Finite Beta Plasma Research Articles

Magnetohydrodynamic theory of field line resonances in the magnetosphere

The linearized ideal magnetohydrodynamic (MHD) equations are cast into a set of global differential equations from which the field line resonance equations of the shear Alfvén waves and slow magnetosonic waves are obtained naturally for finite beta plasmas in general magnetic field geometries with flux surfaces. The coupling between the shear Alfvén waves and the magnetosonic waves is through the combined effects of geodesic magnetic field curvature and plasma pressure. For axisymmetric magnetospheric equilibria there is no coupling between the shear Alfvén waves and the slow magnetosonic waves because the geodesic magnetic field curvature vanishes. We derive the asymptotic singular solutions of the MHD equations near the field line resonant surface. We perform numerical solutions of the field line resonance equations for a dipole magnetic field with constant plasma pressure and density along the magnetic field line. Similar to previous studies, the shear Alfvén wave field line resonant frequency is roughly given by ω ≈ cL−4 ρ−1/2, where the coefficient c is roughly constant with respect to the L shell distance and ρ is the plasma mass density. The slow magnetosonic wave resonant frequency is roughly proportional to P/ρL², where P is the plasma pressure, and is much smaller than the shear Alfvén wave resonant frequency even for equatorial plasma beta as high as unity.

A Simulation of Low-Frequency Electromagnetic Phenomena in Kinetic Plasmas of Three Dimensions

An advanced kinetic simulation method has been developed and implemented in the HIDENEK code to study large space-scale, low-frequency electromagnetic phenomena occurring in inhomogeneous plasmas. The present method is specially designed for high magnetic field (Wcm ⩾ Wpm ), inhomogeneous plasma simulations. The guiding-center approximation with magnetic drifts is adopted to the perpendicular motion of the electrons, whereas the inertia effect is retained in their parallel motion. Also, a slightly backward time-decentered scheme is introduced to the equations of motion and the Maxwell equations. These equations are combined to yield the full-implicit, coupled field-particle equations which allow us to determine the future electromagnetic field in a large time step compared to the electron time scales with the diamagnetic drift and magnetization currents being included. As a demonstration of the present simulation method, three physics applications are shown for the electromagnetic beam-plasma instability, the temperature anisotropy-driven Alfven-ion-cyclotron instability, and the external kink instability of the peaked-density current beam. A remarkable pitch-angle scattering of the ions is observed in the first two applications in association with the plasma instabilities. In the third application to an inhomogeneous, finite-beta plasma of the three dimensions, a helical deformation is shown to take place to the initially straight beam and magnetic axis in an ideal magnetohydrodynamic time scale.

Solitary magnetosonic waves with Landau damping

The effect of Landau damping on nonlinear magnetosonic waves propagating obliquely to the magnetic field in a finite beta plasma has been studied. It has been found that such magnetosonic waves owing to their interaction with resonant particles are governed by a KdV equation with a damping term. This equation has solitary wave solutions whose amplitude decays with time as (1 + τ/τ0)−2. The decay rates of both fast and slow waves have been computed numerically. The decay rates depend on plasma beta and on the angle of propagation and the rates are different for fast and slow magnetosonic waves. At a certain angle of propagation, the decay rates of both modes are equal in the case of a low beta plasma. The fast mode has a higher damping rate for higher beta and becomes practically nonexistent for nearly perpendicular angles of propagation.

Determination of plasma shift in a stellarator from magnetic measurements**Translated by A. Watson and S.A. Edminster, English Translation Section, Division of Languages, IAEA, Vienna.

The poloidal magnetic field created by plasma currents in an ordinary stellarator is calculated. An analytical calculation is performed for a plasma column of ‘average’ circular cross-section in the large-aspect-ratio approximation. The problem of determining the position of the column from poloidal magnetic field and flux measurements is investigated. A formula is obtained which links the plasma shift relative to the geometric axis of the stellarator with measurable quantities, and a simple system of magnetic measurements is proposed. The analysis is performed for a finite-beta plasma with allowance for longitudinal current and the external transverse field.

The effect of random Alfvén waves on the propagation of hydromagnetic waves in a finite-beta plasma

Using first-order smoothing theory, Fourier analysis and perturbation methods, we obtain the evolution equation of the wave spectrum as well as the nonlinear forces generated by random Alfvén waves in a finite-β plasma with phenomenological Landau-damping effects. The effect of microscale random Alfvén waves on the propagation of large-scale hydromagnetic waves is also investigated by solving the mean-field equations. It is shown that parallel-propagating random Alfvén waves are modulationally stable and that obliquely propagating random Alfvén waves can be modulationally unstable when the energy of random waves is converted to slow magneto-acoustic waves that can be Landau-damped, providing a dissipation mechanism for the Alfvén waves.

A tearing mode energy principle for an axisymmetric finite beta plasma

A self-adjoint, gauge invariant stability functional δWTG for determination of a sufficient condition of tearing mode stability is obtained for a plasma with arbitrary current and pressure profiles. The plasma is stable if there is no free energy available to support energy flux into the singular layer. The pressure gradient near the singular layer is found to be stabilizing. The tearing mode energy drive is extracted by casting the problem into its nonlinear context, demanding that both large and small solutions be continuous across the singular layer.

Nonlinear evolution of Alfvén waves in a finite beta plasma

A general form of the derivative nonlinear Schrödinger (DNLS) equation, describing the nonlinear evolution of Alfvén waves propagating parallel to the magnetic field, is derived by using two-fluid equations with electron and ion pressure tensors obtained from Braginskii [in Reviews of Plasma Physics (Consultants Bureau, New York, 1965), Vol. 1, p. 218]. This equation is a mixed version of the nonlinear Schrödinger (NLS) equation and the DNLS, as it contains an additional cubic nonlinear term that is of the same order as the derivative of the nonlinear terms, a term containing the product of a quadratic term, and a first-order derivative. It incorporates the effects of finite beta, which is an important characteristic of space and laboratory plasmas.

Excitation of MHD surface waves propagating with shear Alfven waves in an inhomogeneous cylindrical plasma

Experimental results are presented which show that the MHD surface waves or fast waves of m=+or-1 have been excited in the range of frequency less than ion cyclotron frequency using a helical antenna in a finite beta and cylindrical plasma. Their dispersion relation and spatial structure were successfully confirmed after the wave signals were separated from those of the local shear Alfven waves that propagated together with the surface waves. In order to obtain the dispersion relations from the signals, the authors adopted three kinds of methods: (1) conventional cross-power spectrum density using fast Fourier transform, (2) maximum entropy method, and (3) direct estimation of the phase velocity of wave packets. The observed properties of both the surface waves and the SAWs are in general agreement with theory.

Nonlinear gyrokinetic theory for finite-beta plasmas

A self-consistent and energy-conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell’s equations for finite-beta plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formulation is valid for arbitrary values of k⊥ρi and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques.

Plasma Heating by a Magnetosonic Shock Wave through Resonant Ion Acceleration

Resonant wave-particle interactions in magnetosonic shock waves are studied by theory and simulation. The number of ions trapped by a perpendicular laminar shock in a finite beta plasma is evaluated, and the amount of shock heating due to resonant ion acceleration is analytically obtained in terms of the Alfven Mach number and upstream plasma parameters. Some effects of trapped ions on shock waves are also discussed. In addition, it is shown that a laminar oblique shock can reflect some electrons by a magnetic mirror effect, in spite of a large positive potential in the shock region. These theoretical predictions are confirmed by a fully electromagnetic particle simulation with full ion and electron dynamics in one spatial dimension.

Splitting instability of a high beta tokamak with noncircular cross section

The splitting of a tokamak with elliptical and bean-shaped cross sections is studied for finite beta plasmas. When the plasma beta exceeds a critical value, an elliptical tokamak is subject to an ideal pressure-driven instability, which deforms the ellipse in such a way that a thinned plasma current sheet is formed around the magnetic axis. As a result, magnetic reconnection is nonlinearly driven and the ellipse is split. The bean-shaped tokamak, however, is stable against a splitting perturbation for limited equilibria that could be numerically constructed. An interesting similarity to the energy relaxation process in a force-free plasma, namely, a two-step evolution (initial occurrence of an ideal magnetohydrodynamic instability and subsequent occurrence of driven reconnection), is discussed.

Neoclassical transport coefficients for finite-aspect-ratio and bean-shaped tokamak plasmas

Numerically calculated tokamak equilibria are used to compute banana-plateau transport coefficients for finite-aspect-ratio, finite-beta plasmas. Calculations are presented for the Spherical Torus Experiment (STX) (NTIS Document No. DE 86004663) and the Princeton Beta Experiment (PBX) (NTIS Document No. DE 86011173). In STX, the poloidal variation of B≡‖B‖ over a magnetic surface tends to be reduced in regions of large major radius R. The reduction of radial transport caused by this quasiomnigeneous condition is offset by increased drifts and trapping probabilities for smaller R. Thus the modulation Δ=(Bmax−Bmin)/(Bmax+Bmin) on a magnetic surface becomes the critical parameter determining neoclassical transport. In PBX, the bean-shaped topology of the magnetic surfaces leads to the presence of multiple magnetic wells. Numerical calculations confirm that analytic calculations of neoclassical transport based on the total fraction of circulating particles are valid even when geometrically distinct classes of trapped particles are present.

Theory for resonant ion acceleration by nonlinear magnetosonic fast and slow waves in finite beta plasmas

A Korteweg–de Vries equation that is applicable to both the nonlinear magnetosonic fast and slow waves is derived from a two-fluid model with finite ion and electron pressures. As in the cold plasma theory, the fast wave has a critical angle θc. For propagation angles greater than θc (quasiperpendicular propagation), the fast wave has a positive soliton, whereas for angles smaller than θc, it has a negative soliton. Finite β effects decrease the value of θc. The slow wave has a positive soliton for all angles of propagation. The magnitude of resonant ion acceleration (the vp×B acceleration) by the nonlinear fast and slow waves is evaluated. In the fast wave, the electron pressure makes the acceleration stronger for all propagation angles. The decrease in θc resulting from finite β effects results in broadening of the region of strong acceleration. It is also found that fairly strong ion acceleration can occur in the nonlinear slow wave in high β plasmas. The possibility of unlimited acceleration of ions by quasiperpendicular magnetosonic fast waves is discussed.

Alfvén Solitons

The theory of MHD soliton formation parallel to the magnetic field, as described by the DNLS equation and related models, is reviewed. The known results on the physical relevance, solutions and mathematical properties of the DNLS equation, are briefly described. New results on the extension of the theory to finite beta and three space dimensions are presented. A hybrid fluid and kinetic treatment of the case of finite temperature based on the guiding center plasma model, leads to an additional nonlocal term due to resonant particles. The combination of modulational instability and resonant particle damping leads to the conclusion that parallel weakly dispersive Alfvén waves are nonlinearly damped in a finite beta plasma.

The scattering of energetic particles by waves in a finite beta plasma

Solutions are presented for the dispersion relation of waves propagating parallel to the ambient magnetic field, in a plasma having an arbitrary ratio of thermal to magnetic pressure, beta. Previous results applicable only to wave frequencies much lower than the ion cyclotron frequency are extended to include all frequencies lower than the electron cyclotron frequency. The cyclotron turnovers are found to occur at significantly lower frequencies in finite beta plasmas. Cyclotron damping is significant for small wavelength waves. These results show that in energetic particle scattering by electromagnetic turbulence in a plasma with beta value greater than 40, scattering by waves with wave frequencies greater than the ion cyclotron frequency can be neglected. This condition is satisfied in many astrophysical plasmas.

Neoclassical transport in a multiple-helicity torsatron in the low-collisionality (1/ν) regime

For a sufficiently high number of toroidal field periods (m/ι>l), the magnetic field of a multiple-helicity torsatron can be reduced to a simple form such that the second adiabatic invariant J can be calculated. It is found that the particle and the heat fluxes for a multiple-helicity torsatron in the low-collisionality (1/ν) regime have the same geometric dependences. An optimization of both quantities is carried out for a given equilibrium constraint. It is shown that the transport fluxes can be smaller than those of the conventional stellarator by an order of magnitude. The effect of finite plasma beta on the neoclassical fluxes is also studied.

Drift orbits in the TMX and MFTF-B tandem mirrors

Drift orbits for the TMX and MFTF-B tandem-mirror designs are followed by using a long-thin expansion of the drift equations. Unexpected asymmetries in the field-line curvatures in the yin-yang end-mirror traps, caused by the transition coils between the solenoid and the yin-yang, result in an elliptical distortion of the drift surface with a/b= 1.5 at most, a perhaps tolerable deviation from omnigenity. Yushmanov-trapped particles are no worse than the bulk hot particles. Finite-beta plasma fields, coupled to the asymmetric curvature, produce sizeable banana orbits with widths comparable to the plasma radius, but these orbits are possible for only a few of the particles. Details of the transition through resonance in the solenoid are shown, including the banana shapes of the drift surfaces and the disruption of the surface in the stochastic regime. The orbits in the original design for the A-cell of MFTF-B are the most extreme; in the vacuum fields they all have an extended peanut shape that finally closes only at about 3 m. This shape is strongly non-omnigenous and suggests a hollow plasma-density profile. Finite-beta B⃗×∇B drifts can help to minimize the radial extent of these orbits, but the strength of the vacuum curvatures makes omnigenity only marginally possible. Including B⃗×∇ϕ drifts makes omnigenity even more unlikely for the ions, for which the B⃗×∇B and B⃗×∇ϕ drifts are of opposite sign, and conversely helps to omnigenize the drift surfaces of the ECRH 200-keV electrons. It is argued that not every class of particles can have good, i.e. near-omnigenous drifts, regardless of the ability of ϕ(r) to adjust to limit the radial extent of the orbits. This lack of omnigenity leaves one with no theoretical base for describing the MHD equilibrium in the original designs, but a new magnetic field design for MFTF-B A-cell has apparently completely restored omnigenous orbits.

Parametric amplification of ion-acoustic and Alfven waves

The phenomenon of parametric amplification of propagating ion-acoustic wave and Alfven wave in a finite resistivity and low beta plasma, due to a modulated magnetic field, has been investigated using single mode theory. It is found that the ion-acoustic wave is not amplified in such a medium. For the Alfven waves it is found that the expression for growth rate and the threshold modulation are the same for high resistivity plasma as those for a low resistivity plasma.

Finite beta trapped electron fluid mode

It is shown that finite plasma beta can destabilize relatively long wavelength trapped electron fluid ballooning modes, which are the electromagnetic generalization of the electrostatic ’’ubiquitous’’ mode.

Ideal-MHD stability of finite-beta plasmas

An analytical theory of ideal-MHD ballooning modes that can be excited in finite-β equilibria is carried out on model configurations which include the effects of the increase of the poloidal field toward the outer edge of the plasma column and the dependence of the rate of magnetic shear on the poloidal angle. The relevant growth rates and eigensolutions are, in fact, significantly different from those derived on the basis of ‘low-β’ model configurations that omit one or both of the effects mentioned above, and provide different indications for the expected interaction between ideal-MHD and kinetic modes. For each value of the shear parameter ŝ, the normalized growth rate Γ becomes real at a critical value of the dimensionless pressure gradient parameter G. When the latter is increased at constant ŝ, Γ is found to increase only up to a saturation point, after which it decreases and tends to vanish at a second critical value of G.