller and S. Putterman, J.Acoust. Soc. Am., 92, 2360 (1992). Previous studies of capillary turbulence on the surface of water

19. Scheme of registration of the surface oscillations Time dependence of the registered signal P(t); the driving frequency is equal to  /2  = 27.5 Hz.   2 ~ P  2 / Ф(  ) Thin beam ( ka<< , where a – is the laser spot size) Ф(  )  1 I  ~   4/3 ~   4/3 P  2 Broad beam ( ka>>  ) Ф(  ) ~   4/3 I  ~   4/3 ~ P  2 Apparatus function Ф(  ) = P  2 /   2 I  = = = = k  2 ~   4/3, where   = k   - angle amplitude of the wave. Time, s Power of the reflected laser beam, a.u. Experimental studies of turbulence of capillary waves on the surface of liquid hydrogen (Levchenko at al, 1999)

20. Spectrum of oscillation of liquid hydrogen surface Driving at a single frequency Spectrum of surface oscillations measured in case where the driving frequency is equal to 28 Hz In the frequency range up to 10 kHz the spectrum of oscillations can be approximated as follows I  ~ P  2 = const  -21/6 Spectrum of surface oscillation; driving frequency is equal to135 Hz

21. Excitation at 28 and 48 Hz Excitation by a noisy force P  2 =const  -17/6 In agreement with the results of the weak turbulence theory (by Zakharov, Shafarenko, 1967)

22. Stationary distribution n k for capillary waves calculated for the case of spectrally sharp driving (G.E. Falkovich, A.B. Shafarenko, 1988) Spectral density of the surface elevation shown (noisy driving force; numerical calculations by V.E. Zakharov, A.N. Pushkarev, 1996 ) n k ~k -21/4 (I  ~  -21/6 ) n k ~k -17/4 (I  ~  -17/6 ) Modification of the spectrum of oscillations with changing the spectral characteristics of the driving force.

23. Distribution of P w 2 measured in the case where the amplitude of wave at the driving frequency 135 Hz is equal to 0.016 mm. Arrow marks the high-frequency boundary of the inertial range. Characteristic time of the nonlinear interaction of waves Time of viscous (linear) dumping. 1/  n (  )~|V kkk | 2 n(k) k 2 /  k ~ A 0 2  -1/6 1/  v (  ) ~2 k 2 ~  4/3 n~vn~v Coefficient of nonlinear interaction V kkk ~(  /  3 ) 1/4 k 9/4 Spectrum of waves n(k) = const A 0 2 (k/k p ) -21/4 The dispersion law  (k) ~ k 3/2  b ~ A 0 4/3  p 23/9 Dependence of high-frequency boundary of the inertial range on the amplitude of the wave at driving frequency. The driving frequency is 83 Hz (lower curve), 135 Hz (middle curve), and 290 Hz (upper curve). High frequency cut-off of the power spectrum of turbulence If one supposes that One has

24. An important conclusion is followed from this result: Despite our system has a finite sizes (L is of a few cm order), and, consequently, the intrinsic spectrum of small oscillations of the surface is discrete, the position of the high frequency edge of inertial range is controlled by viscosity, not by discrete character of the spectrum.

25. The recorded signal P(t) after the removal of the driving force (the driving frequency is 98 Hz). Evolution of spectrum of the surface oscillations after the removal of the driving force. Non-stationary processes of turbulence decay

26. What we are waiting for? An initial assumption – the self-similar process of formation and decay Formation Decay The self-similar variable The self-similar function The front position Such scenario of the decay of capillary turbulence is in contradiction with the observations.

27. Relaxation of the surface oscillations after removal of the driving. Upper plot: small cell,  p =98 Hz, Lower plot: large cell,  p =97 Hz. Time, s P(t), arb. units Dependence on time of the amplitude of the recorded signal P(t): curve 1 for a driving frequency  p = 97 Hz; and curve 2 for  p = 173 Hz (large cell). Dependence of the effective relaxation time  of the surface oscillations on the driving frequency  p (circles); and viscous damping time for capillary wave   -1 with frequency  p, calculated from known parameters of liquid hydrogen (line). Non-stationary processes of turbulence decay A.A. Levchenko, M.Yu. Brazhniov, L.P.Mezhov-Deglin, G. V. Kolmakov, A.N. Silchenko, P.V.E. McClintock, JETP Lett. 80 (2) 99-103 (2004); Phys. Rev. Lett. (2004).  p /2 , Hz , s  ~  -4/3

28. Decay of the capillary turbulence on the surface of viscous liquid. t=0 – the moment when the pumping is removed k b (t) – the position of the edge of the inertial range at the moment of time t,  – some exponent, g(  ) – self-similar function,  – self-similar variable The main conclusion: the dissipation is missed in the previous consideration. The main assumption: the evolution is controlled by the position of the inertial interval edge.

29. The position of the inertial range edge k b (t) is governed by the following equation Equation for the self-similar function where

30. The viscosity coefficient and the C constant can be excluded from the equation by using the following substitution: The boundary conditions are: 1) 2)

31. Asymptotes of the self-similar function: (Energy distribution over frequency )

32. “ Local” approximation in the theory of capillary turbulence Steady-state distribution Steady-state spectrum of turbulence in the local model. In this simple model the self-similar representation for the spectrum of capillary waves during the free decay s the following - Is the self-similar variable in scales of frequency

33. The results of numerical integration of the equation (1) for f(x) in cases f(1) = f 1 = 100,10, 1 и 0.1, plotted in logarithmic scales. Dashed lines corresponds to asymptotes x -2/3 и x -3/2. Equation for the self-similar function is the following

34. Numerical studies of the decay of capillary turbulence Evolution with time of the energy distribution of capillary waves E  after step-like switching off the driving force: 1: t = 0 (steady-state spectrum), 2: t = 0.5, 3: t = 2.0 и 4: t = 4.0. Dashed line corresponds to the power spectrum E  ~  -3/2. Time and frequency is shown in dimensionless units.

35. Position of the high-frequency edge of the inertial frequency  b (t) obtained in numerical calculations (squares); and the Plotted for  b (0) = 32.7 and  = 0.187 (solid curve).

36. In experiments on the decay of capillary turbulence on the surface of iquid hydrogen the relative width of the inertial range is  b /  p ~ 50 - 100 Characteristic time of nonlinear interaction of waves with the frequency  Time of viscous damping In case of pumping by a low-frequency noise At the frequencies of the order of the boundary frequency of the inertial range At the pumping frequency

37. Suppression of turbulence at high frequencies by low frequency driving Squared amplitudes of waves at the pumping frequencies  1 /2   = 61 Hz и  2 /2   = 274 Hz. The low-frequency driving is removed at t = 0. Stationary spectra of surface oscillations before and after switching off the low frequency perturbation. а б Squared amplitudes of waves at the multiple frequencies of  2 /2   = 274 Гц. Pumping at  1 /2  = 61 Hz и  2 /2  = 274 Hz.

38. Stationary spectra of capillary turbulence. Red  pumping at two frequencies  1 /2   = 63 Hz and  2 /2   = 178 Hz, blue  pumping at  2 only. Pumping at  1 /2  = 63 Hz and  2 /2  = 178 Hz. Squared amplitudes of waves at the multiple frequencies of the main pumping frequency at switching on (а) and at switching off (б) the additional low-frequency driving. а б

39. The “classical” wave turbulence view. The energy flux to high frequencies So, the energy flux should increase when the driving frequencies is decreased

40. Подавление высокочастотных турбулентных колебаний Calculated evolution of the spectra after switching off the additional low-frequency driving. The inset: the relative changes of the peak amplitudes The reason of the suppression of turbulence at high frequencies is the change of density of states involved in the nonlinear transfer of energy to high frequencies.

42. Впервые изучена динамика установления стационарного турбулентного каскада в системе капиллярных волн на поверхности жидкого водорода при включении/выключении дополнительной накачки с частотой ниже частоты основной накачки. Обнаружено, что при включении дополнительной низкочастотной накачки амплитуды волн в высокочастотной части турбулентного спектра уменьшаются, что приводит к сокращению инерционного интервала частот. При выключении низкочастотной составляющей накачки амплитуды высокочастотных турбулентных осцилляций увеличиваются, инерционный интервал расширяется. Экспериментальные данные и результаты проведённых нами численных расчётов качественно согласуются между собой. Подавление высокочастотных турбулентных колебаний

43. Выводы Таким образом, в работе изучен распад капиллярной турбулентности на поверхности жидкости после ступенчатого выключения внешней возбуждающей силы с учетом вязкого затухания волн на всех частотах. Из результата расчета следует, что распад турбулентности начинается с области высоких частот. При достаточно больших начальных амплитудах возбуждающей силы распределение энергии волн по частотам близко к своему стационарному виду E  ~  -3/2 в широком интервале частот   b, где  b - граничная частота инерционного интервала, в течение относительно длительного времени после выключения внешней силы. После выключения внешней силы граничная частота  b уменьшается, и при больших временах t >> , где t -- характерное время распада турбулентности,  b (t) ~ t -3/4. Полученные результаты качественно согласуются с результатами экспериментов по изучению распада капиллярной турбулентности на поверхности жидкого водорода.

44. CONCLUSIONS Liquid hydrogen is an excellent test fluid for studies of general static and dynamic nonlinear phenomena, and for careful check of predictions of WT theory. At high electric fields a steady solitary-like wave of the surface deformation is formed at the charged surface of liquid hydrogen. Reconstruction of the surface has a soft character and close to the second-order phase transition. The remarkable properties of liquid hydrogen have allowed us to observe and study turbulence of capillary waves a) the power-like spectrum of capillary turbulence in a wide range of frequencies up to 10 kHz; b) The cut-off the power spectrum is observed at high frequencies caused by change of the mechanism of the energy transfer from the nonlinear waves transformation to viscous damping. Free decay of the turbulence has a quasi-adiabatic character. The fast redistribution of energy between waves inside the inertial range damps a propagation of the front through the scales. The turbulent system is in quasi- steady state during the decay. The high frequency edge of the inertial range goes to low frequency. Account for the finite viscous losses at all frequencies changes qualitatively the character of the decay of turbulence.

45. List of Recent Publications 1. A.A. Levchenko, M.Yu. Brazhniov, L.P.Mezhov-Deglin, G. V. Kolmakov, A.N. Silchenko, P.V.E. McClintock, “Decay of the turbulent cascade of capillary waves on the charged surface of liquid hydrogen”, JETP Lett. 80 (2) 99-103 (2004); 2. G. V. Kolmakov, A.A. Levchenko, M.Yu. Brazhniov, L.P.Mezhov-Deglin, A.N. Silchenko, P.V.E. McClintock, “Quasi-adiabatic decay of capillary turbulence on the charged surface of liquid hydrogen”, Phys. Rev. Lett. 93 (7), 074501 (2004). 3. M.Yu. Brazhniov, A.A. Levchenko, and L.P.Mezhov-Deglin, “Excitation and detection of nonlinear waves on a charged surface of liquid hydrogen”, Instrum. Exp. Tech. 45 (6) 758-763 (2002). 4. M.Yu. Brazhniov, G. V. Kolmakov, A.A. Levchenko, and L.P.Mezhov-Deglin, “Linear and nonlinear waves at the charged surface of liquid hydrogen”, Low Temp. Phys. 27 (9-10) 876-882 (2001).

46. Schematic view of the relaxation process We may speak about the inverse front motion during the relaxation, but the motion is defined by both dissipative and nonlinear processes, and it has absolutely different meaning! Results of our estimations for the decay of capillary turbulence are in agreement with the experimental observations Predictions of the theory where the dissipation was neglected

47. Soliton formation on the charged surface of liquid helium (electrons above the surface of helium). P. Leiderer, 1997

48. Выводы Таким образом, в работе изучен распад капиллярной турбулентности на поверхности жидкости после ступенчатого выключения внешней возбуждающей силы с учетом вязкого затухания волн на всех частотах. Из результата расчета следует, что распад турбулентности начинается с области высоких частот. При достаточно больших начальных амплитудах возбуждающей силы распределение энергии волн по частотам близко к своему стационарному виду E  ~  -3/2 в широком интервале частот   b, где  b - граничная частота инерционного интервала, в течение относительно длительного времени после выключения внешней силы. После выключения внешней силы граничная частота  b уменьшается, и при больших временах t >> , где t -- характерное время распада турбулентности,  b (t) ~ t -3/4. Полученные результаты качественно согласуются с результатами экспериментов по изучению распада капиллярной турбулентности на поверхности жидкого водорода.

49. Подавление высокочастотных турбулентных колебаний Частота , Гц P  , отн. ед. Спектрограмма колебаний поверхности жидкого водорода при накачке на частотах  1 /2  = 63 Гц и  2 /2  = 178 Гц. По оси абсцисс  время в секундах, по оси ординат  десятичный логарифм частоты.