4 results
On vertical spinning Alfvén waves in a magnetic flux tube
- L. M. B. C. Campos, N. L. Isaeva
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- Journal:
- Journal of Plasma Physics / Volume 48 / Issue 3 / December 1992
- Published online by Cambridge University Press:
- 13 March 2009, pp. 415-434
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We derive the Alfvén-wave equation for an atmosphere in the presence of a non-uniform vertical magnetic field and the Hall effect, allowing for Alfvén speed and ion gyrofrequency that may vary with altitude; the pair of coupled second-order differential equations for the horizontal wave variables, namely magnetic field or velocity perturbations, is reduced to a single complex, second-order differential equation. The latter is applied to spinning Alfvén waves in a magnetic flux tube, in magnetohydrostatic equilibrium, in an isothermal atmosphere. The exact solution is found in terms of hypergeometric functions, from which it is shown that at ‘high altitude’the magnetic field perturbation tends to grow to a non-small fraction of the background magnetic field. By ‘high-altitude’ is meant far above the critical level, which acts as a reflecting layer for left-polarized waves incident from below, i.e. from the ‘low-altitude’ range. We also obtain the exact solution near the critical level, where the left-polarized wave has a logarithmic singularity, and the right-polarized wave is finite. The latter is plotted in this region of wave frequency comparable to ion gyrofrequency, and it is shown that the Hall effect can cause oscillations of wave amplitude and non-monotonic phases with slope of alternating sign. The latter corresponds to ‘tunnelling’, i.e. waves propagating in opposite directions or trapped in adjoining atmospheric layers; this could explain the appearance of inward- and outward-propagating waves, with almost random phases, in the solar wind beyond the earth, for which the Hall effect on Alfvén waves should be significant.
On the critical layer of Alfvén waves in the solar wind
- L. M. B. C. CAMPOS, N. L. ISAEVA
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- Journal:
- Journal of Plasma Physics / Volume 70 / Issue 3 / June 2004
- Published online by Cambridge University Press:
- 20 May 2004, pp. 271-302
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Alfvén waves are considered in a radial flow and external magnetic field, which relates to some features of the solar wind near the critical point. The Alfvén wave equation for the velocity perturbation is derived, showing that it has in general two singularities (besides the origin and infinity), namely a critical layer (at real distance), where the Alfvén speed equals the mean flow velocity, and a transition level (at imaginary distance), where the spatial derivative of the flow velocity equals the wave frequency. It is shown that in the case of mean flow velocity varying as a power of radial distance the wave field is specified at all distances by a combination of solutions of the Alfvén wave equation around three singularities: a regular singularity at the center, so that ascending power-series solutions exist, some with logarithmic terms; an irregular singularity at infinity, leading to the non-existence of any solution as an ascending Frobenius–Fuchs series, and the existence of two solutions as ascending–descending Laurent series; the region of validity of the preceding solutions is limited by a regular singularity at a finite, non-zero radial distance, which is the critical layer, where the flow velocity and Alfvén speed are equal. The wave field is singular at the critical layer, and has an amplitude jump, which is illustrated by plotting the wave field in the neighborhood of the critical layer, for several values of dimensionless frequency and Alfvén number, combined into a single parameter. When considering Alfvén waves in the solar wind, at least three kinds of boundary conditions could be applied: (i) an initial condition specifying the wave field at the surface of the Sun; (ii) an asymptotic condition excluding wave sources at infinity, by specifying an outward-propagating wave (radiation condition); (iii) a finiteness condition that the wave field be finite at the critical layer. Since the Alfvén wave equation is of second order, only two conditions can in general be applied. It is shown, for example, that (ii) and (iii) are generally incompatible. If the conditions (i) and (iii) are chosen, i.e. an initial wave field is given and the radiation condition of outward propagation at infinity is met, then (ii) will not in general be met; thus the wave field would be singular at the critical layer, in the absence of dissipation, corresponding to the resonance of a linear undamped system. It is shown that in the presence of dissipation, either by fluid viscosity or Ohmic resistivity, the wave field would be finite at the critical layer, corresponding to the resonance of a linear damped system.
On three-dimensional magnetosonic waves in an isothermal atmosphere with a horizontal magnetic field
- L. M. B. C. CAMPOS, R. L. SALDANHA, N. L. ISAEVA
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- Journal:
- Journal of Plasma Physics / Volume 68 / Issue 5 / November 2002
- Published online by Cambridge University Press:
- 01 April 2003, pp. 331-361
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Magnetosonic–gravity waves in an isothermal non-dissipative atmosphere, with a uniform horizontal external magnetic field have been considered in the literature in two cases: (i) ‘one-dimensional’ magnetosonic–gravity waves, in the case of zero horizontal wavenumber and (ii) ‘two-dimensional’ magnetosonic–gravity waves, in which the horizontal wave vector lies in the plane of gravity and the external magnetic field. In the present paper, an extension of case (i) is considered that is distinct from case (ii). This case (iii) is that of magnetosonic–gravity waves with a horizontal wave vector orthogonal to the plane of gravity and the external magnetic field. Since the wave fields depend only on two spatial coordinates and time, the problem could be called ‘two-and-half’-dimensional. The three-dimensional magnetosonic–gravity wave propagates a magnetic field perturbation parallel to the external magnetic field, and velocity perturbations transverse to it. Elimination for the vertical velocity perturbation leads to a second-order wave equation, with four regular singularities. Three regular singularities specify (a) the wave fields at high altitude, where there are two cut-off frequencies involving the acoustic cut-off frequency; (b) the wave fields in the deep layers, where another two cut-off frequencies appear, involving both the acoustic and gravity cut-off frequencies; and (c) the transition between the two regimes, occurring across a critical layer, where one solution of the wave equation vanishes and the other has a logarithmic singularity in the amplitude and also a phase jump. The whole altitude range can be covered using the three pairs of solutions of the wave equation, obtained by expanding in Frobenius–Fuchs series about each regular singularity. The power series solutions are used to plot the wave fields, for several values of the three dimensionless parameters of the problem, namely the plasma $\beta$, frequency and wavenumber. It is shown that the presence of a horizontal wave vector transverse to the plane of gravity and the external magnetic field, can change the properties of the waves significantly: first, the two cut-off frequencies may cease to exist, in which case the full wave frequency spectrum can propagate; secondly, the critical layer occurs at different altitudes for different frequencies, allowing gradual absorption of the waves (e.g. in the solar transition region).
On Alfvén waves in a radial flow and magnetic field
- L. M. B. C. CAMPOS, N. L. ISAEVA
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- Journal:
- Journal of Plasma Physics / Volume 62 / Issue 1 / July 1999
- Published online by Cambridge University Press:
- 04 April 2001, pp. 1-33
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This paper considers Alfvén waves in a radially stratified medium where all background quantities, namely mass density, magnetic field strength and mean flow velocity, depend only on the distance from the centre, the latter two being assumed to lie in the radial direction. It is shown that the radial dependence of Alfvén waves is the same for two cases: (i) when the velocity and magnetic field perturbations are along parallels, in the one-dimensional case of only radial and time dependence; (ii) in the three-dimensional case with dependence on all three spherical coordinates and time, for velocity and magnetic field perturbations with components along parallels and meridians, represented by the radial components of the vorticity and electric current respectively. Elimination between these equations leads to the convected Alfvén-wave equation in the case of uniform flow, and an equation with an additional term in the case of non-uniform flow with mean flow velocity a linear function of distance. The latter case, namely that of non-uniform flow with flow velocity increasing linearly with distance, is analysed in detail; conservation of mass flux requires the mass density to decay as the inverse cube of the distance. The Alfvén-wave equation has a critical layer where the flow velocity equals the Alfvén speed, leading to three sets of two solutions, namely below, above and across the critical layer. The latter is used to specify the wave behaviour in the vicinity of the critical layer, where local partial transmission occurs. The problem has two dimensionless parameters: the frequency and the initial Alfvén number. It is shown, by plotting the wave fields relative to the critical layer, that these two dimensionless parameters appear in a single combination. This simplifies the plotting of the wave fields for several combinations of physical conditions. It is shown in the Appendix that the formulation of the equations of MHD in the original Elsässer (1956) form, often used in the recent literature, does not apply if the background mass density is non-uniform on the scale of a wavelength. The present theory, based on exact solutions of the Alfvén-wave equation for a inhomogeneous moving medium, is unrestricted as to the relative magnitude of the local wavelength and scale of change of properties of the background medium. The present theory shows that the rate-of-decay of wave amplitude is strongly dependent on wave frequency beyond the critical layer, i.e. the process of change with distance of the spectrum of Alfvén waves in the solar wind starts at the critical layer.