Direct Energy Transformation in a System of Nonlinear Second Sound Waves

Direct Energy Transformation in a System of Nonlinear Second Sound Waves

V.B.Efimov1,2 and P.V.E.McClintock2 1) Institute of Solid State Physics RAS, Chernogolovka, Moscow distr., Russia2) Physics department, Lancaster University, Lancaster, UK

The VI-th international conference: Solitons, Collapses and TurbulenceNovosibirsk Akademgorodok, 7 June 2012

Solitons, Collapses and Turbulence

OutlineIntroduction

◦Why superfluid helium?◦Burgers waves

Experiments◦Resonator of second sound wave;◦Low resonance number – a direct energy

cascade of nonlinear waves;◦Far resonances – an inverse energy

cascade;◦Very low resonances – fractional frequency

transformation.Discussion Akademgorodok,

Novosibirsk, 07 June 2012


Akademgorodok, Novosibirsk, 07 June

2012

Sounds in Superfluid Helium

0 1 2 3 40

20

40

60

80

100

120

140

160

180

200

220

240

First and Second sound velosity in 4He at SVP

v1 / 3

v2

v1

v, m

/s

T, K

TStSP

t n

s

2

2

2

2

2

,

SPu )/(1

n

s

CTSu

2

2

The differentiation of two equations with respect to time and substituting into another relationships we obtain for small disturbance (linear waves)

Solutions are wave of density (co-moving motion normal and superfluid

component)

waves of temperature or entropy (counter-movement motion normal and superfluid components, =n+s=const

Laboratory of Quantum Crystals

4

Second sound shock waves in superfluid helium

Temperature dependence of second sound nonlinear coefficient. Right hump is connected with roton waves, left one – with phonon’s temperature excitation

Nonlinear coefficient is positive for first sound

Nonlinear coefficient may be positive as well as negative for second sound. It means, the breakdown appears on front (a2>0) or on backside (for a2<0) of wave depending from T.

0 20 40 60 80 100

-4

-2

0

2

4

6

8

10

12

3

4

5

6

7

8

21

P=3atm., U =7.32V, =10s.1 T=2.103K2 T=2.071K3 T=2.044K4 T=1.999K5 T=1.968K6 T=1.81K7 T=1.783K8 T=1.774K

Arb

. uni

ts.

Time, s.

Why second sound in superfluid helium?

Low velocity of wave of second soundWeak wave dissipation Very strong dependence of velocity from

second sound wave amplitude (δT~mK) Linear dispersion law ω~k

Burgers equation

Dependence of second sound velocity from wave amplitude:

Resonator of second sound waves

We applied harmonic signal to heater U~sin(w*t), P~sin2(w*t)=[1+cos(2w*t)]/2

Resonant frequency fG=c0/4L

Laboratory of Quantum Crystals 7

Resonator In experiments was used high quality

cylindrical resonator (cover and bottom were parallel, wall was with high accurateness cylindrical – in some experiments we used cut off syringe), D~1.5 cm, L=2-8cm.

Heating meander occupied all lid area and we generated one dimensional wave

The eigenfrequencies are multiple harmonics

Quality of resonator 70 mm

The penetration depth is of the order of 2.5 μm for resonance frequency ~ 1000 Hz. Lets us suppose that the part of helium in length of Λ balks a motion of wave. The absolute value of resonator quality is of the order of

Bulk dissipation stays appreciable at frequencies higher 10 kHz

3 March 2010

The mechanical quality of the resonator is determined by parallel of reflecting walls Δl/l ~ 5* 10-4, which corresponds to highest resonance frequency ~400 kHz

Model. The counterflow of the superfluid helium in resonator yields a some viscous friction at penetration depth, which depends from a frequency of a resonance vibrations

The energy loss at motion along walls in layer Λ at distance L, which is defined by interference conditions.. The wave disappears in resonator as only the sum of reflection N multiplies on Δl will be order of wavelength λ=c/f=c/2 πω~Δl*N, where c - is sound velocity. Wave at this case runs a distance

and quality of resonator will be defined as verses value of losses

100 1000 40000 80000

1000

10000

Qua

lity,

f/

f

Frequency, Hz

Q~f 3/2

Phys.Rev. E


Measurements of standing waves at different resonances.

Low frequency f~1000 Hz, N~10


2012

100100 1 ∞

Bad quality Good quality Balk dissipation

10

Energy Transformation in Superfluid Helium-4

11

Experimental studies of one dimensional nonlinear second sound waves in the cylindrical cell

Spectrum of temperature oscillations of the nonlinear second sound waves in a

resonator

Applied signal A=A0*sin(ωt)

100 1000 10000

1E-5

1E-4

1E-3

0.01

Frequency (Hz)

Am

plitu

de

A~f -1.62

0,001 0,002

-0,005

0,000

0,005

0,010

A, a

rb. u

nits

t, ms

-12

-6

0

6

12

Fast Fourier Transformation

Temperature in second sound wave

12

Formation of Kolmogorov’s spectrum

FFT spectrum


Decay of nonlinear energy spectrum

100 1000 10000

1E-5

1E-4

1E-3

0,01

Frequency (Hz)

Am

plitu

de

100 1000 10000

1E-5

1E-4

1E-3

0,01

Frequency (Hz)

Am

plitu

de

We applied harmonic (~sin(ωt)) signal from generator to heater in cylinder resonator. After formation the nonlinear wave spectrum (a) we switched off the pumping signal and have observed transformation of the harmonics with time (b).

a)

b)


Chaos in harmonics after switch off the pump

-8 0 81E-5

1E-4

1E-3

0,01

Change of regime

A=a0*exp(t/5.2)

Am

plitu

de, a

.u.

Time, s

B1 C2 D3 E4 F5 G6 H7 I8 J9 K10 L11 M12 N13 O14

0 1

1E-4

1E-3

Am

plitu

de, a

.u.

Time, s

Stop of energy pumping in the resonator drastically reduces multiple harmonics. The amplitudes of harmonics chaotically derange. Typically time of chaotic energy random walk is order of hundreds wave periods. After 2 seconds the amplitude of the main (first) harmonic reduces with slower rate and can be described by an exponent dependence which correspond to the quality of resonator about 5000.

Chaos in energy spectrum

We found in Fourier Transformation of recording signal change in behavior of harmonic amplitude:

Linear and nonlinear times After ceasing of pumping the main harmonic losses the energy into two

channels the linear process, corresponding to quality of resonator at low excitation and nonlinear energy flux into higher harmonics: . the last term appreciably accelerates the process of vibrations decay of the main harmonic. The prime time of the free decay nonlinear interaction transmits energy into multiple harmonics, amplitudes of which reduce, too.

3 March 2010 Phys.Rev. E

After 2 s the energy flux into multiple harmonics stays vanishing and main harmonic vibration decays with linear time

The difference between should be ascribed to a nonlinear flux into direct energy cascade.



Low frequency f~1000 Hz, N~10;Higher frequency f~5 kHz, N~100


2012

100100 1 ∞

Bad quality Good quality Balk dissipation


Formation of Inverse Cascade

A

f

12

34

5

ω1+ω2→ω3ω3 → ω2+ωf

Dissipation


Formation of Inverse Cascade

A

f

2040

80100

120

104 5

ω f → ω2+ωf

Laboratory of Quantum Crystals

20

Inverse Energy CascadeA

f

2040

80100

120

104 5

ω f → ω2+ωf ω1+ω2→ω3

1000 10000 1000001E-5

1E-4

1E-3

0.01

46 R43 R

35 R31 R

26 R16 R

8 R5 R

51 R

3

2

fd=9594.8 Hz

Am

plitu

de, a

.u.

Frequency, Hz3 March 2010 Chernogolovka, LT seminar

Chernogolovka, June 2008

5.140 5.141 5.142 5.143-0.04

-0.02

0.00

0.02

0.04

0.06

7.590 7.591 7.592 7.593-0.04

-0.02

0.00

0.02

0.04

0.06

8.312 8.313 8.314 8.315-0.04

-0.02

0.00

0.02

0.04

0.06

3.280 3.281 3.282 3.283-0.04

-0.02

0.00

0.02

0.04

0.06

Y A

xis

Title

X Axis Title

B

A B

CD

100 1000 10000

1E-5

1E-4

1E-3

0.01

Frequency (Hz)

Am

plitu

de

A~f -1.62

Direct and inverse cascades

1000 10000 1000001E-5

1E-4

1E-3

0.01

46 R43 R

35 R31 R

26 R16 R

8 R5 R

51 R

3

2

fd=9594.8 Hz

Am

plitu

de, a

.u.

Frequency, Hz

ω1→ω2+ω3

ω1 + ω2 → ω3



2012

Energy transformation at acoustic turbulence

0 20000 40000 60000 80000 100000 1200000.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14 Total Left Right

Ene

rgy,

a.v

.

Time, ms



Low frequency f~1000 Hz, N~10;Higher frequency f~5 kHz, N~100;Very low frequency f~100 Hz, N~1


2012

100100 1 ∞

Bad quality Good quality Bulk dissipation


δT~100 -200 μK

Direct formation of higher harmonics

4 Resonance fG =185.9 HzfG =194.2 Hz (fW =8* fG =4* fD)


2012



2012


Conclusion One-dimensional system of strong

nonlinear wave of second sound allowed to modelling a behaviour of Burgers waves.

The direct cascade of energy flux was observed in discrete system.

At some condition the inverse cascade of energy formats in the system of nonlinear second sound waves

Was observed a new effect of direct formation of multiple harmonics


2012


Direct formation of higher harmonics

fG =198 Hz (fW =6* fG =3* fD)fG =194.2 Hz (fW =8* fG =4* fD)


2012

fG =206.5 Hz (fW =4* fG =2* fD)

Resonance position

50 100 150 200 250 300 3500

2

4

6

8

10

5 643

Upp

=3.6 V f=54.9 Hz U

pp=4.22 V

N o

f pea

ks

f, Hz

Resonance 2

Fractional frequency transformation Standing waves

0 1 2 3 4 5Heater Bolometer

We have 5 waves at distance 6 length of resonatorThe resonance circumstance:

2*N*L=n*λ


THANK YOU FOR YOUR ATTENTION!


2012

Direct Energy Transformation in a System of Nonlinear Second Sound Waves

Documents

Transcript of Direct Energy Transformation in a System of Nonlinear Second Sound Waves