Nazarenko, Warwick Dec 8 2005 Wave turbulence beyond spectra Sergey Nazarenko, Warwick, UK...

Nazarenko, Warwick Dec 8 2005

Wave turbulence beyond spectra

Sergey Nazarenko, Warwick, UK

Collaborators: L. Biven, Y. Choi, Y. Lvov, A.C. Newell, M. Onorato, B. Pokorni, & V.E. Zakharov


Plan of the talk:

Statistical waves – “Wave Turbulence”. Kolmogorov-Zakharov cascades. Non-gaussianity of wave PDF;

intermittency. Discreteness effects; sanpile behaviour

of energy cascade.


What is Wave Turbulence?WT describes a stochastic field of weakly

interacting dispersive waves.


Other Examples of Wave Turbulence:

Sound waves, Plasma waves, Spin waves, Waves in Bose-Einstein condensates, Interstellar turbulence & solar wind, Waves in Semi-conductor Lasers.


How can we describe WT?

Hamilitonian equations for the wave field.

Weak nonlinearity expansion. Separation of the linear and nonlinear timescales.

Statistical averaging, - closure.


Free surface motionr is a 2D vector in horizonal plane; z is the vertical coordinate


Zakharov equation

Deep water waves, 2 = gk, W is complicated (Krasitski 1992)


Frequency renormalization


Statistical variables in WT

Amplitude & phase: ak = Akk ; k = exp(ik).

Stationary distribution of k – unsteady distribution of k ,

Random k - correlated k .


Statistical objects in WT

Spectrum nk = <Ak2>

E.g. Kolmogorov-Zakharov spectrum

N-mode PDF: probability for Ak

2 to be in [sk, sk +dsk] and for k to be in [ξk, ξk+dξk] ,

P (N) {s,ξ} = <δ(s-A2) δ(ξ-)>;

s={s1,s2,…,sN}; A={A1,A2,…,AN}; ξ={ξ1,ξ2,…,ξN}; ={1,2,…,N}.


Random Phase & Amplitude (RPA) wavefield: All the amplitudes and the phase factors

are independent random variables,

The phase factors are uniformly distributed on the unit circle in the complex plane.


RPA fields are not Gaussian.

Gaussian distribution means P(a)(s) ~ e-s/n.

RPA does not fix the amplitude PDF.


Weak nonlinearity expansion

Choose T in between the linear and nonlinear timescales:


Iterations


Evolution of WT statistics

Substitute value of ak(T) into the PDF definition.

Apply RPA to ak(0) . Replace [P(T)-P(0)]/T with ∂t P.


Equation for the N-mode PDF (Choi, Lvov & SN, 2004)

Where Fj is the j-component of the flux,


Use of the N-mode PDF

Validation that RPA holds over the nonlinear evolution time

Non-Gaussian statistics of the wave amplitudes


Single-mode staitstics

Eqn. for the 1-mode PDF (Choi, Lvov, SN, 2003):


Kinetic equation for spectrum

Taking 1st moment of the 1-mode PDF eqn:

Hasselmann, 1963


Kolmogorov-Zakharov spectra

Power-law spectra describing a down-scale energy cascade and an up-scale wave-action cascade

WT may break at a large or small scale


Breakdown of WT

Water surface: wavebreaking means there is no amplitudes higher than critical

Hard breakdown, n~s*, Biven, Newell, SN, 2001

Weak breakdown, n<<s*. Choi, Lvov, SN, 2003


Steady state PDF

Gaussian core, non-gaussian tail

Choi, Lvov, SN, 2003


Direct Numerical Simulations

Truncated (at the 3rd order in amplitude) Euler equations for the free water surface.

Pseudo-spectral method 256X256.


Energy spectrum

Onorato et al’ 02, Dyachenko et al’03, Nakoyama’04, Lvov et al’05


One-mode PDF

Anomalously high amplitude of large waves – Freak Waves


Correlation of ’s.

In agreement with WT, ’s are decorrelated from A’s and among themselves


’s are correlated!

Correct theory is based on random ’s and not random ’s.


Exact 4-wave resonances

Collinear (Dyachenko et al’94): all 4 k’s parallel to each other (unimportant – null interaction).

Symmetric (Kartashova’98): |k1| = |k3|, |k2| = |k4|

or |k1| = |k4|, |k2| = |k3| . Tridents (Lvov et al’05): k1 anti-parallel k3, k2 is

mirror-symmetric with k4 with respect to k1 -k3 axis.


Tridents

Parametrisation (SN’05):


Cascade on quasi-resonances

Cascade starts at resonance broadening << k-grid spacing

It is anisotropic and supported by small fraction of k’s.


Frequency peaks at fixed k.

2nd peak is contribution of k/2 mode

Weak turbulence: 1st peak << 2nd peak

Sometimes 2nd peak gets > 1st peak

Diagnostics of nonlinear activity


Phase runs

Phase runs – diagnostics of nonlinear activity


“Sandpile avalanches”

Nonlinear activity at k1 and k2 are correlated (k2>k1),

It is time delayed and amplified at k2 with respect k1.

“sandpile avalanches”.


Cycle of discrete turbulence Weak turbulence at forcing scale –

no resonance, no cascade. Energy accumulation, growth of

nonlinearity. Nonlinear resonance broadening,

cascade activation – “avalanche”. Avalanche drains energy from the

forcing scale, -> beginning of the cycle.


Summary

Generalised WT description: PDF. Kolmogorov-Zakharov spectrum. RPA validation. Correlations of phases. Anomalous distribution of waves with

high amplitudes Discreteness effects: exact and quasi-

resonances, sanpile behavior of cascade.


Phases vs Phase factors

Illustration through an example :

Phases are correlated, because

Phase factors are statistically independent,


Mean phase

Expression for phase Evolution eqn.

the mean value of the phase is steadily changing over the nonlinear time and it would be incorrect to assume that phases remains uniformly distributed in [-,].


Dispersion of the phase

is always positive and the phase fluctuations experience ultimate growth (linear in steady state)


Essentially RPA fields

The amplitude variables are almost independent is a sense that for each M<<N modes the M-mode amplitude PDF is equal to the product of the one-mode PDF’s up to and corrections.

Nazarenko, Warwick Dec 8 2005 Wave turbulence beyond spectra Sergey Nazarenko, Warwick, UK...

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