ࡱ> }z{|5@Objbj22 8XXW tLttttt$RUtt6tt(Bth |@WZ<b,LT T Trapped lee wave interference in presence of surface friction Ivana Stiperski1, Vanda Grubiai2 1 Meteorological and Hydrological Service, Zagreb, Croatia E-mail: stiperski@cirus.dhz.hr 2 University of Vienna, Vienna, Austria Abstract: Lee-wave interference over twin peaks in the presence of a frictional boundary layer is investigated by means of idealized high-resolution two-dimensional simulations with the NRL COAMPS model. The interference is examined under linear to highly nonlinear flow regimes. The results show that the pattern of positive and negative interference, evident in the wave amplitude and rotor strength, is consistent with the predictions of linear interference theory. Interference is found to most significantly affect the flow in the lee of the downstream ridge, preventing boundary-layer separation there under conditions of destructive interference and amplifying rotor strength under constructive interference in different flow regimes. Only under strongly nonlinear conditions is the rotor strength amplified by constructive interference beyond that obtained in the lee of a single mountain. Keywords: boundary layer separation, critical mountain height, nonlinearity, rotors 1 INTRODUCTION Trapped lee waves form downwind of a mountain ridge under conditions conducive to wave energy confinement within the lower troposphere. The study of nonlinear lee-wave resonances over twin ridges under free slip conditions (Grubiai and Stiperski 2009) shows that the resonance is governed by the ratio of the intrinsic trapped lee wave wavelength ls and the ridge-separation distance V. The positive and negative interference patterns are evident in the wave amplitudes and total wave drag. In the presence of a frictional boundary layer, the adverse pressure gradients associated with nonlinear, trapped lee waves are expected to lead to boundary layer separation and formation of recirculating regions (or rotors) underneath lee wave crests (Doyle and Durran 2002). Here we investigate the effects of a frictional boundary layer on the lee wave resonance and rotor strength over twin peaks.  2 MODEL SETUP Two-dimensional numerical simulations were carried out using the nonhydrostatic COAMPS model initialized with an idealized sounding. The surface wind speed is 10 m s-1, vertical wind shear equal to 6 m s-1 km-1, and the stability N=0.012 s-1 is constant within the troposphere (Fig. 1). Friction at the lower boundary is imposed via a partial-slip condition with surface roughness z0=0.01m. A set of idealized simulations was conducted for single and double bell-shaped obstacles for mountain heights H=300-1500 m and valley widths V=30-65 km. The mountain half-width is 5 km. The simulations were run for 10 hours during which time a quasi-steady state was reached in the simulations. Figure 1. Initial upstream profile of potential temperature (K) and horizontal windspeed (m s-1) 3 RESULTS 3.1. Linear regime H< Hc=600 m For twin peaks lower than Hc the wave interference pattern follows the linear interference theory predictions with constructive and destructive interference occurring, respectively, for V=n*ls, and V=(2n-1)/2* ls, where ls is the intrinsic horizontal wavelength downwind of a single obstacle for the given upstream profile (ls=21.8 km) and n>2 is an integer (Fig. 2). The positive and negative interferences are barely evident in the wave amplitudes and horizontal velocity perturbations in the lee of the first obstacle (A1 and U1 over the valley), but leave a clear imprint in the lee of the second obstacle, where the wave amplitude and horizontal velocity perturbation (A2 and U2) are enhanced or reduced relative to the value in the lee of a single mountain (As). The non-dimensional wavelengths cluster around integer values. The flow structure over the downstream obstacle is different under constructive and destructive interference (not shown). Whereas under constructive interference the flow resembles that over a single ridge, for destructive interference the potential temperature and horizontal wind speed are symmetric around the downstream peak. For H>Hs=370 m, the wave-induced adverse pressure gradients are strong enough to cause boundary-layer separation and rotor formation in the lee of a single obstacle (Us<0). Within the valley, the rotor strength (U1) remains equal to Us for all V, whereas downstream of the secondary obstacle the decrease in amplitude due to destructive interference is strong enough to prevent reversed flow formation, leading to large differences between the flows under constructive and destructive interference. 3.2. Nonlinear regime HcHc=600 m the regime becomes nonlinear as the mountains are high enough for flow reversals to occur underneath the first lee wave crests, downstream of both obstacles irrespective of the interference pattern. In the weakly nonlinear regime, Hc Hn For He"Hn=1000 m, the nonlinearities start to affect the solutions significantly. Even though the interference still retains its linear pattern of occurrence, it is no longer evident in the variations of total drag (Fig.2), and the V/l attains also non-integer values. The flow within the valley, unaffected by the secondary obstacle with the linear and weakly non-linear regimes, is weaker than in the lee of a single obstacle and exhibits some variation with V. Also,, A2>A1 for all V. For H=1000 m constructive interference significantly enhances the rotors developing in the lee of the second ridge, while the rotors within the valley are weaker. For H=1500 m, the enhancement of the rotor circulation is somewhat weaker, possibly due to the nonlinear interaction between the wave field and the PBL flow. Figure 2. Amplitudes: A2 (solid), A1 (dashed); rotor strength U2 (solid), U1 (dashed); normalized wave drag (D); non- dimensional wavelength (V/l), as function of valley width V for H=400 m (left) and H=1000 m (right). Solid red horizontal lines denote values downstream of a single ridge. 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Figure 3. Amplitudes and rotor strength downstream of the first (A1,U1) and second (A2,U2) obstacle as function of mountain height H. Different V for which constructive (orange) and destructive (blue) interference occurs are indicated, with the values for single ridge in red. Vertical lines denote threshold mountain heights Hs=370 m, Hc=600 m, and Hn=1000 m. 4 CONCLUSIONS Figure 3 summarizes the influence of mountain height and interference pattern on the wave amplitudes and rotor strength. The significance of the three threshold mountain heights (Hs, Hc and Hn) is best evident in the rotor strength diagrams. Rotors start to form (Ui<0, i=1,2) once the critical mountain height Hs for boundary layer separation is reached. Up to Hc = 600 m, the wave amplitudes and rotor strength increase linearly with H. Within the valley, the U and A values coincide with those for a single ridge, whereas downwind of the second obstacle a clear distinction between constructive and destructive interference is evident in both of these values. Weaker wave amplitudes under destructive interference require a higher critical mountain height for the rotor formation there. In the strongly nonlinear regime, for H>Hc,, while the distinction between the interference patterns is becoming more apparent over the valley, where an indication of a limit to the rotor strength is also obtained, downstream of the second ridge the distinction is getting smaller and the rotor strength increases with H reduced. REFERENCES Doyle, J.D., and D.R. Durran, 2002: The dynamics of mountain-wave-induced rotors. J. Atmos. Sci., 59, 186 201. Grubiai, V., and I. Stiperski, 2009: Lee wave resonances over double bell-shaped obstacles. J. Atmos. 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OT,gh6$Y溇rp!#@wI 2~0ŔmE@R%mNPJ]CM53 $iV҆j?֬0h$Y涄~ X*&}]_GYxITXLPlEj*,`#D5@M`q -h#M @&t) `SE?WBet𦳑ue@&T `S?h>/4]':eG]v @`*09*!prjCi~k@0%0RMLcXIC6sA`^^~pw:u{  PM @"*[•?Jv}#{kfS!hCnyKW8x chdp(L7|<"ԣS *w/x6gqaZ~ h ci*>:: U<E$"fKs:xݼ>sI",?[#ct<ũgcwoq*\t̼QzA*;cY/iӫ-l|nBKZZ6lS݅_Wx?m_n"SпNa&aora/v>h?osn_>N[UeY{ݞ_m=!ň/1rڼ?vO9>=*p{{X6D㍈Eguw?!b|D;qz۩Z/ÌA 0"!toFg$GO4C2Gn %w ~P? x*[5._-^-| K/HE @g <]pq|@b?4+"`\i߳}0WmVaT;ct){l4]QhhK7Z,D,PGl4;,lPoh6Di>SzuBS4p\6nU~;*n,{cHk$u|Nm5HcSgm6fmt{-`4N=ϫZxoFctׂ؂ G7mKgnm dO ܇''1E-QXyyl!o}LM}g!xz#TwkFg2ޟ=f̴LJY5x)7*In7oTc0~_ͽ忊<9&9ߺa$RP}^"i^u!]*[*L~}Lkwt+SZW$*n>֞6uF]c4=C,_bY߸h$NAcJ[y=}V^J/tE}%`}B 8_7V$ (;eR@=--$.~$4"Oo 6'<xZklTE>wJix.(PRJE aA`Aj #Dd !b M5 @|Ԉib4%@RsC|sf9sbQ9"ZD#<@d\8!Yn̼4ڑJ$r}ELMQOTnTjrPNDkU$lT)bD-*LU->ҁn-y2T_tydnY_@ۣ=~1hMqx"ұb;VrYȼ(Erƾ ePrKh>MY4œs9,X*`N*<3_ݞ1[k8aw!Rv.WxڎJ3t|XMp}LCC7֗֗5TLOŔW:? vP))T1i ȸg}2"i<>ێ9~{7\[|_s};oVϼ}cۊu勒M'2\2/5we:T~i*lrp[ln?3$%i?i"9ϵR&`:;"F=]}BγEq-R&e.eMuRʠ6=f>&Joo }aiÆ=l_?si+ ^kk {a߬=מ"א\oq[xL3(~pE}zPz$=7f\y,8* >^ |Kn/|A^Ky|)Z߂[\"Z k_N S7Ƚ .[/~|p\wo ^ #o&޷&GoUsM!wo )Jπ@{1|/9sD}\3'599n'?L[`Ƚ<$ `?As^ysyn rֵCo|"El8WXo7o~H[t0zݖh7YBs0%z՜N'k;4w_rRf⡙&CSStf=,ܢ4w{hnI@CsҬМ _1&拣W{z-&{q2%ZN|^-%8NfM@g\4z/LPԣoWG='_GLߐ~!?,i;kԸټrο}|ZC]+s_5^s) ۧsƟO׶Sh;~q\t0TH$RDdވ$B\[;k]s!O]v+GNvJr_dt|~OxOT.ѧ7_:}WQߥ{uTiܗhk*/ثYu֚Z8ޞ畼wIC2We}rv]mRǔ׷OzoG)/w"ҿ0rS2e'2`KMg?/r@@@ yNormalCJ_HaJmHsHtHL@L rm Heading 1 $@&a$5aJmHsHtH DA@D Default Paragraph FontRiR Table Normal4 l4a (k( No List 6U@6 C1 Hyperlink >*B*phB'B MComment ReferenceCJaJ@@ MComment Text, CharD!D M Char CharCJ_HaJmHsHtHH2H M Balloon TextCJOJQJ^JaJ\S@B\ 6%Body Text Indent 3 `a$CJaJmH sH tH nSn 6% Table Grid7:V0a$W XW >?bclmX[kl [\prbcopY 00000000000000000000000000000000000000000000>?bclmX[kl [\prbcopY 00000000000000000000000000000000000000000000 M %:'dHJNO !"()+MO*O    Ob$@G J Y + 4 wzg~= V Y 333333333333Ivana Stiperski2RN^`o(^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.2,r/  pQT:D>.TmAC}|HD?DPxO%X@pqp2 dddW @$@4@"$L@(@NUnknownIvana StiperskiGz Times New Roman5Symbol3& z Arial5& zaTahoma q2&2&F::!x24G G CQHP ?'}-H7Sample for complete submission information to ICAM-2009pa08Ivana Stiperski Oh+'0 4@ \ h t 8Sample for complete submission information to ICAM-2009amppa08e fa08 Normal.dotcIvana Stiperski4anMicrosoft Word 10.0@^в@ .@5@F<G Rt C  gl."Systeml r0 l-@Times New Roman- )2 eahTRAPPED LEE WAVE INTCHH=<CHCCCdHIC'HC)2 :ahERFERENCE IN PRESENCCH<CHCHHC'H<HC8CHH)2  ahE OF SURFACE FRICTIOCN<8HI>HHC<H(HC'N 2 ahNWH@Times New Roman- 2 ah W-@Times New Roman- 2 Oah W)@Times New Roman-"2 vahIvana Stiperski)%)%.*% )@Times New Roman- 2 Pt ah1W-v @Times New Roman-.#2 v ah, Vanda Grubii<&)*%<)* %.-- 2 P ah2W- 2 v ah W 2 v ah W& 2  ah W%- 2 ah1W-a2 69ah Meteorological and Hydrological Service, Zagreb, CroatiaJ%%***)%%%)*=(***)%%.%)%%2&)%*7*%% 2 6 ah W%- 2 ahEW3 2 ah-W2 ahmail: ?&@Times New Roman-.2 ahstiperski@cirus.dhz.hr *%! %L% * ** * - 2  ah W%- 2 ah2W-D2 7&ah University of Vienna, Vienna, Austria<))% )*<&*)%<%**%<) % 2 D ah W&- 2 ]Oah W) 2 Oah W)@Times New Roman-2 !OahAbstract6)"%!@Times New Roman-2 !eah:  2 !ahLW-2 !ahee!! 2 !ah-W2 ! ahwave inte6!&!&!2 !Cdahrference over twin peaks in the presence of a frictional boundary layer is investigated by means of !!&""&%!5&&!!%&&!&!!&!!&!!&&!%&%&&!$"%!&%!%!!&'$9!!&& 2 wOahidealized high&!!!!&&%& 2 wah-W 2 wahresolution two!&&&&5& 2 wah-WX2 w3ahdimensional simulations with the NRL COAMPS model. &:!&&&!9&!&&5&&!62-275C+*9&&!@2 w #ahThe interference is examined under r-&!&!!!&!!!&"9&!&&%&! 2 OahlW82 dahinear to highly nonlinear flow&!!&&%&$&&&&!!&62  ah regimes."%9! 2 ah W22 ahThe results show that the -&!!&&&6&!&!O2  -ahpattern of positive and negative interference&!!&&&&%!!&&&!%!%"&!!!&!!2 Oah, "2 wahevident in the !%&!&&& 2 $Oahwave a5"%!)2 $ahamplitude and rotor "9&&&!!&%%&52 $ahstrength, is consistent with!&%&!&&!%5& 2 $ah W2 $ahthe &!"2 $J ahpredictions of &!&!&&&52 $ ahlinear interference theory. &!!&!!!&!!&!&$12 $>ahInterference is found to y&!!!&!!&&&%$I2 zO)ahmost significantly affect the flow in the9&!%&"!&$!""!!&!!%5!&!&!2 zAah lee of !!!&!2 z8ahthe&!2 z ah downstream &&5&!!:!22 zC ahridge, preventing boundary&%! &!%!&&%!%&%&&!$ 2 z ah-W#2 z ahlayer separation"%!!!&!!&%2 z ah there under i!&!! &%&! 2 Oahconditions of !&&&%&&/2 ahdestructive interference&!&!%"&!!"&!!2 Zah and amplifying rotor strength under constructive interference in different flow regimes. !&&"9&$&&&&!&%&&&&!!&&&!%!&!!!&!!&&!!&&6!%:!2 'OahOnly o6&$2 'ahundero&&&! 2 'ah W2 ' ahstrongly &&%$2 ' ahnonlinear &&&&!!2 ' ahconditions!%&&&&2 '2 ah is the rs&!2 'Mahotor strength amplified by constructive interference beyond that obtained in &&!&%&!9&!&&$!&&&!%!&!!!&!!&"$&&&&!%&!&!&%82 }Oahthe lee of a single mountain. &!!!&!&%!9&&&!& 2 }ah W!- 2 Oah W%@Times New Roman- 2 3 O ahKeywords: 2!!2&&@Times New Roman- z2 3 Jahboundary layer separation, critical mountain height, nonlinearity, rotors &%&%&&!&!!!&&&%&! &6%&&&&&!&&%&&&!&!&&- 2 3 G ah W%- 2 Oah W-@Times New Roman- (2  Oah1 INTRODUCTION r* <7<A<<<8 A< 2  ah W-@Times New Roman- 2 T Oah Wha'- 2 Oah WdF2 'ahTrapped lee waves form downwind of a mo4$**%*.%%.:&)% /*?.**;+;)*.*.%/?+^2  7ahuntain ridge under conditions conducive to wave energy *)%).**%.))*%.%*)*** .%*)*)&*%.*/:&)%.%)%*)ha'J2  O*ahconfinement within the lower troposphere. %*))&@%*;)))%+;%*** *)%%22  ahThe study of nonlinear lee4)% )+)*)+))%%%% 2  ah-W#2  ahwave resonances ;%)%& *)&)%% %2  ahover twin ridges *)%;**)% 2  Tahunder *)*%ha')2 o Oahfree slip conditions%%$ *$%*)**) 2 o ah W$o @Times New Roman- .72 o ah(Grubii and Stiperski 2009) <)* &$%)*$.*%!)$****.- 2 o  ah W$2 o ? ahshow )+; 2 o ahsW 2 o  ah that $)%;2 o ah the resonance is governed by th$)%$% *)&)%%$ $)+)%)&*$*($)"2 o <ahe ratio of the %$%*$*%)&ha'-D2 O&ahintrinsic trapped lee wave wavelength )) %,%**%*,%%-;&)%-;&)%&))),@Symbol- 2 ahlW.- 2 ahsW-2  ah and the ,&)*,)%,2 m ahridge *)& 2  ah-WU2 4 1ahseparation distance V. The positive and negative  %*%%*),* %)%&,<,4)%,** )%,&)*,)&)%)&ha'2 5 ONahinterference patterns are evident in the wave amplitudes and total wave drag. )%%&)%%"*%%) "%%"&)*%)")")%#;&)%"&?*)*% "&)*"*%#:&)%"*%)"(2 5 ahIn the presence of s)")%"*% %)%%"*"2 5 aha %#2 5  ahfrictional %*)%ha' 2 Oahboundary layer**))*%(&(% 2 Gah,W2 \ ah the adverse)%%*)% %A2 $ah pressure gradients associated with *% !)%)%*%) %! *%%%*;)2 ahnonlinear, e)**)%%2 ahtrapped %**%*2  ahlee %%2 ahwavese:&)& #2 [ah are expected to%%%)*%%%**2 rah lead %$(ha'q2 ODahto boundary layer separation and formation of recirculating regions ***))*%)&(% %*%%**&)**?%***%%%)%*)%)** 2 ) ah(or rotors) *** 2 ahunder *)*%2 R ahneath )%%) 2 ah W2 ahlee %%(2 ahwave crests (Doyle ;%)&%%  <+(&ha'%2 U Oahand Durran 2002).e%)*<)%)****+2 U ah Here we investigate <%%:%*)% )%%2 U ahthe )% 2 U aheffects of a f%%% *%^2 U 7ahrictional boundary layer on the lee wave resonance and %*)%**))*%(&(%*))%&%:%)%% *)&)%%%)*ha';2 O ahrotor strength over twin peaks. ** %)))+)%;)*%%*  2 aah W%ha' 2 Oah W%ha'- 2 vOah2W* 2 vyah WL2 v ah MODEL SETUP PA<77.87<3 2 vJah W.ha'- 2 Oah Wha'- 2 !Oah W2 !ahTwo4:+ 2 !ah-WX2 !3ahdimensional numerical simulations were carried out a*?&) +)%'**@%%%' @)%*) (;%%'%%%*'*)ha'b2 O:ahusing the nonhydrostatic COAMPS model initialized with an ) *)))%*)***(** %%)7=;J/.*?**%))&%%**;))%*ha'-&2 Oahidealized sounding*%%%%* *))*))2 ah. The 4)%2 y ah surface wi!)&%%:2 ahnd)* 2 Fah W2 aahspeed *%%*2 :ahis 2 qah 10 m s **? - 2 v ah-W- 2  ah1W- 2  ah,W2  ah vertical )&%%ha'-2 @O ahwind shear;)*!)%% 2 @ah W2 @ ahequal to %*)%*2 @ah6 m s *? - 2 ah-W- 2 ah1W-2 @ah km*@- 2 eah-W- 2 wah1W-2 @ah, and %)*.2 @Oahthe stability N=0.012 s)%!%*)</***) - 2 Q ah-W- 2 c ah1W- 2 @~ ah W2 @ ahis !ha':2 Oahconstant within the troposphere%*) &)%;))$)%%*** *)%%2  ah (Fig. 1) $.)$*2 ah. $,2 ahFriction at the lower .%*)$%$)%$+:%ha'=2 O!ahboundary is imposed via a partial**))*%(( (@** %*()%(%)*%% 2 ah-W)2 ahslip condition with *(%*)*+));)(2  ahsurface !*%%%ha'-2 `O ahroughness*))*)& 2 `ah WA 2 `ahzW%- 2 mah0W-2 `!ah=0.01m. /**+?B2 `z ahA set of ide<A &A*B*%&2 `ahalized simulations%%%*A @*%**! 2 ` ah WB2 `M ahwas ;%!ha'A2 O$ahconducted for single and double bell%*)*)%%*:*9 *)%9&)*:**)*%9*% 2 ah-W2 ahshaped !)%*%*92  ahobstacles** %%& 2 3 ah for d:*ha',2 Oahmountain heights H=300?+*)%)))%*) (</*** 2 ah-W2 ah1500 m****)? 2 ah W(,2 ahand valley widths V=30%)*))%&));** (</+* 2 ` ah-W2 { ah65 *)ha'2 Oahkm*? 2 ah.W 2 ah W%2 ahThe mountain half4)%?+*)%))% 2 Uah-W2 rahwidth ;**2 Hahis  :2 ah5 km. The simulations were run **@4)% @)%*)!;%&*)ha'C2 O%ahfor 10 hours during which time a quas***)*) *)));)&)A%%*)% 2 `ahiW 2 wah-W&2 ahsteady state was r %%+( %%;% 2  aheached %%%)%*ha'-2 ?Oahin)%2 ?ah the simulations.)% @*%**  2 ?ah W 2 ?ah W' 2 ?ah W' 2 ?ah W' 2 ?9 ah W'2 ?` ah  2 ?0 ah WW 2 ? ah W-2 ? ahFigure 1. .&)!!&-2 ? ahInitial &!%2 ? ah upstream profile%&!!9&&!- 2 ?ah W-2 ?ahof& 2 ?ah W2 ?  ahpotential &&!&!ha'- 2 Oah W 2 ah Wu 2 vah W' 2 ah W' 2 ah W' 2 ah W' 2 ah W' 2 9 ah W'2 ` ah :2 B ahtemperature (K) and horizontal !9&!!&!6!&&&& &&! 2 ahwindspeed (m s5&&&!!&9@Times New Roman- 2 wah-W- 2 wah1W- 2 ah)W- 2 ah W-ha'- 2 Oah3 * 2 ah Ws2 ahRESULTS <7.=78. 2 ah W 2 ah W' 2 ah W' 2 ah W' 2 ah W' 2 9 ah W' 2 ` ah W'- 2  ah W+ha'- 2 UOah W 2 Uah Wu 2 Uvah W' 2 Uah W' 2 Uah W' 2 Uah W' 2 Uah W' 2 U9 ah W- 2 UL ah W+ha'- .2 Oah3.1. Linear regime H< H**7.%*%%%*C%A/A- 2 ahcW- 2 ah=600/+** 2 |ah W 2 ahmWC 2 ah W.ha'@Times New Roman- 2 Oah W$ha'- 2 aOah Wd42 aahFor twin peaks lower than Ht.*;)*%&) +:%)&)<- 2 nahcW-2 aah the wa )%;&C2 a%ahve interference pattern follows the l)%)%%%)%%*%%)*+; )%22 aK ahinear interference theory )%%)%%%)%&)%*(2 a ahpredictions *%*%** ha'-P2 O.ahwith constructive and destructive interference;*%*) )%)%&)**% )&)%)%%%)%%2 Kah occur*%&)/2  ahring, respectively, for *)%!*%%)%(*2  ahV=<0 2 q ahnW* 2  ah*W(- 2  ahlW.- 2  ahsW-2  ah, and V=(2n%)*<0*) 2 ah-W2 ah1)/2*r*+(- 2 hah l.- 2 ahsW-2 ah, where ;)%&- 2 ahlW.- 2 ahsW- 2 ah Wha'-e2 .O<ahis the intrinsic horizontal wavelength downwind of a single  )%)* %)*%*)%;&*%%)*)*+;*;)**% ))%2 .- ahobstacle** %%% 2 .> ah W2 .U ahfor the *)%,2 .I ahgiven upstream profile))%))* %&@**%2 .?ah (- 2 .rahlW.- 2 ;ahsW-2 . ah=21.8 km) /**)*?ha'2 Onahand n>2 is an integer (Fig. 2). The positive and negative interferences are barely evident in the wave amplitu%)*)/* %))&)%.)*4)%** )%%)*)%)%)%)%%%)%% %%*%%(%**%)))%;%)%&?+)2 ahdes *& ha'-y2 OIahand horizontal velocity perturbations in the lee of the first obstacle (A %)*)*%*)%)%*%(*%)*%*) ))%%%*)%!** %%%;- 2  ah1W-2  ah and U%)*<-2  ah1 -A2  $ahover the valley), but leave a clear *)%)%)&&(*)%%)%%%%%ha'-2 MOahimprint ?*)82 M[ahin the lee of the second obsta))%%&*)& %%***** % 2 M2ahcW%2 MWahle%2 Mah, where ;)%%2 Mahthe )%2 M ahwave r;%)&2 M ahamplitude&?+)*%G2 M4 (ah and horizontal velocity perturbation (A%)*)*%*)%)%*&(*%)*%*);- 2 Znah2W- 2 Zah W-2 Mahand &)*ha'- 2 OahUW<- 2 ah2W- 2 ah)W 2 ah W!2 ahare%% 2 Iah enhanced or r!%))%)%%*!*!e2 G<aheduced relative to the value in the lee of a single mountain%*)%%)!%%)%!*!)%!)%)%!)!)%!%%!*!%! ))%"?+*)%)2 a ah (A!;- 2  ahsW- 2  ah)W2  ah.  2 @ahThe non 4)%!)** 2 `ah-W2 { ahdimensional *?&) +)%ha'M2 O,ahwavelengths cluster around integer values. T;&)%&))* %) %%**)**%)%)&)& 4R2 K/ahhe flow structure over the downstream obstacle )%+; )%)%*)%)%*+;) %&@** %%%2 ahis 2  ah different*%&)2 6ah under))*% 2 ah Wha'J2 lO*ahconstructive and destructive interference %*) )%)%2&)*2*% )%)&2)%%&)%%22 l! ah(not shown))*2 )+;*2 l ah. Wh2O)2 l aher%2 l aheas under %% 2*)*%222 lZ ahconstructive interference %*) )%)&2)%%&)%%22 l ahthe flow )%3+;ha'2 O ahresembles % &?*%!!2 ahthat)&"2 :ah over a single !*)%!%!!*)&"2 Aahridger*)%}2 Lah, for destructive interference the potential temperature and horizontal wind!*!*% )%)%!)%%%)%%!)%!**%*%"&?*%%)%!%)*")*%*)%";)* 2 ah Wha'-2 ,Oahspeedr *%%*2 , ah are r+%%+A2 ,$ahsymmetric around the downstream peak )@@&%+%*))++)%+*+:* %&?+*%%)2 ,S ah. 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