- LETTER
- Open Access
Dispersion relation as a channel of plasma turbulence evolution
- Horia Comişel†^{1, 2},
- Yasuhito Narita†^{3, 4}Email author and
- Uwe Motschmann†^{1, 5}
https://doi.org/10.1186/s40623-015-0191-5
© Comişel et al.; licensee Springer. 2015
- Received: 17 July 2014
- Accepted: 20 January 2015
- Published: 27 February 2015
Abstract
We present evidence from ion-scale plasma turbulence simulations that normal-mode waves and sideband waves co-exist when a plasma evolves from wave fields into turbulence. Ion-scale fluctuations represent normal-mode fluctuations such as Bernstein and cyclotron modes, accompanied by sideband waves having moderate frequency mismatch with the normal modes. The evolution process is studied using the method of energy partition and frequency broadening. Wave evolution into turbulence exhibits distinct stages: growth phase of normal modes, saturation, and decay. The saturation time is of the order of 1000 ion gyroperiods, which is delayed at higher values of the plasma parameter beta. We conclude that the dispersion relations play the central role in plasma turbulence evolution, serving as the energy flux channel in the spectral domain.
Keywords
- Dispersion relation
- Ion-scale plasma turbulence
- Waves
Findings
Space and astrophysical plasmas are collisionless. Their turbulent state, particularly in the ion kinetic regime, is believed to play a fundamental role in transferring energy between an electromagnetic field and a plasma, serving as an effective dissipation mechanism. A useful way to describe plasma turbulence is to view it as a set of normal- and non-normal-mode waves. While normal-mode waves are characterized by dispersion relations associating frequencies with wavevectors and may be sustained for a long time, non-normal-mode waves are not strictly constrained to dispersion relations and may have a relatively shorter lifetime. The wave picture of turbulence should be, naively speaking, valid under a well-established large-scale magnetic field such that perturbative treatment is applicable. In fact, multi-spacecraft observations indicate that the magnetic field fluctuations represent normal modes or linear modes that have been predicted by the linear Vlasov theory (Dudok de Wit et al. 1995; Narita and Glassmeier 2005). Foreshock waves represent the low-frequency whistler mode excited by right-hand ion beam instability (Gary 1991). Magnetosheath fluctuations represent the mirror mode (Gary 1993). Furthermore, fluctuations near the location of magnetic reconnection show characteristics of the whistler mode (Eastwood et al. 2009); the whistler-mode chorus resonates with energetic electrons in the inner magnetosphere (Katoh 2014); and fluctuations in the solar wind show the dispersion relations for kinetic Alfvén waves and ion Bernstein modes (Perschke et al. 2013; Sahraoui 2010).
Of course, not all the fluctuating fields represent normal modes characterized by a dispersion relation, and wave-wave coupling leads to the excitation of sideband waves. A frequency mismatch with the dispersion relation naturally explains the mechanism of wavevector anisotropy in plasma turbulence (Gary 2013). Yet, the spacecraft observations cannot reveal the detailed mechanism of turbulence evolution or information on the time elapsed after pump wave excitation.
Here, we address a question that is essential for understanding the evolution of plasma turbulence - how does a plasma evolve into turbulence? Since the dispersion relations are known to depend on the value of beta, it is natural to anticipate that turbulence evolution might also be influenced by beta. We adopt a numerical approach to answer this question, and use the hybrid plasma code Adaptive Ion Kinetic Electron Fluid (AIKEF) (Müller et al. 2011). This direct numerical simulation extends the earlier works, ref. (Comişel et al. 2013; Verscharen et al. 2012), in two ways: (1) by achieving the longest possible simulation run to resolve turbulence evolution processes and (2) by extending the condition of beta to the highest possible values (up to beta 0.2). Both improvements require substantial computational resources, enabling one to survey systematically the influence of beta on turbulence evolution for, to our knowledge, the first time in the ion kinetic regime. The method using the hybrid code does not assume the existence of normal mode waves in the kinetic regime a priori. Wave fields are produced in the simulation box, and the energy spectra are determined and analyzed in the wavenumber-frequency domain at different times. The concept of energy partition and strength of broadening are used as the analysis tool for the normal-mode waves and sideband waves to track the time evolution of wave fields into turbulence.
Methods
Hybrid simulation
Direct numerical simulation is the best method for studying the role of dispersion relations in turbulence evolution. The AIKEF code has been developed for hybrid plasma simulation, and it solves the set of equations of motion for ions (as particles) and the Maxwell equations self-consistently using the finite-element method. Electrons are treated as a charge-neutralizing massless fluid. No gyration average (or gyro-kinetic treatment) is needed in this simulation. Hence, the hybrid code is advantageous in that one need not assume any statistical properties of the fluctuating field (e.g., Gaussian statistics), wave modes, or propagation directions. The code is disadvantageous with a weakness in that strong electrostatic effects due to charge localization would be immediately canceled out by the mobile electron fluid, but this effect can be safely neglected for our purpose.
Energy conservation under the AIKEF code simulation was verified for freely streaming particles in a periodic simulation box (or domain) in three dimensions with a total length of 32 ion inertial lengths and the ion bulk flow at 8 Alfvén speeds (Müller et al. 2011). Four hundred iterations (or four ion gyroperiods) are needed for one domain crossing. Particle mass (associated with the adaptive mesh refinement, not used in our simulation), particle energy, particle momentum, and electromagnetic energy were recorded at each iteration. These quantities are normalized to their initial values. Energy conservation has been confirmed with errors of less than 1% per gyration. Strictly speaking, the total particle energy decreases at a constant rate of approximately 0.015% per ion gyration, while the electromagnetic energy moderately increases at variable rates of approximately 0.005% per gyration.
where μ _{0} is the permeability of free space, n _{i} is the ion number density, k _{B} is the Boltzmann constant, T _{i} is the ion temperature, and B _{0} is the magnitude of the mean magnetic field. The value of beta is set before the simulation run. The magnetic field magnitude is scaled to unity, and all the quantities relevant to the magnetic field magnitude are scaled according to the dimensionless unit. In our simulations, the ion beta is an input parameter. By combining the ion beta with the number density (which is also an input parameter), the temperature is determined from a given set of beta and density values. This temperature value is further used to generate the thermal population of ions for the simulation runs.
Ion-kinetic waves are generated numerically using the hybrid code in the two-dimensional spatial setup spanning the parallel and the perpendicular directions of the large-scale magnetic field. The simulation in this configuration allows one to track the wave evolution into turbulence directly in the spectral domain (spanning the wavenumbers and frequencies) without any additional complication caused by eddies originating from the advective nature of fluid turbulence. Vectorial quantities such as ion velocity, magnetic field, and electric field are treated as three-dimensional. The wave field is generated on the large possible spatial scale in the simulation box with the smallest possible mesh size to achieve the highest spectral resolution and range. The code is applied to a simulation box with an integration size of 250×250 ion inertial lengths (for protons), a mesh size of a quarter of the inertial length, spatially periodic boundaries, and large-scale constant magnetic field in the z-direction (hereafter the parallel direction). Fluctuating magnetic fields are normalized to the large-scale field magnitude. Initially, a superposition of 1000 Alfvén waves are excited on the large scales where the magnetohydrodynamic picture is valid. The initial or pump waves have random phase distribution with the cutoff wavenumber set to 20% of the inertial length wavenumber (\(\frac {|\vec {k}| V_{\mathrm {A}}}{\Omega _{\mathrm {p}}} < 0.2\), where Ω _{p} and V _{A} denote the ion gyrofrequency and Alfvén speed, respectively). The amplitudes for the pump waves are determined from the model energy spectrum, which is isotropic, and obeys the Kolmogorov power-law scaling with the spectral index −5/3. The total magnetic field fluctuation amplitude (or the standard deviation) is set to 1% of the large-scale magnetic field. During the numerical run, no additional energy is given to the simulation box, and the value of beta is not reset. The equation of motion and the Maxwell equation are solved alternately at a time step of 0.01% of ion gyroperiods for a time length of up to 2,000 gyroperiods. Wave evolution is studied using three simulation runs, each with a different value of beta: β _{i}=0.05, β _{i}=0.1, and β _{i}=0.2. Four-hundred super-particles are set in the computation cell, which corresponds to 8.4×10^{8} particles in the entire simulation box.
Wave analysis
Spectral estimator
The fluctuation data are obtained at each evolution time step for the three field components (x and y are in the direction perpendicular to the mean magnetic field. x is in the direction out of the simulation box or plane, and z is in the direction parallel to the mean field) at all the mesh grid points in the two-dimensional space. The simulation box therefore spans by the y and z directions. The obtained data are then Fourier-transformed into the three-dimensional energy spectra as a function of the frequency ω and the two components of the wavevectors k _{y} and k _{z}. We are interested in turbulence evolution in the direction perpendicular to the mean magnetic field, as the spectral energy is transported primarily in that direction. The parallel component of the wavevector is therefore set to zero in the data analysis, and we obtain the two-dimensional energy spectrum in the domain spanning the perpendicular components of the wavevectors and frequencies. This implies that one averages the three-dimensional spectra over the parallel spatial coordinate z to obtain results with statistical significance, rather than use the data at any specific value of z.
We limit our wave analysis to the propagation direction perpendicular to the mean magnetic field because most of the magnetic fluctuation energy is stored along the axis of the perpendicular wavevectors, as shown in the previous studies (Comişel et al. 2013; Verscharen et al. 2012). Wave analysis in other directions, parallel or oblique to the mean magnetic field, is also possible and worth pursuing. The present analysis provides information on the higher-order picture of turbulence evolution as most of fluctuation energy is in the perpendicular direction.
Partition and broadening
Lower and upper limits of the frequency range used in the moment calculation
IB1 | IB1 | IB2 | IB2 | IC | IC | ||
---|---|---|---|---|---|---|---|
β _{i} | \(\frac {k_{\perp } V_{\mathrm {A}}}{\Omega _{\mathrm {p}}}\) | \(\frac {\omega _{-}}{\Omega _{\mathrm {p}}}\) | \(\frac {\omega _{+}}{\Omega _{\mathrm {p}}}\) | \(\frac {\omega _{-}}{\Omega _{\mathrm {p}}}\) | \(\frac {\omega _{+}}{\Omega _{\mathrm {p}}}\) | \(\frac {\omega _{-}}{\Omega _{\mathrm {p}}}\) | \(\frac {\omega _{+}}{\Omega _{\mathrm {p}}}\) |
0.05 | 1.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.309 | 0.559 |
0.05 | 2.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.445 | 0.695 |
0.05 | 3.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.470 | 0.720 |
0.05 | 4.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.474 | 0.724 |
0.05 | 5.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.475 | 0.725 |
0.05 | 6.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.475 | 0.725 |
0.1 | 1.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.237 | 0.487 |
0.1 | 2.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.350 | 0.600 |
0.1 | 3.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.371 | 0.621 |
0.1 | 4.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.374 | 0.624 |
0.1 | 5.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.375 | 0.625 |
0.1 | 6.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.375 | 0.625 |
0.2 | 1.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.164 | 0.414 |
0.2 | 2.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.255 | 0.505 |
0.2 | 3.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.272 | 0.522 |
0.2 | 4.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.274 | 0.524 |
0.2 | 5.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.275 | 0.525 |
0.2 | 6.0 | 0.875 | 1.125 | 1.875 | 2.125 | 0.275 | 0.525 |
Results
Wavenumber-frequency spectra
Normal modes and sideband waves
In order to resolve the transition to sideband wave growth and the saturation effect, a quantitative analysis is applied to the energy spectra by using the method of energy partition and frequency broadening. Three wave modes are investigated in detail: fundamental the ion Bernstein mode (IB1), second harmonic of ion Bernstein mode (IB2), and oblique ion cyclotron mode (IC). These modes contain the largest fraction of fluctuation energy in the spectra.
The peak time of the energy partition for the three modes
Mode | β _{i} =0 . 05 | β _{i} =0 . 1 | β _{i} =0 . 2 |
---|---|---|---|
IB1 | \(500\, \Omega _{\mathrm {p}}^{-1}\) | \(1000\, \Omega _{\mathrm {p}}^{-1}\) | \(1400\, \Omega _{\mathrm {p}}^{-1}\) |
IB2 | \(700\, \Omega _{\mathrm {p}}^{-1}\) | \(1000\, \Omega _{\mathrm {p}}^{-1}\) | \(1400\, \Omega _{\mathrm {p}}^{-1}\) |
IC | \(700\, \Omega _{\mathrm {p}}^{-1}\) | \(1000\, \Omega _{\mathrm {p}}^{-1}\) | \(1600\, \Omega _{\mathrm {p}}^{-1}\) |
The peak time of the frequency broadening for the three modes
Mode | β _{i} =0 . 05 | β _{i} =0 . 1 | β _{i} =0 . 2 |
---|---|---|---|
IB1 | \(500\, \Omega _{\mathrm {p}}^{-1}\) | \(900\, \Omega _{\mathrm {p}}^{-1}\) | \(1400\, \Omega _{\mathrm {p}}^{-1}\) |
IB2 | \(600\, \Omega _{\mathrm {p}}^{-1}\) | \(1000\, \Omega _{\mathrm {p}}^{-1}\) | \(1400\, \Omega _{\mathrm {p}}^{-1}\) |
IC | \(700\, \Omega _{\mathrm {p}}^{-1}\) | \(1100\, \Omega _{\mathrm {p}}^{-1}\) | \(1500\, \Omega _{\mathrm {p}}^{-1}\) |
Transition time
The transition time is also evaluated for the peak times in the broadening. As in the case for the saturation time, the broadening peak time is dependent on beta: the peak time is increasingly delayed with higher values of beta. The difference in the two times (one is the saturation time in linear-mode fluctuations and the other is the peak time in broadening) is of the order of 50 gyroperiods. The bottom panel in Figure 5 displays the time difference Δ t=t _{L}−t _{NL} (here the subscripts L and NL denote the linear-mode and the sideband or nonlinear waves, respectively). The saturation time data are again averaged over the three wave modes. At lower values of beta (β _{i}=0.05 and β _{i}=0.1), the broadening saturates first and then the linear-mode fluctuation saturates by about 50 gyroperiods. At a higher value of beta (β _{i}=0.2), the order is reversed and the broadening peak time arrives approximately 50 gyroperiods after the saturation of linear-mode fluctuations.
Discussion and conclusions
The dispersion relation plays a central role in wave evolution into turbulence. We conclude that waves evolve into a more turbulent state by exciting both normal-mode waves and sideband waves. Both fluctuation components serve as energy reservoirs. The existence of a peak in the energy partition is suggestive of a scenario in which a turbulent field cannot be constructed by normal modes only, as sideband waves arise naturally with normal-mode waves. It is also worth mentioning that beta influences or regulates the peak time or saturation time; the higher the value of beta is, the later the transition arrives. Beta dependence can be described in such a way that the transition is earlier and clearer when the value of beta is lower. In other words, cold plasmas show a clearer and sharper transition from the early stage to the late stage without the formation of a plateau (neither in the energy partition nor in the frequency broadening). Warm plasmas, in contrast, have delayed transition after the plateau formation.
In our simulation, the particle velocity distribution (of protons) is set to Maxwellian, and furthermore, the system is nearly homogeneous in that the mean magnetic field and plasma density can be regarded as nearly constant. The small-scale fluctuations on the kinetic scales (at wavenumbers higher than that of the ion inertial length) cannot be excited by linear instability processes - neither micro-instability caused by gradients in the velocity coordinate nor macro-instability caused by gradients in the spatial coordinate. To verify how exactly the fluctuation power is transferred to small-wavelength waves by wave-wave interactions (Bernstein and cyclotron modes), a higher-order statistical analysis is needed such as bispectral or bicoherence analysis for three-wave couplings. In the present study, a Maxwellian distribution is used for the initial velocity distribution of protons, and the evolution of wave spectra is associated with not instability but nonlinear wave coupling. This situation is different from that treated in the quasi-linear theory of wave-particle interactions, which is deeply related to the evolution of plasma wave instability.
It is true that ions are heated as waves evolve into turbulence, and in that sense, the value of beta is not strictly constant. In fact, our preceding work using the AIKEF code has already shown that the ion temperature is enhanced by a factor of approximately 1.7 preferentially in the direction perpendicular to the mean magnetic field (Verscharen et al. 2012). The possible effect of the temperature increase is in the determination of the exact scaling or relation between the value of beta and the saturation time. However, the qualitative nature of the relation still holds because all the input values of beta are merely shifted. Tracing the time evolution of ion velocity distribution is an important task that is not feasible with the current computation capacity. In our simulations, computational loads become increasingly demanding as we perform simulations at higher values of beta and for longer runs. To stabilize the simulation run, increasingly more particles must be put into the simulation box to compensate for the increasing mobility of particles (that can easily move from one mesh cell to another), which accordingly takes a huge amount of memory in a high-performance computer.
It is important to emphasize that particle distribution functions should be studied in detail to advance the knowledge derived from our analysis. A quasi-stationary state for wave-particle interactions is obtained in the quasi-linear theory by using the so-called plateau equation, addressing that the particle velocity distribution function exhibits pitch-angle plateau formation such that the energy is no longer transferred between electromagnetic fields and particles (e.g., ref. Marsch and Bourouaine (2011)). It might be possible that the plateau in the time history of the wave spectral energy is associated with the plateau formation in the distribution function.
The presence of inhomogeneity explains the lower limit of the measured broadening.
The detailed evolution profile is diverse. In the early evolution phase, the energy can be stored into the normal-mode waves until a peak is attained. This effect is suggestive of the existence of saturation in exciting the normal mode waves. We propose that dispersion relations should be considered as channels of the energy flux in the spectral domain. The late-stage spectra with enhanced broadening (e.g., bottom panels in Figure 2) are suggestive of a scenario in which fluctuations evolve first as a set of normal mode waves and then into sideband waves. However, this scenario should be regarded as the lowest-order picture in describing wave evolution into turbulence, since not only the partition but also the frequency broadening exhibits decay after saturation. If fluctuations were excited only as sideband waves after the peak, the magnitude of broadening should exhibit a monotonous increase, rather than decrease. A possible reason for the decay of broadening is the wave damping due to cyclotron or Landau resonance processes.
A recent study of the parametric decay instability in laboratory plasmas shows that the three wave coupling is well supported or mediated by dispersion relations, and furthermore, the existence of harmonic branches (such as in the Bernstein mode) even allows many open channels of the spectral energy transfer (Jenkins et al. 2013). In this picture, the dispersion relations of normal-mode waves serve as the stations of wave-wave coupling, and fluctuations are anticipated to live longer when excited as normal-mode waves. To construct a higher-order picture of wave evolution into turbulence, dispersion relations and wave damping need to be taken into account for wave-wave interactions and wave-particle interactions.
A potential application of our discoveries to spacecraft observations is the a remote sensing of wave excitation regions. If the energy partition and frequency broadening are evaluated from observational data, the time elapsed from pump wave excitation can be constrained. The transition time found in our numerically generated waves is of the order of 1000 ion gyroperiods. This time scale corresponds roughly to the convection spatial scale of 400,000 km or 63 R_{e} (in units of Earth radii) in interplanetary space (e.g., solar wind and foreshock) when using the typical values Ω _{p}∼1 rad/s for the gyrofrequency and V∼400 km/s for the flow speed. This spatial scale of approximately 63 R_{e} is by far larger than that of the foreshock or magnetosheath regions, but it may be of significance in the solar wind observation.
Declarations
Acknowledgements
We thank Daniel Verscharen for his assistance in the simulation setup. This work was financially supported by Collaborative Research Center 963, Astrophysical Flow, Instabilities, and Turbulence of the German Science Foundation and FP7-313038/STORM of European Commission. We also acknowledge the North-German Supercomputing Alliance (Norddeutscher Verbund zur Förderung des Hoch- und Höchstleistungsrechnens - HLRN) for supporting direct numerical simulations.
Authors’ Affiliations
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