Vortex-wave Dynamics in the Drift-wave Rossby-wave Problem with Temperature Gradients

W. Horton

The processes governing the propagation of low frequency vortex-wave convective disturbances in the two different physical systems of neutral fluids on rotating planets and plasmas confined by magnetic fields are explored with (i) physical descriptions of the convective transport, (ii) establishing the relevant conservation laws and (iii) computer simulations. The role of a global, ambient temperature gradient in driving the three-dimensional baroclinic instability is compared with the ion temperature gradient instability in magnetically confined plasma. Steady-state power balance and the turbulent viscosities and thermal diffusivities are analyzed using the same class of turbulent transport formulas.


Fusion, Magnetic Confinement---Addendum

H.L. Berk


In the 1990s there has been significant progress in the development of the tokamak concept and serious study by an international research group [the International Thermonuclear Experimental Reactor (ITER) team] on the feasibility and design of an experiment to demonstrate fusion power production. In this addendum we present an overview of the status of tokamak research in relation to achieving this goal.


Steady State Tokamak Equilibria Without Current Drive

K.C. Shaing, A.Y. Aydemir, Y.R. Lin-Liu, and R.L. Miller


Steady-state tokamak equilibria without current drive are found. This is made possible by including the potato bootstrap current close to the magnetic axis. Tokamaks with this class of equilibria do not need seed current or current drive, and are intrinsically steady state.


On Exactly Conservative Integrators

John C. Bowman, B.A. Shadwick, P.J. Morrison


Traditional explicit numerical descretizations of conservative systems generally predict artificial secular drifts of nonlinesar invariants. These algorithms are based on polynomial functions of the time step. We discuss a general approach for developing explicit algorithms that conserve such invariants exactly. We illustrate the method by applying it to the truncated two-dimensional Euler equations.


Studies of Laser-Driven 5 TeV e+ e- Colliders in Strong Quantum Beamstrahlung Regime

M. Xie, T. Tajima, K.? Yokoya, and S. Chattopadhyay


We explore the multidimensional space of beam parameters, looking for preferred regions of operation for a e+e- linear collider at 5 TeV center of mass energy. Due to several major constraints such a collider is pushed into certain regime of high beamstrahlung parameter, U, where beamstrahlung can be suppressed by quantum effect. The collider performance at high U regime is examined with IP simulations using the code CAIN. Given the required beam parameters we then discuss the feasibility of laser-driven accelerations. In particular, we will discuss the capabilities of laser wakefield acceleration and comment on the difficulties and uncertainties associated with the approach. It is hoped that such an exercise will offer valuable guidelines for and insights into the current development of advanced accelerator technologies oriented towards future collider applications.


Space Charge Tracking Code for a Synchrotron Accelerator

M.B. Ottinger and T. Tajima


An algorithm has been developed to compute particle tracking, including self-consistent space charge effects for synchrotron accelerators. In low-energy synchrotrons space charge plays a central role in enhancing emittance of the beam. The space charge effects are modeled by mutually interacting (through the Coulombic force) N cylindrical particles (2-1/2-dimensional dynamics) whose axis is in the direction of the equilibrium particle flow. On the other hand, their interaction with synchrotron lattice magnets is treated with the thin-lens approximation and in a fully 3-dimensional way. Since the existing method to treat space charge fully self-consistently involves 3-D space charge effect computation, the present method allows far more realistic physical parameters and runs in far shorter time (about 1/20). Some examples on space charge induced instabilities are presented.


Analysis of the Bi-modal Nature of Solar Wind-Magnetosphere Coupling

James P. Smith and Wendell Horton


It has been shown that the optimal linear prediction filter relating the solar wind electric field and the geomagnetic activity, as measured by the AL index, is both bi-modal and dependent on the level of activity in the magnetosphere. Further studies truncated the prediction filter to a five parameter model containing two low-pass filtered delta functions of arbitrary amplitude and delay time. The present study elaborates on the nature of the bi-modal response by using the five parameter model to quantify the effects of the level of geomagnetic activity on each of the modes of the filter individually. We find that at all levels of activity, the second mode, occurring at approximately one hour, is relatively unchanged. The first mode, however, has a necessary one parameter dependence on the level of activity in the magnetosphere. The amplitude of the first mode is shown to increase with respect to activity, and this dependence is sufficient in characterizing the changing properties of the magnetosphere with respect to activity.


Nonlinear Theory of Kinetic Instabilities Near Threshold

H.L. Berk, B.N. Breizman, and M.S. Pekker


A universal integral equation has been derived and solved for the nonlinear evolution of collective modes driven by kinetic wave particle resonances just above the threshold for instability. The dominant nonlinearity stems from the dynamics of resonant particles which can be treated perturbatively near the marginal state of the system. With a resonant particle source and classical relaxation processes included, the new equation allows the determination of conditions for a soft nonlinear regime, where the saturation level is proportional to the increment above threshold, or a hard nonlinear regime, where the saturation level is independent of the closeness to threshold. It has been found, both analytically and numerically, that in the hard regime the system exhibits explosive behavior and rapid oscillations of the mode amplitude. When the kinetic response is a requirement for the existence of the mode, this explosive behavior is accompanied by frequency chirping. The universality of the approach suggests that the theory applies to many types of resonant particle driven instabilities.


Microbunching and Coherent Acceleration of Electrons by Subscycle Laser Pulses

B. Rau, T. Tajima, and H. Hojo


The pick-up and acceleration of all plasma electrons irradiated by an intense, subcyclic laser pulse is demonstrated via analytical and numerical calculations. It is shown that the initial low emittance of the plasma electrons is conserved during the process of acceleration, leading to an extremely cold, bunched electron beam. Compression of the electron bunch along the longitudinal coordinate is naturally achieved due to the interaction of electrons and laser pulse. In this paper, we find the localized solutions to Maxwell's equations of a subcyclic laser pulse and use these to determine the acceleration of charged particles and we suggest future application for this acceleration mechanism as low energy particle injector and as electron source for coherent x-ray generation.


Angular momentum injection into a Penning-Malmberg trap

Richard Fitzpatrick & Edmund P. Yu


It is demonstrated using conventional fluid theory that angular momentum can be injected into a single component plasma confined in a Penning-Malmberg trap via an externally generated, oscillating, non-axisymmetric, electric field. The torque exerted on the plasma by the electric field is a highly non-monotonic function of the plasma angular rotation velocity. The torque vs. angular velocity curve is dominated by sharp resonances at which the angular phase velocity of a particular poloidal harmonic of the external field matches the plasma angular rotation velocity. The torque exerted on the plasma by a given poloidal harmonic is negative when the field rotates faster than the plasma, and vice versa. This rather surprising behaviour is shown to be entirely consistent with a standard result in hydrodynamic theory, but is generally not observed in present day experiments.


Studies of Instability and Transport in Tokamak Plasmas with very Weak Magnetic Shear

J.Q. Dong, Y.Z. Zhang, and S.M. Mahajan


Ion temperature gradient (ITG or hi) driven microinstabilities are studied, using kinetic theory, for tokamak plasmas with very weak (positive or negative) magnetic shear (VWS). The gradient of magnetic shear as well as the effects of parallel and perpendicular velocity shear (v||' and vE') are included in the defining equations. Two eigenmodes: the double (D) and the global (G) are found to coexist. Parametric dependence of these instabilities, and of the corresponding quasilinear transport is systematically analyzed. It is shown that, in VWS plasmas, a parallel velocity shear (PVS) may stabilize or destabilize the modes, depending on the individual as well as the relative signs of PVS and of the gradient of magnetic shear. The quasilinear transport induced by the instabilities may be significantly reduced with PVS in VWS plasmas. The vE' values required to completely suppress the instabilities are much lower in VWS plasmas than they are in normal plasmas. Possible correlations with tokamak experiments are discussed.


On the Radial Profile and Scaling of Ion Thermal Conductivity from Toroidal ITG Mode

J.-Y. Kim, Y. Kishimoto, W. Horton T. Tajima and M. Wakatani


The issue of the physics responsible for the radial increase and scaling problems of the ion thermal conductivity ci is presented within the framework of the toroidal ion temperature gradient (ITG) mode. The observed radial increase of ci with its large value near the edge region, which was difficult to explain from the slab-like ITG mode with the gyro-Bohm type ci, can be explained well in terms of the toroidal ITG mode with Bohm-type ci. Against several previous arguments, it is shown how such a Bohm-like scaling is possible from the toroidal ITG mode.


A Photon Accelerator---Large Blueshifting of Femtosecond Pulses in Semiconductors

V.I. Berezhiani, S.M. Mahajan and I.G. Murusidze


The availability of relatively high intensity (I>109 W cm-2) [but moderate (~nJ) total energy], femtosecond laser pulses with wavelengths ranging from the ultraviolet to the mid-infrared has opened the doors for a serious investigation of the nonlinear optical properties of matter on ultrashort time scales in a new parameter regime. Even small intensity-dependent nonlinearities can begin to play a major role in the overall electrodynamics, and in determining the fate of the propagating pulse. It is shown that a femtosecond pulse propagating near a two-photon transition in a semiconductor waveguide can undergo a large blueshift.


Hamiltonian Moment Reduction for Describing Vortices in Shear Flows

S. P. Meacham, P. J. Morrison, and G. R. Flierl


This paper discusses a general method for approximating 2D and quasi-geostrophic 3D fluid flows that are dominated by coherent lumps of vorticity. The method is based upon the noncanonical Hamiltonian structure of the ideal fluid and uses special functionals of the vorticity as dynamical variables. It permits the extraction of exact or approximate finite degree-of-freedom Hamiltonian systems from the partial differential equations that describe vortex dynamics. We give examples in which the functional are chosen to be spatial moments of the vorticity. The method gives rise to constants of motion known as Casimir invariants and provides a classification scheme for the global phase space structure of the reduced finite systems, based upon Lie algebra theory. The method is illustrated by application to the Kida vortex (Kida, 1981) and to the problem of the quasi-geostrophic evolution of an ellipsoid of uniform vorticity, embedded in a background flow containing horizontal and vertical shear (Meacham et al., 1994). The approach provides a simple way of visualizing the structure of the phase space of the Kida problem that allows one to easily classify the types of physical behavior that the vortex may undergo. The dynamics of the ellipsoidal vortex in shear are shown to be Hamiltonian and are represented, without further approximation, beyond the assumption of quasi-geostrophy, by a finite degree-of-freedom system in canonical variables. The derivation presented here is simpler and more complete than the previous derivation which led to a finite degree-of-freedom system that governs the semi-axes and orientation of the ellipsoid. Using the reduced Hamiltonian description, it is shown that one of the possible modes of evolution of the ellipsoidal vortex is chaotic. These chaotic solutions are noteworthy in that they are exact chaotic solutions of a continuum fluid governing equation, the quasi-geostrophic potential vorticity equation.


Spontaneous Hole-Clump Pair Creation in Weakly Unstable Plasmas

H.L. Berk, B.N. Breizman, and N.V. Petviashvili


A numerical simulation of a kinetic instability near threshold shows how a hole and clump spontaneously appear in the particle distribution function. The hole and clump support a pair of Bernstein, Greene, Kruskal (BGK) nonlinear waves that last much longer than the inverse linear damping rate while they are upshifting and downshifting in frequency. The frequency shifting allows a balance between the power nonlinearly extracted from the resonant particles and the power dissipated into the background plasma. These waves eventually decay due to phase space gradient smoothing caused by collisionality.


Generalized superconducting flows---plasma confinement, organization

Swadesh M. Mahajan


Complete expulsion of magnetic vorticity is used to characterize the superconducting flow. It is shown that a simple, intuitive, but speculative generalization can serve as a paradigm for a variety of organized flows.


Effects of Orbit Squeezing on Ion Transport Processes Close to the Magnetic Axis

K.C. Shaing, R.D. Hazeltine and M.C. Zarnstorff


It is shown that ion thermal conductivity close to the magnetic axis in tokamaks is reduced by a factor of |S|5/3 if (Mi/Me)2/3 (Te/Ti)4/3/|S|5/3>>1. Here, S is the orbit squeezing factor, Mi(Me) is the ion (electron) mass, and Ti(Te) is the ion (electron) temperature. The reduction reflects both the increase of the fraction of trapped particles by a factor of |S|1/3, and the decrease of the orbit size in units of the poloidal flux y by a factor of |S|2/3.


Rotation and Locking of Magnetic Islands

F.L. Waelbroeck and R. Fitzpatrick


The polarization drifts arising from the acceleration of plasma flowing alongside an island are shown to be destabilizing. A critical island width is found above which the polarization drifts (inertial forces) defeat the stabilizing effect of the viscous forces and prevent islands from being unlocked by increasing the plasma rotation.


Minimal Model for Transport Barrier Dynamics Based on Ion-Temperature-Gradient Turbulence

G. Hu and C.W. Horton


Low-order mode coupling equations are derived to describe recent computer simulations of the toroidal ion-temperature-gradient turbulent convection with steady and pulsating sheared mass flows in the transport barrier zone. The three convective transport states are identified with the tokamak conf inement regimes called low mode (L-mode), high mode (H-mode), and barrier localized modes (BLMs) when the transport barrier is in the core plasma. The L-mode limit cycle is analytically derived and a bifurcation diagram showing L to H and H to BLM transitions in confinement is constructed numerically. Markovian closure procedures are sought to further reduce the dimensionality of the nonlinear system. First an exact expression is given for the energy transfer rate from the fluctuations to the sheared mass flow through the triplet velocity correlation function. Then the time scale expansion required toderive the Markovian closure formula is given. Markovian closure formulas form the basis for the thermodynamic-like L-H bifurcation models.


Spectrum of the Ballooning Schrodinger Equation

R.L. Dewar


The ballooning Schrodinger equation (BSE) is a model equation for investigating global modes that can, when approximated by a Wentzel--Kramers--Brillouin (WKB) ansatz, be described by a ballooning formalism locally to a field line. This second order differential equation with coefficients periodic in the independent variable qk is assumed to apply even in cases where simple WKB quantization conditions break down, thus providing an alternative to semiclassical quantization. Also, it provides a test bed for developing more advanced WKB methods: e.g. the apparent discontinuity between quantization formulae for ``trapped" and ``passing" modes, whose ray paths have different topologies, is removed by extending the WKB method to include the phenomena of tunnelling and reflection. The BSE is applied to instabilities with shear in the real part of the local frequency, so that the dispersion relation is inherently complex. As the frequency shear is increased, it is found that trapped modes go over to passing modes, reducing the maximum growth rate by averaging over qk.

A Numerical Model of Wave-Induced Fast Particle Transport in a Fusion Plasma - THESIS

Joseph A. Fitzpatrick


For a fusion reaction rate to sustain itself, the charged fusion products (e.g., 3.5 MeV alpha particles from the D-T reaction) must thermalize upon the background plasma, thus directly heating cooler fuel nuclei. Collective loss processes may affect this heating by contributing to alpha loss before the alpha thermalization is complete. One such mechanism arises from the alpha particle excitation of shear Alfven waves, which may be destabilized because the spatial gradient of the alpha distribution allows an energy transfer from the alpha particles to electromagnetic wave energy. Interaction with these destabilized waves can cause rapid alpha particle diffusion and loss. This research considers the feasibility of modeling alpha particle transport with a method that is an extension of a method used to model simpler problems with similar transport mechanisms.

Nonlinear Interaction of Fast Particles with Alfven Waves in Toroidal Plasmas

J. Candy, D. Borba, G. T. A. Huysmans, W. Kerner


A numerical algorithm to study the nonlinear, resonant interaction of fast particles with Alfven waves in tokamak geometry has been developed. The scope of the formalism is wide enough to describe the nonlinear evolution of fishbone modes, toroidicity-induced Alfven eigenmodes and ellipticity-induced Alfven eigenmodes, driven by both passing and trapped fast ions. When the instability is sufficiently weak, it is known that the wave-particle trapping nonlinearity will lead to mode saturation before wave-wave nonlinearities are appreciable. The spectrum of linear modes can thus be calculated using a magnetohydrodynamic normal-mode code, then nonlinearly evolved in time in an efficient way according to a two-time-scale Lagrangian dynamical wave model. The fast particle kinetic equation, including the effect of orbit nonlinearity arising from the mode perturbation, is simultaneously solved for the deviation, df = f - f0, from an initial analytic distribution f0. High statistical resolution allows linear growth rates, frequency shifts, resonance broadening effects, and nonlinear saturation to be calculated quickly and precisely. The results have been applied to an ITER instability scenario. Results show that weakly-damped core-localized modes alone cause negligible alpha transport in ITER-like plasmas - even with growth rates one order of magnitude higher than expected values. However, the possibility of significant transport in reactor-type plasmas due to weakly unstable global modes remains an open question.


Escaping Radio Emission from Pulsars

S.M. Mahajan, G.Z. Machabeli, A.D. Rogava


It is demonstrated that the velocity shear, intrinsic to the e+e- plasma present in the pulsar magnetosphere, can efficiently convert the nonescaping longitudinal Langmuir waves (produced by some kind of a beam or stream instability) into propagating (escaping) electromagnetic waves. It is suggested that this shear induced transformation may be the basic mechanism needed for the eventual generation of the observed pulsar radio emission.


Critical Nonlinear Phenomena for Kinetic Instabilities Near Threshold

B.N. Breizman, H.L. Berk, M.S. Pekker, F. Porcelli, G.V. Stupakov, and K.L. Wong


A universal integral equation has been derived and solved for the nonlinear evolution of collective modes driven by kinetic wave particle resonances just above the threshold for instability. The dominant nonlinearity stems from the dynamics of resonant particles which can be treated perturbatively near the marginal state of the system. With a resonant particle source and classical relaxation processes included, the new equation allows the determination of conditions for a soft nonlinear regime, where the saturation level is proportional to the increment above threshold, or a hard nonlinear regime, where the saturation level is independent of the closeness to threshold. It has been found, both analytically and numerically, that in the hard regime the system exhibits explosive behavior and rapid oscillations of the mode amplitude. When the kinetic response is a requirement for the existence of the mode, this explosive behavior is accompanied by frequency chirping. The universality of the approach suggests that the theory applies to many types of resonant particle driven instabilities, and several specific cases, viz.~energetic particle driven Alfv\'en wave excitation, the fishbone oscillation, and a collective mode in particle accelerators, are discussed.


Plasma Transport Near the Separatrix of a Magnetic Island

R.D. Hazeltine, P. Helander, and P.J. Catto


The simplest non-trivial model of transport across a magnetic island chain in the presence of collisionless streaming along the magnetic field is solved by a Wiener-Hopf procedure. The solution found is valid provided the boundary layer about the island separatrix is narrow compared to the island width. The result demonstrates that when this assumption is satisfied the flattened profile region is reduced by the boundary layer width. The calculation is similar to the recent work by Fitzpatrick [R. Fitzpatrick, Phys. Plasmas 2, 825 (1995)] but is carried out in the collisionless, rather than the collisional, limit of parallel transport, and determines the plasma parameters on the separatrix self-consistently.


Bootstrap Current Close to Magnetic Axis in Tokamaks

K.C. Shaing and R.D. Hazeltine


It is shown that the bootstrap current density close to the magnetic axis in tokamaks does not vanish in simple electron-ion plasmas because the fraction of the trapped particles is finite. The magnitude of the current density could be comparable to that in the outer core region. This will reduce or eliminate the need of the seed current.


Electromagnetic Waves in a Strong Schwarzschild Background

T. Tajima and J.N. Daniel, III


The physics of high frequency electromagnetic waves in a general relativistic plasma with the Schwarzschild metric is studied. Based on the 3+1 formalism, we conformalize Maxwell's equations. The derived dispersion relations for waves in the plasma contain the lapse function in the plasma parameters such as in the plasma frequency and cyclotron frequency, but otherwise look ``flat." Because of this property this formulation is ideal for nonlinear self-consistent particle (PIC) simulation. Some of the physical consequences arising from the general relativistic lapse function as well as from the effects specific to the plasma background distribution (such as density and magnetic field) give rise to nonuniform wave equations and their associated phenomena, such as wave resonance, cutoff, and mode-conversion. These phenomena are expected to characterize the spectroscopy of radiation emitted by the plasma around the black hole. PIC simulation results of electron-positron plasma are also presented.e


Existence and consequences of Coulomb pairing of electrons in a solid

S.M. Mahajan and A. Thyagaraja


It is shown from first principles that, in the periodic potential of a crystalline solid, short-range (i.e., screened) binary Coulomb interactions can lead to a two-electron bound state. It is further suggested that these composite bosonic states ( charge -2e, and typically spin zero) could mediate an effectively attractive interaction between pairs of conduction electrons close to the Fermi level.This necessarily short range attractive interaction, which is crucially dependent on the band structure of the solid, and is complementary to the phonon-mediated one, may provide a source for the existence and properties of short correlation-length electron pairs (analogous to but distinct from Cooper pairs) needed to understand high temperature superconductivity. Several distinctive and observable characteristics of the proposed pairing scheme are discussed.


A Filament Model of MHD Turbulence

V. Petviashvili


Turbulence of ordinary fluid is recognized as chaotic motion with almost no linear features. It is well described in wavenumber space by Kolmogorov's phenomenological theory in wave number k-space: The source of energy should exist in the region of small wavenumbers. Then isotropic energy flux is generated in k-space directed toward a larger k-region where the energy is absorbed by viscosity. The main characteristics of energy spectrum of Kolmogorov turbulence is universal and in good agreement with observations.


Experimental Plasma Astrophysics Using a T3(Table-top Terawatt) Laser

T. Tajima


Lasers that can deliver immense power of Terawatt (10^{12} W) and can still compactly sit on a Table-Top (T^3 lasers) emerged in the 1990s. The advent of these lasers allows us to access to regimes of astronomical physical conditions that once thought impossible to realize in a terrestrial laboratory. We touch on examples that include superhigh pressure materials that may resemble the interior of giant planets and white dwarfs and of relativistic temperature plasmas that may exist in the early cosmological epoch and in the neighborhood of the blackhole event horizon.


Effects of Negative Magnetic Shear on Toroidicity Induced Eigenmode Instability in Tokamak

K. Rajendran, J.Q. Dong, and S.M. Mahajan


The effects of negative magnetic shear on the toroidicity induced (TI) eigenmode are studied with a numerical WKB shooting scheme. The ion temperature gradient (ITG or eta_i), and the parallel velocity shear (PVS) of the ions are included. It is found that for a given set of plasma parameters, the negative magnetic shear causes much stronger damping than the positive shear of equal magnitude. It is also shown that PVS tends to destabilize the TI mode.


Coupling of eta-i and Trapped Electron Modes in Plasmas with negative Magnetic Shear

J.Q. Dong and S.M. Mahajan


In toroidal collisionless plasmas, the ion temperature gradient (ITG or eta_i) and the trapped electron (TE) modes are shown to be weakly (strongly) coupled when both the temperature gradients and the driving mechanism of the trapped electrons are moderate or strong (weak but finite). In the regime of strong coupling, there is a single hybrid mode, unstable for all eta_i in plasmas with positive magnetic shear. For the weak coupling case, two independent unstable modes, one in the ion and the other in the electron diamagnetic direction, are found to coexist. In either situation, the negative magnetic shear exerts a strong stabilizing influence; the stabilizing effect is considerably enhanced by the presence of trapped particles. It is predicted that for a given set of plasma parameters, it will be much harder to simultaneously excite the two modes in a plasma with negative shear. The results of this study are significant for tokamak experiments.


Self-Focused Particle Beam Drivers for Plasma Wakefield Accelerators

B.N. Breizman, P.Z. Chebotaev, A.M. Kudryavtsev, K.V. Lotov, and A.N. Skrinsky


Strong radial forces are experienced by the particle beam that drives the wakefield in plasma-based accelerators. These forces may destroy the beam although, under proper arrangements, they can also keep it in radial equilibrium which allows the beam to maintain the wakefield over a large distance and to provide high energy gain for the accelerated particles. This paper demonstrates the existence of acceptable equilibria for the prebunched beams and addresses the issue of optimum bunch spacing, with implications for forthcoming experiments.


Plasma Analog of Particle-Pair Production

Yu.A. Tsidulko and H.L. Berk


It is shown that the plasma axial shear flow instability satisfies the Klein-Gordon equation. The plasma instability is then shown to be analogous to spontaneous particle-pair production when a potential energy is present that is greater than twice the particle rest mass energy. Stability criteria can be inferred based on field theoretical conservation laws.


Chaos and Structures in the Magnetosphere

W. Horton


The nonlinear plasma transport mechanisms that control the collisionless heating in the Earth's magnetosphere and the onset of geomagnetic substorms are reviewed. In the high pressure plasma trapped in the reversed magnetic field loops on the nightside of the magnetosphere, the key issue of the role of the ion orbital chaos as the mechanism for the plasma sheet energization is examined. The energization rate is governed by a collisionless conductance and the solar wind driven dawn-to-dusk electric field. The low-frequency response function is derived and the fluctuation dissipation theorem is given for the system. Returning to the global picture the collisionless energization rate from the transport physics is the basis for a low-dimensional energy-momentum conserving dynamical model of magnetospheric substorms.