' ' , and z ' components ( u u u'

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1 Mesosale Meteorology: Gravity Waves 3 April 07 Introdution Here, we priarily onsider internal gravity waves, or waves that propagate in a density-stratified fluid (noinally, a stably-stratified fluid, with lower density air residing above higher density air) under the influene of buoyany fores. Indeed, it is buoyany that ats as the restoring fore for internal gravity waves; i.e., a parel perturbed upward or downward will be restored to, or osillate beyond, its initial loation as a funtion of its buoyany relative to its surroundings. Reall that buoyany is defined as: ' v ' ' B g g g The first approxiate for holds in the absene of hydroeteors given greater variation in density due to teperature rather than pressure perturbations. The seond approxiate for holds for dry dynais; e.g., when oisture is altogether negleted. If base-state potential teperature inreases with height, a parel perturbed upward (downward) by an infinitesial distane will have negative (positive) perturbation potential teperature; buoyany restores the parel downward (upward). Matheatial Fraewor Internal gravity waves are one of any allowable solutions to the priitive equations. To onsider the struture and propagation of internal gravity waves in the atosphere without onsidering full atospheri oplexity, we first ae a few siplifying assuptions: The base-state environent is in hydrostati balane (i.e., no vertial parel aelerations). All parel displaeents are adiabati (i.e., onserving potential teperature). The Coriolis and fritional fores are negleted. The Boussinesq approxiation is ade, where density is assued onstant (i.e., 0, or inopressible) exept where it appears with buoyany. This eliinates the restoring fore for aousti/sound waves, thus eliinating these non-eteorologial waves fro the priitive equations. We ust retain density variations with buoyany, however, to obtain internal gravity waves. We assue that the salient wave dynais an be represented by two-diensional planar waves in the x-z diretion. In general, the x diretion an be taen to be any diretion along whih the waves propagate; i.e., internal gravity waves do not always propagate to the east or west. The relevant eteorologial variables an be deoposed into base state and perturbation v oponents ( u u u', w w', z ', p pz p', and z ' ) to aid in linearizing the priitive equations. Note that the density deoposition applies only with buoyany, while we assue no base-state vertial otion suh that w = w only.

2 Given these siplifying assuptions, we an linearize the priitive equations: naely, the x- and z-diretion oentu, ontinuity, and therodynai equations. Doing so, we obtain: u' u' u t x 0 p' x w' w' u t x 0 p' ' g z u' w' 0 x z ' ' u w' 0 t x z This set of four equations ontains four unnowns: u, w, p, and θ. To solve for these variables, we ollapse the syste into a single equation for a single unnown by progressively rewriting eah unnown in ters of one or ore of the other variables. Most oonly, we ollapse this syste into a single equation for w, fro whih we an obtain u, then p and θ. We assue that solutions for eah unnown variable tae a two-diensional plane wave for: f ' x, z, t f ˆ exp ix z t where is the zonal wavenuber, is the vertial wavenuber, and ω is the osillation frequeny. Note that: x z In other words, wavenuber is an inverse funtion of wavelength. Sall wavenubers iply long wavelengths, and vie versa. Note that x + z ωt is defined as the wave phase ϕ. If we obine the above equation set into a single equation for w, we an obtain the dispersion relation, representing an expression for the osillation frequeny ω: u N Here, N is the Brunt-Vaisala frequeny and is equal to the square root of the stati stability N : N g z Stati stability is zero when the environental lapse rate is dry adiabati ( onstant with height) and is a positive value when the environental potential teperature inreases with height. Thus,

3 ω depends on the base-state wind, horizontal and vertial wavelengths (through and ), and the base-state stati stability (through N). The period of osillation for any given wave is equal to: For an internal gravity wave, if ω ~ N, whih is soewhat justified fro the dispersion relation, and assuing 300 K and 6 K -, then τ = s ~ 7.5 in. In other words, internal z gravity waves are high frequeny waves. The phase speed of an internal gravity wave, desribing the speed and diretion of otion of wave phase (e.g., rests and troughs) is given by: px u N pz u N The group veloity of an internal gravity wave, desribing the speed and diretion of otion of the wave and its energy, is given by: gx gz u N 3 N 3 The phase speed is a funtion of wavenuber and does not equal the group veloity. Thus, internal gravity waves are dispersive; e.g., different wavelength waves oprising the full wave for have different propagation harateristis. In other words, a wave s phase propagates in a different sense than does its energy. If we assue two-diensional plane wave solutions for all unnown variables (u, w, p, and θ ; e.g., u' uˆ exp ix z t ), substitute into the siplified priitive equation set, siplify, and solve for the unnowns in ters of eah other, we obtain the polarization relations: i ' w' u' u' N z 3

4 p' N 0 u' We an use these solutions and inforation about the phase speed and group veloity to infer the struture and propagation harateristis of internal gravity waves. This disussion is based upon the diagra on the seond slide of the leture aterials. This disussion is ain to but slightly different than that in the ourse textboo. The onepts here and in the ourse text are generalizable to any orientation of phase line so long as the appropriate signs of and are dedued and the interpretation hanged to ath. Consider phase lines, or lines along whih phase x + z = onstant, sloping up and to the right as depited on the aforeentioned diagra. Manipulating this equation, we find that: z x onstant The slope of the phase lines is thus equal to -/. Given phase lines with positive slope, this eans that and have opposite sign. How do we now whih sign they have? Consider px and pz. If we onsider the positive root for eah and assue N > 0, then px > 0 and pz < 0. This desribes a phase speed direted down and to the east. Realling the definitions of and, this iplies that is positive (λx in the positive x-diretion) and is negative (λz in the negative z-diretion) for the drawn phase lines. We an show that parel osillations are along phase lines, with frequeny given by the dispersion relation. We also an show that the phase speed vetor is perpendiular to the phase lines. Consider its slope (e.g., rise over run): pz px This is the negative of the inverse of the slope of the phase lines theselves. We an also show that the group veloity is along the phase lines and that the group veloity vetor is direted up and to the right. Assuing N > 0 and taing the positive root, gx > 0. For and of opposite sign, and again assuing N > 0 and taing the positive root, gz > 0. The group veloity vetor s slope is given by: gz gx This is idential to the slope of the phase lines theselves. Together, these indiate a group veloity along the phase lines direted up and to the right. The waves propagate up and to the east (as given by the group veloity), while the individual rests and troughs along a wave propagate down and to the east (as given by the phase speed). In general, internal gravity waves propagate upward and away fro their soures, where for this diagra the soure an be envisioned to be at the origin. 4

5 Given and of opposite sign, we now that w has equal sign to u ; i.e., eastward-oving parels asend while westward-oving parels desend. For N > 0 and the positive root (for eastwardoving waves), p has equal sign to u ; eastward-oving parels have high perturbation pressure while westward-oving parels have low perturbation pressure. Finally, given < 0, N > 0, and base-state potential teperature inreasing with height, θ has opposite sign to and is 90 out of phase (due to the leading i) fro u or, equivalently, w. This an be understood physially: an asending (desending) air parel is denser (less dense) than its surroundings. At its axiu upward (downward) displaeent, whih ours when w is equal to zero, the air parel is older (warer) than its surroundings. Duted Mesosale Gravity Waves Gravity waves are ubiquitous atospheri features, and the ajority propagate up and away fro their origins while steadily dereasing in aplitude. Refration and refletion our as any wave approahes the interfae between two distint layers with different refrative indies. For atospheri gases, refrative index is diretly proportional to density. Thus, a siple appliation to the atosphere is that of a wave approahing the interfae between two layers of varying density; e.g., a layer near the ground with higher density beneath a layer with lower density. Radar bea refration is an exaple of these onepts. Dereasing atospheri density with height results in radar bea refration. Subrefration (superrefration) ours when density dereases less (ore) rapidly with height than noral suh that the hange in refrative properties between layers is saller (larger) than noral. Upward-propagating internal gravity waves ay also beoe trapped, or duted, when the hange in refrative properties aross the interfae between high and low density layers is large enough to result in downward wave refration. We now wish to deterine the onditions leading to gravity wave duting, where wave refration at the top and botto of a higher-density layer results in trapped (i.e., priarily horizontally-propagating) waves. While wind disontinuities ay result in duting, the priary ause of duting, and thus our fous here, is on duting fostered by vertial density variations. For sipliity, we assue a gravity wave in the x-z diretion, although gravity waves an and do propagate in any horizontal diretion. The Rihardson nuber is a easure of the ratio of buoyany foring to vertial wind shear, i.e., Ri N u z g z u z Here, N is the stati stability. The denoinator is the squared agnitude of the zonal vertial wind shear. The Rihardson nuber is sall when vertial wind shear is large and large when the stati stability is large (e.g., with a large inrease in potential teperature with height). This Rihardson nuber forally applies only in the ase of subsaturated onditions; for saturated onditions, the stati stability should be replaed by the oist stati stability N, whih is a funtion of θe. 5

6 Duting potential is assessed by oputing the Rihardson nuber over the less dense layer; i.e., for the layer atop the atual duting layer whih has higher density. For lower tropospheri gravity waves, this iplies that the ground is the lower bound on the duting layer. Given this, the following insight an be obtained: For Ri < 0.5, the internal gravity wave is refrated in the duting layer with extration of energy fro the ean flow. Noinally, this desribes an upper layer with stati stability approahing zero (i.e., lapse rate near dry adiabati) and large vertial wind shear. For 0.5 < Ri <, the internal gravity wave is refrated in the duting layer with loss of energy to the ean flow, ore so for larger Ri. Noinally, this desribes an upper layer with soewhat greater stati stability and/or redued vertial wind shear when opared to the previous ase. For Ri >, internal gravity wave refration and duting is unliely; the wave rapidly loses energy to the ean flow. We require that the gravity wave horizontal phase speed p exeed the base-state horizontal wind u at all levels within the duted layer. In other words, a ritial level, where p = u, annot exist within the duted layer. Upon approahing a ritial level, the restoring fore (related to buoyany) beoes infinitesially sall, resulting in parel osillations beoing inreasingly horizontal and the gravity wave beoing part of the ean flow. We now that any wave, inluding gravity waves, an be represented as the superposition of any waves of varying wavelength and aplitude. For duted gravity waves, it an be shown that the wave with the longest wavelength is that whih is ost liely to be duted. Thus, duted gravity waves have wavelengths of tens to hundreds of iloeters, as opared to typial gravity waves with wavelengths one to two orders of agnitude saller. We an also show that the stable duting layer has a iniu depth, given the longest wavelength wave, for duted gravity waves to our: ND p u, or p u D N Here, Brunt-Vaisala frequeny N is evaluated within the stable layer and the depth of the stable layer is given by D. The gravity wave s horizontal phase speed is given by p (assuing no vertial propagation) and the base-state horizontal wind is given by u. The neessary depth is larger for saller stati stability within the stable layer and/or larger differene between the phase speed and the base-state horizontal wind. Gravity Wave Charateristis and Environents Thus, duted gravity waves are ost liely when (a) there is a layer with oparatively high stability and high density beneath a layer with oparatively low stability and low density, (b) the higher density layer is suffiiently thi to allow for duting, and () the base-state horizontal wind speed is saller than the gravity wave s phase speed throughout the higher density layer. In general, duted gravity waves our within stable layers of suffiient depth (and low base-state 6

7 horizontal wind speed) that reside beneath layers with lapse rates that are nearly dry adiabati. A frontal inversion, whether ahead of a retreating war front or behind an advaning old front, an serve as an effetive duting layer. Many esosale gravity waves have been douented when suh environents are olloated with jet streas, where the assoiated horizontal flow ibalane an trigger inertia-gravity waves (for whih the Coriolis fore is also iportant). Duted gravity waves are haraterized by olloated old/high and war/low teperature and pressure anoalies. For an eastward-oving wave with no ean flow, onvergene and asent are axiized east (west) of the high (low) pressure anoaly. Divergene and desent are axiized west (east) of the high (low) pressure anoaly. Vertial otion is zero at the pressure anoalies. Maxiu westerly (easterly) wind speed is found with the high (low) pressure anoaly. Pressure anoalies with duted gravity waves are of agnitude -0 hpa. As noted previously, their horizontal wavelengths an be tens to hundreds of iloeters. We also quantified an approxiate wave period of 7.5 in, whih opares favorably to the periods of observed duted gravity waves that range between 5-30 in. Given typial pressure anoaly agnitudes and wave periods, loal surfae pressure tendeny an be of order 0 hpa h -. There are any potential triggers for gravity waves, duted or otherwise. The ost oon is the geostrophi adjustent proess, wherein gravity waves at to restore geostrophi balane between the ass (or pressure) and wind fields. The wind field adjusts to the pressure field for large-sale disturbanes, the pressure field adjusts to the wind field for sall-sale disturbanes, and the wind and pressure fields utually adjust to eah other for disturbanes of interediate sale. Gravity waves redistribute ass, odulate veloity, or both to restore geostrophi balane. Flow along or over sharply-sloping terrain and the release of shearing instability an also trigger gravity waves. Thunderstors, or deep, oist onvetion, an also result in gravity waves through two eans. In the first, updrafts extending into the high-stability region of the tropopause an trigger buoyany osillations assoiated with gravity waves. These are depited in satellite iagery as pulsating waves that ove radially outward fro the updraft. In the seond, downdrafts and their assoiated density urrents an fore otherwise stable air upward over the even ore stable density urrent, triggering a gravity wave-lie buoyany osillation nown as a bore. The dynais of bores differ slightly fro those of pure gravity waves, however. In onditionally unstable environents, lifting assoiated with the upward rests of duted gravity waves ay be suffiient to bring air to its LCL, if not also its LFC, and thus an also serve as a trigger for thunderstor developent. 7