1 Wave turbulence in resonators Elena Kartashova, RISC , Physical Colloquium

2 Main goals of this talk To put wave trubulence theory into the general physical context To show math. difficulties and give an idea of the solution methods To demostrate how to use the results to describe a real physical phenomenon

3 Background: A Wave Wave height Wavelength -, wave period -, wave frequency -Wave number - Wave energy oscillates as

4 Resonator – finite-size system LINEAR 1D-WAVE violin string or organ pipe is a length of the string, is arbitrary integer. Different initial conditions: initial frequency - half a period (pure tone, one Fourier harmonic) energy concentrated in ONE WAVE initial frequency - arbitrary (overtones, all Fourier harmonics) energy distributed among ALL THE WAVES

5 Enrico Fermi ( ) NONLINEAR 1D-STRING 1938 Nobel Prise (nuclear reaction) 1953 Numerical computations on MANIAC, with Pasta & Ulam 1955 Publication of FPU problem

6 Fermi-Pasta-Ulam problem Number of harmonics for numerical simulations: n=32, 64

7 Numerical results Results: one wave decays into two waves which satisfy time and space synchronization conditions (resonance conditions) and the wave system demonstrates almost exactly periodic behavior: energy goes into other waves and after some time it COMES BACK into the first wave. Expectations due to the Boltzmann theorem: evolution of the INFINITE system from arbitrary initial conditions yields equipartitioned distribution of energy among the waves.

8 Two possible scenarios why the Bolzmann theorem does not work Closeness to an integrable system Discretness of eigen-modes in bounded resonators FPU is not resolved till now. One of the reasons is degeneracy of conservation laws due to 1D and linear dispersion law. On the other side, there are many physically interesting 2D waves in bounded domains with no degeneracy of conservation laws.

9 Nonlinear waves in 2D- media 2D-waves propagate in two directions height of the wave oscillates as Now we have wave vector (not wave number) in resonators are integers dispersion function is

10 Examples 1. Capillary waves (due to surface tension): 2. Water gravity surface waves (due to gravity acceleration): 3. Oceanic planetary waves (due to the Earth rotation), called also Rossby waves: 1. 2.

11 Rossby Waves in the Earth Atmosphere RW detach the masses of cold/warm air that become cyclones and anticyclones and are responsible for day-to- day weather patterns.cyclones anticyclones Radar image of a tropical cyclone.

12 Physical scales when discreteness is important? Capillary waves: < 4cm (desktop experiments) Water gravity waves: ~ m (laboratory experiments, Chanels, Bays, Seas) Oceanic Rossby waves: ~ km (Oceans) Atmospheric Rossby waves: ~ km (Earth Atm.) Main message: is important!

13 Main result Discrete resonances DO exist in many physically relevant 2D-wave systems! What was the problem? Left part has no roots, right part still have them, After taking 8 times power 2, we get terms of commulative degree 16 and some roots are still left. Already at this step, for wave numbers ~ 1000, we need to make computations with integers of order.

14 Powerful trick RESULTS: 1. Math: the problem is equivalent to the Fermat‘s Last theorem 2. Phys: capillary waves do not have 3-wave resonances in a rectangular box 3. Comp: computation time reduced from weeks to 5-15 minutes (for dozens of physically relevant dispersion functions) Last equation is a special case of the Fermat’s Last theorem proven by Euler for n=3.

15 Structure of Resonances atmospheric planetary waves 2500 Fourier harmonics (m, n <= 50) Only 128 are important

16 Important remark To applicate this reasoning to a real physical problem we have answer the question about quasi-resonances: is called resonance width and corresponds to the accuracy of laboratory experiments

17 Intra-seasonal oscillations in the Earth atmosphere linear part nonlinear part The beauty of this example is due to 2 facts: 1.Dynamical system is solvable analytically atmospheric measurements per day are available

18 Statistical Wave Turbulence (STW) Kolmogorov 1941,61 (locally homogeneous & isotropic, existence of an inertial interval -> power-law energy spectrum) Hasselman 1962, Zakharov 1967 (wave kinetic equation ~ Bolzmann equation) Kolmogorov 1961, Arnold 1961, Moser 1962 (KAM-theory: small divisor problem) Zakharov, Shulman 1981 (KAM-theory for SWT ) Zakharov, L’vov, Falkovich 1992 (strict mathematical theory of SWT, without discrete effects )

19 Model of laminated turbulence 1 layer – statistical (classical WTT) 2 layers – statistical and discrete

20 MAIN RESULTS Existence of exact resonances is established for many physically relevant wave systems, computational methods are developed and implemented Dynamics of a wave system in resonator is defined by a few independent wave clusters Model of intra-seasonal oscillations in the Earth atmosphere is developed (important for weather and climate predictability) Some more… Further studies are in PROGRESS (theoretical – with Weizmann Institute, Israel; experimental – with Warwick & Hull Universities, UK)