 # Equilibrium of Tokamak Plasma

## GOVERNING IDEAL MHD EQUATIONS FOR TOKAMAK PLASMAS

For describing plasma, we take velocity moments of collision-less kinetic equations for distribution functions of electrons and thermal ions of the plasma and sum the results over the plasma species to obtain Here (4.1) is the mass conservation equation, p is the plasma mass density, and V the fluid velocity. The equation of plasma motion is given by (4.2), where p is the plasma pressure, and J, В are the plasma current and the magnetic field, respectively. Adiabatic behaviour is assumed for plasma pressure described by (4.3), where у is the adiabaticity index.

The perpendicular Ohm’s law is obtained from the equation of electron motion, Here, we assume a perfectly conducting plasma so that no electric field E can be sustained in the moving fluid reference frame as Eq. (4.4) shows.

Although Eqs. (4.1)—(4.4) are obtained from the kinetic equations, they look similar to hydrodynamic equations describing current-carrying liquid in magnetic field; therefore, they are called ideal magneto-hydrodynamic (MHD) equations [4.1].

For describing electromagnetic fields in the plasma, Maxwell’s equations are used, which in CGS units are:  Scale lengths larger than Debye length are considered with the plasma quasi-neutrality condition, For further analysis, we express plasma current as the sum of the current components parallel and perpendicular to the magnetic field: The second term is determined by the equation of motion (4.2): Next, from Amperes law (4.5), we obtain By substituting Eqs. (4.9) and (4.10) into (4.11) and using (4.7), we arrive at the main governing equation of ideal MHD We introduce equilibrium (subscript 0) and perturbed (denoted by <5) quantities: A static equilibrium means d/df=0, so we obtain the following for the static and axi-symmetric (independent of toroidal angle) equilibrium of tokamak plasma: Therefore, JS0Vp0 = 0,J0Vpo = 0, that is, the plasma pressure is constant along both the magnetic field lines and along the current lines; the plasma expands freely along these lines. Next, because Vp„ is perpendicular to the surface pQ = const, the magnetic field lines and the current lines must lie on the pQ=const surfaces. In a tokamak, the magnetic field lines lie in nested toroidal magnetic surfaces, as shown in Figure 4.1 [4.2]. FIGURE 4.1 Schematic showing magnetic flux surfaces enclosed within each other in torus.

## GRAD-SHAFRANOV EQUATION FOR TOKAMAK PLASMA EQUILIBRIUM

For assessing the equilibrium plasma equation (4.14), we introduce equilibrium poloidal magnetic field flux function у/and equilibrium poloidal current density flux function: The equilibrium equation (4.14) expressed via the poloidal and toroidal components of the current and magnetic field has the following form: Here, we drop the subscript “0” in this section.

By substituting (4.15) into (4.16), we obtain the form of Grad-Shafranov equation [4.3, 4.4] for the poloidal flux i//: For further analysis, we introduce a properly chosen magnetic field coordinate system (see Appendix B) and use the expressions for coordinates from this Appendix to obtain: Equations (4.17) and (4.18) could be analysed by combining the ^-independent terms to obtain: which could also be represented as where the safety factor q was introduced from 4/' = rBT/q = rB{) jq.

Thus, only tw'o of the three flux quantities q, /, and p may be specified independently. Next, we combine the terms <*= cos 8 in (4.17) and (4.18) to obtain: which could be represented as Solving this equation for the Shafranov shift derivative, we find where By introducing internal inductance, and poloidal beta we obtain: ## SOME EXTRAORDINARY TOKAMAK EQUILIBRIA

Plasma equilibria in tokamaks are usually described by the Grad-Shafranov equation pretty well, so this approach is used in a majority of tokamak discharges in present-day machines. There are, however, some tokamak plasma scenarios, parameters of which correspond to pretty unusual limiting cases of the Grad-Shafranov approach. We consider two interesting scenarios here.

### Advanced Tokamak Scenario with Hollow Current Profiles

The first extraordinary equilibrium to be considered is the so-called “advanced tokamak” (AT) scenarios aiming at obtaining internal transport barriers (ITBs) in tokamak plasmas [4.5-4.7]. The ITB scenarios are obtained by applying high power of an auxiliary heating to tokamak plasma early in the discharge and well before the inductive current diffuses to the plasma centre. Under certain conditions, a region of thermal plasma with much improved confinement properties rises at the plasma centre, the ITB. The very high ion temperature and plasma density inside the ITB delivers a very significant fusion performance of the plasma, for example, the record high rate of D-D neutrons, 5.5 x I0l6/s, was achieved on JET with carbon wall in the ITB scenario. Due to the incomplete penetration of the inductive current to the central region at the start of main heating, a hollow current profile is often created with non-monotonic c/(r)-prolilc. In some cases, the plasma forms a “current hole” profile, with zero current value (within the error bars of measurements) over an extended central region of the plasma [4.8]. Figures 4.2-4.5 show an example of AT scenario in JET.

The temporal evolution of the D-D neutrons rate in Figure 4.2 (top) increases faster at /=6s indicating formation of ITB, and the neutron rate doubles by 6.8 s at constant power of NBI and ICRH. Figure 4.3 shows how the magnetic topology changes via the safety factor q(R) when the ITB forms, and Figures 4.4 and 4.5 show the profiles of ion temperature and electron density at the time slice corresponding to Figure 4.3. The flat profiles of plasma in the central region of the plasma are consistent with Eq. (4.14), which requires low values of plasma pressure gradient when the plasma current is small (safety factor is high) as in our case. FIGURE 4.2 Temporal evolution of key discharge parameters in AT discharge JET #58094. From top to bottom: (a) Fusion yield measured via D-D neutrons; (b) NBI and ICRH power wave forms, and (c) isolines showing non-monotonic q(i t) profile. FIGURE 4.3 Profiles of the safety factor q(R) measured in discharge #58094 at two different times. Equilibrium with a deeply reversed magnetic shear is formed prior to ?=6.4s. FIGURE 4.4 Profile of ion temperature T,(R) measured in the discharge #58094 at t=6.4 s. FIGURE 4.5 Profile of electron density ne(R) measured in the discharge #58094 at /=6.4s.

In the case of a “current hole” equilibrium, central region of plasma becomes similar to a stellarator-type equilibrium without current surrounded by a tokamak-type equilibrium with finite current that twists (rotational transform) magnetic field lines. In this case, orbits of charged particles in the current hole region are determined by VB-drifts that cause the charge separation discussed in Chapter 3 followed by the E x В-drift pushing plasma out of the higher magnetic field. However, a finite current density outside the current hole area causes a rotational transform that shortcut the separated charges and prevents the electric drift of the whole plasma.

### Spherical Tokamaks with High-β

The second extraordinary equilibrium is the equilibrium obtained in “spherical tokamaks” (STs) at record high values of plasma Д fi~. In this case, plasma pressure expels magnetic field from the centre of plasma and may form a diamagnetic well. The value of the Shafranov shift becomes comparable to the minor radius of tokamak, and the whole problem of solving the Grad-Shafranov equation could be separated [4.9] into a boundary value problem for plasma at the low field side, and a simplified one-dimensional description of the magnetic flux surfaces - at the high field side and the plasma center. Figure 4.6 shows an example of ST equlibrium with fi (0) -1 Figure 4.7 shows that a magnetic well is formed in this case. FIGURE 4.6 One-dimensional nature of the flux surfaces in the core of the plasma, together with the safety factor q and the flux function as functions of major radius R computed for STPP project [4.10]. FIGURE 4.7 Radial profile of the total equilibrium magnetic field for the case in Figure 4.6.

## REFERENCES

• 1. J.P. Freidberg, Ideal magnetohydrodynamics, Plenum Press, New York (1971).
• 2. V.S. Mukhovatov and V.D. Shafranov, Nucl. Fusion 11 (1971) 605.
• 3. V.D. Shafranov. Soviet Phys. JETP 6 (1958) 545.
• 4. H. Grad and H. Rubin, Hydromagnetic Equilibria and Force-Free Fields, Proceedings of the 2nd UN International Conf. on the Peaceful Uses of Atomic Energy, Geneva 1958 (Columbia University Press, New York, Vol. 31, 1959), 190.
• 5. T.S. Taylor. Plasma Phys. Control. Fusion 39 (1997) B47.
• 6. C.D. Challis et al., Plasma Phys. Control. Fusion 43 (2001) 861.
• 7. R.C. Wolf, Plasma Phys. Control. Fusion 45 (2003) Rl.
• 8. N.C. Hawkes et ah, Phys. Rev. Lett. 87 (2001) 115001.
• 9. S.C. Cowley et ah. Phys. Fluids B3. (1991) 2066.
• 10. H.R. Wilson et ah, The Spherical Tokamak Fusion Power Plant, 19th IAEA Fusion Energy Conference (IAEA, Lyon, France, 2002), No. FT/1-4.