Conservation Of Adiabatic Invariants Research Articles

Improved modelling of SEP event onset within the WSA–Enlil–SEPMOD framework

Multi-spacecraft observations of solar energetic particle (SEP) events not only enable a deeper understanding and development of particle acceleration and transport theories but also provide important constraints for model validation efforts. However, because of computational limitations, a given physics-based SEP model is usually best suited to capture a particular phase of an SEP event, rather than its whole development from onset through decay. For example, magnetohydrodynamic (MHD) models of the heliosphere often incorporate solar transients only at the outer boundary of their so-called coronal domain – usually set at a heliocentric distance of 20–30 R⊙. This means that particle acceleration at coronal mass ejection (CME)-driven shocks is also computed from this boundary onwards, leading to simulated SEP event onsets that can be many hours later than observed, since shock waves can form much lower in the solar corona. In this work, we aim to improve the modelled onset of SEP events by inserting a “fixed source” of particle injection at the outer boundary of the coronal domain of the coupled WSA–Enlil 3D MHD model of the heliosphere. The SEP model that we employ for this effort is Solar Energetic Particle MODel (SEPMOD), a physics-based test-particle code based on a field line tracer and adiabatic invariant conservation. We apply our initial tests and results of SEPMOD’s fixed-source option to the 2021 October 9 SEP event, which was detected at five well-separated locations in the inner heliosphere – Parker Solar Probe, STEREO-A, Solar Orbiter, BepiColombo, and near-Earth spacecraft.

Available energy of magnetically confined plasmas

The concept of the available energy of a collisionless plasma is discussed in the context of magnetic confinement. The available energy quantifies how much of the plasma energy can be converted into fluctuations (including nonlinear ones) and is thus a measure of plasma stability, which can be used to derive linear and nonlinear stability criteria without solving an eigenvalue problem. In a magnetically confined plasma, the available energy is determined by the density and temperature profiles as well as the magnetic geometry. It also depends on what constraints limit the possible forms of plasma motion, such as the conservation of adiabatic invariants and the requirement that the transport be ambipolar. A general method based on Lagrange multipliers is devised to incorporate such constraints in the calculation of the available energy, and several particular cases are discussed for which it can be calculated explicitly. In particular, it is shown that it is impossible to confine a plasma in a Maxwellian ground state relative to perturbations with frequencies exceeding the ion bounce frequency.

Misbehaving High‐Energy Electrons: Evidence in Support of Ubiquitous Wave‐Particle Interactions on Dayside Martian Closed Crustal Magnetic Fields

AbstractMultiple studies have reported either isotropic or trapped pitch angle distributions of high‐energy (>100 eV) electrons on closed crustal field lines on the dayside of Mars. These pitch angle distributions are not to be expected from collisional scattering and conservation of adiabatic invariants alone. We use 2 years of data from the Mars Atmosphere and Volatile EvolutioN mission to analyze the pitch angle distributions of superthermal electrons on dayside‐closed crustal magnetic fields and compare to results from an electron transport model. Low‐energy electrons (10–60 eV) have pitch angle distributions in agreement with modeling results, while high‐energy electrons (100–500 eV) do not. High‐energy electrons have a flux peak at perpendicular pitch angles which suggests there is a ubiquitous energization process occurring on crustal fields. Wave‐particle interactions seem to be the most likely candidate. Trapping of high‐energy electrons may impact the nightside ionosphere dynamics.

Representation of the Third Adiabatic Invariant in Flux Form and Some Consequences of Its Conservation by Examples of Specific Axial Magnetic Systems

The dynamics of invariant magnetic surfaces is considered with examples of axially symmetric magnetic systems. The inner trap (magnetic mirror) and the outer trap (dipole) are distinguished in a system formed by two coils with current. The dynamics of noninteracting particles moving in the plane of the magnetic equator is considered in the drift approximation for the perturbation vector satisfying the third invariant conservation condition. The results are extended to the equator of the geomagnetic trap, in which the external perturbing field is given by the ring current. It is shown that the real effect of the ring current on plasma dynamics in the bounded magnetosphere of the Earth in the approximation of constancy of the third invariant differs from conventional model calculations based on dipole representations of the geomagnetic field. In this case, it is not the magnetic flux that makes practical sense but only its change, which is equal to the difference between the final and initial fluxes calculated by the parameters of the real field. The minimum values of the energies of charged particles in the radiation belts required to satisfy the third condition of adiabatic invariant conservation are estimated for the main phase of a particular magnetic storm.

Electron radiation belt dynamics during magnetic storms and in quiet time

The paper discusses the dynamics of the outer electron belt, adiabatic and nonadiabatic mechanisms of replenishment and losses of energetic electrons. Under undisturbed conditions, the outer electron belt gradually empties: in the inner magnetosphere due to electron precipitation in the atmosphere and in the quasi-trapping region due to losses at the magnetopause because drift shells of electrons are not closed there. The latter process does not occur in normal years due to the masking replenishment by freshly accelerated particles, but in years of extremely low activity it leads to a significant decrease in the electron population of the belt. During the magnetic storm main phase, the first reason for the decrease in the electron flux intensity is the adiabatic cooling associated with conservation of adiabatic invariants and complemented by precipitation of electrons into the atmosphere and their dropout at the magnetopause. Electron flux increases involve EB electron injection by the induction electric field of substorm activation and by the large-scale solar wind electric field, with pitch energy diffusion along with adiabatic heating in the recovery phase. The rate of electron flux recovery after a storm is determined by the ratio of nonadiabatic increases and losses; hence the electron flux represents a continuous series from low to very high values. The combination of these processes determines the individual character of radiation belt development during each magnetic storm and the behavior of the belt in the quiet time.

Динамика электронного радиационного пояса во время магнитных возмущений и в спокойное время

The paper discusses the dynamics of the outer electron belt, adiabatic and nonadiabatic mechanisms of replenishment and losses of energetic electrons. Under undisturbed conditions, the outer electron belt gradually empties: in the inner magnetosphere due to electron precipitation in the atmosphere and in the quasi-trapping region due to losses at the magnetopause because drift shells of electrons are not closed there. The latter process does not occur in normal years due to the masking replenishment by freshly accelerated particles, but in years of extremely low activity it leads to a significant decrease in the electron population of the belt. During the magnetic storm main phase, the first reason for the decrease in the electron flux intensity is the adiabatic cooling associated with conservation of adiabatic invariants and complemented by precipitation of electrons into the atmosphere and their dropout at the magnetopause. Electron flux increases involve EB electron injection by the induction electric field of substorm activation and by the large-scale solar wind electric field, with pitch energy diffusion along with adiabatic heating in the recovery phase. The rate of electron flux recovery after a storm is determined by the ratio of nonadiabatic increases and losses; hence the electron flux represents a continuous series from low to very high values. The combination of these processes determines the individual character of radiation belt development during each magnetic storm and the behavior of the belt in the quiet time.

On the accuracy of conservation of adiabatic invariants in slow-fast Hamiltonian systems

Let the adiabatic invariant of action variable in slow-fast Hamiltonian system with two degrees of freedom have two limiting values along the trajectories as time tends to infinity. The difference of two limits is exponentially small in analytic systems. An iso-energetic reduction and canonical transformations are applied to transform the slow-fast systems to form of systems depending on slowly varying parameters in a complexified phase space. On the basis of this method an estimate for the accuracy of conservation of adiabatic invariant is given for such systems.

THEMIS observations of mirror structures: Magnetic holes and instability threshold

Nonpropagating mirror‐mode structures are commonly observed in many regions of natural plasma such as solar wind, planetary magnetosheaths, in cometary plasma, Io wake, terrestrial ring current and even on the outskirts of solar system. Mirror structures are typically observed in the shape of magnetic holes or peaks. Fast survey mode plasma data from the THEMIS satellites are used to solve the puzzle of how mirror structures in the form of dips can be observed in the regions of mirror stable plasma. THEMIS data also show that for mirror structures with spatial scales that considerably exceed ion Larmor radius the perpendicular temperature anticorrelates with the strength of the magnetic field. This contradiction with the conservation of adiabatic invariants is explained by the role of trapped particles.

Electron dynamics and cross‐shock potential at the quasi‐perpendicular Earth's bow shock

The evolution of the electron distribution function through quasi‐perpendicular collisionless shocks is believed to be dominated by the electron dynamics in the large‐scale coherent and quasi‐stationary magnetic and electric fields. We investigate the electron distributions measured on board Cluster by the Plasma Electron and Current Experiment (PEACE) instrument during three quasi‐perpendicular bow shock crossings. Observed distributions are compared with those predicted by electron dynamics resulting from conservation of the first adiabatic invariant and energy in the de Hoffmann‐Teller frame, for all pitch angles and all types of trajectories (passing and, for the first time, reflected or trapped). The predicted downstream velocity distributions are mapped from upstream measurements using an improved Liouville mapping technique taking into account the overshoots. Furthermore, for one of these crossings we could take advantage of the configuration of the Cluster quartet to compare mapped upstream velocity distributions with those simultaneously measured at a relatively well magnetically connected downstream location. Consequences of energy and adiabatic invariant conservation are found to be compatible with the observed electron distributions, confirming the validity of electron “heating” theories based on DC fields as zeroth‐order approximations, but some systematic deviations are found between the dynamics of low‐ and high‐adiabatic invariant electrons. Our approach also provides a way to estimate the cross‐shock electric potential profile making full use of the electron measurements, and the results are compared to other estimates relying on the steady state dissipationless electron fluid equations. At the temporal resolution of the instruments, the scales associated to the change of the potential generally appear to be comparable to those of the magnetic field, but some differences between the methods appear within the shock transition. It is argued that potentials evaluated from Liouville mapping rely on less restrictive hypotheses and are therefore more reliable. Finally, we show how, in contrast to methods using electron velocity moments, the technique can be used to produce high‐time‐resolution electric potentials and discuss the electric potential profiles through the shock.

Observational appearances of isolated stellar-mass black hole accretion – Theory and observations

General properties of accretion onto isolated stellar-mass black holes in the Galaxy are discussed. An analysis of plasma internal energy growth during the infall is performed. Adiabatic heating of collisionless accretion flow due to magnetic adiabatic invariant conservation is 25% more efficient than in the standard non-magnetized gas case. It is shown that magnetic field line reconnections in discrete current sheets lead to significant nonthermal electron component formation, which leads to a formation of a hard (UV, X-ray, up to gamma), highly variable spectral component in addition to the standard synchrotron optical component first derived by Shvartsman generated by thermal electrons in the magnetic field of the accretion flow. Properties of accretion flow emission variability are discussed. Observation results of two single black hole candidates – gravitational lens MACHO-1999-BLG-22 and radio-loud X-ray source with featureless optical spectrum J1942+10 – in optical band with high temporal resolution are presented and interpreted in the framework of the proposed model.

Low-rate accretion onto isolated stellar-mass black holes

Magnetic field behaviour in a spherically-symmetric accretion flow for parameters typical of single black holes in the Galaxy is discussed. It is shown that in the majority of Galaxy volume, accretion onto single stellar-mass black holes will be spherical and have a low accretion rate ($10^{-6} - 10^{-9}$ of the Eddington rate). An analysis of plasma internal energy growth during the infall is performed. Adiabatic heating of collisionless accretion flow due to magnetic adiabatic invariant conservation is 25% more efficient than in the standard non-magnetized gas case. It is shown that magnetic field line reconnections in discrete current sheets lead to significant nonthermal electron component formation. In a framework of quasi-diffusion acceleration, the energy-radius electron distribution is computed and the function describing the shape of synchrotron radiation spectrum is constructed. It is shown that nonthermal electron emission leads to formation of a hard (UV, X-ray, up to gamma), highly variable spectral component in addition to the standard synchrotron optical component first derived by Shvartsman generated by thermal electrons in the magnetic field of accretion flow. For typical interstellar medium parameters, a black hole at 100 pc distance will be a 16-25$^{\rm m}$ optical source coinciding with the highly variable bright X-ray counterpart, while the variable component of optical emission will be about 18-27$^{\rm m}$. The typical time scale of the variability is $10^{-4}$ sec, with relative flare amplitudes of 0.2-6% in various spectral bands. Possible applications of these results to the problem of search for single black holes are discussed.

Wind observations of extreme ion temperature anisotropies in the lunar wake

We describe Wind observations of two lunar wake encounters which occurred on 12–13 November 1996 and 18 July 2002. The observations were made at downstream distances of around 25 and 15 lunar radii (RL), respectively. Both encounters occurred prior to the spacecraft entering the lunar shadow; one event took place within the magnetosheath. A characteristic feature of the lunar wake is the presence of two counterstreaming ion beams drawn in from either flank. We find that both ion components exhibit an extreme temperature anisotropy, often with T⊥ ∼ 10T∥. The anisotropy is greatest in the central wake region. It appears that the anisotropy arises through the conservation of adiabatic invariants as solar wind plasma expands to fill in the cavity behind the Moon. Despite their large anisotropy, the proton distributions appear stable to the cyclotron instability. Correlated field and flow directional changes show that the wake geometry is dependent upon the prevailing magnetic field orientation.

Profile consistency based on the magnetic entropy concept: theory and observation

The relevance of the magnetic entropy concept for the description of relaxed current density, temperature and pressure profiles has been recently demonstrated in Ohmic TCV discharges. In this paper the parameter space characterizing the tokamak states with stationary magnetic entropy (SME) is extended to sawtoothing auxiliarly heated discharges. The condition of stationary entropy determines the current density profile and the temperature profile through Ohmic relaxation. The compatibility of SME Ohmically relaxed current density profiles with the toroidal force balance (Grad–Shafranov–Schlüter (GSS)) and the power balance basically restricts the profiles of the pressure and of the effective thermal diffusivity. Furthermore, the `ansatz' on the density profile, introduced successfully in the Ohmic case and supported by theoretical considerations on approximate conservation of adiabatic invariants in particle transport (TEP), imposes a final definite form on the pressure profile, consistently with SME and GSS.It is shown that the magnetic configuration of the SME states is invariant with respect to a certain family of transformations involving the auxiliary power PA, the toroidal magnetic field, the loop voltage, the average density and the major and minor radius. The invariance under the admissible transformations is then the appropriate definition of the concept of profile consistency in the presence of auxiliary heating. It will follow that, on the one hand, the insensitivity of the relaxed profiles to external variation of PA is a particular aspect of this invariance. On the other hand, when the value of PA is fixed by some process internal to the plasma, as the threshold of the L–H transition, and the L-state just before the transition is identified as a SME state, PA should scale with the parameters above consistently with the invariant identification of the state, irrespective of the details of the underlying dynamical process. It is shown that this scaling is precisely that observed for the power threshold of the L–H transition, provided that the confinement time of the L- (and SME) state is given by the ITER89-P scaling law. In the rest of the paper the SME point of view is exemplified by comparing the theory with the profiles observed in Ohmic and ECRH discharges in FTU and with the observed critical gradient-dependence of the thermal diffusivity.

Generalization of Berry’s geometric phase, equivalence of the Hamiltonian nature, quantizability and strong stability of linear oscillatory systems, and conservation of adiabatic invariants

A linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters is considered. Its asymptotic solutions are constructed to an arbitrary accuracy in the adiabaticity parameter ε. By extending the phase space of the system not only with the space of parameters, like in the theory of Berry’s phase, but also with the space of their derivatives, it proves possible to represent the phase of the solution as a sum of the dynamic phase and a generalized geometric phase (determined by a contour integral in the space of parameters and derivatives). Hence, it is possible to obtain the asymptotic results to any degree of accuracy in ε, while earlier they were obtainable only in the first-order adiabatic approximation. Namely, for linear oscillatory adiabatic systems, the quantizability, parametric stability, and the Hamiltonian nature of the system are equivalent properties. As a consequence, one can obtain the well-known result regarding conservation with accuracy to any power of ε of adiabatic invariants in a Hamiltonian system on a torus. An important point is that the generalized geometric phase can appear, in contrast to Berry’s phase, with only one real varying parameter.

Radiation belt formation during storm sudden commencements and loss during main phase

Simulation of the March 24, 1991 storm sudden commencement (SSC) has illuminated the rapid formation of new radiation belts on the particle drift time scale. While this event was the most dramatic in terms of radiation belt effects of the last solar maximum, comparable signatures of such events were seen in 1962 by Explorer 15 and in 1986 by DMSP. Several smaller MeV proton events with comparable particle morphology, but less radial transport and energization, were observed during the lifetime of the CRRES satellite (July 1990 – October 1991), which was well instrumented for both particle and field measurements inside geosynchronous orbit. Typically a solar proton event is accompanied by an SSC in the CRRES data set, while the converse is not always true. An SSC accompanied by solar protons produces a trapped population which remains on closed drift orbits until ring current buildup disrupts trapping, either by violation of the adiabatic trapping criterion or generation of waves whose frequency is comparable to periodic particle motion. Thus, new radiation belts formed around L = 4 by SSC injection are short-lived compared to the March 24, 1991 storm, wherein solar protons were transported radially inward to L = 2.5, with greater energization corresponding to first adiabatic invariant conservation than for the weaker events.

On the conservation of adiabatic invariants for a system of coupled rotators

We study the accuracy of the conservation of adiabatic invariants in a model of n weakly coupled rotators. Most attention is devoted to n = 2 and frequency ω = ( ω 1, ω 2), with ω 2 ω 1 quadratic irrational. We apply a heuristic approximation scheme, going back to Jeans and to Landau and Teller, and perform a very accurate numerical check of the result, observing a quite remarkable agreement.

Modeling of sawtooth destabilization during radio-frequency heating experiments in tokamak plasmas

Sawtooth oscillations in tokamaks have been stabilized using ion cyclotron resonance heating (ICRH), but often reappear while ICRH continues. It is shown that the reappearance of sawteeth during one particular ICRH discharge in the Joint European Torus (JET) [Campbell et al., Phys. Rev. Lett. 60, 2148 (1988)] was correlated with a change of sign in the energy δW associated with m=1 internal kink displacements. To compute δW, a new analytical model is used for the distribution function of heated minority ions, which is consistent with Fokker–Planck simulations of ICRH. Minority ions have a stabilizing influence, arising from third adiabatic invariant conservation, but also contribute to a destabilizing shift of magnetic flux surfaces. As the minor radius of the q=1 surface rises, the stabilizing influence of minority ions diminishes, and the shape of the plasma cross section becomes increasingly important. It is shown that an increase in ICRH power can destabilize the kink mode: this is consistent with observations of sawteeth in JET discharges with varying levels of ICRH. It is suggested that the sawtooth-free period could be prolonged by minimizing the vertical extent of the ICRH power deposition profile.1996 American Institute of Physics

Long-Term Conservation of Adiabatic Invariants by Using Symplectic Integrators

Abstract The advantage of symplectic integrators compared with non-symplectic Runge-Kutta integrators is shown in the computation of adiabatic invariants. Symplectic integrators do not produce a secular growth of the error, contrary to non-symplectic Runge-Kutta integrators. Furthermore, we compared the performance of 2nd, 4th, and 6th order symplectic integrators and obtained the numerical results that show SI6 has the best performance when the numerical error in the adiabatic invariants must be below 0.03%. The model systems that we used are a one-dimensional harmonic oscillator with a slowly varying frequency and a one-particle system in a slowly varying isochrone potential.

Condition of adiabatic invariant conservation in radiative transfer theory

Condition of adiabatic invariant conservation in radiative transfer theory