Identification of slow magnetosonic wave trains and their evolution in 3-D compressible turbulence simulation

In solar wind, dissipation of slow-mode magnetosonic waves may play a significant role in heating the solar wind, and these modes contribute essentially to the solar wind compressible turbulence. Most previous identifications of slow waves utilized the characteristic negative correlation between δ |B| and δρ. However, that criterion does not well identify quasi-parallel slow waves, for which δ |B| is negligible compared to δρ. Here we present a new method of identification, which will be used in 3-D compressible simulation. It is based on two criteria: (1) that VpB0 (phase speed projected along B0) is around ±cs, and that (2) there exists a clear correlation of δv‖ and δρ. Our research demonstrates that if vA > cs, slow waves possess correlation between δv‖ and δρ, with δρ/δv‖ ≈±ρ0/cs. This method helps us to distinguish slow-mode waves from fast and Alfvén waves, both of which do not have this polarity relation. The criteria are insensitive to the propagation angle θkB , defined as the angle between wave vector k and B0; they can be applied with a wide range of β if only vA > cs. In our numerical simulation, we have identified four cases of slow wave trains with this method. The slow wave trains seem to deform, probably caused by interaction with other waves; as a result, fast or Alfvén waves may be produced during the interaction and seem to propagate bidirectionally away. Our identification and analysis of the wave trains provide useful methods for investigations of compressible turbulence in the solar wind or in similar environments, and will thus deepen understandings of slow waves in the turbulence.


Introduction
The turbulent solar wind contains various magnetohydrodynamic (MHD) and kinetic waves (Tu and Marsch, 1995;Goldstein et al., 1995;Bruno and Carbone, 2005;Marsch, 2006;He et al., 2011;Podesta and Gary, 2011;He et al., 2012;Salem et al., 2012).Alfvén waves hardly dissipate, and therefore this wave mode is observed ubiquitously (Belcher and Davis, 1971;Wang et al., 2012).Fast magnetosonic waves are compressible and are thus easily damped for a plasma β of order unity.Moreover, their energy propagates in divergent directions from the source, and therefore their wave energy dilutes by spreading all around, making fast waves more difficult to observe in the solar wind (Tu and Marsch, 1995).Though slow waves are compressible and thought to be prone to damping, in the solar wind they tend to possess k small enough to avoid strong damping (Chen et al., 2012).Moreover, their wave energy is mainly confined within magnetic flux tubes and transfers approximately along the magnetic field.Sequentially it is more likely to observe slow waves.Kellogg and Horbury (2005) researched density oscillations and found a compressible mode of waves that may be interpreted as slow waves.Howes et al. (2012) found negative correlations of δn and δB in data from the Wind spacecraft, and suggested that the principal compressible component of inertial range turbulence is the kinetic slow mode.Yao et al. (2013) (Y13) reported MHD slow waves observed with the Wind spacecraft.Their study described an event with slow waves embedded in Alfvén fluctuations, where the plasma β ≈ 1.These cases were identified using the criteria given by Gary and Winske (1992), i.e., compressibility C p L. Zhang et al.: Slow MHD wave trains in 3-D compressible simulation and dimensionless cross-helicity σ c .However, these criteria require a wave train with a pure mode, and may fail when waves of different modes mix up.Though cases reported by Y13 did agree well with this requirement, the usability of these criteria is limited.(Various linear spectral models (e.g., Klein et al., 2012) can be used for the problem.) Recently simulations of these wave modes in MHD turbulence have been carried out.Simulations of compressible turbulence are especially important in that they promote our understandings of wave features and wave-wave interactions in the solar wind turbulence.To our knowledge, most of them focus on the statistical properties of compressible modes regarded as components of MHD turbulence.Most relevant research and reviews -e.g., Cho et al. (2002), Cho andLazarian (2003, 2005), Vestuto et al. (2003), Brunt and Mac Low (2004), Elmegreen and Scalo (2004), Zhou et al. (2004), Hnat et al. (2005), Kowal et al. (2007), Shivamoggi (2008), Kowal and Lazarian (2010), Tofflemire et al. (2011), and Brandenburg and Lazarian (2013) -have concentrated on spectral properties and/or anisotropy of different MHD wave modes.Among these, Hnat et al. (2005), Shivamoggi (2008), and Kowal and Lazarian (2010) focused on compressible modes and investigated their characteristics.There was also research investigating the correlations of quantities in turbulences as well as dominating modes in of MHD turbulences.Passot and Vazquez-Semadeni (2003) have found that which wave mode dominates the contribution to density fluctuations in turbulences is decided by θ kB and Alfvénic Mach numbers M A .When M A is low, the slow mode is more dominant, especially when θ kB is close to 90 • .Wisniewski et al. (2013) researched the ratio of energy of different modes as well as anisotropy.However, the local behaviors of MHD waves are seldom discussed.
Among these, it is worth highlighting the works of Cho et al. (2002) and Cho andLazarian (2003, 2005).Those studies mainly covered the anisotropy of MHD waves turbulence spectra, but also extended their discussion to include the generation of fast/slow waves in ambient Alfvénic fluctuations.It was shown that fast/slow waves can be generated in pure Alfvén wave background (explicitly reported by Cho and Lazarian, 2005), but these articles did not include direct identifications of the generated MHD waves.Instead, only global, statistical parameters were extracted using a "projection" method.In order to "decompose" wave modes, complex amplitudes δv(k) were projected on eigenvectors determined by the background magnetic field B 0 and wave vector k.The Alfvén mode has velocity oscillations in the B 0 ×k direction, while the fast mode oscillates quasi-perpendicularly and slow mode the in quasi-parallel direction, both in the plane in which both B 0 and k lie.
In this article we intend to identify slow wave trains from our simulations of MHD turbulences and describing their evolution and possible interaction with other types of MHD waves.Our goal differs from previous studies in that we provide direct and descriptive examples of slow waves, focusing on local behaviors and deformation of the associated wave trains.Our criteria are insensitive to θ kB and more versatile than the correlation between δρ and δB .
In Sect. 2 we introduce our numerical model.In Sect. 3 we give a brief prescription of the identification of MHD waves and verify our new methods.In Sect. 4 we present our slow wave cases and compare them with the case of Y13.Finally, in Sect. 5 we summarize the results and present some discussions.

Model description
In this section, we briefly introduce the codes that we used in our compressible turbulence simulation.We have conducted an MHD simulation with codes employing a splittingbased finite-volume scheme (Feng et al., 2011;Zhang et al., 2011;Yang et al., 2013), where the magnetic field is computed with the constrained transport (CT) algorithm (Tóth, 2000) and the fluid part with a Gudonov-type central scheme (Ziegler, 2004;Fuchs et al., 2009).The codes are based on the PARAMESH package (MacNeice et al., 2000) and provide compressible solutions.(See Fig. 1.)The model is defined in a three-dimensional rectangular coordinate system, with 256 × 256 × 256 grid points along the x, y, and z directions, respectively.The corresponding computational region ranges from 0 to 2π in each direction.The model describes an ideal MHD system whose adiabatic index γ = 5/3, without explicit kinetic or magnetic viscosity terms.
All data and analyses presented in this article make use of a dimensionless unit system to simplify and clarify the illustration of the method.However, in order to interpret physical significances, a set of conversion factors should be set.In the solar wind at 1 AU, the typical Alfvén speed is about 100 km s −1 , and we denote it as 2 dimensionless velocity units.Hence the speed unit is v 0 = 50 km s −1 .Typical solar wind contains about 5 protons per cubic centime-ter, and so we assign the mass density unit to be ρ 0 = 8 × 10 −21 kg m −3 .Their combination gives a unit of magnetic field B 0 = 5.013 nT.Take a period of t 0 = 20 s and the corresponding length is L 0 = 1 Mm.
A simulation of decaying turbulences is performed.As initial conditions, we adopt random "broadband initial conditions" of Matthaeus et al. (1996).These initial conditions consist of a randomly superposed wave packet of velocity fluctuations u(k) with |u(k)| 2 ∝ 1/(1 + (k/k knee ) q ) in the k range of 1 < k < 8, where k knee is chosen to be 3.The parameter q decides the slope of the initial turbulence spectrum.For oscillations we set v 2 x = 0.5 and v 2 y = 0.5, as well as v x = v y = 0.There are no oscillations of v z and B. We uniformly set the mass density ρ 0 = 1, temperature T 0 = 0.6, and magnetic field strength B z0 = 2, B x0 = B y0 = 0 in initial conditions, so that the initial Alfvén velocity (v A = B 0 / √ ρ 0 , in dimensionless unit system) is 2, and the speed of sound (c s = √ γ p 0 /ρ 0 ) is 1.Hence, the plasma β = 0.3 everywhere.These initial conditions intend to simulate the solar wind with in situ compressible turbulence and represent the local solar wind observations, yet they do not reflect radial expansion nor large-scale shear or stream interactions (Matthaeus et al., 1996).Hence, the simulation is limited to describing the local behaviors and cannot describe largescale phenomena, such as the radial evolution of solar wind turbulence.
We employ periodic boundary conditions on all six boundaries, considering the two following issues.Firstly, there is no rigid wall in the solar wind environment that our model simulates.Secondly, periodic boundary conditions correspond to wave modes propagating in an infinite space, so that the wave vector k, which stems from Fourier analysis, makes sense.

Wave mode diagnostics
In this section we present our methods to identify slow wave trains in the simulation data.The identification is mainly accomplished with help of time-space slices along B 0 .We show that the slow wave trains exhibit a strong correlation between δρ and δv , and such correlated structures propagate at a characteristic velocity.The theoretically predicted ratio δρ/δv changes little with varying θ kB , and thus the determination of the wave vectors can be bypassed.Though we fix on a single β in the test case, these criteria are quite credible whenever v A is quite above c s .
In Sect.3.1, we describe the methods to identify slow waves, and we show the necessary features of other modes as well.The identification mainly involves the dispersion relation and polarity relations, where the angle θ kB functions as a kernel parameter, yet the results are insensitive to it.In Sect.3.2, we justify our method and check its robustness and applicability.We use a randomly generated wave packet to verify the correlation of δρ(t) and δv (t).The correla-tion only depends on the sign of cos θ kB , regardless of the absolute value of cos θ kB .The justification ensures that the method is reliable and available for analysis of time series.

Identification and features of MHD waves
Firstly, it is worth revisiting theoretical solutions of MHD waves, i.e., dispersion and polarity relations.These results are built on two hypotheses: (i) linearized oscillations, i.e., fluctuations, are small enough that higher order terms can be neglected, and (ii) there is only one monochromatic plane wave of one mode.With these hypotheses one can decompose a physical quantity ψ into a constant "background" part and an oscillating part: where the background ψ 0 and the complex amplitude δ ψ are constant.When only the oscillating part is discussed we will simply use δ ψ instead.
Single-fluid MHD equations can only support three modes of waves: fast, slow, and Alfvén mode.If the speed of sound c s , Alfvén speed v A , and θ kB are given, the phase velocities of fast and slow modes are determined according to where the plus sign is for fast waves and minus for slow waves.The dispersion relation simply reads ω = V p k.
For fast and slow mode, polarity relations in the form given by, for example, Olbers and Richter (1973) and Marsch (1986) are presented here (with some symbols altered and unit system changed so that the permeability of free space µ 0 is 1): where is the relative density fluctuation amplitude.Here it is plausible to omit the expression of δ pth from the solutions, since it is merely a passive quantity directly related to δ ρ (e.g., Zank and Matthaeus, 1993).From the form, it is worth noting that both δv and δB lie in the plane defined by B 0 and k (e.g., take B 0 = (0, 0, B 0 ) and k = (k x , 0, k z ), and the fluctuations will lie in the x-z plane).In that plane the parallel and perpendicular directions can be defined relative to B 0 .From Eq. (3b) the parallel and perpendicular components of δv are For an Alfvén wave, the dispersion relation reads where the plus sign corresponds to the case k > 0, and vice versa for the minus sign.The Alfvén wave has δ ρ = δ p = 0. Its velocity and magnetic field oscillate in k × B 0 direction, with with the minus sign for k > 0 and vice versa for k < 0.
Let the phase velocity of the wave be V p and the propagation velocity of wave phase along B 0 be V pz .Then, since olarity of velocity and magnetic field oscillations Since V pz depends only on θ kB instead of ω, Eq. ( 7) also applies for wave packets with all components sharing the same θ kB .For fast, slow, and Alfvén modes we plot V pz versus θ kB in Fig. 2a.Along B 0 , Alfvén mode travels at a speed of 2, independent of θ kB .For fast mode, V pz is greater than or equal to 2, more approximated to 2 at smaller θ kB .For slow mode, V pz does not vary much from 1.As a result, slowmode structures stand out due to the distinguishing velocity along B 0 , and the analysis of phase-propagating speed V pz still works for general wave packets.
Next we analyze characteristic velocity oscillations of fast and slow waves.Since the wave energy density can be calculated as ρ (δv) 2 , the amplitudes of δv can represent wave energy.In our case v A = 2 and c s = 1, and from Eqs. (4a) and (4b) the inequality δv ⊥ /δv < 0.12 is valid, i.e., the oscillations of slow waves keep within 7 • (i.e., arctan 0.12) relative to B 0 (see Fig. 3).Oscillations in parallel direction thus contribute at least 98 % to the wave energy (because this percentage ρ 0 δv 2 /(ρ 0 δv 2 ) can be written as 1/(1 + δv 2 ⊥ /δv 2 ), which is greater than 1/(1 + 0.12 2 ) ≈ 0.986).For fast mode, one can infer in the same fashion that the perpendicular direction contains almost all of the wave energy.Therefore, it is suitable to assign the parallel direction as the characteristic direction of velocity oscillations of slow mode, and the perpendicular one for the fast mode.
From Eqs. (3a), (4a) and (4b), one can derive that Considering the characteristic velocity oscillation above, we take the former equation for the analysis of slow wave and the latter of fast wave.Figure 2b clearly presents δ ρ/δ v⊥ of fast mode and δ ρ/δ v of slow mode.For slow mode, the polarity of δ ρ/δ v depends on the signs of k , and the ratio is not sensitive to θ kB in each branch.In our case with v A > c s , maximal compression per unit "characteristic oscillation" occurs when θ kB = 0 or 180 • , where The ratio of the minimum to the maximum of δ ρ/δ v is v A / v 2 A + c 2 s (in our case ≈ 0.894).If v A is much larger than c s , it is safe to assume For fast mode, when θ kB is near 0 or 180 • , there is no obvious compression, but when θ kB is close to 90 • , the ratio (Eq.8b) approaches ρ 0 / v 2 A + c 2 s .As above, the diagnosis of slow wave utilizes two criteria: wave phase speed along B 0 and δρ/δv .The criteria work well in cases with v A > c s , yet better when the ratio v A /c s is greater (i.e., with smaller β), where all the approximations applied here will be slightly more precise.In order to employ them in data analysis, we must derive a relation between δρ(t) and δv (t).In other words, we should further justify that also holds well.The plus sign is for wave packets consisting of slow wave with k > 0, and the minus sign for k < 0. We also compute The quantity δb = δB/ √ ρ 0 denotes fluctuations of the magnetic fields, normalized to have velocity units.Gary and Winske (1992) pointed out that σ c and C distinguish wave modes sharply: in our case, slow waves should have C p 1 and negative σ c , Alfvén waves and "small-angled" fast waves should have C p ≈ 0 and σ c near to −1, and "large-angled" fast waves should have C p < 1 and σ c between −1 and 0 (see Table 1, from Table 1a of Gary and Winske, 1992).These conclusions are for cases where k • B 0 > 0. For cases where k • B 0 < 0, σ c will change signs, and C p will remain unchanged.The computation of the two parameters is identical to that of Y13 (cf.Y13's Eqs. 1 and 2).
In summary, MHD waves can be categorized into three types: (i) slow waves, (ii) Alfvén waves and quasi-parallel propagating fast waves (AW-like), and (iii) oblique and quasi-perpendicular propagating fast waves (fast-like).With v A = 2, c s = 1 and ρ 0 = 1 (typical for β < 1), their behaviors can be listed and compared in Table 1.For completeness, the non-propagating entropy mode is also supplied.This mode has inhomogeneity of density and temperature, but keeps pressure, velocity and magnetic field strength all uniform.Since quasi-parallel phase-propagating slow waves show little δB, it may be problematic to analyze correlations between δρ and δB in such situations.However, since in the solar wind compressible fluctuations have larger k ⊥ than k (Chen et al., 2012), such quasi-parallel cases may be neglected.

Justification of methods
The criteria described in Sect.3.1 are based on the hypothesis that only a monochromatic wave of a given mode is involved.This is hardly the case in reality, where turbulent fluctuations have a wide range of frequencies and/or wave vectors, and different kinds of modes may be superposed.
To check the performances of the criteria in complicated cases, we have constructed a slow-mode wave packet to investigate the correlation of δρ and δv .The packet consists of N wave modes, i.e., for δρ and δv δv (x, t) For each mode i, we generate complex amplitude v i with random moduli and initial phase.Each wave mode has a random k, whose magnitude and direction are randomly decided with only one restriction θ kB < 89 • .This is intent to guarantee that k keeps the same sign in the wave packet.The corresponding ω is determined by k from dispersion relation.
Hence we can calculate ρi from v i with Eq. (8a), which describes the polarity relation between ρ and v .To test Eq.( 10), we computed time profiles at the point x = (0, 0, 0) of δv and δρ.The parameters are taken as in our initial condition: v A = 2, c s = 1, and ρ 0 = 1, so we expect the ratio to be about 1.For each of N = 80 modes, amplitudes of δv i are randomly selected in the range from 0 to 1, initial phase from 0 to 2π , azimuthal angle of k ⊥ (relative to B 0 ) from 0 to 2π , and |k| from 0 to 2 in order to implement the random superposition of wave modes and extend the range of frequencies.The plots of δv and δρ almost overlap in Fig. 4, which implies the ratio ≈ 1.In this way the method is justified.
If other modes coexist, the correlation between δρ and δv may be (or not) destroyed, depending on the type of the coexisting wave.Fast and Alfvén waves do not have obvious δv .As a result, observed δv is contributed by slow waves.The correlation is possibly affected when the coexisting mode is either "large-angled" fast wave or slow wave propagating in the opposite direction.At the same time, however, whenever only Alfvén waves and/or parallel propagating fast waves coexist, the correlation will remain intact.To analyze mixed www.ann-geophys.net/33/13/2015/Ann.Geophys., 33, 13-23, 2015 (i) Slow (ii) AW-like (iii) Fast-like (iv) Entropy wave forms, linear spectral models (e.g., Klein et al., 2012) may be used along the methods presented above to separate different modes with help from eigenmode solutions.

Results
In this section we analyze the data from compressible simulations with the methods above and diagnose possible slow wave cases, four of which are presented with one in detail and the others in brief.The data are extracted in time-space slices with coordinates x and y fixed, so that the slices lie in the z direction.The primary characteristics of the cases are listed in the Table 2. Propagating speeds are computed by fitting the track of phase in the (z, t) diagram.Averaged values and oscillations are evaluated with "wave points" (defined in the next paragraph).
For case 1 (see the white lines in Fig. 1), profiles of characteristic quantities are plotted in Fig. 5. Red squares in panel a track the slow wave train.The points tracked are maxima of density ρ.Panel b plots a z-t profile of v z with the same tracked points marked.Since slow waves usually appear shortly in turbulences, their widths are confined so that we can analyze the region near to the marked points.For each time t with the wave peak marked at z = z p (t), we take the points where z p − 0.2 ≤ z < z p + 0.2 and define them as "wave points".An estimation of background quantities can be obtained from the average of all wave points through the time span, giving B ≈ (−0.1, −0.5, 2.0) and v ≈ (−0.4,−0.3, 0.04) in (x, y, z) coordinates.The angle between B and z axis is only 14 • .In our qualitative analy- sis it is reasonable to approximate the z direction as the parallel direction.The background velocity vz is not large enough to force the subtraction of this velocity from the wave speed along the z slice.Meanwhile, ρ and B z change little from the initial condition, and thus the initial values are still valid as background values.The root-mean-square value of δv z is about 0.2, and δρ ≈ 0.2.This confirms to the polarity relations, and the root-mean-square values show that the wave train is linear in amplitude.Panels a and b exhibit characteristic properties of a slow wave train: (1) an obvious density change (∼ 20 %), (2) typical propagating velocity (1.04 ± 0.049) around c s = 1, and (3) good correlation between fluctuations of ρ and v (correlation coefficient = 0.91).
Equation ( 10) serves as another important criterion for slow-mode waves.To check against it, we have analyzed the correlation between δv z and δρ (Fig. 5d).As a reference, a blue line is plotted corresponding to δρ/δv = 1.The points gather around the blue line, highlighting a typical strong correlation for slow waves.From the distribution the averages and oscillations are also calculated, so the averaged wave energy density can be calculated as Ēw = ρ (δv z ) 2 , and is listed in Table 2.
To explain Y13's observation, we further compute some features employed in Y13's analysis.We take points where 0 ≤ z < 4.2 on the slice at t = 0.71.The running smooth window used to compute C p and σ c has a width of 0.5, chosen as about a typical scale of the wave (see Fig. 5a).The results are presented in Fig. 6, where panels a and b are plotted without smoothing.Panel b shows the correlation of v x and B x as the result of searching for possible Alfvénic structures.Since the x direction can be interpreted as a perpendicular direction, the Alfvén mode or small-angled fast mode both have δv ⊥ ≈ ∓δB ⊥ / √ ρ (in the case of Alfvén waves, the equal sign holds strictly).In the segment marked with blue vertical lines in Fig. 6b, a negative correlation between B x and v x can be seen, and the amplitudes of their change are almost equal.In the same plot, the segment z ≤ 1 shows almost overlapping profiles of B x and v x .The two evidences may suggest an Alfvénic background.In this way, an Alfvén or small-angled fast-mode background is expected.For comparison with Gary and Winske (1992), panels c and d give parameters of the wave train.The wave points show C 1  11b).(d) Dimensionless cross-helicity σ c computed with Eq. (11a).Horizontal axes represent the z coordinate along the slice, which is taken at x = 4.10, y = 3.14, and 0 ≤ z < 4.2.In (a) and (b), both vertical axes are adjusted so that they share the same scale.Red lines mark "wave points", and the vertical blue lines mark a region of negative correlation of v x and B x .and σ c ≈ −0.5, which agrees well with the criteria.Therefore the slow-mode wave train is again identified.
Thus we identified a slow wave train embedded in Alfvénic-like structures, which explains Y13's observation.However, the wave train does not propagate forever.This is also described in Fig. 5a and b.At t = 0.87, the density peak in panel a becomes hardly recognizable and starts to blur.At the same moment, the v z structure in panel b does not propagate any further.The slow wave train seems to be deformed.In Fig. 5a, a fan-like structure appears, hinting at a possible generation of at least two waves and propagation in oppo- site directions.To check them, we plot dv x /dz in panel c and marked in both wave trains in blue and cyan squares at local minima of the profile, respectively.These waves have the speed of fast waves, and moreover they cause density change.They are possibly "large-angled" fast waves.
To understand what happens to the wave train when it deforms, we take another slice along the x axis at the same time interval and with z = 2.3 and y = 3.14, so that it passes the site where the slow wave train begins to deform.We plot the corresponding profiles in Fig. 7, with maxima of |∂v x /∂x| tracked, and we have them marked in panels a to c.The series of the maxima can be regarded as wave trains.The blue wave train exhibits some density fluctuations (see panel c), which are much smaller than in slow mode.To understand its wave mode, we perform two fits (see panels d and e).Its behaviors resemble fast waves with θ kB ∼ 40 • (cf.Figs.2a and 3).For the cyan wave train, the oscillations of B x and v x are similar (see panel b).A scatterplot of B x and v x near the cyan train is provided in panel f, displaying a positive correlation.The two trains are not slow waves, the blue one behaves "fastlike", and the cyan one behaves somehow "Alfvén-like".It is notable that they meet at z ≈ 4.1 and t = 0.87, the very point where the slow wave train deforms.
Hence, the deforming of the slow wave train in case 1 can be described in such a scenario: a slow wave train interacts with at least two non-slow counterparts (one of them probably being "large-angled" fast waves), and deforms into at least two non-slow ones.
For other cases, only overviews of the slow wave trains are presented.In case 2 (Fig. 8), two slow wave trains propagate together.However, no clear wave form between the peaks is revealed.In contrast to case 1, the two wave trains remain quite a long time, especially for the wave train at larger z (case 2R).Moreover, waves in case 2 propagate "down- wards" to smaller z, while the counterpart in case 1 propagates "upwards" to larger z.Accordingly, δρ and δv show negative correlation in case 2. Case 3 is a rather short case, whose profiles are shown in Fig. 9.Because we take fewer points along the wave train, the correlation coefficient r appears larger than that of cases 1 and 2. Before the slow wave train is formed, a fast wave train exists (see the cyan markers in panel c).At t ∼ 0.47, it seems to bifurcate, with most of its energy being transferred into the slow-mode train.A minor part continues propagating as fast or Alfvén mode, which is barely detectable in panel c after t = 0.47.This scenario resembles the one discussed by Kudoh and Shibata (1999).After t = 0.67, the slow wave train appears to excite another sharp wave train (panel c).However, the slow wave train does not fully deform.Case 4 is also a rather short case with pro- files shown in Fig. 10.The excited wave is so weak that it can barely be detected (see panel c).However, the energy of the deformed slow wave train may be contributed to the complicated velocity structure shown at the top of Fig. 10b.In all cases, the slow wave trains seem to be destroyed as they interact with other modes at the same point.After the destruction, new wave trains might be formed and shape "fan-like" structures shown in time-space slices of density.Such structures are best shown in case 1, but still detectable in cases 2 to 4 (cf.Figs.8a, 9a, and 10a).

Summary and discussions
This article establishes criteria to identify slow wave trains in MHD simulation data, and presents four cases of such slow wave trains.Several of their physical properties are extracted.The main results are summarized here.
1.The identification of slow wave trains is achieved by studying data along a slice approximately aligned to the background magnetic field.Typical slow wave behaviors include V pz ≈ ±c s and quite strong correlation of δρ and δv , with their ratio ∼ ±ρ 0 /c s .They are consistent with the derivation from dispersion and polarity relations of linear MHD oscillations.These properties are dominated only by the sign of cos θ kB and insensitive to its absolute value, allowing us to bypass the calculation of wave vector k.The criteria are theoretically reliable if v A is well above c s , and they have passed a test with randomly generated wave packets.
2. Four cases of linear slow wave train are analyzed.They comply well with the criteria.For case 1, the parameters σ c and C p are computed, and they support the identification.3. Near the slow wave train of case 1, an Alfvénic structure exists.This explains Y13's observation as a local slow wave train generated in an Alfvénic environment, except that the existence of a pressure-balanced structure cannot be clearly confirmed.
4. Such slow wave trains are inclined to dissolve.They deform suddenly, probably when they interact with other waves, and their density structures will expand so that they dissolve like a fan in time-space slice plots.The boundary of the fan propagates at a typical speed of fast wave, and the plasma there shares properties of the fast wave or Alfvén wave.This is clearly shown by case 1, but in other cases the fan-like density structures are also detectable (though barely).
We have computed the wave energy density of the four wave trains, and found that the wave energy densities are about 0.03 to 0.04 except in case 2. All points in the computational region at t = 0.71 give ρ 0 (δv) 2 ≈ 0.39.Since the zones of compressible modes occupy a small portion of the computational region, and energy densities of slow wave trains are much smaller than the background fluctuations, it is reasonable to roughly estimate the energy density of Alfvén mode as 0.4.Since all four cases appear at time spans near t = 0.71, this background energy density applies to all cases in a rough estimation.Therefore, the typical wave energy density of slow wave trains amounts about 0.1 in all fluctuations.(For case 2, this proportion might be 0.2.) There are certain limitations in this work.Firstly, this research is limited within MHD descriptions.Kinetic magnetosonic waves behave differently from MHD eigenmodes (Klein et al., 2012).Therefore, kinetic effects (not included in our MHD investigation) may limit applications of the reported MHD methods to solar wind observations.Slow mode waves generally suffer more damping than Alfvén waves do.Damping rates of slow modes vary with θ kB , with the quasiperpendicular ones less damped (Barnes, 1966;Schekochihin et al., 2009).In the solar wind, the compressive fluctuations are found to be anisotropic with k ⊥ larger than k , based on the structure function analysis (Chen et al., 2012).This explains why slow-mode fluctuations exist in the solar wind even though they suffer some damping.Nevertheless, our MHD model is limited in that it does not cover slow wave damping.Secondly, this work mainly focuses on phenomenal features of slow waves.However, their genesis has not yet been discussed, because our initial condition is a rather complex one and we need a method to trace the development of wave trains.With such a method, it would be possible to perform further investigations, such as analysis of fan structures in density profile, mechanisms of wave interactions, and relations between such wave trains and structures (e.g., discontinuities or intermittencies), Thirdly, though in principle our numerical simulation is able to produce a time series of 3-D data, we are forced to select only the essential information for the analysis because of the formidable computational cost when high-resolution (x, y, z, t) data are produced and analyzed.Local analysis of such wave trains, which utilizes information about the three-dimensional and temporal evolution at local space-time points, may be conducted in future research.The analyses presented in this article employ slices and thus provide a workable compromise.Nevertheless, this research has to bypass some aspects such as determination of wave vectors, which may be essential for further analysis.

Figure 1 .
Figure 1.Density distribution (color) and in-plane velocities (arrows) in the plane y = 3.14 at t = 0.71.As a reference, an arrow is drawn to denote a velocity of 1 dimensionless unit.Between the white lines is the slice used in analyses of the slow wave case 1.

Figure 2 .
Figure 2. Theoretically predicted features of MHD waves.(a) Magnitudes of wave phase speed along z direction V pz versus angle θ kB .(b) Compression due to fast and slow waves versus different θ kB , with vertical axis representing δ ρ/δ v for slow mode (red), and δ ρ/δ v⊥ for fast mode (blue).Both horizontal axes represent θ kB in degrees.Data are computed with v A = 2, c s = 1, and ρ 0 = 1.

Figure 3 .
Figure 3. Polarity relations when v A = 2, c s = 1 and ρ 0 = 1.(a) to (d) show directions of velocity oscillations of fast and slow modes with different θ kB .(e) to (h) show directions of magnetic field oscillations of fast and slow modes.In each panel, the arrow length of δB is scaled proportional to δv.

Figure 4 .
Figure 4. Time profiles of δv (t) and δρ(t) of a composed wave packet containing slow waves with random amplitudes, random wave vectors (k > 0), and random initial phase.The vertical axis is for oscillating quantities, and the horizontal axis represents time t.

Figure 5 .
Figure 5. Temporal evolution of quantities near propagating region of slow wave case 1.(a) Temporal profiles of density ρ, with slow wave train displayed.(b) Temporal profiles of z component of velocity v z , with the same (z, t) points as shown in (a).(c) Temporal profiles of dv x /dz, with two secondary wave trains displayed.(d) Distribution of δv z and δρ at points near the slow wave train displayed in (a).In (a) to (c) the horizontal axis represents coordinate z in the slice.The vertical axis marks the times of corresponding profiles.The colors plot the corresponding physical quantities.In (d) the blue line plots δρ/δv z = 1 as a reference; colors of markers show their time, with red ones earlier and green ones later.The fluctuation values are obtained by subtraction of the average of the quantities at these points.

Figure 6 .
Figure 6.Features of points near density maximum in case 1, at t = 0.71.(a) Instant values of thermal pressure p th = ρT (black) and magnetic pressure p B = B 2 /2 (green).(b) Instant values of perpendicular components of velocity v x (black) and magnetic field B x (green).(c) Compressibility C computed with Eq. (11b).(d) Dimensionless cross-helicity σ c computed with Eq. (11a).Horizontal axes represent the z coordinate along the slice, which is taken at x = 4.10, y = 3.14, and 0 ≤ z < 4.2.In (a) and (b), both vertical axes are adjusted so that they share the same scale.Red lines mark "wave points", and the vertical blue lines mark a region of negative correlation of v x and B x .

Figure 7 .
Figure 7. Temporal evolution of quantities along the x slice in case 1, taken to intersect the z slice in Fig. 5. (a) Temporal profiles of v x , with two wave trains displayed.(b) Temporal profiles of B x .(c) Temporal profiles of ρ.(d) Distribution of δρ and δv x at points near the blue (left) wave train.(e) Distribution of δB y (profile not plotted) and δv x at points near the blue wave train.(f) Distribution of δB x and δv x at points near the cyan (right) wave train.In (d) to (f), the points are selected as ones within 0.1 unit length to the tracked points.Redder markers stand for earlier time points and greener for later.

Figure 8 .
Figure 8. Features of slow wave case 2. In (a), the orange markers show case 2L and the red ones case 2R.

Figure 9 .
Figure 9. Features of slow wave case 3, in the same format as Fig. 5.

Figure 10 .
Figure 10.Features of slow wave case 4, in the same format as Fig. 5.In (a) and (b), tracked points are maxima of |∂ρ/∂z|.

Table 1 .
Features of MHD waves.

Table 2 .
Overview of the cases analyzed.