3.1 The perpendicular cascade of energy

Figure 4. Plots of the perpendicular magnetic energy $E_{B_{\bot }(k_{x},k_{y})}$ (arbitrary units) on a log scale in the perpendicular Fourier plane $(k_{x},k_{y})$ (a) at the initial time $t/T_{c}=0$ , (b) after the first strong Alfvén wavepacket collision at $t/T_{c}=1$ and (c) after the second strong Alfvén wavepacket collision at $t/T_{c}=2$ .

To explore the nonlinear cascade of smaller perpendicular scales in this strong Alfvén wavepacket collision simulation, we plot in figure 4 the perpendicular magnetic energy of fluctuations in perpendicular Fourier space integrated over $z$ , $E_{B_{\bot }(k_{x},k_{y})}\equiv \int _{-L_{z}/2}^{L_{z}/2}\,\text{d}z|\unicode[STIX]{x1D6FF}B_{\bot }(k_{x},k_{y})|^{2}/8\unicode[STIX]{x03C0}$ in arbitrary units. In figure 4(a) is the perpendicular magnetic energy of the initial Alfvén wavepackets at $t=0$ , showing all of the energy is contained within the three perpendicular Fourier modes $(k_{x}\unicode[STIX]{x1D70C}_{i},k_{y}\unicode[STIX]{x1D70C}_{i})=(1,0)$ , $(0,1)$ and $(-1,0)$ . Note that AstroGK uses a reality condition imposed on the complex Fourier coefficients such that the magnetic field can be described using only Fourier modes in the upper half-plane $k_{y}\geqslant 0$ . The reality condition requires $\unicode[STIX]{x1D6FF}\hat{B}(k_{x},k_{y})=\unicode[STIX]{x1D6FF}\hat{B}^{\ast }(-k_{x},-k_{y})$ , so if there is power in the $(1,0)$ mode, there must be equivalent power in the $(-1,0)$ mode, as seen in figure 4(a).

After the first collision at $t/T_{c}=1$ , figure 4(b) shows clearly that the strong nonlinear interactions have transferred significant energy to many other modes in the perpendicular Fourier plane $(k_{x},k_{y})$ . Note that the perpendicular wavenumber is given by $k_{\bot }=\sqrt{k_{x}^{2}+k_{y}^{2}}$ , so that modes further away from the origin represent smaller scale fluctuations in the perpendicular plane. The plot after the second collision at $t/T_{c}=2$ in figure 4(c) shows that successive collisions continue to facilitate the perpendicular cascade of energy. These results demonstrate that the finding from the idealized periodic case – that Alfvén wave collisions mediate the nonlinear transfer of energy to smaller perpendicular scales – indeed persists in the more realistic case of localized Alfvén wavepacket collisions, answering the first key question posed in the introduction.

3.2 Current sheet development

Figure 5. Plot of the normalized parallel current density $j_{z}/j_{0}$ at $z=0$ during the first collision at (a) $t/T_{c}=0.3$ and (b) $t/T_{c}=0.5$ , at $z=L_{z}/2$ during second collision at (c) $t/T_{c}=1.3$ and (d) $t/T_{c}=1.5$ and at $z=0$ during the third collision at (a) $t/T_{c}=2.3$ and (b) $t/T_{c}=2.5$ .

For plane Alfvén wave collisions in the idealized periodic case, plots of the parallel current density $j_{z}$ in the $(x,y)$ plane perpendicular to the equilibrium magnetic field demonstrate that strong Alfvén wave collisions self-consistently generate current sheets (Howes Reference Howes2016). These current sheets extended the full parallel length of the original Alfvén waves, with widths in the perpendicular plane of approximately the perpendicular wavelength of the original interacting Alfvén waves, but with a much smaller thickness in the perpendicular plane. Here we use plots of the $(x,y)$ plane perpendicular to the equilibrium magnetic field to determine whether this current sheet formation persists in the more realistic case of strong collisions between localized Alfvén wavepackets.

Figure 6. Plot of the normalized parallel current density $j_{z}/j_{0}$ of colliding Alfvén wavepackets before the first collision at $t=0$ for (a) the unipolar wave at $z=-L_{z}/4$ and (b) the dipolar wave at $z=+L_{z}/4$ , after the first collision when the wavepackets have separated for (c) the unipolar wave at $z=+L_{z}/4$ and $t/T_{c}=0.98$ and (d) the dipolar wave at $z=-L_{z}/4$ and $t/T_{c}=0.96$ and after the second collision for (e) the unipolar wave at $z=-L_{z}/4$ and $t/T_{c}=2.1$ and ( f) the dipolar wave at $z=+L_{z}/4$ and $t/T_{c}=2.0$ .

First, we examine the development of the parallel current density $j_{z}$ in the $(x,y)$ plane during the course of each collision. In figure 5, we plot the normalized parallel current density $j_{z}/j_{0}$ at $z=0$ (a) early in the first collision at $t/T_{c}=0.3$ and (b) later in the same collision at $t/T_{c}=0.5$ , where we remind the reader that the midpoint of each collision occurs at $t/T_{c}=0.5,1.5,2.5$ . The nearly circular $j_{z}$ pattern in figure 5(a) is largely due to the linear superposition of the current of each of the initial Alfvén wavepackets (the top row of figure 6 shows the current patterns of each of the initial Alfvén wavepackets). The nonlinear interaction of the Alfvén wavepackets during the first collision leads to the thinning of these initial circular current patterns into a more sheet-like morphology in figure 5(b). Figure 5(c,d) shows the evolution of the second collision in the plane $z=L_{z}/2$ (at the parallel boundary of the periodic domain) at (c) $t/T_{c}=1.3$ and (d) $t/T_{c}=1.5$ . Here we see the current sheets becoming yet more elongated and intense through the course of the second collision. Figure 5(e,f) shows plots of $j_{z}/j_{0}$ during the third collision at $z=0$ , with further thinning and intensification of the current sheets. Note that the three-dimensional isocontours of $j_{z}$ in figure 2 show that the parallel extent of the current sheets is the same as the parallel length of the original wavepackets.

These results show clearly that strong collisions of localized Alfvén wavepackets indeed self-consistently generate intermittent current sheets, answering the second question in the introduction.

Another important question, however, is whether the current sheets that develop during the wavepacket collisions persist within each wavepacket after the collision. In figure 6, we show the initial pattern at $t=0$ of the normalized parallel current density $j_{z}/j_{0}$ for (a) the unipolar wave at $z=-L_{z}/4$ with only variation in the $x$ direction and (b) the dipolar wave at $z=+L_{z}/4$ with only variation in the $y$ direction. After the first collision, we show $j_{z}$ for (c) the unipolar wave at $z=+L_{z}/4$ and  $t/T_{c}=0.98$ and (d) the dipolar wave at $z=-L_{z}/4$ and $t/T_{c}=0.96$ . At this time when the wavepackets are no longer overlapping in $z$ , one can clearly see the distortion of current density pattern, due to the previous collision, has led to a thinning and intensification of the current density into a more sheet-like morphology that persists after the wavepackets have separated. After the second collision, we plot (e) the unipolar wave at $z=-L_{z}/4$ and $t/T_{c}=2.1$ and ( f) the dipolar wave at $z=+L_{z}/4$ and  $t/T_{c}=2.0$ . Here we can see that the current sheets that develop as a consequence of the strong Alfvén wavepacket collisions indeed persist within the wavepackets after the collision is over, showing that these interactions in the more realistic case may explain the ubiquitous observations of intermittent current sheets in plasma turbulence.

3.3 Evolution of energy

A simple physical interpretation of the evolution of the energy in this localized Alfvén wavepacket collision can be developed using the equations of incompressible MHD for guidance. These equations, written in terms of Elsasser variables, take the form

(3.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{z}^{\pm }}{\unicode[STIX]{x2202}t}\mp \boldsymbol{v}_{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{z}^{\pm }=-\boldsymbol{z}^{\mp }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{z}^{\pm }-\unicode[STIX]{x1D735}P/(n_{0i}m_{i}+n_{0e}m_{e}),\end{eqnarray}$$

and $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{z}^{\pm }=0$ . Here $\boldsymbol{v}_{A}=\boldsymbol{B}_{0}/\sqrt{4\unicode[STIX]{x03C0}(n_{0i}m_{i}+n_{0e}m_{e})}$ is the Alfvén velocity due to the equilibrium field $\boldsymbol{B}_{0}=B_{0}\hat{\boldsymbol{z}}$ where $\boldsymbol{B}=\boldsymbol{B}_{0}+\unicode[STIX]{x1D6FF}\boldsymbol{B}$ , $P$ is total pressure (thermal plus magnetic) and $n_{0i}m_{i}+n_{0e}m_{e}$ is mass density. Recall that the Elsasser variables are defined by $\boldsymbol{z}^{\pm }=\boldsymbol{u}\pm \unicode[STIX]{x1D6FF}\boldsymbol{B}/\sqrt{4\unicode[STIX]{x03C0}(n_{0i}m_{i}+n_{0e}m_{e})}$ , representing waves that propagate up or down the mean magnetic field. The nonlinear term, $\boldsymbol{z}^{\mp }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{z}^{\pm }$ , governs the nonlinear interactions of the counterpropagating Alfvén wave collisions.

Figure 7. Plot of the evolution of energy transfer between $(k_{x},k_{y})$ modes. Note that the $(1,0)$ mode is plotted as the same line as the cyan $(-1,0)$ mode, indicating they are identical.

In figure 7, we plot the temporal evolution of the magnetic energy $E_{B_{\bot }(k_{x},k_{y})}$ in some of the lowest perpendicular Fourier modes $(k_{x},k_{y})$ , illustrating some of the key properties of Alfvén wavepacket collisions. The first main point is that there is no nonlinear transfer of energy among Fourier modes until the two wavepackets begin to overlap in space along $z$ (first vertical dashed line). As expected from theoretical considerations of incompressible MHD turbulence (Iroshnikov Reference Iroshnikov1963; Kraichnan Reference Kraichnan1965; Sridhar & Goldreich Reference Sridhar and Goldreich1994; Montgomery & Matthaeus Reference Montgomery and Matthaeus1995; Ng & Bhattacharjee Reference Ng and Bhattacharjee1996; Galtier et al. Reference Galtier, Nazarenko, Newell and Pouquet2000; Howes & Nielson Reference Howes and Nielson2013), the nonlinear interaction is zero unless both $z^{-}$ (the unipolar wavepacket) and $z^{+}$ (the dipolar wavepacket) are both non-zero at the same point in space, as can be seen by inspection of the nonlinear term $\boldsymbol{z}^{\mp }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{z}^{\pm }$ . Between collisions, when the wavepackets have no overlap in $z$ , there is no nonlinear transfer of energy among different $(k_{x},k_{y})$ modes, as clearly seen in figure 7. The second main point is that the perpendicular Fourier modes $(k_{x}L_{\bot },k_{y}L_{\bot })$ of the initial Alfvén wavepackets, $(1,0)$ and $(0,1)$ generally lose energy to nonlinearly generated perpendicular Fourier modes during each collision. This is the transfer of energy to smaller perpendicular scales, illustrated by the plots of perpendicular magnetic energy in $(k_{x},k_{y})$ Fourier space in figure 4. A more quantitative examination contrasting the nonlinear transfer of energy between the idealized plane Alfvén wave collisions in a periodic geometry and localized Alfvén wavepackets collisions, in both the weakly and strongly nonlinear limits, will be presented in a subsequent paper (Verniero & Howes Reference Verniero and Howes2018).

3.4 Wavepacket collisions

The three-dimensional visualization of this localized Alfvén wavepacket collision in figure 2 presents a concise overview of many of the properties examined above. The series of panels (a) through (c) show clearly the distortion of the original wavepackets that arises during the collision and persists after the wavepackets have separated again. This is a physical visualization of the nonlinear cascade of energy to smaller scales, the key characteristic of turbulence in space and astrophysical plasmas. Although a little difficult to see clearly in the three-dimensional projection in figure 2, the nonlinear evolution of this strong Alfvén wavepacket collision also leads to the self-consistent generation of a sheet-like morphology for the regions of intense parallel current density $j_{z}$ . This thinning and intensifying of the current into sheets is more clearly seen in the perpendicular cross-section at $z=L_{z}/2$ (the right-hand $z$ boundary of the simulation domain) in figure 2(d); the same cross-section at $z=L_{z}/2$ is also shown in figure 5(d).

In addition to the isocontours of the normalized parallel current density $j_{z}/j_{0}$ , we also plot the paths of a sample of magnetic field lines as they traverse the domain. Beginning at points on a $5\times 5$ grid at $z=-L_{z}/2$ , we trace the field lines in the $+z$ direction. Although the perpendicular magnetic field fluctuation associated with each Alfvén wavepacket is small compared to the equilibrium magnetic field, $\unicode[STIX]{x1D6FF}B_{\bot }\ll B_{0}$ , one can still see the distortion of the field lines as they pass through each wavepacket. In the space between wavepackets, both before and after each collision, the magnetic field lines are straight and uniform, further illustrating the point that nonlinear evolution ceases when the wavepackets are separated, even in a gyrokinetic simulation where dispersive and kinetic effects are resolved.

It is worthwhile also noting the important point that, even after each strong Alfvén wavepacket collision, the individual wavepackets continue to propagate along the equilibrium magnetic field and they remain localized in $z$ . There is a very small spreading of the wavepacket, hardly noticeable in figure 2, due to the fact that Alfvén waves become dispersive at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gtrsim 1$ , with an increasing parallel group velocity. Therefore, some of the higher $k_{\bot }$ modes that are nonlinearly generated by the strong Alfvén wavepacket collision in this gyrokinetic simulation – a numerical approach that resolves these dispersive kinetic effects at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gtrsim 1$ – will propagate a little faster than Alfvén waves in the MHD limit $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ that have a parallel group velocity at the Alfvén speed $v_{g\Vert }=v_{A}$ , causing a slight spreading out of the localized wavepacket.

3.5 Relation to similar wavepacket collision simulations

A recent study by Pezzi et al. (Reference Pezzi, Parashar, Servidio, Valentini, Vásconez, Yang, Malara, Matthaeus and Veltri2017) has used numerical simulations to tackle what they term the ‘Parker–Moffatt problem.’ The basic question is inspired by the properties of incompressible MHD equations given in § 3.1: do two initially separated, localized Alfvén wavepackets interact nonlinearly only during the time that they overlap, and then cease evolving after they have passed through each other and become separated again? The simulation results shown in figure 2, and examined in further detail in the subsections above, definitively answer this question – yes.

The results of the gyrokinetic simulation of strong Alfvén wavepacket collisions presented here, however, stand in stark contrast to the results of the study by Pezzi et al. (Reference Pezzi, Parashar, Servidio, Valentini, Vásconez, Yang, Malara, Matthaeus and Veltri2017), which find that after the collision, in both their compressible MHD and Hybrid Vlasov–Maxwell simulations, the wavepackets do not separate cleanly and continue to evolve, as illustrated in figure 1 of their paper. What is the cause for the dramatic contradiction in these results?

We argue here that the limitation of the Pezzi et al. (Reference Pezzi, Parashar, Servidio, Valentini, Vásconez, Yang, Malara, Matthaeus and Veltri2017) simulations to two spatial dimensions is the root of these striking differences. Due to the large computational cost of running the Hybrid Vlasov–Maxwell (HVM) code (Valentini et al. Reference Valentini, Trávníček, Califano, Hellinger and Mangeney2007) in three spatial dimensions, the authors chose to approximate the three-dimensional problem by using an oblique two-dimensional approach (Howes Reference Howes2014), where spatial variation is allowed in two dimensions in the $(x,y)$ plane, but the equilibrium magnetic field is slightly tilted away from the normal to the plane, $\boldsymbol{B}_{0}=B_{0}(\sin \unicode[STIX]{x1D703}\hat{\boldsymbol{x}}+\cos \unicode[STIX]{x1D703}\hat{\boldsymbol{z}})$ with $\unicode[STIX]{x1D703}=6^{\circ }$ , yielding a small component of the equilibrium magnetic field in the $x$ direction. Since plane Alfvén waves only propagate if there is a non-zero component of the wavevector parallel to the equilibrium magnetic field, $k_{\Vert }\neq 0$ , this trick enables the $x$ component of a wavevector to have a small component parallel to $\boldsymbol{B}_{0}$ , enabling propagating Alfvén waves to be simulated.

The limitation to two spatial dimensions has two important effects: (i) it precludes the possibility of initializing two perpendicularly polarized, counterpropagating wavepackets (as initialized here), which is the essential set-up of the Parker–Moffatt problem to be studied; and (ii) it significantly limits the nonlinear couplings that can arise, constraining the results in an unphysical way (Howes Reference Howes2014).

Although Pezzi et al. (Reference Pezzi, Parashar, Servidio, Valentini, Vásconez, Yang, Malara, Matthaeus and Veltri2017) state that they set up two Alfvénic  wavepackets, several pieces of evidence suggest that their initial wavepackets are indeed not Alfvénic. First, they initialize the fluctuations by imposing the constraint that fluctuations are strictly transverse to the equilibrium magnetic field $\boldsymbol{B}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{B}=0$ . Although this is a necessary condition for an Alfvén wave, it is not sufficient: the polarization of the magnetic field fluctuation for the Alfvén wave with wavevector $\boldsymbol{k}$ must also be oriented in the direction given by $\boldsymbol{B}_{0}\times \boldsymbol{k}$ (Maron & Goldreich Reference Maron and Goldreich2001; Howes & Nielson Reference Howes and Nielson2013; Howes Reference Howes2014). Any component of the magnetic field fluctuation in the other direction of the plane given by $\boldsymbol{B}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{B}=0$ represents not an Alfvén wave, but a different mode: in incompressible MHD, it would belong to the pseudo-Alfvén wave (Maron & Goldreich Reference Maron and Goldreich2001; Howes & Nielson Reference Howes and Nielson2013; Howes Reference Howes2014), and in compressible MHD, it would represent some combination of the fast and slow magnetosonic modes (Cho & Lazarian Reference Cho and Lazarian2003). That their initialization in fact excites some magnetosonic modes is supported by two results in their paper. First, their initialization excites density fluctuations even before the wavepackets collide, a feature they attribute to magnetic pressure fluctuations due to finite-amplitude magnetic fluctuations, but that could simply be because their initialization directly included these linear magnetosonic modes. Second, even before the waves collide in the MHD simulation, shown in the upper left panel of their figure 1, one can see small disturbances well ahead of and behind each wavepacket, suggesting that magnetosonic modes, which have a different group velocity from the Alfvén mode, are dispersively spreading out the wavepacket, a linear effect.

The second impact of the limitation to two spatial dimensions is that the nonlinear couplings possible in the oblique two-dimensional configuration are unphysically constrained relative to the three-dimensional case, as explained in detail in Howes (Reference Howes2014). This constraint can artificially enhance the nonlinear coupling to magnetosonic modes relative to the coupling to Alfvénic modes, dramatically altering the nonlinear evolution.

Our perspective is that any investigation of the so-called Parker–Moffatt problem is most cleanly performed in three spatial dimensions. Our results show definitively that when two localized Alfvén wavepackets collide, the interaction occurs only while they overlap, ceasing when the wavepackets separate cleanly, leaving distorted wavepackets that have essentially the same finite extent along the equilibrium magnetic field. Before concluding, it is also worthwhile emphasizing that our results were obtained with a gyrokinetic simulation that resolves all of the low-frequency – not including cyclotron and fast magnetosonic physics (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006) – compressible and kinetic effects of a weakly collisional plasma.