Generalized Grad-Shafranov Equation Research Articles

On the Equilibrium and Stability of ITER Relevant Plasmas

We present recent results on steady states of translational symmetric and axisymmetric ITER relevant plasmas with incompressible sheared flow in connection with a generalized Grad-Shafranov (GGS) equation and on their stability. On the basis of a vari- ety of ITER pertinent solutions of the GGS equation constructed it turns out that the flow affects noticeably the equilibrium quantities in qualitative agreement with the advanced confinement regimes phenomenology. Also, application of a sufficient condition for linear stability indicates that stabilization is mainly caused by the variation of the magnetic field in the direction perpendicular to the magnetic surfaces, related to the magnetic shear, depends on the plasma shaping and is sensitive to the up-down asymmetry. In certain cases the sheared flow has a weaker stabilizing effect enhanced by the equilibrium nonlinearity.

Magnetohydrodynamic equilibria with incompressible flows: Symmetry approach

We identify and discuss a family of azimuthally symmetric, incompressible, magnetohydrodynamic plasma equilibria with poloidal and toroidal flows in terms of solutions of the Generalized Grad Shafranov (GGS) equation. These solutions are derived by exploiting the incompressibility assumption, in order to rewrite the GGS equation in terms of a different dependent variable, and the continuous Lie symmetry properties of the resulting equation and, in particular, a special type of “weak” symmetries.

Analytical up-down asymmetric equilibria with non-parallel flows

Generic linear axisymmetric equilibria with plasma flow nonparallel to the magnetic field are obtained on the basis of a generalized Grad-Shafranov equation by employing an ansatz reducing the problem to a set of ordinary differential equations which can be solved recursively. In particular, an ITER like equilibrium with reversed magnetic shear and peaked current density is constructed and its characteristics are studied in connection with the flow. Also for parallel flows, the linear stability is examined by means of a sufficient condition. The results indicate that the flow may have a stabilizing effect.

Two-dimensional nonlinear cylindrical equilibria with reversed magnetic shear and sheared flow

Nonlinear translational symmetric equilibria with up to quartic flux terms in free functions, reversed magnetic shear, and sheared flow are constructed in two ways: (i) quasi-analytically by an ansatz, which reduces the pertinent generalized Grad–Shafranov equation to a set of ordinary differential equations and algebraic constraints which is then solved numerically, and (ii) completely numerically by prescribing analytically a boundary having an X-point. This latter case presented in Sec. 4 is relevant to the International Thermonuclear Experimental Reactor project. The equilibrium characteristics are then examined by means of pressure, safety factor, current density, and electric field. For flows parallel to the magnetic field, the stability of the equilibria constructed is also examined by applying a sufficient condition. It turns out that the equilibrium nonlinearity has a stabilizing impact, which is slightly enhanced by the sheared flow. In addition, the results indicate that the stability is affected by the up-down asymmetry.

Nonlinear translational symmetric equilibria relevant to the L–H transition

AbstractNonlinear z-independent solutions to a generalized Grad–Shafranov equation (GSE) with up to quartic flux terms in the free functions and incompressible plasma flow non-parallel to the magnetic field are constructed quasi-analytically. Through an ansatz, the GSE is transformed to a set of three ordinary differential equations and a constraint for three functions of the coordinate x, in Cartesian coordinates (x,y), which then are solved numerically. Equilibrium configurations for certain values of the integration constants are displayed. Examination of their characteristics in connection with the impact of nonlinearity and sheared flow indicates that these equilibria are consistent with the L–H transition phenomenology. For flows parallel to the magnetic field, one equilibrium corresponding to the H state is potentially stable in the sense that a sufficient condition for linear stability is satisfied in an appreciable part of the plasma while another solution corresponding to the L state does not satisfy the condition. The results indicate that the sheared flow in conjunction with the equilibrium nonlinearity plays a stabilizing role.

Plasma Equilibrium in Presence of the Poloidal Flow in Tokamaks

Main purposes of this contribution are analysis of dynamic equilibrium properties of a large aspect ratio and low beta tokamaks, in particular deriving a modified relation for the Shafranov shift in the presence of poloidal flow, and demonstrate it experimentally. Poloidal flow can produce modifications in the equilibrium properties. By increasing Alfvenic Mach number from zero, flow produce outward force, and plasma shifted in outward direction. If the poloidal Alfvenic Mach number equal to one, singularity will observe in the solution of generalized Grad-Shafranov equation. Also inversion of Shafranov shift in the transition of flow speed between sub-Alfvenic to super-Alfvenic speeds can be observed due to inward force produced by flow.

Symmetric and asymmetric equilibria with non-parallel flows

Several classes of analytic solutions to a generalized Grad-Shafranov equation with incompressible plasma flow non-parallel to the magnetic field are constructed. The solutions include higher transcendental functions such as the Meijer G-function and describe D-shaped and diverted configurations with either a single or double X-points. Their characteristics are examined in particular with respect to the flow parameters associated with the electric field. It turns out that the electric field makes the safety factor flatter and increases the magnitude and shear of the toroidal velocity in qualitative agreement with experimental evidence on the formation of internal transport barriers in tokamaks, thus indicating a potential stabilizing effect of the electric field.

Semi-analytic approach to diverted tokamak equilibria with incompressible toroidal and poloidal flows

Generalized Grad–Shafranov equation for tokamak equilibrium with incompressible toroidal and poloidal flows is solved to obtain a double-null diverted configuration based on an approach presented before (Shi 2008 Plasma Phys. Control. Fusion 50 085006). This solution consists of only two terms of base functions obtained from the variable-separating method and suitable for describing both the internal region within the separatrix and a scrape-off layer region. Dependences of the main equilibrium properties, such as the magnetic field, plasma pressure and the equilibrium radial electric field and the plasma current on flows are revealed. In particular, we find that the presence of poloidal flow causes a deviation of the current surface from the magnetic surface and the sheared poloidal flow produces a non-zero toroidal current component that possibly affects the peeling–ballooning stability of the pedestal.

RETRACTED ARTICLE: Effects of Alfvenic Poloidal Flow and External Vertical Field on Plasma Position in IR-T1 Tokamak

We analyzed dynamic equilibrium properties of a large aspect ratio and low Beta tokamaks, in particular deriving a modified relation for the Shafranov shift in the presence of poloidal flow and external vertical field, and demonstrate it experimentally on the IR-T1 tokamak. Poloidal flow can produce modifications in the equilibrium properties. By increasing Alfvenic Mach number from zero, flow produce outward force, and plasma shifted in outward direction. If the poloidal Alfvenic Mach number equal to one, singularity will observe in the solution of generalized Grad-Shafranov equation. Also inversion of Shafranov shift in the transition of flow speed between sub-Alfvenic to super-Alfvenic speeds can be observed due to inward force produced by flow.

Mathematical and physical aspects of Kappa velocity distribution

One major characteristic associated with collisionless space plasmas is the development of non-Maxwellian velocity distribution that in many circumstances can be represented by the κ function characterized by the κ parameter. This paper discusses the mathematical character and physical origin of the κ function by first showing that the κ velocity function may be expressed in terms of exponential functions multiplied by the kinetic energy and its higher orders. The possible development of κ velocity distribution is illustrated by the problem of low-frequency waves and instabilities in uniform magnetized plasmas with bi-Maxwellian distribution. It is observed that the background and perturbed distribution functions bear the same forms as the zeroth- and higher-order terms of the κ function expanded in the limit of κ→∞. The consequence of assuming κ velocity distribution in inhomogeneous plasmas is illustrated by the Vlasov-Maxwell equilibrium problems that show the nonthermal equilibrium characteristic of nonuniform plasmas. A generalized Grad-Shafranov equation is proposed for two-dimensional Vlasov equilibria with κ velocity distribution.

Four motional invariants in axisymmetric tori equilibria

In addition to the standard set (ε,μ,pφ) of three invariants in axisymmetric tori, there exists a fourth independent radial drift invariant Ir. For confined particles, the net radial drift has to be zero, whereby the drift orbit average Ir=⟨r¯0⟩ of the gyro center radial Clebsch coordinate is constant. To lowest order in the banana width, the radial invariant is the gyro center radial coordinate r¯0(x,v), and to this order the gyro center moves on a magnetic flux surface. The gyro center orbit projected on the (r,z) plane determines the radial invariant and first order banana width corrections to Ir are calculated. The radial drift invariant exists for trapped as well as passing particles. The new invariant is applied to construct Vlasov equilibria, where the magnetic field satisfies a generalized Grad-Shafranov equation with a poloidal plasma current and a bridge to ideal magnetohydrodynamic equilibria is found. For equilibria with sufficiently small banana widths and radial drift excursions, the approximation Ir≈r¯0(x,v) can be used for the equilibrium state.

Magnetic divertor design in GAMMA10 central cell

A divertor magnetic configuration is an alternative method of obtaining interchange mode stability in the central cell of the GAMMA10 tandem mirror (Cho et al 2005 Phys. Rev. Lett. 94 085002). Various divertor configurations are investigated in the central cell of GAMMA10 to determine the best configuration. The equilibrium in the divertor configuration is calculated using the generalized Grad–Shafranov equation with anisotropic plasma pressure to estimate how high a plasma pressure is sustained by the magnetic divertor.

Equilibrium of field reversed configurations with rotation. IV. Two space dimensions and many ion species

In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] a generalized Grad–Shafranov equation for the plasma flux function was derived which provides a complete description of equilibria of field reversed configurations with rotation. In this paper this fundamental equation is solved for two space dimensions and many ion species. The following fusion fuels are considered: D–T, D–He3, and p–B11. Using periodic boundary conditions the original differential equation is converted to an equivalent integral equation which involves a Green’s function. The integral equation is solved by iteration. Approximate solutions are found for all the fusion fuels considered using a two-dimensional equilibrium model for one type of ion [A. Qerushi and N. Rostoker, Phys. Plasmas 9, 5001 (2002)]. They are used as starting trial functions of the iterations. They turn out to be so close to the real solutions that only a few iterations are needed.

Equilibrium of field reversed configurations with rotation. III. Two space dimensions and one type of ion

A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is presented. In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that a complete description of equilibria for FRCs with rotation is provided by a generalized Grad–Shafranov equation for the plasma flux function. In this paper it is shown how to solve that fundamental equation for the case of two space dimensions and one type of ion. Periodic boundary conditions and a Green’s function are used to convert the original differential equation to an equivalent integral equation. The integral equation is solved by iteration. An iteration algorithm is described which converges to a solution of the generalized Grad–Shafranov equation starting with a one-dimensional trial function. Analytic one-dimensional solutions are shown to be a limiting case of two-dimensional solutions when the applied magnetic field is constant. In addition to rapid convergence for a complex nonlinear problem, the Green’s function method guarantees that the boundary conditions are satisfied in every iteration.

Equilibrium of field reversed configurations with rotation. II. One space dimension and many ion species

In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that a complete description of equilibria of field reversed configurations with rotation can be obtained by solving a generalized Grad–Shafranov equation for the flux function. In this paper we show how to solve this equation in the case of one space dimension and many ion species. The following fusion fuels are considered: D–T, D–He3, and p-B11. Using a Green’s function the generalized Grad–Shafranov equation is converted to an equivalent integral equation. The integral equation can be solved by iteration. Approximate analytic solutions for a plasma with many ion species are found. They are used as starting trial functions of the iterations. They turn out to be so close to the true solutions that only a few iterations are needed.

Equilibrium of field reversed configurations with rotation. I. One space dimension and one type of ion

Self-consistent solutions of the Vlasov–Maxwell equations are obtained. They involve rigid rotor distributions. This selection is justified on physical grounds. For this selection the Vlasov equation can be replaced by moment equations which terminate without any additional assumptions. For one-dimensional equilibria with one type of ion these equations have exact solutions. A complete equilibrium solution appropriate to a field reversed configuration with rotation can be obtained by solving a generalized Grad–Shafranov equation for the flux function. From this solution all other physical quantities can be determined. A Green’s function method is developed to solve this equation, which provides a basis for an iterative solution. This method has the advantage that at every iteration the boundary conditions are satisfied. In this paper cylindrical geometry with one space dimension and one type of ion is considered, where analytic solutions are available. The convergence of the Green’s function method is established. For this nonlinear problem there is usually more than one solution for completely specified boundary conditions (bifurcation). The present method selects one solution. It is applicable to equilibria with many ion species and to two dimensions.

General formulation and exact solution for two-dimensional field-aligned magnetohydrodynamic equilibrium flows

A general formulation is presented for steady field-aligned magnetohydrodynamic (MHD) equilibrium flows with isotropic or gyrotropic pressures. Closure to the anisotropic MHD model is provided by a pair of double-polytropic energy equations, for which double-adiabatic and double-isothermal conditions are special limits of the model. For the latter case, a MHD counterpart of Bernoulli’s equation is derived. The study is then focused on the two-dimensional (∂/∂y=0 but By≠0) problems, for which a generalized Grad–Shafranov equation is developed for field-aligned MHD flow equilibria with isotropic or gyrotropic pressures. The formulation is put in a form that allows self-consistent solutions to be constructed numerically in a way similar to the static case; examples of such MHD equilibria are shown. An asymptotic formulation is also developed for stretched gyrotropic plasma configurations, which, however, is not applicable to two-dimensional planar configurations with regions of weak magnetic field strength, such as the geomagnetic tail.

Steady MHD flows with an ignorable co-ordinate and the potential transonic flow equation

The paper explores the interrelationship between the generalized Grad-Shafranov equation, or trans-field force balance equation, for steady MHD flows with an ingnorable co-ordinate, and work by Imai on field-aligned MHD flows. The development of Imai, assumes at the outset that the fluid velocity V is parallel to the magnetic field B, and exploits an analogy with steady compressible irrotational flow in ordinary fluid dynamics. In Imai's analysis the magnetic induction B is written in the form B = σb, where , and MA is the appropriate Alfvén Mach number. Gauss' law Δ. B = Δ. (σb) = 0 then plays a role analogous to the mass continuity equation in ordinary fluid dynamics, where σ corresponds to the density of the pseudo-fluid. Imai's analysis leads to a transonic equation for the field potential φ defined by b = Δφ. For a restricted class of flows the trans-field force balance equation formulation also leads to the transonic potential flow equation, but the assumption of an ignorable co-ordinate allows for the possibility of non-field-aligned flows with non-zero electric field potential ΦE The characteristics of the generalized Grad—Shafranov equation are related to the Mach cone and the group velocity surface for linear magnetosonic waves. The corresponding forms of the characteristics for the potential transonic flow equation in the (x, y) plane and in the (bx, by) hodograph plane are discussed. Sample solutions of the potential transonic flow equation for radial, helical and spiral flows are obtained by means of the hodograph transformation, and are used to illustrate the differences between hyperbolic and elliptic flows. The potential transonic flow equation is obtained for the case of an ignorable co-ordinate z of a rectangular Cartesian co-ordinate system (x, y, z), and also for the case of flows with an ignorable co-ordinate of a spherical polar co-ordinate system (r, θ, ω). Astrophysical applications are briefly discussed.

Exact solutions for helical magnetohydrodynamic equilibria. II. Nonstatic and nonbarotropic solutions

In the framework of the analytical study of magnetohydrodynamic (MHD) equilibria with flow and nonuniform density, a general family of well-behaved exact solutions of the generalized Grad–Shafranov equation and of the whole set of time-independent MHD equations completed by the nonbarotropic ideal gas equation of state is obtained, both in helical and axial symmetry. The helical equilibrium solutions are suggested to be relevant to describe the helical morphology of some astrophysical jets.

Electromagnetically driven relativistic jets - A class of self-consistent numerical solutions

We explore the generalized Grad-Shafranov (GS) equation (Lovelace et al. 1986), which describes the cross-field balance of the magnetic flux surfaces in a stationary, axisymmetric, cold relativistic MHD wind. We construct a family of self-consistent, jet-type, non-self-similar solutions in which flux surfaces thread the equatorial plane vertically and are eventually collimated into nested cylinders. These solutions conserve the total energy and angular momentum along each flux surface and satisfy the relativistic GS equation in the cross-field direction. We find that the total specific energy in the flow depends primarily on a dimensionless combination of the poloidal magnetic flux, the rotation, and the mass loading rate, which reduces to Michel's (1969) magnetization parameter for radial winds. The final width of the flow and its ratio of kinetic energy flux to Poynting flux depend sensitively on the pressure distribution of the ambient medium, which is required to contain the jet at its outer edge when no jet-confirming external toroidal magnetic field is present. Furthermore, we show that the magnetic pressure in the flow can be much higher near the axis than close to the edge, reflecting the 'pinching' effect of the toroidal magnetic field.