Plasma velocity in hydromagnetic dynamos

Plasma velocity in hydromagnetic dynamos Manuel Núñez Citation: Journal of Mathematical Physics 43, 3202 (2002); doi: 10.1063/1.1473679 View online: https://doi.org/10.1063/1.1473679 View Table of Contents: http://aip.scitation.org/toc/jmp/43/6 Published by the American Institute of Physics Plasma velocity in hydromagnetic dynamos Manuel Nu ´n ˜eza) Departamento de Ana ´lisis Matema ´tico, Universidad de Valladolid, 47005 Valladolid, Spain 共Received 12 February 2002; accepted for publication 25 February 2002兲 Hydromagnetic dynamos are plasma configurations generating for some time an exponentially increasing magnetic field. By using a number of functional inequalities, we estimate the rate of increase of magnetic energy in terms of the plasma resistivity and diferent norms on the plasma velocity. Our bounds are proved to be optimal as far as the powers of the relevant magnitudes are concerned. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1473679兴 I. INTRODUCTION A hydromagnetic dynamo in a plasma is a configuration allowing for a finite time an exponential growth of the magnetic field. The behavior of the main magnitudes in an incompressible plasma is governed by the magnetohydrodynamic 共MHD兲system: the velocity u, magnetic field B, kinetic pressure p, viscosity ␯ and resistivity ␩ satisfy, after the usual normalizations, ⳵ u ⳵ t⫽ ␯ ⌬u⫺u•ⵜu⫹B•ⵜB⫺ⵜp⫺ⵜ 冉 B2 2 冊 ,共1兲 ⳵ B ⳵ t⫽ ␩ ⌬B⫺u•ⵜB⫹B•ⵜu,共2兲 ⵜ•u⫽ⵜ•B⫽0. 共3兲 The MHD system, for any boundary conditions allowing no input of energy from the outside, is dissipative 共see e.g. Ref. 1兲. Therefore, any growth of magnetic energy must ultimately be done at the expense of the kinetic one, i.e., of the plasma velocity. Once this velocity is taken for granted, the magnetic field is governed by the induction equation 共2兲, and the magnetic energy by the integral identity obtained making the scalar product of 共2兲and B: 1 2 ⳵ ⳵ t 冕 ⍀B2dV⫽ ␩ 冕 ⍀ ⌬B•BdV⫹ 冕 ⍀B•ⵜu•BdV⫺ 冕 ⍀u•ⵜB•BdV.共4兲 If we assume u•n 兩 ⳵ ⍀⫽0共i.e., the fluid does not cross the boundary兲, the last integral vanishes. As for the term ␩ 冕 ⍀ ⌬B•BdV⫽⫺ ␩ 冕 ⍀ 兩 ⵜB 兩 2dV⫹ ␩ 2 冕 ⳵ ⍀ ⳵ B2 ⳵ nd ␴ , provided there is no input of magnetic energy from the outside, 冕 ⳵ ⍀ ⳵ B2 ⳵ nd ␴ ⭐0, 共5兲 yields the fundamental energy inequality a兲Electronic mail: [email protected] JOURNAL OF MATHEMATICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 2002 32020022-2488/2002/43(6)/3202/5/$19.00 © 2002 American Institute of Physics 1 2 ⳵ ⳵ t 冕 ⍀B2dV⭐⫺ ␩ 冕 ⍀ 兩 ⵜB 兩 2dV⫹ 冕 ⍀B•ⵜu•BdV.共6兲 Condition 共5兲holds 共with an equality兲for Dirichlet (B 兩 ⳵ ⍀⫽0) or perfect conductor (B•n 兩 ⳵ ⍀ ⫽0; (ⵜ⫻B)⫻n 兩 ⳵ ⍀⫽0) conditions. We will assume either periodic boundary conditions in a box ⍀with 冕 ⍀udV⫽ 冕 ⍀BdV⫽0,共7兲 or u•n 兩 ⳵ ⍀⫽B•n 兩 ⳵ ⍀⫽0, 共8兲 in a smooth N-dimensional domain ⍀. Thus we will take 共6兲as the starting inequality. The first term on the right-hand side of 共6兲accounts for the diffusive effects of the resistivity, while the second is an advective term showing the transport of the magnetic field by the flow. In fact, in ideal plasmas ( ␩ ⫽0) the magnetic field lines are transported by the plasma as material points and the field strength may be enhanced by this process. From here one may ignore the diffusive term and bound the advective one by 冏 冕 ⍀B•ⵜu•BdV 冏 ⭐1 2 储 ⵜu⫹共ⵜu兲t 储 ⬁ 冕 ⍀B2dV,共9兲 where 储储 ⬁means the maximum norm and ( )tthe transposed matrix 共see, e.g., Ref. 2兲. Therefore, the growth parameter ␥ satisfies ␥ ⭐1 2 储 ⵜu⫹共ⵜu兲t 储 ⬁.共10兲 This estimate goes back to Backus.3Thus the maximal exponential growth rate does not exceed the largest eigenvalue of the strain matrix 1 2(ⵜu⫹(ⵜu)t). This elementary inequality has some merits: the main one is that it does not depend on the resistivity and therefore it holds even when ␩ →0. A velocity configuration yielding a dynamo even when ␩ →0关inf ␩ →0 ␥ ( ␩ )⬎0兴is called a fast dynamo;4this is an extensively studied subject. On the minus side, we first note that anything involving the gradient of the velocity is somewhat unsatisfactory. This is so because in many turbulent flows there exist sharp changes in the velocity vector, whereas the velocity size remains moderate. Indeed, on general principles one may reject an extremely large plasma velocity, but there is no physical reason to exclude rapid variations of the flow: thus any norm on the velocity itself may be much smaller than the norm of the gradient. Moreover, the maximum norm is the worst possible: it could happen that the plasma remains almost quiescent except for a tiny portion which alone ensures that the maximum of the strain matrix is large. One does not expect the magnetic energy of the whole domain to be governed by a minute portion of the plasma. We will see that 共9兲may be significantly improved. II. THE MAIN ESTIMATES Certain subspaces of the Sobolev space H1(⍀) possess the property that 储 f 储 H1⭐k 储 ⵜf 储 2, i.e., the L2-norm of fis dominated by the norm of its gradient. These are the so-called Poincare ´ inequalities. One of the most general descriptions of spaces where one of these inequalities holds is as follows 共see Refs. 1 and 5兲: let pbe a continuous seminorm 关i.e., a continuous norm, except for the fact that p(f)⫽0 does not imply f⫽0兴on H1(⍀) such that for every constant function 3203J. Math. Phys., Vol. 43, No. 6, June 2002 Plasma velocity in hydromagnetic dynamos g⫽0, p(g)⫽0. Then any subspace Hof H1(⍀), such that p(f)⫽0 for all f苸H, satisfies a Poincare ´inequality. Among the many examples of such seminorms, we will use the following ones: p共f兲⫽ 冏 冕 ⍀fdV 冏 ,共11兲 p共f兲⫽ 冕 ⳵ ⍀ 兩 f•n 兩 d ␴ .共12兲 Equation 共11兲covers periodic problems because of the zero mean condition 共7兲, while 共12兲covers the remaining cases, since 共8兲holds. That the seminorm pof 共12兲is continuous follows from the fact that the trace of any function f苸H1(⍀) at the boundary belongs to L1( ⳵ ⍀)关and even to L2( ⳵ ⍀)]. Our main tool will be a weak version of the Gagliardo–Nirenberg inequality 共Ref. 6, pp. 65–68兲: denoting as usual by 储储 pthe norm in Lp(⍀), 储 f 储 p⭐C 储 f 储 H1 ␶ 储 f 储 2 1⫺ ␶ ,共13兲 where Cis a constant depending only on the domain, ␶ ⫽(N/2)⫺(N/p). This holds provided p ⭓2, (N/2)⫺(N/p)⬍1, i.e., p⬍2N/(N⫺2). Thus, for N⫽3, 2⭐p⬍6; for N⫽2, any p⭓2is admissible. Since 冕 ⍀B•ⵜu•BdV⫽⫺ 冕 ⍀B•ⵜB•udV,共14兲 by the inequality of Cauchy–Schwarz 冏 冕 ⍀B•ⵜu•BdV 冏 ⭐ 冕 ⍀ 兩 B 兩兩 ⵜB 兩兩 u 兩 dV⭐ 储 B 储 p 储 ⵜB 储 2 储 u 储 q,共15兲 for any positive p,qsuch that 1/p⫹1/q⫽1 2; hence p,q⭓2. By 共13兲, 储 B 储 p⭐C 储 B 储 H1 共N/2兲⫺共N/p兲 储 B 储 2 1⫺共N/2兲⫹共N/p兲⫽C 储 B 储 H1 N/q 储 B 储 2 1⫺共N/q兲,共16兲 provided p⬍2N/(N⫺2), i.e., q⬎N. Thus, 冏 冕 ⍀B•ⵜu•BdV 冏 ⭐C 储 B 储 H1 N/q 储 B 储 2 1⫺共N/q兲 储 ⵜB 储 2 储 u 储 q.共17兲 Let us use now the Poincare ´inequality, written as 储 B 储 H1⭐k 储 ⵜB 储 2. We have 冏 冕 ⍀B•ⵜu•BdV 冏 ⭐CkN/q 储 ⵜB 储 2 1⫹共N/q兲 储 B 储 2 1⫺共N/q兲 储 u 储 q.共18兲 Let us denote r⫽1 2⫺N 2q.共19兲 3204 J. Math. Phys., Vol. 43, No. 6, June 2002 Manuel Nu ´n ˜ez We may write the right-hand term as CkN/q共 储 ⵜB 储 2 2兲1⫺r共 储 B 储 2 2兲r 储 u 储 q⫽共 ␣ 储 ⵜB 储 2 2兲1⫺r共共CkN/q 储 u 储 q兲1/r ␣ ⫺(1⫺r)/r 储 B 储 2 2兲r,共20兲 where ␣ is a positive constant to be determined later. By using the classical inequality xry1⫺r⭐rx⫹共1⫺r兲y, for x,y⬎0共which amounts to the convexity of the exponential function兲,wefind 冏 冕 ⍀B•ⵜu•BdV 冏 ⭐共1⫺r兲 ␣ 储 ⵜB 储 2 2⫹rC1/rkN/qr ␣ ⫺(1⫺r)/r 储 u 储 q 1/r 储 B 储 2 2.共21兲 Take now ␣ ⫽ ␩ /(1⫺r). Then the term in ⵜBcancels with the dissipative term in 共6兲, and we are left with 1 2 d dt 储 B 储 2 2⭐r共1⫺r兲(1⫺r)/r共CkN/q ␩ ⫺(1⫺r) 储 u 储 q兲1/r 储 B 储 2 2.共22兲 Therefore, if there exists a magnetic dynamo of exponential growth rate ␥ , for any q⬎N, ␥ ⭐2r共1⫺r兲(1⫺r)/r共CkN/q ␩ ⫺(1⫺r) 储 u 储 q兲1/r,共23兲 where ris given by 共19兲.Cand kare universal constants. The estimate improves with large qand ␩ , and becomes singular as ␩ →0orr→0共i.e., q→N兲. For q→⬁it becomes ␥ ⭐1 2C ␩ ⫺1 储 u 储 ⬁ 2,共24兲 which improves the Backus bound 共10兲in the sense that it does not need the velocity gradient, although the resistivity occurs. The estimate 共23兲is satisfactory in the sense that it involves an integral norm of the velocity and therefore it is a measure of its mean size: it shows that the dynamo cannot be governed by what happens in small regions of the plasma, although these may be relevant in the process of stretching which is basic in the dynamo process. However, the physically most important norm of the velocity is the kinetic energy 储 u 储 2, which is not reached by 共23兲. For N⫽2 it lies at the lower limit and the constants blow there; for N⫽3 it is well beyond reach. To see that this is a physical fact and not merely the result of poor bounds, we will prove that 共23兲is a sharp inequality as concerns the order of the magnitudes. III. COUNTEREXAMPLES FOR LOWER ORDER NORMS We will consider an initial condition formed by velocity and field depending only on the radius, and radially directed. Then B•ⵜuis also radially directed and the term B•ⵜu•Bis precisely 兩 B 兩 2 兩 ⵜu 兩 . Specifically, assume B⫽B(R)er,erthe unit radial vector, Bdecreasing linearly from B⫽hat r⫽0toB⫽0atr⫽L:B(r)⫽h⫺hr/Lfor r苸关0,L兴,B(r)⫽0 for r⬎L. Take u ⫽B. These magnitudes are not really smooth, as they fail to be differentiable at r⫽0 and r ⫽L, but they can be uniformly approximated by smooth functions such that the values of all the integrals tend to the respective values for our chosen functions. First, since the Jacobian in dimension Ndepends on rlike rN⫺1, the norm 储 u 储 qbehaves like hLN/q, and 储 B 储 2like hLN/2. 兩 ⵜu 兩 is identical to h/Lfor r苸关0,L兴, and zero otherwise; thus 冕 ⍀ 兩 B 兩 2 兩 ⵜu 兩 dV⫽hL⫺1 冕 ⍀ 兩 B 兩 2dV⬃hL⫺1h2LN⫽h3LN⫺1.共25兲 On the other hand, 3205J. Math. Phys., Vol. 43, No. 6, June 2002 Plasma velocity in hydromagnetic dynamos ␩ 冕 ⍀ 兩 ⵜB 兩 2dV⬃ ␩ h2LN⫺2,共26兲 so that the behavior of the right-hand side of Eq. 共6兲, as a function of hand L,is ⫺ ␩ h2LN⫺2⫹h3LN⫺1.共27兲 Therefore, any exponential growth rate ␥ should be of the order of 共27兲divided by h2LN, i.e., ␥ ⬃⫺ ␩ L⫺2⫹hL⫺1,共28兲 while 储 u 储 q⬃hLN/q. Assume q⬍N, and take ssuch that 1⬍s⬍N/q. Choose h⫽L⫺s. Then, for Lsmall, ␥ ⬃⫺ ␩ L⫺2⫹L⫺1⫺s⬃L⫺1⫺s.共29兲 While 储 u 储 q⬃LN/q⫺stends to zero with L, ␥ →⬁. Thus there is no possible bound of ␥ in terms of 储 u 储 q. For q⫽N, we must avoid the possible indetermination in 共29兲occurring for h⫽L⫺1. Therefore, we take h⫽L⫺1logL⫺1. For Lsmall enough, ⫺ ␩ L⫺2⫹L⫺2logL⫺1⬃L⫺2logL⫺1,共30兲 whereas 储 u 储 q⬃logL⫺1. Since obviously, for any power n, 共logL⫺1兲nⰆL⫺2logL⫺1,共31兲 when L→0, there cannot be any bound of ␥ in terms of any power of 储 u 储 q. Logically the method fails for q⬎N, because any attempt of setting h⫽L⫺swould yield a negative power of Lat both sides; we could choose an adequate power on the right-hand side to make the magnitudes comparable. Notice that our test functions are localized in a neigborhood of 0and therefore satisfy our boundary conditions. IV. CONCLUSIONS Defining hydromagnetic dynamos as plasma configurations producing an exponential growth of the magnetic field for some time, it is desirable to bound the possible growth rates in terms of the size of the plasma velocity. Classical inequalities involve the maximum norms of the velocity gradient, which are unsuitable for several reasons. We prove a bound of the growth rate by a power of the Lq-norm of the velocity and the conductivity, for any qstrictly larger than the space dimension N. The estimate blows up in the limit q⫽Nas well as in the ideal limit of zero resistivity. It is shown by examples that there cannot be analogous bounds for q⭐N. ACKNOWLEDGMENT This work was partially supported by the Ministry of Science of Spain under Grant No. BMF2000-0814. 1R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics 共Springer, New York, 1988兲. 2M. Nu ´n ˜ ez, J. Math. Phys. 38, 1583 共1997兲. 3G. Backus, Ann. Phys. 共N.Y.兲4,372共1958兲. 4S. Childress and A. D. Gilbert, Stretch, Twist and Fold: the Fast Dynamo,共Springer, New York, 1995兲. 5J. Deny and J. L. Lions, Ann. Inst. Fourier 5, 305 共1954兲. 6V. G. Maz’ja, Sobolev Spaces 共Springer, Berlin, 1985兲. 3206 J. Math. Phys., Vol. 43, No. 6, June 2002 Manuel Nu ´n ˜ez