The Parametric Decay Instability of Alfvén Waves in Turbulent Plasmas and the Applications in the Solar Wind

Simulations are performed using the ideal MHD module of PLUTO code (Mignone et al. 2007). Two sets of simulations are performed that are categorized as turbulence simulations and non-turbulence simulations. The simulation domain is a 3D box elongated along the background magnetic field: ${{\boldsymbol{B}}}_{0}={B}_{0}\hat{{\boldsymbol{x}}}$. Simulation domain is $[8\pi ,2\pi ,2\pi ]$ and the grid size is $[1024,256,256]$ in $x,y,z$ direction, respectively. Periodic boundary conditions are applied in all directions. In normalized units, ${B}_{0}=1,{\rho }_{0}=1$, Alfvén speed v_A = 1, sound speed ${c}_{s}=\sqrt{\gamma \beta /2}$, where β is the ratio between thermal pressure and magnetic pressure, and $\gamma =5/3$ is the adiabatic index. Time is normalized by ${\tau }_{A}$, where ${\tau }_{A}=8\pi /{v}_{A}$ is Alfvén crossing time of the simulation domain along the $\hat{{\boldsymbol{x}}}$ direction.

For the turbulence simulations, we adopt a two-step process. First, we produce a turbulent background in the simulation domain following the procedure described in Makwana et al. (2015). Six linearly polarized Alfvén waves are injected into the domain, with their magnetic field and velocity perturbations in the following form:

$$\begin{eqnarray}\delta {{\boldsymbol{B}}}_{\mathrm{turb}} & = & \displaystyle \sum _{j,k}\,\delta {B}_{\mathrm{turb}}\cos ({{jk}}_{x}x+{{lk}}_{z}z+{\phi }_{j,l})\hat{{\boldsymbol{y}}}\\ & & +\displaystyle \sum _{m,n}\,\delta {B}_{\mathrm{turb}}\cos ({{mk}}_{x}x+{{nk}}_{y}y+{\phi }_{m,n})\hat{{\boldsymbol{z}}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\delta {{\boldsymbol{v}}}_{\mathrm{turb}} & = & -\displaystyle \sum _{j,k}\,{sgn}(j)\delta {v}_{\mathrm{turb}}\cos ({{jk}}_{x}x+{{lk}}_{z}z+{\phi }_{j,l})\hat{{\boldsymbol{y}}}\\ & & -\displaystyle \sum _{m,n}\,{sgn}(m)\delta {v}_{\mathrm{turb}}\cos ({{mk}}_{x}x+{{nk}}_{y}y+{\phi }_{m,n})\hat{{\boldsymbol{z}}}\end{eqnarray} \tag{ 2 }$$

where (j, l) = (1, 1), (1, 2), (−2, 3), (m, n) = (−1, 1), (−1, −2), (2, −3), and ${k}_{x,y,z}=2\pi /{L}_{x,y,z}$. Random phases are given by ϕ, and sgn is the sign function. There are three waves propagating obliquely in the positive $\hat{{\boldsymbol{x}}}$ and negative $\hat{{\boldsymbol{x}}}$ direction, respectively. The amplitudes $\delta {B}_{\mathrm{turb}}$ and $\delta {v}_{\mathrm{turb}}$ characterize the initial strength of the fluctuations, and they also determine the magnitude of turbulence at later times when an Alfvén wave is injected. These counter-propagating Alfvén waves will cascade and form a decaying turbulent state. The turbulence is typically established after one Alfvén time (see Makwana et al. 2015 for details about the turbulence properties). Second, at t = 1.2, a circularly polarized Alfvén wave is injected into the domain with

$$\begin{eqnarray}&&\delta {{\boldsymbol{B}}}_{\mathrm{cir}}=\delta B\cos (4x)\hat{{\boldsymbol{y}}}+\delta B\sin (4x)\hat{{\boldsymbol{z}}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\delta {{\boldsymbol{v}}}_{\mathrm{cir}}=-\delta {{\boldsymbol{B}}}_{\mathrm{cir}},\end{eqnarray} \tag{ 4 }$$

where $\delta B$ is the amplitude of the injected Alfvén wave and it has a wavenumber ${k}_{x0}=4$. Simulations are typically extended to four Alfvén times to study the evolution of PDI. The power spectrum of B_z at different times are shown in Figure 1. Before the Alfvén wave injection, the system has decayed to a turbulent state (blue line). After the injection of an Alfvén wave, a discrete mode at k = 4 with a relatively high power (corresponding to the injected Alfvén wave) is now visible in the spectrum (green line).

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To our knowledge, all previous PDI studies were in the non-turbulence situation (e.g., Del Zanna et al. 2001). Typically, a circularly polarized Alfvén wave described by Equations (3) and (4) is injected into the simulation domain with a uniform background in density, pressure, and magnetic field, plus a very small random density perturbation ($\delta \rho /{\rho }_{0}\sim {10}^{-2}$). This is what we used as well in non-turbulence simulations.

The PDI of Alfvén waves is characterized by an exponential growth stage of the density fluctuations before saturation. In the 1D limit, theoretical growth rates can be obtained from solving the dispersion relation derived by Derby (1978) and Goldstein (1978):

$$\begin{eqnarray}&&({\omega }^{2}-\beta {k}^{2})(\omega +k+2)(\omega +k-2)(\omega -k)\\ &&\quad ={\eta }^{2}{k}^{2}({\omega }^{3}+k{\omega }^{2}-3\omega +k),\end{eqnarray} \tag{ 5 }$$

where ω and k are normalized by the frequency and wavenumber of the injected Alfvén wave and $\eta =\delta B/{B}_{0}$. Note that β in Equation (5) is defined as ${c}_{s}^{2}/{v}_{A}^{2}$. Furthermore, for a given β and η, in multi-dimensional studies such as ours, we find that many oblique modes with different ω and ${\boldsymbol{k}}$ are excited due to PDI.

Figure 2(a) shows, in the non-turbulence simulations, the rms density fluctuation evolution for different amplitudes $\delta B$ of the injected Alfvén waves. The plasma $\beta =0.5$ for all the simulations. An exponential growth stage is clearly seen for all simulations in Figure 2(a), with the growth rate becoming larger when $\delta B$ is larger. Note that even though the initial random density fluctuation amplitude is ∼0.01, it quickly drops below 10⁻³, out of which PDI gradually grows. The growth of the rms density is due to a single growing mode.

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For the turbulence simulations with $\delta {B}_{\mathrm{turb}}=\delta {v}_{\mathrm{turb}}=0.05$, the rms density fluctuation evolution is shown in Figure 2(b). The density fluctuation increases rapidly and remains nearly constant at ∼0.01 during the cascade of Alfvén waves to develop turbulence. At t = 1.2, when a circularly polarized Alfvén wave is injected into the domain, the density fluctuation shows a sudden jump (most likely due to the ponderomotive effect of the injected wave), followed by an exponential growth stage. This is the signature that the injected Alfvén wave undergoes the PDI. The dependence of the PDI's growth rate on the amplitude of the injected waves can be seen from the slope changes during the exponential stage.

Figure 3 shows the evolution of density distribution in the $\{x,y\}$ plane ($z=\pi$) for a turbulence case with $\delta B=0.2$ in Figure 2. After the injection of the circularly polarized Alfvén wave (Figures 3(c) and (d)), the density mode shows a dominant mode with k_x = 5 and ${k}_{y}={k}_{z}=0$, which is exactly what the 1D linear theory of PDI predicts with this parameter set . In addition, oblique slow modes with non-zero k_y and k_z are also developed, though with smaller growth rates. These results demonstrate that the PDI of a single Alfvén wave still occurs with the background turbulence, with both parallel and obliquely propagating slow modes (see below for a detailed mode analysis).

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To examine the mode growth in detail, we have carried out the FFT analysis of the density fluctuations and found that some modes exhibit exponential growth. Three of them ($\{{k}_{x},{k}_{y},{k}_{z}\}\,=\,\{5,0,0\},\{5,0,1\},\{5,1,2\}$) are shown in Figure 4. We calculate the growth rates of the three modes in the time range between t = 1.6 and t = 2.4. The growth rates for the three modes are 0.024, 0.017, and 0.017, respectively, as shown in the legend of Figure 4. In particular, the mode $\{{k}_{x},{k}_{y},{k}_{z}\}\,=\,\{5,0,0\}$ (predicted by the linear theory) has the highest growth rate and is responsible for most of the density rms growth seen in Figure 2(b). Figure 4 also shows two other modes ($\{{k}_{x},{k}_{y},{k}_{z}\}\,=\,\{4,0,0\},\{6,0,0\}$), which have less energy than these PDI modes.

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To quantitatively study the influence of turbulence on the growth rate of PDI, we have investigated two types of situations: one has the same background turbulence but different injected single Alfvén wave amplitudes and the other is the same injected Alfvén wave amplitudes but different background turbulence levels. We now discuss them in detail.

First, we present results from a suite of 3D simulations with the same turbulence background and the same single Alfvén wave injection time (t = 1.2) except that we vary the injected Alfvén wave amplitude, ranging from $\delta B/B=0.1$ to 0.5. The plasma $\beta =0.5$. The 1D linear theory predicts that the PDI mode with an integer wavenumber should have k_x = 5 for all these parameters. From our nonlinear 3D turbulence simulations, we confirm that indeed in all cases the maximum growth mode has k_x = 5. We then utilize the FFT analysis to select all the density modes with k_x = 5 and integrate over k_y and k_z. The growth rates are obtained during the exponential growth stage of their time histories.

Figure 5 shows the growth rate of PDI for different $\delta B$ from three different situations: the 1D linear theory, non-turbulence simulations, and turbulence simulations with $\delta {B}_{\mathrm{turb}}=0.05$. The growth rates of non-turbulence simulations match the theoretical results with an error of less than 10%. The growth rates of turbulence simulations, however, are about half of the theoretical results. This suggests that the background turbulence has decreased the growth rate of PDI of Alfvén waves.

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One interesting point worth noting is that, at $\delta B=0.1$, the linear theory predicts there is no growth rate at the integer mode k = 5, though finite PDI growth is predicted for a (non-integer) wavenumber very close to k = 5. Interestingly, both the turbulence and non-turbulence simulations show the growth of the k_x = 5 mode. We speculate that this is because the turbulence and/or the background fluctuations give rise to a "resonant broadening effect" (Dupree 1966) that produces a finite growth rate at k_x = 5.

Next, we show the evolution of PDI modes under different turbulence levels ($\delta {B}_{\mathrm{turb}}=0.05,0.1,0.15,0.2$) and different amplitudes ($\delta B=0.1$ to 0.5) of injected Alfvén waves in Figure 6. While the existence and the growth rate of PDI are relatively easy to quantify in many runs, it becomes difficult when the turbulence level is high and the PDI stage is relatively short. Figure 7 singles out the case with a fixed injected Alfvén wave amplitude ($\delta B=0.4$) while varying the background turbulence level. This figure gives a clearer sense on the effects of the background turbulence. Overall, we find several trends. (1) The background turbulence levels do influence the growth rate of PDI. The growth rate decreases with increasing turbulence background. It is difficult to quantify the reduction for all cases due to the relatively short growth stage of PDI with high turbulence levels. (2) The background turbulence levels do influence the saturation level of PDI. Higher turbulence background gives a lower saturation level of density variations from PDI. (3) Higher turbulence level also shortens the duration of the PDI stage. (4) In all the runs we have made so far, PDI nonetheless still gets excited. This demonstrates the robustness of this instability.

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Density fluctuations arise during the growth of PDI of Alfvén waves. To understand the nature of these fluctuations better, we employ the spatial–temporal FFT analysis to determine whether the density variations follow dispersion relations. For example, using the 3D simulation shown in Figure 2(b) with $\delta B=0.2$, we put 50 time frames of 3D data together between t = 2 and 2.4 with a time resolution of $\delta t=0.008$ to form a big 4D matrix. The FFT analysis of the matrix gives power P versus ω and k: $P(\omega ,{k}_{x},{k}_{y},{k}_{z})$. To get a clear view of the modes with the most power, we specify ${k}_{y},{k}_{z}$ and get a contour figure in the $\omega \mbox{--}{k}_{x}$ plane. These modes are then compared with the linearized ideal MHD modes for the same choices of k_y and k_z.

Figure 8(a) shows the contour figure of FFT results of B_z in the $\omega \mbox{--}{k}_{x}$ plane (with ${k}_{y}=0,{k}_{z}=0$), as well as the theoretical dispersion relation of three MHD waves. The Alfvén wave dispersion curve crosses the high power region exactly, indicating that this mode (${k}_{x}=4,{k}_{y}={k}_{z}=0$) is an Alfvén wave, which is actually the initial circularly polarized pump Alfvén wave. This result confirms that our method is reliable. To analyze the nature of the compressible modes, we perform the spatial–temporal FFT of density fluctuations $\delta \rho$ during the same time domain. Figures 8(b) and (c) show the contours of two modes with high power, along with the theoretical dispersion relations of MHD waves. The mode in Figure 8(b) propagates parallel to the background magnetic field and is either a slow wave or a sound wave (they are degenerate in this situation). Figure 8(c) shows a mode with finite k_y and k_z, and it matches well with the dispersion relation of a slow wave. The growth rate of the slow modes in Figures 8(b) and (c) is relatively large (see Figure 4). These results show that slow MHD waves are indeed generated via PDI.

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