## 1. Introduction

[2] Ultralow frequency (ULF) compressional waves with periods between 2 and 20 minutes and short parallel wavelengths have been observed by many authors in different regions of the Earth's magnetosphere where β_{i} (the ratio of ion to magnetic field pressure) is on the order of unity [*Takahashi et al.*, 1987a, 1987b; *Zhu and Kivelson*, 1994, 1991]. While drift-mirror modes are typically invoked to explain some of these observations, the observed temperature anisotropies [*Zhu and Kivelson*, 1991; *Takahashi et al.*, 1987a] often, however, did not exceed the threshold values necessary for drift-mirror modes to be unstable [*Hasegawa*, 1969]. Recently, drift compressional modes (DCM), first studied by *Rosenbluth* [1981] in the context of magnetic mirror fusion research, have been proposed as an alternative excitation mechanism for these ULF compressional wave events [*Crabtree et al.*, 2003]. DCM are similar to drift-mirror modes; that is, they are slow magnetic compressional waves that occur in high β plasmas. DCM, however, have the important property that they do not require anisotropy to be driven unstable. The underlying instability mechanism is a kinetic wave-particle resonance between the perturbed magnetic field compression δ*B*_{∥} and the magnetic gradient-curvature drifts of ions; tapping the free energy associated with plasma nonuniformities.

[3] Previous theoretical treatments of compressional modes [*Cheng and Lin*, 1987; *Ng et al.*, 1984; *Crabtree et al.*, 2003] generally invoke the following assumptions: 1) δ*B*_{∥} is the dominant wave perturbation, and 2) the ion gyroradius (ρ_{i}) is small; i.e., *k*_{⊥}ρ_{i} ≪ 1 with *k*_{⊥} being the wave number perpendicular to the confining magnetic field. While 1 is motivated by observations, its validity regime has never been theoretically established. Specifically, one needs to delineate the conditions under which couplings to the electrostatic as well as field-line-bending shear Alfvén perturbations can be systematically neglected. As to 2, note that in the *k*_{⊥}ρ_{i} ≪ 1 limit, the complex mode frequency scales with *k*_{⊥}ρ_{i} [*Crabtree et al.*, 2003]; that is, the DCM growth rate increases linearly with *k*_{⊥}ρ_{i}. Thus, finite *k*_{⊥}ρ_{i} effects need to be included in order to assess the crucial issue of the parameter regime for the most unstable modes. These considerations constitute the primary motivation of the present work.

[4] In the present work, we have employed the linear gyrokinetic equations [*Chen and Hasegawa*, 1991; *Antonsen and Lane*, 1980; *Horton et al.*, 1983] and carried out a non-local eigenmode stability analysis along the field line. The derived dispersion relation is valid for finite *k*_{⊥}ρ_{i} and indicates that the most unstable DCM tends to occur at short azimuthal wavelengths; *k*_{ϕ}ρ_{i} ∼ O(1). Our theoretical analysis, furthermore, clearly establishes that couplings between δ*B*_{∥} and the electrostatic and shear Alfvén perturbations can be systematically neglected due to, respectively, cold electrons and narrow mode localization along the field line. We find that DCM can become unstable when the density gradient and proton temperature gradient are in opposite directions or when there is drift reversal, i.e., when the proton magnetic guiding center drift is opposite to its diamagnetic drift. We also find that under certain conditions the magnetic field model of *Tsyganenko* [2002a, 2002b] allows the drift reversal of protons on the inner edge of the magneto-tail current system and, thus, renders DCM unstable.