Miguel D. Bustamante School of Mathematical Sciences ...€¦ · Conclusions 21/05/2013 Miguel D....
Transcript of Miguel D. Bustamante School of Mathematical Sciences ...€¦ · Conclusions 21/05/2013 Miguel D....
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Miguel D. Bustamante School of Mathematical Sciences
University College Dublin
Collaborator: B. Quinn
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Outline Some (accepted?) statements:
Strong turbulence High nonlinearity
Wave turbulence small amplitudes & resonant N-wave
interactions (plus quasi-resonant) Evidence against these statements: Discovery of a new nonlinear transfer mechanism, stronger
at intermediate values of nonlinearity Mechanism favours transfers towards non-resonant triads
rather than quasi-resonant / resonant triads Robust result, backed up with
Direct Numerical Simulations of nonlinear wave systems
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Interaction Representation Equations for generic finite-sized 3-wave system:
𝜕𝑏𝒌𝜕𝑡
= 𝑉12,𝒌(1)
𝛿(𝒌 − 𝒌1 − 𝒌2) 𝑏𝒌𝟏 𝑏𝒌𝟐 𝑒𝑖(𝜔𝒌−𝜔1−𝜔2)𝑡
𝒌𝟏,𝒌𝟐∈ ℤ𝟐
+ 𝑉12,𝒌2 𝛿 𝒌 + 𝒌1 − 𝒌2 𝑏𝒌𝟏 𝑏𝒌𝟐 𝑒
𝑖(𝜔𝒌+𝜔1−𝜔2)𝑡
𝒌𝟏,𝒌𝟐∈ ℤ𝟐
+ 𝑉12,𝒌3 𝛿 𝒌 + 𝒌1 + 𝒌2 𝑏𝒌𝟏 𝑏𝒌𝟐 𝑒
𝑖(𝜔𝒌+𝜔1+𝜔2)𝑡
𝒌𝟏,𝒌𝟐∈ ℤ𝟐
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Robust Instability Mechanism (1/3) 𝜕𝑏𝒌𝜕𝑡
= 𝑉12,𝒌(1)
𝛿(𝒌 − 𝒌1 − 𝒌2) 𝑏𝒌𝟏 𝑏𝒌𝟐 𝑒𝑖(𝜔𝒌−𝜔1−𝜔2)𝑡
𝒌𝟏,𝒌𝟐∈ ℤ𝟐
+ 𝐞𝐭𝐜…
Limit of small amplitudes (|𝑏𝒌𝒋| ≪ 1, for all 𝑗):
phases 𝑒𝑖(𝜔𝒌−𝜔1−𝜔2)𝑡 rotate faster than amplitudes 𝑏𝒌𝒋
each term in the RHS averages to zero at
intermediate time scales
(Exception: exact resonances 𝜔𝒌 − 𝜔1 − 𝜔2 = 0, but these are irrelevant for the mechanism of this talk)
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Robust Instability Mechanism (2/3) 𝜕𝑏𝒌𝜕𝑡
= 𝑉12,𝒌(1)
𝛿(𝒌 − 𝒌1 − 𝒌2) 𝑏𝒌𝟏 𝑏𝒌𝟐 𝑒𝑖(𝜔𝒌−𝜔1−𝜔2)𝑡
𝒌𝟏,𝒌𝟐∈ ℤ𝟐
+ 𝐞𝐭𝐜…
Increase initial amplitudes 𝑏𝒌𝒋 from infinitesimally
small to finite values:
𝑏𝒌𝒋’s nonlinear oscillation frequency, 𝛤, grows
proportional to the amplitudes, until resonance occurs:
nonlinear 𝛤 ~ 𝜔𝒌 − 𝜔1 − 𝜔2 linear
some terms in the RHS will contain zero modes
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Robust Instability Mechanism (3/3) 𝜕𝑏𝒌𝜕𝑡
= 𝑉12,𝒌(1)
𝛿(𝒌 − 𝒌1 − 𝒌2) 𝑏𝒌𝟏 𝑏𝒌𝟐 𝑒𝑖(𝜔𝒌−𝜔1−𝜔2)𝑡
𝒌𝟏,𝒌𝟐∈ ℤ𝟐
+ 𝐞𝐭𝐜…
Zero modes in RHS:
Some amplitudes 𝑏𝒌 grow linearly in time, without
bound: 𝑏𝒌 𝑡 ~ 𝑐 𝑡, 𝑐 = const. , even from zero i.c.
Far more robust than modulational instability!!!
Can this mechanism be observed/modelled?
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Quantitative Studies of Strong Transfer Mechanism Finite-dimensional ODE model:
Two triads connected via two common modes
Initial conditions: 𝑏𝒌 0 = 𝐴 𝑏𝒌ref 0 , 𝐴: arbitrary const.
Energy flows from source triad to target triad
Physical mechanism and “linear-nonlinear” resonance
Full direct numerical simulation of a PDE model
Initial conditions: 𝑏𝒌 0 = 𝐴 𝑏𝒌ref 0 , 𝐴: arbitrary const.
Study turbulent cascades & transfer efficiency as a function of 𝐴
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ODE Model: The “Atom” Wave-vectors: 𝒌1 + 𝒌2 = 𝒌3, 𝒌2 + 𝒌3 = 𝒌4 Frequency mismatches:
𝛿𝑆 = 𝜔1 + 𝜔2 − 𝜔3 𝛿𝑇 = 𝜔2 +𝜔3 − 𝜔4 Take 𝛿𝑆 = 0 for simplicity (not essential) Equations of motion: 𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3
𝐵 2 = 𝑆2 𝐵1∗ 𝐵3+ 𝑇1 𝐵3
∗ 𝐵4 𝑒𝑖𝛿𝑇 𝑡
𝐵 3 = 𝑆3 𝐵1 𝐵2+ 𝑇2 𝐵2∗ 𝐵4 𝑒
𝑖𝛿𝑇 𝑡
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡, 𝐵𝑗 𝑡 ∈ ℂ
Initial Conditions:
𝐵1 0 , 𝐵2 0 , 𝐵3 0 ≠ 0 𝐵4 0 = 0
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ODE Model: The “Atom” 𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3
𝐵 2 = 𝑆2 𝐵1∗ 𝐵3+ 𝑇1 𝐵3
∗ 𝐵4 𝑒𝑖𝛿𝑇 𝑡
𝐵 3 = 𝑆3 𝐵1 𝐵2+𝑇2 𝐵2∗ 𝐵4 𝑒
𝑖𝛿𝑇 𝑡
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡, 𝐵𝑗 𝑡 ∈ ℂ
𝐵1 0 , 𝐵2 0 , 𝐵3 0 ≠ 0 𝐵4 0 = 0
8-dimensional phase space
2 quadratic conservation laws & 2 slave variables Effectively 4 degrees of freedom
Boundedness: ∃ positive-definite conservation law
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ODE Model: The “Atom” 𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3
𝐵 2 = 𝑆2 𝐵1∗ 𝐵3+𝑇1 𝐵3
∗ 𝐵4 𝑒𝑖𝛿𝑇 𝑡
𝐵 3 = 𝑆3 𝐵1 𝐵2+ 𝑇2 𝐵2∗ 𝐵4 𝑒
𝑖𝛿𝑇 𝑡
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡, 𝐵𝑗 𝑡 ∈ ℂ
Initially 𝐵4 0 = 0, 𝐵1 0 , 𝐵2 0 , 𝐵3 0 ≠ 0
Assume |𝐵4 𝑡 | remains small for all times system is further approximated by:
𝐵 1 = 𝑆1 𝐵2∗ 𝐵3 𝐵 2 = 𝑆2 𝐵1
∗ 𝐵3 𝐵 3 = 𝑆3 𝐵1 𝐵2
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡
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ODE Model: The “Atom”
Assume 𝐵4 𝑡 is small:
𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3 𝐵 2 = 𝑆2 𝐵1∗ 𝐵3 𝐵 3 = 𝑆3 𝐵1 𝐵2
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡
𝐵1, 𝐵2, 𝐵3 satisfy the usual integrable triad equations
𝐵4(𝑡) obtained by quadratures after 𝐵2, 𝐵3 are known
Triad: Jacobi Elliptic functions 𝐛𝐨𝐮𝐧𝐝𝐞𝐝, quasi-periodic motion
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ODE Model: The “Atom” 𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3 𝐵 2 = 𝑆2 𝐵1
∗ 𝐵3 𝐵 3 = 𝑆3 𝐵1 𝐵2
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡
Amplitude-Phase representation: 𝐵𝑗 𝑡 = 𝐵𝑗 𝑡 𝑒𝑖 𝜑𝑗(𝑡)
𝐵1 𝑡 , 𝐵2 𝑡 , 𝐵3 𝑡 , 𝜑 𝑡 = 𝜑1 𝑡 + 𝜑2 𝑡 − 𝜑3 𝑡 are periodic functions with nonlinear frequency
Γ = Γ( 𝐵1 0 , 𝐵2 0 , 𝐵3 0 ; 𝜑 0 )
Homogeneity: Γ 𝐴 𝑥, 𝐴 𝑦,𝐴 𝑧; 𝛼 = 𝐴 Γ 𝑥, 𝑦, 𝑧; 𝛼
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ODE Model: The “Atom” 𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3 𝐵 2 = 𝑆2 𝐵1
∗ 𝐵3 𝐵 3 = 𝑆3 𝐵1 𝐵2
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡
𝐵2 𝑡 = 𝐵2 𝑡 𝑒𝑖 (𝜑2per
𝑡 +Ω2 𝑡)
𝐵3 𝑡 = 𝐵3 𝑡 𝑒𝑖 (𝜑3per
𝑡 +Ω3 𝑡) (exact triad solutions)
𝐵2 𝑡 , 𝐵3 𝑡 , 𝜑2per
𝑡 ,𝜑3per
𝑡 are periodic: nonlinear frequency Γ ~ amplitudes (homogeneity)
Ω2, Ω3: precession frequencies, also ~ amplitudes
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ODE Model: The “Atom” 𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3 𝐵 2 = 𝑆2 𝐵1
∗ 𝐵3 𝐵 3 = 𝑆3 𝐵1 𝐵2
𝑩 𝟒 = 𝑻𝟑 𝑩𝟐 𝑩𝟑 𝒆−𝒊𝜹𝑻 𝒕
Solving by quadratures:
𝐵4 𝑡 = 𝑇3 𝑓per 𝜏𝑡
0 𝑒𝑖 Ω2+Ω3−𝛿𝑇 𝜏 𝑑𝜏
𝑓per(𝑡) ∈ ℂ: periodic, nonlinear frequency Γ
Unbounded growth if resonance occurs:
𝑛 Γ + Ω2 + Ω3 − 𝛿𝑇 = 0, for some 𝑛 ∈ ℤ
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ODE Model: The “Atom” 𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3 𝐵 2 = 𝑆2 𝐵1
∗ 𝐵3 𝐵 3 = 𝑆3 𝐵1 𝐵2
𝑩 𝟒 = 𝑻𝟑 𝑩𝟐 𝑩𝟑 𝒆−𝒊𝜹𝑻 𝒕
𝐵4 𝑡 = 𝑇3 𝑓per 𝜏𝑡
0 𝑒𝑖 Ω2+Ω3−𝛿𝑇 𝜏 𝑑𝜏
Fine-tuning initial conditions via simple re-scaling:
𝐵𝑗 0 𝐴 𝐵𝑗 0
Instability if 𝐴 = 𝐴𝑛 ≡𝛿𝑇
𝑛 Γref+Ω2ref+Ω3
ref , for some 𝑛 ∈ ℤ
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ODE Model: Numerical Study 𝐵 1 = 𝑆1 𝐵2
∗ 𝐵3
𝐵 2 = 𝑆2 𝐵1∗ 𝐵3+𝑇1 𝐵3
∗ 𝐵4 𝑒𝑖𝛿𝑇 𝑡
𝐵 3 = 𝑆3 𝐵1 𝐵2+ 𝑇2 𝐵2∗ 𝐵4 𝑒
𝑖𝛿𝑇 𝑡
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡
𝛿𝑇 = − 8
9,
𝑆1 = 1, 𝑆2 = 9,𝑆3 = −8 𝑇1 = −1, 𝑇2 =
83 , 𝑇2 = −
95
𝐵1 0 = 0.007772 𝐴 𝐵2 0 = 0.038582 𝐴 𝐵3 0 = − 0.0358876 𝑖 𝐴 𝐵4 0 = 0
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𝐴−1 = 4.40
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ODE Model: Numerical Study Introduce an 휀-family of systems:
𝐵 1 = 𝑆1 𝐵2∗ 𝐵3
𝐵 2 = 𝑆2 𝐵1∗ 𝐵3+ 휀 𝑇1 𝐵3
∗ 𝐵4 𝑒𝑖𝛿𝑇 𝑡
𝐵 3 = 𝑆3 𝐵1 𝐵2+ 휀 𝑇2 𝐵2∗ 𝐵4 𝑒
𝑖𝛿𝑇 𝑡
𝐵 4 = 𝑇3 𝐵2 𝐵3 𝑒−𝑖𝛿𝑇 𝑡
where 0 ≤ 휀 ≤ 1
휀 = 0: Integrable system, with new resonant instability
휀 = 1: Full original system
Study transfer efficiency as function of 𝐴 & 휀
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ODE Model: Analysis of Results
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Transfer Efficiency as function of 𝐴 & 휀
Predicted Resonances 𝐴−3, 𝐴−2, 𝐴−1
𝐴𝑛 =𝛿𝑇
𝑛 Γref +Ω2ref +Ω3
ref
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Efficiency Plateau: 𝜔2 + 𝜔3 −𝜔4 ≤ Γ, 휀 ≫ 1
ODE Model: Analysis of Results
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Transfer Efficiency as function of 𝐴 & 휀
Predicted Resonances 𝐴−3, 𝐴−2, 𝐴−1
𝐴𝑛 =𝛿𝑇
𝑛 Γref +Ω2ref +Ω3
ref
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Efficiency Plateau: 𝜔2 + 𝜔3 −𝜔4 ≤ Γ, 휀 ≫ 1
ODE Model: Analysis of Results
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Transfer Efficiency as function of 𝐴 & 휀
Predicted Resonances 𝐴−3, 𝐴−2, 𝐴−1
𝐴𝑛 =𝛿𝑇
𝑛 Γref +Ω2ref +Ω3
ref
Resonances in 휀: Source vs. Target interaction coefficients
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ODE Model: Analysis of Results
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Nature of the Instability
Stiff Ridge: • Persistence of invariant manifold • Unstable periodic orbit
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ODE Model: Analysis of Results
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Nature of the Instability
Stiff Ridge: • Persistence of invariant manifold • Unstable periodic orbit
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ODE Model: Analysis of Results
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Nature of the Instability
Three Lyapunov exponents
Ratios (-1):(-2):(3)
Stiff Ridge: • Persistence of invariant manifold • Unstable periodic orbit
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Barotropic Vorticity Equation
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Direct Numerical Simulations
Pseudospectral
Resolution 128 x 128
Initial conditions at large scales, with an overall re-scaling factor A in front
Dissipation at small scales: ENSTROPHY direct cascade
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Barotropic Vorticity Equation
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Efficiencies
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Barotropic Vorticity Equation
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Efficiency as a function of A
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Barotropic Vorticity Equation
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VIDEOS
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Conclusions
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Robust energy transfer mechanism towards non-resonant triads
Analytically derived and verified numerically: ODE “Atom” model
Direct numerical simulations of a full PDE model
Implications of this mechanism: Understanding turbulent cascades as a natural selection
mechanism of triads
Yet another mechanism of rogue wave generation (triggered either by forcing or dissipation)
New paradigm for a complete theory of wave turbulence
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Thank You!
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This Paper: ArXiv version http://arxiv.org/abs/1305.5517
References: