# Magnetosonic resonances in the magnetospheric plasma

- A. S. Leonovich
^{1}Email author and - D. A. Kozlov
^{1}

**65**:1

**DOI: **10.5047/eps.2012.07.002

© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012

**Received: **10 February 2012

**Accepted: **7 July 2012

**Published: **10 June 2013

## Abstract

A problem of coupling between fast and slow magnetosonic waves in Earth’s magnetosphere (magnetosonic resonance) is examined. Propagation both slow magnetosonic wave and Alfven wave can easily be canalized along the magnetic field line direction. The main difference between the two is that slow magnetosonic waves dissipate strongly due to their interaction with the background plasma ions, whose temperature is above the electron temperature. In Earth’s magnetosphere, however, there is a region where the dissipation of slow magnetosonic waves can be weak—the inner plasmasphere. The slow magnetosonic waves generated there can be registered directly. In other regions, with strong dissipation of slow magnetosonic waves, their signature may be detected through their impact on the Alfven resonance at frequencies for which the resonant Alfven and slow magnetosonic waves exist simultaneously in the magnetosphere. Owing to their strong coupling with the background plasma ions, resonant slow magnetosonic waves can transfer the energy and impulse from the solar wind to the magnetospheric plasma ions via fast magnetosonic waves penetrating into the tail lobes. A problem of resonant conversion of fast magnetosonic waves into slow magnetosonic oscillations in a magnetosphere with dipole-like magnetic field is also examined.

### Key words

Inhomogeneous plasma magnetosonic waves resonance magnetosphere## 1. Introduction

The magnetosonic resonance has the same physical nature as the well-known Alfven resonance (Tamao, 1965), which in magnetospheric physics is conventionally called ‘field line resonance’ (Chen and Hasegawa, 1974; Radoski, 1974; Southwood, 1974). In the Alfven resonance, a monochromatic fast magnetosonic (FMS) wave propagating in an inhomogeneous plasma with magnetic field, drives an Alfven wave at the resonance magnetic shell, where its frequency is the same as the local frequency of Alfven oscillations. This coupling is due to the Alfven waves propagating practically along magnetic field lines, their frequency, for a fixed wavelength, is determined by the magnitude of the Alfven speed. Many—both theoretical (Inhester, 1987; Lee and Lysak, 1989; Leonovich and Mazur, 1989; Rankin et al., 2006), and experimental (Cheng et al., 1998; Rankin et al., 2005; Agapitov et al., 2009) (see also review by Pilipenko, 1990)—papers have scrutinised the Alfven resonance during magnetospheric phenomena.

Slow magnetosonic (SMS) waves are in many aspects similar to Alfven waves: both the modes are guided by magnetic field lines. This results in SMS waves at the resonance magnetic shells being capable of being driven by FMS waves travelling in an inhomogeneous plasma (Yumoto, 1985). However, investigations into SMS waves in the magnetosphere are much fewer than those devoted to the Alfven resonance. Noteworthy are a significant number of papers dealing with the resonant coupling of the Alfven and SMS waves in a curvilinear magnetic field (Southwood and Saunders, 1985; Walker, 1987; Cheng and Lin 1987; Ohtani et al., 1989; Klimushkin, 1998; Cheremnykh et al., 2004; Klimushkin and Mager, 2008).

All of these studies were performed for small-scale MHD modes with large azimuthal wave numbers *m* ≫ 1. The source of such waves should be located on the same magnetic shells where they are generated. Since the magneto-sphere is an opacity region for the fast magnetosonic waves with *m* ≫ 1, their amplitude in the magnetosphere decreases exponentially, on a small scale, with distance from the region of its generation. The most common source of fast magnetosonic waves is believed to be either shear plasma flow at the magnetopause, or oscillations in the solar wind. Only fast magnetosonic waves with small *m* ~ 1 can penetrate inside the magnetosphere while retaining sufficient amplitude. Such waves can drive the Alfven and SMS oscillations at the resonant magnetic shells. It is exactly these resonant waves that are discussed in this paper. The Alfven resonance is a well-studied phenomenon by now, therefore we will focus on studying the magnetosonic resonance.

However, papers studying the resonance of the fast and slow magnetosonic waves are extremely few. This results, first of all, from the fact that they are difficult to detect during observations. Unlike the Alfven waves, the SMS waves are highly dissipative. The SMS wave travel speed in most of the magnetosphere is close to the thermal velocity of plasma ions to which they easily transfer their energy. There is only one exception. In the inner plasmasphere, for the magnetic shells *L* < 2, where the background plasma electrons are hotter than the ions, these waves can travel almost without dissipation. Therefore, charged particle concentration oscillations due to the solar terminator movement over the ionosphere are regularly observed here (Afraimovich et al., 2009), with parameters allowing them to be regarded as SMS waves (Leonovich et al., 2010).

Despite—or even thanks to—the rather intensive decay of SMS waves, however, they can play an important role in their coupling with other modes of MHD oscillations, as well as interacting with background plasma ions. In solar investigations, these oscillations are often invoked for interpreting oscillations observed in the active regions (Miles and Roberts, 1989; Gonzalez and Gratton, 1991), as well as a heating mechanism for the solar corona (Nakariakov et al., 1999).

This paper will consider several problems dealing with the magnetosonic resonance and its possible signatures in the Earth’s magnetosphere.

The paper is organized as follows. In Section 2, we introduce the basic equations and the equilibrium condition of the plasma configuration under consideration. Section 3 discusses the magnetosonic and Alfven resonances in a one-dimensional inhomogeneous medium in the form of a smoothly varying transition layer. This layer may be considered as a model for describing the process of FMS wave incidence and reflexion from the magnetopause and plasma-pause. Section 4 solves the problem of impulse transfer from the solar wind into the geotail lobes via FMS waves. The momentum transfer to the background plasma ions is via SMS waves excited by FMS waves on the resonance magnetic shells. In Section 5, the full spatial structure of resonant SMS waves is obtained in a two-dimensionally in-homogeneous magnetosphere with a dipole magnetic field. The main features of the resonant SMS waves that can be used for their detection during observations are summarized in the Conclusion.

## 2. Basic Equations

*γ*= 5/3 is the adiabatic constant. Assuming plasma to consist of singly-ionized hydrogen ions and electrons as well as being quasi-neutral (

*n*

_{ e }=

*n*

_{ i }=

*n*), its parameters in the one-fluid approximation have to be understood as follows: —mass-average velocity, .

*t*= 0)

## 3. Magnetosonic Resonance in 1-D Inhomogeneous Plasma

*x*,

*y*,

*z*) for solving the problem. We will consider a plasma configuration in which the magnetic field is directed along the

*z*axis, with the plasma parameters varying in the

*x*axis direction. The

*y*axis makes it a right-hand system of coordinates. Figure 1 shows the characteristic distributions of the Alfven and SMS-wave speeds in the plasma configuration in question.

*v*

_{ x }= ∂

*ζ*/∂

*t*—the plasma velocity vector component in the

*x*direction within a wave, where

*ζ*is the plasma element displacement. We will consider a simple harmonic wave, which in the

*y*and

*z*directions is a plane wave of the form exp (

*i*

*k*

_{ y }

*y*+

*i*

*k*

_{ z }

*z*−

*i*

*ω*

*t*), where

*k*

_{ y },

*k*

_{ z }are the respective wave vector components,

*ω*is wave frequency. Linearizing the set of Eqs. (1)–(4) and expressing the other components of the oscillation field through

*ζ*produces:

*ζ*, we have the equation

*x*component squared of the wave vector when the solution can be presented as

*ζ*~ exp(

*i*∫

*k*

_{ x }

*dx*).

The behaviour of the
function is important for the problem to be stated correctly. We wish to explore the process of incidence and reflexion of a magnetosonic wave on a smoothly varying transition layer. If the wave source is to the right of the transition layer, the solution of the problem should be the superposition of the incident and reflected waves of finite amplitude when *x* → ∞. Two variants of the distribution of the
function are possible— numbered 1 and 2 in Fig. 1. An analysis of (10) reveals that, for monotone increasing *A*(*x*), when the x coordinate varies from +∞ to −∞, the
function passes through zero twice at points which we will denote as *x*_{01}, *x*_{02}, between which the opacity region (where
) is located. This behaviour of
is illustrated by curve 1 in Fig. 1.

There are also two singular points in Eq. (9) at which the coefficient attached to the higher derivative tends to zero. One is the Alfven resonance point, *x*_{
A
}, defined by equality Ω^{2}(*X*_{
A
}) = 0, located in the opacity region in the interval (*x*_{01}, *x*_{02}). The second is the point of magnetosonic resonance, *x*_{
S
}, where the denominator in expression (10) tends to zero, producing the local dispersion equation for SMS waves when
, where
. The point *x*_{
S
} is located more to the left of the turning point *x*_{01}, the transparency region for SMS waves being located between the two. To the left of *x*_{
S
}, there is an opacity region expanding to −∞ in the *x* direction.

*β** =

*S*

^{2}/

*A*

^{2}, and are the two roots of the biquadratic equation

*A*(

*x*) as a function of the form

*A*

_{±}is the Alfven speed as

*x*→ ±∞, Δ is the typical thickness of the transition layer. In the chosen model of the Alfven speed

*A*

_{+}= 1 and Δ = 1 (see Fig. 1). The ratio

*A*

_{Δ}/

*A*

_{+}= 30 is chosen large enough for both types of resonant surfaces—those for the Alfven and SMS waves—to be able to exist in this system, for a wide enough spread of the plasma temperature values (from

*β** = 0.01 to

*β** ~ 1).

*x*=

*x*

_{ S }. Let us linearize the coefficient attached to the higher derivative in (9), representing , where is the characteristic scale of the variation near

*x*=

*x*

_{ S }. In order to regularize the singularity in the solution of (9) we will formally introduce a damping decrement,

*γ*

_{s}, for SMS waves near the resonant surface

*x*=

*x*

_{ S }, making a substitution,

*ω*→

*ω*+

*i*

*γ*

_{s}, in the denominator of (10). Equation (9) may then be rewritten as

*x*=

*x*

_{ S }, where

*ε*

_{s}=

*k*

_{ y }

*a*

_{ s }

*γ*

_{s}/

*k*

_{ z }

*C*

_{ s }(

*x*

_{ s }) is the regularized factor determined by the decrement of SMS waves, where the subscript

*S*indicates that the values of the parameters are taken for

*x*=

*x*

_{ S }. Its solution is

*I*

_{0}(

*z*),

*K*

_{0}(

*z*) is the modified Bessel functions. If the wave source is to the right of the transition layer, then, using an asymptotic representation of (14) in the opacity region at , we find

*C*

_{1}=0. To the right of resonance point

*ξ*

_{s}= 0 the solution (14) has the form

*ξ*

_{s}→ ∞, has an asymptotic representation, . When

*ξ*

_{s}→ 0, the solution (14)–(15) is

There are essential differences, however. The above solution describes a wave incident on the resonance surface. The wave reflected from this surface is absent—it is described by function *I*_{0}(*z*), whose coefficient *C*_{1} = 0. This means that the total energy of a wave incident on the resonance surface is completely absorbed in its neighbourhood, irrespective of the dissipation mechanism involved. If the transparency region for SMS waves expands ad infinitum to the right (which corresponds to the second condition (11)), such waves will be completely absorbed near the resonance surface. If the parameters of the wave under consideration are such (the first condition (11)) that the incident wave “leaks” through the barrier of opacity region (*x*_{01}, *x*_{02}) into the SMS-wave transparency region (*x*_{
s
}, *x*_{01}), then the energy of the wave penetrating through this barrier is absorbed completely in the neighbourhood of the resonance surface. Note the energy absorption of the incident wave does not exceed 50% in the neighbourhood of the Alfven resonance surface located deep within the opacity region (*x*_{01}, *x*_{02}) (Leonovich et al., 2010).

If the wave source is to the left of the resonance surface (in the opacity region *x* < *x*_{
s
}(*ξ*_{s} < 0)), then in the SMS wave transparency region, *ξ*_{s} > 0, expanding ad infinitum, the solution of (9) must describe a wave escaping from the resonance surface. There is no incident wave on the resonance surface, as described by the function *K*_{0}(*z*), to the right of *ξ*_{s} = 0: *C*_{2} = 0. Since the function *I*_{0}(*z*) is regular on the resonance surface, magnetosonic resonance is also absent.

Using the ideal MHD approximation to study the SMS waves, their very high dissipation ought to be taken into account. To this end, their decrement can be introduced in equations, near the resonance surfaces for SMS waves as was done above. The magnitude of the decrement crucially depends on the plasma ion to electron temperature ratio. The dependence of the SMS wave decrement on the plasma nonisothermality level is specified in Appendix A based on the kinetic theory equations.

Detecting the presence of resonant SMS waves in a plasma of high dissipation level is a rather difficult problem. The oscillation amplitude increases only little at the resonance surface. There is another possibility, however. Let the parameters of the FMS wave incident on the inhomogeneous plasma layer be such that the resonance surfaces for both the Alfven and SMS waves exist simultaneously. Due to weakly localized resonant SMS oscillations, they can— when dissipation is large enough—affect the behaviour of oscillations near the Alfven resonance surface. The presence of magnetosonic resonance in the plasma system under study may be detected via the behaviour of the oscillation hodograph when moving through these surfaces.

A distinctive feature of resonant Alfven oscillations is the change of the hodograph rotation direction of the transverse magnetic field vector **B**_{⊥} = (*B*_{
x
}, *B*_{
y
}) as we pass through the resonant surface. It follows from the sign reversal in ∂*ζ*/∂*x*. In a case with small decrements *γ*_{a}, *γ*_{s} ≪ *ω*, this rule is valid when we pass through each resonant surface *x* = *x*_{
A
} and *x* = *x*_{
S
}. We will look, however, at what happens when decrements *γ*_{
A
} and *γ*_{
s
} are not too small.

*ζ*/∂

*x*as calculated for

*ε*

_{ A }=

*γ*

_{ A }/

*ω*= 0.1 and three values of

*ε*

_{ s }=

*γ*

_{ s }/

*ω*= 0.01; 0.1; 1. Here the behaviour of the hodographs is conventionally presented in the plane (

*B*

_{ y },

*B*

_{ x }). For small

*ε*

_{ s }= 0.01 (

*T*

_{ e }/

*T*

_{ i }≫ 1) the behaviour of the hodograph is as expected. When

*ε*

_{ s }increases to 0.1 (

*T*

_{ e }~

*T*

_{ i }) the points where the hodograph rotation direction changes shift away from the resonant surfaces about as far as the distance between them. With

*ε*

_{ s }increasing further to 1 (

*T*

_{ e }/

*T*

_{ i }≈ 0.1), the hodograph rotation direction does not change at all. This example demonstrates that the presence in the system of strongly damped resonant SMS oscillations can change the behaviour of the field components essentially, even in the neighbourhood of the resonant surface for the Alfven waves.

Almost everywhere in the Earth’s magnetosphere, *T*_{
e
} ≪ *T*_{
i
}. As follows from the above calculations, this means that SMS waves decay fast (*γ**s* ~ *ω*). Therefore, the resonance peaks are poorly-expressed for these waves and difficult to detect in observations. If the resonance peaks for the Alfven waves are also present in the system, however, the above features of the hodograph behavior may indicate the presence of resonant SMS waves. The only exception is the region inside the plasmasphere on the magnetic shells *L* < 2, where *T*_{
e
} ≈ 2*T*_{
i
} (Titheridge, 1998). The decrement of SMS waves is relatively small here so that resonant SMS oscillations can be observed directly. It is, however, hard to imagine FMS waves capable of reaching magnetic shells that are so close to Earth.

## 4. Transfer of Momentum from the Solar Wind Into Geotail via Magnetosonic Waves

Let us now consider the specific problem of momentum transfer from the solar wind into geotail lobes via magnetosonic waves.

The magnetosheath plasma flow is turbulent. Such plasma oscillations can be regarded as a stochastic flow of magnetosonic waves, partly directed towards the magneto-sphere. Leonovich et al. (2003) have shown that up to 50% of the energy of this wave flux can penetrate into the geotail. The integrated energy of the wave flux penetrating into the magnetosphere during a typical time interval between two successive substorms is two orders of magnitude larger than the total energy of magnetospheric convection and can be used to maintain it. This is only a potential capacity, however.

For each harmonic of magnetosonic waves there is a surface in the magnetosphere from which it is completely reflected. Therefore, if no appreciable absorption of their energy takes place while magnetosonic oscillations travel from the magnetopause to the turning point, they must be reflected back into the solar wind in almost their entirety. The energy of MHD oscillations is known to be efficiently absorbed at resonance surfaces for the Alfven and slow magnetosonic waves (Leonovich and Kozlov, 2009). The SMS waves are especially interesting in this regard. Due to their very dissipative nature they are weakly localized across magnetic shells and can interact with ions of the bulk of the background plasma distribution function. To check this possibility, we employ quasilinear theory to calculate the velocity the background plasma acquires when interacting with the flux of fast magnetosonic waves penetrating into the magnetosphere from the magnetosheath.

We introduce a cylindrical coordinate system (*r*, *φ*, *z*) in which the origin *r* = 0 coincides with the axis of the plasma cylinder. The background magnetic field is directed along the *z* axis. We assume that plasma in the magnetosheath moves along the *z* axis at velocity *v*_{0}, while plasma is motionless in the geotail, in the absence of waves (see Fig. 3 ). Transition from the magnetospheric parameters to the magnetosheath parameters occurs in a narrow transition layer of thickness Δ_{
r
} ≪ *r*_{
m
}, where *r*_{
m
} is the characteristic radius of the geotail. We set such a plasma density distribution over the radius that its maximum is reached on the axis of the plasma cylinder, falling to a minimum toward its boundary. Magnetic field in the magnetotail is stronger than in the solar wind. The distribution of the Alfven speed
over the radius is presented in Fig. 4. Such a distribution is typical of plasma parameters in the geotail lobes.

*i*

*k*

_{ z }

*z*+

*im*

*φ*−

*i*

*ω*

*t*), where

*m*= 0, 1, 2, 3 … is the azimuthal wave number, produces the following equation for displacement

*ζ*:

*d*

*ζ*/

*dr*determining the maximum oscillation amplitude on the resonance surfaces. The resonance surfaces for the SMS wave are determined by the intersection points of functions Re and , where the real part of the denominator in (10) vanishes. As was noted above, the decrement of SMS waves strongly depends on the plasma ion to electron temperature ratio. In the solar wind the plasma electrons are hotter than the ions (

*T*

_{ e }≈ 3

*T*

_{ i }), therefore, we assume in the magnetosheath. In the tail lobes, on the contrary, the plasma ions are hotter than the electrons (

*T*

_{ i }≈ 8

*T*

_{ e }), which corresponds to (see Appendix A). The decrement of SMS waves chosen for the magnetosheath changes to the one typical of the magnetosphere in the same transition layer as the other plasma parameters.

*t*→ ∞) equation for the ion distribution function

*f*(

*v*

_{ǁ}) in the presence of SMS waves has the following form (see Appendix B)

*n*

_{0}is the plasma ion concentration, is the thermal velocity of plasma ions on the magnetic shell under scrutiny,

*v*

_{ǁ}is ion velocity along magnetic field lines, is the averaged amplitude of the radial magnetic field of oscillation,

*m*is azimuthal wave number.

*v*

_{ǁ}yields

*t*→ ∞, a “plateau” must appear in the distribution function in the intervals of

*v*

_{ǁ}where .

There are three areas where a “plateau” forms in the distribution function
and
. The
correspond to the maximum and minimum velocities of the SMS waves for which the local dispersion equation
holds true in our model. The value
is reached on the axis of the plasma cylinder,
in the vicinity of the magnetopause. The solar wind is transparent in the intervals of parallel wave numbers *k*_{
z
} < min(*k*_{1}, *k*_{2}) and *k*_{
z
} > max(*k*_{1}, *k*_{2}), where *k*_{1,2} = *ω*/*v*_{1,2}, *v*_{1} = *v*_{0} + *S*_{
w
}, *v*_{2} = *v*_{0} − *S*_{
w
}, *S*_{
w
} is sound velocity in the solar wind. In our model of the medium the sound velocity in the magnetosheath *S*_{
w
} = 177 km/s. For solar wind plasma flows with *v*_{0} > 200 km/s we have *v*_{1} > *v*_{2} > 0 and the solar wind is opaque when 0 < *k*_{1} < *k*_{z} < *k*_{2}. Considering the resonance conditions for plasma particles interacting with waves (*k*_{
z
} = *ω*/*v*_{ǁ}), we find that the distribution function remains unchanged in the range *v*_{2} < *v*_{ǁ} < *v*_{1}. The area
corresponds to “downstream”, while the two other areas to “upstream” FMS waves in the solar wind.

*j*= 1, 2, 3 is the number of an area with a “plateau”, and the values correspond to the maximum and minimum value of the parallel velocity of particles in each of these intervals.

*v*

_{ǁ}= 0) parts of is zero. Figure 6 shows the distribution of

*v*

_{0}(

*r*) as calculated for the parameters of the cylindrical model of the geotail we use in this study, for different solar wind velocities in the magnetosheath.

Figure 6(a) presents the solar wind velocity *v*_{0}(*r*) profiles taking into account the transition layer, and plasma velocity profiles in the geotail lobes
calculated for two limiting cases. The first of these (curves 4 and 5 in Fig. 6(a)) assumes that all waves in the magnetosheath move “downstream” and no plateaux form in the ranges
and
. Obviously, in this case the impulse transferred by MHD waves to ions in the geotail lobes is tailward
. In the second limiting case, the “downstream” and “upstream” fluxes of waves are equal. It is evident from Fig. 5(a), that in this case the impulse transferred to plasma ions is Earthward. From satellite observations of solar wind oscillations, it is difficult to determine which portion of the wave flux is “downstream” or “upstream”.

The most probable seems to be an intermediate case between the two, when the “downstream” waves in the magnetosheath occupy a broader part of the spectrum than do the “upstream” waves. The summarized plasma velocity distribution in the case when the “upstream” waves are absent from the range is displayed in Fig. 6(b). Evidently, in this case the impulse transferred to ions in the regions adjacent to the transition layer reverses the plasma flow motion back to Earth, whereas closer to the cylinder axis the motion becomes tailward again. Note that the model in question is inapplicable to those inner parts of the geotail where the plasma sheet lies. As will be seen later, another reason why the obtained results cannot be used for the inner regions of the geotail is that the characteristic time for the asymptotic regime of the plasma flow to set in there is too long.

*τ*needed for the completely motionless plasma to switch to the asymptotic regime of its motion is determined by the amplitude of MHD waves transferring the momentum from the solar wind into the magnetosphere. To estimate this time, let us replace the time derivative in (22) with

*τ*

^{−1}, resulting in

*C*is a constant determined by the average oscillation amplitude, and

*Φ*(

*k*

_{ t },

*ω*) is a step function determining the upper and lower limits of the spectrum, as well as the wave range for which the solar wind is an opacity region. This function well describes properties of the spectrum of magnetosonic waves observed in the solar wind (Matthaeus and Goldstein, 1982; Marsh and Tu, 1990; Goldstein et al., 1995). The function Φ(

*k*

_{ t },

*ω*) may be written as

*x*) is the Heaviside step function, and is the range of parallel wave numbers corresponding to solar wind opacity for FMS waves. Constant

*C*in (24) is determined by the inverse Fourier transformation

*B*

_{ r }component of the solar wind oscillation field at

*r*= 2

*r*

_{ m }. In our calculations we assume .

*τ*needed for the asymptotic regime to set in in the geotail plasma flow, as calculated using formula (23), in which the diffusion coefficient is determined by (21), and the spectrum of FMS fluctuations in the magnetosheath (24) corresponds to the plasma flow profiles in Fig. 6(b). It is evident that the values of

*τ*comparable with the time during which the geotail can be regarded as a stable enough plasma configuration (average interval between two successive substorms ~3–6 h) is achieved in the ranges 0.8

*r*

_{ m }<

*r*<

*r*

_{ m }(for the

*v*

_{0}= 400 km/s solar wind) and 0.85

*r*

_{ m }<

*r*<

*r*

_{ m }(for the

*v*

_{0}= 200, 800 km/s solar wind). It is in this range of magnetic shells that a maximum concentration of resonance surfaces for SMS waves is reached in our model geotail.

The obtained values of *τ* can be regarded as the upper limit of the time needed for the asymptotic regime to set in in the plasma flow. Time *τ* decreases quadratically when the amplitude of turbulent plasma oscillations in the magnetosheath increases. Moreover, a more accurate approach to solving the initial problem (19) must take into account contribution from MHD oscillations related to the evolution of a Kelvin-Helmholtz instability at the magnetopause. The solar wind being opaque for such oscillations (Leonovich, 2011a, b), the problem would be formulated in a different manner than in this work.

These oscillations do not provide a significant contribution to the oscillation amplitude in the solar wind far from the plasmapause. They can, however, produce a significant additional contribution to the oscillation amplitude in the magnetotail, in the region adjacent to the magnetopause. To determine this contribution it is necessary to specify the amplitude of these oscillations and solve the problem of determining their spatial structure. In contrast to oscillations in the magnetosheath, the amplitude of these oscillations is uncertain, varying strongly as per the solar wind parameters. Therefore, to avoid unnecessary complications of the problem, the Kelvin-Helmholtz instability-related oscillations were not taken into account in the above suggested approach. If the flux of unstable waves in the geotail is assumed to be comparable with the wave flux discussed in this work, we may expect a 1.5–2 fold increase in the amplitude of the resonant SMS waves. As follows from the above calculations, this means a 2–3-fold decrease in the characteristic time *τ* as well as a somewhat wider range of magnetic shells on which the asymptotic regime of magnetospheric convection can set in.

## 5. Magnetosonic Resonance in a 2-D Inhomogeneous Dipole-Like Magnetosphere

*x*

^{1},

*x*

^{2},

*x*

^{3}), in which the coordinate

*x*

^{3}is along the field line,

*x*

^{1}is across the magnetic shells, and the azimuthal

*x*

^{2}coordinate completes the right hand coordinate system. The squared length element in this coordinate system is found as

*g*

_{ i }(

*i*= 1, 2, 3) are metric coefficients. We assume that the plasma and magnetic field are homogeneous along the azimuthal coordinate

*x*

^{2}.

**E**= (

*E*

_{1},

*E*

_{2}, 0) this expansion has the form

_{⊥}= (Δ

_{1}, Δ

_{2}) is the transverse 2-dimensional gradient,

*φ*and ψ are the scalar and vector potentials, respectively. Under proper gauge calibration, the vector potential has a longitudinal (field-aligned) component only, ψ = (0,0,

*ψ*

_{3}=

*ψ*). Using the linearized system (1)–(4) we express the perturbed magnetic field components through the potentials

*φ*and

*ψ*as

*φ*and

*ψ*(see Leonovich et al, 2006)

*φ*, and magnetosonic modes are characterized by the longitudinal component

*ψ*of the vector potential. The solution of the dispersion equation (28) can be represented as

*S*≪

*A*,

*A*≪

*S*, or holds, the following approximate dispersion equations can be obtained: for the FMS waves, where and for the SMS waves, where .

*ψ*describes both the fast and slow magnetosonic modes, in the linear approximation this potential can be decomposed as the sum of the component

*ψ*

_{ F }, related to the FMS wave, and

*ψ*

_{ S }, related to the SMS wave, that is

*ψ*=

*ψ*

_{ F }+

*ψ*

_{ S }. Away from the resonance surface, the main contribution to potential

*ψ*comes almost exclusively from the FMS oscillations (

*ψ*≈

*ψ*

_{ F }). Neglecting the small component (~

*S*/

*A*≪ 1) related to the derivatives along the longitudinal coordinate

*x*

^{3}in the operator in (27), we obtain an equation that describes the FMS wave field far from the resonant surface:

*φ*= 0 in its right-hand part. Therefore, in the vicinity of the resonant surface we obtain the equation for the resonant SMS oscillations:

*m*. FMS oscillations with

*m*≫ 1 practically do not penetrate into the magnetosphere. Only oscillations with

*m*~ 1 on resonant shells have an amplitude sufficient to drive SMS waves effectively. Therefore, we shall consider oscillations with

*m*~ 1.

*ℓ*is the coordinate measured along the field line from the equator, ,

_{1}

*ψ*

_{ s }/

*ψ*

_{s}≫ ǀ∇

_{3}

*ψ*

_{s}/

*ψ*

_{s}ǀ. Therefore, a solution to (30) may be sought using the method of different scales, representing the potential

*ψ*

_{S}as

*U*(

*x*

^{1}) describes, in the main order, the small-scale transverse structure of oscillations along the

*x*

^{1}coordinate, whereas the function

*S*(

*x*

^{1},

*x*

^{3}) describes the oscillation structure along magnetic field lines. The typical scale of

*S*(

*x*

^{1},

*x*

^{3}) along

*x*

^{1}is assumed to be much larger than the scale of

*U*(

*x*

^{1}). The small correction term

*h*(

*x*

^{1},

*x*

^{3}) describes the oscillation structure in higher orders of the perturbation theory.

*S*(

*x*

^{1},

*x*

^{3}) satisfy the homogeneous boundary conditions in the ionosphere: . The solution of (33), with such boundary conditions, is a series of eigenfunctions

*S*

_{ N }(

*x*

^{1},

*ℓ*) and corresponding eigenfrequencies

*Ω*

_{ SN }(

*x*

^{1}), where

*N*= 1, 2, 3 … is the longitudinal wavenumber. In the two first orders of the WKB approximation, the solution of (33) satisfying the above boundary conditions has the form

_{ SN }=

*π*

*N*/

*t*

_{ s },

*N*magnetosonic harmonics. For a numerical solution, we use the coordinate system (

*a*,

*φ*,

*θ*) related to the dipole magnetic field lines (see Fig. 8). The plasma distribution is set using a self-consistent model of the dipole magnetosphere (Leonovich et al., 2004). The radial distributions of the Alfven and magnetosonic speeds in the equatorial magnetospheric plane derived from this model are shown in Fig. 9(a). Such a distribution of plasma parameters is typical of the Earth’s dayside magnetosphere.

All the following calculations concern the magnetic shell corresponding to the geosynchronous orbit, *a* = 6.6*R*_{
E
}. Figure 9(b) shows the radial distributions of the eigenfrequencies of the first three harmonics of standing SMS waves, obtained from a numerical solution of (33) for the ionosphere under homogeneous boundary conditions. The same figure displays the distribution of transit time *t*_{
S
} determining, in the WKB approximation, the frequencies of standing SMS waves. It is easily verifiable that the numerically calculated frequencies of the first harmonics differ significantly from the WKB ones. They occupy the lowest-frequency part of the spectrum of MHD oscillations observed in Earth’s magnetosphere (*f* ≲ 1 mHz).

*S*

_{ N }(

*x*

^{1},

*x*

^{3}) results in a number of important consequences. First, resonant SMS oscillations are impossible to detect on the ground or by a low-orbit satellite. Second, the ionosphere cannot be an absorber of the resonant SMS wave energy. SMS wave damping in the magnetosphere is caused by their resonant interaction with the background plasma particles.

Note that such a structure of standing SMS waves is only typical of long magnetic field lines in the outer magneto-sphere. In the inner plasmasphere (on magnetic shells *L* < 2), the distribution of standing SMS wave amplitudes on short field lines is such as to feature sharp peaks at the ionospheric *F*_{2}-region altitudes (Leonovich et al., 2010). Moreover, the plasma ion to electron temperature ratio in this plasmaspheric region is such that SMS waves exhibit weak enough dissipation (*T*_{
e
} > 2*T*_{
i
} and *γ* /*ω* ~ 10). Therefore, standing SMS waves can exist in the plasmasphere long enough, thus making their registration possible—based, for example, on observations of the ionospheric total electron concentration variations as detected by the GPS network receivers (Afraimovich et al., 2009).

*h*

_{ N }(

*x*

^{1},

*x*

^{3}) in (32) satisfies the following ionospheric boundary condition (see (31))

*U*

_{ N }(

*x*

^{1}):

*γ*

_{ N }, for each of the harmonics of standing SMS waves is determined, near the resonance surface, by the plasma ion to electron temperature ratio.

_{ SN }(

*x*

^{1}) change monotonically, so that a linear dependence

_{ SN }(

*x*

^{1}) in the vicinity of the resonant surface.

_{ SN }variation at Substituting (38) into (37) and introducing the dimensionless variable , where , we obtain an equation describing the transverse structure of magnetosonic resonance

*G*

_{ N }= Γ

_{ N }

*λ*

_{ SN }

*L*. These coefficients may be considered as constants because they vary insignificantly within the localization region of the desired solution

*U*

_{ N }(

*ξ*).

*ξ*→ 0 the bulk of the integrand (40) accumulates in the domain

*k*≫ 1, making it possible to set

*ζ*(

*k*) ≈

*ζ*(∞) in the exponent, while neglecting all the terms but

*k*

^{2}in the denominator. This yields

*ξ*ǀ → ∞ the bulk of the integrand (40) accumulates in the domain

*k*≪; 1, making it possible to set

*k*= 0 in

*ζ*(

*k*) and in the denominator. The integral is then easily calculated

*ξ*ǀ

^{−1}. This behavior satisfies the boundary conditions on the

*x*

^{1}coordinate—resonant oscillations have a finite amplitude far from the resonance surface. The magnetic field components of the oscillation near the resonance surface are described by following expressions

*B*

_{3N}has the strongest singularity, ∝

*ξ*

^{−1}. The radial magnetic component

*B*

_{1N}has a weaker logarithmic singularity, and the azimuthal component

*B*

_{2N}is regular. Figure 10(b) shows the radial amplitude-phase structure of the physical components of the wave magnetic field , and of the fundamental harmonic (

*N*= 1) of SMS waves near the resonant magnetic shell

*a*= 6.6

*R*

_{ E }. The response to the FMS wave is normalized in such a manner as to make the peak value ǀ

*B*

_{ z }ǀ = 1 at the resonance surface.

The initial oscillation phase is chosen to be zero in an asymptotically distant region right of the resonant shell. For numerical calculations the damping rate and the imaginary correction factor were chosen to be rather small, *ε* = 10^{−2}, to expose the resonant structure. The amplitude of the resonant SMS oscillations is controlled by the FMS wave amplitude and the SMS damping rate. When *γ*_{
N
} and *ε* increase, the maximum amplitude decreases and the resonant peak widens. Passing through the resonant peak the phase of the compressional *B*_{
z
} component changes approximately by *π*, the phase of the *B*_{
x
} component by ~ *π*/2, while the phase of the *B*_{
y
} component remains practically the same.

## 6. Conclusion

In this paper we would like to emphasize the possibility of resonant conversion of large-scale fast magnetosonic waves into localized slow magnetosonic oscillations. Several magnetospheric processes are discussed in which magnetosonic resonance may play an appreciable role.

1. The problem of the magnetosonic waves incidence on and reflexion from the plasma transition layer is solved. The conditions for the Alfven and magnetosonic resonances are realized when FMS waves pass from the solar wind into the magnetosphere through the magnetopause, as well as in the regions of well-developed ring current in the magnetosphere. Some field components of resonant oscillations have singularities on the resonance surfaces, in the absence of dissipation.

To regularize singularities near the resonant surfaces, effective decrements are introduced both for the Alfven and SMS waves as imaginary additions to the oscillation frequency. Dissipation of the Alfven oscillations is small enough. The decrement of the SMS oscillations, in contrast, can be rather large. Its value is determined by the plasma ion to electron temperature ratio near the resonant surface for the SMS wave. Oscillations reaching the resonant surface for SMS waves are absorbed completely in its neighbourhood.

It was shown that, if dissipation of oscillations near the resonant surfaces for SMS waves is large enough, the presence of strongly decaying resonant SMS oscillations in the system changes the wave field substantially. When *γ*_{
s
} ~ *ω* this influence extends up to the resonant surface for the Alfven waves. Specifically, the hodograph rotation direction is not reversed for monochromatic oscillations when we pass through the Alfven resonant surface. This phenomenon is applicable for identifying the presence of a resonance surface for SMS waves in the plasma configuration under consideration.

2. Magnetosonic resonance may be used to transfer the momentum from fast magnetosonic waves penetrating into the geotail lobes from the magnetosheath, to the background plasma. A rather wide (in frequency and wave numbers) range of waves is shown to exist for which the conditions for magnetosonic resonance are satisfied in the geotail. The highest concentration of resonance shells is achieved in the geotail regions adjacent to the magnetopause. The framework of quasilinear theory is used to obtain an approximate solution to the equation describing the evolution of the plasma ion distribution function under the impact of an MHD wave flux. It is shown that, on the time asymptotic (when *t* → ∞), an Earthward plasma flow moving at 50–150 km/s is established in the geotail lobe regions adjoining the magnetopause.

The characteristic time *τ* needed for the asymptotic regime to set in in the plasma flow in the regions adjoining the magnetopause was found to be comparable with the mean time interval ~3–6 h, during which the geotail configuration can be regarded as stable. It increased sharply on the inner magnetic shells, however. Thus, the FMS wave flux moving from the magnetosheath into the magnetopause transfers its momentum to plasma ions in the geotail lobes which is capable of forming an Earthward flow of magnetospheric convection. This mechanism may explain the formation of an Earthward flow of magnetospheric convection in the geotail lobes (on open field lines) during prolonged periods of the Northern IMF component.

3. The problem of the spatial distribution of resonant SMS oscillation field in an axi-symmetric magnetosphere with a dipole-like magnetic field is solved. These oscillations are peculiar in that their longitudinal structure represents a wave standing along field lines due to boundary conditions on the ends of field lines crossing the highly conductive ionosphere twice. On the long field lines examined in this work (at the *L* ~ 5–10 magnetic shells) the oscillation amplitude is maximum near the equatorial surface decreasing rapidly away from it. Therefore, resonant SMS oscillations can only be observed near the magnetospheric equatorial plane.

The frequency spectra of the fundamental harmonics of standing Alfven and SMS waves differ by about two orders of magnitude, so that an effective coupling between these two branches of MHD oscillations is impossible in a mildly disturbed magnetosphere. If dissipation of SMS waves is not too high—allowing their resonant structure to be conspicuous enough—it is the longitudinal components of the magnetic field and velocity of the oscillations that have the largest amplitude. Passing through the resonant peak, the phase of the compressional component *B*_{ǁ} changes by *π*.

## Declarations

### Acknowledgments

This work was partially supported by Program of presidium of Russian Academy of Sciences #22 and grants 12-02-0031-a, 13-05-90436-Ukr_f_a from the Russian Foundation for Basic Research.

## Authors’ Affiliations

## References

- Abramowitz, M. and I. A. Stegun (editors),
*Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series -55*, 832 pp., USA, 1964. - Afraimovich, E. L., I. K. Edemskiy, A. S. Leonovich, L. A. Leonovich, S. V. Voeykov, and Y. V. Yasyukevich, MHD nature of night-time MSTIDs excited by the solar terminator,
*Geophys. Res. Lett.*,**36**, L15106, doi:10.1029/2009GL039803, 2009.View ArticleGoogle Scholar - Agapitov, O., K.-H. Glassmeier, F. Plaschke, H.-U. Auster, D. Constantinescu, V. Angelopoulos, W. Magnes, R. Nakamura, C. W. Carlson, S. Frey, and J. P. McFadden, Surface waves and field line resonances: A THEMIS case study,
*J. Geophys. Res.*,**114**, A00C27, doi:10.1029/2008JA013553, 2009.Google Scholar - Akhiezer, A. I., I. A. Akhiezer, R. V. Polovin, A. G. Sitenko, and K. N. Stepanov,
*Plasma Electrodynamics*, Nauka, Moscow, 237 pp., 1974 (in Russian).Google Scholar - Borovsky, J. E., M. F. Thomsen, R. C. Elphic, T. E. Cayton, and D. J. McComac, The transport of plasma sheet material from the distant tail to geosynchronous orbit,
*J. Geophys. Res.*,**103**, 20297–20331, 1998.View ArticleGoogle Scholar - Chen, L. and A. Hasegawa, A theory of long period magnetic pulsation. 1. Steady state excitation of field line resonances,
*J. Geophys. Res.*,**79**, 1024–1032, 1974.View ArticleGoogle Scholar - Cheng, C. Z. and C. S. Lin, Eigenmode analysis of compressional waves in the magnetosphere,
*Geophys. Res. Lett.*,**14**, 884–887, 1987.View ArticleGoogle Scholar - Cheng, C.-C., J.-K. Chao, and T.-S. Hsu, Evidence of the coupling of a fast magnetospheric cavity mode to field line resonances,
*Earth Planets Space*,**50**, 683–697, 1998.View ArticleGoogle Scholar - Cheremnykh, O. K., A. S. Parnowski, and O. S. Burdo, Ballooning modes in the inner magnetosphere of the Earth,
*Planet. Space Sci.*,**52**, 1217–1229, 2004.View ArticleGoogle Scholar - Goldstein, M. L., D. A. Roberts, and W. H. Matthaeus, Magnetohydrody-namic turbulence in the solar wind,
*Ann. Rev. Astron. Astrophys.*,**33**, 283–325, 1995.View ArticleGoogle Scholar - Gonzalez, A. G. and J. Gratton, Magnetoacoustic surface gravity waves,
*Sol. Phys.*,**134**, 211–232, 1991.View ArticleGoogle Scholar - Inhester, B., Numerical modelling of hydromagnetic wave coupling in the magnetosphere,
*J. Geophys. Res.*,**92**, 4751–4756, 1987.View ArticleGoogle Scholar - Klimushkin, D. Yu., Theory of azimuthally small-scale hydromagnetic waves in the axisymmetric magnetosphere with finite plasma pressure,
*Ann. Geophys.*,**16**, 303–321, 1998.View ArticleGoogle Scholar - Klimushkin, D. Yu. and P. N. Mager, On the spatial structure and dispersion of slow magnetosonic modes coupled with Alfven modes in planetary magnetospheres due to field line curvature,
*Planet. Space Sci.*,**56**, 1273–1279, 2008.View ArticleGoogle Scholar - Korn, G. A. and T. M. Korn,
*Mathematical Handbook for Scientists and Engineers*, McGraw-Hill Book Company, 1968. - Lee, D.-H. and R. L. Lysak, Monochromatic ULF wave coupling in the dipole model: The impulsive excitation,
*J. Geophys. Res.*,**94**, 17097–17109, 1989.View ArticleGoogle Scholar - Leonovich, A. S., MHD-instability of the magnetotail: Global modes,
*Planet. Space Sci.*,**59**, 402–411, 2011a.View ArticleGoogle Scholar - Leonovich, A. S., A theory of MHD instability of an inhomogeneous plasma jet,
*J. Plasma Phys.*,**77**, 315–337, 2011b.View ArticleGoogle Scholar - Leonovich, A. S. and D. A. Kozlov, Alfvénic and magnetosonic resonances in a nonisothermal plasma,
*Plasma Phys. Control. Fus.*,**51**, 085007, doi:10.1088/0741-3335/51/8/085007, 2009.View ArticleGoogle Scholar - Leonovich, A. S. and V. A. Mazur, Resonance excitation of standing Alfvén waves in an axisymmetric magnetosphere (monochromatic oscillations),
*Planet. Space Sci.*,**37**, 1095–1108, 1989.View ArticleGoogle Scholar - Leonovich, A. S. and V. A. Mazur, Penetration to the Earth’s surface of standing Alfvén waves excited by external currents in the ionosphere,
*Ann. Geophys.*,**14**, 545–556, 1996.View ArticleGoogle Scholar - Leonovich, A. S. and V. A. Mazur, A model equation for monochromatic standing Alfvén waves in the axially symmetric magnetosphere,
*J. Geophys. Res.*,**102**, 11443–11456, 1997.View ArticleGoogle Scholar - Leonovich, A. S. and V. V. Mishin, An energy flux of magnetosonic waves from the solar wind into magnetosphere,
*Geomagnetizm i aeronomiya*,**39**, 52–58, 1999 (in Russian).Google Scholar - Leonovich, A. S. and V. A. Mazur, Structure of magnetosonic eigenoscillations of an axisymmetric magnetosphere,
*J. Geophys. Res.*,**105**, 27707–27716, 2000.View ArticleGoogle Scholar - Leonovich, A. S. and V. A. Mazur, On the spectrum of magnetosonic eigenoscillations of an axisymmetric magnetosphere,
*J. Geophys. Res.*,**106**, 3919–3928, 2001.View ArticleGoogle Scholar - Leonovich, A. S., V. V. Mishin, and J. B. Cao, Penetration of magnetosonic waves into the magnetosphere: Influence of a transition layer,
*Ann. Geophys.*,**21**, 1083–1093, 2003.View ArticleGoogle Scholar - Leonovich, A. S., V. A. Mazur, and J. B. Cao, Self-consistent model of a dipole-like magnetosphere with an azimuthal solar wind flow,
*J. Plasma Phys.*,**70**, 99–111, 2004.View ArticleGoogle Scholar - Leonovich, A. S., D. A. Kozlov, and V. A. Pilipenko, Magnetosonic resonance in a dipole-like magnetosphere,
*Ann. Geophys.*,**24**, 2277–2289, 2006.View ArticleGoogle Scholar - Leonovich, A. S., D. A. Kozlov, and I. K. Edemskiy, Standing slow magnetosonic waves in a dipole-like plasmasphere,
*Planet. Space Sci.*,**58**, 1425–1433, 2010.View ArticleGoogle Scholar - Marsh, E. and C.-Y. Tu, Spectral and spatial evolution of compressive turbulence in the inner solar wind,
*J. Geophys. Res.*,**95**, 11945–11956, 1990.View ArticleGoogle Scholar - Matthaeus, W. H. and M. L. Goldstein, Measurement of the rugged invariants of magnetohydrodynamics turbulence in the solar wind,
*J. Geophys. Res.*,**87**, 6011–6028, 1982.View ArticleGoogle Scholar - Mazur, V. A. and A. S. Leonovich, ULF hydromagnetic oscillations with the discrete spectrum as eigenmodes of MHD-resonator in the near-Earth part of the plasma sheet,
*Ann. Geophys.*,**24**, 1639–1648, 2006.View ArticleGoogle Scholar - Miles, A. J. and B. Roberts, On the properties of magnetoacoustic surface waves,
*Sol. Phys.*,**119**, 257–278, doi:10.1007/BF00146179, 1989.View ArticleGoogle Scholar - Nakariakov, V. M., L. Ofman, E. E. DeLuca, B. Roberts, and J. M. Davila, TRACE observation of damped coronal loop oscillations: Implications for coronal heating,
*Science*,**285**, 862, doi:10.1126/science.285.5429.862, 1999.View ArticleGoogle Scholar - Ohtani, S., A. Miura, and T. Tamao, Coupling between Alfven and slow magnetosonic waves in an inhomogeneous finite-
*β*plasma—I. Coupled equations and physical mechanism,*Planet. Space Sci.*,**37**, 567–577, 1989.View ArticleGoogle Scholar - Pilipenko, V. A., ULF waves on the ground and in space,
*J. Atmos. Terr. Phys.*,**52**, 1193–1209, 1990.View ArticleGoogle Scholar - Radoski, H. R., A theory of latitude dependent geomagnetic micropulsations: The asymptotic fields,
*J. Geophys. Res.*,**79**, 595–604, 1974.View ArticleGoogle Scholar - Rankin, R., K. Kabin, J. Y. Lu, I. R. Mann, R. Marchand, I. J. Rae, V. T. Tikhonchuk, and E. F. Donovan, Magnetospheric field-line resonances: Ground-based observations and modeling,
*J. Geophys. Res.*,**110**, A10S09, doi:10.1029/2004JA010919, 2005.Google Scholar - Rankin, R., K. Kabin, and R. Merchand, Alfvénic field line resonances in arbitrary magnetic field topology,
*Adv. Space Res.*,**38**, 1720–1729, 2006.View ArticleGoogle Scholar - Sergeev, V. A. and N. A. Tsyganenko,
*The Earth’s Magnetosphere*, Moscow, Nauka, 1980 (in Russian).Google Scholar - Sizonenko, V. L. and K. N. Stepanov, Quasilinear theory of plasma oscillations with linear dispersion,
*Ukrainian J. Phys.*,**13**, 876–878, 1968.Google Scholar - Southwood, D. J., Some features of field line resonances in the magnetosphere,
*Planet. Space Sci.*,**22**, 483–492, 1974.View ArticleGoogle Scholar - Southwood, D. J. and M. A. Saunders, Curvature coupling of slow and Alfvén MHD waves in a magnetotail field configuration,
*Planet. Space Sci.*,**33**, 127–134, 1985.View ArticleGoogle Scholar - Tamao, T., Transmission and coupling resonance of hydromagnetic disturbances in the non-uniform Earth’s magnetosphere,
*Sci. Rep. Tohoku Univ.*,**5, 17**, 43–54, 1965.Google Scholar - Titheridge, J. E., Temperatures in the upper ionosphere and plasmasphere,
*J. Geophys. Res.*,**103**, 2261–2277, 1998.View ArticleGoogle Scholar - Walker, A. D. M., Theory of magnetospheric standing hydromagnetic waves with large azimuthal wave number. 1. Coupled magnetosonic and Alfvén waves,
*J. Geophys. Res.*,**92**, 10039–10045, 1987.View ArticleGoogle Scholar - Yumoto, K., Characteristics of localized resonance coupling oscillations of the slow magnetosonic wave in a non-uniform plasma,
*Planet. Space Sci.*,**33**, 1029–1036, 1985.View ArticleGoogle Scholar