Prepared for the U.s. Department of Energy, under Contract De-ac02-76ch03073 Princeton Plasma Physics Laboratory Princeton University, Princeton, New Jersey

A mechanism is proposed and evaluated for driving rotation in tokamak plasmas by minority ioncyclotron heating, even though this process introduces negligible angular momentum. The mechanism has two elements: First, angular momentum transport is governed by a diffusion equation with a non-slip boundary condition at the sepatratrix. Second, Monte-Carlo calculations show that energized particles will provide a torque density source which has a zero volume integral but separated positive and negative regions. With such a source, a solution of the diffusion equation predicts the on-axis rotation frequency Ω to be Ω = (4qmaxW J* ) (eBR3a2ne(2π)2)-1(τM/τE) where |J*| ≈ 5-10 is a nondimensional rotation frequency calculated by the Monte-Carlo ORBIT code. Overall, agreement with experiment is good, when the resonance is on the low-field-side of the magnetic axis. The rotation becomes more counter-current and reverses sign on the high field side for a no-slip boundary. The velocity shear layer position is controllable and of sufficient magnitude to affect microinstabilities. Introduction. Control of plasma rotation is an effective method for optimizing magnetic fusion plasmas. Differential rotation increases the stability of fine scale modes, which cause turbulent transport, as well as of large-scale distortions of the entire plasma. In the case of turbulent modes, differential rotation breaks up their structure and prevents growth [1, 2]. Large-scale modes acquire increased stability when, by differential rotation, magnetic distortions which are fixed in the frame of the plasma appear as time-dependent fluctuations in the frame of a conducting shell which surrounds the plasma. Consequently, with sufficient differential rotation, these fluctuations can not penetrate the shell, increasing the maximum pressure that can be stably confined [3, 4] . The physics of plasma rotation and the generation and transport of angular momentum density is therefore interesting both as a fundamental physics process and as the basis for a plasma control tool. Review articles by Ida [5] and Chan [6] give a comprehensive account of radial electric field and plasma rotation observations and a detailed discussion of the interaction of radiofrequency heating methods with plasma rotation, respectively. Rotational response of plasmas to angular momentum input is observed to have a momentum confinement time τM comparable to the observed energy confinement time τE (c.f. Sec. 4.2 of [5] and [7-9]) and an angular momentum diffusivity profile similar to the anomalous heat diffusvity profile. Recently, observations of Alcator C-Mod plasmas have discovered that plasma heating by the fast-wave, minority-ion-cyclotron process can cause an appreciable co-current toroidal rotation to develop in the vicinity of the magnetic axis, even though the heating method provides negligible angular momentum [10-12] Alcator C-Mod observations have further established that the central rotation velocity increases roughly linearly with the plasma energy content and that the rotation is strongly peaked toward the plasma center when the ion-cyclotron resonance is close to the magnetic axis. The rotation profile broadens as the cyclotron resonant surface moves to larger minor radius. The sense of rotation is co-current when the ion-cyclotron resonance lies on the low-field-side of the magnetic axis. The co-current rotation reported in ohmically-heated Alcator C-Mod plasmas [13] lies outside the scope of this work but could possibly be understood in terms of a modification of the no-slip boundary condition introduced below. How can a plasma develop an angular momentum content when none is supplied ? This paper proposes and evaluates a mechanism which resolves the apparent conflict. The argument has two parts. First, it is assumed that angular momentum transport is governed by a diffusion equation that has a no-slip boundary condition at the separatrix and a torque density source term as discused below. If the torque-density source term has two separated regions, one with positive and the other with negative torque density, but is constrained to have zero volumeintegrated torque, then the solution of the angular momentum diffusion equation will yield a finite central rotation rate The physics picture is that angular momentum generated in the outer part of the plasma diffuses to the surface and is lost faster than that supplied to the inner part. The second part of the argument rests on an evaluation of the torque density applied to the bulk plasma arising from the slowing down of ions accelerated by the minority-ioncyclotron process. The cyclotron acceleration process itself introduces no angular momentum. The motivating physics picture is that, as a result of finite banana widths and collisions, a fast ion which is born on an initial magnetic surface will slow down and return to the bulk plasma over a distribution of magnetic surfaces. This constitutes a radial current in the fast particles.A neutralizing radial current then flows in the bulk plasma which produces a jrBθR torque density. This is just the separated region of torque density needed to drive rotation. However this simple picture must be augmented by collisional transfer of mechanical angular momentum from the fast particles to the bulk plasma, which is of the same magnitude as the jrBθR torque density. Thus a precise calculation of all sources of torque density that rigorously accounts for angular momentum is required to determine whether torque density will be applied to the bulk plasma and to determine its sense. The Monte-Carlo code ORBIT [14, 15] has been modified to rigorously acount for collisional momentum exchange between energetic particles and a bulk plasma as well as providing for stochastic energization by perpendicular energy diffusion. The present work differs from previous theoretical models [16] in its rigorous accounting of angular momentum, the role of radial currents associated with energetic-ion banana diffusion, and the use of a diffusive transport equation to describe plasma response to torques. The manuscript first describes our models for fast wave propagation and ion-cyclotron heating. Next, we develop a solution to the angular momentum diffusion equation in general axisymmetric geometry that defines the integrated collisional and jrBθR torque densities that the ORBIT code must compute. Additions to ORBIT for this work are summarized. Results give plasma rotation curves parametrized by location of the ion-cyclotron resonance. A discussion of their sensivity to input parameters, correspondence to experiment, and a conclusion follow. Two-Component Plasma Model. The starting point for our model is to separate the plasma into two components: a high-energy tail created by minority ion-cyclotron heating whose evolution will be followed by the Monte-Carlo ORBIT code and a bulk plasma, which responds to applied torque density via a diffusive angular momentum transport equation with a model momentum diffusvity profile χM = a2qn/CnτM that spatially depends on q. Here τM denotes the momentum confinement time, which is taken comparable to the energy confinement time τE [5, 9]. The motivating physics comes from the observation that if one interprets the almost linear dependence of tokamak energy confinement time on a q-dependent diffusivity, then n≥2. We will focus on n=2 and for which C2 = 2(1+κ-2) qmax based on an analytic power balance model. Fast Wave Propagation. An important aspect of fast wave heating is that refraction focuses the waves onto the magnetic axis region and continues to maintain high wave intensities near the midplane for major radius values less than the magnetic axis. Calculations by the TORIC code [17] , portrayed in Fig. 1 illustrate this. Qualitatively, one can capture this aspect of fast wave heating by defining an intense wave region as portrayed in Fig. 1. Particles will undergo ion-cyclotron energization only if their orbits cross the cyclotron resonance surface within the intense wave region. This has the consequence of limiting the range of magnetic surfaces where ion-cyclotron heating can take place and generating regions of high rotational shear, especially when the cyclotron resonance lies to the high-field-side of the magnetic axis. The boundary ±zo of the intense field region has been taken to be zo = zmax R – Ra < zmax R – Ra R – Ra > zmax (1) with Ra the magnetic axis major radius and zmax = 7 cm for Alcator C-Mod example of Fig. 1. Ion-Cyclotron Heating. Two models for ion-cyclotron heating have been used. Model 1 instantaneously energizes a particle from the bulk plasma to a specified energy Eo. This initial creation is rigorously constrained to introduce zero net angular momentum and canonical angular momentum for each particle, as is appropriate for ion-cyclotron heating, and is effected by starting energetic particles with their banana tips lying on the cyclotron surface within the intense wave region.A distribution with off-midplane, banana-tip height with z, dN /dz = 1– (z / zo) 2 2zo – z 2 –1 / 2 is used so that only particles in the intense wave illustrated in Figure 1 are created. The energetic ions are then followed until they lose all their energy by the Monte-Carlo ORBIT code [14, 15], which includes ion-ion pitch-angle-scattering collisions [18] as well as ion and electron energy drag collisions . These collisions return energetic particles to the bulk plasma distributed over a region comparable to the banana full width about the originating magnetic surface. Our assumption that the fast waves transfer no net angular momentum to the energetic particles is rigorous for fast-waves with k|| = n/R = 0. For realistic values n ≈ ±10, it can be shown that angular momentum input remains negligible for a balanced n-spectrum. Ion-cyclotron Model 2 introduces ion-cycltron heating by giving a particle a stochastic kick in perpendicular energy ∆E⊥ each time it passes through the cyclotron resonance surface. The kicks are given by <(∆E⊥)2> = 2 E⊥ Es <∆E⊥> = Es (2) where Es = c⊥ 4π νoq Rc Eo Mp 1 / 2 2(E – E⊥) + T 1 / 2 F(z) αc α + αc 1 / 2 (3)