Magnetosonic resonances in the magnetospheric plasma

Let us consider the problem of magnetosonic resonance in a simple one-dimensionally inhomogeneous transitional plasma layer. Such a layer—separating the magnetosphere from the solar wind—is formed at the magetopause, for example. Let us introduce a Cartesian system of co-ordinates (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.

Let us denote 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 (ik _y y+ik _z z−iωt), 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:

(6)

(7)

(8)

where

is the Alfven speed, is sound speed in plasma. For displacement ζ, we have the equation

(9)

where ,

(10)

It is evident from (9) that, in the WKB approximation, is the 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 x₀₁, x₀₂, 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 Ω²(X _A ) = 0, located in the opacity region in the interval (x₀₁, x₀₂). 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 x₀₁, 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.

Curve 2 in Fig. 1 corresponds to the case when the transparency region for SMS waves extends up to ∞, and there is no resonance surface for the Alfven waves. Analyzing expression (10) helps define the two ranges of wave field and plasma parameters corresponding to curves 1 and 2 in Fig. 1, when

(11)

where β* = S²/A², and are the two roots of the biquadratic equation

corresponding to the solutions with the positive/negative radical. The first describes a FMS wave, and the second a SMS wave incident on the transition layer.

Let us describe the Alfven speed profile A(x) as a function of the form

(12)

where 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).

Let us define the solution of (9) near the resonant surface 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 + iγ_s, in the denominator of (10). Equation (9) may then be rewritten as

(13)

nearx = 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

(14)

where I₀(z), K₀(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 C₁ =0. To the right of resonance point ξ_s = 0 the solution (14) has the form

(15)

where is the Hankel function of the second kind, which, as ξ_s → ∞, has an asymptotic representation, . When ξ_s → 0, the solution (14)–(15) is

(16)

i.e. it has the same logarithmic singularity on the resonance surface, as in the region of the Alfven resonance.

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 I₀(z), whose coefficient C₁ = 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 (x₀₁, x₀₂) into the SMS-wave transparency region (x _s , x₀₁), 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 (x₀₁, x₀₂) (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 K₀(z), to the right of ξ_s = 0: C₂ = 0. Since the function I₀(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.

Figure 2 demonstrates the distribution of ∂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 (γs ~ 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 ≈ 2T _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.

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.

Let us consider a model magnetotail in the form of an in-homogeneous plasma cylinder as shown in Figs. 3–4. The plasma distribution over radius corresponds to the geotail lobes. This model does not explicitly take into account the plasma sheet. Its presence is simulated by the radial distribution of the Alfven and SMS velocity. Moving away from the cylinder axis, they change from values characteristic of the plasma sheet to those typical of the geotail lobes. The radial distribution of the Alfven speed is plotted in Mazur and Leonovich (2006) based on satellite data for the distribution of plasma concentration and magnetic field in the magnetosphere (Sergeev and Tsyganenko, 1980; Borovsky et al., 1998). Since the main results of that study concern the region of open field lines, the presence of plasma sheet should not be an essential element in the calculations.

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 v₀, 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.

As can be seen here, a maximum concentration of resonance surfaces for SMS waves in the geotail lobes is reached near the transitional layer. Using the same system of MHD equations (1)–(4) for cylindrical harmonics of the form exp(ik _z z + imφ − iωt), where m = 0, 1, 2, 3 … is the azimuthal wave number, produces the following equation for displacement ζ:

(17)

where is the Doppler-modified oscillation frequency,

(18)

and . We substitute in the denominator of (18) to take into account SMS wave dissipation.

Figure 5 shows the radial structure of two monochromatic harmonics. There are resonance surfaces for SMS waves in the magnetosphere (Fig. 5(a)) for one of them, but not for the other (Fig. 5(b)). This figure presents the unity-normalized structure of the derivative dζ/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 ≈ 3T _i ), therefore, we assume in the magnetosheath. In the tail lobes, on the contrary, the plasma ions are hotter than the electrons (T _i ≈ 8T _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.

Consider the problem of the ion distribution function in magnetospheric plasma transformed under the impact of the MHD wave flux from the magnetosheath. We use kinetic theory equations in the locally quasilinear approximation. This means that, on each resonance shell inside the geotail, we will consider the distribution function in the same manner as if it were defined over the entire space, while assuming the MHD oscillation field to correspond to this shell. We assume the plasma to consist of the hydrogen ions and electrons. The asymptotic (when t → ∞) equation for the ion distribution function f(v_ǁ) in the presence of SMS waves has the following form (see Appendix B)

(19)

where we use the Maxwell distribution function

(20)

as the initial condition for solving (19). Presumably, the plasma acquires this state in the geomagnetic tail lobes during the substorm recovery phase. Here n₀ is the plasma ion concentration, is the thermal velocity of plasma ions on the magnetic shell under scrutiny,

(21)

is the ion diffusion coefficient in the velocity space, 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.

Multiplying (19) by on the left and integrating over v_ǁ yields

(22)

Hence, if a new equilibrium state is reached at the asymptotic 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(k₁, k₂) and k _z > max(k₁, k₂), where k_1,2 = ω/v_1,2, v₁ = v₀ + S _w , v₂ = v₀ − 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 v₀ > 200 km/s we have v₁ > v₂ > 0 and the solar wind is opaque when 0 < k₁ < k_z < k₂. 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 v₂ < v_ǁ < v₁. The area corresponds to “downstream”, while the two other areas to “upstream” FMS waves in the solar wind.

The level of the plateau in each of these areas is determined by the condition that the total number of particles should remain the same and can be expressed through the following relation

where 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.

The average velocity of plasma resulting from its interaction with MHD waves is determined by the equation

Obviously, the contribution from symmetric (with respect to v_ǁ = 0) parts of is zero. Figure 6 shows the distribution of v₀(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 v₀(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.

The characteristic time 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

(23)

For the function, let us choose the Maxwell distribution of the form (20).

To calculate the diffusion coefficient * , it is necessary to specify the spectrum of MHD waves in the magnetosheath. There are many onboard observations of variations in the solar wind parameters. They are generally stochastic oscillations with a “white noise” spectrum (Matthaeus and Goldstein, 1982). Here we shall use the model spectrum function constructed in Leonovich and Mishin (1999)

(24)

where 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

where Θ(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

where is the mean square of the amplitude of the B _r component of the solar wind oscillation field at r = 2r _m . In our calculations we assume .

Figure 7 shows the distribution over the radius of the characteristic time 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 0.8r _m < r < r _m (for the v₀ = 400 km/s solar wind) and 0.85r _m < r < r _m (for the v₀ = 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.

Let us now address the features of magnetosonic resonance in a dipole-like magnetosphere (Fig. 8). Unlike the media models discussed above, resonance in this case arises on closed field lines. The result is that, due to the boundary conditions imposed on the ionosphere, SMS oscillations form standing waves along field lines.

Let us introduce a curvilinear orthogonal coordinate system (x¹, x², x³), in which the coordinate x³ is along the field line, x¹ is across the magnetic shells, and the azimuthal x² coordinate completes the right hand coordinate system. The squared length element in this coordinate system is found as

where g _i (i = 1, 2, 3) are metric coefficients. We assume that the plasma and magnetic field are homogeneous along the azimuthal coordinate x².

It is convenient to describe the MHD field components via electromagnetic potentials. According to the Helmholtz expansion theorem (Korn and Korn, 1968), an arbitrary vector field, at any point of which its first derivative is determined, can be represented as a sum of the potential and vortex fields. For the two-dimensional vector E = (E₁, E₂, 0) this expansion has the form

where Δ_⊥ = (Δ₁, Δ₂) 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, ψ₃ = ψ). Using the linearized system (1)–(4) we express the perturbed magnetic field components through the potentials φ and ψ as

(25)

After some transformation, the system of linearized equations (1)–(4) is reduced to a system of two related equations for the potentials φ and ψ (see Leonovich et al, 2006)

(26)

(27)

where . In (26) we introduced the toroidal and poloidal longitudinal operators, as follows

and in (27) the longitudinal operator

and the differential operators analogous to the Laplace operator:

In a homogeneous plasma, the right-hand parts of these equations vanish. The operator in the left-hand part of (26) then provides the dispersion equation for the Alfven waves , where is the field-aligned component of the wave vector. The operator in the left-hand part of (27) yields the dispersion equation for the slow and fast magnetosonic waves:

(28)

where is the squared total wave vector, and is the squared transverse wave vector component. Thus, Alfven oscillations are described by the scalar potential 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

Here the plus/minus sign corresponds to the FMS/SMS waves. If one of the inequalities 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 .

In an inhomogeneous plasma, the right-hand parts of (26) and (27) describe the interaction of the Alfven and magnetosonic modes. Although the potential 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 x³ in the operator in (27), we obtain an equation that describes the FMS wave field far from the resonant surface:

(29)

An approximate solution of (29) was found in Leonovich and Mazur (2000, 2001) where feedback from the mode driven by the FMS wave was shown to be small, thus making it possible to use the decoupled equation (29) to describe FMS oscillations throughout the entire region of their existence, even inside the resonant region.

For the magnetosphere the typical eigenfrequencies of fundamental harmonics of standing Alfven and SMS waves, as determined by the background plasma, differ considerably. This means that interaction between the Alfven and SMS waves, possible in a finite-pressure plasma embedded in a curved magnetic field (Southwood and Saunders, 1985), is negligible. While examining the SMS oscillation structure described by Eq. (27) one may put φ = 0 in its right-hand part. Therefore, in the vicinity of the resonant surface we obtain the equation for the resonant SMS oscillations:

(30)

The right-hand part of (30) represents the driver— monochromatic FMS wave field—that will be treated as a function known from the solution of (29). At the frequencies in question, the magnetosphere as a whole is an opacity region for FMS. If we assume the source of FMS oscillations to be either outside, or at the boundary of, the magnetosphere, their amplitude decreases exponentially inside the magnetosphere on a scale proportional to 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.

The boundary condition for SMS waves on the ionosphere, taking into account its finite conductivity, has the form (Leonovich and Mazur, 1996; Leonovich et al., 2006)

(31)

where the signs “±” refer to the intersection points of the field lines with the Northern and Southern ionospheres; ℓ is the coordinate measured along the field line from the equator, ,

is the height-integrated Pedersen conductivity of the ionosphere.

As we will see, the typical scale of resonant SMS oscillations across magnetic shells is much smaller than their longitudinal wave length, ǀ∇₁ψ _s /ψ_s ≫ ǀ∇₃ψ_s/ψ_sǀ. Therefore, a solution to (30) may be sought using the method of different scales, representing the potential ψ_S as

(32)

where the function U (x¹) describes, in the main order, the small-scale transverse structure of oscillations along the x¹ coordinate, whereas the function S(x¹, x³) describes the oscillation structure along magnetic field lines. The typical scale of S(x¹,x³) along x¹ is assumed to be much larger than the scale of U(x¹). The small correction term h(x¹, x³) describes the oscillation structure in higher orders of the perturbation theory.

An equation for the longitudinal structure can be obtained if one retains in (30) only the main-order terms of perturbation theory:

(33)

where . We assume that, in the main order, the functions S(x¹, x³) satisfy the homogeneous boundary conditions in the ionosphere: . The solution of (33), with such boundary conditions, is a series of eigenfunctions S _N (x¹,ℓ) and corresponding eigenfrequencies Ω _SN ( x¹), 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

(34)

where Ω _SN = πN/t _s ,

(35)

is the travel time along a field line between the magnetocon-jugate ionospheres at SMS wave speed. The eigenfunctions (34) are normalized by the following condition

(36)

Only numerical solutions can be found to equations (33), however, describing the longitudinal structure of the fundamental and low-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 = 6.6R _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).

Figure 10(a) shows the field-aligned structures of the first three harmonics of standing SMS waves. The fundamental harmonics of standing SMS waves differ from their WKB representation (34) radically. The main peculiarity of fundamental SMS harmonics is a rapid decrease in amplitude when approaching the ionosphere. Such a structure of S _N (x¹, x³) 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 F₂-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 > 2T _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).

Let us now address the structure of resonant SMS oscillations across magnetic shells. Let us pre-multiply (30) by and integrate along the field line between the conjugate ionospheres. The correction term h _N (x¹, x³) in (32) satisfies the following ionospheric boundary condition (see (31))

Given this boundary condition and (33), we obtain the following equation for function U _N (x¹):

(37)

where

Here, the damping decrement, 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.

Let the function Ω _SN (x¹) change monotonically, so that a linear dependence

(38)

be used to approximate Ω _SN (x¹) in the vicinity of the resonant surface.

This approximation is valid at , where is the typical scale of the Ω _SN variation at Substituting (38) into (37) and introducing the dimensionless variable , where , we obtain an equation describing the transverse structure of magnetosonic resonance

(39)

The coefficients of this equation are: is the dimensionless width of the resonance, , and 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).

Solution to (39) has the form (see Leonovich and Mazur, 1997)

(40)

where

As 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 k² in the denominator. This yields

(41)

In the asymptotic ǀξǀ → ∞ 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

Thus, the amplitude of the resonant SMS oscillations away from the resonant plane decreases asymptotically as ∝ ǀξǀ⁻¹. This behavior satisfies the boundary conditions on the x¹ 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

where . The longitudinal (compressional) magnetic component 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 = 6.6R _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.