Abstract
A threewave coupling model with complex linear frequencies is investigated for the nonlinear interaction in a triad that has linearly unstable and stable modes. Time scales associated with linear and nonlinear physics are identified and compared with features of the frequency spectrum. From appropriate time scales, the frequency spectra are well characterized even in the transition to the steady state. The nonlinear time scales that best match spectral features are the nonlinear frequency of the fixed point and a frequency that depends on the amplitude displacement from the fixed point through the largeamplitude Jacobian elliptic solution. Two limited efforts to model the effect of other triads suggest robustness in the single triad results.
This work is support by the grant DEFG0289ER53291 of Department of Energy.
I. INTRODUCTION
II. MODELEQUATION
III. TRIAD FIXED POINT AND STABILITY
A. Resonant triad Δω = 0
B. Nonresonant triad Δω ≠ 0
IV. SATURATION AND RELAXATION OF A TRIAD
V. FREQUENCY SPECTRUM OF A STABLE TRIAD
VI. PERTURBED TRIAD INTERACTION
VII. CONCLUSION
Key Topics
 Turbulent flows
 20.0
 Turbulence simulations
 19.0
 Energy transfer
 12.0
 Plasma turbulence
 10.0
 Plasma waves
 8.0
Figures
(Color online) Stability diagram of Δω = −2.5 and Im ω_{1} = 0.3. The horizontal and vertical axes represent γ_{2} and γ_{3}. The thick blue line (dashed) shows the stability condition Δγ = γ_{ c } = 3Γ_{3}/2Γ_{2} in Eq. (20). The thick red line (solid) shows Re σ = 0. Triad (I) (+), Triad (II) (Δ), and Triad (III) (⋄) are shown for Sec. IV.
(Color online) Stability diagram of Δω = −2.5 and Im ω_{1} = 0.3. The horizontal and vertical axes represent γ_{2} and γ_{3}. The thick blue line (dashed) shows the stability condition Δγ = γ_{ c } = 3Γ_{3}/2Γ_{2} in Eq. (20). The thick red line (solid) shows Re σ = 0. Triad (I) (+), Triad (II) (Δ), and Triad (III) (⋄) are shown for Sec. IV.
(Color online) The evolution of the mode energies (left) and the triad phase (right), from the numerical simulations of Eq. (1) for Triad (I). The black (solid), red (dotted), and blue (dashed) traces represent modes 1, 2, and 3. The red horizontal line indicates the triad phase of the fixed point, Δψ_{0}.
(Color online) The evolution of the mode energies (left) and the triad phase (right), from the numerical simulations of Eq. (1) for Triad (I). The black (solid), red (dotted), and blue (dashed) traces represent modes 1, 2, and 3. The red horizontal line indicates the triad phase of the fixed point, Δψ_{0}.
(Color online) The evolution of the mode energies (left) and the triad phase (right), from the numerical simulations of Eq. (1) for Triad (II) (top) and Triad (III) (bottom). The legends in the plots are described in Fig. 2.
(Color online) The evolution of the mode energies (left) and the triad phase (right), from the numerical simulations of Eq. (1) for Triad (II) (top) and Triad (III) (bottom). The legends in the plots are described in Fig. 2.
(Color online) Time evolution of three phases ψ_{ i } [black (+), red (*), blue (⋄)] and the triad phase Δψ [green (△)] of the unstable Triad (II) (left) and the stable Triad (III) (right) at t = 90−95.
(Color online) Time evolution of three phases ψ_{ i } [black (+), red (*), blue (⋄)] and the triad phase Δψ [green (△)] of the unstable Triad (II) (left) and the stable Triad (III) (right) at t = 90−95.
(Color online) The frequency spectra of the mode ψ_{1} (black, solid), the mode ψ_{2} (red, dotted) and the mode ψ_{3} (blue, dashed) are plotted at t = (a) 100, (b) 125, (c) 150, (d) 200, (e) 250. The vertical lines represent (black, solid), (red, dotted), (blue, dashed), σ_{ f } (green, long dash), and ω_{amp} (black, dash dot). The arrows and the shaded region refer to the interval bounded by .
(Color online) The frequency spectra of the mode ψ_{1} (black, solid), the mode ψ_{2} (red, dotted) and the mode ψ_{3} (blue, dashed) are plotted at t = (a) 100, (b) 125, (c) 150, (d) 200, (e) 250. The vertical lines represent (black, solid), (red, dotted), (blue, dashed), σ_{ f } (green, long dash), and ω_{amp} (black, dash dot). The arrows and the shaded region refer to the interval bounded by .
(Color online) The evolution of (a) amplitudes φ_{ i } (b) the triad phase Δψ (c) n _{ i } (d) m _{ i } are shown with the initial condition of Ψ = Ψ_{0} and Δψ = −Δψ_{0}. Black (solid), red (dotted), and blue (dashed) lines in (a,c,d) represent the modes i = 1, 2, 3.
(Color online) The evolution of (a) amplitudes φ_{ i } (b) the triad phase Δψ (c) n _{ i } (d) m _{ i } are shown with the initial condition of Ψ = Ψ_{0} and Δψ = −Δψ_{0}. Black (solid), red (dotted), and blue (dashed) lines in (a,c,d) represent the modes i = 1, 2, 3.
(Color online) ((a) and (b)) The frequency spectra at t = 0,75 in Fig. 6(c) (Re φ_{1}, Im φ_{1}) at t = 0,75. The legends for the spectra are the same as in Fig. 5 and the red dashed line in (c) and (d) represent the mode 1 at the fixed point.
(Color online) ((a) and (b)) The frequency spectra at t = 0,75 in Fig. 6(c) (Re φ_{1}, Im φ_{1}) at t = 0,75. The legends for the spectra are the same as in Fig. 5 and the red dashed line in (c) and (d) represent the mode 1 at the fixed point.
(Color online) [β = i] (a and d) Φ_{ i } (b and e) Δψ and (c and f) the frequency spectra at t = 325(c) and 200(f). The top panel (a, b, c) and the bottom panel (d, e, f) corresponds to the strength of the perturbation f _{0} = 0.1 and f _{0} = 0.5. Each plot has the same legends as described in Figs. 2 and 5.
(Color online) [β = i] (a and d) Φ_{ i } (b and e) Δψ and (c and f) the frequency spectra at t = 325(c) and 200(f). The top panel (a, b, c) and the bottom panel (d, e, f) corresponds to the strength of the perturbation f _{0} = 0.1 and f _{0} = 0.5. Each plot has the same legends as described in Figs. 2 and 5.
(Color online) [β = 1] (a and d) Φ_{ i } (b and e) Δψ and (c and f) the frequency spectra at t = 325. The top panel (a, b, c) and the bottom panel (d, e, f) corresponds to the strength of the perturbation f _{0} = 0.1 and f _{0} = 0.2. Each plot has the same legends as described in Figs. 2 and 5.
(Color online) [β = 1] (a and d) Φ_{ i } (b and e) Δψ and (c and f) the frequency spectra at t = 325. The top panel (a, b, c) and the bottom panel (d, e, f) corresponds to the strength of the perturbation f _{0} = 0.1 and f _{0} = 0.2. Each plot has the same legends as described in Figs. 2 and 5.
(Color online) Diagram of two triad connection.
(Color online) Diagram of two triad connection.
(Color online) In the case of tan Δψ_{1} = tan Δψ_{2} and σ_{1} < σ_{2}, the amplitudes of (a) the modes 1 (black, solid), 2 (red, dotted), 3 (blue, dashed) of triad 1 and (b) the modes 1′ (black, solid), 2′ (red, dotted), 3′ (blue, dashed) of triad 2 are shown in time.
(Color online) In the case of tan Δψ_{1} = tan Δψ_{2} and σ_{1} < σ_{2}, the amplitudes of (a) the modes 1 (black, solid), 2 (red, dotted), 3 (blue, dashed) of triad 1 and (b) the modes 1′ (black, solid), 2′ (red, dotted), 3′ (blue, dashed) of triad 2 are shown in time.
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