Fundamentals of Plasma Physics

12.5 Generalization of tearing to sheared magnetic fields 375

this equation provides the essential dynamics offluid vortices that are antisymmetric with
respect toxand driven by the torque associated with the non-conservative nature, i.e., curl,
of theJ×Bforce. This is essentially the same as Eq.(12.35) and the rest of the analysis is
the same as for the sheet current problem except that nowkyis used instead ofkandb′ 0 is
used instead ofB′ 0 y.Sinceb 0 (x)only differs fromBy 0 (x)by a constant,b′ 0 =By′ 0. Thus,
using Eq.(12.54) the growth rate will be

γ=0.55(∆′)^4 /^5

[

η

μ 0

] 3 / 5 [(

kyBy′ 0

) 2

ρ 0 μ 0

] 1 / 5

. (12.80)

The global system has to be periodic in both theyandzdirection in order for well-defined
kyandkzto exist. In particular, the physical arrangement and dimensions of the global
system determine the quantized spectra ofkyandkzand so determine the allowed planes
wherek·B 0 can vanish. As suggested earlier, the allowed planes can be considered
as ‘cleavage’ planes where the magnetic field can most easily become unglued from the
plasma.
Let us express this result in the context of toroidal geometry such as that of atokamak.
This is done by lettingBz 0 correspond to the toroidal fieldBφandBycorrespond toBθ
the poloidal field. The Alfvén time is now defined in terms ofBφas

τ−A^1 =

Bφ

a

√

ρ 0 μ 0

(12.81)

whereais the minor radius. The safety factor, a measure of the twist, is definedas

q=

aBφ

RBθ

(12.82)

so that

Bθ=

aBφ

Rq

(12.83)

and

By′ 0 →−

aBφ

Rq^2

q′. (12.84)

Thus it is possible to replaceky→m/aand

(

kyBy′ 0

) 2

ρ 0 μ 0

→

(

ma

Rq^2

q′

) 2

1

τA

. (12.85)

At the tearing layerk·B=0or

m

a

Bθ+

n

R

Bφ=0 (12.86)

so
q=−

m

n

(12.87)

and Eq.(12.85) becomes
(
kyB′y 0

) 2

ρ 0 μ 0

→

(

na

R

q′

q

) 2

1

τA

. (12.88)