We consider a one-dimensionally non-uniform cylinder model of the magnetosphere where the magnetic field lines are concentric circles with radius *R.* All the plasma parameters (particle number density *n*(*x*), temperature *T*(*x*), and ambient magnetic field *B*(*x*)) vary across the magnetic shells only. Although this model is rather different from the real magnetosphere, it retains all the basic features necessary for the ballooning mode formation: magnetic field line curvature and inhomogeneous plasma pressure. The radius *R* has the role of the radial coordinate *x*. The coordinate *y* along the cylinder axis corresponds to the azimuthal coordinate in the magnetosphere, the coordinate *l*_{∥} is along the magnetic field line.

Following Chen and Hasegawa (1991), we assume that the magnetospheric plasma consists of a core component which is mainly responsible for the mass density, but has a small plasma pressure, and a hot component which has a relatively small concentration, but large temperature, thus providing a high plasma pressure and large plasma to magnetic pressure ratio, *β* ∼ 1. The main contribution to the plasma pressure is by hot ions. The equilibrium distribution is assumed to be Maxwellian and isotropic, *F = n*(2*πT*)^{−3/2}*e*^{−ε/T}, where *ε* = *υ*^{2}/2 is the particle energy per unit mass. In this case, the equilibrium condition is written as

where *κ*_{
B
} = *B′/B* and *κ*_{
P
} = *P′/P* are the normalized magnetic field and pressure gradients, *κ*_{
R
}*= R*^{−1} is the field line curvature, prime denotes differentiation over the radial coordinate *x*. Note that the equilibrium current is related to the pressure gradient as , that is, *κ*_{
P
} = *jB/cP*; a negative *κ*_{
P
} value corresponds to the westward current. In the isotropic case, specific modes related to temperature anisotropy, such as the drift anisotropy mode (Pokhotelov *et al.*, 1985) and the drift mirror mode (Woch *et al.*, 1988; Klimushkin and Mager, 2012), are ruled out.

The dependence of the wave variables on the spatio-temporal coordinates has the form exp [−*iωt* + *ik*_{
y
}*y + ik*_{
∥
}*l*_{∥}], where *ω* is the wave frequency, *k*_{
y
} and *k*_{∥} are the azimuthal and parallel components of the wave vector. The radial component of the wave vector is assumed to be zero. The ballooning approximation assumes that waves with *ω* ≪ *k*_{
y
}υ_{
A
} are considered, which rules out the fast magnetosonic mode.

In the gyrokinetic approach, the wave’s electromagnetic field can be described by three variables (Chen and Hasegawa, 1991). The first one (Ψ) is related to the longitudinal vector potential as follows *A*_{∥} = −(*ic*/*ω*)∂Ψ*/*∂*l*_{∥}. The second variable (*b*) is proportional to the longitudinal magnetic field of the wave *b*_{∥} as *b* = *ωb*_{∥}*/c*. The third variable is electrostatic potential *φ*. It was shown (e.g., by Chen and Hasegawa, 1991) that a small fraction of cold electrons (that is, electrons with a lower thermal speed than the parallel phase velocity of the wave) shorts out the longitudinal wave’s electric field (*E*_{∥} = −*ik*_{∥} (*φ* − Ψ) = 0); thus, the potential *φ* is not an independent variable, and the variables Ψ and *b* fully describe the electromagnetic field of the perturbation. If the electrostatic component of perturbation is considered, a great variety of wave branches and instabilities appear (Marchenko *et al.*, 1988), but all of them become irrelevant if at least small fraction of cold electrons occurs in plasma.

The variables Ψ and *b* are related by the gyrokinetic equations (Antonsen and Lane, 1980; Catto *et al.*, 1981; Chen and Hasegawa, 1991) which can be conveniently written as follows:

Here the compressional mode operator is

where is an argument of the Bessel function *J*_{1}, *ω*_{
c
} is gyrofrequency. The operator is

Here

is the particle drift frequency in an inhomogeneous magnetic field.

The transverse Alfvén ballooning operator is

where

Finally, the coupling operator is

The terms *a*_{
M
}, *a*_{
A
}, *a*_{
c
} are responsible for wave-particle interaction. In our study, we search for hydrodynamic instability, therefore relatively weak resonant kinetic effects are neglected. In the quasi-MHD limit, the wave frequency is higher than the magnetic drift frequency, *ω* ≫ *ω*_{
d
}, the wave phase velocity is much larger than the ion thermal speed, *ω/k*_{∥} ≫ υ_{∥}, and the wave perpendicular scale is much larger than the ion Larmor radius, . Then the Alfvén ballooning operator is recast as

where the second and third terms in brackets (proportional to *β*) can be called the ballooning terms. The coupling and the compressional mode operators are written as

Here we have introduced the plasma Larmor drift frequency

Note that this value is different from the diamagnetic drift frequency obtained in the two-fluid MHD approach (see, e.g., equation (8) in Miura *et al.*, 1989).

Expressing *b* from the first equation of the system (2) and substituting it into the second equation, we obtain

It is worth noting that (as seen from Eqs. (5) and (6)) the second term in brackets (caused by the mode coupling) generally has the same order of magnitude as the ballooning additions to the Alfvén operator, which justifies the necessity of taking into account the coupling between the Alfvén and drift compressional modes. Further, Eq. (9) leads to the dispersion relation for the coupled Alfvén ballooning and drift compressional modes in a non-uniform plasma:

where and Ω_{
M
} = ω_{*}*β*/(1 + *β*) (the drift compressional frequency) is the frequency which turns the operator *L*_{
M
} into zero.