Nonlinear Landau Damping of Alfvén Waves in a High β Plasma

1982 ◽  
Vol 37 (8) ◽  
pp. 809-815 ◽  
Author(s):  
Heinrich J. Völk ◽  
Catherine J. Cesarsky
Keyword(s):  
Wave Length ◽  
Wave Spectrum ◽  
Nonlinear Damping ◽  
Alfven Waves ◽  
Alfvén Waves ◽  
Propagating Waves ◽  
Damping Rate ◽  
Nonlinear Landau Damping ◽  
The Waves ◽  
The Magnetic Field

A study is made of the nonlinear damping of parallel propagating Alfvén waves in a high β plasma. Two circularly polarized parallel propagating waves give rise to a beat wave, which in general contains both a longitudinal electric field component and a longitudinal gradient in the magnetic field strength. The wave damping is due to the interactions of thermal particles with these fields. If the amplitudes of the waves are low, a given wave (ω1, k1) is damped by the presence of all longer wavelength waves; thus, if the amplitudes of the waves in the wave spectrum increase with wave length, the effect of the longest waves is dominant.However, when the amplitude of the waves is sufficiently high, the particles are trapped in the wave packets, and the damping rate may be considerably reduced. We calculate the induced electrostatic field, and examine the trapping of thermal particles in a pair of waves. Finally, we give examples of modified damping rates of a wave in the presence of a spectrum of waves, and show that, when the trapping is effective, the waves are mostly damped by their interactions with waves of comparable wavelengths

scholarly journals Investigating the damping rate of phase-mixed Alfvén waves

2019 ◽  
Vol 632 ◽  
pp. A93 ◽  
Author(s):  
A. P. K. Prokopyszyn ◽  
A. W. Hood
Keyword(s):  
Wave Energy ◽  
Alfven Waves ◽  
Alfvén Waves ◽  
Coronal Loops ◽  
Phase Mixing ◽  
Heating Mechanism ◽  
At Resonance ◽  
Damping Rate ◽  
Field Lines ◽  
The Waves

Context. This paper investigates the effectiveness of phase mixing as a coronal heating mechanism. A key quantity is the wave damping rate, γ, defined as the ratio of the heating rate to the wave energy. Aims. We investigate whether or not laminar phase-mixed Alfvén waves can have a large enough value of γ to heat the corona. We also investigate the degree to which the γ of standing Alfvén waves which have reached steady-state can be approximated with a relatively simple equation. Further foci of this study are the cause of the reduction of γ in response to leakage of waves out of a loop, the quantity of this reduction, and how increasing the number of excited harmonics affects γ. Methods. We calculated an upper bound for γ and compared this with the γ required to heat the corona. Analytic results were verified numerically. Results. We find that at observed frequencies γ is too small to heat the corona by approximately three orders of magnitude. Therefore, we believe that laminar phase mixing is not a viable stand-alone heating mechanism for coronal loops. To arrive at this conclusion, several assumptions were made. The assumptions are discussed in Sect. 2. A key assumption is that we model the waves as strictly laminar. We show that γ is largest at resonance. Equation (37) provides a good estimate for the damping rate (within approximately 10% accuracy) for resonant field lines. However, away from resonance, the equation provides a poor estimate, predicting γ to be orders of magnitude too large. We find that leakage acts to reduce γ but plays a negligible role if γ is of the order required to heat the corona. If the wave energy follows a power spectrum with slope −5/3 then γ grows logarithmically with the number of excited harmonics. If the number of excited harmonics is increased by much more than 100, then the heating is mainly caused by gradients that are parallel to the field rather than perpendicular to it. Therefore, in this case, the system is not heated mainly by phase mixing.

scholarly journals On viscous and resistive dissipation of Alfvén waves in an isothermal atmosphere

1997 ◽  
Vol 20 (3) ◽  
pp. 605-610
Author(s):  
Hadi Yahya Alkahby
Keyword(s):  
Transition Layer ◽  
Kinematic Viscosity ◽  
Magnetic Energy ◽  
Alfven Waves ◽  
Alfvén Waves ◽  
Lower Region ◽  
Propagating Waves ◽  
The Waves ◽  
Isothermal Atmosphere ◽  
Viscosity Changes

In this article we will investigate reflection and dissipation of Alfvén waves, resulting from a uniform vertical magnetic field, in a viscous, resistive and isothermal atmosphere. It is shown that the atmosphere may be divided into two distinct regions connected by an absorbing and reflecting transition layer. In the transition layer the reflection, dissipation and absorption of the magnetic energy of the waves take place and in it the kinematic viscosity changes from small to large values. In the lower region the effect of the resistive diffusivity and kinematic viscosity changes from small to large values. In the lower region the effect of the resistive diffusivity and kinematic viscosity is negligible and in it the solution can be represented as a linear combination of two, incident and reflected, propagating waves with different wavelengths and different dissipative factors. In the upper region the effect of the resistive diffusivity and kinematic viscosity is large and the solution, which satisfies the prescribed boundary conditions, will behave as a constant. The reflection coefficient, the dissipative factors are determined and the conclusions are discussed in connection with solar heating.

Nonlinear Landau damping of purely perpendicular Bernstein modes

1986 ◽  
Vol 36 (1) ◽  
pp. 111-125 ◽  
Author(s):  
Spilios Riyopoulos
Keyword(s):  
Cyclotron Frequency ◽  
Nonlinear Damping ◽  
Landau Damping ◽  
Linear Dispersion ◽  
Linear Dispersion Relation ◽  
Small Wave ◽  
Damping Rate ◽  
Vlasov’S Equation ◽  
Nonlinear Landau Damping ◽  
The Magnetic Field

The linear dispersion relation for Bernstein modes, obtained by the integration of Vlasov's equation along the unperturbed cyclotron orbits, predicts that the modes propagating perpendicularly to the magnetic field are undamped. However, when the frequency is close to a multiple of the cyclotron frequency, most of the particles become trapped for small wave amplitude and the unperturbed orbit approximation breaks down. The trapped particle trajectories are calculated analytically here using a resonant Hamiltonian approximation. Integration, consistent with the wave, along the orbits yields the nonlinear damping rate in a manner similar to that used by O'Neil for the damping of unmagnetized electrostatic modes. The results can be extended for the general case of almost perpendicular, short-wavelength electrostatic modes near cyclotron harmonics.

scholarly journals On resistive dissipation of Alfvén waves in an isothermal atmosphere

1996 ◽  
Vol 19 (3) ◽  
pp. 587-594
Author(s):  
H. Y. Alkahby
Keyword(s):  
Transition Region ◽  
Magnetic Energy ◽  
Alfven Waves ◽  
Scale Height ◽  
Wave Number ◽  
Alfvén Waves ◽  
Propagating Waves ◽  
The Waves ◽  
Isothermal Atmosphere ◽  
Density Scale

In this paper we will examine the reflection and dissipation of Alfvén waves, resulting from a uniform vertical magnetic field, in an inviscid, resistive and isothermal atmosphere. An equation for the damping length distance that wave can travel at Alfvén speed is derived. This equation shows that the damping length is proportional to the wave number and the density scale height and it is valid not only for Alfvén waves but also for any wave that travels at Alfvén speed. Moreover, it is shown that the atmosphere may be divided into two distinct regions connected by an absorbing and reflecting transition region. In the lower region the solution can be represented as a linear combination of two, incident and reflected, propagating waves with the same wavelengths and the same dissipative factors. In the upper region the effect of the resistive diffusivity and Alfvén speed is large and the solution, which satisfies the prescribed boundary conditions, either decays with altitude or behaves as a constant. In the transition region the reflection, dissipation and absorption of the magnetic energy of the waves take place. The reflection coefficient, the dissipative factors, which are proportional to the damping length, are determined and the conclusions are discussed in connection with heating of the solar atmosphere.

scholarly journals Nonlinear Alfvén waves, discontinuities, proton perpendicular acceleration, and magnetic holes/decreases in interplanetary space and the magnetosphere: intermediate shocks?

2005 ◽  
Vol 12 (3) ◽  
pp. 321-336 ◽  
Author(s):  
B. T. Tsurutani ◽  
G. S. Lakhina ◽  
J. S. Pickett ◽  
F. L. Guarnieri ◽  
N. Lin ◽  
...  
Keyword(s):  
Interplanetary Space ◽  
Alfven Waves ◽  
Alfven Wave ◽  
Alfvén Waves ◽  
Reverse Shock ◽  
Propagating Waves ◽  
Mirror Mode ◽  
Interaction Regions ◽  
Electron Holes ◽  
Proton Cyclotron

Abstract. Alfvén waves, discontinuities, proton perpendicular acceleration and magnetic decreases (MDs) in interplanetary space are shown to be interrelated. Discontinuities are the phase-steepened edges of Alfvén waves. Magnetic decreases are caused by a diamagnetic effect from perpendicularly accelerated (to the magnetic field) protons. The ion acceleration is associated with the dissipation of phase-steepened Alfvén waves, presumably through the Ponderomotive Force. Proton perpendicular heating, through instabilities, lead to the generation of both proton cyclotron waves and mirror mode structures. Electromagnetic and electrostatic electron waves are detected as well. The Alfvén waves are thus found to be both dispersive and dissipative, conditions indicting that they may be intermediate shocks. The resultant "turbulence" created by the Alfvén wave dissipation is quite complex. There are both propagating (waves) and nonpropagating (mirror mode structures and MDs) byproducts. Arguments are presented to indicate that similar processes associated with Alfvén waves are occurring in the magnetosphere. In the magnetosphere, the "turbulence" is even further complicated by the damping of obliquely propagating proton cyclotron waves and the formation of electron holes, a form of solitary waves. Interplanetary Alfvén waves are shown to rapidly phase-steepen at a distance of 1AU from the Sun. A steepening rate of ~35 times per wavelength is indicated by Cluster-ACE measurements. Interplanetary (reverse) shock compression of Alfvén waves is noted to cause the rapid formation of MDs on the sunward side of corotating interaction regions (CIRs). Although much has been learned about the Alfvén wave phase-steepening processfrom space plasma observations, many facets are still not understood. Several of these topics are discussed for the interested researcher. Computer simulations and theoretical developments will be particularly useful in making further progress in this exciting new area.

scholarly journals Wave particle interactions in the high-altitude polar cusp: a Cluster case study

2005 ◽  
Vol 23 (12) ◽  
pp. 3699-3713 ◽  
Author(s):  
B. Grison ◽  
F. Sahraoui ◽  
B. Lavraud ◽  
T. Chust ◽  
N. Cornilleau-Wehrlin ◽  
...  
Keyword(s):  
Electromagnetic Wave ◽  
High Altitude ◽  
Wave Spectrum ◽  
Alfven Waves ◽  
Wave Activity ◽  
Alfven Wave ◽  
Alfvén Waves ◽  
Plasma Convection ◽  
Field Lines ◽  
Kinetic Alfvén Waves

Abstract. On 23 March 2002, the four Cluster spacecraft crossed in close configuration (~100 km separation) the high-altitude (10 RE) cusp region. During a large part of the crossing, the STAFF and EFW instruments have detected strong electromagnetic wave activity at low frequencies, especially when intense field-aligned proton fluxes were detected by the CIS/HIA instrument. In all likelihood, such fluxes correspond to newly-reconnected field lines. A focus on one of these ion injection periods highlights the interaction between waves and protons. The wave activity has been investigated using the k-filtering technique. Experimental dispersion relations have been built in the plasma frame for the two most energetic wave modes. Results show that kinetic Alfvén waves dominate the electromagnetic wave spectrum up to 1 Hz (in the spacecraft frame). Above 0.8 Hz, intense Bernstein waves are also observed. The close simultaneity observed between the wave and particle events is discussed as an evidence for local wave generation. A mechanism based on current instabilities is consistent with the observations of the kinetic Alfvén waves. A weak ion heating along the recently-opened field lines is also suggested from the examination of the ion distribution functions. During an injection event, a large plasma convection motion, indicative of a reconnection site location, is shown to be consistent with the velocity perturbation induced by the large-scale Alfvén wave simultaneously detected.

Concerning the Dynamics of Energetic Protons in Coronal Magnetic Loops: Dispersion Effects of Alfven Waves

1985 ◽  
Vol 107 ◽  
pp. 559-559
Author(s):  
V. A. Mazur ◽  
A. V. Stepanov
Keyword(s):  
Magnetic Field ◽  
Wave Dispersion ◽  
Alfven Waves ◽  
Alfven Wave ◽  
Wave Number ◽  
Alfvén Waves ◽  
Coronal Loops ◽  
Coronal Magnetic Loops ◽  
The Magnetic Field ◽  
Solar Coronal

It is shown that the existence of plasma density inhomogeneities (ducts) elongated along the magnetic field in coronal loops, and of Alfven wave dispersion, associated with the taking into account of gyrotropy U ≡ ω/ωi ≪ 1 (Leonovich et al., 1983), leads to the possibility of a quasi-longitudinal k⊥ < √U k‖ propagation (wave guiding) of Alfven waves. Here ω is the frequency of Alfven waves, ωi is the proton gyrofrequency, and k is the wave number. It is found that with the parameter ξ = ω2 R/ωi A > 1, where R is the inhomogeneity scale of a loop across the magnetic field, and A is the Alfven wave velocity, refraction of Alfven waves does not lead, as contrasted to Wentzel's inference (1976), to the waves going out of the regime of quasi-longitudinal propagation. As the result, the amplification of Alfven waves in solar coronal loops can be important. A study is made of the cyclotron instability of Alfven waves under solar coronal conditions.

scholarly journals Dynamics of Alfvén waves in the night-side ionospheric Alfvén resonator at mid-latitudes

2005 ◽  
Vol 23 (2) ◽  
pp. 499-507 ◽  
Author(s):  
V. V. Alpatov ◽  
M. G. Deminov ◽  
D. S. Faermark ◽  
I. A. Grebnev ◽  
M. J. Kosch
Keyword(s):  
Pulse Duration ◽  
Alfven Waves ◽  
Fundamental Period ◽  
Alfvén Waves ◽  
Shear Mode ◽  
Q Factor ◽  
Trapped Waves ◽  
Alfven Resonator ◽  
Ionospheric Alfven Resonator ◽  
The Waves

Abstract. A numerical solution of the problem on dynamics of shear-mode Alfvén waves in the ionospheric Alfvén resonator (IAR) region at middle latitudes at nighttime is presented for a case when a source emits a single pulse of duration τ into the resonator region. It is obtained that a part of the pulse energy is trapped by the IAR. As a result, there occur Alfvén waves trapped by the resonator which are being damped. It is established that the amplitude of the trapped waves depends essentially on the emitted pulse duration τ and it is maximum at τ=(3/4)T, where T is the IAR fundamental period. The maximum amplitude of these waves does not exceed 30% of the initial pulse even under optimum conditions. Relatively low efficiency of trapping the shear-mode Alfvén waves is caused by a difference between the optimum duration of the pulse and the fundamental period of the resonator. The period of oscillations of the trapped waves is approximately equal to T, irrespective of the pulse duration τ. The characteristic time of damping of the trapped waves τdec is proportional to T, therefore the resonator Q-factor for such waves is independent of T. For a periodic source the amplitude-frequency characteristic of the IAR has a local minimum at the frequency π/ω=(3/4)T, and the waves of such frequency do not accumulate energy in the resonator region. At the fundamental frequency ω=2π/T the amplitude of the waves coming from the periodic source can be amplified in the resonator region by more than 50%. This alone is a basic difference between efficiencies of pulse and periodic sources of Alfvén waves. Explicit dependences of the IAR characteristics (T, τdec, Q-factor and eigenfrequencies) on the altitudinal distribution of Alfvén velocity are presented which are analytical approximations of numerical results.

Unified theory of damping of linear surface Alfvén waves in inhomogeneous incompressible plasmas

1996 ◽  
Vol 56 (1) ◽  
pp. 107-125 ◽  
Author(s):  
M. S. Ruderman ◽  
M. Goossens
Keyword(s):  
Dispersion Relation ◽  
Resonant Absorption ◽  
Alfven Waves ◽  
Alfvén Waves ◽  
Dimensionless Number ◽  
Inhomogeneous Layer ◽  
Wide Range ◽  
Damping Rate ◽  
Nonmonotonic Variation ◽  
Incompressible Mhd

The viscous damping of surface Alfvén waves in a non-uniform plasma is studied in the context of linear and incompressible MHD. It is shown that damping due to resonant absorption and damping on a true discontinuity are two limiting cases of the continuous variation of the damping rate with respect to the dimensionless number Rg = Δλ2Re, where Δ is the relative variation of the local Alfvén velocity, λ is the ratio of the thickness of the inhomogeneous layer to the wavelength, and Re is the viscous Reynolds number. The analysis is restricted to waves with wavelengths that are long in comparison with the extent of the non-uniform layer (λ ≪ 1), and to Reynolds numbers that are sufficiently large that the waves are only slightly damped during one wave period. The dispersion relation is obtained and first investigated analytically for the limiting cases of very small (Rg ≪ 1) and very large (Rg ≫ 1) values of Rg, For very small values of Rg, the damping rate agrees with that found for a true discontinuity, while for very large values of Rg, it agrees with the damping rate due to resonant absorption. The dispersion relation is subsequently studied numerically over a wide range of values of Rg, revealing a continuous but nonmonotonic variation of the damping rate with respect to Rg.

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