Abstract
Alfvén waves are lowfrequency transverse waves propagating in a magnetized plasma. We define the Alfvén frequency as , where is the wave number, is the Alfvén speed, and is the angle between the wave vector and the ambient magnetic field. There are partially ionized plasmas in laboratory, space, and astrophysical plasma systems, such as in the solar chromosphere, interstellar clouds, and the earth ionosphere. The presence of neutral particles may modify the wave frequency and cause damping of Alfvén waves. The effects on Alfvén waves depend on two parameters: (1) , the ratio of neutral density , and ion density ; (2) , the ratio of neutral collisional frequency by ions to the Alfvén frequency . Most of the previous studies examined only the limiting case with a relatively large neutral collisional frequency or . In the present paper, the dispersion relation for Alfvén waves is solved for all values of and . Approximate solutions in the limit as well as are obtained. It is found for the first time that there is a “forbidden zone (FZ)” in the parameter space, where the real frequency of Alfvén waves becomes zero. We also solve the wavenumber k from the dispersion equation for a fixed frequency and find the existence of a “heavy damping zone (HDZ).” We then examine the presence of FZ and HDZ for Alfvén waves in the ionosphere and in the solar chromosphere.
This work was partially supported by a grant from National Cheng Kung University, grants from the National Science Council to Academia Sinica (NSC 1012628M001007MY3) and to the National Central University (NSC 992111M008017MY3 and NSC 1022111M008001), and by the Chinese Academy of Sciences under Grant Nos. KZCX2YWQN512 and KZCX2YWN28.
I. INTRODUCTION
II. FORMULATIONS
A. Basic equations for fluid with ions and neutrals
B. Dispersion equation and eigenmodes for Alfvén waves
III. SOLUTIONS FOR ALFVÉN WAVE FREQUENCY: THE PRESENCE OF A FORBIDDEN ZONE
IV. SOLUTION FOR ALFVÉN WAVENUMBER: THE PRESENCE OF HEAVY DAMPING ZONE
V. APPLICATION TO ALFVÉN WAVES IN THE IONOSPHERE AND SOLAR CHROMOSPHERE
VI. SUMMARY
Key Topics
 Plasma waves
 47.0
 Solar chromosphere
 13.0
 Plasma collisions
 11.0
 Plasma ionization
 9.0
 Ionospheric plasmas
 8.0
Figures
Geometry for propagating Alfvén wave. The wave vector is in the direction, and the ambient magnetic field is in the plane. The perturbed velocities , , and magnetic field are in the direction.
Geometry for propagating Alfvén wave. The wave vector is in the direction, and the ambient magnetic field is in the plane. The perturbed velocities , , and magnetic field are in the direction.
The solutions of Alfvén waves can be classified into four regions in the plane, where is the ratio of ion and neutral density and is the normalized collisional frequency for neutral particles by ions. In Region 1 (red) and Region 3 (blue), there are two Alfvén modes ( and ) propagating in opposite direction and are purely damping mode with zero real frequency ( ). In the “forbidden zone (FZ)” (Region 2, yellow), the real frequency of all three modes is zero. The green area is the transition zone between different regions.
The solutions of Alfvén waves can be classified into four regions in the plane, where is the ratio of ion and neutral density and is the normalized collisional frequency for neutral particles by ions. In Region 1 (red) and Region 3 (blue), there are two Alfvén modes ( and ) propagating in opposite direction and are purely damping mode with zero real frequency ( ). In the “forbidden zone (FZ)” (Region 2, yellow), the real frequency of all three modes is zero. The green area is the transition zone between different regions.
The real part ( ) and imaginary part ( ) of the wave frequency of three eigenmodes are plotted as a function of the density ratio for . The real frequency is shown by the blue (red) line in the top figure. The damping rates and have the same value and are shown by the same blue line in the bottom figure. The yellow area is FZ with . As the value of approaches the yellow zone from the left, the real frequency of and decreases sharply to zero. On the right hand side of the yellow region, the real frequency of modes 1 and 2 can be expressed as .
The real part ( ) and imaginary part ( ) of the wave frequency of three eigenmodes are plotted as a function of the density ratio for . The real frequency is shown by the blue (red) line in the top figure. The damping rates and have the same value and are shown by the same blue line in the bottom figure. The yellow area is FZ with . As the value of approaches the yellow zone from the left, the real frequency of and decreases sharply to zero. On the right hand side of the yellow region, the real frequency of modes 1 and 2 can be expressed as .
Same as Fig. 3 except that FZ (yellow) is very narrow.
The real ( ) and imaginary ( ) part of wave frequency for three eigenmodes are plotted as a function of for .
The real ( ) and imaginary ( ) part of wave frequency for three eigenmodes are plotted as a function of for .
Same as Fig. 3 .
The eigenamplitudes, and , and phases of two propagating Alfvén modes with and as a function of for (left) and (right). For , and on the left side of forbidden zone, while . For , and the ion and neutral velocity of mode 1 (mode 2) are out of phase (in phase) with .
The eigenamplitudes, and , and phases of two propagating Alfvén modes with and as a function of for (left) and (right). For , and on the left side of forbidden zone, while . For , and the ion and neutral velocity of mode 1 (mode 2) are out of phase (in phase) with .
The real and imaginary parts of the wavenumbers, and , are plotted as a function of for The curve of for is nearly the same as the curve for in (a). The yellow sections of lines in (a) and (b) correspond to those with values in FZ of Fig. 2 .
The real and imaginary parts of the wavenumbers, and , are plotted as a function of for The curve of for is nearly the same as the curve for in (a). The yellow sections of lines in (a) and (b) correspond to those with values in FZ of Fig. 2 .
The contours of (a) and (b) are plotted in the plane. The contour with is shown by the black line. The region with is termed “the heavy damping zone (HDZ).”
The contours of (a) and (b) are plotted in the plane. The contour with is shown by the black line. The region with is termed “the heavy damping zone (HDZ).”
(a)–(c) The spatial profiles of of Alfvén wave injected from the left boundary with and the spatial profile of in (d). The yellow region corresponds to FZ for . (e)–(h) The spatial profiles of for four simulation cares with .
(a)–(c) The spatial profiles of of Alfvén wave injected from the left boundary with and the spatial profile of in (d). The yellow region corresponds to FZ for . (e)–(h) The spatial profiles of for four simulation cares with .
The vertical profiles of (a) neutral and ion density for daytime and nighttime ionosphere, indicated by solid and dashed lines, (b) the collision frequency for a neutral particle to collide with ions in the daytime and nighttime ionosphere, (c), (e) the ratio of neutral density to ions density , and (d), (f) the parameter for the daytime and nighttime ionosphere in blue line and black line for and . The FZ (HDZ) for shear Alfvén waves is enclosed by red solid (blue dashed) line.
The vertical profiles of (a) neutral and ion density for daytime and nighttime ionosphere, indicated by solid and dashed lines, (b) the collision frequency for a neutral particle to collide with ions in the daytime and nighttime ionosphere, (c), (e) the ratio of neutral density to ions density , and (d), (f) the parameter for the daytime and nighttime ionosphere in blue line and black line for and . The FZ (HDZ) for shear Alfvén waves is enclosed by red solid (blue dashed) line.
(a) The neutral and ion density profiles in the solar chromosphere, indicated by black and blue lines; (b) the vertical profile of collisional frequency for a neutral particle to collide with ions in the solar chromosphere; (c) profile of the ratio of neutral density to ion density, ; and (d) profile of for , , and (solid, dashed, and pointdashed lines). The FZ (HDZ) of Alfvén waves is enclosed by red solid (blue dashed) line.
(a) The neutral and ion density profiles in the solar chromosphere, indicated by black and blue lines; (b) the vertical profile of collisional frequency for a neutral particle to collide with ions in the solar chromosphere; (c) profile of the ratio of neutral density to ion density, ; and (d) profile of for , , and (solid, dashed, and pointdashed lines). The FZ (HDZ) of Alfvén waves is enclosed by red solid (blue dashed) line.
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