Nonlinear & Stochastic Growth Processes in Beam-Plasma Systems: Recent Work on Type III Bursts

Transcription

1 Nonlinear & Stochastic Growth Processes in Beam-Plasma Systems: Recent Work on Type III Bursts Iver H. Cairns 1, Daniel B. Graham 1,2, Bo Li 1, A. Layden 1, B. Layden (1 = U. Sydney, 2 = Swed. Int. Sp. Phys.) Thredbo, 2/6/13

2 Outline 1. Introduction: Wave processes, Type III bursts, bursty Langmuir waves, beam persistence 2. Stochastic Growth Theory (SGT) 3. Electrostatic Decay (radiation processes not discussed) 4. Wave Collapse & Modulational Instability 5. Eigenmodes 6. Conclusions

3 1. Introduction to electron beam-plasma interactions in the corona & solar wind Beam growth near k b ω p /v b Saturation movement in k space until loss = gain. Quasilinear diffusion relaxes beam & waves to smaller k Density inhomogeneities refraction, diffusion in k, SGT, & trapping (eigenmodes) Nonlinear mechanisms wave-wave processes, wave collapse & modulational.. radiation processes Nonlinear QL n e k b 3

4 Why wave-wave rather than wave-particle processes? W ave-wave process M1 ± M2 M3 versus waveparticle process M1 ± p ± M3? W-W is more collective / coherent than W-P. Expect nonlinear rates in ratio Γ ww /Γ wp T M2 / T p W-W dominates when mode M2 has nonthermal level. Except need Γ ww min ( ω M1, ω M2, ω M3 ), Γ wp min ( ω M1, ω M3 ) for processes to be well defined. W-P can dominate when W-W doesn t satisfy this. E.G., ES decay will dominate STI except when Γ ES ω S ω p since STI constraint Γ wp ω p is satisfied. 4

5 5

6 Type III radio bursts Electron beams propagate outward from Sun along B, generating Langmuir waves. Spectrogram recorded by STEREO A and B on 2011 February 26. Multiple type III events are observed. Some Langmuir wave energy converted to f p and 2f p radio waves. Inhomogenous plasma: large-scale n e gradient and turbulence. Beam persists for > λ D Image from

7 Why bursty waves & persistent beam? [Lin et al., 1981] Type III solar burst Electron beam Langmuir waves Radio waves: f p & 2f p Earth s foreshock radiation Type II solar bursts Why do waves become bursty and electron beams persist?

8 Langmuir waves in type III source regions Two example of Langmuir events in type III sources: Beats nonlinear processes? Localization trapping, eigenmodes, wave collapse?

9 2. Stochastic Growth Theory (SGT) Waves grow amid ambient fluctuations that perturb wave-particle coupling in an interaction zone. Growth rate fluctuates gain G = dt γ random walks. If N fl» 1 then Central Limit Theorem implies lognormal statistics: [Robinson et al., 1993; C. & Robinson, 1994] P( G) P(log E) exp[ ( G 2 G 0 ) / 2σ 2 ]

10 Type III bursts achieving a SGT state Type III Langmuir waves Linear instability & quasilinear relaxation in inhomogeneous plasma Strong evidence for SGT Also evidence for a nonlinear process at high E 3 mv/m. Log E (V/m) Reasonable threshold for ES decay [Robinson et al., 1993; C. & Robinson, 1995]

11 Data Collapse: SGT If SGT then X = (G <G>) / σ G & P(X) = (2π) -1/2 exp(-x 2 / 2). No free parameters. 8 distinct wave phenomena (pulsars to Langmuir waves and mirror mode waves) Quasilinear simulations (filled circles). All consistent with SGT! Multiple mechanisms? [C & Grubits, 2002;...Robinson et al., 2006]

12 3. Electrostatic Decay in Type III Source Regions

13 3.1.1 Electrostatic Decay: Theory Langmuir waves generated by electron beams have wave number These Langmuir waves can decay into backward propagating Langmuir waves: L L + S. By assuming the linear dispersion relations and the wave numbers are: where Note: k L > k L, k 0 /2 so to lower k, not larger.

14 3.1.2 Doppler-shifted frequencies Langmuir waves are convected past STEREO at v sw. Therefore STEREO will observe waves at Doppler-shifted frequencies: The predicted Doppler-shifted frequency difference is: For ion-acoustic waves, the predicted Doppler-shifted frequency is: [e.g., Cairns and Robinson, 1992b; Henri et al., 2009]

15 3.1.3 Langmuir/z-mode waves In a magnetized thermal plasma Langmuir waves connect to the magnetoionic z-mode wave to form the Langmuir/z-mode wave. [Willes & Cairns, 2000; Layden et al., 2013] Langmuir portion of the mode is electrostatic (E parallel to B 0 ). Z-mode portion is electromagnetic (E perpendicular to B 0 ). F = E perp 2 /E tot 2 is the proportion of perpendicular energy density to total energy density.

16 3.1.4 Decay of Langmuir/z-mode waves For k b > k 0 Langmuir waves decay into Langmuir-like waves, implying small F. For k b k 0 Langmuir waves decay into z-mode-like waves, implying large F. [A. Layden et al., PRL, 2013] For decay of Langmuir waves to z-mode-like waves the Doppler-shifted frequency difference is: Z-mode waves can form for k b > k 0 if multiple decays occur (i.e., an ES backscatter followed by a decay to z-mode waves).

17 Unmagnetized Magnetized

18 3.2 Event selection We analyse Langmuir waveforms in type III source regions between 2009 June and 2012 February; a total of 596 events were selected. We also divide events into F < 0.2 and F > 0.2, corresponding to weak and strong perpendicular fields. Figure of F versus v b /c for Langmuir events with multiple distinct spectral peaks (86 events for F < 0.2, 145 events for F > 0.2). weaker E perp are generally observed at lower v b /c.

19 3.3 Electrostatic decay for F < 0.2 STEREO events on 2011 January 22. Panels: waveforms of E par, wavelet transforms, and power spectra. Left: 0.5s before ES decay. Right: during ES decay. Observed and expected frequency differences agree (360±80Hz versus 300±90Hz). 11:38: UT 11:38: UT [Graham and Cairns, JGR, 2013, submitted]

20 ES decay (F < 0.2) general analysis Expected Δf calculated using: Observed and expected Δf agree well. When intense ion-acoustic waves are observed then Δf = f S as expected for ES decay. These results provide strong evidence for ES decay in type III source regions. [Graham and Cairns, JGR, 2013, submitted]

21 3.4 Decay for F > 0.2: Example Example of a decay event with strong perpendicular fields. Higher frequency peak is E par. Lower frequency peak is E perp. Observed frequency difference is consistent with decay to low-k Langmuir-z mode waves. [Graham and Cairns, JGR, 2013, submitted]

22 Decay to Langmuir/z waves (F > 0.2): General Analysis Expected Δf calculated using: Observed and expected Δf agree well. When intense S waves are observed Δf = f S, as expected for ES decay to z-mode waves. These results provide strong evidence for 3-wave decay to Langmuir/z-mode waves. [Graham and Cairns, JGR, 2013, submitted]

23 3.5 Multiple backscatters example v b /c = (Parker spiral) k b /k 0 = 2.8. Based on best fit v b /c = and T e = 1.8e5 K k b /k 0 = 3.1, consistent with three decays.

24 ES Decay: Nonlinearity & Threshold [Graham & Cairns, in prep., 2013] Clear evidence for nonlinear threshold near P L ~ 4 x 10-4 Theoretical threshold energy density Observations almost all above predicted threshold decay should proceed as observed.

25 3.6 Summary ES decay of Langmuir-like waves to Langmuir-like and z-mode-like waves is commonly observed (~ 40 % of TDS events). z mode waves near k =0 are commonly observed (~ 25 % of TDS events). Evidence for multiple decays is observed. Evidence for a nonlinear threshold in Langmuir energy, above which S-wave products are observed. Observed fields for decay events exceed predicted threshold. Very strong evidence that ES decay proceeds in type III sources. [Graham and Cairns, JGR, 2013, submitted; ib id, in prep, 2013.]

26 4. Does Wave Collapse Occur in Type III Source Regions

27 4.1 Wave packets undergoing collapse? Wave packet collapse occurs when nonlinear self-focusing exceeds linear dispersion. The collapse threshold is defined as: Both theoretical work and 3D numerical simulations show that Θ 230 is required for collapse to occur. [e.g., Robinson, 1997; Graham et al., 2011a,b]

28 4.2 Approximate method The quantities l/λ D and W max are calculated by assuming STEREO transits the wave packet at a characteristic length from the center. W max and W(l) are calculated from the electric field envelope (red) for the three perpendicular E fields. l/λ D is calculated from the width of the electric field envelope (where W(l) = W max /2), the solar wind speed, and T e = 1.5 x 10 5 K.

29 Statistical analysis for Approximate Method no collapse Threshold The characteristic length l and energy density W(l) at l from TDS data are compared with the collapse threshold Θ. 167 Langmuir packets from eight different type III bursts are considered. Threshold None of the Langmuir packets identified exceed the collapse threshold. For most packets the peak energy density is over an order of magnitude too small for collapse to occur.

30 4.3 Detailed fitting Collapse theory and simulations show wave packets to be well fitting by potentials Lorentzian Gaussian We fit the potentials to the three orthogonal fields simultaneously. We fit the fields given by E = -grad Φ to the observed waveforms. [e.g., Robinson, 1997; Graham et al., 2011a,b; 2012]

31 Excellent fits to 3D Langmuir fields The Gaussian potential provides the better fit to the STEREO data. Very good Gaussian fits are found. Fits Θ = 12 << 230 Θ = 0.67 << 230 no collapse - Observed fields - Lorentzian fit - Gaussian fit [Graham et al., 2012]

32 Detailed fits no collapsing wavepackets Detailed fitting is applied to packets observed on 2011 February 26. Good agreement between good fits and approximate method are found. When good fits are found and l i /l < 2 the packets are below threshold and collapse cannot occur. Θ > 230 only when fits imply l i /l > 3. These estimates are unreasonable. - Approximate method - Good fits - Average fits - Poor fits [Graham et al., JGR, 2012]

33 4.4 Other arguments against modulational instability Analysis of nonlinear dispersion equation separate regions for modulational instabilities (MI) and decay: MI if k < k 0 /2= ω p V S /3 V 2 e, decay if k > k 0 /2. MI in solar wind type III sources if v b > c/2, but < c/10 usually. expect decay in type IIIs, at least for beam-driven waves MI is phase-coherent but ES decay can be random-phase Need Γ net > ω for MI to proceed. Observed ω too large and predicted Γ net too small MI cannot proceed in observed type III sources. [Zakharov et al., 1985; Cairns et al., 1998, Cairns, 2000, Graham et al., 2012a,b, contrary to Thejappa et al., 2010, 2011]

34 5. Langmuir Eigenmodes of Density Wells in Type III Source Regions

35 5.1 Eigenmode theory Langmuir eigenmodes are modeled using the first Zakharov equation: [Zakharov, 1972] We assume This yields the Schrodinger equation: [e.g. Alinejad et al., 2007] Here this equation is solved for the waveform using measured density profiles & then tested.

36 5.2 Eigenmode theory Mexican Hat & Parabolic density wells considered: and find similar eigenmode solutions for both wells. Parabolic wells yield Hermite-Gauss functions of the Ergun et al. [2008] theory. Requirement ω n > 0 for the nth eigenmode to form leads to. Here a and l are the depth and length scale of the well.

37 5.3 STEREO data STEREO s Time Domain Sampler (TDS) records electric fields in three orthogonal directions. Low frequency fields observed on all three antennas correspond to density perturbations. [Kellogg et al, 2009; Henri et al., 2011] This allows the small scale density perturbations and Langmuir waves to be investigated simultaneously. [TDS data: a localized Langmuir waveform and the associated density profile.]

38 5.4.1 Example: 20:24: UT on 2012 February, STEREO B. (a) Observed waveform (black), (c) eigenmode fit for observed density well (red), (e) Ergun et al. fit. (green) (b) Observed density profile is well modeled by the Mexican Hat function. (f) Spectra agree well. Observed waveform is the lowest order eigenmode solution for density profile.

39 5.4.2 Example: 20:32: UT, 2012 February 11, STEREO B. Observed density profile is well modeled by two Mexican hat functions. (b) Find two eigenmode solutions with comparable eigenfrequencies for the observed density profile. (f) Spectra agree OK. (a), (c), and (e) the lowest order eigenmode reproduces the observed waveform well.

40 5.5 Statistical analysis: 11 February 2012 type III [l δn length scale of density well, l W length scale of eigenmodes, W max = ε 0 E max2 /4 n e k B T e ] During the Feb. 11 type III burst 35 waveforms (~80%) consistent with eigenmodes were observed. (a) most waveforms consistent with only the lowestorder eigenmode forming in the density well. (b) depths of density wells are consistent with the ponderomotive force expelling plasma. Events are inconsistent with wave collapse.

41 6. Conclusions I. Field distributions, burstiness, and persistence of type III Langmuir waves often are consistent with SGT (ISEE-3). II. STEREO analyses strong evidence for ES decay to Langmuir or Langmuir/z-mode waves in ~ 40% of events: Low-f density responses, wave matching, threshold, etc. III. STEREO analyses no evidence for wave collapse and modulational instability (thresholds and wavenumber conditions not met). IV. 3D Langmuir waveforms consistent with Zakharov Eqs. V. STEREO analyses localized Langmuir waveforms are the lowest-order eigenmodes of inferred density depletions. VI. ES decay is fundamentally modified in magnetized plasma. [Robinson et al., 1993; Cairns & R., 1995; Henri et al., 2009; Graham et al., 2012a,b; Layden et al., 2013; Graham and Cairns, PRL and JGR, submitted, 2013a,b]

42 SGT in Earth s Foreshock [C. & R, 1997, 1999] 1. Consistent with SGT prediction of Gaussian in X = A log E. 2. Evidence for 2 acceleration regions or wave growth mechanisms - change in slope of < log E>(D f ).

43 5.1.5 Doppler-shifted frequencies versus v b /c f L d f L d Plot of Dopplershifted frequencies versus v b /c for nominal solar wind conditions Dashed line is f p. Z-mode waves have frequencies near f p, so are difficult to distinguish from f L d. v b /c is estimated by assuming electrons travel at constant speed along a Parker spiral.

44 3.1 Multiple backscatters example 1 v b /c = (Parker spiral) k b /k 0 = 2.6. Based on best fit v b /c = and T e = 9.0e4 K k b /k 0 = 2.1, consistent with two decays.

45 4. Quasilinear / Nonlinear Simulations of SGT Want to understand evolution to SGT. Here, simulate evolution to SGT state: 1-D electron-langmuir quasilinear system Separately driven long wavelength density irregularities (ion acoustic waves). Include decay L L + S and f p and 2f p radiation processes [Li et al., PRL, 2006; PoP, 2006]

46 Burstiness develops in inhomogenous case Total occupation number N(x,t) = dv N(v,x,t) Bursty: inhomogenous case Smooth: homogenous case [Li et al., PRL, 2006; PoP, 2006]

47 Statistics Field statistics consistent with SGT. Fluctuations in free energy also lognormal & consistent with SGT. First simulation showing evolution to an SGT state. [Li et al., PRL, 2006; PoP, 2006]

48 SGT and Type III Bursts E Time E ( V/m) [Robinson et al., 1993; C. & Robinson, 1995]

49 5. New results and issues related to SGT 1. Small deviations from lognormal for pure SGT [Krasnoselskikh et al., 2007]? 2. Different classes of wavepackets have different statistics [Nulsen et al., JGR, 2007]. 3. Several mechanisms for achieving SGT?

50 Wave Collapse E 2 wave refraction ponderomotive force n Wave collapse or modulational instabilities? Eigenmodes Nonlinear electrostatic decay L L + S? Radiation via LMC or nonlinear processes? [C. & R., 1995]