A solvable model is developed for the linearized sausage mode within the context of resistive magnetohydrodynamics. The model is based on the assumption that the fluidmotion of the plasma is self‐similar, as well as several assumptions pertinent to the limit of wavelength long compared to the pinch radius. The perturbations to the magnetic field are not assumed to be self‐similar, but rather are calculated. Effects arising from time dependences of the z‐independent perturbed state, e.g., current rising as tα, Ohmic heating, and time variation of the pinch radius, are included in the analysis. The formalism appears to provide a good representation of ‘‘global’’ modes that involve coherent sausage distortion of the entire cross section of the pinch, but excludes modes that are localized radially, and higher radial eigenmodes. For this and other reasons, it is expected that the model underestimates the maximum instability growth rates, but is reasonable for global sausage modes. The net effect of resistivity and time variation of the unperturbed state is to decrease the growth rate if α≲1, but never by more than a factor of about 2. The effect is to increase the growth rate if α≳1.

1.
J. D.
Sethian
,
A. E.
Robson
,
K. A.
Gerber
, and
A. W.
DeSilva
,
Phys. Rev. Lett.
59
,
892
,
1790
(
1987
).
2.
J. E. Hammel and D. W. Scudder, in Proceedings of the 14th European Conference on Controlled Fusion and Plasma Physics, Madrid, 1987 (E.P.S., Petit-Lancy, Switzerland, 1987), Part 2, p. 450.
3.
I. D.
Culverwell
and
M.
Coppins
,
Phys. Fluids B
2
,
129
(
1990
).
4.
A. H. Glasser and R. A. Nebel, in Second International Conference on High Density Pinches, Laguna Beach, 1989 (AIP, New York, 1989), p. 226.
5.
M.
Coppins
,
Phys. Fluids B
1
,
591
(
1989
).
6.
M. Coppins and J. Scheffel, in Ref. 4, p. 211.
7.
T. D. Arber and M. Coppins, in Ref. 4, p. 220.
8.
F. L.
Cochran
and
A. E.
Robson
,
Phys. Fluids B
2
,
123
(
1990
).
9.
I. R. Lindemuth, in Ref. 4, p. 327.
10.
S. I.
Braginskii
,
Sov. Phys. JETP
6
,
494
(
1958
).
11.
M. G.
Haines
,
Proc. Phys. Soc. London
76
,
250
(
1960
).
12.
P.
Rosenau
,
R. A.
Nebel
, and
H. R.
Lewis
,
Phys. Fluids B
1
,
1233
(
1989
).
13.
L. R.
Lindemuth
,
G. H.
McCall
, and
R. A.
Nebel
,
Phys. Rev. Lett.
62
,
264
(
1989
).
14.
W. H.
Bennett
,
Phys. Rev.
45
,
890
(
1934
).
15.
L. Spitzer, Physics of Fully Ionized Gases (Interscience, New York, 1962).
16.
R. J.
Tayler
,
Rev. Mod. Phys.
32
,
907
(
1960
).
17.
To be precise, it has been well known for decades that the growth rate is of the order of kvA, but exact growth rates from ideal MHD for diffuse profiles have recently been calculated by
M.
Coppins
,
Plasma Phys. Controlled Fusion
30
,
201
(
1988
).
18.
M.
Coppins
,
I. D.
Culverwell
, and
M. G.
Haines
,
Phys. Fluids
31
,
2688
(
1988
).
19.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 634.
20.
S.
Lundquist
,
Ark. Fys.
5
,
297
(
1952
).
21.
I. D. Culverwell, Ph.D. thesis, Imperial College, London, 1989.
22.
I. D. Culverwell (private communication, 1989).
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