{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 146 "Investigate the effects \+ of a finite-correlation-time noise on a Langevin equation.\nUsing Krom mes' symmetrization trick to get Galilean invariance." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Equations and notati on used here follow the paper" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 198 "\"Non-white noise and a multiple rate Ma rkovian closure theories for turbulence\", by Gregory W. Hammett and J ohn C. Bowman. Submitted to Physics of Fluids, Feb. 2002, and availab le at www.arxiv.org." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 204 "In particular, this maple script was used to check and produce some of the results in Section 3 on the effects of finite-cor relation-time noise on a Langevin equation, and to make the plots foun d there. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "Note that some of the notation used here is a little bit diffe rent than in the paper. In particular, the various rates are denoted \+ by eta in the paper but by nu here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "rednoise k12 like rednoisek11, but with some additional plots for the paper.\n " }}{PARA 0 "" 0 "" {TEXT -1 115 "rednoisek11 like rednoisek10, but us ing the direct solution instead of the iterative solution for nu_eff(n u, nu_f)\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "restart; kern elopts(version); interface(version); ssystem(\"date\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%XMaple~7.00,~IBM~INTEL~LINUX,~May~28~2001~Build ~ID~96223G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pMaple~Worksheet~Inter face,~Maple~7.00,~IBM~INTEL~LINUX,~May~28~2001~Build~ID~96223G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!Q>Tue~Feb~12~12:29:25~GMT~2002| +6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "# We are working w ith the Langevin equation of the following form.\n\ndiff(psi(t),t) = - nu * psi + conjugate(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG 6$-%$psiG6#%\"tGF*,&*&%#nuG\"\"\"F(F.!\"\"-%*conjugateG6#%\"fGF." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "#\n# where nu is the decay \+ rate and conjugate(f) is the noise (forcing) term.\n# Note that the co njugate(f) convention is being used, to make it similar to \n# the pra ctice in the DIA/EDQNM/RMC.\n#\n# Note that the damping rate is denote d by eta in the paper, but by nu here.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 " # Next is the 2-time correlation function for comp lex arguments that I found both by\n # Fourier transforming and integ rating around the poles, and by a longer calculation.\n # Also note t hat I am using the DIA convention that conjugate(f) appears in the \n \+ # Langevin equation, so the resonance condition is omega+omega_f=0.\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "c2 := proc(tau,nu,nu_f ) ( (nu_f+conjugate(nu_f))\n /(conjugate(nu_f)-nu) *( exp(-nu*tau) \+ /(nu+conjugate(nu))/(nu+nu_f)\n -exp(-conjug ate(nu_f)*tau)/(nu_f+conjugate(nu_f))/(conjugate(nu)+conjugate(nu_f)) \+ ) ) end ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#c2Gf*6%%$tauG%#nuG%%n u_fG6\"F*F**(,&9&\"\"\"-%*conjugateG6#F-F.F.,&F/F.9%!\"\"F4,&*(-%$expG 6#,$*&F3F.9$F.F4F.,&F3F.-F06#F3F.F4,&F3F.F-F.F4F.*(-F86#,$*&F/F.FF.F/F.F4F4F.F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "c0 := proc(nu,nu_f) ( (nu + conjugate(nu)+nu_f+conjugate(nu_f)) \n /(nu+conjugate(nu)) / (nu + nu_f) / conjugate(nu+nu_f) ) \+ end ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c0Gf*6$%#nuG%%nu_fG6\"F)F) **,*9$\"\"\"-%*conjugateG6#F,F-9%F--F/6#F1F-F-,&F,F-F.F-!\"\",&F,F-F1F -F5-F/6#F6F5F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c2(ta u,nu,nu_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%%nu_fG\"\"\"-%*con jugateG6#F%F&F&,&F'F&%#nuG!\"\"F,,&*(-%$expG6#,$*&F+F&%$tauGF&F,F&,&F+ F&-F(6#F+F&F,,&F+F&F%F&F,F&*(-F06#,$*&F'F&F4F&F,F&F$F,,&F6F&F'F&F,F,F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "normal(c2(0,nu,nu_f)); \+ c0(nu,nu_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,*-%*conjugateG6#%%n u_fG\"\"\"F(F)-F&6#%#nuGF)F,F)F),&F*F)F%F)!\"\",&F,F)F*F)F.,&F,F)F(F)F ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,*-%*conjugateG6#%%nu_fG\"\"\"F (F)-F&6#%#nuGF)F,F)F),&F,F)F*F)!\"\",&F,F)F(F)F.-F&6#F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "numer(normal(c2(0,nu,nu_f))) - nume r(c0(nu,nu_f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "denom(normal(c2(0,nu,nu_f))) - expa nd(denom(c0(nu,nu_f)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*(,&-%*co njugateG6#%#nuG\"\"\"-F'6#%%nu_fGF*F*,&F)F*F&F*F*,&F)F*F-F*F*F**&)F)\" \"#F*F&F*!\"\"*&F1F*F+F*F3*(F)F*F-F*F&F*F3*(F)F*F-F*F+F*F3*&)F&F2F*F)F *F3*(F&F*F)F*F+F*F3*&F8F*F-F*F3*(F&F*F-F*F+F*F3" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "c2(0,g+I*w,g_f+I *w_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,(%$g_fG\"\"\"*&^#F&F&%$w_ fGF&F&-%*conjugateG6#,&F%F&F'F&F&F&,(F*F&%\"gG!\"\"*&^#F0F&%\"wGF&F&F0 ,&*&F&F&*&,(F/F&*&F(F&F3F&F&-F+6#,&F/F&F8F&F&F&,*F/F&F8F&F%F&F'F&F&F0F &*&F&F&*&F$F&,&F9F&F*F&F&F0F0F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(**%$g_fG\"\"\",& F%F&%\"gG!\"\"F&,&*$)F'\"\"#F&F&*$),&%$w_fGF)%\"wGF)F-F&F&F),&*(F(F),& F(F&F%F&F&,&*$)F5F-F&F&*$),&F2F&F1F&F-F&F&F)#F&F-*&#F&F-F&*(F%F)F5F&,& F7F&F.F&F)F&F)F&F-*,F-F&F%F&F0F&F*F),&*(F(F)F;F&F6F)#F)F-**F " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"gG\"\"\"%$g_fGF&F&F%!\"\",.*$)F%\"\"# F&F&*(F,F&F'F&F%F&F&*$)F'F,F&F&*$)%$w_fGF,F&F&*(F,F&F2F&%\"wGF&F&*$)F4 F,F&F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "# This has a n ice simple form, and shows c2(t=0) is purely real, as it should be.\n# Use this to redefine c2 so c2(tau=0)=1:\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 346 "c2 := proc(tau,nu,nu_f) \n local g, g_f, c0; \n g:=Re(nu) ;\n g_f:=Re(nu_f) ;\n c0:=(1/g + 1/g_f)/2/ ((g+g_f)^2 + (Im(nu)+Im(nu_f))^2); \n1/c0*1/(conjugate(nu_f)-nu) \+ *( exp(-nu*tau) /(nu+conjugate(nu))/(nu+nu_f)\n \+ -exp(-conjugate(nu_f)*tau)/(nu_f+conjugate(nu_f))/(conjugate(nu)+conju gate(nu_f) ) ) end ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#c2Gf*6%% $tauG%#nuG%%nu_fG6%%\"gG%$g_fG%#c0G6\"F.C&>8$-%#ReG6#9%>8%-F36#9&>8&,$ *&,&*&\"\"\"FAF1!\"\"FA*&FAFAF7FBFAFA,&*$),&F1FAF7FA\"\"#FAFA*$),&-%#I mGF4FA-FMF9FAFHFAFAFB#FAFH*(F " 0 "" {MPLTEXT 1 0 14 "c2(0,nu,nu_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* *,&*&\"\"\"F'-%#ReG6#%#nuG!\"\"F'*&F'F'-F)6#%%nu_fGF,F'F,,&*$),&F(F'F. F'\"\"#F'F'*$),&-%#ImGF*F'-F:F/F'F5F'F'F',&-%*conjugateGF/F'F+F,F,,&*& F'F'*&,&F+F'-F>F*F'F',&F+F'F0F'F'F,F'*&F'F'*&,&F0F'F=F'F',&FCF'F=F'F'F ,F,F'F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "c2(0,1,2); # ve rify that this rescaled c2 is 1 at tau=0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "c2(0,2.0+3. 0*I, 4.0+7.0*I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+++++5!\"*$!+ :YQ:'*!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "c2(1,1-I,2-I); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&^$#\"#_\"#:#!$/\"F'\"\"\",&*&^$# \"\"$\"#E#F*\"#8F*-%$expG6#^$!\"\"F*F*F**&^$#!\"$F&#F*F0F*-F46#^$!\"#F 7F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 387 "# recursive def inition of nu_eff with symmetrized weight\nnu_eff_rec := proc(nu,nu_f, nu_eff) \n ( nu*conjugate(nu_f)*(nu+nu_f + conjugate(nu)+conjugat e(nu_f)) \n + conjugate(nu_eff) * (nu * conjugate(nu_f) - nu_f*c onjugate(nu)) )\n / ( (nu +conjugate(nu_eff)) * (nu + nu_f + conju gate(nu) + conjugate(nu_f)) \n + (conjugate(nu_f)+conjugate(nu))*(n u_f+conjugate(nu_f)) ) end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+nu_e ff_recGf*6%%#nuG%%nu_fG%'nu_effG6\"F*F**&,&*(9$\"\"\"-%*conjugateG6#9% F/,*F.F/-F16#F.F/F3F/F0F/F/F/*&-F16#9&F/,&*&F.F/F0F/F/*&F3F/F5F/!\"\"F /F/F/,&*&,&F.F/F8F/F/F4F/F/*&,&F0F/F5F/F/,&F3F/F0F/F/F/F>F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "nu_eff_rec(nu,nu_f,nu_eff); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*(%#nuG\"\"\"-%*conjugateG6#%% nu_fGF',*F(F'F+F'-F)6#F&F'F&F'F'F'*&-F)6#%'nu_effGF',&*&F&F'F(F'F'*&F- F'F+F'!\"\"F'F'F',&*&,&F&F'F0F'F'F,F'F'*&,&F-F'F(F'F',&F+F'F(F'F'F'F6 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "nu_eff_rec(1.0, 0.2+5*I , 0.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+9(\\X1#!\"*$!+)e^Ht#F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 695 "# A complete definiti on of nu_eff (which iterates internally till it converges):\nnu_effc : = proc(nu, nu_f)\n local nu_eff, nu_eff0, error_rms, i;\n if n ot (type(nu,complexcons) and type(nu_f,complexcons)) then\n RE TURN('procname(nu, nu_f)'); # if arguments aren't (complex) numbers, r eturn unevaluated.\n fi;\n nu_eff := 0.0; # Initial guess\n \+ for i from 1 to 100 do \n nu_eff0 := nu_eff; nu_eff := eval f(nu_eff_rec(nu, nu_f, nu_eff));\n error_rms := abs(nu_eff0-nu_ eff);\n if ( i > 5 and error_rms < 1.0e-8 ) then break fi;\n \+ od;\n if(error_rms>1.0e-8) then print(\"Warning in nu_eff: erro r, # of iterations =\",error_rms, i); fi;\n nu_eff\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(nu_effcGf*6$%#nuG%%nu_fG6&%'nu_effG%(nu_e ff0G%*error_rmsG%\"iG6\"F.C'@$43-%%typeG6$9$%,complexconsG-F46$9%F7-%' RETURNG6#.-9!6$F6F:>8$$\"\"!FE?(8'\"\"\"FH\"$+\"%%trueGC&>8%FC>FC-%&ev alfG6#-%+nu_eff_recG6%F6F:FC>8&-%$absG6#,&FMFHFC!\"\"@$32\"\"&FG2FV$\" #5!\"*[@$2F[oFV-%&printG6%QMWarning~in~nu_eff:~~error,~#~of~iterations ~=F.FVFGFCF.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1202 "# Dir ect evalution of nu_eff from nu and nu_f:\n\nnu_effd := proc(nu,nu_f) \+ \n local g, w, g_f, w_f , g_eff, w_eff;\n g := Re(nu) ; \+ w := Im(nu);\n g_f := Re(nu_f) ; w_f := Im(nu_f);\ng_eff := -1/ 2*(g^4+g_f^2*w^2+w^2*g^2+6*g^2*g_f^2+4*g^3*g_f+4*g*g_f^3-2*g*g_f*w^2+g _f^4-4*g*w*w_f*g_f+w_f^2*g_f^2+g^2*w_f^2+2*g^2*w*w_f-2*w_f^2*g_f*g+2*g _f^2*w*w_f-sqrt((g+g_f)^2*(g^6-2*w_f^4*g_f*g+w^4*g^2+w^4*g_f^2+2*w_f^2 *g_f^4+4*g_f^4*w*w_f+4*g_f^2*w^3*w_f+6*w^2*w_f^2*g_f^2+4*w*w_f^3*g_f^2 -12*g*g_f*w_f^2*w^2-8*g*g_f*w_f*w^3-8*g*g_f*w*w_f^3+2*g^4*w_f^2+4*g^4* w*w_f+6*g^2*w_f^2*w^2+4*g^2*w_f*w^3+4*g^2*w*w_f^3+2*g^4*w^2+2*g_f^4*w^ 2+g^2*w_f^4+w_f^4*g_f^2+g_f^6-2*w^4*g*g_f-4*g_f^2*g^2*w^2+8*g*g_f^3*w^ 2+8*g^3*g_f*w^2+31*g^2*g_f^4+44*g^3*g_f^3+8*w_f^2*g_f*g^3-4*w_f^2*g_f^ 2*g^2+8*w_f^2*g_f^3*g+31*g^4*g_f^2-8*w_f*g_f^2*g^2*w+16*w_f*g_f^3*g*w+ 16*g^3*g_f*w*w_f+10*g^5*g_f+10*g*g_f^5)))*(g+g_f)/(g^4+g_f^2*w^2+w^2*g ^2+6*g^2*g_f^2+4*g^3*g_f+4*g*g_f^3-2*g*g_f*w^2+g_f^4-4*g*w*w_f*g_f+w_f ^2*g_f^2+g^2*w_f^2+2*g^2*w*w_f-2*w_f^2*g_f*g+2*g_f^2*w*w_f) \+ ;\nw_eff := -(g*w*g_eff-g_eff*w_f*g_f+g^2*w_f+g*w_f*g_eff+w_f*g_f*g-g* g_f*w-g_f^2*w-g_f*w*g_eff)/(g^2+g_f^2+2*g*g_f) ;\nevalf(evalc(g_eff+I* w_eff));\nend ;\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(nu_effdGf*6$%# nuG%%nu_fG6(%\"gG%\"wG%$g_fG%$w_fG%&g_effG%&w_effG6\"F0C)>8$-%#ReG6#9$ >8%-%#ImGF6>8&-F56#9%>8'-F;F?>8(,$*(,@*$)F3\"\"%\"\"\"FL*&)F=\"\"#FL)F 9FOFLFL*&FPFL)F3FOFLFL*(\"\"'FLFRFLFNFLFL*(FKFL)F3\"\"$FLF=FLFL*(FKFLF 3FL)F=FWFLFL**FOFLF3FLF=FLFPFL!\"\"*$)F=FKFLFL*,FKFLF3FLF9FLFBFLF=FLFe n*&)FBFOFLFNFLFL*&FRFLFjnFLFL**FOFLFRFLF9FLFBFLFL**FOFLFjnFLF=FLF3FLFe n**FOFLFNFLF9FLFBFLFL-%%sqrtG6#*&),&F3FLF=FLFOFL,fo*(FJFLF9FLFBFLFK**F OFL)F9FKFLF3FLF=FLFen**\"\")FLF3FLFYFLFPFLFL**FKFLF9FL)FBFWFLFNFLFL**F joFLFjnFLF=FLFVFLFL**FKFLFjnFLFNFLFRFLFen*,\"#7FLF3FLF=FLFjnFLFPFLFen* ,FjoFLF3FLF=FLFBFL)F9FWFLFen*,FjoFLF3FLF=FLF9FLF\\pFLFen*,FjoFLFBFLFNF LFRFLF9FLFen*,\"#;FLFBFLFYFLF3FLF9FLFL*,FfpFLFVFLF=FLF9FLFBFLFL*$)F3FT FLFL*$)F=FTFLFL**FOFL)FBFKFLF=FLF3FLFen**FTFLFRFLFjnFLFPFLFL**FKFLFRFL FBFLFbpFLFL**FKFLFRFLF9FLF\\pFLFL**FjoFLFVFLF=FLFPFLFL**FjoFLFjnFLFYFL F3FLFL*&FhoFLFRFLFL*&FhoFLFNFLFL*&FRFLF]qFLFL*&F]qFLFNFLFL*(FOFLFjnFLF gnFLFL*(FOFLFJFLFjnFLFL*(FOFLFJFLFPFLFL*(FOFLFgnFLFPFLFL*(\"#JFLFRFLFg nFLFL*(\"#WFLFVFLFYFLFL*(F\\rFLFJFLFNFLFL*(\"#5FL)F3\"\"&FLF=FLFL*(Far FLF3FL)F=FcrFLFL**FTFLFPFLFjnFLFNFLFL**FKFLFNFLFbpFLFBFLFL**FKFLFgnFLF 9FLFBFLFL**FKFLFNFLFRFLFPFLFenFLFenFLFdoFL,>FIFLFMFLFQFL*(FTFLFRFLFNFL FL*(FKFLFVFLF=FLFL*(FKFLF3FLFYFLFL**FOFLF3FLF=FLFPFLFenFfnFL*,FKFLF3FL F9FLFBFLF=FLFenFinFLF[oFL**FOFLFRFLF9FLFBFLFL**FOFLFjnFLF=FLF3FLFen**F OFLFNFLF9FLFBFLFLFen#FenFO>8),$*&,2*(F3FLF9FLFEFLFL*(FEFLFBFLF=FLFen*& FRFLFBFLFL*(F3FLFBFLFEFLFL*(FBFLF=FLF3FLFL*(F3FLF=FLF9FLFen*&FNFLF9FLF en*(F=FLF9FLFEFLFenFL,(*$FRFLFL*$FNFLFL*(FOFLF3FLF=FLFLFenFen-%&evalfG 6#-%&evalcG6#,&FEFL*&^#FLFLFesFLFLF0F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "nu_effd(1, 4+10*I); nu_effc(1, 4+10*I);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"+\"G%)3I\"!\"*$!*k)Q*Q%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+zU)3I\"!\"*$!+?')Q*Q%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "nu_effd(1,1); nu_effc(1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*iN@9%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +@c8UT!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "nu_effd(1,1+16 *I); nu_effc(1,1+16*I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+0%QS7 (!\"*$!\")\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+1%QS7(!\"*$!+ r******zF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "nu_effd(1,1+5 0*I); nu_effc(1,1+50*I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+0o*R S#!\")$!#D\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%QMWarning~in~nu_eff :~~error,~#~of~iterations~=6\"$\"+`;ja8!#7\"$,\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+y\\*RS#!\")$!+^?P)\\#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "tau_eff := proc(nu,nu_f) 1/nu_effd(nu,nu_f) end \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(tau_effGf*6$%#nuG%%nu_fG6\"F)F )*&\"\"\"F+-%(nu_effdG6$9$9%!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "tau_eff(1.0,1.0+I);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#^$$\"+++++5!\"*F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 370 "# O lder, more complex definition of tau_eff.\ntau_effa := proc(nu,nu_f) \+ \n local g, g_f, c0;\n g:=Re(nu) ;\n g_f:=Re(nu_f) ;\n \+ c0:=(1/g + 1/g_f)/2/((g+g_f)^2 + (Im(nu)-Im(nu_f))^2); \n1/c 0*1/(nu_f-nu) *( 1/nu/(nu+conjugate(nu))/(nu+conjugate(nu_f))\n \+ -1/nu_f/(nu_f+conjugate(nu_f))/(conju gate(nu)+nu_f) ) end ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)tau_ef faGf*6$%#nuG%%nu_fG6%%\"gG%$g_fG%#c0G6\"F-C&>8$-%#ReG6#9$>8%-F26#9%>8& ,$*&,&*&\"\"\"F@F0!\"\"F@*&F@F@F6FAF@F@,&*$),&F0F@F6F@\"\"#F@F@*$),&-% #ImGF3F@-FLF8FAFGF@F@FA#F@FG*(F;FA,&F9F@F4FAFA,&*&F@F@*(F4F@,&F4F@-%*c onjugateGF3F@F@,&F4F@-FVF8F@F@FAF@*&F@F@*(F9F@,&F9F@FXF@F@,&FUF@F9F@F@ FAFAF@F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 614 "# Older, m ore complex, definition of tau_eff, now including Galilean invariance. \n# Perhaps the best fit?\ntau_effg := proc(nu,nu_f) \n local g, \+ g_f, c0, w, dw, tau_eff;\n w := Im(nu);\n dw := Im(nu_f)-Im( nu);\n g:=Re(nu) ;\n g_f:=Re(nu_f) ;\n c0:=(1/g + 1/g_f )/2/((g+g_f)^2 + (dw)^2); \n tau_eff := 1/c0*1/(g_f-g+I*dw)/ 2*( 1/g/g/(g+g_f-I*dw)\n -1/g_f /(g_f+I*dw)/(g+g_f+I*dw) );\n # That is tau_eff in the w=0 frame. Now correct it:\n tau_eff := 1/(1/tau_eff + I*w);\n # prin t(\"c0, tau_eff = \", c0, tau_eff);\n tau_eff \nend ;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)tau_effgGf*6$%#nuG%%nu_fG6(%\"gG%$g _fG%#c0G%\"wG%#dwG%(tau_effG6\"F0C*>8'-%#ImG6#9$>8(,&-F56#9%\"\"\"F4! \"\">8$-%#ReGF6>8%-FCF<>8&,$*&,&*&F>F>FAF?F>*&F>F>FEF?F>F>,&*$),&FAF>F EF>\"\"#F>F>*$)F9FRF>F>F?#F>FR>8),$*(FHF?,(FEF>FAF?*&^#F>F>F9F>F>F?,&* &F>F>*&)FAFRF>,(FAF>FEF>*&^#F?F>F9F>F>F>F?F>*&F>F>*(FEF>,&FEF>FenF>F>, (FAF>FEF>FenF>F>F?F?F>FU>FW*&F>F>,&*&F>F>FWF?F>*&FfnF>F3F>F>F?FWF0F0F0 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "# Try a redefinition o f the simple version of tau_eff to handle the weak-turbulence problem \n# but further modified to handle a kind of Galilean invariance:\ntau _eff_wt := proc(nu,nu_f)\n local w, nu_eff;\n w := Im(nu);\n nu_ eff := w*I+1/(1/Re(nu) + 1/(nu_f-w*I)) ;\n # print(\"w, nu_eff =\", \+ w, nu_eff);\n 1/nu_eff\n end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+ tau_eff_wtGf*6$%#nuG%%nu_fG6$%\"wG%'nu_effG6\"F,C%>8$-%#ImG6#9$>8%,&*& ^#\"\"\"F9F/F9F9*&F9F9,&*&F9F9-%#ReGF2!\"\"F9*&F9F9,&9%F9*&^#F?F9F/F9F 9F?F9F?F9*&F9F9F5F?F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "# 1/tau_eff(g+I*w,g_f+I*w_f); # In this case, it is 1/tau_eff whi ch has a simpler form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# evalc(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "# simplify(% ); # Thus 1/tau_eff is fairly simple in this case, though tau_eff look s more complicated,\n# if evalc is used in its evaluation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# simplify(1/%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "# simplify(expand(evalc(%)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "nu:=1 ; nu_f := 4;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "nu_eff := 1/tau_eff(nu,nu_f);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "p1 := plot(Re(c2(tau,nu,nu_f)), tau=0..5, `C(t au)`=-1..1, color=red):\np2 := plot(Im(c2(tau,nu,nu_f)), tau=0..5, \+ `C(tau)`=-1..1, color=green):\np1b := plot(Re(exp(-tau*nu_eff)), tau= 0..5, `C(tau)`=-1..1, color=red, linestyle=3):\np2b := plot(Im(exp(-ta u*nu_eff)), tau=0..5, `C(tau)`=-1..1, color=green, linestyle=3):\np1c := plot(Re(exp(-tau*nu)), tau=0..5, `C(tau)`=-1..1, color=black, linestyle=2):\np2c := plot(Im(exp(-tau*nu)), tau=0..5, `C(tau)`= -1..1, color=blue, linestyle=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plots[display](\{p1,p2,p1c,p2c,p1b,p2b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nuG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nu_f G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'nu_effG$\"+\"=@c,(!#5" }} {PARA 13 "" 1 "" {GLPLOT2D 556 556 556 {PLOTDATA 2 "6*-%'CURVESG6%7S7$ $\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=F(7$$\"3umm\"H2P\"Q?F-F(7$$\"3MLL$eRwX5 $F-F(7$$\"32ML$3x%3yTF-F(7$$\"3fmm\"z%4\\Y_F-F(7$$\"3aLLeR-/PiF-F(7$$ \"3\\***\\il'pisF-F(7$$\"3>MLe*)>VB$)F-F(7$$\"3X++DJbw!Q*F-F(7$$\"3%om mTIOo/\"!#jU6FIF(7$$\"38++]i^Z]7FIF(7$$\"33++](=h(e8FIF (7$$\"30++]P[6j9FIF(7$$\"3VL$e*[z(yb\"FIF(7$$\"3wmm;a/cq;FIF(7$$\"3%om mmJFIF(7$$\"3L+]i!f#=$3#F IF(7$$\"3>+](=xpe=#FIF(7$$\"37nm\"H28IH#FIF(7$$\"3wm;zpSS\"R#FIF(7$$\" 3GLL3_?`(\\#FIF(7$$\"3fL$e*)>pxg#FIF(7$$\"33+]Pf4t.FFIF(7$$\"3uLLe*Gst !GFIF(7$$\"30+++DRW9HFIF(7$$\"3:++DJE>>IFIF(7$$\"3F+]i!RU07$FIF(7$$\"3 +++v=S2LKFIF(7$$\"3Jmmm\"p)=MLFIF(7$$\"3B++](=]@W$FIF(7$$\"36L$e*[$z*R NFIF(7$$\"3e++]iC$pk$FIF(7$$\"3[m;H2qcZPFIF(7$$\"3O+]7.\"fF&QFIF(7$$\" 3Xmm;/OgbRFIF(7$$\"3x**\\ilAFjSFIF(7$$\"3zLLL$)*pp;%FIF(7$$\"3(RL$3xe, tUFIF(7$$\"3Dn;HdO=yVFIF(7$$\"3a+++D>#[Z%FIF(7$$\"3SnmT&G!e&e%FIF(7$$ \"3#RLLL)Qk%o%FIF(7$$\"37+]iSjE!z%FIF(7$$\"3c+]P40O\"*[FIF(7$$\"\"&F)F (-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%*LINESTYLEG6#\"\"#-F$6%7U7$F($ \"\"\"F)7$$\"3VmmmT&)G\\a!#>$\"3Be,(4Y_'p%*F-7$F+$\"3IL)[e'F-7$F8$\"3\"4w)\\1)Hw\"fF-7$F;$\"3oD$3?kb&f`F- 7$F>$\"3Arb6*G+r$[F-7$FA$\"3>LjNAtG]VF-7$FD$\"3=N'RI5*z8RF-7$FG$\"331n ;))fY5NF-7$FK$\"304,D-Qy*=$F-7$FN$\"3k0(H'GpojGF-7$FQ$\"3&G4nbG*ypDF-7 $FT$\"3\"o8k!>-9:BF-7$FW$\"3JBLOTY#e5#F-7$FZ$\"3M5'fM\"fT\")=F-7$Fgn$ \"3f&\\t'Qs756 ]\"*Fju7$F\\p$\"3))R78JIyG#)Fju7$F_p$\"3G-it!yw)ptFju7$Fbp$\"3k7vI()Q_ &p'Fju7$Fep$\"3pu!y8$Fju7$Fjq$\"3&e,\"QCER,HFju7$F]r$\"3n!)p:&*z42EFju7 $F`r$\"3DozF-M]dBFju7$Fcr$\"3D-AUP/6A@Fju7$Ffr$\"3GU#*oy2r9>Fju7$Fir$ \"3<,O7*zm#>5Fju7$ F[t$\"3.$*)=%GJ-O#*!#?7$F^t$\"3^6vFi^l7$ Fdt$\"3+na3**p%zt'Fi^l-Fgt6&FitF)F)F)F]u-F$6$7YFdu7$$\"3gmmTN@Ki8Fju$ \"33Bt=V7P'***F-7$$\"3@LL$3FWYs#Fju$\"36QPF')z!e)**F-7$$\"3$)***\\iSmp 3%Fju$\"3!3SFCVu(o**F-7$Fhu$\"3DFn3h)3d%**F-7$$\"3l****\\7G$R<)Fju$\"3 z27-YM6$))*F-7$F+$\"3+FJ^'fg5!)*F-7$Fav$\"3J$\\pgNg(>'*F-7$F/$\"3'\\6 \"p_)o(*R*F-7$F2$\"3\\E1T*[o>\"))F-7$F5$\"3D>X=&*z7`\")F-7$F8$\"3:#zdv j69[(F-7$F;$\"3A#p<>WN5(oF-7$F>$\"3v')p]Qa)pE'F-7$FA$\"34V%G3*y*4o&F-7 $FD$\"38'4)G**o=S^F-7$FG$\"3]Nsb*[***HYF-7$FK$\"3n+z&evO&=UF-7$FN$\"3? 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0 "> " 0 "" {MPLTEXT 1 0 496 "p1 := plot(Re(c2(tau,nu,nu_f) ), tau=0..5, `C(tau)`=-1..1, color=red):\np2 := plot(Im(c2(tau,nu,n u_f)), tau=0..5, `C(tau)`=-1..1, color=green):\np1b := plot(Re(exp(- tau*nu_eff)), tau=0..5, `C(tau)`=-1..1, color=red, linestyle=3):\np2b := plot(Im(exp(-tau*nu_eff)), tau=0..5, `C(tau)`=-1..1, color=green, linestyle=3):\np1c := plot(Re(exp(-tau*nu)), tau=0..5, `C(tau)`= -1..1, color=black, linestyle=2):\np2c := plot(Im(exp(-tau*nu)), \+ tau=0..5, `C(tau)`=-1..1, color=blue, linestyle=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plots[display](\{p1,p2,p1c,p2c,p1b,p2b\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nuG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nu_fG^$\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'nu_effG^$$\"+\"o')G<(!#5$!+*f`#R6F(" }}{PARA 13 "" 1 "" {GLPLOT2D 556 556 556 {PLOTDATA 2 "6*-%'CURVESG6$7co7$$\"\"!F)F(7$$\"+ sUkCF!#6$\"+)[dLV&!#77$$\"+X&)G\\aF-$\"+wh!y2\"F-7$$\"+$\"+?4C0DF-7$$\"+#R 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color=green):\n p1b := plot(Re(exp(-tau*nu_eff)), tau=0..5, `C(tau)`=-1..1, color=red , linestyle=3):\np2b := plot(Im(exp(-tau*nu_eff)), tau=0..5, `C(tau)` =-1..1, color=green, linestyle=3):\np1c := plot(Re(exp(-tau*nu)), \+ tau=0..5, `C(tau)`=-1..1, color=black, linestyle=2):\np2c := plot(Im( exp(-tau*nu)), tau=0..5, `C(tau)`=-1..1, color=blue, linestyle=2) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plots[display](\{p1,p2,p1c,p2c ,p1b,p2b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nuG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nu_fG^$\"\"\"\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'nu_effG^$$\"+1%QS7(!\"*$!+++++!)F(" }}{PARA 13 "" 1 "" {GLPLOT2D 556 556 556 {PLOTDATA 2 "6*-%'CURVESG6%7hp7$$\"\"!F)F(7$$ \"+S`!eS$!#7$\"+:D+fE!#67$$\"+!o5;\"oF-$\"+Z+g)=&F07$$\"+-;u@5F0$\"+#* ej\"f(F07$$\"+O@Ki8F0$\"+*3$)4()*F07$$\"+qE!Hq\"F0$\"+FR&H?\"!#57$$\"+ /K[V?F0$\"+v@-29FE7$$\"+QP1%Q#F0$\"+yif*f\"FE7$$\"+sUkCFF0$\"+10(4y\"F E7$$F,F0$\"+a!*H6@FE7$$\"+3k'p3%F0$\"+$FE7$$\"+>FE$\"+6&yda 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lay](\{p1,p2,p1c,p2c,p1b,p2b\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#nuG^$\"\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nu_fG^$\"\" \"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'nu_effG$\"+0%QS7(!\"*" } }{PARA 13 "" 1 "" {GLPLOT2D 556 556 556 {PLOTDATA 2 "6*-%'CURVESG6$7es 7$$\"\"!F)$\"\"\"F)7$$\"+sUkCF!#6$\"+J9+k(*!#57$$\"+X&)G\\aF/$\"+J_\"Q 3*F27$$\"+D_F27$$\"+#R(*Rc\"F2$\"+1`&=q$F27$$\"+Ls1,=F2$\"+;*yt6#F2 7$$\"+uq8Q?F2$\"+7of4`F/7$$\"+*)>Wr@F2$!+/lq0MF/7$$\"+0pu/BF2$!+_SH&= \"F27$$\"+?=0QCF2$!+coH%*>F27$$\"+NnNrDF2$!+&o1$fFF27$$\"+ml'z$GF2$!+# *y\"z7%F27$$\"+(RwX5$F2$!+3HXS_F27$$\"+!\\`HP$F2$!+@tukgF27$$\"+%eI8k$ F2$!+WDfqlF27$$\"+e[U3PF2$!+cm@YmF27$$\"+J\">bx$F2$!+)3?:q'F27$$\"+0Mh UQF2$!++IhOnF27$$\"+ywq4RF2$!+1Gm^nF27$$\"+^>!o(RF2$!+WH*ou'F27$$\"+Di *Q/%F2$!+Q9eAnF27$$\"+)\\!*46%F2$!+>%f!zmF27$$\"+sZ3yTF2$!+>%4nh'F27$$ \"+;j=XWF2$!+IZ0\">'F27$$\"+hyG7ZF2$!+%)e04bF27$$\"+0%*Qz\\F2$!+/_s9YF 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=-1..1, color=black, linestyle=2):\np2c := plot(Im(exp(-tau*nu)), \+ tau=0..15, `C(tau)`=-1..1, color=blue, linestyle=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plots[display](\{p1,p2,p1c,p2c,p1b,p2b\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nuG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nu_fG^$$\"#D!\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'nu_effG^$$\"+r;A$z\"!#5$!+,m=:AF(" }}{PARA 13 "" 1 "" {GLPLOT2D 556 556 556 {PLOTDATA 2 "6*-%'CURVESG6%7S7$$\"\"!F)$\"\"\"F)7$$\"+DJdp K!#5$\"+Ud#eS*F/7$$\"+>7T9hF/$\"+@jWz))F/7$$\"+)=HPJ*F/$\"+qVS#G)F/7$$ \"+JaU`7!\"*$\"+yB7\"o(F/7$$\"+%GZRd\"F?$\"+Aq<(3(F/7$$\"+s?6r=F?$\"+* *[RARF/ 7$$\"+)[D9v$F?$\"+^,BSMF/7$$\"+iNGwSF?$\"+m7Z\")HF/7$$\"+7XM*Q%F?$\"+^ @HkDF/7$$\"+ZQjtYF?$\"+?762AF/7$$\"+i8o6]F?$\"+!pS&4=F/7$$\"+]>0)H&F?$ \"+6')*e\\\"F/7$$\"+A!p6j&F?$\"+jXld6F/7$$\"+vS.EfF?$\"+YFZ=))!#67$$\" +sxa\\iF?$\"+j'GF/'Fgq7$$\"+;$4wb'F?$\"+o@EPOFgq7$$\"+>#R!zoF?$\"+;BCn 8Fgq7$$\"+4A@urF?$!+RX`*3&!#77$$\"+chf#\\(F?$!+nm!yJ#Fgq7$$\"+(f2L#yF? 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