Turbulence quantique versus classique

Texte intégral

(1)Turbulence quantique versus classique Julien Salort. To cite this version: Julien Salort. Turbulence quantique versus classique. Physique [physics]. Université de Grenoble, 2011. Français. �NNT : 2011GRENY051�. �tel-00648090�. HAL Id: tel-00648090 https://tel.archives-ouvertes.fr/tel-00648090 Submitted on 5 Dec 2011. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(22) 5VSCVMFODF DMBTTJRVF -B USBOTGPSN©F EF 'PVSJFS EF M‫©ہ‬RVBUJPO QFSNFU EF SFUSPVWFS MB E©QFOEBODF FO k −5/3 EF MB UI©PSJF EF ,PMNPHPSPW %F GB§PO H©O©SBMF PO BQQFMMF GPODUJPO EF TUSVDUVSF E‫ہ‬PSESF n MB GPODUJPO SF n (r ) E©৯OJF DPNNF SF n (r ) = ⟨δv(r )n ⟩. . -B UI©PSJF EF ,PMNPHPSPW QFSNFU EF QS©EJSF VOF FYQSFTTJPO H©O©SBMF EF MB NªNF GPSNF RVF M‫©ہ‬RVB UJPO ⟨δv(r )n ⟩ ∼ r n/3 . &O QBSUJDVMJFS VO DBMDVM FYBDU CBT© TVS VO CJMBO ©DIFMMF QBS ©DIFMMF QFSNFU E‫ہ‬PCUFOJS VOF SFMBUJPO FOUSF MB GPODUJPO EF TUSVDUVSF E‫ہ‬PSESF FU MF UBVY E‫©ہ‬OFSHJF ǫ USBWFST MFT ©DIFMMFT 4 ⟨δv(r )3 ⟩ = − ǫr 5. . -B GPODUJPO EF TUSVDUVSF E‫ہ‬PSESF DBSBDU©SJTF MB EJTTZN©USJF EF MB EJTUSJCVUJPO EF δv(r ) &O F৮FU TJ MB EJTUSJCVUJPO ©UBJU TZN©USJRVF D‫ہ‬FTU EJSF p(δv(r )) = p(−δv(r )) MFT NPNFOUT E‫ہ‬PSESF JNQBJS TFSBJFOU OVMT 0O QFVU JOUSPEVJSF VOF HSBOEFVS OPO EJNFOTJPOO©F QPVS DBSBDU©SJTFS MB EJTTZN©USJF BQQFM©F TLFX OFTT EF MB EJTUSJCVUJPO FU OPU©F S(r ) S(r ) =. ⟨δv(r )3 ⟩. ⟨δv(r )2 ⟩3/2. . 6OF WBMFVS OPO OVMMF EF S(r ) JOEJRVF RVF MB EJTUSJCVUJPO O‫ہ‬FTU QBT TZN©USJRVF 4J MB TLFXOFTT FTU O©HBUJWF DFMB TJHOJ৯F RVF EFT ©W©OFNFOUT δv < 0 EF HSBOEF BNQMJUVEF TPOU QMVT QSPCBCMFT RVF EFT ©W©OFNFOUT δv > 0 EF NªNF BNQMJUVEF %BOT DF DBT TJ PO USBDF M‫ہ‬IJTUPHSBNNF EFT JODS©NFOUT EF WJUFTTF PV CJFO MB GPODUJPO EF QSPCBCJMJU© p(δv) PO QPVSSB PCTFSWFS M‫ہ‬BJMF EF HBVDIF FTU QMVT ©MFW©F RVF M‫ہ‬BJMF EF ESPJUF WPJS QBS FYFNQMF MFT TZNCPMFT TVS MB ৯HVSF -B SFMBUJPO FTU BQQFM©F k MPJ EFT { &MMF FTU TPVWFOU QS©TFOU©F DPNNF MB TFVMF SFMBUJPO FYBDUF EF MB UVSCVMFODF &MMF JNQMJRVF RVF MB TLFXOFTT EF MB EJTUSJCVUJPO EF δv(r ) FTU O©HBUJWF DFMB TJHOJ৯F RVF M‫ہ‬IJTUPHSBNNF EF δv(r ) B VOF BJMF QMVT MPOHVF EV DÚ© HBVDIF -B NFTVSF FYQ©SJNFOUBMF EF MB EJTTZN©USJF FU OPUBNNFOU MB DPO৯SNBUJPO EV TJHOF O©HBUJG EF MB TLFXOFTT QFSNFU EF W©SJ৯FS M‫ہ‬FYJTUFODF E‫ہ‬VOF DBTDBEF NFTVSF EV ৰVY E‫©ہ‬OFSHJF USBWFST MFT ©DIFMMFT FU TPO TFOT EF QSPQBHBUJPOr EFT HSBOEFT ©DIFMMFT WFST MFT QFUJUFT ©DIFMMFT TJ MF TJHOF FTU O©HBUJG FU EBOT M‫ہ‬BVUSF TFOT EBOT MF DBT DPOUSBJSF . SYSYl pāaC~,jCRNc 3uUāaCL3NjI3c %F OPNCSFVTFT NFTVSFT EBOT EFT DPOEJUJPOT EJWFSTFT POU DPO৯SN© MB SFMBUJPO FO k −5/3 QPVS MB EFOTJU© TQFDUSBMF EF QVJTTBODF EF MB WJUFTTF &R EBOT MB MJNJUF EFT US¨T HSBOET OPNCSFT EF 3FZOPMET 5PVT MFT TQFDUSFT PCUFOVT TF SBNÖFOU TVS VOF NªNF DPVSCF DF RVJ TVHH¨SF RVF MB DPOTUBOUF EF ,PMNPHPSPW C k FTU VOJWFSTFMMF WPJS ৯HVSF 6OF SFWVF EFT WBMFVST NFTVS©FT QPVS C k EBOT EJWFST ©DPVMFNFOUT QFVU ªUSF USPVW©F EBOT <4SF> $F S©TVMUBU FTU DPNQBUJCMF BWFD MB UI©PSJF EF ,PMNPHPSPW FU JM USBEVJU M‫ہ‬JOWBSJBODF E‫©ہ‬DIFMMF EBOT MB [POF JOFSUJFMMF $FQFOEBOU JM OF QSPVWF QBT M‫ہ‬FYJTUFODF EF MB DBTDBEF EF 3JDIBSETPO FU O‫ہ‬JOEJRVF QBT MF TFOT EF QSPQBHBUJPO EF M‫©ہ‬OFSHJF 1PVS BMMFS QMVT MPJO PO QFVU DBMDVMFS MFT GPODUJPOT EF TUSVDUVSF FU EF W©SJ৯FS FYQ©SJNFOUBMFNFOU MB MPJ EFT $PNNF PO QFVU MF WPJS TVS MB ৯HVSF HBVDIF MB EJTUSJCVUJPO EF δv FTU EJTTZN©USJRVF QPVS r /L 0 < 1 MFT ©W©OFNFOUT δv < 0 EF HSBOEF BNQMJUVEF TPOU QMVT QSPCBCMFT RVF MFT ©W©OFNFOUT δv > 0 SFM¨WFNFOU EF M‫ہ‬BJMF HBVDIF EPOD ⟨δv(r )3 ⟩ < 0 -B TLFXOFTT EF MB EJTUSJCVUJPO EF δv(r ) FTU O©HBUJWF DF r0O OPUFSB RVF EBOT EFT DBT EJ৮©SFOUT EF DFMVJ EF MB UVSCVMFODF % IPNPHÖF FU JTPUSPQF EPOU JM FTU RVFTUJPO JDJ FU OPUBNNFOU EBOT MF DBT E‫ہ‬VOF UVSCVMFODF CJEJNFOTJPOOFMMF JM QFVU FYJTUFS VOF DBTDBEF E‫©ہ‬OFSHJF JOWFSTF EBOT MBRVFMMF M‫©ہ‬OFSHJF TF QSPQBHF EFT QFUJUFT ©DIFMMFT WFST MFT HSBOEFT ©DIFMMFT. .

(23) )ZESPEZOBNJRVF DMBTTJRVF FU RVBOUJRVF. 'JH %FOTJU© TQFDUSBMF EF QVJTTBODF EV TJHOBM EF WJUFTTF PCUFOV EBOT EJऋ©SFOUT ©DPVMFNFOUT &O DPVMFVST BWFD VO E©DBMBHF BSCJUSBJSF QPVS MB MJTJCJMJU© MFT TQFDUSFT PCUFOVT EBOT M‫ڣ‬FYQ©SJFODF (3F$ <1JF> KFU E‫ڣ‬I©MJVN HB[FVY DSZPH©OJRVF . .

(24) 5VSCVMFODF DMBTTJRVF RVJ NPOUSF RVF M‫©ہ‬OFSHJF TF QSPQBHF EFT HSBOEFT WFST MFT QFUJUFT ©DIFMMFT 0O QFVU BMPST USBDFS ⟨δv(r )3 ⟩ BWFD VOF DPNQFOTBUJPO QBS MB MPJ EFT 0O PCUJFOU VO k QMBUFBV { JDJ TVS QMVT E‫ہ‬VOF E©DBEF E‫©ہ‬DIFMMFT ৯HVSF ESPJUF 100 10. −1. 10. −2. 101. p(δv(r )). − 45. ⟨δv(r )3 ⟩ r. × vL30. SNT. 100. 10−3 10−1 10−4 10−5 −6. r /L 0 ∼ 10−1 r /L 0 ∼ 1 −4. −2. 0 2 p δv(r )/ ⟨δv 2 ⟩. 4. 6. 10−2 −3 10. 10−2. 10−1. 100. r /L 0. 'JH .FTVSF FYQ©SJNFOUBMF EFT JODS©NFOUT EF WJUFTTF EBOT MF NJOJKFU EF $IBOBM FU DPMMBCPSBUFVST Rλ = 989 HBV࠺F EJTUSJCVUJPO EF M‫ڣ‬JODS©NFOU EF WJUFTTF δv -B DPVSCF r /L 0 ∼ 10−1 FTU E©DBM©F E‫ڣ‬VOF E©DBEF WFST MF IBVU QPVS MB MJTJCJMJU© ESPJUF 'PODUJPO EF TUSVDUVSF E‫ڣ‬PSESF DPNQFOT©F QBS MB MPJ EFT . %FT NFTVSFT EF EJTTZN©USJF EF MB EJTUSJCVUJPO EF δv(r ) POU ©U© S©BMJT©FT EBOT EF OPNCSFVY ©DPVMF NFOUT ञFMRVFT WBMFVST EF TLFXOFTT QPVS EFT OPNCSFT EF 3FZOPMET CBT©T TVS MB NJDSP©DIFMMF R λ DPNQSJT FOUSF FU TPOU EPOO©FT QBS FYFNQMF EBOT <$IF> -FT EJ৮©SFOUFT EPOO©FT DPOEVJTFOU UPVUFT VOF TLFXOFTT EF M‫ہ‬PSESF EF QPVS MFT ©DIFMMFT JOFSUJFMMFT WPJS ৯HVSF . SYSYk n 03I© 03 FRILR<RaRq SO:S 4J PO SFHBSEF MB EJTUSJCVUJPO EFT JODS©NFOUT EF WJUFTTF EFT ©DIFMMFT QMVT QFUJUFT ৯HVSF PO QFVU T‫ہ‬BQFSDFWPJS RVF M‫ہ‬IJTUPHSBNNF B EFT k BJMFT { EF QMVT FO QMVT SFMFW©FT D‫ہ‬FTU EJSF RV‫ہ‬JM Z B VOF QSPCBCJMJU© JNQPSUBOUF E‫ہ‬PCTFSWFS EFT ৰVDUVBUJPOT JNQPSUBOUFT EF MB WJUFTTF QFUJUF ©DIFMMF $F QI©OPNÖF FTU BQQFM© k JOUFSNJऒFODF { 6OF DPOT©RVFODF RVBOUJUBUJWF EF M‫ہ‬JOUFSNJऒFODF FTU RVF MFT GPODUJPOT EF TUSVDUVSF OF TVJWFOU QBT MB QS©EJDUJPO EF ,PMNPHPSPW &R %F GB§PO QMVT H©O©SBMF PO QFVU DPOTJE©SFS O©BONPJOT RV‫ہ‬FMMFT TVJWFOU VOF MPJ EF MB GPSNF ⟨δv(r )n ⟩ ∼ r ζ(n) . P¹ ζ(n) FTU M‫ہ‬FYQPTBOU EF MB GPODUJPO EF TUSVDUVSF E‫ہ‬PSESF n -PSTRVF ζ(n) T‫©ہ‬DBSUF EF MB QS©EJDUJPO ζ(n) = n/3 PO EJU RVF MFT FYQPTBOUT TPOU k BOPSNBVY { -‫ہ‬JOUFS NJऒFODF FU MFT FYQPTBOUT EFT GPODUJPOT EF TUSVDUVSF POU GBJU M‫ہ‬PCKFU EF OPNCSFVY USBWBVY RVJ POU DPOEVJU QSPQPTFS EJ৮©SFOUT NPE¨MFT 0O QFVU DJUFS QBS FYFNQMF MF NPE¨MF EF ;IFO4V 4IF &NNBOVFM -©WªRVF <4-> RVJ QS©EJU M‫ہ‬FYQSFTTJPO TVJWBOUF QPVS M‫ہ‬FYQPTBOU EFT GPODUJPOT EF TUSVDUVSF " µ ¶ # 2 n/3 n ζ(n) = + 2 1 − 9 3. . .

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