Gap Flows through Idealized Topography. Part I: Forcing by ...durrand/pdfs/AMS/2004Gabersek_Durran.pdf · Gap Flows through Idealized Topography. Part I: Forcing by Large-Scale Winds

2846 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 2004 American Meteorological Society

Gap Flows through Idealized Topography. Part I: Forcing by Large-Scale Windsin the Nonrotating Limit

SASA GABERSEK* AND DALE R. DURRAN

Department of Atmospheric Sciences, University of Washington, Seattle, Washington

(Manuscript received 3 September 2003, in final form 26 April 2004)

ABSTRACT

Gap winds produced by a uniform airstream flowing over an isolated flat-top ridge cut by a straight narrowgap are investigated by numerical simulation. On the scale of the entire barrier, the proportion of the oncomingflow that passes through the gap is relatively independent of the nondimensional mountain height e, even overthat range of e for which there is the previously documented transition from a ‘‘flow over the ridge’’ regime toa ‘‘flow around’’ regime.

The kinematics and dynamics of the gap flow itself were investigated by examining mass and momentumbudgets for control volumes at the entrance, central, and exit regions of the gap. These analyses suggest threebasic behaviors: the linear regime (small e) in which there is essentially no enhancement of the gap flow; themountain wave regime (e ; 1.5) in which vertical mass and momentum fluxes play a crucial role in creatingvery strong winds near the exit of the gap; and the upstream-blocking regime (e ; 5) in which lateral convergencegenerates the strongest winds near the entrance of the gap.

Trajectory analysis of the flow in the strongest events, the mountain wave events, confirms the importanceof net subsidence in creating high wind speeds. Neglect of vertical motion in applications of Bernoulli’s equationto gap flows is shown to lead to unreasonable wind speed predictions whenever the temperature at the gap exitexceeds that at the gap entrance. The distribution of the Bernoulli function on an isentropic surface shows acorrespondence between regions of high Bernoulli function and high wind speeds in the gap-exit jet similar tothat previously documented for shallow-water flow.

1. Introduction

Strong surface winds can develop in several wayswhen stably stratified air interacts with a mountain bar-rier. In one prototypical case—downslope windstorms—the winds blow down the lee slope of a mountain ridge.Downslope windstorms occur only when there is a sig-nificant cross-barrier flow. In a second prototypicalcase—gap winds—the winds blow through a gap in theridge. Gap winds typically occur when there is a sig-nificant drop in atmospheric pressure between the en-trance and exit regions of the gap. Gap winds, whichhave received considerably less theoretical attentionthan downslope winds, are the focus of this paper.

The pressure differences responsible for gap winds canbe generated by several types of synoptic-scale weatherpatterns. Low-level cross-barrier pressure gradients maybe present even when the large-scale flow surrounding

* Current affiliation: Faculty of Mathematics and Physics, Uni-versity of Ljubljana, Ljubljana, Slovenia.

Corresponding author address: Dr. Dale R. Durran, Dept. of At-mospheric Sciences, University of Washington, P.O. Box 351640,Seattle, WA 98195-1640.E-mail: [email protected]

the mountain is essentially stagnant provided there aresignificant low-level temperature differences in the airmasses on each side of the mountain. Gap flow undersuch conditions occurs frequently during the summer inthe Columbia River Gorge (between the states of Wash-ington and Oregon in the United States). When synoptic-scale pressure gradients exist above mountain-top levelthey are usually in approximate geostrophic balance withthe larger-scale flow. Geostrophically balanced pressuregradients associated with the large-scale flow have beenidentified as the primary agent in creating gap flows inlocations such as the Shelikof Strait in Alaska (Lackmannand Overland 1989) and Lake Tornetrask in Scandinavia(Smedman et al. 1996) through a process known as ‘‘pres-sure driven channeling’’ (Whiteman and Doran 1993).Perhaps the most common type of gap flow occurs inconnection with cold-air surges, when both significantcross-mountain winds and cross-mountain temperaturedifferences may be present. Examples of this type includethe wintertime easterlies in the Columbia River Gorge,as well as easterly gap winds in Howe Sound, BritishColumbia, (Jackson and Steyn 1994a,b), the Strait of Juande Fuca (Overland 1984; Colle and Mass 2000), and theStrait of Gibraltar (Scorer 1952; Dorman et al. 1995),and northerly winds through Chivela Pass into the Gulfof Tehuantepec, Mexico (Steenburgh et al. 1998).

1 DECEMBER 2004 2847G A B E R S E K A N D D U R R A N

This is the first of a pair of papers that will attemptto present a relatively comprehensive analysis of thosegap flows that are dynamically forced by the interactionof a large-scale flow with the topography. The morecomplicated problem of gap winds driven by the com-bined influences of cross-mountain winds and temper-ature differences is left for future study.

Many previous theoretical studies of gap flow haveused shallow-water theory to examine the response offlow within a channel to changes in the channel widthand channel depth (Baines 1995, chapters 2 and 3).While such studies are very useful for investigating theresponse to small-scale features within the gap, the po-tential for mesoscale circulations near the gap-entranceand -exit regions to dominate the total cross-gap pres-sure gradient (Colman and Dierking 1992; Colle andMass 1998a,b) provides the motivation for our approachin Part I of this paper, which is to examine the processesthat occur along the entire length of the gap, includingthe entrance and exit regions. In particular, this paperfocuses on the regimes of free-slip gap flow generatedby mesoscale pressure perturbations arising from theinteraction of a uniform cross-mountain flow with amountain barrier cut by a straight, sea level gap. Coriolisforces are neglected. The influence of surface frictionand the relative importance of mesoscale and geostroph-ically balanced synoptic-scale pressure perturbations(pressure-driven channeling) in determining the strengthand structure of gap flow will be considered in Part II.

Idealized simulations of gap winds generated by auniformly stratified airstream flowing perpendicular tothe long axis of the barrier (and parallel to the axis ofthe gap or mountain pass) have been conducted by Saito(1993) and Zangl (2002), both of whom found that sig-nificant gap flows could develop in response to meso-scale pressure gradients produced by the large-scalecross-ridge flow. The focus in this paper is on the de-tailed analysis of the kinematics and dynamics that areresponsible for the generation of the high gap winds thatcan develop in such cross-mountain flows, including

• the extent to which air is deflected through the gapor around the ends of a ridge as a function of thenondimensional mountain height,

• the variations, with respect to the nondimensionalmountain height, in the portion of the gap withinwhich the maximum flow acceleration occurs and thedynamical processes responsible for that acceleration,and

• the application of Bernoulli’s theorem to those caseswith the strongest gap winds.

2. Model description

The calculations presented in this paper were con-ducted using a numerical model to simulate nonhydro-static compressible flow governed by the equations

]u ]u ]p9 u 2 u 1 ]Tiji i1 u 1 c u 2 d g 5 2 , (1)j p i3]t ]x ]x u r ]xj i j

]u ]u ]Bj1 u 5 , (2)j

]t ]x ]xj j

]p9 R ]u Rp ]Bj j1 5 . (3)

]t c ]x c u ]xy j y j

Here (x1, x2, x3) 5 (x, y, z) is the spatial position vector,(u1, u2, u3) 5 (u, y, w) is the velocity vector, u is thepotential temperature, r is the density, cp and cy are thespecific heats of air at constant pressure and constantvolume, and R is the gas constant for dry air. Pressurep appears in (1)–(3) through the Exner function p 5(p/p0) . The thermodynamic variables p and u are di-R/cp

vided such that p 5 (z) 1 p9(x, y, z, t) and u 5 (z)p u1 u9(x, y, z, t), where the reference state ( , ) is inp uhydrostatic balance (i.e., cp ] /]z 5 2g). Finally, Tiju pand Bj are the turbulent subgrid-scale fluxes of mo-mentum and heat, parameterized in terms of an eddydiffusivity K following Lilly (1962) as

]uT 5 rKD , B 5 K . (4)i j i j j ]xj

Here the Prandtl number has been taken as unity,

]u ]u 2 ]uji lD 5 1 2 d ,i j i j]x ]x 3 ]xj i l

and K is proportional to (1 2 Ri)1/2, in which Ri is theRichardson number

21/22g ]u2Ri 5 D .O O i j1 2u ]x i j3

The numerical techniques used to solve (1)–(3) aredescribed in Durran and Klemp (1983). The model in-corporates the topography using a terrain-following co-ordinate (Gal-Chen and Sommerville 1975) and in-cludes two-way interactive nesting (Skamarock andKlemp 1993). At the lateral boundaries of the coarsestgrid, a one-way wave equation is applied to the normalvelocity using a constant outward-directed phase speed.The linear radiation condition of Klemp and Durran(1983) and Bougeault (1983) is applied at the top bound-ary as modified for local evaluation by Durran (1999).At the bottom boundary, T13, T23, B3, and the velocitycomponent normal to the topography are set to zero.

The topography in our experiments is an elongatedflat-topped ridge parallel to the y axis with a gap per-pendicular to the ridgeline defined by the product h(x, y)5 r(x, y)g(y). The shape of the ridge into which thegap is incised is given by the formula

4h ps0 1 1 cos , if s # 4a;1 2 [ ]16 4ar(x, y) 5 (5)0, otherwise,

where


max(0, |x | 2 b), if |y | # c;s 5

2 2 1/25max{0, [x 1 ( |y | 2 c) ] 2 b}, otherwise.

The ridge is centered at (x, y) 5 (0, 0) and is flat toppedwith uniform height h0 over a distance 2b along the xaxis and a distance 2(b 1 c) along the y axis. The endsof the flat-topped region are semicircular, with radiusb. The slopes of the ridge have an approximate half-width a. The gap is carved out of the ridge by multi-plying r(x, y) by

0, if |y | # d /2; p( |y | 2 d /2) d dg(y) 5 sin , if , |y | # e 1 ; (6)[ ]2e 2 2

1, otherwise.

In (6), d is the width of the floor of the gap, and e isthe horizontal distance over which the sidewalls risefrom the floor of the gap to the ridgeline. In the sim-ulations presented in Part I of this paper, h0 5 1.4 km,a 5 b 5 10 km, c 5 85 km, e 5 5 km, and unlessotherwise noted, d 5 10 km.

Some previous studies, both laboratory (Baines 1979)and numerical (Saito 1993), have economized by study-ing gap flow in a narrow channel perpendicular to theridge axis with channel walls along the centerline of thegap and across the adjacent ridge. We also conductedsimulations in such narrow domains (with slab-sym-metric boundary conditions in the gap and on a sectionof the uniform ridge) and found a tendency for the low-level flow upstream of the gap to become blocked whenNh0/U was large. As will be shown in sections 4 and5, such blocking can play an important role in the dy-namics of gap flow, but the physical relevance of up-stream blocking in the essentially two-dimensional ge-ometry of a channel flow is open to question. Epifanioand Durran (2001) have shown that once mountainwaves begin to break, the flow over the centerline of avery long, but finite ridge is not well approximated bya purely two-dimensional simulation. In order to faith-fully represent the nature of any upstream blocking (andin Part II, to accommodate large-scale winds at arbitraryangles to the ridgeline) the simulations described in thispaper employ multiply nested grids to compute the flowthrough the gap in an isolated ridge. This configurationallows any tendency toward upstream blocking to bemitigated by the possibility of flow around each end ofthe ridge.

In these simulations the aspect ratio of the the ridge,defined as the length of the ridge at half height dividedby the width of the ridge at half height, is b 5 (b 1c)/(a 1 b) 5 4.75. Epifanio and Durran (2001) foundthat for values of Nh0/U . O(1), the flow over thecenterline of a very long ridge (b 5 12) is much betterapproximated by that over a ridge with b 5 5 than bythe flow in a purely two-dimensional (x 2 z) domain.Thus, the results reported in this paper for gaps in ridges

with b ; 5 are expected to be more representative ofthe flow through gaps in extremely long ridges thanresults obtained using the channel geometry.

Each of the nested grids covered a square domainwith the gap at its center. The spatial and temporal res-olution was refined by a factor of 3 on each nest. Thefinest grid, on which Dx 5 1.5 km, occupied a square271 km on a side, which was just large enough to includethe entire mountain. The intermediate grid, on whichDx 5 4.5 km, covered a square 405 km on a side. Theouter grid, on which Dx 5 13.5 km, extended over asquare 1269 km on a side. The depth of the domain zT

was 13 km. The vertical grid spacing is variable, startingat 100 m in the layer 0 # z # 3 km, then smoothlyincreasing to 250 m over layer 3 # z # 4 km, andremaining constant at 250 m above 4 km. This verticallystretched grid allows us to efficiently resolve both thelow-level gap flow and any mountain waves that mightdevelop aloft.

The horizontal wind field is ‘‘turned on’’ instanta-neously, and integrated to a nondimensional time T [Ut/a 5 40, by which point the flow in the vicinity ofthe topography reaches a nearly steady state. Other ini-tialization techniques, such as a gradually ramping upof the velocity, were tested, but all approaches gaveessentially the same nearly steady solution. All numer-ical calculations were performed with single-precisionarithmetic because of computational costs. Solutions ob-tained using double precision differ from the single-precision simulations only in the lee of the obstacle inregions of highly turbulent flow, but there was no dif-ference in the basic character of the quasi-steady so-lutions.

3. The barrier-scale response

Using linear theory, Smith (1989) investigated thedevelopment of stagnation points in flow around an iso-lated three-dimensional barrier. For flows with constantN and U impinging on simple convex barriers of heighth0, he noted that stagnation was favored in cases withlarger values of e 5 Nh0/U, and that in addition, thebehavior of the stagnation points depended on the ratioof the cross-flow extent of the ridge to its along-flowextent b. For ridges oriented perpendicular to the flow(b . 1), stagnation is favored in the wave-breakingregion above the lee slope. For ridges aligned parallelto the flow (b , 1), stagnation is favored on the wind-ward slope in connection with the splitting and deflec-tion of the flow around each side of the ridge. Manynumerical studies have subsequently confirmed that theparameters e and b are sufficient to characterize thesusceptibility of cross-barrier flows with constant N andU to wave breaking and flow splitting.

When the ridge is cut by a gap, a new path becomesavailable along which fluid parcels may travel to the leeside of the barrier. In this section we consider two basicquestions that arise when there is a gap in the topog-


FIG. 1. Horizontal streamlines and normalized perturbation velocity (u 2 U)/U (shaded contours) at z 5 300 m andT 5 Ut/a 5 40, for flow over a ridge with a gap when e equals (a) 0.25, (b) 1.4, (c) 2.8, and (d) 5.0. The contourinterval is 0.5; dark (light) shading corresponds to negative (positive) values. Terrain contours are every 300 m.

raphy. The first question is: Do the barrier-scale flowperturbations generated by a ridge with a narrow gapdiffer from those that develop when no gap is present?The second question is: How does the fraction of theoncoming flow that is channeled through the gap varyas a function of e?

Representative examples (selected from a much largerset of simulations) of the low-level flow around ridgeswith and without gaps at various values of e are shownin Figs. 1 and 2. The ridges were identical in all thesimulations and defined according to (5). When a gapwas present, it was defined by (6). The static stabilitywas N 5 0.01 s21, and the variations in e were achievedby changing the speed of the upstream flow, such thatvalues of e equal to (0.25, 1.4, 2.8, 5) were obtainedusing values of U equal to (56, 10, 5, 2.8) m s21. Eachpanel in Figs. 1 and 2 shows streamlines and the nor-malized perturbation velocity [(u 2 U)/U] on the sur-

face z 5 300 m; no data are plotted where the elevationof the topography exceeds 300 m.

When e 5 0.25 (Figs. 1a and 2a), mountain wavesare present over the ridge, but there is no wave breakingand the flow is similar to that obtained for very smalle (not shown). The waves are clearly visible in the is-entrope displacements in the vertical cross section alongthe gap axis plotted in Fig. 3a. Also plotted in Fig. 3are contours of the normalized x-component perturba-tion velocity field using the same shading and contourintervals used in Figs. 1 and 2. For e 5 0.25, the lateraldeviation of the streamlines when they encounter theridge or the gap is small, which is consistent with theresults of Epifanio and Durran (2001), who found thatonly modest lateral flow deviations occurred aroundlong uniform ridges unless the crest was high enoughto trigger wave breaking over the lee slope. Figures 1aand 3a show a slight enhancement of the wind within


FIG. 2. Horizontal streamlines and normalized perturbation velocity, as in Fig. 1, for flow over a ridge without a gapfor e equal to (a) 0.25, (b) 1.4, (c) 2.8, and (d) 5.0.

and downstream of the gap, but there is no distinct jetof high winds emanating from the gap.

When e 5 1.4 (Figs. 1b and 2b), wave breaking oc-curs over the lee slopes of the ridge, creating a narrowzone of high winds that ends abruptly in a feature anal-ogous to a hydraulic jump. A turbulent wake of decel-erated flow is present downstream of the jump. In thesimulation without a gap the central portion of the wakecontains a well-organized current of reversed flow backtoward the mountain, whereas in the simulation with agap, the wake is split by a jet flowing rapidly away fromthe mountain. This jet is the continuation of the accel-erated gap flow, and it is flanked by a pair of vorticesin which the circulation is opposite to that typicallyfound in vortices forming in lee of a barrier without agap. The vertical cross section along the gap axis (Fig.3b) shows the high-amplitude wave aloft, a zone ofstagnant or reversed flow (the region inside the thirdlevel of dark shading) centered near (x, z) 5 (50, 1.2),

and the extension of the gap flow well downstream fromthe ridge. Although the shape of the topography is some-what different, the presence of wave breaking over thegap itself is consistent with the simulations of Zangl(2002), who in contrast to Saito (1993), found high gapwinds developing beneath a wave-breaking region ex-tending across the gap from the adjacent ridges. Up-stream of the barrier, the 300-m flow is mostly blockedand deflected around the ends of the ridge. The presenceof the gap enhances the upstream blocking, except inthe localized region just upstream of the gap.

In comparison to the e 5 1.4 simulations, when e 52.8 the mountain waves over ridge are weaker (cf. Fig.3c), wave breaking is reduced, and the high winds donot extend down the lee slope to the 300-m level onwhich the data are plotted in Figs. 1c and 2c. Down-stream of the ridge without a gap, a typical pair of leevortices produces reversed (‘‘easterly’’) flow along thecenterline, but when a gap is present, a pronounced jet


FIG. 3. Potential temperature and horizontal velocity in a y–z plane along the centerline of the gap for the four casesshown in Fig. 1: e equal to (a) 0.25, (b) 1.4, (c) 2.8, and (d) 5.0. The heavy lines are the 275-, 280-, and 285-Kisentropes; the normalized perturbation velocity field is contoured every 0.5 and shaded as in Fig. 1. The dashed lineshows the profile of the adjacent ridge.

of enhanced ‘‘westerly’’ winds penetrates over 100 kmdownstream, splitting the wake into four distinct vor-tices. The gap-wind jet is flanked by a pair of smallervortices that rotate opposite to the main pair of vorticesthat fill the wake farther downstream. Finally, there isslightly less deceleration immediately upstream of thecentral portion of the ridge when the gap is present.

The lee wave amplitude is negligible when e 5 5(Fig. 3d). The lee vortex patterns in the gap and no-gapsimulations remain similar to those in the e 5 2.8 case,but the jet emanating from the gap is much weaker; infact the gap winds in the exit region are weaker thanthe undisturbed upstream flow (Figs. 1d and 2d). Ac-celerated winds are nevertheless found within the gapitself, and the lateral confluence feeding air into the gap

is more pronounced than in the simulations with smallervalues of e.

A closer look at the flow within the gap in the pre-ceding simulations is provided in Fig. 4, which showsthe normalized pressure perturbation [p 2 (z)]/(er0U 2)pat z 5 300 m (here r0 is a representative surface density),along with the normalized perturbation wind speed pre-viously plotted in Figs. 1 and 2. Note that the regionof highest perturbation wind speed shifts position as afunction of e. When e 5 0.25, the normalized pertur-bation winds are too weak to show up at the contourinterval plotted in Fig. 4; nevertheless, other plots usingfiner contour intervals show that the highest perturbationwinds are downstream of the centerline but still withinthe gap. As e increases to 0.5 (not shown) and 1.4, the


FIG. 4. Normalized pressure perturbation (p 2 )/(er0U 2) (solid black lines for positive, dashed black lines forpnegative values, thick solid line represents zero perturbation) and normalized perturbation velocity (u 2 U )/U (shadedcontours) at z 5 300 m and Ut/a 5 40, for (a) e 5 0.25, (b) e 5 1.4, (c) e 5 2.8, and (d) e 5 5.0. The contourinterval for pressure perturbation (italics) is 0.25 (cases e 5 0.25 and e 5 1.4) and 0.2 (cases e 5 2.8 and e 5 5.0).For velocity perturbation, the contour interval is 0.4, dark (light) shading corresponds to positive (negative) values,speeds in the interval [20.2, 0.2] are not shaded. Terrain contours are every 300 m.

FIG. 5. Cross-mountain pressure drag D as a function of Ut/a forthe simulations shown in Figs. 1 and 2. Pairs of curves are plottedfor each value of e, using filled circles for cases with a continuousridge and open squares for cases with a gap.

normalized perturbation wind maxima strengthen andshift completely downstream of the ridge. Further in-creases in e move the perturbation maximum wind speedback upstream; it appears back inside the gap when e5 2.8, and it shifts even farther upstream, to the gap-entrance region, when e 5 5.0. The locations of the

maximum wind speed perturbations are approximatelycoincident with the locations of the maximum windspeed perturbations are approximately coincident withthe locations of the minima in the 300-m pressure field.Note that for most values of e, the pressure within thegap is lower than the ambient pressure upstream; how-ever when e 5 1.4, the pressure in the upstream halfof the gap is higher than the pressure upstream, and thepressure gradient contributing to the acceleration of thegap flow is concentrated in the gap-exit region.

The normalized x component of the pressure drag onthe topography,

`1 ]hD 5 p9 dx dy,E2r NUh L ]x0 0 y 2`

for each of the preceding simulations is plotted as afunction of nondimensional time Ut/a in Fig. 5. Herep9 is the perturbation pressure and Ly is the length ofthe ridge, taken as 2(b 1 c) 5 190 km [see (5)]. Thenormalization factor is a scale for the pressure dragassociated with linear flow over a ridge of height h0 andlength Ly, but it is not the precise drag for linear flowover the flat-top ridge defined by (5). Data for simu-lations with and without a gap are indicated by opensquares or filled circles, respectively. The normalizeddrag in the weakly nonlinear e 5 0.25 cases becomesquite steady at values of about unity after Ut/a 5 20,


FIG. 6. A control volume for low-level mass-flux calculations.

with the drag slightly stronger in the case with no gap.The slight decrease in drag when the gap is present isdue to the reduction in the total area of the obstacle onwhich the pressure drag can act.

In contrast to previous results for pressure drag acrossridges with well-defined peaks (Olafsson and Bougeault1997; Epifanio and Durran 2001), the maximum valueof the normalized drag for the flat-top ridge is achievedat relatively small e, for example, among our simula-tions with e in the set {0.25, 0.5, 1.0}, the largest value,D 5 2.0, was obtained for e 5 0.5. As suggested bythe remaining pairs of curves in Fig. 5, in the morenonlinear cases, the drag undergoes a strong initial tran-sient and then settles down but never becomes com-pletely steady. The e 5 1.4 cases become quasi-steadywith values of D around 1.1, with slightly higher dragfor the ridge pierced by the gap. As noted previously,the simulation with the gap produces a little more up-stream blocking along the ridges on each side of thegap, and it is the upstream pressure perturbations as-sociated with this blocking that appear to be responsiblefor the slightly enhanced drag. In the e 5 5.0 case, thedrag is much weaker; it is virtually independent of thepresence of the gap, and it decreases gradually in as-sociation with a long time-scale evolution of the wake.The e 5 2.8 cases exhibit the most complex behavior.After achieving an almost steady value of 0.55 betweenUt/a 5 15 and 35, the drag begins to undergo fluctu-ations with a period of roughly 20 Ut/a. These fluctu-ations are associated with vacillations in the lee waveand wake structure that are a subject of continued in-vestigation. The phase/onset time of the oscillation issensitive to the presence or absence of the gap.

Now consider the question of how the fraction of theoncoming flow channeled through the gap varies as afunction of e. A quantitative measure of low-level flowdeflection through the gap or around the ends of theridge can be obtained by evaluating the mass fluxesthrough the sides of the control volume shown in Fig.6. The lower boundary of the control volume followsthe terrain, the sides are vertical planes and the topboundary follows a surface of constant potential tem-perature (ut 5 277 K). The downstream boundary isparallel to the y axis and is located where the topographyfirst rises to its full height, at x 5 2b 5 210 km. Thedownstream boundary is sufficiently upstream, with re-spect to the centerline of the ridge, that wave breakingnever occurs within the control volume; therefore theflow along the top of the control volume is isentropic.The lateral sides are parallel to the x axis, intersectingthe north and south ends of the uniform section of theridge at y 5 6c 5 685 km. The upstream boundaryis placed at x 5 xi 5 2200 km, which is sufficientlyfar upstream to ensure that all significant deflection ofthe flow around the ends of the mountain is includedin the mass fluxes through the lateral sides of the controlvolume. The mass flux budget is evaluated at nondi-mensional time T 5 40, at which point the flow through

the control volume in each simulation is almost com-pletely steady.

Since at steady state, both the lower and upper bound-aries of the control volume act as material surfaces, themass entering the control volume through its upstreamface is either deflected laterally around the ends of theobstacle, lifted over the ridge crest, or channeledthrough the gap. Let zt(x, y) be the height of the ut

isentropic surface. The fluxes through each face of thecontrol volume shown in Fig. 6 may be evaluated as

c z (x ,y)t i

f 5 ru(x , y, z) dy dz,i E E i

2c 0

0 z (x,c)t

f 5 ry(x, c, z) dx dz,d1 E Ex h(x,c)i

0 z (x,2c)t

f 5 2 ry(x, 2c, z) dx dz,d2 E Ex h(x,2c)i

c z (2b,y)t

f 5 ru(2b, y, z) dy dz,c1 E Ee1d /2 h0

2e2d /2 z (2b,y)t

f 5 ru(2b, y, z) dy dz,c2 E E2c h0

e1d /2 z (2b,y)t

f 5 ru(2b, y, z) dy dz.g E E2e2d /2 h(2b,y)

Normalized fluxes may now be defined representingthe fraction of the mass entering the control volume thatis deflected around the ends of the ridge Fd 5 (fd1 1fd2)/fi, lifted over the crest Fc 5 (fc1 1 fc2)/fi orchanneled through the gap Fg 5 fg/fi. These normal-ized mass fluxes are plotted as a function of e in Fig.7a, which clearly shows the expected shift between aflow-over regime at small e and a flow-around regimeat large e. Somewhat surprisingly, the fraction of the


FIG. 7. Partitioning of the low-level mass flux as a function of e.(a) Laterally deflected flux Fd (diamonds), gap flow Fg (shadedsquares), and cross-crest flow Fc (circles) for the control volumeshown in Fig. 6 when the bottom of the gap d is 10 km wide. (b) Asin (a), except that the control volume is topped by a horizontal planeat the elevation of the ridge crest and the circles represent the verticalflux through this plane. (c) Comparison of Fg for d 5 10 km (shadedsquares) and d 5 40 km (open squares).

FIG. 8. Control volumes for analysis of the low-level mass budgetin the gap. Arrows representing mass fluxes for the entrance regionare also shown.

total mass flux that exits through the gap Fg is relativelyindependent of e. For the topography shown in Fig. 1,the gap flow Fg varies from 0.18 to a minimum of 0.11at e 5 1.4 and then gradually increases back to 0.14.In contrast, over the same range of e both the portionthe flow deviating laterally around the ends of the ridgeFd and that flowing over the crest Fc change by roughly100% as the flow-over regime (small e) gives way tothe flow-around regime (large e).

The preceding analysis does not uniquely differentiatebetween the gap flow itself and the air that passes overthe gap above the height of the ridge. Does the per-centage of air passing through the gap itself exhibit agreater dependence on e? To address this question, thetop of the control volume defined in Fig. 6 was replacedwith a horizontal plane at the height of the ridge crest,and the portion of the flow passing over the ridge wasredefined as the vertical flux through this horizontalplane normalized by the total incoming flux upstream.The variation of these ‘‘below crest’’ fluxes as a functionof e is plotted in Fig. 7b, which shows the same basicbehavior revealed in the previous analysis: as e increas-es, there is a clear shift between a flow-over regime toa flow-around regime while the percentage of flow pass-ing through the gap remains relatively constant. Oneminor piece of additional information available in Fig.7b is that the net vertical flux through the top of thevolume drops almost to zero for e $ 2.8.

The robustness of the result that Fg does not expe-rience a regime change as a function e was verified ina second series of simulations in which the width of thebottom of the gap d was increased from 10 to 40 km

[see (6)] without modifying any of the other parametersdefining the shape of the topography. This increase ind widens the total y–z cross section of the gap by afactor of 2.8. The fraction of the incoming mass fluxthat is channeled through the gap in both the d 5 10and d 5 40 km simulations is plotted as a function ofe in Fig. 7c. In both cases, the fraction of the flow thatpasses through the gap is relatively independent of e.Not surprisingly, the fraction of the total flow channeledthrough the gap increases as the width of the gap in-creases. Furthermore, for a given value of e, the ratioof Fg for the d 5 40 km case to that in the d 5 10case is approximately equal to the ratio of the cross-sectional areas of the gaps in the two simulations.

4. Flow through the gap: Kinematics

Having examined the barrier-scale response, let usnow consider the kinematics of the gap flow itself,which may be revealed by analyzing the mass fluxesthrough the series of three control volumes orientedalong the gap shown in Fig. 8. The top of all threevolumes is bounded by the horizontal plane z 5 1200m, which lies 200 m below the ridgetop. The width ofeach volume is the minimum of either 20 km [d 1 2eas defined in connection with (6)] or the actual widthof the gap. The entrance volume occupies the region240 # x # 210 km along the windward slopes of theridge, the exit volume is aligned with the lee slopes overthe interval 10 # x # 40 km, and the central volumeis found where the gap width is independent of x in theinterval 210 5 2b # x # b 5 10 km.

Define the area-integrated mass fluxes , ,x xf f→en →c

, and as the integral of ru over the y–z surfacesx xf fc→ ex→

that constitute the upstream sides of the entrance, centraland exit volumes, and the downstream face of the exitvolume, respectively. Let , , and be the integralz z zf f fen c ex

of rw over x–y surfaces at the top of the entrance, cen-tral, and exit volumes. Finally, let and be they yf fen ex


FIG. 9. Normalized mass fluxes through the control volumes shownin Fig. 8, for simulations with different e. Vertical dashed lines denotethe x-coordinate locations of the boundaries of the individual controlvolumes. For each simulation, the leftmost pair of lines delimits theentrance volume, the center pair the central volume, and the rightmostpair the exit volume. The cross-ridge topographic profile associatedwith these control volumes is plotted below each set of mass-fluxdata, along with the normalized surface wind speed. Squares, dia-monds, and circles denote the normalized values of f x, f y, and f z,respectively.

net area-averaged lateral mass flux out of the entranceand exit volumes, defined for example for the entrancevolume such that 5 fN 2 fS, where fN and fS areyf en

the integrals of ry over the ‘‘northern’’ and ‘‘southern’’faces indicated in Fig. 8. At steady state, conservationof mass requires

x x z yf 5 f 2 f 2 f , (7)→c →en en en

x x zf 5 f 2 f , (8)c→ →c c

x x z yf 5 f 2 f 2 f . (9)ex→ c→ ex ex

Note that the sidewalls of the gap prevent any lateralmass fluxes from entering the central volume, and theyreduce the x–y cross-sectional area of the central volumeby a factor of 0.64 relative to the area of the upstreamface of the entrance volume (and the downstream faceto the exit volume).

The steady-state mass balance for four simulationswith e 5 0.25, 1.4, 2.8, and 5.0 is displayed in Fig. 9.In all cases the mass budget closes to within 5% of thelargest term. For each simulation all terms appearing in(7)–(9) are normalized by division by max( | | ,xf→en

| | , | | , | | ) and plotted at representativex x xf f f→c c→ ex→

locations along the x axis. The along-gap fluxes f x areplotted at the x coordinate of the y–z face through whichthe flux is transmitted, whereas the fluxes f y and f z

are plotted at the x coordinate of the center of the surfacethrough which they are transmitted. The ridge profileand the normalized surface wind speed u(x, 0, 50)/[U(11 e)] are also displayed for each simulation. Three basicregimes of mass transport through the gap are apparentin Fig. 9.

In the first regime, which applies to the case e 5 0.25,the air flows up and over the topography with only

minimal lateral divergence; there is no wave breakingand almost no amplification of the gap flow. The var-iation in the average along-gap wind speed can be de-duced from the along-gap mass fluxes in this case asfollows. The mass flux out the downstream face of theentrance volume is decreased from the upstreamxf→c

value by almost the full factor of 0.64 by whichxf→en

the cross-sectional area of flow is reduced within thegap. The decrease in mass flux is slightly less than 64%because there is a slight acceleration of the along-gapwind component within the entrance volume. The massbalance required by (7) is achieved primarily by re-moving mass through the top boundary of the entrancevolume ( . 0); lateral convergence provides only azf en

small contribution to the total mass balance. Within thecentral volume there is very slight acceleration of theflow and enhancement of the along-gap flux duexf c→

to a weak downward flux . The mass flux out exitzf c

volume is increased relative to by downwardx xf fex→ c→

fluxes through the top of the exit volume ( , 0);zf ex

lateral divergence out the sides of the exit volume isweak. The increase in relative to is due almostx xf fex→ c→

entirely to the increase in cross-sectional area of the exitvolume downstream of the gap. Since is onlyxf ex→

slightly greater than , there is almost no net en-xf→en

hancement of the along-gap wind speed.The second gap-flow regime, which will be called the

mountain wave regime, is illustrated by the e 5 1.4case; it is characterized by a monotone increase in massflux through each of the control volumes leading to asignificant enhancement of the gap wind. Despite thereduction in the y–z cross-sectional area across the en-trance volume, exceeds due to both lateralx xf f→c →en

convergence ( , 0) and downward transport ( ,y zf fen en

0). Downward mass fluxes into the central volume pro-duce a further enhancement of relative to . Fi-x xf fc→ →c

nally, very strong downward fluxes in the exit regionoffset modest lateral divergence to accelerate the along-gap flow to the point where exceeds by rough-x xf fex→ →en

ly a factor of 5.A third gap-flow regime, the upstream-blocking re-

gime is apparent in the e 5 5.0 simulation, in whichthe largest increase in the along-gap mass flux occursin the entrance volume due primarily to strong lateralconvergence. Modest downward transport also plays arole in increasing , but there is little subsequentxf→c

change in the along-gap mass flux through central andexit regions. Since ø despite the factor of 1.6x xf fex→ c→

increase in the cross-sectional area between the up-stream and downstream faces of the exit volume, theaverage wind speed decreases within the exit volume.

The behavior of the averaged along-gap wind speeddeduced from the preceding mass budgets is reflectedin the distributions of normalized 50-m wind speedshown at the bottom of Fig. 9. These wind speeds, takenfrom the lowest grid level in the numerical simulation,are normalized by U(1 1 e), which is a characteristicscale for the maximum horizontal wind speed that would


FIG. 10. Control volumes for analysis of the low-level momentumbudget within the gap. Arrows representing advection through thefaces of the entrance volume are also shown.

FIG. 11. Normalized momentum forcing in the control volumesshown in Fig. 10, for simulations with different e. Vertical dashedlines denote the x-coordinate locations of the boundaries of the in-dividual control volumes. For each simulation, the leftmost pair oflines delimits the entrance volume, the center pair the central volume,and the rightmost pair the exit volume. The cross-ridge topographicprofile associated with these control volumes is plotted below eachset of mass-flux data, along with the normalized surface wind speed.Squares, diamonds, circles, and triangles denote the normalized vol-ume integrals of ]ru2/]x, ]ruy/]y, ]ruw/]z, and ]p/]x, respectively.

be obtained in the linear mountain wave solution. Al-though the flow in the e 5 0.25 case is almost linear,the surface winds within the gap itself are weaker thanthe maximum winds in the mountain wave aloft, and[u(x, 0,50)/U](1 1 e)21 remains approximately equal to(1 1 e)21. In particular, there is neither significantblocking near the gap entrance, nor significant accel-eration of the surface winds within the gap. In contrast,upstream blocking reduces the normalized surface windto near zero in the three cases with larger values of e,and there is significant subsequent acceleration of thewinds farther along the gap. In the mountain wave re-gime (e 5 1.4) this acceleration occurs along the entirelength of the gap and is strongest in the exit region. Inthe upstream-blocking regime (e 5 5.0), accelerationoccurs only in the entrance region, and the flow decel-erates as it passes through the exit region. The simu-lation with e 5 2.8 is an intermediate case with char-acteristics of both the mountain wave and upstreamblocking: the flow accelerates rapidly near the gap en-trance, but continued acceleration occurs through thecentral region of the gap to balance a downward massflux .zf c

5. Flow through the gap: Dynamics

The mass budgets calculated in section 4 provide in-formation about the basic kinematics of the flow throughthe gap. Insight into the gap-flow dynamics can be ob-tained by examining the momentum budgets for threecontrol volumes placed along the axis of gap as shownin Fig. 10. These control volumes are similar to thoseused in the mass-budget analysis except the cross-flowdimension of each volume was reduced to the 10-km-wide region | y | # d/2 along which the bottom of thegap is completely flat, and the top of each volume waslowered to z 5 500 m to focus on flow near the surface.

At steady state the x-momentum equation may bewritten in flux form as

2]ru ]ruy ]ruw ]p1 1 1 5 0, (10)

]x ]y ]z ]x

where the divergence of the subgrid-scale fluxes hasbeen neglected because it is negligible in comparisonto the other retained terms. (These are free-slip simu-lations and any wave-breaking regions lie outside thesecontrol volumes). The momentum budgets for a seriesof simulations with different e were computed once theflow in the gap reached an essentially steady state byintegrating each of the terms in (10) over the three sub-volumes shown in Fig. 10. Because of the flux form of(10) these volume integrals reduce to differences in theadvective momentum fluxes through opposing faces, ordifferences in the pressure on the opposing faces of eachcontrol volume. In all cases the momentum budget ob-tained from this procedure closes to within 10% of thelargest individual term.1

Each term in the volume integral of (10) is plottedin Fig. 11 for a series of simulations with e 5 0.25, 1.4,2.8, and 5.0. The dimensional integrals from each sim-ulation are normalized to fall in the interval [21, 1] bydividing them by the magnitude of the largest individualintegral from that simulation. These normalized resultsare plotted at the x coordinate of the centroid of theirrespective volumes (i.e., the centroids of the entrance,central, or exit regions). The topography and normalizedsurface wind speed u(x, 0,50)/[U(1 1 e)] are also plottedbelow the momentum budget data for each simulation

1 The sole exception occurs in the exit volume of the e 5 1.4simulation, in which the residual is 14.6% of the largest term becauseof transience due to wave breaking. Nevertheless, the interpretationof the budget remains clear cut because in this case the residual isonly about 15% of all the individual terms in the budget.


as a reference. The volume integral of ]ru2/]x is denotedby a square, with positive values indicating net accel-eration along the gap axis. Negative values of the re-maining volume integrals indicate a contribution towardacceleration along the gap axis; they are the integralsof of ]ruy/]y (lateral-momentum divergence), denotedby diamonds; ]ruw/]z (vertical-momentum divergence),denoted by circles; and ]p/]x (pressure gradient), de-noted by triangles.

As noted in the previous analysis of mass fluxes, thereis little amplification of the gap flow in the e 5 0.25simulation. The volume-averaged acceleration in the en-trance region is almost exactly offset by deceleration inthe exit region; although there is also a contributiontoward acceleration in the central region that producesa modest net increase in the along-gap winds. Perhapsthe most interesting aspect of the momentum budget inthe e 5 0.25 case is the relative unimportance of thepressure gradient force, particularly in the entrance andcentral regions.

The momentum budget in the e 5 1.4 simulation, themountain wave regime, shows acceleration of the along-gap winds in all three subvolumes, with the rate of ac-celeration increasing downstream. Vertical and lateralmomentum flux convergence and pressure gradient forc-es all contribute toward the acceleration of the flow inthe entrance and central regions. Lateral momentum fluxdivergence acts to reduce the acceleration in the exitvolume, but it is more than offset by strong pressuregradient forces and strong downward momentum trans-port. The mountain wave regime is the only case inwhich a net acceleration of the gap wind occurs withinthe exit region, and that acceleration is quite intense.

In contrast to the mountain wave regime (e 5 1.4),in which vertical momentum flux convergence plays acrucial role in amplifying the gap winds, vertical mo-mentum transport is essentially zero in the upstream-blocking regime (e 5 5.0). Lateral momentum flux con-vergence and pressure gradient forces accelerate the gapflow in the entrance region and retard the flow in theexit region. The e 5 2.8 simulation once again appearsas a hybrid between the two gap-wind regimes. Themomentum budgets in the entrance and exit volumesare similar to those in the upstream-blocking regime,whereas downward and lateral momentum fluxes to-gether with the pressure gradient force produce signif-icant accelerations within the central volume.

6. Flow through the gap: Trajectories andBernoulli’s equation

It has often been suggested that to a first approxi-mation the dynamics of gap flow may be interpreted asa rough balance between the along-gap pressure gra-dient, along-gap accelerations, and surface friction(Mass et al. 1995, and references therein). The influenceof surface friction will be discussed in Part II; as apreliminary step in the assessment of the preceding gap-

flow paradigm, we consider the free-slip case, which isoften analyzed using Bernoulli’s equation (Reed 1981).Under the assumption that the air parcel trajectories arehorizontal and that the vertical and cross-gap wind com-ponents are negligible, Bernoulli’s equation for incom-pressible flow through a gap parallel to the x axis be-comes

2 2u u p 2 pex en en ex5 1 , (11)2 2 r

where the subscripts ‘‘en’’ and ‘‘ex’’ denote quantitiessampled at the gap-entrance and -exit regions, respec-tively. As noted by Mass et al. (1995), (11) typicallyoverpredicts the actual acceleration experienced by airparcels passing through the gap.

It does not seem to have been recognized that a correctapplication of the compressible Bernoulli equation ac-tually leads to a rather different relation for horizontalflow through a gap. Accounting for the compressibilityof the atmosphere, Bernoulli’s equation implies that

1B 5 c T 1 u u 1 gzp i i2

is conserved following a fluid parcel in a steady inviscidflow. Here cpT 5 cyT 1 p/r, and cp and cy are the specificheats of air at constant pressure and constant volume.As in the approximations leading to (11), suppose theflow is level and that all velocity perturbations are dom-inated by the component of the flow along the gap, thenconservation of B implies

2 2u uex en5 1 c (T 2 T ). (12)p en ex2 2

It follows that for purely horizontal flows, the gap windsin the exit region can only exceed those in the entranceregion when the temperature at the exit is colder thanthe temperature upstream. Since O(cpT) k O(u2) thedecrease in temperature required to produce a significantincrease in wind speed is only a fraction of a degree;however, observations of flows through essentially levelgaps often show the temperature increases between theentrance and exit regions.

The key element for the correct application of Ber-noulli’s equation to the strongest gap-wind simulationsin this paper is the inclusion of vertical motion. Thestrongest gap winds occur in the e 5 1.4 simulations,in which the 10 m s21 upstream flow accelerates to 22m s21. In comparison the winds in the upstream-block-ing regime (e 5 5.0) increase from 2.8 m s21 upstreamto 7 m s21 within the gap. Consistent with the factorsthat produce different flow regimes over a specificmountain range in the real world, the 1.4-km height ofthe ridge was held fixed between the different simula-tions, and the values of e characteristic of the flow-blocking regime were associated with relatively weakupstream winds. Thus, although the gap-induced nor-malized velocity perturbations u[U(1 1 e)]21 have sim-


FIG. 12. Variation of key parameters along a trajectory above the centerline of the gap thatexits at the height z 5 300 m: (a) air parcel elevation (thin black) and ridge profile (thick gray);(b) temperature (thin black) and temperature under the assumption that u is constant along thetrajectory (thick gray); (c) actual along-gap wind component (thin black) and along-gap windcomponent under the assumptions that B and u are conserved along the trajectory and that y andw are negligible (thick gray); (d) B* 5 B 2 cpT0, the Bernoulli function minus the enthalpy ofthe parcel at the upstream end of the trajectory (thin black) and the constant reference lineB*0(thick gray); and (e) potential temperature.

ilar magnitudes in all the simulations, the potential forthe local amplification processes to generate damagingwinds is greatest in the mountain wave regime.

Figure 12 shows the variation of several key param-eters along a trajectory above the centerline of the gapin the e 5 1.4 simulation. The trajectory lies along theline y 5 0; the x coordinate of the parcel is labeledalong the bottom axis. Figure 12a shows the elevationof the parcel, along with the cross section of the ridge.The parcel, which exits the gap (at x 5 40 km) at aheight of 300 m, originates from an elevation of roughly870 m in the upstream flow, and ascends to 1239 m inthe gap-entrance region. The adiabatic heating and cool-ing associated with these vertical motions is mirroredin the parcel temperature (Fig. 12b). Note that the tem-perature of the parcel increases by roughly 5 K as itmoves along the trajectory. Despite the increase in en-thalpy, the parcel accelerates from 6.4 m s21 over thegap entrance to 21 m s21 in the exit region because ofthe conversion of potential to kinetic energy during its939-m descent (Fig. 12c). Because of this descent, thereis an increase in atmospheric pressure along the trajec-tory between the gap-entrance and -exit regions. Thusif (11) were applied along the actual trajectory, insteadof at constant height, it would erroneously predict thatair parcels decelerate as they pass through the gap.

To what extent can the acceleration of the gap windbe predicted from Bernoulli’s equation? As indicated inFigs. 12d and 12e, neither the Bernoulli function B northe potential temperature are exactly conserved alongthe air parcel trajectory. The small increase in the po-tential temperature along the trajectory is due to theaction of the fourth-order smoother on steep gradientsin the numerically simulated potential temperature field.If the potential temperature data are adjusted to removethe effects of the fourth-order smoother, the simulatedwind speed may be shown to be in very good agreementwith that which would be predicted based on conser-vation of the Bernoulli function. To demonstrate suchagreement, B and T were recomputed along the trajec-tory as if u were exactly conserved, and the results wereplotted as the thick gray curves in Figs. 12d and 12b.With u held constant, B is constant along the trajectory(thick gray curve in Fig. 12d), yet there is essentiallyno change in the along-trajectory fluctuation in T shownin Fig. 12b. Finally, assuming a constant value for B,the variation in u along the trajectory was evaluatedfrom the along-trajectory values of T and z and plottedas the thick gray curve in Fig. 12c. Comparing the actualvariations in u along the trajectory (thin black curve)with those deduced from conservation of the Bernoullifunction assuming adiabatic flow (thick gray curve), it


FIG. 13. Normalized perturbation fields on the u 5 276 K surface (a) height z, (b) pressure , (c) Bernoulli function , and (d) cross-p Bmountain wind speed u. Light (dark) shading denotes positive (negative) perturbations; the zero contour is the heavy black and white dashedline. Contour intervals are (a) 0.2, (b) 0.01, (c) 0.01, and (d) 0.5. The thick black line denotes the area where the constant potential temperaturesurface intersects the ground.

is apparent the two are almost identical. Thus for thistrajectory, which lies outside the wave-breaking region,the analysis of air parcel accelerations via Bernoulli’sequation yields quantitatively correct results after com-pensating for the small impact of fourth-order smooth-ing on the potential temperature.

This same type of Bernoulli function analysis showsthat descent continues to play a crucial role in producingacceleration along those trajectories that exit the gapvery near the surface. In particular, an air parcel orig-inating at z 5 283 m upstream subsequently decelerateswhile rising to a height of 724 m in the gap-entranceregion before finally accelerating as it descends to exitthe gap at z 5 50 m, the lowest level at which ther-modynamic and horizontal wind data are carried in thenumerical model.

7. Bernoulli function distribution in jets andwakes

Pan and Smith (1999) performed shallow-water sim-ulations in which a uniform upstream flow encountereda long ridge pierced by a series of gaps. A gap-windjet flanked by regions of decelerated flow was foundover the level terrain in the lee of each gap. Since theBernoulli function for the shallow-water flow over flatbottom topography is simply

12B 5 rU 1 rgH,sw 2

where U is the horizontal flow speed and H the fluiddepth, Pan and Smith (1999) noted that the variationsin Bsw depend solely on the variations in the horizontalspeed of the fluid and the fluid depth. Furthermore, sincethe fluid depth tends to equilibrate downstream of theobstacle, they found that the variations in U 2 were di-rectly proportional to the values of Bsw within the wakeregion. The gap-wind jet appeared in a region of rela-tively high Bsw; the slower winds within the wake oneach side of the jet were associated with lower valuesof Bsw. This spatial gradient in Bsw was generated bydissipation and Bernoulli loss in jumps in the lee of theridges on each side of the gap, whereas there was nodissipation or Bernoulli loss within the gap flow itself.

In the mountain wave regime (e 5 1.4), which pro-duces the strongest gap winds, there is no tendency forthe gap-wind jet to coincide with a region of high Ber-noulli function if both fields are compared on a constantlevel surface. If, however, these fields are plotted on anisentropic surface (which is a better analog to the freesurface in a shallow-water model because for inviscidflow the isentropic surface is a material surface), theregion of strong gap flow does indeed coincide with aregion of high Bernoulli function, as shown in Figs. 13cand 13d. The normalized perturbation vertical displace-ment z 5 (z 2 z0)/z0 of the 276-K isentropic surface isplotted in Fig. 13a, where z0 5 1.14 km is the undis-turbed height of that isentropic surface in the upstreamflow. The 276-K u surface ascends so that z increases


to roughly 1.4 throughout a broad region immediatelyupstream of the gap; the surface continues to ascend asit passes over the crest of the ridge, and then plungesto the ground along the lee slope on each side of thegap. Within the gap-exit jet itself, the 276-K u surfacenever intersects the ground, but z falls below 0.4. Notsurprisingly, this descent is associated with an increasein the atmospheric pressure along the isentropic surface,which is shown in Fig. 13b by the field of normalizedperturbation pressure 5 (p 2 p0)/p0 (where p0 ispthe value of the Exner-function pressure on the undis-turbed 276-K surface).

As in the shallow-water case examined by Pan andSmith, the Bernoulli function in the region of the gapflow is essentially unmodified from its upstream value,whereas the lower values of the Bernoulli functionthroughout the remainder of the wake appear to havebeen produced by dissipation in the wave-breaking re-gions over and leeward of the higher topography. Thisis apparent in Fig. 13c, which shows contours of thenormalized perturbation Bernoulli function B 5 (B 2B0)/B0 (where B0 is the Bernoulli function on the un-disturbed 276-K isentropic surface). In fact, B remainsin the range [20.005, 0.005] along an air parcel trajec-tory through the gap, but B drops below 20.03 down-stream of the wave-breaking regions.

The normalized perturbation x component of the ve-locity u 5 (u 2 u0)/u0 (where u0 is the value of u onthe undisturbed 276-K u surface) is plotted in Fig. 13d.Clearly high values of u are found in the regions ofhighest B. Within the gap flow at x 5 45 km plotted inFig. 13c, u exceeds 1.0, implying the wind speeds aremore than double their value in the undisturbed flow.Farther downstream the winds decrease, but u still re-mains positive, indicating air parcels have undergonenet acceleration. These high winds are associated witha cross-wake maximum in B, but in contrast to the sin-gle-layer shallow-water results, the lee-side pressure onthe isentropic surface is higher than the pressure up-stream (see Fig. 13b). Indeed even within the wake, thepressure and the enthalpy are slightly higher along thejet axis [since the data are plotted on an isentropic sur-face, the perturbation enthalpy cp(T 2 T0) is justcpu0p0 , so its spatial distribution is identical to thatpfor ]. Rather than a pressure minimum, the gap-windpjet downstream of x 5 100 km is associated with a cross-wake minimum in z.

The results shown in Fig. 13 are consistent with thoseexamined in more quantitative detail in Fig. 12 by fol-lowing an air parcel trajectory that originated at the u5 275.4 K level along the centerline of the gap. Bothu and B are almost conserved along trajectories passingthrough the center of the gap-wind jet; the Bernoullifunction is able to remain constant despite increases inboth enthalpy and wind speed, because the air parcelsundergo significant descent.

8. Conclusions

This paper has examined gap winds generated by anairstream with uniform static stability and horizontalwind speed impinging on an isolated ridge cut by arelatively narrow gap. Both the barrier-scale responseand the flow within the gap itself have been investigatedas a function of the normalized mountain height e 5Nh0/U.

On the scale of the entire barrier, the previously welldocumented transition from a flow-over regime for e ,O(1) to a flow-around regime for e . O(1) was clearlyevident in our numerical simulations. The fraction ofthe oncoming flow passing through the gap did not,however, reveal any such regime change. In comparisonto the dramatic e-dependent changes in the percentageof flow going over or around the ridge, the percentageof the total flow passing through the gap was found tobe relatively independent of e. This result, which wasobtained for gaps that transmitted up to 40% of theoncoming low-level flow, must not be applicable to gapsthat are so wide that the flow through the gap becomesindistinguishable from the flow around the interior endof each adjacent ridge. Unfortunately, limitations on ourcomputational resources prevented us from determiningthe width at which the change to a flow-around regimebegins to increase the proportion of air channeledthrough the gap.

The kinematics of the flow within the gap itself wereinvestigated by examining mass budgets through threecontrol volumes located in the entrance, central, andexit regions of the gap. Three basic gap-flow regimeswere encountered for different ranges of e. In the linearregime, which included the e 5 0.25 case, there wasalmost no enhancement of the gap flow. The e 5 1.4case was representative of the mountain wave regime,in which there is a monotonic increase in the along-gapmass fluxes through all three control volumes and aparticularly strong increase in the mass flux and windspeed within the exit volume due to downward transportby the mountain wave above the lee slopes of the to-pography. The upstream-blocking regime is well illus-trated by the e 5 5.0 case, in which the largest increasein the along-gap mass flux occurs in the entrance volumedue to lateral convergence. In contrast to the mountainwave regime, in which the highest gap winds appeardownstream near the exit, the highest winds occur inthe upstream portion of the gap in the upstream-blockingregime.

The dynamical processes associated with each flowregime were determined by examining the momentumbudgets for control volumes in the entrance, central, andexit regions. The momentum budgets confirm the crucialimportance of vertical transport in creating the highwinds in the gap-exit region in the mountain wave re-gime. On the other hand, vertical momentum transportplays almost no role in generating the high winds in theupstream-blocking regime, which are produced by lat-


eral momentum flux convergence and pressure gradientforces in the entrance volume.

In a previous study of airflow with constant N and Uperpendicular to a ridge with a gap, Zangl (2002) iden-tified two flow regimes, linear and nonlinear, and sug-gested that in the nonlinear regime confluence on theupstream side of the mountain is negligible and ‘‘thelow-level pressure difference across the mountain ridgeprimarily drives the gap flow.’’ Zangl’s topography roseto a single peak, rather than the flat-top ridge used inthis study. As a consequence, the dynamical processesactive at the gap-entrance and -exit regions are not easilyseparated in his results. His nonlinear regime is com-parable to the mountain wave regime identified in thispaper, although our Eulerian momentum budgets forflow in the mountain wave regime show that the pressuregradient force is not the single dominant factor accel-erating the gap winds. We also found upstream conflu-ence to be important for gap-wind acceleration in theupstream-blocking regime, but the prototypical exam-ples of this type of flow occur at larger values of e thanthe maximum of 3.0 considered by Zangl, so our find-ings are not inconsistent with his results.

In our simulations, the variations in e were obtainedby holding h0 and N fixed while varying U. To the extentthat a constant N and U profile may be offered as amodel for the real atmosphere, the strategy of varyinge by changing U mimics the situation involving airflowacross a given mountain barrier in the real world. Themountain height is obviously fixed, and the troposphericstatic stability averaged through a deep layer is nevergreatly different from 0.01 s21. The primary variationsin the deep-layer-averaged e are due to changes in thecross-mountain wind speed.

Since the gap winds that develop in the upstream-blocking regime involve accelerations with respect to aweak mean flow, the upstream-blocking regime is notlikely to be associated with damaging winds in real-world applications (2.8 m s21 winds upstream accelerateto 7 m s21 in the gap in the e 5 5.0 case). The mountainwave regime appears far more likely to serve as a modelfor those severe gap winds that develop in response tostrong cross-mountain flow in real-world events. Indeed,previous work by Colman and Dierking (1992) andColle and Mass (1998b) has suggested that mountainwaves play a major role in enhancing gap winds alongthe Taku River in Alaska and through the Stampede Gapin Washington State.

The dynamics of potentially severe gap winds pro-duced in the mountain wave regime were further in-vestigated by evaluating the Bernoulli function alongair parcel trajectories for the e 5 1.4 simulation. Com-plementing the control-volume budgets for mass andmomentum, the trajectory analysis confirmed the linkbetween descent and acceleration in the gap flow, evenfor parcels exiting the gap at heights as low as 50 m.

Previous attempts to diagnose the strength of gapwinds using Bernoulli’s equation for an incompressible

fluid under the assumption of level flow (11) have typ-ically led to serious overprediction of the wind speed.However, (11) is defective because it neglects the in-fluence of changes in atmospheric density. The correctexpression for conservation of the Bernoulli functionfor level flow in a compressible atmosphere is (12),which implies that constant-level gap winds will un-dergo acceleration only if the temperature at the gapexit is lower than the temperature at the gap entrance.Observations often show higher temperatures at the exitthan at the entrance during gap-flow events, and this isalso the case in the e 5 1.4 simulation. Gap-wind ac-celeration is sustained despite the increase in temper-ature between the entrance and the exit because the airparcels undergo net descent during their passage throughthe gap.

Comparison of the Bernoulli function distribution onan isentropic surface in the high-wind e 5 1.4 caseshows similarities to the shallow-water results obtainedby Pan and Smith (1999) in that the regions of highestBernoulli function are coincident with the highest gapwinds, and the B field is essentially unmodified withinthe gap-exit jet but is reduced on each side of the jetby dissipation in the wave-breaking regions in the leeof the higher topography. The flow undergoes substan-tial subsidence as it accelerates near the gap exit to morethan double the speed of the upstream winds. It is thissubsidence, rather than the change in pressure (whichincreases along the isentropic trajectory) that gives themost direct analog to the downward free-surface dis-placements in the Bernoulli equation analysis of shal-low-water gap flow.

Surface friction is a potentially important factor lim-iting the maximum velocities in actual gap-wind events.The influence of surface friction together with the im-pact of pressure-driven channeling associated with geo-strophic flows crossing the ridge at various angles willbe investigated in Part II of this paper. In Part II, it willbe demonstrated that these additional factors do notchange the central conclusion of this paper, that at leastin the mountain wave regime, vertical fluxes of mo-mentum and mass play a crucial role in the formationof strong gap winds.

Acknowledgments. This research was supported byNational Science Foundation Grants ATM-9817728 andATM-0137335.

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