Inmeteorology,lee waves areatmospheric stationary waves. The most common form ismountain waves, which are atmospheric internalgravity waves. These were discovered in 1933 by two Germanglider pilots,Hans Deutschmann andWolf Hirth, above theGiant Mountains.^[1][2][3]They areperiodic changes ofatmospheric pressure,temperature andorthometric height in acurrent ofair caused by vertical displacement, for exampleorographic lift when thewind blows over amountain ormountain range. They can also be caused by the surface wind blowing over anescarpment orplateau,^[4] or even by upper winds deflected over athermalupdraft orcloud street.

The vertical motion forces periodic changes inspeed anddirection of the air within this air current. They always occur in groups on thelee side of theterrain that triggers them. Sometimes, mountain waves can help to enhance precipitation amounts downwind of mountain ranges.^[5] Usually aturbulentvortex, with itsaxis of rotation parallel to the mountain range, is generated around the firsttrough; this is called arotor. The strongest lee waves are produced when thelapse rate shows a stable layer above the obstruction, with an unstable layer above and below.^[4]

Strong winds (with wind gusts over 100 miles per hour (160 km/h)) can be created in the foothills of large mountain ranges by mountain waves.^[6][7][8][9] These strong winds can contribute to unexpected wildfire growth and spread (including the2016 Great Smoky Mountains wildfires when sparks from a wildfire in the Smoky Mountains were blown into the Gatlinburg and Pigeon Forge areas).^[10]

Lee waves are a form ofinternal gravity waves produced when a stable,stratified flow is forced over an obstacle. This disturbance elevates air parcels above their level ofneutral buoyancy. Buoyancy restoring forces therefore act to excite a verticaloscillation of the perturbed air parcels at theBrunt-Väisäla frequency, which for the atmosphere is:

${\displaystyle N=\sqrt{\frac{g}{{\theta }_0}\frac{d{\theta }_0}{dz}}}$, where${\displaystyle {\theta }_0(z)}$ is the vertical profile ofpotential temperature.

Oscillations tilted off the vertical axis at an angle of${\displaystyle \varphi }$ will occur at a lowerfrequency of${\displaystyle N\cos \varphi }$. These air parcel oscillations occur in concert, parallel to the wave fronts (lines of constantphase). These wave fronts represent extrema in the perturbedpressure field (i.e., lines of lowest and highest pressure), while the areas between wave fronts represent extrema in the perturbedbuoyancy field (i.e., areas most rapidly gaining or losing buoyancy).

Energy is transmitted along the wave fronts (parallel to air parcel oscillations), which is the direction of the wavegroup velocity. In contrast, the phase propagation (orphase speed) of the waves points perpendicular to energy transmission (orgroup velocity).^[11][12]

Both lee waves and the rotor may be indicated by specificwave cloud formations if there is sufficient moisture in the atmosphere, and sufficient vertical displacement to cool the air to thedew point. Waves may also form in dry air without cloud markers.^[4] Wave clouds do not move downwind as clouds usually do, but remain fixed in position relative to the obstruction that forms them.

• Around thecrest of the wave,adiabatic expansion cooling can form a cloud inshape of alens (lenticularis). Multiple lenticular clouds can be stacked on top of each other if there are alternating layers of relatively dry and moist air aloft.
• The rotor may generatecumulus orcumulus fractus in its upwelling portion, also known as a "roll cloud". The rotor cloud looks like a line of cumulus. It forms on the lee side and parallel to the ridge line. Its base is near the height of the mountain peak, though the top can extend well above the peak and can merge with the lenticular clouds above. Rotor clouds have ragged leeward edges and are dangerously turbulent.^[4]
• Afoehn wall cloud may exist at the lee side of the mountains, however this is not a reliable indication of the presence of lee waves.
• Apileus or cap cloud, similar to a lenticular cloud, may form above the mountain or cumulus cloud generating the wave.
• Adiabatic compression heating in the trough of each wave oscillation may alsoevaporatecumulus orstratus clouds in theairmass, creating a "wave window" or "Foehn gap".

Lee waves provide a possibility forgliders to gainaltitude or fly long distances whensoaring. World record wave flight performances for speed, distance or altitude have been made in the lee of theSierra Nevada,Alps,PatagonicAndes, andSouthern Alps mountain ranges.^[13] ThePerlan Project is working to demonstrate the viability of climbing above thetropopause in an unpowered glider using lee waves, making the transition intostratospheric standing waves. They did this for the first time on August 30, 2006 inArgentina, climbing to an altitude of 15,460 metres (50,720 ft).^[14][15] TheMountain Wave Project of theOrganisation Scientifique et Technique du Vol à Voile focusses on analysis and classification of lee waves and associated rotors.^[16][17][18]

The conditions favoring strong lee waves suitable for soaring are:

• A gradual increase in windspeed with altitude
• Wind direction within 30° of perpendicular to the mountain ridgeline
• Strong low-altitude winds in a stable atmosphere
• Ridgetop winds of at least 20 knots

The rotor turbulence may be harmful for other smallaircraft such asballoons,hang gliders andparagliders. It can even be a hazard for large aircraft; the phenomenon is believed responsible for manyaviation accidents and incidents, including the in-flight breakup ofBOAC Flight 911, aBoeing 707, nearMount Fuji,Japan in 1966, and the in-flight separation of an engine on anEvergreen International AirlinesBoeing 747 cargo jet nearAnchorage, Alaska in 1993.^[19]

The rising air of the wave, which allows gliders to climb to great heights, can also result in high-altitude upset in jet aircraft trying to maintain level cruising flight inlee waves. Rising, descending or turbulent air, in or above the lee waves, can causeoverspeed,stall or loss of control.

There are a variety of distinctive types of waves which form under different atmospheric conditions.

• Wind shear can also create waves. This occurs when anatmospheric inversion separates two layers with a marked difference in wind direction. If the wind encounters distortions in the inversion layer caused bythermals coming up from below, it will create significant shear waves in the lee of the distortions that can be used for soaring.^[20]

• Hydraulic jump induced waves are a type of wave that forms when there exists a lower layer of air which is dense, yet thin relative to the size of the mountain. After flowing over the mountain, a type of shock wave forms at the trough of the flow, and a sharp vertical discontinuity called thehydraulic jump forms which can be several times higher than the mountain. The hydraulic jump is similar to a rotor in that it is very turbulent, yet it is not as spatially localized as a rotor. The hydraulic jump itself acts as an obstruction for the stable layer of air moving above it, thereby triggering wave. Hydraulic jumps can be distinguished by their towering roll clouds, and have been observed on theSierra Nevada range^[21] as well as mountain ranges in southern California.
• Hydrostatic waves are vertically propagating waves which form over spatially large obstructions. In hydrostatic equilibrium, the pressure of a fluid can depend only on altitude, not on horizontal displacement. Hydrostatic waves get their name from the fact that they approximately obey the laws of hydrostatics, i.e. pressure amplitudes vary primarily in the vertical direction instead of the horizontal. Whereas conventional, non-hydrostatic waves are characterized by horizontal undulations of lift and sink, largely independent of altitude, hydrostatic waves are characterized by undulations of lift and sink at different altitudes over the same ground position.
• Kelvin–Helmholtz instability can occur when velocity shear is present within a continuous fluid or when there is sufficient velocity difference across the interface between two fluids.
• Rossby waves (or planetary waves) are large-scale motions in the atmosphere whose restoring force is the variation in Coriolis effect with latitude.

• Gravity wave
• Nor'west arch

• Alexander, P.; Luna, D.; Llamedo, P.; de la Torre, A. (2010-02-19)."A gravity waves study close to the Andes mountains in Patagonia and Antarctica with GPS radio occultation observations".Annales Geophysicae.28 (2):587–595.Bibcode:2010AnGeo..28..587A.doi:10.5194/angeo-28-587-2010.hdl:11336/61424.ISSN 0992-7689.

• Grimshaw, R., (2002).Environmental Stratified Flows. Boston: Kluwer Academic Publishers.
• Jacobson, M., (1999).Fundamentals of Atmospheric Modeling. Cambridge, UK: Cambridge University Press.
• Nappo, C., (2002).An Introduction to Atmospheric Gravity Waves. Boston: Academic Press.
• Pielke, R., (2002).Mesoscale Meteorological Modeling. Boston: Academic Press.
• Turner, B., (1979).Buoyancy Effects in Fluids. Cambridge, UK: Cambridge University Press.
• Whiteman, C., (2000).Mountain Meteorology. Oxford, UK: Oxford University Press.

Wikimedia Commons has media related toOrographic waves.

• Mountain Wave Project official website
• Chronological collection of meteorological data, satellite pics and cloud images of mountain waves in Bariloche, Argentina (in Spanish)
• On High Winds and Foehn Warming associated with Mountain-Wave Events in the Western Foothills of the Southern Appalachian Mountains
• An Examination of the Areal Extent of High Winds due to Mountain Waves along the Western Foothills of the Southern Appalachian Mountains
• SOUTHTRAC (Transport and Composition of the Southern Hemisphere Upper Troposphere and Lower Stratosphere) Campaign in Southern Argentina

• Atmospheric dynamics
• Cumulus
• Gliding meteorology
• Gravity waves
• Mesoscale meteorology
• Mountain meteorology
• Sailing

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