Mountain Waves

Learning Goal 3d. Describe the characteristics and causes of mountain waves, relate them to winds and stability using the Froude-number tool, and describe how they affect flight

Intensity of Waves

When the air is relatively cold near the ground and warmer aloft, the air is statically stable (see Learning Goal 3b). If statically stable air blows toward a mountain range, then some of the lower altitude air might be blocked (or can go around the sides of isolated mountains as sketched in the first figure below), while the air closer to the mountain top can flow over the top. But this top air is deflected upward by the mountain. As this deflected air blows toward the other side of the mountain, it sinks back down to its original altitude, and can even overshoot downward. The resulting airflow traces out a wave downwind of the mountain range, as sketched in the figures below. Such terrain-induced vertical oscillations of the air are called mountain waves.

A variety of wave motions are possible as shown in the figure below. The type of motion depends on the relative strengths of the static stability (S) and the mean wind speed (M). The air has a "natural wavelength" that depends on M and S. Faster wind speeds (M) create longer wavelengths. Greater static stability (S) creates shorter wavelengths. You can use M and S in the Froude-number graphical tool below to estimate the natural wavelength (indicated in yellow at the top of the top graph).

For the example in that tool, if M = 20 km/h and S = 4°C/km, then follow the 20 km/h grey horizontal line until it hits the diagonal blue line for S = 4°C/km. From that point, follow the thin green lines vertically upward in the graph until you get to the top axis, which shows the natural wavelength of air for that case is roughly 200 km.

The ratio of natural wavelength to mountain width is called the Froude number (Fr) , which is dimensionless. If the natural wavelength of air matches the mountain width (W), then the most intense mountain waves can occur with the most violent up- and down-drafts.

The figure above shows how the natural wavelength of the waves (distance from crest to crest) varies with the Froude Number. ("Froude" rhymes with "food".)

Small Fr << 1 happens when winds (M) are light and/or static stability (S) is strong. For this situation, the natural wavelenth is shorter than the mountain width, causing weak mountain waves downwind of the mountain range (see Fig. a above).
The case of Fr = 1 causes the most violent waves with strongest up and downdrafts (see Fig b above). For this case, the natural wavelength of the wave matches the width of the mountain.
Large Fr >> 1 causes weaker amplitude, longer wavelength waves, with the possibility of a region of reversed wind flow near the ground close to the mountain (see Fig c).
Fr = infinity (for fast winds and weak static stability) creates strong turbulent wakes behind the mountain, but not much mountain waves.

How to find the Froude number

The Froude number is defined only for positive static stabilities S. Indeed, mountain waves can occur only when the air is statically stable. The reason is that if statically stable air is moved up, it tries to return to its starting altitude but overshoots too far down. Then it tries to return up to its starting altitude, but overshoots too far up. Namely, the air parcel oscillates up and down as it blows downwind of the mountain.

You can use the following tool to find the Froude number. (Since this Froude-number tool uses the static stability S as one of the inputs, I've copied the Static Stability graph from Learning Goal 3b and put it under the Froude number tool below.) The Froude-number and Static Stabilty tools will be provided on exams if needed to answer the exam questions.

(If the final exam has a question requiring you to estimate the Froude number, then this tool will be provided on the exam.)

As promised, here is a copy of the Static Stability tool from Learning Goal 3b:

static stability tool

Effects of Mountain Waves on Flight

Although scientifically a wave is a smooth periodic motion that is different from the quasi-random nature of turbulence, in aviation these motions are often reported as mountain-wave turbulence. Flying through these waves can cause the aircraft to be pushed up and down in a regular cycle, which pilots report as chop. Mountain waves can create strong up and downdrafts that can move your aircraft above or below your assigned or desired altitude — so the pilot must make frequent corrections to try to maintain altitude. Strong mountain waves can move the aircraft so quickly that it is impossible to maintain the correct altitude, and can even cause structural damage to the aircraft.

Sailplane pilots seek non-turbulent mountain waves because in it are regions of sustained updraft.

TIME: Such mountain waves allow sailplane pilots to stay aloft for longer durations.
DISTANCE: Also, if the wind is flowing over a long mountain range (instead of an isolated hill), then a geographically long region of updraft can exist. This allows sailplane pilots to travel long horizontal distances while remaining in the updraft.
ALTITUDE: If conditions are right, then the waves can exist from mountain top up into the stratosphere, allowing sailplanes to set altitude records. (A Windward Performance Perlan II glider set an unofficial altitude record of 22,646 m = 74,298 ft on 2 September 2018).

See more info for soaring in Learning Goal 2f. In the figure above, the updraft regions in the wave are to the left of the "wave crests". Also, note that these regions of wave lift tend to tilt upwind (they get closer toward the mountain location) as sailplanes fly toward higher altitudes.

When mountain waves form, they can also create wind shears that cause shear turbulence that does indeed have quasi-random motions. Thus, mountain waves often have turbulence and waves superimposed on each other, creating a very bumpy ride.

Lenticular clouds & Rotor clouds

If there is a layer of humid air near the altitude of the mountain wave, then sometimes a cloud called a lenticular cloud or mountain wave cloud or lee-wave cloud will form at the crests of the waves (see Learning Goal 1b for more photos of mountain-wave clouds). These provide visible evidence of the presence of mountain waves. But most mountain waves do not form lenticular clouds.

If you see a rotor clouds (the ragged clouds below the lenticular clouds in the figures below), that indicates extremely violent turbulence with up and downward motions that can exceed the climb speed of the aircraft. If the rotor is near the ground near an airport, then pilots should avoid that airport and land somewhere else.

Extra info for experts; not needed for this course.
The equation for the Froude number (Fr) is:
Fr = c · M / [ W · sqrt(S) ]
for wind speed M in km/h, mountain width W in km, and static stability S in °C/km (see the Static Stability Tool in learning goal 3b). For these units, the constant is c = 21.5. This equation was used to create the Froude Number tool in the image earlier in this section.

Key words: mountain wave, natural wavelength, Froude number, mountain-wave turbulence, chop, lenticular cloud, mountain-wave cloud, lee-wave cloud, rotor cloud