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Wave Wave turbulenceturbulence in in resonatorsresonators

Wave Wave turbulenceturbulence in in resonatorsresonatorsElena Kartashova, RISCElena Kartashova, RISC

18.10.07, 18.10.07, Physical ColloquiumPhysical Colloquium

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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

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Background: A Wave

aWave height

Wavelength -

, wave period -

kT

, wave frequency -k

k T

2Wave number

-2k

Wave energy kE2a

)sin( tkx oscillates as

oscillates as

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Resonator – finite-size system

LINEAR 1D-WAVEviolin 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

,ckk L

nkn

nL

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Enrico Fermi (1901-1954)

NONLINEAR 1D-STRING 1938 Nobel Prise

(nuclear reaction)

1953 Numerical computations on

MANIAC, with Pasta & Ulam1955 Publication of

FPU problem

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Fermi-Pasta-Ulam problem

Number of harmonicsfor numerical simulations:n=32, 64

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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.

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Two possible scenarioswhy 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.

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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

),( txa

)sin( txk

),( 21 kkk

21,kk

),( 21 kkk

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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.

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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.

Radar image of a tropical cyclone.

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Physical scales when discreteness is important?

• Capillary waves: < 4cm (desktop experiments)

• Water gravity waves: ~ 0.1 - 1000m (laboratory experiments, Chanels, Bays, Seas)

• Oceanic Rossby waves: ~ 100-200 km (Oceans)

• Atmospheric Rossby waves: ~ 1000-5000 km (Earth Atm.)

Main message: is important!

/L

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Main resultDiscrete 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.

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Powerful trick

RESULTS: 1. Math: the problem is equivalent to the Fermat‘s Last theorem2. Phys: capillary waves do not have 3-wave resonances in a rectangular box3. 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.

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Structure of Resonancesatmospheric planetary waves

2500 Fourierharmonics (m, n <= 50)

Only 128are important

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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

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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 2. 10.000 atmospheric measurements per day are available

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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 )

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Model of laminated turbulence

1 layer –statistical(classical WTT)

2 layers –statisticaland discrete

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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)