Simple examples of MHD equilibria

Department of Physics Seminar. grade: Nuclear engineering Simple examples of MHD equilibria Author: Ingrid Vavtar Mentor: prof. ddr. Tomaž Gyergyek Ljubljana, 017 Summary: In this seminar paper I will represent the magnetohydrodynamics description (MHD) of the plasma. In the first part I will focus on the MHD equations and their properties. In the second part of the seminar paper the basic properties of the MHD equilibrium model will be presented; I will focus on radial pressure balance in cylindrical system, namely θ-pinch, Z-pinch and screw pinch. 1

TABLE OF CONTENTS 1 INTRODUCTION... MHD equations... 3.1 Basic scaling relations for MHD... 3. Summary of the MHD equations... 4 3 THE MHD EQUILIBRIUM MODEL... 5 3.1 General properties... 6 3.1.1 Flux surfaces... 6 3.1. Current surfaces... 6 3.1.3 Magnetic pressure and tension... 7 3. Radial pressure balance... 8 3..1 The θ-pinch... 8 3.. The Z-pinch... 8 3..3 The screw pinch... 9 4 CONCLUSION... 10 5 SOURCES AND LITERATURE... 10 1 INTRODUCTION Fusion is a form of nuclear energy which involves the merging (i.e., the fusing) of light elements, mainly hydrogen (H) and its isotopes deuterium (D) and tritium (T). The main application of fusion energy is the production of electricity, because the laws of physics have shown that a large amount of kinetic energy is released every time a nuclear fusion reaction occurs. To achieve fusion reaction it is necessary that the material (fuel) is in the form of plasma (plasma is hot gas made of electrons and bare nuclei) and that fusion plasma achieves a temperature of about 15 kev and a pressure of approximately 7 atm. Fusion plasma is a fully ionized gas whose behaviour is dominated by long-range electric and magnetic fields. A major consequence of this behaviour is that plasma is an exceptionally good conductor of electricity. The reason is associated with the fact that at a high temperature and because of the low density of plasma, there are rarely any Coulomb collisions between electrons and ions and thus there is very little resistance to the current flow. In terms of fusion reactor it is the high conductivity that opens up the possibility of confinement by magnetic fields. Carrying out a workable fusion reactor requires good knowledge of plasma properties, therefore my focus in this seminar paper will be on the description of the plasma. I will introduce a magnetic hydrodynamic description of the plasma, the adequacy thereof and equilibrium magnetic configuration on fusion reactors. Plasma can be described with a self-consistent plasma model, where it is considered that the charges in the plasma which move under the influence of electric and magnetic fields affect

those fields back. In developing self-consistent models we should be aware that various levels of description are possible. The most accurate models are kinetic models which strive to determine the distribution functions of ions and electrons and then the macroscopic quantities are determined as the moments of the distribution functions. They are the most accurate and describe almost all physical phenomena in plasma but are mathematically very demanding. The next level of description corresponds to macroscopic fluid models, where the great simplification in comparison to kinetic models is that all the unknowns are functions of only space and time, but not of the velocity. In comparison with the kinetic models they are less accurate and they cannot describe some phenomena which can be described in kinetic models, but they are mathematically less demanding. In a simplified two-fluid model it is assumed that plasma contains two fluids ; electrons and ions. The two-fluid model consists of the conservation of mass, momentum, energy for electrons and ions plus Maxwell s equations and describes the self-consistent behaviour of a fusion plasma including all of the main phenomena such as macroscopic equilibrium and stability, transport, heating and current drive. The third level of description is magnetohydrodynamics (MHD) on which I will focus in the following. The MHD model is a reduction of the two-fluid model to one-fluid model derived by focusing attention on the length and time scales characteristic of macroscopic behavior. MHD equations The major issue in which self-consistency plays a crucial role is the macroscopic equilibrium. It needs to be known how a magnetic field can produce forces to hold plasma in a stable, macroscopic equilibrium thereby allowing fusion reactions to take place in a continuous, steady state mode of operation. The analysis of macroscopic equilibrium and stability is based on a single-fluid model known as MHD. The derivation of the MHD model from the two-fluid model is relatively simple, but requires a few steps. In the following I will present MHD equations and their characteristics..1 Basic scaling relations for MHD It is necessary to define the characteristic spatial and temporal criteria to allow comparison orders of magnitude of individual parts of equations in the two-fluid model and to allow neglecting small parts in equations. The appropriate length scale L is the plasma radius (L~a) while the appropriate time scale τ is the ion thermal transit time across the plasma ( τ~a/v Ti ), which leads to a characteristic velocity u (u~l/τ~v Ti ). v Ti is the fastest macroscopic speed that the plasma can achieve, a.k.a. the ion sound speed. The ion sound speed v Ti assimilated to the average ion velocity in two-fluid model u i. In a fusion plasma there is E B drift (because of a uniform perpendicular electric field superimposes a constant drift velocity on the gyro motion known as the E B drift and is perpendicular to both field, electric and magnetic; it is also independent of the mass and charge, so electrons and ions drift with same velocity.). Even E B drift speed should not exceed the ion sound speed (u e ~u i ~E /B ~v Ti ). Since the E B drift velocity is identical for both electrons and ions the current densities J = e(n i u i n e u e ) cancels to leading order. It follows J /env Ti 1. The next request is that the plasma pressure is assumed to be finite compared to the magnetic pressure, which requests that β p/( B μ 0 )~1. 3

Inequalities which must be satisfied are also small ion gyro radius r Li a and low collision frequency ν ei v Ti /a.. Summary of the MHD equations The MHD equations which are collected together in the following are the basis for studies of macroscopic balance in fusion plasma studies. First equation of MHD model is mass equation: dρ dt + ρ v = 0. (1) The momentum equation is the most important one in MHD as it describes the basic force balance of the plasma: ρ dv dt = J B p. () The left side of the equation is a product of density and acceleration of plasma which represents the inertial force which is important in determining the dynamical behaviour of the plasma. On the right side of the equation are the magnetic field force used to confine the plasma and pressure gradient force that causes a hot core of plasma to expand outwards. In a steady state situation without flow the inertial force is zero and equilibrium is achieved when the magnetic force balances the pressure gradient force. In terms of scaling it can be seen that the current density J must satisfy J~ p J for the two forces to balance. This implies that ~ r Li 1 as ab e 0 nv Ti a previously asserted. [1] If the MHD equilibrium is achieved, the resistive Ohm s law can be derived from equations of conservation of momentum in two-fluid model: E + v B = ηj. (3) A more detailed kinetic analysis shows that the resistance of the plasma varies considerably in the direction which is parallel to B, as in the direction perpendicular to it. The resistance η in fact corresponds to the rectangular part of the resistance; parallel part of the resistance is about half the size. Taking into account the resistance of plasma in Ohm s law brings two important effects. First, use of the resistive Ohm s law allows a treatment of a large number of plasma instability. The time scale of these instabilities, however, is much larger than the time scale for MHD, which was defined at the beginning, so that instabilities do not lead to loss of plasma in its entirety, but only to increased transport losses. Secondly, even if there were no instabilities as a result of the final resistance of the plasma it is a fact that the resistance of the plasma is only a dissipative mechanism in the equation of motion. This causes the diffusion of the particles and the magnetic field which are the two main mechanisms of transport losses. However, these appear as slower than the characteristic MHD time constant. In the MHD regime the energy equation is simplified considerably, because all sources and sinks of energy can be neglected, since all the processes of heating and cooling are much slower than the MHD time constant τ, which leads to the single-fluid energy equation: 4

d dt ( p ρ γ) = 0, (4) where γ gives the ratio of specific heats and his value is γ = 5 3. Some simplifications of Maxwell s equations can be made. Taking into account that the speed of macroscopic plasma flows is much smaller than the speed of lights (v Ti c), the displacement current 1 E can be neglected. Because characteristic length scale of MHD easily c t satisfies the inequality λ De a, the E term in Poisson s equation can be neglected leading to the quasi-neutrality relation: n e n i. Quasi-neutrality implies that ε 0 E en e (not that E = 0). These two simplifications reduce Maxwell s equations from the exact relativistic Lorentz-invariant form to a self-consistent, low-frequency, Galilean-invariant form: E = B t, (5) B = μ 0 J, B = 0. In the equations above single fluid variables, mass density ρ, macroscopic velocity v, and the pressure p appear. J denotes the current densities, B the magnetic field and E the electric field. 3 THE MHD EQUILIBRIUM MODEL An important application of the MHD model is concerned with the calculation of equilibrium. In MHD equilibrium model we are looking for an answer to the question of how does a combination of externally applied and internally induced magnetic fields need to act to provide an equilibrium force balance that holds the plasma together at the desired location in the vacuum chamber. The equilibria of interest must correspond to the confined equilibria, where the purpose of the magnetic field is to isolate the plasma from the wall of the vacuum chamber which keeps the plasma hot and the wall cool. The MHD equilibrium in a toroidal geometry can be separated into two pieces, namely the radial pressure balance (along the minor radius r) and toroidal force balance (along the major radius R) (see Figure 1). The plasma is a hot core of gas that tends to expand uniformly along the minor radius r. Two basic magnetic configurations that can produce radial pressure balance are the θ-pinch and Z-pinch plus combinations thereof. These configurations will be discussed below. A problem of toroidal force balance arises solely because of the toroidal geometry. Because of toroidicity, unavoidable forces are generated by both the toroidal and poloidal magnetic fields that tend to push the plasma horizontally outwards along the direction of the major radius R. The toroidal force balance is outside the frameworks of this seminar paper. 5

Figure 1: On the left radial MHD balance and on the right torodal MHD balance. Reproduced from [1]. To focus only on the equilibrium the MHD model needs to be simplified. The simplifications result from two basic assumptions. First assumption is that for plasma in equilibrium all quantities are independent of time, which results in = 0. The second assumption is that the t plasma is treated as a stationary liquid, v = 0. With these assumptions the MHD equations are converted in the following form: J B = p, B = μ 0 J, (6) B = 0. These equations describe the equilibrium properties of all magnetic configurations of fusion interest. 3.1 General properties 3.1.1 Flux surfaces The first property concerns the concept of flux surfaces. Vector p is, by definition, perpendicular to the surface where p is a constant. So if the first equation from simplified MHD model (6) is multiplied with the magnetic field B it immediately follows that: B p = 0. (7) In equation (7) the rule for mixed product of three vectors is taken into account. The implication is that the magnetic field lines must lie on isobaric surfaces, so that there is no component of B perpendicular to the surface. These surfaces are called flux surfaces and are forming a set of closed, nested, toroidal surfaces. 3.1. Current surfaces Current surfaces need to be proceeded similarly as flux surfaces above. Momentum equation (6) forming the dot product with current densities J results in: J p = 0. (8) The current lines, like magnetic field, lie in the constant pressure surfaces (see Figure ) and there is also no component of J perpendicular to the pressure contours, which implies that 6

current flows between flux surfaces and not across them. Vectors B and J thus both lie on the isobaric surfaces, but we cannot conclude anything about what is the angle between them. Figure : Vectors B and J both lie on the isobaric surfaces. On right reproduced from [1], on left reproduced from []. 3.1.3 Magnetic pressure and tension Magnetic pressure and tension are two ways in which the magnetic field can act to hold the plasma in equilibrium force balance. Magnetic pressure uniformly presses on the plasma from all sides and tension extends plasma in the direction of magnetic field lines. Combining first two equations (6) and the vector identity ( B ) = B ( B) + (B )B yields to: (p + B ) B κ = 0, (9) μ 0 μ 0 where is the perpendicular component of the gradient operator and κ is the curvature vector. Equation (9) describes pressure balance perpendicular to the magnetic field. The quantity p represents the plasma pressure, quantity B B μ 0 represents the magnetic pressure and the last term μ 0 κ represents the tension force created by the curvature of the field lines. So the magnetic field distresses plasma in the direction perpendicular to the magnetic field and stretches it in the parallel direction (see Figure 3). [1] Figure 3: Magnetic field distresses plasma in the direction perpendicular to the magnetic field and stretches it in the parallel direction. Reproduced from []. 7

3. Radial pressure balance For further analysis simpler cylindrical geometry will be used, where the torus will be stretched in the cylinder. This way the problem is much more simplified because it becomes onedimensional (it is necessary to observe only the radial compression of the plasma). 3..1 The θ-pinch Consider a cylindrical column of plasma in a conductive, thin, metal panel, where the axis of the plasma column and metal panel coincide. Electric current is running through this panel in the poloidal direction. In the plasma a current is induced in the opposite direction and it creates a magnetic field in the axial direction, i.e., toroidal direction. Tags poloidal and toroidal are understood in the terms of a toroid which is extended into the cylinder. The geometry and field components of a θ-pinch are illustrated in Figure 4. Figure 4: Schematic diagram of a θ-pinch on left and typical θ-pinch profiles on right. Reproduced from [1]. The non-trivial unkowns in the problem are p = p(r), B = B z (r)e z and J = J θ (r)e θ, where current density in the θ direction is the origin of the name θ-pinch. First, by symmetry the condition. B = 0 is automatically satisfied. Second, Ampère s law reduces to μ 0 J θ = db z dr. Substituting J θ into the momentum equation leads to a simple relation between p and B z, which can be written as d (p + B z ) = 0 and can be easily integrated: dr μ 0 p(r) + B z (r) = B 0. (10) μ 0 μ 0 In equation (10) the part B 0 = const. is the externally applied magnetic pressure. Equation μ 0 (10) represents the basic radial pressure balance relation for a θ-pinch. It states that at any radial position r the sum of the local plasma pressure plus internal magnetic pressure is equal to the applied magnetic pressure. Equation (10) can be solved with different radial dependence of the pressure in a magnetic field. We are interested in a balance where the plasma is confined which means that p(r) has the maximum at r = 0 and then monotonically decreases towards the edge of the plasma column at r = a. The radius of the metal panel should be b and b > a. A typical example is illustrated in Figure 4. [1] 3.. The Z-pinch The Z-pinch is the complementary configuration of the θ-pinch. Like in the case of θ-pinch, it is considered a cylindrical column of plasma, but now the electric current in the column flows 8

in a toroidal direction, i.e., in z direction, which produces a purely poloidal magnetic field in the θ direction. It is given the name Z-pinch because of the current density in the z direction. The geometry and field components of a Z-pinch are illustrated in Figure 5 on the left side. Figure 5: On the left is shematic diagram o a Z-pinch, in the middle typical Z-pinch profiles, on the right the radial forces on a Z-pinch. Reproduced from [1]. In this configuration the non-trivial unknowns are given by p = p(r), B = B θ (r)e θ and J = J z (r)e z. Even in this case it is so because the symmetry. B = 0. Since the magnetic field B θ on the closed circular lines is constant, Ampère s law yields a relation for J z given by μ 0 J z = 1 d r dr (rb θ). Z-pinch pressure balance relation can be written as: d dr (p + B θ ) + B θ = 0, (11) μ 0 rμ 0 if we expand rb θ derivative from pressure balance relation which is obtained by substituting the expression p = J z B θ into momentum equation. In order, from left to right, the terms correspond to the plasma pressure, the magnetic pressure and the magnetic tension. In this case the equation (11) cannot be directly integrated, therefore we need to invent a physically acceptable, radial dependence density of the electric current flow J z (r) through the plasma column from which poloidal magnetic field B θ (r) and pressure profile p(r) can be then calculated. Typical Z-pinch profiles are illustrated in Figure 5 in the middle. In equation (11) the positive force dp wants to expand the plasma column in the radial direction and the negative dr forces on the right side are pressing on it (see Figure 5 on the right). It can be seen that only the tension force is confining the outer portion of the plasma. The conclusion is that the Z-pinch configuration is capable of providing radial pressure balance by means of the magnetic tension force. [1] 3..3 The screw pinch The screw pinch is a combination of a θ-pinch and Z-pinch. In its configuration the magnetic lines twisted around the surface giving the appearance of a screw thread and hence the name screw pinch. Using a very similar method to the one in θ-pinch and Z-pinch, the desired expression for radial pressure balance in a generalized screw pinch yields: d dr (p + B θ + B z ) + B θ = 0. (1) μ 0 μ 0 rμ 0 9

It is a differential equation with three variables, two of which we can choose on our own. Both are usually a magnetic field. This can be done in an experiment, for example, if we create magnetic fields in axial and poloidal direction. Both can be independently modified. [1] 4 CONCLUSION In the seminar paper, magnetic hydrodynamic description (MHD) of the plasma is discussed. The first part of the seminar paper is focused on the MHD equations and its properties. The MHD model is reduction of the two-fluid model to the one-fluid model derived by focusing attention on the length and time scales characteristic of macroscopic bahavior. MHD model contains the mass and momentum equations, the resistive Ohm s law, energy equation and three Maxwell s equations. An important application of the MHD model is the analysis of equilibrium. The second part of this seminar paper was focused on it. To focus only on the equilbrium, the MHD model was simplified. The MHD equilibrium in a toroidal geometry can be separated into two pieces, namely radial pressure balance and toroidal force balance. In the seminar paper, only radial pressure balance was presented. The analysis of general properties like flux surfaces, current surfaces and magnetic pressure and tension points to the notion that pressure contours from a set of closed-nested toroidal surfaces. Both the magnetic field lines and current density lines lie on these surfaces. The magnetic forces that hold the plasma together arise from a combination of magnetic pressure and magnetic tension. Finally, several basic configurations like θ-pinch, Z-pinch and screw pinch were researched using the simplified MHD model. The conclusion is that the θ-pinch and Z-pinch configurations are capable of providing radial presure balance and that the screw pinch is a configuration that consists of an arbitrary combination of θ-pinch and Z-pinch fields. 5 SOURCES AND LITERATURE [1] J. P. Freidberg. Plasma physics and fusion energy. Cambridge University Press, New York 007. [] J. A. Bittencourt. Fundamentas of plasma physics 3 rd edition. Springer-Verlag, New York 004. [3] Glossary of terms in nuclear technology and radiation protection. Nuclear society of Slovenia, Ljubljana 01. [4] T. Gyergyek. Lecture notes: Physics and engineering of fusion reactors. Faculty of mathematics and physics, University of Ljubljana, Ljubljana 016. [5] J. P. Freidberg. Ideal MHD. Cambridge University Press, New York 014. [6] H. P. Goedbloed, S. Poedts. Princiles of magnetohydrodynamics. Cambridge University Press, New York 004. 10