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Comment on “Symmetry analysis of the nonlinear MHD equations” by Wu
13 March 1995
PHYSICS
kI~SF.VIklI
LETTERS
A
Physics Letters A 198 ( 1995) 467,468
Comment on “Symmetry analysis of the nonlinear MHD equations” by Wu J.P. Goedbloed ‘, A, Lifschitz b A FOM-Instituut voor Plasmafysica, Nieuwegein. The Netherlands h Department ofMathematics, University of Illinois, Chicago, IL 60607, USA Received
24 August
1994; revised manuscript received 13 January 1995; accepted Communicated by C.R. Doering
for publication
18 January
1995
Abstract In a recent paper by Wu [ Phys. Lett. A 185 ( 1994) 3041 the nonlinear resistive MHD equations are analyzed with respect to their symmetry properties. Although the general approach of the paper is in the framework of existing methodology, the explicit solutions found are unacceptable from a physical standpoint because of their strong singularities. Moreover, the illustrations exhibiting magnetic islands are spurious.
In a recent
paper
by Wu [ l]
the nonlinear
resis-
tive MHD equations
are analyzed with respect to their symmetry properties. The analysis is restricted to incompressible quasi-stationary states with a magnetic field and flow velocity confined to the plane perpendicular to the direction of the assumed translation symmetry. For this case, the symmetry group is constructed and, subsequently, reduced to a one-parameter optimal system described by PDEs in two independent variables and, next, to a two-parameter optimal system described by ODES. A particular example of the latter system is chosen, which should illustrate the power of the method since it exhibits quite unexpected features like magnetic islands (Fig. 1 of Ref. [ 1 ] ) . The symmetry analysis given in the paper is in agreement with previously known results for nonlinear resistive MHD equations (see, for example, Ref. [ 21). At the same time, the particular solutions found in Ref. [ l] cannot be accepted from a physical standpoint because of their strong singularities. Moreover, the illustrations are misleading and wrong. 0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00074-7
The first step in the analysis boils down to the elimination of time in a trivial manner, $(x,y,t)
= f(f)
+$(x,y).
f(t)
= -Eat,
,~(x,y,t)
=g(f)
+$(x,y),
g(t)
arbitrary,
(1)
where the magnetic flux and the velocity stream function (indicated by the symbols u and w in Ref. [ 11) are denoted by + and x, respectively. This results in the basic equations p{02/V, ,$} = {V2& J},
(2a)
{$L 2:) = rlv2&
(2b)
where the density p and the resistivity v are just constant parameters and curly brackets denote Poisson brackets. The particular symmetry class chosen by Wu represents solutions in the form of functions of r (or 19) and X = 0 + y In r, where r and 8 are the polar coordinates of the poloidal x-y plane and y is a free pa-
468
J.P. Goedbloed, A. Lifschitz / Physics Letters A I98 (I 995) 467,468
rameter (denoted by B in Ref. [ I ] ) . Hence we look for solutions of the form 4 =a[U(X)
+lnrl,
f=/3[V(X)
+Inrl,
(3)
so that the PDEs (2) are reduced to nonlinear ODES for U(X) and V(X). In hydrodynamics solutions of this kind are known as spiral viscous flows (see, for example, Ref. [ 31, Section 8 1) Note that Eq. (2a) reduces to the usual GradShafranov equation V’$ = F(G) when inertia is neglected (p = 0). Having obtained a solution, Eq. (2b) is then easily solved separately. For the time being, let us keep both p and 7 though. Introducing new variables P z U’ and Q = V’, where the prime indicates differentiation with respect to X, and new parameters E = r]( 1 + 7’) /p, 6 = pj12/a2, the basic nonlinear ODES may be written as P’ + P’ + Y = &$P”f2PP’feP’2),
(44
Q = P + EP’,
(4b)
where v is an integration constant. This problem is easily solved when the r.h.s. of Eq. (4a) vanishes, i.e. when ~6 = 0, either because of the neglect of inertia (6 = O), as in Ref. [ 11, or because of the neglect of resistivity (E = 0). In that case, it is logical to require periodicity in X, so that v = n* where n is an integer (or a half-integer) and the solution becomes P = -n tan( nX - C). The solution to the original problem then becomes
x = b + g(t) +p[ln]rcos(nX-C>/
-Entan(nX-C)],
(5)
which is a slight generalisation of the expressions given in Ref. [ 11. The solutions (5) exhibit singular behaviour at the spirals 5 - nX - C - (I+ i) rr = 0, where the expressions for the magnetic flux, the field, and the current blow up as $ N ln],$l, B w l/S, j N l/S2, respectively. Since even the magnetic energy diverges, these solutions cannot be accepted as approximations to a
I
ii
/ /’
-0.1
-0.05
-0.1
Fig. I. Flux contours
0
0.05
0.1
for n = 1, y = 2, C = 0.
real physical situation where finite inertia or resistivity (~8 # 0 in Eq. (4) ) would limit the actual solution. Note that the level lines of 9 also correspond to spirals, Ircos[n(ylnr
+ 0) - C] / = const.
(6)
These do not coincide with those of X except at the singularities 5 = 0. It is clear that no closed islandtype contours are obtained. A particular example is shown in Fig. 1, which is to be compared with Fig. 1 of Ref. [ 11. Clearly, illustrations, given in Ref. [ 11, with the myriads of islands, are not representations of the solutions (5) but just the result of unresolved graphics. It is easy to show that islands are impossible because of the symmetry class chosen. (In fact, the few dots on the singular spiral in our Fig. 1 are a similar manifestation of unresolved graphics, which gives rise to the same intricate but irrelevant patterns when magnified.)
References [ 1 J J. Wu, Phys. Lett. A 185 (1994) 304. [2] J.C. Fuchs and E.W. Richter, J. Phys. A 20 (1987) 3135. [ 3 1 G. Birkhoff, Hydrodynamics, revised edition (Princeton Univ. Press, Princeton, 1960).
PHYSICS
kI~SF.VIklI
LETTERS
A
Physics Letters A 198 ( 1995) 467,468
Comment on “Symmetry analysis of the nonlinear MHD equations” by Wu J.P. Goedbloed ‘, A, Lifschitz b A FOM-Instituut voor Plasmafysica, Nieuwegein. The Netherlands h Department ofMathematics, University of Illinois, Chicago, IL 60607, USA Received
24 August
1994; revised manuscript received 13 January 1995; accepted Communicated by C.R. Doering
for publication
18 January
1995
Abstract In a recent paper by Wu [ Phys. Lett. A 185 ( 1994) 3041 the nonlinear resistive MHD equations are analyzed with respect to their symmetry properties. Although the general approach of the paper is in the framework of existing methodology, the explicit solutions found are unacceptable from a physical standpoint because of their strong singularities. Moreover, the illustrations exhibiting magnetic islands are spurious.
In a recent
paper
by Wu [ l]
the nonlinear
resis-
tive MHD equations
are analyzed with respect to their symmetry properties. The analysis is restricted to incompressible quasi-stationary states with a magnetic field and flow velocity confined to the plane perpendicular to the direction of the assumed translation symmetry. For this case, the symmetry group is constructed and, subsequently, reduced to a one-parameter optimal system described by PDEs in two independent variables and, next, to a two-parameter optimal system described by ODES. A particular example of the latter system is chosen, which should illustrate the power of the method since it exhibits quite unexpected features like magnetic islands (Fig. 1 of Ref. [ 1 ] ) . The symmetry analysis given in the paper is in agreement with previously known results for nonlinear resistive MHD equations (see, for example, Ref. [ 21). At the same time, the particular solutions found in Ref. [ l] cannot be accepted from a physical standpoint because of their strong singularities. Moreover, the illustrations are misleading and wrong. 0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00074-7
The first step in the analysis boils down to the elimination of time in a trivial manner, $(x,y,t)
= f(f)
+$(x,y).
f(t)
= -Eat,
,~(x,y,t)
=g(f)
+$(x,y),
g(t)
arbitrary,
(1)
where the magnetic flux and the velocity stream function (indicated by the symbols u and w in Ref. [ 11) are denoted by + and x, respectively. This results in the basic equations p{02/V, ,$} = {V2& J},
(2a)
{$L 2:) = rlv2&
(2b)
where the density p and the resistivity v are just constant parameters and curly brackets denote Poisson brackets. The particular symmetry class chosen by Wu represents solutions in the form of functions of r (or 19) and X = 0 + y In r, where r and 8 are the polar coordinates of the poloidal x-y plane and y is a free pa-
468
J.P. Goedbloed, A. Lifschitz / Physics Letters A I98 (I 995) 467,468
rameter (denoted by B in Ref. [ I ] ) . Hence we look for solutions of the form 4 =a[U(X)
+lnrl,
f=/3[V(X)
+Inrl,
(3)
so that the PDEs (2) are reduced to nonlinear ODES for U(X) and V(X). In hydrodynamics solutions of this kind are known as spiral viscous flows (see, for example, Ref. [ 31, Section 8 1) Note that Eq. (2a) reduces to the usual GradShafranov equation V’$ = F(G) when inertia is neglected (p = 0). Having obtained a solution, Eq. (2b) is then easily solved separately. For the time being, let us keep both p and 7 though. Introducing new variables P z U’ and Q = V’, where the prime indicates differentiation with respect to X, and new parameters E = r]( 1 + 7’) /p, 6 = pj12/a2, the basic nonlinear ODES may be written as P’ + P’ + Y = &$P”f2PP’feP’2),
(44
Q = P + EP’,
(4b)
where v is an integration constant. This problem is easily solved when the r.h.s. of Eq. (4a) vanishes, i.e. when ~6 = 0, either because of the neglect of inertia (6 = O), as in Ref. [ 11, or because of the neglect of resistivity (E = 0). In that case, it is logical to require periodicity in X, so that v = n* where n is an integer (or a half-integer) and the solution becomes P = -n tan( nX - C). The solution to the original problem then becomes
x = b + g(t) +p[ln]rcos(nX-C>/
-Entan(nX-C)],
(5)
which is a slight generalisation of the expressions given in Ref. [ 11. The solutions (5) exhibit singular behaviour at the spirals 5 - nX - C - (I+ i) rr = 0, where the expressions for the magnetic flux, the field, and the current blow up as $ N ln],$l, B w l/S, j N l/S2, respectively. Since even the magnetic energy diverges, these solutions cannot be accepted as approximations to a
I
ii
/ /’
-0.1
-0.05
-0.1
Fig. I. Flux contours
0
0.05
0.1
for n = 1, y = 2, C = 0.
real physical situation where finite inertia or resistivity (~8 # 0 in Eq. (4) ) would limit the actual solution. Note that the level lines of 9 also correspond to spirals, Ircos[n(ylnr
+ 0) - C] / = const.
(6)
These do not coincide with those of X except at the singularities 5 = 0. It is clear that no closed islandtype contours are obtained. A particular example is shown in Fig. 1, which is to be compared with Fig. 1 of Ref. [ 11. Clearly, illustrations, given in Ref. [ 11, with the myriads of islands, are not representations of the solutions (5) but just the result of unresolved graphics. It is easy to show that islands are impossible because of the symmetry class chosen. (In fact, the few dots on the singular spiral in our Fig. 1 are a similar manifestation of unresolved graphics, which gives rise to the same intricate but irrelevant patterns when magnified.)
References [ 1 J J. Wu, Phys. Lett. A 185 (1994) 304. [2] J.C. Fuchs and E.W. Richter, J. Phys. A 20 (1987) 3135. [ 3 1 G. Birkhoff, Hydrodynamics, revised edition (Princeton Univ. Press, Princeton, 1960).