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Submitted on 5 Dec 2011

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Julien Salort

To cite this version:

Julien Salort. Turbulence quantique versus classique. Physique [physics]. Université de Grenoble, 2011. Français. �NNT : 2011GRENY051�. �tel-00648090�

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P¹k FTU VO WFDUFVS EہPOEF _2π^k _=〈v^f_〉 EBOT MF DBESF EF MہIZQPUI¨TF EF 5BZMPS EF UVSCVMFODF HFM©FǫFTU MF ৰVY NBTTJRVF Eہ©OFSHJF   USBWFST MFT ©DIFMMFT FUCk FTU VOF DPOTUBOUF $FQFOEBOU DFऒF QS©EJDUJPO OF USBEVJU QBT MF TFOT EF QSPQBHBUJPO EF Mہ©OFSHJF   USBWFST MFT ©DIFMMFT EFT HSBOEFT WFST MFT QFUJUFT TJ PO JOWFSTBJU MF TFOT EV UFNQT QPVS MF TJHOBMv(t) PO PCUJFOESBJU MB NªNF EFOTJU© TQFDUSBMF EF QVJTTBODF

1PVS BMMFS QMVT MPJO PO QFVU JOUSPEVJSF VOF BVUSF RVBOUJU© TUBUJTUJRVF MFT GPODUJPOT EF TUSVDUVSF EFT JODS©NFOUT EF WJUFTTF -ہJODS©NFOU EF WJUFTTFδv(r;x) JMMVTUS© TVS MB ৯HVSF FTU E©৯OJ DPNNF

δv(r;x)=v(x+r)−v(x)

P¹v(x)FTU MF TJHOBM TQBUJBM EF WJUFTTF PCUFOV   QBSUJS EFv(t)  MہBJEF EF MہIZQPUI¨TF EF UVSCVMFODF HFM©F

0 10 20 30 40 50 60 70 80 90 100

−2

−1 0 1 2

v(x)

v(x+r) δv(r)

v(x)

x x+r

'JH %©ऌOJUJPO EF MڣJODS©NFOU EF WJUFTTFδv(x)

1IZTJRVFNFOU EBOT MF DBT EہVO ©DPVMFNFOU UVSCVMFOU IPNPH¨OF FU JTPUSPQF PO QFVU DPOTJE©SFS RVF δv(r;x)DBSBDU©SJTF VO UPVSCJMMPO EF UBJMMFr TJUV© FOx

r

~v(x+r)

~ v(x)

FU EPOD^p〈δv(r;x)²〉FTU MB WJUFTTF UZQJRVF EFT UPVSCJMMPOT EF UBJMMFr 〈.〉SFQS©TFOUF MB NPZFOOF TVJWBOU x %BOT MB TVJUF PO PNFU MB E©QFOEBODF FYQMJDJUF FOx

$POTJE©SPOT VO UPVSCJMMPO EF UBJMMFr FU EF WJUFTTF UZQJRVFδv(r) *M TہBHJU EہVOF TUSVDUVSFJOTUBCMF 0O QFVU BMPST TVQQPTFS RVF TPO UFNQT EF WJF DPSSFTQPOE UZQJRVFNFOU BV UFNQT EF SFUPVSOFNFOUr/p

〈δv(r)²〉 0O QFVU EPOD FO E©EVJSF VO PSESF EF HSBOEFVS BऒFOEV QPVS MF UBVY Eہ©OFSHJF   USBWFST MFT ©DIFMMFTǫ

ǫ=‰OFSHJF DJO©UJRVF NBTTJRVF 5FNQT EF SFUPVSOFNFOU ^≈

〈δv(r)²〉 r/p

〈δv(r)²〉≈〈δv(r)²〉^3/2 r 0O PCUJFOU EPOD MB SFMBUJPO

〈δv(r)²〉 ∼(ǫr)^2/3

5VSCVMFODF DMBTTJRVF -B USBOTGPSN©F EF 'PVSJFS EF Mہ©RVBUJPOQFSNFU EF SFUSPVWFS MB E©QFOEBODF FOk^−5/3 EF MB UI©PSJF EF ,PMNPHPSPW

%F GB§PO H©O©SBMF PO BQQFMMF GPODUJPO EF TUSVDUVSF EہPSESFn MB GPODUJPOSFn(r)E©৯OJF DPNNF SFn(r)= 〈δv(r)ⁿ〉

-B UI©PSJF EF ,PMNPHPSPW QFSNFU EF QS©EJSF VOF FYQSFTTJPO H©O©SBMF EF MB NªNF GPSNF RVF Mہ©RVB 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 Eہ©OFSHJFǫ  USBWFST MFT ©DIFMMFT

〈δv(r)³〉 = −4 5ǫr

-B GPODUJPO EF TUSVDUVSF EہPSESF DBSBDU©SJTF MB EJTTZN©USJF EF MB EJTUSJCVUJPO EFδv(r) &O F৮FU TJ MB EJTUSJCVUJPO ©UBJU TZN©USJRVF DہFTU EJSFp(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©FTLFX OFTTEF MB EJTUSJCVUJPO FU OPU©FS(r)

S(r)= 〈δv(r)³〉

〈δv(r)²〉^3/2

6OF WBMFVS OPO OVMMF EFS(r)JOEJRVF RVF MB EJTUSJCVUJPO OہFTU QBT TZN©USJRVF 4J MBTLFXOFTTFTU O©HBUJWF DFMB TJHOJ৯F RVF EFT ©W©OFNFOUTδv<0EF HSBOEF BNQMJUVEF TPOU QMVT QSPCBCMFT RVF EFT ©W©OFNFOUT δv>0EF 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 SFMBUJPOFTU BQQFM©F k MPJ EFT { &MMF FTU TPVWFOU QS©TFOU©F DPNNF MB TFVMF SFMBUJPOFYBDUF EF MB UVSCVMFODF &MMF JNQMJRVF RVF MBTLFXOFTTEF MB EJTUSJCVUJPO EFδv(r)FTU O©HBUJWF DFMB TJHOJ৯F RVF MہIJTUPHSBNNF EFδv(r)B VOF BJMF QMVT MPOHVF EV D´U© HBVDIF -B NFTVSF FYQ©SJNFOUBMF EF MB EJTTZN©USJF FU OPUBNNFOU MB DPO৯SNBUJPO EV TJHOF O©HBUJG EF MBTLFXOFTT QFSNFU EF W©SJ৯FS MہFYJTUFODF EہVOF DBTDBEF NFTVSF EV ৰVY Eہ©OFSHJF   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 FOk^−5/3QPVS MB EFOTJU©

TQFDUSBMF EF QVJTTBODF EF MB WJUFTTF &R EBOT MB MJNJUF EFT US¨T HSBOET OPNCSFT EF 3FZOPMET 5PVT MFT TQFDUSFT PCUFOVT TF SBN¨OFOU 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 QPVSC_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 Eہ©DIFMMF EBOT MB [POF JOFSUJFMMF $FQFOEBOU JM OF QSPVWF QBT MہFYJTUFODF EF MB DBTDBEF EF 3JDIBSETPO FU OہJOEJRVF QBT MF TFOT EF QSPQBHBUJPO EF Mہ©OFSHJF

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 EFδv FTU EJTTZN©USJRVF QPVS r/L0<1 MFT ©W©OFNFOUTδv<0EF HSBOEF BNQMJUVEF TPOU QMVT QSPCBCMFT RVF MFT ©W©OFNFOUT δv>0 SFM¨WFNFOU EF MہBJMF HBVDIF EPOD〈δv(r)³〉 <0 -BTLFXOFTTEF MB EJTUSJCVUJPO EFδv(r)FTU O©HBUJWF DF

r0O OPUFSB RVF EBOT EFT DBT EJ৮©SFOUT EF DFMVJ EF MB UVSCVMFODF % IPNPH¨OF FU JTPUSPQF EPOU JM FTU RVFTUJPO JDJ FU OPUBNNFOU EBOT MF DBT EہVOF UVSCVMFODF CJEJNFOTJPOOFMMF JM QFVU FYJTUFS VOF DBTDBEF Eہ©OFSHJF JOWFSTF EBOT MBRVFMMF Mہ©OFSHJF TF QSPQBHF EFT QFUJUFT ©DIFMMFT WFST MFT HSBOEFT ©DIFMMFT

'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

5VSCVMFODF DMBTTJRVF RVJ NPOUSF RVF Mہ©OFSHJF 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 Eہ©DIFMMFT ৯HVSFESPJUF

−6 −4 −2 0 2 4 6

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

δv(r)/p

〈δv²〉 p(δv(r))

r/L0∼10⁻¹ r/L0∼1

10⁻³ 10⁻² 10⁻¹ 10⁰

10⁻² 10⁻¹ 10⁰ 10¹

r/L0

−⁵

4

〈^δv(r)³〉

r × ^L0

v³SNT

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%FT NFTVSFT EF EJTTZN©USJF EF MB EJTUSJCVUJPO EFδv(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©DIFMMFRλ DPNQSJT FOUSF FU TPOU EPOO©FT QBS FYFNQMF EBOT <$IF> -FT EJ৮©SFOUFT EPOO©FT DPOEVJTFOU UPVUFT   VOFTLFXOFTTEF 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©OPN¨OF 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) P¹ζ(n)FTU MہFYQPTBOU EF MB GPODUJPO EF TUSVDUVSF EہPSESFn

-PSTRVFζ(n)Tہ©DBSUF EF MB QS©EJDUJPOζ(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 ζ(n)=n

9+2

"

1− µ2

3

¶n/3#

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−10 −5 0 5 10

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

δv/p

〈δv²〉

N(δv)

r/L₀=10⁻³ r/L₀=10⁻² r/L₀=10⁻¹ r/L₀=10⁰

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$FऒF N©UIPEF DPOTJTUF   E©৯OJS MہFYQPTBOU   QBSUJS EFT JODS©NFOUT EF WJUFTTF BCTPMVT

〈|δv(r)|ⁿ〉 ∼r^ζ(n)

FU   ©DSJSF MFT GPODUJPOT EF TUSVDUVSF PCUFOVFT DPNNF EFT GPODUJPOT EF_〈|δv|³〉 EہP¹

〈|δv(r)|ⁿ〉 = 〈|δv(r)|³〉^ζ(n)/ζ(3).

0O QSFOEζ(3)=1FU PO E©UFSNJOFζ(n)DPNNF MB WBMFVS EF QMBUFBV EF MB E©SJW©F MPHBSJUINJRVF ζ(n)=d log〈|δv|ⁿ〉

d log〈|δv|³〉.

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0 2 4 6

0 0.5 1 1.5 2 2.5

n

ζ(n)

n/3 ,PMNPHPSPW 4IF -©WªRVF .JOJKFU ^Rλ=989

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'MVJEF T <,> P <CBS> ν=η/ρ<NT>

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