Spatiotemporal Analysis of Waves in Compressively Driven Magnetohydrodynamics Turbulence

The 3D CMHD model is given by the mass continuity equation, the momentum equation, the induction equation for magnetic field and the polytropic state equation as,

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+{\boldsymbol{\nabla }}\cdot (\rho {\boldsymbol{u}})=0,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\boldsymbol{u}}}{\partial t}+{\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}{\boldsymbol{u}}=-\displaystyle \frac{{\boldsymbol{\nabla }}P}{\rho }+\displaystyle \frac{{\boldsymbol{J}}\times {\boldsymbol{B}}}{\rho }\\ \quad +\,\nu \left[{{\rm{\nabla }}}^{2}{\boldsymbol{u}}+\displaystyle \frac{{\boldsymbol{\nabla }}({\boldsymbol{\nabla }}\cdot {\boldsymbol{u}})}{3}\right],\end{array}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}={\boldsymbol{\nabla }}\times ({\boldsymbol{u}}\times {\boldsymbol{B}})+\eta {{\rm{\nabla }}}^{2}{\boldsymbol{B}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{P}{{\rho }^{\gamma }}=\mathrm{constant},\end{eqnarray} \tag{ 4 }$$

where u is the velocity field, B = B₀ + b is the total magnetic field (with a fluctuating part b and a mean field B ₀). Note that the change of the magnetic field B₀ involves changing the relative amplitude between the initial fluctuations δb and B₀ (without modifying the initial conditions). In addition, ρ is the mass density, P is the scalar isotropic pressure, J = ∇ × B is the electric current, γ = 5/3 is the polytropic index, and ν and η are the kinematic viscosity and magnetic diffusivity, respectively. The main purpose of these last terms is to dissipate energy at scales smaller than MHD scales, while allowing us to study with an adequate scale separation compressible effects at the largest scales. In the present study, we take the viscosity and magnetic diffusivity to be independent of the mass density.

The set of Equations (1)–(4) can be written in a dimensionless form in terms of a characteristic length scale L₀, a mean scalar density ρ₀ and pressure P₀, and a typical magnetic and velocity field magnitude b_rms and ${u}_{\mathrm{rms}}={b}_{\mathrm{rms}}/\sqrt{4\pi {\rho }_{0}}$ (i.e., the r.m.s. Alfvén velocity), respectively. Then, the unit time is t₀ = L₀/u_rms, which for MHD becomes the Alfvén crossing time. The resulting dimensionless equations are,

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+{\boldsymbol{\nabla }}\cdot (\rho {\boldsymbol{u}})=0,\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\boldsymbol{u}}}{\partial t}+{\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}{\boldsymbol{u}}=-\displaystyle \frac{1}{\gamma {M}_{s}^{2}}\displaystyle \frac{{\boldsymbol{\nabla }}P}{\rho }+\displaystyle \frac{{\boldsymbol{J}}\times {\boldsymbol{B}}}{\rho }\\ \quad +\,\nu ^{\prime} \left[{{\rm{\nabla }}}^{2}{\boldsymbol{u}}+\displaystyle \frac{{\boldsymbol{\nabla }}({\boldsymbol{\nabla }}\cdot {\boldsymbol{u}})}{3}\right],\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}={\boldsymbol{\nabla }}\times ({\boldsymbol{u}}\times {\boldsymbol{B}})+\eta ^{\prime} {{\rm{\nabla }}}^{2}{\boldsymbol{B}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{P}{{\rho }^{\gamma }}=\mathrm{constant},\end{eqnarray} \tag{ 8 }$$

where M_s = u_rms/C_s is the Mach number, ${C}_{s}^{2}=\gamma {P}_{0}/{\rho }_{0}$ is the sound speed, and $\nu ^{\prime}$ and $\eta ^{\prime}$ are the dimensionless viscosity and magnetic diffusivity, respectively (written in terms of the number of the kinetic and magnetic Reynolds numbers).

Considering a static equilibrium (u₀ = 0) with homogeneous external magnetic field ${{\boldsymbol{B}}}_{0}={B}_{0}\hat{z}$, a constant density ρ₀, and a constant pressure P₀, we linearize Equations (5)–(8) and obtain the dispersion relation ω( k ) of small amplitude waves propagation in a CMHD plasma. It is straightforward to show that there are three independent propagating modes (or waves), which correspond to the so-called Alfvén waves (A), fast (F), and slow (S) magnetosonic waves (e.g., Fitzpatrick 2014),

$$\begin{eqnarray}&&{\omega }_{A}^{2}(k)={k}_{\parallel }^{2}{u}_{A}^{2},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\omega }_{F,S}^{2}(k)={k}^{2}{u}_{A}^{2}\left[\displaystyle \frac{(1+\beta )}{2}\pm \sqrt{\tfrac{{\left(1+\beta \right)}^{2}}{4}-\beta {\displaystyle \frac{{k}_{\parallel }}{k}}^{2}}\right],\end{eqnarray} \tag{ 10 }$$

where $\beta ={({C}_{s}/{u}_{A})}^{2}$ is the plasma beta, i.e., the ratio of plasma pressure to magnetic pressure (or as a function of input parameters in the run β = 1/(M_s B₀)²); with ${u}_{A}={B}_{0}/\sqrt{4\pi {\rho }_{0}}$ the Alfvén velocity, C_s as defined above (note that the change of the initial M_s modify the plasma β), and $k=| {\boldsymbol{k}}| =\sqrt{{{\boldsymbol{k}}}_{\parallel }^{2}+{{\boldsymbol{k}}}_{\perp }^{2}}$ with ∥ and ⊥ the wavenumber component along and perpendicular to the external magnetic field, respectively. It is worth noting that B₀ is a parameter and has no relation with the initial fluctuations of the system. On one hand, Alfvén waves are incompressible fluctuations transverse to the external magnetic guide field B ₀. On the other hand, both fast and slow modes, unlike Alfvén modes, carry density fluctuations and their magnetic field perturbations have longitudinal and transverse components. In the case of fast modes, the magnetic field and the plasma are compressed by the wave motion, such that the restoring force is large and hence the frequency and the propagation speed are high. While, for slow modes, the magnetic field oscillation and the pressure variation are anticorrelated with each other such that the restoring force acting on the medium is weaker than that for the fast mode. For this reason the frequency and the propagation speed are the lowest among the three MHD waves modes. Note that for the perpendicular propagation (i.e., k_∥ = 0 and k_⊥ ≠ 0), the Alfvén and slow modes become nonpropagating modes (i.e., ω_{A, S} = 0) and are degenerate, but they can be distinguished using their different polarization, since δB_∥A = 0 and δB_∥S ≠ 0.

As one of the goals of the present paper is to identify the various possible waves and structures in the simulations, we adopt the assumption that energy concentrated closely to the linear-dispersion relation can be explained by linear and weak turbulence theories; while any spread around, or away from, those linear curves is a sign of strong turbulence that requires fully nonlinear theories to be understood (Andrés et al. 2017).

We carried out DNSs of the compressible fluid equations in the presence of a mean-magnetic field B ₀ in a periodic 3D box of size 2π with linear spatial resolutions of 128³ and 256³ grid points. The CMHD Equations (5)–(8) were numerically solved using the Fourier pseudospectral code GHOST (Gómez et al. 2006; Mininni et al. 2011b). The discrete time integration used is a second-order Runge–Kutta method. The scheme ensures exact energy conservation for the continuous time, spatially discrete equations (Mininni et al. 2011b). For simplicity, we used identical dimensionless viscosity and magnetic diffusivity, $\nu ^{\prime} =\eta ^{\prime} =1.25\times 10-3$ (i.e., the magnetic Prandtl number is P_m = 1).

Random initial conditions were used for the velocity and magnetic field and null density fluctuations in the whole space. For all times t > 0, the velocity field and the magnetic vector potential were forced by a mechanical forcing F and electromotive forcing , respectively. The mechanical and electromotive forcings are uncorrelated and they inject neither kinetic nor magnetic helicity. At t = 0, for each forcing function, a random 3D isotropic field f_k is generated in Fourier space, by filling the components of all modes in a spherical shell with 2 < k < 5 with amplitude f, and a random phase ϕ_k for each wavevector k . We used an amplitude f = 0.2 for the mechanical and electromotive forcings. For the kinetic forcing, we introduced a parameter that allow us to determine its compressibility, so we can go from a pure incompressible forced system to a pure compressible forcing controlling this factor. The forcing is given by the equation (Cerretani & Dmitruk 2019),

$$\begin{eqnarray}&&{{\bf{F}}}_{k}=i{\boldsymbol{k}}\times \hat{z}{\psi }_{1}-i{\boldsymbol{k}}{\psi }_{2},\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}&&{\psi }_{1}=(1-{f}_{0}){f}_{1}({\boldsymbol{k}}),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\psi }_{2}={f}_{0}{f}_{2}({\boldsymbol{k}}).\end{eqnarray} \tag{ 13 }$$

Here f_i ( k ) is a function that is 1 if k is within the forced wavenumber range and is 0 out of that range. By modifying the value of the parameter f₀, the fluid can be compressibly excited (f₀ = 1) generating thrusts parallel to the direction of the movement and, alternatively, the incompressible modes are generated using f₀ = 0 pushing in the direction perpendicular to the fluid movement. By means of this forcing, we perfomed three groups of simulations: one with purely compressible forcing, another with purely compressible forcing and finally a mixed forcing that injects energy in the same amount to compressible and incompressible modes. In addition, for each of these three groups, four simulations were carried out for different values of the Mach number (M_s ) and the mean-magnetic field (B₀). We selected two different values of the sonic Mach number being 0.25 and 0.55, while for the magnetic field values, B₀ = 2 and B₀ = 8 were used. In Table 1, we summarized the parameters of our simulations.

The spatiotemporal spectrum allows identification of waves in a turbulent flow and reveals aspects of its dynamics. The technique consists of calculating the complete spectrum in wavenumber and frequency for all available Fourier modes in a numerical simulation or an experiment (Clark di Leoni et al. 2015a; Sahraoui et al. 2003a). As a result, it can separate between modes that satisfy a given dispersion relation (and are thus associated with waves) from those associated with nonlinear structures or turbulent eddies, and quantify the amount of energy carried by each of them at different spatial and temporal scales. The method we use does not require the preexistence of wave modes or eddies. Quantifying the relative importance of each of them and understanding the physics that controls it is the main outcome expected from the present analysis. In the following, the spatiotemporal magnetic energy spectral density tensor is defined as:

$$\begin{eqnarray}&&{E}_{{ij}}({\boldsymbol{k}},\omega )=\displaystyle \frac{1}{2}{\hat{{\boldsymbol{B}}}}_{i}^{* }({\boldsymbol{k}},\omega ){\hat{{\boldsymbol{B}}}}_{j}({\boldsymbol{k}},\omega ),\end{eqnarray} \tag{ 14 }$$

where ${\hat{{\boldsymbol{B}}}}_{i}({\boldsymbol{k}},\omega )$ is the Fourier transform in space and time of the i-component of the magnetic field B ( x , t) and the asterisk implies the complex conjugate. The magnetic energy is associated with the trace of E_ij ( k , ω).

As the external magnetic field B ₀ in the simulations points in $\hat{z}$, in practice we will consider either i = j = y or i = j = z, to identify different waves based on their polarization (either transverse or longitudinal with respect to the guide field). In all cases, the acquisition frequency was at least two times larger than the frequency of the fastest wave, and the total time of acquisition was larger than the period of the slowest wave in the system. It is worth mentioning that spatiotemporal spectra have been used before in numerical simulations and experiments of rotating turbulence (Clark di Leoni et al. 2014), stratified turbulence (Clark di Leoni & Mininni 2015), quantum turbulence (Clark di Leoni et al. 2015b), IMHD turbulence simulations (Meyrand et al. 2015, 2016; Lugones et al. 2016), and in spacecraft observations (Sahraoui et al. 2003a, 2010). In the present paper, we use the technique to investigate the interaction between waves in CMHD turbulence extending our previous study (Andrés et al. 2017).

Figures 1, 2, and 3 show histograms of density fluctuations for different Mach numbers M_s , mean-magnetic fields B₀ and values of the compressibility amplitude of the forcing f₀. For the present analysis, we use some statistics tools (such as the variance and quantiles) in order to improve the description of density variations. In the case of B₀ = 0, numerical results indicate an increase in density fluctuations as the Mach number increases. For example, we find that for fixed f₀ = 0 and M_s = 0.25, the 90% of the data is located around the interval ρ = [0.96 − 1.03] while for M_s = 0.55 this happens around ρ = [0.89 − 1.09]. For f₀ = 0.5 and f₀ = 1, results are analogous. In particular, for M_s = 0.25 in the vast majority of cases, the density does not vary being the simulation that most closely resembles to an incompressible plasma. The same occurs when the compressibility amplitude of the forcing is increased, and although the differences are less noticeable, we observe that for M_s = 0.55 there is a slight increase in the density fluctuations with a variance ranging from 3.8 × 10⁻³ (f₀ = 0) to 5.1 × 10⁻³ (f₀ = 1).

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Turning on the mean-magnetic field B₀, we can notice a significant increase in the variance of about an order of magnitude, ranging from 4.5 × 10⁻⁴ (B₀ = 0) to 4.8 × 10⁻³ (B₀ = 2) for fixed f₀ = 0 and M_s = 0.25. When we increase B₀, the density fluctuations grows with a variance of 4.8 × 10⁻³ (B₀ = 2) and 7.8 × 10⁻³ (B₀ = 8). Therefore, the 90% of the data is located around the intervals ρ = [0.88 − 1.11] (B₀ = 2) and ρ = [0.86 − 1.15] (B₀ = 8). It is worth mentioning that Yang et al. (2016) studied the energy transfer across scales in 3D CMHD turbulence and showed that kinetic and magnetic energies cascade conservatively from large to small scales in cases with varying degrees of compression. To do this, the density fluctuation was defined as δ ρ/〈ρ〉 and analyzed obtaining values close to the ones reported here. Therefore, we conclude that our results are compatible with Yang et al. (2016). However, for different values of M_s and f₀ and fixed B₀, the growth of density fluctuations is more subtle. For instance, in the case of B₀ = 2 and M_s = 0.55, we have an increase of fluctuations as we vary f₀ with a variance of 3.8 × 10⁻³ (f₀ = 0), 4.9 × 10⁻³ (f₀ = 0.5) and 5.1 × 10⁻³ (f₀ = 1). For B₀ = 8 the results are similar as shown in Figure 3. Therefore, this analysis allows us to show that we have larger fluctuations as we increase sonic Mach number (approaching to M_s = 1 and therefore, at speeds comparable to the speed of sound), the compressibility amplitude of the forcing and the mean-magnetic field.

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In order to study the types of fluctuations present in a spatial spectrum, we computed the spatial isotropic spectrum for the kinetic energy. Besides, we computed the compressible and incompressible kinetic spectra of the flow using the usual Helmholtz decomposition (Joseph 2006). Figure 4 shows the spatial spectrum for M_s = 0.25, 0.55, B₀ = 2, 8, and f₀ = 0, 0.5, 1. Also, for comparative purpose, in dotted line we include the scaling law k^−5/3, predicted by Kolmogorov theory for incompressible, isotropic and homogeneous turbulent flows and the scaling law k⁻² predicted by Burgers (1948) for highly compressible, isothermal, turbulent flows. On the one hand, in the case M_s = 0.25 and B₀ = 2, 8 for a large range of wavenumbers, an inertial range compatible with a ∼k^−5/3 can be observed for the incompressible and compressible kinetic energy. For similar values of the Mach number, Yang et al. (2016) also showed that the solenoidal velocity exhibit a k^−5/3 spectrum. Whereas, when we take M_s = 0.55 and same values of B₀, we obtain an inertial range compatible with a ∼k⁻² power law. Our results are in good agreement with Federrath et al. (2010) who studied density and velocity fluctuations using high resolution simulations and data to analyze the interstellar turbulence; in particular, molecular clouds for compressive and solenoidal forcing. This means that we find important differences in the shape of the spectrum when increasing the Mach number, while the increase in the magnetic field does not imply significant changes. Note also that, although the vast majority of the kinetic energy is located in its incompressible component and dominates over the compressible one, it is known that the small compressible component can still affect the flow dynamics in this regime. For instance, DNSs showed that proton acceleration is significantly enhanced when compared to the incompressible case (González et al. 2016). Also, recently it has been showed that small compressibility in the plasma increases the energy dissipation rate as the cascade enters into the sub-ion scales (Andrés et al. 2019), affects the anisotropic nature of solar wind magnetic turbulence fluctuations (Kiyani et al. 2012), and increases the turbulent energy cascade rate in the inner heliosphere (Andrés et al. 2021).

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To determine the presence of Alfvén or magnetosonic waves and which (if any) dominates the dynamics, we use a spatiotemporal analysis (see, Clark di Leoni et al. 2015a). To perform this type of analysis, the field must be stored with very-high-frequency cadence in order to resolve the waves in time and space; in particular, dt = 1 × 10⁻⁵ was taken as the temporal sampling rate. Since the spatiotemporal spectrum is four dimensional (with k = (k_x , k_y , k_z ) and the frequency ω), we fix two components of k to plot the remaining component against the frequency.

Figures 5 and 6 show the spatiotemporal spectrum of the perpendicular magnetic field fluctuation E_xx (k_x = 0, k_y = 0, k_z ) (with k_z = k_∥) for fixed k_x = k_y = 0 for the set of simulations I2/M2/C2 and I4/M4/C4, respectively. The dispersion relation for Alfvén and magnetosonic waves given by Equations (9) and (10) are shown in blue dashed, green dashed–dotted and orange solid lines, respectively. In the case of runs I2/M2/C2, the wave modes that are excited correspond to fast magnetosonic and Alfvén branches (note that for k_⊥ = 0, these branches overlap) and there is no apparent traces of slow magnetosonic waves. For runs I4/M4/C4, the energy also accumulates mainly in Alfvén and fast branches; however, there is a small portion of energy spread along the slow magnetosonic modes at high parallel wavenumbers. This small portion of energy in the slow modes decrease as the compressibility amplitude of the forcing f₀ grows in apparent contradiction with what one would expect, since the slow mode is of a compressible type. A possible way to understand this is that the generation of the portion of energy in the slow mode could originate in a three-mode interaction (e.g., Shebalin et al. 1983; Cho & Lazarian 2003) between two Alfvén modes and one slow mode. As the Alfvén modes are more prominent with the smaller amplitude of the compressible parameter forcing f₀, this three-mode interaction would be more efficient as f₀ is decreased. In addition, we can observe that when f₀ = 1 at high wavenumbers, the energy reaches its minimum with a maximum in k_∥ < 25 and ω/B₀ < 20. It is worth noting that, as we increase the Mach number, the energy near the Alfvén/fast branches also increase by approximately an order of magnitude.

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From these last two spectra, the question of which branch dominates the dynamics of the system may arise: is it Alfvén or fast magnetosonic waves? Looking back to spatial spectra and the fact that the incompressible component dominates over the compressible one, we came to the conclusion that in both spatiotemporal spectra, energy accumulates mainly around Alfvén modes. For spatiotemporal spectrum E_xx (k_x = 0, k_y = 15, k_z ) for fixed k_y = 15, shown in Figure 7, a great amount of energy accumulates around the Alfvén wave branch. Although some energy is also present in the vicinity of the fast magnetosonic branch and along slow modes at high parallel wavenumbers, fast waves do not dominate the dynamics as predicted using the weak wave turbulence theory (Chandran 2005, 2008). Instead, Alfvén waves dominate the linear dynamics as we suspected from the spatial spectra results. As we mentioned above, Makwana & Yan (2020) have analyzed the properties of the different MHD modes in solenoidally and compressively driven turbulence. In that case, unlike our results, the compressive magnetosonic modes were dominated in the magnetic field fluctuations. However, higher values of the Mach number (close to the unity) were used increasing the compressibility of the system, so it would be useful to compare these results with future studies. Besides, lower magnetic energy compared to the case k_y = 0 is observed for the different values of f₀. This is because, when we choose k_y = 15, we are moving along the inertial range and, according to previous results (Andrés et al. 2017), the presence of waves tends to decrease.

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Fast magnetosonic waves can be separated from the Alfvén waves by looking at the spatiotemporal spectrum of parallel magnetic field fluctuation E_zz (k_x = 0, k_y , k_z = 0) shown in Figures 8 and 9 for the set of simulations I1/M1/C1 and I2/M2/C2. The Alfvén waves do not contribute to the parallel component of the magnetic field energy since their magnetic perturbations are perpendicular to the guide field. In both figures, we find that the energy accumulates in two regions: at high frequencies near the fast magnetosonic branch and at low frequencies near slow modes with ω/B₀ = 0. We can observe that as we increase the mean-magnetic field, we go from having most of the energy spread across the spectrum to being located around the fast and slow modes. In general, the increase of the mean-magnetic field leads the system to obey the linear theory or equivalently there is more energy located in the linear waves. A possible explanation is that, when we increase the magnetic field B₀, the anisotropic turbulence grows, the energy is concentrated on nonpropagating modes (with ω = 0) and produce a cascade in the wavevectors perpendicular to the mean-magnetic field direction only (Matthaeus et al. 1996; Oughton et al. 1998, 2015).

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In order to quantify the amount of energy near the different wave modes, we use an integration method available in the literature. Clark di Leoni et al. (2015a) propose to calculate the ratio of the energy accumulated near these modes to the total energy in the same wavenumber,

$$\begin{eqnarray}&&F({k}_{z})=\displaystyle \frac{{E}_{{xx}}({k}_{x}=0,{k}_{y}=0,{k}_{z},\omega ={\omega }_{a,f,s})}{{{\rm{\Sigma }}}_{j}{E}_{{xx}}({k}_{x}=0,{k}_{y}=0,{k}_{z},{\omega }_{j})},\end{eqnarray} \tag{ 15 }$$

with ω_a,f,s the frequencies that satisfy a certain dispersion relation (we have used the E_xx (k_z ) component as an illustration example only). Figure 10 shows the energy near Alfvén and slow magnetosonic waves for runs I2/M2/C2. We can observe that energy is located mainly in the Alfvén branch with a maximum in the smallest wavenumbers. Analyzing runs I4/M4/C4 (shown in Figure 11), for f₀ = 0 and large scales of k_∥, we obtain an increase of energy around slow modes. However, the vast majority of the energy is located around the Alfvén branch with a maximum in k_∥ < 60 and f₀ = 0.5. Figure 12 shows that, for runs I1/M1/C1, the amount of energy located both in the fast branch and in the slow one is similar. Whereas, for runs I2/M2/C2 (Figure 13), there is a greater amount of energy around the fast magnetosonic branch with a maximum in k_∥ < 20. In all cases, as we increase the value of f₀ (and therefore the compressive forcing), no significant growth of energy was found in the different branches. On the other hand, Makwana & Yan (2020), reported that the type of driving is a major factor affecting the composition of the MHD modes. In particular, compressive driving leads to a larger proportion of slow and fast magnetosonic modes. It is worth mentioning that we have also used an alternative criterion to compute the amount of energy in each wave branch reported by Cerretani & Dmitruk (2019) and Cerretani (2018). This method consists of performing the mean squared differences between the frequencies that follow a certain dispersion relation ω_a,f,s and the spectrum, e.g., E_xx (k_x = 0, k_y = 0, k_z ). The results (not shown here) are similar to the ones present here and, therefore, both methods are compatible.

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Finally, we investigate the spatiotemporal spectrum of density defined as, E(k, ω) ∼ δ ρ^* δ ρ, where δρ = ρ − ρ₀ represents the density fluctuations. Figure 14 shows the spatiotemporal spectrum E(k_x = 0, k_y , k_∥ = 0) of density fluctuations for runs I3/M3/C3. We observe that, energy is spread along both fast and slow magnetosonic modes. This means that, we recover the presence of waves as seen for the spatiotemporal spectrum for magnetic field fluctuations. In the case of the slow branch, energy accumulates mostly in modes with low k_y and low ω/B₀, typically k_y < 40 and ω/B₀ < 10 and increasing with the value f₀. We also measure the energy near the wave modes using Equation (15). Spangler (2003) presents some theoretical results on the relation between density and magnetic field fluctuations. For instance, Bhattacharjee et al. (1998) showed that if the background plasma had zero-order spatial gradients, $\delta \rho /\overline{\rho }\sim \delta b/{B}_{0}$. This theoretical work was inspired by a number of investigations which have utilized the Helios spacecraft data set to study the dependence of pressure or density fluctuations on velocity and magnetic field fluctuations (Tu & Marsch 1994; Bavassano & Bruno 1995; Bavassano et al. 1995; Klein et al. 1993). In the present paper, we will focus on analyzing the amount of energy located around wave modes for density and magnetic field fluctuations as shown in Figure 15. A similar amount of energy was found for both density and magnetic field fluctuations. In this case, the energy reaches a maximum for k_y < 20. Although there might be a relation between these two quantities due to their similar energetic behaviors, we plan to use different methods that allow us to more accurately compare density and magnetic field fluctuations in a future work.

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