Properties of magnetohydrodynamic modes in compressively driven plasma turbulence

We study properties of magnetohydrodynamic (MHD) eigenmodes by decomposing the data of MHD simulations into linear MHD modes - namely the Alfven, slow magnetosonic, and fast magnetosonic modes. We drive turbulence with a mixture of solenoidal and compressive driving, while varying the Alfven Mach number (MA) and plasma beta. We find that the proportion of fast and slow modes in the mode mixture increases with increasing compressive forcing. This proportion of the magnetosonic modes can also become the dominant fraction in the mode mixture. The anisotropy of the modes is analyzed by means of their structure functions. The Alfven mode anisotropy is consistent with the Goldreich-Sridhar theory. We find a transition from weak to strong Alfvenic turbulence as we go from low to high MA. The slow mode properties are similar to the Alfven mode. On the other hand the isotropic nature of fast modes is verified in the cases where the fast mode is a significant fraction of the mode mixture. The fast mode behavior does not show any transition in going from low to high MA. We find indications that there is some interaction between the different modes and the properties of the dominant mode can affect the properties of the weaker modes. This work identifies the conditions under which magnetosonic modes can be a major fraction of turbulent astrophysical plasmas and their coexistence with an Alfvenic cascade will have important implications for cosmic ray scattering and transport.


I. INTRODUCTION
Plasma turbulence plays an important role in various astrophysical processes. It is important in solar wind heating and acceleration [1], it regulates the star formation processes [2], it scatters the cosmic rays [3] amongst other things. The properties of turbulence depend on the underlying modes it is made up of. The magnetohydrodynamic (MHD) system of equations of a 3D, homogenous, uniform, isothermal plasma with a uniform background magnetic field allows for three separate linear eigenmodes -the Alfvén mode [4], the slow magnetosonic mode, and the fast magnetosonic mode [5]. Alfvénic turbulence (turbulence consisting of mostly Alfvén modes nonlinearly interacting with each other) is thought to be quite important in solar turbulence as Alfvén waves have been observed in the solar wind [6]. Alfvénic turbulence has been studied for several decades and several theories have been developed to describe it. The Alfvén modes are incompressible solutions to the linearized MHD equations. The presence of a mean magnetic field in MHD turbulence can lead to an anisotropic energy cascade. In the regime of strong turbulence, a critical balance is conjectured to be reached between the linear interaction time of wave packets with their nonlinear cascade time [7]. In the limit of weak turbulence, the resonant three-wave couplings involve only the non-propagating Alfvén modes and produce a cascade in the wavevectors perpendicular to the local mean magnetic field direction only [8].
There has not been a comprehensive theory of turbulence consisting of the compressible MHD modes, namely the slow and fast magnetosonic modes. Turbulence in the * kirit.makwana@desy.de interstellar medium is identified by the measurement of density fluctuations in it, indicating the presence of compressible turbulence [9]. Numerical simulations of compressible turbulence to identify the inertial range scalings are more difficult and complex, giving varying results depending on the plasma parameters like Mach number [10][11][12]. Since astrophysical plasma turbulence is expected to be compressible, the magnetosonic modes should be considered when studying such turbulence. Whether the cascades of these different MHD modes are independent of each other and what is their nature are still open questions. The slow modes are cascaded by the shear-Alfvén modes and hence are expected to behave like the Alfvén modes [13]. Earlier studies have shown that the energy spectrum and anisotropy of slow modes is quite similar to Alfvén modes [14]. They also indicated that there is not much interaction between the Alfvén and the magnetosonic modes. On the other hand, fast modes have been seen to be quite different in their spectrum and anisotropy characteristics. Unlike Alfvén modes which preferentially cascade in the field perpendicular direction, fast modes seem to show an isotropic cascade. This has also led to several important implications for astrophysical turbulence. Based on this it has been shown that fast modes could be the most effective scatterers of cosmic ray particles [15,16].
Particle scattering and diffusion critically depends on the nature of these MHD modes. While fast modes can play an important role in scattering of cosmic rays, simulations have shown that the fast modes might only be a marginal component of compressible turbulence [17]. However, these simulations have been driven incompressively by solenoidal forcing [10,14,17,18]. So a natural question to ask is whether and how changing the nature of the forcing affects the mode composition of tur-bulence. We try to answer this by compressively driving turbulence in a variety of different plasma parameter regimes. Some earlier studies have driven turbulence by keeping a mixture of solenoidal and compressive velocity field at large scales [19] or by decomposing the driving force into solenoidal and compressive components [20]. We adopt a similar forcing but focus on the MHD mode decomposition. We find that the nature of the forcing significantly affects the composition of the turbulence in terms of the MHD modes. Another phenomenon less explored in numerical simulations is low M A (ratio of r.m.s velocity to Alfvén speed) turbulence. Theoretically, the Alfvénic turbulent cascade is expected to be weak up to some scales and then transition to a state of strong turbulence, mediated by the critical balance condition, at smaller scales. This transition from weak to strong turbulence has only been recently simulated in decaying turbulence [21]. We explore the M A dependence of this transition in our compressively driven simulations and find that the nature of Alfvénic turbulence can significantly change with M A .
The isotropic cascade properties of the fast mode have been studied at limited resolution previously [14] and the spectrum of this cascade was cautiously claimed to be k −3/2 . Higher resolution studies are needed to verify this behavior through a well-resolved inertial range. We perform higher resolution studies and find the isotropic nature of the fast modes throughout the inertial range, suggesting no scale-dependent anisotropy. Another related question is whether fast modes also show an M A dependent behavior like Alfvén modes in terms of weak or strong turbulence? We do not find any such dependence for the fast modes. Our results also suggest that these different mode cascades are not completely independent of each other, depending on which mode is dominant. This study shows that the nature of turbulence can be different depending on a variety of parameters and this has important implications for understanding the effect of this turbulence on related problems.

II. SIMULATION SETUP AND MODE DECOMPOSITION
The simulations are performed by using the PLUTO code [22]. The ideal MHD equations are solved with no explicit resistivity or viscosity, only with numerical dissipation. The isothermal equation of state is used. The inbuilt HLLD Riemann solver [23] is utilized in conjunction with a WENO3 reconstruction scheme [24]. The time stepping is done by a 3 rd order Runge-Kutta scheme. The simulation box is a cube of length L x = L y = L z = 1. A uniform magnetic field B 0 = 0.1 is put in along the zdirection of the box. In the normalized units of PLUTO this implies that the Alfvén velocity is also v A = 0.1. The sound speed (c s ) is varied to vary the plasma β defined Turbulence is driven by using a readily available forc-ing module in PLUTO. This drives turbulence by adding a force F turb in both the momentum and energy equations. This force is modeled as an Ornstein-Uhlenbeck (OU) process [25,26]. This process is a stochastic differential equation governing the evolution of the force F turb , given by, Here dF is a force added at every time step to the existing force F turb (k) at wavevector k. The P ζ (k) operator is a projection operator which separates the solenoidal and compressive parts of the force.
(2) When ζ = 1 the forcing is purely solenoidal, when ζ = 0 it is purely compressive, and intermediate values give a mixture of solenoidal and compressive forcing. The forcing is limited to range of wavenumbers k min ≤ k ≤ k max . We ignore the 2π factor in the definition of the wavenumber and since the box length is also normalized to unity, the smallest wavenumber possible in our simulation is 1. Thus k min = 1 and k max = 3 in our simulations. We vary the Mach number M A of the turbulence by varying the energy injected into the turbulence E inj . We also vary the plasma β by changing the isothermal sound speed c s . We utilize solenoidal driving (ζ = 1) and mostly compressive driving (ζ = 0.1). Two different grid resolutions are utilized, 512 3 and 1024 3 . Table. 1 shows the different simulations we have analyzed and their parameters. The first letter "S" or "C" in the simulation ID represents whether it is solenoidally or compressively driven. The alphabet "a" or "b" denotes the resolution (512 3 or 1024 3 respectively). Table. 1 also lists the correlation time of the forcing T . This is close to the usual value of eddy turnover time at the injection scale (L inj ), [27]. The simulations are run for several 10's of eddy correlation time T so that there are many snapshots of a statistical steady state.
The mode decomposition is a linear eigenmode decomposition where the MHD state vector comprising of the density, velocity, and magnetic fluctuations is decomposed into a linear combination of the two Alfvén mode eigenvectors, the two fast magnetosonic mode eigenvectors and the two slow magnetosonic mode eigenvectors. We follow the prescription of [14] to decompose the MHD data into the MHD eigenmodes. At each Fourier wavevector k the MHD state is arranged in a column vector consisting of (ρ k , v k , b k ), where ρ is the density, v is the perturbed velocity field vector (normalized to the Alfvén speed), and b is the perturbed magnetic field vector (normalized to the mean field B 0 ). Here the mean density ρ 0 and mean magnetic field B 0 have been subtracted and only the fluctuating components are kept. The magnetic field components are not truly independent due to the divergence free condition k · b k = 0. Therefore only 2 components of the b field are kept to reduce the MHD state vector to 6 components. These two components are selected to be closest to the magnetic field vector of the Alfvén mode and the slow (or fast) mode. The 6 eigenmode vectors are placed in a 6 × 6 matrix A, while the MHD state is a column vector b, solving for the linear amplitudes, x, of the modes by solving the linear matrix equation Ax = b. Once the Fourier amplitudes of the different modes are obtained in this manner, an inverse Fourier transform gives us data cubes in the physical space consisting entirely of single, specific MHD modes. The perpendicular propagation, k = 0, is a special case where the Alfven and slow modes become degenerate and non-propagating. The parallel propagation case (k ⊥ = 0) is also special as here the Alfvén mode is degenerate with either the slow or fast mode depending on whether v A < c s or v A > c s .
Using this decomposition, the density, velocity and magnetic fields obtained from MHD simulations are decomposed into these three modes, where the subscripts A, S, F refer to the Alfvén, slow, and fast modes respectively. To measure the relative presence of the different modes (for ex. the magnetic field of Alfvén mode) in the turbulence, the relative fraction of "energy" in that mode is calculated by taking, where the sum over the Fourier modes excludes the k = 0 and k ⊥ = 0 modes to count only non-degenerate modes, while the sum over m represents sum over the three MHD modes. P M EA stands for the percentage of magnetic energy in Alfvén modes. This estimates the fraction of Alfvénic magnetic fluctuations in the total mode mixture. Likewise P KEF stands for the percentage in velocity field of fast modes, and so on. These mode energy fractions change as a function of time as shown in Fig. 1. It shows the energy fractions as a function of time in the simulation C1a. This simulation is first run at a lower resolution of 128 3 in order to reach a steady state in energy quickly. Then the 512 3 simulation is launched using a data cube from the 128 3 simulation as the initial condition with trilinear interpolation. We see that it takes some time initially for the different mode fractions to attain a steady value. The Alfvén mode shows a very similar level of its kinetic and magnetic fluctuations, as expected from its eigenmodes. The dominant contribution comes from the kinetic component of the slow modes, while its magnetic component is comparable to the Alfvén modes. The fast mode also shows a strong kinetic component and a weaker magnetic component.
These mode energy fractions are averaged over the steady state snapshots of each 512 3 simulation and shown in Fig. 2. In the solenoidally driven simulations, the Alfvén and slow modes form the major fraction, with very little contribution from fast modes. Going from S1 to S4 as the Alfvén Mach number increases, there is a slight increase in the fraction of Alfvén modes. The Alfvén modes have roughly equal energies in the velocity and magnetic fields while the slow mode has a stronger component of velocity fields. A striking feature of Fig. 2 is that the fast mode has a significantly large proportion in the compressively driven simulations, which has not been observed before. Comparing S1a and C1a, the percentage of fast mode fluctuations rises almost 7 times going from solenoidal driving to compressive driving. On the other hand, comparing C1a, C2a, and C4a shows that changing the M A is not affecting the mode fractions significantly (except for a gradual increase in Alfvén mode propor- Simulation IDs tion). Comparing CB0a with CB1a shows that kinetic fluctuations of slow modes decrease while their magnetic fraction increases as the plasma β increases, taking the slow mode magnetic and kinetic fluctuations closer to equipartition. This is understood from the fact that as β → ∞ the slow mode dispersion becomes like the Alfvén mode which implies equipartition between kinetic and magnetic fluctuations. Also as β increases the fraction of fast mode increases in the kinetic fluctuations, while decreasing in the magnetic fluctuations. This is expected from the fast mode eigenvector (Eq. A30 of Ref. [14]) as in the high β → ∞ limit we have v k,F ∝ αk and ω k,F ∼ kc s . This gives |b k,F | ∼ |v k,F |B 0 /c s and hence If we consider only the velocity fluctuations then in all the compressively driven simulations, the total fraction of slow and fast modes is larger than the fraction of Alfvén mode. Since the Alfvén mode is incompressible while the fast and slow magnetosonic modes are compressible, this means that compressive driving expectedly makes the compressible velocity components dominant. If we consider only the magnetic fluctuations, then in simulations C1a, C2a, and C4a, the slow plus fast fraction is larger than the Alfvén fraction. Therefore, even magnetic fluctuations are dominated by the compressible magnetosonic modes in compressively driven turbulence of plasma β close to unity.
The mode fractions only give us a crude estimate of the strengths of various modes. We take a look at the per-pendicular wavenumber energy spectrum to get a sense of the wavenumber distribution of the mode energies. This is shown in Fig. 3. The perpendicular direction is taken w.r.t. the z direction which is the mean field direction. The spectrum in the perpendicular x − y plane is averaged over the angle θ between k ⊥ vector and x axis, E(k ⊥ ) = dθk ⊥ E(k ⊥ , θ). This spectrum is very similar to the 1D wavenumber spectrum averaged over all three directions, E(k). The k = 0 wavenumber is removed from the data cube when calculating this spectrum in order to be consistent with Fig. 2. The energy spectra of the Alfvén and slow modes show very similar behavior across all the cases.The solenoidally driven low M A case S1a is a case where the Alfvén cascade is weak as we will see later. The fast mode in this case is very weak energetically compared to the slow and Alfvén modes, with a very steep spectrum. In the solenoidally driven trans-Alfvénic case S4a, the Alfvén and slow modes show a spectrum close to k −3/2 ⊥ , which is indicative of strong turbulence. The magnetic field spectrum of fast modes is close to k −3/2 ⊥ while the velocity field spectrum is between to k −5/3 ⊥ and k −2 ⊥ . This is similar to earlier results of fast mode spectra [14].
In the compressively driven cases of C1a and C4a we see that the fast mode energy level increases compared to the solenoidally driven cases at large scales close to the driving scales. In C4a the Alfvén and slow modes still show very similar spectra, close to k −3/2 ⊥ . The fast mode velocity field in both C1a and C4a shows a spectrum of k −2 ⊥ , while the magnetic field spectrum is between k −5/3 ⊥ and k −2 ⊥ . This spectrum is steeper than the k −3/2 spectrum claimed in Ref. [14]. This hints at the possible role of shocks in dissipating the fast mode cascade and making it steeper [28]. We find that the velocities in the C1a fast mode data cube are sub-sonic and sub-Alfvénic. Nevertheless we do see sharp, shock-like, gradient fronts in these fast mode data cubes, which are not present in the Alfvén or slow mode data cubes. This is probably due to the nonlinear steepening of the dispersive fast modes. We also perform simulations of single fast modes with a low amplitude such that their velocity amplitude is subsonic and sub-Alfvénic and find that they too steepen into shock fronts. Such a process could be the reason behind the steeper slope of fast mode spectrum. As the slope of the fast mode spectrum is steeper, even though it is dominant close to the driving scale, its energy component drops off compared to the Alfvén and slow modes at smaller scales.
We can also see signatures of different modes in the frequency spectra. For this 1 dimensional slices along the x (perpendicular to mean field) and z (parallel to mean field) axes are taken. These slices are output at a high frequency at a time interval of ∆t = 0.002τ A . Then doing a 2D Fourier transform along the x(z) and time axes gives us the power distribution in k ⊥ (k )-ω space. Fig. 4 shows this power spectrum in the k −ω space. The white dashed line follows the ω = ±k v A line, which is the dis- 2k v A . This is the relation for a fast mode where c s = v A (approximately true for these simulations) and k ⊥ = k . Fig. 4(a) shows the frequency spectrum from the velocity fluctuations in simulation S2a, in which the Alfvén and slow mode contributions dominate significantly over fast modes as seen in Fig. 2. We see that the power is concentrated close to the Alfvénic dispersion. Fig. 4(b) shows the frequency spectrum for velocity field in simulation C2a which also shows a branch of power concentrated at higher frequen-cies ω which are close to the fast mode dispersion. The fast mode is also a significant proportion of the mode mixture in C2a simulation (Fig. 2) and this reflects in the frequency characteristics. In the next section we focus on the anisotropy characteristics of the Alfvén and slow modes.

III. ALFVÉN AND SLOW MODES
We are interested in the nature of cascade of the different MHD modes, especially in its anisotropy. For this we analyze the energy spectrum in the k -k ⊥ space. Here the parallel direction is along the mean field, i.e. along the z direction. The 2D spectrum is defined as x + k 2 y , and θ is the angle between k ⊥ and x axis. Fig. 5 shows these spectra for velocity field of the Alfvén modes in simulations S1a to S4a with increasing M A . We see that for S1a the energy is distributed along the k ⊥ axis close to k = 0. There is very little cascade along the parallel direction in the sense that as energy spreads to higher k ⊥ , there is no spread to higher k . The situation is similar for S2a simulation. As the M A increases the cascade slowly spreads in the parallel direction, slightly in the S3a simulation and more prominently in the S4a simulation. This is an indication of the turbulence transitioning from weak to strong as M A increases.
We further analyze the anisotropy of the Alfvén modes by using structure functions. The anisotropic structure function is defined as This involves an ensemble average over a number of pairs of points which are separated by distance l in the magnetic field parallel direction (b) and distance l ⊥ in the field perpendicular direction (b ⊥ ). The magnetic field directionb is the local mean magnetic field. To obtain this, first for each parallel and perpendicular distance pair (l , l ⊥ ) a distance l = l 2 + l 2 ⊥ is calculated. Then a random point is selected in the data cube and a sphere is taken around this point with a diameter l. The local mean magnetic field direction is calculated by taking an average of a few (≥ 5) random points located in this sphere. We have verified that the results do not change when taking more points. This gives us the local mean field directionb and a random vector perpendicular tob is also constructed,b ⊥ . This allows us to select 2 points on this sphere that are separated by l b + l ⊥b⊥ . An ensemble average over thousands of such pairs gives us SF 2 (l , l ⊥ ).
The isocontours of the structure function are plotted in Fig. 6. A second order smoothing is applied a few times on the structure function to make the contours smoother, without changing their behavior. These are derived from the magnetic and velocity fields of the decomposed Alfvén mode in the 4 different simulations S1b, S4b, C2a, and CB1a, averaged over several time snapshots. The driving occurs up to length scale of 0.33 units in terms of box length, and so we only focus on l and l ⊥ up to 0.2. The anisotropy of the Alfvén modes is clearly visible across the different simulations and is quite similar. The parallel length scales are larger than the perpendicular length scales. Moreover, this ratio l /l ⊥ changes with l , with this anisotropy increasing as we proceed to smaller scales making this a scale-dependent anisotropy. Comparing S4b and C2a simulations shows that the anisotropy of Alfvén modes is not affected by the type of driving. Changing the plasma β in simulation CB1a also does not seem to change the anisotropy. The simulation S1a shows a structure function of the velocity field that is highly elongated along the l direction. Similar behavior is seen for the larger l ⊥ values in C2a and CB1a velocity field structure functions. These are low M A simulations. We observe that at low M A the velocity field is highly correlated along the magnetic field direction, giving rise to an almost l -invariant structure function. This is indicative of a weak nature of cascade at low M A . As M A increases smaller-scale structure develops in the parallel direction. The trans-Alfvénic simulation S4b shows structure function that has smaller l scales. We need to further look at the variation of l with l ⊥ to get a quantitative understanding of the anisotropy of Alfvén modes.
We also analyzed the 2D structure functions of the slow modes. They show a behavior similar to the Alfvén modes with an anisotropy showing l > l ⊥ at all scales.
From the 2D structure functions we can extract the relation of l versus l ⊥ by taking the cuts of the isocontours of constant structure function at the l and l ⊥ axis. For this the structure function along the two axis (l = 0 and l ⊥ = 0) is calculated with very high statistics such that a smooth interpolation can be used to obtain a relation between the l and l ⊥ . This measure gives a better picture of the anisotropy scaling, shown in Fig. 7. It shows this measure derived from the magnetic fields of the Alfvén modes. We see that the anisotropy depends on the M A for the Alfvén modes. For simulation S1b the l is almost constant (very weakly changing) at large l ⊥ implying that eddies form at smaller perpendicular length scales but maintain the same parallel length scales, which is an indication of weak turbulence. For the simulation S3b also a similar behavior is observed at large l ⊥ , but the l starts reducing as l ⊥ gets smaller. As the Mach number increases in simulation S4b, the l starts trending closer to the Goldreich-Sridhar scaling of l ∼ l 2/3 ⊥ . The C4b simulation shows a power law with the Goldreich-Sridhar scaling for a significant range of 0.02 l ⊥ 0.1. Comparing S1b and C1b shows not much difference between solenoidal and compressive driving. Comparing CB0a with CB1a shows that the plasma β does not have a significant effect on the anisotropy. This shows that at low M A there is a large range of scales in l ⊥ where l remains unchanging with l ⊥ , indicating no cascade in parallel direction. As M A increases, this range decreases and we start seeing a transition to the Goldreich-Sridhar scaling.    8 shows the variation of l with l ⊥ derived from the magnetic fields of the slow modes. It shows a behavior similar to the Alfvén modes. For simulations S1b, S3b, C1b, and CB1a, the l drops very slowly with decreasing l ⊥ , which is similar to the weak nature of Alfvén modes in Fig. 7. Simulations CB0a, C4b, and S4b show a behavior close to Goldreich-Sridhar scaling. There is some isotropic scaling also seen in simulation C4b. This could be due to coupling with the fast mode which is strong in this case and shows isotropic scaling as we will see later.
The Alfvén cascade is expected to transition from weak to strong at a transition scale λ CB . The weak turbulence spectrum of Alfvén modes is E(k ⊥ ) ∼ ( /τ A ) 1/2 k −2 ⊥ [29,30], where is the energy injection rate. Then, λ CB is the scale where the linear interaction time, τ A = L/v A , balances the nonlinear interaction time, τ nl = λ CB /δv λ CB . From the weak turbulence spectrum, the velocity strength goes as δv l ⊥ ∼ l 1/2 ⊥ . If we assume a We try to estimate this transition scale by making use of the structure function anisotropy. As seen in Fig. 7 the l is expected to be flat as a function of l ⊥ in the weak turbulence regime (at large l ⊥ ) and tend towards the Goldreich-Sridhar slope of l ∼ l 2/3 ⊥ in the strong regime. Therefore, we fit a power low of the form l = Cl α ⊥ at each l ⊥ in a window of l ⊥ − ∆ to l ⊥ + ∆ around it. We take ∆ = 5 grid points and the scaling exponent α is calculated at each l ⊥ . As expected at large scales close to the driving scale the exponent is very close to 0 and it increases as l ⊥ reduces. It crosses the Goldreich-Sridhar value of 2/3 for the first time at some l ⊥ value which can be taken as the transition scale λ CB . So starting from large l ⊥ as we go down to smaller l ⊥ , we define the transition scale λ CB as the l ⊥ at which α goes over a threshold value close to 2/3 for the first time. This way λ CB is identified for the Alfvén mode in all the different simulations. The variation of this transition scale as a function of M A for each of these simulations is shown in Fig. 9. It is broadly indicative of the M 2 A variation expected from theory. At low M A the λ CB is at small enough scales such that they are close to the dissipative scales. Therefore, we see a plateau at low M A . However, the trend is clearer from the simulations with M A 0.5. This is an important verification of the existence of the weak regime of Alfvén turbulence. This important feature needs to be taken into account when developing models of interstellar medium turbulence for cosmic ray scattering. Next we focus on the properties of the fast mode.

IV. FAST MODES
FIG. 10. k − k ⊥ spectrum of the velocity field of fast modes for the 4 simulations S1a, S4a, C1a, and C4a.
In Fig. 3 the fast modes showed a different spectrum compared to the Alfvén and slow modes. Fig. 10 shows the fast mode wavenumber spectrum as a function of k ⊥ and k . Contrasting it with Fig. 5 we see a different nature of the cascade here. For the Alfvén mode the cascade was clearly anisotropic with energy cascade taking place mostly in the direction of larger k ⊥ . However, the spread of energy for the fast mode appears very close to isotropic as there is almost uniform distribution of power in the parallel and perpendicular wavenumbers. In the low-M A case of S1a where the fast mode has a tiny fraction, the energy distribution seems isotropic at low k with a small anisotropic cascade along the k ⊥ direction for higher k ⊥ . However, for the other three simulations S4a, C1a, and C4a, the cascade is extending radially. Another aspect is that the cascade for Alfvén mode is changing with M A in Fig. 5, with almost no cascade in the parallel direction for S1a case due to the weak nature of turbulence. Here in both C1a and C4a simulations the fast mode shows an isotropic cascade. The isotropic nature of the fast mode cascade is similar even with solenoidal and compressive driving. This shows that the isotropic nature of fast mode cascade is a robust feature.
The 2D structure function isocontours of the velocity and magnetic fields of fast modes are shown in Fig. 11. These contours also show the isotropic nature for the various simulations. For the C2a and C4b simulations, we can see the isotropic contours extending almost up to l ⊥ /L ∼ 0.2 which is close to the driving scale of 0.3. Also in the 1024 3 simulations, this isotropy extends well up to scales smaller than l ⊥ /L ∼ 0.05. From these results we can say that the isotropy does indeed extend throughout the inertial range. The magnetic field contours in C2a show a slight anisotropy at large l . However, the contours from the velocity field are highly isotropic. From Fig. 2 we know that in simulation C2a, the fast mode fraction is larger in the velocity field than in the magnetic field, so the velocity field shows a clearer isotropy. Cases CB0a and S2b show a little anisotropy similar to Alfvén modes. These are cases where the fast mode fraction is smaller (Fig. 2) compared to other cases. This indicates that the properties of the mode might be influenced by how strong it is compared to the other modes.
If it is weak then its properties might get influenced by the dominant mode properties. This points to the cascades of these modes not being entirely independent of each other, with some interaction between them. We calculate the variation of l with l ⊥ for the fast modes now. Fig. 12(a) shows this for the fast mode magnetic field while 12(b) is from the velocity field. The magnetic field of the simulations S1b, S3b, C1b,CB0a, C4b, and CB1a follows a scaling very close to the isotropic scaling. In simulation S4b the relation is closer to GS95 scaling, but this is a case of weak fast modes. For the velocity field in almost all the simulations the relationship l = l ⊥ is followed very closely. This verifies that the isotropic nature of fast modes is a robust feature and it extends throughout the inertial range. It is the same for both low and high M A , as well as high β and low β, as long as the driving generates a significant fraction of fast modes.
Considering the fact that the cascade of energy for Alfvén and slow modes is stronger in the perpendicular direction while for the fast mode it is isotropic, it would be worthwhile to compare their spectrum in the parallel direction. Fig. 13 compares their parallel spectra for the cases S2a and C2a. In the case of S2a the cascade of all three modes is very weak in the parallel direction as their spectra are very steep compared to the k −2 reference slope. In the case of C2a, the Alfvén and slow modes are still expected to have a weak cascade. However, the fast mode will be more energetic since this is compressively driven and will be isotropic. Thus, the fast mode parallel spectrum shows a k −2 slope while the Alfvén and slow modes on the other hand retain a weak cascade with a spectrum steeper than k −2 . As a result, even at high k (k > 20 in Fig. 13) the fast mode has a slightly greater energy that the Alfvén and slow modes. Thus, in this regime of compressively driven, weak Alfvén turbulence, the fast mode can be dominant to the Alfvén and slow modes even at large parallel wavenumbers. This can lead to very different and interesting turbulence characteristics.

V. SUMMARY AND DISCUSSION
We have analyzed the properties of the different MHD modes in solenoidally and compressively driven turbulence which is highly relevant for astrophysical plasmas. One of the important aims of studying different MHD modes is for their different properties in particle acceleration and scattering, so these modes have been studied separately in this context. But an important open question was what are the relative fractions of these modes in astrophysical plasma turbulence and what do they depend on.
Here we found that the type of driving is a major factor affecting the composition of these modes. Understandably, compressive driving leads to a larger proportion of the slow plus fast magnetosonic modes, but more specifically the fast magnetosonic modes. Moreover, the compressive modes can also dominate in the magnetic field fluctuations. This is crucial for the cosmic ray transport and acceleration in turbulence since fast modes dominate the particle scattering [15,16]. A signature from polarization analysis (SPA) method to identify such modes from synchrotron polarization maps has been invented and this study holds important implications for such measurements [33].
The nature of turbulent cascade is also a highly rel-  evant feature for a variety of astrophysical phenomena. We identify many important properties of the Alfvén and slow mode turbulent cascade from these simulations. We see the anisotropic nature of the cascade as predicted by Goldreich-Sridhar theory. We also see indications of a weak regime of Alfvénic turbulence that extends from L inj up to L inj M 2 A . This is an important numerical test of a theoretically predicted regime.
On the other hand, the isotropic nature of the fast mode cascade which was earlier seen in low resolution studies is now verified to extend throughout the inertial range through higher resolution studies. This seems a robust feature which does not show a weak-strong transition unlike the Alfvén and slow modes and does not depend on plasma β or M A . The spectrum of the fast modes can be steeper than k −3/2 and closer to k −2 when the fast mode dominates and can be dissipated by shocks. This has implications for the cutoff scale and damping of fast modes.
Particularly interesting is the case of low M A , compressively driven turbulence. Fast modes do not exhibit any transition from the large scales to small scales and have an extended inertial range with isotropic cascade in contrast to the case of Alfvén and slow modes. As a result, the energy in fast modes becomes dominant even at small scales. Thus cosmic ray scattering and acceleration remain effective counterintuitively in this regime through both the gyroresonance and transit-time damping interactions with fast modes.