Turbulence in the sub-Alfvénic solar wind driven by reflection of low-frequency Alfvén waves
aa r X i v : . [ a s t r o - ph . S R ] M a y D raft version N ovember
17, 2018
Preprint typeset using L A TEX style emulateapj v. 08 / / TURBULENCE IN THE SUB-ALFV ´ENIC SOLAR WIND DRIVEN BY REFLECTION OF LOW-FREQUENCY ALFV ´ENWAVES
A. V erdini , M. V elli and E. B uchlin Draft version November 17, 2018
AbstractWe study the formation and evolution of a turbulent spectrum of Alfv´en waves driven by reflection o ff thesolar wind density gradients, starting from the coronal base up to 17 solar radii, well beyond the Alfv´eniccritical point. The background solar wind is assigned and 2D shell models are used to describe nonlinearinteractions. We find that the turbulent spectra are influenced by the nature of reflected waves. Close to thebase, these give rise to a flatter and steeper spectrum for the outgoing and reflected waves respectively. Athigher heliocentric distance both spectra evolve toward an asymptotic Kolmogorov spectrum. The turbulentdissipation is found to account for at least half of the heating required to sustain the background imposed solarwind and its shape is found to be determined by the reflection-determined turbulent heating below 1.5 solarradii. Therefore reflection and reflection-driven turbulence are shown to play a key role in the acceleration ofthe fast solar wind and origin of the turbulent spectrum found at 0.3 AU in the heliosphere. Subject headings:
MHD — waves — turbulence — solar wind INTRODUCTION
Recent high resolution observations from Hinode(De Pontieu et al. 2007) and the Coronal Multi-channelPolarimeter (Tomczyk et al. 2007) have shown that thesolar atmosphere is pervaded by Alfv´enic (or kink-like, e.g.Van Doorsselaere et al. 2008) oscillations: observed in jets,spicules or in coronal loops, velocity and magnetic fieldoscillations ( δ b , δ u ) are coupled and propagate at speedsclose to the Alfv´en speed. Far from the sun, between 0.3 AUand several AU, in situ data show that in the frequency range10 − Hz . f . − Hz, fluctuations in magnetc field andvelocity δ b and δ u possess many of the properties of outward-propagating ”spherically polarized” Alfv´en waves, namely:quasi incompressibility, correlated oscillations, and a con-stant (total) magnetic field intensity, while at the same timerevealing their turbulent nature through a well-developedpower-law frequency spectrum, with a break separatingdi ff erent power law slopes of -1 and -1.6 which movesto lower frequencies with increasing distance from the sun(Bavassano et al. 1982; Tu et al. 1984). The δ u · δ b correlation,upon which the propagation direction determination is made,depends on the frequency considered and varies with distance(Bavassano et al. 2000b,a) and latitude (Grappin 2002), typi-cally in the range 1 / < | ( δ u · δ b / p πρ ) / ( δ u + δ b / πρ ) | < inward travelling wave-mode component, requiredfor nonlinear couplings between incompressible fluctuations,must indeed be present.This component might be generated locally between 0.3 AUand 1 AU by shear, compressible or pick-up ions interactions,or it could be already formed in the sub-Alfv´enic corona andlater on nonlinearly advected into the heliosphere by the solar Observatoire Royale de Belgique, 3 Avenue Circulaire, 1180, Bruxelles,Belgium; e-mail: [email protected] Dipartimento di Astronomia e Scienza dello Spazio, Univ. di Firenze,Largo E. Fermi 3, 50125, Firenze, Italy Jet Propulsion Laboratory, California Institute of Technology, 4800 OakGrove Drive, Pasadena, CA 91109, USA Institut d’Astrophysique Spatiale, CNRS - Universit´e Paris Sud, Bˆat.121, 91405, Orsay Cedex, France wind, the hypothesis we consider here.The dynamics inside the Alfv´enic point region is of primaryimportance to understand the origin of the spectrum one findsat 0.3 AU and whether it has any role in accelerating thesolar wind. The variation of the propagation speed inducedby density gradients in the stratified corona and acceleratingsolar wind causes outward Alfv´en waves to be reflected,predominantly at lower frequencies, hence triggering theincompressible cascade. The power dissipated by the cascadecontributes to coronal heating, also modifying the overallturbulent pressure gradient, fundamental to the accelerationof the fast solar wind.While there are several studies on the linear propagationand reflection of Alfv´en waves in the sub-Alfv´enic corona andsolar wind (Hollweg 1978; Heinemann & Olbert 1980; Velli1993; Hollweg & Isenberg 2007), excepting phenomenolog-ical models with an essentially dimensional estimate of therole of turbulent heating (Hollweg et al. 1982; Dmitruk et al.2001; Cranmer & van Ballegooijen 2005; Verdini & Velli2007) very few of them have considered nonlinear interac-tions. Velli et al. (1989, 1990) studied the turbulent cascadesustained by reflected waves in the super-Alfv´enic solar wind,while Dmitruk et al. (2002) considered the same mechanismin the sub-Alfv´enic corona below 3 R ⊙ , hence neglecting thesolar wind.In the present letter we will extend these studies follow-ing the development of the turbulent cascade from the baseof the corona up to 17 R ⊙ , well beyond the Alfv´enic crit-ical point (located at about 13 R ⊙ in the solar wind modeladopted). Direct numerical simulations are still prohibitivelycostly in terms of computational times, so nonlinear interac-tions are simulated using a 2D shell model (Buchlin & Velli2007) which simplifies nonlinear interactions but still allows4 decades in the perpendicular wavenumber space to be cov-ered while rigorously treating the propagation and reflectionof waves along the radial mean magnetic field. MODEL DESCRIPTION
The equations describing the propagation of Alfv´en wavesin an inhomogeneous medium are derived from magnetohy- Verdini et al.drodynamics (MHD), assuming that the large scale fields arestationary and separating the time-fluctuating fields from thelarge-scale averages (Heinemann & Olbert 1980; Velli 1993).Therefore the large scale magnetic field, bulk wind flow anddensity ( B , U , ρ respectively) appear as specified coe ffi cientsin the MHD equations for the fluctuations.We consider a magnetic flux tube centered in a po-lar coronal hole, which expands super-radially with a(non-dimensional) area A ( r ) = r f ( r ) first prescribed by(Kopp & Holzer 1976; Munro & Jackson 1977). Distancesare normalized to the solar radius, and the coe ffi cients aregiven by r = . , σ = . , f max = .
26 (respectively thelocation, width, and asymptotic value of the super-radial ex-pansion) so that A(1) =
1. The field becomes B ( r ) = B ⊙ / A ( r )where we take B ⊙ =
10 G. The wind speed and density U ( r ) , ρ ( r ) are obtained solving the 1D momentum equationwith an assigned temperature T ( r ) , T ⊙ = K and a nu-merical density at the coronal base n ⊙ = cm − (seeVerdini & Velli 2007 and references therein for details on theequation and on the temperature profile). The resulting windis supersonic far from the sun with U ≈
750 km / s and n ≈ − . The Alfv´en critical point ( r a ) is at about13 R ⊙ , the sonic critical point is at about 1 . R ⊙ , the Alfv´enspeed V a = B / p πρ at the base is V a , ⊙ ≈ − andhas a maximum V a , max ≈ / s at r = . R ⊙ . At the endof the domain (17 R ⊙ ) U ≈
740 km / s and V a ≈
630 km / s.We assume δ u to be incompressible and transverse with re-spect to B . The momentum and induction equations for δ u and δ b are written in terms of the Els¨asser fields Z ± = δ u ∓ δ b / p πρ , corresponding to Alfv´en waves which propa-gate respectively outwards and inwards in the solar wind ref-erence frame.Substituting the nonlinear terms, which act in planes per-pendicular to the radial direction, with a 2D MHD shell modelrepresentation (in the form given by Biskamp 1994) and as-suming radial propagation finally yields the model equationsfor Z ± = Z ± ( r , k ⊥ ) = Z ± n ( r ): ∂ Z ± n ∂ t + ( U ± V a ) ∂ Z ± n ∂ r +
12 ( U ∓ V a ) d log V a d r + d log A d r ! Z ± n −
12 ( U ∓ V a ) d log V a d r ! Z ∓ n = − k n (cid:0) ν + Z ± n + ν − Z ∓ n (cid:1) + ik n ( T ± n ) ∗ . (1)The complex scalar values, u n = ( Z + n + Z − n ) / b n = ( Z − n − Z + n ) /
2, represent the velocity and magnetic (in velocity units)field fluctuations corresponding to the scale λ n = λ − n = π/ k n , n is the shell index, and T ± n accounts for nonlinear in-teractions of the form Z + l Z − m , with l , m = n ± , n ±
2. Finally, ν ± = ( ν ± η ) / η = ν ).Simulations are carried out with the code S hell -A tm (Buchlin & Velli 2007). The advection terms in eq. 1 are com-puted with a second order upwind scheme (Fromm scheme)which allows a good conservation of the phase of the fluc-tuation. Time is advanced with a third order Runge-Kuttafor the nonlinear part of the equations. The radial domainis decomposed in ∼ ,
000 planes over a non-uniform gridwhile 21 shells are used for the nonlinear interactions. Trans-parent boundary conditions are imposed at the top for bothwaves, and at the bottom for the Z − . Here all the gradi-ents are artificially set to zero, in order to avoid reflections. modified to include the wind spherical expansion Energy is injected in the domain imposing the amplitude Z + n , ⊙ = f n ( t ) at the first 3 shells corresponding to length scaleof the order of 8.000-34.000 km ( λ , ⊙ = . R ⊙ in the shellmodel), with f ( t ) a function with a time correlation and pe-riodicity τ ∗ ≈ low-frequency fluctuations are injected for long time series (as in the presentsimulation). Simulations last about 20 crossing time scale, τ cr = R R ⊙ R ⊙ ≈ τ cr , during which the system has an ap-proximate stationary state. REFLECTION AND NONLINEARITIES
Given that we start with an initial flux of Z + at the coro-nal base, reflection is the only trigger for nonlinear interac-tions. In Figure 1 the root mean squared (rms) amplitudes | Z ± | = pP n | Z ± n | are plotted as a function of time and he-liocentric distance. Two components of the reflected wavesare clearly visible in the Z − contours: a “classical” compo-nent Z − class , which propagate with the expected U − V a phasespeed (negative below the Alfv´enic critical point, r a ≈ R ⊙ )and an “anomalous” component Z − anom , which travels with thesame speed as the mother wave, U + V a . As shown analyti-cally and numerically (Velli et al. 1990; Hollweg & Isenberg2007) this “anomalous” component is the direct product of re-flection. Generally in presence of density gradients, for smallvalues of the ratio ǫ = α/ω , typical of the upper corona, with α = τ − R = | ( U ∓ V a ) V ′ a / V a | in eq. 1, each field can be de-composed in a primary and secondary component. A Z + pri-mary component is injected at the base, while the Z − is madeup of only the secondary component, given by reflection. Ineach plane, as the Z + arrives, the secondary component canbe seen as the result of a forcing term given by ≈ α Z + , henceproducing a wave which travels with the same phase speedand of the above amplitude. The value Z − = ǫ Z + follows nat-urally by finding the “forced” solution to the linearized equa-tion for Z − . At later time, as the Z + has propagated away, theforcing disappears and the secondary component propagatesbackward with the classical phase speed. When a Z + pulsewave is excited in the corona, Z − appears as a halo spreadingbackward from the mother Z + wave. Nonlinear interactionsmodifies the above picture, acting as a local (in a given plane)source for the Z − n which is uncorrelated with respect to thatgiven by reflection and hence generating waves propagatingwith classical phase speed U − V a , i.e., a primary component.Inward propagating waves have a very long propagation timeat r a that slows down the overall relaxation toward a steadystate: for example in Figure 1 the increase of the Z − ampli-tude at t ≈ τ cr results from the superposition of the Z − anom with the backward propagating Z − class produced at t ≈ τ cr .The di ff erent nature of the reflected waves influences thespectral energy transfer. Generally speaking, while the waveamplitudes increase with distance the nonlinear timescales τ ± nl = ( k n Z ∓ n ) − decrease, following the reflection coe ffi cientand the flux tube expansion. At small wavenumbers τ − nl < τ + nl ,the energy transfer is more e ffi cient for the reflected waves,although the Z + ’s contribute more to the total energy dis-sipation. As a matter of fact, the Z − ’s develop an inertialrange in the whole domain and the resulting cascade is more“aged”, their dynamics is governed by the nonlinear interac-tions which have shorter timescales (as Z + > Z − ) and act forlonger periods, the Z − having a smaller propagation speed.eflection-driven turbulence 3 F ig . 1.— Root mean squared amplitudes of the mother wave Z + and the reflected waves Z − in km s − (left and right panels respectively) as a function ofheliocentric distance and time (in unit of τ cr ), along with the outgoing and ingoing characteristics U ± V a (red curves). Two di ff erent paths of the Z − can bedistinguished in the right panel: one associated to its phase speed ( U − V a , negative below the Alfv´enic critical point at ≈ R ⊙ ) and the other associated toreflection from the Z + which forces the fluctuations to follow the outgoing wave (with phase speed U + V a ).F ig . 2.— Energy spectra E ± k (solid and dashed lines respectively) as a func-tion of the perpendicular wavenumber in four di ff erent planes, as indicatedat the bottom left of each panel ( r = R / R ⊙ ). Spectra are averaged in timeand normalized to ≈ . × cm s − , wavenumbers to R − ⊙ . Thin lines arepower-law with the indicated slopes, plotted for reference. The contribution of coherent interactions Z + Z − anom in shapingthe energy spectra E ± k = | Z ± n | / (4 k n ) can be seen in the lowcorona, before 1 . R ⊙ (left panels of Figure 2). In fact at1 . R ⊙ , assuming waves of period T = ǫ ≈ . ǫ max = .
16 at the coronal base), hence in presence of lowerfrequency fluctuations (injected by the forcing) reflection isrelatively high. At about 1 . R ⊙ , V a has a maximum and re-flection vanishes, so that the Z − are not produced locally butrather propagating from above. This inhibits the spectral en-ergy flux of Z + that becomes negative at large scales. Theinverse spectral transfer is responsible of the flatter part ofthe E + spectrum one finds at small k ⊥ . Further out reflec-tion is negligible and the spectra are the remnant of those pro-duced in the inner layers, showing a slow evolution towardthe asymptotic state in which E + & E − . The spectral energyflux is given by the interctions Z + Z − class that are subject to the“Alfv´en e ff ect” (a longer cascade timescale due to the nonlin-ear interactions between wavepackets propagating in oppositedirections), producing the same spectral slopes -5 / E ± (right panels of Figure 2). Note that the largest perpendicular scale is proportional to theflux tube width, λ ( r ) = λ , ⊙ √ A ( r ), hence the same numberof shells spans a k ⊥ interval that shifts to smaller values withincreasing r .The asymptotic slope − / a = / , a = − a , a = − / Π ± n = − Im h k n Z + n (cid:16) a Z + n + Z − n + + a Z − n + Z + n + (cid:17) + k n − Z + n + (cid:16) a Z + n − Z − n + a Z − n − Z + n (cid:17)i (2)When reflection is negligible one can assume that the Z + and Z − are uncorrelated h Z + Z −∗ i t ≈
0. Assuming a powerlaw for the spectral energies, E ± n ∝ ( k n / k ) p ± , substituting Z ± n = k n E ± n ) / in eq. 2 one finds that Π ± n is indepen-dent of the shell index n when p + = p − = − /
3. For agiven plane, it implies a constant normalized cross helicity σ c = ( E + − E − ) / ( E + + E − ) in the inertial range, with a crosshelicity spectrum H c = E + − E − of the same slope as E + , incontrast to EDQNM closure models, which, including non-local interactions in Fourier space, predict a distinct steeperspectrum, H c ∝ k − (Grappin et al. 1983). From the evolutionof the spectra (right panels) one can see that the normalizedcross helicity decreases with r because of two factors, thegeneral increase of the total energy and the decrease of H c atall scales. The former is due to the approximate conservationof the total wave action density. The later results fromthe competition of the linear coupling (reflection), whichforces Z − . Z + for low frequency fluctuations, with thenonlinear coupling, which damps the non dominant wavepopulation, the Z − (Dmitruk et al. 2001). Indeed the E ± spectra look the same at small wavenumbers (correspond-ing to n = , ,
2) in which low frequency fluctuations reside.When reflection is high the two fields are strongly coupledand one can assume that the growth of Z − n has contributionsboth fromt the nonlinear cascade and direct generationthrough reflection ≈ ǫ n Z + n . The reflected contribution shares Verdini et al. F ig . 3.— Time averaged heating per unit mass as a function of distance fordi ff erent root mean squared amplitudes at the coronal base Z + ⊙ . Also plottedin red is the heating function necessary to sustain the background specifiedsolar wind. H is in units of ≈ × cm s − . the phase properties of Z + n so interactions due to reflectionare coherent below 1 . R ⊙ and the cascade may be isotropicfor small wavenumbers in the k ⊥ − k || plane. For the first 7shells (2 orders of magnitude), one would find ω n = k n V a , ǫ n ∝ λ − n and the following scaling for the spectra E + ∝ k − , E − ∝ k − (Velli et al. 1990). If the nonlinear cascade is somehow inhibited, keeping fixed the total energy in the parallelwavenumbers, low frequencies are confined to smaller n and ǫ n ∝ λ − np with p <
1, yielding a steeper (flatter) spectrumfor E + ( E − ) compared to the isotropic case. In the presentsimulations, ǫ . R ⊙ ∝ λ . for n < p + = . p − = . Z ± n ) ∗ and summed over the shellindex. The total energy dissipation per unit mass H = Q /ρ = / ν P n k n ( | Z + n | + | Z − n | ) increases in thelow corona and decreases exponentially in the sub-Alfv´enicregion of the wind, as expected. For δ u ⊙ ≈
49 km s − (solidline in Figure 3), it is also very close to the heating required tosustain the specified background wind (red line in Figure 3):the location and height of the peak coincide, although moreenergy is dissipated in the upper corona. This would producea faster wind, not altering too much the mass flux, since thepeak of the dissipation is close to the sonic critical point(Hansteen et al. 1999).For smaller values of δ u ⊙ a better agreement is found in thedecreasing part, the rapid increase in the low corona is stillreproduced but the peak intensity is not attained. Despite thefact that the spectra and the spectral fluxes possess the sameproperties, for δ u ⊙ ≈
45 and 42 km s − the peak intensitydecreases by a factor 1 / / δ u is determined by theinjection of Z + at the coronal base but also by the responseof the atmosphere and by the nonlinear interactions (the levelof Z − ). On the countrary the peak dissipation seems to scalelinearly with the rms amplitude Z + ⊙ although further studies(dependence on the frequency, on the nonlinear interactionsin the shell model, on the imposed wind) are necessary todefine the scaling precisely. DISCUSSION
We have studied the propagation, reflection and nonlinearinteraction of Alfv´en waves from the base of the corona upto 17 solar radii, well beyond the Alfv´enic critical point. Forthe first time 2D shell models have been applied to accountfor nonlinear interactions in magnetically open regions on thesun, such as coronal holes. Thanks to such a simplification,compared to MHD or Reduced MHD direct numerical simu-lations, it is possible to follow the development of a turbulentspectrum in the expanding solar wind, where waves are con-tinuously reflected by the gradients in mean fields. Reflectedwaves are made of two components, one propagating withthe characteristic phase speed U − V a ( Z − class ) and the otherfollowing the path of the outgoing wave with speed U + V a ( Z − anom ), a confirmation of previous linear results (Velli et al.1989; Hollweg & Isenberg 2007) which hold in a similar wayalso in the nonlinear regime. For typical coronal parameterswe find Z − = ǫ ( r , ω ) Z + ∝ V ′ a Z + , in contrast to Z − ∝ λ V ′ a (independet of Z + ) found in a strong turbulence regime(Dmitruk et al. 2002) in which Z − , λ →
0. Di ff erencesarise because the above limits impose a time scale ordering τ − nl << τ + nl . τ R < τ cr which is not satisfied in our simulation,basically because Z − λ → Z − anom Z + , which give E + ∝ k − . , and the one from theincoherent nonlinear interaction Z − class Z + , giving E ± ∝ k − / (in this shell model, which includes nonlinear interactionsonly locally in Fourier space). The resulting spectra changewith distance, starting from a coherent-interaction domi-nated spectrum at the coronal base and evolving toward theasymptotic Kolmogorov spectra at greater distance, wherereflection is negligible. According to this model, outside theAlfv´enic critical point, the turbulent spectra have alreadylost any feature acquired in the low corona. Note that thisis referred to perpendicular wavenumber spectra and not tothe frequency spectra, which on the countrary are almostunchanged, since their evolution is limited to the first solarradius above the coronal base.Turbulent dissipation is remarkably high in the low corona.Depending on the injected energy an almost complete orpartial matching is found with the “theoretical” heating, thatis the one required to form the imposed background solarwind. The best agreement is found for δ u ⊙ ≈
50 km s − which is at the limit of observational constraints (Chae et al.1998). Nonetheless, even for more conservative values δ u ⊙ ≈
40 km s − , turbulent dissipation accounts for half ofthe above theoretical heating, maintaining the same profile(i.e. a peak at the sonic point). This implies that the role ofcoherent interactions is fundamental in shaping the heatingfunction and that turbulence and turbulent heating can not beneglected when studying the acceleration of the solar wind.The peak dissipation seems to scale linearly with the rmsamplitude Z + ⊙ although a proof of the precise scaling wouldrequire further studies. We finally observe that if the lowerboundary is shifted to the base of the chromosphere, henceincluding the transition region, a stronger dissipation rate isexpected to be found at the transition region and in the lowcorona. The Alfv´en speed is smaller below the transitionregion but its gradients are higher, increasing the amount ofenergy residing in the anomalous reflected component which,having more time to interact with its mother wave, mighteflection-driven turbulence 5increase the spectral energy transfer. Acknowledgments.
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