Torque Balance
According to Equations (3.185) and (3.186),

(7.25) 
In other words, the shift in the EcrossB frequency at the rational surface,
, that develops in response to the electromagnetic
torque exerted at the surface by the errorfield (see Section 3.13), is mirrored by an equal shift in the ion fluid rotation frequency (as well as in the electron fluid rotation frequency). This is the
essence of the noslip constraint, (3.185), which follows from Equations (2.321) and (2.322)
because the torque modifies the EcrossB velocity at the rational surface, but does not affect the diamagnetic velocities.
Equations (3.190) and (3.191) yield
where we have set (because the errorfield is static). Here,
,
, and
.
Furthermore, the dimensionless neoclassical viscosity parameter
is
defined in Equation (2.204),
is the ion poloidal flowdamping timescale, the fraction of trapped particles [see Equation (2.202)], the ionion collision time [see Equation (2.21)], the safetyfactor profile (see Section 3.2), and
the magnetic shear at the
rational surface [see Equation (5.28)]. Moreover, denotes a Bessel function, whereas
and denote the th zeros of the and Bessel functions, respectively.
Figure: 7.3
Torque balance curve in a lowfield tokamak fusion reactor with
. The intersection of the curve with the horizontal line of height 2.0 shows the possible steadystate values of when
. There are three solutions. Two (indicated by the circle markers) are dynamically stable, and one (indicated by the cross marker) is dynamically unstable.

Let

(7.32) 
where
is the EcrossB frequency at the rational surface in the absence of the errorfield.
It is helpful to define
.
Equations (7.25)–(7.27) can be combined to give

(7.33) 
where .
Here, use has been made of the results [7]
Equations (7.9), (7.10), and (7.33) yield the torque balance equation [4,6]

(7.36) 
where
The lefthand side of Equation (7.36) represents an electromagnetic braking torque that develops at the rational
surface, and acts to halt the local electron fluid rotation. On the other hand, the righthand side of the equation
represents a viscous restoring torque that opposes any changes in the electron fluid rotation at the rational surface.
Equation (7.36) can be rearranged to give

(7.41) 
where

(7.42) 
Figure: 7.4
Torque balance curve in a highfield tokamak fusion reactor with
. The intersection of the curve with the horizontal line of height shows the possible steadystate values of when
. The circle and
cross markers indicate the highslip and lowslip solutions, respectively.

Figure 7.3 shows the torque balance function, , calculated for a lowfield tokamak fusion reactor with
.
The parameters used in the calculation are ,
, ,
, and
, as well as
, , and
. (See Tables 5.1 and 6.1.) Note that
and
are specified in Equations (7.16) and
(7.17), respectively. The parameter takes the value
. As indicated in the figure,
when
there are three possible values of the normalized EcrossB frequency at the rational
surface, , that satisfy Equation (7.41). However, as is
easily demonstrated, the
middle solution is dynamically unstable [4]. We thus conclude that there are two dynamically stable
branches of solutions to the torque balance equation, (7.36). On the highslip branch (i.e., the rightmost solution in the figure), the electron
fluid at the rational surface rotates with respect to the errorfield, generating a shielding current that suppresses
driven magnetic reconnection. On the lowslip branch (i.e., the leftmost solution in the figure), the
electron fluid rotation at the rational surface is arrested, there is no shielding current, and driven magnetic
reconnection proceeds unhindered.
Figure 7.4 shows the torque balance function, , calculated for a highfield tokamak fusion reactor with
.
The parameters used in the calculation are
, and
, as well as
, , and
. (See Tables 5.1 and 6.1.) Note that
and
are specified in Equations (7.22) and
(7.23), respectively. The parameter takes the value
.
As indicated in the figure, when
attains the critical value , the highslip and the dynamically
unstable solutions of the torque balance equation merge together and annihilate one another. For
,
only the lowslip solution branch exists. Thus, if the system is initially on the highslip solution branch, and the
errorfield amplitude is raised such that
exceeds the critical value 3.271, then there is a bifurcation
from the highslip to the lowslip solution branch. This bifurcation is associated with the sudden arrest of the
electron fluid rotation at the rational surface, the collapse of the shielding current, and the onset of driven magnetic
reconnection [4,6]. Of course, the bifurcation corresponds to the errorfield penetration phenomenon discussed in
Section 7.1.
Figure 7.5:
Analytically determined errorfield penetration threshold in a lowfield tokamak fusion reactor. The topleft panel shows the critical radial field needed to trigger penetration as a function of the unperturbed electron fluid rotation frequency. The bottomleft
panel shows
electron fluid rotation frequency before (solid curve) and after (dashed curve) penetration. The topright panel
shows the
shielding factors before (solid curve) and after (dashed curve) penetration. The bottomright panel shows the
reduced width of the driven locked island chain before (solid curve) and after (dashed curve) penetration.
