Effect Of Inner Scale Atmospheric Spectrum Models On Scintillation In All Optical Turbulence Regimes

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1 Uiversit of etra Foria Eetroi Teses a Dissertatios Masters Tesis Ope Aess Effet Of er Sae Atosperi Spetru Moes O Sitiatio A Optia Turbuee egies 7 Keet Maer Uiversit of etra Foria Fi siiar wors at: ttp:stars.ibrar.uf.euet Uiversit of etra Foria ibraries ttp:ibrar.uf.eu Part of te Mateatis oos STAS itatio Maer Keet "Effet Of er Sae Atosperi Spetru Moes O Sitiatio A Optia Turbuee egies" 7. Eetroi Teses a Dissertatios. 8. ttp:stars.ibrar.uf.euet8 Tis Masters Tesis Ope Aess is brougt to ou for free a ope aess b STAS. t as bee aepte for iusio i Eetroi Teses a Dissertatios b a autorize aiistrator of STAS. For ore iforatio pease otat ee.otso@uf.eu.

2 EFFET OF NNE SAE ATMOSPHE SPETUM MODES ON SNTATON N A OPTA TUBUENE EGMES b KENNETH J. MAYE B.S. Miiga Teoogia Uiversit 8 M.A. Foria stitute of Teoog 8 A tesis subitte i partia fufiet of te requireets for te egree of Master of Siee i te Departet of Mateatis i te oege of Siees at te Uiversit of etra Foria Orao Foria Sprig Ter 7

3 7 Keet J. Maer ii

4 ABSTAT Eperieta stuies ave sow tat a bup ours i te atosperi spetru just prior to turbuee e issipatio. wea optia turbuee tis bup affets auate sitiatio. Te purpose of tis tesis was to eterie if a o-bup atosperi power spetru a be use to oe sitiatio for pae waves a speria waves i oerate to strog optia turbuee regies. Sitiatio epressios were eveope fro a effetive vo Kara spetru usig a approa siiar to tat use b Arews et a. 8 i eveopig epressios fro a effetive oifie bup spetru. Te effetive spetru etes te tov approiatio ito a optia turbuee regies usig fiter futios to eiiate i-rage turbuet e size effets to te sitiatio ie. Fiter utoffs were estabise b atig to ow wea a saturate sitiatio resuts. Te resutig ew epressios tra tose erive fro te effetive bup spetru fair ose. etree strog turbuee ifferees are iia. iii

5 AKNOWEDGMENTS Ta ou to Dr. Youg for our patiee a efforts to Dr. Arews for awas beig tere to ep a to Dr. Moaptra for guiig pat. iv

6 TABE OF ONTENTS ST OF FGUES...viii ST OF TABES... i ST OF AONYMS.... NTODUTON.... ATMOSPHE SPETUMS.... Koogorov.... Tatarsii.... Vo Kara.... Moifie Bup.... SNTATON N WEAK FUTUATONS Pae Wave Speria Wave.... SNTATON N SATUATON EGME.... Pae Wave.... Speria Wave.... SNTATON N MODEATE TUBUENE.... Eistig Pae Wave Equatios.... Eistig Speria Wave Equatios.... New Moerate Turbuee Equatios..... Pae Wave arge-sae og-rraiae Variae... 7 v

7 ... Sa-Sae og-rraiae Variae Asptoti oparisos Wea Turbuee Saturate Turbuee A Turbuee egies..... Speria Wave.... OMPASONS opariso Of Te Pae Wave Sitiatio e Moes opariso Of Te Speria Wave Sitiatio e Moes... APPENDX A: WEAK TUBUENE KOMOGOOV SPETUM PANE WAVE... 7 APPENDX B: WEAK TUBUENE VON KAMAN PANE WAVE... APPENDX : WEAK TUBUENE MODFED BUMP PANE WAVE... APPENDX D: WEAK TUBUENE KOMOGOOV SPHEA WAVE... APPENDX E: WEAK TUBUENE VON KAMAN SPHEA WAVE... APPENDX F: WEAK TUBUENE MODFED BUMP SPHEA WAVE... APPENDX G: SATUATED TUBUENE KOMOGOOV PANE WAVE... 8 APPENDX H: SATUATED TUBUENE VON KAMAN PANE WAVE... 8 APPENDX : SATUATED TUBUENE MODFED BUMP PANE WAVE... APPENDX J: SATUATED TUBUENE KOMOGOOV SPHEA WAVE... APPENDX K: SATUATED TUBUENE VON KAMAN SPHEA WAVE... 8 APPENDX : SATUATED TUBUENE MODFED BUMP SPHEA WAVE... APPENDX M: MODEATE TUBUENE KOMOGOOV PANE WAVE... APPENDX N: MODEATE TUBUENE MODFED BUMP PANE WAVE... vi

8 APPENDX O: MODEATE TUBUENE KOMOGOOV SPHEA WAVE... APPENDX P: MODEATE TUBUENE MODFED BUMP SPHEA WAVE... APPENDX : MODEATE TUBUENE VON KAMAN PANE WAVE... 7 APPENDX : MODEATE TUBUENE VON KAMAN SPHEA WAVE... APPENDX S: ASYMPTOT OMPASONS... ST OF EFEENES... vii

9 ST OF FGUES Figure - Spetra oes of refrative-ie futuatios... Figure - Sae spetra oes of refrative-ie futuatios... 7 Figure Effetive Moifie Sitiatio e vs. For Various er Sae Vaues... Figure Pae Wave Sitiatio base o te vo Kara Spetru λ.88µ... 8 Figure Pae Wave Sitiatio base o te vo Kara Spetru λ.µ... 8 Figure Sitiatio e base o varie spetra s 7 λ.88µ... Figure 7 - Sitiatio e base o varie spetra s λ.88µ... Figure 8 - Sitiatio e base o varie spetra s 7 λ.µ... Figure - Sitiatio e base o varie spetra s λ.µ... Figure Speria Wave Sitiatio e base o te vo Kara Spetru... Figure - Sitiatio e base o varie spetra s λ.88µ... Figure - Sitiatio e base o varie spetra s λ.µ... viii

10 ST OF TABES Tabe Eistig Sitiatio Epressios... i

11 ST OF AONYMS a Asptoti opariso ostats eterie fro ow asptoti beavior of te sitiatio ie efrative ie struture paraetera easure of ie of refratio futuatio stregt D p ρ Wave struture futio for a pae wave. f A fator tat esribes basi power aw for ier sae oifiatios F Pase frot raius of urvature of a bea at te reeiver G arge-sae turbuee e fiter G Sa-sae turbuee e fiter β < No-iesioa paraeter β > λ Optia wave uber Propagatio pat egt er iro sae size Outer aro sae e size No-iesioa paraeter ratio of ier sae to Frese zoe No-iesioa paraeter ratio of ier sae to Frese zoe W Bea raius at te reeiver Assoiate wit arge-sae turbuee effets Assoiate wit sa-sae turbuee effets Φ Power spetru oe for refrative ie futuatios Φ Effetive atosperi spetru e No-iesioa utoff frequeies for te fiter futio. Saar spatia wave uber. er sae wave uber paraeter. er sae wave uber paraeter Outer sae wave uber paraeter

12 Λ Θ Θ ρ λ Spatia freque utoff for arge-sae turbuet es Spatia freque utoff for sa-sae turbuet es waveegt Frese ratio of te bea at te reeiver urvature paraeter of a bea at te reeiver Θ Pae wave spatia oeree raius Sitiatio ie: oraize variae of irraiae og irraiae variae arge-sae og-irraiae variae Sa-sae og-irraiae variae p. tov Variae s tov variae for a pae wave wit ier sae tov variae for a speria wave wit ier sae arge sae sitiatio ie Sa sae sitiatio ie i

13 . NTODUTON assia teoretia atosperi turbuee oes begiig i te s assue tat turbuet power is iitia geerate o arge sae e sizes a tat issipative fores ause te turbuet power to be trasferre to saer a saer saes uti te es issipate. Eperieta stuies ave sow tat at ig wave ubers sa sae sizes a bup ours i te atosperi spetru just prior to e issipatio. wea optia turbuee tis bup affets auate sitiatio wi is a easure of irraiae futuatios eperiee b a optia wave propagatig troug a rao eiu su as te Eart s atospere. Most teoretia stuies o bea statistis a resutig atosperi spetra oes iuig Koogorov Tatarsii a vo Kara o ot refet tese observe bups. Arews et a. eveope a oifie bup spetru tat aouts for te bup. Arews et a. 8 aso eveope epressios fro a effetive oifie spetru tat etes vaiit of te tov approiatio ito a optia turbuee regies usig fiter futios to eiiate irage turbuet e size effets to te sitiatio ie. Fiter utoffs were estabise b atig to ow wea a saturate sitiatio resuts. Te itet of tis tesis is to eterie if a siper o-bup atosperi power spetru a be use to aurate auate pae a speria wave sitiatio i oerate to strog optia turbuee regies. Sitiatio epressios for oerate to strog optia turbuee regies were eveope fro a effetive vo Kara spetru usig a approa siiar to tat use b Arews et a. 8. Te resutig ew epressios were opare to tose erive fro te effetive oifie spetru. Te foowig tabe Tabe ists eistig pae a speria wave sitiatio epressios base o Koogorov Moifie bup a vo Kara atosperi power

14 spetrus for bot te wea a saturatio futuatio regies. Tabe aso ists sitiatio epressios base o effetive Koogorov a bup spetrus tat were reet auate for oerate futuatio regies. A effetive vo Kara spetru is eveope i tis tesis as we as orrespoig epressios for te sitiatio ie. Mateatia erivatios of a of tese epressios are sow i te Appei setio. Note referee equatio tat eistig saturate turbuee sitiatio epressios base o te oifie atosperi power spetra were reate wit te assuptio tat D D χ D s D s were D is te wave struture futio D χ is te og-apitue struture futio a D is te pase struture futio. Te sae assuptio was ae i tis tesis to s revise eistig saturate turbuee sitiatio epressios base o te vo Kara atosperi power spetru referee equatios a. Te revisios were eessar for eveopig ew effetive vo Kara base sitiatio epressios a for oparig tese ew epressios to te eistig effetive Moifie spetru erive epressios.

15 Tabe Eistig Sitiatio Epressios Turbuee Spetru Wave Tpe Sitiatio Epressio Wea Koogorov Pae Speria. { } [ ] Vo Kara Pae [ ].8 si ta Speria.8 {. si[ ta ]} Moifie Pae Speria.8 si ta.7 si ta si ta..8. Saturatio Koogorov Pae si ta.7.. si ta si ta..8 Speria. Vo Kara Pae. Speria. Moifie Pae. Speria 7. [ ] [.8.. ] [.. ] ρ Moerate Koogorov Pae ep.... Moifie Pae Speria Speria ep. ep ρ were: p si ta si ta.... si ta. [.. ] s ep s were: Vo Kara Pae Tesis Speria Tesis s.8. si ta si ta si ta 7 7

16 . Koogorov. ATMOSPHE SPETUMS te ear s Koogorov eveope a atosperi turbuee teor tat eerg is itroue ito arge air asses eiter b wi sear or ovetio. Tese air asses uer te ifuee of iertia fores brea up ito a otiuu of saer a saer es fro a iitia outer aro sae e size wi is usua o te orer of oe to oe ure eters to a fia ier iro sae size o te orer of iieters. For sae sizes ess ta reaiig eerg is issipate as eat. Ea e size as a sigt ifferet ie of refratio. arge sae sizes te to refrat fous optia waves wereas sa sae sizes te to iffrat te waves. Te eerg of tese ie of refratio futuatios optia turbuee is esribe b power spetra esit futios. Te Koogorov power spetru for refrative ie futuatios is efie b Φ. << << were is te refrative ie struture paraeter a easure of ie of refratio futuatio stregt a is te saar spatia wave uber.. Tatarsii Te Koogorov spetru is teoretia vai o over te iertia subrage << <<. Oter spetra oes ave bee propose for auatios we ier sae a outer sae effets aot be igore. 7 Tatarsii 7 suggeste a equatio first

17 propose b Noviov for veoit futuatios tat uses a Gaussia futio to ete te Koogorov Spetru ito te issipatio rage > Φ. ep << were. is a ier sae wave uber paraeter. f << or tis equatio reues to te Koogorov spetru.. Vo Kara Te Tatarsii Spetru a be etee ito te rage < usig te vo Kara spetru Φ. ep were is te outer sae wave uber paraeter. f >> or tis equatio reues to te Tatarsii spetru. Note tat i te issipatio rage tere is o psia reaso to oose te vo Kara spetru over te Tatarsii spetru. However to aow for oparisos of sitiatio epressios area eveope fro te vo Kara spetru tis tesis paper wi refer to vo Kara as oppose to Tatarsii wit te assuptio tat.. Moifie Bup Hi perfore aases tat e to a oe tat refets observe bups i eperieta ata. Arews eveope te foowig aati approiatio to te Hi spetru iuig a outer sae paraeter ae te oifie atosperi spetru. Φ ep [ ].8.

18 were. is a ier sae wave uber paraeter. Te foowig two figures opare te tree atosperi spetra oes 8. Figure sows tat a ozero ier sae reues spetru vaues at ig wave ubers > over tat preite b te Koogorov spetru. Note tat at ow wave ubers < siiar spetru vaue reutio is ause b te presee of a fiite outer sae. a Φ Atosperi Spetru Moes ertia Subrage Koogorov vo Kara Moifie Figure - Spetra oes of refrative-ie futuatios Bup Figure sows a bup for te Moifie spetru tat is ot refete i te Koogorov or vo Kara spetra oes.

19 Atosperi Spetru Moes. Φ Koogorov Spetru vo Kara Spetru Moifie Spetru.... Figure - Sae spetra oes of refrative-ie futuatios 7

20 . SNTATON N WEAK FUTUATONS Sitiatio ie is efie as te oraize variae of irraiae 8 < > - < > < > < > < > were represets optia wave irraiae a < > eotes esebe average. Sitiatio is ofte epresse i ters of te tov variae. were is te propagatio pat egt optia wave uber λ wit λ beig waveegt. For pae wave sitiatio i wea futuatio usig te Koogorov spetru. Geera << is assoiate wit wea optia turbuee >> is assoiate wit saturate turbuee a ~ is assoiate wit oerate futuatios. is efie b 8 Uer wea futuatio oitios te o-ais sitiatio opoet for a bea wave Λ Φ Θ 8 ep os were Λ W is te Frese ratio of te bea at te reeiver Θ is a urvature F paraeter of a bea at te reeiver Θ Θ W is te bea raius at te reeiver a F is te pase frot raius of urvature at te reeiver.. Pae Wave For a pae wave Λ Θ Θ Θ equatio reues to 8

21 Φ 8 os 7 Perforig te itegratio i equatio 7 wit Koogorov vo Kara a oifie atosperi spetra epressios give i equatios a resuts i te foowig wea optia turbuee sitiatio ie epressios see appeies A B a. Koogorov 8 vo Kara.8 si ta Moifie.8 [ A]. A si ta.7 si ta.7 si ta were. a. are o-iesioa paraeters ratios of ier sae to Frese zoe. Note tat as sitiatio ie base o te vo Kara a oifie spetra reue to tat base o te Koogorov spetru.. Speria Wave For a speria wave Λ Θ Θ Θ equatio reues to 8 Φ os Perforig te itegratio i equatio wit Koogorov vo Kara a oifie atosperi spetra epressios give i equatios a resuts i te foowig sitiatio ie epressios for speria waves i wea optia turbuee see appeies D E a F. o

22 Koogorov. vo Kara.8. si ta Moifie.8. [ D]. D si ta. si ta. si ta

23 . SNTATON N SATUATON EGME. Pae Wave te saturatio regie te sitiatio ie for a uboue pae wave a be epresse i 8 te for were Φ si ep Ds β β < > β for a pae wave ote β for a speria wave D s is te pase struture futio for a pae wave Assuig D D D D p χ s s were D p is te wave struture futio for a pae wave D χ is te og-apitue struture futio for a pae wave Perforig te itegratio i equatio wit Koogorov vo Kara a oifie atosperi spetra epressios give i equatios a resuts i te foowig saturate optia turbuee sitiatio ie epressios see appeies G H a. Koogorov 8. 8 >>. vo Kara >> 7

24 Moifie 8. >> 8. Speria Wave For β perforig te itegratio i equatio wit Koogorov vo Kara a oifie atosperi spetra epressios resuts i te foowig speria wave saturate optia turbuee sitiatio ie epressios see appeies J K a. Koogorov 8. vo Kara 7 Moifie 8. 7.

25 . SNTATON N MODEATE TUBUENE We ier ier-sae effets are tae ito aout te spetru oe uer wea irraiae futuatios is esribe b 8 eferee equatios a : Koogorov Φ. f f a vo Kara f ep Moifie f ep [.8. ] As optia turbuee stregt ireases o ver arge a ver sa turbuee e sizes otribute sigifiat to overa optia refratio a iffratio. 8 As a resut Arews et a. 8 were abe to ete vaiit of te tov approiatio ito a optia turbuee regies b usig fiter futios to eiiate itereiate sae sizes as optia turbuee stregt ireases. 8 arge sae a sa sae freque utoffs were estabise b atig to ow wea a saturatio resuts. Arews et a. 8 eveope te foowig effetive atosperi spetru Φ wi iues a apitue spatia fiter futio for eiiatig i-rage es e Φ [ G G ]. e << << were G a G are arge-sae a sa-sae turbuee e fiters respetive G f ep

26 G f is a fator tat esribes basi power aw for ier sae oifiatios is te spatia freque utoff for te arge-sae turbuet es is te spatia freque utoff for te sa-sae turbuet es. Eistig Pae Wave Equatios Usig te effetive atosperi spetru oe a estabisig arge sae a sa sae freque utoffs fro te previous iste wea a saturatio resuts Arews et a. 8 eveope te foowig sitiatio ie epressios for pae wave propagatio i a optia turbuee regies see appeies M a N. Effetive Koogorov 8 Effetive Moifie 8.. ep. <.. ρ ep 7. ρ were is te arge-sae og-irraiae variae.... [ B] [ B] p is te tov variae for a pae wave wit ier sae 7 7

27 si ta.8 [ ]. p.7 si ta.7 si ta Figure pots te effetive oifie spetru sitiatio ie agaist for various ier sae vaues. t sows tat as ier sae ireases sitiatio ireases. Pae Wave Sitiatio - Effetive Moifie Spetru SNTATON NDEX Koogorov λ.88µ - - o o o o o Figure Effetive Moifie Sitiatio e Equatio 7 vs. For Various er Sae Vaues. Eistig Speria Wave Equatios Usig te effetive atosperi spetru oe a estabisig arge sae a sa sae freque utoffs fro te previous iste wea a saturatio resuts Arews et a. 8 eveope te foowig sitiatio ie epressios for speria wave propagatio i a optia turbuee regies see appeies O a P.

28 Koogorov 8 ep..8.. [ ] 8 [ ] s Moifie 8 ep.. s were. [ E] 8. E s.8. [ F] F si ta. si ta. si ta. New Moerate Turbuee Equatios Atoug sitiatio ie epressios ave bee eveope base o te Koogorov a oifie spetra tat are vai i a optia turbuee regies fiter futios orrespoig to te vo Kara spetru ave ot et bee eveope. Tis setio uses a siiar approa to tat use b Arews et a. 8 to estabis tese fiters. Te resutig sitiatio ie wi be opare to tat eveope for te oifie spetru to eterie te iportae of usig a bup spetru as turbuee stregt ireases. A ore etaie breaow of te ateatis esribe b tis setio is give i Appeies a.

29 .. Pae Wave... arge-sae og-rraiae Variae Usig te effetive atosperi spetru equatio te arge-sae og-irraiae variae for a pae wave i te presee of a fiite ier sae is 8 z 8. G os z were G f ep ep ep spetru. for te vo Kara B assuig a Taor epasio estiate of os a b substitutig z a equatio reues to Upo itegratio. G.. as were is a o-iesioa utoff freque for te fiter futio.... Sa-Sae og-rraiae Variae Usig te effetive atosperi spetru equatio te sa-sae og-irraiae variae for a pae wave i te presee of a fiite ier sae is 8 7

30 8 z z G os. 8 z G. 8 Substitutig z a equatio reues to G. were G Evauatig te itegra eas to Asptoti oparisos Tis tesis taes te sae approa as use b Arews Piips a Hope to perfor asptoti oparisos of ow sitiatio beavior i wea a saturate regies to eterie freque utoffs for te fiter futios. We assue te foowig futioa for of a [see Appei S]. o ρ o o ρ ρ << >> ρ ρ ρ ρ ρ >> << ρ ρ 7

31 were a are ostats to be eterie.... Wea Turbuee Te og irraiae variae is efie uer te tov approiatio for wea turbuee b 8 8 Φ z os z 8 Substitutig z a te equatio reues to Upo evauatig te itegra [ os ] Φ Uer wea irraiae futuatios 8 << Equatig equatios a a usig equatios a resuts i..7 << Assuig te is a approiate soutio. Substitutig ito equatios a 7 ρ o

32 ρ Tus a are ow estabise usig beavior of te sitiatio ie i te wea turbuee regie.... Saturate Turbuee First perfor a asptoti opariso for sa-sae turbuet e effets. Noraize irraiae 8 a be epresse as were is assoiate wit arge-sae turbuet e affets is assoiate wit sa-sae e effets a are statistia iepeet rao quatities a <><>. Give tese oitios <> <><>. Te seo oet of irraiae taes te for 8 < > < >< > were a are te arge sae a sa sae oraize variaes respetive. Base o equatios a te ipie sitiatio ie is Per equatio 7 te asptoti beavior of te sitiatio ie i te saturatio regie is esribe b. >> wi approaes a asptoti iit of uit. Terefore i saturate turbuee 7 strog futuatios epet te arge-sae sitiatio ters to ie out.

33 8 Te sa sae irraiae is give b 8 ep As a resut i strog futuatios ep We << ρ << te spatia oeree raius of a pae wave ρ a be approiate b ρ.. ρ ea tat i te presee of a fiite ier sae. 7 equatio. saturate turbuee te sa-sae utoff is give b equatio 7. Terefore i ρ saturate turbuee [ ]..7.7 Now eterie a asptoti opariso for arge-sae turbuet e effets. For saturate optia turbuee 8 ep ep Aso 8 ep Terefore

34 ep ep 7 Uer wea futuatio << ep 8 ea fro equatio tat. as. As a resut. For a vo Kara spetru te pae wave spatia oeree raius is. ρ. [ ]. ρ eferee equatio assuig >> ρ ρ o ρ o Sove for b equatig a equatio 7.

35 ...8 Tus a are ow estabise usig beavior of te sitiatio ie i te saturate turbuee regie.... A Turbuee egies Appig ostats a to equatios 7 a resuts i te foowig epressios for preitig arge a sa sae og irraiae beavior i a turbuee regies iuig ier sae effets:.... To app ier-sae effets to te vo Kara erive sitiatio ie et ref. equatio pae wave i wea turbuee p.8 si ta ipose p ea fro equatio tat. 7. Te

36 .7 8. p p Aso assue tat uer strog futuatios ier-sae effets te to iiis su tat.7 ρ. ea tat. equatio. Te uer strog futuatios ρ.7 As a resut for a futuatios.7 p Te sa-sae og-irraiae variae beoes. p 7 p.7. p p. p 7. Terefore referee equatio 7 ep sitiatio ie for use i a optia turbuee regies is p a equatio te resutig. p ep.. 7. p were.8 si ta p

37 .. Speria Wave Usig te effetive atosperi spetru equatio te arge-sae og-irraiae variae for a speria wave i te presee of a fiite ier sae is give b 8 z 8. G os z z 7 Assuig os a substitutig z a resuts i! tegratig equatio 7 resuts i. G 7.. µ 7 te sa-sae og-irraiae variae et z 8. G os z z 8. G z >> 7 Tis is vai uer oerate-to-strog futuatios a eas to te sae approiatio erive b te pae wave soutio equatio. 7. Assue i equatios a 7 tat Uer wea futuatio <<. As a resut

38 ep 7 Terefore i wea turbuee..7 << Assuig te 8 is a approiate soutio For saturate turbuee assuig ρ Perforig oparisos to te strog turbuee regie usig te sae approa as for te pae wave auatios resuts i. s ep 8. s were s.8. si ta

39 . OMPASONS. opariso Of Te Pae Wave Sitiatio e Moes Figures a sitiatio epressios base o te vo Kara spetru for wea a saturate turbuee regies are opare to te ew erive sitiatio ie base o a effetive vo Kara spetru. For assue oitios λ.88µ figure λ.µ figure a te ew pae wave epressios tra previous wea a saturate turbuee urves fair ose proviig a visua iiatio tat auatios a assuptios use to eveop it were orret. wea turbuee < te ew epressios agrees wit te wea turbuee ie to witi % for λ.88µ figure a to witi % for λ.µ figure. Note tat i oerate turbuee as te waveegt ireases te effet of te bup ereases. Aso ote tat i te oerate turbuee regie of approiate < < te ew epressios provies a soot trasitio betwee sitiatio epressios preite b eistig wea a saturate turbuee sitiatio epressios. For > figures a bot agree wit te saturate turbuee ie to witi %. 7

40 Pae Wave Sitiatio - vo Kara a Effetive vo Kara Spetrus λ.88µ - - SNTATON NDEX % % vo Kara Wea Effetive vo Kara vo Kara Saturate Figure Pae Wave Sitiatio base o te vo Kara Spetru wea equatio eiu [effetive spetru] equatio 7 saturate equatio 7 λ.88µ. Pae Wave Sitiatio - vo Kara a Effetive vo Kara Spetrus λ.µ - - SNTATON NDEX..8. %. % vo Kara Wea Effetive vo Kara vo Kara Saturate Figure Pae Wave Sitiatio base o te vo Kara Spetru wea equatio eiu [effetive spetru] equatio 7 saturate equatio 7 λ.µ 8

41 figures a 7 sitiatio epressios are opare base o tree atosperi spetra s effetive Koogorov effetive oifie bup a te ew erive effetive vo Kara. For assue oitios λ.88µ 7 figure a figure te ew effetive vo Kara urves apture te effet of ier sae wi is te irease i sitiatio over a o-zero ier sae epressio su as Koogorov but o ot rea te pea of te bup spetru i te fousig regie. figure te effetive oifie urve peas at. wit a preite sitiatio of.8. Te orrespoig effetive vo Kara erive sitiatio is. or approiate 7% ess. figure 7 te effetive oifie urve peas at. wit a preite sitiatio of.. Te orrespoig effetive vo Kara erive preitio is. or approiate % ess. Note tat i figures a 7 as gets saer sitiatio preite b te ew erive effetive vo Kara spetru approaes tat erive b te effetive vo Kara spetru. etree strog turbuee te ifferee betwee sitiatio erive b te ew effetive vo Kara epressio versus te effetive Koogorov epressio is iia.

42 Pae Wave Sitiatio - Effetive Moifie vo Kara a Koogorov Spetrus SNTATON NDEX. λ.88µ %.... Effetive Koogorov Effetive vo Kara Effetive Moifie Bup Figure Sitiatio e base o varie spetra s equatios 7 a 7 7 λ.88µ SNTATON NDEX Pae Wave Sitiatio - Effetive Moifie vo Kara a Koogorov Spetrus % λ.88µ - - Effetive Koogorov Effetive vo Kara Effetive Moifie Bup Figure 7 - Sitiatio e base o varie spetra s equatios 7 a 7 λ.88µ

43 Figures 8 a sow te effet of a age i waveegt. figure 8 te sae oitios are assue as i figure eept tat te waveegt as bee irease fro λ.88µ to λ.µ. Te resutig ifferee betwee te effetive vo Kara te oifie erive sitiatio at te pea of te urve is reue fro 7% i figure to 8%. Siiar i figure assuig te sae oitios as i figure 7 eept for a irease fro λ.88µ to λ.µ te resutig ifferee betwee te effetive vo Kara te oifie erive sitiatio at te pea of te urve as bee reue fro % i figure 7 to %. Note tat urves i figures 8 a pea at ower sitiatio vaues ta orrespoig figures a 7 a ote tat tese peas our i te oerate turbuet regies. Terefore for te oitios give i oerate turbuee irease waveegt size iiises te bup effet. SNTATON NDEX... Pae Wave Sitiatio - Effetive Moifie vo Kara a Koogorov Spetrus 8 % λ.µ Effetive Koogorov Effetive vo Kara Effetive Moifie Bup Figure 8 - Sitiatio e base o varie spetra s equatios 7 a 7 7 λ.µ

44 . Pae Wave Sitiatio - Effetive Moifie vo Kara a Koogorov Spetrus % λ.µ - - SNTATON NDEX... Effetive Koogorov Effetive vo Kara Effetive Moifie Bup Figure - Sitiatio e base o varie spetra s equatios 7 a 7 λ.µ. opariso Of Te Speria Wave Sitiatio e Moes Figure speria wave sitiatio epressios base o te vo Kara spetru for wea a saturate turbuee regies are opare to te ew erive speria wave sitiatio ie. As wit te ew pae wave epressio te ew speria wave epressio tras previous wea a saturate turbuee urves fair ose proviig a visua iiatio tat auatios a assuptios use to eveop it were orret. wea turbuee < te ew epressio agrees wit te wea sitiatio epressio to witi %. For > te ew epressio agrees wit te saturate sitiatio epressio to witi %. te oerate turbuee regie < < te ew epressio provies a soot trasitio betwee sitiatio epressios preite b eistig wea a saturate turbuee sitiatio epressios.

45 Speria Wave Sitiatio - vo Kara a Effetive vo Kara Spetrus SNTATON NDEX λ.88µ - - % % vo Kara Wea Effetive vo Kara vo Kara Saturate Figure Speria Wave Sitiatio e base o te vo Kara Spetru wea equatio eiu [effetive spetru] equatio 8 saturate equatio figures a sitiatio ifferees are opare betwee speria wave oes base o effetive Koogorov effetive oifie bup a te ew erive effetive vo Kara spetra s. Assue oitios are λ.88µ figure λ.µ figure a. figure te effetive oifie urve peas at. wit a preite sitiatio of.. Te orrespoig effetive vo Kara erive preitio is. or approiate % ess. figure te effetive oifie urve peas at.8 wit a preite sitiatio of.. Te orrespoig effetive vo Kara erive preitio is. or approiate % ess As wit te ew effetive vo Kara pae wave oe te ew epressio aptures a irease i sitiatio wit ier sae but oes ot rea te pea of te bup spetru i te fousig regie. Note tat urves i figures pea at ower sitiatio vaues ta i figures a ote tat tese peas our i te oerate turbuet

46 regies. Terefore i oerate turbuee for te oitios give irease waveegt size iiises te bup effet. etree strog turbuee te ifferee betwee sitiatio erive b te ew effetive vo Kara epressio versus te effetive Koogorov epressio is iia. Speria Wave Sitiatio - Effetive Moifie vo Kara a Koogorov Spetrus λ.88µ - - SNTATON NDEX % Effetive Koogorov Effetive vo Kara Effetive Moifie Bup Figure - Sitiatio e base o varie spetra s equatios 8 a 8 λ.88µ

47 Speria Wave Sitiatio - Effetive Moifie vo Kara a Koogorov Spetrus SNTATON NDEX..... % λ.µ - -. Effetive Koogorov Effetive vo Kara Effetive Moifie Bup Figure - Sitiatio e base o varie spetra s equatios 8 a 8 λ.µ

48 7. ONUSON tis tesis pae a speria wave sitiatio epressios were eveope fro a effetive vo Kara spetru for use i varig futuatio regies. Tis effetive spetru was use to ete te tov approiatio ito a optia turbuee regies b usig fiter futios to eiiate i-rage turbuet e size effets to te sitiatio ie. Fiter utoffs were estabise b atig to ow wea a saturate sitiatio resuts. Atoug stuies iiate tat usig a bup spetru is iportat for eveopig aurate sitiatio epressios i wea turbuee sitiatio epressios base o te effetive vo Kara spetru tra tose base o te effetive oifie bup spetru 8 fair ose. oerate turbuee te ew epressios provie a soot trasitio betwee sitiatio epressios preite b eistig wea a saturate turbuee sitiatio epressios. Aso i oerate turbuee irease waveegt size iiises te bup effet. etree strog turbuee te ifferee betwee te oes is iia.

49 APPENDX A: WEAK TUBUENE KOMOGOOV SPETUM PANE WAVE 7

50 ep Te Vo Kara Spetru φ. reues to te Koogorov Spetru φ. if ier sae goes to zero a outer sae goes to ifiit. Terefore te sitiatio ie erive b te vo Kara spetru see appei B reues to tat erive b te Koogorov spetru if ier sae goes to zero a outer sae goes to ifiit. For a pae wave equatio f p.8 si ta A te..8 si ta.8 si A p Terefore te sitiatio ie for a pae wave i wea turbuee base o te Koogorov spetru is. 8

51 APPENDX B: WEAK TUBUENE VON KAMAN PANE WAVE

52 Uer wea futuatio oitios te o-ais sitiatio ie for a pae wave is efie b equatio 7: For te vo Kara spetru Φ 8 os B. ep 8 ep. os ep. eep os. e. e i ep ep i ep ep i B were ep Φ. equatio os e[ep i] eogizig µ ep µ 8 Γ u µ µ ; µ ; B were µ µ

53 µ 8 8 µ µ resuts i ; ; Γ u ; ; [ i ] p. e Γ u ; ; u ; ; i. e u BAssuig Γ Γ z u a; ; z Γ a Γ a B te Γ Γ u ; ; Γ Γ Γ Γ Γ 7 Γ.8. B Equatio B a be reue to [. i ] p. e. [. ][ i ]. e 8.7 e i 8.7 [ i ] [ i ] e i i

54 e i i i i e B7 i i Upo eetig te ter ot rea i p 8.7 e i i i.7 e B8 i Evauate i i i poar ooriates: [ i ta ] ep[ i ta ] i r ep { } i ep[ ta ] i ep[ i ta ] B Aso i r ep[ i ta ] ep[ i ta ] Te i i i ep [ i ta ] ep B i ep i B A i ep[ i ta i ] i

55 os[ ta ] si[ ta ] si [ ta ] [ ta ] si B were epi os igorig te iagiar ter Equatio B8 a be rewritte as were p.7 si[ ta ].7.8 si ta.8 si[ ta ].8 [ ] si [ ta ].8 si ta B.

56 .8 Terefore te pae wave epressio i wea futuatios usig te vo Kara spetru is.8 si ta

57 APPENDX : WEAK TUBUENE MODFED BUMP PANE WAVE

58 Uer wea futuatio oitios te o-ais sitiatio ie for a pae wave is efie b equatio 7 Φ os 8 For te oifie bup spetru [ ] os ep were [ ] ep..8. Φ equatio Tis equatio a be spit ito itervas. et..8 Β. Deterie a geera soutio for Β te sove for ea iterva separate. ]} os[ { ep. 8 Β Β os ep. 8 Β os ep. i Β ep e ep. i Β ep ep e. i Β ep ep e. i Β ep ep e. i Β ep ep e. i Β ep ep. were

59 [ ] os e ep i eogizig µ ep resuts i were µ 8 Κ Γ µ u µ ; µ ; µ µ µ 8 8 µ [ ] µ [ ] [ ].Β e Γ[ ] u[ ];[ ]; Γ[ ] u[ ];[ ]; i.β Γ[ ] e u[ ];[ ]; u[ ];[ ]; i eogizig te were Γ Γ z u a; ; z Γ a Γ a 7 [ [ ] Γ [ ] Γ Γ [ ] ] Γ u [ ];[ ]; 7 a ] [ ] a [ ] [ ]

60 Equatio a be reue to were [ ].Β Γ[ ] Γ[ ] Γ e Γ Γ[ ] [ ] [ ] Γ[ ] Γ[ ] i Γ Γ[ ] Γ[ ].Β Γ[ [ ] i Γ[ ] ] e Γ[ ] Γ[ ] [ ].Β Γ[ ] Γ[ ] e Γ[ ] [ ] [ ] [ i ] [ ] [ ].Β Γ[ ] Γ[ ] e [ ] Γ[ ] [ ] [ i ].Β Γ[ ] Γ[ ] [ ] e [ i ] Γ[ ] [ ].Β Γ[ ] e [ i ].8.Β Γ[ ] Γ[ ] e Γ[ ] [ ] [ i ].Β Γ[ ] e [ i ].Β Γ[ ] e.β.β. Γ[ ] e [ ] i [ ] i [ i ] [ ] i i i [ ] i [ ] [ ] i Γ[ ] e 8 i i 8

61 .8 Upo eetig ter ot rea i.β Γ[ ] e i i [ i [ ] ].Β Γ[ ] e[ ] i Evauate i i [ ] i poar ooriates: i r ep [ i ta ] ep[ i ta ] Terefore { } ] [ ] [ i ep[ i ta ] [ ] ep[ i[ ] ta ] Aso i r ep[ i ta ] ep[ i ta ] Terefore i i ep [ i ta ] ep i Te i ep i a

62 [ ] [ ] i ep[ i [ ] ta i ] i [ ] os[ [ ] ta ] [ ] si[ [ ] ta ] were epi os igorig te iagiar ter Terefore te geera soutio is.β Γ [ ] [ ] si[ [ ] ta ] Te fia soutio is obtaie fro te above b sovig.8. Β for ea iterva iepeet te aig te togeter. terva Β : terva.β Γ [ ] [ ] si[ [ ] ta ] []. Γ [ ] [ ] si [ ] ta. Γ [ ] si ta [ ta ]..78 si [ ] si ta.8 [ ] si ta.8

63 [ ] si ta.8 si[ ta ].8.8 si[ ta ] terva.8 Β.8 : terva []..8 Γ[ ] [ ] si[ [ ] ta ] [ ]..8 si[ ta ] Γ [ 8..8 [ ]. si ta 8 [ ] si ta.8 [ ] si ta.8 si[ ta ].8 [ ] si ta.8 [ ] si ta.8

64 [ ] si ta.8.7 [ ].7si ta.8. terva -. Β. terva [ ].. Γ[ ] [ ] si[ [ ] ta ] [ ] [ ta ] 8.. Γ si 8.. [ ]. si ta [ ] 8 si ta. 8 [ ] si ta. - [ ] si ta. - [ ] si ta. - [ ] si ta. - [ ].7si ta

65 Now obie te tree itervas were Β.8..8 si[ ta ].8.8.7si [ ta ]. [ ta ]..7si si ta.8 si ta si ta si ta si ta si ta..8 si ta.7.7 si ta si ta. 8 Terefore te pae wave epressio i wea futuatios usig te oifie bup spetru is.8 si ta.7 si ta.7 si ta.

66 APPENDX D: WEAK TUBUENE KOMOGOOV SPHEA WAVE

67 Te Vo Kara Spetru φ. ep Koogorov Spetru φ. reues to te if ier sae goes to zero a outer sae goes to ifiit. Terefore te sitiatio ie erive b te vo Kara spetru see appei E reues to tat erive b te Koogorov spetru if ier sae goes to zero a outer sae goes to ifiit. For a speria wave equatio sp.8. si ta D f.8. si ta.8.si. D sp Terefore te sitiatio ie for a speria wave i wea turbuee base o te Koogorov spetru is.

68 APPENDX E: WEAK TUBUENE VON KAMAN SPHEA WAVE

69 7 Uer wea futuatio oitios te o-ais sitiatio ie for a speria wave is efie b equatio Φ os 8 E For te vo Kara spetru os ep. 8 os ep. i ep e ep. i ep ep e. i ep ep e. i ep ep e. [ ] i ep ep e. E were ep. Φ equatio ] e[ep os i eogizig ; ; ep 8 Γ Κ u µ µ µ µ µ E were µ µ

70 µ 8 8 µ µ resuts i. e Γ u ; ; Γ u ;;[ [ i ] ; ; u ; ; [ i ]. e u EAssuig u a; ; z Γ Γ z Γ a Γ a E Equatio E a be reue to te Γ Γ u ; ; Γ Γ Γ Γ Γ Γ 7 Γ. E were. e[. ]. [ i ] [. ]{ [ i ] }. e { [ i ] } 8.7 e 8.7 e [ i ] e [ i ] 7.7 e [ i ] E7 8

71 ..8 Evauate e [ i ] Wit bioia epasio < e [ i ] e [ i ] e i [ ] E8 were i aitio Γ[ ] Γ p [ p ] a p p a a Γ E As a resut e i Γ Γ were a p otig tat [ ] e i E Γ a a! E

72 e Γ Γ Γ Γ i e! Γ i! Γ Γ Γ Γ e! i Γ Γ! e Γ Γ! i Γ Γ e Γ Γ! i Γ Γ! e Γ Γ! i Γ Γ Γ Γ Γ e! i Γ Γ Γ Γ e i! Γ Γ Γ e i E! Γ Equatio E7 beoes Γ Γ 7.7 e i! Γ 7.7 e i F ; E! were Evauate: F ; osier F ; ; Γ Γ a b F a b; ; E Γ a Γ b a b z a b z F a b; ; z E!! For F;;!! Tus for F ; ; F;; sae as

73 For F ;;!!! Γ! Γ Γ! Γ Γ Γ Γ! Γ Tus for : F ; ; sae as For F ;;! Γ Γ Γ! Γ!! Γ! Γ Γ! Γ Tus for : F ; ;. Note tat o we &. atioae as foows: opare to Terefore we a we F ; E

74 E7 F ;!! i F ; i i! E8 ea fro tat a b z F a b; ; z.! te were i i F ;;! i i i E i F a;; a a Equatio E a be re-writte as a a 7.7 e! i F ; 7.7 e i i

75 e 7.7 i i e.8 i i i E Gettig ri of te iagiar ter e.8 i i E Evauate i i i poar ooriates: [ ] [ ] ta ep ta ep i i r i Terefore ta ep i i ta ep i ta ep i E Aso [ ] [ ] ta ep ta ep i i r i. Terefore [ ] ep ta ep i i i i E

76 ep ep i i i E ta ep i i i i ta si ta si ta si 7. ta si. E were epi os igorig te iagiar ter Tus for a Speria Wave Sitiatio i Wea Turbuee usig vo Kara Spetru equatio E reues to ta si..8 E

77 APPENDX F: WEAK TUBUENE MODFED BUMP SPHEA WAVE

78 Uer wea futuatio oitios te o-ais sitiatio ie for a speria wave is efie b equatio : Φ os 8 F For te oifie spetru [ ] os ep F were [ ] ep..8. Φ equatio Tis equatio a be spit ito itervas. et..8 Β. Deterie a geera soutio for Β te sove for ea iterva separate. sp Β os ep. 8 Β os ep. i Β ep e ep. i Β ep ep e. i Β ep ep e. i ] ep ep e. Β i Β ep ep e. i Β ep ep e. F

79 were os e[ep i] eogizig µ ep 8 Κ Γ µ u µ ; µ ; µ F were µ µ µ 8 8 µ [ ] µ [ ] [ ] resuts i.β e Γ[ ] u[ ];[ ]; Γ[ ] u[ ];[ ]; i sp.β Γ[ ] e u[ ];[ ]; u[ ];[ ]; [ i ] F Appig u a; ; z Γ Γ z Γ a Γ a F te 7

80 [ ] Γ[ ] Γ[ ] u [ ];[ ]; F7 Γ Γ[ ] were a ] [ ] a [ ] [ ] equatio F a be reue to sp.β Γ[ ] Γ[ ] Γ[ ] e Γ Γ[ ] [ ] Γ[ ] Γ[ ] i Γ Γ[ ].Β Γ[ ] e [ ] Γ[ ] Γ[ ] [ i ] [ ] Γ[ ] Γ[ ] [ ] [ ].Β Γ[ ] Γ[ ] e Γ[ ] [ ] [ ] [ ] [ ] { [ i ] } [ ].Β Γ[ ] Γ[ ] e [ ] Γ[ ] [ ] { [ i ] }.Β Γ[ ] Γ[ ] e Γ[ ] [ ] { [ i ] }.Β.8 Γ[ ] Γ[ ] e Γ[ ] [ ] { [ i ] } 8

81 .Β.Β.Β Γ[ ] e Γ[ ] e Γ[ ] e [ ] { [ i ] } [ ] { [ i ] } [ i ] [ ] [.Β Γ[ ] e i ] F8 were..8 Evauate e [ ] [ ] i : Wit bioia epasio < e [ ] [ ] [ [ ] i ] e [ i ] e i [ ] F were i [ ] aitio a a p a p Γ[ ] Γ p Γ[ p ] F

82 As a resut e [ ] i Γ[ ] Γ Γ[ ] F ote tat were a p a a! F Terefore e [ ] [ ] e! [ i ] Equatio F8 beoes Β Γ. [ ] e i Γ[ ] Γ Γ[ ] [ ] Γ[ ] Γ e i! Γ[ ] [ ] Γ[ ] Γ e i! Γ[ ] Γ [ ] Γ[ ] Γ e i Γ! Γ[ ]! [ ] [ ] e Γ Γ i Γ! Γ[ ] [ ] [ ] e Γ Γ i Γ! Γ[ ]! [ ] [ ] e Γ Γ i Γ! Γ[ ] Γ [ ] Γ[ ] Γ e i Γ! Γ[ ] [ ] Γ[ ] Γ e i! Γ[ ] [ ] Γ[ ] Γ e i! Γ[ ] Γ[ ] Γ Γ[ ] F [ ] i F sp! 7

83 Evauate [ ] Γ Γ i :! Γ Note tat Γ Γ a b F a b; ; F Γ a Γ b opare to Γ Γ Γ As a resut: -a-b -a -b or reaso: Γ Γ : fro above a: a a a b: a - b b b orporatig tis ito equatio F [ ]. Β Γ[ ] { e sp i F ;}! [ ].Β Γ[ ] {e i F ; }! Evauate: F ; F osier F ; ; a b z a b z F a b; ; z F7!! For F;;!! Tus for F ; ; F;; sae as 7

84 For F ;;! Γ Γ Γ! Γ!! Γ! Γ Γ! Γ Tus for : F ; ; sae as For F ;;! Γ Γ Γ! Γ!! Γ! Γ Γ! Γ Tus for : F ; ;. Note tat o we &. atioae as foows: opare to Terefore we a we F ; F8 7

85 F F ; [ ] i F ;!! i [ ] i F! eaig fro equatio F tat a b z F a b; ; z! [ ]! i i F [ ] ;; [ ] i [ ] i F were F a;; a a a [ ] a [ ] [ ] [ ] Equatio F a be re-writte as [ ].Β Γ[ ] { e sp i F ;}! [ ]. Γ[ ] e[ B i F ; ]!.B [ ] i Γ[ ] e i 7

86 7 Γ e ] [. ] [ i i B Γ ] e [. ] [ i i i B F Get ri of te iagiar ter: Γ e ] [. ] [ sp i i B F Evauate ] [ i i i poar ooriates: ] ta ep[ ] ta ep[ i i r i Terefore ] [ ] [ ta ep i i ] [ ] [ ta ep i ta ep ] [ i F Aso ] ta ep[ ] ta ep[ i i r i. Terefore [ ] ep ta ep i i i i F

87 7 ep ep i i i F ta ep ] [ ] [ i i i i ta os ] [ ta si ] [ ta si ] [ ta si ] [ ] [ F7 were epi os igorig te iagiar ter. Tus te geera soutio is Γ ta si ] [. ] [ ] [ r sp B F8 Te fia soutio is obtaie fro te above b sovig..8 Β for ea iterva iepeet te aig te togeter: terva Β.8 Γ

88 7 terva Γ ta si ] [. ] [ ] [ B Γ ta si. Γ ta si. Γ ta si... Γ. ta si... Γ. ta si....8 ta si..8 F terva.8 Β : terva Γ ta si ] [. ] [ ] [ B Γ ta 8 si..8 ] [ ] [ Γ ta si Γ ta si.

89 77 Γ ta si. ] ta si [. Γ Γ ta si. Γ. ta si. Γ. ta si. Γ. ta si.. ta si. Γ. ta si. Γ. ta si.

90 78 ]. ta si [. Γ... ta. si ta.si..8 F terva. Β : terva Γ ta si ] [. ] [ ] [ B Γ ta si Γ ta si. 8 8 Γ ta si. Γ. ta si. 8 Γ 8. ta si.

91 7 Γ. ta si. Γ. ta si. Γ. ta si. Γ. ta si. Γ. ta si ta.8 si ta.8si..8 F sp terva terva terva 8. ta si..8 sp. ta.si..8. ta.8si..8

92 8. ta.8si.. ta.si..8 ta si..8 ]. ] ta.8 si[. ] ta. si[. ta si..8 ]. ta.8si ta.si ta si..8 F Tus for a speria wave sitiatio i wea turbuee usig oifie spetru ]. ta.8si ta.si ta si..8 sp

93 APPENDX G: SATUATED TUBUENE KOMOGOOV PANE WAVE 8

94 8 eferee equatio i te saturatio regie te sitiatio ie for a uboue pae wave a be epresse i te for D s Φ ep si G For te Koogorov spetru ier sae ot iue p s D D. As a resut D p ep si G Assue te foowig approiatio for te pae wave struture futio base o te Koogorov spetru. ρ ρ D p << << ρ G were ρ Te. ep si..8 ep si [ ]. ep si. ep si β β ] [ ] [. ep si. ep si 8. ep si G were

95 8.8 β < β > β Assue for << tat si... 7!!! 7 si si β β Te equatio G a be writte as p 8. ep 8. ep 8 8. ep ep.8 G

96 were. equatio.8 Evauate { } ep s were s. 8 Substitutig t t t t { s } t ep{ st} t t t ep{ st}t ep G Terefore Note tat Γ t ep{ st} t G7 s ep Γ { s } G8 Te p G 8 7 8

97 Appig u µ µ u [ F ν µ ; µ βu] G ν β µ Te µ.7 u [ F ν µ ; µ β ] µ p u.7. [ [.7] 7 F ; ] 8.8 G Terefore for a pae wave i saturate turbuee usig te Koogorov spetru.8 >> p 8

98 APPENDX H: SATUATED TUBUENE VON KAMAN PANE WAVE 8

99 87 eferee equatio i te saturatio regie te sitiatio ie for a uboue pae wave a be epresse i te for D s Φ ep si H For te vo Kara spetru ier sae iue assue D D s ρ ρ. As a resut D p ep si H Assue te foowig approiatio for te pae wave struture futios base o te vo Kara spetru.8 ρ D p H were ρ Te Φ.8 ep si Φ. ep si Φ.8. ep si [ ] Φ 7 7. ep si [ ] Φ... ep si [ ] Φ.77 ep si [ ] Φ.77 ep si Φ.77 ep si β β Φ ] [ ] [.77 ep si

100 88 Φ.77 ep si Φ.77 ep si Φ.77 ep si Φ.77 ep si Φ.77 ep si H were... is a o-iesioa ier sae paraeter.8 β < β > β for a pae wave Assue for << tat si... 7!!! 7 si si

101 8 β β Te Φ.77 ep.77 ep 8.77 ep 8.77 ep 8.77 ep ep ep ep..77 ep..77 ep ep.8 7 ep.8 ep.8 H were

102 ep. equatio.8.77 ep Evauate eogizig µ were te µ ep 8 Γ µ ; ; u µ µ µ µ µ µ µ 8 8 Appig tis to equatio H te ep Γ u; ; ep Γ u; ; Γ u; ; ε ε

103 .8 Γ u; ; ε.8 Γ u; ;.77 ε H Evauate ; ;.77 u u a; ; z Γ Γ z Γ a Γ a Terefore u ; ; [ ] Γ Γ { [ ] } Γ Γ Γ Γ Γ { [ ] } Γ Γ Γ Γ Γ Γ.7 Γ H7 As a resut Γ.8 Γ {.7 Γ } ε 7 ε Γ Γ.8 Γ.7 7 Γ Γ Γ ε ε H8

104 Assuig ifiite outer sae te ε.77.8 Γ.77.8 Γ H f >> te H Appig ] ; [ u F u u β µ µ ν µ β µ ν µ H were

105 u µ µ β ν esuts i. [ F ; ]. [ F ; ]]. [.8]. H Terefore for a pae wave i saturate turbuee usig te vo Kara spetru. >>

106 APPENDX : SATUATED TUBUENE MODFED BUMP PANE WAVE

107 eferee equatio i te saturatio regie te sitiatio ie for a uboue pae wave a be epresse i te for D s Φ ep si For te oifie spetru ier sae iue assue D D s ρ ρ. As a resut D p ep si Assue te foowig approiatio for te pae wave struture futios base o te oifie spetru [ ] D ρ ρ ρ ρ ρ ] [.7 ρ.7 ρ were ρ Te.7 ep si Φ.8.7 ep si [ ] Φ.8. ep si [ ] Φ. ep si Φ. ep si

108 [ ] [ ] β β Φ. ep si Φ. ep si Φ. ep si were.8.8. β < β > β for a pae wave Assue for << tat si... 7!!! 7 si si β β

109 7 Te Φ. ep Φ. ep 8 eferee equatio for te oifie spetru [ ] ep..8. Φ Assuig ifiite outer sae [ ] ep..8. Φ 7 Equatio a be writte as [ ] { } 7 ep ep 8 Tis equatio a be spit ito itervas. et..8 Β. Deterie a geera soutio for Β te sove for ea iterva separate. [ ] Β. ep ep. 8 Β. ep ep. 8 Β. ep ep. 8 Β. ep. 7

110 8 Β 7 7. ep.8 were.8 Substitutig t t t t t t t t Β 7 7. ep.8 t t t Β. ep. eogizig { } s t st t ep Γ Te Γ Β B Γ

111 f >>.. a B Γ B Γ eier:..8 Β Sove for ea terva iepeet te a te togeter: i.e. ter ter ter First iterva : B iterva Γ ;] [.88 7 F

112 .8.8. were ] ; [ u F u u β µ µ ν µ β µ ν µ Seo iterva: B 8. iterva B Γ ]. 7. Γ Sie iterva << iterva Tir iterva: B-. 7 iterva B Γ Γ Sie iterva << iterva 7

113 ea tat iterva iterva iterva.. Terefore for a pae wave i saturate turbuee usig te oifie bup spetru:. 8

114 APPENDX J: SATUATED TUBUENE KOMOGOOV SPHEA WAVE

115 eferee equatio i te saturatio regie te sitiatio ie for a uboue pae wave a be epresse i te for D s Φ ep si J For te Koogorov spetru ier sae ot iue p s D D. As a resut D p ep si J For speria wave overt p D to sp D te wave struture futio for a speria wave Assue te foowig approiatios for te pae wave a speria wave struture futios base o te Koogorov spetru [page aser boo]. ρ ρ D p J. ρ ρ D sp J ρ ρ ρ ρ sp p D D J were ρ As a resut D sp. ep si.. ep si.8.. ep si. ep si [ ]. ep si

116 [ ]. ep si. ep si. ep si β β ] [ ] [. ep si ] [ ] [. ep si. ep si ep si { 8 } { 8. ep si 8 8 { 8 8. ep si [ ] 8. ep si.888 ep si J were.8 β < β > β speria wave Now evauate: si Assue for << tat

117 si... 7!!! 7 si si β β Equatio J a be re-writte as.888 ep.888 ep.888 ep ep ep ep ep. 8 [ ].888 ep.8.8 [ ].888 ep.8 [ ].888 ep.8 J7 were

118 φ. equatio.8 Evauate ep{ s } were s.888 Substitutig t t t t { s } t ep{ st} t t t ep{ st}t ep J8 Note tat Γ t ep{ st} t J s were 7 Te As a resut ep Γ { s } [ ] [ ].8 [ ] [ ] J

119 [ ]... J eogizig tat u µ µ u [ F ν µ ; µ βu] ν J β µ were u µ µ β ν µ Te. F [ ; ]...7 J Terefore for a speria wave i saturate turbuee usig te Koogorov spetru. 7 7

120 APPENDX K: SATUATED TUBUENE VON KAMAN SPHEA WAVE 8

121 eferee equatio i te saturatio regie te sitiatio ie for a uboue pae wave a be epresse i te for D s Φ ep si K For te vo Kara spetru ier sae iue assue D D s ρ ρ. As a resut D p ep si K For speria wave overt p D to sp D te wave struture futio for a speria wave Assue te foowig approiatios for te pae wave a speria wave struture futios base o te vo Kara spetru ρ ρ ρ ρ D p K..7. ρ ρ ρ ρ D sp K...8 ρ ρ ρ sp sp p D D D K were ρ As a resut D sp. ep si Φ.. ep si Φ.8.. ep si [ ] Φ. ep si

122 [ ] Φ... ep si [ ] Φ.77 ep si K were..8 β < β > For a speria wave β. Equatio K reues to Φ.77 ep si [ ] [ ] β β Φ [.77 ep si [ ] [ ] Φ [.77 ep si Φ.77 ep si Φ.77 ep si Φ } { } {.77 ep si Φ.77 ep si [ ] Φ.77 ep si Φ.77 ep si

123 [ ] Φ. ep si K7 Assue for << tat si... 7!!! 7 si si β β Equatio K7 a be re-writte as [ ] Φ. ep. ep 8. ep 8. ep 8 K8 te Vo Kara Spetru assue. Te ep. ep. ep. K a

124 7. ep ep. 8 [ ]. ep. [ ] 7. ep..8. ep.8 K were.8 Substitutig t t t t t t t t. ep.8 t t t. ep.8 K Appig { } s t st t ep Γ K were resuts i [ ]..8 Γ [ ]..8 K f 7 >> te..8 K

125 a [ ] K eogizig tat u µ µ u [ F ν µ ; µ βu] ν K β µ Te.8. { F [ ; ]}.8... K7 Terefore for a speria wave i saturate turbuee usig te vo Kara spetru. >>

126 APPENDX : SATUATED TUBUENE MODFED BUMP SPHEA WAVE

127 eferee equatio i te saturatio regie te sitiatio ie for a uboue pae wave a be epresse i te for Φ si ep Ds For te oifie spetru ier sae iue assue D s ρ D ρ. As a resut For speria wave overt ep Dp si D p to D sp te wave struture futio for a speria wave Assue te foowig approiatios for te pae wave a speria wave struture futios base o te vo Kara spetru Dsp ρ. ρ [.ρ..88.ρ.. ρ [.8 ρ [.8. ] ].8.8ρ..88. ρ ] D ρ.7 ρ [.ρ..88.7ρ.7.7 ρ ρ [.8 ].8.ρ..88 ].7 D p ρ Dsp ρ.dsp ρ. were ρ

128 As resut D sp. ep si.. ep si.8.. ep si [ ]. ep si [ ].8. ep si [ ]. ep si [ ] [ ] β β. ep si [ ] [ ]. ep si [ ] [ ]. ep si. ep si. ep si { } } {. ep si. ep si [ ]. ep si. ep si [ ]. ep si were.8

129 7..8. β < β > β For a speria wave Assue for << tat si... 7!!! 7 si si β β Equatio a be approiate as. ep. ep 8 7 eferee equatio for te oifie spetru [ ] ep..8. Φ 8 Assuig ifiite outer sae

130 8 [ ] ep..8. Φ Equatio 7 a be writte as [ ] { } ep ep Tis equatio a be spit ito itervas. et..8 Β. Deterie a geera soutio for Β te sove for ea iterva separate. Β. ep ep. 8 Β. ep ep. 8 Β. ep ep. 8 [ ] Β. ep. [ ] Β. ep.8 Β. ep.8 were.8 Substitutig t t t t Te t t t t Β. ep.8 t t t Β 7. ep. eogizig { } s t st t ep Γ were 7

131 Equatio beoes [ ] Γ Β [ ] Γ Β [ ] B Γ [ ] B Γ f >>.. a [ ] B Γ [ ] B Γ B Γ B Γ B Γ eier:..8 Β

132 Sove for ea iterva iepeet te a te togeter i.e. iterva - iterva First iterva: B iterva 7. Γ iterva 7. 7.Γ { F [ ;]} u µ µ u were [ F ν µ ; µ βu] ν β µ u µ µ β ν µ

133 Seo iterva: B 8. Ter B Γ Γ Sie iterva << iterva Sie iterva << iterva assue iterva Tir iterva: B-. 7 iterva B Γ Γ Sie iterva << iterva 7 ea tat iterva iterva iterva 7.. Terefore for a speria wave i saturate turbuee usig te oifie bup spetru: 7. 8

134 APPENDX M: MODEATE TUBUENE KOMOGOOV PANE WAVE

135 arge-sae og-rraiae Variae Usig te effetive atosperi spetru referee equatios a te arge-sae ogirraiae variae for a pae wave i te presee of a fiite ier sae is 8 z z G os. 8 M et z os z its Taor epasio estiate. Te approiatio is vai for te arge-sae og-irraiae variae beause fiter futio G eiiates ig spatia freque otributios i oerate-to-strog futuatios. osequet z z G. 8 z z G. M Substitutig z z G. G. G. ep.

136 ep ep ep ep. ep ep.7 ep M were G f ep ep. eogizig e s s Γ M were s - 7 esuts i

137 Γ M Sa-Sae og-rraiae Variae Usig te effetive atosperi spetru referee equatios a te sa-sae ogirraiae variae for a pae wave i te presee of a fiite ier sae is 8 z 8. G os z M oerate to strog irraiae futuatios >> et Substitutig 8. G z M7 z z G G

138 [ ]. [ ].7 M8 were G. Asptoti oparisos For ostats a assue te foowig futioa for of a [see Appei S]. ρ o ρ o ρ o ρ ρ << >> M

139 7 ρ ρ ρ >> << ρ ρ M Wea Turbuee Te og irraiae variae is efie uer te tov approiatio for wea turbuee b 8 z z Φ os 8 M were. Φ zero ier sae Substitutig z z z z Te [ ] Φ os 8 [ ] Φ os 8 [ ] os. 8

140 [ os ] [ os ] [ os ] [ os ] [ os ]. M were. Usig te reatiosip si u [ os u ] eogizig te previous equatio reues to. si M u si u Γ uos a u u a M were u - a eas to 8

141 . Γ os.γ os.γ M Uer wea irraiae futuatios 8 << M Equatig equatios M a M a usig equatios M8 a M resuts i..7 << M7 Assuig te is a approiate soutio. Puggig tis ito equatios M a M ρ o M8 M ρ Tus a are ow estabise usig beavior of te sitiatio ie i te wea turbuee regie. Saturate Turbuee

142 First perfor a asptoti opariso for sa-sae turbuet e effets. Noraize irraiae 8 a be epresse as were is assoiate wit arge-sae turbuet e affets is assoiate wit sa-sae e effets a are statistia iepeet rao quatities a <><>. Give tese oitios <> <><>. Te seo oet of irraiae taes te for 8 < > < >< > M were a are te arge sae a sa sae oraize variaes of a respetive. Base o equatios a Mte ipie sitiatio ie is M Per equatio te asptoti beavior of te sitiatio ie i te saturatio regie is. esribe b 8 wi approaes a asptoti iit of oe. Terefore i saturate turbuee M strog futuatios epet te arge-sae sitiatio ters to ie out. M Te sa sae irraiae is give b 8 ep As a resut i strog futuatios M ep M M

143 Te pae wave spatia oeree raius ρ we is ostat a << << ρ a be approiate b ρ. ρ. M7 ρ Puggig tis ito te tov Variae.... ρ M8 saturate turbuee equatig equatio M to equatio M8 resuts i.7.7 ρ [. ].7.7. were equatio M ρ.7 M ρ. equatio M8 Now eterie a asptoti opariso for arge-sae turbuet e effets. For saturate optia turbuee 8 ep M ep

144 Aso 8 ep Terefore Uer wea futuatio << ea fro Equatio M tat Te M ep ep M ep M... ρ. o M were

145 ρ o equatio M. ρ equatio M8 opare tis to te eistig sitiatio epressio equatio for a pae wave i saturate turbuee M Tus a are ow estabise usig beavior of te sitiatio ie i te saturate turbuee regie. A Turbuee egies Appig ostats a resuts i te foowig epressios for preitig sitiatio beavior i a turbuee regies iuig ier sae effets:.. ρ o. ρ o. ρ o.7.

146 .7. << >> M a. 7 [ ] { }... << >> M7 ea fro equatio M tat ep Terefore for a pae wave i oerate turbuee usig a effetive Koogorov spetru.7. ep.. < M8

147 APPENDX N: MODEATE TUBUENE MODFED BUMP PANE WAVE

148 arge-sae og-rraiae Variae Usig te effetive atosperi spetru referee equatios a te arge-sae ogirraiae variae for a pae wave i te presee of a fiite ier sae is 8 z z G os. 8 N et z os z its Taor epasio estiate. Te approiatio is vai for te arge-sae og-irraiae variae beause fiter futio G eiiates ig spatia freque otributios i oerate-to-strog futuatios. osequet z z G. 8 z z G. N Substitutig z z te G. G. G.

149 7..8 ep...8 ep...8 ep ep ep...8 ep...8 ep.7 ep.7.8 ep. ep ep.7 ep.8 7 ep. N were ep f G [ ] ep..8 ep..8 ep.

150 8 eogizig Γ s s e N were s resuts i Γ Γ Γ..8.7 Γ Γ Γ..8.7 Γ Γ Γ

151 µ N were 7..7 µ Sa-Sae og-rraiae Variae Usig te effetive atosperi spetru referee equatios a te sa-sae ogirraiae variae for a pae wave i te presee of a fiite ier sae is 8 z z G os. 8 N oerate to strog irraiae futuatios >> et z G. 8 N7 Substitutig z z G. 8 G...

152 [ ]. [ ].7 N8 were G. Asptoti oparisos For ostats a assue te foowig futioa for of a [see Appei S]. ρ o ρ o ρ o ρ ρ << >> N

153 ρ ρ ρ >> << ρ ρ N Wea Turbuee Te og irraiae variae is efie uer te tov approiatio for wea turbuee b 8 z z Φ os 8 N were. Φ zero ier sae Substitutig z z z z Te [ ] Φ os 8 [ ] Φ os 8 [ ] os. 8

154 [ os ] 7 [ os ] [ os ] [ os ] [ os ]. N were. Usig te reatiosip si u [ os u ] eogizig te previous equatio reues to. si N u si u Γ uos a u u a N were u - a eas to

155 . Γ os.γ os.γ N Assue i equatio N tat N Uer wea irraiae futuatios 8 << N7 Equatig equatios N a N7 a usig equatios N8 a N resuts i..7 << N8 Assuig te. is a approiate soutio. Puggig tis ito equatios N a N ρ o..8 N

156 . N ρ Tus a are ow estabise usig beavior of te sitiatio ie i te wea turbuee regie. Saturate Turbuee First perfor a asptoti opariso for sa-sae turbuet e effets. Noraize irraiae 8 a be epresse as were is assoiate wit arge-sae turbuet e affets is assoiate wit sa-sae e effets a are statistia iepeet rao quatities a <><>. Give tese oitios <> <><>. Te seo oet of irraiae taes te for 8 < > < >< > N were a are te arge sae a sa sae oraize variaes of a respetive. Base o equatios a Nte ipie sitiatio ie is N Per equatio 8 te asptoti beavior of te sitiatio ie i te saturatio regie is. esribe b >> wi approaes a asptoti iit of oe. Terefore i saturate turbuee N strog futuatios epet te arge-sae sitiatio ters to ie out. N