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On the Existence of Eigenmodes of Linear Quasi-Periodic Differential Equations and their Relation to the MHD Continuum * A. Salat Max-Planck-Institut für Plasmaphysik, Garching

Z. Naturforsch. 37a, 830-839 (1982); received January 18, 1982

To Professor Arnulf Schlüter on his 60th Birthday The existence of quasi-periodic eigensolutions of a linear second order ordinary differential

equation with quasi-periodic coefficient f{a>it, io%t) is investigated numerically and graphically. For sufficiently incommensurate frequencies coi, a>2 a doubly indexed infinite sequence of eigen-values and eigenmodes is obtained.

The equation considered is a model for the magneto-hydrodvnamic "continuum" in general toroidal geometry. The result suggests that continuum modes exist at least on sufficiently ir-rational magnetic surfaces.

1. Introduction

For the linear second order ordinary differential equation with periodic coefficient f(t)=f(t + Ji), (Hill's equation)

y(t) + [Uf(t)]y = 0 (1.1)

it is well known [1] that with mild assumptions for f(t), there exists an infinite sequence of character-istic values or eigenvalues X = Xn, n = 0 , l , . . . , such that the solutionsy =yn{t) have the same periodicity as the coefficient f(t).

As a generalization of Hill's equation we consider the differential equation with quasi-periodic coef-ficient / :

y(«) + [ * + / K < , o > 2 0 ] y = o , (1.2)

where / is periodic with respect to both arguments /(0,9?) =f(9 + 2n,

A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations 831

of periodic differential equations to systems of quasi-periodic differential equations so difficult and in-complete [3 — 6]. We shall discuss in Section 3 below some results relevant to us.

The paper is organized as follows: Section 2 presents a short discussion of the physical problem which prompted the investigation. For a quasi-periodic (5-function-type/(wj t, co21) (1.2) is trans-formed into a recurrence relation in Sect. 3, and the general features of the numerical solution [7] to-gether with pertinent analytic results [5, 6] are re-called. In Sect. 4 numerically obtained eigenvalues and eigensolutions are presented and discussed. Sec-tion 5 contains a critical discussion.

2. The MHD Continuum in General Toroidal Geometry

For a plasma confined in a toroidal equilibrium configuration and described by magneto-hydro-dynamic equations the linearized equations of mo-tion may be put in the form [8] :

d X = A X + B - Y , (4.1a) dtp

L Y = K X , (4.1b)

where X and Y are vectors with two and four com-ponents, respectively and together describe the per-turbed fluid motion and the perturbed magnetic field. A, B, K and L are matrix operators contain-ing derivatives in the magnetic surfaces rp = const. Derivatives out of the surfaces are explicitly indi-cated in (4.1a). The operator L is particular in that it only contains derivatives along the magnetic field lines on y = const. The existence of equilibria with a continuous set of nested toroidal magnetic surfaces is non-trivial in the general case but is as-sumed here.

If (4.1 b) can be solved for Y an equation for the radial variation of X is obtained. With suitable boundary conditions a set of discrete eigenvalues co2 may be determined, where an ansatz ~ exp (i co r) is made for the time dependence of the perturba-tions. If however

L(t,

832 A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations 832

with particular 2 -T-quasi-periodic functions f(t) = f(Q(t),

A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations 833

0.000000 0.200000 0.400000 0.600000 0.800000 1.000000 1.200000 1 .«10000 1.600000 1.800000 2.000000

222222 77 77575 772222227 777777 77 7 ^ 77222222 4

753 77 77 222222 7 222744 7777 77 222222 77 22227 7 77 7 22222227 2222227 7777 222222 77 22222222 7 7 77 222222 4 2222222227 7 72222227 2222222222 4 7 7222227774 772222222227 7 222227 77 777 72222222777 4 222227 47 77222222237 222227

77 722222223 222223 77 77 2222227 4222227 4

7777 2222227 7 222227 77777 2222227 722227

773775 47222227 7222277 77774 77 7222227 4722227 7 4 7222227 4222277

7 7222227 222277 7 7222227722273 4

7 77 22222 22227 4 77722222222277

4 7732222222277 22222222222227 47772222222233

33372222222777 7 22222 22277

7 222227722277 4

4 4 4

4 4 4

4 7 722227 722277 77 722225 722277 3377 4 722227 722277 777775 4 4 7222274 722277

3333 4 722227 7722777 333 722227 5522577

777 7222277 7722777 77 7222277 772277 477 7222777 772277

4 4 77 7722777 772277 77 7722777 4 772277

4 47 7722777 777777 4 7 7722773 777777

4 7 7722777 777777 7 7777777 777777

7 7777777 4 777575 7 7777777 777757 7 7777777 777777

77777777 777777 43777775 777777

55555 777755

777 777 737 777 774 733 733 3 3777

7 777 777

i 777 77 7 777 777

: 777 777 77 774 774

777 774 777 777

77 77 77777 77 y

77 77 77 77 774 7M 774

774 774 774 774 777 774

4 477 4 437

477 73 777

4 4 4 4 4 4 4 4 4 4 4

4 4 4 7 4

I 7 7 7

0.00000000 0.04000000 o.oeoooooo 0.12000000 0.16000000 0.20000000 0.24000000 0.28000000 0.72000000 0.76000000 0.40000000 0.44000000 0.48000000 0.52000000 0.56000000 0.60000000 0.64000000 0.68000000 0.72000000 0.76000000 0.80000000 0.84000000 0.88000000 0.92000000 0.96000000 1.00000000 1.0000000 1.08000000 1.12000000 1.16000000 1.20000000 1.24000000 1.28000000 1.72000000 1.76000000 1.40000000 1.44000000 1.48000000 1.52000000 1.56000000 1.60000000 1.64000000 1.68000000 1.72000000 1.76000000 1 .80000000 1.84000000 1.88000000 1.92000000 1 .«6000000 2.00000000

Fig. la. Unstable bands in the ((0,(02) grid for (o 1 = 1 and 2V= 104 iterations. Higher numbers correspond to weaker instability. Amplitudes Fi = F% = 0.2.

Fig. l b . Enlarged section of Fig. la , F\ = F2 — 0.5. Lines no (o + ni a>. + n2U>2 = 0 are indexed as (no, wi, 1/12)

1.700000 1.710000 I . W T O 1.770000 1.740000 1.750000 1.760000 1.770000 1.780000 1.700000 1.800000

i."v»ooooo 1 .«»100000 1.50200000 1.50700000 1.50400000 wxmxm 1.50600000 1.50̂00000 1.50800000 1.50900000 1.51000000 1.51100000 1.51200000 1.51700000 1.51400000 1.51500000 1.51600000 1.51700000 1.51800000 1.51900000 1.52000000 1.52100000 1.52200000 1.52700000 1.52400000 1.52500000 1.52600000 1.52^00000 1.52800000 1.52°00000 1.57000000 1.57100000 1.57200000 1.57700000 1 .=57400000 1.57500000 1.57600000 1.5T700000 1.57800000 1.57000000 I.54OOOCCO 1.54100000 1.54200000 1.54700000 1.54400000 1.54500000 1.54600000 1.54'700000 1.54800000 1.54000000 1.^5000000

834 A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations 834

general, however, this representation is not possible, see [3], not even for the first order equation y = f(oo, displaying the functional dependence of y(t) on the two sub-argu-ments (JO1 t and co2 t. "Trivial" eigenfunctions of this type (see below) are shown in Figures 2 a, 3 a.

Another useful representation is the pair of phase space diagrams y/co versus y, plotted at t — n r1 and t = nt2 , n = 1, 2 , . . . For eigenmodes a closed curve

2a) 2b) 2c) Fig. 2. "Trivial" eigenmode y for periodic case jFi = 0.1, — 0, toi = l , a>2 = l / / 2 . Eigenvalue Q = 0.28675534, initial direction r — 0.500, initial phase c = 0. y as function of o>2< and coi t (Fig. a) and phase space diagrams (y, y/co) at multiples of periods n (Fig. b) and r2 (Fig. c). N = 8000.

3a) 3b) Fig. 3. Same as Fig. 2 with r = 0.111.

A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations

results for n—>00, or rather a curve with disconti-nuities in y since in our case only y is continuous but y in general is not (except if y = 0). Figures 2 b, 2 c, 3 b, 3 c, for example, correspond to the cases 2 a and 3 a.

For all solutions which are not eigenmodes the graphs show at twodimensional distribution of scat-tered points [7] instead of curves except for sub-harmonics of eigensolutions, i. e. solutions with quasi-periods m2n, m ^ 2. Such subharmonics, however, may be identified by the fact that for them y has more than one branch.

In order to search for eigenmodes the following procedure has been applied. A coj-ojg-quasi-periodic eigensolution was constructed for a eoj-periodic f(t). In small steps a co2-periodic contribution was then added to f(i) and the eigenvalue parameter co2 was adjusted each time so that the solution looked as like as eigenmodes as possible. In the Appendix the construction of the initial "trivial" eigenmodes is explained. Such an eigenmode with eigenvalue co = Q = 0.28675534 is shown in Figs. 2 a — 2 c and 3 a — 3 c for two values of the initial value parameter, r = 0.111 and r = 0.500, respectively, for later refer-ence. The amplitude is F1 = 0.1 and the frequencies are co1 = l , co2= 1/Y2. The same frequencies are used throughout in the following for all cases con-sidered ; see Sect. 5 for discussion.

Unfortunately, it turns out that the intended procedure does not work. Once the originally vanish-ing amplitude F2 reaches a few percent of F1 even the best fit of co does not yield well defined curves but some structure of finite width. Figure 4 shows how poorly one such "optimum" fit, i t) is 2ji-periodic in co* t, £ = 1,2. In-spection, however, shows [6] that for v given by (4.1) the eigenvalue parameter (O is exactly in one

835

of the "forbidden" gaps, Eq. (3.7), which for our (5-functiontype f(t) were shown in the last section to be connected with unboundedness of the solu-tions.

Although the foregoing results seem to give evi-dence against the existence of eigenmodes, such modes may nevertheless be found. Consider again (3.1) with coj-periodic coefficient, equivalent to Hill's equation. It is well known [1] that ^-peri-odic eigenmodes occur exactly if the eigenvalue parameter is at the boundary between bounded and unbounded behaviour of the solutions, i. e. at the boundary of gaps which for small amplitude are situated approximately at co =

Fig. 6. Eigenmode for = ^ 2 = 0.1, « 2 = 1/1/2, r = 0.111, c = 0 with eigenvalue ü" = 0.27504376.

5a) 5b) 5c)

Fig. 5. Eigenmode for Fi — F2 = 0.1, w2 = l/j/2, r = 0.500, c = 0 with eigenvalue ü' = 0.27495798.

836 A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations

4a)

Fig. 4. "Optimum" solution for Fi

4b) 4c)

= F2 — 0.1, cü2 = 1/1/2, r — 0, c = 0 for co = 0.27504376. N = 16000.

A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations 837

7a) 7b) Fig. 7. Eigenmode for Fy = F% = 0.1, a>2 = 1//2, r =

7c) 0.6085, c = 0.5 jr with eigenvalue Q " c = 0.27504378.

known a detailed investigation of the behaviour of y(t) in the region of the left and right boundaries of the gap was made. In particular, max | y (t) \ was determined on a two-dimensional co — r ( = initial value) grid. In the most promising regions of this grid the solutions were visually checked to see how far they corresponded to an eigenmode. And indeed an eigenmode was found on each boundary of the gap. the two eigenvalues are = ß ' = 0.27495798±3-1(T9 (see Fig. 5 a - 5 c ) and Q " = 0.27504376± 7 • 10-9 (see Fig. 6 a — 6 c). The corresponding initial value parameters, see Eq. (3.6), are r— 0.500 + 5 • 10~4 and r " = 0 . 1 1 1 ± 1 1 0 " 3 respectively. The parameters co and r, in particular co, have to be determined more and more precisely if one wants to go to higher and higher numbers N of iterations. A r = 1 . 2 8 x l 0 5 was used to determine the eigen-values above. In Figs. 5a — 6c . /V=16 000 and only every 2nd iteration is plotted.

The modes in Figs. 5 and 6 are the generalization of the modes from the periodic case F2 = 0, Figs. 2 and 3, to the quasi-periodic case F2 = Ft. Both eigen-values Q', Q " are roughly 4°/o smaller than Q. If F2 (or Fj) is decreased both Q' and Q" increase and the width of the gap Q " — Q' shrinks until at F2 = 0 both eigenvalues coalesce into co = Q, the gap disappears and the eigenmode is degenerate with respect to the initial value r. In general the modes have a discontinuity in y at t = nT1 and t = nT2. The modes should be indexed with ( — 2,2) cor-responding to the gap index (/^ , n2).

In order to check the effect of the phase difference c (see Eq. (3.2)) on the eigenvalues, the search for eigenmodes was repeated with c changed from zero

to 0.5 JI i. e. At = 0.5 Tx , on the right hand boundary of the same gap as before. Figures 7 a — 7 c show the resulting eigenmode. Its eigenvalue co = Qc" = 0.27504378 ± 1.6-10 - 8 agrees with the previous value Q" for c = 0 within the limits of accuracy aspired. Hence, as expected for incommensurate cOj and co2, the effect of the initial phase difference disappears for sufficiently large N. The "proper" initial value is rc" = 0.6085 ± 1 • 10 -4.

From the discussion above it is obvious that the existence of eigenmodes is not restricted to the par-ticular values of the amplitudes, gap indices and phase Fi, /ij, c, i = 1, 2 used. Eigenmodes with other parameters have indeed been constructed. It might happen, however, that for large amplitudes some of the eigenvalues disappear when different gaps over-lap.

5. Discussion and Conclusions

It has been shown within the limits of numerical and graphical methods, that the quasi-periodic dif-ferential equation (3.1), (3.2) possesses a doubly indexed infinite sequence of eigenvalues co = ßm>M and co = Q'm

838 A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations 838

periods 2 r1 and/or 2 r2 was investigated more in passing. They exist at the edges of gaps with half-integer m and/or n.

A particular point which deserves discussion is the choice of q = OJ1/CO2 . In the paper q = \2 is used throughout, i. e. an irrational number, as intended. On the other hand, in the computer all numbers are truncated so that q becomes rational. During the computations, however, it was monitored whether two ^-function pulses ever coincided again if two pulses did so initially; in other words, whether c o j co2 = NJN2 for Nt,N2 N0 , where 2 T2 N0 = TX(N0 + 1), i.e. for N^S/\e\>l. Thus, the criterion of "phase scrambling" leads to the re-quirement of an exceedingly large number of itera-tions if q is very close to a rational number m/n with small m, n. This was indeed observed numeri-cally. For the case FX = F2 = 0.5, co1 = 1, co2 = (4 + l x l O - 8 ) 1 / 2 and c = 0 an eigenvalue was found at co = 1.126872 while for c = 0.5 n it changed to co = 1.137703, even at N=2-105 iterations. In contrast, the eigenvalues for co = l / ] /2 were independent of c up to at least 7 decimal places. Such problems with rational versus irrational numbers have their coun-terpart in the requirement of "strong incommen-surability" ' (Ot + n2 co2 | Q (n) where ß is a sufficiently fast decreasing function of n = max (! j, ! n2 |) in the analytic treatment of (3.1) (see [5, 6 ] ) .

In applying the foregoing considerations to the problem of the MHD continuum in general toroidal geometry (see Sect. 1 and 2) it seems plausible that eigenmodes exist on "sufficiently irrational" surfaces while their existence for other values of q is less certain.

Acknowledgements

The author would like to thank Dr. J. Tataronis for many helpful discussions.

Appendix

In the periodic case, say F2 = 0, the solution y, y can readily be obtained from Eqs. (3.4), (3.5) after one full period r1 = 2T 1 as a function of the initial values y (0), y (0). The result is

Y ( r 1 ) = A - Y ( 0 ) ( A l )

with An = cos 2 z — / sin z cos z, A12 = sin 2 z — / sin2 z , (A 2) A21 = — sin 2 z — f sin2 z — /2 sin z cos z , A22 = cos 2 z + f sin z cos z — f2 sin2 z ,

where Y ={y,y/A))T, f = Fjw, Z = JICO/(D1.

(A3)

According to the Floquet theory [2] the solution of (3.1) in the present case is of the form

y{t) =eivtu{oo1t) +cc (A4)

with u(Q + 2n) =u{9), provided v=f=nx

A. Salat • Eigenmodes of Linear Quasi-Periodical Differential Equations 839

here: the usually existing finite co region of un-bounded solutions has collapsed and disappeared. This degeneracy, which does not occur in Mathieu's equation, is due to the infinite number of harmonics of equal amplitude which build up the (3-functions.

The case X — — 1 implies Fi = i 2 co ctg z (A 7)

which is satisfied by two infinite sets of co = an , ßn which are situated pairwise to the left and right of (2 n + l)coj2, n — 0 , 1 , . . . Between each conjugate pair the solutions are unbounded. It is easily checked

[1] W. Magnus and S. Winkler, Hill's Equation, John Wiley, New York 1966.

[2] V. A. Yakubovich and V. M. Starzhinskii, Linear Dif-ferential Equations with Periodic Coefficients, John Wiley, New York 1975.

[3] N. N. Bogoljubov, Yu. A. Mitropolskii, and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Chapter 5 and Appendix VII, Springer-Verlag, Berlin 1976.

[4] G. R. Sell, in Lecture Notes in Mathematics; Proc. Conf. on Ordinary and Partial Differential Equations, Dundee (1974), Eds. A. Dold and B. Eckmann, Springer-Verlag, Berlin 1974.

[5] E. I. Dinaburg and Ya. G. Sinai, Funct. Analysis Appl. 9, 279 (1975).

[6] H. Rüssmann, in Nonlinear Dynamics; Int. Conf. on Nonlinear Dynamics, New York 1979; Ed. R. H. G. Helleman, The New York Academy of Sciences, New York 1980.

[7] A. Salat and J. Tataronis, Stochasticity and Order in a Linear Quasi-Periodic Differential Equation, IPP-Report 6/204 (1981).

[8] J. A. Tataronis, J. N. Talmadge, and J. L. Shohet, Alfven Wave Heating in General Toroidal Geometry, University of Wisconsin Report ECE-78-10 (1978).

with Eqs. ( A l ) , (A 2) that for