 Vol.2, No.2, 95-97 (2010) Natural Science http://dx.doi.org/10.4236/ns.2010.22015 Copyright © 2010 SciRes. OPEN ACCESS Magnetohydrodynamic equations for toroidal plasmas M. Asif Department of Physics, COMSATS Institute of Information Technology, Lahore, Pakistan; dr.muhammad.asif@gmail.com Received 23 November 2009; revised 4 December 2009; accepted 30 December 2009. ABSTRACT A set of reduced MHD equations is derived us-ing the standard energy balance equation. By applying assumption of internal energy, i.e. constuR 2, a set of reduced magnetohydro-dynamic equations are obtained for large aspect ratio, high  tokamaks. These equations in-clude all terms of the same or der as the toroidal effect and only involve three variables, namely the ﬂux, stream function and internal energy. Keywords: Magnetohydrodynamics; Aspect Ratio; Viscosity 1. INTRODUCTION Magnetohydrodynamic (MHD) instabilities , such as tearing modes, play an important role in plasma behav-iour of tokamaks. They may inﬂuence both the particle and the energy conﬁnement and are considered to be one of the main reasons for disruptions. Hence they have been studied by many authors [2-4]. The method of re-duced magnetohydrodynamic based on a large-aspect ratio expansion has provided a powerful method for lin-ear and nonlinear numerical computations. In previous studies [2-4], the density of plasma is as-sumed to be constant. This assumption is valid in cylin-drical geometry because the divergence of velocity is of the order 3 with  being the ratio of the minor ra-dius to the length of cylinder. Thus we can neglect the divergence of velocity in the continuity equation. By applying assumption of density, i.e. 2R const, Ren et al.  has derived the reduced MHD Equations in toroidal geometry for large aspect ratio, low  toka-maks. Since the divergence of velocity is of the order 2 in toroidal geometry the same order as the toroidal effect, and cannot be neglected. Hence the assumption of constant density does not agree satisfactorily with the continuity equation in toroidal geometry. The present paper extends this work to high  tokamaks. In this paper, by using a new assumption about inter-nal energy that is, 2Ru const a set of reduced MHD Equations is derived. Where R is major radius and u is an internal energy. The reduced MHD Equations involve three variables: the ﬂux, stream function and internal energy. The Equations can be used to calculate the nonlinear evolution of tearing modes for toroidal plas-mas. 2. REDUCED MHD EQUATIONS The basic MHD Equations are of the form 35.022dP Pdt (1) PJBt  (2) JB (3) BEt (4) EBJ (5) In MHD, if 1Pu, denote internal energy than standard energy Balance Equation has the form .0du udt (6) For the sake of simplicity, we adopt the quasi-cylin-drical coordinate ,,r with  being the toroidal angle. By assuming the inverse aspect ratio  to be a small quantity, the ordering of the high  tokamaks . 11,,OBrr 1,,,,,OBJ utR 2,,OBJ M. Asif / Natural Science 2 (2010) 95-97 Copyright © 2010 SciRes. OPEN ACCESS 96 where 00BRBR is the externally applied magnetic ﬁeld with 0R being the coordinate of the geometric center of plasma and 00RRBB. B is the toroidal magnetic ﬁeld produced by plasma current. J is the toroidal current and  the toroidal velocity. The sub-script  denotes perpendicular to . The u is the internal energy assumed to be of the order , for high  tokamaks. For simplicity we use B to express RB, B to RB, J to RJ,  to R. Thus B is of the order 1, J is of zeroth order while B and are of the order . Furthermore we assume  to be of the order 2 , for high  tokamaks . Introducing the vector potential A for magnetic ﬁel- d B, that is,  AeAeAA rr (7) Then B can be expressed as  rARArRBeAeARArrr1 (8) From Eq.(8) one can see that A is of zeroth order and rA and A are of the order 2 . Compared with equilibrium magnetic ﬁled expressed as B BB   , whereA is the ﬂux function. Later it will be shown that the effect of rA and A is of the order of 2 and can be neglected so that within our approximation B can be determined by A only. Substituting Eqs.(8 ) and (5) into Eq.(4) we obtain ,JBtAJBtA (9) where  is the gauge potential. Eq.(9) can be deduced as 42BOR Hence we can introduce stream function .U for  0,UB (10) where U is the order  and  can be expressed as 230RUOR  (11) where the relation 00BBR is used. The divergence of  is obtained as 2301...RU OR  (12) This Eq.(12) shows that the divergence of  is of the order 2 . Taking the  component of Eq.(9) and substituting Eqs.(10) and (12) into Eq.(9), we obtain the ﬂux evolution equation 220..ARBU OtR (13) Taking the curl of B, we get the expression of cur-rent BRJ  .2222 .OARR   (14) where the -derivatives in operator are of the higher order and can be dropped. From the internal energy evolution Eq.(6) by substituting Eq.(11), we obtain 2dRu Odt (15) The term on the right-hand side of Eq.(15) can be ne-glected because the term on the left-hand side is of ze-roth order. This indicates that if we assume the internal energy to be constant, Eq.(15) cannot be satisﬁed. We assume constuRuR  0202 (16) This assumption satisﬁes Eq.(1 5) and includes the tor-oidal effect. Taking the curl of momentum Eq.(2) after multiplying by 2R we can eliminate J and B. Then we get 2222000.1 .dW RRBJR uOdt RR(17) where 22,WR U 20..dRUdtt R The -derivatives in the Laplacian are also of the higher order and can be neglected. Using the Assump-tion of Eq.(16) we can directly obtain the parallel vis-cosity equation. Then a set of reduced MHD equations can be written as 20.ARBUtR (18)  .1. 202020uRRRJBRRdtdW (19) 20dRudt  (20) ARRJ  22. (21) 22,WR U (22) M. Asif / Natural Science 2 (2010) 95-97 Copyright © 2010 SciRes. OPEN ACCESS 97The Assumption 16 that the product of the square of the major radius and the internal energy is a constant, is quite stringent as taking this to be a constant, the ﬁrst driving term in the Grad-Shafranov equation becomes just proportional to the ﬂux derivative of the logarithm of the major radius, which shows a rather weak depend-ence. Therefore, this assumption seems to drastically narrow down the range of equilibrium conﬁgurations to which it is applicable. On the other hand, it has been observed  that a high density region appears near the inside limiter, which means that the density proﬁle at the inside and outside of plasma along a ﬂux surface is asymmetric. The pressure is calculated as the product of experimental temperature and density. Since the internal energy is related to the pressure as 1Pu, we can say that the pressure distribution is nonuniform poloidally and the pressure is higher at the inside of plasma than at the outside. The result is, however, consistent with our expectation. On the other hand, density and pressure proﬁle widths are clearly correlated . 3. CONCLUSIONS In summary, we derived the reduced MHD Equations (18-22) by using the Assumption 16 about the internal energy in a large aspect ratio limit. These equations in clude all terms of the same order as the toroidal effect and only involve three variables, namely the ﬂux, stream function and internal energy. 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