Power exhaust, by which we mean the safe removal of power from a burning plasma, is an essential requirement for the successful operation of any fusion reactor. Specifically, plasma thermal energy must be conveyed across the first wall without undue damage to plasma facing components (divertor and limiter tiles) by heat load related plasma–surface interactions (ablation, melting, erosion). Unlike other ‘technological’ problems related to fusion reactor design, e.g. tritium retention in plasma facing materials, neutron damage to structural components or non-inductive current drive, power exhaust is intimately linked to plasma confinement and thus a perennial concern for any fusion reactor. While only a minor issue in existing tokamaks, it will be critical for ITER (the next step plasma-burning experiment) and even more so for DEMO (the demonstration fusion power plant). Even non-burning, superconducting machines, such as EAST, KSTAR, JT60-SA, W7-X, etc. will be forced to tackle this problem due to their long pulse capabilities.

This monograph is an attempt to draw a unified and up-to-date picture of power exhaust in fusion plasmas, focusing primarily on the leading tokamak concept. Emphasis is placed on rigorous theoretical development, supplemented by numerical simulations when appropriate, which are then employed to explain and model a range of experimental observations. The objective is not just to provide the reader with a reliable map of the conquered territory and a guided tour over its many hills and valleys, but also to supply him or her with the tools necessary to embark on independent, and hopefully fruitful, journeys into the uncharted regions, the white spaces on the map, la terra incognita.

By definition, all exothermal reactors, including any fusion reactor one may envisage (tokamak, stellarator, etc.), produce both energy and spent reactants, or ash. In order for the reactor to operate in steady-state, (i) fresh fuel must be added at the rate at which it is consumed, (ii) this fuel must be heated, ideally by the reactions themselves, (iii) fuel must be confined, by whatever means are available, for sufficiently long to allow the exothermic processes to continue, (iv) the energy and ash must be removed from the system at the rate at which they are created, (v) the impurities released from the reactor walls must likewise be removed at the rate at which they are produced, and (vi) the reactor itself, primarily its walls, must not be damaged by all the exhaust processes. Translating the above to a D–T burning tokamak, conditions (i)–(iii) may be labelled loosely as the ignition criteria, and conditions (iv)–(vi) as the exhaust criteria. Taken together they constitute the criteria of mutual compatibility between the burning plasma and first wall materials/components.

Considerably more energy, one might add – in the planned ITER experiment, Qα (1.5) is expected to reach a maximum value of 2 and a steady-state value of 1. How is it possible to achieve this extraordinary level of plasma energy confinement in a laboratory environment? The theoretical framework developed in the previous chapter provides a simple answer: by embedding the plasma in a strong ambient magnetic field, such that the thermal pressure, p, is supported by the magnetic pressure, B2/2µ0, see (2.198) and (2.247). Indeed, after no less than half a century of international research, the strategy of magnetic confinement appears to offer the most promising route to realizing the long-held dream of constructing a technologically feasible and commercially viable fusion reactor. Yet, it is not immediately obvious what such a reactor should look like, e.g. what is the optimal magnetic geometry? How strong does the ambient magnetic field have to be? Can this field be generated by the plasma itself or are external coils and/or antennas required? etc.

Dynamical equilibrium does not guarantee dynamical stability. In general, an equilibrium is said to be stable if the system remains bounded, i.e. confined to its neighbourhood, after being subjected to a small perturbation. There are many different classifications of stability, e.g. linear stability implies small perturbations, while non-linear stability allows for perturbations of arbitrary size. The former is equivalent to spectral stability, which occurs when all eigenvalues of the linearized dynamical operator have real parts which are positive or zero. The most general formulation of linear stability is the energy principle, which states that an equilibrium point is stable when it represents a minimum of the potential energy of the system. Most dynamical systems have both stable and unstable equilibria, e.g. the lowest and highest position of a pendulum. Having identified an equilibrium point, one should next investigate its stability properties and, if the point proves unstable, the physical mechanism responsible for the instability.

Hydrodynamic waves and instabilities

By way of introduction to plasma instabilities, let us consider the stability properties of a stratified neutral fluid in the presence of a gravitational field g = gêg.

In the previous chapter, we investigated the relatively slow, transport-ordered, evolution of quasi-equilibrium plasma quantities due to radial collisional flows. In this chapter we consider similar evolution due to radial flows associated with plasma turbulence. As always, we will first try to shed some light on the issue by investigating the relatively simple case of hydrodynamic turbulence, Section 6.1, then proceed to turbulence in magneto-hydrodynamics, Section 6.2, and finally, turn to the dominant process in magnetized plasmas, namely, drift-wave turbulence, Section 6.3.

Hydrodynamic turbulence

What is turbulence? We know it when we see it, yet it is not easy to define. A compact, yet accurate, definition has been formulated by Corrsin (1961),

Having examined the equilibrium, stability and transport properties of magnetized plasmas, we are finally ready to tackle our ultimate aim, i.e. the exhaust of particles and power from the plasma itself and the limits imposed by the resulting plasma wall loads and thermo-mechanical material constraints on the reactor performance. Geographically, this question naturally leads us towards the plasma boundary, commonly known as the scrape-off layer (SOL), defined as the region of open field lines between the last closed flux surface (LCFS) and the vessel wall, which is the only place in a fusion reactor where the stellar world of hot (keV) plasmas meets the earthly world of cold (sub-eV) solids. As the name indicates the LCFS represents the transition between the regions of closed and open magnetic field lines, i.e. between the edge and SOL regions. This surface can be created either by inserting a solid object, known as the limiter, into the plasma, or by shaping the poloidal magnetic field with external current carrying coils to create a poloidal field null, or X-point, and a magnetic separatrix, and thus to divert the SOL plasma into a specifically designed structure, known as the divertor, see Section 7.1.3. Since the location and degree of plasma contact with the wall is determined by transport processes in the SOL, the understanding of these processes is critical for translating a plasma load limit (per component surface area) into a plasma exhaust limit (per plasma surface area at the LCFS), recall (1.16).

And even today, intact atoms make up no more than a tiny fraction of the observable universe. As a result, the effort to understand a broad spectrum of phenomena – from the way stars and planets coalesce out of plasma discs, to the evolution of galaxies, to the magnetic maelstroms around black holes and the firing of interstellar bullets we call “cosmic rays” – requires the perspective of plasma science.

What is surprising, however, is the remarkable ubiquity of magnetic fields in the universe, and the relationship of those fields to plasmas on size and intensity scales that range over dozens of orders of magnitude. There is no obvious reason that so many objects and aggregations of objects – including individual stars and planets, the interstellar medium, whole galaxies, and even the vast evacuated spans between galaxies – should have associated long-lived magnetic fields. But they do, and the dynamics of these bodies are strongly affected by the interplay of plasmas and fields.