Relaxation of Turbulence Behind Interplanetary Shocks

A. Pitna, J. Safrankova, Z. NemecekCharles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

Abstract. Relaxation of solar wind turbulence has been a topic of interest fordecades. Many phenomenological laws for the energy decay have been derived andsubsequently confirmed in MHD simulation studies. However, in-situ observationsof decay laws is an inherently difficult task. In this study, we use interplanetaryshocks because (a) they enhance the level of turbulent fluctuations and (b) theypropagate into a pristine solar wind. Using a few simplifying assumptions, we studythe evolution of large-scale energy-containing eddies. We analyze a relaxation ofthe turbulent energy enhanced by interplanetary shocks and discuss its observedturbulent decay.

Introduction

Although the solar wind is the most studied magnetized plasma, there are still many unresolved issuesconcerning the expanding solar plasma, mainly its origin and non-adiabatic radial temperature profile[Marsch et al., 1982; Gazis, 1984; Goldstein et al., 1996; Matthaeus et al., 1999b,a]. Moreover, MHDturbulence plays a crucial role in the evolution of solar wind fluctuations [Zank et al., 1996; MacBrideet al., 2008]. The first evidence for the turbulent nature of the fluctuations was reported by Coleman[1968]. He analyzed magnetic field and plasma velocity measurements from the Mariner 2 spacecraft andhe found that a level of turbulent fluctuations was sufficient to heat the protons at 1 Astronomical Unit(AU). Many other authors attributed the excess of the proton temperature to the transfer of energy fromthe large scale MHD fluctuations into the ion kinetic scales, li ' (λi, ρi), where λi is the ion inertial lengthand ρi is the proton thermal gyroradius. The first phenomenological description of a turbulent cascadewas presented by Iroshnikov [1963] and Kraichnan [1965]. According to their theory of turbulence, thepower spectrum density (PSD) of magnetic and velocity fluctuations in the inertial range (where thecascade takes place) has a power law behaviour, PSD(f) ∝ fα, with α = −3/2, f being the spacecraftframe frequency. Goldreich and Sridhar [1995] derived another theory of MHD turbulence, where thePSD of magnetic field fluctuations scales differently in perpendicular, α⊥ = −5/3, and parallel, α‖ = −2,directions relative to a local magnetic field [Horbury et al., 2008]. For more details, see comprehensivereviews on solar wind turbulence by Tu and Marsch [1995] and Bruno and Carbone [2013].

In the theory of hydrodynamic turbulence, there are two important spatial scales that define therange where turbulence operates. First, the outer scale, L, denotes the scale on which the energy isinjected into the system. Second, the dissipation scale, ld, is the scale where the kinetic energy containedwithin the turbulent eddies transforms into heat. In MHD turbulence, this picture is still valid, but the(weakly) collisionless solar wind indicates that the dissipation mechanism is an unresolved problem upto now. Observations of the Cluster spacecraft show that the PSD of magnetic fluctuations betweenion and electron spatial scales is a power law, suggesting another turbulent cascade [Alexandrova et al.,2009; Sahraoui et al., 2009].

It is believed that large-scale (l ∼> L) magnetic and plasma fluctuations originate in the solar coronaand could also be produced by the high-velocity shears between the fast and slow solar wind [Robertset al., 1992]. A transition between the injection and inertial ranges of turbulence, at frequency fL, isobserved as a spectral break in the power spectrum of magnetic field fluctuations [Bruno et al., 2009].At spacecraft-frame frequencies f < fL, where fL lies between 10−4 and 10−3 Hz at 1 AU, the spectrumis a power law with α = −1. The Taylor hypothesis [Taylor, 1938] allows us to determine the spatialscale corresponding to fL as lL = vsw/fL, vsw being the bulk solar wind speed.

As mentioned above, the energy contained in large-scale fluctuations is transferred towards smallerscales via a turbulent cascade and this process leads to a decrease of the level of fluctuations in time.Kolmogorov [1941] showed that the decay of the total energy in these fluctuations, E, over time, t,should have a power-law behavior, E ∝ t−n, with n = 10/7. This phenomenological result is valid forhydrodynamic turbulence, while for MHD turbulence, the decay is still a power law buth with n < 10/7.Biskamp [2003] summarized the decay laws for various MHD cases of turbulence that differ in the valuesof magnetic helicity HM and the energy ratio, Γ = EK/EM (EK is the kinetic energy of the turbulenteddies and EM is the magnetic energy of turbulent fluctuations). The most relevant scenario for the

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PITNA ET AL.: RELAXATION OF TURBULENCE BEHIND INTERPLANETARY SHOCKS

solar wind is that the magnetic helicity, HM , is finite but greater than zero and 0� Γ < 1. The energydecay laws should be E ∝ t−1/2, whereas EK ∝ t−1 [Biskamp and Muller, 1999]. Biskamp and Muller[1999] and Biskamp [2003] questioned the validity of a power-law behavior when 0 < t ' 2, due to theeffect of the initial conditions, and they concluded that it is valid asymptotically for t ∼> 10, where thetime t is in units of the eddy turnover time, defined as τnl = 1/(ku), u is the root mean square turbulentamplitude and k is the wave vector [Matthaeus et al., 2014].

An optimal way of investigation of the decay laws in the solar wind is to use a straight-line config-uration of three spacecraft, so that the radial expansion of the solar wind and the decay of turbulentfluctuations can be studied directly [Bruno et al., 2009]. Another possibility is to analyze data comingfrom a spacecraft moving radially to (or from) the Sun, which observes the same plasma stream at differ-ent heliocentric distances [Bavassano et al., 1982]. Finally, a novel approach is to use interplanetary (IP)shocks because they are naturally occurring in the solar wind with a cadence of about one per day1[Webband Howard, 2012].

IP shocks have been intensively studied for decades and many aspects of shock physics are understood[see Balogh and Treumann, 2013, for a comprehensive review], nevertheless, many open questions remain.We know that an IP shock dissipates the bulk kinetic energy straight into heat in a thin region aroundthe shock ramp (a few hundred kms wide). Moreover, after the IP shock passage (in the downstreamregion), the power within the turbulent fluctuations is enhanced approximately tenfold relative to itsupstream values [Luttrell et al., 1984; Pitna et al., 2016].

In this preliminary study, we investigate the decay of the turbulent energy behind IP shocks. Weperform a statistical analysis of a small set of IP shocks (13) and we present the implications of our studyfor the MHD decay laws. We discuss a relaxation of turbulence of both magnetic and kinetic energyconstituents behind IP shocks.

Data used

We used the data from the BMSW instrument on board Spektr-R. A description of the BMSWinstrument has been reported by Safrankova et al. [2013] and Zastenker et al. [2013], thus only a briefreminder is given here. The principle of measurements lies in the use of six Faraday Cups (FCs) ina special configuration. The instrument operates in two modes, adaptive and sweep modes. The formermode used in this paper allows us to determine the plasma moments with a high cadence. This isachieved by acquiring three points of the distribution function2. One FC exhibits a control grid withoutan applied voltage and on control grids of next two FCs, a voltage is applied so that their currents arefractions (0.3 and 0.7) of the first FC. The time resolution of the density, thermal and solar wind speedsis 31 ms. For a comparison, we used the data from the Solar Wind Experiment (SWE) instrument onboard Wind [Ogilvie et al., 1995] for the density and solar wind velocity estimates. However, Windrotates every 3 seconds, therefore the measured cadence is much lower than that of BMSW, namely 92 s.Finally, we used the data from the Magnetic Field Investigation (MFI) Wind instrument [Lepping et al.,1995] with the resolution of ≈ 0.1 s.

In our statistics, the number of cases is limited mainly by the BMSW data coverage. We analyzedthe BMSW data between September 9, 2011 and June 23, 2015, finding only 13 shocks with a sufficientlyhigh-time resolution and with an adequately long time of downstream measurements. For these IPshocks, we analyzed the kinetic energy from BMSW and Wind and the magnetic energy from Wind.

Data analysis

First, we present a method of an analysis of downstream IP shock fluctuations. The instruments(BMSW,SWE) measure the density, N , and the bulk and thermal solar wind speeds, vsw and vth,

respectively, and the measurements of the magnetic field, ~B, are from MFI.A unit of time that is relevant for the MHD decay analysis is the eddy turnover time, τnl [Zhou

et al., 2004]. The time measured at the spacecraft, tsp, is transformed into the time in units of τnl, tnl,according to

tnl = tsp ·Ksh/τnl, (1)

where Ksh is the multiplying factor accounting for the fact that the IP shock propagates into the un-shocked region slower than the upstream speed of the solar wind. Assuming that (a) the IP shock

1But strongly dependent on the solar cycle.

2Assuming the Maxwellian proton particle distribution function.

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PITNA ET AL.: RELAXATION OF TURBULENCE BEHIND INTERPLANETARY SHOCKS

propagates into a stationary medium and (b) the IP shock speed along its normal is a constant, thenKsh = vsh/(vd − vsh), where vsh is the shock speed and vd is the speed of the solar wind downstream ofthe shock in the spacecraft frame of reference3. The eddy turnover time is computed as

τnl =vd,0∆t

2π

√2∑i=0

var∆t(Vi)

, (2)

where ∆t is the temporal scale for estimations of the moments of any physical quantity of interest, vd,0 isthe immediate downstream solar wind speed, Vi, i=0,1,2 are GSE components of the velocity measuredby the spacecraft, and var∆t denotes the variance of the designed quantity over the time scale ∆t.

In order to estimate the energy contained within turbulent fluctuations, we assume that the velocityand magnetic fields could be expressed as ~V = ~v0 + ~v1 and ~B = ~b0 +~b1, where ~v0 is the average solarwind speed, ~v1 the velocity field of turbulent fluctuations, ~b0 the ambient magnetic field, and ~b1 thefluctuating magnetic field, respectively. Then, we express the kinetic, EK , and magnetic, EM , energiesas

EK =1

2

∫V

ρv21dV and EM =

1

2

∫V

b21/(2µ0)dV, (3)

and we compute them from the measurements as

EK =1

2ρ∆t

√√√√ 2∑i=0

var∆t(Vi) and EM =1

2µ0

√√√√ 2∑i=0

var∆t(Bi), (4)

where ρ∆t is the average density, Bi is defined as Vi, and µ0 is the vacuum permeability. All variancesand the mean values were computed on a time scale of ∆t = 17 minutes.

Case study of an IP shock

As an example of an IP shock analysis, we present the IP shock that was detected by BMSW onJune 22, 2015 at 0536 UT (0505 UT at Wind). Its basic parameters are: the magnetosonic Mach number,Mms = 2.6, the angle between the shock normal and upstream magnetic field, θBn = 78◦, the shock speedin the spacecraft frame, vsh = 483 km·s−1, the ratio of the downstream to upstream density, ρd/ρu = 1.8,the difference between downstream and upstream solar wind speeds, vd − vu = 60 km · s−1 and the ratiobetween downstream and upstream magnitudes of the magnetic field, Bd/Bu = 1.7 (propagated IMFfrom Wind).

For this particular IP shock, we have 12 hours of BMSW and WIND data measurements downstreamof the shock, after it, a second IP shock was detected4. We analyze the data immediately upstream(≈ 1 hour) and 12 hours downstream and calculate the kinetic and magnetic energies using eq. (4). Thetime shift between two consecutive evaluations is 1 min. We rescale the spacecraft time tsp by eq. (1)with Ksh = 5.1 and τnl = 1.7 hours. Figure 1 shows the evolution of the kinetic and magnetic energiesas a function of tnl. The black and green lines showing the kinetic energies from BMSW and SWE,respectively, resemble themselves rather closely. The blue line (magnetic energy computed from Wind)is above the green and black lines indicating that the energy contained in magnetic fluctuations is higherthan that in the velocity fluctuations, i.e., Γ < 1.

In Figure 1, despite large fluctuations, a decreasing trend with time is quite visible in all threequantities. For this reason, we performed a statistical study, although a number of appropriate IPshocks in the analysis is relatively small.

Superposed epoch analysis

As already mentioned, in our limited dataset, most of the shocks have short intervals of downstreamdata (1–2 hours in the spacecraft time). On the other hand, a number of eddy turnover times could beas high as 10 even for 1 hour, so it could be still enough time for observations of the energy decay.

An important point about the validity of a power law as a good model for ‘young’ turbulence wasalready commented. The turbulent energy in the upstream solar wind is decaying by some rate, an IP

3For the sake of simplicity, we assumed that the shock normal is parallel to upstream and downstream solar wind speeds.

4It is detected by BMSW on July 22, 2015 at 1828 UT.

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PITNA ET AL.: RELAXATION OF TURBULENCE BEHIND INTERPLANETARY SHOCKS

0 10 20 30tnl

10-13

10-12

10-11

E [

J⋅m

-3]

Figure 1. Time evolution of the magnetic (red), EM , and kinetic (green), EK , energies behind the IPshock registered on June 22, 2015 at Wind. The black diamond dotted line is the kinetic energy computedfrom BMSW measurements. In the x-axis, the time tnl is expressed in units of the eddy turnover time,τnl, with tnl = 0 for the IP shock passage.

shock enhances turbulence and the energy within the fluctuations increases. The question is what isthe proper interpretation of the ‘time’ in a power law model t−n immediately downstream of the shock?There are at least two possibilities: (a) the IP shock ‘resets’ the time and we start from tnl ≈ 0; and(b) there is one master power law E = E0t

−nnl and we jump into different ‘times’ given by a downstream

energy level. We argue that the later is closer to the truth because downstream turbulence seems to bewell developed.

We perform a superposed epoch analysis using the method described in the previous section foreach IP shock. Because the downstream energy levels of different shocks vary more than a decade, wenormalize the energy profiles by the energy immediately downstream of the shock, E0. Another reasonfor such normalization is that the expected decreasing trend should be more clear. In Figure 2, thetrends of these normalized energy profiles are reported for each IP shock. The kinetic energies estimatedat both spacecraft (panels b and c in Figure 2) have a slight tendency to decrease in time confirming thetrends observed in the case study. The decrease of the magnetic energy (panel 2a) is less evident, butnotable. On the last panel (d) of Figure 2, we show the Alfven ratio, rA, defined as the ratio between thekinetic and magnetic energies. The profiles are normalized by the first values immediately downstreamof the shocks.

Discussion and Summary

Let us discuss assumptions and approximations made through an estimation of kinetic and mag-netic energies. The most important assumption is a stationarity of solar wind conditions for the wholedownstream periods. We do not know a priori whether the observed decrease of a level of the turbulentenergy is caused by its actual decrease or by a lower energy level of the yet unshocked plasma. However,it is reasonable to assume that for an interval spanning only a few hours, a probability of encounteringdiscontinuities that significantly change the energy levels is relatively low.

According to eq. (1), a rough estimate of the downstream time interval affected by the shock canbe made. The mean value of Ksh is about 6 and the standard travel time for the solar wind plasma toreach the Earth is around 3–4 days. From this, it follows that one should not consider the measurementsolder that ≈ 12 hours after the passage of an IP shock because this solar wind was not affected by theshock. On the other hand, a longer data interval gives a higher chance to observe the decay of theturbulent energy. Therefore, we made a compromise and analyzed usually 2–3 hour-long time intervalsand Figures 1 and 2 confirm a validity of such approach.

The large fluctuations in time profiles of the kinetic and magnetic energies are caused by the crossingsof various magnetic and plasma discontinuities. The energy levels prior to and after them could be thesame but a jump of the magnetic or velocity fields artificially increases the level (eq. 4). Nevertheless,an overall decreasing trend is visible in Figure 2, although the observed decay of the turbulent energy isnot statistically significant due to a small number of samples.

Finally, we mention the role of the inverse turbulent cascade that can take place in the solar wind.The inverse cascade means that if we inject a physical quantity of interest, Q(k), k being the wave

83

PITNA ET AL.: RELAXATION OF TURBULENCE BEHIND INTERPLANETARY SHOCKS

(a) (b)

0 5 10 15 20 25tnl

0.01

0.10

1.00

10.00

100.00

Eb/E

b,0

Wind

0 5 10 15 20 25tnl

0.01

0.10

1.00

10.00

100.00

Ek/

Ek,

0

Wind

(c) (d)

0 5 10 15 20 25tnl

0.01

0.10

1.00

10.00

100.00

Ek/

Ek,

0

Spektr-R

0 5 10 15 20 25tnl

0.01

0.10

1.00

10.00

100.00

r A/r

A,0

Wind

Figure 2. Superposed epoch analysis of the magnetic (a) and kinetic (b) energies behind the IP shocksat Wind. The panel (c) shows the kinetic energy from the BMSW measurements and the panel (d)shows the Alfven ratio of the kinetic and magnetic energies from the WIND measurements. The x-axisis marked in the units of the eddy turnover time, with tnl = 0 for the IP shock passage. Different colorsrepresent various IP shocks.

number, into the system at wave number k0, it will diffuse in k-space towards lower k. In 2D and3D MHD turbulence, the only ideal invariant that shows an inverse cascade is the magnetic city, HM ,the total energy and cross-helicity cascade directly [Biskamp, 2003]. In situ observations show that themagnetic helicity in the solar wind is not a zero [Matthaeus et al., 1982]. This implies that the inversecascade plays some role in the turbulent evolution of the solar wind and its role in the energy decayshould be examined. Magnetic and kinetic energies cascade to lower scales (higher k) and are eventuallydissipated, however, the magnetic helicity cascades to higher scales (lower k). On the other hand, anIP shock “injects” the energy in a wide range of scales (from kinetic up to inertial) [Pitna et al., 2016].It is not clear, what is the impact of the energy injected by an IP shock on the inverse cascade. Onecould argue that the “original” injection scale and the inverse cascade are unaffected by the IP shockpassage and the only difference is that the turbulent energy at the lower k-end of the inertial range isenhanced. Another possibility is that the injected energy cascades in both direct and inverse directions.In any case, we can use the results of Christensson et al. [2001]. They performed a 3D MHD simulationwith a focuss on the inverse cascade. They found that a freely decaying MHD turbulence with a finitemagnetic helicity shows both direct and inverse cascades of the magnetic energy. They also found thatthe asymptotic decay rates of magnetic and kinetic energies are close to t−0.7 and t−1.1, respectively.This is consistent with the panels (a)–(c) in Figure 2. However, normalized rA should increase slightlyin time but this is not observed in the panel (d) in Figure 2.

As a conclusion, we analyzed a relaxation of the turbulent energy behind IP shocks and found thatthe turbulent energy decay can be observed. The slight decrease in the kinetic energy suggests that themagnetic energy decreases slower in time than the kinetic energy, in agreement with Biskamp and Muller[1999]. It seems that an enhanced superposed epoch analysis could allow us to estimate the energy decayfrom the observed data with a reasonable significance. In a following study, we plan to increase thenumber of analyzed IP shocks.

Acknowledgments. The present work was supported by the Charles University Grant Agency underContract 1922214.

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