Advanced Tokamak Stability Theory by Linjin Zheng

Author:Linjin Zheng
Language: eng
Format: epub, pdf
ISBN: 9781627054232
Publisher: IOP Publishing
Published: 2015-03-10T00:00:00+00:00

The solution of this equation is simply . The next order equation is as follows

(3.43)

Keeping the second term in the equation of this order is based on the consideration that it is of order , which is much larger than , as compared to the rest of the terms on the left-hand side. Noting that in the lowest order can be proved a posteriori to be independent of θ and , the field line average () of (3.43) yields . Therefore, the solution of (3.43) can be obtained as

(3.44)

We now move on to solve (3.32). In the lowest order one can obtain

(3.45)

Using (3.45), in the first order (3.32) can be solved

(3.46)

where . In the next order (3.32) gives the solubility condition for :

(3.47)

Here, equations (3.45)–(3.44) have been used, Q is defined in (3.10) in section 3.1 with the replacement

(3.48)

for the parallel inertia parameter inside the definition of M, and .

We need to solve (3.47) to obtain the solution for the resistive or inertia region. Letting

(3.49)

where , one can transform (3.47) to the Kummer equation [10]:

(3.50)

The solution, which remains finite and thus satisfies the boundary condition at , is

(3.51)

where , , Γ is the gamma function and is the Kummer function

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