April 1994

T. Satomura and P. Boueeault

Numerical

Simulation

172

of Lee Wave Events over the Pvrenees

By TakehikoSatomura

fuIeteorologicalResearch Inst'itute, 1-1, Nagamine, Tsukuba, Ibaraki 305, Japan

andPhilippe

Bougeault

M 6t6o-France,Centre N ational de Recherches
M 6.t6orolog,iques
f2, au. G. Conolis, 31057 Toulouse,France(Manuscript rece'iued
31 August 1993, in reuisedform 12 January 1991r)

Abstract
A two-dimensional, non hydrostatic, compressible model is used to simulate the airflow over the
Pyrenees in two lee wave events (IOP-3 and IOP-9) during the PYREX program. For each case, themean flow is initialized by a single upstream sounding. The bottom topography used in the model istaken from the reai mountain profiie afber smoothing out the wavelength shorter than 10 km.
In all cases, the model simuiates well several characteristics of the lee waVes: the wavelength,
amplitude, and position. This is a strong indication that the lee waves observed in these events were
excited by non linear wave-wave interactions of long mountain waves, because the topographic forcing
in the model has negligible spectral amplitude at the wavelength of the lee waves (- 10 km) . Sensitivity
experiments show that the phases of the lee waves are very sensitive to a small change of the mean
wind and that their amplitudes are increased by the addition of smaller-scale topography.
In both cases,the simulated downward momentum fluxes agree well with the observed fluxes around
4km height: -15 x 104N/m in IOP-3 and f 10 x 10aN/m in the third day of IOP-9. The simulated
fluxes are, however, almost constant through the troposphere. while the observations show rapid
decreaseof absolute values above 4 km height. This overestimation of the simulated momentum fluxes
in the upper half of the troposphere is caused by the overestimation of the amplitude of long mountainwaves at these heights; these long mountain waves control the vertical profile of the momentum flux
there. It is suggested that the time evoiution of the mean wind and the lateral momentum flux
divergence found in the reai atmosphere give rise to this overestimation.

1. Introduction
Gravity waves induced by mountains have been
attracting many researcherssince they induce many
severe and non-severe phenomena in the atmosphere: Foehn or downsiope windstorms on theearth's surface, hanging ciouds, roll clouds or clearair turbuience around the mountain height level, etc.
Deceleration of the mean wind is also one of theimportant effects of orographicaily induced gravity
waves becausethe change of the mean wind results
in an alteration of the mean (2.e., large-scale) pressure distribution. It generally occurs at higher levels
where the gravity waves dissipate for reasons suchas critical levei or overturning, and transfer their
momentum to the mean wind.
In recent years. global circulation models havebeen found to over-estimate westerlv wind in the@1994, Meteorological Society of Japan

northern hemisphere (seePalmer et a1.,1986,for exampie). Many researchershave tried to reduce thissystematic westerly bias. Two methods to reduce it
are now commonly used: envelope mountain andgravity-wave-drag parameterization. The methodof envelope mountain augments mountain heights
by considering blocking effects of subgrid scale valleys (e.9., Wallace et al., 1983). On the other hand,
the gravity-wave-drag parameterization is intended
to represent the effects of subgrid scale mountainwaves. Thanks to these parameterizations. the westeriy bias in general circulation models was reduced
mainly in the stratosphere (Palmer el al.. 1986: IVIcFarlane, 1987). So far, most of the gravity-wavedrag parameterization schemeswere based on thehydrostatic two-dimensional mountain wave theory.
In the troposphere, another type of mountainwave is commonJ.y observed: non-hydrostatic gravity waves trapped in iower atmosphere called 1ee174

Journal of the Meteorological Society of Japanwaves. Ii has been explored since the ver1, beginning of mountain wave research (Scorer, 1949, forexample) and continue to be an interesting subject
for theoretical, observational and numerical studies(e.9., Vergeiner. 1971; Smith, 1979; Sharman and
Wurtele. 1983). Very recently, Iee waves have drawn
the new attention from the view point of the atmospheric momentum budget (Iwasaki et al.. 1989:
Durran, 1991; Kim and Arakawa. 1991). Iwasaki
et al. (1,989)introduced a new type of gravity-wavedrag parameterization to improve the troposphericwesterly bias by including the effects of trapped
gravity waves. Using simple two-dimensional numerical models, Kim and Arakawa (1991) and Durran (1991) showedthe importance ofnonhydrostatic
trapped gravity waves on the momentum budget
in the troposphere. Because lee waves are trappedwaves, thev are able to keep significant amplitudes
even far downstream of the mountain. For the same
reason) however, they seem to carry little momentum upward be;rond the duct region. Therefore, thedetailed processeswhereby trapped gravity waves
affect the mean flow are not yet clear, and studies
looking into the substance of the momentum transport in the troposphere are still required.
Nleanwhile, efforts to understand the mountaineffects by observational study have also been made.
The 1970 Colorado Lee Wave Program targeted the
Rocky Mountains (e.g., Lllly and Kennedy, 1973),
the ALPEX program spread observational nets
around and overthe Alps (e.g.,Kuettner, 1986),and
the PYREX program measured the dynamical influence of the Pyrenees mountains (Bougeault el ol.,
1990. 1993). But, in general, observational data aregathered only at a single point or on a few aircraft
tracks, and it is difficult to reconstruct the two- or
three-dimensional structure of the mountain waves
in detail by the observations alone.
On the other hand, numerical modeling offers awealth of consistent information. Therefore, numerical studies of mountain waves initialized by the observed data are abie to give supplementary information about these phenomena. This has been attempted in severalstudies since the pioneering workof Klemp and Lilly (1978) in the case of mountainwaves over the eastern slope of the Colorado Rockies. Recently, Hoinka and Clark (1991) and Clark
and Nliller (1991) used a three-dimensional nested
nonhydrostatic modei in the case of ALPEX. They
pointed out a strong horizontal variabiiity of boththe momentum flux and the pressure drag. While
the momentum flux in the model was much iarger
than the observed one (Hoinka and Clark, 1991),
Clark and Miller (199i) concluded that hydrostatic
simulations are suficient to describe the basic dynamics in their ALPEX case, except for strong dependence of the drag and momentum fluxes on resolution. The lee waves which the newest gravity-

Vo1.72, No. 2

wave-drag parameterization uses as physicai substance were, however, not studied in their papers.
In the case of Engiand, Shutts (1992) and Shutts
and Broad (1993) studied lee wavesby using a threedimensional nonhydrostatic model. The comparisonof their simulated waves .'vith observations reveaieda high level of agreement, especially in the case ofnorthern trngland (Shutts and Broad, 1993). However, they could not compare the simulated momentum flux to the one calculated from observation. because the momentum flux based on the observation
was so sensitive to the averaging length that they
could not obtain reliable values. In the case of adownsiope windstorm in Western Australia, Block1ey and Lyons (1994) simulated lee waves over the
Darling Scarp by using a two-dimensional nonhydrostatic model. Whiie they studied nonhydrostatic
effectson the windstorm and effectsof the mountainshape on pressuredrag, they did not compared theirresults with observation in detail.
In the present study, we perform numericalsimuiations with a two-dimensional nonhydrostatic
model to get two-dimensional information for two
lee wave events observed in the PYREX program(Bougeault et al., 1990, 1993) and we compare
simulated results with observational data in detail.
While only one lee wave event over the Pyrenees
was observed during the ALPEX program (Hoinka,1984), the PYREX program captured several such
events (Bougeault et al., 1993) in a dense observationai network using various instruments. This
program, accordingly, offers a good database for researchersinvestigating (nonhydrostatic) lee wave effects on the atmosphere.
2. Model Description
2. 1 Gouetn'ing equations
The numerical model used in this studv is thatdevelopedby Satomura (1989) and Sasaki and Satomura (1991). The momentum, thermodynamic andcontinuity equations are written in Cartesian coordinates as:

0u, 0u,0^,0r - .
"tozg+ cp\u - a)
E
a,
Ari
A

- ? y - ! { o _ ( % * q e ) } 'e r )
3 0r;

ae
&

' "r

0r ,
-

-

At

where

^ r

'

E*, \"^

Ett ) I'\0r;

a @ - 0 )- a ( K * a 0 \\ri

6r_r\ os 0r1)(, ,)\'O(il - r) , Ra(II - r) 0u1
n

0*j

cu

)tj

Ra(I *r\0 ( X^ ad\ ,^ ",c,(9'a)O x . 1\ o s O r i /

-

|

^

I r \ - ' u lApril 1994

T. Satomura and P. Boueeault

rr+tr:('rna/cr
\Po,/

and
dII

s

az

c-ct

In the above, ui (i : 1. 2. 3) are the tlneed i m e n s i o n aw
l i n d v e c t o r su . u . u . r e s p e c t i v e i y , 9i s
the potential temperature deviation from a reference
state O which is a function of z only, E is the subgrid scale turbulent kinetic energy! K^ is the eddy
mixing coefficient for momentum and oe : llI.5,
p is the pressure)po : 105Pa. R6 is the gas constant for dry air, c, and cu are the specific heat ofdry air at constant pressureand at constant volume,
respectiveiy. Here the summation convention (e.g.,Aris, 1962) has been used with respect to the indicesi and j.
The eddy mixing coefficient is determined by theE-e ciosure model (Rodi, 1985) with a slight modification proposed by Detering and Etling (1985):( 5 ' = ) - p- c - €\ 2 4 ):+*O f * u" O,I+j
O:ri \oE
Ori/
0e
At

0e
-'0r.,

A /K^0e\^t2"'E0r, \ ou fu1 f^ € . + CrE(P + G)(t + Cri?r)( 25 )

where -E and € arg the subgrid scaleturbulent kinetic
energy and its dissipation ratio, respectively.

P : 2 K ^ S i t S ,-t
^

-tr
(

_

'

)uj

L/0u;

Eu'\

2 \Ez,

0r, )

t

'

t

(2 . 6 )it ai.
J(2.7)

l-t ^ K - I 0 ( O + 0 )
"t O
Ar,
(

- )\
'". "t" _

-

PiC

I o

G

t

(2 8)
l

o

t

tor u1' Lu2' ) u3''

( 2.s)

toru12+u22<u32,p2

K- : C,,€(2.10)

andoa :0.741, oe : I.3, Cp:0.0256.
Ct :7.14, Cz:1.9, C3: 0.8.

(2.11)In this study, a two-dimensional version of themodel is used: every derivative with respect to they-direction (i,.e.,r 2) is set to zero in Eqs. (2.t)-(2.6).
Whereas the model equations have been describedin Cartesian coordinates. the ecuations are solved intransformed coordinates to apply them to irregularterrain surface. The transformed coordinates are
non-orthogonal curvilinear coordinates generated by
the computer as a pre-processingof the model. The
generation method is an elliptic generation system
proposed by Shieh (1984), where coordinate lines
are orthogonal to the terrain surface. The maximum
deviation of the coordinate line from the verticai is,
however, only of the order of 10 m, because we usevery gently-sloping terrain in this paper (seebelow).
Wb use a discretization method centered both intime and in space. with a grid system where all
variables are located at the same points except the
Exner functions r and II , which are staggered in thevertical. A time-splitting method similar to Klemp
and Wilhelmson (1978) is used for computational efficiency. An implicit method is also used to gain further efficiency in vertical derivatives of terms reiatedto sound waves and to viscous terms. A fourth-order
artificial viscous term is added to Eqs. (2.1), (2.2),(2.4) and (2.5) to damp smali-scale computationai
modes.
Durran and Kiemp (1982) showedthat latent heatreleasestrongly modifies the structure of mountainwaves. Some stratiform clouds were observed on
the windward side of the Pyreneesmountains during
the observational periods we consider (Atti6 et al..
1991), but the effect of condensation does not seemto be important in these cases (Richard, personalcommunication). For simplicity, we do not consider
moisture effects at a1l in the present work.
2.2 Boundary condit'ions
At the lateral inflow boundary, the velocitl' normd to the boundary does not changein time (i.e.,
0ul0t:0).
An Orlanski (1976) type extrapoiation(\'{i11erand Thorpe, 1981) is used at outflow boundary for u. Other variables are specified to be thesame as the inner point both at inflow and outflow.
The upper boundary is a free-slip rigid wal1 withno heat flu-x. The lower boundary is rigid rough terrain with a roughness length zs -- 0.5 m. The surface temperature is fixed at its initial value, which isthe same as the initial air temperature at the sameheight. The lowest layer is assumedto be a constantflux layer and turbulent variables in the lowest layer
are estimated by the method of Busch et al. (1976).
3. Descriptionof the experiments

3.1 Numerical aspects
The integration domain is divided into 325 grids
in the r-direction, with a centered area of constantI km resolution having 304 points. In order to keep
lateral boundaries sufficiently far from the regionof interest and also to save the computer resources,
grid intervals on both sides of the 1 km grid region
are stretched in the power of two. The outermost
grid interval becomes 128 km and this grid system

776

Vbl. 72, No. 2

Journal of the Meteorological Society of Japan

42. 0

4000

43

44.A ("N)

tr

3000
-.lJ

-q
Ol 2 0 0 0

-'-l

0)

r

1000

0
A<-

50

1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0( k m )--)Bfig. la. The profile of the model terrain. The dotted line with small rectangles is the topography observedby the airborne radar.

Frg. 1b. The topography around the Pyrenees mountains. The contour interval is 500 m, the line A-B
is the section used in our 2-D model. Characters T and Z indicate two upper-air sounding stations,
Toulouse and Zaragoza, respectively.

Aprii 1994

T. Satomura and P. Boueeault

177

(a)

(c)

, \ \ \ \ \ \ t , , ,sE t5./10/90 R

oHTU

t

UV K:3OP M E R( H P R I

/

l

l

l

"

l

t

t

t

t

l

"'

8L--sASE 15/10./90 R

oHTU

VENT BSOHPF

0HTU

VENT SOOHPF(d)
I t t r t l t tt t l I t t t

l

oHlU

t

"

"t "

(b)

BnSEl5l10/90 B

t

v

/

-

4

2

\

€ L _ B R s Et 5 l l o l 9 0

qR

Fig. 2. Analysis of (a) surface wind, (b) sea surface pressure, (c) wind on 850 hPa isobaric surface and (d)wind on 500 hPa surface at 0600 UTC f5 October 1990 over south Europe. Dotted lines are contoursof surface topography at 500 m intervais. The contour interval of sea surface pressure is 2 hPa. One
grid iength of wind arrow is 10m/sec in (a), and is 20m/sec in (c) and (d).

gives 815 km horizontal domain length in total. As
a result of this stretching, the total -ass of a[mospherein the model increasesand, therefore, deceleration of horizontal veiocity in time due to the surface
friction is reduced. In nested models, such an abruptchange of grid interval as described above produces
erroneous wave reflection (see for example, Phillips
and Shukla, 1973;Kurihara et a1.,1979). To examine the effects of grid stretching in our model, we
completed a simulation using a uniform grid inter-val (1km) model with the same domain length as
the stretched one. The comparison of the results inuniform grid modei (not shown) with those in thestretched modei reveals that these two models givealmost the same results untii 7 hours integration.
In the z-direction, the domain height is 22 km andis divided into 47 layers whose depths change withthe height: from 30m near the ground to 500m in
the free atmosphere. Above a height of 12 km, we
put a sponge layer to avoid contamination by the re-178

Journal of the Meteorological Society of Japan

flection of vertically-propagating wavesat the upperboundary. The short and long time steps in the time
splitting procedure are 2 s and 10 s, respectively.
The model uses a smoothed terrain shape shownin Fig. 1a. This shape was originally taken fromthe topography in the central transect (Bougeault
et al.. 1993) shown in Fig.lb at 10km resoiution
data. The terrain data were interpolated to a 1 km
mesh by cubic spline functions. and the Nleseta in
Spain was wiped out, as shown in Fig. la, to avoid
terrain height variation near the model boundary.
3.2 Synoptic situations and, 'in'it'ial cond'it'ions ofeach experiment
There were 10 Intensive Observation Periods(IOPs, in the foilowing) during the PYREX experiment. From these 10 IOPs, we chose two offour caseswhere significant lee waveswere observed:
IOP-3 and the third day of IOP-9. Lee waves werealso observed on the first and the second days of
IOP-9, but these cases are not suitable for our 2dimensional simulation because the synoptic conditions in these casesindicate poor 2-dimensionality.
In the following sub-sections,a brief description of
the synoptic situations of these days and the model
initial conditions are given (see Bougeault el al..
1993, for brief descriptions of all IOPs).
3.2.1 IOP-?
This IOP started at 1800 (all time in UTC) on
14 October 1990 and ended at 1200 on 15 October1990. As shown in Fig.2, the synoptic wind aroundthe Pyreneeswas from southwest through the whole
troposphere. In the layer below 850 hPa, however,the southwest wind was blocked and divided by the
Pyrenees. A cyclone rvas deveJ.opingover the eastern Atlantic Ocean. As a consequence)the verticai
wind shear over the Pyrenees strengthened during
the IOP-3 (not shown). This transient characteristic of the synoptic situation has been ignored in our
simulation.
The potential temperature and wind soundings
over Zaragozaat 0600 on 15 October 1990 are shownby soiid iines in Fig. 3a and 3b, respectively. The
temperature profi.ie shows a strong inversion below
900 m height, and a neariy neutral layer of 1200m
depth above the inversion. The wind profile shows
three strong shear layers (8000-9000m. 3700-4600
m and below 700 m) and two moderate shear layers(4600-8000m and 2000 3700m).
The initial profiIes used for experiment P3X1 (thereference experiment for the IOP-3) are shown bydashed lines in Fig.3a and 3b. The initial temperature profile foilows approximately the observation of Zaragoza below 10500m and is extrapolated
to the modei top by using information from theMadrid sounding (upstream) at the same observation time. The tropopause is put at 12 krn height
and the stratosphere is assumed to be isothermal.

Vol. 72, No. 2

The initiai wind profile is modified beiow 2000 m
and above8500m height as shown in Fig.3b. In theupper layer, the wind veiocity foilows airpiane observations executed on the morning of 15 October. We
have decided to slow down the wind velocity in thehigher layers because the soundings were contaminated b;' the mountain wave at these altitudes (balioons did not rise vertically upstream of the Pyrenees, but flew over the Pyrenees). In contrast, theaircraft observations provide averaged wind values
over 200 km length crossing the Pyrenees. They are
more suitable for the model initial profile than rawinsonde soundings in the upper troposphere in thiscase. In the lower layers, we took account of thestrong blocking effects by the mountain. which produce a weak wind region in the lower layers on the
upstream side, as shown in Fig. 2a and 2c. Only
three-dimensional models can manipulate this effectcorrectly.
The bottom topography in P3X1 is very smoothand the drag effect by the small-scale topography
is not fuily represented by only the surface roughnesszs:[.$m. Therefore,in the experiment P3X2,
the bottom topography is modified to study the effects of smaller-scaleterrain on mountain waves under the same atmospheric conditions as in P3X1.
The smaller-scale terrain is generated by a pseudorandom number generator in a computer Iibrary andthe maximum of the random number is scaledto one
quarter of the original mountain height. The random terrain is also restricted to have no ampiitude
at very short wavelengths (< 4km) becauseterrain
forcing of very short wavelengths excites computational modes in the model, and to have no amplitudein the region where the originai mountain is lower
than 300 m. The topograph;r created as describedabove is shown in Fig.4 and is used in P3X2. The
initial wind and temperature are the same as P3X1.
In the experiment P3X3, the initial wind is decreased by 1 m/sec against P3X1, while the initial
temperature profile and the bottom topography are
the same as P3X1. Because the upstream sounding used to make the initial profile representsonly
a very local condition, it is possible that the initial
profile is not suitable for simulating the observed
event. This experiment P3X3 is designed to study
the dependenceof the simulated resuits on a smallchange of the initial velocity profile.
3.2.2 The thi,rd,day of IOP-9
The anticyclone intensifi.ed over the eastern Atlantic during the night from 15 to 16 November.
This resulted in nearly meridional winds at all levels. As shown in Fig.5c and 5d, wind around the
Pyrenees at 0600 on 16 November was northerly at
850 hPa and even a weak easterly component waspresent at 500 hPa. At the surface, strong blocking
was noticed, as shown in Fig.5a and 5b. This syn-

Aprii 1994

T. Satomura and P. Boueeault

179

(u)
9 q / ' t a / ) az 6 a ainJl

ZARAG.rSA

aaaa
a

r
2aao
r
I
I

,
[,

1g a a Q

,F
e

(

8A@@

'o)
T
*

I

,/
6aaz

4AAA

2040

(b)
9A/iA/'a46,40(m)
1a O @ On

ZARAGOSA

12A@A

1A A A O

(
-

SAAO

(

.;

<,

T urw
/

{"
i".

t @

-5

tZ

l5

2A

5 0 { m , /s e c )uFig. 3. Upper-air sounding over Zaragoza at 0600 UTC 15 October 1990 (solid line) and initial profile ofthe model (dashed line) for P3X1. (a) Potential temperature d, (b) wind component parallel to themodel plane (crossing the mountain). Wind velocity is positive when it blows from the south.

180

Journal of the lvleteorological Society of Japan

4 2. 4

4000

c

I+J

. U

Vbl. 72, No. 2

44.0 ('N)

3000

lJ

( :
O)
.-{ 2 0 0 0
C)
;E

1000

0
50

i - 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0( k m )XFis. -1. The nrofile of the model terrain for P3X2. The dotted line is the topography used in P3X1.

optic pattern did not vary signifi.cantlythrough theday.
The potential temperature and wind soundings
over Toulouse at 0600 on 16 November are shown in
Fig.6a and 6b, respectively. The temperature profi.leshows a stable layer below 4000 m height, including a thin (<-1000 m in depth) neutral Iayer around
1500m height. The stabiiity gradually decreased
with height in the troposphere. The observed windshows a rapid increaseto 3500 m height followed bya constant velocity layer of 2000m depth. Above
5500m height, the wind increases lineariy to thetropopause (-13 km).
Initial profiles for experiment P9D3 are shown blr
dashed lines in Fig.6a and 6b. Both initial temperature and wind profiles approximateiy follow theobservation of Toulouse 0600 UTC 16 November.
except for the smoothing out of small fluctuations.l
4. Results
In this section, we present the results of numerical simulation afber 6 hours integration. and compare them with observations. This integration time
is chosenbecausethe simulated mountain waves become quasi-steady.
4,.1IOP-?
/t.1.1P3X1
The potential temperature field after 6 hours ofintegration is shown in Fig.7 (only a central part of
the fulI domain is shown). This figure clearly shows
lThe neutral layer around 1500 m height (just below themountain height) is a local feature because it is not observedby the sounding over Pau (about 150 km west by south from
Toulouse). The results with this neutral layer in the initial
profile (not shown) differ only a little from the results without
this layer (r.e., P9DS).

three lee waves in the middle troposphere, followedby two smaller undulations in the lower atmosphere.
The large lee waves have wavelengths of 12-13 km
and amplitudes of -0.5 km. The trailing lee waves
have smalier ampiitudes (<-0.5 km) and shorter
These waves are superimwavelengths (-7km).
posed on a large amplitude (-2.5 km in peak-topeak) long wavelength (>-80 km) mountain wave
(called hereafter the principal wave).
While the reason why the large amplitude lee
waves are restricted to extend to only three wavelengths is not clear, the generation mechanism of
1eewaves in this case is certainly as follows: these1eewaves are produced by the non-linearity of theprincipal wave such as an oscillatory bore generation process (Christie, 1989), becausewe use a verysmooth mountain shape which directly excites negligible amplitude around 12 km waveiength. This suggestion was confi.rmed by an additional experimentwith reduced mountain height (10 m maximum) andthe same atmospheric initial conditions. In this additional experiment, no such lee wave appeared in asimulation (not shown).
Some potential temperature undulations caused
by iee wavesreach the surfaceafter the 210 km point.
Undulations by lee waves near the surface are aLso
seen both in the turbulence kinetic energy .E (Fig.
8) and the horizontal wind (Fig. 9). If the synoptic
wind is not exactiy steady. which is a natural situation in the real atmosphere, the lee waves would
neither be steady nor stationary. Such movement ofIee waves would cause a drift of the high turbulence
region near the surface, and, therefore it is probably
responsiblefor the oscillations of friction velocity observed by the sodar at Lannemezan, as pointed byBougeault et al. (1993).
The turbulence kinetic energy (Fig.8) has a max-

April 1994

181

T. Satomura and P. Bouseault

(a)

(c)

/r-l)

BFSE 16./1 1./90 noHTU

PMER{ HPq ]Fig. 5. Same as Fig. 2, except at 0600UTC 16 November1990.

imum of l}m2fsec2 at some hundreds of meters
above the ground on the lee side of the mountain.
This vaiue and its position are comparable to the observed maximum of 5.7 m2f sec2about 500 m heightnear the 190km point (not shown). It should also
be noted that in the middle and the upper troposphere the turbulence kinetic energy is less than 0.1
m2f sec2. This fact indicates that overturning of theprincipal wave does not occur in the model. Thisagreeswith the reports of on-board scientists of research airplanes (Atti6 ei al., 1991), and contrasts
with the severe downslope windstorm event in central Colorado on 11 January 1972 where strong turbulence was reported through the troposphere (Lillv,

1978).
Although the horizontal wind is deceleratedin theupper troposphere as gravity-wave-drag parameterizations in iarge-scalemodels expect) the horizontal
velocity varies significantly both in the vertical andin the horizontal direction as shown in Fig. 9. A
maximum wind of more than 32mf sec occurs justbehind the mountain peak around 3 to 4 km height.
extending down cioseto the surface. On the lee side,
the horizontal wind exhibits undulations producing
multiple strong wind areas on the iee slope. In the
middle tropospherer a rather complicated structure
due to the superposition of lee waves on the principal wave is simulated, while the upstream side ist82

Journal of the Nleteorological Society of Japan

Vol. 72, No.

(u)(ml
I 4000 r

cO,/II/16 06:00

T0ULgUSE.{s'l 3000')'l 2000/

11 0 0 0l 0000

!

9000

/

"f

-c

8000

(.) ? 0 0 0
I
6000

/
,s

50 0 040 0 0
3 00 0

"/
2000

(
,/

I 000
0
2'70

/
330

e
(b)
{m)
14 0 0 0 -90/ll/16 06:00

T0UL0USE

13 0 0 0
r 200011000
I 0000
9 00 0

-c

' Uo'
=

I 000

"(t

70 0 0

\.
\

6000
5000

\6J

4000

?\ -

3000

\ --\\,€

2000
10 0 0

\

0
-40

-20

-30

-10

I0 (m,/sec)uFig. 6. Same as Fig. 3, except for upper-air sounding over Toulouse at 0600 UTC 16 November1990(soiidline) and initial profile of the model (dashed line) for P9D3.

April 1994

183

T. Satomura and P. Bouseault

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r2.0
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8.0
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5.0

4.0

3.0

2.4

1 . 0

0 . 0 '70.

120 130 140 r50

160 l?0

180 190 20a 210 224 230 240 2s0 260

X
Fig. 7. Potential temperature around the mountain afber 6 hours of integration in P3X1. The abscissa is
the z-direction in km and the ordinate is the heieht in km. The contour interval is 2 K.

(k)(k.r,)

TIME:

6h 0min

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Fig. 8. Same as Fig. 7, except for subgrid scale turbuient kinetic energy calculated by E-e model. The
contour interval is logarithmic (0.1, 0.5, 1, 5, 10) in m2/sec2.

181

Journal of the Nleteorological Society of Japan

(u)

(km)

TIME:

Vol. 72, No. 2

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Fig. 9. Same as Fig.7, except for horizontat wind. The contour interval is 2 m/sec.

acceleratedand the lee side is deceleratedin general.
Combining Figs.S and g, we see that the mainstream separatesfrom the surface behind the pointat 210-220 km, and some weak positive or negative
velocity appears below a few hundreds meters above
the ground. This weak wind correspondsweli withthe sheltered area observed by the surface mesonetwork (Bougeault et al., 1993).
Horizontal variations of u' : u-u and wt : w -A
on four isobaric surfacesare shown in Fig. 10a and b,
respectively, where the overbar represents the horizontal average. This figure clearly shows that themodel simulates the lee wave's amplitude and wavelength quite well, while their position in the model isabout one wavelength (-10 km) behind the observation. This differencein the phase is not considereda
serious probiem, becausea small changeof the mean
horizontal velocity improved this phase difference asshown in the later sub-section (experiment P3X3).
On the other hand, the amplitude of the principal
wave in horizontal velocity (Fig. 10a) is too large at
345 hPa and 217hPa.
We have computed from the model outputs thevertical flux of horizontal momentum.

M":

rL"
t -

I

pu'w'dr,

JLr

and that by long mountain wave,

rd1)

fLz

M-: I

p'i'tit'd:r,

(J '\

JLt

where d' and {u' are the horizontal and the vertical
perturbation velocity that have waveiengths longer
than 17 km. and the line integration is completed
from ,L1 : 75 km to L2 : 245km. These fluxes
are shown in Fig. 11 (see Appendix for the model
performance in case of the linear mountain wave).
Figure 11 obviously shows that J.ongmountain waves
control the vertical momentum flux. This agrees
with a theoretical consideration that trapped waves
do not transport momentum flux vertically.
In Fig. 11, the value of M, observed by the airplanes is also shown (reproduced from Table 4 inBougeault et al., 1993). The observed momentum flllx rapidly increasesfrom -1.5 x 105N/m to-0.15 x 105N/m above 4km height. This rapid increase (or reduction of the absolute value) of observed momentum flux in the vertical direction is an
expected profile when lee waves are fully developed(Durran, 1991; Kim and Arakawa, 1991). The momentum flux in the model. however, keeps a nearly
constant value of -1.5 x i05 N/m up to 10 km. Thisis about a 3 times overestimation, except at 4 km
height.
W-econsider that the rather straight profile of tU"
in the model is a naturai result in our 2-dimensionai
model, becausethe linear theorv states that the mo-185

T. Satomura and P. Bougeault

April 1994

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186

Journal of the Meteorological Society of Japan- 2 2 A S A 0 - 2 A 0 A A O- 1 8 0 9 6 A - 1 6 6 6 4 A - 1 4 A O A . A- 1 2 A A A O- l A A A O A - B A A O A _ 6 A 0 A A

Vol. 72. No. 2

IQOAA

V e r t i c a lM o m e n t u m F l u x ( N / m )Fig. 11. Vertical profile of momentum flix M, in the model (so1id line) and observed b5r airplanes (shortdashed line). The long dashed tine is the vertical momentum flux by lonq mountain waves.

mentum flux must be constant if the mean velocity
U l0 in the inviscid atmosphere(e2.. Eliassenand
Palm. 1961). The wave characteristics in tiris casewould be described approximately by the linear theory and M" should be approximateiy constant because the principal wave does not produce strong
turbulence due to the wave overturning which causes
the strong viscosity and, consequently.the deviationfrom the iinear inviscid theory. The overestimationof M" is also consistent with the overestimation of
the amplitude of the principal wave. The reason
why the observed principal wave has much smaller
amplitude than the simulated one will be discussed
later.
The pressuredrag in the model is about 2Paxl75
km. This value is about twice the momentum flrix
M, at 4 km height. This fact that only a small fraction of the surface pressure drag propagates to thefree atmosphere agrees with the results of Hoinka
and Clark (1991). The excess of pressure dragshould be dissipated in the layer between the surface and the 4km height, especially in the boundary layer. On the other hand, the observed pressure drag is about 6Pa (Bougeault et ol., 1993).

three times larger than the simulated value. We
attribute this difference to the fact that observedpressure drag includes the geostrophic componentof the pressure difference. The magnitude of thisgeostrophic component of the pressure difference is.
however, very difficult to identify in the observation
across the mountain. Therefore a straightforward
comparison between the observedpressure drag andthe one calcuiated by a 2-dimensional modei which
includes no Coriolis effect is meaninsless.
4.1.2 P3X2
The horizontal variation of wl of P3X2 is shownin Fig. 12. In contrast to P3X1, the bottom topography in P3X2 directly excites mountain waves ofwavelength around 12 km with certain amplitude.
Lee waves in P3X2 are, therefore, stronger thanthose in P3X1, and more than four significant waves
appear. The phase difference between the simulated
and observed 1ee waves is practically the same asin P3X1, and the profile of the momentum flux M,(not shown) is also almost the same as P3X1.
There is another way to incorporate the drag effects of the smail topography: increasing the surface
roughness. We examined its effect by increasing zs

187

T. Satomura and P. Bouseault

April 1994

w'
o:.a

(oN)

"*'-w't,tV v v
vivf

1 ,Q O A

r55

r98

285 {lar)X
Fig. 12. Same as Fig. 10b, except for P3X2.

to 10 m, but the results were again almost the sameas in P3X1. Therefore, neither the smailer-scaletopography in this case nor a larger surface roughness can change the basic characteristics of mountain waves, though the former creates small-scalefeatures in the horizontal variations ofthe waves andexcites stronger lee waves. The small sensitivity oflee waves to zs is also found in Shutts and Broad(1 e e 3 ) .
4.1.3 P3X3
Horizontal variation of w' of P3X3 is shown in
Fig. 13. Comparing Fig. 10b, it is found that only
1m/sec modification of the mean velocity improves
the phase of the simulated lee waves by up to rf 2,
though the amplitudes of the iee waves are about
half of P3X1. On the other hand, the amplitude of
the principal wave is reduced oniy a littie and thewavelength of the lee waves is unchanged.
Looking into Fig. 10a, it is noticed that the simuiated mean horizontal velocity in the lower (higher)
Iayer in P3X1 is 3-5 m/sec slower (faster) than theobserved velocity. Therefore, we consider that thediscrepancy of the phase of the lee wave between thesimulation in P3X1 and the observation is explainedby the difference of the mean velocity between thereal atmosphere and the simulation.
If we want to make the simuiated results moreclose to the observation, we have to tune the initial horizontal velocity by trial and error. We do
not adjust. however, the initial profile further because we think it would not be worth doing in our
two-dimensional model which can represent neither
three-dimensional effects such as the flow deviationin the lower iayer rior the time evolution of the synoptic situation.
The vertical momentum flux M" (not shown) differs very little from that of P3X1 except the absolutevalue of the mean value decreasedto 1.3-1.4 x 105
N/m. Considering the decreasedlee-waveamplitudein P3X3, this indicates again that the principai wave
transports the major part of the vertical momentum
flux.
4.2 P9D3
The potential temperature field after 6 hours ofintegration in the case of P9D3 is shown in Fig.
14. This figure shows a well-established train oflee waves beiow 6 km height. These lee waves havewavelength of -13 km and a maximum amplitude

188

Journal of the Meteoroloeical Societv of Jaoan

Voi. 72. No. 2

43.8 ("N)

X
Fig. 13. Same as Fig. 10b, except for P3X3.

of -0.7km. Above that height, almost no wave activity is found owing to the less stable layer (e.e.
small Scorer parameter; see next section) from 7 to
10km height. The wave crest just downstream ofthe mountain peek is very steep and an overturningor strong turbuience is expected.
Indeed, the simul.ated turbulence kinetic energy
shown in Fig.15 presents a large and strong turbuience area which extends up to a height of 5 km
and spreads downstream about 70 km in length. It
has a maximum of I0m2fsecz around 2km heightover 130-140km points. The area of turbulencecorresponds well with the on-board scientists' reports (Atti6 et al., I99l) marked by crossesin Fig.
15. The spread of the area of strong turbulence
to downstream is probably caused by advection of
the strong turbulence energy which originates in the
wave-overturning area over 130-140km points.
The ma-ximum value and its position of simulated
turbulence kinetic energ-yare aiso comparable to theobserved maxima of 7.2m2fsec2 at -2000m heightover -135km point and 6.8m2fsec2 at -2500m
height over -135km point (not shown). It should
also be noted that in the middle and the upper tro-

posphere the turbulence kinetic energy is less than
0.1m2/secz in the simulation and this agreeswith
the observation.
As seen by the horizontal variations of u and trr
shown in Fig. 16a and b, respectively,the model simulates the wavelength of the lee waveswell. Both theamplitudes of lee waves and of the principal wave
are, however, overestimated compared with the airplane observation.
The amplitude overestimation of the principal
wave produces the overestimation of the vertical momentum flux as seenin Fig. 17. In this figure, we findthat the momentum flux deduced from the observation decreasedfrom 10 x 104N/m at 3.5 km to zeroaround 5.5 km, then it increasedto -2 x 104N/m.
while the simulated values decreasedfrom 9.5 x 104
N/m at 4 km to 6 x 104N/m at 6 km and then became constant. It is noteworthy that lee waves contribute negligibly to the momentum flux becausethe
profiles of momentum fluxes M, and.M, are almost
the same above 4 km height. Below the 4 km height,
however, these two profiies separate out, probabiyowing to the strong dissipation of lee waves by the
turbulence in these heishts.

April 1994

189

T. Satomura and P. Boueeault

T I M E=(o )(k..')

6h Omin: 2. 4

10.0
9.0
8.0
?.0

320.v,1

v.L

5.0
4.0
3.0
2.0

o'ori

80

90

loo

lIo

120

l3o

l40

150

l6!

]70

180

190

20o

2|a

22a

23o

24D

250

260

2?0

(km)

X
Fig. 14. Same as Fig.7, except for P9D3

(k)

(km)

TIME:

6h Ornin

t3,0
12,0
It.0
1 0 .0
9.0
8.0
7.0
6.0

Z

5.0

{.0
3.0
2.4
1.0
"'-?o

180

Bo

190

200

210

22A

230

2{0

250

x
Fig. 15. Subgrid scale turbulent kinetic energ.ycalculated by the E-e model after 6 hours of integration in
P9D3. The abscissais the r-direction in km and the ordinate is the height in km. The contour interval
is logarithmic (0.1, 0.5. 1, 5, 10) in m2/sec2. Airplane tracks are shown by dashed lines. Approximate
locations of turbulence reported by on-board scientists are shown by crosses. Parts of dashed line witha character 'S' are approximate areas of sporadic turbulence reported by on-board scientists.

190

Journal of the \4eteorological Society of Japan

Vol. 72, No. 2

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April 1994

191

T. Satomura and P. Bouseault

P

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1

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4AAAA

60000

B A A A A 1 0 @ @ A 01 2 @ A O A 1 4 A A O @ 1 6 0 0 5 4 I A A A A A 2 A A A A A 2 2 @ A A A

V e r t i c a lM o m e n t u m F l u x ( N / m )Fig. 17. Sameas Fig. 11, exceptfor P9D3.
5. Summary and Discussion
Two lee-wave events observed during PYREX
were simulated using a two-dimensional, nonhydrostatic model initiated by upstream soundings. In
spite of a very smooth bottom topography, the characteristics of lee waves were well simulated in bothcases. We conciuded that these lee waves were generated through noniinear interactions of the principal wave, because the smooth mountain we used
can directly excite mountain waves of the observed
wavelength in negligible amplitude.
The upward momentum flux did not correspond with the airplane observation: the simuiatedflux was almost constant through the troposphere
whereas the observation showed a rapid decrease
above 4km height in both cases. We also foundthat, in our two-dimensional model, mountain waves
which have wavelengths longer than the lee waves
controi the vertical profile of the momentum flux:
lee waves transport relatively very small amount of
momentum flux in the vertical direction in almost
al1 parts of the atmosphere. The exception was thelayer belou' 4 km height in the case of P9D3. Because the strong turbulent dissipation was calculated to be around 2 km height in this case. thisnon-zero momentum flux of lee waves appears to becaused by the strong dissipation.
Figure 18 showsprofiles ofthe square of Scorerparameter, 12,based on the initial conditions of P3X1
and P9D3, defined by:-o 1;z
" [J2
t 4

-

_

r d2u
(J dz2'

with buoyancy frequency -|y' and the wind component perpendicuLar to the mountain U. in bothcases, the Scorer parameter becomes very small inthe middle troposphere. According to the linear
theory (Smith, 1979, for example), these profiles of
Scorer parameter are favorable to the occurrence ofsignifi.cant lee waves. In fact Tannhauser and Atti6(1993) used the same vertical profiles of wind andtemperature as in the case of P3X1 of this paperand succeededin calculating the wavelength of thelee wave in good agreement with observation.
Looking to Fig. 18, we notice a slight differencein the Scorer parameter profile between P3X1 andP9D3: in the caseof P9D3, it reducesrapidly around
4 km height and keeps small values above) whereasin P3Xl it diminishes gradually from 5.5 km to 7.5
km height and still has noticeable values occasionally above 8 km height. A steep (gentle) gradientof Scorer parameter indicates a strong (weak) trapping. This agrees with the fact that practicaily no

192

Journal of the Meteorological Society of Japan

Vol. 72. No. 2

(km)
T3
12

v

.2

.4

.5

.B

1.A 1.2 1.4 r6

1.8 2.A 2.2 2.4 2.6 2.8 3.Ai'(kn-')
Fig. 18. !'erticalprofiles of 12 in the case of IOP-3

These profiies are caiculatedby the initial(solid iine) and the third

conditions

wave activity was found in the middle and the upper
troposphere in the case of the third day of IOP-9,
while a significant activity was present in the caseof IOP-3.
Although the modei simulates well the overali
characteristicsof mountain waves observedin IOP-3
and in the third day of IOP-9, the vertical momentum flux differs from observation. In order to simuIate the vertical momentum flux well. it is necessary
to simulate the principal wave well (which transports the major part of the upward momentum fluxin the cases examined in this paper). The modei
clearly overestimates the principal wave in the upper troposphere.
The cause of this problem is believed to be thefollowing: in the case of IOP-3 of PYREX, Beau
(1992) and Bougeault et al. (1993) found that boththe ageostrophic pressure force and the horizontal
momentum advection through the lateral boundaries of a control box encompassing the mountain
range were necessary to explain the horizontal momentum budget in their 3-dimensional, hydrostatic
model. Furthermore, this effect was substantial in
the upper troposphere. The importance of the horizontal momentum advection denotes that moun-day of IOP-9(dashed line).

used in P3X1 and P9D3, respectively.

tain waves observed during IOP-3 have a significant degree of three-dimensionality, even though
the Pyrenees mountains seem to be approximately
two-dimensional. The horizontal momentum budget in other cases is not available, but the threedimensionality Iikely contaminates the vertical momentum flux profiie also in the case of IOP-9.
In addition to the horizontal advection, Beau
(1992) and Bougeault et al. (1993) pointed out thatthe situation was not stationary in the upper troposphere. As shown recentiy by Lott and Teitelbaum (1993a, 1993b), momentum fluxes of untrapped mountain waves under unsteady wind conditions are smaller than those under steady conditions.
These effectswhich cannot be incorporated in thetwo-dimensional stationary model used in this paper
probably explain the difference of the observed verticai momentum flux with the simulated flu-x bothin P3X1 and in P9D3. Thus, it is suggested that
one must be carefui in concluding that the horizontal mean wind will be decelerated albeit the upward momentum flux divergence is observed in one
cross section of mountain as in the case of IOP-3.
Still, nonhydrostatic 3-dimensional simulations andApril 1994

T. Satomura and P. Boueeault

193

(km)
l6

10

_c
q)

=

. 4
Ratio

. 6

. 8

of

Vertical

1 . O

1 . 2

McmenLLm

FIux

Fig. A1. Vertical profile of momentum flux ratio of model results to the linear theory. Numbers labeledon the lines are the inteqration time in hours.

the detailed analysis of the results are indispensable
to clarify the effects of the non-steadinessand thehorizontal momentum advection on the horizontal
momentum budget. They are lefb to the future.

,,:

Acknowledgments
The PYREX experiment is a cooperative programsupported by M6t6o-FYance,the Spanish Institutefor meteorology, the CNRS/INSU (ARAT, PAMOS
and PAtr{OY programs), the CNES, EDF, R6gion
Nlidi-Pyr6n6es and the DLR.
We thank R. Smith of Yale Univ. and J. Stein of
CNRM for helpful discussionsand correspondence.
We have greatly benefited from the aircraft momentum flux computations done by both K.R. Hoinka
of DLR and J.L. Atti6 of Univ. Paul Sabatier.
Most of this work was performed during a visit ofT.S. to CNRX{, which was made possibie by speciai
grants from STA, Japan, and CNRS, FYance: The
author gratefull;r acknowledges this support.
Appendix
Cornparisonof lrlorrlerltunl
rnod,el uith the linear theory

constant mean wind speed U and the Brunt-Vaisalafrequency l{ over a beli-shaped mountain as (seeSmith, 1979,for exampie),

flur

in

the

In this appendix. a comparison of simulated momentum flux with the linear theory is presented toshow the modei's ability to reproduce the vertical
momentum flux of the mountain wave. The iinear theory predicts the vertical momentum flux at aI*

putwtdr :- [r r xun'

(A.1)

where ps is the surface air density, n is the circularconstant. h is the mountain height. This equation
implicitly assumes that the mountain is wide andthe atmosphere is hydrostatic.
The ratio of the model result to the linear theory (A.1) is shown in Fig. A1 in the caseof U : 15
m/sec, l/ : 0.01sec-1, the mountain height h: I0
m and the half width of the mountain is 10 km. The
model is run with the free-slip lower boundary condition, the model horizontal grid interval is 2.5 km
and the vertical grid interval varies from 10 m to 500
m. The sponge layer is placed above 12 km height.
This figure demonstratesthat the model used in thispaper agreesvery well with the linear theory after 6hours integration.
References
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286 pp.
Atti6, J.L., B. Benech, A. Druilhet and P. Durand,
J.991;Comptes rendus scientifiquesdes exp6riences
a6roport6esr6alis6esavec I'ARAT et le Nleriin VI

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Journal of the Meteorological Societ;' of Japanpendant PYREX, Techn'icalReportN" 91-2, Laboratoire d'A6rologie, Universit6 Paule Sabatier.
Beau, L, 1992; Evaluation des param6trisation de l'effet
orographique sous-maiile dans les modbles de circulation g6n6rale A,l'aide de PERIDOT l0km, Note detrauail de I'ENM, ENNI, CNRM.
Bougeault, P., A. Jansa, B. Benech, B. Carrisimo,
J. Pelon and E. Richard. 1990; N{omentum budget
over the Pyr6n6es: The PYREX experiment, Bzll.
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R. Benoit, P. Bessemoulin,J.L.'Caccia, J. Campins,
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Blockley, J.A. and T.J. Lyons, 1994; Airflow over a twodimensional escarpment. III: Nonhydrostatic flow,
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atmosphere, J. Atmos. Scz., 46, 1462-7491.
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Meteor. Soc., L17, 527-552.
Detering, H.W. and D. Etling, 1985; Appiication of the
E-a turbulence model to the atmospheric boundarv
Iayer, Boundary layer Meteor.,33, 113-133.
Durran D.R., 1991; Orographic wave drag on the lower
troposphere: The importance of trapped waves.
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Durran D.R. and J.B. Klemp. 1982; The effects of moisture on trapped mountain J.eewaves, J. Atmos. Sci.,
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in stationary mountain waves, Geofys. Publ.,22, No.
2

o.

1

02

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