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Chaos viscosity and turbulent viscosity
Physica A 274 (1999) 476 – 483
www.elsevier.com/locate/physa
Chaos viscosity and turbulent viscosity Hiroshi Shibata ∗ Department of General Education, Kumamoto Institute of Technology, Kumamoto 860-0082, Japan Received 7 June 1999
Abstract Chaos or turbulence induces a viscosity by the contraction of motion. The driven damped pendulum which shows chaos is rewritten in the form that has viscosity derived from the nonlinear term. It is shown that the uctuation–dissipation theory holds between the dissipation and the random force derived from the nonlinear term. Then we see that the power spectrum which takes account of the chaos viscosity has smaller low-frequency power and larger high-frequency power compared to the original power spectrum. Further the turbulence viscosity is calculated c for Kuramoto–Sivashinsky equation. 1999 Elsevier Science B.V. All rights reserved. Keywords: Chaos viscosity; Turbulent viscosity; Driven damped pendulum; Kuramoto–Sivashinsky equation; Fluctuation–dissipation theory
1. Introduction It is well known that turbulence gives rise to heat [1–5]. But this phenomenon has not yet been described well. Recently, Mori proposed that the time reversible nonlinear
ux can be converted into friction and random force [6]. And this friction is much larger than the molecular viscosity. This demands that the uctuation–dissipation theory (FDT) holds between the dissipation and the random force derived from the nonlinear
ux. In this paper, this proposal is applied to the driven damped pendulum and Kuramoto– Sivashinsky (KS) equation. It is shown that this proposal can be applied to concrete systems. First the original nonlinear equation is rewritten to the one that has the viscosity by chaos or turbulence. Then the power spectra obtained from the original equation and the rewritten one are compared. When we assume that turbulence induces viscosity, the long time span motion is changed into short time noise. So the heat production ∗
Fax: +81-96-326-3000. E-mail address: [email protected] (H. Shibata)
c 0378-4371/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII : S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 4 1 2 - 4
H. Shibata / Physica A 274 (1999) 476 – 483
477
occurs. This change of motion by chaos or turbulence is detected through the change of the power spectrum. This paper is organized as follows. In Section 2 the chaos viscosity is derived for the driven damped pendulum. Then whether the FDT holds or not is studied. In Section 3 the power spectrum which has the chaos viscosity is shown and compared to the orignal power spectrum. In Section 4 KS equation is taken up and the turbulent viscosity for it is calculated. Conclusions are given in Section 5. 2. Chaos viscosity for driven damped pendulum In this section driven damped pendulum is taken up and the chaos viscosity for it is given. Further the uctuation–dissipation theory (FDT) [7–13] between the friction constant and the random force is shown to hold. Driven damped pendulum is a type which shows chaotic behavior [14 –16]. It takes the form q p d p = −sin q − 0 p + b cos ; (1) dt !0
where 0 is the friction constant due to molecular viscosity and b is the amplitude of external force, and !0 is the angular velocity of the phase . First the equation of motion has to be modi ed as chaos viscosity exists. In Eq. (1) the nonlinear term ‘−sin q’ has to be decomposed into the friction term and the random force term. In order to execute it, we use the following formula: Z t @F(q(t − s)) dq(t − s) ; (2) ds F(q(t)) = F(q(0)) + @q(t − s) d(t − s) 0 where F(q(t)) is a function of q(t). Using formula (2), the nonlinear term,−sin q(t), is rewritten as Z t − sin q(t) = −sin q(0) − ds cos q(t − s)p(t − s) : (3) 0
Then in order to extract the memory function, we extend cos q(t − s) around −s as cos q(t − s) = cos q(−s) + (t n terms) ; where n = 1; 2; 3; : : : : Eq. (2) is rewritten further to Z − sin q(t) = −sin q(0) − where
Z r(t) ≡ −
t 0
t 0
ds cos q(−s)p(t − s) + r(t) ;
ds(t n terms)p(t − s) :
(4)
(5)
(6)
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H. Shibata / Physica A 274 (1999) 476 – 483
Using Eq. (5), the equation of motion is described as
q
p
Z t d p= 0 : ds cos q(−s)p(t − s) − p + b cos + r(t) dt −sin q(0) − 0 !0 (7) Note that Eq. (7) is equivalent to Eq. (1). Assuming the contraction of motion by chaos, we obtain the equation of motion
q
p
d p = −sin q(0) − ( 0 + [0])p(t) + r(t) + b cos ; dt 0 !
(8)
that includes the chaos viscosity as Z [0] =
∞
Z
∞
ds 1 :
(9)
r(t) corresponds to random force and it is considered that the FDT holds between and r(t) when system (1) is in chaos. The FDT is written down as
[0]
0
Z [0] =
∞ 0
ds cos q(−s)
0
Z ds1 hR(s1 )R(0)i
∞
0
Z ds2 1
∞
0
dsp(s)2 ;
(10)
where R(t) = r(t) − sin q(0) and h· · ·i means the long time average. We investigate whether the FDT holds or not. We expand the r.h.s. of Eq. (10) as Z
∞ 0
Z dt1 hR(t1 )R(0)i Z
=
0
∞
0
∞
Z dt2 1 Z
dt1 hsin q(t1 ) sin q(0)i
+ [0]
Z 0
∞
∞
0 ∞
0
Z dtq(t) sin q(t)
0
dtp(t)2 Z
dt2 1 ∞
∞
0
dtp(t)2 :
dtp(t)2 (11)
On the other hand, it can be shown easily that hp(0)2 i = hq(0) sin q(0)i from Eq. (8) by partial integral. In the chaotic region the rst term of the r.h.s. of Eq. (11) vanishes for its mixing property. In the regular region it takes nite value. So the FDT holds when the system is chaotic, i.e., the Lyapunov exponent is large and the FDT does not hold when the system is nonchaotic.
H. Shibata / Physica A 274 (1999) 476 – 483
479
3. Power spectrum for chaos viscosity In this section the chaos viscosity is described by the power spectrum S(!) of momentum p(t) is de ned as n 4 2k 4X j + hp(0)2 i ; hp(j)p(0)i cos (12a) S[k] ≡ n n n j=1
2k : n The evolution equation for momentum p is described by Z t 0 ds (t − s)p(s) + r(t) + b cos (!0 t) ; p(t) ˙ = −sin q(0) − p(t) −
(12b)
![k] ≡
(13)
0
where we assume that r(t) is a random force. We multiply p(0) on both sides of Eq. (13) and take the long time average. Then the evolution equation for (t) = hp(t)p(0)i becomes Z t d (t) = 0 (t) − ds (t − s)hp(s)p(0)i − C ; (14) dt 0 where C = hsin q(0)p(0)i: Solving Eq. (14) we get S[k] as S[k] =
{ 4n hp(n)p(0)i + 4C k; 0 }( 0 + 12 Re [!] + (0)) { 0 + 12 Re [!] + (0)}2 + (! + 12 Im [!])2
0 + 12 Re [!] + (0) 4 + hp(0)2 i 1 − 0 1 n { + 2 Re [!] + (0)}2 + (! + 12 Im
! [!])2
; (15)
where Re
[!] =
n X
cos
j=1
Im
[!] = −
n X j=1
2k j n
sin
2k j n
cos q(j) ;
(16a)
cos q(j)
(16b)
for k 6= 0 and (0) = cos q(0) :
(16c)
We set 0 , !0 , and n at 0.25, 0.65, and 256, respectively. Result (15) does not include the ensemble average. So we take the long time average for [!] and (0). Then
480
H. Shibata / Physica A 274 (1999) 476 – 483
Fig. 1. Power spectra (a) and (b) are obtained from Eqs. (1) and (17), respectively. C and [0] are calculated by the use of 106 ensembles of one trajectory.
[!] → 0 and S[k] =
(0) → [0]. In this limit Eq. (15) takes the form as
{ 4n hp(n)p(0)i + 4C k; 0 }( 0 + [0]) ( 0 + [0])2 + !2 4
0 + [0] + hp(0)2 i 1 − 0 : n ( + [0])2 + !2
(17)
Fig. 1 shows the power spectra obtained from Eqs. (1) and (17). The number of ensemble is 106 for C and [0]. The power spectrum obtained from Eq. (1) exhibits slightly high peak around at 23 , which can be considered as the reminiscence of the 3 band periodic motion before the crisis. Compared to this, power spectrum (17) shows smaller low-frequency modes and larger high-frequency modes. This means that the chaos viscosity decreases the low-frequency modes and increases the high-frequency modes. In addition to this, we obtain S[0] = 2:41 from Eq. (1) and S[0] = 1:17 from Eq. (17).
4. Turbulent viscosity for Kuramoto–Sivashinsky equation In this section Kuramoto–Sivashinsky (KS) equation is taken up as an example which leads to turbulent viscosity. Turbulent viscosity is derived through the ordinary matrix form. KS equation is written as @t + + @xx + @xxxx + 2 @x = 0 ;
(18)
where is the damping constant and there is an unstable mode for ¡ 14 [17–20]. @t and @x means the partial di erentiation with respect to t and x. The control parameters are the length L of the system and the damping constant . We set periodic boundary condition. We divide the length L to N sites and write this as (t) = (
1 (t);
2 (t);
3 (t); : : : ;
j (t); : : : ;
N (t))
:
(19)
H. Shibata / Physica A 274 (1999) 476 – 483
481
Eq. (18) is rewritten to nite-di erence equation d j (t) + dt + +
j+1 (t)
+
j (t)
+
j−1 (t)
3
j+2 (t)
−4
j+1 (t)
j (t)( j+1 (t)
− x
+
+ 6 j (t) − 4 (x)4
j−1 (t))
j+1 (t)
j−1 (t)
+
− 2 j (t) + (x)2
j−1 (t)
j−2 (t)
=0;
(20)
where j is the index of space site which runs from 1 to N . The nonlinear term which leads to turbulent viscosity in Eq. (20) is [ j (t)(
j+1 (t)
−
j−1 (t))]=x:
(21)
So the nonlinear term vector is written down as f ( (t)) = (f1 ( (t)); f2 ( (t)); f3 ( (t)); : : : ; fi ( (t)); : : : ; fN ( (t))) ; where
fi ( (t)) =
i (t)
i+1 (t)
− x
i−1 (t)
:
(23)
Here, we have to extend formula (2) to Z t @f ( (t − s)) d (t − s) : ds f ( (t)) = f ( (0)) + @ (t − s) d(t − s) 0 [@f ( (t − s))]=@ (t − s) is a @f1 @ 1 @f 2 @ 1 @f ( (t − s)) =· @ (t − s) · · @fN @ 1
(22)
(24)
matrix which is expressed as @f1 @f1 @f1 ··· @ 2 @ 3 @ N @f2 @f2 @f2 ··· @ 2 @ 3 @ N ; · · ···· · · ···· · · ···· @fN @fN @fN ··· @ 2 @ 3 @ N
(25)
where @fi = @ j
i+1 (t
− s) − x
i−1 (t
− s)
i; j −
i (t
− s) i; j+1 + x
i (t
− s) i; j−1 : x
(26)
As executed in Section 2, we extract the terms which are proportional to (t − s) from [@f ( (t − s))]=[@ (t − s)][d (t − s)]=[d(t − s)]. The results are written as
= ( 1 ; 2 ; 3 ; : : : ; i ; : : : ; N ) ;
(27)
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H. Shibata / Physica A 274 (1999) 476 – 483
where
Z
i ≡
i (s)
∞
0
Z ds i (s)
0
∞
ds 1 :
(28)
is the memory function for the turbulent viscosity and is expressed as 2 1 d i−1 1 d i+1 8 2 + (s) = + + (−s) − i i x dt t=−s x dt t=−s 3 (x)3 (x)5 2 6 2 − − i+1 (−s) + 3x (x)3 (x)5 2 6 2 − + + i−1 (−s) 3x (x)3 (x)5 +
i (−s) i−1 (−s) (x)2
−
i (−s) i+1 (−s) (x)2
:
(29)
The values of i s will be published elsewhere. 5. Conclusions The method to calculate the chaos viscosity and the turbulent viscosity was shown. The nonlinear ux that is time reversible is decomposed into dissipation and random force. For driven damped pendulum the memory function that gives the chaos viscosity is calculated as (s) = cos q(−s). For Kuramoto–Sivashinsky equation the memory function that gives the turbulent viscosity is calculated as =(
1;
where i (s) =
2;
3; : : : ;
i; : : : ;
N)
;
2 1 d i−1 1 d i+1 8 2 + + + (−s) − i x dt t=−s x dt t=−s 3 (x)3 (x)5 2 6 2 − − i+1 (−s) + 3x (x)3 (x)5 2 6 2 − + + i−1 (−s) 3x (x)3 (x)5 +
i (−s) i−1 (−s) (x)2
−
i (−s) i+1 (−s) (x)2
:
The original equations that describe chaos or turbulence include only molecular viscosity. So we have to modify these equations when the chaos or turbulence occurs. This process is comparatively easy. The crucial problem is whether this theory really describes the actual physical systems, i.e., chaos viscosity or turbulent viscosity, or not. This remains as an important future problem.
H. Shibata / Physica A 274 (1999) 476 – 483
483
Acknowledgements The author would like to thank Prof. Atsuyama for fruitful discussions and especially Prof. Mori who set the problems about the chaos viscosity and the turbulent viscosity. Appendix. Integral on the chaotic attractor We often encounter the integral on the attractor as Z ∞ dsf(−s) ; 0
(A.1)
where s is time. We cannot execute this integral for chaos systems or turbulence systems. But we can consider that the initial time of this integral is in nite and the initial point is on the chaotic attractor. Then (A.1) is converted to Z ∞ dsf(s) : (A.2) 0
Especially ∞ in the integral (A.2) is converted to some nite value if (A.2) converges to some value in nite time. This is important because we numerically execute integration for nonintegrable systems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
J. Boussinesq, Mem. Acad. Sci. Paris 23 (1877) 1. W. Heisenberg, Z. Phys. 124 (1948) 628. W. Heisenberg, Proc. Roy. Soc. London A 195 (1948) 402. C.E. Leith, J. Atmos. Sci. 28 (1971) 145. U. Frisch, Turbulence, Cambridge University Press, Cambridge, MA, 1995. H. Mori, H. Fujisaka, Transport and Entropy Production due to Chaos or Turbulence, preprint. H.B. Callen, T.A. Welton, Phys. Rev. 83 (1951) 34. H. Takahashi, J. Phys. Soc. Japan 7 (1952) 439. R. Kubo, J. Phys. Soc. Japan 12 (1957) 507. S.R. de Groot, P. Mazur, Nonequilibrium Thermodynamics, North-Holland, Amsterdam, 1962. H. Mori, Prog. Theor. Phys. 33 (1965) 423. R. Kubo, Rep. Prog. Phys. 29 Part I (1996) 255. S.W. Lovesey, Condensed Matter Physics: Dynamics Correlations, 2nd Edition, Benjamin=Cummings, Menlo Park. CA. E.A. Jackson, Perspectives of Nonlinear Dynamics, Vol. 1, Cambridge University Press, Cambridge, 1989. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, MA 1993. H. Mori, Y. Kuramoto, Dissipative Structures and Chaos, Springer, Berlin, 1998. Y. Kuramoto, T. Tsuzuki, Prog. Theor. Phys. 55 (1976) 356. G.I. Sivashinsky, Acta Astronaut. 4 (1977) 1177. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851.
www.elsevier.com/locate/physa
Chaos viscosity and turbulent viscosity Hiroshi Shibata ∗ Department of General Education, Kumamoto Institute of Technology, Kumamoto 860-0082, Japan Received 7 June 1999
Abstract Chaos or turbulence induces a viscosity by the contraction of motion. The driven damped pendulum which shows chaos is rewritten in the form that has viscosity derived from the nonlinear term. It is shown that the uctuation–dissipation theory holds between the dissipation and the random force derived from the nonlinear term. Then we see that the power spectrum which takes account of the chaos viscosity has smaller low-frequency power and larger high-frequency power compared to the original power spectrum. Further the turbulence viscosity is calculated c for Kuramoto–Sivashinsky equation. 1999 Elsevier Science B.V. All rights reserved. Keywords: Chaos viscosity; Turbulent viscosity; Driven damped pendulum; Kuramoto–Sivashinsky equation; Fluctuation–dissipation theory
1. Introduction It is well known that turbulence gives rise to heat [1–5]. But this phenomenon has not yet been described well. Recently, Mori proposed that the time reversible nonlinear
ux can be converted into friction and random force [6]. And this friction is much larger than the molecular viscosity. This demands that the uctuation–dissipation theory (FDT) holds between the dissipation and the random force derived from the nonlinear
ux. In this paper, this proposal is applied to the driven damped pendulum and Kuramoto– Sivashinsky (KS) equation. It is shown that this proposal can be applied to concrete systems. First the original nonlinear equation is rewritten to the one that has the viscosity by chaos or turbulence. Then the power spectra obtained from the original equation and the rewritten one are compared. When we assume that turbulence induces viscosity, the long time span motion is changed into short time noise. So the heat production ∗
Fax: +81-96-326-3000. E-mail address: [email protected] (H. Shibata)
c 0378-4371/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII : S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 4 1 2 - 4
H. Shibata / Physica A 274 (1999) 476 – 483
477
occurs. This change of motion by chaos or turbulence is detected through the change of the power spectrum. This paper is organized as follows. In Section 2 the chaos viscosity is derived for the driven damped pendulum. Then whether the FDT holds or not is studied. In Section 3 the power spectrum which has the chaos viscosity is shown and compared to the orignal power spectrum. In Section 4 KS equation is taken up and the turbulent viscosity for it is calculated. Conclusions are given in Section 5. 2. Chaos viscosity for driven damped pendulum In this section driven damped pendulum is taken up and the chaos viscosity for it is given. Further the uctuation–dissipation theory (FDT) [7–13] between the friction constant and the random force is shown to hold. Driven damped pendulum is a type which shows chaotic behavior [14 –16]. It takes the form q p d p = −sin q − 0 p + b cos ; (1) dt !0
where 0 is the friction constant due to molecular viscosity and b is the amplitude of external force, and !0 is the angular velocity of the phase . First the equation of motion has to be modi ed as chaos viscosity exists. In Eq. (1) the nonlinear term ‘−sin q’ has to be decomposed into the friction term and the random force term. In order to execute it, we use the following formula: Z t @F(q(t − s)) dq(t − s) ; (2) ds F(q(t)) = F(q(0)) + @q(t − s) d(t − s) 0 where F(q(t)) is a function of q(t). Using formula (2), the nonlinear term,−sin q(t), is rewritten as Z t − sin q(t) = −sin q(0) − ds cos q(t − s)p(t − s) : (3) 0
Then in order to extract the memory function, we extend cos q(t − s) around −s as cos q(t − s) = cos q(−s) + (t n terms) ; where n = 1; 2; 3; : : : : Eq. (2) is rewritten further to Z − sin q(t) = −sin q(0) − where
Z r(t) ≡ −
t 0
t 0
ds cos q(−s)p(t − s) + r(t) ;
ds(t n terms)p(t − s) :
(4)
(5)
(6)
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H. Shibata / Physica A 274 (1999) 476 – 483
Using Eq. (5), the equation of motion is described as
q
p
Z t d p= 0 : ds cos q(−s)p(t − s) − p + b cos + r(t) dt −sin q(0) − 0 !0 (7) Note that Eq. (7) is equivalent to Eq. (1). Assuming the contraction of motion by chaos, we obtain the equation of motion
q
p
d p = −sin q(0) − ( 0 + [0])p(t) + r(t) + b cos ; dt 0 !
(8)
that includes the chaos viscosity as Z [0] =
∞
Z
∞
ds 1 :
(9)
r(t) corresponds to random force and it is considered that the FDT holds between and r(t) when system (1) is in chaos. The FDT is written down as
[0]
0
Z [0] =
∞ 0
ds cos q(−s)
0
Z ds1 hR(s1 )R(0)i
∞
0
Z ds2 1
∞
0
dsp(s)2 ;
(10)
where R(t) = r(t) − sin q(0) and h· · ·i means the long time average. We investigate whether the FDT holds or not. We expand the r.h.s. of Eq. (10) as Z
∞ 0
Z dt1 hR(t1 )R(0)i Z
=
0
∞
0
∞
Z dt2 1 Z
dt1 hsin q(t1 ) sin q(0)i
+ [0]
Z 0
∞
∞
0 ∞
0
Z dtq(t) sin q(t)
0
dtp(t)2 Z
dt2 1 ∞
∞
0
dtp(t)2 :
dtp(t)2 (11)
On the other hand, it can be shown easily that hp(0)2 i = hq(0) sin q(0)i from Eq. (8) by partial integral. In the chaotic region the rst term of the r.h.s. of Eq. (11) vanishes for its mixing property. In the regular region it takes nite value. So the FDT holds when the system is chaotic, i.e., the Lyapunov exponent is large and the FDT does not hold when the system is nonchaotic.
H. Shibata / Physica A 274 (1999) 476 – 483
479
3. Power spectrum for chaos viscosity In this section the chaos viscosity is described by the power spectrum S(!) of momentum p(t) is de ned as n 4 2k 4X j + hp(0)2 i ; hp(j)p(0)i cos (12a) S[k] ≡ n n n j=1
2k : n The evolution equation for momentum p is described by Z t 0 ds (t − s)p(s) + r(t) + b cos (!0 t) ; p(t) ˙ = −sin q(0) − p(t) −
(12b)
![k] ≡
(13)
0
where we assume that r(t) is a random force. We multiply p(0) on both sides of Eq. (13) and take the long time average. Then the evolution equation for (t) = hp(t)p(0)i becomes Z t d (t) = 0 (t) − ds (t − s)hp(s)p(0)i − C ; (14) dt 0 where C = hsin q(0)p(0)i: Solving Eq. (14) we get S[k] as S[k] =
{ 4n hp(n)p(0)i + 4C k; 0 }( 0 + 12 Re [!] + (0)) { 0 + 12 Re [!] + (0)}2 + (! + 12 Im [!])2
0 + 12 Re [!] + (0) 4 + hp(0)2 i 1 − 0 1 n { + 2 Re [!] + (0)}2 + (! + 12 Im
! [!])2
; (15)
where Re
[!] =
n X
cos
j=1
Im
[!] = −
n X j=1
2k j n
sin
2k j n
cos q(j) ;
(16a)
cos q(j)
(16b)
for k 6= 0 and (0) = cos q(0) :
(16c)
We set 0 , !0 , and n at 0.25, 0.65, and 256, respectively. Result (15) does not include the ensemble average. So we take the long time average for [!] and (0). Then
480
H. Shibata / Physica A 274 (1999) 476 – 483
Fig. 1. Power spectra (a) and (b) are obtained from Eqs. (1) and (17), respectively. C and [0] are calculated by the use of 106 ensembles of one trajectory.
[!] → 0 and S[k] =
(0) → [0]. In this limit Eq. (15) takes the form as
{ 4n hp(n)p(0)i + 4C k; 0 }( 0 + [0]) ( 0 + [0])2 + !2 4
0 + [0] + hp(0)2 i 1 − 0 : n ( + [0])2 + !2
(17)
Fig. 1 shows the power spectra obtained from Eqs. (1) and (17). The number of ensemble is 106 for C and [0]. The power spectrum obtained from Eq. (1) exhibits slightly high peak around at 23 , which can be considered as the reminiscence of the 3 band periodic motion before the crisis. Compared to this, power spectrum (17) shows smaller low-frequency modes and larger high-frequency modes. This means that the chaos viscosity decreases the low-frequency modes and increases the high-frequency modes. In addition to this, we obtain S[0] = 2:41 from Eq. (1) and S[0] = 1:17 from Eq. (17).
4. Turbulent viscosity for Kuramoto–Sivashinsky equation In this section Kuramoto–Sivashinsky (KS) equation is taken up as an example which leads to turbulent viscosity. Turbulent viscosity is derived through the ordinary matrix form. KS equation is written as @t + + @xx + @xxxx + 2 @x = 0 ;
(18)
where is the damping constant and there is an unstable mode for ¡ 14 [17–20]. @t and @x means the partial di erentiation with respect to t and x. The control parameters are the length L of the system and the damping constant . We set periodic boundary condition. We divide the length L to N sites and write this as (t) = (
1 (t);
2 (t);
3 (t); : : : ;
j (t); : : : ;
N (t))
:
(19)
H. Shibata / Physica A 274 (1999) 476 – 483
481
Eq. (18) is rewritten to nite-di erence equation d j (t) + dt + +
j+1 (t)
+
j (t)
+
j−1 (t)
3
j+2 (t)
−4
j+1 (t)
j (t)( j+1 (t)
− x
+
+ 6 j (t) − 4 (x)4
j−1 (t))
j+1 (t)
j−1 (t)
+
− 2 j (t) + (x)2
j−1 (t)
j−2 (t)
=0;
(20)
where j is the index of space site which runs from 1 to N . The nonlinear term which leads to turbulent viscosity in Eq. (20) is [ j (t)(
j+1 (t)
−
j−1 (t))]=x:
(21)
So the nonlinear term vector is written down as f ( (t)) = (f1 ( (t)); f2 ( (t)); f3 ( (t)); : : : ; fi ( (t)); : : : ; fN ( (t))) ; where
fi ( (t)) =
i (t)
i+1 (t)
− x
i−1 (t)
:
(23)
Here, we have to extend formula (2) to Z t @f ( (t − s)) d (t − s) : ds f ( (t)) = f ( (0)) + @ (t − s) d(t − s) 0 [@f ( (t − s))]=@ (t − s) is a @f1 @ 1 @f 2 @ 1 @f ( (t − s)) =· @ (t − s) · · @fN @ 1
(22)
(24)
matrix which is expressed as @f1 @f1 @f1 ··· @ 2 @ 3 @ N @f2 @f2 @f2 ··· @ 2 @ 3 @ N ; · · ···· · · ···· · · ···· @fN @fN @fN ··· @ 2 @ 3 @ N
(25)
where @fi = @ j
i+1 (t
− s) − x
i−1 (t
− s)
i; j −
i (t
− s) i; j+1 + x
i (t
− s) i; j−1 : x
(26)
As executed in Section 2, we extract the terms which are proportional to (t − s) from [@f ( (t − s))]=[@ (t − s)][d (t − s)]=[d(t − s)]. The results are written as
= ( 1 ; 2 ; 3 ; : : : ; i ; : : : ; N ) ;
(27)
482
H. Shibata / Physica A 274 (1999) 476 – 483
where
Z
i ≡
i (s)
∞
0
Z ds i (s)
0
∞
ds 1 :
(28)
is the memory function for the turbulent viscosity and is expressed as 2 1 d i−1 1 d i+1 8 2 + (s) = + + (−s) − i i x dt t=−s x dt t=−s 3 (x)3 (x)5 2 6 2 − − i+1 (−s) + 3x (x)3 (x)5 2 6 2 − + + i−1 (−s) 3x (x)3 (x)5 +
i (−s) i−1 (−s) (x)2
−
i (−s) i+1 (−s) (x)2
:
(29)
The values of i s will be published elsewhere. 5. Conclusions The method to calculate the chaos viscosity and the turbulent viscosity was shown. The nonlinear ux that is time reversible is decomposed into dissipation and random force. For driven damped pendulum the memory function that gives the chaos viscosity is calculated as (s) = cos q(−s). For Kuramoto–Sivashinsky equation the memory function that gives the turbulent viscosity is calculated as =(
1;
where i (s) =
2;
3; : : : ;
i; : : : ;
N)
;
2 1 d i−1 1 d i+1 8 2 + + + (−s) − i x dt t=−s x dt t=−s 3 (x)3 (x)5 2 6 2 − − i+1 (−s) + 3x (x)3 (x)5 2 6 2 − + + i−1 (−s) 3x (x)3 (x)5 +
i (−s) i−1 (−s) (x)2
−
i (−s) i+1 (−s) (x)2
:
The original equations that describe chaos or turbulence include only molecular viscosity. So we have to modify these equations when the chaos or turbulence occurs. This process is comparatively easy. The crucial problem is whether this theory really describes the actual physical systems, i.e., chaos viscosity or turbulent viscosity, or not. This remains as an important future problem.
H. Shibata / Physica A 274 (1999) 476 – 483
483
Acknowledgements The author would like to thank Prof. Atsuyama for fruitful discussions and especially Prof. Mori who set the problems about the chaos viscosity and the turbulent viscosity. Appendix. Integral on the chaotic attractor We often encounter the integral on the attractor as Z ∞ dsf(−s) ; 0
(A.1)
where s is time. We cannot execute this integral for chaos systems or turbulence systems. But we can consider that the initial time of this integral is in nite and the initial point is on the chaotic attractor. Then (A.1) is converted to Z ∞ dsf(s) : (A.2) 0
Especially ∞ in the integral (A.2) is converted to some nite value if (A.2) converges to some value in nite time. This is important because we numerically execute integration for nonintegrable systems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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