THE INTERACTION OF TURBULENCE WITH PARALLEL AND ...

The Astrophysical Journal, 833:218 (16pp), 2016 December 20

doi:10.3847/1538-4357/833/2/218

© 2016. The American Astronomical Society. All rights reserved.

THE INTERACTION OF TURBULENCE WITH PARALLEL AND PERPENDICULAR SHOCKS: THEORY AND OBSERVATIONS AT 1 au 1

L. Adhikari1, G. P. Zank1,2, P. Hunana1,2, and Q. Hu1,2

Center for Space Plasma and Aeronomic Research (CSPAR), University of Alabama in Huntsville, Huntsville, AL 35899, USA 2 Department of Space Science, University of Alabama in Huntsville, Huntsville, AL 35899, USA Received 2016 July 22; revised 2016 October 17; accepted 2016 October 18; published 2016 December 19

ABSTRACT Shocks are thought to be responsible for the amplification of turbulence as well as for generating turbulence throughout the heliosphere. We study the interaction of turbulence with parallel and perpendicular shock waves using the six-coupled-equation turbulence transport model of Zank et al. We model a 1D stationary shock wave using a hyperbolic tangent function and the Rankine–Hugoniot conditions for both a reduced model with four coupled equations and the full model. Eight quasi-parallel and five quasi-perpendicular events in the WIND spacecraft data sets are identified, and we compute the fluctuating magnetic and kinetic energy, the energy in forward and backward propagating modes, the total turbulent energy, the normalized residual energy, and the normalized cross helicity upstream and downstream of the observed shocks. We compare the observed fitted values upstream and downstream of the shock with numerical solutions to our model equations. The comparison shows that our theoretical results are in reasonable agreement with observations for both quasi-parallel and perpendicular shocks. We find that (1) the total turbulent energy, the energy in forward and backward propagating modes, and the normalized residual energy increase across the shock, (2) the normalized cross helicity increases or decreases across the shock, and (3) the correlation length increases upstream and downstream of the shock, and slightly flattens or decreases across the shock. Key words: shock waves – turbulence shocks are driven by ICMEs (Jian et al. 2011). Interplanetary shocks accelerate energetic particles (Gosling 1983; Giacalone et al. 1994; Zank et al. 2000, 2006, 2007; Desai et al. 2004; Simnett et al. 2005; Lee et al. 2012; Reames 2013; Neergaard Parker et al. 2014; Verkhoglyadova et al. 2015), and produce ions with energies often exceeding 1 MeV. These particles may pose significant hazards to satellite technology and human activity in space (Reames 1999; Zank et al. 2007; Verkhoglyadova et al. 2015). Shocks can reduce the intensity of cosmic rays, and also accelerate pickup ions (Gloeckler et al. 1994; Giacalone & Jokipii 1995; Zank et al. 1996b; Zilbersher & Gedalin 1997). The rate of particle acceleration is controlled by the magnetic field strength, the shock strength, and turbulence levels (Zank et al. 2000) as well as shock obliquity (Zank et al. 2006). Therefore, the acceleration of charged particles is determined in part by where the shock forms since the acceleration rate is largest near the Sun (Zank et al. 2000). The maximum energy to which a particle is accelerated can be determined by a balance of the particle acceleration rate and the dynamical timescale of the shocks (Zank et al. 2000; Li et al. 2003; Rice et al. 2003). ICME-driven shocks form near the Sun, so particles can be accelerated for a longer time. Shocks driven by stream interaction form at larger distances from the Sun and particles experience acceleration for a shorter time (BlancoCano et al. 2016), so the particle acceleration rate is much slower (Zank et al. 2000). Shocks driven by CMEs are faster and stronger, and show a larger distribution of shock parameters, than those driven by stream interaction (Kilpua et al. 2015). During the formation of quasi-perpendicular shocks, a foot develops upstream of the main shock ramp, and an overshoot forms behind the ramp. The magnetic profile of the shock is closely related to the reflection of incoming ions at the shock.

1. INTRODUCTION The solar wind is highly magnetized plasma in which turbulence is ubiquitous (Bruno & Carbone 2005, 2013). Turbulence underlies several important phenomena in the solar wind, such as acceleration of the solar wind and coronal heating (Leer et al. 1982; Matthaeus et al. 1999a; Dmitruk et al. 2001, 2002; Oughton et al. 2001; Suzuki & Inutsuka 2005, 2006; Verdini et al. 2010; van Ballegooijen et al. 2011; Asgari-Targhi et al. 2013; Lionello et al. 2014; Matsumoto & Suzuki 2014), solar wind heating (Freeman 1988; Gazis et al. 1994; Williams et al. 1995; Matthaeus et al. 1999b; Smith et al. 2001, 2006a, 2006b; Isenberg et al. 2003, 2010; Isenberg 2005; Breech et al. 2008; Ng et al. 2010; Oughton et al. 2011; Usmanov et al. 2011; Adhikari 2015; Adhikari et al. 2015a, 2015b), and the scattering of the solar energetic particles (Li et al. 2003; Zank et al. 2007). To understand these phenomena, it is necessary to understand the transport of lowfrequency turbulence in inhomogeneous astrophysical flows such as the solar wind. Interplanetary shocks are common in the solar wind, and are thought to be responsible for the amplification of turbulence as it is convected through a shock, as well as generating turbulence throughout the heliosphere. Interplanetary shocks are generated in the heliosphere by the interaction of solar wind disturbances with the solar wind (Burlaga 1971; Richter et al. 1985), e.g., the interaction of fast solar wind streams with slow preceding plasma or fast interplanetary coronal mass ejections (ICMEs). However, to form a shock in the heliosphere, the difference in speed between two solar wind streams or the ICME and the preceding plasma should exceed the magnetosonic speed. Shocks are much more prevalent during solar maximum than solar minimum. During solar minimum most shocks are driven by stream interactions (Jian et al. 2011, 2013), whereas during solar maximum most 1

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The shock parameter θBn, which is the angle between the shock normal and the upstream magnetic field, distinguishes the properties of the shock and indirectly the nature of the downstream turbulent plasma flow. It is thought that θBn influences the mechanism of charged particle acceleration by the shock (Zank et al. 2006). The value of θBn depends on the configuration of the interplanetary magnetic field along which the shock is propagating. Shocks are called quasi-parallel if θBn„45° and quasi-perpendicular if θBn>45°. We analyze five quasi-perpendicular and eight quasi-parallel shocks, and study the turbulent quantities upstream and downstream of the shock. Quasi-perpendicular shocks, for which the dispersive scale lengths are small and the associated cross-field currents are strong, produce significant levels of downstream plasma wave turbulence. Turbulence limits shock steepening and therefore the structure of shocks depends on the properties of the associated turbulence. Macek et al. (2015) find that turbulence behind quasi-perpendicular shocks is more intermittent than that behind quasi-parallel shocks. In this manuscript, we apply the model equations of Zank et al. (2012)3 to study the interaction of turbulence with parallel and perpendicular shocks. The study is based on both theoretical and observational analysis. From the model equations of Zank et al. (2012), we derive two 1D steadystate model equations, one corresponding to six and the other to four turbulence transport equations in a Cartesian coordinate system for parallel and perpendicular shocks. We solve the four and six coupled turbulence transport equations for parallel and perpendicular shocks numerically using a Runge–Kutta fourthorder method. We then compare the numerical solutions to observations made by the WIND spacecraft. In Section 2, we introduce the four coupled model equations that are coupled to equations that describe the parallel and perpendicular shock equations. We describe the shock structure in terms of a hyperbolic tangent function for which we prescribe the shock thickness. Section 3 presents and discusses the observations; here we compare the numerical solutions of áb 2ñ1 2 to the observed áb 2ñ1 2 of the quasi-parallel and quasi-perpendicular shock events, where áb 2ñ1 2 is the variance in the fluctuating magnetic field. Section 4 presents discussion and conclusions. We compare solutions derived from the four- and six-coupledequation turbulence transport models in Appendix A (quasiparallel shocks) and Appendix B (quasi-perpendicular shocks) and compare both sets of solutions to observations (Appendices C and D). We solve for the fluctuating kinetic energy áu2ñ, the energy in forward and backward propagating modes g and f, the total turbulent energy ET, the normalized residual energy σD, and the cross helicity σc.

corresponding to forward and backward propagating modes, i.e., λ− and λ+, respectively, and the correlation length of the residual energy λD can be expressed in terms of correlation lengths corresponding to the velocity fluctuations λu, magnetic field fluctuations λb, and the mixed correlation λub (see also Adhikari 2015). For the special case λu=λb=λub=λ, we find that λ+ = λ−=λ and λD=2λ, which we apply to derive a reduced set of four coupled turbulence transport equations. By introducing a stationary shock wave in the form of a hyperbolic tangent function of x, we can study the interaction of turbulence with parallel and perpendicular shocks. 2.1. Parallel Shock Model For a parallel shock, the magnetic field B and the flow velocity U are parallel to each other i.e., U=Ux and B=Bx. The reduced 1D steady-state coupled turbulence transport equations for a parallel shock can be written in the form (Zank et al. 2012; Adhikari et al. 2015a, 2015b) ⎡f ⎛ ¶f 1 ⎞ ⎤ ¶U + ⎢ + ⎜2a - ⎟ ED ⎥ ⎝ ⎣2 ¶x 2 ⎠ ⎦ ¶x ¶VA fg1 2 ( f - ED) = - 2 + ; l ¶x

(U - VA)

⎡g ⎛ ¶g 1 ⎞ ⎤ ¶U + ⎢ + ⎜2a - ⎟ ED ⎥ ⎝ ⎣2 ¶x 2 ⎠ ⎦ ¶x 1 2 ¶VA gf (g - ED) = - 2 ; l ¶x

(1 )

(U + VA)

U

(2 )

⎤ ¶ED 1 ¶U ⎡ ⎛⎜ 1⎞ + ⎢ ⎝2a - ⎠⎟ ( f + g) + ED⎥ ⎦ ¶x 2 ¶x ⎣ 2 ¶f ⎤ VA ⎡ ¶g + -g ⎥ ⎢f ¶x ⎦ 2 fg ⎣ ¶x +

⎡ f1 ¶VA f - g = - ED ⎢ ⎣ ¶x 2 U

¶l = f1 ¶x

2

2

+ g1 2 ⎤ ⎥; ⎦ l

+ g1 2 ,

(3 ) (4 )

where f and g are the energies in backward and forward propagating modes respectively, ED is the residual energy, and λ is the correlation length. Note that in this case the magnetic field is directed outward. The parameter U is the solar wind velocity, and VA is the Alfvén velocity. The parameter a=1/2 or 1/3 for 2D or 3D mixing tensors, respectively. We solve the coupled Equations (1)–(4) for a=1/3.

2. SHOCK MODEL

2.2. Perpendicular Shock Model

The six coupled equations of the model of Zank et al. (2012) are complex, so we reduce them to a simpler set of four coupled model equations that describe the interaction of turbulence with parallel and perpendicular shock waves. In Appendices C and D, we solve the full six coupled equations to illustrate how the results differ from those of the four coupled equations. According to Dosch et al. (2013), the correlation lengths

For a perpendicular shock, the flow velocity U and the magnetic field B are perpendicular to each other, e.g., U=Ux and B=Bz. If the flow velocity is in the x direction, the magnetic field is in the z direction. Therefore, the 1D turbulence transport equations in a Cartesian coordinate system x do not contain the Alfvén velocity. The four coupled equations therefore become (Zank et al. 2012; Adhikari et al. 2015a, 2015b)

3

Zank et al. (2012) developed six coupled equations to describe the evolution of the energy in forward and backward propagating modes, the residual energy, the correlation length corresponding to forward and backward propagating modes, and the correlation length of residual energy.

U

2

⎡f ⎛ ¶f 1 ⎞ ⎤ ¶U fg1 2 + ⎢ + ⎜2a - ⎟ ED ⎥ = -2 ; ⎝ ⎣2 ¶x 2 ⎠ ⎦ ¶x l

(5 )

The Astrophysical Journal, 833:218 (16pp), 2016 December 20

U

⎡g ⎛ 1 ⎞ ⎤ ¶U gf 1 2 ¶g + ⎢ + ⎜2a - ⎟ ED ⎥ = -2 ; ⎝ ⎣2 ¶x 2 ⎠ ⎦ ¶x l U

⎤ ¶ED 1 ¶U ⎡ ⎛⎜ 1⎞ + ⎢ ⎝2a - ⎠⎟ ( f + g) + ED⎥ ⎦ ¶x 2 ¶x ⎣ 2 1 2 1 2 ⎡f +g ⎤ = - ED ⎢ ⎥; ⎦ ⎣ l U

¶l = f1 ¶x

2

+ g1 2.

Adhikari et al.

statistically that the thickness of a quasi-perpendicular shock decreases with increasing Mach number of the shock. The overall shock thickness can be written in terms of the proton gyroradius rg (=mpv/qB), where mp is the proton mass, v is the solar wind velocity, q is the charge of a proton, and B is the magnetic field. Kan et al. (1991) studied the dependence of shock thickness on the shock normal. They find that the shock thickness Δ increases rapidly from Δ/(c/ωpi)∼5 for a quasiperpendicular shock to Δ/(c/ωpi)∼15 for quasi-parallel shocks, where c/ωpi is the inertial length. In our study, we use a shock thickness of 5rg for perpendicular shocks and 10rg for parallel shocks. Obviously, these are crude estimates and particular shocks can have a thickness different than these values. However, in general a parallel shock is always thicker than a perpendicular shock. In the case of a perpendicular shock, reflected protons provide the dissipation, i.e., specular reflection of some incident ions by the cross-shock potential. The specularly reflected protons have a velocity approximately equal to the incident flow speed. The reflected ions form a foot ahead of the ramp. This part of the overall shock structure roughly determines the thickness of a perpendicular collisionless shock (e.g., Lowe & Burgess 2003). The dissipation mechanism at a parallel shock is quite different because reflected particles do not automatically return to the shock front. However, a beam of particles propagates away from the shock in the upstream region, which excites fluctuations that scatter the proton beam so that the protons form a diffuse ion population that is convected back to the shock. Consequently, the dissipation process at a quasi-parallel shock is extended over a much wider region than for a perpendicular shock. The parallel shock structure therefore has a transition that can be modeled as many proton gyroradii thick (much thicker than the perpendicular case). These properties distinguish parallel and perpendicular shocks. Equation (12) can be modified to relate the densities in the upstream and downstream regions, i.e., ρ1 and ρ2, respectively (Vinas & Scudder 1986), although now with ρ2>ρ1. The Alfvén velocity can also be written in terms of a hyperbolic tangent function. The Alfvén velocity is given by

(6 )

(7 ) (8 )

The coupled turbulence transport Equations (5)–(8) have a simpler form than the coupled transport Equations (1)–(4). The coupled Equations (1)–(4) and (5)–(8) are solved using a Runge–Kutta fourth-order method for different initial conditions. The other turbulent quantities can be calculated as (Zank et al. 2012) f+g g-f ; EC = ; 2 2 E E sc = C ; sD = D ; ET ET

ET =

áu 2 ñ =

f + g + 2ED ; 2

b2 m0 r

=

f + g - 2ED , 2

(9 ) (10) (11)

where ET is the total turbulent energy, EC is the cross helicity, σc is the normalized cross helicity, σD is the normalized residual energy, áu2ñ is the fluctuating kinetic energy, and áb 2ñ is the fluctuating magnetic energy. 2.3. Shock Wave: Solar Wind Velocity and Alfvén Velocity Burgers’ well known equation, a conservation equation, describes a shock layer in a flow. The steady-state solution of Burgers’ equation gives a hyperbolic tangent function (e.g., Zank 2014), and yields a transition of the flow from higher to lower speed. Many previous works have used the hyperbolic tangent function to describe a shock wave, including, for example, Bale et al. (2005), who used a tanh functional fit to describe the overall structure of observed quasi-perpendicular shocks. A simple velocity function that describes a steady shock is a hyperbolic tangent function of x (e.g., Webb & McKenzie 1984), U=

⎛ x - x0 ⎞ 1 1 (U1 + U2) - (U1 - U2) tanh ⎜ ⎟, ⎝ Dsh ⎠ 2 2

VA =

B , m0 r

where B is the magnetic field, μ0 is the permeability of free space, and ρ is the density. By mass conservation ρU=m, allowing the above equation to be written as

(12)

where U1 and U2 are the velocities upstream and downstream of the shock. Note that U1 and U2 are asymptotic values. Of course, all the other corresponding asymptotic upstream and downstream planar variables determined by the Rankine– Hugoniot conditions can be expressed using a form of the tanh function. One drawback about using a simple tanh profile is that it cannot capture additional fine structure in the shock, such as the ramp or post-shock overshoots. For our initial study, these additional complications are not especially important. The parameter x0 is the position of the shock and Dsh is its thickness. The thickness of a shock is determined by the primary dissipation mechanism that is responsible for heating the downstream plasma. Hobara et al. (2010) have shown

VA =

B m0 m

U,

where m is a constant. This shows that the Alfvén velocity is proportional to the product of the magnetic field and the square root of the velocity, which is a function of a hyperbolic tangent function. For a parallel shock, the magnetic field is constant across a shock, i.e., B1=B2, where B1 and B2 are the magnetic field strengths in the upstream and downstream regions of the shock. However, the magnetic field increases across a perpendicular shock, i.e., B2>B1, and B2/B1=r, where r is the shock compression ratio. By using these properties of the 3

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Table 1 Quasi-parallel and Quasi-perpendicular Events Quasi-parallel Shocks DOY:YR

Tsh

210:2000 62:2001 215:2001 213:2002 214:2004 266:2004 16:2005 168:2012

10:01:33 UT 11:29:15 UT 7:18:45 UT 5:19:27 UT 1:44:36 UT 5:53:36 UT 9:27:24 UT 9:3:33 UT

Quasi-perpendicular Shocks Tup

Tdown

DOY:YR

Tsh

2 2 3 3 1 3 1 2

2 2 3 3 1 3 1 2

260:2011 67:2012 305:2012 46:2014 58:2014

2:57:12 UT 3:28:42 UT 14:28:24 UT 12:46:36 UT 15:50:15 UT

hr hr hr hr hr hr hr hr

hr hr hr hr hr hr hr hr

Tup

Tdown

2 2 3 1 1

2 2 3 1 1

hr hr hr hr hr

hr hr hr hr hr

Note. DOY:YR is the day of year and the year. Tsh is the position of a shock. Tup is the time interval ahead of it and Tdown is the time interval behind it. The theoretical and observational turbulent quantities are calculated between these time periods.

magnetic field in the above equation, we get VA2 = VA1

r

and

VA2 = VA1 r

data sets into one by overlapping the times that are closest to each other. We then consider a moving interval (of ∼15 s) upstream and downstream of the shock. We remove bad data points and calculate the turbulent quantities for the outwardly or inwardly directed magnetic field in each interval. We calculate various turbulent quantities with respect to the direction of the magnetic field. In the case of perpendicular shocks, the y-component of the magnetic field By is used to determine the direction of the magnetic field, whereas for parallel shocks, the x-component of the magnetic field Bx is used. In this analysis, we use two criteria: (i) there should be at least three good data points in each interval, and (ii) the ratio of the mean square fluctuations and the square of the mean field must be small, i.e., ádx 2ñ X 2  1. Here ádx 2ñ represents the mean square fluctuations of the field in each interval, and X is the average upstream or downstream field of the shock. The parameter X takes constant upstream and downstream values. The purpose of the criterion ádx 2ñ X 2  1 is to exclude large quasi-periodic downstream (and upstream) fluctuations that may correspond to overshoots and undershoots associated with shock structure that are not accounted for in the Rankine–Hugoniot analysis (Russell et al. 1982; Bavassano-Cattaneo et al. 1986; Mellott & Livesey 1987). The data have greater scatter downstream of the shock than upstream. A possible reason for the large downstream scatter may be due to embedded structures and large downstream oscillations. These are difficult to remove clearly from the data. Simulations and observations suggest that shock waves can generate high levels of downstream vortical turbulence, including magnetic islands (Zank et al. 2015; Zheng & Hu 2016). Therefore, the data associated with different downstream structures have to be removed. In this work, we use the criterion ádx 2ñ X 2  1 to remove data associated with large structures. In future, we hope to develop a better technique to study the observed downstream turbulent parameters. First, we calculate the observed values for every ∼15 s interval if and only if criterion (i) is satisfied. We then smooth the calculated turbulent quantities in the upstream and downstream regions for every 1 minute, where we use criterion (ii). Recall that we calculate the fluctuations in the velocity and magnetic field. These fluctuating quantities do not depend on a frame transformation, i.e., the fluctuating quantities are the same in the spacecraft frame and in the shock frame. To

(13)

for a parallel and perpendicular shocks, respectively. Here, VA1 and VA2 are the Alfvén velocites upstream and downstream of the shock. Equation (13) illustrates that the Alfvén velocity decreases across a parallel shock and increases across a perpendicular shock by the square root of the shock compression ratio. 3. OBSERVATIONS In this section we calculate various turbulent quantities upstream and downstream of selected shocks using magnetometer and plasma data sets with 3 s resolution from the WIND spacecraft, and compare the theoretical and observed values. We consider five quasi-perpendicular and eight quasiparallel shock events from the Heliospheric Shock Database, generated and maintained at the University of Helsinki, and calculate turbulent quantities upstream and downstream of the shock separately. The quasi-perpendicular and quasi-parallel shocks are listed in Table 1. Note that these shocks are forward shocks. A shock is defined as a forward shock if it propagates away from the Sun in the rest frame of the solar wind. It is found that the shocks at 1 au are mostly forward shocks, and their source is ICMEs. Reverse shocks, i.e., sunward propagating shocks, can also be found at 1 au, and the source of such reverse shocks is the interaction between fast and slow solar wind. Time periods Tup and Tdown shown in Table 1 are the upstream and downstream periods over which we calculate various turbulent quantities. These upstream and downstream time periods are chosen on the basis of conservation of mass. Sometimes data upstream and downstream of a shock do not show smooth behavior and exhibit sudden increases or decreases. Within the time periods shown in Table 1, the solar wind parameters vary smoothly and satisfy the conservation of mass. In Table 1, the column Tsh indicates the location of shocks for different events, DOY is the day of year and YR is the year. We use the same procedure as used by Zank et al. (1996a) and Adhikari et al. (2014, 2015a, 2015b) to calculate the turbulent quantities in the upstream and downstream regions. Initially, we obtain magnetometer and plasma data sets with 3 s resolution separately from the WIND spacecraft. We combine these two 4

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Adhikari et al. Table 2 Initial Conditions

Events 210:2000 62:2001 215:2001 213:2002 214:2004 266:2004 16:2005 168:2012

f(km2 s−2)

g(km2 s−2)

ED (km2 s−2)

λ (106 km)

r

Dsh

U (km s−1)

Vsh (km s−1)

ρ (cm−3)

MA

θBn

47.54 17.23 9.76 216.78 86.39 186.71 29.49 35.21

10.75 60.58 3.64 25.17 16.89 22.87 10.77 6.14

2.27 −3.47 1.18 −6.93 −5.04 2.7955 4.44 −3.2824

1.2 1.2 1.2 1.2 1.2 0.6 2.4 2.4

1.49 2.02 2.91 2.21 2.21 2.03 1.48 2.5

10rg 10rg 10rg 10rg 10rg 10rg 10rg 10rg

130.34 99.85 110.01 161.77 159.27 142.39 110.69 146.79

495 502 450 443 590 533 569 451

19.89 3.04 8.79 2.03 3.80 3.67 3.07 3.08

3.5 2.12 2.56 2.8 3.25 2.5 1.84 4.45

30° 32° 27° 7° 8° 16° 18° 15°

Note. Initial conditions are shown for the energy in backward and forward propagating modes f and g, the residual energy ED, the correlation length λ, and other shock parameters: r is the shock compression ratio, Dsh is the thickness of a shock, U is the upstream solar wind velocity in the shock frame, rg is the gyroradius of the magnetic field, Vsh is the shock speed, ρ is the solar wind density, MA is the Alfvén Mach number, and θBn is the obliquity angle.

Figure 1. Comparison between the theoretical and observed square root of the fluctuating magnetic energy áb 2ñ1 2 as a function of X/Xsh or T/Tsh upstream and downstream of the quasi-parallel shocks shown in the title of each plot. The black solid curves are the theoretical áb 2ñ1 2 , the solid straight red lines the fitting to the observed áb 2ñ1 2 upstream and downstream of the shock, and the scatter “+” symbols the observed áb 2ñ1 2 . The parameters Xsh and Tsh are the spatial and temporal positions of the shock.

calculate the Elsässer variables z = u  b m0 r (Elsässer 1950), we also need the solar wind density ρ. The solar wind density is frame-independent as well. The parameter u is the velocity fluctuation in the solar wind, and b is the magnetic

field fluctuation. We show the comparison between the theoretical and observed fluctuating magnetic energy for different events in Sections 3.1 and 3.2, and the other quantities in Appendices A and B. 5

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Table 3 Initial Conditions for the Energy in Forward and Backward Propagating Modes g and f, the Residual Energy ED, the Correlation Length λ, and the Shock Parameters Events 260:2011 67:2012 305:2012 46:2014 58:2014

f(km2 s−2)

g(km2 s−2)

ED (km2 s−2)

λ (106 km)

r

Dsh

U (km s−1)

Vsh (km s−1)

ρ (cm−3)

θBn

5.03 4.76 2.19 9.56 2.31

16.81 14.59 5.65 38.57 4.07

−0.55 0.44 0.78 0.13 0.19

2.4 1.2 0.6 0.6 0.6

2.24 1.82 2.36 2.25 2.25

5rg 5rg 5rg 5rg 5rg

149.09 147.06 94.58 124.17 163.57

509 485 391 469 469

2.96 6.34 3.00 5.83 13.40

87° 89° 86° 88° 81°

Note. Initial conditions are shown for the energy in forward and backward propagating modes g and f, the residual energy ED, the correlation length λ, and other shock parameters: r is the shock compression ratio, Dsh is the thickness of a shock, U is the upstream solar wind velocity in the shock frame, rg is the magnetic gyroradius, Vsh is the shock speed, ρ is the upstream solar wind density, and θBn is the obliquity angle.

3.1. Comparison to Quasi-parallel Shocks

Figure 1 shows that the black curve is approximately constant (or decaying slowly) upstream of the shock in all events, then increases across the shock, and then very slowly decreases downstream with increasing X/Xsh or T/Tsh. A similar trend is also shown by the observed values (“+” symbols) and the red lines. The theoretical and the least-squares fitted áb 2ñ1 2 all agree rather well, with the exception of day 16 of 2005, downstream (bottom left). In this case, the observed downstream value is nearly three times larger than the theoretical result. The theoretical and observational results can be brought into closer accord by either reducing the threshold of the assumed criterion ádx 2ñ X 2  1 or increasing the shock compression ratio. Our choice of a shock compression ratio r = 1.48 for the model solutions was because this value was given by the shock database http://ipshocks.fi/ database. These values can be inaccurate owing to the general uncertainty in solving the Rankine–Hugoniot conditions (Szabo 1994; Hu et al. 2013). Accordingly, we show an additional dashed curve on assuming a stronger compression ratio r = 2.0, which shows better agreement between observation and theory. In our theoretical model, the turbulent energy decays freely upstream and downstream of the shock. The decay in energy is due to the nonlinear interaction of forward and backward propagating modes. In this work, the only source of turbulence is due to the shock wave, and thus it applies only at the shock position, and the turbulence is zero upstream and downstream of the shock. In this paper, we have neglected the possible contribution of upstream instabilities, particularly at quasi-parallel shocks, to the generation of turbulence (e.g., Rice et al. 2003; Zank et al. 2006; Verkhoglyadova et al. 2015). In the absence of upstream and downstream sources, the decay in the theoretical áb 2ñ1 2 (black curves) in these regions is very small (or approximately constant) in contrast to the larger decay in the observed values (red curves). However, the theoretical results depend on the assumed initial values. The initial values for f, g, and ED are obtained by averaging the upstream values, but for the correlation length we choose characteristic interplanetary values. Ideally, we would calculate the correlation length observationally and use it as an initial value, but this is beyond the scope of the manuscript. Obviously, the decay rate in energy can be increased by reducing the correlation length. For the present, the theoretical result captures the observed basic behavior of áb 2ñ1 2 upstream, across, and downstream of the shock quite satisfactorily. A comparison between the theoretical and observed fluctuating kinetic energy, the energy in forward and backward propagating modes, the total turbulent energy, the normalized residual energy, and the cross helicity is shown in Appendix A.

Here we compare the theoretical and observed fluctuating magnetic energy áb 2ñ (in the square-root form) upstream and downstream of the shock. We solve the four coupled Equations (1)–(4) for different initial conditions and use Equation (11) to calculate áb 2ñ1 2 , which we compare with the observed áb 2ñ1 2 of eight quasi-parallel shocks of day 210 of 2000, 62 of 2001, 215 of 2001, 213 of 2002, 214 of 2004, 266 of 2004, 16 of 2005, and 168 of 2012. Table 2 shows the initial conditions for the energy in backward and forward propagating modes, the residual energy, and the correlation length. It also shows the shock parameters, i.e., the compression ratio r, the thickness of a shock Dsh, the solar wind velocity U, the solar wind density ρ, and the Alfvén Mach number MA, which we use in the equations of the turbulence transport model. Here the velocity U is calculated in the shock frame. It is obtained by subtracting the shock speed from the normal solar wind velocity, i.e., U = Vn - Vsh , where Vn is the projection of the solar wind velocity along the normal direction of the shock, and Vsh is the shock speed. The normal solar wind velocity Vn is given by V ·n, where V =(Vx, Vy, Vz) and n=(nx, ny, nz). Figure 1 compares theoretical and observed values of áb 2ñ1 2 as a function of X/Xsh or T/Tsh from upstream to downstream of the shock for different events as identified in the title of each plot. The parameters X and T are spatial and temporal positions. We first find the temporal positions Tsh, Tup, and Tdown using data sets from the WIND spacecraft shown in Table 1. We then use X=U×T to find a spatial position for use in the theoretical steady-state model equations. The parameter U is the velocity of the solar wind in the shock frame. In Figure 1, the black curves describe the theoretical áb 2ñ1 2 , the solid straight red lines the least-squares fittings to the observed áb 2ñ1 2 upstream and downstream of the shock, and the scatter “+” symbols the observed áb 2ñ1 2 . The top three plots of Figure 1 (left to right) show the fluctuating magnetic energy upstream and downstream of the shocks for day 210 of 2000, 62 of 2001, and 215 of 2001; the middle three plots from left to right show day 213 of 2002, 214 of 2004, and 266 of 2004, and the bottom two plots from left to right show day 16 of 2005 and 168 of 2012. Note that the fluctuating magnetic energy is calculated from 2 hr upstream to 2 hr downstream of the shock for the shocks of day 210 of 2000, 62 of 2001, and 168 of 2012. In the case of the shocks of day 215 of 2001, 213 of 2002, and 266 of 2004, the fluctuating magnetic energy is calculated from 3 hr upstream to 3 hr downstream, and for the shocks of day 214 of 2004 and 16 of 2005, it is calculated from 1 hr upstream to 1 hr downstream of the shock. 6

The Astrophysical Journal, 833:218 (16pp), 2016 December 20

Adhikari et al.

Figure 2. Comparison between the theoretical and observed square root of the fluctuating magnetic energy áb 2ñ1 2 upstream and downstream of the quasiperpendicular shocks shown in the title of each figure. The black solid curves describe the theoretical áb 2ñ1 2 . The solid straight red lines describe the fitting to the observed áb 2ñ1 2 upstream and downstream of the shock. The scatter “+” symbols denote the observed áb 2ñ1 2 .

3.2. Comparison to Quasi-perpendicular Shocks

and backward propagating modes, the total turbulent energy, the normalized residual energy, and the cross helicity is shown in Appendix B.

In this section we solve the perpendicular shock model Equations (5)–(8) for the initial conditions of f, g, ED, and λ shown in Table 3, and compare the theoretical and observed áb 2ñ1 2 for different quasi-perpendicular shocks of day 260 of 2011, 67 of 2012, 305 of 2012, 46 of 2014, and 58 of 2014 (Figure 2). Table 3 also shows other shock parameters: the compression ratio r, the thickness of a shock Dsh, the solar wind velocity U, and the solar wind density ρ. We use these shock parameter values in the turbulence transport equations. The description of Figure 2 is similar to Figure 1. The top three plots of Figure 2 (left to right) describe áb 2ñ1 2 upstream and downstream of the shocks of day 260 of 2011, 67 of 2012, and 305 of 2012, and the bottom two plots of Figure 2 (left to right) show shocks on days 46 of 2014 and 58 of 2014. Here, the fluctuating magnetic energy is calculated from 2 hr upstream to 2 hr downstream of the shocks of day 260 of 2011 and 67 of 2012, from 3 hr upstream to 3 hr downstream of the shocks of day 305 of 2012, and from 1 hr upstream to 1 hr downstream of the shocks of day 46 of 2014 and 58 of 2014. The black curves of all events indicate that the magnetic energy is approximately constant (or decaying slowly) upstream and downstream of the shock, with an increase across the shock. The comparison between the black curves, the straight red lines, and the scatter diagram shows that the theoretical and fitted magnetic energies are in reasonable agreement with each other. We see that the shock of day 260 of 2011 (Figure 2, top left) has an observed value of áb 2ñ1 2 larger than the theoretical result, most likely for the same reasons as discussed for Figure 1. Better agreement can be obtained by assuming a stronger compression ratio r = 3.8 (dashed curve), although this differs from the value quoted in the shock database http:// ipshocks.fi/database. The comparison between the theoretical and observed fluctuating kinetic energy, the energy in forward

4. DISCUSSION AND CONCLUSION In this manuscript we have analyzed the interaction of turbulence with parallel and perpendicular shocks theoretically and observationally. We derived the six or four coupled equations for the 1D steady-state parallel and perpendicular shock model from the model of Zank et al. (2012), and solved the coupled turbulence transport equations using a Runge– Kutta fourth-order method. We compared the numerical solutions with the observations using data sets from the WIND spacecraft with 3 s resolution. An important difference between the parallel and perpendicular shock model equations is that the parallel shock model equations contain the Alfvén velocity, and the perpendicular shock model equations do not. We calculated various turbulent quantities: the fluctuating magnetic energy, the fluctuating kinetic energy, the energy in forward propagating modes, the energy in backward propagating modes, the total turbulent energy, the correlation length, the normalized cross helicity, and the normalized residual energy from upstream to downstream of the shock. The observed turbulent quantities were calculated separately in the upstream and downstream regions, i.e., Tup„T