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An approximate equation for nonlinear cross-field instability
Volume 113A, number 1
PHYSICS LETTERS
25 November 1985
AN A P P R O X I M A T E E Q U A T I O N F O R N O N L I N E A R C R O S S - F I E L D I N S T A B I L I T Y Takuji K A W A H A R A and Sadayoshi T O H
Department of Physics, Facultyof Science, Kyoto University,Kyoto 606, Japan Received 4 October 1984; revised manuscript received 15 March 1985; accepted for publication 23 September 1985
A simple equation which approximately governs a nonlinear cross-field instability is derived in terms of an iterative perturbation procedure. A numerical inital value problem reveals that the nonlinear diffusion term involved in the equation causes an explosiveevolution of waves at a finite time.
The aim of this article is to point out that the following equation governs approximately the nonlinear cross-field instability in a weakly ionized plasma
u t +OtUxx +3Uxx x +?Uxxxx +6(u2/2)xx = 0 ,
(1)
where a, 3, % and 6 are constants to be determined shortly and subscripts t and x denote corresponding derivatives. The linear part of this equation is of the same form as that of the equation whose solution properties have recently been investigated in relation to formations of soliton-like pulses [I ], but the nonlinear term of convective type is replaced by the nonlinear diffusion term in eq. (1). This type of equation (3 = 0) has also been derived by Kuramoto [2] in a general treatment of phase dynamics in reaction-diffusion systems. Nonlinear diffusion terms arise also in a higher order approximation of unstable waves on a fluid flowing down an inclined plane [3] or nonlinear Marangoni convection [4]. Nonlinear cross-field instability in a weakly ionized plasma has been investigated from various aspects [ 5 - 7 ] . Amongst them, the present study is motivated by the work of Sato and Tsuda [6]. They solved the initial value problem numerically by integrating a set of simplified equations relevant to the cross-field instability and showed that the instability triggered according to the linear instability explosively develops into a strong turbulence. In a linear approximation with long wavelength, the dispersion relation for the crossfield instability agrees with that for an equation which brings about formations of soliton-like pulses [ 1]. This 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
fact motivated us to investigate what type of equation governs approximately such explosive evolutions. We start with the set of equations derived in ref. [6] (eqs. (13) and (14)) and further simplify this set to derive a single equation of the form (1) by an iteration procedure. Initial value problem of this equation is solved numerically in order to get an insight into the properties of solutions to this equation. The set of equations with which we start is given as follows
Pr + Pn + e(o~ -- A l e - l p n n + A2 ((onn + P~bnn + Pn0n ) = 0 ,
(2)
BI (Pn + e(on) + B2e-l pnn + B3(Onn + P(onn + Pn(°n) = 0 ,
(3)
where r/, z, p(r/, ~'), and (o(rt, r) denote respectively the dimensionless distance, time, charge density, and potential, and A 1, A 2, B1, B2, B3 and e are system parameters. Here we do not go into details of the physical background. The derivation of this set of equations and the physical meaning are detailed in ref. [6]. (Some typographical errors in the signs of eqs. (13) and (14) of their original paper were corrected afterward.) As a preliminary to the subsequent consideration, we first derive the linear dispersion relation for the set of eqs. (2) and (3). Substituting p, (Ooc exp(ikr/+ o~') into (2) and (3), we get after linearization 21
Volume 113A, number 1
PHYSICS LETTERS
o = {i[B 3 - B 2 - ( . 4 2 + A 1 ) B 1 ] k 3 + (A1B 3 + A 2 B 2 ) e-lk4}/(ikB1 e - B 3 k 2 ) .
(4)
If we consider the case with relatively long wavelengths and assume that the wavenumber k is small, the linear dispersion relation can be expanded into a power series o f k leading up to O(k 4) to o = ctk 2 + i/3k 3 - 7k 4 ,
(5)
ing on the magnitudes of the nonlinearity and the parameters ct,/3, and 7. If we expand the dependent variable O as O = p(0) +/.tp(1) +//20(2) + ... and introduce the slow variables r/n =/.tnr/and r n = lanr(n = 1,2 .... ) in terms o f the small parameter/~, we can sum up several typical asymptotic equations in the lowest-order approximation as follows. (i) For relatively strong nonlinearity with O = 0 ( 1 ) , we have 0(o)+~(0) r2
where
tt Prl l'O 1
+6(0(0)0 (0))
(B 3 - B 2 ) ( B 3 - . 4 2 B 1 ) ,
0~1)+^.-1~(1) abt
7 = B] -3 e -3 (B3 - B2)(B3 - A2B1)B3 •
dpn = _ e - 1 0 n + (B 3 - B2)B] -1e-20nn - B 3 (B3 - B2)B-f2e-30nnn + B 2 (B 3 _ B2 ) B]-3 e-40nnnn + B3B-{ 1 e-1 (00n)n ,
(7) where the terms up to O(~ 6) are retained. On substituting @n from (7) into eq. (2) and retaining terms up to O(/06), we obtain a single equation for 0 as follows (8)
where t~,/3, and 7 are the same as those given in (6), and 6 = B] -1 e -1 (B 3 - A2B1). For those values o f system parameters used in the calculations in ref. [6] which have physical relevance to the cross-field instability, ct, 7, and 8 are positive and/3 is negative. Eq. (8) has been derived by the iteration in expectation o f a balance between the lowest nonlinearity and the fourth-order diffusion term, so that the terms in eq. (8) are not of the same order. On the other hand, by means o f the reductive perturbation method, we can derive several approximate equations depend22
•
P'O l'r/1
+/3p~l~r/lr/
1
+ 6(0(1)0(1~)r/
1
=0
(6)
We now derive a single equation from eqs. (2) and (3) by iteration. Since we are concerned with long wavelengths, we may assume that the spatial derivative is small (order k). Thus we assume blab7 = k ~ O(/~), where/a is a smallness parameter. Furthermore we assume that 0 and ¢ are of O(/02). From eq. (3), we represent ¢n in terms of 0 by iteration,
P~ + ~Onn +/30~nn + "YPnn,~n+ 6 (port)n = 0 ,
=0
r/l-r/1
(ii) When 0 = 0((~), the lowest equation is a linear equation 0(1 )__ + ~p(1)n 1 . , = O. In the case when a = O(tO, we obtain
o~= Bi-1 e -1 [B 3 - B 2 - (.4 2 + A 1 ) B 1 ] , /3 = _ a i - 2 e - 2
25 November 1985
"
(9)
(iii) When 0 = O(/02), we have linear equations 0 (2) + ctp(2) = 0 if a = O(1) and p(2) + a/a_lp(2 ) + T2 2~1rll ~'a n1~1 2 /30(_:) _ = 0 if t~ = O(/~). In the case when t~ = O(ta ), TI17.llTI1 we obtain (2) + . . . . 2 ^(2) r4
~
/a't/1 r/1
+ "yp?~
+ 6 (0(2)0 (2)) r/1 ~/1 r/1
= O.
"t/1 :r/l
(10) As is seen from eqs (9) and (10), a balancing of terms is possible if t~ is small and of O(/a) or O(~u2). The coefficient ,v given in (6) represents negative diffusion (instability) if a > 0. Thus asymptotically eqs. (9) and (10) describe the asymptotic behaviors close to the marginal condition (a = 0). We are interested in equations like (9) and (10), since especially the latter, eq. (10), also arises in nonlinear phase dynamics [2]. Eq. (8) formally includes eqs. (9) and (10) so that we hereafter regard the terms in eq. (8) or eq. (1) are o f the same order and consider some features of its solution. Eq. (1) with periodic boundary conditions on the spatial interval [0, 2] was solved for an initial condition cos lrx by a finite difference scheme in space and the R u n g e - K u t t a - G i l l scheme in time. The parameter values used in the present numerical calculations are t~ = 1.0X 1 0 - 2 , 7 = 5 . 0 6 6 X 1 0 - 6 , 6 = 1.0 and /3= 0 or /3 = 4.0 X 10 - 4 . The results for/3 = 0 and/3 = 4.0 X 10 - 4 are shown in figs. 1 and 2. Each figure shows a superposition of successive time steps just before the moment when the numerical solution indicates explosive increase in amplitude. After then the solution will lose its meaning as a solution to the finite difference
Volume 113A, number 1
PHYSICS LETTERS
The numerical results suggest that the solution has a singularity at some finite time, say t 0. We may assume that the solution consists of a superposition o f a regular (finite and smooth) part and a singular part. The singular part may be written as
8 m
U
25 November 1985
6
Us = It0 - t l - P F ( x l t 0 - tl - q ) .
4m
:LA 0
X
2
Fig. 1. Time evolution of waveforms for the nondispersive case. Superposition of four successive time steps at t = 1.02, 1.04, 1.06, 1.08 just around the moment when an explosion takes place. Initial condition is cos ~rx. (a = 1.0 X 10-2 , • = 0, 3' = 5.066 x 10-6, ~ = 1 . 0 ) . equation or the differential equation. Explosions were also observed for other initial conditions with smaller amplitudes than that shown in fig. 1.
10
Introducing this into eq. (1) and balancing the dominant terms, we can determine p and q. We obtain p = 1/2 and q = 1/4 for/~ = 0 and p = q = 1/3 for large/3. The numerical result in fig. 1 (0 = 0) shows an explosive evolution like u s = It 0 - t 1-1/2 (where t o = 1.087 by extrapolation) and agrees with the explosive evolution due to the similarity solution. The case in fig. 2 shows oscillatory growth but it also agrees with u s = It 0 - t1-1/3 (where t o = 2.4941). Fig. 3 shows time evolutions o f this explosive process. In both cases, the wave evolutions show a rapid increase o f the amplitude and an explosion occurs at a finite time. When/3 = 0, eq. (1) is symmetric with respect to the sign change o f the spatial coordinate x, so that fig. 1 shows a symmetric evolution o f waveforms. On the other hand, for dispersive case with nonzero/~, the symmetry in x is no longer valid and waveforms in fig. 2 show a different type explosive evolution. By means o f a conventional procedure, we may find particular stationary solutions for some special cases o f eq. (1). For/3 = 0, eq. (1) admits a solarity pulse o f the form u = u 0 - 3(u 0 + tx/6) sech 2 -[[1 - (u06 + a ) / 4 ? ] l / 2 x ) , (11)
U
where u 0 is an arbitrary constant. For/~ 4= O, stationary solutions may have oscillatory shock structure analogously to the case o f the K o r t e w e g - d e V r i e s Burgers equation. On the other hand, if the dispersion /~ dominates and 3' = 0, we have a step solution o f the form
5
u = -or~6 + (u 0 + otiS) tanh [+ (u06 + or) x/2~] .
-5 I
0
]-
1
X
2 _ _
2
Fig. 2. Time evolution of waveforms for the dispersive ease. Superposition of five successive time steps at t = 2.4, 2.45, 2.46, 2.47, 2.48. The parameter values are the same as in fig. 1 except for# = 4.0 X 10-4.
(12)
Although the numerical results appear to be reminiscent o f the particular solutions, i.e., pulse-like structures are embedded in fig. 1 and step-like structures in fig. 2, i t i s inferred that the particular solutions like (11) cannot be stable. If the curvature o f u 2 once becomes large at certain spatial region, that part o f wave evolves explosively due to nonlinear diffusion. Such an explosive growth in amplitude cannot be quenched 23
Volume 113A, number 1
PHYSICS LETTERS
(a) i
I1
-.....
10
1
10 -~
to-t
10-'
(b)
I
!1
10
--.'L
I
1 10-~
10-
to-
10-'
Fig. 3. Explosive evolution of wave amplitude (u versus to - t) for cases with (a) # = 0 (for the other parameter values see fig. 1; to = 1.087) and Co) # = 4.0 X 10-4 (see fig. 2; to = 2.4941). Dashed lines indicate the theoretically estimated slopes - 1/2 and - 1/3 for cases (a) and (b) respectively. by a balance with any linear term. It has been pointed out b y K u r a m o t o [2] that cnoidal waves rather than solitary waves can become solutions for/3 = 0. It might be possible that a subtle balance is attained for some particular initial conditions leading to cnoidal wave solutions in periodic systems. Generally speaking, however, it is unlikely that such a balance is achieved for arbitrary initial conditions unless some
24
25 November 1985
other nonlinear effects that balance with the nonlinear diffusion term are introduced. Sato and Tsuda [6] solved the set o f eqs. (2) and (3) b y a spatial Fourier transformation and observed a turbulent state after explosion. Every Fourier component once attained almost the same amount of energy presumably corresponding to explosive evolution in physical space and then exhibited a turbulent-like energy exchange. The set o f eqs. (2) and (3) includes higher order nonlinear effects that were omitted in the derivation of eq. (1) and it might be possible that these higher order effects admit the validity o f eqs. (2) and (3) for longer time duration than eq. (1). Numerical calculations due to Fourier transformation can be continued formally even after the stage o f explosion where finite difference method (or differential equation) loses its validity. Therefore, it will be interesting to seek what type o f equations can describe explosive evolutions and developments into the turbulent state. In conclusion, we may say that eq. (1) describes a certain stage o f the nonlinear cross-field instability and its explosive wave evolutions. But it is valid only just before the m o m e n t when the explosive singularity takes place. When the changes in amplitudes become very violent, higher order terms omitted in deriving eq. (1) will be significant and should be included properly. The authors would like to thank an anonymous referee for valuable comments.
References [1] T. Kawahara, Phys. Rev. Lett. 51 (1983) 381; S. Toh and T. Kawahara, J. Phys. Soc. Japan 54 (1985). [2] Y. Kuramoto, Prog. Theor. Phys. 71 (1984) 1182. [3] A. Pumlr, P. Manneville and Y. Pomeau, J. Fluid Mech. 135 (1983) 27. [4] G.I. Sivashinsky, Physica 4D (1982) 227. [5] A. Simon, Phys. Fluids 6 (1963) 382. [6] T. Sato and T. Tsuda, Phys. Fluids 10 (1967) 1262. [7] T. Sato, Phys. Fluids 14 (1971) 2426.
PHYSICS LETTERS
25 November 1985
AN A P P R O X I M A T E E Q U A T I O N F O R N O N L I N E A R C R O S S - F I E L D I N S T A B I L I T Y Takuji K A W A H A R A and Sadayoshi T O H
Department of Physics, Facultyof Science, Kyoto University,Kyoto 606, Japan Received 4 October 1984; revised manuscript received 15 March 1985; accepted for publication 23 September 1985
A simple equation which approximately governs a nonlinear cross-field instability is derived in terms of an iterative perturbation procedure. A numerical inital value problem reveals that the nonlinear diffusion term involved in the equation causes an explosiveevolution of waves at a finite time.
The aim of this article is to point out that the following equation governs approximately the nonlinear cross-field instability in a weakly ionized plasma
u t +OtUxx +3Uxx x +?Uxxxx +6(u2/2)xx = 0 ,
(1)
where a, 3, % and 6 are constants to be determined shortly and subscripts t and x denote corresponding derivatives. The linear part of this equation is of the same form as that of the equation whose solution properties have recently been investigated in relation to formations of soliton-like pulses [I ], but the nonlinear term of convective type is replaced by the nonlinear diffusion term in eq. (1). This type of equation (3 = 0) has also been derived by Kuramoto [2] in a general treatment of phase dynamics in reaction-diffusion systems. Nonlinear diffusion terms arise also in a higher order approximation of unstable waves on a fluid flowing down an inclined plane [3] or nonlinear Marangoni convection [4]. Nonlinear cross-field instability in a weakly ionized plasma has been investigated from various aspects [ 5 - 7 ] . Amongst them, the present study is motivated by the work of Sato and Tsuda [6]. They solved the initial value problem numerically by integrating a set of simplified equations relevant to the cross-field instability and showed that the instability triggered according to the linear instability explosively develops into a strong turbulence. In a linear approximation with long wavelength, the dispersion relation for the crossfield instability agrees with that for an equation which brings about formations of soliton-like pulses [ 1]. This 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
fact motivated us to investigate what type of equation governs approximately such explosive evolutions. We start with the set of equations derived in ref. [6] (eqs. (13) and (14)) and further simplify this set to derive a single equation of the form (1) by an iteration procedure. Initial value problem of this equation is solved numerically in order to get an insight into the properties of solutions to this equation. The set of equations with which we start is given as follows
Pr + Pn + e(o~ -- A l e - l p n n + A2 ((onn + P~bnn + Pn0n ) = 0 ,
(2)
BI (Pn + e(on) + B2e-l pnn + B3(Onn + P(onn + Pn(°n) = 0 ,
(3)
where r/, z, p(r/, ~'), and (o(rt, r) denote respectively the dimensionless distance, time, charge density, and potential, and A 1, A 2, B1, B2, B3 and e are system parameters. Here we do not go into details of the physical background. The derivation of this set of equations and the physical meaning are detailed in ref. [6]. (Some typographical errors in the signs of eqs. (13) and (14) of their original paper were corrected afterward.) As a preliminary to the subsequent consideration, we first derive the linear dispersion relation for the set of eqs. (2) and (3). Substituting p, (Ooc exp(ikr/+ o~') into (2) and (3), we get after linearization 21
Volume 113A, number 1
PHYSICS LETTERS
o = {i[B 3 - B 2 - ( . 4 2 + A 1 ) B 1 ] k 3 + (A1B 3 + A 2 B 2 ) e-lk4}/(ikB1 e - B 3 k 2 ) .
(4)
If we consider the case with relatively long wavelengths and assume that the wavenumber k is small, the linear dispersion relation can be expanded into a power series o f k leading up to O(k 4) to o = ctk 2 + i/3k 3 - 7k 4 ,
(5)
ing on the magnitudes of the nonlinearity and the parameters ct,/3, and 7. If we expand the dependent variable O as O = p(0) +/.tp(1) +//20(2) + ... and introduce the slow variables r/n =/.tnr/and r n = lanr(n = 1,2 .... ) in terms o f the small parameter/~, we can sum up several typical asymptotic equations in the lowest-order approximation as follows. (i) For relatively strong nonlinearity with O = 0 ( 1 ) , we have 0(o)+~(0) r2
where
tt Prl l'O 1
+6(0(0)0 (0))
(B 3 - B 2 ) ( B 3 - . 4 2 B 1 ) ,
0~1)+^.-1~(1) abt
7 = B] -3 e -3 (B3 - B2)(B3 - A2B1)B3 •
dpn = _ e - 1 0 n + (B 3 - B2)B] -1e-20nn - B 3 (B3 - B2)B-f2e-30nnn + B 2 (B 3 _ B2 ) B]-3 e-40nnnn + B3B-{ 1 e-1 (00n)n ,
(7) where the terms up to O(~ 6) are retained. On substituting @n from (7) into eq. (2) and retaining terms up to O(/06), we obtain a single equation for 0 as follows (8)
where t~,/3, and 7 are the same as those given in (6), and 6 = B] -1 e -1 (B 3 - A2B1). For those values o f system parameters used in the calculations in ref. [6] which have physical relevance to the cross-field instability, ct, 7, and 8 are positive and/3 is negative. Eq. (8) has been derived by the iteration in expectation o f a balance between the lowest nonlinearity and the fourth-order diffusion term, so that the terms in eq. (8) are not of the same order. On the other hand, by means o f the reductive perturbation method, we can derive several approximate equations depend22
•
P'O l'r/1
+/3p~l~r/lr/
1
+ 6(0(1)0(1~)r/
1
=0
(6)
We now derive a single equation from eqs. (2) and (3) by iteration. Since we are concerned with long wavelengths, we may assume that the spatial derivative is small (order k). Thus we assume blab7 = k ~ O(/~), where/a is a smallness parameter. Furthermore we assume that 0 and ¢ are of O(/02). From eq. (3), we represent ¢n in terms of 0 by iteration,
P~ + ~Onn +/30~nn + "YPnn,~n+ 6 (port)n = 0 ,
=0
r/l-r/1
(ii) When 0 = 0((~), the lowest equation is a linear equation 0(1 )__ + ~p(1)n 1 . , = O. In the case when a = O(tO, we obtain
o~= Bi-1 e -1 [B 3 - B 2 - (.4 2 + A 1 ) B 1 ] , /3 = _ a i - 2 e - 2
25 November 1985
"
(9)
(iii) When 0 = O(/02), we have linear equations 0 (2) + ctp(2) = 0 if a = O(1) and p(2) + a/a_lp(2 ) + T2 2~1rll ~'a n1~1 2 /30(_:) _ = 0 if t~ = O(/~). In the case when t~ = O(ta ), TI17.llTI1 we obtain (2) + . . . . 2 ^(2) r4
~
/a't/1 r/1
+ "yp?~
+ 6 (0(2)0 (2)) r/1 ~/1 r/1
= O.
"t/1 :r/l
(10) As is seen from eqs (9) and (10), a balancing of terms is possible if t~ is small and of O(/a) or O(~u2). The coefficient ,v given in (6) represents negative diffusion (instability) if a > 0. Thus asymptotically eqs. (9) and (10) describe the asymptotic behaviors close to the marginal condition (a = 0). We are interested in equations like (9) and (10), since especially the latter, eq. (10), also arises in nonlinear phase dynamics [2]. Eq. (8) formally includes eqs. (9) and (10) so that we hereafter regard the terms in eq. (8) or eq. (1) are o f the same order and consider some features of its solution. Eq. (1) with periodic boundary conditions on the spatial interval [0, 2] was solved for an initial condition cos lrx by a finite difference scheme in space and the R u n g e - K u t t a - G i l l scheme in time. The parameter values used in the present numerical calculations are t~ = 1.0X 1 0 - 2 , 7 = 5 . 0 6 6 X 1 0 - 6 , 6 = 1.0 and /3= 0 or /3 = 4.0 X 10 - 4 . The results for/3 = 0 and/3 = 4.0 X 10 - 4 are shown in figs. 1 and 2. Each figure shows a superposition of successive time steps just before the moment when the numerical solution indicates explosive increase in amplitude. After then the solution will lose its meaning as a solution to the finite difference
Volume 113A, number 1
PHYSICS LETTERS
The numerical results suggest that the solution has a singularity at some finite time, say t 0. We may assume that the solution consists of a superposition o f a regular (finite and smooth) part and a singular part. The singular part may be written as
8 m
U
25 November 1985
6
Us = It0 - t l - P F ( x l t 0 - tl - q ) .
4m
:LA 0
X
2
Fig. 1. Time evolution of waveforms for the nondispersive case. Superposition of four successive time steps at t = 1.02, 1.04, 1.06, 1.08 just around the moment when an explosion takes place. Initial condition is cos ~rx. (a = 1.0 X 10-2 , • = 0, 3' = 5.066 x 10-6, ~ = 1 . 0 ) . equation or the differential equation. Explosions were also observed for other initial conditions with smaller amplitudes than that shown in fig. 1.
10
Introducing this into eq. (1) and balancing the dominant terms, we can determine p and q. We obtain p = 1/2 and q = 1/4 for/~ = 0 and p = q = 1/3 for large/3. The numerical result in fig. 1 (0 = 0) shows an explosive evolution like u s = It 0 - t 1-1/2 (where t o = 1.087 by extrapolation) and agrees with the explosive evolution due to the similarity solution. The case in fig. 2 shows oscillatory growth but it also agrees with u s = It 0 - t1-1/3 (where t o = 2.4941). Fig. 3 shows time evolutions o f this explosive process. In both cases, the wave evolutions show a rapid increase o f the amplitude and an explosion occurs at a finite time. When/3 = 0, eq. (1) is symmetric with respect to the sign change o f the spatial coordinate x, so that fig. 1 shows a symmetric evolution o f waveforms. On the other hand, for dispersive case with nonzero/~, the symmetry in x is no longer valid and waveforms in fig. 2 show a different type explosive evolution. By means o f a conventional procedure, we may find particular stationary solutions for some special cases o f eq. (1). For/3 = 0, eq. (1) admits a solarity pulse o f the form u = u 0 - 3(u 0 + tx/6) sech 2 -[[1 - (u06 + a ) / 4 ? ] l / 2 x ) , (11)
U
where u 0 is an arbitrary constant. For/~ 4= O, stationary solutions may have oscillatory shock structure analogously to the case o f the K o r t e w e g - d e V r i e s Burgers equation. On the other hand, if the dispersion /~ dominates and 3' = 0, we have a step solution o f the form
5
u = -or~6 + (u 0 + otiS) tanh [+ (u06 + or) x/2~] .
-5 I
0
]-
1
X
2 _ _
2
Fig. 2. Time evolution of waveforms for the dispersive ease. Superposition of five successive time steps at t = 2.4, 2.45, 2.46, 2.47, 2.48. The parameter values are the same as in fig. 1 except for# = 4.0 X 10-4.
(12)
Although the numerical results appear to be reminiscent o f the particular solutions, i.e., pulse-like structures are embedded in fig. 1 and step-like structures in fig. 2, i t i s inferred that the particular solutions like (11) cannot be stable. If the curvature o f u 2 once becomes large at certain spatial region, that part o f wave evolves explosively due to nonlinear diffusion. Such an explosive growth in amplitude cannot be quenched 23
Volume 113A, number 1
PHYSICS LETTERS
(a) i
I1
-.....
10
1
10 -~
to-t
10-'
(b)
I
!1
10
--.'L
I
1 10-~
10-
to-
10-'
Fig. 3. Explosive evolution of wave amplitude (u versus to - t) for cases with (a) # = 0 (for the other parameter values see fig. 1; to = 1.087) and Co) # = 4.0 X 10-4 (see fig. 2; to = 2.4941). Dashed lines indicate the theoretically estimated slopes - 1/2 and - 1/3 for cases (a) and (b) respectively. by a balance with any linear term. It has been pointed out b y K u r a m o t o [2] that cnoidal waves rather than solitary waves can become solutions for/3 = 0. It might be possible that a subtle balance is attained for some particular initial conditions leading to cnoidal wave solutions in periodic systems. Generally speaking, however, it is unlikely that such a balance is achieved for arbitrary initial conditions unless some
24
25 November 1985
other nonlinear effects that balance with the nonlinear diffusion term are introduced. Sato and Tsuda [6] solved the set o f eqs. (2) and (3) b y a spatial Fourier transformation and observed a turbulent state after explosion. Every Fourier component once attained almost the same amount of energy presumably corresponding to explosive evolution in physical space and then exhibited a turbulent-like energy exchange. The set o f eqs. (2) and (3) includes higher order nonlinear effects that were omitted in the derivation of eq. (1) and it might be possible that these higher order effects admit the validity o f eqs. (2) and (3) for longer time duration than eq. (1). Numerical calculations due to Fourier transformation can be continued formally even after the stage o f explosion where finite difference method (or differential equation) loses its validity. Therefore, it will be interesting to seek what type o f equations can describe explosive evolutions and developments into the turbulent state. In conclusion, we may say that eq. (1) describes a certain stage o f the nonlinear cross-field instability and its explosive wave evolutions. But it is valid only just before the m o m e n t when the explosive singularity takes place. When the changes in amplitudes become very violent, higher order terms omitted in deriving eq. (1) will be significant and should be included properly. The authors would like to thank an anonymous referee for valuable comments.
References [1] T. Kawahara, Phys. Rev. Lett. 51 (1983) 381; S. Toh and T. Kawahara, J. Phys. Soc. Japan 54 (1985). [2] Y. Kuramoto, Prog. Theor. Phys. 71 (1984) 1182. [3] A. Pumlr, P. Manneville and Y. Pomeau, J. Fluid Mech. 135 (1983) 27. [4] G.I. Sivashinsky, Physica 4D (1982) 227. [5] A. Simon, Phys. Fluids 6 (1963) 382. [6] T. Sato and T. Tsuda, Phys. Fluids 10 (1967) 1262. [7] T. Sato, Phys. Fluids 14 (1971) 2426.