A dispersion relation for ultralow frequency drift compressional modes in high pressure plasmas with finite gyroradius is derived from the linearized gyrokinetic-Maxwell equations with cold electrons and narrow eigenmode localization width along the field line. The dispersion relation demonstrates instability under two different conditions: 1) when the density gradient and proton temperature gradient are in opposite directions, or 2) when the magnetic guiding center drift is reversed with respect to the proton diamagnetic drift, i.e., drift reversal, which could occur during periods of strong magnetospheric disturbance. Furthermore, it is found that the most unstable modes have short azimuthal wavelengths comparable to the proton gyroradius.
 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 βi (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  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 δB∥ and the magnetic gradient-curvature drifts of ions; tapping the free energy associated with plasma nonuniformities.
 Previous theoretical treatments of compressional modes [Cheng and Lin, 1987; Ng et al., 1984; Crabtree et al., 2003] generally invoke the following assumptions: 1) δB∥ is the dominant wave perturbation, and 2) the ion gyroradius (ρi) is small; i.e., k⊥ρi ≪ 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⊥ρi ≪ 1 limit, the complex mode frequency scales with k⊥ρi [Crabtree et al., 2003]; that is, the DCM growth rate increases linearly with k⊥ρi. Thus, finite k⊥ρi 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.
 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⊥ρi and indicates that the most unstable DCM tends to occur at short azimuthal wavelengths; kϕρi ∼ O(1). Our theoretical analysis, furthermore, clearly establishes that couplings between δB∥ 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.
2. Theoretical Formulation
 Here we consider a two component isotropic plasma consisting of trapped protons and electrons in an axisymmetric magnetic field. A typical proton energy of Ti ≈ 10 KeV and electron energy of Te ≈ 1 KeV give a small parameter τ = Te/Ti ∼ O(δ) ∼ O(10−1) to be used in the following ordering scheme. We take the proton plasma pressure to be on the same order as the magnetic pressure and define the dimensionless parameter βi ≡ 2μ0nTi/B02 ∼ O(1) where B0 is the minimum B along the field line. We consider k⊥ρi ∼ O(1) and the frequency of interest is on the order of the proton magnetic precessional drift frequency and the proton diamagnetic drift frequency. That is, ω ∼ ω*i ∼ ωDi where ωDi = kϕvDi, ω*i = kϕv*i and vDi and v*i are, respectively, the magnetic drift and diamagnetic drift velocities [Hazeltine and Waelbroeck, 1998]. To be more quantitative, let us assume, for the outer magnetosphere, a typical proton energy of 10 KeV (or thermal speed of 0.15 RE/s) and a gradient scale length Ln ∼ L ∼ 8RE. Taking kϕρi ≃ 1, the corresponding DCM frequencies are then estimated to be 19 mHz; consistent with the range of observed frequencies.
 We further assume a narrow mode width along the magnetic field, specifically k∥Ln ∼ O(δ−1/2). Using Ln ∼ 8RE the mode width Δ ∼ 1/k∥ is approximately 2.5RE. Thus, 10 KeV protons have a bounce frequency, ωbi, of approximately 60 mHz making ω/ωbi ∼ O(δ1/2) and ω/ωbe ∼ O(δ). This ordering allows for a bounce-averaged solution to the linearized gyrokinetic equation (GKE) [Rutherford and Frieman, 1968; Wong et al., 2001]. It is important to point out here that because the eigenmode structure is still an unknown the ordering k∥Ln ∼ O(δ−1/2) must be demonstrated to be consistent with the solutions a posteriori.
 The three components of the perturbed electromagnetic field are δϕ the perturbed electrostatic potential, δψ a quantity related to the perturbed parallel component of the magnetic vector potential (iωδA∥ = ∂δψ/∂l), and δB∥ the parallel component of the perturbed magnetic field. The field variables δϕ and δψ are measured in units of Te/e and δB∥ in units of B0. We adopt the following dimensionless units: l the distance along the magnetic field line is measured by Ln, B is measured by B0 the minimum B along the field line, the frequencies are all measured in units of vti/Ln where vti = ; the velocity variables are taken to be the energy ε = msv2/(2Ts), the pitch angle λ = v⊥2/(v2B) and the sign of the parallel velocity = ±1. Using the WKB assumption and high azimuthal wave numbers for the perturbed quantities (δϕ, δψ, δB∥, and δF) the perturbed distribution function, δF, for species s of charge qs is given by
u∥ = ηs = d log(Ts)/d log(n), Lk = k⊥ρs cos (ξ), and ξ is the gyrophase. Here J0,1 = J0,1(k⊥ρs) are the finite gyroradius Bessel functions.
 The bounce-averaged solution to the GKE for the protons to O(δ) may be written,
where the represents bounce-averaging defined by, = (1/τb) ∮ dl(⋯)/ where τb = ∮ dl and the integrals are taken between turning points. The GKE for electrons may also be solved by bounce-averaging. Using the small electron gyroradius to expand the Bessel functions and the cold electrons to remove terms like ω*e/ω the electron solution accurate to O(δ) is δKe = . If we measured the potentials with Ti/e instead of Te/e the terms involving δϕ and δψ in the proton solution (equation (4)) would be O(1). However, in the relevant Maxwell equations these terms would then be cancelled by their electron counterparts.
 It is convenient to follow Chen and Hasegawa  and define two quantities using the solutions to the GKE: δKs1 = (1/2) ∑δKs() and δKs2 = (1/2)∑δKs(). In our ordering scheme δKi,e2 is O(δ) or smaller, and δKi,e1 = δKi,e. Defining the following notation for the velocity integrals, , we can write the three coupled field equations: 1) the quasi-neutrality condition
2) the perpendicular component of Ampére's law
and 3) the vorticity equation
 Note that in equation (7) the term involving the trapped electron contribution, δKe, is multiplied by the factor ωDe/ω ∼ O(δ), thus the trapped electron contribution will be small. The trapped protons only contribute through δB∥ because the terms involving δϕ and δψ are multiplied by τ (see equation (4)). In Mϕψ the first order terms cancel and the remaining terms are order τ ∼ O(δ), thus the coupling of the shear Alfvén branch to the electrostatic branch, Mϕψ, is small. On the other hand the coupling to the compressional branch, MBψ, is important, having terms of O(1). Writing the vorticity equation to O(δ) we find
Note that formally, Ln/LB = ∂l log B ∼ O(δ−1/2) for high β geometries (e.g., take B ∝ l2, then ∂l log B ≈ 2) and that to leading order the right hand side only has terms involving δB∥. Equation (8) demonstrates that the compressional component drives δψ, and that if the compressional mode is highly localized, k∥Ln ∼ O(δ−1/2), then δψ and δB∥ are of the same order. Physically, highly localized modes require more energy to bend the magnetic field lines and, therefore, suppress coupling between δB∥ and δψ ∝ δB⊥.
 The quasi-neutrality condition to O(δ) is derived to be
where δΦ∥ = δϕ − δψ. Since no terms in equation (9) are O(δ−1) we conclude that δϕ is on the order of δB∥ and ordering out terms with δϕ in the vorticity equation is justified. Recalling that δϕ and δψ are measured in units of Te/e and that τ ∼ O(δ), it can be understood that the cold electrons suppress coupling between δB∥ and δΦ∥ by shorting out the parallel electric field to O(δ). Finally, the perpendicular component of Ampére's law, equation (6), then yields the following DCM dispersion relation, accurate to O(δ):
We emphasize, again, that equation (10) represents the lowest-order dispersion relation where we have systematically suppressed the higher-order couplings to the electrostatic and field-line-bending shear Alfvén perturbations.
 In Crabtree et al.  numerical solutions of the small gyroradius limit of the integral equation were found to be highly localized around the minimum B value supporting the a priori ordering scheme used here that k∥Ln ∼ O(δ−1/2). Furthermore, by carefully noting the dependence on l in equation (10) it can be shown that δB∥ ∝ 1/B3/2. Thus k∥ ∼ ∂l log δB∥ ∼ (3/2)∂l log B, and as already noted for high-β geometries ∂l log B ∼ 2, thus the ordering k∥Ln ∼ 3 ∼ O(δ−1/2) is consistent.
3. Results and Discussion
 First, by ignoring the field line dependence of δB∥ in equation (10) we can write down a local DCM dispersion relation in the equatorial plane. Holding kϕρi constant (note that in this paper we consider k⊥ ≃ kϕ), a Nyquist analysis may be performed similar to the one detailed in Crabtree et al. . The results of this analysis are presented in Figure 1 for kϕρi = 0.05 and 0.78. For both unstable regions of parameter space (ηi < 0 and kϕρi/ωDi < 0) we find that for larger values of kϕρi a larger βi is required to reach the instability threshold and the size of the unstable region of parameter space increases.
 We solve equation (10) numerically using the Tsyganenko 2001 magnetic field model [Tsyganenko, 2002a, 2002b] with the IGRF internal magnetic field model which in the limit of large kϕ is approximately axisymmetric. In Figure 2, the growth rate γ and real frequency ωr are shown (for two values of kϕρi) versus ηi for two different flux tubes. The first flux tube is located on the noon-midnight meridian at x = −8RE under moderately quiet conditions. For this flux-tube, all protons drift westward, (i.e., same as the diamagnetic drift) and, thus for instability we must have ηi < 0. The second flux tube is also located on the noon-midnight meridian but at x = −11.5RE and under very active conditions. For this flux-tube, particles trapped near the equatorial plane (λ ≃ 1) drift eastward (i.e., drift reversed). Here, the magnetic field is increasing tailward causing the ∇B drift to be opposite to the diamagnetic drift. This situation arises on the earthward edge of the plasma sheet when the magnetotail current is intense.
 The real part of four unstable eigenfunctions are shown in Figure 3 as a function of distance along the field line. In the top panel, the flux-tube at x = −8RE under moderately quiet conditions is used. The first unstable eigenfunction with kϕρi = 0.1 and ηi = −1 is narrowly localized about the equatorial plane (l = 0) and is found to be approximately δB∥ ≃ 1/B3/2 supporting our ordering k∥Ln ∼ O(δ−1/2). The other eigenfunction with kϕρi = 1.2 and ηi = −4.0 also supports our ordering k∥Ln ∼ O(δ−1/2). However, the maximum amplitude is now located approximately one RE away from the equatorial plane where the local value of kϕρi corresponds to the maximum local growth rate. The bottom panel shows two unstable eigenfunctions for the drift reversed flux-tube. The first with kϕρi = 0.6 and ηi = 2.5 peaks at the equatorial plane, and the second with kϕρi = 1.2 and ηi = 7 peaks slightly away from the equatorial plane. This flux-tube has an extremely narrow magnetic well around the equatorial plane. Particles with turning points inside of 2RE of the equatorial plane drift eastward and those with turning points outside of 2RE drift westward kinking the wave function at these points. The narrowness of this mode also supports our approximation k∥Ln ∼ O(δ−1/2).
 In conclusion, we have systematically demonstrated that DCM can effectively decouple from electrostatic waves (due to cold electrons shorting out the parallel electric field) and shear Alfvén waves (due to narrow mode localization widths that suppress the field-line-bending shear Alfvén wave) in high-β plasmas. In Figure 4 the value of kϕρi at the equatorial plane is computed which gives the maximum growth rate as a function of ηi for the previously mentioned moderately quiet flux-tube at x = −8RE. One sees that in general kϕρi ∼ O(1), as expected, gives the most unstable perturbation. This is consistent with observations of high m compressional events in high β plasmas with low anisotropy. DCM is thus a viable alternative explanation of these events to drift-Mirror modes. Finally, it was demonstrated that the Tsyganenko 2001 series of models contain flux-tubes that exhibit drift reversal at the inner edge of the plasma sheet which can be unstable to DCM.
 C. C. is supported in part by the U.S. D.O.E. Fusion Energy Postdoctoral Research Program administered by the Oak Ridge Institute for Science and Education. L. C. is supported by NSF Grant ATM-0335279 and U.S. D.O.E. grant DE-FG-94ER54736.