Three-dimensional hybrid simulations have been carried out to verify the hypothesis of simultaneous multi-channel decay of
a large-amplitude Alfvén wave in a
low-beta plasma, e.g., in the shock-upstream region or the solar corona.
Obliquely propagating daughter modes
are excited along the perpendicular direction to the mean magnetic field at the same parallel wavenumbers and frequencies as the daughter modes driven by the field-aligned decay. We find that
the transversal spectrum of waves
is controlled by the multi-channel coupling of the decay process
in low-beta plasmas and
originates in the dispersion state of the shear Alfvén wave.
Introduction
Quasi-parallel-propagating Alfvén waves with circular polarization are
frequently observed in various space plasma domains,
in particular in the vicinity of the Earth's bow shock
or other planetary bow shocks see, e.g.,, where the shock-reflected ions transfer energy from the ion beam into electromagnetic waves
through the ion beam instability (often right-hand resonance
in which the gyration of the beam ions is observed
as right-hand polarized in the rest frame of the core or thermal plasma).
The beam-excited waves attain large amplitudes up to
the saturation level of the beam instability and develop
into a stage of wave–wave interactions see, e.g.,.
Parametric instability is thought to play an important role
in distributing the fluctuation energy of
large-amplitude Alfvén waves into different modes
and different wavelengths see, e.g.,.
The decay instability is a type of parametric instability
and is expected in both low- and high-beta plasmas see, e.g.,.
We point out that the decay instability in three-dimensional nature
has multiple channels at once and can excite
daughter waves in both the Alfvén mode and
sound (or ion acoustic) mode
without further cascade of the
daughter waves or before the cascade occurs
and the wave develops into turbulence.
The reason for the multi-channel coupling
lies in the fact that the decay instability
in three-dimensional low-beta plasmas occurs simultaneously
in various directions
imposed by the mean magnetic field
and
has a degeneration over the directions
perpendicular to the mean field.
In the analytical dispersion analysis, one typically
sets a priori the propagation direction of the daughter wave
and studies the growth rate of the decay instability
in the wavenumber domain see, e.g.,.
The situation is different
in a higher-dimensional setup; that is,
the daughter waves can be excited
in various directions at once
in the spirit of Boltzmann's equal probability
over all the possible channels.
The degeneration of the decay instability
over the perpendicular directions is illustrated
in Fig. .
Wave–wave coupling in the decay instability
forms a parallelogram
in the frequency–wavenumber domain (parallel
to the mean magnetic field) such that
both the energy quantum ℏω
and the momentum quantum ℏk are
conserved during the wave decay.
The pump Alfvén wave (propagating parallel
to the mean field, denoted by A1, the first-order
magnetic field after the zero-th order
mean field), decays into
a backward or anti-parallel-propagating
Alfvén mode (A2, the second-order field)
and a forward-propagating sound wave S2.
Wave decay in the parallel direction has a degeneracy
over the perpendicular directions. The degeneracy
can be resolved by plotting parallelogram-type
wave couplings in the wave–vector domain spanning
the parallel component k‖ and the perpendicular component k⟂.
That is, the momentum quantum is conserved
in the vectorial sense, ℏk=const.
One may write the individual realization of the
wave decay as “0” for the exactly parallel decay,
“+” and “-” for a slightly oblique decay,
“++” and “--” for an even more oblique decay, and so on.
Three-wave couplings of the decay instability
(parallelogram formation)
in the frequency–wavenumber domain parallel to
the mean magnetic field (a)
and in the wave–vector domain spanning parallel and perpendicular
directions (b).
Earlier studies on the nonlinear interaction between Alfvén waves
and obliquely propagating waves
see, e.g., report that the growth rates of the decay instability
in the oblique direction to the mean magnetic field
are typically smaller than the case of field-aligned parametric instabilities.
Numerical simulations such as multi-dimensional MHD (magnetohydrodynamics)
and hybrid plasma simulations
lead to the conclusion with respect to the circular polarization
that the parametric decay of large-amplitude Alfvén waves
develops particularly along
the propagation direction of the pump wave
see, e.g.,.
The nature of Alfvén wave decay into oblique-propagation angles
has been studied extensively.
, for example,
argue in their analytical dispersion analysis with circularly polarized
Alfvén waves
that the oblique-propagation decay instability can compete
against the field-aligned decay in the limit of small propagation or decay
angles to the mean magnetic field in the low-beta plasma.
confirm in their MHD two-dimensional simulations
that obliquely propagating daughter waves are excited by the
decay of a field-aligned Alfvén wave for the low-beta regime.
The results by are based on
a β value of 0.5, but nevertheless they demonstrate
that the field-aligned decay instability is persistent
and dominates the oblique-decay instabilities.
discovered by two-dimensional hybrid simulations
that linearly polarized Alfvén waves with an oblique direction of propagation
decay in low-beta plasmas in a broad spectrum of coupled Alfvén waves
and density fluctuations perpendicular to the direction of the mean magnetic field.
also noticed that the magnetic daughter
modes follow the dispersion relation for a
shear Alfvén wave,
ω=k∥VA, where ω, k∥,
and VA are the frequency, parallel wavenumber, and Alfvén velocity, respectively.
Here we present a scenario that decay instability can occur simultaneously
at various angles to the mean magnetic field, generating
a number of second-order fluctuations or waves
(after the pump wave as the first-order fluctuation).
We refer to the simultaneous decay as “multi-channel couplings”
following the notion in scattering theory, for example, coupled channels in nuclear reactions see, e.g.,.
Our goal is to study the hypothesis or the scenario of the
multi-channel coupling by running a three-dimensional hybrid plasma simulation
in a low-beta plasma setup.
Hybrid plasma simulationSimulation run
A hybrid plasma simulation is carried out to verify the hypothesis of the multi-channel coupling
in the low-beta decay instability. We use the AIKEF hybrid code
in a three-dimensional spatial configuration.
The size of the simulation box in each direction is L=288di, and
(576)3 computational cells are set, containing 1000 super-particles in each cell.
The length scale is normalized to the ion inertia length di=VA/Ωp,
where VA and Ωp are the Alfvén velocity and the ion gyro-frequency (for protons), respectively.
The values of ion and electron betas used in the simulation are βi=βe=0.01.
This low value of beta (the equivalent fluid beta in the simulation is
β=0.02) ensures a faster linear growth and an earlier nonlinear
saturation of the parametric instabilities.
The magnetic-field amplitude of the Alfvén pump wave (normalized to the value of the background magnetic field) has a value of 0.2. The simulation is halted
at a time of tΩp=600, shortly after the saturation of the decay instability.
Figure displays the time evolution of the
fluctuation energy for the field-aligned pump wave (Alfvén mode A1)
which represents the Fourier mode (m‖,m⟂)=(10,0),
(m∥(⟂)=k∥(⟂)L/2π),
and the daughters A2(0),A2(-),A2(--),A3(---) and
S2(0), S2(-), S2(--), S3(---), respectively,
in the parallel and perpendicular wavenumber domains.
Counter-propagating Alfvén daughter
modes (to the pump wave propagation direction) and sound daughter modes
start to develop simultaneously, and the fluctuation energy increases
both along the mean magnetic field and in the oblique directions
to the mean field. The mode number for the parallel-propagating Alfvén
lower sideband mode is m∥=-9, while the corresponding sound mode
has a mode number of m∥=19, satisfying the three-wave coupling rule.
An exponential growth of the fluctuation energy
represents the linear stage of the decay instability.
Both parallel (or field-aligned) propagating waves and obliquely propagating
waves grow simultaneously.
The slope of the wave growth is nearly the same between
the magnetic mode and the sound mode at lower propagation angles
to the mean magnetic field (typically up to 40∘).
The decay instability then becomes saturated at a time of tΩp=300.
It is interesting to note that the pump Alfvén wave still dominates
the fluctuation energy in the system.
The low level of the amplitude of the magnetic field and density daughter modes in Fig. 2 could
be a consequence of the lower level of the fluctuation background developed in the three-dimensional system
with respect to that one developed in a two-dimensional system with equivalent characteristics.
The obliquely propagating modes are strong enough and compete against
the parallel-propagating modes in the linear stage.
At later times (tΩp>300) the fluctuation energy decreases
at larger propagation angles to the mean field.
Time evolution of the magnetic field (a) and
density (b) fluctuations: Alfvén field-aligned pump A1,
Alfvén daughter A2 at selected perpendicular Fourier modes,
and corresponding longitudinal daughter modes S2 provided by the three-dimensional hybrid simulation. Symbols of “A2(0)”, “A2(-)”, “A2(--)”, “A2(---)”, and “S2(0)”, “S2(-)”, “S2(--)”, “S2(---)” denote the mode m numbers (-9; 0, 3, 8, 12) and (19; 0, 3, 8, 12), respectively. The propagation angles θ of the density
daughter modes have values close to 10∘ (dotted),
20∘ (dashed), and 30∘ (dashed dotted).
Multi-channel coupling in the decay instability is identified
in the spectral domain over the perpendicular wavenumbers.
The energy spectra for the magnetic field fluctuations, δB2(ω,k), and that of the density, δρ2(ω,k),
are plotted as slices in the frequency–wavenumber domain parallel to the mean magnetic field (Fig. a) and in the wave–vector domain spanning the perpendicular component k⟂ and the parallel component k∥ (Fig. b).
The spectral analysis is performed after the saturation of the instability at
the latest time tΩp≈600 of the simulation.
The pump wave is in the Alfvén mode
and propagates along the mean magnetic field (k⟂=0).
Daughter Alfvén waves and daughter sound waves appear in the wavenumber–frequency spectrum
at k∥VA/Ωp∼-0.2 and k∥VA/Ωp∼0.4, respectively.
The magnetic field and density fluctuations in
Fig. (bottom panel) are analyzed at
the frequency of the field-aligned daughter modes
and differ in the perpendicular wavenumbers (or in the propagation angles).
The obliquely propagating Alfvén modes share the same frequency as that of the field-aligned Alfvén daughter wave. These fluctuations along the perpendicular direction to the mean magnetic field are
daughter waves in the Alfvén mode resulting from the multi-channel coupling of the decay process. The daughter sound modes with oblique propagation have the same frequency as the parallel-propagating daughter mode
(ω∼k∥vAβ≈0.05Ωp)
and thus fulfill the three-wave coupling conservation law. One can mention that in the limit of low-beta plasmas (β≪1), these oblique compressive waves are still normal modes of plasma.
Spectrum for the magnetic field δB2(ω,k)
and density ρ2(ω,k) fluctuations at the latest time of the simulation (tΩp≈600). Slices at wavenumber
k⟂=0(a) display the Alfvén
pump wave A1 (k∥VA/Ωp∼0.2), Alfvén A2(0) (k∥VA/Ωp∼-0.2), and
ion acoustic S2(0) (k∥VA/Ωp∼0.4)
daughter modes in the frequency–wavenumber domain parallel to the mean magnetic field.
Slices at the frequency of the Alfvén A2(0) and density S2(0) field-aligned modes are drawn in the perpendicular and parallel wavenumber domain in panel (b). The vertical extension
on the left (k∥VA/Ωp∼-0.2) is for the Alfvén mode
and that on the right (k∥VA/Ωp∼0.4) is for the sound
mode.
Growth rate estimate
Growth rates for the individual decay instability
can be computed from the hybrid simulation
and are compared to that of the analytical dispersion analysis of
the two-fluid model developed by .
Briefly, the instability analysis of consists in the following.
By using the bi-fluid magnetohydrodynamic model, a linear
perturbation of linear-mode waves is applied to study the stability
of the large-amplitude circularly polarized Alfvén wave.
Sideband electromagnetic waves are driven at upper and lower sideband frequencies obeying resonant wave–wave interaction rules. A set of linear equations in the perturbed quantities is obtained, satisfying the resonance condition, while higher sidebands are neglected.
We consider the oblique modes at non-zero propagation angles
to the mean magnetic field.
The determinant of the matrix system is
on order 78 in terms of the frequency normalized
by using the Alfvén speed VA
and the ion gyro-frequency Ωp for protons,
ω^=ω/(k0VA)=ωr^+iγ^.
The elements of six-by-six order matrices can be found in the Appendix in .
The dispersion relation ω(k) and the growth rate γ(k)
are obtained by solving the determinant of the matrix.
We choose a low-beta value of 0.02 in the semi-analytical calculation.
The solutions are numerically solved by using the symbolic calculator (Mathematica).
We denote that the pump wave has the wavenumber k0 and the frequency ω0,
the longitudinal daughter wave k and ω,
and the Alfvén (sideband) waves k± and ω±.
We assume that all the participating waves (pump wave, longitudinal daughter wave,
and sideband wave) satisfy the coupling rules, that is,
1k±=k0±k2ω±=ω0±ω.
Figure displays the growth rate γ^
as a function of the wavenumber for the parameters
used in the hybrid simulation.
Seven values of the propagation angle are chosen for the comparison:
θ={0,10,20,30,45,60,65∘}.
The maximum growth rate falls at k/k0∼1.9
for the field-aligned decay (θ=0∘).
The solutions of the dispersion relation indicate that
the growth rates slightly decrease at larger propagation angles.
The result of the hybrid simulation is overall consistent with
the analytical dispersion analysis. The observed growth rates are overplotted
in Fig. and fit
reasonably well with the analytical prediction. At larger oblique angles, the prediction has larger
deviations with respect to the simulation results.
Kinetic effects are most probably responsible for such
discrepancies between the MHD model and the hybrid simulation.
The field-aligned decay weakly prevails, but the low and moderate
oblique decay instabilities have growth rates close to the former one,
confirming the increasing trend with beta decreasing
see, e.g.,.
Therefore, the multi-channel coupling decay instability is
regarded as a logical extension of the decay instability theory
developed by to simultaneous, independent,
multiple decay channels.
Growth rates of decay instability in the wavenumber domain of the daughter
modes for propagation angles of 0∘ (solid), 10∘ (dotted),
20∘ (dashed),
and 30∘ (dashed dotted) estimated by the analytical dispersion analysis (a).
Overplotted by diamond, triangle, square, and cross symbols are the growth rates obtained in the three-dimensional hybrid simulation for the above propagation angles, respectively.
Panel (b) shows the same comparison at larger propagation angles,
45, 60, and 65∘.
Discussion and conclusions
The damping of the oblique daughter waves observed in Fig. 2 can be explained
in terms of the wave-particle processes (e.g., pitch-angle scattering, cyclotron
resonance, or Landau damping) in competition with the plasma turbulence evolution.
We have computed the time evolution of the spectrum of the magnetic field and density fluctuations. In the late stages of the fluctuation evolution,
the one-dimensional reduced spectra (not shown) along the perpendicular wavenumber
reveal turbulence cascades (by the fact that a power-law spectrum is being formed) starting from maxima at k⟂=0 specifically for
low-beta plasma turbulence.
The spectral slope along the parallel wavenumber is flatter in the low wavenumber range and becomes steeper at larger values beyond the pump wavenumber. This spectral anisotropy at low and moderate wavenumbers favors the parallel-propagating modes which remain persistent at the latest stage of the decay process.
The oblique electromagnetic daughter waves
in Fig. 3 are followed by corresponding fluctuations in the
wavenumber–frequency density spectrum (not shown).
in their analytical study found that the coupling between
the electrostatic and electromagnetic solutions increases with the increase in the propagation angle of the daughter wave.
On the basis of this remark, we naively interpret this result by interchanging
the density daughter modes with the sideband electromagnetic waves.
At large oblique propagation angles, the density daughter modes are electromagnetic
waves with fluctuation in both density and electromagnetic fields, while
the sideband waves become electrostatic waves. Thus, the three-wave coupling rule is fulfilled. The obliquely propagating modes develop into a turbulent cascade
which saturates the instability at a stage when the Alfvén pump wave still dominates the
spectrum.
A modulation-like process is observed at early times close to the
pump wavenumber and saturates before the saturation of the decay instability.
The modulation is exclusively oblique and can be reminiscent of the beat instability.
This additional parametric instability and the consequences for the decay instability and its early saturation are still under study.
first noticed the occurrence of
a perpendicular spectrum of daughter waves developed at the
decay of a linearly polarized Alfvén pump wave with oblique direction of propagation to the background magnetic field. The authors
concluded that the generation of obliquely propagating daughter waves is a consequence of the obliqueness of the Alfvén pump wave, i.e., its finite perpendicular wavenumber, in the framework of the three-wave coupling process.
Our three-dimensional treatment is simplified with respect to that former study in the sense that the pump wave is circularly polarized and propagates parallel to the mean magnetic field.
Thus we can make useful the predictions of the MHD analytical analysis
for the dispersion relations and the growth rates of the decay instability at different propagation angles.
The dispersion relation for left-handed circular polarization,
(kVA/Ωp)2=(ω/Ωp)2/(1-ω/Ωp),
is satisfied for both the Alfvén pump wave (as an imposed initial condition) and
Alfvén daughter modes. The value of the pump wavenumber, k0VA/Ωp=0.21, is fairly high with respect to typical MHD scales, but still it can be assumed to be in the dispersionless range on the first order.
The analytical dispersion analysis at weak and moderate oblique propagation angles provides the largest growth rates at approximately the same parallel wavenumber as that one predicted for the field-aligned decay.
This result can be easily understood by introducing the three-wave coupling equations and the dispersion relations:
3k∥-=k0-k∥,4k⟂-=-k⟂,5ω-=ω0-ω,
where (k0,ω0=k0VA), (k∥-,k⟂-,ω-=-k∥-VA), and (k∥,k⟂,ω)
are the wavenumbers, frequencies, and dispersion relations characterizing the Alfvén pump, Alfvén daughter, and ion acoustic waves, respectively.
The MHD slow mode still preserves the field-aligned dispersion relation at small and moderate propagation angles in the limit of low-beta plasmas, namely,
ωslow=k∥VAβ (here, β=cs2/VA2).
Therefore, the sound daughter modes with oblique propagation are normal modes of plasma like their counterpart oblique Alfvén modes, and they fulfill similar dispersion relations, i.e.,
ω=k∥cs.
Equations (3)–(5), the dispersion relations, and some elementary algebra
provide the parallel wavenumber of the oblique ion acoustic wave,
k∥=2k01+cs/VA≡k(0),
where k0 is the wavenumber of the sound daughter driven by the field-aligned decay see, e.g.,.
The overall decay process is not controlled by the field-aligned decay but by the dispersion relation of the participating waves which drives the oblique decay to share
identical parallel wavenumbers to those attained in the parallel decay.
The above rough evaluation
is validated by both the analytical analysis in the MHD framework and
by the hybrid simulation applied in the present study.
In the analytical analysis, the evolution of the daughter waves is determined by the interaction with the field-aligned pump wave by constructing a system of quasi-linear equations which exhibits wave–wave couplings generating obliquely propagating waves as the daughter component. The growth rate of the oblique modes generally has a maximum value at the
parallel wavenumber prescribed by the maximum growth rate of the field-aligned decay instability and depends on the propagation angle.
However, we still point out that the field-aligned
decay remains in any case the fastest one, as shown in Fig. 4.
Also, this dynamics is different from the case of an oblique pump wave propagating at
a given theta angle with respect to B0, where the oblique mode's decay rate gamma is
found to scale cosθkB, so controlled by the k|| projection of the initial oblique
wave vector .
This suggests then that there are two possible ways of generating oblique modes from
the parametric decay: (1) from a purely parallel mother wave, as in this study; and (2) from
an oblique pump wave see, e.g.,.
In both cases the oblique
modes grow at a rate that is smaller than the parallel decay.
In our study the central role is played by the conservation of frequencies and the dispersion relations of the
components involved in the three-wave coupling. show that in their analysis, the central role in driving the transversal spectrum is played by the conservation of the momentum and the non-zero perpendicular projection of the pump wave vector.
An obliquely propagating (pump) wave is expected to be more compressive, and it can generate a broadband spectrum of compressive fluctuations.
Thus, the configuration with an oblique pump wave should be more
efficient in driving oblique modes, even though the daughter waves in our study are driven earlier after a short time of nonlinear evolution. On the other hand, one should not ignore the major differences in the involved setups, e.g., the different polarization of the pump wave and different βe values (see discussion below).
Besides the plasma beta, the wavenumber of the pump wave may play a significant role in the process of driving the transverse broadband modulation.
In both studies, the medium is still
weakly dispersive (k0VA/Ωp∼0.2). In a more dispersive medium such as that one used by , the dispersion relations probably cannot construct a transversal spectrum of waves for the decay of a field-aligned Alfvén wave.
Unlike MHD simulations, where the instability can
saturate only through the steepening of the excited sound waves, in hybrid simulations it saturates
via particle trapping and phase-space modulation see, e.g.,. A
consequence of this dynamics is a significant perturbation of the ion velocity distribution function,
leading also to the generation of field-aligned beams see, e.g.,.
In low-beta plasmas, the decay instability dominates over the other parametric instabilities.
Plasma beta values for both ions and electrons also play an important role in the dynamics of the parametric decay. For instance, βe is responsible for activating different saturation mechanisms.
reported that for cold fluid electrons (βe∼0), the MHD saturation mechanism is recovered. At larger values (e.g., βe∼0.1), the trapping and beam formation is in use.
The electron plasma beta βe in our simulation has an intermediate value of 0.01.
The parallel distribution in the phase space z-v∥ (not shown) at a time still close to the linear phase of the instability (tΩp=300) suggests that particles are confined and accelerated in different regions along the parallel axis to the mean magnetic field. The phase-space modulation can be a signature for the instability saturation. There is no evidence of a field-aligned velocity beam at the latest time of the simulation. The accelerated particles are thermalized via pitch-angle diffusion by the
developed oblique modes.
A consequence of the oblique parametric decay in three-dimensional plasmas is a more efficient heating of the ions; see, e.g., the pitch angle scattering study by . Their conclusion
based on a hybrid simulation by using similar parameters to the actual run was that the plateau levels observed in the proton velocity distribution functions are driven by the pitch-angle diffusion of the ions by obliquely propagating modes.
The spectral analysis performed in Fig. 3 in the frequency–wave-vector domain over a broad range of oblique propagation angles proves
that the multi-channel decay of the parallel-propagating Alfvén pump wave is the source
of obliquely propagating daughter waves assumed to satisfy the resonance condition with ions. The former and actual results reconfirm in particular the role of oblique compressive daughter waves in heating the protons parallel and perpendicular to the mean magnetic field according to the in situ observation in solar wind plasmas
see, e.g.,.
We have studied by three-dimensional hybrid simulations the decay of a
large-amplitude Alfvén pump wave
expected to achieve the largest
growth rate for parallel propagation and circular
polarization.
Our conclusions are drawn below.
The parallel-propagating Alfvén wave decays into a transversal spectrum
of daughter waves. The oblique decay is controlled by the growth rates
of the decay instability which are significantly larger at small and moderate propagation angles in low-beta plasmas, in agreement with the analytic analysis.
The transversal spectrum of daughter waves is controlled by the dispersion relation under the constraint of the three-wave coupling process. Thus, the perpendicular alignment to the mean magnetic field
originates in the dispersion state of the genuine circularly polarized Alfvén wave.
Kinetic effects are assumed to become more important at larger propagation angles. Their role correlated with the value of the plasma β parameter will be the subject of a forthcoming study on the decay instability in three-dimensional plasmas.
Data availability
Data from our hybrid simulations are stored at the
Institut fuer Theoretische Physik – Technische Universitaet Braunschweig. Data can be obtained by writing to the following email addresses: h.comisel@tu-braunschweig.de or comisel@spacescience.ro.
Author contributions
HC worked on theory development, simulations, and manuscript writing.
YN worked on discussion and manuscript writing. UM worked on theory development, discussion, and coordination of the simulation study.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
Horia Comişel acknowledges Yasuhiro Nariyuki for fruitful discussions and the hospitality at the University of Toyama.
We also acknowledge support by the German Research Foundation and the Open Access Publication Funds of the Technical University of Braunschweig.
The work conducted by Horia Comişel in Bucharest is supported by
ESA project MAGICS, PRODEX contract C4000127660.
The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. for funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS at the Jülich Supercomputing Centre (JSC). Furthermore, we are grateful to the anonymous referees for their valuable suggestions and evaluation of this paper.
Financial support
This research has been supported by the DFG (grant no. MO539/20-1).This open-access publication was funded by Technische Universität Braunschweig.
Review statement
This paper was edited by Vincent Maget and reviewed by two anonymous referees.
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