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Stabilization of magnetohydrodynamic instabilities by force-free magnetic fields
Physica 100C (1980) 273-275 © North-Holland Publishing Company
STABILIZATION OF MAGNETOHYDRODYNAMIC INSTABILITIES BY FORCE-FREE MAGNETIC FIELDS IV. THE BOUNDARY CONDITIONS FOR A PLASMA-PLASMA INTERFACE
J. P. GOEDBLOED Association Euratom-FOM, FOM-lnstituut veer Plasmafysica, Ri/nhuizen, Nieuwegein, The Netherlands
Received 16 November 1979
An error in the boundary condition for a plasma-plasma interface used in a previous analysis is corrected. A new expression is derived which is symmetric in the variables of the two fluids.
In a series of papers [1-3] the stability of plasmas surrounded by force-free magnetic fields has been analysed. It was recently discovered by the author that one of the basic equations used in these papers is wrong, viz. the boundary condition for plasma-plasma boundaries as expressed by eq. (7) of ref. 1. Fortunately, the error in the boundary condition applied in these papers vanishes for the cases considered, viz. plane slab and cylindrical geometry. For toroidal systems the err.or would not have cancelled. Consider a system of two plasmas of different properties, like the system considered in refs. 1 - 3 which consisted of a dense plasma surrounded by a tenuous plasma with a force-free magnetic field created by parallel currents. The surface separating the two plasmas is a surface of discontinuity where the velocity u, the pressure p, and the magnetic field B may jump, but the jumps are restricted to satisfy the following conditions:
~[u] = 0,
(1)
n . [B~ = o,
(2)
~p -b ½B2~ = O.
(3)
n.
The brackets denote a jump of the quantities inside the bracket in the direction of the normal n to the surface: [[f~ -=f - f , where variables with a hat refer to 273
the exterior plasma (the plasma with the force-free field in the quoted papers). In linear stability theory the conditions ( 1 ) - ( 3 ) apply to both the equilibrium variables no, B0, and P0 (o0 = 0 then) and to the perturbed variables, which are all expressible in terms of the plasma displacement vector ~: u ~
aUat,
(4)
n ~- n o - - ( V ~ ) " n o + n o n 0 • ( V ~ ) " n O,
(s)
B~Bo+Q,
Q - V x (~XBo),
(6)
7P0 V" ~,
(7)
P ~ PO - ~" V P o --
where ~ indicates that only terms up to first order in are kept. These expressions also hold for the exterior plasma, where all variables should be supplied with hats. For a vacuum outer region; on the other hand, only the variable ~ ~ 8 0 + ~ would have physical signi ficance. Therefore, boundary conditions for the perturbed variables in a plasma-vacuum system would basically involve the plasma displacement ~ and the perturbation ~ of the vacumn magnetic field at the perturbed boundary. Since 0 is not defined in the vacuum only the jump conditions (2) and (3) remain, giving two relations between ~ and ~. One relation expresses the continuity of the normal component of
Z P. Goedbloed/Magnetohydrodynamic
274
the magnetic field perturbation: n o • VX
(8)
(~X I~0)= n 0 • O,
which may be transformed to an expression in terms of the normal components of ~ and 0 only [4] :
(89
B" V~jn- n " (V/~)"n~n = On"
The other relation expresses the continuity of the perturbation of the total pressure:
-VpoV' + Be" (Q +
VBo)=he" (0 + (9)
These equations were derived for a plasma-vacuum system [5], but they also hold for a plasma surrounded by a tenuous (P0 ~ 0) force-free region. I n the case of a plasma-plamaa boundary we have 0 - V X (~ X B0) so that ~ takes the place o f 0 as the basic variable. Now the jump condition (1) gives n o • ~ = n o • ~,
(10)
i.e. continuity of the normal displacement, whereas the jump condition (2) which led to the boundary condition (8) becomes superfluous as it is implied in eq. (10). For the pressure balance equation one has to add pressure terms of the exterior fluid to the boun-
\
instabilities
dary condition (9). One may then be tempted to infer from the continuity of the Lagrangian perturbation of the total pressure that the RHS of the boundary condition should be just the same expression as the LHS of eq. (9) with P0, B0, ~, Q replaced by ~0,/~0, ~, 0. In fact, this is the mistake we made in ref. 1 (eq. (7)), where we wrote ~. V/~0 rather than ~. V/~0 in the bracket of the RHS of eq. (9). (This was not a printing error.) The point is that although -TP0 V" ~ + B 0 • (Q + ~ . VBo) is the Lagrangian perturbation of the total pressure of the inner fluid, and - 7 # 0 V. ~ +/~0" (~ + ~" V/~0) is the Lagrangian perturbation of the total pressure of the exterior fluid, the two pressures are not evaluated at the same position since the tangential com ponents of ~ need not be continuous. For the sake of symmetry between inner and outer fluid it is therefore to be preferred to express the perturbation at the per. turbed boundary at the position r 1 = r 0 + (n o • ~)n 0 since the normal components of ~ are continuous. Writing , / = n o • ~n0 we notice that the perturbed quan. tries at r 1 may be expressed in terms of the perturbed quantities at r 0 (which, in turn, are known from the integration of the plasma dynamical equations) by means of a Taylor series, of which only terms up to first order are needed: f ( r 1) "~-"f ( r O) + rl * V f ( r O)
Io(,o) + A(ro) +
Vro(,o).
Hence, the expressions for the perturbed fluid and magnetic pressures at the position r I read: P(rl) ~ PO(rO) -- ~" 'VPo -- 7Po V" ~ ÷ n O• ~n0 • VP 0
\
= po(ro) - 'Po v" \1
where we have used the fact that the component of tangential to the boundary surface is perpendicular to the gradient of the equilibrium pressure, ~t "VP0 = 0, and
\
½B ,I)
\ \ perturbed surface unperturbed surface
F~.L
½So(,o) + Be" Q +,,o" e'o" v(½B .
Pressure balance at r 1 then gives the required boundary condition: -TP0 V" ~ + B 0 • Q + n o • ~n0 • V(½B0 2)
=-Ti~OV" ~ + ~0" ~ + no. ~no • V (½/~), (11)
Z P. Goedbloed/Magnetohydrodynamic
which is nicely symmetric now in the variables P0, B0, ~, Q, and/)0, B0, ~, ~. Notice that we now have appearing in the RI-IS instead of ~ as in eq. (9). For P0 = 0 we get eq. (9) back by the use of the relation t" [[V(p + ½B2)] = O,
(12)
when t is a unit vector tangential to the boundary surface. Eq. (12) is obtained from eq. (3) by realizing that the pressure jump condition applies anywhere in the boundary surface, so that the gradient of the total pressure cannot jump in a tangential direction. This is a crucial observation for toroidal geometries where tangential gradients of the equilibrium magnetic field appear. For highly symmetric systems, like plane slab and cylindrical geometry, such gradients do not occur so that the error in the boundary condition of ref. 1 could go undetected.
instabilities
275
Acknowledgement The author is indebted to Dr. J. Rein for constructive criticism. This work was performed as part of the research programme of the association agreement of Euratom and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelij~ Onderzoek" (ZWO) and Euratom.
References [1] [2] [3] [4]
J.P. Goedbloed, Physica 53 (1971) 412. J.P. Goedbloed, Physica 53 (1971)501. J.P. Goedbloed, Physica 53 (1971) 535. J.P. Freidberg and F. A. Haas, Phys. Fluids 16 (1973) 1909. [5] I.B. Bernstein, E. A. Frieman, M. D.Kruskal and R. M. Kuiszud, Proc. Roy. Soc. A224 (1958) 1.
STABILIZATION OF MAGNETOHYDRODYNAMIC INSTABILITIES BY FORCE-FREE MAGNETIC FIELDS IV. THE BOUNDARY CONDITIONS FOR A PLASMA-PLASMA INTERFACE
J. P. GOEDBLOED Association Euratom-FOM, FOM-lnstituut veer Plasmafysica, Ri/nhuizen, Nieuwegein, The Netherlands
Received 16 November 1979
An error in the boundary condition for a plasma-plasma interface used in a previous analysis is corrected. A new expression is derived which is symmetric in the variables of the two fluids.
In a series of papers [1-3] the stability of plasmas surrounded by force-free magnetic fields has been analysed. It was recently discovered by the author that one of the basic equations used in these papers is wrong, viz. the boundary condition for plasma-plasma boundaries as expressed by eq. (7) of ref. 1. Fortunately, the error in the boundary condition applied in these papers vanishes for the cases considered, viz. plane slab and cylindrical geometry. For toroidal systems the err.or would not have cancelled. Consider a system of two plasmas of different properties, like the system considered in refs. 1 - 3 which consisted of a dense plasma surrounded by a tenuous plasma with a force-free magnetic field created by parallel currents. The surface separating the two plasmas is a surface of discontinuity where the velocity u, the pressure p, and the magnetic field B may jump, but the jumps are restricted to satisfy the following conditions:
~[u] = 0,
(1)
n . [B~ = o,
(2)
~p -b ½B2~ = O.
(3)
n.
The brackets denote a jump of the quantities inside the bracket in the direction of the normal n to the surface: [[f~ -=f - f , where variables with a hat refer to 273
the exterior plasma (the plasma with the force-free field in the quoted papers). In linear stability theory the conditions ( 1 ) - ( 3 ) apply to both the equilibrium variables no, B0, and P0 (o0 = 0 then) and to the perturbed variables, which are all expressible in terms of the plasma displacement vector ~: u ~
aUat,
(4)
n ~- n o - - ( V ~ ) " n o + n o n 0 • ( V ~ ) " n O,
(s)
B~Bo+Q,
Q - V x (~XBo),
(6)
7P0 V" ~,
(7)
P ~ PO - ~" V P o --
where ~ indicates that only terms up to first order in are kept. These expressions also hold for the exterior plasma, where all variables should be supplied with hats. For a vacuum outer region; on the other hand, only the variable ~ ~ 8 0 + ~ would have physical signi ficance. Therefore, boundary conditions for the perturbed variables in a plasma-vacuum system would basically involve the plasma displacement ~ and the perturbation ~ of the vacumn magnetic field at the perturbed boundary. Since 0 is not defined in the vacuum only the jump conditions (2) and (3) remain, giving two relations between ~ and ~. One relation expresses the continuity of the normal component of
Z P. Goedbloed/Magnetohydrodynamic
274
the magnetic field perturbation: n o • VX
(8)
(~X I~0)= n 0 • O,
which may be transformed to an expression in terms of the normal components of ~ and 0 only [4] :
(89
B" V~jn- n " (V/~)"n~n = On"
The other relation expresses the continuity of the perturbation of the total pressure:
-VpoV' + Be" (Q +
VBo)=he" (0 + (9)
These equations were derived for a plasma-vacuum system [5], but they also hold for a plasma surrounded by a tenuous (P0 ~ 0) force-free region. I n the case of a plasma-plamaa boundary we have 0 - V X (~ X B0) so that ~ takes the place o f 0 as the basic variable. Now the jump condition (1) gives n o • ~ = n o • ~,
(10)
i.e. continuity of the normal displacement, whereas the jump condition (2) which led to the boundary condition (8) becomes superfluous as it is implied in eq. (10). For the pressure balance equation one has to add pressure terms of the exterior fluid to the boun-
\
instabilities
dary condition (9). One may then be tempted to infer from the continuity of the Lagrangian perturbation of the total pressure that the RHS of the boundary condition should be just the same expression as the LHS of eq. (9) with P0, B0, ~, Q replaced by ~0,/~0, ~, 0. In fact, this is the mistake we made in ref. 1 (eq. (7)), where we wrote ~. V/~0 rather than ~. V/~0 in the bracket of the RHS of eq. (9). (This was not a printing error.) The point is that although -TP0 V" ~ + B 0 • (Q + ~ . VBo) is the Lagrangian perturbation of the total pressure of the inner fluid, and - 7 # 0 V. ~ +/~0" (~ + ~" V/~0) is the Lagrangian perturbation of the total pressure of the exterior fluid, the two pressures are not evaluated at the same position since the tangential com ponents of ~ need not be continuous. For the sake of symmetry between inner and outer fluid it is therefore to be preferred to express the perturbation at the per. turbed boundary at the position r 1 = r 0 + (n o • ~)n 0 since the normal components of ~ are continuous. Writing , / = n o • ~n0 we notice that the perturbed quan. tries at r 1 may be expressed in terms of the perturbed quantities at r 0 (which, in turn, are known from the integration of the plasma dynamical equations) by means of a Taylor series, of which only terms up to first order are needed: f ( r 1) "~-"f ( r O) + rl * V f ( r O)
Io(,o) + A(ro) +
Vro(,o).
Hence, the expressions for the perturbed fluid and magnetic pressures at the position r I read: P(rl) ~ PO(rO) -- ~" 'VPo -- 7Po V" ~ ÷ n O• ~n0 • VP 0
\
= po(ro) - 'Po v" \1
where we have used the fact that the component of tangential to the boundary surface is perpendicular to the gradient of the equilibrium pressure, ~t "VP0 = 0, and
\
½B ,I)
\ \ perturbed surface unperturbed surface
F~.L
½So(,o) + Be" Q +,,o" e'o" v(½B .
Pressure balance at r 1 then gives the required boundary condition: -TP0 V" ~ + B 0 • Q + n o • ~n0 • V(½B0 2)
=-Ti~OV" ~ + ~0" ~ + no. ~no • V (½/~), (11)
Z P. Goedbloed/Magnetohydrodynamic
which is nicely symmetric now in the variables P0, B0, ~, Q, and/)0, B0, ~, ~. Notice that we now have appearing in the RI-IS instead of ~ as in eq. (9). For P0 = 0 we get eq. (9) back by the use of the relation t" [[V(p + ½B2)] = O,
(12)
when t is a unit vector tangential to the boundary surface. Eq. (12) is obtained from eq. (3) by realizing that the pressure jump condition applies anywhere in the boundary surface, so that the gradient of the total pressure cannot jump in a tangential direction. This is a crucial observation for toroidal geometries where tangential gradients of the equilibrium magnetic field appear. For highly symmetric systems, like plane slab and cylindrical geometry, such gradients do not occur so that the error in the boundary condition of ref. 1 could go undetected.
instabilities
275
Acknowledgement The author is indebted to Dr. J. Rein for constructive criticism. This work was performed as part of the research programme of the association agreement of Euratom and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelij~ Onderzoek" (ZWO) and Euratom.
References [1] [2] [3] [4]
J.P. Goedbloed, Physica 53 (1971) 412. J.P. Goedbloed, Physica 53 (1971)501. J.P. Goedbloed, Physica 53 (1971) 535. J.P. Freidberg and F. A. Haas, Phys. Fluids 16 (1973) 1909. [5] I.B. Bernstein, E. A. Frieman, M. D.Kruskal and R. M. Kuiszud, Proc. Roy. Soc. A224 (1958) 1.