Mirror Instability Threshold Research Articles

Kinetic Simulations of Proton Mirror Instability: Phase Relations and Thermodynamics

Mirror-mode waves driven by the large temperature anisotropy of T ⊥/T ∣∣ > 1 have been widely observed in the solar wind, planetary magnetosheaths, heliosheath, etc. Recent studies have shown that the phase relations and thermodynamics of the mirror waves observed in the terrestrial magnetosheath may well be interpreted by the linear mixed kinetic–MHD theory of proton mirror instability. In particular, the energy laws possess the form of double-polytropic closures with the thermodynamic exponents being functions of β ⊥,∣∣ = p ⊥,∣∣/(B 2/2μ 0). In this study, we examine the time evolution of proton mirror instability based on the hybrid particle simulations for twenty sets of β ⊥,∣∣ values. Quantitative comparisons between the kinetic simulations, linear Vlasov theory, and observations are made in terms of the growth rates, phase relations, thermodynamic conditions, etc., which show high agreements. In particular, the dependences of various compressibility and thermodynamic exponents on β ⊥,∣∣ are confirmed by the kinetic simulations, which show that the polytropic exponents are in the ranges of γ ⊥ = 0.64 ± 0.21, and γ ∣∣ = 1.07 ± 0.12 consistent with the theoretical predictions and mirror observations of γ ⊥ < 1 and γ ∣∣ ≳ 1. It is shown that the observed features, including the various perturbations and wavelengths, may indeed be reproduced by the nonlinear simulations. The saturated temperature anisotropy β ⊥/β ∣∣ and plasma β ∣∣ show an anticorrelation, which may well be fitted by the modified mirror instability threshold of with γ ⊥ ≈ 0.8, γ ∣∣ ≈ 1.3, and the saturated magnetic field of δ B/B ≈ 0.26 ∼ 0.97 increases with increasing values of β ≈ 1.6 ∼ 8.3.

Proton Temperature Anisotropy Variations in Inner Heliosphere Estimated with the First Parker Solar Probe Observations

We report proton temperature anisotropy variations in the inner heliosphere with Parker Solar Probe (PSP) observations. Using a linear fitting method, we derive proton temperature anisotropy with temperatures measured by the Solar Probe Cup (SPC) from the SWEAP instrument suite and magnetic field observations from the FIELDS instrument suite. The observed radial dependence of temperature variations in the fast solar wind implies stronger perpendicular heating and parallel cooling than previous results from Helios measurements made at larger radial distances. The anti-correlation between proton temperature anisotropy and parallel plasma beta is retained in fast solar wind. However, the temperature anisotropies of the slow solar wind seem to be well constrained by the mirror and parallel firehose instabilities. The perpendicular heating of the slow solar wind inside 0.24 AU may contribute to its same trend up against mirror instability thresholds as fast solar wind. These results suggest that we may see stronger anisotropy heating than expected in inner heliosphere.

Thermodynamic Properties of Mirror Structures in the Magnetosheath: MMS Observations and Double-polytropic MHD Simulations

Abstract Mirror structures are manifested as a wave train of magnetic peaks and/or dips, which are anticorrelated with plasma density. The evolution behavior of the ion temperatures perpendicular and parallel to the local magnetic field, T ⊥ and T ∥, in the magnetic peaks and dips is rare to be theoretically studied. In this paper, the thermodynamic properties of mirror structures are investigated using two-dimensional magnetohydrodynamic simulations with double-polytropic laws. Two polytropic exponents, γ ∥ and γ ⊥, are used as parameters to describe various thermodynamic conditions in the anisotropic plasma—for example, γ ∥ = 1, γ ⊥ = 1 for double-isothermal and γ ∥ = 3, γ ⊥ = 2 for double-adiabatic. Using empirical values of γ ∥ = 1.14 and γ ⊥ = 0.94 for magnetosheath plasma, the variations of T ∥ and T ⊥ in the magnetic dips and peaks observed by the Magnetospheric Multiscale Mission in the Earth’s magnetosheath can be reproduced. This consistent result cannot be achieved by use of γ ∥ = 3, γ ⊥ = 2 and γ ∥ = 0.5, γ ⊥ = 2, where the latter exponents can lead to the same mirror instability threshold as kinetic theory. In magnetic dips, T ∥ is found to be increased for γ ∥ &gt; 1 but decreased for γ ∥ &lt; 1, while T ⊥ is decreased for γ ⊥ &gt; 1 but increased for γ ⊥ &lt; 1. For magnetic peaks, the variation features of T ∥ and T ⊥ are opposite to that of the magnetic dips. It is concluded that the thermodynamic properties of mirror structures are further from double-adiabatic but closer to double-isothermal conditions.

Simulation of magnetic holes formation in the magnetosheath

Magnetic holes have been frequently observed in the Earth's magnetosheath and are believed to be the consequence of the nonlinear evolution of the mirror instability. Mirror mode perturbations mainly form as magnetic holes in regions where the plasma is marginally mirror stable with respect to the linear instability criterion. We present an expanding box particle-in-cell simulation to mimic the changing conditions in the magnetosheath as the plasma is convected through it that produces mirror mode magnetic holes. We show that in the initial nonlinear evolution, where the plasma conditions are mirror unstable, the magnetic peaks are dominant, while later, as the plasma relaxes toward marginal stability, the fluctuations evolve into deep magnetic holes. While the averaged plasma parameters in the simulation remain close to the mirror instability threshold, the local plasma in the magnetic holes is highly unstable to mirror instability and locally mirror stable in the magnetic peaks.

Diamagnetic “bubble” equilibria in linear traps

The plasma equilibrium in a linear trap at β ≈ 1 (or above the mirror-instability threshold) under the topology-conservation constraint evolves into a kind of diamagnetic “bubble.” This can take two forms: either the plasma body greatly expands in radius while containing the same magnetic flux, or, if the plasma radius is limited, the plasma distribution across flux-tubes changes, so that the same cross-section contains a greatly reduced flux. If the magnetic field of the trap is quasi-uniform around its minimum, the bubble can be made roughly cylindrical, with radius much larger than the radius of the corresponding vacuum flux-tube, and with non-paraxial ends. Then the effective mirror ratio of the diamagnetic trap becomes very large, but the cross-field transport increases. The confinement time can be found from solution of the system of equilibrium and transport equations and is shown to be τE≈τ∥τ⊥. If the cross-field confinement is not too degraded by turbulence, this estimate in principle allows construction of relatively compact fusion reactors with lengths in the range of a few tens of meters. In many ways, the described diamagnetic confinement and the corresponding reactor parameters are similar to those claimed by the field-reversed configurations.

Inertial-range kinetic turbulence in pressure-anisotropic astrophysical plasmas

A theoretical framework for low-frequency electromagnetic (drift-)kinetic turbulence in a collisionless, multi-species plasma is presented. The result generalises reduced magnetohydrodynamics (RMHD) and kinetic RMHD (Schekochihinet al.,Astrophys. J. Suppl. Ser., vol. 182, 2009, pp. 310–377) to the case where the mean distribution function of the plasma is pressure-anisotropic and different ion species are allowed to drift with respect to each other – a situation routinely encountered in the solar wind and presumably ubiquitous in hot dilute astrophysical plasmas such as the intracluster medium. Two main objectives are achieved. First, in a non-Maxwellian plasma, the relationships between fluctuating fields (e.g. the Alfvén ratio) are order-unity modified compared to the more commonly considered Maxwellian case, and so a quantitative theory is developed to support quantitative measurements now possible in the solar wind. Beyond these order-unity corrections, the main physical feature of low-frequency plasma turbulence survives the generalisation to non-Maxwellian distributions: Alfvénic and compressive fluctuations are energetically decoupled, with the latter passively advected by the former; the Alfvénic cascade is fluid, satisfying RMHD equations (with the Alfvén speed modified by pressure anisotropy and species drifts), whereas the compressive cascade is kinetic and subject to collisionless damping (and for a bi-Maxwellian plasma splits into three independent collisionless cascades). Secondly, the organising principle of this turbulence is elucidated in the form of a conservation law for the appropriately generalised kinetic free energy. It is shown that non-Maxwellian features in the distribution function reduce the rate of phase mixing and the efficacy of magnetic stresses, and that these changes influence the partitioning of free energy amongst the various cascade channels. As the firehose or mirror instability thresholds are approached, the dynamics of the plasma are modified so as to reduce the energetic cost of bending magnetic-field lines or of compressing/rarefying them. Finally, it is shown that this theory can be derived as a long-wavelength limit of non-Maxwellian slab gyrokinetics.

Variational approach for static mirror structures

Anisotropic static plasma equilibria where the parallel and perpendicular pressures are only functions of the amplitude of the local magnetic field are shown to be amenable to a variational principle with a free energy density given by the parallel tension. This approach is used to demonstrate that two-dimensional small-amplitude static magnetic holes constructed from a Grad-Shafranov type equation slightly below the (subcritical) mirror instability threshold identify with lump solitons of KPII equation, but turn out to be unstable. Differently, large-amplitude magnetic structures, which are stable as they realize a minimum of the free energy, are computed using a gradient method within two-dimensional numerical simulations where the regularizing effect of finite Larmor radius corrections is retained. Interestingly, these structures transform from stripes to bubbles when the angle of the magnetic field with the coordinate plane is increased.

Subcritical mirror structures in an anisotropic plasma

Based on Grad-Shafranov-like equations, a gyrotropic plasma where the pressures in the static regime are only functions of the amplitude of the local magnetic field is shown to be amenable to a variational principle with a free energy density given by the parallel tension. This approach is used to demonstrate that small- amplitude static holes constructed slightly below the mirror instability threshold identify with lump solitons of KPII equation and turn out to be unstable. It is also shown that regularizing effects such as finite Larmor radius corrections cannot be ignored in the description of large-amplitude mirror structures.

General formulation for magnetohydrodynamic wave propagation, fire-hose, and mirror instabilities in Harris-type current sheets

Harris-type current sheets with the magnetic field model of B⇀=Bx(z)x̂+By(z)ŷ have many important applications to space, astrophysical, and laboratory plasmas for which the temperature or pressure usually exhibits the gyrotropic form of p↔=p∥b̂b̂+p⊥(I↔−b̂b̂). Here, p∥ and p⊥ are, respectively, to be the pressure component along and perpendicular to the local magnetic field, b̂=B→/B. This study presents the general formulation for magnetohydrodynamic (MHD) wave propagation, fire-hose, and mirror instabilities in general Harris-type current sheets. The wave equations are expressed in terms of the four MHD characteristic speeds of fast, intermediate, slow, and cusp waves, and in the local (k∥,k⊥,z) coordinates. Here, k∥ and k⊥ are, respectively, to be the wave vector along and perpendicular to the local magnetic field. The parameter regimes for the existence of discrete and resonant modes are identified, which may become unstable at the local fire-hose and mirror instability thresholds. Numerical solutions for discrete eigenmodes are shown for stable and unstable cases. The results have important implications for the anomalous heating and stability of thin current sheets.

Magnetic reconnection associated fluctuations in the deep magnetotail: ARTEMIS results

Abstract. On the basis of ARTEMIS two-probe mission magnetic reconnection (MR) outflow associated magnetic fluctuations and turbulence are analyzed on 19 February 2011. In the deep-tail, at distances between X = 45 – 51 RE, evidence for reconnection associated plasma sheet thinning was found, accompanied by heating of the plasma sheet. Correlated flow and field reversals and the large-scale Hall-effect signatures indicated the presence of the reconnection X-line. Within fast reconnection plasma outflows, magnetic fluctuations exhibit the same spectral scaling features and kinked spectra as magnetic fluctuations in the solar wind or in various parts of geospace. It was shown that the proton scale magnetic fluctuations are constrained by oblique firehose, proton cyclotron and mirror instability thresholds. For parallel plasma β|| &gt; 1, where the thresholds converge, perpendicular magnetic fluctuations are enhanced. Magnetic compressibility decreases with the distance to the neutral sheet, however, near the instability thresholds it is comparable to the values obtained in the solar wind.

Case studies of mirror-mode structures observed by THEMIS in the near-Earth tail during substorms

[1] An examination of the magnetic field and plasma observed by the inner THEMIS-D spacecraft (P3) close to the equatorial plane at ∼11RE at local midnight reveals the occurrence of mirror-mode structures. These structures have the same characteristic waveform seen in other regions. The examination of the mirror-mode instability shows that inside these structures the threshold of mirror instability is marginally reached, while the surrounding plasma is mirror stable. The observed mirror structures occur in the dipolarized magnetic field following a substorm-related dipolarization. It is found that after the dipolarization front, the local ions become more anisotropic and initial magnetic holes form inside this anisotropic plasma before the fully-fledged mirror structures are observed. The ions become less anisotropic afterward, but the strong field depression in the magnetic holes enhances the effective plasma beta so that the mirror instability threshold is marginally reached. Thus, the dipolarization process provides the large-amplitude magnetic field fluctuations and the anisotropic plasma environment for mirror structures to grow. The isolated large-amplitude mirror-mode structures survive in the mirror-stable plasma even through the plasma becomes less anisotropic. It is also found that the width of magnetotail mirror-structures is smaller than one gyroradius of a plasma sheet proton, which is different from the width of mirror structures in other regions. These mirror structures appear to have a strong correlation with electron anisotropy changes. These observations suggest that electron kinetics may also play a role during the growth and saturation of mirror instability in the near-Earth tail.

Equilibrium of a High-β Plasma with Sloshing Ions above the Mirror Instability Threshold

Within paraxial approximation it is shown that in an anisotropic plasma confined an open-ended system with β above the mirror instability threshold a “magnetic hole” is formed near the turning point of the sloshing ions.

Formation of a magnetic hole above the mirror-instability threshold in a plasma with sloshing ions

Within the framework of paraxial approximation it is shown that in an anisotropic plasma with sloshing ions confined an open-ended system a magnetic hole is formed near the turning point of the sloshing ions above the threshold of the mirror instability. The magnetic field experiences a jump at the hole boundary from the side of the magnetic mirror. For a small excess over the mirror instability threshold, the surface of the discontinuity has the shape of a truncated paraboloid, and the magnitude of the magnetic field jump at the system axis is proportional to the radius of the hole and gradually decreases to zero away of the axis. It is argued that disappearance of the magnetic hole because of the widening of the sloshing ions angular spread in the course of the neutral beam injection results in abrupt anticorrelated changes of the diamagnetic signals measured near the turning point of the sloshing ions and near the midplane of the gas-dynamic trap.

Magnetic Fluctuation Power Near Proton Temperature Anisotropy Instability Thresholds in the Solar Wind

The proton temperature anisotropy in the solar wind is known to be constrained by the theoretical thresholds for pressure-anisotropy-driven instabilities. Here, we use approximately 1x10;{6} independent measurements of gyroscale magnetic fluctuations in the solar wind to show for the first time that these fluctuations are enhanced along the temperature anisotropy thresholds of the mirror, proton oblique firehose, and ion cyclotron instabilities. In addition, the measured magnetic compressibility is enhanced at high plasma beta (beta_{ parallel} greater, similar1) along the mirror instability threshold but small elsewhere, consistent with expectations of the mirror mode. We also show that the short wavelength magnetic fluctuation power is a strong function of collisionality, which relaxes the temperature anisotropy away from the instability conditions and reduces correspondingly the fluctuation power.

Mirror structures above and below the linear instability threshold: Cluster observations, fluid model and hybrid simulations

Abstract. Using 5 years of Cluster data, we present a detailed statistical analysis of magnetic fluctuations associated with mirror structures in the magnetosheath. We especially focus on the shape of these fluctuations which, in addition to quasi-sinusoidal forms, also display deep holes and high peaks. The occurrence frequency and the most probable location of the various types of structures is discussed, together with their relation to local plasma parameters. While these properties have previously been correlated to the β of the plasma, we emphasize here the influence of the distance to the linear mirror instability threshold. This enables us to interpret the observations of mirror structures in a stable plasma in terms of bistability and subcritical bifurcation. The data analysis is supplemented by the prediction of a quasi-static anisotropic MHD model and hybrid numerical simulations in an expanding box aimed at mimicking the magnetosheath plasma. This leads us to suggest a scenario for the formation and evolution of mirror structures.

Nonlinear mirror waves in non‐Maxwellian space plasmas

A theory of finite‐amplitude mirror type waves in non‐Maxwellian space plasmas is developed. The collisionless kinetic theory in a guiding center approximation, modified for accounting of the finite ion Larmor radius effects, is used as the starting point. The model equation governing the nonlinear dynamics of mirror waves near instability threshold is derived. In the linear approximation it describes the classical mirror instability that is valid for a wide class of the velocity distribution functions. In the nonlinear regime the mirror waves form solitary structures that have the shape of magnetic holes. The formation of such structures and their nonlinear dynamics has been analyzed both analytically and numerically. It is suggested that the main nonlinear mechanism responsible for mirror instability saturation is associated with modification (flattening) of the shape of the background ion distribution function in the region of small parallel particle velocities. The width of this region is of the order of the particle trapping zone in the mirror hole. Near the mirror instability threshold the saturation arises before its width reaches the ion thermal velocity. The nonlinear mode coupling effects in this approximation are smaller and unable to take control over evolution of the space profile of saturated mirror waves or lead to their magnetic collapse. This results in the appearance of quasi‐stable solitary mirror structures having the form of deep magnetic depressions. A phenomenological description of this process is formulated. The relevance of the theoretical results to recent satellite observations is stressed.

Properties of magnetosheath mirror modes observed by Cluster and their response to changes in plasma parameters

Mirror modes are large amplitude nonpropagating compressive structures frequently observed in the magnetosheath. They appear in the form of quasi‐sinusoidal oscillations in the magnetic field, profound magnetic decreases (dips) or magnetic enhancements (peaks), accompanied by a corresponding anticorrelated signature in plasma density. In this study we present an analysis of the properties of mirror mode structures in the magnetosheath of the Earth based on a database of Cluster observations and also a detailed case study of one magnetosheath traversal. We focused primarily on the identification of conditions associated with the magnetic dips and magnetic peaks. It is shown that the character of mirror structures is related to the local degree of instability of the plasma with respect to the mirror instability threshold: peaks are typically observed in an unstable plasma, while mirror structures observed deep within the stable region appear almost exclusively as dips. This observation is found to be consistent with recent theoretical and numerical studies. An abrupt transition of mirror structures from peaks to dips at an approximate distance of 2 Earth radii from the magnetopause was identified by multispacecraft analysis and we interpret this effect as a consequence of plasma expansion in the vicinity of the magnetopause locally changing the plasma conditions towards a more stable state.

Magnetic holes in plasmas close to the mirror instability

Abstract. Non-propagating magnetic hole solutions in anisotropic plasmas near the mirror instability threshold are investigated in numerical simulations of a fluid model that incorporates linear Landau damping and finite Larmor radius corrections calculated in the gyrokinetic approximation. This FLR-Landau fluid model reproduces the subcritical mirror bifurcation recently identified on the Vlasov-Maxwell system, both by theory and numerics. Stable magnetic hole solutions that display a polarization different from that of Hall-MHD solitons are indeed obtained slighlty below threshold, while magnetic patterns and spatio-temporal chaos emerge when the system is maintained in a mirror unstable regime.

Mirror instability including finite Larmor radius effects

We present a fully kinetic theory of the magnetic mirror instability accounting for finite ion Larmor radius (FLR) effects. Including FLR effects leads to a substantial modification of both the instability growth rate and the instability threshold. For wavelengths the order of the ion Larmor radius the effective elasticity of the magnetic field lines increases substantially. The latter results in an increase of the mirror instability threshold. A compact analytical expression is obtained for the growth rate of the fastest growing mode in the fully kinetic regime. In the presence of FLR effects, a non-coplanar component of the magnetic field perturbation is generated which is occasionally observed in satellite data.

Mirror Instability in Space Plasmas: Finite Ion Larmor Radius Effects

A fully kinetic theory of the magnetic mirror instability accounting for the finite ion Larmor radius (FLR) effect is developed. It is shown that the FLR effect leads to a substantial modification of both the instability growth rate and the instability threshold. It is found that for wavelengths of the order of the ion Larmor radius the effective elasticity of the magnetic field lines increases substantially. The latter results in a subsequent increase of the mirror instability threshold. An analytical expression for the growth rate of the fastest growing mode in the fully kinetic regime is derived. Furthermore, it is shown that the presence of FLR effect leads to the generation of the non-coplanar components of the magnetic field perturbations occasionally observed in satellite data.