Nazarenko, Warwick Dec 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

Nazarenko, Warwick Dec Plan of the talk: Statistical waves – “Wave Turbulence”. Kolmogorov-Zakharov cascades. Non-gaussianity of wave PDF; intermittency. Discreteness effects; sanpile behaviour of energy cascade.

Nazarenko, Warwick Dec What is Wave Turbulence? WT describes a stochastic field of weakly interacting dispersive waves.

Nazarenko, Warwick Dec 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.

Nazarenko, Warwick Dec How can we describe WT? Hamilitonian equations for the wave field. Weak nonlinearity expansion. Separation of the linear and nonlinear timescales. Statistical averaging, - closure.

Nazarenko, Warwick Dec Free surface motion r is a 2D vector in horizonal plane; z is the vertical coordinate

Nazarenko, Warwick Dec Zakharov equation Deep water waves,  2 = gk, W is complicated (Krasitski 1992)

Nazarenko, Warwick Dec Frequency renormalization

Nazarenko, Warwick Dec Statistical variables in WT Amplitude & phase: a k = A k  k ;  k = exp(i  k ). Stationary distribution of  k – unsteady distribution of  k, Random  k - correlated  k.

Nazarenko, Warwick Dec Statistical objects in WT Spectrum n k = E.g. Kolmogorov-Zakharov spectrum N-mode PDF: probability for A k 2 to be in [s k, s k +ds k ] and for  k to be in [ξ k, ξ k +dξ k ], P (N) {s,ξ} = ; s={s 1,s 2,…,s N }; A={A 1,A 2,…,A N }; ξ={ξ 1,ξ 2,…,ξ N };  ={  1,  2,…,  N }.

Nazarenko, Warwick Dec 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.

Nazarenko, Warwick Dec RPA fields are not Gaussian. Gaussian distribution means P (a) (s) ~ e -s/n. RPA does not fix the amplitude PDF.

Nazarenko, Warwick Dec Weak nonlinearity expansion Choose T in between the linear and nonlinear timescales:

Nazarenko, Warwick Dec Iterations

Nazarenko, Warwick Dec Evolution of WT statistics Substitute value of a k (T) into the PDF definition. Apply RPA to a k (0). Replace [P(T)-P(0)]/T with ∂ t P.

Nazarenko, Warwick Dec Equation for the N-mode PDF (Choi, Lvov & SN, 2004) Where F j is the j-component of the flux,

Nazarenko, Warwick Dec Use of the N-mode PDF Validation that RPA holds over the nonlinear evolution time Non-Gaussian statistics of the wave amplitudes

Nazarenko, Warwick Dec Single-mode staitstics Eqn. for the 1-mode PDF (Choi, Lvov, SN, 2003):

Nazarenko, Warwick Dec Kinetic equation for spectrum Taking 1 st moment of the 1-mode PDF eqn: Hasselmann, 1963

Nazarenko, Warwick Dec 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

Nazarenko, Warwick Dec 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

Nazarenko, Warwick Dec Steady state PDF Gaussian core, non-gaussian tail Choi, Lvov, SN, 2003

Nazarenko, Warwick Dec Direct Numerical Simulations Truncated (at the 3 rd order in amplitude) Euler equations for the free water surface. Pseudo-spectral method 256X256.

Nazarenko, Warwick Dec Energy spectrum Onorato et al’ 02, Dyachenko et al’03, Nakoyama’04, Lvov et al’05

Nazarenko, Warwick Dec One-mode PDF Anomalously high amplitude of large waves – Freak Waves

Nazarenko, Warwick Dec Correlation of  ’s. In agreement with WT,  ’s are decorrelated from A’s and among themselves

Nazarenko, Warwick Dec  ’s are correlated! Correct theory is based on random  ’s and not random  ’s.

Nazarenko, Warwick Dec Exact 4-wave resonances Collinear (Dyachenko et al’94): all 4 k’s parallel to each other (unimportant – null interaction). Symmetric (Kartashova’98): |k 1 | = |k 3 |, |k 2 | = |k 4 | or |k 1 | = |k 4 |, |k 2 | = |k 3 |. Tridents (Lvov et al’05): k 1 anti-parallel k 3, k 2 is mirror- symmetric with k 4 with respect to k 1 -k 3 axis.

Nazarenko, Warwick Dec Tridents Parametrisation (SN’05):

Nazarenko, Warwick Dec Cascade on quasi-resonances Cascade starts at resonance broadening << k- grid spacing It is anisotropic and supported by small fraction of k’s.

Nazarenko, Warwick Dec Frequency peaks at fixed k. 2 nd peak is contribution of k/2 mode Weak turbulence: 1 st peak << 2 nd peak Sometimes 2 nd peak gets > 1 st peak Diagnostics of nonlinear activity

Nazarenko, Warwick Dec Phase runs Phase runs – diagnostics of nonlinear activity

Nazarenko, Warwick Dec “Sandpile avalanches” Nonlinear activity at k 1 and k 2 are correlated (k 2 >k 1 ), It is time delayed and amplified at k 2 with respect k 1. “sandpile avalanches”.

Nazarenko, Warwick Dec 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.

Nazarenko, Warwick Dec 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.

Nazarenko, Warwick Dec Phases vs Phase factors Illustration through an example : Phases are correlated, because Phase factors are statistically independent,

Nazarenko, Warwick Dec 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 [- ,  ].

Nazarenko, Warwick Dec Dispersion of the phase is always positive and the phase fluctuations experience ultimate growth (linear in steady state)

Nazarenko, Warwick Dec 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.