Eamensarbete vd Instttonen för Geovetenskaper ISSN 650-6553 Nr 99 Development and valdaton of a new mass-consstent model sng terrannflenced coordnates Lns Magnsson 60 Wnd Speed MIUU model m/s 9 50 8.5 8 40 7.5 km 30 7 6.5 0 6 0 5.5 5 0 0 30 40 50 60 60 Wnd Speed model m/s 9 50 8.5 8 40 7.5 km 30 7 6.5 0 6 0 5.5 5 0 0 30 40 50 60 Tme =8, Hegt =67.4, =4, R=0.3905
Abstract Smlatons of te wnd clmate n comple terran ma be sefl n man cases, e.g. for wnd energ mappng. In ts std a new mass-consstent model (MCM), te -model, was developed and te ablt of te model was eamned. In te model an ntal wnd feld s adsted to flfll te reqrement of beng non-dvergent at all ponts. Te advance of te - model compared wt prevos MCM:s s te se of a terran-nflenced coordnate sstem. Ecept te wnd feld, te model parameters nclde constants, one for eac drecton. Tose constants ave no obvos pscal meanng and ave to be determned emprcall. To determne te ablt and qalt of te -model, te reslts were compared wt reslts from te mesoscale MIUU-model. Frstl, comparsons were made for a Gass-saped ll, to fnd statons wc are not cagt b te -model, e.g. wakes and termal effects. Drng datme te reslts from te -model were good bt te model fals drng ngttme. From te comparsons between te models te mportance of te -constants were stded. Secondl, comparsons between te models were made for real terran. Wnd data from te MIUU-model wt resolton 5 km was sed as npt data and was nterpolated to a km grd and made non-dvergent b te -model. To std te qalt of te reslts, te were compared wt smlatons from te MIUU-model wt resolton km. Te reslts are qte accrate, after adstng for a dfference n mean wnd speed between MIUU-model rns on km and 5 km resolton. Good reslts from te -model were reaced f a clmate average wnd speed was calclated from several smlatons wt dfferent wnd drectons. Especall f te mean wnd speed for te doman n te -model was modfed to te same level as n te MIUU km. Te -model ma be a sefl tool as te reslts were fond to be reasonable good for man cases. Bt te ser mst be aware of statons wen te model fals. Ftre stdes cold be done to nvestgate f te -model s seable for resoltons down to 00 meters.
Sammanfattnng av Utvecklng oc tvärderng av en n Mass- Consstent Model med terrängnflerat koordnatsstem Modellerng av vndklmat komple terräng är användbart många sammanang, t e vd vndkarterng för vndenerg. I den är stden tvecklas oc ndersöks användbareten av en sk. Mass-Consstent Model, -modellen. Modellen bgger på att ett ntalt vndfält steras för att ppflla kontntetsekvatonen alla pnkter. För att göra vndfältet dvergensfrtt används en metod som bgger på varatonskalkl. Fördelen med denna na modell ämfört med tdgare är användandet av ett terrängnflerat koordnatsstem. I teorn för -modellen nförs en parameter. Då denna nte ar någon sälvklar fskalsk betdelse beöver den bestämmas emprskt. För att ndersöka kvaltén os -modellen gordes ämförelser med den mesoskalga MIUUmodellen. Det första steget var att ämföra körnngar över en Gassformad klle, detta för att ämföra modellerna oc fnna statoner som -modellen nte löser pp. Eempel på sådana är termska effekter oc vakar. Resltaten nder dagtd var bra medan nder nattetd var det stora skllnader mellan modellerna. Utfrån resltaten knde betdelsen av -parametern stderas. Nästa steg var att ämföra med verklg terräng. Detta gordes för ett område Norrbotten. Här användes vnddata från MIUU-modellen med pplösnng 5 km som ndata för att beräkna vnden på en skala km. För att ndersöka kvaltén os -modellen användes data från MIUU-modellen med pplösnng km som ämförelse. Resltaten avseende vndvaratonerna terrängen är tllfredställande, dock med något för öga vndastgeter -modellen. Detta vsade sg bero på för ögre medelvnd MIUU 5 km än MIUU km. Jämförelse mellan modellerna gordes även för Sorva-dalen Lappland vlken omges av bergg terräng. Resltaten är var sämre avseende medelvndarna, men med bättre resltat avseende vndrktnngarna. Bra resltat för -modellen nåddes då resltat från flera smlerngar slogs samman tll ett medelvärde. Framförallt blev resltatet bra då medelvnden sterades tll samma nvå som MIUU km. Sammanfattnngsvs kan sägas att resltaten från -modellen är rmlga många statoner men att det är vktgt att veta vlka statoner den nte fngerar. Framtda ndersöknngar bör göras för att ndersöka om modellen är användbar för pplösnngar ner tll ca 00 meter. 3
Contents. INTRODUCTION... 6. FLOW OVER HILLS... 7 3. DESCRIPTION OF THE MIUU MODEL... 9 4. THEORY...0 4. CONSERVATION OF MASS...0 4. CALCULUS OF VARIATION... 4.3 TREATMENT OF THE CARTESIAN PROBLEM... 4.4 TERRAIN-INFLUENCED COORDINATE SYSTEM...3 4.5 TREATMENT OF THE PROBLEM IN TERRAIN FOLLOWING COORDINATE SYSTEM...4 4.6 DISCRETISATION OF THE PROBLEM...5 4.7 BOUNDARY CONDITIONS...6 4.8 -CONSTANTS...6 5. DESCRIPTION OF -MODEL...7 5. COLLECTING DATA...8 5. PREPARING DATA FOR -MODEL FROM MIUU 5 KM DATA...8 5.3 SOLVING THE EQUATION SYSTEM...9 6. SIMULATED FLOW OVER HILL WITH -MODEL...0 6. MODEL SETUP...0 6. RESULTS... 6.3 DIVERGENCE... 7. SIMULATION ARTIFICIAL HILL...4 7. MODEL SETUP...4 7. RESULTS CASE...6 7.3 RESULTS FOR CASE...3 7.4 RESULTS AT OTHER LEVELS...33 8. -CONSTANTS...33 8. PROPERTIES OF -CONSTANTS...33 8. CHOICE OF Z...35 8.3 IMPROVED USE OF -CONSTANTS...36 9. SIMULATION AAPUA...37 9. MODEL SETUP...37 9. RESULTS...39 0. SIMULATION SUORVA...44 0. MODEL SETUP...44 0. RESULTS...47. COMPOSITE SIMULATIONS...50. CONCLUSIONS...55. FUTURE DEVELOPMENTS...56.3 USING THE -MODEL WITH HIGHER RESOLUTION...56 3. REFERENCES...57 APPENDIX A. LIST OF SYMBOLS...59 4
APPENDIX B. SOURCE CODE FOR THE NON-DIVERGENT PROCESS...60 APPENDIX C. THE LINEAR EQUATION SYSTEM...64 5
. Introdcton Te researc on mesoscale models n comple terran as been gong on for man ears and man dfferent models ave been developed. De to te scale of nterest, dfferent smplfcatons of te basc eqatons of fld mecancs ave been done. For flow over lls bot analtcal and nmercal models ave been developed. Knowledge n te sbect s mportant for eample n ar-pollton stdes and wnd energ mappng. Tere are varos categores of nmercal flow over ll models. Frstl, te can be dvded nto two sbgrops, stead state models and prognostc models (Pnard, 999). Prognostc models take care of te basc eqatons of fld mecancs (conservaton of mass, momentm and energ) and are tme dependent. An eample of a prognostc mesoscale model s te MIUU-model, developed at te Department of Eart Scences, Uppsala Unverst. Te model s drostatc, non-lnear and ses ger order closre scemes. Te MIUU-model s descrbed n Secton 3. In te stead-state models te assmpton du/dt=0 (acceleraton term) s made (Pnard, 999). Stead state models can be dvded nto two sbgrops, mass conservng onl and bot mass and momentm conservng models. A teor for a two dmensonal lnear mass and momentm conservaton model was ntrodced b Jackson and Hnt (975). Te teor dvdes te regon above te ll nto two parts. One nner regon were te trblence plas an mportant role on te mean flow and one oter regon domnated b nertal forces. Te eqatons for te pertrbaton were derved after lnearaton and oter assmptons and solved wt Forer transforms. Ts report wll focs on mass conservaton models (MCM), also called mass-consstent models. MCM:s consst of two parts. Frstl, nterpolatng estng meteorologcal data nto a tree-dmensonal grd generates te ntal wnd feld. Secondl, te wnd feld s made nondvergent to flfl te reqrement of conservaton of mass. For ts, tere est dfferent metods. One metod, bldng on te Sasak (970) approac, ses calcls of varaton to fnd te mnmal vale of an ntegral (see Secton 4). In te late 70 s, two dfferent models sng ts approac, MATHEW (Serman, 978) and MASCON (Dckerson, 978), were developed. MASCON treats te sb-nverson laer as a sngle laer wt varable egt of te nverson. Ts teor gves an eqaton of constrant, wc makes t possble to se calcls of varaton to fnd te wnd feld. MATHEW treats a tree dmensonal problem nstead of sng te egt of bondar laer. In Serman (978) MATHEW was sed for assmlatng a few observed data at grond to a doman. Te ntal pper wnd s calclated from a power law and te geostropc wnd. Te teor of MATHEW s descrbed n Secton 4.3. 6
In earler papers, measrements were sed as npt data. Tese were nterpolated nto te doman grd. In Serman (978) te reslts of MATHEW was not verfed wt ndependent feld data becase all measrements were sed as npt data. In Walmsle et al (990), a model called NOABLE*, bldng on te deas from Serman (978), was compared wt Mason-Kng Model D, MS-Mcro/ and BZ-WASP. Tese are tree models bldng on te teor developed b Jackson and Hnt (975). Te comparsons also nclded measrements. Te test area was Blasaval Hll, on te eastern sde of te sland of Nort Ust n te Oter Hebrdes of Scotland. Te models were n qte good agreement wt eac oter and generall fell wtn te observed range of varaton ±6% as compared to observaton. Comparsons between te models sowed tat te tree models bldng on Jackson and Hnt (975) were somewat better tan NOABLE*. In Go and Paltkof (990) NOABLE* was compared wt te MCM-model COMPLEX and measrements for tree dfferent stes n te UK. Te compared te models wt measrements also sed as npt data. Becase adstments were done b te models te reslts dd not eactl matc te measrements. Te also fond tat NOABLE* overestmates wnd speed at te top of te ll and nderestmate te wnd n low terran. In ts proect te target was to fnd ot weter t s possble to estmate te wnd clmate wt a smple nmercal model wt g resolton. Terefore a new MCM-model, -model, was developed, bldng on te deas from Serman (978). Te maor mprovement wt te new -model s te se of terran-nflenced coordnates and also te se of data from te MIUU-model as npt data nstead of measrements. Te reslts from te new model are compared wt reslts from te MIUU-model (smlatons not sed as npt data) to determne te qalt of te -model and to fnd cases wen ts model does not gve a relable reslt. To make t possble to fnd tose statons, te teor of flow over ll was stded.. Flow over lls Ts sort revew from Stll (988) capter 4..3, eplanng some general featres regardng flow over ll s made n order to make t possble to eplan te reslts from te - model and also te dfferences between te -model and te MIUU-model. Te flow over a ll ma dffer qte a lot dependng on stratfcaton, wnd speed and sape of te ll. Man flow over ll featres are mpossble to captre wt te -model bt ma be fond n te MIUU-model reslts, de to ts more etensve pscs. In a statcall stable stratfcaton, an ar parcel lfted starts to oscllate wt te Brnt-Väsälä freqenc: N = g θ 0 θ (.) 7
Wen te parcel also moves wt te flow, a wave-pattern s ndced, so called lee waves. To descrbe te condtons n stable stratfcatons, te dmensonless Frode Nmber s ntrodced: U Fr = (.) NH were H s te tpcal egt scale of te ll or montan. Fgre - a-e. Te streamlnes for te flow over ll for dfferent Fr. After Stll (988), p 60 For dfferent Fr te streamlnes over te ll wll look qte dfferent. For strongl stable stratfcaton (Fgre - a) wt lgt wnd, te boanc s domnatng over dnamc forces. Ts leads to tat te ar s forced arond te ll rater tan over t. In front of te ll some ar s blocked. For ger Fr (Fgre - b) some of te ar s passng over te ll. Bend te ll lee waves are formed b te pertrbatons cased b te ll. Some ar s also streamng arond te 8
ll. Te wave formaton s most ntense for Fr= (Fgre - c) wen te natral wavelengt of te ar s te same as tat of te ll. Te teor bend lee waves s eplaned n Holton (99). For Frode nmber above (Fgre - d) te dnamcal effect domnates over boanc effect. Ts cases a cavt wt reverse srface wnd drectons mmedatel bend te ll. For netral stratfcaton, wc gves nfnte Frode nmber, Fr s not a vald parameter. Te wnd feld n front of te ll and above, s dstrbed ot to a dstance of tree tmes te se of te ll. Bend te ll tere s a trblent wake regon wt low mean wnd speed. Above te top of te ll tere s a speed-p effect, also called over-speedng. Te sape of te wnd profle above te ll s a combnaton between te speed-p effect and te frcton at te grond. Ts combnaton gves a wnd mamm at some egt above grond. Tere are several analtcal models descrbng te vertcal profle; te mamm wnd speed and te egt for te mamm. Te reslt from one analtcal model (see Stll, 988, pp 605) gves te magntde of te speed-p for a tree dmensonal ll as percentage of te ndstrbed wnd speed, M.6 ll ll = were M W ll s te speed-p magntde, ll s ½ te egt of te ll and W ½ s te alf wdt of te ll (dstance between te top and te pont were te elevaton as decreased to alf of ts mamm). In Sectons 7-, te reslts from te -model and te MIUU-model wll be compared and te dfferences fond wll be related to te dfferent effects dscssed n ts secton. 3. Descrpton of te MIUU model Te MIUU (sort for Meteorologcal Insttte, Uppsala Unverst) model s a mesoscale model developed at te Department of Eart Scences, Uppsala Unverst drng te latest 0 ears (e.g. Enger 990, Ternström, 987, Ternström et al., 988, Abodn and Enger, 00, Andrén, 990, Mor, 003, Enger and Grsogono, 998). Te model s tree dmensonal, tme dependent and drostatc. Te coordnate sstem s te terran followng one descrbed n Secton 4.4 and te model ses a ger order closre sceme. Prognostc eqatons are sed for orontal wnd, potental temperatre, specfc mdt and trblent knetc energ. For pressre, vertcal veloct and oter varables dagnostc eqatons are sed for eac tme step. Te vertcal grd s staggered, wt te wnd components, temperatre and mdt, on te man vertcal levels and te trblent energ defned n te vertcal levels n between. Te orontal resolton can be specfed eqal at all grd ponts, telescopc wt te gest resolton n te center of te doman or ser-defned wt nform resolton n te center of te doman and sparser n te otermost grd ponts. Vertcal levels are logartmcall spaced near te grond and lnear for g levels. Te nmbers of vertcal levels n te present smlatons are 9 wt te lowest on te egt of 0 and te top at 0000 meters. 9
Te model as for eample been sed to std terran-ndced flow, montan waves, dsperson and wnd-energ potental. For comple terran, te MIUU-model as been sed earler wt satsfactor reslts. In Bergström and Källstrand (000), reslts from te MIUUmodel are compared wt measrements from 4 dfferent stes n te Sorva valle. Te reslt from te comparson sows tat te MIUU-model ncel reprodces te wnd feld bot n te valle and at te peak of one of te montans near te valle. Te MIUU model data wll n ts std be sed as tre vales n comparson wt te new -model. For te smlatons wt te MIUU-model te ntal wnd feld was calclated from a pressre feld correspondng to a specfed geostropc wnd speed and drecton. Wen te model starts rnnng te radaton balance satsfes te mdngt condtons. Te reslts from te model are stored as te average felds once ever or. Reslts from te frst ors of te model smlatons are not n eqlbrm, bt or s close to eqlbrm and ts sed for comparsons n ts report. 4. Teor Wnd feld smlatons wt a prognostc mesoscale model sc as te MIUU-model needs mc compter tme. A more economc wa s to se a smpler model. In ts secton te teor bend a mass-consstent model (MCM) wll be dscssed and modfed to se a terran-nflenced coordnate sstem. Te problem to solve n a MCM-model s to adst a wnd feld so tat t satsfes te conservaton of mass n ever grd pont and te adstment of te orgnal feld wll be mnmed. Te reason for mnmng te adstment s tat te most reasonable solton wc s non-dvergent s te solton closest to te npt data. To make t eas to mplement te terran t s preferable to se a terran-nflenced coordnate sstem. Te dervaton of te solton to te problem s based on calcls of varaton wc s a matematcal tecnqe to fnd a mnmal (or mamal) solton for an ntegral. Ts teor appled to a Cartesan sstem s earler descrbed n Serman (978). 4. Conservaton of mass One of te basc eqatons n fld dnamcs s conservaton of mass n te sstem. ρ = t r [ ( ρu )] (4.) Under condton tat te crclaton s mc less tan scale dept of te atmospere t s possble to se te sallow convecton contnt eqaton (Pelke, 984): U r = 0 (4. ) Ts s workng nder te assmpton of ncompressblt and omogenet (constant denst). Ts assmpton s sed n man mesoscale models. 0
4. Calcls of varaton Calcls of varaton s one of te man tecnqes n matematcs and pscs. For a dervaton of te teor, see Smmons (999) capter. Te dea s to fnd te fncton () so tat te ntegral I as a mnmm solton. { ( ), ' ( ) ; } I = f d (4.) were '( ) = d d f s a known fncton and te lmts are fed. Te mnmal solton of te ntegral I can be fond b Eler s eqaton: f d d f ' = 0 (4.3) Te teor s powerfl and applcable n varos problems. For eample, calcls of varaton s te basc dea for Hamltonan and Lagrangan mecancs. Te teor works n te same manner for more tan one nknown fncton: f d d f ' = 0 (4.3 ) If s workng nder te condton g(,)=0 (e.g te moton s lmted to a plane or conservaton of mass), a new fncton can be ntrodced, F = f g, were s Lagrange mltpler. s a varable tat descrbes te adstment tat as to be done to satsf te condton. Te mnmm of te ntegral s ten fond from: F d d F ' = 0 (4.3 ) It gves s eqatons pls te constran eqaton wc togeter close te sstem of eqatons. 4.3 Treatment of te Cartesan problem In order to make te wnd feld mass-consstent n a Cartesan coordnate sstem, te ke s to mnme te ntegral n Eq. (4.4). Te reason w te ntegral sall be mnmed s tat te resltng wnd feld solton sold be te solton closest to te npt wnd feld. Becase te resltng wnd feld s non-dvergent, tat part of te ntegral s ero, see Eq. (6d).
( ) ( ) ( ) = ddd w v w w v v I ~ ~ ~ (4.4) (Dfference between te felds) (Contnt eqaton),, are constants named Gassan Precson Modl and are descrbed n Secton 4.8. Splttng te ntegral n components gves: ( ) ( ) ' ~ ';, F = (4.5a) ( ) ( ) ' ~ ';, v v v v v F v = (4.5b) ( ) ( ) ' ~ ',, w w w w w F w = (4.5c) Ts step s not descrbed n an of te reference papers. Bt to reac te reslt below wt te Eler eqaton ts step s needed. Te teor of varatonal calcls sas tat te wnd feld gvng te mnmal solton for te ntegral n Eq. (4.4) can be fond sng Eler eqaton (Eq. 4.3 ). Tat gves rse to te followng tree eqatons for te adsted wnd feld. d d ~ = (4.6a) d d v v ~ = (4.6b) d d w w ~ = (4.6c) Te adsted components mst satsf te contnt eqaton wc s te constran eqaton. Eq. (4.6 a-c) dfferentated and nserted nto te contnt eqaton gves: ~ ~ ~ 0 w v w v = = wc can be rewrtten to: = w v ~ ~ ~ (4.6d) To fnd te adsted wnd feld te problem to solve s actall te eqaton sstem (4.6a-d). In Eq. (4.6d) s te onl nknown varable and terefore Eq. (4.6d) can be solved ndependentl. Bt te eqaton s a dfferental eqaton on a form called Possons eqaton. Ts dfferental eqaton s dffclt to solve (n man cases mpossble) analtcall and terefore as to be solved nmercall. Wen Eq. (4.6d) s solved te adsted wnd feld can easl be fond from eqatons (4.6a-c).
3 4.4 Terran-nflenced coordnate sstem Te terran-nflenced coordnate sstem descrbed below ses te conventonal orontal Cartesan coordnates bt a terran-nflenced vertcal coordnate η, wc s b defnton: ( ) ( ) [ ] ( ) [ ] s s G G,,,, = η (4.7) were s s te top egt of te model and G s te egt of grond. Ts sstem s terran followng near grond and plane at te top; η = 0 at = G and η = s at = s. Te base vectors are non-ortogonal. (Pelke, 984) Fgre 4-. Sows te vertcal levels above a ll. After Pelke (984) Te sstem as followng propertes: * = v* = v [ ] G G G G G s s w s s v s s w = η η * (4.8) In te followng tet te smbols, v and w* are sed as smbols for te terran followng components. Te contnt eqaton: ( ) [ ] 0 * * = G G s s ρ (4.9) Under assmpton of ncompressble gas and orontal omogenet n denst te contnt eqaton becomes: 0 * = v s s w v G G G G η (4.9 )
4 4.5 Treatment of te problem n terran followng coordnate sstem One problem wen sng te Cartesan coordnate sstem s to ncorporate a comple terran nto a model. Terefore, t s easer to se a terran followng coordnate sstem. For te coordnate sstem descrbed n Secton 4.4, te correspondng ntegral to Eq. (4.4) s: ( ) ( ) ( ) = η η ddd v s w v w w v v I g g g ~ ~ ~ * * * (4.0) Te steps to fnd te adsted wnd feld are te same as n Secton 4.3. Te ntegral fncton dvded nto components: ( ) [ ] Z F = ' ~ (4.a) ( ) [ ] v Z v v v F v = ' ~ (4.b) ( ) ' ~ * * * w w w F w = (4.c) tere, for smplct: s Z g g = (4.a) s Z g g = (4.b) Usng Eler eqaton (Eq. 4.3 ) on te eqatons (4.a-c) gves: Z d d ~ = (4.4a) Z d d v v ~ = (4.4b) η d d w w * * ~ = (4.4c) Tese eqatons dfferentated and nserted n te contnt eqaton (Eq. 4.9 ) and after some re-arrangng gves:
5 [ ] [ ] = v Z Z w v Z Z Z Z Z Z ~ ~ ~ ~ ~ * η η (4.4d) (4.4a-d) s an eqaton sstem of,v,w* and. Te sstem can be solved b frst solvng te dfferental eqaton (Eq. 4.4d), becase tat eqaton s onl dependent on. Eq. (4.4d) can be smplfed b sng te can rle for dfferentals. Te reason w not dong so s tat t gves a small nmercal error. 4.6 Dscretsaton of te problem Eq. (4.4d) s on te form 3 3 k k k = were 3 s te tree dmensonal second order dfferental operator and s te orontal frst order dfferental operator. k,,3 are constants. Wen sng fnte dfferences on Eq. (4.4d) te nmercal problem s set on te form of A = b, wc s a basc lnear algebra problem. Te tree-dmensonal matr of can be rewrtten as a vector b pttng ever element n te matr n te vector (see Append B, fcnhlvektor). Tat gves te dmenson of A as two. Te se of A wll be te sqare of te lengt of (nmber of elements). Te matr s ver sparse wt onl 7 nonero elements n eac row (see below). An eample of te eqaton sstem s sown n Append C. From Kapota and Eppel (986) te frst and second dervatves for an arbtrar fncton p can be epressed trog: ( ) ( ) = p p p p (4.5a) ( ) [ ] ( ) = p p p p (4.5b) s te dstance between p - and p. Usng tese dscretsatons of te dervatves does not make t necessar to ave an eqal spaced grd. Usng te fnte dfferences on te left and sde of (4.4d) gves te followng eqaton sstem (4.6): ( ) ( ) ( ) ( ) ( ),,,,,,,,,, k k k k k k Z Z Z Z B B B
6 k k k Z Z,,,, (4.6) O ( ) ( ) ( ) ( ) ( ),,,,,,,,,, k k k k k k Z Z Z Z A A A Te non-ero elements n A are algned at tese postons (see Append C): (Y nmber of elements n, Z nmber of elements n k) O n,n A n,n-yz A n,n-z A n,n- B n,nyz B n,ny B n,n Te dscretsaton of te rgt and sde of (4.4d) gves: ( ) ( ) ( ) ( ) ( ) ( ) k k k k k k k k k k k v Z Z w w w v v v ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~,, (4.7) 4.7 Bondar condtons From Serman (978) te bondar mst satsf ( ) 0 = n δ. Ts means tat as to be ero on te bondar, or tat te adstment n normal drecton to te bondar as to be ero (d/dn=0). Tose two bondar condtons ave dfferent pscal meanngs. =0 s called open bondar and allows adstment n normal drecton on te bondar bt not n non-normal drecton. Te oter condton s called closed bondar. Closed bondar n combnaton wt ero wnd on te bondar gves no flow trog te bondar. Closed bondares are sed for te bottom of te model and open for te oter ones. How te bondar condtons appear n te eqaton sstem s sown n Append C. 4.8 -constants In Serman (978) te defnton for Gassan precson modl s =/σ, were σ s te devaton n te wnd feld. Te devaton can be epressed as observatonal devaton, spatal devaton or devaton from te desred feld. In Gao and Paltkof (990) te -constant s connected wt te stratfcaton. Te connecton s possble to do becase a large does not allow adstment n te vertcal wnd (and n te same manner for te orontal wnd). Domnant adstment n te orontal wnd s related to stable stratfcaton and adstment
n vertcal wnd s related to nstable condtons. = = sall represent te netral condton wen te adstment s evenl dstrbted. Ts s a large smplfcaton becase tere are oter penomena de to stratfcaton tat s not possble to resolve n ts knd of model, e.g. wakes, blockng and lee waves. In te dervaton of Eq. (4a-d), s treated as a constant. It mples tat te dvergence wll not be ero f s allowed to var n te doman. Te reslts of dfferent -vales wll be dscssed n Secton 8. 5. Descrpton of -model Te -model code s created n Matlab (Te Matworks, Inc). Te advantages of sng Matlab are te andlng of matrces and all plottng fnctons. Te matr A (see Secton 4.6) as to be defned as a sparse matr de to te memor capact. Oter blt n fnctons sed n te model are fnctons for nterpolaton and te \-operator (Pärt-Enander and Söberg, 00) for solvng eqaton sstems. Te model can be dvded nto tree parts; collectng data from te MIUU-model, preparaton of data for te -model from MIUU 5 km (5 km orontal resolton) and fnall solvng te eqaton sstem. Te descrpton of collectng data and preparaton of data, wll manl descrbe te case ten te -model s rnnng wt MIUU 5 km data as npt feld, and wt MIUU km ( km orontal resolton) terran map. Fgre 5- sows te strctre for te two processes. 7
Collectng data from MIUU-model Inpt parameters: Inpt doman corners Nmber of vertcal levels for npt data Dataset Tme Get terran from fles Get orsonal grd from fles Get vertcal levels from fles Fnd ot te grd ponts correspondng to te doman corners Ct ot te grd and terran for te npt doman Get te wnd feld from MIUUmodel res-fles Preparng data for -model from MIUU 5km data Inpt parameters: -constants Nmber of vertcal levels n -model Collected data from MIUU-model Set nmber of grdponts n eac drecton Creates grd for te -model doman Ct ot terran for -model from MIUU km terrran map Get vertcal levels (same as MIUU 5 km) Set mamm model egt Set constants Cange te colmns and rows n all matrces Create matrces Z and Z Interpolate MIUU 5 km data to - model grd Transform te Cartesan vertcal veloct to terran followng coordnates sstem Fgre 5-. Te steps for collectng and preparng data for te -model. 5. Collectng data Before rnnng te -model, data from te MIUU-model as to be collected. Te wnd data s saved n fles for ever or of smlaton. Te grd data, vertcal levels and g vales are saved n separate fles. Before collectng all ts data some parameters ave to be set. Tose are corners of te npt data doman, nmber of vertcal levels tat wll be sed, name of dataset and for wc or te data sold be read. Te vales for te corners ave to be set n Sweds Natonal Coordnates. Te corners are sed to calclate te locaton of new doman from te MIUU-model data. Ten new varables are created for te grd and te terran ( g -vales) of te new doman and te wnd feld s loaded from MIUU-model reslt fles. Te wnd components are saved n a tree dmensonal matr, one for eac component. Data s collected bot for km and 5 km rns wt te MIUU-model. 5. Preparng data for -model from MIUU 5 km data Before startng te smlatons wt te -model, wc wll make te wnd feld nondvergent, a frst gess of te wnd components as to be done. 8
Te npt parameters for te preparaton process are te -constants and te data collected from te MIUU-model. Te nmber of grd ponts and te doman for te -model s determned from te se of te npt data doman. Te doman for te -model as to be smaller tan te npt data doman becase te wnd data can onl be nterpolated to a smaller doman wt relable reslt. Te terran map from MIUU km s sed for te - model. Te vertcal levels for te -model are set to te same vertcal levels as n te MIUU 5 km model. Te top of te -model s determned from nmbers of vertcal grd ponts. Te frst gess for te wnd feld s created b sng te blt n Matlab fncton nterp3 wc nterpolates te 5 km spaced data to te new km spaced grd. Te tpe of nterpolaton sed ere s lnear nterpolaton. Splne nterpolaton as been tested bt can sometmes gve strange reslts, especall near borders. Te vertcal wnd component s not nterpolated. Instead te Cartesan vertcal veloct s set to ero and te terran followng vertcal wnd veloct s calclated from Eq. (4.8). Ts treatment of te vertcal wnd wll be dscssed n Secton 6.. 5.3 Solvng te eqaton sstem To make te wnd feld non-dvergent te eqaton sstem (4a-d) as to be solved. In Append B te sorce code for te sbrotne fcnkorm and te nderlng fnctons are sown. Frstl, te rgt-and sde of Eq. (4.4d) s created (fcnb) b calclatng te dvergence of te npt wnd feld from Eq. (4.7) (see sbrotne fcnukont). Te dvergence s set to ero for all bondar ponts. Te reason for ts s to form te bondar condtons (see below). To make t possble to se te blt n eqaton sstem \-operator n Matlab, te rgt-and sde as to be a vector. Terefore te elements n te dvergence matr ave been pt nto a vector. Ts s made n te sbrotne fcnhlvektor. Te matr A s created n sbrotne fcnmat. Te matr s two dmensonal and ts se eqals te nmber of grd ponts sqared. For a doman se of 663 te se of te matr s 85583 85583. To make t possble to store te matr n te compter memor, te matr s declared as a sparse matr n Matlab, mplng tat compter memor s onl allocated for te non-ero elements. For all bondar ponts ecept te lower bondar te vale on te dagonal s set to. Ts n combnaton wt ero-vales n te rgt-and sde of Eq. (4.4d) gves te bondar condton =0. At te lower bondar te dagonal nmbers are also set to bt ere te vale to te rgt of te dagonal s set to -. Wt te rgt-and sde of Eq. (4.4d) eqal to ero, ts gves te bondar condton - =0. For all oter rows te elements are gven accordng to Eq. (4.6). An eample of te matr A s sown n Append C. Te dfferental eqaton (Eq. 4.4d) s ten solved wt te \-operator wc mples tat te eqaton sstem s sall solved b Gassan elmnaton wt LU (LowerUpper)- factoraton (for detals see an lteratre n scentfc comptng). Te reslt from te solved eqaton sstem s a vector contanng a -vale for all grd ponts n te doman. To transform te elements of te vector wt -vales n to a matr, te fncton FknLmat s sed. Ts fncton does te reversed procedre from te fncton fcnhlvektor. 9
Wen te vales of are determned at all grd ponts te new wnd feld can be calclated from eqatons (4.4a-c) (see fcnun, fcnvn and fcnwn). To get te Cartesan vertcal veloct Eq. (4.8) s sed (fcnwtw). 6. Model setp 6. Smlated flow over ll wt -model To std te resltng wnd feld from te -model, smlatons were performed for an artfcal ll. Te eqaton for te terran was gven b: g ma ep ( s( X ) ( Y ) ) = (6.) c c Ts eqaton gves a -D Gassan dstrbton. ma s te mamm egt of te ll. S s a parameter for te steepness of te ll. Pont (X c,y c ) s te central pont of te ll. Here te mamm egt was set to 300 meter and te steepness parameter to 0.08. Ts gves a mamm terran gradent of 6.7 cm/m. Te sape of te ll s sown n Fgre 6-. Terran 300 50 00 50 00 50 0 3.5.5 0 4.5 0.5 0.5 m m Fgre 6-. Te sape of te Gassan ll wt te mamm egt 300 meters..5 3 0 4 Te npt wnd feld was set to = w = 0 and v=0 m/s for egts above 50 m. For egts below 50 meters te logartmc wnd law was sed wt 0 =0.0. Te se of w = 0 does not mpl tat w* necessarl wll be eqal to ero, see Eq. (4.8). Te coce of w=0 nstead of w*=0 wll reslt n a larger dvergence. Usng w* = 0 means tat 0
te ar flow follows te terran and dvergence wll onl be created b te compresson of te terran followng coordnate sstem. Usng w = 0 leads to tat te ar s ntall blowng nto te ll, wc wll gve a larger dvergence. If w*=0 s sed te adstment wll be too small n te orontal drecton. Ts treatment of te vertcal wnd s one of te ke ponts to get a pscall relevant reslt from te -model. Te -constants was set to = = and =4. Ts coce gves more adstment n te orontal drectons tan n te vertcal, see Secton 8. Te grd se was 330 wt a orontal spacng of 000 meters. In vertcal, te spacng vares wt egt. Te levels are 0,, 4, 0, 0, 50, 00, 50, 00, 300, 400 and above 400 m ever 00 meter p to 00 meters. 6. Reslts All reslts from te -model are plotted n a terran followng level. Ts means tat te alttde s not te egt above sea level. Te terran followng level s comparable to egt over terran bt s not eactl te same (see Fgre 4-). (m) 0 4 3.5.5 0.5 (a) component m/s 0.6 0.4 0. 0 0. 0.4 0.6 (m) 0 4 3.5.5 0.5 (b) v component m/s.5 0.5 0 9.5 3 (m) 0 4 3 (m) 0 4 (m) 0 4 3.5.5 (c) w component m/s 0.4 0. 0 (m) 0 4 3.5.5 (d) Wnd Speed m/s.5 0.5 0. 0 0.5 0.4 0.5 9.5 3 3 (m) 0 4 (m) 0 4 Fgre 6-a-d. Te adsted - (a), v- (b), w-(c) component and wnd speed(d) at Hegt=50 m. Fgre 6-a sows te adsted -component at 50 meters egt. It sows tat te flow s gong arond te ll. Te adstment on ts level s as a mamm 7 % of te wnd speed n
te orgnal feld bt t s dependent on te coce of and te sape of te ll. Te smmetr s a reslt of te omogeneos npt wnd feld. Te pscs n te model s smmetrc n te wa tat smmetrc dvergence mples smmetrc adstment. Fgre 6-b sows te v component at 50 meters egt. Jst above te ll tere s a speedp of te flow. In front of and bend te ll te wnd speed s decreased. Te speed-p effect s also dependent of te coce of. Te fact tat te mamm s located eactl above te ll s not tre n realt. Instead te mamm s sall located bend te ll (see smlaton reslts from MIUU-model n Secton 7). Ts dfference s one of te error sorces n te model. Te mnmm n front of and bend te ll s of te same magntde, wc s also a problem. In ver stable stratfcaton, ar s blocked n front of te ll, wc gves a stronger mnmm tere. Drng nstable or weakl stable stratfcaton, a wake bend te ll gves a larger area wt low speed tan n front of te ll for ts case. Fgre 6-c sows te Cartesan vertcal veloct at 50 meters egt. It sows tat n front of te ll te ar s ascendng and descendng bend te ll. Te wnd speed sowed n Fg 6-d s calclated n te followng wa: U = v (6.) Te pctre looks nearl te same as te v-component. Te reason for ts s te fact tat te v-component s mc larger tan te -component. Te wnd speed s te parameter tat wll be most etensvel sed n te comparsons between te -model and te MIUU-model. 6.3 Dvergence Te dstrbton of te dvergence n te npt feld s plotted n Fgre 6-3 and sows te smmetr dscssed above.
0 4 3 Dvergence n te npt feld s 0 4.5.5.5 0.5 (m).5 0 0.5.5 0.5.5 0.5.5.5 3 (m) 0 4 Fgre 6-3. Te dvergence n te npt feld for level 6 (50 m). De to te teoretcal treatment of te flow, (Eq. 4.4d), te dvergence as to be ver close to ero. Bt f te resltng wnd feld s nserted n te contnt eqaton, te error wll be mc larger as seen n Table 6-, colmn. Ts s cased b te treatment of te dscretsatons of dfferentals. Usng te frst order dfferental operator (Eq. 4.5a) two tmes wll not gve te same reslt as sng te second order (Eq. 4.5b) one tme. Te dscretsaton of te dfferental eqaton (Eq. 4.6) ses te second order operator. Afterwards te frst order operator s sed to fnd te adsted feld. Te fact tat te contnt eqaton also ses te frst order s te reason for ts error. Te dvergence n te npt feld s confned to te lowest 6 levels. Te reason for ts s tat tere s a wnd sear followng te logartmc wnd law for tese levels. Table 6- sows te mamm dvergence for eac level ecept te bottom and te top. Te vales for te resltng feld are calclated b pttng te adsted wnd feld nto te contnt eqaton, wc gves te problem dscssed above. Te reslts n te rgt-and colmn are calclated b pttng te calclated -vales nto Eq. (4.4d). 3
Table 6-. Te dvergence for level to 9. Te reason for te dfference between colmn and 3 s dscssed above. Level Hegt (m) Inpt feld Resltng feld Dvergence from eqn 4d 0. 0.05.78e-04 3 4 0.03 0.033.06e-04 4 0 0.0094 0.0033.73e-05 5 0 0.0046 0.003 3.67e-06 6 50 0.005 0.0006 8.0e-07 7 00 9.49e-09 0.0008 4.9e-07 8 50.33e-08 0.0000 3.45e-07 9 00.57e-08 8.84e-005.77e-07 0 300 3.79e-09 6.85e-005 9.0e-08 400 3.79e-09 5.75e-005 4.47e-08 600 3.5e-09 3.99e-005.48e-08 3 800.63e-09.e-005.56e-08 4 000.63e-09.3e-005.53e-08 5 00.084e-09 7.3e-006.0e-08 6 400.084e-09 4.38e-006 7.3e-09 7 600 5.4e-00.59e-006 4.58e-09 8 800 5.4e-00.45e-006.9e-09 9 000 5.4e-00 6.6e-007.9e-09 7. Smlaton Artfcal ll To determne te qalt and ablt of te -model, a large nmber of dfferent smlatons ave been made. Te reslts of te smlatons were ten compared wt reslts from te MIUU-model. Te model comparsons were also sed to fnd te optmal vales of for dfferent meteorologcal condtons. 7. Model setp To make te comparson smple, te terran was created as a -dmensonal Gassan dstrbton (Eq. 6.). See Fgre 6- for an eample of a ll created wt ts eqaton. Tree dfferent sapes of te ll are sed n ts test. Hll and 3 ave te same area bt te egts are 600 and 00 meters respectvel. Hll covers a larger area and te egt s 300 meters. Table 7-. Parameters for te dfferent lls sed n te smlatons. Mamm egt Steepness parameter Mamm gradent Hll 300 0.0 3.6 cm/m Hll 600 0.08 3.4 cm/m Hll 3 00 0.08. cm/m 4
To eamne te mportance of te -constants, dfferent -vales were sed n te smlatons, wle and were set to for all smlatons (see Secton 8.). Smlatons wt te MIUU-model were made for te same terran as te -model and resolton was set to km. As ntal condtons te geostropc wnd was set to 9 m/s and te radaton condton was comparable to tat n Aprl n sotern Sweden. Te reslts were gven as 36 dfferent orl averages. Ts makes t possble to do an analss coverng bot da and ngt condtons. Comparsons between te models were made for ever second or between and 36. Te npt wnd feld to te -model was mplemented n two dfferent was. In Case, te vertcal wnd profle from pont (0, 0) n te MIUU-model (cf. Fg 7.) was cosen. Te pont s located pstream te ll and cosen so tat t s not affected b te ll. Ts wnd profle was sed as an npt wnd feld profle n ever grd pont n te -model. Te vertcal wnd w was set to ero, wen te terran-nflenced w* was calclated. Ts coce of npt wnd feld can be related to te case were te wnd feld does not know te presence of te ll;.e. s not affected b te ll. It s comparable to te se of model data calclated wt a lower resolton terran map. Vales for sed n ts case were 0.5,,, 3, 4, 5, 6, 7. In Case, vertcal wnd profles from ever tent pont of te MIUU-data were sed as npt. Ts data was nterpolated to all grd ponts b lnear nterpolaton. Splne nterpolaton was tested bt wt bad reslts, especall near borders. De to te nterpolaton, te model doman n ts case was restrcted to 44. Ts npt feld knows te presence of te ll, becase te MIUU-model ses te same terran map as te -model. Ts case can be compared wt sng measrements as npt data. Vales for sed n ts case were,,.5, 3, 3.5, 4, 5. Te -model was rn n Case wt a grd se of 660 and n Case wt 440. In bot cases te resolton was km, te same as n te MIUU-model. Te tme needed for one smlaton for Case s arond 90 seconds on an Intel Pentm 4 3.00 GH wt 04 Mb RAM. If notng else s sad, te reslts aregven for 67.4 meters egt, wc s level 8 n bot models. Te reslts from te -model are compared wt te reslts from te MIUU model. Te man parameter compared s te orontal wnd speed; te absolte vale of - and v- components. For comparsons, te dfference n wnd speed between te models s calclated n ever pont and normaled wt te mean speed of te MIUU model for te present level. It gves te new parameter D, defned as: 5
U U MIUU D = (7.) U MIUU mean In te same wa te normaled dfference for te npt wnd feld compared wt te MIUU-model reslt s calclated wc gves te parameter D. To get a vale of te overall qalt for a certan level, te mean of te absolte vale of D s calclated for a level as: I J (, ) D = = D = 0 0 IJ (7.) n 7. Reslts for Case In Table 7-, dfferent qalt parameters are sown for Hll and Case. In colmn, te vales of D s sown for dfferent ors. Te vale gves a measre of te overall agreement between te -model and te MIUU-model. Te rato sown n colmn 3 gves a vale for te mprovement of te wnd feld wt te -model smlaton. In colmn 4 and 5 te mamm and mnmm vale of D for te level are sown. Ts gves te span for te dstrbton of te dfferences between te -model and te MIUU-model. A dstrbton for D s plotted n Fgre 7-4. In colmn 6 te Rcardsons nmber (R) s sown for level 8 (67 m) and at pont (0,0). Te defnton of R s: g θ R = θ (7.3) U Te gradents n R are appromated b te dfferences between te present level and te level above. All vales for D sown for eac or are calclated for te smlaton wt te optmal coce of as fond accordng to te mnmm of D. Ts vale for s sown n colmn 7. Anoter mportant parameter to std s te speed p-effect on te top of te ll. Optmal to get a mamm vale of te wnd speed s sown n te last colmn. 6
Table 7-. Dfferent qalt parameters for Hll, Case and level 8 (67 m). Hor D Rato Ma D Mn D R Optmal D for D D n Optmal for mamm wnd speed 0.055 0.9 0.46-0.54 -.54 5 5 4 0.034 0.89 0.35-0.36 -.0 4 4 6 0.0 0.8 0.33-0.3-0.96 4 3 8 0.05 0.79 0.33-0.3-0.3 4 3 0 0.05 0.8 0.43-0.50 0.8 5 4 0.036 0.94 0.34-0.58 0.36 4 4 4 0.063 0.98 0.55-0.57 0.39 3 4 6 0.09 0.99 0.80-0.39 0.39 3 8 0.73.00.03-0.07 0.40 3 30 0.70.00.03-0.6 0.4 3 3 0.9 0.93 0.67 -.8 0.04 7 7 34 0.44 0.94 0.40 -.47-4.57 7 7 36 0.05 0.94 0.40-0.8-0.6 3 3 Table 7- sows tat te -model works qte well drng afternoons and evenngs bt fals drng ngttme and n te mornngs. Between or 4 and 30 te mprovement rato s nearl wc means tat te -model does not mprove te wnd feld compared to te nterpolated data. As te -model does not andle termal effects, te calclated wnd from te model s nearl te same da and ngt. Te onl dfferences come from te npt wnd feld. Te MIUUmodel andles stratfcaton effects and gves dfferent reslts for ngt and da. Ts s sown for Case n Fgre 7- and 7-3. Te terran sed for tese reslts are Hll. Te smlaton n Fgre 7- s for datme (or 6) wt nstable stratfcaton. Te smlatons from te two models look te same n front of and above te ll. Bend te ll a wake s appearng n te MIUU-model. Ts s a non-lnear effect not resolved b te smple pscs sed n te -model. Anoter dfference, ard to see n Fgre 7-, s tat te mamm wnd speed n te MIUU-model smlatons s located somewat bend te top of te ll nstead of rgt above te top as n te -model. 7
60 (a) Wnd Speed MIUU model m/s 8 50 7.5 40 7 6.5 30 6 0 5.5 5 0 4.5 0 0 30 40 50 60 4 60 (b) Wnd Speed model m/s 8 50 7.5 40 7 6.5 30 6 0 5.5 5 0 4.5 4 0 0 30 40 50 60 Tme =6, Hegt =67.4, =4, R=0.95886 Fgre 7-. Smlatons b MIUU-model (a) and -model (b) for datme. Hll and Case. 8
60 (a) Wnd Speed MIUU model m/s 8 50 7 40 30 6 5 4 0 3 0 0 0 30 40 50 60 60 (b) Wnd Speed model m/s 8 50 7 40 30 6 5 4 0 3 0 0 0 30 40 50 60 Tme =8, Hegt =67.4, =3, R=0.40077 Fgre 7-3. Smlatons b MIUU-model (a) and -model (b) for ngttme. Notce tat te wnd speed goes down to nearl ero n te MIUU-model smlaton. Hll and Case. 9
In Fgre 7-3 (sowng te wnd speed for ngttme condtons), te reslts from te two models dffer sgnfcantl. Te termal effects are domnatng n te MIUU-model smlatons. Frstl, ar s blocked n front of te ll wc s cased b stable stratfcaton (see Fgre -a). Besde te ll, ar flows ot from te ll as an effect of coolng of ar close to te srface of te ll. Ts cooler ar s denser tan te srrondng ar and te flow wll be downslope on bot sdes of te ll. Tose two effects ave notng to do wt conservaton of mass wen ar s flowng over te ll and te are terefore not cagt b te -model. Te mprovement of te wnd feld b rnnng te -model can be llstrated b comparng te dstrbton of D n te npt feld and te -model feld. 4 Dstrbton of D model Inpt feld 0 Relatve Freqenc % 8 6 4 0 0. 0.5 0. 0.05 0 0.05 0. 0.5 0. D Fgre 7-4. Te dstrbton of 8 and =5. D n te npt feld and te -model feld. Te smlatons s for Hll, or Fgre 7-4 sows tat te ponts wt large D ave dsappeared after rnnng te -model, especall on te negatve sde. Ts s cased b te creaton of te speed-p mamm above te top of te ll. Te mprovement on te postve sde s cased b te areas wt low speed bot n front of and bend te ll created b te -model. Wen comparng all ors, te conclson s tat te most accrate reslts from te -model appears between or and or 0. All oters ors ave strong termal effects wc are not captred b te -model. 30
7.3 Reslts for Case Table 7-3. Dfferent qalt parameters for Case and level 8 (67 m). Hor D Rato Ma D Mn D R Optmal D for D D n Optmal for mamm wnd speed 0.0 0.83 0.47-0.67 -.54 4 5 4 0.08 0.8 0.9-0.47 -.0 3.5 4 6 0.03 0.75 0.6-0.43-0.96 3 3 8 0.00 0.70 0. -0.40-0.3 3 3 0 0.06 0.75 0.30-0.59 0.8 3.5 4 0.00 0.87 0.3-0.76 0.36 3.5 4 4 0.038 0.89 0.39-0.98 0.39 6 4 6 0.057 0.94 0.55 -.07 0.39 6 3 8 0.08 0.98 0.6-0.93 0.40 6 30 0.080 0.96 0.5-0.8 0.4 6 3 0.095 0.89 0.8 -.65 0.04 6 7 34 0.065 0.90 0.76 -.4-4.57 6 7 36 0.05 0.80 0.40-0.9-0.6.5 3 Table 7-3 sows te same parameters as n Table 7- bt for Case. Te D for Case s clearl less tan for Case becase te ger accrac of te npt wnd feld. Bt te mprovement rato s stll good wc ndcates tat t s meanngfl to rn te -model. In Fgre 7-5 two dfferent ors from Case are plotted, or 8 (a-c) and or 30 (d-f). In row, te plots sow te nterpolated wnd feld wc s sed as npt data. In te second row, te wnd felds from te MIUU-model are sown and n te trd row te reslts from te -model are sown. For or 8 te reslt looks good. Te onl dfference of mportance s tat te mamm wnd speed n te MIUU-model reslt s fond bend te top. For or 30 te same problems as n Case appear. Bt becase te nterpolated wnd feld ncldes nformaton abot te area wt blocked ar and tat te -model generates te speed-p effect on te lltop, te reslts look better tan for Case. 3
km m/s (a) Wnd speed Interpolated wnd feld 9 40 8.5 35 8 30 7.5 5 7 0 6.5 5 6 0 5.5 5 5 0 0 30 40 km 40 35 30 5 0 5 0 5 m/s (d) Wnd speed Interpolated wnd feld 0 0 30 40 7 6 5 4 3 km 40 35 30 5 0 5 0 5 (b) Wnd Speed MIUU model 0 0 30 40 m/s 9 8.5 8 7.5 7 6.5 6 5.5 5 km 40 35 30 5 0 5 0 5 (e) Wnd Speed MIUU model 0 0 30 40 m/s 7 6 5 4 3 km 40 35 30 5 0 5 0 5 (c) Wnd Speed model 0 0 30 40 Tme =8, Hegt =67.4, =3 m/s 9 8.5 8 7.5 7 6.5 6 5.5 5 km 40 35 30 5 0 5 0 5 (f) Wnd Speed model 0 0 30 40 Tme =30, Hegt =67.4, =6 Fgre 7-5. Interpolated wnd speed (a, d), MIUU-model reslts (b, e) and -model reslts (c, f) for or 8 and 30 and level 8 (67 m). m/s 7 6 5 4 3 3
7.4 Reslts at oter levels Te nflence of te ll s largest close to te grond and decreases wt egt. Ts s descrbed n Stll (988), capter 4..3. Te adstment of te wnd feld n te -model s also largest near grond. Ts s cased b te fact tat te dvergence s largest near te grond. Wen nvestgatng te qalt of te -model for dfferent egts, t s fond tat te vales of optmal ncrease wt egt. Ts ncrease s most domnant n Case, bt s also seen n Case. Te optmal for dfferent levels for or 8 s sown n Table 7-4. Table 7-4. Optmal for dfferent levels and or 8. Terran sed s Hll. Level Case Case R 0.5-0.0 3-0.03 4 3-0.06 5 4.5-0.0 6 4.5-0.6 7 5.5-0.5 Te dfference n optmal for dfferent egts ma depend on te frcton from te grond. Te wnd profle sed for te npt wnd feld s calclated wt frcton. Bt te - model adsts te wnd speed wtot accontng for te frcton. Tat ma be te reason for te overestmate of wnd speed near grond n te -model and terefore a lower vale of s te optmm coce. (A lower vale of gves a smaller adstment n orontal drecton, see Secton 8.) Anoter effect varng wt egt s te locaton relatve to te lltop of te wnd mamm as fond n te MIUU-model reslts. Near grond te mamm s fond closer to te top of te ll. For eample, at or 8 te mamm n te MIUU-feld s fond one grd pont (33, 3) closer to te top of te ll at low levels (below m) tan te laers above (-50 m), were te mamm s fond at pont (33, 3). In te -model te wnd speed mamm s fond at pont (3, 3) for all levels. 8. -constants In Secton 4.8 te teoretcal backgrond of te -constants was dscssed. In ts secton te effect of dfferent -constants wll be stded. 8. Propertes of -constants Frstl, tests wt = = = and = = = were performed. Te reslts were dentcal. Ts leads to te conclson tat t s te rato / wc s te mportant parameter ( = = ). Ts s an mportant conclson and conseqentl onl as been vared n te followng tests and s kept to n all cases. 33
In Serman (978), = and =00 were sed. Bt tese vales gve nnatral reslts n te -model. Walmsle et al (990) ses = and =0.5. Gao and Paltkof (990), ave made rns for a lot of -vales. Te consder tat =0. corresponds to ver nstable and =0 (bot wt =) to be sed for ver stable stratfcaton. Dfferent beavors, as reslt of two dfferent are sown n Fgre 8-. Smlatons for te two dfferent vales of were made wt te same ll and npt wnd data as n Secton 6. Te absolte vales of te adstment are calclated for eac pont of te doman. Te adstment decreases wt ncreasng egt. Fgre 8- sows te egt were te adstment passes te tresold 0.3 m/s. (a) Level for adstment >0.3 = (b) Level for adstment >0.3 =4 500 500 000 000 Z (m) 500 Z (m) 500 0 4 0 3 Y (m) X (m) 3 0 4 0 4 0 3 Y (m) X (m) 3 0 4 Fgre 8-a-b. Level wt adstment >0.3 m/s for = (a) and =4 (b) For te reslts sown n Fgre 8-a te vale of was set to. Te adstment as two peaks, one n front of te ll and one bend. Ts adstment belongs to adstment n vertcal wnd speed. For te reslts sown n Fgre 8-b te vale of was set to 4 (same as te smlaton descrbed n Secton 6). Te plot sows one peak above te ll. Ts peak belongs to te adstment n te -drecton. Te adstment n Fgre 8-b also covers a larger area tan n (a). Ts s cased b larger adstment n te -drecton wt =4. Te reslts from (a) and (b) togeter sow tat a ger vale on gves more adstment n orontal drecton and tat te adstment goes p to a ger level wt low. Anoter wa to determne adstment n te wnd feld for dfferent s to calclate te dfference between te mamm and mnmm wnd speed from te -model smlatons n Secton 7 and Case. Te npt wnd speed at eac egt s eqal for ever pont. After rnnng te -model te dfference between te mnmm of orontal wnd speed n front of te ll and te speed-p above te top s dependent of te vale of. Ts dependence for dfferent egts s sown n Fgre 8-. 34
0 9 8.8 m 67.4 m 57. m 875.4 m MammMnmm as fncton of 7 MammMnmm (m/s) 6 5 4 3 0 0 3 4 5 6 7 Fgre 8-. Te dfference between mamm and mnmm n wnd speed from te -model for dfferent levels. Terran sed s Hll. Fgre 8- sows tat te adstment n orontal wnd goes to ero wen goes to ero. Te reason for ts, as descrbed above, s tat te adstment domnates n vertcal drecton for low. Te fgre also sows tat te adstment s largest near grond and decreases wt ncreasng egt. For te gest sown level (875 m) te adstment as a mamm for =. Ts s cased b te fact tat a large vale of gves onl small adstments n vertcal veloct. Ts means tat no orontal adstments are needed for g levels. 8. Coce of Becase te lack of an obvos pscal mplementaton of -constants, tese ave to be emprcall cosen. From te comparsons between te MIUU-model and te -model n Secton 7 t s ard to see dependence between optmal and R (stablt). Te reason for ts s tat t s ard to determne te optmal for stable condtons wen te -model reslts does not sow te same wnd speed patterns as te MIUU-model (see Fgre 7-3). Instead, te qalt of te npt data s mportant. In Case, wc ses npt data wt low qalt, te optmal s arond 4 and for Case te optmal vale s 3.5. Tese vales are for Hll wt mamm terran gradent 3.4 cm/m. For smlatons wt Hll, wc as a smaller terran gradent (3.6 cm/m), ger vales for are needed for Case. Te optmal vale for ts case and level 8 (67 m) s between 5 and 6. For Case te optmal vale s 3. For Hll 3, wc as a mamm terran gradent of. cm/m, te optmal vale for Case s between 3 and 4. Terefore, t looks lke te optmal vale for s not dependent on te sape of te ll. However, more smlatons 35