View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Juelich Shared Electronic Resources MHD INSTABILITIES IN TOKAMAKS H.J. de Blank FOM Institute DIFFER – Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM, P.O. Box 1207, 3430 BE Nieuwegein, The Netherlands, www.differ.nl. ABSTRACT not part of ideal MHD: electric resistivity, drift waves, and energetic (not thermalized) particles [1]. Yet, these insta- A general introduction to ideal magnetohydrodynamic bilities often look very much like MHD instabilities if one (MHD) stability of tokamak plasmas is given, using linear considers the plasma motion, electric currents, and magnetic perturbations of the ideal MHD equations. Subsequently the field perturbations. The second purpose of this lecture is Energy Principle for ideal MHD instabilities is derived. The therefore, to learn about the structure of MHD instabilities. specific instabilities which are then discussed are loosely di- In experiments, this helps to understand how a variety of in- vided into two categories. Under the name “current driven stabilities show up in diagnostic signals. instabilities”, external and internal kink modes, which are modes with a large radial extent, are discussed. The in- B. Ideal MHD ternal m = 1 kink mode is responsible for sawtooth col- lapses and fishbone oscillations in tokamaks. Under the Starting point is the set of equations of resistive MHD: header “pressure driven instabilities”, more localized modes ∂ρ are presented. These modes may limit the pressure gradient = (ρU) , ∂t −∇· in the plasma without causing sizeable disruptions. The bal- ∂p looning limit and the Mercier criterion are presented. The = U p γp U , γ = 5/3 , ∂t − ·∇ − ∇· Troyon limit is mentioned as a synthesis of several of these ∂U stability boundaries. ρ + U U = j B p , ∂t ·∇ × − ∇ ∂B = E , I. GENERAL THEORY OF MHD INSTABILITIES ∂t −∇ × E = ηj U B , − × A. The stability problem j = B . ∇ × In magnetically confined plasmas, the optimization of the In addition there is Gauss’ law B = 0 which, once sat- plasma density and temperature for fusion energy produc- isfied, is conserved by Faraday’s∇· law. In hot plasmas, the tion has lead to a wide range of plasma instabilities. The electric resistivity η is negligible for sufficiently fast plasma adaptation of current and pressure profiles to avoid one type processes. Taking η = 0 in Ohm’s law, we obtain the of instability can lead to yet another type of instability. ideal MHD model. Introducing the total time derivative The fastest instabilities in magnetically confined plasmas are d/dt ∂/∂t + U , the ideal MHD equations can be usually MHD instabilities, and part of this lecture describes written≡ as · ∇ how to avoid them. dU γ 1 2 The main question in MHD stability theory is to con- ρ = B B sρ + 2 B , (1) sider an MHD equilibrium (measured or computed), and to dt ·∇ − ∇ dρ predict if it is stable or unstable. The obvious approach is = ρ U , (2) to simulate the evolution of the plasma numerically. The dt − ∇· ds p simulation may show growing instabilities and their long- = 0 , s , (3) dt ργ term fate: saturation, triggering of other instabilities, or tur- ≡ dB bulence. However, this essentially nonlinear modelling is = B U B U . (4) computationally expensive, especially when a wide range of dt ·∇ − ∇· length or time scales are involved. where s is the entropy density of the plasma. The mo- The present lecture focusses on linear instabilities in- mentum balance equation (1) is central to the MHD physics: stead. This approach is systematic and decides if infinites- it gives the evolution of the plasma flow U in terms of the imal perturbations of an equilibrium are stable (wave-like, density ρ, the magnetic field B, and the entropy s (or pres- oscillating) or unstable (exponentially growing). sure p = sργ ). However, many other (usually slower) instabilities have An important property of the MHD model is that the been discovered that depend on physics ingredients that are other three equations (the mass continuity equation (2), the 93 energy equation of state (3), and Faraday’s law (4)), give the seem to be dangerous to plasma confinement. However, as evolution of ρ, s (or p), and B due to the plasma flow as we shall see later, MHD instabilities may involve “magnetic local conservation laws. resonant surfaces” in the plasma, where the plasma motion This is most easily seen for the energy equation (3), induces a narrow layer with very high current density. In which merely states that the entropy density s is conserved such a layer, even very low resistivity may be sufficient to in each point as it moves along with the plasma velocity U. cause magnetic reconnection. Although this reconnection of For the other conservation laws, we first specify how an in- field lines is confined to a thin layer, these field lines may finitesimal line element dx moves with the plasma flow U, extend into regions where they are far apart. Therefore, lo- calized reconnection may have global consequences for par- d d dx dx = (x + dx) ticle and energy confinement. dt dt − dt The topological constraints can prevent ideal MHD in- = U(x + dx) U(x) − stabilities altogether, even if there is plenty of (magnetic and = dx U . kinetic) free energy to drive instabilities, as is often the case ·∇ in magnetically confined (fusion-) plasmas. In such cases By constructing infinitesimal surface and volume elements there may be a much slower, resistive MHD-instability, for 2 3 out of line elements, d x = dx1 dx2 and d x = dx1 which magnetic reconnection (again, taking place in a thin dx dx , we obtain × × 2 · 3 resonant layer) is essential. One such instability is the tear- d ing mode, presented in another lecture [2]. d2x = (d2x ) U dt − × ∇ × Another important property of the ideal MHD system is = d2x U ( U) d2x , (5) that it can be derived from Hamilton’s principle: the plasma ∇· − ∇ · motion U(x, t) that makes the action d d3x = d3x U . (6) dt ∇· t1 S = L dt Combining expression (6) with (2), one finds the mass con- Zt0 servation law in integral form, stationary, where the Lagrangian is [3,4] d 3 ρ d x = 0 , (7) 3 1 2 p 1 2 dt L = d x ρU B , (9) 2 − γ 1 − 2 Z Z − for any volume that moves with the plasma flow. In the same is the true dynamical motion that satisfies the MHD equa- way we can combine Eqs. (4) and (5) to give tions. Here, it is understood that the plasma motion deter- mines the evolution of ρ, p, and B through Eqs. (2), (3), d B d2x = 0 . (8) and (4), respectively. dt Z In order to demonstrate that Hamilton’s principle for This equation states that the magnetic flux through an arbi- the Lagrangian (9) indeed produces the MHD momentum trary surface that moves with the plasma is conserved. equation (1), we investigate how the action S changes if To understand the consequences of this flux conserva- the MHD fields are perturbed. Since all MHD quantities tion law, consider the surface of a thin tube that surrounds respond to the plasma motion, the primary perturbation is a stretch of magnetic field line. By construction, there is an arbitrary infinitesimal displacement ξ(x, t) of the plasma zero magnetic flux crossing the surface. In addition, Gauss’ fluid. We introduce the operator δξX, the Lagrangian pertur- law states that the flux that enters one end of the tube equals bation of a variable X, which is the change in the quantity the flux that leaves the other end. Let this tube flow with while following the perturbed plasma motion. For instance, the plasma velocity as time proceeds. The flux conservation δξx = ξ. It is helpful to introduce also the Eulerian pertur- E law dictates that the flux that crosses the tube will remain bation δξ δξ ξ , which gives the perturbation at a ≡ − ·∇E zero and that the flux through the tube will remain the same. fixed point in space, δξ x = 0. It therefore commutates with Hence, also at later times, the moving tube will precisely sur- partial space and time derivatives, round a magnetic field line. We can therefore say that mag- ∂ ∂ netic field lines move with the plasma flow in ideal MHD. It δE = δE , δE = δE . ξ ξ ξ ∂t ∂t ξ follows that in an ideal MHD plasma, magnetic field lines ∇ ∇ cannot be created or annihilates, nor can they break up and While the Lagrangian perturbation does not commutate with reconnect. The magnetic topology is conserved, “frozen in and ∂/∂t, it commutates with the total derivative instead, ∇ the fluid”, so to speak. d d dξ Strictly speaking, ideal MHD instabilities cannot δ = δ , since δ U = . ξ dt dt ξ ξ dt change the magnetic topology of nested toroidal surfaces in a tokamak plasma. Thus, in a very hot (e.g. fusion-) plasma An infinitesimal line element varies as δξdx = dx ξ. With with negligible resistivity, ideal MHD instabilities may not these tools, one can obtain the perturbed density,·∇ pressure, 94 and magnetic field from Eqs. (7), (3), and (8) respectively, Since Eq. (14) is linear in ξ, it determines eigenfunctions iωt ξ(x, t) = ξ(x)e− δξρ = ρ ξ , (10) The force operator F possesses the important property − ∇· δ p = γp ξ , (δ s = 0) (11) that it is self-adjoint, i.e., given any two vector fields ξ and ξ − ∇· ξ δ B = B ξ B ξ . (12) ζ, the operator satisfies ξ ·∇ − ∇· 3 3 3 3 The perturbedvolume element is given by δξ d x = d x ξ.