Gravity-wave refraction bythree-dimensional winds: neweffects and a newparametrization schemeOliver Bühler & Alex HashaCourant Institute of Mathematical SciencesCenter for Atmosphere Ocean Science

Gravity wavesGravity wavesevolve locallybut they affect thecirculation globallyThink locally act globally

Current theory paper in JFM 2005J. Fluid Mech. (2005), vol. 534, pp. 67–95. c○ 2005 Cambridge University Pressdoi:10.1017/S0022112005004374Printed in the United Kingdom67Wave capture and wave–vortex dualityBy O L I V E R B Ü H L E R 1 AND M I C H A E L E. M c I N T Y R E 21 Center for Atmosphere Ocean Science at the Courant Institute of Mathematical Sciences,New York University, New York, NY 10012, USA2 Centre for Atmospheric Science at the Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Wilberforce Rd., Cambridge CB3 0WA, UK(Received 7 July 2004 and in revised form 7 January 2005)New and unexpected results are presented regarding the nonlinear interactionsbetween a wavepacket and a vortical mean flow, with an eye towards internal wavedynamics in the atmosphere and oceans and the problem of ‘missing forces’ inatmospheric gravity-wave parametrizations. The present results centre around a prewave-breakingscenario termed ‘wave capture’, which differs significantly from the

Inertia-gravity wavesundulating materialstratification surfaces(isentropes/isopycnals)surfaces are flat at restw ′ ∝ exp(i[kx + mz − ˆωt])zxlinear particle trajectoriesscale-free dispersion relationˆω 2 = (N 2 − f 2 )f 2 ≤ ˆω 2 ≤ N 2k 2k 2 + m 2 + f 2Momentum fluxu ′ w ′ < 0 (unlike surface waves)Internal waves make a significant contribution to atmosphericangular-momentum fluxes... convergence yields wave dragOliver Bühler, Courant Institute, New York University, 2005

Wave breakingLarge-amplitude waves overturn and break nonlinearly, leading to 3dturbulence, irreversible mixing, and convergence of wave momentum fluxbeforeafter!(x=0.6, z)10.90.80.70.60.50.40.30.20.100 0.2 0.4 0.6 0.8 1z!(x, z=0.6)0.90.850.80.750.70.650.60.550.50 0.2 0.4 0.6 0.8 1xFigure 3: Diffusive scheme applied to the unstable buoyancy profile: b(x, z) = 0.3 exp[−((x−0.5) 2 +(z −0.5) 2 )/0.09] cos[2π(x+z)]. (a) (b) are surface plots of the potential temperaturedistribution before and after mixing. (c) and (d) give potential temperature profiles on thetransects x = 0.6 and z = 0.6 respectively. The blue curve is the profile before mixing, andthe green curve is the profile after mixing.Marcus Roper,GFD report 2005both sides of (21) by b and integrating over the entire of the fluid domain, we have∫ ∫ ( )Oliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005The classical gravity wave drag pictureRay-tracing theory for steadywavetrain as function of altitude zHorizontal mean-flow inhomogeneityis ignored (“co-dimension 1”)Vertical flux of horizontalpseudomomentum equals vertical fluxof horizontal mean momentumVertical flux of horizontalpseudomomentum is constant unlesswaves are dissipating or breakingThis results in the classical forcebalance: there is anequal-and-opposite action at adistance between mountain drag andwave drag

Oliver Bühler, Courant Institute, New York University, 2005Wave drag on a rotating planet“gyroscopic pumping”Retrograde forceifk/ˆω < 0Convergence of angular-momentum flux acts aseffective force on zonal-mean flowPrograde forceifk/ˆω > 0Retrograde force-> polewardmotionAll Rossby wavesWinter gravitywavesPrograde force-> equatorwardmotionSummer gravitywaves

Oliver Bühler, Courant Institute, New York University, 2005Vertical propagation of topographicRossby and gravity wavesGW f 2 ≤ ˆω 2 ≤ N 2 RW ˆω ≤ 0Absolute frequency:ω = ˆω + UkTopographic waves have zero absolute frequency:ω = 0Topographic waves areretrograde if U>0 nearsurfaceVertical critical layer occurswhere shear pushes intrinsicfrequency towards lowerlimitSummer zero-wind surfaceacts to filter topographicwaves

Wave-driven global circulationPrograde and retrogrademesospheric gravity and Rossbywavedrag:Murgatroyd--Singleton circulationfrom summer to winter poleSummerWinterRetrograde stratospheric Rossbywavedrag:Brewer--Dobson circulation fromequator to poleSummer polar mesosphere issunniest place on EarthAlso the coldest (-163 o Celsius)Fermi review,ME McIntyre 1993Oliver Bühler, Courant Institute, New York University, 2005

Rough guide to gravity wave scalesTo resolve a scale need 10 grid points across itGravity wave scales(excludes breaking details)Model resolution(grid size * 10)Horizontalscale10 - 1000km1000 kmVerticalscale0.1 - 10 km 10 kmTime scale10 mins - 1day200 minsGWs parametrized because need a factor 100 higher resolution.How long will that take?Oliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005Computer power and resolution increaseMoore’s law: computing power doubles every 18 months.To reduce grid size by a factor 10 requires 10x10x10x10=10000 timesmore computing power in a three-dimensional time-dependent simulation(grid points x time steps with CFL condition)Let N by number of necessary doubling events:2 N = 10 4 N log 2 = 4 log 10 N = 13.3T = N ∗ 18 months = 19.95 yearsTo increase resolution by a linear factor of 10 takes 20 yearsMoore is not enough..........

Oliver Bühler, Courant Institute, New York University, 2005Multiscale-based career adviceFor gravity waves need factor 100This will take 40 years, ok for a career!

Oliver Bühler, Courant Institute, New York University, 2005Columnar gravity wave wave parametrizationParametrization is is applied appliedindependently in in each each vertical model modelcolumnTime-dependence is is ignoredVertical wave wave propagation and andvertical mean-flow derivatives are aretaken taken into into account accountHorizontal wave wave propagation propagation and andhorizontal horizontal mean-flow mean-flow derivatives derivatives are areignored:ignored:nonorefractionrefractionbybymeanmeanflowflowMany effects are neglected. WhichMany effects are neglected. Whichare the important ones?are the important ones?Some neglected effects are known toSome neglected effects are known tobe important.be important.For instance, intermittency.For instance, intermittency.

Known unknown: intermittency1416 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 SVOLUME 60FIG. 4. Wave packets due to hypothetical intermittent wave source at y 0 0. (left) Steady, nonintermittentsource. (middle) Intermittent source with 50%. (right) Intermittent source with 25%. The waveamplitude increases by a factor of 2 from wave to wave, keeping the expected wave energy flux the samein all three cases.Bühler, 2003 JASs 00 00 2Oliver Bühler, Courant Institute, New York University, 2005Intermittency increases waveamplitudes at fixed meanwave activity fluxWave breaking is highlyamplitude-dependent“Intermittency factor”adjusted to increasepredicted amplitudes toobtain breakingBetter to model wavegeneration as stochasticprocess (future work) E 00rameterized flux E{F}. So all three wave trains aremapped onto the first wave train for the purpose ofparameterization with equal expected flux.This is despite the fact that the wave amplitudes aremarkedly different in the different cases. This is importantfor nonlinear effects, such as the amplitude-deingthe same expected value and variance. This correspondsto a source spectrum defined as1 B(0)E E E 1 . (28)

Unknown unknowns in parametrizationRequirements for better parametrizationTheoreticalconstraintsalmost none;plenty of ideasComputationalconstraintsMust be in-column& inexpensiveObservationalconstraintsMust not requiremore informationThree-dimensional refraction is a candidateOliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005Claim: singular geometric perturbationKey to strong effects:ignoring the horizontalrefraction is a singularperturbation3d ray tracing admitsexponentially fast wavebreaking without criticallayers (Jones 1969, Badulin &Shira 1993). Wave capture.3d wave-mean interactionsexhibit new features such asremote recoil and missingforces (Bühler & McIntyre2003) Wave-vortex duality.What are the new effects?

An adventure in ray tracingWavepackets alonggroup-velocity rayWavepackets are the fundamentalsolutions of ray tracingWavetrains can be built fromwavepacketsAmplitude along non-intersectingrays is determined bywave-action conservationOliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005Warm-up example: shallow water systemSingle layer of hydrostaticincompressible fluidVariablesMasscontinuityMomentumconservationhx = (x, y) u = (u, v)Ddepth h**Dt** = ∂ ∂tDh**Dt** + h∇ · u = 0Du**Dt** + g∇h = 0+ (u · ∇)

Oliver Bühler, Courant Institute, New York University, 2005Linear small-scale wavepacket(Example in shallow water)u = U + u ′ + O(a 2 ) D t u ′ + g∇h ′ = 0 D t = ∂ ∂th = H + h ′ + O(a 2 )D t h ′ + H∇ · u ′ = 0+ (U · ∇)a ≪ 1Slowly varying wavetrain h ′ = Ha(µx, µt) exp(iθ), µ ≪ 1k = ∇θ, ω = −θ t ,Dispersion relationω = U · k + ˆωbackground flow, if anyˆω 2 = gH|k| 2Wavepacket:phase lines

Oliver Bühler, Courant Institute, New York University, 2005Geometric ray tracing - phasesΩ(k, x, t) = U · k + ˆωphase linesof a wavepacketdxdt = +∂Ω ∂k and dkdt = −∂Ω ∂xGroup velocityu g = dxdt = U + û gRay time derivativeddt = ∂ ∂t + (u g · ∇)Simple case ˆω(k):dk idt = −∂U j∂x ik jWavenumber changesdue to backgroundinhomogeneity--> refraction

Physical ray tracing - amplitudesWave energy“Mean” is the averageover rapidly varyingwave phaseE = 1 )(u2 H ′2 + gh ′2 /Hh = h + h ′h ′ = 0Wave actionA = Ḙ ω∂A∂t + ∇ · (Au g) = 0Amplitude prediction fromwave action conservationAnother importantwave property:Pseudomomentump = k APseudomomentum changes with wavenumberdue to refractionOliver Bühler, Courant Institute, New York University, 2005

only, i.e. theflow U(x, t).ion equationUnderstanding wavenumber refraction(1.6). d side. Thisfraction, i.e.ground flow≡calar φ(x, t)dk idt = −∂U jk j∂x iderivative following a particle:(1.7)dtm the utimeg = dxdt = U + û g= −∇U · k,is understood and k contracts with U on the right-hand sidthe instantaneous change in k = ∇θ due to mean-flow refracWavenumbertial advection of the phase lines θ = const. by the backgrouto the evolution equation is equivalent for to the gradient of a passive scala( ∂∂t + (U · ∇) )Passive tracersuch thatD t φ = 0φ = 0 ⇒ D t (∇φ) = −∇U · (∇φ).Wave captureof course, that the time derivative along a ray differs from tIntrinsic differenceddt − D t = (û g · ∇).So k behaves like the gradient of a passive tracer only to the extent thatbetween these two time derivatives measures does the not misfit matter. This indicates that ucircumstances the well-studied properties of passive advection can be relevsive wave dynamics. For instance, in the passive advection case it is well knOliver Bühler, Courant Institute, New York University, 2005dkdt = −∇U · kk and ∇φ evolve similarly(i.e. wave phase and passive tracer evolve similarly)

Wavepacket exposed to pure strain inWave capture analogy andwith wave–vortex passive duality advectionWavepacket issqueezed in x andstretched in y.Action is constantp = k AWavenumbervector k isincreases in sizePseudomomentump increases as wellet exposed Is there to pure a “Batchelor” horizontal strain regime contracting for wave alongphase?the x-axis andaxis. The wavecrests align with the extension axis and their spacinOliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005Not in shallow water...Answer is no in shallow water with sub-critical steady background flowω = U · k + ˆω = const.ˆω = √ gH|k||k| ≤ω√ gH − |U|wavenumbers are bounded unlessU 2 > gH(geophysically lessrelevant regime)Same answer for rotating shallow water, but not for 3d flow!

Boussinesq system(no Coriolis force in talk, but in paper)Variables x = (x, y, z) u = (u, v, w)Incompressible∇ · u = 0MomentumconservationDu**Dt**+ ∇P = bẑbuoyancyaccelerationStratificationD**Dt** (b + N 2 z) = Db**Dt** + N 2 w = 0constant value defines 3dstratification surfacesOliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005Plane Boussinesq gravity waves∇ · u = 0 implies k · u ′ = 0Dispersion relationWavenumber vectorˆω 2 = N 2 k 2 + l 2k = (k, l, m)k 2 + l 2 + m 2Frequency is independent ofκ = √ k 2 + l 2 + m 2Group velocity magnitude|û g | 2 = N 2 − ˆω 2κ 21) Unbounded wavenumber growth is possible at fixed frequency2) Group velocity inversely proportional to wavenumber at fixed frequency

U y )/(2U x )).) Also indicated are captured wavepackets at the orientation of theode in the first two cases and in that of maximal transient amplification in theimensional grounds that the intrinsic group velocity ĉ g = ∂ ˆω/∂k must be inverselyse. roportional The large arrows to κ indicate for fixed k H . ˆω. Left: Explicitly, a hyperbolic the case components (γ = 0.5α) similar of ĉ gtoarethe topic of this paper first in a fairly specific and simplified setting,so thatre general commentsThree-dimensional√later**. |ĉFor large-scalerayatmospherictracingflows the mostbackground(k, l) Nvelocity 2 − ˆω 2fieldˆω 2 − fis 2 g | 2 = (N 2 − ˆω 2 )(ˆω 2 − f 2 )√strain axes close like a pair of scissors. The advectednon-divergentwavecrests alignˆω with 2 andκ 2 ,thehas negligible verticalk*), i.e. H ˆωκ N 2 − f 2 and ŵ g = −sgn(m) ˆω2 − f 2 N 2 − ˆω 2ˆωκ N 2 − f 2 , (3.3)(in which γ = 0). If γ > 0 then the axis of extension is turned counterclockwiseα) whilst the axis of contraction is turned clockwise by the same angle. With(û g , ˆv g ) =k H ˆωκ N − f ˆωκ N − fconfirming the inverse proportionality of ĉ g to κ and the possibility of wave caWith U of the form (2.4), the background velocity gradient tensor isand the growing horizontal wavenumber vector k H , which is always perpendicts,becomes perpendicular to the extension axis. Middle: the scissors shut insimple shear case Horizontal γ = α. Right: background an elliptic case flow(γ = 1.5α). The ellipses haveo that(γ + α)/(γ − α), which equals the maximum transient amplification factor for|ĉ g | 2 =⎛ (N 2 − ˆω 2 )(ˆω 2 − f 2 ⎞)ˆω 2 κ 2 ,⎛⎞U(3.4)x V x W x U x V x 0onfirming the inverse proportionality ∇U ≡ ⎝ of U y ĉ g to V y κ and W y the ⎠ possibility = ⎝ U y of−U wave x capture. 0 ⎠With U ofGradientthe form (2.4),of backgroundthe backgroundflow treatedvelocityas steady fortensorsimplicitytational strain reversed. This is a consequence U z of kVbeing z Wthe z gradient of U ismodern geometric language, k is a 1-form rather than a 1-vector.z V z 0⎞ant for our purposes is the long-time behaviour of k H (t), whichU x is governedV x 0ntial stretching rates given by the matrix eigenvalues ± √ D, U y where −UD x is 0 ⎠ (3.5)t Ux 2 + V x U y , i.e.U z V z 0U = (U, V, 0) and U x + V y = 0. (2.1)ient is then⎛∇U = ⎝ U ⎛ ⎞ ⎛ ⎞x V x 0U y V y 0 ⎠ = ⎝ α ⎛U x V x W x β + γ 0∇U ≡ ⎝ U y V y W y⎠β −=⎝applicable, we require large Richardson γ number, −α 0U z V z W zU z V z 0δ µ 0say, where suffixes denote partial derivatives and where, for ray theory to be con( ) 2 ( ) 2D = Ux 2 Vx + U y Vx − U y+−. (3.8)Ri ≡ N 2 /(Uz 2 + Vz 2 ) ≫ 1 .⎠ , (2.2)ay, where suffixes denote partial derivatives and where, for ray theory to be consistentlypplicable, µ} are functions we require 2large of time Richardson along 2 number, the ray. stream Inlines the in simplest group-velocity setting, frameNotice that the evolution of the horizontal wavenumber vector k Hhowever,nzero me-dependence curl always Wavenumber diminishes of these evolution stretching Riparameters. ≡decouplesNrate. 2 /(UAs 2 illustrated intoThishorizontalcorrespondsand verticalto acomponentsz + Vz 2 ) ≫in figure 1 . 2a, linear (3.6) layernttially otice shear/strain that time (J69) the evolution with( stretching flow)of U the rate = (the streamlines are hyperbolic and there is a wavenumber eigenmode thathorizontal √ UD. 0 + Theαx growing wavenumber + (β )eigenmode( − γ)y )vector is+ ex-δtrom all ere initial that (x, conditions; gand V = V 0 + (β +dk H = (k, l) decouplesof y, the z) vertical are asymptotically k Ux Vmeasured wavenumber = −k H from (t) ∝ m. (−V Thus the x , Ustarting x (2.7) + √ D) splits exp( location √ x k dinto D t). gand the two of subsystems the ray at t = 0.e of wave capture. ( dt ) l ( U y ) ( −U)x lgure this2b) simplest then d g the k flow case is aU parallel Ux is not shear V constant along the ray, though ∇U is.= −x flow with k a linearly growing d gandof (2.2) dt iseasy lthat2d in sub-problem (1.6) U y −U the x evolution of advection l ofbydt the m = horizontal −U zk − V z l , (3.7)suchwhich that asymptotically can be studied k H (t) ∝ in (−V sequence.x , U x ) |V x − U y | t, i.e. classical wavenumberehaviour. If D < 0 (figure 2c) then the streamlines are closed ellipses,) decouples from that area-preserving of the vertical flow wavenumber m, i.e.the hichwavenumber can studied evolution in sequence. is bounded, though temporary amplificationGenerically, U x etc. are functions of time along the ray. For the sake of simplicity, weow neglect Oliver this time-dependence. Bühler, Courant ThisInstitute, a severe simplification New York and University, we hope to report 2005, up to a factor equal to the aspect ratio of the ellipse. For D 0 andinitial k H (0), the asymptotic orientation and growth rate of k H dependk ) ( α β + γ ) ( k ) dmpositive for generic case with open 2d= (k, l) dfrom that of the vertical wavenumber m. Thus (2.7) splits into the two subsystdt m = −U zk − V z l ,Generically, U x etc. are functions of time along the ray. For the sake of simpnow neglect this time-dependence. This is a severe simplification and we hopelater on work that goes beyond it, following the Haynes & Anglade work alrea

Streamlines in group-velocity frameBühler & McIntyreHyperbolic D>0 Parabolic D=0 Elliptic DInstitute,0 then the axisNewofYorkextensionUniversity,is turned counterclockw2005

V x + U y22−V x − U y22. (3.8)Growing mode in three dimensionsiminishes the stretching rate. As illustrated in figure 2a,hyperbolic and there is a wavenumber eigenmode thatwith stretching rate √ WaveD. capture The growing and wave–vortex eigenmode duality is exns;asymptotically k9umber, implying zero H (t) ∝ (−Vvertical shear x , Uin x + √ D) exp( √ D t).that direction. Specifically, at large timee flow is a parallel shearm(t) =flow− U zk(t) with + V z l(t)√a linearly+ O(1)growing, (3.9)otically k H (t) ∝ (−V x , U x ) |V x − U D y | t, i.e. classical0 k(t) (figure and l(t) correspond to the growing eigenmode, k H ≡ (k, l) ∝ (−V x , U x +xp( √ 2c) then the streamlines are closed ellipses,Intrinsic group velocity decreases as wavenumberevolution D t). isFor bounded, a captured though wavepacket temporary we therefore amplificationgrows; the wavepacket becomes frozen have into the the scaling flowrelationual to the(Jones aspect 1969, ratio of Badulin the ellipse. & Shrira For D1993) 0 andasymptotic orientation and growth |k H /m| rate ∼ H/L of k (3.10)H dependThe reinforces the analogy between phase andwe er of have magnitude, a robustifbehaviour H, L are vertical in which andthe horizontal wavenumber length scales characteristic ofpassive tracer behaviour; reasonably to expectforgets field. Inabout the real theatmosphere initial conditions we typically at large have time, H/Landexponential straining to persist once it∼ f/N.gets startedIf for the sake ofion entdetermined we take |k H by /m| the = H/L local= velocity f/N precisely, gradient inalone.the wave-capture limit, then fromof the vertical wavenumber m in the case D > 0, whichˆω(t) −2 (H and therefore Wave κ, →with 12 f Exponentially −2 stretching + N −2) ; rate fast √ hence D. Turning wave ˆω(t) 2 breaking → 2f 2 (3.11)exhibitcapture:exponential growth scenariofurther approximate using N 2 at large≫ f 2 without time, unless critical so layers!. It is noteworthy that values of ˆω close to fof (3.7b) is zero for the growing eigenmode in horizontalr to be commonplace in the oceans and the atmosphere (e.g. Garrett and MunkOliver Bühler, Courant Institute, New York University, 2005

Numerical exampleSnapshots taken fromnumerical simulation ofmeandering jet streamInterpreted based on wavestrainingPlougonven & SnyderFigure GRL, 4. 2005Horizontal and vertical cross-sections of thehorizontal divergence at lower (dx = 100km, dz = 500m,upper Oliver panel) Bühler, and higher Courant (dx = 25km, Institute, dz = 125mNew lower York University, 2005

Theory example: wave capture byblocking dipoleUpstream wind10 Bühler & McIntyreUclockwise vortexfixed wavepacketat t1drifting wavepacketat t2>t1stagnationpointω = 0u g (t1) = 0 u g (t2) > 0Strained wavepacket drifts towardsstagnation point where it must break(similar to horizontal critical layer)Figure 2. Contourscounter-clockwisevortexside-steps complications due to the possible divergence or non-uniqueness of global horizontalOliver momentum Bühler, integrals Courant for layerwise Institute, non-divergent New York flows on University, horizontal surfaces 2005(e.g.

Oliver Bühler, Courant Institute, New York University, 2005Pseudomomentum surge and mean flowp = k Agrows exponentiallyduring wave capture(action is conserved,but wavenumber grows)Standard dissipative wave-mean interaction paradigm:mean-flow momentum + wave pseudomomentum = constant.Does the exponential pseudomomentum surge lead to adramatic local mean-flow response ?No, because the standard paradigm does not hold for horizontalwavepacket refraction...Need to investigate O(a 2 ) wave–mean interaction theory with refraction

ct that it allows deducing changes in uWave-mean interaction theorycaused by p, at leastetermined by its circulation on mean isentropes. Physically,mean sound and gravity waves, as in the standard quasi-geotheory for ‘balanced’ large-scale geophysical flows (e.g. SalmSlowly varying Lagrangian mean flow with strongstratification is layerwise 2d, layerwise non-divergent, andat is governed by (Bretherton 1969)he mean-flow response under the exclusion of such waves as thd in theO(a geophysical 2 ) regime relevant here it approximately sati∇ · u L = 0 and w L = 0.ow uCan L is show then that determined (AM 1978, BM by 98, the B00, circulation BM05) integrals (5.2) or equivalently ( by ) the distribution of the scalar∂ {ẑQ L = ẑ · ∇ × (u L − p) such that D L Q L ∂t + uL · ∇ · ∇ × [u L } − p] = 0 = 0,(Lagrangian and Eulerian meanflows are equal to leadingorder for Boussinesqwavepackets, but this versionholds more generally)the mean material derivative following u L .† The key to the pHence only the vertical curl of pseudomomentum affects themean flow3-5.4) is that for small wave amplitude p can be computed athe linear equations alone and the mean-flow response can thethese relations. We will now turn to such small-amplitude cothat the above equations are valid a finite amplitude.Oliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005Vertical pseudomomentum curlNeglect intrinsic group velocityp = k A = ∇θ Aand thereforehorizontal projectionarea-preserving mapẑ · ∇ × p = lA x − kA y∝= dA dθ = const.as both A and the wavephase are advected byarea-preserving flowExponential surge in pseudomomentumbut not in its vertical curl

Bretherton`s flow (1969)O(a 2 )Large-scale dipolar return flowat second order in wave amplitudeFar-field mean velocity is nondivergentand decays withsquare of distance to wavepacketThe impulse (ie the skew linearmoment of vorticity) of thislayerwise 2d flow is welldefined, but not its momentumFeynman:“children on a slide”Oliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005Wave-vortex dualityWavepacketDual vortex dipoleSame mean flow fieldwave pseudomomentum = dual vortex impulse.....suggests new thinking of interactions....

Straining of wavepacket and vorticesWave capture and wave–vortex duality 3Figure 1. Left: wavepacket exposed to pure horizontal strain contracting along the x-axis andextending along the y-axis. The wavecrests align with the extension axis and their spacing isdecreased, so that the wavenumber vector k points at right angles to the extension axis andgrows in magnitude, as suggested by the large arrow. Right: a pair of oppositely signed vorticesexposed to same strain. The arrow now indicates the vortex pair’s Kelvin impulse.Wavepacket and vortex dipole are strained in the same wayHere we show in detail how the resolution of the paradox lies precisely in the remotenessof the associated recoil and in the presence of missing forces and momentum fluxes —missing, Oliver that Bühler, is, from the Courant above picture Institute, — which operate New York throughout University, the evolution 2005 of the

q = ẑ · ∇ × (u − p H ) ; D q = 0 .dimensional accountsnondissipativeforFar-fieldvortex the exponentialrecoilpair motion,effectssketched rate ofhave beeninbut increasepushedfigure also 3b, ofhow theoutwithpacket-integratedof sight,velocities the individualto infinity.falling ratesIn componentquare the situation of distance. of figure For convenience 1a. we call the layerwise-2-dimensionalindicated llection to wave ofin Pseudomomentum wavepackets propagationfigure 3a∂u the L and + impulse = conserved∂x + Bretherton ∂vL vortices and∂y = 0 flow and within horizontalof∂v the a L closed refraction.wavepacket;∂x − ∂uL system∂y = qL +the whose For insẑ ·part∇ × p H ,nfinity, the−(∂u wavepacket all the L j /∂x flux will i terms )p also Hj be vanish that called fast accounts theenough wavepacket to for make the return noexponential contributionflow. In B69 rate oly, fieldthis pseudomomentum wasremains ( ∂vq L =∂x − ∂u)Lgeneralizingcomputed true(5.5),ineven termsfor hence in of qapplying L thedistributions usual situation bothEulerian-mean withto the Bretherton ofnonvanishingfigure velocityproblem 1a. u,q L ≡ 0ng ince Potential over withextensions a wave q L , vorticity iperiod and pto arbitrary at H fixed all compactwavefields x . It was orandpointed sufficiently = ẑ · ∇ × (u L − p∂yvortices. out in evanescent B69 that H ) theGLM theorysimilarly the Eulerian-mean used hereIfcompact we now define or flowevanescent utheinimpulse figureare 3a I and those is formally itsquadratic density well i by defined, u L which, by anent ses, volume boundaries are O(rintegral −2 ) where recede — unlike r∫ 2 ∫= ∫to xthe 2 + infinity, momentum y 2 . So byall adding — the andflux (7.5a) that in terms to the (7.5b) rayationImpulse atvanish fasave simply theinfinity.impulse of I(t) Remarkably,u=is just equal i (x, to t) dxdydz thethispacket-integratedremains where truehorizontal i = even (y, −x, for 0) q L q,L. That is, B69 showed thatskew linear moment of PV( P H∂v, −x, 0)∂x − ∂u+ I ) = constant , ∫ ∫ ∫(7.6)Pseudomomentum dxdydz = P H ≡ p∂yH dxdydz (5.1)ng individual only flux rates terms of change not aresimilarly nonzero onlycompact when there or is horizonere,hence creation or destruction of horizontal pseudomomentum:evanescent aorizontal projection∫ ∫ ∫of the pseudomomentum density p, with p itselfeoretic dP and H approximation integrating, by the westandard formula= − (∇dtp = Ḙ H u L have simply) · p H dxdydz (7.7)∫ ∫ ∫ k = Ak Refraction . terms(5.2)dIω P H + I = constant ,1) thedt impulse = integral(∇ H uis L )absolutely· p H dxdydzconvergent,.essentially because(7.8)compactred thatsupport.I , thoughTonotsufficientits rateapproximation,of change, dependsthe integrandon theischoicezero atofnd below the wavepacket. At the intervening z values, the return flow isn q L tal hasrefraction nonvanishingsomewhere, monopole moment. hence creation or destructionnal outside Oliver theBühler, wavepacket Courant implying Institute, (∂v/∂x −New ∂u/∂y) York ≡University, ẑ · ∇ × u = 02005For an arbitrary collection of wavepackets and vorticmonopole moment, since with q L , i and p H all compaeven in monopolar cases, are O(r −2 ) where r 2 = x 2 + ywhile the compensating individual rates of change are no

The resolutionDipole straining increases wavepacket pseudomomentum.Bretherton flow advects vortex dipole and reduces impulse.Both compensate and the sum of P + I is conserved!Looks preciselylike dipole-dipoleleap-froginteraction...no accident!Non-local interactionat a distanceOliver Bühler, Courant Institute, New York University, 2005

Oliver Bühler, Courant Institute, New York University, 2005Duality and dissipationWavepacketVortex dipoledissipationDissipationmakes dualvortex real, butDissipation itself does not accelerate the mean flow!

Oliver Bühler, Courant Institute, New York University, 2005Towards a new parametrization schemeJoint work with John Scinocca (CCCma, Univ. Victoria, CA)1. based on existing columnar parametrization scheme2. wave dissipation and breaking part unchanged, only nondissipativepropagation part is changed3. requires no new assumptions on gravity wave launchspectrum (have only weak observational constraints)4. requires horizontal derivatives of model fields, whichis a bit non-trivial in operational GCMs because ofparallel architecture

Oliver Bühler, Courant Institute, New York University, 2005The key differenceOld scheme is based on constant pseudomomentum fluxNew scheme is based on constant wave action fluxpseudomomentum flux(z) = k(z) wave action fluxchanges in k(z) due to horizontal refraction change thepseudomomentum flux(z) and hence producenew mean-flow forces

Concluding imageWave action fluxreplacespseudomomentumfluxRefraction leads towave-meanmomentum exchangeswithout dissipationMountain drag doesnot simply equal wavedrag anymore(it never did)Oliver Bühler, Courant Institute, New York University, 2005