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2 Review of Linear Algebra 21 Matrices The mxn matrix A is de ned as r all a12 aln A1 a a a Ar 7 21 22 Zn 7 2 7 c c c A7 7 7A1 A2 Am r am1 am2 amn An Where aJ is the i jth element ofthe matrixA i 1 2 mj 1 2 n A a1 a2 am is the 1m ith row vector i 1 2 m an a A 21 is the mx1jth column vectorj 1 2 n amjj The A matrix is sguare ifm n smrlretric ifm n and aJ a for all ij diagonalifmnandauOforalli j1J 2n an identig matrix denoted by In ifm n and aJ 0 for all i j ij 1 2 n 1 for al ij an anullmatrix denoted by 0 ifaJ 0 for all i 1 2 mj 1 2 n The Amatrixis ifaJ E R i 1 2 mj 1 2 n The Amatrixis complex ifaJ a 1 2 B or 316 Ri 1 2 mj 1 2 n Ifall elements 1J 1 2 B ofmatrix A are replaced by their complex conjugate mu 1 2 B the resulting matrix is called the complex conjugate of A 22 Matrix Operations Addition Let A and B be mxn matrices Then C A B ifand only if cJ aJ bu i 1 2 mj 1 2 n Where C is a mxn matrix Multiplication Let A be a mxk matrix and B be a kxn matrix Then D A B if and only if dJ 21 as b5 for all i 1 2 mj 1 2 n Where D is a mxn matrix M In general B BA AB0 ispossib1evvithA 0andB 0 Example 2 4 72 A B 1 2 1 72 CDCE ispossib1evvithC 0andD 0Examp1e 2 3 1 1 72 1 C D E 6 9 1 2 3 2 Transpose The nxm matrix B is the transpose of the mxn matrix A if and only if b aJ for alli1 2 mj12 n ThisisdenotedbyBAT Note ATT A ABTATBT ABTBTAT 23 Linear Dependence of Vectors Let X x1 x2 xm be a mvector in Rm either a lxm row vector or a mxl column vector De nition A set ofvectors x1 x2 x in Rm is linearly dependent ifthere exist numbers A1 A2 A not all zeros such that EFL n 7 x 0 De nition A set of vectors is linearly independent if it is not linearly dependent De nition A vector yis a linear combination ofthe vectors x1 x2 xn ify 21 n 0 x for some 16 Ri l 2 n De nition A set of linear combinations of the vectors x1 x2 xn is the set H n 0 a E i n The set V is said to be spanned or generated by the vectors x1 x2 x The set V is a vector space 24 Rank of a Matrix Let A be a mxn matrix De nition The row rank of A is the maximum number of linearly independent rows of A The column rank of A is the maximum number of linearly independent columns of A Theorem 1 Rank theorem T e row rank ofA is equal to the column rank ofA It is called simply the rank ofA denoted by rankA It follows that rankA g minm n rankA E g minrankA rankB De nition a mxm matrix A is sipgular if rankA lt m nonsing ar if rankA m 25 Basis Let V be a subset ofthe vector space Rm Let r rankV the maximum number of linearly independent vectors in V De nition A basis of V is a set of r linearly independent vectors in V that spans V De nition The dimension of V denoted by dimV is the number of vectors in any basis of V e rowvectors of A A39 1 rowAE1V m 0 A Ar E R 0 E R i l 2 m De nition The row space of a mxn matrix A denoted by rowA is the vector space spanned b 2 m De nition The column space of a mxn matrix A denoted by colA is the vector space spanned by the columnvectors ofA AJ j l 2 n colA 211 n 71AJ AJE E Rm 116 Rj l 2 n A basis of col A is given by all the linearly independent column vectors of A dim col A is the number of linearly independent column vectors of A De nition The null space of a mxn matrix A denoted by nullA is nullA X A X 0 X E R a subspace of R Theorem 2 Fundamental theorem Let A be a mxn matrix Then dimnulA n rankA The theorem states that the number of linearly dependent columns of A dimnullA is equal to the number of columns of A n minus the number of linearly independent columns of A rankA 26 Solving Linear Equations Consider a system of m equations in n unknown A X b where A E R is a mxn known matriX X E R is a nxl vector of unknown variables andb E Rm is a mxl known vector Let Al b be a mxnl augmented matriX obtained by adding the column vector b to the columns of A Theorem 3 The system of equations A X b has a no solution if and only if rankA lt rankAl b b a unique solution if and only ifrankA rankAl b n c an in nite number of solutions if and only if rankA rankAl b lt n Proof Let r rankA It follows that rankAl b 2 rankA r a Case where rankAl b gt rankA A basis ofcolAl b is A1E A b where Af are the linearly independent columns ofA And a basis ofcolA is A1 Af Assume that there eXists a solution It follows that b E colA implying that there eXist B39s such that b 21 V B A But this means that the columnvectors A1 A b are linearly dependent a contradiction This implies that no solution can eXist b Case where rankAl b rankA Let A1 A be a basis for colA where A1 AE are the linearly independent columns ofA Note that A1 AS is also a basis for colAl b This implies that b is a linear combination of A1 A and thus of A1 An It follows that there eXist y39s such that b EFL V y A implying the eXistence of a so ution bl Case where rankA n Assume that there eXist 2 solutions X1 and X2 It follows that AX1 b and A X2 b This implies that AX1 X2 0 ie that X1 X2 6 nullA But dimnulA n rankA 0 yielding X1 X2 This proves the eXistence ofa unique AE solutron b2 Case where rankA lt n It means that dimnullA 2 1 Let y X0 6 nullA for all y E R v 0 It follows that A y X0 0 Let X3 be a solution A X3 b It follows that A X3 y X0 b implying that X3 y X0 is also a solution for all y 0 This proves the eXistence of an in nite number of solutions 27 Implications Let A X b where A is a mxn known matriX and b a mx 1 known vector We have rankA g minm n l A x b has a unigue solution only ifm 2 n ie only ifthe number of equations is at least as large as the number of unknowns 2 lfm lt n ie ifthe number ofequations is less than the number ofunknowns then either there is no solution when b is not a linear combination of the columns of A ie rankAl b gt rankA or there is an in nite number of solutions when b is a linear combination of the columns ofA ie rankAl b rankA lt n 3 If m lt n and b 0 then there is an in nite number of solutions as rankAl b rankA lt n 4 A X b has at most one solution if rankA n g m It has one solution ifb is linear combination of the columns of A It has no solution if b is not a linear combination of the columns of A 5 A X b has at least one solution if rankA m g n It has one solution if rankA rankAl b m n It has an in nite number of solutions if rankA rankAl b m lt n 28 Inverse of 3 Matrix Let A be a mxm matrix De nition The matrix Bis the inverse ofA ifA B B A lm It is denoted by B A39l lfA391 exists then A is said to be invertible A has at most one inverse A391391 A A B391 B391 A391 AT 71A71T Theorem 4 A matrix A is invertible if and only if it is nonsingular ie if rankA m oof Assume that rankA m It follows that dimcolA m implying that A is a basis for Rm Consider the mxl unit vector e e1 e2 emT where eJ l wheni wheni j Since e E R there exist bJg s such that e Zj Af bJ 1 1 2 m Consider the mxm matrix B bJ ij l 2 m Note that Im e1 e2 em It follows that Im A B ie that A is invertible and that B A39l Assume that A is invertible ie that there exists a matrix B such that A B lm We have rankA B s rankA g m Since ranklm m it follows that rankA m implying that A is nonsingular 29 Determinants Let A be a mxm matrix De nition The determinant of A denoted by detA is given by detA 2H m aJ cu for alli l 2 m EFL Vm aJ cu for allj l 2 m where cJ is the i jth cofactor of A de ned as cJ l detsubmatrix of A obtained after deleting the ith row and jth column Note The mxm matrix C cu ij l 2 m is called the cofactor matrix ofA detA detAT detA B detA detB 211 m aJ ckJ detA ifki 0 if k 1 Theorem 5 A is invertible if and only if detA 0 If A is invertible then detA ldetA391 Assume that A is invertible Then A A391 lm It follows that detA detA39l detlm 1 This implies that detA 0 and that detA39l ldetA Assume that A is not invertible Then A is singular and rankA r lt m A basis of colA is 1 given by the mxr matrix It follows that there exists a rxm matrix B0 such that I I 0 B0 A 0 B0 Tlus canbe alternatively written as A 0 0 B where B B B1 1 I 0 I 0 being any mrxm matrix Thus detA det 6 0 detB But det 3 0 implying that detA 0 210 Inverting a Matrix Let A be a mxm matrix Theorem 6 If A is invertible then A391 ldetA CT where C is the cofactor matrix of A Proof A ldetA CT ldetA EFL m aJ ckJ ldetA detA Im 1 211 Solving a System of Linear Equations Consider the system of m equations in m unknowns A x b where A is mxm matrix b is a mx 1 vector Theorem 7 If rankA m the system of equations A x b has for solution x A391 b Proof A391 exists ifrankA m Premultiplying by A391 gives A391 A x A391 b But A391 A lm by de nition implying that x A391 b Solving the system of equations A x b can then be done rst by evaluating A and then by multiplying A39l b Theorem 8 Cramer s Rule The solution of the system of equations is given by x detAdetA i l 2 m where A the matrix A a er replacing the ith column by the vector b Proof x A391 b ldetA CT b Let e be the mxl unit vector Where e e1 e2 emT satisfying e 1 When i j and eJ 0 When i j We have x eT x eT ldetA CT b ldetA 21 ck bk detAdetA 212 Eigen Analysis Let A be a mxm real matrix Consider the system of equations A v 7 v Where 7 is a scalar and v is a mxl column vector satisfying v 0 This can be alternatively Written as AAlm v0v 0 This system of equations has a solution for v if and only if dimnullA A Im gt 0 This corresponds to rankA 7 1 lt m implying detA m This equation is calledthe characteristic equation of A It is a mth order polynomial equation in 7 2121 Eigenvalues De nition The solutions of detA 7 1m 0 for 7 are the characteristics roots or eigenvalues of A There are m not necessarily distinct characteristic rots of any mxm matrix A Denote them by 1 7M2 7m The characteristic roots can be real or complex when 7 aJ i b Where i 1 2 7L1 0 A2 Let A be the mxm diagonal matrix of characteristic roots of A 0 0 7 The m ofA is traceA EFL m an EFL m The determinant of A is detA DH m 2 It follows that rank of A is equal to the number of nonzero characteristic roots 2122 Eigenvectors De nition The ith eigenvector of A is the mx 1 column v satisfying AA 1m v 0i 1 2 Thus v E nullA 7 1 Let V V1 V2 vm be the mxm matrix ofeigenvectors ofA Note V can be a complex matrix Theorem 9 If the roots 7 7L2 km of the mxm matrix A are distinct then its eigenvector matrix V is nonsingular Proof Assume that the mxm matrix V is a singular matrix ofrank r lt m Let v1 V2 v denote the r linearly independent columns of V It follows that V can be expressed as a linear combination ofthe vectors in v1 V2 v ie that there exists scalars c39s such that vm EFL V c v Where c1 c2 c 0 We have AM vm A vm A EFL V c v EFL V c A v EFL V c J v Also A vm AM EFL V c v Subtracting these two representations of7 1 VM yields EFL V c 7 7 1 V 0 This can be written as 7 7 7 0 0 c 0 7 77 0 c v1 v2 Vr 2 39H 2 0 Al 0 0 7 7 7 1 or By de nition the matrix V1 V2 v has rank r and thus has a null space ofdimension 7 1 7 Am 0 7 2 77m 0 zero Also under distinct roots the rxr matrix 0 0 7 7 7 has rank r and thus a null space of dimension zero This implies from Al that c1 c2 c 0 a contradiction Thus V must be nonsingular Note lfv E nullA 7 1 then v v E nullA 7 lm for all y E R v 7 0 Thus ifv is an eigenvector of A so is v v This suggests de ning a normalized eigenvector as the eigenvector satisfying VT V l which corresponds to choosing the eigenvector of unit length De nitions Two mxl column vectors V1 and V2 6 Rm are said to be orthogonal ifvlT V2 0 A mxm real matrix A is orthogonal ifit satis es AT A39l Theorem 10 If the mxm matrix A is symmetric with distinct eigenvalues 7 1 7 then the normalized eigenvector matrix V v1 V2 Vm is orthogonal ie satis es VT V39l Proof UsingA V 7 v we have VT A V 7 VT v and VT A V 7 VT V after switching subscripts for all ij l 2 m But the symmetry ofA implies that VT A V VT A v It follows that 7 7 VT V 0 Given distinct roots 7 7 7 for i j implying that V V 0 for 1 7 The V39s being normalized eigenvectors it follows that VT V 7 rm This yields VT 7 VI M Theorem 10 remains valid even if the eigenvalues of A are not distinct For example it is true that any real smmetric matrix A always has a linear independent set of eigenvectors such that the eigenvector matrix V is orthogonal and satis es VT V39l Let A be a mxm real matrix The eigen decomposition ofA yields A V V A where V is the matrix of normalized eigenvectors of A and A is the diagonal matrix of eigenvalues of A If A has distinct eigenvalues then theorem 9 implies that it can be written as A 7 V A V39l And ifA is real and symmetric then A can be written as A A VT This is called the sing ar value decomposition of A Theorem 11 If A is a symmetric mxm matrix then all eigenvalues of A are real m Assume that 7 a i b is a complex eigenvalue ofA where i l 2 with associated complex eigenvector v Let W a i b be the complex conjugate of 7 and xquot be the complex conjugate ofx Then we have A v 7 v and A vquot N vquot It follows that vquotT A v 7 vquotT v and vT A v W vT v Subtracting the latter expression from the former yields vquotT A v vT A vquot 7 7 quotvT vquot But the symmetry of A implies that vquotT A v vT A vquot 0 Thus 7 7 quotvT vquot 0 Since v 0 it follows that vT vquot gt 0 implying that 7 W ie that 7 is real lfA is a symmetric matrix then the eigenvectors ofA can also be taken to be real 213 Principal Components Consider the linear mode A x where y is a mx 1 vector of m observations on a quotdependent variable A is a mxn real matrix of m observations on n quotexplanatory variables and x is a nx 1 vector of parameters From the eigen decomposition of the nxn matrix AT A we have AT A V V A where V is the nxn matrix of eigenvectors of AT A and A is the nxn diagonal matrix of eigenvalues of AT A The matrix AT A being real and symmetric we have AT A V A VT where VT V Given V VT In it follows that the linear model can be alternatively written as yAVVT x Z where B A V and z VT x The columns of the mxn matrix B are called principal components Note their special property A1 0 0 0 A 0 BTBVTATAVVTVAVTVA 2 0 0 A This implies that the principal components the columns of B are orthogonal to each other B T B 1 0 for all i j and that the length of the ith principal component is related to the corresponding characteristic root 7 since B T BiE 7 i l 2 n 214 Quadratic Forms Let A be a mxm real symmetric matrix De nition uT A u is aguadratic form where u E Rm is a mxl vector De nitions The matrix A is negative semide nite ifuT A u g 0 for all mxl vectors u E Rm negative de nite ifuTA u lt 0 for all mxl vectors u E Rm u 0 positive semide nite ifuT A u 2 0 for all mxl vectors u E Rm positive de nite ifuT A u gt 0 for all mxl vectors u E Rm u 0 Theorem 12 The mxm symmetric matrix A is positive negative de nite if and only if all eigenvalues of A are positive negative It is positive negative semide nite if and only if all eigenvalues of A are nonnegative nonpositive Proof The matrix A being real and symmetric we have A V A VT Where V is the mxm nonsingular matrix of eigenvectors of A and A is the mxm diagonal matrix of eigenvalues of A ItfollowsthatuTAuuT VAVTu LettinghVTuWehaveuTAuhT A h Where h 111 h2 hmT E Rm is amx1 vector But hT A h 2quotth 7 It follows that uT A u 2quotth 7 0 for allhifandonlyiflx1 0 for alli lt0 for allh 0ifandonlyif7h lt0 foralli 20 for allhifandonlyiflxI 20 for alli gt0 for allh 0ifandonlyif7h gt0 foralli Finally V being nonsingular h 0 if and only ifu 0 De nition Let A be a mxm matrix A successive principal minor of order k is the determinant of the matrix obtained a er deleting from AroWs k1 k2 m and columns k1 k2 m A principal minor of order k is the determinant of the matrix obtained a er deleting mk rows and the same numbered columns from A Determinant Rules The matrix A is positive de nite if and only if all its successive principal minors of order k are positive gt 0 k 1 m The matrix A is negative de nite if and only if all its successive principal minors of order k multipliedby 1 give a positive number gt 0 k 1 m The matrix A is positive semide nite if and only i all its principal minors of order k are non negative 2 0 k 1 m The matrix A is negative semide nite if and only if all its principal minors of order k multiplied by 1 give a nonnegative number 2 0 k 1 m 215 Dynamics De nition The modulus of a 1 2 b is W a2 b2 2 Where a b E R De Moivre formula my a 4 2 b w cost e 4 2 sint 9 Where 9 arctgba or ba tg9 Let A be a mxm rea1 matrix Let 7 a 1 2 b be the ith eigenvalue ofA i 1 2 m We Will assume that the eigenvalues of A are distinct De nition The dominant root of A is the eigenvalue of A With the largest modulus Without a loss of generality we order the roots according to their modulus 17M gt M21 gt gt pm where M is the dominant root Consider the d apnic system xt A X Where XL is a mxl vector ofstate variables at time t A is a mxm rea1 matrix and t 1 2 Note Assuming that I A is a nonsingular matrix the above system is equivalent to the more general dynamic sys em Y b A Yul 10 where b is a mxl vector and xt yt 1 A 1 b Noting that the vector 1 A 1 b can be interpreted as an equilibrium value for yt it follows that xt yt 1 A 1 b is simply measuring the deviation of yt from its equilibrium value From eigen analysis we have A V V A where V is a mxm eigenvector matrix ofA and A is a mxm diagonal eigenvalue matrix of A Given distinct eigenvalues theorem 9 implies that V391 exists yielding A V A VI The dynamic system can altematively written as xt tx0 whereA AAAttimes for allt 1 2 NotethatA AAAVAV391VA V391 v A V391 v At v4 It follows that xtVA V391x0 or V39lxtA V391x0 for all t 1 2 Let zt V391 xt where zt Zn z2t zmT is a mxl vector of canonical variables at time t The dynamic system expressedin terms of the canonical variables zt can then be written as ztAt 39Zg where xt V z for all t 1 2 Since A is a diagonal matrix we have my 0 0 At 0 My 0 0 0 0 Amy implying that all dynamics can be captured by the following equation zt M zm i 12 m for allt 1 2 Case 1 lkll gt 1 the dominant root is outside the unit circle If is real then lime A1 is unbounded If is complex then My W l cost 9 1 2 sint 9 which is also unbounded as t gt 00 Thus unless 210 0 Zn becomes unbounded over time whenever the dominant root is loutside the unit circle lkll gt 1 Case 2 lkll lt 1 all roots are in the unit circle as lM lt 1 for all i l 2 m a A1 is w This implies that lime All 0 yielding lime zt 0 for all 20 and thus lime xt 0 for all x0 b A1 is complex Given MY lkll cost e 1 2 sint 9 this implies that lime Mt 0 39el 39 ime zt or al 20 and thus lime xt or 1x0 Thus when all roots are in the unit circle ml lt 1 for all i all canonical variables and state variables approach their long run equilibrium 0 over time Since xt V zt it follows that Q state variables xt must approach their long run eguilibrium 0 over time The path to long run equilibrium is either exponential if M is real and positive oscillatory if M is real and negative or cyclical if M is complex where the period of the cycle is 21r9 9 arctgba being expressed in radians Case 3 lMl l the dominant root is on the unit circle a ll 1 This implies that lime Alt 1 and lime M 0 fori 2 3 m It follows that lime Zn 210 and lime zt 0 for i 2 3 m This yields lime xt V lime zt V1E 210 showing that the long run equilibrium of XL is proportional to the eigenvector V1 corresponding to the unit root ll b M 1 This implies that lime Neil and lime M 0 fori 2 3 m It follows that lime zu u and lime zt 0 for i 2 3 m This 39elds lime xt V lime zt VIE rzm showing that the long run dynamics ofxt is oscillatogy c M a 1 2 b Uni a2 b2 2 1 This implies that lime Mt lime M1 l cost 9 1 2 sint e hme cost e 1 2 sint e and hme M 0 for i 2 3 m It follows that the long run equilibrium ofzu Alt zm is cyclical while lime zt 0 for i 2 3 m Since lime xt V lime zt V1E lime Zn the long run dmamics ofxt is also cyclical lfe is expressed in radians the period ofthe cycle is 27r9 where 9 arctgba w The above analysis can be generalized to the situation where the eigenvalues of A are not distinct in which case the matrix A is said to be defective The generalization involves replacing the diagonal matrix A by a block diagonal matrix called the Jordan tri the matrix V by a matrix of generalized eigenvectors which is non singular Then it can be shown that the results obtained in case 1 and case 2 above still hold Case 3 becomes more complex PRL 94 055002 2005 PHYSICAL REVIEW LETTERS week ending 11 FEBRUARY 2005 Most Electron Heat Transport Is Not Anomalous It Is a Paleoclassical Process in Toroidal Plasmas I D Callen University of Wisconsin Madison WI 537061609 USA Received 9 February 2004 revised manuscript received 20 April 2004 published 9 February 2005 It is hypothesized that radial electron hmt transport in magnetically con ned toroidal plasmas results from paleoclassical Coulomb collision processes parallel electron heat conduction and magnetic eld diffusion In such plasmas the electron temperature is equilibrated along magnetic eld lines a long length L gtgt poloidal periodicity length arRoq which is the minimum of the electron collision length and an effective eld line length Thus diffusing eld lines induce a radial electron heat diffusivity M E LaTROq N 10 gtgt 1 times the magnetic eld diffusivity nMO E u5cwz DOI 10 1 lOSPhysRevLett94055002 For more than three decades the outstanding and perva sive mystery in pursuit of magnetic fusion has been 174 what causes radial across magnetic eld lines electron heat transport in magnetically con ned toroidal plasmas such as tokamaks Because the experimentally inferred electron heat transport exceeds the theoretical classical 5 gyromotioninduced and neoclassical 5 drift orbitinduced collisional transport by factors of about 1 and 102 respectively it is called anomalous Since this is often the dominant radial transport process resolv ing this conundrum is very important both for understand ing plasma con nement in present experiments and for developing accurate plasma performance predictions for the planned international thermonuclear experimental re actor ITER 6 Salient properties of radial electron heat transport ob served in tokamak plasmas over the past three decades are 174 1 The experimentally inferred effective radial electron heat diffusivity e is of order 2 mZs to within a factor of 10 in a wide variety of tokamak plasmas 2 The inferred e is typically 3730 times the magnetic eld diffusivity IIMO 3 The 8 usually increases from the hot plasma core toward the cooler edge 4 Tokamak plasmas heat up to the low collisionality bananaplateau regime 5 5 In high density Ohmically heated plasmas e is inversely proportional to the electron density socalled Alcator scaling 7 Some mysterious properties are 6 The 8 can be up to an order of magnitude smaller in the vicinity of low order rational surfaces 8 and internal transport barriers often form there 910 7 The 8 can also be smaller just inside a magnetic separatrix at the plasma edge 10 This Letter provides a new model for all these characteristics of radial electron heat transport based on the dominant Coulomb collision processes in lowcollisionality toroidal plasmas The earliest in time and most primitive paleo domi nant Coulombcollisioninduced transport processes in magnetically con ned plasmas will be called paleoclassi cal processes They occur on the electron collision time 0031 9007 05 94505500242300 0550021 PACS numbers 5225Fi 5235Vd 5255Dy 5255Fa scale 1 18 The dominant transport processes are parallel electron heat conduction and magnetic eld diffusion Classical and neoclassical diffusivities 5 develop on the same time scale but are smaller than the magnetic eld diffusivity for most low electron pressure plasmas On the 1 18 time scale electron heat conduction equi librates the electron temperature over parallel to the magnetic eld B distances of order the electron colli sion length A8 E UTE 18 in which UTE E2Teme12 Magnetic eld diffusion see 8 below is induced by the plasma resistivity 1 It causes magnetic eld lines to diffuse perpendicular to B with a diffusion coef cient D7 2 quotno10 E yakup2 N Ax2At which implies a diffusive radial step Ax 2 68 E cwp the electromagnetic em skin depth in which up EneeZmeeo12 is the electron plasma frequency in a collision time At 2 1118 Thus it will be hypothesized that paleoclassical processes equilibrate Te over a collision length A5 typically N200 In along a eld line that is diffusing radially about cwp N1 mmiall in a collision time 1118 N10 23 The parallel equilibration can be limited by the nite length of rational eld lines in toroidal magnetic systems and by the parallel length over which eld lines are diffusing In axisymmetric toroidal magnetic con nement systems the helical eld lines form nested toroidal surfaces called magnetic ux surfaces The winding number of eld lines in tokamaks is de ned by a safety factorquot for kink stability qr in which r is the cylindricallike radial label of the ux surface it typically ranges from order unity in the plasma core to 3 at the edge Flux surfaces are rational or irrational depending on whether or not 1 is the ratio of integers m n mn 20 mm rational surface irrational surface 1 The irrational surfaces form a dense set while the rational surfaces are radially isolated from each other 2005 The American Physical Society PRL 94 055002 2005 PHYSICAL REVIEW LETTERS week ending 11 FEBRUARY 2005 Rational surfaces are of interest because their helical magnetic eld lines close on themselves after m toroidal or n poloidal transits The length of such eld lines on a qr qk E mn rational surface in a large aspect ratio 5 E rRO ltlt 1tokamak is 267 2 277R0m E Z Roqw in which R0 is the major radius of the torus Magnetic eld lines diffuse for radial distances x off a rational surface greater than the em skin depth le gt 68 but are not created and do not diffuse for M lt Seisee discus sion after 7 below In a sheared magnetic eld the half length 5 along B see 6 below over which eld lines are diffusing is obtained ll from 1 gt kX 6 with kX keeLS k9 E nqr and lLS 2 rq R0q2 5 E LSke 68 Setting 7 5 determines a maximum It typically B 10 for eld lines diffusing radially over their entire length 1 c dq nm 2 652 1 2 lt2 ax x eq 0p dr r The length of such rational helical eld lines is 26mm 2 ZwRqumax maximum field line length 3 However in the vicinity of a low order rational surface q E m n eg 32 the relevant n is n and the eld line length is short 26 2 277R0q n The effective radial electron heat diffusivity e can now be estimated phenomenologically As a magnetic eld line diffuses radially it will be hypothesized to carry with it the electron heat contained on the eld line The half length L over which the electron temperature is equilibrated is L min max A8 na equilibration length 4 Because L is longer than the poloidal periodicity half length of magnetic eld lines wROq the paleoclassical superscript pc electron heat diffusivity in a torus is a multiple M larger than the magnetic eld diffusivity X1 NMl MEL 5 M0 7713011 which except for constants is this Letter s main result To be more precise the effects of paleoclassical pro cesses in the vicinity of a medium order lt n S nmax N 10 rational helical eld line need to be quanti ed Because the relevant properties of magnetic eld lines are their poloidal toroidal and helical periodicities eld curvature and torsion can be neglected However magnetic shear is important Thus a simple sheared slab model can be used to represent the magnetic eld in the vicinity of a helical eld line on a rational surface B Bo Z xL5 y Bo Z z gtlt VLI 6 A companion more detailed paper ll uses axisymmet ric toroidal geometry magnetic ux coordinates Here BO constant is the magnetic eld strength z E Vz is a unit vector along the rational eld line and 11 E BoxzZLS is the magnetic ux function associated with the small magnetic shear For a large aspect ratio toka mak the shear length is L5 2 ROqs with s E rq q Magnetic shear is caused by a parallel current owing in the plasma J V X BMO zvzlIILO ZBO MOLS E Jz z The preceding properties are evaluated on the rational surface where x E r E m vanishes The relevant magnetic eld evolution equation can be obtained from Earaday s law together with a plasma Ohm s law of E EV X B 1 meneezdJdt which in cludes electron inertia BBBt EV X E V X V X B E V X 1 MOSEdJdt in which ddt E BBt V V Using the pre Maxwell Ampere s law V X B 0 and the fact that for the B in 6 BBBt EV gtlt 8 8t z setting the coef cient of the z component inside the curl in the resultant equation to a constant yields the evolution equation for magnetic ux 11 7 2 2 d l inll 2 8 1 sewdt MOV ab at 7 Here the constant of the spatial integration is B PBt E t the inductive axial toroidal electric eld which is the source of magnetic ux 1 and hence eld lines This term represents the effect of the magnetic ux change in the central solenoid of a tokamak it is negative so the Poynting ux is in the Ex directionEto balance in equi librium the resistivityinduced magnetic ux diffusion rst term on the right in 7 and thereby produce a sta tionary magnetic eld on the resistive time scale Resistive diffusion of the magnetic ux 1 is induced by the effect of the parallel plasma resistivity 1 gt 1 on the parallel current density Jz E 1M0V2 Since TIMLO N 1186 for scale lengths x less than 68 in 7 where the SEVZ term dominates on the left the mag netic eld diffusivity becomes negligible the solution for 1 becomes spatially constant ll and hence no magnetic eld lines are produced ie z gtlt VLI gt 0 or diffuse in this region The paleoclassical analysis will thus be re stricted to M gt 68 and interpret that 1 represents diffusing eld lines only for M gt 68 S 6m Neglecting the SEVZ operator assuming x2 gt 6 in 7 and the advective V VLI term which is negligible ll compared to the ux motion induced by D77 the equation for LIx t becomes a simple diffusion equation 2 ngi Dn g at 8x 8t 20 Since the time scales of interest are long compared to the electron gyrofrequency the electron kinetics is usually described by a gyroaveraged kinetic equation which is called a driftkinetic equation 5 In the usual driftkinetic equation magnetic ux surfaces and hence eld lines are assumed to be stationaryibut 8 indicates 11 obeys a diffusion equation In particular consider the evolution of the small magnetic ux bundle of eld lines llx t traversed by the gyromotion of an electron gyrat 0550022 PRL 94 055002 2005 PHYSICAL REVIEW LETTERS week ending 11 FEBRUARY 2005 ing at its gyroradius Q E viwce N 01 run around a magnetic eld line Substituting 1 gt 110 llx t with 110 E BoxzZLS into 8 and using the equilibrium relation DnBZLIOsz E nl Jz E E B PBt one obtains 85118t DnBZSLIsz 9 Since the electron gyroradius is small Qe ltlt 55 61 can be taken to be a unit delta function initially at x x0 gt 68 for this initial condition the solution of 9 is mm t e X X 24Dv 477Dnt12 t2 0 10 This small initially localized ux diffuses radially with a radial spreading that grows linearly with time x 7 ml 2 f dx x 7 x026 1x t 2Dt 11 Thus because of D as time progresses a bundle of eld lines initially located at x x0 assumes a probability distribution given by 10 whose radial spread grows ac cording to 11Eeven for a stationary magnetic eld Because of symmetry in z pz mvz quz is a con stant of collisionless electron motion For the B in 6 A 711 hence pz mvz E gell However the mag netic ux 1 traversed by the electron gyromotion diffuses radially A key hypothesis of the paleoclassical model is that as such bundles of eld lines diffuseradially they carry electron guiding centers with them This causes the radial coordinate x or 11 of the electron guiding center to become a stochastic variable in the driftkinetic equa tion which implies for x2 gt 6 2 S 73mm an electron guiding center FokkerPlanck diffusion coef cient my dltx X02gt 7 T E dt 2D 12 and a vanishing drag coef cient AxAt 0 These stochastic electron diffusion effects are taken into account 1213 by adding a spatial FokkerPlanck diffu sion operator to the usual driftkinetic equation 5 afat UHBfBelw Cf DU 13 In this magnetic elddiffusionmodi ed driftkinetic equation MDKE fx v t gt fLI 7 vHv v t is the distribution function 1 is the radial eld line label of the electron guiding center position 6 is the distance along a eld line 1 E v BB is the particle speed along B and C is the Coulomb collision operator Particle drifts off eld lines have been neglected because in magnetized axisym metric toroidal plasmas their radial excursions Ax N QqelZ are small compared to the magnetic eld diffu sion scale length 68 E cwp for the usual low electron pressure situations where 38 lt eqz Finally the spatial FokkerPlanck operator is 1213 02 Ax2 02 8x2 ZAZ f 8x2 D f 14 DU 2 The lowest order approximation to the MDKE 13 includes parallel freestreaming and Coulomb collisions UHBfOaelw CUO 15 For long collision lengths A8 compared to the periodicity length of a helical eld line 7 its solution is a Maxwellian distribution constant along magnetic eld lines and hence a function of the local ux function 11 2 f0lI v t quotEUl rm f 28412 2Twt 16 Physically the electron temperature T8 is equilibrated along helical eld lines by parallel electron heat conduc tion For A8 lt 7 this process limits the parallel length over which this solution applies to A8 Thus the Maxwellian f0 in 16 is only applicable for S fM E min A8 The dependence of ne and Te on time t in 16 allows for their transport time scale evolution To obtain an electron energy balance equation one takes the energy fd3v maul2 moment of 13 using f zfo ln toroidal geometry the helical periodicity length 267 of a rational eld line is n times longer than the poloidal periodicity length 277R0q To take account of these peri odicity effects in the sheared slab model and in particular the n times a helical eld line wraps around the poloidal periodicity direction the DUO operator is also operated on by faquot 16271301 Formally in axisymmetric toroidal geometry this factor emerges 11 from a ballooning rep resentation 1114 used to preserve the poloidal and hel ical periodicities for these utelike responses in the vicinity of a rational surface Since T8 is only equilibrated over the parallel distance KM and the maximum length of diffusing eld lines is max this parallel integration is limited to EL to L where L is de ned in 4 and the net effect is the multiplier M de ned in 5 Assuming for simplicity that only an electron temperature gradient is present one thus obtains 3 8T t 02 3 5 2X Teltx0 X5 EMD 8x n5 2 2 8t n 17 in which Q8 is the collisional electron heating The relevant D7 and 1 for a toroidal plasma parallel neoclassical superscript nc electrical resistivity are 1115 no no 2 Z D 217i 17quot z f 18 L0 710 xi 13Z 4 Va Here Z gt Zaff E zinizlzne is the effective ion charge the reference perpendicular resistivity is 1IoMo E meVeMoneez 14 gtlt 103ZTe Vl32 mZS and the parallel electron viscosity 15 LeV8 z 71111 xEfchzgt115ffc in the banana 0550023 PRL 94 055002 2005 PHYSICAL REVIEW LETTERS week ending 11 FEBRUARY 2005 collisionality regime with ft 2146E 0632 and c i t39 Equation 17 is a diffusiontype equation for the elec tron temperature Te with a paleoclassical diffusion coef cient EC Since EC N MDn Te relaxes a factor oforder M faster than magnetic ux 1 does cf 8 and 17 In typical toroidal plasmas where he gt 67m EC 32nmax quothf10 For shorter he or near a low order rational surface the parallel equilibration length is limited by these effects 11 as indicated in 4 Because the distance between a low order rational surface q E m n and the nearest rational surface with n nmax is 6x 2 1n nmaxq 77681 12n or at a minimum in q 6Xmm 2 2n 237765q 13 only the lowest n rational surfaces eg 11 32 21 will be isolated enough radially from other n S nmax rational surfaces for EC to be small Nn IIILO in their vicinity The salient and mysterious properties of anomalous radial electron heat transport identi ed in the second para graph can be interpreted in terms of the paleoclassical model developed in 4 5 17 and 18 as follows 1 MagnitudeiFor a typical Ohmically heated Tokamak fusion test reactor TFTR plasma 16 Te 2 12 keV n8 2 3 gtlt1019 m 3 Zaff 2 2 R0 2 25 In 1 2 16 and lq 2 04 m at the plasma halfradius ra 2 0408 05 which yields quotno10 2 0067 mZs n cno 2 22 cwp 210 3 m nmax 11 and he 2 300 m gt wRanmax 2 140 In so that L wRanmax M nmax 11 and the estimated EC is 25 mZs N ZXP Since this EC 0lt TQ32 it becomes less than 1 mZs for T8 B 2 keViand it may then be smaller than possible microturbulence induced transport 2 Ratio to DviFor this TFTR case one has ECD 32M 17 gtgt 1 3 Radial variationiIn the usually applicable colli sionless paleoclassical regimequot AZ gt wRanmax EC 0lt TQM2 increases as T8 decreases from the hot plasma core toward the cooler edge 4 Collisionality regimeiTokamak plasmas will Ohmically heat until T8 is limited by EC which is relevant only if he gt 771301 bananaplateau collisionality regime 50 5 Density scalingiln collisional high density plasmas with 71301 lt he lt wRanmax EC 0lt UTERoq gtlt chop2 Olt1ne and 7E821124ng027ne1020m 3 gtlt aZROq for Tel2 2 500 eV12 which is an Alcatorlike energy con nement scaling law 6 Low order rational surfaces 7As indicated in 4 L and hence EC are much smaller near low order n 1 2 rational surfaces particularly when 1 is near a minimum 7 Near separatrixiOn closed eld lines just inside a magnetic separatrix q and q are large and nmax and EC are reduced 11 Perhaps the most remarkable paleoclasssical predictions are the reduced 8 values near low order rational surfaces in agreement with some key experimental results For example Rijnhuizen Tokamak Project RTP experiments 8 showed that as highly localized electron cyclotron heating ECH was moved radially outward the central Te had a stairstepquot behavior it decreased abruptly as the ECH passed each low order rational surface which indicated low 8 at such surfaces Alsojumps in T8 over a narrow radial region approximately predicted by 26xmin have been observed in evolving D IID Lmode plasmas 17 as an offaxis minimum in qr passes low order rational surfaces Finally the paleoclassical model predicts a EC pro le magnitude and barrier width in reasonable agreement with JT 60U experiments 9 that used reversed central shear q lt 0 and apparently qmjn 31 to pro duce a large Te gradient in an electron internal transport barrier The author thanks his colleagues at University of WisconsinMadison especially C C Hegna VV Mirnov S C Prager K C Shaing and C R Sovinec and P B Parks and R E Waltz of General Atomics for their insightful questions and comments on this work He is also grateful to the US Department of Energy for support of his research over the past three decades Electronic address callenengrwiscedu URL wwwcaewisceducallen 1 LA Artsimovich et all Nuclear Fusion Special Supplement IAEA Vienna 1969 Vol I p 17 H P Furth Nucl Fusion 15 487 1975 JD Callen Phys Fluids B 4 2142 1992 BA Carreras IEEE Trans Plasma Sci 25 1281 1997 5 F L Hinton and R D Hazeltine Rev Mod Phys 48 239 1976 R Aymar et al Nucl Fusion 41 1301 2001 M Greenwald et al Phys Rev Lett 53 352 1985 GMD Hogeweij et al Nucl Fusion 38 1881 1998 T Fujita et al Phys Rev Lett 78 2377 1997 10 E Doyle et al Nucl Fusion 42 333 2002 11 J D Callen Paleoclassical Transport in Low Collisionality Toroidal Plasmasquot to be published 12 S Chandrasekhar Rev Mod Phys 15 1 1943 13 NO Van Kampen Stochastic Processes in Physics and Chemistry NorthHolland Amsterdam 1981 14 1W Connor R Hastie and 1B Taylor Phys Rev Lett 40 396 1978 15 SP Hirshman and DJ Sigmar Nucl Fusion 21 1079 1981 16 E D Fredrickson et al Nucl Fusion 27 1897 1987 17 M E Austin private communication 2004 M E Austin et all Bull Am Phys Soc 46 No 8 251 2004 EEE EEEE 0550024
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