Now, according to the analysis of Section 2.25, the dominant terms in the previous equation are and . However, if we either take the poloidal flux-surface integral, , or the toroidal flux-surface integral, , of this equation then the term is completely annihilated (because is a single-valued function of and ), and the term is largely annihilated. In fact, as was demonstrated in the previous section, the residual term is quadratic in perturbed quantities, and localized in the vicinity of the rational surface. In these circumstance, it makes sense to include contributions from the other smaller terms in Equation (3.156). We shall calculate these contributions using the lowest-order (i.e., neglecting the contribution of the tearing perturbation) ion flow,Here, and are the ion poloidal and toroidal angular velocity profiles, respectively.
Taking , where the flux-surface integration operator is defined in Equation (3.131), we find that where use has been made of Equations (3.133) and (3.157). Note that the advective inertia terms in Equation (3.156) make no contribution to the previous equation because . Here, . Moreover, . Here, we have added a source term, , to the previous equation in order to account for the equilibrium flow. Let be the unperturbed (by the tearing mode) ion poloidal angular velocity profile. It follows that
Taking , we find that where use has been made of Equations (3.134) and (3.157). We have again added a source term, , to the previous equation in order to account for the equilibrium flow. Let be the unperturbed (by the tearing mode) ion toroidal angular velocity profile. It follows that
Equations (3.161) and (3.165) are subject to the boundary conditionsThe boundary conditions (3.166) merely ensure that the ion angular velocities remain finite at the magnetic axis. On the other hand, the boundary conditions (3.167) are a consequence of the action of charge exchange with electrically neutral particles emitted isotropically from the wall in the edge regions of the plasma [2,4,8]. Charge exchange with neutrals gives rise to dominant damping torques acting at the edge of the plasma that relax the edge ion angular velocities toward particular values. Moreover, the electromagnetic torques that develop at the rational surface are not large enough, compared with the charge-exchange torques, to significantly modify the edge ion angular velocities .