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 (*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

where *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}.

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** = (*E*_{1}, *E*_{2}, 0) this expansion has the form

where Δ_{⊥} = (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

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)

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:

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

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:

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)

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∇_{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

where the function *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.

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

where . We assume that, in the main order, the functions *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

where Ω_{
SN
} = *π**N*/*t*_{
s
},

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

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* = 6.6*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).

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
}(*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).

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
}(*x*^{1}, *x*^{3}) 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
}(*x*^{1}):

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

be used to approximate Ω_{
SN
}(*x*^{1}) 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

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)

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 *k*^{2} in the denominator. This yields

In the asymptotic ǀ*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

Thus, the amplitude of the resonant SMS oscillations away from the resonant plane decreases asymptotically as ∝ ǀ*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

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* = 6.6*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.