Turbulent Mixing and Beyond Workshop

Transcription

1 from V2 ology V2 Turbulence F. Fraternale 1, L. Gallana 1, M. Iovieno 1, J.D. Richardson 2, D. Tordella 1 1 Dipartimento di Ingegneria Meccanica e Aerospaziale (DIMEAS) Politecnico di Torino, Italy 2 Kavli Institute for Astrophysics Space Research Massachusetts Institute of Technology (MIT), Cambridge, USA Turbulent Mixing Beyond Workshop ITCP Trieste, 4 9 August 214

2 from V2 ology V2 Table of contents from V2 3. ology V

3 from V2 ology V2 is flying now at 15.6km/s, 14.7 AU from Earth, in the Heliosheath, the outermost layer of the heliosphere where the solar wind is slowed by the pressure of interstellar gas Termination Shock was passed on Sep 5, 27 source: M. Opher et al. Interstellar Mission source: A hypothesis for the magnetic field in the Heliosheath M. Opher et al, ApJ 734, 211 Is the magnetic field in the Heliosheath laminar or a turbulent sea of bubbles?

4 L.L. Orionis colliding with the Orion Nebula. Image from the Hubble Space Telescope, February 1995 (Credit: NASA, The Hubble Heritage Team (STScI/AURA)) from V2 ology V2

5 Turbulence Year 1979: V B V2 normalized (1979) 8 4 ology V2-4 VR BR -8 (x µ)/σ from V2 4-4 VT BT VN BN time (doy) Velocity magnetic field from V2, period 1979 (DOY 1 18). RTN heliographic reference frame.

6 Turbulence Year 1979: V B 2 from V2 1 ology V2 VA ) (km/s) (v µv ), ( B 4πρ V2 in Alfvenic units (1979) VR BR -1 1 VT BT VN BN τ (doy)

7 from V2 ology V2 Year 1979: V B moments PDFs µ σ 2 Sk Ku V R V T V N B R B T B N units: km/s, nt n i (cm 3 ).23 T (K) 2738 β p.225 V A (km/s) 47.7 c s (km/s) 19.3 f ci (Hz) f pi (Hz) 11 f (Hz).44 r i (km) 158 λ D (m) 5.5 PDF (m/s) -1 σ σ PDF (nt) V2 velocity Radial Normal Tangential ( ) V2 mag. V - µ /σ Radial Normal Tangential ( ) B - µ/σ PDF of V B stardized components comparison with a Normal distribution Evidence of anisotropy

8 Year 1979: V B moments PDFs 1.1 velocity module module χ 2 (k=3) from V2 PDF.1 ology V2.1 V (x i μ i ) 2 /σ 2 i PDF of stardized modules comparison with a χ 2 distribution. High intermittency? Evidence of high Ku(> 3) The origin of intermittency : advected coherent structures (flux tubes, etc), stochastic Alfvénic fluctuations generated at solar corona frozen in the wind? Intermittency interests a broad range of scales

9 from V2 ology V2 Ri i /σ 2 i Autocorrelations Autocorrelations - V2 () V R V R B RB R V T V T B TB T V N V N B NB N τ (days) R ii(τ) =< x(t)x(t + τ) >

10 Cross-correlations tensor: off-diagonal terms from V2 ology V2 Rij /σi σj B R V T B TV R B R V N B NV R B TV N B N V T Cross-correlations - V2 () τ (days) R ii(τ) =< x(t)y(t + τ) >

11 Cross-correlations tensor: diagonal terms from V2 ology V2 Ri i /σ 2 i.4 BRVR BTVT BNVN Summary: τ (days) Averages are computed on 5797 points for V, 1248 points for B, spanning the whole 18 days period Evidence of a 25 days periodicity. Minimum of solar activity in 1979 High cross-correlation V R B R not in-phase High cross-correlation V R B T not in-phase Low Alfvénic one-point correlation (this is often the case in the slow-wind periods)

12 Data reconstruction techniques from V2 ology V2 V2 mag. are discontinuous irregularly spaced. In the whole year 1979 there is 45% of missing velocity, 25.4% in the period here considered (DOY 1 18). About mag., the percentage is 23.8%. These values are about 97% in 212. To perform an accurate spectral analysis on these kind of sets, a reconstruction technique may be matory. In the following, the effect of two interpolation/recovery ologies on averaged turbulent spectra will be discussed. Linear interpolation Maximum likelihood reconstruction realizations constrained by 1 1 & Press, ApJ 398, 1992

13 from V2 ology V2 Data reconstruction techniques To discuss the effects of averaging,interpolating applying windowing techniques, two 1D sequences of have been generated from imposed spectral properties: Synt 1 E 3D (n/n ) = Synt 2 E 3D (n/n ) = (n/n)β (n/n ) α+β (n/n)β (n/n ) [ 1 exp( n ntot α+β γ + ɛ) ] β = 2, α = 5/3, n = 11, γ = 1 4, ɛ = 1 1 The Synt 1 sequence reproduces the Kolmogorov inertial range of canonical, while Synt 2 reproduces both the inertial the dissipative part of the spectrum. Synthetic are scaled on a 18 days time grid ( t = 1 s, n tot = 15552) The same gaps of V2 velocity are projected on these sequences analysis is performed. Parameters: L g, L s

14 from V2 ology V2 Effect of no/linear interpolation on Synt 1 V 2 (Km 2 /s 2 /Hz) V 2 (Km 2 /s 2 /Hz) gaps of V2 () Synthetic 1, effects of averaging with no recovery Ns = 84 (Ls = 6 8 hrs ) Ns = 84 (Ls = 6 8 hrs ), Hann Ns = 32 (Ls = 3 4 hrs ) α: Ns = 32 (Ls = 3 4 hrs ), Hann original: slope α: gaps of V2 () Lg =1 hr Synthetic 1, effects of linear recovery, no windowing α: α: α: Ns =48(Ls =6hrs) Ns =2(Ls =12hrs) Ns =68(Ls =24hrs) original: slope f (Hz) V 2 (Km 2 /s 2 /Hz) V 2 (Km 2 /s 2 /Hz) gaps of V2 () Ls = 24 hrs Synthetic 1, effects of linear recovery, Hann windowing Lg =.5 hrs Lg = 1 hrs Lg = 4 hrs original: slope α: α: -1.9 α: -2.5 gaps of V2 () Lg = 1 hr α: α: α: Synthetic 1, effects of linear recovery, Hann windowing Ns = 48 (Ls = 6 hrs ) Ns = 2 (Ls = 12 hrs ) Ns = 68 (Ls = 24 hrs ) original: slope f (Hz)

15 from V2 ology V2 Effect of no/linear interpolation on Synt 2 V 2 (Km 2 /s 2 /Hz) gaps of V2 () α: α: α: Synthetic 2, effects of linear recovery, no windowing Lg =.5 hrs (Ls = 24 hrs) Lg = 1 hr (Ls = 24 hrs) Lg = 4 hrs (Ls = 24 hrs) original spectrum f (Hz) V 2 (Km 2 /s 2 /Hz) gaps of V2 () fit: fit: fit: Synthetic 2, effects of linear recovery, Hann windowing Lg =.5 hrs (Ls = 24 hrs) Lg = 1 hr (Ls = 24 hrs) Lg = 4 hr (Ls = 24 hrs) original spectrum f (Hz) Effect of segmentation: increase in slope of about 5% in the inertial range. Effect of linear interpolation: function of L g (length of filled gaps). This interpolation transfers energy to the low frequences, resulting in an increase (about 6%) in the slope, especially in the high-frequency range (f > 1 3 Hz).

16 Effect of no/linear interpolation on Synt 2 from V2 ology V2 Effect of windowing: the Hann window function allows to eliminate spurious energy due to discontinuities ( 1/f) at the boundary of each segment. The effect is minimal at low wavenumbers. In the high-frequency range, on the one h a significant increase ( up to 23%) of the slope is found to be a function of L g, on the other h any change in slope of the real spectrum can be followed. Energy correction factor for Hann: Without windowing, the segmentation error doesn t allow to represent the correct slope, in the general case (see the analysis on Synt 2 ). These cases can be recognized by a flattening in the high-frequency range of the spectrum. Averaging long segments helps.

17 1 6 V2 velocity spectra at 5 AU (pre-jupiter) V2 velocity 1 6 V2 velocity from V2 ology V2 VR (Km 2 /s 2 /Hz) VN (Km 2 /s 2 /Hz) α: α: α: -1.6 α: no recov, Ls = 3 4 hrs no recov, Ls = 3 4 hrs, Hann lin. recov, Lg =.5 hrs, Ls = 12 hrs lin. recov,lg =. 5hrs, Ls = 12 hrs, Hann α: -1.6 α: α: α: V2 velocity no recov, Ls = 3 4 hrs no recov, Ls = 3 4 hrs, Hann lin. recov, Lg =.5 hrs, Ls = 12 hrs lin. recov,lg =. 5hrs, Ls = 12 hrs, Hann f (Hz) VT (Km 2 /s 2 /Hz) E (Km 2 /s 2 /Hz) α: α: α: α: no recov, Ls = 3 4 hrs no recov, Ls = 3 4 hrs, Hann lin. recov, Lg =.5 hrs, Ls = 12 hrs lin. recov,lg =. 5hrs, Ls = 12 hrs, Hann V2 kinetic energy α: -1.6 α: α: α: -1.5 no recov, Ls = 3 4 hrs no recov, Ls = 3 4 hrs, Hann lin. recov, Lg =.5 hrs, Ls = 12 hrs lin. recov,lg =. 5hrs, Ls = 12 hrs, Hann f (Hz)

18 from V2 ology V2 B 2 R (nt2 /Hz) B 2 N (nt2 /Hz) V2 spectra at 5 AU (pre-jupiter) V2 mag. α: α: α: α: no recov, Ls = 2 3 hrs no recov, Ls = 2 3 hrs, Hann lin. recov, Lg =.5 hrs, Ls = 48 hrs lin. recov,lg =. 5hrs, Ls = 48 hrs, Hann V2 mag f (Hz) B 2 T (nt2 /Hz) B 2 (nt 2 /Hz) α: α: -1.9 α: α: -1.9 no recov, Ls = hrs no recov, Ls = 2 3 hrs, Hann lin. recov, Lg =.5 hrs, Ls = 48 hrs lin. recov,lg =. 5hrs, Ls = 48 hrs, Hann 1-2 V2 mag. α: -1.9 α: α: α: no recov, Ls = 2 3 hrs no recov, Ls = 2 3 hrs, Hann lin. recov, Lg =.5 hrs, Ls = 48 hrs lin. recov,lg =. 5hrs, Ls = 48 hrs, Hann V2 mag. α: α: α: α: no recovery, Ls = 2 4 hrs no recovery, Ls = 2 4 hrs, Hann linear rec, Lg =.5 hrs,ls = 48 hrs linear rec, Lg =.5 hrs,ls = 48 hrs, Hann f (Hz)

19 from V2 ology V2 Velocity: V2 spectra at 5 AU (pre-jupiter) The observed frequency range constitute the inertial range All computed slopes (1 4 < f < Hz) are flatter than the Kolmogorov one: α = 1.53 ±.7 Computed slopes may be slightly overestimated A peak is located at f =.26 Hz for T N components: is it physical or instrumentation-related? (no relation with f ci, f pi, f )) Magnetic field: Computed slopes (1 4 < f < ) are lower than the reference one: α = 1.81 ±.9 Observed steepening for f > Hz should not be linked to interpolation issues: the situation recalls that of Synt 2 case, blue (no recovery) violet (small gaps filled) give the same result. Anisotropy is higher with respect to the velocity field

20 from V2 ology V2 G.B. &W.H. Press Minimum variance (interpolation): y = s + n irreg. spaced vector with errors n s = M i=1 d iy i + x s =true value at a particular point ŝ = S T [S + N] 1 y ŝ =min. variance estimate for s Assuming stationary process: S ij = s i s j = f(t i t j ) is the correlation matrix, estimated from N ii = n 2 i is the errors diagonal matrix n i in new points The min. variance estimation is not, however, a typical realization of the underlying process. Minimum variance + Gaussian process To obtain a typical realization, a Gaussian process is added to the min. var. estimate: s = u + ŝ If realizations constrained to are desired: u = V diag(λ 1/2 1,..., λ 1/2 M )r where λ i = eig(q), Q = [S 1 +N 1 ] 1, r = r(µ =, σ 2 = 1)

21 from V2 ology V2 B 2 R (nt2 /Hz) * & Press, ApJ, 398, 1992 V 2 (Km 2 /s 2 /Hz) gaps of V2 () * & Press, ApJ, 398, 1992 f (Hz) no recov, Ls = 2 4 hrs, Hann lin. rec., Lg =.5 hrs, Ls= 48 hrs, Hann R&P recov*,lg= 4 hrs,ls = 48 hrs, Hann f (Hz) R&P reconstruction Synthetic 1, effects of R&P* recovery lin. recov, Lg = 4 hrs, Ls = 24 hrs, Hann R&P recov, Lg = 4 hrs, Ls = 24 hrs, Hann R&P recov, Lg = 4 hrs, Ls = 96 hrs, Hann original: slope V2 mag. α: α: α: VR (Km 2 /s 2 /Hz) * & Press, ApJ, 398, 1992 V2 velocity no recov, Ls = 3 4 hrs,hann lin. recov,lg =.5 hrs, Ls = 12 hrs, Hann R&P recov*,lg = 4 hrs, Ls = 12 hrs, Hann f (Hz)

22 from V2 ology V2 Final considerations future development Kolmogorov (-5/3) or Iroshnikov Kraichnan (-3/2) cascade? Debated question. Many works suggested K41 as the more consistent for SW (Goldstein, GRL 1995), but in recent works values close to IK for K41 for mag. field are observed (Safranova et al. PRL 213 at 1AU, Podesta et al. ApJ 27, 1AU) We provide spectra at 5 AU, supporting these recent observations. The high frequency range. Break frequency/ies, dissipation or further cascades. Different mechanisms had been proposed to explain the steepening of collisionless SW spectra in the high freq. range (foreshock waves, Lau damping of KAW, wave dispersion). Up to what regime will we be able to observe, in the Heliosheath region? Heliosheath are very sparse. How to get reliable spectra when a loss of 95% is present? We have started to apply the Compressive Sensing technique to this problem. CS is a very recent paradigm for acquisition, providing reconstruction for a broad class of sparse signals.