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Mountain wave schematic. The wind flows towards a mountain and produces a first oscillation (A). A second wave occurs farther and higher with lenticular clouds stuck on top of the flow (B).

In meteorology, lee waves, are atmospheric standing waves. The most common form is mountain waves, which are atmospheric internal gravity waves, and were discovered in 1933 by German glider pilots above the Riesengebirge.^[1] They are periodic changes of Atmospheric pressure, temperature and orthometric height in a 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,^[2] or even by upper winds deflected over a thermal updraft or cloud street.

The vertical motion forces periodic changes in speed and direction of the air within this air current. They always occur in groups on the lee side of the terrain that triggers them. Usually a turbulent horizontal vortex is generated around the first trough, the so called 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.^[2]

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 below the dew point. Waves may also form in dry air without cloud markers.^[2] 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 of the wave, Adiabatic expansion cooling can form a cloud in shape of a lens (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 or cumulus fractus in its upwelling portion, also known as a "roll cloud".^[2]
• 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 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 also evaporate cumulus or stratus clouds in the airmass, creating a "wave window" or "Foehn gap".

A Wave window over the Bald Eagle Valley of central Pennsylvania as seen from a 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

Lee waves provide a possibility for gliders to gain altitude or fly long distances when soaring. World record wave flight performances for speed, distance or altitude have been made in the lee of the Sierra Nevada, Alps, Patagonic Andes, and Southern Alps mountain ranges.^[3] 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 stratospheric standing waves. They did this for the first time on August 30, 2006 in Argentina, climbing to an altitude of 50,671 feet (15,447 m).^[4]

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 balloons, hang gliders and para gliders. It can even be a hazard for large aircraft; the phenomenon is believed responsible for many aviation accidents and incidents including the in-flight break up of BOAC Flight 911, a Boeing 707, near Mt. 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.^[5]

The rising air of the wave, which allows gliders to fly at high speed, 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 or stall, resulting in mach tuck and loss of control, especially when the aircraft is operated near the "coffin corner".

There are a variety of distinctive types of Lee waves which form under different atmospheric conditions. Some less commonly known types are as follows:

• Hydraulic Jump Induced Waves: this type of wave 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 distinguished by their towering roll clouds, and have been observed on the Sierra Nevada range ^[6] as well as mountain ranges in southern California.

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  Hydrostatic wave (schematic drawing)
  Hydrostatic Waves: 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

• Grimshaw, R., (2002). Environmental Stratified Flows. Boston: Kluwer Academic Publishers.
• Jacobson, M., (1999). Fundamentals of Atmospheric Modeling. Cambridge: 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: Cambridge University Press.
• Whiteman, C., (2000). Mountain Meteorology. Oxford Oxfordshire: Oxford University Press.

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