### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Equilibrium Geotail Plasma Models
- 3. Compressional Eigenmode Equation
- 4. Earthward Gradients of the Pressure Distribution Functions with Solar Wind Variation
- 5. Conclusions
- Acknowledgments
- References

[1] The stability of low-frequency drift magnetosonic waves in the nightside magnetosphere between the outer ring current and the distant neutral line is investigated. As low-frequency Pi2 oscillations are seen before, during, and after substorm onset, these compressional waves are important for understanding the connection between the two substorm onset mechanisms of (1) the near geosynchronous orbit (NGO) mechanism and (2) the near-Earth neutral line (NENL) mechanism. It is found that there are two different regions of parameter space where drift-compressional modes can be unstable: (1) when pressure gradients become sufficiently steep to reverse the magnetic-guiding center drift and (2) when the temperature gradient is in the opposite direction to the density gradient. Nonlocal bounce-averaged eigenmode equations are solved with a simple analytic model of the magnetospheric magnetic field and with the *Tsyganenko and Stern* [1996] magnetic field model. Resulting growth rates, frequencies, and eigenmode structures are reported.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Equilibrium Geotail Plasma Models
- 3. Compressional Eigenmode Equation
- 4. Earthward Gradients of the Pressure Distribution Functions with Solar Wind Variation
- 5. Conclusions
- Acknowledgments
- References

[2] For decades the detailed observational data from both ground-based magnetometer stations and nightside spacecraft measurements of plasma particles and fields have been analyzed for the purpose of understanding substorm dynamics. While many mechanisms have been proposed and investigated over the years, processes in two distinct regions now seem to be intimately linked to substorms. The first, historically, is the formation of a neutral line by a *B*_{n} + δ*B*_{z} < 0 perturbation in the midtail region *x* ≃ −25 *R*_{E}, called the NENL model. Here *B*_{n}(*x*) is the northward ambient field on the axis of the geomagnetic tail, and δ*B*_{z} is a southward perturbation which, when larger than *B*_{n}(*x*), forms the neutral line. The second mechanism, for which there is growing observational evidence [*Maynard et al.*, 1996; *Erickson et al.*, 2000], is characterized by the onset of low-frequency Pi2 precursor oscillations and the sudden brightening of auroral arcs associated with flux tubes that map to equatorial crossing points in the region *x* = −6*R*_{E} to −12*R*_{E}, dependent on the state of the magnetosphere. Substorms with these onset characteristics have been named NGO (near geosynchronous orbit) events, although the Keplerian orbits with a 24-hour period play no role in the plasma electrodynamics. The large number of geosynchronous spacecraft provide excellent data coverage of this region. To understand the relationship between these two regions we have investigated the stability of drift-kinetic compressional waves in the region between geosynchronous orbit *x* = −6*R*_{E} and the NENL region *x* = −25*R*_{E}. To treat these waves correctly it is necessary to consider that in one wave period, thermal ions execute one to several mirroring orbits along the magnetic field, making the response of the ions to the wave fields nonlocal.

[3] There is strong evidence for the existence of an x-point in the midtail region [*Nagai et al.*, 1998; *Baker et al.*, 1999], which is generally accepted in the literature. However, there is an open issue, namely, whether or not this x-point is the trigger for substorm onset (NENL) or whether it occurs after substorm onset. *Nagai et al.* [1998], using Pi2 oscillations to define substorm onset, argued that the timing evidence supports the NENL model for substorm onset. In contrast, *Lui et al.* [1998] used data from the UVI instrument on POLAR and tail measurements from GEOTAIL and came to a different conclusion. They found that the overall plasma flow pattern during the course of substorms remains ill defined. They showed one event (28 April 1996) in detail and constructed synoptic charts for both the total flow velocities and the perpendicular flow velocities in the equatorial plane for the relevant space-time domain for 102 events with good Geotail coverage and correlated it with UVI images of auroral breakups. They concluded that strong flows do not necessarily precede substorm onset, supporting the synthesis model of *Lui* [1991] rather than supporting the NENL model for substorm onset.

[4] The synthesis model of *Lui* [1991] combines the observational evidence for both the formation of a near-Earth neutral line and an instability at near geosynchronous orbit. In their model a configurational instability exists at the inner edge of the plasma sheet, which generates a finite amplitude compressional/rarefraction wave that travels to the midtail region, where the amplitude is large enough for the fluctuating δ*B*_{∥} to produce the necessary negative *B*_{z} = 〈*B*_{z}〉 + δ*B*_{∥} to initiate magnetic reconnection and the global reconfiguration of the magnetosphere that is associated with substorms.

[5] Further evidence that the initiation of substorm onset is not due to the formation of a NENL is that ground-based auroral observations show that the arc brightening at the expansion onset is on field lines that map to ∼ −10 *R*_{E} [*Lopez and Lui*, 1990; *Samson*, 1994; *Frank et al.*, 1998, 2000; *Frank and Sigwarth*, 2000], rather than to the midtail region where x-points have been observed.

[6] *Pu et al.* [1999] proposed a different synthesis of the NENL substorm model and the cross-field current disruption model, by observing that the fast Earthward plasma flow produced by magnetic reconnection in the midtail can be abruptly decelerated at the inner edge of the plasma sheet, compressing the plasma and creating the large pressure gradients necessary to drive a ballooning instability. We use a one-dimensional (1-D) transport equation to further investigate the process of pressure gradient steepening due to braking of the Earthward flows. *Ohtani* [2001] gives a good review of the various synthesis models and the observational evidence used to support them. He points out that the most popular substorm synthesis models have the NENL creating the high-speed flows that then cause current disruption, dipolarization, and Pi2 oscillations in the near-geosynchronous region but points out that the issue is not closed as there are key open issues such as coherency of flows and timing issues related to multisatellite observations that leave open possibilities such as those proposed by *Lui* [1991].

[7] *Kepko and Kivelson* [1999, 2001] looked at correlations of Pi2 waves and variable Earthward flows. They found that the wave forms were correlated and defined a class of Pi2 waves that are directly driven by variable Earthward flow. *Sigsbee et al.* [2002] showed that low-frequency magnetic fluctuations in the Pi2 frequency band (2–25mHz) [*Kivelson*, 1995] preceded substorm onset and were associated with increased Earthward **E** × **B** flows greater than 100 km/s. They reported that the Pi2 pulsations are observed before the initiation of strong Earthward flow but also that the largest flows were observed after substorm onset, finding a large class of Pi2 oscillations that were not directly driven. Thus it is also possible that a marginally stable state exists at the inner boundary of the plasma sheet, which is driven unstable by strong Earthward flows. Once this mode is made unstable, a strong enough fluctuating δ*B*_{∥} is created to initiate reconnection and the largest of the flows that are seen. To decide between these scenarios, the issue of timing must be resolved.

[8] Here we introduce a one-dimensional transport model for the transient steepening of the pressure gradient in the magnetotail due to changes in the thermal sources deep in the tail. The model shows that pressure pulses steepen as the flow brakes against the stronger near-Earth magnetic field. Such pressure pulses are a natural source of transient steepening of the pressure gradient that can lead to magnetic driven pulses with guiding center drift reversal as shown by the models of *Li et al.* [1998]. *Li et al.* [1998] proposed a model with a Gaussian magnetic pulse that caused particles to reverse the direction of their magnetic drift and found this to be a mechanism for dispersionless injection. We show in section 3.3 that when particles reverse the direction of their magnetic drift, compressional/rarefraction waves become unstable due to wave-particle interactions.

[9] Another related candidate for these observed magnetic oscillations is the drift mirror mode [*Hasegawa*, 1969; *Chen and Hasegawa*, 1991] in which an instability due to the temperature anisotropy (*T*_{⟂} > *T*_{∥}) drives the unstable mode. The classical mirror mode instability has serious problems with self-consistency, as explained by *Hasegawa* [1969], due to its derivation from double adiabatic MHD, which neglects the parallel heat flux; nevertheless this is an important mode. Here in this work we do not focus on the temperature anisotropy to drive the instability but rather include the resonant effects that the waves have on the particles to drive a high-β mode that is analogous to the typical electrostatic drift-waves.

[10] To evaluate the frequency of such marginal modes, the long orbits of the mirroring ions in a flux tube must be considered. The MHD model is not generally valid for low frequencies ω < ∣ω_{*pi}∣, ω_{bi}, where ω_{*pi} is the diamagnetic drift frequency formed by a pressure gradient and ω_{bi} = *v*_{ti}/*L*_{∥}^{eff} ≈ 20mHz is the frequency of the ion bounce motion between the southern and northern ionosphere, with *L*_{∥}^{eff} being the approximate length of the field line. Thus a bounce-averaged eigenmode equation must be solved [*Horton et al.*, 2001]. The stability analysis is local to the flux tube, and we investigate which flux tubes tend to be the first to go unstable. In our earlier analysis based on the high-frequency Kruskal-Oberman kinetic energy principle [*Kruskal and Oberman*, 1958], the answer was found that the transitional region β ∼ 1 to 10 was the first to go unstable with respect to kinetic interchange modes. We now investigate which flux tubes are the first to go unstable with respect to drift-kinetic compressional waves.

[11] *Rosenbluth* [1981] pointed out that for compressional modes, the role of magnetic drift reversal of the guiding center flux is important for high-β plasma stability. Drift reversal can occur when the ∇*B* drift is large and opposite to the curvature drift, which occurs for high-β equilibria. A Nyquist analysis shows that the dimensionless stability parameter is

where ω_{*i} is the pitch angle-independent diamagnetic drift frequency from the density gradient, and _{Di}(β) is the bounce-averaged magnetic guiding center drift frequency which is proportional to the kinetic energy *K* = *mv*^{2}/2. Here we use the maximum kinetic energy *K*_{max} with appreciable flux to define _{Di max}. The relevant dimensionless factor *R*(β) as a function of β = 2μ_{0}*p*/*B*^{2} is an important quantity in determining stability. For the *Tsyganenko and Stern* [1996] magnetic field model we report here the dimensionless value of *R*(β) as a function of both *x*[*R*_{E}], between the NGO region and the NENL region, and β_{min} = 2μ_{0}*p*/*B*_{min}^{2} the ratio of plasma kinetic energy density to magnetic field energy density, where *B*_{min} is the magnitude of the magnetic field at the equatorial plane, which is the minimum value of *B* for that flux tube.

### 2. Equilibrium Geotail Plasma Models

- Top of page
- Abstract
- 1. Introduction
- 2. Equilibrium Geotail Plasma Models
- 3. Compressional Eigenmode Equation
- 4. Earthward Gradients of the Pressure Distribution Functions with Solar Wind Variation
- 5. Conclusions
- Acknowledgments
- References

[13] A useful equilibrium model that we will employ in this paper is obtained by adding the field due to a constant current in the **ŷ** direction to the field due to a two-dimensional dipole. Here and in the rest of the paper we use GSM coordinates, where the positive *◯* direction points towards the Sun, the direction points northward, and *ŷ* points from dawn to dusk. We call this model the constant current model (CCM) [*Horton et al.*, 2001; *Wong et al.*, 2001]. The 2-D magnetotail model is derived from **B** = **ŷ** × ∇Ψ with

This model has three parameters: (1) *B*_{0}*r*_{0}^{2}, which represents the strength of the dipole field, (2) *B*_{x}′, which is directly proportional to the current (μ_{0}*J*_{y} = ∇^{2}Ψ), and (3) *B*_{n}, which gives a constant *B*_{z} throughout the model and physically represents the contribution from currents outside of our region of interest. Specifically, the value of *B*_{n} is correlated with the IMF *B*_{z}. To define an isotropic pressure, we assume an equilibrium (∇*p* = **J** × **B**), choose a point *x*_{tail} sufficiently far down the tail where the pressure is low, and integrate d*p*/d*x* towards the Earth along the geotail axis. For the constant current model this gives

Here we are considering points only in the magnetotail such that *x*_{E} > *x* ≥ *x*_{tail}, where *x*_{E} = −1 *R*_{E} and *x*_{tail} = −30 *R*_{E}; then *p*(*x*, *z* = 0) is always positive and increasing towards the Earth. The value for *p*_{tail} = *p*(*x* = *x*_{tail}, *z* = 0) can be chosen to fit data for the pressure at *x*_{tail} = −30 *R*_{E}. For example, ISEE-2 gives a particle pressure of 0.06 nPa at −30 *R*_{E}. In this model the Earthward pressure gradient is *dp*/*dx* = *B*_{x}′ [*B*_{n} + *B*_{0}*r*_{0}^{2}/*x*^{2}]/μ_{0}; a steeper transient gradient will be computed in section 4. For ∣*x*∣ ≫ (*B*_{0}*r*_{0}^{2}/*B*_{n})^{1/2} the pressure gradient is *p*′ = *B*_{n}*B*_{x}′/μ_{0} and for |*x*| ≪ (*B*_{0}*r*_{0}^{2}/*B*_{n})^{1/2} it is *p*′ = *B*_{x}′*B*_{0}*r*_{0}^{2}/(μ_{0}*x*^{2}). The equilibrium is given by the Grad-Shafranov equation with d*p*/dΨ = *B*_{x}′/μ_{0} for d*p*/*d*Ψ = *J*_{y} = const. Several other useful analytic computations were performed for this model by *Horton et al.* [2001] and will not be repeated here.

[14] Figure 1 shows magnetic field line traces (level curves of Ψ) and *B* and *R*_{c}, as functions of *s*, the distance along the field line, computed with the constant current model (CCM) model and with *Tsyganenko and Stern*'s [1996] model. The function *B*(*z*) in the two-dimensional dipole used in the CCM model falls off more slowly as a function of *x*/*R*_{E} than the real three-dimensional dipole field; thus the model cannot simultaneously match global quantities over 20 *R*_{E} down the tail and “local” quantities such as the *B* profile along a magnetic field. The 2-D dipole is very useful due to translational symmetry in the y direction and is used extensively in particle simulations [*Pritchett and Coroniti*, 1997; *Pritchett and Coroniti*, 1999; *Horton et al.*, 2001]. This model was found to be MHD stable but kinetically unstable, with waves that oscillate at ω ≃ ω_{*pi} = (*k*_{y}*T*_{i})/(*qB*_{min}*L*_{p}), where for η_{i} = *L*_{n}/*L*_{Ti} = 2/3 the density gradient scale length and the pressure gradient scale length are related by *L*_{pi} = 3*L*_{n}/5. Thus we can take the waves to be of the form δ*B*_{∥} = δ*B*_{∥}(*s*)exp(*ik*_{y}*y* − *i*ω*t*) + c.c. In the equatorial plane the electric field associated with these waves is given by *ik*_{y}*E*_{x} = *i*ωδ*B*_{∥}, and the displacement is **ξ** = **E** × **B**/(*i*ω*B*^{2}).

[15] The magnetic guiding-center curvature and ∇*B* drift frequencies are given, respectively, by

Here λ = sin^{2}(α) = μ*B*_{min}/*K* is the dimensionless pitch angle variable in the equatorial plane with the pitch angle given by α, **κ** = · ∇ is the curvature vector, *K* is the kinetic energy of the particle, and ∣**κ**∣ = 1/*R*_{c} is the inverse of the radius of curvature *R*_{c}(*x*) which is a function of position. For ρ_{i} = 300 km the wavelength is 2π/*k*_{y} = 20 ρ_{i} = 6 · 10^{6} m ≈ 1*R*_{E}, which is approximately the limit of the local approximation. Notice that *k*_{y}*T*_{j}/(*q*_{j}*B*_{min}) can be rewritten as *k*_{y}ρ_{j}*v*_{tj}; with typical magnetospheric parameters of a 5 KeV proton (corresponding to a thermal velocity of ≃ 2.7 *R*_{E}/min) and *k*_{y}ρ_{i} = 0.5 and a 0.2 *R*_{E} radius of curvature, this gives an estimate of 3mHz for drift frequencies. In a vacuum field the ∇*B* drift frequency reduces to ω_{∇B} (*T*_{⟂}/2*T*_{∥})ω_{κ} [*Goldston and Rutherford*, 1995]. Thus near the Earth where the magnetic field is primarily due to the Earth's dipole field, the plasma current is small, and consequently the pressure is low (β ≪ 1) and the curvature and ∇*B* drift are in the same direction. However, at high beta one can use **B** × (**J** × **B** − ∇**p**) = **0** to derive

and thus

so that reversal of the ∇*B* drift direction occurs for β ≳ *L*_{p}/*R*_{c}. Using this relation we can repartition the total magnetic drift ω_{Di} into one term that represents the vacuum effect on particles ω_{κ′} and another that is due to the pressure gradients. This separation gives ω_{Di} = ω_{κ′} + ω_{∇p} with

which is useful for the high-β stability formulas. The drift ω′_{κ}/*k*_{y} is westward, and the drift ω_{∇p}/*k*_{y} is eastward for Earthward increasing pressure profiles. For λ*B*/*B*_{min} = 1 the opposing drifts cancel when *R*_{c}μ_{0}∇*p*/*B*^{2} ≃ 1. In terms of the plasma pressure β = 2μ_{0}*p*/*B*^{2}, this drift reversal of the total guiding center drift velocity occurs for β > 2*L*_{p}/*R*_{c} where *L*_{p} is the pressure gradient scale length. Generally, the drift reverse β value β_{dr} = 2 *L*_{p}/*R*_{c} is very high since *L*_{p} ≫ *R*_{c} except during transient periods.

[16] The response of the plasma to a wave depends on the orbits of the particles over a wave period. For low frequencies the bouncing motion of the particles due to the mirror force becomes important, and thus the bounce average of the magnetic drift velocities arises. The bounce average of a quantity is denoted by an overline and is defined as

where the limits of the integrals are to be determined by the mirroring motion of the particle with pitch angle at the equatorial plane given by λ and . These integrals rely on particles being trapped and maintaining a constant magnetic moment over several wave periods.

[17] Figure 2 shows the bounce-averaged drifts as a function of λ for three different positions, computed from the *Tsyganenko and Stern* [1996] magnetic field model, and Figure 3 shows the same calculation using the constant current model (CCM). These figures show that the constant-current model equilibrium agrees well with the standard empirical magnetic field model for a particular flux tube. The characteristic property of the NGO region to notice on these plots is that Earthward of the NGO region, the magnetic curvature drift and the magnetic ∇*B* drift are in the same direction, and tailward they are in the opposite direction, as is expected from equation (6). However, neither of these two models exhibit drift reversal as defined previously. Also shown in Figure 2 is the stability parameter . The minimum of *R* is found to be in the NGO region.

[18] Figure 4 shows the relevant frequencies for the kinetic stability analysis as a function of *x*/*R*_{E} in the nightside equatorial plane, computed from *Tsyganenko and Stern*'s [1996] model, and Figure 5 shows the same computation using the constant current model. Bounce-averaged magnetic drift components for particles with three different equatorial pitch angles are shown, along with the diamagnetic drift frequency for a pressure gradient scale length of *L*_{p} = 2.0*R*_{E}. It is seen that . It can also be seen from these plots that the _{∇Bi} component becomes negative tailward of the NGO region, where ω_{bi} = 2π*v*_{ti}/τ(λ) is the bounce frequency and *v*_{ti} is the ion thermal velocity. The polarization of the fluctuations is δ*B*_{∥} ≫ δ*B*_{⟂}, with a small δ*E*_{⟂} ∼ *v*_{di}δ*B*_{∥} ≪ *v*_{ti}δ*B*_{∥} in the direction perpendicular to that of the wave propagation.

[19] Here we took the particles to have a Maxwellian velocity distribution *f*_{j}(**x**, **v**) = *n*_{j} (**x**)(*m*_{j}/2π*T*_{j}(**x**))^{3/2}exp(−*K*/*T*_{j} (**x**)). In a future work we will consider a kappa distribution. For the local Maxwellian, the diamagnetic drift frequency is

For an adiabatically compressed energy distribution, we have *f*_{j} = *f*_{j}(*KV*^{2/3}), where *V* = ∫*ds*/*B* is the flux tube volume. Then the effective value of η_{j} = *L*_{n}/*L*_{Tj} is η_{j} = 2/3.

### 3. Compressional Eigenmode Equation

- Top of page
- Abstract
- 1. Introduction
- 2. Equilibrium Geotail Plasma Models
- 3. Compressional Eigenmode Equation
- 4. Earthward Gradients of the Pressure Distribution Functions with Solar Wind Variation
- 5. Conclusions
- Acknowledgments
- References

[20] The most useful way to express the three coupled equations for the three electromagnetic field components is to use a variational functional ℒ for field components that are analogous to those used in MHD calculations. The variational form that describes stability of low-frequency waves in the bounce averaged limit is given by *Horton et al.* [2001] and is repeated here:

where �� is the perturbed particle energy,

and the overline represents bounce averaging as defined by equation (9). Taking variations of this integral equation with respect to ϕ and the two components of **ξ**, which in our case are only a function of *s*, the distance along the field line, yields the set of integral-differential eigenvalue equations to be solved. In equation (11)

Here we consider modes dominated by δ*B*_{∥}, which implies *Q*_{L} ≈ δ*B*_{∥} ≈ −*B*∇ · **ξ**_{⟂} [*Rosenbluth*, 1981], from the converging/diverging cross field flows that compress and expand the magnetic field. The kinetic energy from the flows is . From Faraday's law for the frozen-in flux equation δ**B** = ∇ × (**ξ** × **B**), we get δ*B*_{∥} = − *iBk*_{y}ξ_{y} for the most unstable displacements with *k*_{y} ≪ *k*_{x}. To maintain ∇ · **B** = **0**, especially for the modes that we find which are so strongly localized in the equatorial plane, there must be a small δ**B**_{⟂} perturbation. Noting that ∂*B*_{∥}/∂*s* ∼ ∂*B*_{z}/∂*z* ∼ −*ik*_{y}*B*_{y}, we find that in the large *k*_{y} limit only a small perpendicular perturbation is necessary to satisfy ∇ · **B** = **0**.

[21] Taking �� = μδ*B*_{∥}, *Q*_{L} = δ*B*_{∥}, and ξ_{y} = *ik*_{y}^{−1}δ*B*_{∥}/*B* and neglecting all other terms with ϕ or ξ^{ψ}, the variational form reduces to

for compressional drift modes. Here we have replaced the velocity integral by an integral over the dimensionless energy ϵ = *K*/*T*_{a} and the pitch angle λ.

[22] For modes with odd symmetry δ*B*_{∥}(−*s*) = −δ*B*_{∥}(*s*), the bounce integral vanishes and the waves are given by ω^{2} = *k*_{y}^{2}*B*^{2}/ρ_{m}μ_{0} ∼ *k*_{y}^{2}*v*_{i}^{2}/β_{i}. In the absence of density and temperature gradients, *Chen and Hasegawa* [1991] used the principal value of the integral and the limit ∣γ/ω∣ < 1 to show that the bounce-averaged terms in equation (15) are all stabilizing, and thus modes with odd symmetry are the most unstable. They relied on pressure anisotropy to drive a drift mirror instability. Here, we do not consider the pressure anisotropy but rely on the wave-particle resonance to drive an instability. In this case modes with odd symmetry are all stable oscillations, making the modes with even symmetry the most unstable.

[23] Taking the variation of ℒ in equation (15) we arrive at an integral eigenvalue equation, which can be written,

where

is the bounce-averaging operator that describes the response of the field to the mirroring particles and acts to the right in equation (16), and

represents the dielectric response of particles with a particular pitch angle given by λ. To solve equation (16) the field variables are discretized along a particular flux tube, which creates a matrix eigenvalue problem. In this discrete representation the bounce-averaging operation is a matrix operator on the magnetic fluctuations,

where both the operator *M*_{i,j}(ω) and the eigenvalue Λ(ω) are dependent on the frequency ω, the operator being dependent on the frequency through the dielectric response accounted for in *G*^{j}(ω, λ). Thus the problem to be solved is nontrivial, and an iterative procedure is used. An initial frequency ω^{(0)} is assumed, and the eigenvalues and eigenvectors of *M*_{i, j}(ω^{(0)}) are computed, Λ^{(0)} and δ*B*_{∥}^{(0)}. Then the eigenvectors at this order are used to form a dispersion relation

from which the next-order frequency ω^{(1)} can be found from the root of the dispersion relation. Next, we can again find the eigenvectors and eigenvalues of *M*_{i, j}(ω^{(1)}), namely Λ^{(1)} and δ*B*_{∥}^{(1)}, and form a new dispersion relation. This iteration procedure terminates when the frequency ω^{(n)} is a solution to the dispersion relation at order *n*.

[24] The parameter vector used to specify the problem is given by *P*^{L} = (*k*_{y}ρ_{i}, η_{i}, η_{e}, *T*_{e}/*T*_{i}, *L*_{n}/*R*_{c}, *R*_{c}/*L*_{B}, β_{min}), plus the parameters needed to specify the magnetic field model from which we compute such quantities as *R*_{c}/*L*_{B}, β_{min}, and the details of the flux tube. Here η_{i, e} = *L*_{n}/*L*_{Ti, Te} is the ratio of the density gradient scale length to the temperature gradient scale length for each species. For the *Tsyganenko and Stern* [1996] model the parameters are *P*^{T96} = (*P*_{dyn}, *DST*, *B*_{y}^{IMF}, *B*_{z}^{IMF}), and for the constant current model the parameters are *P*^{CCM} = (*B*_{0}*r*_{0}^{2}, *B*_{x}′, *B*_{n}). The final parameters that need to be specified are the (*x*, *y*) coordinates (in units of *R*_{E}) in the equatorial plane for the flux tube.

[25] In terms of the normalized drift wave units ω ω [*c*_{s}/*L*_{n}], and *k*_{y} *k*_{y} [ρ_{i}^{−1}], the left-hand side of equation (16) is small compared with δ*B*_{∥}, namely,

and thus it does not appear in the 3 × 3 drift kinetic matrix formulation of low-frequency drift stability theory (see, for example, *Horton et al.* [1985]).

#### 3.1. Mode Frequencies Above the Guiding Center Drift Frequency

[26] The high-frequency approximation is ω ≫ _{D}, which yields *G*^{j}(ω, λ) = (15/4)(1 − ω_{*j}(1 + 2η_{j})/ω). In this limit only purely oscillating modes exist, related to the familiar magnetoacoustic mode with the dispersion relation

where and Γ^{HF} is given by the eigenvalues of the integral operator

Notice that in this high-frequency limit, the quantity _{Dj} does not appear, so that the integral operator in equation (22) does not have any species-dependent quantities. The β_{j} and ω_{*j} absorb all of the species dependence. The difference between this mode and the usual magnetoacoustic mode is that in equation (21), the adiabatic index is given by the eigenvalues Γ^{HF} of the bounce-averaging integral operator. In the high β ≫ 1 limit and taking ω to be large compared with ω_{*j}, we find the approximate solution to equation (21) to be given by

[27] Figures 6 and 7 show ω for the magnetoacoustic mode computed from equation (21), compared with the approximate formula of equation (23). The agreement is found to be good between equation (23) and the numerical solution of equation (21). Also shown in Figure 6 are the eigenvalues Γ^{HF} of the integral operator of equation (22) as a function of β. The β dependence is found to be relatively weak, varying approximately 20% as β varies over an order of magnitude. Also shown are the eigenfunctions of the integral operator as a function of *s*, the distance along the field line. The eigenfunctions on flux tubes closest to the Earth (lowest β) tend to vary approximately as (*B*_{min}/*B*(*s*))^{3/2}, and the eigenfunctions on flux tubes at higher β tend to vary more as (*B*_{min}/*B*(*s*))^{5/2}. These eigenfunctions and eigenvalues are computed with both *Tsyganenko and Stern*'s [1996] model and the CCM model through the tail/dipole transition region as a function of β. The value of β is computed from equilibrium assumptions with the magnetic field models. In some cases we use the Lysak model [*Lysak*, 1993] for the density profile *n*(*s*) in order to compute the Alfvén velocity.

#### 3.2. Mode Frequencies Below the Guiding Center Drift Frequency

[28] The low-frequency modes occur for ω ≪ _{D}, which yields to first order. Unlike the high-frequency limit in equation (21), in the low-frequency limit the left-hand side ω^{2}/*k*_{y}^{2}*V*_{A}^{2} is negligible. The dispersion relation is given by

Here Γ^{LF} are the eigenvalues of the integral operator

The integral operator is again independent of species since ω_{*i} = − ω_{*e}*T*_{i}/*T*_{e} and . Notice that if *T*_{e} = *T*_{i}, then the terms dependent on ω in equation (24) exactly cancel and it is necessary to take the expansion to next order.

[29] Figures 8 and 9 show the eigenvalues and eigenvectors of the low-frequency bounce-averaged operator of equation (25), as well as the shifted frequency computed from equation (24). *Tsyganenko and Stern*'s [1996] magnetic field model as well as the CCM model were used to compute all quantities, and *T*_{e} was taken to be negligible to prevent the previously mentioned cancellation. The computed frequency shift which is a rearrangement of the terms in equation (24) taking *T*_{e} = 0,

is seen to increase at lower β. The frequency ω_{k} is closest to the diamagnetic drift frequency ω_{*pi} and well below the bounce-averaged guiding center drift frequency. The eigenfunctions are seen to be strongly ballooning, meaning the perturbation is very strongly peaked at the equatorial plane. The eigenvalues of the operator are seen to follow the general form of the computed stability parameter *R*(β), and the approximation *R*(β) ≃ 2Γ^{LF}(β) is reasonable, as can be seen by comparing *R*(β) from Figures 2 and 3 with Γ^{LF}(β) from Figures 8 and 9.

#### 3.3. The ∇*B* and Curvature Drift Resonance Response

##### 3.3.1. Nyquist Analysis of the Variational Form Dispersion Relation

[31] To analytically perform the Nyquist analysis on the variational form given in equation (16), we ignore the inertial term, considering ω ≪ *k*_{y}*V*_{A}, take the perturbing field to be flute-like δ*B*_{∥}(*s*) = const., ignore the pitch angle dependence of the bounce averaged guiding center drift, and consider only the ions by taking the limit *T*_{e}/*T*_{i} ≪ 1. Then we obtain the complex dispersion relation

for the compressional mode. In this limit *G*^{j}(ω) = *G*(ω) can be written as

in order to make clear the four frequencies for which Im(*C*) = 0, namely, ω = 0, ω_{I}, ±∞, where

Here *R*_{c} = *R*_{c}(β) and using the constant current model deep in the tail we have *R*_{c}^{−1}*B*_{x}′/*B*_{n} when d*B*_{x}/*B*_{n} d*z* ≫ d*B*_{n}/*B*_{n}d*x* from ∇*B*/*B*. This gives roughly *R*_{c}(β)^{−1} = (1 − *R*_{c}β′)*R*_{c}^{−1} = *R*_{c}^{−1} (1 − *R*_{c}β/*L*_{p}). For β > *L*_{p}/*R*_{c}, the ∇*B*/*B* drift overcomes the curvature drift. Correspondingly, the real part of *C* at each of these critical frequencies is given by

where . The imaginary part of *C*(ω) for real ω can be written by distorting the ϵ integration contour below the singularity at _{Di}:

Here Θ(ω/_{Di}) is the step function, such that when ω and _{Di} have the same sign the function has the value unity and otherwise is zero. An instability exists when *C*(ω = 0) and *C*(ω = ω_{I}) have opposite signs and *C*(ω) between these two frequencies is imaginary so that a contour enclosing the upper half plane in the complex ω plane corresponds to a contour that encircles the origin in the complex *C* plane. The system is marginally stable when the two values of *C*(ω) are both zero or when *C*(ω) between these two frequencies is purely real. In this analysis there are three stability parameters: β, ω_{*i}/_{Di}, and η_{i}. By combining β and ω_{*i}/_{Di} to form the parameter *R*(β) we can construct a simple picture.

[32] From equations (30)–(32) it can be seen that an instability exists when *R*(β) < 0, which is in the region of drift reversal, for a particular small range of η_{i} values. Another instability region exists for η_{i} < 0, or when the temperature gradient is opposite to the density gradient. For η_{i} = −1 it is clear that *C*(ω = 0) > 0 so that for *R*(β) > 1 we find *C*(ω = ω_{I}) < 0 and the flux tube is unstable. There is some evidence (X. Garner, personal communication, 2002), based on data from *Paterson et al.* [1998], for such a negative η_{i} to exist.

[33] To determine whether or not Im*C*(ω) > 0 for these regions we must determine if _{Di} and ω_{I} have the same sign, which would imply

Thus if we are considering η_{i} < 0 then the above condition holds true as long as ω_{*i}/_{Di} > 0 (which is true for the unstable region we are considering). If we are considering omega;_{*i}/_{Di} < 0 then it is easy to see that we must have η_{i} > 2/3 for instability.

[34] (Figure 10) shows the boundary between instability and stability as a function of η_{i} and *R*(β) in two different regions: Figure 10a shows the stability boundaries in the drift reversal region, and Figure 10b shows the stability boundaries in the negative η_{i} region.

##### 3.3.2. Growth Rate Estimates

[35] From equation (15) we now estimate, by making a local approximation, that the compressional drift waves have

with

where *C* = 1 + μ_{0} δ*p*_{⟂}/*B*δ*B*_{∥} is the complex response function considered in detail in the previous section. Notice that in equation (36) we have used a value of ϵ = 3; this value is near the maximum contribution from the integral ∫ dϵ ϵ^{5/2}*exp*(−ϵ), and agreement with actual results was quite good. Instability occurs from ω_{0}Im(*C*(ω_{0})) > 0 for Re *C*(ω_{0}) > 0 and from new low-frequency roots of *C*(ω) = 0. The approximate root of *C*(ω) = 0 are

In the limit , then ω_{k} ∼ ω_{*i}(1 + 3η_{i}/2); and in the opposite limit β ≪ _{Di}/ω_{*i}, then ω_{k} ∼ ω_{Di}. Figure 4 shows the relevant frequencies used in this estimation as a function of *x*[*R*_{E}] at the edge of the plasma sheet. The magnetic drift frequencies were obtained by bounce-averaging equations (3)–(4) with the use of the *Tsyganenko and Stern* [1996] magnetic field model. For example, for *x* = −10*R*_{E}, *L*_{pi} = 2*R*_{E} and *k*_{y}ρ_{i} = 0.5, we find that 5 KeV particles have ω_{*i} = 5 *k*_{y}ρ_{i}*v*_{i}/(3*L*_{pi}) = 3mHz, ω_{bi} = 16mHz, and _{Di} = mHz. For ρ_{i} = 300 km the wavelength is 2π/*k*_{y} = 20 ρ_{i} = 6 · 10^{6} m ≈ 1*R*_{E}, which is approximately the limit of the local approximation. Thus since , we approximate ω ≃ ω_{*i} ≃ 3mHz.

[36] For these modes δ*E* = δ*E*_{x} = (ω/*k*_{y})δ*B*_{∥} = *v*_{Di}δ*B*_{∥}, so for δ*B*_{∥} = 10nT we find δ*E*_{x} ∼ 10^{−4} V/m ∼ 100 μV/m. The Poynting flux is **S** = δ**E** × δ**B**/μ_{0} = (ω/*k*_{y})δ*B*_{∥}^{2}**ŷ**/μ_{0} = 1μ W/m^{2}. The displacement vector is then ξ ≈ (1/6)*R*_{E}. The nonlinear limit is reached when δ*B*_{∥}/*B* ∼ *k*_{y} ξ_{y} ∼ 1.

[37] Using equation (37) as the real part of the true root, we can then make the approximation that γ/ω ≪ 1 and obtain an approximate formula for the growth rate as

Here ω_{k} is the solution to *C*(ω_{k}) = 0 given by equation (37). Figure 13 shows the growth rates and real parts of the frequencies of the full solution to the non-local dispersion relation, compared with the approximate formulas given by equations (37) and (38) for a flux tube that exhibits drift reversal _{Di} < 0. The agreement is quite good for this case and is seen to be only slightly worse for cases with larger growth rates.

[38] The energy associated with the compressional drift wave is

where

with energy released for *C*(ω) < 0 due to the spatial gradients rather than the anisotropy as in a mirror mode. The energy release is not as large as a typical ballooning/interchange event. For example, for δ*B*_{∥} = 1nT over a region of volume *R*_{E}^{3}, there is δ*W*^{comp} = 2 × 10^{8} J. The waves are most dangerous when they induce a local neutral line due to *B*_{n}(*x*) + δ*B*_{z} (*x*, *y*) < 0.

[39] These modes have ∣γ_{k}/ω_{k}∣ < 1 and can produce local mirror trapping of ions at finite amplitude. These modes are due to the ω = _{Di}ϵ resonance and do not require temperature anisotropy, as do the drift-mirror modes. The drift-mirror mode [*Chen and Hasegawa*, 1991; *Kulsrud*, 1983] goes unstable when 1 + β_{⟂} − β_{⟂}^{2}/β_{∥} < 0, and its growth rate is given by

#### 3.4. Numerical Growth Rates

[40] Figure 11 shows the real and imaginary parts of the solutions, ω, to the dispersion relation given in equation (16), for different positions and different values of η_{i}, in the region of parameter space where η_{i} < 0. These solutions take into account the non-local behavior of the perturbation δ*B*_{∥} by solving the full matrix problem using the technique described with equations (19) and (20). Comparing the growth rates in Figure 11 to the marginal stability conditions in Figure 10, one sees that for *x* = −5*R*_{E} we have *R*(β) ≃ 1.0 and the compressional mode is unstable for −4 < η_{i} < −1. As we consider flux tubes farther in the tail, the value of *R*(β) increases so that smaller values of negative η_{i} will lead to instability over a smaller region of η_{i} space, as seen in both by the marginal stability conditions shown in Figure 10 and the actual growth rates shown in Figure 11. The results of the local Nyquist analysis accurately predict the boundaries of marginal stability for this region of parameter space. The growth rates are not large (γ/ω < 1), and the direction that these waves travel is from dawn to dusk ( = ω/*k*_{y} > 0). As β gets larger the growth rate becomes comparable to the real part of the frequency. Shown in Figure 12 are two eigenfunctions corresponding to two different parameter vectors. These perturbations are seen to be strongly ballooning, i.e., the maximum perturbation is confined to the equatorial region (*Z* = 0).

[41] Neither the *Tsyganenko and Stern* [1996] model nor the constant current model exhibit magnetic guiding center drift reversal. We verify that an instability exists when the magnetic guiding center drift reverses direction, by arbitrarily reversing the bounce-averaged drift velocity computed from the magnetic field models. An example of a model that does have drift reversal is given by *Li et al.* [1998] where a large localized magnetic pulse is applied to the Earth's dipole field. Figure 13 shows that for a flux tube at *x* = −5*R*_{E}, corresponding to *R*(β) ≃ −1 a weakly unstable mode exists (γ/ω ≪ 1) for values of η_{i} predicted by the local Nyquist analysis. Also shown in this figure is the growth-rate estimate given by equation (38). These modes travel from dusk to dawn ( = ω/*k*_{y} < 0) in the direction of drifting protons. Since compressional disturbances are driven by the arrival of interplanetary shocks, the existence of marginally stable compressional drift waves may be of importance. The eigenfunctions corresponding to these two solutions to the dispersion relation are seen to also be strongly ballooning, with the maximum perturbation being in the equatorial plane. The growth rates for this magnetic field model were found to be very small; however these growth rates are model-dependent and may be much larger for a truly drift-reversed magnetic field model.

### 4. Earthward Gradients of the Pressure Distribution Functions with Solar Wind Variation

- Top of page
- Abstract
- 1. Introduction
- 2. Equilibrium Geotail Plasma Models
- 3. Compressional Eigenmode Equation
- 4. Earthward Gradients of the Pressure Distribution Functions with Solar Wind Variation
- 5. Conclusions
- Acknowledgments
- References

[42] In the absence of turbulence the Earthward convection of the plasma flux tubes compresses the plasma against the Earth's dipolar magnetic field. In the MHD limit there is no heat loss from the flux tubes, and the pressure increases adiabatically with *p*^{ad} ≡ *p* (*V*_{0}/*V*(*x*))^{Γ}, where *V*(*x*) = ∫_{0}^{L}*ds*/*B* and *L* = *L*(*x*) denotes the position of the ionosphere measured along *B* from the equatorial plane position (*x*, *y* = 0, *z* = 0). In Figure 14 we show *V*(*x*) and *p*^{ad}(*x*) = *p*_{0} (*V*_{0}/*V*(*x*))^{Γ} for Γ = 5/3 and *V*_{0} = *V*(*x* = −10). The sharp increase of the pressure gradient in the transition region *x* = −6 to −12*R*_{E} is thought to drive drift-Alfvén ballooning mode instabilities over a spectrum of wavelengths *k*_{y}ρ_{i} ≤ 1 and to produce a turbulent diffusivity *D*_{Ψ} that limits the pressure gradient. In this section we compute the Earthward gradient of the pressure distribution function relative to the adiabatic distribution. For higher energy ions with ρ_{i} > *R*_{c}, the chaotic motion of these ions breaks the magnetic moment invariant, giving a fairly isotropic distribution.

[43] The distant neutral line (DNL) is a source region for entry of new solar wind ions into the geotail, which we model as *S*_{DNL}(*x*, *t*) = *S*_{DNL}(*t*)δ(*x* − *x*_{DNL}). Near the geosynchronous orbit region there is a sink of ions with respect to the geotail plasma. The ions cross from the geotail region into the inner magnetosphere through the Alfvén layer seperatrix. The ions are also lost through the duskside low-latitude boundary layer. We write this net loss region as *S*_{NE}(*t*)δ(*x* − *x*_{NE}) for the near-Earth (NE) loss region.

[44] Now the transport of ions is through the Earthward convective flow , and the turbulent velocity space diffusive flux is −*D*_{Ψ}∂*p*^{ad}/∂Ψ. Thus a one-dimensional model for the ion pressure is

where *D*_{Ψ} is the turbulent diffusivity due to low-frequency drift wave turbulence. For *D*_{Ψ} = 0 and *x*_{DNL} < *x* < *x*_{NE}, the solution of equation (42) is *p*^{ad}(*x*, *t*) = *p*^{ad}(Ψ − ∫_{0}^{t}*E*_{y}(*t*′) *dt*′), where *d*Ψ = *B*_{z}*dx* = *E*_{y}*dt*. This solution is the adiabatic convection of flux tubes in a time-varying convection electric field. We estimate the value of *D*_{Ψ} through

where 〈δ*x*^{2}〉 is the dominant wavelength in the fluctuation spectrum and τ_{c} is the correlation time.

[45] The associated local **E** × **B** turbulent diffusivity is

For 〈δ*x*^{2}〉 ∝ ρ_{i}*L*_{p} ≈ (300 km)(10^{4} km) = 3 · 10^{6} km^{2} and τ_{c} ≈ 100 s we estimate that *D*_{x} = 3 · 10^{4} km^{2}/s. This compares with Borovsky's MHD estimate of *D*_{⟂} ≈ 10^{7} km^{2}/s. The corresponding *D*_{Ψ} = 0.3 R_{E}^{2}(nT)^{2}/s for *B*_{z} = 20nT gives a solution of equation (42),

where *D*_{Ψ} = 〈*B*^{2}*D*_{⟂}〉. Thus we estimate the characteristic drift wave frequency as

in the westward direction for 1/*L*_{pi} = (1 + η_{i})/*L*_{n} > 0. For *L*_{p}^{−1} = ∂_{x} ln*p* from equation (45) used in equation (46) we get

Typically, the fastest growing waves have *k*_{y} ρ_{i} ≈ 0.4 to 1.0.

[46] Equation (42) was solved numerically using a grid in Ψ space with constant spacing. *Tsyganenko and Stern*'s [1996] magnetic field model was used to compute the Δ*x*_{i}'s corresponding to ΔΨ = 1nT · *R*_{E}^{2}. This choice of independent variable translates to variable spaced grid in *x* with many points placed closer to the Earth where the pressure pulses are predicted to pile up and steepen. Second-order operator splitting was used so that a flux-corrected transport algorithm could be used to transport the pressure and a traditional finite-difference method to handle the source terms and the diffusion term.

[47] Figure 15 shows five solutions to equation (42) for different times. An adiabatic pressure profile was taken as the initial condition and the tailward endpoint was taken to be time dependent, corresponding to a time-dependent deposition of ions in the distant tail. The tailward source of ions is given mathematically by

where *p*^{ad}(*x* = *x*_{DNL}) is the adiabatic pressure at the distant neutral line taken here to be 0.06 nPa, Θ(*t* − τ/2) is the step function such that when *t* > τ/2 the oscillating source is turned off.

This choice of the argument of the step function ensures that the pressure pulse is entirely positive. The source pressure *p*^{src} = 1 nPa/s gives the height of the pressure pulse with a period of τ = 60 s. Given these parameters, the energy density deposited over the duration of the pulse is computed to be 19 nPa. The pressure pulse then drifts Earthward by **E** × **B** convection, with a constant *E*_{y} = 1mV/m applied along the tail, until it reaches the dipole braking region where the drift slows, owing to the increase in *B*, and the pressure piles up. The axial *B*_{z}(*x*, 0, 0) is taken from the *Tsyganenko and Stern* [1996] model with *PS* = 0, *P*_{dyn} = 3.0 nPa, DST = −50nT, *B*_{y}^{IMF} = 0, and *B*_{z}^{IMF} = −5.0nT for this simulation. The turbulent diffusivity was taken to be zero, 0.05 R_{E}^{2}(nT)^{2}/*s*, and 0.3 R_{E}^{2}(nT)^{2}/*s*. The pressure pulse is significantly washed out for larger values of the turbulent diffusivity, though there is still some steepening.

[48] In Figure 15e, which occurs at 891 seconds after the tailward source begins its half period oscillation, we see a sharp trailing edge pressure gradient. Such a pressure steepening could potentially be large enough to reverse the direction of the magnetic guiding center drift due the mechanism explained through equation (6) and shown by *Li et al.* [1998]. In fact, the pressure gradient shown in the trailing edge of the pulse in Figure 15 is over 5 times the adiabatic pressure gradient, which is the same factor used to artificially produce drift reversal in *Tsyganenko and Stern* [1996]. This drift reversal will then cause the drift compressional mode to go unstable. The steep transient gradient will also drive the ballooning/interchange mode unstable as explained by *Horton et al.* [2001], which is the primary energy release mechanism in this NGO region.

### 5. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Equilibrium Geotail Plasma Models
- 3. Compressional Eigenmode Equation
- 4. Earthward Gradients of the Pressure Distribution Functions with Solar Wind Variation
- 5. Conclusions
- Acknowledgments
- References

[49] We have derived the kinetic equations for the low-frequency compressional/rarefraction drift-waves in the near-geosynchronous orbit (NGO) region. These waves are thought to be responsible for the shorter period magnetic oscillations reported in the CRRES data [*Maynard et al.*, 1996]. At an early stage before the dipolarization event, there are longer period (60–90 s) predominantly δ*E*_{y} oscillations that are interpreted as the drift-Alfvén ballooning modes. The 30 s magnetic oscillations that occur immediately following dipolarization are consistent with the properties of the drift compressional waves described herein. *Sigsbee et al.* [2002] reported that strong compressional fluctuations of the magnetic field in the range of 7 to 30mHz were observed at the distance of −10 to −13*R*_{E} during the 26 April 1995 substorm. At Geotail the timing they reported is that dipolarization and compressional waves occur together and the large Earthward flows are a few minutes later. In the 10 minute interval following the initialization of dipolarization, the *B*_{z} (GSM) has large fluctuations (δ*B*_{z} ∼ 〈*B*_{z}〉), with 〈*B*_{z}〉 increasing from 5 nT to 20 nT and the total B dropping from 45nT to 20nT during the event. The Pi2 band from 7 to 30mHz (40 s to 150 s period) is strongly excited during this event.

[50] To understand this low-frequency compressional wave, the complex mirror orbits on the flux tubes need to be considered. To account for these orbits we have introduced matrix operators that act on the electromagnetic fields. We have computed the eigenvalues and eigenfunctions of these operators and used these to form dispersion relations. We found that an unstable state due to the ω = _{Di}ϵ resonance can exist for a pressure gradient profile in which the guiding center particle drift is in the ∇*B* drift direction (eastward) due to the large β*R*_{c}/*L*_{p} (large β and *R*_{c}/*L*_{p}). We find that for typical time-averaged magnetic field models, particles do not reverse the direction of the westward bounce-averaged guiding center drift velocity. In these cases, however, a density gradient that is in the opposite direction to the temperature gradient can support the growth of compressional waves. Theoretically, such an inversion of the density gradient may occur as a transient response to a sharp change in the convective flux of *nE*_{y}/*B*. We have found purely oscillating solutions when the frequency of the oscillation is well below and also well above the bounce-averaged guiding-center drift frequency; these results follow closely with traditional compressional waves.

[51] We have numerically solved the nonlocal dispersion relation for the δ*B*_{∥}(*s*) eigenmodes for frequencies below the ion transit frequency, where the wave-particle resonance is responsible for the instability. We report these growth rates γ, real part of the frequency (ω), and growth per wave period γ/ω < 1 as a function of η_{i} for different positions along the geotail axis, using the *Tsyganenko and Stern* [1996] magnetic field model and a simple constant current magnetic field model to compute the properties of the different flux-tubes as well as the structure of the δ*B*_{∥}(*s*) drift compressional eigenmode. We find that the real part of the frequencies that we found correspond to the Pi2 frequency range of a few mHz. We found these magnetic perturbations to be very strongly ballooning, with the maximum perturbation being at the equatorial plane. Our results for the growth rate reproduce well the stability boundaries determined in the local approximation from the Nyquist analysis for marginal stability. We have also presented approximate formulas for these growth rates for both η_{i} > 0 and η_{i} < 0.

[52] We have proposed a transport model for the Earthward transport of the pressure distribution function. In this model, it is seen that sharp pressure gradients can form in the dipole/tail transition region from a small oscillation in the deep tail boundary conditions. The maximum steepness of the pressure gradient is limited by the turbulent diffusivity *D*_{⟂}. We propose that this transient steepening of the pressure gradient could cause particles to reverse the direction of their magnetic drift, which, in turn induces an unstable compressional perturbation. Drift reversal from strong ∇*B* has been invoked by *Li et al.* [1998] as the mechanism for the dispersionless injection of energetic electrons.

[53] In this work we have considered modes dominated by δ*B*_{∥}. When the ambient pressure gradient is steep enough, the coupling to other components of the electromagnetic field increases in importance. The ballooning/interchange mode characterized by a strong *E*_{y} perturbation and weak δ*B*_{∥} becomes active. The nonlocal behavior of these other modes and their coupling to the drift-compressional modes requires the extension of this work to the full 3 × 3 electromagnetic eigenmode equations. This full complex problem will be pursued in a later work.

[54] Previously we found interchange/ballooning modes to be most unstable in the *x* = −6 to −12*R*_{E} region [*Horton et al.*, 1999, 2001; *Wong et al.*, 2001]. As is well known, the interchange/ballooning modes also have a compressional component, δ*B*_{∥}, due to inhomogeneities of the plasma. There are still two distinct drift modes, the drift interchange/ballooning mode and the compressional drift mode, with the compressional magnetoacoustic drift-waves having a higher frequency than the convective interchange mode [*Mikhailovskii*, 1992]. Convective interchange motions of the flux tubes closer to the Earth are stabilized strongly by the energy required to bend the magnetic field, and flux tubes tailward of this region were found to be strongly stabilized by plasma compression. In contrast to the convective interchange mode instability, the compressional modes are unstable at high β from the plasma compression. The condition for the compressional modes to be unstable is complex and presented in this work. One mechanism is the ∇*B* drift reversal. We find the compressional mode is most unstable when the ∇*B* drift is large and opposite to the curvature drift. In the region Earthward of the transition region, the two drifts are in the same direction, and on the tailward side of this region the ∇*B* drift is in the opposite direction to the curvature drift, though it is not large enough to exhibit drift reversal. In our simulation of the transport of the pressure distribution, it was in this transition region, between dipolelike and taillike magnetic field configurations, where the steepest gradients were observed. The other mechanism for compressional instability is inverted density and temperature gradients.

[55] In conclusion, the role the compressional drift wave plays may be required to understand the Pi2 signals associated with substorm dynamics. The mode has some features, such as propagation in the ion diamagnetic direction, in common with the kinetic drift ballooning/interchange mode. The polarization and the stability conditions of the two modes are different as developed and discussed here. Integrated transport codes containing both the effects of the compressional drift modes and the ballooning interchange modes need to be developed to provide a clear interpretation of the complex signatures of observed low-frequency wave signals in the NGO to midtail region.