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{{Short description|Atmospheric stationary oscillations}}
[[File:Vol d'onde.svg|right|thumb|The wind flows towards a mountain and produces a first oscillation (A) followed by more waves. The following waves will have lower amplitude because of the natural damping. [[Lenticular cloud]]s stuck on top of the flow (A) and (B) will appear immobile despite the strong wind.]]
[[File:Lenticular clouds 1.jpg|thumb|Lenticular clouds]]
In [[meteorology]], '''lee waves''' are [[Earth's atmosphere|atmospheric]] stationary waves. The most common form is '''mountain waves''', which are atmospheric internal [[gravity wave]]s.  These were discovered in 1933 by two German [[glider pilot]]s, [[:de:Hans_Deutschmann|Hans Deutschmann]] and [[Wolf Hirth]], above the [[Giant Mountains]].<ref>On 10 March 1933, German glider pilot Hans Deutschmann (1911–1942) was flying over the Giant Mountains in Silesia when an updraft lifted his plane by a kilometre. The event was observed, and correctly interpreted, by German engineer and glider pilot Wolf Hirth (1900–1959), who wrote about it in:  Wolf Hirth, ''Die hohe Schule des Segelfluges'' [The advanced school of glider flight] (Berlin, Germany:  Klasing & Co., 1933).  The phenomenon was subsequently studied by German glider pilot and atmospheric physicist Joachim P. Küttner (1909 -2011) in: Küttner, J. (1938) "Moazagotl und Föhnwelle" (Lenticular clouds and foehn waves), ''Beiträge zur Physik der Atmosphäre'', '''25''', 79–114, and Kuettner, J. (1959) "The rotor flow in the lee of mountains." GRD [Geophysics Research Directorate] Research Notes No. 6, AFCRC[Air Force Cambridge Research Center]-TN-58-626, ASTIA [Armed Services Technical Information Agency] Document No. AD-208862.</ref><ref>{{cite magazine
|title=Modeling and Classification of Mountain Waves
|last=Tokgozlu
|first=A
|author2=Rasulov, M. |author3=Aslan, Z.
|date=January 2005
|volume=29
|issue=1
|page=22
|magazine=Technical Soaring
|issn=0744-8996
}}</ref><ref>{{cite web
| url = http://www.nateferguson.com/glider.html
| title = Article about wave lift
| access-date = 2006-09-28
}}</ref>
They are [[Frequency|periodic]] changes of [[atmospheric pressure]], [[temperature]] and [[orthometric height]] in a [[Air current|current]] of [[air]] caused by vertical displacement, for example [[orographic lift]] when the [[wind]] blows over a [[mountain]] or [[mountain range]]. They can also be caused by the surface wind blowing over an [[escarpment]] or [[plateau]],<ref name=Pagen/> or even by upper winds deflected over a [[thermal]] [[updraft]] or [[cloud street]].
The vertical motion forces periodic changes in [[speed]] and [[Boxing the compass|direction]] of the air within this air current. They always occur in groups on the [[Windward and leeward|lee]] side of the [[terrain]] that triggers them. Sometimes, mountain waves can help to enhance precipitation amounts downwind of mountain ranges.<ref>{{cite journal|author1=David M. Gaffin |author2=Stephen S. Parker |author3=Paul D. Kirkwood |title=An Unexpectedly Heavy and Complex Snowfall Event across the Southern Appalachian Region|journal=Weather and Forecasting|date=2003|volume=18|issue=2|pages=224–235|doi=10.1175/1520-0434(2003)018<0224:AUHACS>2.0.CO;2|bibcode=2003WtFor..18..224G|doi-access=free}}</ref> Usually a [[turbulent]] [[vortex]], with its [[rotation|axis of rotation]] parallel to the mountain range, is generated around the first [[Crest (physics)|trough]]; this is called a '''rotor'''. The strongest lee waves are produced when the [[lapse rate]] shows a stable layer above the obstruction, with an unstable layer above and below.<ref name=Pagen>{{cite book | last = Pagen | first = Dennis | title = Understanding the Sky | publisher = Sport Aviation Pubns | location = City | year = 1992 | isbn = 978-0-936310-10-7 | pages= 169–175 | quote= This is the ideal case, for an unstable layer below and above the stable layer create what can be described as a springboard for the stable layer to bounce on once the mountain begins the oscillation.}}</ref>
Strong winds (with wind gusts over {{convert|100|mph|km/h}}) can be created in the foothills of large mountain ranges by mountain waves.<ref name=WAFarticle>{{cite journal|author=David M. Gaffin|title=On High Winds and Foehn Warming Associated with Mountain-Wave Events in the Western Foothills of the Southern Appalachian Mountains|journal=Weather and Forecasting|date=2009|volume=24|issue=1|pages=53–75|doi=10.1175/2008WAF2007096.1|bibcode=2009WtFor..24...53G|doi-access=free}}</ref><ref>{{cite journal|author=M. N. Raphael|title=The Santa Ana winds of California|journal=Earth Interactions|date=2003|volume=7|issue=8|page=1 |doi=10.1175/1087-3562(2003)007<0001:TSAWOC>2.0.CO;2|bibcode=2003EaInt...7h...1R |doi-access=free}}</ref><ref>{{cite journal|author=Warren Blier|title=The Sundowner Winds of Santa Barbara, California|journal=Weather and Forecasting|date=1998|volume=13|issue=3|pages=702–716|doi=10.1175/1520-0434(1998)013<0702:TSWOSB>2.0.CO;2|bibcode=1978JAtS...35...59L|doi-access=free}}</ref><ref>{{cite journal|author=D. K. Lilly |title=A Severe Downslope Windstorm and Aircraft Turbulence Event Induced by a Mountain Wave|journal=Journal of the Atmospheric Sciences|date=1978|volume=35|issue=1|pages=59–77|doi=10.1175/1520-0469(1978)035<0059:ASDWAA>2.0.CO;2|bibcode=1978JAtS...35...59L |doi-access=free}}</ref> These strong winds can contribute to unexpected wildfire growth and spread (including the [[2016 Great Smoky Mountains wildfires]] when sparks from a wildfire in the Smoky Mountains were blown into the Gatlinburg and Pigeon Forge areas).<ref name=AMSpresentation>{{cite journal|author1=Ryan Shadbolt |author2=Joseph Charney |author3=Hannah Fromm|url=https://ams.confex.com/ams/2019Annual/meetingapp.cgi/Paper/353193|title=A mesoscale simulation of a mountain wave wind event associated with the Chimney Tops 2 fire (2016)|publisher=American Meteorological Society|date=2019|issue=Special Symposium on Mesoscale Meteorological Extremes: Understanding, Prediction, and Projection|pages=5 pp}}</ref>
== Basic theory ==
[[File:Lee wave GFD lab.JPG|thumb|A fluid dynamics lab experiment illustrates flow past a mountain-shaped obstacle. Downstream wave crests radiate upwards with their group velocity pointing about 45° from horizontal. A downslope jet can be seen in the lee of the mountain, an area of lower pressure, enhanced turbulence, and periodic vertical displacement of fluid parcels. Vertical dye lines indicate effects are also felt upstream of the mountain, an area of higher pressure.]]
Lee waves are a form of [[Internal wave|internal gravity waves]] produced when a stable, [[Atmospheric stratification|stratified]] flow is forced over an obstacle. This disturbance elevates air parcels above their level of [[neutral buoyancy]]. Buoyancy restoring forces therefore act to excite a vertical [[oscillation]] of the perturbed air parcels at the [[Brunt–Väisälä frequency|Brunt-Väisäla frequency]], which for the atmosphere is:
<math>N = \sqrt{{g \over \theta_0}{d\theta_0 \over dz}}</math>, where <math>\theta_0(z)</math> is the vertical profile of [[potential temperature]].
Oscillations tilted off the vertical axis at an angle of <math>\phi</math> will occur at a lower [[frequency]] of <math>N\cos{\phi}</math>. These air parcel oscillations occur in concert, parallel to the wave fronts (lines of constant [[Phase (waves)|phase]]). These wave fronts represent extrema in the perturbed [[pressure]] field (i.e., lines of lowest and highest pressure), while the areas between wave fronts represent extrema in the perturbed [[buoyancy]] 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 wave [[group velocity]]. In contrast, the phase propagation (or [[Phase velocity|phase speed]]) of the waves points perpendicular to energy transmission (or [[group velocity]]).<ref>{{Cite book|title=Atmosphere-ocean dynamics|last=Gill|first=Adrian E.|publisher=Academic Press|year=1982|isbn=9780122835223|edition=1|location=San Diego, CA|url-access=registration|url=https://archive.org/details/atmosphereoceand0000gill}}</ref><ref>{{Cite book|title=Atmospheric Processes over Complex Terrain|last=Durran|first=Dale R.|date=1990-01-01|publisher=American Meteorological Society|isbn=9781935704256|editor-last=Blumen|editor-first=William|series=Meteorological Monographs|pages=59–81|language=en|doi=10.1007/978-1-935704-25-6_4|chapter = Mountain Waves and Downslope Winds}}</ref>
==Clouds==
[[File:Wave win.jpg|thumb|A wave window over the [[Bald Eagle Valley]] of central [[Pennsylvania]] as seen from a [[Glider (sailplane)|glider]] looking north. The wind flow is from upper left to lower right. The [[Allegheny Front]] is under the left edge of the window, the rising air is at the right edge, and the distance between them is 3–4 km.]]
Both lee waves and the rotor may be indicated by specific [[wave cloud]] formations if there is sufficient moisture in the atmosphere, and sufficient vertical displacement to cool the air to the [[dew point]]. Waves may also form in dry air without cloud markers.<ref name=Pagen/> Wave clouds do not move downwind as clouds usually do, but remain fixed in position relative to the obstruction that forms them.
* Around the [[Crest (physics)|crest]] of the wave, [[Adiabatic process#Adiabatic heating and cooling|adiabatic expansion cooling]] can form a cloud in [[shape]] of a [[Lens (geometry)|lens]] ([[lenticular cloud|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 generate  [[cumulus cloud|cumulus]] or [[cumulus 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.<ref name=Pagen/>
* A [[foehn]] wall cloud may exist at the lee side of the mountains, however this is not a reliable indication of the presence of lee waves.
* A [[pileus (meteorology)|pileus]] or cap cloud, similar to a lenticular cloud, may form above the mountain or cumulus cloud generating the wave.
* [[Adiabatic process#Adiabatic heating and cooling|Adiabatic compression heating]] in the trough of each wave oscillation may also [[evaporate]] [[cumulus cloud|cumulus]] or [[stratus cloud]]s in the [[airmass]], creating a "wave window" or "Foehn gap".
==Aviation==
Lee waves provide a possibility for [[Glider (sailplane)|gliders]] to gain [[altitude]] or fly long distances when [[Gliding|soaring]]. World record wave flight performances for speed, distance or altitude have been made in the lee of the [[Sierra Nevada (U.S.)|Sierra Nevada]], [[Alps]], [[Patagonia|Patagonic]] [[Andes]], and [[Southern Alps (New Zealand)|Southern Alps]] mountain ranges.<ref>[http://records.fai.org/gliding/ FAI gliding records] {{webarchive|url=https://web.archive.org/web/20061205015245/http://records.fai.org/gliding/ |date=2006-12-05 }}</ref> The [[Perlan Project]] is working to demonstrate the viability of climbing above the [[tropopause]] in an unpowered glider using lee waves, making the transition into [[stratosphere|stratospheric]] standing waves. They did this for the first time on August 30, 2006 in [[Argentina]], climbing to an altitude of {{convert|15460|m|ft}}.<ref>{{cite web |url=http://www.fai.org/fai-record-file/?recordId=14043 |title=Fai Record File |access-date=2015-01-27 |url-status=dead |archive-url=https://web.archive.org/web/20150413093412/http://www.fai.org/fai-record-file/?recordId=14043 |archive-date=2015-04-13 }}</ref><ref>[http://www.perlanproject.com/ Perlan Project]</ref> The [[Mountain Wave Project]] of the [[Organisation Scientifique et Technique du Vol à Voile]] focusses on analysis and classification of lee waves and associated rotors.<ref>[http://www.pa.op.dlr.de/ostiv/projects.htm OSTIV-Mountain Wave Project]</ref><ref name="MWP">[http://mwp.flightplanner.info/Defaultengl.htm] {{Webarchive|url=https://web.archive.org/web/20160303204340/http://mwp.flightplanner.info/Defaultengl.htm|date=2016-03-03}} – accessed 2009-11-03</ref><ref>{{cite magazine
|title=Leewaves in the Andes Region, Mountain Wave Project (MWP) of OSTIV
|last=Lindemann
|first=C
|author2=Heise, R. |author3=Herold, W-D.
|date=July 2008
|volume=32
|issue=3
|page=93
|magazine=Technical Soaring
|issn=0744-8996
}}</ref>
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 small [[aircraft]] such as [[Balloon (aeronautics)|balloons]], [[Hang gliding|hang gliders]] and [[Paragliding|paraglider]]s. It can even be a hazard for large aircraft; the phenomenon is believed responsible for many [[aviation accidents and incidents]], including the in-flight breakup of [[BOAC Flight 911]], a [[Boeing 707]], near [[Mount Fuji]], [[Japan]] in 1966, and the in-flight separation of an engine on an [[Evergreen International Airlines]] [[Boeing 747]] cargo jet near [[Anchorage, Alaska]] in 1993.<ref>[https://www.ntsb.gov/investigations/AccidentReports/Pages/AAR9306.aspx NTSB Accident Report AAR-93-06]</ref>
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 in [[lee waves]]. Rising, descending or turbulent air, in or above the lee waves, can cause [[Overspeed (aircraft)|overspeed]], [[Stall (flight)|stall]] or loss of control.
==Other varieties of atmospheric waves==
[[File:Hydrostatic.JPG|thumb|Hydrostatic wave (schematic drawing)]]
There are a variety of distinctive types of waves which form under different atmospheric conditions.
* ''[[Wind shear]]'' can also create waves. This occurs when an [[atmospheric inversion]] separates two layers with a marked difference in wind direction. If the wind encounters distortions in the inversion layer caused by [[thermal]]s coming up from below, it will create significant shear waves in the lee of the distortions that can be used for soaring.<ref>{{cite book
|last = Eckey
|first = Bernard
|title = Advanced Soaring Made Easy
|publisher = Eqip Verbung & Verlag GmbH
|year = 2007
|isbn = 978-3-9808838-2-5
}}</ref>
* ''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 the [[hydraulic 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 the [[Sierra Nevada (U.S.)|Sierra Nevada]] range<ref name="Kuettner+Hertenstein">[http://ams.confex.com/ams/pdfpapers/40363.pdf Observations of Mountain-Induced Rotors and Related Hypotheses: a Review] by Joachim Kuettner and Rolf F. Hertenstein</ref> 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 wave]]s'' (or planetary waves) are large-scale motions in the atmosphere whose restoring force is the variation in Coriolis effect with latitude.
==See also==
* [[Gravity wave]]
* [[Nor'west arch]]
==References==
{{reflist}}
* {{Cite journal |last1=Alexander |first1=P. |last2=Luna |first2=D. |last3=Llamedo |first3=P. |last4=de la Torre |first4=A. |date=2010-02-19 |title=A gravity waves study close to the Andes mountains in Patagonia and Antarctica with GPS radio occultation observations |url=https://angeo.copernicus.org/articles/28/587/2010/ |journal=Annales Geophysicae |language=English |volume=28 |issue=2 |pages=587–595 |doi=10.5194/angeo-28-587-2010 |doi-access=free |bibcode=2010AnGeo..28..587A |issn=0992-7689|hdl=11336/61424 |hdl-access=free }}
==Further reading==
* 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.
== External links ==
{{Commons category|Orographic waves}}
* [http://www.mountain-wave-project.com Mountain Wave Project official website]
* [https://web.archive.org/web/20110516090742/http://inglaner.com/meteorologia_onda.htm Chronological collection of meteorological data, satellite pics and cloud images of mountain waves in Bariloche, Argentina (in Spanish)]
* [https://journals.ametsoc.org/doi/full/10.1175/2008WAF2007096.1 On High Winds and Foehn Warming associated with Mountain-Wave Events in the Western Foothills of the Southern Appalachian Mountains]
* [http://nwafiles.nwas.org/digest/papers/2011/Vol35No1/Pg47-Gaffin-etal.pdf An Examination of the Areal Extent of High Winds due to Mountain Waves along the Western Foothills of the Southern Appalachian Mountains]
* [http://wp.df.uba.ar/atmosfera/southtrac SOUTHTRAC (Transport and Composition of the Southern Hemisphere Upper Troposphere and Lower Stratosphere) Campaign in Southern Argentina]
[[Category:Atmospheric dynamics]]
[[Category:Cumulus]]
[[Category:Gliding meteorology]]
[[Category:Gravity waves]]
[[Category:Mesoscale meteorology]]
[[Category:Mountain meteorology]]
[[Category:Sailing]]