Shielding of the error field by a liquid metal wall in tokamaks

It is shown that the error field in a tokamak can be shielded by a flowing liquid metal wall. In particular, a flowing liquid metal wall can prevent resonance amplification of the error field by the plasma near its no-wall stability limit. PACS numbers: 52.35.Py, 52.55.Fa, 52.55.Hc The resistive wall mode stability is an issue of concern for tokamak confinement [1]. Experimentally, it was observed that plasma rotation can stabilize resistive wall modes [2–4]. Theoretically, it was found that mode coupling to the sound wave and to the shear Alfvén resonances in rapidly rotating plasmas can lead to stabilization of the resistive wall modes [5–7]. Unfortunately, recent experiments in DIII-D show that the rotation damps even in the presence of continuous unbalanced neutral beam injection [3]. This indicates that there is a strong braking torque that slows down the plasma rotation. Theoretically, an error field amplification mechanism has been proposed to explain the rotation braking [8, 9]. Error-fieldinduced rotation braking has been verified experimentally [10]. When the error field is reduced, the plasma rotation is found to last longer. A review of various theoretical models has been given in [11]. In this letter, we propose an alternative approach to ameliorate the effects of the error field. We show that the error field, which is otherwise amplified, can be shielded by a flowing liquid metal wall. In particular, we prove that the strongly peaked torque barrier on the plasma rotation at the no-wall stability limit can be removed by a flowing liquid metal wall. To investigate the shielding effect from a liquid metal wall, we use a thin-wall approximation and model the error field by a thin current-carrying layer. Without loss of generality, the error-field current layer is assumed to be located immediately outside the liquid metal wall. The system is as follows: the plasma torus is surrounded by an inner vacuum region; the inner vacuum region is enclosed by a thin liquid metal wall; immediately outside the liquid metal wall is a thin error-field current layer, and outside the error-field current layer is an outer vacuum region, which extends to infinity. We use the coordinate system (ψ , θ , φ), where ψ represents the radial coordinate, θ is the poloidal angle and φ is the axisymmetric toroidal angle. The equilibrium magnetic field is expressed as B = ∇φ ×∇ψ + g(ψ)∇φ. The Jacobian of this coordinate system is J = 1/(∇ψ ×∇θ · ∇φ). Let ψa , ψb−, ψb andψb+ designate the radial locations for the interface between the plasma and the inner vacuum region, the interface between the inner vacuum region and the liquid metal wall, the interface between the liquid metal wall and the error-field current layer, and the interface between the error-field current layer and outside vacuum region, respectively. Since the error field is small compared with the toroidally axisymmetric equilibrium field, linear perturbation theory can be used to evaluate its effect. The solution procedure is as follows: we first describe the solutions in the vacuum and plasma regions. These solutions are then matched to each other across the various interfaces. Instead of calculating the torque on the plasma, we calculate the torque on the wall and the error-field current layer; this is opposite to the torque on the plasma that causes braking of plasma rotation. For the sake of conciseness, we outline the general solutions for the inner and outer vacuum regions simultaneously. The vacuum regions are described by the Laplace equation