Journ,al of th,e lv[eteoroLogzcaL
Societg af Japan. \o1. 7,1. No. 1. pp. 147 153, 1996

NOTES AND CORRESPONDENCE

Supplement to
ttNumerical Simulation of Lee-Wave Events over the Pyreneest'

By Takehiko Satornurar
Nleteorol,ogico,l,Reseo,rchInstit.ute1-1. Naqamine,Tsu.ku,ba.
Ibo,raki,?0,5.
,lapan(Man,uscriptrer:ezr:ed
21 ,luly 1995,in re.rised.form1 Decent,lte.r
1995)

Abstractl'he vt'r'tical flux of horizontal momentum bv three-dimensional rnountain r.vavesis examined using alinear theolr'. .\nalr-tic solutions fbr trvo cliffert:nt wirrd <lirections ar{r olrtairxrl using nurnerical Fourit:r
analvsis. Although the mountain exanrincd in this rrote is a fairlv tq-o-rlirnensiorralone. ther resrrltsshos'that. at tlrc tropopause lcvc1.thc rrromentum flux greatlv decreasesto Il4-Il:lof its near-srrrfacevtrluceven o\'('r thc <t:rrtct arca of the rnourrtain u'herr the mean wind crossesthe morrntain at '15 degrees. It it
also suggcstcd thart the decrease of thc rrrorrcnturn flux observerl in IOP-ll of PYREX is ciruscclbv boththe firrite le.ngthof the PyreneesmoLlntainsairclthe slant attac:kanqle of the rnean r.incl.

1. Introduction
Irr this uote. the vertical flux ofhorizontal rnornentunr lrv linear gravit)' waves ovur three-dirnensional
nlourrtainsivill be calculatedto clarifv the effectsof
the n'inrl rlircction and of the finite length of thc
on tlie \.ertical profile of the flux.
rrrorrrrtains
In a prt'r'iorls paper (Satomura and Bongcirult,
199,1:lrcrcaftcrSB9.1).the rnountain \ravesobscn.cd
in tllc trvo lee-s'aveeventsduring the PYREX prograrri (Bor.rgearrlt
cf al.. 1990. 1993) u.eresinurlated
non-h)'drostaticnrodcl. Therbv a trvo-dirrrcrrsiorralmodel useclin SB9-1simulated rvell thc rvarclelgth.
the amplitricle arrd thc position of ther observecl
lee u'aves. and tllev inclicated that the lee u'ar-ts
observedin the thirrl intensive observatiorrpcriorl(IOP-3) lvere excite(l lrv non-linear u''ave-waveiuteractions of long ilrollntailr \\:aves.In spite of thcstagrccrncnts,thc vertical clistribution of tlic nromcntum flux b-v tire simulatr:rl rrountain riaves did notagree with the observation: tlre sirrmlatedfhx u'asalmost constant through tlie troposphere. u'hereas
the obserrvationshow'eclrapirl dccrcaseof lhe absoiute r.alues.
linder the stationarv and trvo-dirnensiorralconciitiorrs. the r.ertical flux of irorizorrtal rnomenlull
1

Present affiliation: Dept. of Geophys.. Kr''oto Univ. Kit :rshir:rkau'a C)iu'akecho. Sakvo-ku. K1-oto 606-0 1. .lapan
l ocietv of JapanC ) 1 9 9 6 . N l e t e o r o J o g i c aS

bv srnall arnplitude gravitv waves rrnrst bc constarrt
iri inr.iscitl fluicis u,ithout a critical lervel (Eliassern
anrl Palrn. 1960). Since the Lrng anrl verticalh'proptrg:rt,ingmountairi \\,ave(callecla principal \\.avein S139,1)
cliclnot producestrong trrrbulenccirr IOP-ll<luring the PYR.EX bv the simulation (neither bv olr
scrvation).the aboveresultsof SB9.1u-ereconsist,enlu'ith that non-accelerationtireorem. On the othertirnerciependentmodel
hancl, tlie three-clirnensionalof Rearr (1992) and Borrgeauk e,t o,l. (i993) simrrlated tire momentlulr flrrx u'c11. Consiclcring abovc
in SB94 that a reasongirresults. it rvassrrggeste.d
ing rise to the clisagreenentabor-rtthc rrrorncrrtturi
flrix is the tlrree-rliniensionalitvof the enr.ironmeutzrl
fielc1srrchas thc rrrountainshape and the directiorrof tlrt: rnearr."r,indin tlie real atmosphere.
\lorrntain wavesover three-dinrensionalellipticalnrorrnta,inshave been exarnincd bv scveral stuclics.
Illunien and Dietze (1981. 19E2)solved mount:rin$.ave equations over multiple elliptical mountains
in a shear flou, and calcr-rlatedthe momentum flux.
Smith (1989) showccla cliagrarnof critical rnountairr
height for stagnation as a function of liorizontal aspect ratio using the linear theorv. His cliagrarnindicates that mountains of high aspect ratio (> 5) act
as a two-dimensional u'a11.Using a hvdrostatic numerical rnodel. Stein (1992) examinednon-lineareffects on critical mountain hcight for stagrration andconfirmed the diagram of Smith (1989).

148

Journal of the Nleteorological Society of Japan\bl. 7-1.No. 1

1990.10.15062

200 -200

Fig. 1. Hodographof the horizontalwind at
0600UTC 15 October 1990at Zarag;ozo.
Spairr.The angle0" indicatesthat thewind crossr:sthe Pyreneesmountain atright anglesfrom Spain to Franceside.
Numbers labelled on the dotted lirre
sho'w.theheightin km.

According to the diagram of Smith (1989). the
Py'reneesare fairly two-dimensional mourrtains because the P1'reneeshaye high aspect ratio of about
5 or more: the scale of the Pyrenees is about
350 kmx70 km at 1000 m height contour (seeFig.
1b in SB94 for exarnple). This expectation is contradictory to the suggestionin SB94. Abovc studies,
however. examined mountain rn'avesunder thc condition that the mean u.ind approachesthe rnountainat right angles. which rarely occurs in the real atmosphere.
In IOP-3, indeed, the air flowed over the Pyrenees at other than right angles in the most partof the troposphere. as shown in Fig. 1. This rn'ind
hodograph at an upstream-side obserr,ationstation.
Zaragoza. Spain. reveals the fact that the air approached the P1'reneesat an angle of - -30 degrees(nearI1'south-westerly)except at 4-5.5 km height.
In particular. the attack angle was nearlv -45 degreesat 2 km heiglrt u'here the peaks ofthe Pvrenees
mountains stick out. Though the mountain in'avesin
a slant mean u.ind will have intermediate characteristics between the casesu'here the wind crossesthe
mountain at the attack angle of 0o and of tg0',
quantitative discussionsof such waves have not yet
been made.
Therefore, in this note. attention n'ill be given
to casesrn'herethe uniform mean wind approaches
the mountain aslant. Also. a cause of the observed
decreaseof the momentum flux will be siver
2. Basic equations
Consider the stead,v flou'' of a verticall). halfunbounded, stratified Boussinesq fluid. over smallamplitude topography z : h(r,9). The perturbations to the verticallv uniforni background u,'ind,pressure and potential temperature fields are de-Fig. 2. Bird's-e1'evierv of thc bottom topographf q'ith a - b : 20 and I : 100.

scribed bv the linearized equations:, , 0 u ' -I , 0 u ' ' r ' n t 7 9 9 0 r ' : U .
i^
^
o.r
ou
o.r
or'
,,0,'' ,,0t"

(21)

(

(

I

^

o:r

/t t\

:U.

fr(/n6;^oaou, , 0 u ' ' , , 0 u ' ' - r ' , , ( 7 g g0;r ' : g 0'
t ^
^
^

(2.3)

. . 0 0 '- - - 0 e ',( 24 )

0u' - r 0u' - t / .

( 25 )

t

o.r

uI

{ ^

I ^

Ax

L/A

0u'
;--r

^

or

ua

oz

lu'

tTtJtt, t l o , ,: U
Qz

^

uz

u,'here(U. I,/.0)and (.rL',u' . tl') are the constantmean
wind velocitl-and the perturbation v"'indvelocity inthe (r. y. z) direction, respectively:cp is the specific heat for constant pressure; d' is the perturbatiori potential temperature: @00and dOsf dz are
the reference potentiai temperature and the background vertical potential temperature glraclient.respectivelr':tr' : (plpoiRlc:, - II is the petrt'rbatio'
Exner furrction. in rvhich p is the pressure.ps6 is thc
referencepressure, fi is the dry gas constant and^ dn/.
r'rOno- :. -1g l l u'

'

0u\

l.

)on)

Introducing scale parameters of the velocity' anrl
the time as L'6 and 1/N. respectively. variables arenorr-dimensionalizedas:(r,y.z): (t.y.z)/;I,\ t - . V .r r .' , ' '. u J) : ( { . Y . , . i ' . i r ' ) U s .
where
rnYr z- _

9 dO
orrE'

is the square of the buoyancy frequenc.vancl'| " -_ 'unis the Scorer parameter of the uniform atrnosphere.
T h e e q u a t i o n s( 2 . 1 ) - ( 2 . 5 ) c a n b e r e d u c e d t o asingle non-dimensionalizedequation for the verticalvelocity:

149

T. Satomura

February 1996

(a)

(b)

.a'1
'Y*_

='-l I\l'-.*g J-o-llAX =

4.438-92

3.8gE-92

Fig. 3. Perturbation vertical velocity d'at 2 :0.4 x 2n for (a) a - 0, and (b) a - rl4. Solid andldashed lines indicate positive and negative values. respectivell'. The contour interval is (a) 20 ^ 10a.
The v"'hite arrows indicate mean u'ind directions. The mountain is located at theand (b) 30 x 10
center of the domain( k L r+ M ) 2 w " . - ( x 2+ 1 2 ){ ( k L I + l v ) 2- 1} , : 0 ,( 2 . 6 )ancl lve
rvhere the subscript z indicates .z-derivati.".eassuniethe perturbations to be representedas a double Fourier integral
r i ' ( r ,y , . ) :|JTo examine the effectsof thc three-dimensionalityof a nearly tn'o-dimensional mountain range, a" t a b l e - s l r a p , ' dr"r o u n t a i n

"i
l)

-

1+\rlo)'(/ ! J - :' )\ ) f. / u-./\l
I I r t n , ' (] '!lr+tnnn( )t,/' r+'" ( k , 1 . z l e i ( k ' + t i ld k d l .
|
J

-

a

With 1v*2taken as constant and the linearized lower
boundary conditionu ( k . 1 . 0 ): i ( A U + / Y ) h ( k , l ) ,(27)

the solution to (2.6) is ivritten in physical spaceas:Li(r.y.z):
rf+'xo k) r t t .
i ( k u + t v ) h l k . l ) e i " r ze i ( k x + L d
| |

(2.8)

rvhere,-2

tk2+l2)\I-IALI-lvl2l( . k t '+ I V ) 2,?e\

For rn2 < 0 the positive irnaginary root of (2.9)
must be chosento eliminate the non-physical gror,vthof the disturbance aniplitude with height, and form2 > 0 the sign of m must be chosen to be thesame as the sign of (kU * lV) in order to satisfl.
the radiation condition aloft (see Smith. 1980, for
lr:0)

1

-

|

\

b

/)

l.

\

h

/)

.

(2.10)
is considered. rn'herether non-dimensional parameters are o, : b : 20 arrd d : 100. Tliese pararneters give a long sierrcicrnrountairi range as showrt
in Fig. 2. Applf ing the Scoreparameterobservedirr
the rniddle troposphere during the IOP-3 of PYRtrX(-0.6 kni-1r SB94), this mountain has half widthsof 30 km in the crossing-rnountain direction and
150 krn in the along-mortntain clirection. These va1ues are equivalent to the Pvrenees mountains. and
it is expected to behar.elike a two-dimensional wall
becauseof its high aspect ratio.
\\rlien the wind crossesthe mountain at right angles. mountain u'avesgenerated by the above tableshaped mountain show the best two-dimensionaliti,'.
However, it rarelv occurs in the real atmosphere.
During the IOP-3 of PYREX also. the wind approached the Pyrenees mountains aslant as shownin Fig. 1. Therefore we will examine two cases of150

Journtrl of the N'leteorologicalSocietl' of Japan

.8

I

H

N

x

.6

I

]J

-d

o

//

)t
6

xl
.6

-.5

-.4

-.3

-.2

-.1

0

;-x

uw
Fig. 4. \rertical profile of the norr-clirnensionai vertical nrornenturrrflux ,\1-1 at
y - 0 ( l i n en u r n b enr : 0 ) . 2 0 ( n : i ) .
4 0 ( n : 2 ) . 6 0 ( n - 3 ) , 8 0( n : 4 ) a n d
100 (n - 5) for o : 0. The ordinateis
the non-dirncnsionalheight divided by'2r.

attack anglesof the u,ind: o : tan t (VlU):
0 arrdii 14.
In the fbllowing secrtions,anall'tical solutions are
obtained using a fast Fourier transforrl (FFT) algorithrn in the horizontal direrctiorrs at spercified
heiglrts. Tiris implies the liorizontally perioclic nature in a square clomain. The sides of the clornainintiris studv are 1024 x 102-1in the rion-dirncnsional
rurits clivicledbv 256 grids for the discretc FFT.
The effects of c1'clic sidc boundarv conditrons on
the nronienturn flux are rregligible bccause the solutions do not change practicallr. irr a rvide dornainof 2048 x 20-18sides.
3. Results
As shou,'nin Fig. 3a, the gravitv r'".avesover thetable-shaped mountain for a - 0 are not rnuch
cliffererit frorrr those over a circular mountairr (er.
Srnith. 1980). On the other hancl, r'herr n : Tl4(Fig. 3b). u.e notice \\aves are geni:rated over two
distinctir-e areas: thc iower-ieft corner of the rnouritain lr'hich is the leading up-slope lbr this u.inri direction. and the uppcr-right corlrer wliicli is the firral
dorvn-slope. The wave-train is less reurarkable thanthat for ci : 0.
!'ertical profiles of the non-diniensional rrrornenturn flux.

.t/r(y. 2) : i1n,:

rLxl2

l
J -r*

irEdi.

1z(3 1)\ c - ' 17, 1 . N o . 1

rvhcre -L1 is thc dorriain lerrgth irr tlxr .r:-tlilt'ttion.
ale calculated at y - 0 (at the center of the rlorurt a i n ) , 2 0 . 4 0 , 6 0 , 8 0 a r r d 1 0 0 ( a t t h e s h o u k l c ro f t i r i rrnountain) for cr : 0 arr: shon'n in Fig. .1. Sirrce
the non-dirnensiorral liciglrt 1.0 x 2r crirresporrcls-10 kni appll-ing ls -0.6 krn-1 obserl'edin IOP-3
of PYR.EX (5894), Fig. -1arid tbllowirrgcorresponding figures shorn'veltical variation of .1-f; tirrcngh
thc troposphere. It is sho"vn in Fig. -1that l1-y tlecreasesonly' 5 -20 '/c near the celrter ol lire nrountain (lirie rrurriberrr- 0 - 2). Sirrcc tirc' rrrornenturn flux of steaclr'-two-dirnensional liuear rrrouritairr
u'avesmust be constant in the r.erticaldirec'tiorr.thissnrilll change of l1-1 irrclicatesthat our ttrblc rrroun
rrear
tain has nearly tu'o-dimensional characterristic's
the center if n : 0. In the ofl'-cerrterarea. lrt.rut:r,er'.
,11,adecreases
to about .10% (rr:3) evenin tirrs ctise.
Tlre morrreriturrr flux ,/ifu urhen o : tr 14 is slior'vn
in Fig. 5. In this case.r'f/-aat the top of the dorrrain
is as srnall as 1/rl - f 13 of its l'alue al the iou'est
altitude rrearthe center of the rnountairr(rr- 0. +1
and t2) rvherethe \vavesare rreariv two-clinrelsir.rrral
if n - 0. l-urther. l/-a is not syrnrnetric u'ith u - 0.
Leaving torvard the upstrearndirection (l.e. ,! < {))fiorn the center ol the rnountain. the alrsolute r-:rlr.re(Fig. 5a). In contriLst.it
il1.rI ternclsto clecre:rser
tends to increaseon the other side of the rnorurtairr(i > 0) as shor'vnin Fig. irb arrd to be nearl1' r't.rrstarrt in the vertical. Thc irrcreascof ,\1;r tori'arrl
thc dou-nstrcarn directiorr is rnost 1ikel1,causerl llgravity \vavesgencrateclupstrc.arnside of thc rrrorrutain surnmeclup in tlie clorvnstrearnarea.
4. Discussion
Tl'noI'eaturesare obvious frorri the results. Irirstlr'.
the miignitr.rdeof thc urorrrcritlrilrflux at tlie top oftlre dcrniainis about I14 - 113 of the iiear surfac'evalLreover the center of the table-shapecinlountfuri
u''lrerra :7i14. This clecreaseof the flrrx is rrearlrtire sarne as that over a circular bell-shripetl rnountain. Onc nrar'.holvcver.notice that the irrtcglatiorr
l i n e i n , l f u ( 1 . e . . : r - a x i si)s r r o t p a r a l l e lt o t h e r v i r x l
direction ancl suppose ttiis r'",ouldcairse the rerhrction of ,\fu. To clarit.r'tiris lroint. tlrt' h,rrizontal
distributiorr of uu' at .i : 0.8 r 2;i is shorvrrirr Fig.
6. Bccausethe monienturl flux -11.yis tire line integration ol 'ldr. it rvil1rernain large at this height er.err
if the integration line is not parallel to the u'ind dir e c t i o nv n h e na - 0 ( F i g . 6 a ) . W h e n n : ; r / 1(Fig.
6b). horvever.the flrrx will be srnall near the cenierof the rnountain cveu if integration line is parallel tothe ',vind directiorr. Indered,the results of the line integration reveal that the rnonrerrturnflux dtrr:reases
at thc top of the domain to about 80 % of the nearsurfacelalue ifn : 0 arrd the integration line crosses
tlre rvind at an angle of r 14. but the urornenturn flux
decreasesto about 30 % oI tlie near surface value if
a : r14 and the integration linc is parallel to theFebnrarl' 1996

T. Satonrura

(a)

151

(b)

P

N

X

N

.6

v . 6

JJ

-

.lJ

u, .4
.-1
q)

ur .4
.rl

-

o

o
.8

-.5

-.4

0
.7

---F

.)

-.4

-.3

-.2

uw

uw

'2), iiO (n :Fig. 5. (a) Sanre as Fig. ,1. except at y : 0 ( n - 0 ) , - 2 0 ( n : - 1 ) . - 4 0 ( n :
-4)
-100
-5)
:
:
:
(rz
(n
an<l
for trrl4. (b) Sanre as Fig. 4, except fbr rr r l 4 .

(a)

UW

3).

80

(b)

&
Fig. 6. Sarneas Fig. 3, exceptfor the correlationt i i a t 2 : 0 . 8 x 2 r . T h e c o n t o u ri n t e r v a il s 5 x 1 0 - { .

u'ind (rrot shorn'n).Therefbre. the non-parallel integration line of r'f/; is rrot the cause of the redr-rctionof ,411l. but the shift of the maximum of -0ur tothe dor.nstrearn direction along the rnourrtairicauses
this reduction.
One may further notice that the width of themountain iri the cross-strearndircctiorr decreasesto
sin a and may supposethat this causesthe reductionof Iily when n : Tl4. This possibilitv is also de-nied bv calculations with a defbrnied rnountain (theextent is shortenedby'sin(r/4) aud the half-n'idth iselongatedbr' 1/ sin(r/4)) and n - 0, becausellf.ai
at the top of the dornain differs onh' a small arnount
O5 %) from that at the bottom (rrot sliown).
Therefbre. '"vecorrcluclethat both the finite lengthof the mouritairr arid the slarrt attack angle of thenrean lr,ind (. I 0) rnainlv cause of the decreaseof the niomenturn flux. On the other hand. the di-t52

Journal of the \'Ieteorological Societv of Japan

rection of the line of flux integration plavs a minor
role on it. Consequently, these results suggest theobserveclreduction of the momentum flux (to aboutll3 of that irr lower altitude) in IOP-3 (Bougeault
et al.. 1993: SB94) was mainly caused by both thefinitc lengtli of the Pyreneesmountain and the slant
attack angle of the mean u,'ind.
The second featurc is that even for tlie gravitv
waves generated by fairlv two-dirnensional rriountains as the Pyrenees one should not use the lirie
averaged flux .4ft as an indicator of the depositionof wave momentum. The important quantitl,' for the
momenturn budget is. as i,r,eilknown. the momentum
flux averaged in a liorizontal area.

=
xrs

f l"^aoar.

(r1)

where S : Lx x try is a horizontal plane of thedonrain and Iy is the <iornain length in the gdirection. Not the line-averaged momentum flux
,lfg bLlt this area-averagedflux ,tfs is constant for
stationarv three-diniensional rnountain wa."'esin aunilbrm flou'2 becausethe non-accelerationtheorem(Eliassen and Palm. 1960) holds for ,41s. Therefore.
the vertical variations of the line-a.,'eragecl
momentum flux ,l,fs shown in Fig. 5 do not indicate theoverall liorizontal rnornenturn change.
This point is evident rvhen tlic mountain shapeis highly three-dinensional (circular bell-shaped
rnountain, for exarriple). Nevertheless, it is notervorth-v-that ,[fr greatlv decreasesover tire fairly
two-dirnensional rnountains of large horizontal aspect ratio (> 5) if the wind doesnot crossthe rnountain at the right angles. This happens frequentlv irr
the real atrnosphere. In field experiments, the momenturn flux in the uppcr air is observedb-v-aircraft
which is a fundarnentalll' line-averagedrnomentunr
flux. Therefore. one must pay great attention tousing the observed rnolnentum flux profile to estimating the wliole effects of the mountainous areaon the atmospliere.
5. Conclusions
The vertical flux of horizontal momentum bygravity waves over a table-shaped mountailt wascomputed using linear equations. The rnairr result
is that the momentum flux at the top of the domain (-lg km height) is about Il4 - 1/3 of therrear surface value when the wind crossesthe rnountain with the angle of rl4. The flux also shows thestrong asynimetry respect to the center of the mountain in this case. Further. it was suggested that\\'e also confirmed that ,,\,19is constant in our calculatron.

Vol. 74. No. 1

the decrease of momentum flux observed in IOP-3 of
PYRtrX was rnairrly.caused lty both the finite lengthof the Pyrenees mountains and the 5161f 6f 1661'r n oloof the mean wind.
Acknowledgrnents
The author greatly acknon'ledges helpful cliscussions u,.itir the rrreurber of App1. Nleteclr. Res. Dep.
of \IRI, and also useful cornrrrents of anonvnrolts revieu,ers.
The PYREX experinrcrrt is a cooperatir-e prograrn
supported bir \l6t6o-Frarrcc. the Spanish lrrstitutefor I'Ieteorologi,, CNRS/INSU. CNtrS. trDF. R6gion
Nlidi-Pyr6n6es and DLR.
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orographique sous-maille dans les moddle de circulation g6n6ra1ed 1'aiciede PtrRIDOT 10 km, Note deTrauazlle de I'ENLI. ENNI, CNRNI, 153 pp.
Blumen. W. and S.C. Dietze, 19811An analysisof threedimensional rnountain waves in a stratified shear
flo'n''../. Atmos.,9ci., 38, 1949 1963.
Blumen, \\r. and S.C. Dietze. 1982;An analysisof threedinensional rnountain waves in a stratified shear
flor': Part II, ,1.Atmos.Scz., 39. 2712 2720
Bougeault. P., A. Jansa. B. Berrech, B. Carrisirno.
J. Pelon and E. Richard. 1990; Nlomentum budget
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11, 395 ,118.
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22, I 23.
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Soc.Japan.72.773 195.
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flow past an isolated mountain. Tellus. 32.348 364.
Sniith. R.B., 1989; \Iountain-induced stagnation points
in hvdrostatic flow, Tellus. 41A.270 274.
Stein. J., 1992; Contribution ) 1'6trrdedcs rdgimes h1drostatiques d'6coulements orographiques. Ph. D.
Thesis. Univ. Paul Sabatier. Toulouse. France.

Febmarv 1996

153

T. Satomura

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