Gravity Wave Refraction by Three-Dimensionally Varying ...obuhler/Oliver_Buhler/... · changes due to refraction are indeed possible, and they fall outside the columnar pseudomomentum

$Page 1: Gravity Wave Refraction by Three-Dimensionally Varying ...obuhler/Oliver_Buhler/... · changes due to refraction are indeed possible, and they fall outside the columnar pseudomomentum$
Gravity Wave Refraction by Three-Dimensionally Varying Winds and theGlobal Transport of Angular Momentum

ALEXANDER HASHA AND OLIVER BÜHLER

Courant Institute of Mathematical Sciences, New York, New York

JOHN SCINOCCA

Canadian Centre for Climate Modelling and Analysis, University of Victoria, Victoria, British Columbia, Canada

(Manuscript received 11 July 2007, in final form 8 January 2008)

ABSTRACT

Operational gravity wave parameterization schemes in GCMs are columnar; that is, they are based on aray-tracing model for gravity wave propagation that neglects horizontal propagation as well as refraction byhorizontally inhomogeneous basic flows. Despite the enormous conceptual and numerical simplificationsthat these approximations provide, it has never been clearly established whether horizontal propagation andrefraction are indeed negligible for atmospheric climate dynamics. In this study, a three-dimensional ray-tracing scheme for internal gravity waves that allows wave refraction and horizontal propagation in spheri-cal geometry is formulated. Various issues to do with three-dimensional wave dynamics and wave–meaninteractions are discussed, and then the scheme is applied to offline computations using GCM data andlaunch spectra provided by an operational columnar gravity wave parameterization scheme for topographicwaves. This allows for side-by-side testing and evaluation of momentum fluxes in the new scheme againstthose of the parameterization scheme. In particular, the wave-induced vertical flux of angular momentumis computed and compared with the predictions of the columnar parameterization scheme. Consistent witha scaling argument, significant changes in the angular momentum flux due to three-dimensional refractionand horizontal propagation are confined to waves near the inertial frequency.

1. Introduction

In columnar ray-tracing schemes both the horizontallocation as well as the horizontal wavenumber vectorare constant along a ray. In addition, for a steady non-dissipative wave train the net vertical flux of horizontalpseudomomentum is constant along a ray tube. Be-cause the vertical flux of horizontal pseudomomentumequals the wave-induced vertical flux of mean horizon-tal momentum (e.g., Andrews and McIntyre 1978, sec-tion 5.2), this leads to the standard paradigm of colum-nar wave–mean interaction, which is the “pseudomo-mentum rule.” According to this rule persistent,cumulative forcing of the mean flow occurs only where

waves dissipate and then the mean force is equal to therate of dissipation of the horizontal pseudomomentum.

Columnar parameterization schemes for gravitywaves (or other processes) offer enormous conceptualand numerical simplifications, the latter being of par-ticular importance for the current generation of highlyparallelized GCMs. Nevertheless, it has never beenclearly established whether horizontal propagation andrefraction are, indeed, negligible for atmospheric cli-mate dynamics. This is an interesting question not onlyfor parameterization efforts but also for understandingthe dynamics of explicit gravity waves that are increas-ingly produced and observed in high-resolution GCMs(e.g., Plougonven and Snyder 2005). Such explicitly re-solved waves may also affect data assimilation proce-dures.

The present study makes a step toward estimatinghow three-dimensional refraction of internal gravitywaves affects the global vertical transport of angularmomentum. To this end, we built an offline ray-tracing

Corresponding author address: Alexander Hasha, Courant In-stitute of Mathematical Sciences, Room 1108, 251 Mercer Street,New York, NY 10012.E-mail: [email protected]

2892 J 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 VOLUME 65

DOI: 10.1175/2007JAS2561.1

© 2008 American Meteorological Society

JAS2561

model in spherical geometry and applied it to outputfrom the Third Generation Atmospheric General Cir-culation Model (AGCM3) of the Canadian Centre forClimate Modeling and Analysis.

The AGCM3 includes a columnar gravity wave pa-rameterization scheme for topographic waves and mul-tiple runs of the ray tracer were computed using thesame topographic waves at the lower boundary as theparameterization scheme. In addition, further runs withartificially lowered intrinsic frequencies were computedso as to assess the sensitivity of the angular momentumtransport to the details of the wave spectrum. Overall,the scientific aims of this study were twofold: to clarifythe fundamental issues to do with three-dimensionalrefraction and wave–mean interactions in atmosphericmodels and to produce a clear, quantitative comparisonbetween a three-dimensional ray-tracing model and acolumnar parameterization scheme in the context ofthe angular momentum budget of the atmosphere.

We briefly summarize the scientific background ofthree-dimensional ray tracing, which was already de-scribed analytically in Jones (1969). Subsequent nu-merical studies noted that inclusion of three-dimen-sional effects in the middle atmosphere could lead tosignificant changes in the wave dynamics at least forsome wave rays, especially those with reduced verticalgroup velocities (e.g., Dunkerton and Butchart 1984;Dunkerton 1984). Meridional propagation alone leadsto well-known changes in the wave structure due to thelatitude dependence of the Coriolis parameter, andsuch changes are important for the interpretation of theobserved latitudinal structure of wave energy spectra(Bühler 2003). However, such latitude-dependent ef-fects would not change the vertical flux of zonal angularmomentum associated with a wave train. On the otherhand, it is well known that horizontal refraction by zon-ally asymmetric mean fields would alter this flux (thestandard argument is presented in section 2).

What is less well known is how such horizontal re-fraction affects the wave–mean interactions. Recently,this problem was considered theoretically by Bühlerand McIntyre (2003, 2005, hereafter BM05), who dem-onstrated that the refractive changes in the wave-induced momentum fluxes go hand in hand with per-sistent changes in the horizontally inhomogeneousmean flow that caused the refraction. This extends thepioneering earlier work of Bretherton (1969). In par-ticular, for simple idealized scenarios a new conserva-tion law for the sum of horizontal pseudomomentumand the impulse of the layerwise distribution of poten-tial vorticity was derived in BM05. For instance, thisconservation law allows understanding how total hori-zontal momentum (or angular momentum in the

spherical case) is conserved when three-dimensional re-fraction is present. Persistent, cumulative mean-flowchanges due to refraction are indeed possible, and theyfall outside the columnar pseudomomentum rule be-cause they do not rely on wave dissipation in any ob-vious manner. These considerations motivated thepresent study.

The plan for the paper is as follows: The basic theoryfor three-dimensional wave dynamics is laid out in sec-tion 2, and the numerical ray-tracing model is describedin section 3. The model was carefully tested for consis-tency with the spherical coordinates and in columnarmode it was calibrated against the topographic gravitywave scheme used in AGCM3. This close comparisongave us confidence that the ray tracing scheme pro-duces realistic fluxes of angular momentum. The nu-merical experiments and results are discussed in sec-tions 4 and 5.

2. Three-dimensional gravity wave dynamics

For simplicity, we first work in Cartesian geometryand later describe the adaptation to spherical geometryin section 2c.

a. Ray tracing and refraction

The ray-tracing equations describe the linear evolu-tion of a slowly varying wave train containing small-scale waves that propagate relative to a large-scale ba-sic flow (e.g., Whitham 1974; Lighthill 1978). This is thenatural setting for unresolved, subgrid-scale internalwaves. Thus, all wave fields are taken to be propor-tional to the real part of a(x, t) exp[i!(x, t)], where a K1 is the slowly varying wave amplitude and ! is therapidly varying wave phase whose derivatives; that is,the wavenumber vector k " (k, l, m) " !! and theabsolute frequency # " $!t are again slowly varying.A suitable dispersion relation # " %(k, x, t) then linksk and # and this serves as a first-order PDE for !whose characteristics are the ray-tracing equations

dxdt

" &!k!'k, x, t(;dkdt

" $!x!'k, x, t(, '1(

which form a Hamiltonian system of ODEs for the ca-nonical variables (x, k) based on the Hamiltonian func-tion %. Physically, d/dt " )/)t & cg • ! is the time de-rivative following a ray and cg " !k% is the absolutegroup velocity. As is well known, the spatial and tem-poral symmetries of the basic flow induce conservationlaws along the rays; that is, if %(k, x, t) is independentof one of (x, y, z, t), then the corresponding member of(k, l, m, #) is constant along a ray. In a columnar pa-

SEPTEMBER 2008 H A S H A E T A L . 2893

rameterization scheme the ray is constrained not totravel horizontally, and therefore for consistency thehorizontal gradient of background fields must be ig-nored. In this context !(k, x, t) is formally independentof x and y and, therefore, (k, l) are both constant alonga ray.

In the absence of forcing and dissipation the wavetrain amplitude along ray tubes formed by noninter-secting rays is governed by the conservation law for theO(a2) wave action density A; that is,

!A!t

" ! " #Acg$ %dAdt

" A! " cg % 0. #2$

Here A % E/&̂, where E is the phase-averaged distur-bance energy density per unit volume (Bretherton andGarrett 1968) and &̂ is the intrinsic frequency. The waveaction is conserved by virtue of the assumed scale sepa-ration between the small-scale waves and the large-scale basic flow.

Of particular interest is the pseudomomentum vectorp % kA, which arises naturally in wave–mean interac-tion theory. For instance, p appears naturally in aver-aged expressions for the potential vorticity (Bühler andMcIntyre 1998, 2005) and, as noted above, the verticalflux of horizontal pseudomomentum pH % (k, l)A, say,equals the wave-induced vertical flux of mean horizon-tal momentum (e.g., Andrews and McIntyre 1978, sec-tion 5.2). In columnar parameterization schemes thehorizontal components of pseudomomentum are con-served by construction because dk/dt % dl/dt % 0 insuch schemes and, therefore, horizontal pseudomomen-tum conservation is inherited from wave action conser-vation. This leads to the pseudomomentum rule dis-cussed in section 1.

On the other hand, by combining (1) and (2) it isclear in principle that a component of p is conservedonly if the corresponding component of k is invariantalong a ray, that is, only if the basic flow has a symmetryin the corresponding spatial direction. In particular,zonal pseudomomentum p % kA is conserved only ifthe basic flow is zonally symmetric. This elementaryfact implies that the wave-induced vertical flux of meanzonal momentum along a ray tube is not constant inprinciple, even if the wave field is steady and nondissi-pating. This does not contradict the usual nonaccelera-tion conditions based on the Eliassen–Palm flux theo-rems because these conditions are always derived underthe assumption that the basic flow is zonally symmetric(e.g., Andrews et al. 1987). Indeed, there is no contra-diction with momentum conservation either becausemore detailed studies of the wave–mean interactionsthat accompany such nondissipative changes in pseudo-

momentum illustrate that the changes in the pseudo-momentum flux are precisely compensated for bywave-induced changes in the basic flow (Bühler andMcIntyre 2003; BM05).

A specific example of how pseudomomentum can becreated or destroyed by a three-dimensional basic flowis wave refraction by a shear flow. With a basic velocityfield U(x, t) % (U, V, W), the dispersion function ! hasthe generic Doppler-shifting form

# % $̂ " U " k, #3$

where &̂ is the intrinsic frequency, and the group ve-locity cg % !k&̂ " U now consists of advection by thebasic flow plus intrinsic propagation relative to it. Forsimplicity, we restrict &̂ to be a function of k only, butthe general case will be treated in section 3. In this case,the symmetries of ! are those of U, and we have theexplicit refraction equations

dkdt

% '#!U$ " k !dki

dt% '

!U!xi

k '!V!xi

l '!W!xi

m.

#4$

Clearly, any dependence of U on the zonal coordinatex (say, due to synoptic-scale vortices, meandering jetstreams, or stratospheric Rossby waves) induces con-comitant changes in k and therefore in the vertical fluxof zonal momentum.

Actually, slightly more can be predicted from (4) be-cause refraction affects k (and also p) in a manner par-tially analogous to the evolution of the gradient of apassive tracer advected by U. Specifically, for a passivetracer ( we have

D%

Dt% 0 "

D#!%$

Dt% '#!U$ " #!%$, #5$

where D/Dt is the material derivative based on U. Thus,the evolution of k and !( differ only due to advectionby the intrinsic group velocity, which is the differencebetween advection along group-velocity rays in (4) andalong material trajectories of the basic flow in (5). Thispartial analogy can be exploited in a number of ways(e.g., Badulin and Shrira 1993; BM05). In particular, byaligning k with a right eigenvector of !U it is possibleto sustain unbounded exponential growth in time of allthree components of k (Jones 1969). This leads to apeculiar three-dimensional version of a critical layer inwhich the local amplitude also grows exponentiallywhile the intrinsic group velocity, which for gravitywaves is inversely proportional to |k | at fixed &̂, decaysexponentially to zero. In this scenario the wave patternis eventually passively advected by the basic flow; the


transition to this passive advection regime and the con-comitant wave–mean interactions have been termed“wave capture” in BM05. Such dynamics is of courseabsent in columnar parameterization schemes, al-though some evidence for the occurrence of this kind ofeffect is available from direct numerical simulation ofgravity waves with high-resolution GCMs, as reportedin Plougonven and Snyder (2005).

Two simple scaling arguments demarcate the intrin-sic frequency range for which three-dimensional refrac-tion can be expected to be important. This will be thecase if three-dimensional refraction can induce signifi-cant relative changes in the horizontal wavenumberskH ! (k, l), and corresponding changes in the wave-induced vertical fluxes of horizontal momentum alongthe ray tube, before the ray encounters a critical layeror is otherwise dissipated. We assume that the basicflow is described by a layerwise horizontal velocity fieldsuch that |U | ! | (U, V, 0) | " U and L and H arehorizontal and vertical length scales such that |!HU | "U /L and |Uz | " U /H. Then, for a ray propagatingthrough a depth of atmosphere #r during a residencetime #t, we have

!kH

|kH | "UL

!t "UL ! !r

dr"dt". $6%

Using &̂ defined by the dispersion relation for gravitywaves, given in Eq. (10) below, and f defined as themagnitude of the inertial frequency, the vertical groupvelocity scales as

drdt

" $#̂ ' | f |%3"2

so that as &̂ ! f, dr/dt ! 0. This implies a longer resi-dence time #t at fixed #r for low frequency waves. Onthe other hand, the relative importance of changes inhorizontal and vertical wavenumber can be estimatedfrom (1b) as

|kH |'1 |dkH "dt ||m |'1 |dm"dt |

"|m ||kH |

U "LU "H

"|m ||kH |

HL

. $7%

Under quasigeostrophic scaling the typical aspect ratioof the basic flow is the Prandtl ratio H/L ! | f | /N, andhence horizontal refraction is comparable to verticalrefraction when |m | / |kH | $ | f | /N. Again, this scalingsupports the importance of three-dimensional effectsfor low frequency waves. Making use again of theanelastic dispersion relationship for internal gravitywaves, the ratio (7) is unity if &̂ ! (2 | f | . The ratiotends to zero for &̂ k | f | , suggesting that vertical re-fraction dominates horizontal refraction in the rest ofthe frequency spectrum. Both of these scaling argu-

ments suggest that three-dimensional refraction is mostimportant for the near-inertial part of the atmosphericwave spectrum. This dependency of horizontal refrac-tion and propagation on intrinsic frequency has beennoted in numerous ray tracing studies, such as Dunker-ton (1984), Eckermann (1992), and Hertzog et al.(2001). The robust nature of the dependence is furthersupported by the experimental results presented in sec-tion 4.

Additionally, allowing horizontal propagation canhave a large impact whenever the absolute horizontalgroup velocity of a wave is large. In such cases, the pathof a three-dimensionally propagating ray quickly di-verges from that of its columnar analog, and it movesthrough a different environment. Even if horizontal re-fraction does not play a large role, the final destinationof the wave’s pseudomomentum flux could be signifi-cantly changed by three-dimensional effects. Topo-graphic waves in two dimensions, for which columnarpropagation schemes were originally developed, haveabsolute horizontal group velocities near zero, so thiseffect should be of little importance for them. But itcould play a significant role for many waves in thenonorographic spectrum and for certain three-dimensional topographic waves (e.g., Smith 1980;Shutts 1998).

b. Local amplitude prediction

The local wave amplitude is important because wavedissipation due to nonlinear wave breaking must bemodeled in parameterization schemes and incipientwave breaking is sensitive to local wave amplitudes. Ina columnar scheme (2) is applied to a steady and hori-zontally homogeneous wave train, in which case it re-duces to Awg ! A0wg0 in which wg is the vertical groupvelocity and the index denotes the known conditions atthe lower boundary. From this equation A, and, there-fore the local wave amplitude a, can be inferred usingthe usual polarization relations. One-dimensional caus-tics correspond to wg ! 0 and these occur at reflectionand critical layers where ray-tracing predicts infiniteamplitudes. Now, in a three-dimensional setting, it issimplest to retain the assumption of a steady wave trainand then (2) reduces to ! • (Acg) ! 0. In principle, thisequation can be solved along a bundle of rays forminga ray tube with constant net wave-action flux along thecross section of the tube.

The kinematic ray theory of Hayes (1970) achievesthis by adding differential equations for the spatial gra-dients of k to the ray tracing set and using these quan-tities to compute ! • cg along a single ray. The localamplitudes along the ray may then be calculated, pro-


vided the gradient of k is specified at the lower bound-ary. This requirement can pose a difficulty in manypractical applications of ray tracing where the boundaryconditions may not be specified with sufficient resolu-tion to constrain !k.

Furthermore, even if !k is well known at the bound-ary, the use of the Hayes ray equations is complicatedby the ubiquitous appearance of caustics. At causticsneighboring rays intersect, the ray tube area vanisheslocally, ray theory breaks down, and the predicted am-plitude is infinite. Though it is possible to formulate theray-tracing equations in a way that prevents singulari-ties at caustics (e.g., Broutman et al. 2004, 2001), thereis no simple way of computing the correct amplitudenear a caustic in a general three-dimensional basicstate. As is well known from more elaborate ray theory,the correct wave amplitude at such a caustic is not in-finite and can be computed, in principle, by resolvingthe singularity of ray theory with a more detailed localtheory. In idealized geometries, the solution involvesAiry functions across the caustic (Lighthill 1978). Aseries of studies (Broutman 1984, 1986; Eckermann andMarks 1996; Sonmor and Klaassen 2000; Walterscheid2000) has used the Hayes equations to investigate theformation of caustics in idealized background flows. Allnoted that the location and severity of amplitude am-plification at caustics is extremely sensitive to theboundary conditions on !k and the details of the back-ground flow. The Hayes amplitude equations are there-fore inappropriate for applications in which these de-tails are poorly constrained, as they are in the presentcase. Therefore, after some experimentation withHayes’s extended equations, we restricted our study tothe crude approximation in which the local wave actiondensity is computed by assuming that the cross-sectional area of the ray tubes is constant (e.g., Marksand Eckermann 1995).

c. Spherical geometry

The ray-tracing equations (1)–(2) apply in sphericalgeometry provided that due care is taken when adapt-ing the relevant vector expressions to the nonconstantbasis vectors of the spherical coordinate system. Ourresulting ray-tracing equations differ from the equa-tions used in the Gravity Wave Regional or Global RayTracer (GROGRAT) model of Marks and Eckermann(1995) by several metric terms that become relevant ifrays travel substantial horizontal distances. Our deriva-tion is detailed in the model description in section 3 andthe appendix.

As for the wave–mean interaction problem in spheri-cal geometry, the computation of the detailed structure

of a wave field and the corresponding local response ofthe mean flow is a subtle and difficult problem. Wehope to report some interesting progress in this area inthe near future, but in the present study we concernourselves only with the computation of the aggregatevertical transport of angular momentum around theearth’s rotation axis by an ensemble of ray tubes. Thenet angular momentum transport by an individual raytube can be equated to the cross-sectional integral ofthe vertical flux of angular pseudomomentum, whichhas density kAr cos ! (Jones 1969). Here k is the localzonal wavenumber, r is the distance to the center of theearth, and ! the latitude such that r cos ! is the distancefrom the rotation axis. The net vertical flux is

P " !! kr cos!Awg dS # Fkr cos!, $8%

where the integral is over the horizontal cross section ofthe ray tube with area element dS. The second expres-sion involves the cross-sectionally integrated wave ac-tion flux

F " !! Awg dS $9%

and the useful approximate equality holds if k is essen-tially constant over the horizontal extent of the raytube. For a steady nondissipating wave train F is con-stant along a ray tube and, therefore, P & kr cos! in thiscase. Again, it follows from (1) that for a zonally sym-metric basic state the net flux of angular pseudomo-mentum P is constant, and this shows that the combi-nation kr cos! must be constant along a ray in a zonallysymmetric atmosphere. This is a good test for a ray-tracing scheme in spherical geometry because (k, r, !)all vary individually in this case. Finally, if the basicflow is not zonally symmetric, then P will not be con-stant along the tube and, therefore, the wave-inducedflux of angular momentum will be variable in altitudeeven without wave dissipation. This is analogous to thenonconstant flux of zonal momentum in the Cartesiangeometry discussed in section 2a.

3. Model description

A global ray-tracing model in spherical coordinateshas been developed in Matlab. It computes the propa-gation, refraction, and dissipation of ray tubes throughsteady, three- dimensionally varying stratification andhorizontal winds U " (U, V, 0) in a compressible deepatmosphere. The nonessential restriction to steady ba-sic flows is common to all parameterization schemes.The model uses the dispersion relationship for inertia–


gravity waves in a rotating stratified compressible at-mosphere, which is

!̂ ! "!N2#k2 " l2$ " f 2#m2 " "2$

k2 " l2 " m2 " "2 . #10$

Here N is the buoyancy frequency, f the local Coriolisparameter, and % ! &0.5'&1('/(z. This general disper-sion relationship, also used in the GROGRAT ray trac-ing model of Marks and Eckermann (1995), is accurateover the entire frequency range | f | ) *̂ ) N. The signconvention *̂ + 0 has been chosen, which implies thatm ) 0 corresponds to gravity waves with upward propa-gating group velocity, k + 0 denotes waves with positivezonal intrinsic group velocity—and therefore with pro-grade zonal momentum flux—and l + 0 denotes north-ward intrinsic group velocity.

a. Spherical ray tracing

As noted before, finding the correct ray-tracingequations in spherical coordinates is a slightly subtlematter. For instance, the wavenumber vector in thesecoordinates is k ! k!̂ " l"̂ " mr̂, where !̂, "̂, and r̂ arethe usual zonal, meridional, and radial unit vectors. Thecomponents of k in spherical coordinates evolve notonly due to changes in k but also due to the spatialvariation of the coordinate frame. Furthermore, as de-scribed in Francis (1972), the curvature of the earthintroduces a gentle refraction of k that becomes signif-icant when rays propagate horizontally on global scales.The appendix demonstrates how to compute the cor-rect ray-tracing equations for an arbitrary dispersionrelation in curvilinear coordinates. The equations re-sulting from (10) are

d#

dt!

1r cos$

"U "k#N2 & !̂2$

!̂%#, #11$

d$

dt!

1r "V "

l#N2 & !̂2$

!̂%#, #12$

drdt

! &m#!̂2 & f 2$

!̂%, #13$

dkdt

! &k

r cos$U# &

lr cos$

V#

&1

2!̂%"#k2 " l2$

r cos$

&

&##N2$ &

#!̂2 & f 2$

r cos$

&

&##"2$#

&kr

drdt

" k tan$d$

dt, #14$

dldt

! &kr

U# &lr

V# &1

2!̂%"#k2 " l2$

r&

&$#N2$

&#!̂2 & f 2$

r&

&$#"2$ &

m2 " "2

r&

&$#f 2$#

&lr

drdt

& k sin$d#

dt, #15$

dmdt

! &kUr & lVr &1

2!̂% "#k2 " l2$&

&r#N2$

& #!̂2 & f 2$&

&r#"2$#" k cos$

d#

dt" l

d$

dt,

#16$

where

% ! k2 " l2 " m2 " "2. #17$

To facilitate comparison with columnar gravity waveparameterization schemes, the ray tracer can also berun in columnar mode in which

d#

dt!

d$

dt!

dkdt

!dldt

! 0,

and solutions are obtained by integrating only Eqs. (13)and (16).

The last two terms in each of (14), (15), (16) wereabsent in the GROGRAT model of Marks and Ecker-mann (1995). These terms are necessary, for instance,to ensure that kr cos, (and therefore the angular mo-mentum flux) is constant along a ray propagating in aquiescent basic flow. As an example, consider a quies-cent, nonrotating, and spherically symmetric atmo-sphere. Ray solutions in this environment must respectthe atmosphere’s spherical symmetry since without ro-tation there is no difference between the zonal and me-ridional directions. This means that the horizontal pro-jection of rays follows great circles, and this is true ifthese terms are included. On the other hand, if theseterms are absent, a ray launched eastward in the mid-latitudes will propagate along a line of constant lati-tude, violating spherical symmetry. Of course, these dif-ferences only matter if rays travel substantial horizontaldistances or if the rays stray close to the poles, as notedin Dunkerton and Butchart (1984).

b. Numerical implementation

To initialize the model, the steady background fields(U, V, N, %) are given on a regular (-, ,, r) grid. Raysare initialized by specifying the launch location, hori-zontal wavenumber, and intrinsic frequency *̂ + 0. Thevertical wavenumber m ) 0 is then computed from (10)


consistent with upward vertical propagation. The localwave amplitude in terms of vertical particle displace-ments and the cross-sectionally integrated wave actionflux F are also specified. Though the ray-tracing equa-tions are valid for time-dependent basic flows, themodel currently does not allow for them. In this setting,t is not a real physical time but merely a parameteriza-tion of distance along a ray tube: it measures theamount of time it would take for the steady wave fieldto develop from undisturbed initial conditions due toforcing at the lower boundary.

The gridded background fields are extended to theray location by local interpolation in (!, ", r) using thetricubic scheme of Lekien and Marsden (2005). Thescheme is local in the sense that it requires only datafrom the corners of the grid box containing the ray andsmooth in the sense that the function values and all firstpartial derivatives agree on the grid-cell boundaries.The function values and a set of seven partial andmixed partial derivatives must be prescribed at eachgrid point. These derivatives are computed by second-order finite differences. The grid must be regularlyspaced in (!, ", r) for the interpolation to be smooth, sodatasets that use pressure levels and/or Gaussian gridsare regridded to meet this requirement.

The ray-tracing equations are integrated using thebuilt-in Matlab function ode45. This ODE solver usesan explicit Runge–Kutta (4, 5) formula, the Dormand–Prince pair, and an adaptive step size algorithm to keeprelative error within prescribed tolerances. For the ex-periments reported in this paper, the relative error tol-erance is 10#3. In addition, we found it essential toinclude redundant equations for the background fieldsas dependent variables in the ray-tracing system in or-der for the adaptive time stepping to work consistently.Otherwise, it would not be the case that d$/dt % 0 forsteady background fields, for instance. The GROGRATmodel of Marks and Eckermann (1995) accomplishesthis by including an equation for d$/dt, but we pre-ferred to leave conservation of $ as an independentcheck on the validity of the solutions.

Ray integrations are terminated when a ray leavesthe model domain or the wave action flux falls below aprescribed tolerance. The wave action flux toleranceused in the following experiments was 1% of the totalwave action flux launched, divided by the number ofrays.

Some assumption was also required to model the be-havior of rays undergoing vertical reflection. As dis-cussed in section 2b, the detailed calculation of thewave field in the neighborhood of a caustic is a subtleand poorly constrained problem. In particular, the am-plitude depends on the phase of the incoming and out-

going waves. Some ray tracing codes, such as Marks andEckermann (1995), terminate rays upon vertical reflec-tion because of a local breakdown of the WKB approxi-mation. They were primarily concerned with the char-acteristics of waves that entered the stratosphere andnot vertically trapped tropospheric waves, so the latterwere removed using this WKB criterion. This proce-dure is equivalent to assuming that the angular momen-tum flux of the upward traveling wave is perfectly can-celed by the flux of downward flux of the reflectedwave, which should be accurate in the absence of dis-sipation when horizontal propagation and refractionare neglected. However, this cancellation is not to beexpected in the present case, when kH evolves along theray. It is well known (e.g., Lighthill 1978) that raytheory can be “healed” by the consideration of extraterms, which become significant only in the neighbor-hood of a reflection. The solutions obtained by thosemethods partially justify the continuation of the ray-tracing equations through reflection points, so verticalreflection does not automatically trigger ray termina-tion in our model.

c. Nonlinear wave saturation

Wave saturation schemes are heuristic methods bywhich nonlinear wave breaking can be modeled withina linear ray-tracing computation. Numerous saturationschemes exist in the literature and, to facilitate com-parison of our model to the topographic wave param-eterization of AGCM3, its saturation scheme has beenused here in a modified form that applies throughoutthe entire frequency range | f | & $̂ & N. Full details ofthis scheme can be found in McFarlane (1987). Basi-cally, it is assumed that max|u'z | /N ! Fc, where u' is thecomponent of the disturbance horizontal velocity in thedirection of the horizontal wavenumber vector kH, themaximum is over a wave cycle, and the threshold valueof the Froude number Fc % (1⁄2. This can be convertedinto a saturation threshold for the vertical particle dis-placement )' % Re[)̂(*0/* exp(i")] by using the stan-dard polarization relations. This yields

"̂ ! "̂sat %FcN|m |#̂! $+#̂2 # f 2,

$0+N2 # #̂2,

+18,

for the density-adjusted amplitude )̂. The vertical ac-tion flux density is related to )̂ by

Awg % #$0

+#̂2 # f 2,m

2+k2 - l2,"̂2; +19,

therefore, (18) implies a local saturation threshold onthis flux.


However, the model only carries the cross-sectionallyintegrated flux F of (9) along the ray tube, so, as dis-cussed in section 2b, at this step we must make thecrude assumption that the cross-sectional area is con-stant along a ray tube. This means that F ! Awg and,therefore, (18) implies a threshold directly on F. Now,at each time step along the ray the value of "̂ requiredto hold the wave action flux F constant is computedusing Eq. (19). The saturation amplitude "̂sat is thencomputed using Eq. (18) and, if the local amplitudesatisfies "̂ # "̂sat, then no dissipation occurs and F main-tains its previous value. However, if the amplitude ex-ceeds the saturation amplitude, then the saturation "̂ $"̂sat is applied and a new, reduced value of F is com-puted. It is clear that F can only decrease along the ray.On the other hand, the flux of angular pseudomomen-tum P $ Fkr cos % may increase or decrease along a ray,depending on the refraction by the basic flow.

We exploit a side effect to allow for vertical wavereflection without saturation, despite the fact that raytheory predicts infinite amplitude when m ! 0. Thepoint is that both "̂ and "̂sat diverge when m ! 0 but theratio "̂2 ⁄ "̂2

sat ! |m | (m2 & '2) goes to zero. In otherwords, the numerical scheme continues the ray throughthe reflection process without activating the saturationscheme.

4. Experiments and results

AGCM3 provided the background fields on a T47Gaussian grid with 31 vertical levels. The model do-main extends from the ground to 40 500-m altitude. Be-cause the ray tracer’s interpolation scheme requires aregular spatial grid, the raw GCM data were regriddedusing two-dimensional cubic spline interpolation in thehorizontal on each level, followed by one-dimensionalcubic spline interpolation along each vertical column.The '2 field was computed before regridding using cu-bic splines along each vertical column to compute thevertical derivative of density.

AGCM3 uses the topographic gravity wave param-eterization scheme of Scinocca and Mc-Farlane (2000).The scheme employs two waves at launch to representanisotropy in the topographic wave field. These wavesare launched from each surface grid point over landwith zero absolute frequency at each model time step.Properties of this scheme employed for the presentstudy are derived from instantaneous output of theprameterization every six model hours. Included in thisoutput is the launch value of the vertical displacementamplitude "̂0.

The ray tracer was used for offline computationsbased on the output of the AGCM3 model. Specifically,

we compute the global wave–induced angular momen-tum transport for a realistic topographic launch spec-trum with and without three-dimensional propagation.For topographic waves with ( $ 0, the intrinsic fre-quency at the launch level is

!̂ $ )Uk ) Vl. *20+

For typical topographic wavelengths, (̂ falls in the in-termediate range | f | K (̂ K N, for which it is expectedthat three-dimensional refraction effects have a weakimpact. This exercise is intended to demonstrate that,although the assumption of horizontal homogeneity inthe parameterization of topographic waves is incorrectin principle, it has a weak impact on the global wave-induced vertical transport of angular momentum.

It is also interesting to test the dependence of three-dimensional refraction effects on intrinsic frequency.From the viewpoint of testing their impact on angularmomentum transport, it would be ideal to conduct asimilar comparison of a physically realistic nonoro-graphic parameterization with fluxes computed fromthree-dimensional ray tracing solutions. The authorshope to undertake such a project in the future. How-ever, AGCM3 does not include a nonorographic pa-rameterization scheme.

Thus, as a simple test of the frequency dependence ofangular momentum transport, an artificial, ad hoc, lowfrequency spectrum was created by remapping the in-trinsic frequencies of the AGCM3 launch spectrumfrom the interval | f | # (̂ # N onto the interval | f | #(̂ # 3| f | , while holding k, l, "̂, F and the launch Froudenumber constant. Specifically, we use the power-lawmapping

!̂"

| f | $ ! !̂

| f |"s

with s $ln3

lnN# | f | .

While this remapping is unphysical, as a thought ex-periment it has the advantage of redistributing the fre-quency content of the topographic spectrum whileholding constant all other properties that could poten-tially impact angular momentum transport (i.e., hori-zontal wavenumber, the quantity and spatial distribu-tion of vertical wave action flux, and launch stability).

Another important difference between the topo-graphic launch spectrum and the remapped spectrum isthat the topographic waves satisfy Eq. (20) so that ( $0, whereas the remapped waves do not. We note inpassing that a third computation was performed with asecond spectrum in which the launch values of k and lwere rescaled to maintain ( $ 0. The results obtainedwere qualitatively similar to those of the first remap-ping, so this indicates a certain robustness of the effectswe describe here.


a. Ray examples

Examining individual ray solutions provides a quali-tative illustration of the dependence of three-dimen-sional propagation effects on intrinsic frequency. Asmentioned above, previous ray tracing studies, such asDunkerton (1984), have described similar solutions, butwe include these here for purposes of illustration.

Figures 1 and 2 show the detailed evolution of twowaves with altitude in the topographic and low fre-quency cases. Both figures show the vertical evolutionof k, the zonal velocity U at the location of the ray, andthe cross-sectionally integrated vertical angular pseudo-momentum flux P from (8) for both three-dimensionaland columnar propagation. The topographic case isshown in Fig. 1. Here, the relative change in k is smallover the depth of the atmosphere. Furthermore, be-cause the rays do not propagate far horizontally, thezonal velocity profile experienced by the three-dimensional ray is almost identical to that experiencedby the columnar ray. Not surprisingly, both topographicrays have avoided saturation in both cases.

The profiles for the low frequency rays launchedfrom the same site, shown in Fig. 2, show a large impact

from three-dimensional refraction. There is a largerelative change in k over the depth of the atmosphere.Because the three-dimensional rays propagate signifi-cant horizontal distances, the zonal velocity profile thatthey experience differs significantly from the columnarcase. The changed environment leads to the depositionof wave momentum flux at a significantly different al-titude.

b. Comparison with GCM parameterization scheme

Though individual solutions of the ray-tracing equa-tions show that three-dimensional propagation can leadto significantly different outcomes for individual raytubes compared with columnar propagation, by them-selves they give no information about how the new ef-fects impact global, aggregate forcing of mean winds byinternal gravity waves.

To investigate this question, the ray tracer was usedto compute the trajectory of every wave in the AGCM3parameterization launch spectrum. Though the wavesare launched from just above ground level in the GCM,they are launched from the GCM model level nearestto 4000 m in these computations. Thus, the experiments

FIG. 1. Vertical profiles of zonal wavenumber k, vertical angular pseudomomentum flux P,and zonal velocity U along the ray trajectory of a topographic internal gravity wave. This wavewas launched from 39°N, 96°E with a horizontal wavelength of 104 km; !̂/f " 6.9 and !̂/N "0.048.


with 3D propagation are equivalent to switching on 3Deffects once waves have propagated above 4000 m. TheGCM parameterization includes detailed considerationof near-ground dynamics such as blocking and down-slope windstorm flow. Launching from the middle ofthe troposphere avoids consideration of these boundarylayer issues. The ray tracing solutions of all launchedrays are added to construct vertical profiles of globallyintegrated angular pseudomomentum flux, which isequivalent to the flux of mean angular momentum.

c. Results for topographic waves

Figure 3 shows the January mean globally integratedvertical angular pseudomomentum flux profiles com-puted by the ray tracer with and without three-dimensional refraction. The daily variability of the pro-file at three reference levels is indicated by error barsspanning an interval containing 90% of the variabilityfrom time step to time step. To validate the ray-tracingscheme, the same flux profile computed from the out-put of the AGCM3 parameterization scheme is alsoshown. The AGCM3 scheme is equivalent to the ray-tracing model in columnar propagation mode, exceptfor differences in the representation of the background

fields arising from the use of different interpolationmethods and the modification of the saturation schemeto handle rotation and nonhydrostatic waves.

It is evident that the introduction of three-dimen-sional refraction does not have a strong impact on theglobal vertical transport of angular pseudomomentumby topographic gravity waves. Both profiles reach aminimum flux of approximately !2.0 " 1019 kg m2 s!2

near the tropopause, rapidly increasing to approxi-mately !3.2 " 1018 kg m2 s!2 at 20 000 m, then gentlyincreasing to approximately !9 " 1017 kg m2 s!2 at thetop of the model domain. This profile shape corre-sponds to a strong retrograde forcing of the lowerstratosphere, relatively little forcing of the middlestratosphere, and approximately 5% of the launchedretrograde angular pseudomomentum flux reaching theupper stratosphere or mesosphere. The relative differ-ence between the two profiles averages 4% throughmost of the domain and increases to 13% at 40 000 m.The angular pseudomomentum flux escaping the top ofthe model domain is more negative for three-dimen-sional propagation than for columnar propagation, cor-responding to a slightly stronger retrograde forcing ofthe upper atmosphere. A globally integrated vertical

FIG. 2. As in Fig. 1 but of the low frequency counterpart of the wave; #̂/f $ 1.5 and#̂/N $ 0.011.


angular pseudomomentum flux O(2 ! 1019 kg m2 s"2)is equivalent to an average vertical pseudomomentumflux density of 8 ! 10"3 kg m"1 s"2.

The qualitative shape of the vertical flux profile canbe explained by considering the mean vertical profilesof zonal velocity and density, shown in Fig. 4b. Thelaunch spectrum is divided between waves carrying pro-grade angular pseudomomentum flux, for which k # 0,and those carrying retrograde angular momentum flux,for which k # 0. Figure 4a shows how the vertical fluxprofile is decomposed into a contribution from retro-grade waves and one from prograde waves.

Broadly speaking, prograde waves are generated bywinds with a westerly component near the ground, andprograde waves are generated by winds with an easterlycomponent. Two processes combine to dissipate themajority of both prograde and retrograde waves below40 000 m. First, rays can be dissipated at critical surfaceson which $̂ % "U • k ! | f | . Second, the oscillationamplitude increases like &"1/2 to conserve action flux asdensity decays. Whether the ray encounters a criticalsurface or not, increasing amplitudes eventually lead tosaturation of the wave field by nonlinear instabilities.

As the waves propagate upward, prograde waves aretypically annihilated at critical surfaces as the zonalwind shifts from easterly to westerly leading into the jet.In Fig. 4, one can see that the contribution from pro-

grade waves is almost completely suppressed by 13 000m, where the westerly jet reaches its maximum. On theother hand, the positive shear environment suppressesbreaking for retrograde waves, and there is only weakdecay of their contribution over the same interval.Above the jet, the negative shear environment beginsto Doppler shift retrograde waves back toward lowerintrinsic phase velocities, making them more suscep-tible to saturation as the atmospheric density continuesto decay. Between 15 000 and 20 000 m, the majority ofthe pseudomomentum flux in retrograde waves is dis-sipated by saturation processes. The remaining flux isassociated with waves with amplitudes sufficiently lowto escape significant saturation between 20 000 and40 000 m.

d. Results for low-frequency waves

As expected, the vertical angular pseudomomentumflux profiles computed for the low-frequency spectrumshow a much greater impact from three-dimensionalpropagation and refraction. Figure 5 shows the angularpseudomomentum flux profiles for the same monthcomputed with the hypothetical low-frequency launchspectrum. The profile for three-dimensional rays be-comes more negative than the columnar profile imme-

FIG. 3. January mean vertical profile of globally integrated ver-tical angular pseudomomentum flux computed by summing thecontribution of every ray in the AGCM3 launch spectrum. Thesolid line shows the profile computed from three-dimensionallypropagating rays. The dashed line is the result from rays com-puted in column mode. Error bars denote the range containing90% of the variability of the profiles from time step to time step.Asterisks denote the same profile computed from the AGCM3parameterization scheme output.

FIG. 4. (a) The decomposition of the vertical angular pseudo-momentum flux profile for topographic waves into a contributionfrom prograde waves with k0 ' 0 and retrograde waves with k0 '0. The solid line is the total flux profile, the dotted line indicatescontributions from retrograde waves, and the dashed–dotted lineindicates contributions from prograde waves. (b) The Januaryglobal mean vertical profile of zonal velocity and the amplitudeamplification factor (&0/&)1/2 are plotted on the same axis. Thesolid line represents zonal wind, and the dashed line indicates theamplification factor.


diately above the launch level, and retains this biasthrough the entire depth of the model domain. Thethree-dimensional profile is 41% more negative on av-erage and transmits 81% more retrograde angular

pseudomomentum flux to the upper atmosphere. Thedaily variance of the flux profile is increased in thethree-dimensional case by an average of 30%. The in-crease in variance is primarily due to increased variabil-ity of the angular momentum flux above 20 000 m.

A heuristic argument to explain why three-dimen-sional refraction should lead to increased negative an-gular pseudomomentum flux can be based on the anal-ogy between k and the gradient of a passive tracerdiscussed below (5). Haynes and Anglade (1997) dem-onstrated that passive advection in typical stratosphericflows has a robust statistical tendency to increase themagnitude of the tracer gradient in a fashion compat-ible with the usual results from random advectiontheory. As this result did not rely on the physics under-lying the advecting velocity field, and this is the onlysense in which wave vector refraction differs fromtracer gradient evolution, it seems likely that this cu-mulative gradient stretching is inherited by k in theray-tracing problem. Though individual rays may expe-rience a decrease in |k | , the statistical tendency for |k |to increase is apparent. Indeed, a similar cascade toshorter horizontal wavelengths has been observed in astudy of internal waves in the ocean propagatingthrough a background drawn from the Garrett–Munkspectrum (see Flatté et al. 1985; Henyey et al. 1986).Thus, as prograde waves are filtered out at low alti-tudes, it can be expected that retrograde waves will

FIG. 6. Scatterplot describing the relative change in zonal wavenumber k from launch to thetop of the model domain: (a) Launch k0 vs k at the top of the model domain for all rays thatreach the top of the model domain in both column and 3D mode; (b) the same plot restrictedto waves with large wave action flux that account for 90% of the action flux out of the modeldomain. Data points are taken from every 10th time step to improve the plot’s readability.

FIG. 5. January mean vertical profile of globally integrated ver-tical angular pseudomomentum flux for low-frequency waves,computed as in Fig. 3. The solid line shows the profile computedfrom three- dimensionally propagating rays. The dashed line is theresult from rays computed in column mode. Error bars denote therange containing 90% of the variability of the profiles from timestep to time step.


have more negative k on average when three-dimen-sional refraction is allowed.

Figures 6 and 7 indicate that increased retrogradeflux P ! kFr cos" in the three-dimensional case is dueto the tendency for k to become more negative com-bined with a tendency for three-dimensional rays totransmit more wave action flux. In Fig. 6a, the value ofk for the three-dimensional ray at the top of the modeldomain has been plotted against its launch value k0 forall rays that escape to the upper atmosphere in boththree-dimensional and columnar mode. The medianrelative change (k # k0)/k0 is #17%. This tendency ismore pronounced when we consider only the strongestrays. Figure 6b shows rays with the largest action flux atthe top of the model domain, which explain 90% of theflux transmitted to the upper atmosphere. For theserays, the median relative change of k is #32%.

Meridional propagation contributes very little to thechange in angular pseudomomentum flux. The result-ing change in r cos " has a mean of less than 5%.

Figure 7 compares the wave action flux at the top ofthe model domain for individual rays with columnarand three-dimensional propagation. The flux increases52% with three-dimensional propagation, which to-gether with the mean decrease of k contributes to theincrease of retrograde momentum flux to the upperatmosphere. Figures 6 and 7 demonstrate that the ten-dencies just described are statistical and that there isgreat variation in the evolution of these quantities be-tween individual rays.

5. Concluding comments

Consistent with the scaling argument presented insection 2, we have found that significant changes in

angular momentum fluxes due to three-dimensional re-fraction and horizontal propagation are more or lessconfined to low frequency waves. This result indicatesthat columnar parameterization schemes for topo-graphic waves should produce results that are little af-fected by the neglect of three-dimensional effects,which is encouraging. There is, however, the caveat thatwe used data at the fairly low horizontal resolution ofT47. It remains to be seen what the impact of increasedhorizontal resolution would be and, also, how three-dimensional effects would change the parameterizationof nontopographic waves, especially in the low fre-quency range. We hope to follow up on these questionsin the near future.

Acknowledgments. The authors thank Stephen Eck-ermann for providing the source code to GROGRAT,which was an extremely useful resource. AH’s work issupported by a National Science Foundation GraduateResearch Fellowship and a National Defense Scienceand Engineering Graduate Fellowship. OB gratefullyacknowledges financial support for this work under theNational Science Foundation Grants OCE-0324934 andDMS-0604519. We are grateful for the comments oftwo anonymous referees that significantly improved anearlier version of this paper.

APPENDIX

Ray Tracing on a Sphere

If the wavelengths of internal gravity waves are suchthat the curvature of the earth is negligible on the scaleof a wavelength, then the dispersion relation $ ! %(k,x) computed in Cartesian coordinates may be retained,simply substituting k ! k • !̂, l ! k • "̂ and m ! k • r̂.The components of k in the underlying Cartesian coor-dinates will be denoted in index notation ki. In general,the dispersion relation may be written in Cartesian co-ordinates

!̂ ! " &k # r̂'x, y, z(, k # "̂'x, y, z(, k # !̂'x, y, z(,

r'x, y, z(, $'x, y, z(, %'x, y, z() , 'A1(

! %̃'k1, k2, k3, x, y, z(. 'A2(

As the unit vectors of the spherical coordinate systemare slowly varying on the scale of a wavelength, one canuse this dispersion relationship to obtain ray-tracingequations. This approach was used in Francis (1972)and further justified in Dong and Yeh (1993), and weuse it now to demonstrate how the use of sphericalcoordinates introduces metric terms to the refraction of

FIG. 7. Scatterplot comparing the action flux exiting the top ofthe model domain in column mode and 3D mode for all rays thatexit in either 3D mode or column mode. Data points are takenfrom every 10th time step to improve the plot’s readability.


the wavenumber for a general dispersion relationshipof the form in Eq. (A1).

We then have

dxi

dt!

!"̃

!ki,

!!"

!m!

!ki#kjr̂j$ %

!"

!k!

!ki#kj#̂j$ %

!"

!l!

!ki#kj$̂j$,

!!"

!mr̂i %

!"

!k#̂i %

!"

!l$̂i.

Therefore,

cgr !dxdt

% r !!"

!m,

and likewise cg& ! "k, cg' ! "l.However, simply taking the partial derivative of the

dispersion relationship with respect to the spatial vari-ables will not yield the correct refraction equations. Wefind

dmdt

!ddt

k % r̂ !dkdt

% r̂ % #cg % !$r̂ % k. #A3$

By the chain rule,

dki

dt! (

!"̃

!xi,

! (!"

!r!r!xi

(!"

!#

!#

!xi(

!"

!$

!$

!xi(

!"

!m!

!xi#k % r̂$

(!"

!k!

!xi#k % "̂$ (

!"

!l!

!xi#k % #̂$.

This equation combined with the expression for cg andEq. (A3) implies that

ddt

#k % r̂$ !!"

!k#"̂ % !r̂ ( r̂ % !"̂ $ % k

%!"

!l##̂ % !r̂ ( r̂ % !#̂$ % k (

!"

!rr̂ % !r

(!"

!#r̂ % !#̂ (

!"

!$r̂ % !$̂.

An equation of this form would be obtained for anycurvilinear coordinate system. The metric terms in pa-rentheses are a property of the coordinate system. Forspherical coordinates,

!r̂ !1r

#"̂"̂ % #̂#̂$, #A4$

!"̂ !tan$

r"̂#̂ (

1r

"̂r̂, #A5$

!#̂ !tan$

r"̂"̂ (

1r#̂ r̂. #A6$

In this notation, the vector to the left contracts with thegradient. Simplifying, we find

dmdt

! (!"

!r%

kr

!"

!k%

lr

!"

!l. #A7$

Similar computations give

dkdt

! (1

r cos$

!"

!#(

kr

!"

!m%

k tan$

r!"

!l, #A8$

dldt

! (1r

!"

!$(

1r

!"

!m(

k tan$

r!"

!k. #A9$

REFERENCES

Andrews, D. G., and M. McIntyre, 1978: On wave-action and itsrelatives. J. Fluid Mech., 89, 647–664.

——, J. R. Holton, and C. Leovy, 1987: Middle Atmosphere Dy-namics. Academic Press, 489 pp.

Badulin, S. I., and V. I. Shrira, 1993: On the irreversibility of in-ternal waves dynamics due to wave trapping by mean flowinhomogeneities. Part 1: Local analysis. J. Fluid Mech., 251,21–53.

Bretherton, F. P., 1969: On the mean motion induced by internalgravity waves. J. Fluid Mech., 36, 785–803.

——, and C. Garrett, 1968: Wavetrains in inhomogeneous movingmedia. Proc. Roy. Soc. London, A302, 529–554.

Broutman, D., 1984: The focusing of short internal waves by aninertial wave. Geophys. Astrophys. Fluid Dyn., 30, 199–225.

——, 1986: On internal wave caustics. J. Phys. Oceanogr., 16,1625–1635.

——, J. Rottman, and S. Eckermann, 2001: A hybrid method foranalyzing wave propagation from a localized source, withapplication to mountain waves. Quart. J. Roy. Meteor. Soc.,127, 129–146.

——, ——, and S. D. Eckermann, 2004: Ray methods for internalwaves in the atmosphere and ocean. Annu. Rev. Fluid Mech.,36, 233–253.

Bühler, O., 2003: Equatorward propagation of inertia–gravitywaves due to steady and intermittent wave sources. J. Atmos.Sci., 60, 1410–1419.

——, and M. McIntyre, 1998: On nondissipative wave–mean in-teractions in the atmosphere or oceans. J. Fluid Mech., 354,301–343.

——, and ——, 2003: Remote recoil: A new wave–mean interac-tion effect. J. Fluid Mech., 492, 207–230.

——, and ——, 2005: Wave capture and wave–vortex duality. J.Fluid Mech., 534, 67–95.

Dong, B., and K. C. Yeh, 1993: Propagation and dispersion ofatmospheric wave packets around a spherical globe. Ann.Geophys., 11, 317–326.

Dunkerton, T. J., 1984: Inertia–gravity waves in the stratosphere.J. Atmos. Sci., 41, 3396–3404.

——, and N. Butchart, 1984: Propagation and selective transmis-sion of internal gravity waves in a sudden warming. J. Atmos.Sci., 41, 1443–1460.

Eckermann, S., 1992: Ray-tracing simulation of the global propa-


gation of inertia gravity waves through the zonally averagedmiddle atmosphere. J. Geophys. Res., 97, 15 849–15 866.

——, and C. Marks, 1996: An idealized ray model of gravitywave–tidal interactions. J. Geophys. Res., 101 (D16), 21 195–21 212.

Flatté, S., F. Henyey, and J. Wright, 1985: Eikonal calculations ofshort-wavelength internal-wave spectra. J. Geophys. Res., 90(C10), 7265–7272.

Francis, S. H., 1972: Propagation of internal acoustic-gravitywaves around a spherical earth. J. Geophys. Res., 77, 4221–4226.

Hayes, W., 1970: Kinematic wave theory. Proc. Roy. Soc. London,A320, 209–226.

Haynes, P., and J. Anglade, 1997: The vertical-scale cascade inatmospheric tracers due to large-scale differential advection.J. Atmos. Sci., 54, 1121–1136.

Henyey, F., J. Wright, and S. Flatté, 1986: Energy and action flowthrough the internal wave field: An eikonal approach. J. Geo-phys. Res., 91 (C7), 8487–8495.

Hertzog, A., C. Souprayen, and A. Hauchecorne, 2001: Observa-tion and backward trajectory of an inertio-gravity wave in thelower stratosphere. Ann. Geophys., 19, 1141–1155.

Jones, W. L., 1969: Ray tracing for internal gravity waves. J. Geo-phys. Res., 74, 2028–2033.

Lekien, F., and J. Marsden, 2005: Tricubic interpolation in threedimensions. Int. J. Numer. Meth. Eng., 63, 455–471.

Lighthill, J., 1978: Waves in Fluids. Cambridge University Press,469 pp.

Marks, C., and S. Eckermann, 1995: A three-dimensional nonhy-drostatic ray-tracing model for gravity waves: Formulationand preliminary results for the middle atmosphere. J. Atmos.Sci., 52, 1959–1984.

McFarlane, N., 1987: The effect of orographically excited gravitywave drag on the general circulation of the lower strato-sphere and troposphere. J. Atmos. Sci., 44, 1775–1800.

Plougonven, R., and C. Snyder, 2005: Gravity waves excited byjets: Propagation versus generation. Geophys. Res. Lett., 32,L18802, doi:10.1029/2005GL023730.

Scinocca, J., and N. McFarlane, 2000: The parametrization of draginduced by stratified flow over anisotropic orography. Quart.J. Roy. Meteor. Soc., 126, 2353–2394.

Shutts, G., 1998: Stationery gravity-wave structure in flows withdirectional wind shear. Quart. J. Roy. Meteor. Soc., 124, 1421–1442.

Smith, R., 1980: Linear theory of stratified hydrostatic flow pastan isolated mountain. Tellus, 32, 348–364.

Sonmor, L., and G. Klaassen, 2000: Mechanisms of gravity wavefocusing in the middle atmosphere. J. Atmos. Sci., 57, 493–510.

Walterscheid, R., 2000: Propagation of small-scale gravity wavesthrough large-scale internal wave fields: Eikonal effects atlow-frequency approximation critical levels. J. Geophys. Res.,105 (D14), 18 027–18 038.

Whitham, G., 1974: Linear and Nonlinear Waves. Pure and Ap-plied Mathematics Series, Wiley-Interscience, 620 pp.


Documents

Gravity Wave Refraction by Three-Dimensionally Varying ...obuhler/Oliver_Buhler/... · changes due to refraction are indeed possible, and they fall outside the columnar pseudomomentum