Infinitedimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.


 Reynold Bell
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1 Infinitedimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach adviso: Pofesso O. I. ogoyavlenskij. In this pape we establish äcklund tansfomations between solutions of seveal cases of classical isotopic MHD and plasma equilibia and coesponding anisotopic equilibia. The tansfomations appea to be infinitedimensional and allow building solutions with vaiety of physical popeties. We also pesent a new infinitedimensional Lie goup of intinsic symmeties of anisotopic plasma equilibia equations, simila to those fo the isotopic case.
2 Contents. Contents..... Isotopic and anisotopic MHD plasma equilibia Isotopic plasma equilibium equations Anisotopic plasma equilibium equations äcklund tansfomation connecting solutions of MHD and AMHD equilibium systems of equations Discussion and impotant cases Infinitedimensional Lie goup of tansfomations of anisotopic MHD equilibia... 9 Refeences...
3 . Isotopic and anisotopic MHD plasma equilibia.. Isotopic plasma equilibium equations. The system of equations of ideal isotopic magnetohydodynamic (MHD) equilibium is div ( ρ V ) = 0, V ρ V (cul V) (cul ) gad P ρ gad = 0, (.) µ cul ( V ) = 0, div = 0. Hee V is plasma velocity, is magnetic field, ρ  plasma density, µ  magnetic pemeability coefficient, P plasma pessue. The coefficient µ can be emoved by scaling, theefoe fom now on we assume µ=. The most impotant eductions of this system ae listed below. The system ( cul ) = gad P, (.) div = 0 is a called Plasma Equilibium equations; it is a eduction of (.) fo the case of motionless plasma. The system of focefee plasma equilibium equations cul = α( x), (.3) div = 0, is in tun a eduction of (.) when (cul ) and ae collinea. The simplest case of (.3) is α(x)=0, which gives ise to pue magnetic field equations cul = 0, (.4) div = 0. All of the above eductions, as well as the oiginal MHD equilibium system, will be studied in this pape. 3
4 .. Anisotopic plasma equilibium equations. The system of equations of anisotopic magnetohydodynamic (AMHD) equilibium is div ( ρ V ) = 0, V ρ V (cul V) (cul ) gad P ρ gad = 0, (.5) µ cul ( V ) = 0, div = 0. The coefficient µ can be emoved by scaling, theefoe we put µ=. This system is analogous to (.) in eveything except the fact that the pessue P is now a 3 3 tenso. In the case of small Lamo adius, pessue tenso has only two independent components: p p P = I p + ( ), (.6) hee I is unit tenso. Using vecto calculus identities, one can epesent gadient of tenso pessue as follows: gad P = gad p + τ ( cul ) + τ gad + ( gad τ), (.7) p p τ =. Hence the equation of consevation of momentum in (.5) can be ewitten as ρ V (cul V) ( τ) (cul ) = V (.8) gad p +τ gad +ρ gad + ( gad τ). The motionless case of (.8) is ( τ) (cul ) = gad p + τ gad + ( gad τ). (.9) 4
5 . äcklund tansfomation connecting solutions of MHD and AMHD equilibium systems of equations. Conside the system (.). Let Ψ be a function constant both on the magnetic field lines and on the steamlines of plasma, and let the density ρ = ρ(ψ). Suppose {, V, ρ, P} is a solution of (.). We intoduce two vecto fields = f Ψ) ; V = ( Ψ) V. (.) ( g Let us now show that it is possible to make these new vecto fields satisfy the AMHD equilibium equations (.5). Indeed, div = f ( Ψ) div + gad f ( Ψ) = 0, (.) div V = g ( Ψ) div V + gad g( Ψ) V = 0, (.3) cul ( V ) = f g cul ( V ) + ( f g) gad Ψ ( V ) = 0, (.4) We have to satisfy the equiement div ( ρ V ) = 0, which can be done by choosing ρ = ρ (Ψ), and also the momentum consevation equation (.8) ρ V (cul V ) ( τ ) (cul ) = V gad p + τ gad + ρ gad + ( gad τ). Let us expand left and ighthand sides of (.6) sepaately. Fom (.), theefoe cul = f ( Ψ) cul + gad f ( Ψ), (.5) (.6) cul = f cul + ff gad Ψ, (.7) V culv = g V culv + gg gad Ψ. (.8) V The lefthand side of (.6) then takes the fom ( ρ gg V ( τ ) ff ) gad Ψ ρ g V (cul V) ( τ) f (cul ) + and the ighthand side: V gad p + τ gad + ρ gad + ( gad τ). To make them agee, we make assumptions τ = τ ), ( Ψ ρ g ( τ ) f = ρ, Then, afte using (.), the lefthand side of (.6) takes the fom ( ρ gg V ( τ ) ff ) gad Ψ V ( τ ) gad + ρ gad f P +, (.9) (.0), 5
6 o ( τ ) f V gad P + ρ + ρ ρ gg ( τ) f V ( τ ) ff gad Ψ. (.) The ighthand side ewites as f g V gad p +τ +ρ ( f τ + g V ρ ) gad Ψ. (.) Velocity and magnetic field depend not only on the magnetic suface vaiable Ψ, theefoe in ode to make the (.) and (.) equal we must fist demand so V P+ρ = M( Ψ ), p +τ f +ρ g V = N( Ψ), N ( Ψ) = M ( Ψ) ( τ ) f, V V P= M( Ψ) ρ, p = N( Ψ) τ f ρ g. Setting tems of (.) and (.) containing espectively velocity and magnetic field to be equal, we get ( ) f τ = τ ff, ρ g ρ =ρ gg ( τ) f. (.3) The fist condition fom (.3) esults in a fomula fo τ : 0 τ = C. f (.4) Using this, we calculate: ρ = C0 ρ / g, N ( Ψ ) = M ( Ψ) C0. (.5) The second equation of (.3) then becomes an identity, and thus we have established a tansfomation fom a class of solutions of isotopic MHD equilibium equations into a class of solutions of AMHD equilibium equations. This esult can be fomulated as a theoem. 6
7 Theoem.. Let {, V, ρ, P} be a solution to isotopic MHD equilibium equations (.) with popeties ρ = ρ(ψ), P= M( Ψ) ρ V, whee Ψ is a function constant both on the magnetic field lines and on plasma steamlines, and M is an abitay function of Ψ. Then the tansfomation = f( Ψ), V = g( Ψ) V, ρ p C0 ρ = g(ψ) = C 0 P + C C f ( Ψ ), 0, τ = + ( C0 f ( Ψ) ) defines a solution to anisotopic MHD equilibium equations (.5)(.7). Hee f(ψ), g(ψ) ae abitay continuously diffeentiable functions, and C 0, C ae abitay constants. 7
8 3. Discussion and impotant cases. Theoem (.) can be used to build solutions of AMHD equilibium and anisotopic plasma equilibium equations, stating fom isotopic MHD equilibium, plasma equilibium, and even vacuum magnetic field configuations. The theoems below descibe exact pocedues of such constuction. Theoem 3.. Let {, p} be an abitay solution to isotopic plasma equilibium equations (.). Then the tansfomations = f ( ), p C 0 τ =, f ( p) p = C0p + C + ( C 0 f( p) ) define a solution {, p, τ } to anisotopic plasma equilibium equations (.9). Hee f(p) is an abitay continuously diffeentiable function of pessue p; C 0, C ae abitay constants. Theoem 3.. Let be an abitay solution to the equations fo pue magnetic field in vacuum (.4): cul = 0, div = 0, and let p be a function constant on magnetic field lines. Then the fomulas = f ( ), τ p = C 0 f ( p) p = C + ( C 0 f( p) ) define a solution {, p, τ } to anisotopic plasma equilibium equations (.9). Hee f(p) is an abitay continuously diffeentiable function of p; C 0, C ae abitay constants. Theoems 3., 3. can be applied to constuct a wide vaiety of anisotopic plasma equilibium solutions of diffeent topologies. Indeed, with the help of Theoem 3., using any hamonic function φ: φ=0 we can build an anisotopic plasma equilibium. 8
9 4. Infinitedimensional Lie goup of tansfomations of anisotopic MHD equilibia. The system of anisotopic magnetohydodynamics (AMHD) equilibium equations can be epesented as (see (.5), (.7), (.8)) ρ V (cul V) ( τ) (cul ) = V (4.) gad p + τ gad + ρ gad + ( gad τ), div ( ρ V ) = 0, div = 0, (4.) cul ( V ) = 0, p p τ =. It is possible to show that these equations possess symmeties that ae vey simila to those fo isotopic plasma equilibia found by O. I. ogoyavlenskij in [], and ae indeed a natual genealization of them fo the anisotopic case. This new class of symmeties allows building ich families of AMHD equilibium solutions fom single known solutions, and the popeties of the new solutions can be chosen appopiately in quite wide ange, depending on an application. (4.3) (4.4) Theoem 4.. Let {V(), (), p (), ρ(), τ()} be a solution of (4.)(4.4), whee the density ρ() and the function τ() ae constant on both magnetic field lines and steamlines. Then {V (), (), p (), ρ (), τ ()} is also a solution of the same system, whee ρ ( ) = m ( ) ρ( ), τ ( ) = n ( ) τ( ), (4.5) ( ) b() τ() a() V () = () + V(), m() ρ() m() (4.6) a() b() ρ() () =± () + V(), n() n() τ() (4.7) p ( ) C p ( ) + ( C( ) ( ) ). (4.8) = Hee a(), b(), m(), n() ae abitay diffeentiable functions constant on both magnetic field lines and steamlines (i.e. on magnetic sufaces Ψ = const, if they exist); a() and b() satisfying a ( ) b ( ) = C = const. 9
10 Poof of the theoem. y the assumption of the theoem, functions a(), b(), m(), n(), ρ(), τ() ae constant on steamlines and magnetic field lines. Using this fact and the identities (4.5)(4.8), we can veify that equations (4.) hold: ( m b ) ( m b ) div( ρ V) = gad () ρ() () τ() + () ρ() () τ() div ( m a ) ( m a ) + gad () ρ() () V+ () ρ() () divv = 0, a() a() b() ρ() b() ρ() div =± gad + div + gad V+ div V = 0. n() n() n() τ() n() τ() The equation (4.3) is veified in the same manne: a() a() b() τ() b() ρ() cul ( V ) =± cul ( V ) m() n() m() ρ() n() τ() a () b () a () b () =± ( cul( V ) ) ± ( V ) gad = 0. m() n() m() n() This is tue because the vectos ( V and ) a () b () gad ae both othogonal to magnetic m() n() field lines and steamlines (which do not coincide), and theefoe paallel. (Note: in the case V ( V ) = 0, when the pevious statement is incoect, identically). We also have to veify that equation (4.) holds fo the tansfomed plasma paametes. It. can be ewitten as ρ V (cul V) ( τ) (cul ) V (4.9) =ρ gad ( τ) gad + gad p +. We denote A = ρ V, M = τ, whee A = b() τ() + a() ρ() V, (4.0) ( a b ) M =± () τ() + () ρ() V. (4.) It is known that fo any function f() and a vecto field a the following elation holds: a ( fa) cul ( fa) = f a (cul a) + gad f f ( a gad f) a. (4.) If we wite the equation (4.9) in tems of A and M and use (4.), it takes the fom A M A (cul A) M (cul M) = gad p + +. (4.3) Substituting hee the fomulae (4.5)(4.8), (4.0)(4.), we get 0
11 V V ( a b ) ρ (cul ) ( a b ) ( τ) (cul ) V + gad ( ) gad ( ) ( ) a b ρ a b τ (4.4) A M = gad C p ( ) + C +. Using the identity a () b () = C, the fact A M ( = C ρ V ( τ) ) and the fomula (4.) which is satisfied by {V(), (), p (), ρ(), τ()}, we tansfom (4.4) to the equivalent fom V V C gad p +τ gad +ρ gad + C gad ρ C gad ( ) τ ( V ( ) ) C ρ τ = gad C p ( ) + C +, which is an identity. Theoem is poven. (4.4)
12 Refeences.. O. I. ogoyavlenskij. Phys. Rev. E 6 (6), (000).. O. I. ogoyavlenskij. Phys. Lett. A 9, 56 (00).
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