THE GEOMETRY OF VORTEX FILAMENTS FOR MHD IN MINKOWSKI 3-SPACE

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1 International Journal of Pure and Applied Mathematics Volume 91 No , ISSN: (printed version); ISSN: (on-line version) url: doi: PAijpam.eu THE GEOMETRY OF VORTEX FILAMENTS FOR MHD IN MINKOWSKI 3-SPACE Nevin Gürbüz Mathematics and Computer Sciences Department Eskişehir Osmangazi University Eskişehir, TURKEY Abstract: In this article, we study geometrical constraints on the magnetic vortex filaments and Beltrami magnetic fields. Later, we give the relation between abnormalities and hydrodynamics for non-null curves. In last section, we derive geometrical constraints on MHD for null curves. 1. Introduction The geometry of vortex filaments has interesting applications in plasma physics [1], [2]. Recently, Schief denoted the effects of curvature and torsion of lines on the plasma physical phenomena [3], [4]. Andrade investigated the effects of curvature and torsion on vortex filaments in magnetohydrodynamic dynamos (MHD) with the aid of the Gauss-Mainardi-Codazzi equations [5], [6]. Bjorgum studied Beltrami magnetic fields and flows [7]. The paper is organized as follows: In Section 1, we give some definitions. In Section 2, we research geometrical constraints on the magnetic vortex filaments and Beltrami magnetic fields for non-null curves. In Section 3, we study the relation between abnormalities and hydrodynamics for non-null curves. In last section, we derive geometrical constraints on MHD for null curves. Received: October 28, 2013 c 2014 Academic Publications, Ltd. url:

2 220 N. Gürbüz Let γ(s) a unit speed curve in Minkowski 3-space, s being the arclength parameter. Consider the Frenet frame {t = γ,n,w} attached to the curve γ = γ(s) such that is t is the unit tangent vector field, n is the principal normal vector field and w is the binormal vector field. The Frenet -Serret formulas is given by t = ε 2 κn, n = ε 1 κt ε 3 τw, w = ε 2 τn (1) wheret,t = ε 1,n,n = ε 2,w,w = ε 3. is the semi Riemannian connection on M and κ = κ(s) and τ = τ(s) are the curvature and the torsion functions of γ, respectively The vector product given by [8] w t = ε 3 n, t n = ε 2 w, n w = ε 1 t (2) There are three types of vector fields in Minkowski 3-space: spacelike, timelike, and null. Let X be a three dimensional vector field in in Minkowski 3- space R 3 1. If X,X = 0 and X 0, X is called null vector. A null curve γ :[a,b] R 3 1 has acausal structureif all its tangent vectors arenull. Thereexists a local frame X = (t = γ,n,w), called Cartan equations satisfying t,t = 0, n,n = 0, t,w = 0,w,w = 1, t,n = 1. The vector product is defined : t w = t, t n = w, w n = n [9], [10]. Intrinsic derivatives of the Null Frenet type orthonormal triad are given as t = κw, n = τw, w = τt κn (3) where κ and τ are the curvature and torsion functions of γ, respectively. The null localized induced equation(lie) for null curves is given as [9], [10] δt = δ2 t δs 2 2. Geometrical Constraints on MHD Dynamos for Non-Null Curves δ δs, δ δn and δ show directional derivatives in the tangential, principal normal and binormal directions in E 3 1. {t,n,w} denote the directional derivatives of the orthonormal triad in the n-and w-directions in R 3 1. The gradient operator is grad = t δ δs +n δ δn +w δ (4) and ψ ns = n, δt δn, ψws = w, δt are the quantities which first introduced by Bjorgum in R 3 [6]. Using (1) and (2), we compute divt = t δ δs +n δ δn +w δ,t = ψ ns +ψ ws,

3 THE GEOMETRY OF VORTEX FILAMENTS where divn = curlt = divw = t δ δs +n δ,n δn +w δ t δ δs +n δ δn +w δ,w = κ+ w, δn = t δ δs +n δ δn +w δ,t n, δn = κw +ε 3 Ω s t (5) Ω s = curlt,t = ε 2 [ w, δt n, δt ] (6) δn is called the non-null abnormality of t-field curln = t δ δs +n δ δn +w δ,n = ε 1 ε 3 divwt+ε 2 Ω n n+ε 1 ε 2 ψ ns w where Ω n = curln,n = ε 2 (τ +ε 1 ε 3 t, δn ) (7) is the non-null abnormality of the n-field. curlw = t δ δs +n δ δn +w δ,w = ε 3 Ω w w ε 1 ε 3 ψ ws n+ε 1 ε 2 (κ+divn)t where Ω w = curlw,w = ε 3 (τ ε 1 ε 2 t, ) (8) δn is the non-null abnormality of the w-field. Ω s +Ω n ε 1 Ω w = 2ε 2 τ Theorem. For non-null curves, using the magnetic helicity equation, the componentsofthemagneticfieldofthevortexfilamentsundergeodesicmotionsω s = 0 are = 2ε 3τ B s κ Proof. We consider the magnetic field of vortex filament as B = B s t + w. Expansion of LHS of the magnetic helicity is given for non-null curves as following: B s t+ B s t+ w+ w=λ(b s t+ w)

4 222 N. Gürbüz δb s B = [ε 1 ε 2 (κ+divn) +ε 3 Ω s B s ]t+[ε 3 ε δ 1ε 3 ψ ws ε 3 δs ]n +[ε 3 Ω w +κb s ]w = λ(b s t+ w) ε 1 ε 2 (κ+divn) +ε 3 Ω s B s = λb s (9) δ ε 3 δs ε δb s 3 = ε 1ε 3 ψ ws (10) ε 3 Ω w +κb s = λ (11) From (9),(11) B s = ε 3Ω w λ κ = Ω s ε 3 λ B s κ+divn Magnetic vortex lines have relations divn κ 1. Thus (13) yields = Ω s ε 3 λ B s κ (12) (13) (14) For Ω n = 0, Ω s ε 1 Ω w = 2ε 2 τ (15) From (12),(14) and hypothesis theorem Ω s = 0, it is obtained 2ε 1 ε 2 ε 3 τ λ κ = ε 3 κ λ (16) With aid (16), λ 2 2τλ+ε 3 κ 2 = 0 (17) (15) gives (14) yields λ=τ ±τ 1 ε 3κ 2 τ 2 = 2ε 3τ B s κ (18) From (18), the ratio of the components of the magnetic field is constant. Using (10), we express δ δs = δb s (19)

5 THE GEOMETRY OF VORTEX FILAMENTS Maxwell equation in E 3 1 is given (19) and (20) give δb s,b = ε 1 δs +ε δ 3 = 0 (20) = ε 3 δ δs, B s = ε 1 δ and δ 2 ε 1 δs 2 +ε δ = 0 (21) (21) is a Minkowski Laplacian-like equation 2.1. Beltrami Flows and Fields for Non-Null Curves in Minkowski 3-Space Beltrami magnetic fields and flows have an important role in magnetodynamic [11 ], plasma physics, and other contexts [12]. For instance, Etnyre and Ghrist denoted geometric properties of Beltrami fields [13]. They also stated connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in three dimension [13,14]. Dombre and Urish mentioned the role of ABC fields on the Euclidean 3-torus with aid Beltrami fields [15]. For this reason, it is useful to study Beltrami magnetic dynamos as an application of the Gauss-Weingarten equations for non-null curves. Beltrami magnetic flow are defined v = mv (22) where v=v t t is non-null flow velocity in R 3 1. Together (5) and (2) give (23) yield v t (κw +ε 3 Ω s t) ε 2 w δv t δn +ε 3n δv t = mv tt (23) Ω s = ε 3 m δv t δn = ε 2κv t δv t = 0 As similar, non-null Beltrami dynamos are given B = mb (24)

6 224 N. Gürbüz where B = B(s)t Beltrami field. The expansion of the left-hand side (LHS) of (24) yields (25) yield three equations B(κw +ε 3 Ω s t) ε 2 w δb δn +ε 3n δb (26) shows, hydrodynamical motion is not geodesic. From (26), = mbt (25) δb κb ε 2 δn = 0 (26) Ω s = ε 3 m = constant (27) δb = 0 B(w) = e ε 2 (κ)(n)dn (28) Equation (28) denotes that the non-null Beltrami field B depends on just n- direction. 3. The Relation between Abnormalities and this Hydrodynamics for Non-Null Curves We can express the vortex filament rotation of non-null curves as following: v = Ω (29) where is the non-null gradient operator, Ω is defined as rotation or curl the of the velocity field and v = κw is the curvature binormal vector.. Our aim is to study the relation between abnormalities and this hydrodynamics for non-null curves. From the Eqs.(29), we have v, v = v,ω. (30) v, v = κw, (κw) (31) = κw,κ(ε 3 Ω w w ε 1 ε 3 ψ ws n+ε 1 ε 2 (κ+divn)t = κ 2 Ω w Together (30) and (31) give, w,ω = κ 2 Ω w (32)

7 THE GEOMETRY OF VORTEX FILAMENTS The non-null vorticity is computed as Ω = t ( δκ s w +κ s )+ w κ (δκ w +κ ) (33) = ε 1 ε 2 (κ+divn)t ε 3 (ε 1 ψ ws +κ s )n ε 1 ε 2 κτw (33) reduces to w,ω = ε 1 ε 2 ε 3 κτ = κτ (34) A comparison of Eqs.(32) and (34) shows that Ω w = τ (35) 4. Geometrical Constraints on the Magnetic Vortex Filaments for Null Curves We take the null gradient vector = grad = t δ δs + n δ δn + w δ. δ δs. We define the quantities ψ ns and ψ ws which first introduced by Bjorgum in R 3 ψ ns = n, δt, ψ ws = w, δt δn (36) We reproduce some formulas derived in this section, see [16]. divt = t,κw+ n, δt + w, δt = ψ ns +ψ ws, δn (37) divn = t δ δs +n δ δn +w δ,n = t,τw+ w, δn = w, δn (38) divw = t δ δs +n δ δn +w δ,w = κ+ n, δn (39) curlt = κt+nφ s curln = Φ n t (κ+divb)n ψ ns w curlw = ψ ws t+ndivn+φ w w where respectively, Φ s = curlt,t = w, δt n, δt, (40) δn

8 226 N. Gürbüz Φ n = curln,n = τ + t, δn Φ w = curlw,w = κ+ t, δn are called the abnormalities of t- field, n- field and w- field in R 3 1. The Gauss-Weingarten equations are δ t n w =, (41) (Φ n +τ) 0 ψ ws 0 Φ n +τ divn divn ψ ws 0 Using Φ n = 0, the Gauss-Weingarten equations Eq.(43) turns into t n w (42), (43) δκ = ψ2 ws +κ 2 + δψws δs κτ δτ = τ2 + δ(divn) δs +κτ +divnψ ws (44) δτ δs = κdivn 2τψ ws. Theorem. Using the magnetic helicity equation, the components of the magnetic field of the vortex filaments are obtained as constant for null curves. B s = τ κ Proof. The left expansion the null magnetic helicity is B s t+ B s t+ w+ w=λ(b s t+ w) B = B s ( κt+nφ s )+(w t) δb s +(Φ w w tψ ws +ndivn) t δ δs = ( κb s + δb s ψ ws δ δs )t+(b sφ s +divn )n+ Φ w w = λ(b s t+ w) We obtain PDE equations as following: B s Φ s +divn = 0 B s = Φ s divn Φ w = λ λ=φ w

9 THE GEOMETRY OF VORTEX FILAMENTS For Φ n = 0, δ δs δb s = ψ ws (κ+λ)b s (45) Φ s +Φ w = κ+τ Φ s = (κ+τ) λ = (κ+τ) λ B s divn (46) Φ = t ( δκ s w +κ s )+ w κ (δκ w +κ ) (47) = δκ δs t+κ2 w +ψ ws n divnt From Eq.(47), we have Φ w = κ = λ And Φ s = τ = τ B s divn Magnetic vortex lines have divn κ 1.Therefore it is obtained (48) Using (44) and ψ ws = 0 is expressed B s = τ κ δ δs δb s = 2κB s. (49) References [1] J. Brat, Plasma Loops in the Solar Corona, Cambridge University Press, [2] R. Ricca, The effect of torsion on the motion of a helical vortex filament, J. Fluid Mech., 237 (1994), 241. [3] W.K. Schief, Hidden integrability in ideal magnetohydrodynamics, The Pohlmeyer-Lund-Regge Model, Phys. Plasmas, 10 (2003),

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