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[1] The nonlinear saturation of the firehose instability in the high plasma pressure central plasma sheet is shown to produce a wide spectrum of Alfvénic fluctuations in the range of Pi-2 geomagnetic pulsations. The wave energy sources are the small p_{∥}/p_{⊥} > 1 + B^{2}/μ_{0}p_{⊥} anisotropies which are created by Earthward ion convection at constant first and second adiabatic invariants. In the nonlinear state, the field-line curvature force is weaker than the linear force. This weakening of the driving force limits the amplitude of the Alfvénic fluctuations. Away from the equatorial plane, the plasma is firehose stable, but carries large magnetic fluctuations.

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[2] Geomagnetic pulsations of the Pi-2 type are defined as magnetic fluctuations with periods in the range 40–200 s with irregular wave forms. The geomagnetic Pi-2 oscillations are known to be intimately correlated with substorm growth and onsets [Sigsbee et al., 2002] and bursty bulk flows [Kepko and Kivelson, 2001].

[3] During the time of enhanced Earthward plasma convection, there are numerous nonequilibrium features that develop in the ion phase space density distributions, including currents and anisotropies, that may serve to trigger low and ultra-low frequency instabilities. Here we argue that ion pressure anisotropies p_{∥} > p_{⊥} associated with convection or bursty bulk flows drive nonlinear Pi-2 type fluctuations in the geomagnetic tail. We find that the nonlinear firehose driven fluctuations have a rich k_{∥}ω-spectrum and that the local nonlinear firehose stability parameter

fluctuates with only short-lived excursions into the unstable domain σ(t) < 0 occurring at the equatorial plane where the curvature = (b · ∇)b vector has its maximum value.

[4]Kaufmann et al. [2000] use one year of one-minute Geotail data to analyze the firehose parameter A = μ_{0}(p_{∥} − p_{⊥})/B^{2} as a function of β. Kaufmann et al. [2000] report A ranging from −0.1 to +0.4 with the values closer to zero occurring in the high-β bins. The number of data samples in the high-beta bins, β > 30 and 10 < β < 30, is, however, low. Kaufmann et al. [2000] conclude that while A from the data is a few tenths, that the methodology used underestimates the value of A.

[5]Chen and Wolf [1999] develop a model for bursty bulk flows that follows an Earthward accelerated flux tube that develops a firehose instability due to the faster increase of p_{∥} from the shortening of the magnetic field line lengths than of p_{⊥} from B. Ji and Wolf [2003a] follow up the Chen and Wolf model with a detailed Lagrangian simulation model that couples the firehose dynamics to the magnetoacoustic wave dynamics. They find a shock front propagating to the Earth and the development of the firehose instability at the largest k_{∥} = π/Δ in the simulation. They pose the problem of finding the kinetic theory physics that limits the ∣k_{∥}∣ for the growth rate and finding the nonlinear saturation level.

2. Nonlinear Firehose Model

[6] In this letter we present a kinetically-modified Eulerian ion fluid description of the nonlinear firehose instability for the magnetotail problem. We use a simple field-line model with an anisotropic pressure.

[7] At latitude θ, the σ(t, θ) → 1 with the fluctuation energy w_{B} = (δB_{x}^{2}/2μ_{0}) and w_{K} = ρ(E_{y}^{2}/B^{2}) flowing into the region from the magnetic equatorial plane. Outside the high plasma pressure region β = 2μ_{0}p_{⊥}/B^{2} ≫ 1 the oscillations are large amplitude Alfvén waves at finite k_{∥}ρ_{i} where ρ_{i} = (m_{i}T_{i})^{1/2}/eB is the ion gyroradius. The finite ion gyroradius effects and the finite ion inertial scale c/ω_{pi} effects determine the maximum growth rate γ_{max} = [γ(k_{z})] of the firehose instability. The calculation of these dispersion terms is well known and gives rise to the following linear dispersion relation

where we neglect the Landau damping contributions from the warm plasma kinetic dispersion relation [Stix, 1992; Gary, 1993]. For k_{∥}ρ_{i} = (k_{∥}c/ω_{pi})(β_{i}/2)^{1/2} ≪ 1 the linear dispersion term in ω/ω_{ci} is negligible and equation (2) yields the MHD firehose instability ω^{2} = k_{∥}^{2}v_{A}^{2}σ with σ defined in equation (1). For p_{∥}/p_{⊥} > 1 + 2/β_{i} the MHD growth rate is = ∣k_{∥}∣v_{A} increasing monotonically with k_{∥}. From the kinetic dispersion relation there is a well-defined maximum growth rate at k_{∥}ρ_{i} = (1 − T_{⊥}/T_{∥} − 2/β_{i∥})^{1/2} where the maximum growth rate

We have written an initial value code using the dipole magnetic field lines and typically taking the mass density ρ(s) B(s) to study the linear and nonlinear kinetically-modified Alfvénic-Firehose turbulence occurring in the night-side magnetotail. The nonlinear equation for the complex displacement field ξ(z, t) is

where σ_{nℓ} is given in equation (1) with B^{2} = B_{n}^{2}(1 + ∣∂_{z}ξ∣^{2}) and ν = ω_{ci}ρ_{i}^{2} = cT/eB follows from equation (2). The complex valued fields are ξ = ξ_{x} + iξ_{y} and δB/B_{n} = ∂_{z}ξ_{x} + i∂_{z}ξ_{y} with the real δB^{2} = B_{n}^{2} + δB*δB used in equation (1). The last term in equation (4) describes the Faraday rotation of the polarization vector of the Alfvén wave. The phase rotation δϕ in time Δt is δ = νk_{∥}^{2}Δt = k_{∥}^{2}(cT_{i∥}/eB)Δt.

[8] The derivation of equation (4) follows from the acceleration equation

where P = (p_{∥} − p_{⊥})(BB/B^{2}) + p_{⊥}I, B = B_{n}_{x} + δB_{x}_{x}, j_{y} = = and δB_{x} = B_{z}∂_{z}ξ from Faraday's law ∂_{z}E_{y} = ∂B_{x}/∂t and Ohm's law E_{y} − v_{x}B_{z} = E_{y} − B_{z}(∂ξ/∂t) = 0. The inertial acceleration from the j_{y}B_{z} force gives ∂_{t}^{2}ξ = v_{A}^{2}∂_{z}^{2}ξ. The nonlinear pressure gradient force follows from the calculation of

which reduces to

after using B = B_{n}(1 + ∣∂_{z}ξ∣^{2})^{1/2} and B_{n} = B_{z} = constant.

[9] The nonlinear driving term at small δB_{x}/B = ∂_{z}ξ ≪ 1 is proportional to the field-line displacement ξ. At large amplitudes δB_{x}/B 1 the force is greatly weakened, eventually decreasing with ξ as ∂_{z}^{2}ξ/∣∂_{z}ξ∣^{2}. The effect is familiar as the decrease of curvature for a string y = y(x) given by the curvature formula y″/(1+y′^{2})^{3/2}. This may be called the magnetic field line crinkling.

[10] The nonlinear partial differential equation (4) develops mode coupling from the nonlinear curvature force. The nonlinear partial differential equation for U(, τ) = B_{z}ξ(z/L, tv_{A0}/L) with τ = tB_{z}/(μ_{0}ρ_{0})^{1/2}L where L is the scale length (∼R_{E}) of the field line variation is

with boundary conditions U(z = ±L, τ) = 0. Here ν = T_{∥i}/eBLv_{A0} = ρ_{ix}v_{∥i}/Lv_{A}.

[11] The total energy is

where K = dθρ(∂ξ/∂t)^{2}, W_{B} = dθδ^{2}/2μ_{0} and W_{P} = −A/2ℓndθ. We use the energy invariant to test the accuracy of the numerical solutions.

[12] The nonlinear potential is of the type that occurs in the derivative nonlinear Schrödinger equation (DNLS) which governs dispersive Alfvén waves [Horton and Ichikawa, 1996]. In perturbative theory the nonlinear force weakens as ≃ [A − 1 − A∣∂_{z}ξ∣^{2}](∂_{z}^{2}ξ). Here we show a typical example of the solution of equation (8). In a future work, we will develop solutions for the magnetotail that include parametric decays of these fluctuations into Alfvén waves and acoustic waves. A single k_{∥}-mode calculation shows that the saturated amplitude varies as ξ_{max} = k_{∥}^{−1}(A^{1/2} − 1)^{1/2} and has the nonlinear frequency ω_{A}(A − 1)^{1/2}.

[13] In addition to the intrinsic nonlinear force derived in equation (8) that saturates the instability and produces the nonlinear oscillations shown in Figures 1–4 there is the quasilinear change of the pressure anisotropy due to reaction of the magnetic fluctuations on the background ion phase space distribution function [Gary, 1993; Sagdeev and Galeev, 1969]. Here we ignore the quasilinear change of the pressures arguing that on the background evolution time scale there are processes building up the anisotropy competing with the quasilinear relaxation of the anisotropy. Thus, the turbulence shown here for a fixed value of A = μ_{0}(p_{∥} − p_{⊥})/B_{n}^{2} given here overestimates the amplitude obtained with addition of the quasilinear background transport. In the simulation we model this effect rather crudely by setting p_{∥} = p_{⊥} after some number of nonlinear oscillations. There follows a period of large amplitude Alfvén wave oscillations which may be subject to parametric instabilities for A above a critical value.

[14] The simulations are performed as an initial value problem with equatorial field-line crossing at x ∼ 8 R_{E} with β_{⊥}(0) = 2μ_{0}p_{⊥}/B_{n}^{2} = 5 and p_{∥}/p_{⊥} = 1.44 so that A = 1.1 or σ(t = 0) = −0.1 and ν = 0.01. We use a 4th–5th over adaptive RK integrator to advance the finite differential equations on 129 grid points along the magnetic field line. The local equatorial field strength is B_{n} ≃ 100 nT. The local ion gyroradius ρ_{i} ≅ 200 km.

3. Simulation Results

[15]Figure 1a shows the fastest growing eigenfunction at 10t_{A} ≃ 820 s as a function position along the field line. Figure 1b shows that there is a well-defined eigenmode frequency (ω = 2πf ∼ 8/t_{A} ∼ 5–10 mHz), Figure 1c shows the spectrum of k_{∥} = 2πn/L_{∥} values in the eigenmode. Figure 1d gives the profile of σ(θ, t = 10t_{A}). We have not calculated analytically the eigenmodes. The local equatorial plane frequency and growth rate for the most unstable k_{∥} are ω ≃ γ_{max} = 10(v_{A0}/L), but the observed eigenmode growth rate is much smaller than γ_{max} due to the localization of σ(θ).

[16]Figures 2a and 2b show the magnetic fluctuations δB_{x}(t) and its frequency spectrum δB_{x}^{2}(ω). Figure 2c gives the electric field fluctuations E_{y}(t). Figure 2d shows the nonlinear state of the plasma defined by ξ(t) = ξ_{x}(t) + iξ_{y}(t) for a time interval Δt = 15t_{A} ≃ 20 min in the saturated state. There are several characteristic frequencies in the dynamics. In this example, there is a short period signal T_{1} ≃ 100 s and a long nonlinear period T_{0} ≃ 300 s.

[17] The time series in Figure 2 for the electric and magnetic fields show a chaotic structure even though the power is concentrated in two frequencies. Such spectra are typical of lower-order dynamical systems suggesting a search for a low-order model derived from the pde. There are bursts of energy releases from the nonlinear dynamics that are one aspect of a self-organized criticality (SOC) system. The second aspect of space-scale invariance of an SOC system is not satisfied due to the key role of the ion gyroradius in defining the fast-growing linear mode at k_{∥,max}. The turbulent fluctuations mode-couple to both shorter and longer wavelength fluctuations. For σ = 1 there are soliton solutions to the associated DNLS equation [Horton and Ichikawa, 1996] derived by factoring the Alfvén equation into uncoupled right and left propagating waves. In the present problem the driving by the pressure anisotropy disrupts those solitons and couples the parallel (right) and antiparallel (left) propagating Alfvén waves. For equation (8) we find coherent, localized solutions not unlike solitons for ν = 0.03.

[18]Figure 3 shows the three energy components in equation (9) for the solutions in Figure 2. After the exponential growth the turbulence saturates with the magnetic fluctuation energy W_{B} ≃ −W_{P} the source of instability and both large compared with kinetic energy K. In the simulation the total energy is well conserved as follows analytically from the model. Thus, there is an efficient conversion from the anisotropy thermal energy reservoir into magnetic energy. The increase of the magnetic turbulence with the anisotropy parameter A is shown in Figure 4. At small A − 1 values the system shows coherent structures and the error bars may underestimate the actual error.

4. Conclusions

[19] In conclusion, the presence of small fractional pressure anisotropies in the high-beta magnetotail plasma produce an intermittent spectrum of Alfvénic fluctuations that are in the range of the Pi-2 signals commonly associated with bursty bulk flows [Kepko and Kivelson, 1999, 2001] and substorm dynamics. The kinetic ballooning pressure gradient instability [Roux et al., 1991] remains a possible cause of these fluctuations. The nonlinear firehose model also produces candidate magnetic fluctuations and requires the kinetic dispersion term to give the maximum growth rate, the wave dispersion and polarization. The nonlinear model is integrated and shows irregular wave forms with intermittent turbulent energy releases. Further research that includes the driving sources of the anisotropy and the quasilinear velocity scattering of the ions from the magnetic fluctuations is required for a more detailed picture of the long-time evolution of the system.

[20] In a related work Ji and Wolf [2003b] argue that the appearance of the anisotropy is a product of the near-Earth neutral line formation. Consequently, the observations of Pi-2 turbulence described here along the auroral field lines may be a signature of the flow braking from fast Earthward flows produced by a near-Earth neutral line. It remains for future 2D theory and simulations to find the balance of the quasilinear reduction of the anisotropy and its production by Earthward flows.

[21] Finally, the problem of correlating the observed Pi2 oscillations with the various theoretical models remains.

Acknowledgments

[22] The authors thank Richard Wolf and Shuo Ji for numerous useful discussions along the path to this solution. The work was supported by the National Science Foundation ATM-0229863.