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Singular nonlinear equations and explosive recurrent events
29 July 1996
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 218 (1996) 64-69
Singular nonlinear equations and explosive recurrent events A.C. Coppi”, B. Coppi b a Department of Mathematics, Yale Universily New Haven, CT 06511, USA b Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 14 February 1996; revised manuscript received 1 May 1996; accepted for publication 1 May 1996 Communicated by M. Porkolab
Abstract Events of relatively brief duration (involving for example accelerated or heated particles, excitation of fluctuations, radiation emission) that are related to the onset of different kinds of explosive instabilities can recur at regular intervals or randomly. An analytical model is introduced, to reproduce these events, consisting of a set of nonlinear differential equations which involve a characteristic singularity. It is this feature that models explosive or quasi-explosive events either for a “primary” factor (e.g. the population of heated or accelerated particles) or for the relevant plasma fluctuations that are excited when the primary factor exceeds an appropriate threshold value.
1. Introduction There are important plasma processes that involve explosive instabilities [l-3] where the magnitude of a given physical quantity can, in principle, reach relatively high values within a relatively narrow time interval. Here we consider the particular case where a given physical factor (“primary”) such as the electron temperature in a magnetically confined plasma can periodically undergo an explosive instability which can be quenched by the onset of fluctuations that are excited when this factor exceeds an appropriate threshold. These fluctuations are excited in characteristic bursts that are rather frequently observed experimentally [ 4-61. In this Letter we propose a new set of equations, with a characteristic singularity, which can model the realistic circumstance where the factor that drives the onset of fluctuations is subject to an explosive instability. Thus a “double bursting” can occur of both the driving factor and of the induced fluctuations. The
conditions where the growth of these fluctuations can prevent the transition to an explosive state of the driving factor are identified. The equations that we introduce constitute a noncanonical Hamiltonian system. A parallel set of equations that we discuss briefly is one in which the fluctuation level itself is subject to an explosive instability. We consider two illustrative cases. One is inspired by the process of electron (Ohmic) heating that occurs in a magnetically confined plasma, when a constant (in time) electric field is applied along the magnetic field, and the plasma electrical conductivity is resulting from electron-ion collisions. The other case is that of a plasma in which the electron heating is produced by the slowing down of the charged particles produced by fusion reactions [ 31. In both cases, as the electron temperature increases the rate of heating also increases. The rate of heating corresponds to two distinct values of a “singularity parameter” that characterizes the explosive nature of the fluctuation driving factor.
0375~9601/%/$12.00 Copyright 8 1996 Published by Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00378-7
A.C. Coppi, B. Coppi/Physics
In the first case, the “singular cycle” of nonlinear oscillations is reached with a relatively wide variation of values of the Hamiltonian, up to infinity. In the second case, where the rate of increase of the electron heating with temperature is higher, the singular cycle is reached earlier, with modest variation of the Hamiltonian’s values. The Hamiltonian value corresponding to the “singular cycle” is in fact finite in this case. We note that, when the considered physical system is in an oscillation that approaches the singular cycle, it is susceptible to small perturbations that can lead it into an explosive state. In particular, this occurrence can be a random event. A special case of this circumstance is represented by the existence of a small periodic modulation of the source of the driving factor where the period of this modulation is not commensurate with that of the basic fluctuation bursts. The onset of random bursting of the fluctuation level is produced by the introduction of a small periodic modulation of the threshold for the instability driven by the primary factor when this instability is itself of the explosive type.
Letters A 218 (1996) 64-69
g
= M(D)N(A),
(4)
where N(A = A,)
and M(D)
= 0,
dN dA < 0,
(5)
is a positive function of D with
dM z > 0.
(6)
The product M(D) N( A) represents the “source function” of the driving factor D. Thus D(t) decreases when the induced fluctuation amplitudes exceed an appropriate value. It is evident that this choice of the source function does not exhaust the spectrum of physical possibilities. In particular, the source function may not separate into the product of M(D) and N(A) or may have an explicit time dependence. In other words, there are significant cases where a non-Hamiltonian [ 71 system of equations should be considered. To be specific, we consider M( 0) = D’++
2. Relevant class of equations
65
(7)
and We refer to a spatially homogeneous plasma, that may be considered as the result of an average over small scale variations of the relevant physical parameters. We indicate the primary factor that can drive the onset of fluctuations by the function D(t) and the amplitude of these fluctuations by A(t). The value of the threshold for D(t) above which A(t) begins to grow is indicated by Ds. Next, we model the evolution of D ( f ) and A(t) by a pair of coupled nonlinear equations. The first is 2
N(A)
= (T - A’,
where (Yand Y are positive parameters chosen on the basis of physical considerations. It is convenient to introduce the function
(9)
= F(D)G(A), P=l-
where F(D=Ds)
=O,
dF
dD
>O, D=Ds
and G(A)
The second equation is
$
I/a
(>
(10)
and (2)
1 D= (1-P)“’ A
is a positive function of A with
dG dA > 0.
(8)
(3)
(11)
It is evident that L3 z D/OS when p -+ 1, corresponding to the “explosive limit”. The singularity appearing in JZq. ( 11) is, in fact, the dominant characteristic of the model equations we consider.
66
A.C. Coppi, B. Coppi/Physics
Then Eq. (4) can be rewritten as
Letters A 218 (1996) 64-69
1
b=
(23)
(1 -P)“’
dP --S-/p, dt
(12)
where S E crD~‘“lcu and iv E AYD~‘a/a. introduce the function
If we
dP dt
-=s-/p.
We note that if the (secondary) instability that is involved in the onset of the considered fluctuations is nonlinear we may consider
(13) G(A) M A2 Eq. ( 1) can be rewritten as z
(25)
as a significant case. Then
= F(D(P)).
Then we may introduce the relevant Hamiltonian by writing Eqs. (12) and (14) as dP
8H
-=-dt
(15)
aQ
and dQ
aH
-=--. dt
(16)
c?P
Proceeding, we choose very simple expressions for P(D) and G(A), F(D)
- 1
= ;
(17)
and the set of equations to be solved consists of Eq. (26) and dP -dt =S-
1 (1
(27)
_Q)v’
Therefore a characteristic singularity is introduced in each of the two equations. In most of this paper we assume the “saturation” of the source term, represented by A” in Eq. (24)) to be proportional to the square of the amplitude of the fluctuations [ 9- 111, consistent with standard quasilinear considerations, and hence take Y = 2. 3. Appropriate Hamiltonians
=a,
G(A)
(18)
which correspond to a linear approximation of the equations for the instability that describes the onset of the considered fluctuations. Then Q=lnA
(l-p)”
dr=
H(eQ)
= K(P) + V(Q),
1
dp = S _ evQ, dt
dK dP=
1
(1-P)”
dV
-I9
(B - l)A,
(29)
- = e*Q - S. dQ
(21)
The constants of integration are chosen so that K( P = 0) = 0 and V(Q = Qo) = 0 where exp(2Qa) = S. The interval -cc < P < 1 is considered. When (Y> 1 we obtain
A
=
-l,
(20)
or alternatively, g
(28)
(19)
and the final form of the equations [ 81 that we analyze in the following is
dQ
Referring to Eqs. (20) and (21)) we see that the relevant Hamiltonian is given by
(22)
K(P)
= -P
(30)
1
+ -n-1
A.C. Coppi, B. Coppi/Physics
Letters A 218 (1996) 64-69
and when (Y< 1 we obtain K(P) =-P+
$$I
- (1-
P)‘_“I.
(32)
In addition, for both cases V(Q) = i[ezQ - S-S(2Q-Ins)].
(33)
We observe that when (r < 1 the “explosive state” (PMax= 1 limit) is reached with relatively small positive values of H. Conversely when (Y> 1, corresponding to a weaker rate of increase of the driving factor source as a function of D, the pkax = 1 limit is reached only with H at infinity. Finally, when cy = 1 K(P) =-P+ln
i--$ (
.
(34)
1
It is instructive to analyze the characteristics of the H = const orbits in the P, Q plane and we consider at first cy < 1. This corresponds to the case where the driving factor is the electron temperature and is sustained by the slowing-down on the electrons of the charged particles produced by fusion reactions as we shall show. In particular, we note that the limiting value of H corresponding to pMax= 1 and Q = Qc is HL=-.
cy 1-ff
(35)
Thus, if for instance (Y= 213 or HI_ = 2, that is a case of special interest to model the effects of the heating of fusion products in a well-confined plasma, we notice that, in general, the limit pMax= 1 can be approached even for small values of H. In particular, if we take Jhlax=l-E
?? =[a(l-E)+E-H(l-cu)]l’(‘-n)
(36)
we see that E remains small even for quite small values of H. This fact is connected with the strong dependence on D of the source of the driving factor corresponding to (Y< 1. We also note that the minimum value of P for H = HL is P = -pb where
pD=
1 l-n
The intersections of the “limit orbit” with the P = 0 axis, where cYH/LYP= 0, are given by the equation
f(2Qr.J =e2QL-S2QL-
S(l+lnS)+&)
=O, (39)
which, as can be verified graphically, has two solutions, a positive and a negative one. Therefore, for given values of (Yand S, the limit orbit can be identified quite well. A representative numerical solution of Eqs. (22) and (24) yielding A = A(t) andB = b(t) is given in Fig. 1. In the case where the fluctuation level A can undergo an explosive instability and is related to Q by Eq. (26), the component V(Q) of the Hamiltonian is, for v = 1, V(Q) = -SQ + 1 - ln( 1 - Q).
(40)
ii”
( >
- 1.
(37)
The corresponding value of Is is Bo=l-cu.
Fig. 1. Numericalsolution of Eqs. (22) and (24) that describe the time dependence of the driving factor, l?(t) , and of the fluctuation amplitude,A( I), with S = 1, a = 2/3, Y = 2 and pku = 0.975, for which the correspondingrelevantHamiltoniandefined by Eq. (28) has the value H = 1.2. The dashed line represents A(t) and the solid line representsB(I).
(38)
The component K(P) can be constructed in the same way as was done earlier. We notice that when cy > 1 and v = 2 the explosive limit for l3 is reached before that of A. Therefore we consider Y = 1 as a more mathematically interesting choice to analyze.
A.C. Coppi. B. Coppi/Physics
68
Letters A 218 (19%) 64-69
ar,
4. Relevant physical processes and random events
cx T3’*Cn ’ at e
In a D-T fusion burning plasma, the produced 3.5 MeV a-particles, produced by the D-T reactions, slow down by losing energy to the electron population. Thus the electron thermal energy balance equation in which we neglect, for simplicity, all forms of energy loss, becomes
Now we know that the electron temperature is subject to so-called sawtooth oscillations resulting from the excitation of fluctuations driven by the plasma pressure gradients in the center of the plasma column when the magnetic field winding parameter Lexceeds unity. Thus the value of the singularity parameter cy corresponding to this case is (YN 513. As indicated earlier, there are significant circumstances where non-Hamiltonian systems of equations have to be considered to describe the time evolution of the quantities D and A. A particularly interesting case is where a periodic component, represented for instance by S(t) = So + SI cos wst, is introduced in the source for D. Then we may define the function
in, $
= EanDnT(av)&.
(41)
Here E, II 3.5 MeV, ((Tv)~ is the D-T fusion reactivity, T, is the &CtrOU temperatUre, ne = nD + nT the electron density, and no and nr the deuteron and triton densities. If the ion and the electron temperatures are close to each other, (uv)~ can be considered to be roughly proportional to T,“‘*. The quantity C, represents the fraction of the a-particle original energy that contributes to increase the electron temperature. Thus C, includes the effects of excited fluctuations that decrease the rate of a-particle’s energy deposition on the electrons. The coefficient of the singularity parameter a introduced in EZq.(3) for the model equation that may represent this case is therefore a 21 2/3. Thus, in adopting the model equation (4)) we consider the (realistic) case where the amplitude of the modes that interact with the a-particle population increases as both the plasma temperature and the rate of Lu-particleproduction simultaneously increase. We note in fact that the so-called fishbone modes that scatter a high energy particle population in a thermal plasma are characteristic modes of the thermal plasma itself [ 101 and owe their growth rate to the resonant interaction of the mode with the high energy particle population. The other case mentioned earlier is that of a magnetically confined plasma in which a constant electric field El1 is applied along the magnetic field. In this situation, the electron thermal energy balance equation in the central part of the plasma column can be written as (42) where ~11is the longitudinalelectrical resistibility, considered to result only from collisional effects, and Cn is the fraction of the Ohmic power that contributes to increase the central temperature. Thus we may argue that
Pr f P - 2
(43)
sinwst
W
(44)
and consider, instead of Eqs. (20) and (2 1) ,
f+-+l
(45)
and -dh dt
=so -eYQ,
with a
B(Pr)=l/
.
1-F$--2sinost (
>
(47)
Clearly D( &) becomes singular when Pr = 1 (St /us) sin wt. When the frequency ws is not related to the intrinsic frequency of the system that is found for St = 0, the explosive state can be reached unpredictably thus representing the rather common phenomenon of random bursting. A relevant example is given in Fig. 2 where the numerical solution of E!qs. (22) and (24) is given with S replaced by the function S(t) introduced earlier. We may now consider the “dual” case where A is subject to an explosive instability but D is not. In this case we consider a “jittery” threshold Ds for the instability and take Ds = Ds( t) = 0; + 0; cos &Dt.
(48)
A.C. Coppi, B. Coppi/ Physics Letters A 218 (1996) M-69
A = l/[ 1 - Qr - Ebsin(wbt)/wo].
69
(52)
This expression for A^clearly shows the role that ED has in the relevant singularity. Acknowledgement
We wish to thank B. Basu for an analysis of his that has steered us away from inadequate model equations for random bursting. This work was sponsored in part by the U.S. Department of Energy.
30
40
50
So
70
so
90
100
Fig. 2. Example of “random bursting,” appearing in the solution of Eqs. (22) and (24) where S is replaced by S(t) = &+Sl cosost. In particular, & = 1, a = 2/3, Y = 2, S1 = 0.25 with initial conditions&=0.7andA=l,atr=O.
The relevant set of model equations becomes
ds=,_, dt
’
which produces sharp bursts of the fluctuation level rather than of the driving factor. With the variable Q as defined by Eq. (26), we note also that Eq. (49) can be rewritten as d@-1,
(51)
where Qr = Q - ?? o sin or>tlwD and is solved in combination with Eq. (50) where
References [ 1] B. Coppi, M. Rosenbluth aad R. Sudan, Ann. Physics 55 (1969) 207. [2] B. Coppi aud A. Friedland, Astrophysical 169 (1971) 379; B. Coppi, Astrophysical 195 (1975) 545. [3] B. Coppi, G. Cenacchi and A. Taroni, in: Plasma physics and controlled nuclear fusion 1978, Vol. 1 (Publ. IAEA, Vienna, 1979) p. 487. [4] W.W. Heidbrink, H.H. Duong, J. Maason, E. Wilfrid, C. Obermaa and E.J. Strait, Phys. Huids B 5 (1993) 2176. [5] R. Potteiette, M. Malingre, A. Bahnsen and M. Jespersen, Geophys. Res. Lett. 14 (1987) 515. [6] R.L. Arnoldy, K.A. Lynch, P.M. Kintner, J. Vago, S. Chesney, T.E. Moore and C.J. Pollock, Geophys. Res. Lett. 19 (1992) 413. [7] B. Basu and B. Coppi, Phys. Plasmas 2 (1995) 14. [ 81 AC. Coppi and B. Coppi, Bull. Am. Phys. Sot. 40 (1995) 1657. [9] L. Chen, R. White and M. Rosenbluth, Phys. Rev. Letters 55 (1984) 1122. [lo] B. Coppi and E Porcelli, Phys. Rev. Letters57 (1986) 2272. [ 111 B. Coppi, S. Migliuolo and E Porcelli, Phys. Fluids 31 (1988) 1630.
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 218 (1996) 64-69
Singular nonlinear equations and explosive recurrent events A.C. Coppi”, B. Coppi b a Department of Mathematics, Yale Universily New Haven, CT 06511, USA b Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 14 February 1996; revised manuscript received 1 May 1996; accepted for publication 1 May 1996 Communicated by M. Porkolab
Abstract Events of relatively brief duration (involving for example accelerated or heated particles, excitation of fluctuations, radiation emission) that are related to the onset of different kinds of explosive instabilities can recur at regular intervals or randomly. An analytical model is introduced, to reproduce these events, consisting of a set of nonlinear differential equations which involve a characteristic singularity. It is this feature that models explosive or quasi-explosive events either for a “primary” factor (e.g. the population of heated or accelerated particles) or for the relevant plasma fluctuations that are excited when the primary factor exceeds an appropriate threshold value.
1. Introduction There are important plasma processes that involve explosive instabilities [l-3] where the magnitude of a given physical quantity can, in principle, reach relatively high values within a relatively narrow time interval. Here we consider the particular case where a given physical factor (“primary”) such as the electron temperature in a magnetically confined plasma can periodically undergo an explosive instability which can be quenched by the onset of fluctuations that are excited when this factor exceeds an appropriate threshold. These fluctuations are excited in characteristic bursts that are rather frequently observed experimentally [ 4-61. In this Letter we propose a new set of equations, with a characteristic singularity, which can model the realistic circumstance where the factor that drives the onset of fluctuations is subject to an explosive instability. Thus a “double bursting” can occur of both the driving factor and of the induced fluctuations. The
conditions where the growth of these fluctuations can prevent the transition to an explosive state of the driving factor are identified. The equations that we introduce constitute a noncanonical Hamiltonian system. A parallel set of equations that we discuss briefly is one in which the fluctuation level itself is subject to an explosive instability. We consider two illustrative cases. One is inspired by the process of electron (Ohmic) heating that occurs in a magnetically confined plasma, when a constant (in time) electric field is applied along the magnetic field, and the plasma electrical conductivity is resulting from electron-ion collisions. The other case is that of a plasma in which the electron heating is produced by the slowing down of the charged particles produced by fusion reactions [ 31. In both cases, as the electron temperature increases the rate of heating also increases. The rate of heating corresponds to two distinct values of a “singularity parameter” that characterizes the explosive nature of the fluctuation driving factor.
0375~9601/%/$12.00 Copyright 8 1996 Published by Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00378-7
A.C. Coppi, B. Coppi/Physics
In the first case, the “singular cycle” of nonlinear oscillations is reached with a relatively wide variation of values of the Hamiltonian, up to infinity. In the second case, where the rate of increase of the electron heating with temperature is higher, the singular cycle is reached earlier, with modest variation of the Hamiltonian’s values. The Hamiltonian value corresponding to the “singular cycle” is in fact finite in this case. We note that, when the considered physical system is in an oscillation that approaches the singular cycle, it is susceptible to small perturbations that can lead it into an explosive state. In particular, this occurrence can be a random event. A special case of this circumstance is represented by the existence of a small periodic modulation of the source of the driving factor where the period of this modulation is not commensurate with that of the basic fluctuation bursts. The onset of random bursting of the fluctuation level is produced by the introduction of a small periodic modulation of the threshold for the instability driven by the primary factor when this instability is itself of the explosive type.
Letters A 218 (1996) 64-69
g
= M(D)N(A),
(4)
where N(A = A,)
and M(D)
= 0,
dN dA < 0,
(5)
is a positive function of D with
dM z > 0.
(6)
The product M(D) N( A) represents the “source function” of the driving factor D. Thus D(t) decreases when the induced fluctuation amplitudes exceed an appropriate value. It is evident that this choice of the source function does not exhaust the spectrum of physical possibilities. In particular, the source function may not separate into the product of M(D) and N(A) or may have an explicit time dependence. In other words, there are significant cases where a non-Hamiltonian [ 71 system of equations should be considered. To be specific, we consider M( 0) = D’++
2. Relevant class of equations
65
(7)
and We refer to a spatially homogeneous plasma, that may be considered as the result of an average over small scale variations of the relevant physical parameters. We indicate the primary factor that can drive the onset of fluctuations by the function D(t) and the amplitude of these fluctuations by A(t). The value of the threshold for D(t) above which A(t) begins to grow is indicated by Ds. Next, we model the evolution of D ( f ) and A(t) by a pair of coupled nonlinear equations. The first is 2
N(A)
= (T - A’,
where (Yand Y are positive parameters chosen on the basis of physical considerations. It is convenient to introduce the function
(9)
= F(D)G(A), P=l-
where F(D=Ds)
=O,
dF
dD
>O, D=Ds
and G(A)
The second equation is
$
I/a
(>
(10)
and (2)
1 D= (1-P)“’ A
is a positive function of A with
dG dA > 0.
(8)
(3)
(11)
It is evident that L3 z D/OS when p -+ 1, corresponding to the “explosive limit”. The singularity appearing in JZq. ( 11) is, in fact, the dominant characteristic of the model equations we consider.
66
A.C. Coppi, B. Coppi/Physics
Then Eq. (4) can be rewritten as
Letters A 218 (1996) 64-69
1
b=
(23)
(1 -P)“’
dP --S-/p, dt
(12)
where S E crD~‘“lcu and iv E AYD~‘a/a. introduce the function
If we
dP dt
-=s-/p.
We note that if the (secondary) instability that is involved in the onset of the considered fluctuations is nonlinear we may consider
(13) G(A) M A2 Eq. ( 1) can be rewritten as z
(25)
as a significant case. Then
= F(D(P)).
Then we may introduce the relevant Hamiltonian by writing Eqs. (12) and (14) as dP
8H
-=-dt
(15)
aQ
and dQ
aH
-=--. dt
(16)
c?P
Proceeding, we choose very simple expressions for P(D) and G(A), F(D)
- 1
= ;
(17)
and the set of equations to be solved consists of Eq. (26) and dP -dt =S-
1 (1
(27)
_Q)v’
Therefore a characteristic singularity is introduced in each of the two equations. In most of this paper we assume the “saturation” of the source term, represented by A” in Eq. (24)) to be proportional to the square of the amplitude of the fluctuations [ 9- 111, consistent with standard quasilinear considerations, and hence take Y = 2. 3. Appropriate Hamiltonians
=a,
G(A)
(18)
which correspond to a linear approximation of the equations for the instability that describes the onset of the considered fluctuations. Then Q=lnA
(l-p)”
dr=
H(eQ)
= K(P) + V(Q),
1
dp = S _ evQ, dt
dK dP=
1
(1-P)”
dV
-I9
(B - l)A,
(29)
- = e*Q - S. dQ
(21)
The constants of integration are chosen so that K( P = 0) = 0 and V(Q = Qo) = 0 where exp(2Qa) = S. The interval -cc < P < 1 is considered. When (Y> 1 we obtain
A
=
-l,
(20)
or alternatively, g
(28)
(19)
and the final form of the equations [ 81 that we analyze in the following is
dQ
Referring to Eqs. (20) and (21)) we see that the relevant Hamiltonian is given by
(22)
K(P)
= -P
(30)
1
+ -n-1
A.C. Coppi, B. Coppi/Physics
Letters A 218 (1996) 64-69
and when (Y< 1 we obtain K(P) =-P+
$$I
- (1-
P)‘_“I.
(32)
In addition, for both cases V(Q) = i[ezQ - S-S(2Q-Ins)].
(33)
We observe that when (r < 1 the “explosive state” (PMax= 1 limit) is reached with relatively small positive values of H. Conversely when (Y> 1, corresponding to a weaker rate of increase of the driving factor source as a function of D, the pkax = 1 limit is reached only with H at infinity. Finally, when cy = 1 K(P) =-P+ln
i--$ (
.
(34)
1
It is instructive to analyze the characteristics of the H = const orbits in the P, Q plane and we consider at first cy < 1. This corresponds to the case where the driving factor is the electron temperature and is sustained by the slowing-down on the electrons of the charged particles produced by fusion reactions as we shall show. In particular, we note that the limiting value of H corresponding to pMax= 1 and Q = Qc is HL=-.
cy 1-ff
(35)
Thus, if for instance (Y= 213 or HI_ = 2, that is a case of special interest to model the effects of the heating of fusion products in a well-confined plasma, we notice that, in general, the limit pMax= 1 can be approached even for small values of H. In particular, if we take Jhlax=l-E
?? =[a(l-E)+E-H(l-cu)]l’(‘-n)
(36)
we see that E remains small even for quite small values of H. This fact is connected with the strong dependence on D of the source of the driving factor corresponding to (Y< 1. We also note that the minimum value of P for H = HL is P = -pb where
pD=
1 l-n
The intersections of the “limit orbit” with the P = 0 axis, where cYH/LYP= 0, are given by the equation
f(2Qr.J =e2QL-S2QL-
S(l+lnS)+&)
=O, (39)
which, as can be verified graphically, has two solutions, a positive and a negative one. Therefore, for given values of (Yand S, the limit orbit can be identified quite well. A representative numerical solution of Eqs. (22) and (24) yielding A = A(t) andB = b(t) is given in Fig. 1. In the case where the fluctuation level A can undergo an explosive instability and is related to Q by Eq. (26), the component V(Q) of the Hamiltonian is, for v = 1, V(Q) = -SQ + 1 - ln( 1 - Q).
(40)
ii”
( >
- 1.
(37)
The corresponding value of Is is Bo=l-cu.
Fig. 1. Numericalsolution of Eqs. (22) and (24) that describe the time dependence of the driving factor, l?(t) , and of the fluctuation amplitude,A( I), with S = 1, a = 2/3, Y = 2 and pku = 0.975, for which the correspondingrelevantHamiltoniandefined by Eq. (28) has the value H = 1.2. The dashed line represents A(t) and the solid line representsB(I).
(38)
The component K(P) can be constructed in the same way as was done earlier. We notice that when cy > 1 and v = 2 the explosive limit for l3 is reached before that of A. Therefore we consider Y = 1 as a more mathematically interesting choice to analyze.
A.C. Coppi. B. Coppi/Physics
68
Letters A 218 (19%) 64-69
ar,
4. Relevant physical processes and random events
cx T3’*Cn ’ at e
In a D-T fusion burning plasma, the produced 3.5 MeV a-particles, produced by the D-T reactions, slow down by losing energy to the electron population. Thus the electron thermal energy balance equation in which we neglect, for simplicity, all forms of energy loss, becomes
Now we know that the electron temperature is subject to so-called sawtooth oscillations resulting from the excitation of fluctuations driven by the plasma pressure gradients in the center of the plasma column when the magnetic field winding parameter Lexceeds unity. Thus the value of the singularity parameter cy corresponding to this case is (YN 513. As indicated earlier, there are significant circumstances where non-Hamiltonian systems of equations have to be considered to describe the time evolution of the quantities D and A. A particularly interesting case is where a periodic component, represented for instance by S(t) = So + SI cos wst, is introduced in the source for D. Then we may define the function
in, $
= EanDnT(av)&.
(41)
Here E, II 3.5 MeV, ((Tv)~ is the D-T fusion reactivity, T, is the &CtrOU temperatUre, ne = nD + nT the electron density, and no and nr the deuteron and triton densities. If the ion and the electron temperatures are close to each other, (uv)~ can be considered to be roughly proportional to T,“‘*. The quantity C, represents the fraction of the a-particle original energy that contributes to increase the electron temperature. Thus C, includes the effects of excited fluctuations that decrease the rate of a-particle’s energy deposition on the electrons. The coefficient of the singularity parameter a introduced in EZq.(3) for the model equation that may represent this case is therefore a 21 2/3. Thus, in adopting the model equation (4)) we consider the (realistic) case where the amplitude of the modes that interact with the a-particle population increases as both the plasma temperature and the rate of Lu-particleproduction simultaneously increase. We note in fact that the so-called fishbone modes that scatter a high energy particle population in a thermal plasma are characteristic modes of the thermal plasma itself [ 101 and owe their growth rate to the resonant interaction of the mode with the high energy particle population. The other case mentioned earlier is that of a magnetically confined plasma in which a constant electric field El1 is applied along the magnetic field. In this situation, the electron thermal energy balance equation in the central part of the plasma column can be written as (42) where ~11is the longitudinalelectrical resistibility, considered to result only from collisional effects, and Cn is the fraction of the Ohmic power that contributes to increase the central temperature. Thus we may argue that
Pr f P - 2
(43)
sinwst
W
(44)
and consider, instead of Eqs. (20) and (2 1) ,
f+-+l
(45)
and -dh dt
=so -eYQ,
with a
B(Pr)=l/
.
1-F$--2sinost (
>
(47)
Clearly D( &) becomes singular when Pr = 1 (St /us) sin wt. When the frequency ws is not related to the intrinsic frequency of the system that is found for St = 0, the explosive state can be reached unpredictably thus representing the rather common phenomenon of random bursting. A relevant example is given in Fig. 2 where the numerical solution of E!qs. (22) and (24) is given with S replaced by the function S(t) introduced earlier. We may now consider the “dual” case where A is subject to an explosive instability but D is not. In this case we consider a “jittery” threshold Ds for the instability and take Ds = Ds( t) = 0; + 0; cos &Dt.
(48)
A.C. Coppi, B. Coppi/ Physics Letters A 218 (1996) M-69
A = l/[ 1 - Qr - Ebsin(wbt)/wo].
69
(52)
This expression for A^clearly shows the role that ED has in the relevant singularity. Acknowledgement
We wish to thank B. Basu for an analysis of his that has steered us away from inadequate model equations for random bursting. This work was sponsored in part by the U.S. Department of Energy.
30
40
50
So
70
so
90
100
Fig. 2. Example of “random bursting,” appearing in the solution of Eqs. (22) and (24) where S is replaced by S(t) = &+Sl cosost. In particular, & = 1, a = 2/3, Y = 2, S1 = 0.25 with initial conditions&=0.7andA=l,atr=O.
The relevant set of model equations becomes
ds=,_, dt
’
which produces sharp bursts of the fluctuation level rather than of the driving factor. With the variable Q as defined by Eq. (26), we note also that Eq. (49) can be rewritten as d@-1,
(51)
where Qr = Q - ?? o sin or>tlwD and is solved in combination with Eq. (50) where
References [ 1] B. Coppi, M. Rosenbluth aad R. Sudan, Ann. Physics 55 (1969) 207. [2] B. Coppi aud A. Friedland, Astrophysical 169 (1971) 379; B. Coppi, Astrophysical 195 (1975) 545. [3] B. Coppi, G. Cenacchi and A. Taroni, in: Plasma physics and controlled nuclear fusion 1978, Vol. 1 (Publ. IAEA, Vienna, 1979) p. 487. [4] W.W. Heidbrink, H.H. Duong, J. Maason, E. Wilfrid, C. Obermaa and E.J. Strait, Phys. Huids B 5 (1993) 2176. [5] R. Potteiette, M. Malingre, A. Bahnsen and M. Jespersen, Geophys. Res. Lett. 14 (1987) 515. [6] R.L. Arnoldy, K.A. Lynch, P.M. Kintner, J. Vago, S. Chesney, T.E. Moore and C.J. Pollock, Geophys. Res. Lett. 19 (1992) 413. [7] B. Basu and B. Coppi, Phys. Plasmas 2 (1995) 14. [ 81 AC. Coppi and B. Coppi, Bull. Am. Phys. Sot. 40 (1995) 1657. [9] L. Chen, R. White and M. Rosenbluth, Phys. Rev. Letters 55 (1984) 1122. [lo] B. Coppi and E Porcelli, Phys. Rev. Letters57 (1986) 2272. [ 111 B. Coppi, S. Migliuolo and E Porcelli, Phys. Fluids 31 (1988) 1630.