This article is an attempt to describe some of the features of soarable waves using words and diagrams alone. Some of the observations from satellites and reports from glider pilots show that even the most sophisti- cated mathematical models do not yAt give a complete description of all the varieties of wave patterns.
There are three main factors controlling the development of soarable
waves: these are the static stability of the air, the wind velocity and the
nature of the terrain.
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If the air is so moist that upward motion lcads to condensation of water vapour into cloud then the stability will be reduced and calculation of ihe periods of oscillation becomes more elaborate. To avoid this many mathematical modellers specified a dry atmosphere.
Wind Velocity. Some horirontal motion is necessary to force the air over a ridge. If the horizontal wind was 10 m/s (nearly 20kt) then the standard atmosphere with its oscillation period of 590 seconds would travel 5900m during the time needed to complete one full cycle. The air in the inversion layer would only need to travel 2800m to complete a full cycle. These distances can be thought of as the "natural" wavelegths of the particular layers at that wind speed. Clearly the stronger tho horizontal wind the longer would be these "natural' wavelengths.
The "natural" wavelength ot a particular layer of air is not the ~ee wavelength. In the atmosphere one can select many layers each with a different stability and wind velocity and hence a different "natural" wavelength. For lee waves to develop it is generally necessary for the "natural" wavelength to be longer at high levels than at low levels. In simple cases the lee wavelength is found to lie somewhere between the longest and shortest "natural" wavelength in a deep layer of air below the stratosphere. The stratosphere complicates the calculations and used to be neglected in the simplest lee wave models. More elaborate models showed that there were a number of occasions when neglect of the stratosphere did not invalidate the predictions. Simple models are still useful provided one knows their limitations.
Changes of wind speed with height. If one assumes that the stability remalns unchanged from the surface up to a great height then the "natural" wavelength at any level is controlled by the horizontal wind velocity. Fig. 2 shows how the propagation of wave energy from the surface is affected by changes of wind speed with height. In each diagram the wind speed at a particular level is represented by the length of a horizontal arrow on the left hand side.
In case A the wind is constant with height and so the "natural" wavelength is the same at all levels. Wave energy radiating from a point on the ground is shown as straight lines. Each line represents a different wavelength. Long waves are radiated almost vertically while short waves radiate at an angle nearer the horizontal.
In case B the wind speed is shown increasing with height; the
"natural" wavelength will increase with height too. The result is that the
rays of wave energy are bent over in the direction of the wind. The
shorter waves are turned back first, the longer waves later. It the winds
aloft were strong enough practically all the wave energy would be
reflected back. This would confine the energy within a limited depth of
the atmosphere and such waVes are called "trapped waves". When the
wave energy is trapped resonance can develop at certain wavelengths.
In the simplest cases only one wavelength achieves resonance and
becomes amplified to form a long train of lee waves extending far down-
stream from the point of origin. In more complicated cases there can be
relatively short waves at low levels with a longer wavelength higher up.
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If there is insufficient incrcease of wind with height some energy at the longer wavelengths will not be trapped. This is sometimes termed the "leaky' mode; a lee wave train can still develop but is not likely to extend so far downstream.
Case C is just the reverse of B. Now the wind speed (and the "natural" wavelength) decrease with height. The rays of wave energy are shown bending upwards. It would be very unusual to observe a steady decrease of wind speed from the surface to high levels but it is quite common to find a rapid dccrease at high levels just above the belt of strong winds known as a jet stream. It is possible for the wind direction to be reversed above a layer containing lee waves. When this happens the smooth flow can break down into violent turbulence.
A small decrease of wind speed with height may just cause the wave to become steeper like an ocean roller steepening as it approaches the beach. The shape of the wave may be controlled by other factors and, as is illustrated later, the horizontal wind speed is itself modified by the shape of the wave.
Researchers who make mathematical models of wave flow need to take account of the way waVes can alter both wind speed and air stability as they develop. Such effects arebest displayed as a moving picture on a video screen.
The simplest lee wave model. Any mathematical model made before computers were readily aVailable had to be very simple. The first model was produced by Scorer (1949). It only considered two dimensional flow and assumed that the air was dry and the streamlines had become steady. The atmosphere was silnplified to just two layers, each with a constant "natural" wavelength, separated by an interface which could undulate up and down to follow the shape of the streamlines of lee waves.
The depth of the lower layer could be varied but the upper layer had no fixed limit. It was assumed to be deep enough for the leee waves to die away to nothing before reaching the top.
Wave flow was set off by a redge whose shape was defined by a convenient mathematical formula which produces a bell shaped obstruction to the horizontal airflow. The height and width of the ridge could be altered independently; this effectively altered the aspect ratio of the ridge without changing its basic curves.
Predictions from the two layer model. The model would not predict lee waves unless the "natural" wavelength of the upper layer was longer than in the lower layer. The depth of the lower layer had to exceed a minimum value which was always more than a quarter wavelength in that layer. To illustrate this the two wavelengths chosen are 4.4km for the lower layer and 14km for the upper layer. The minimum depth needed in the lower layer works out as 1170m.
Fig 3 shows how the lee wavelength would change as the depth of the lower layer is increased. The pecked line shows the limiting depth. The vertical scale gives the height of the lower layer while the horizontal scale shows the wavclcnyth. Just above the minimum the lee wavelength is at its longest, almost the same as the "natural" wavelength of the upper layer. As the lower layer deepens the lee wavnlength shortens so that near the 3km level the lee wavelength has been reduced to less than 6km.
Fig 4 shows the effect of increasing the depth of the lower layer on the amplitude of the lee wave. The vertical scale gives the depth of the lower layer as before but the horizontal scale now shows the wave amplitude on a scale of zero to 1.0. (The actual amplitude depends on the dimensions of the ridge as well.)
Again the pecked line shows the minimum depth. Once the lower layer exceeds this depth the amplitude of the lee wave increases rapidly to reach a peak near the 1.5km level. As the depth of the lower layer increases beyond this the wave amplitude starts to die away.
Fig 5 shows how the amplitude of the wave would vary with height if all the factors were kept constant. In this case the top of the lower layer (marked "interface") is fixed at 2km on the upstrearn side of the ridge The curve shows the amplitude increasing from zero at the surface to reach its maximum near 1500m. Above this height the amplitude slowly decreases to become very small at the 5km level.
This decrease in amplitude depends on the difference between the lee wavelength and the "natural" wavelength in the upper layer. The greater the difference in wavelengths the more rapidly the amplitude should decrease with height.
Ridge width and wave amplitude.Although the lee wavelength is not affected by the dimensions of the ridge the amplitude of the wave is sensitive to these factors. Fig 6 shows how the width of the ridge affects the wave amplitude. The lee wavelength is the same in all examples In 6A the ridge is too narrow for the wavelength and the amplitude is small.
In 6B the lee wavelength and the ridge width fit, the system is in tune and the wave will develop its maximum amplitude. In 6C the ridge is much too wide, it is still falling away at the point where the streamline starts to rise again. This much reduces the amplitude.
In 6D a second ridge has been added one wavelength downwind. The second ridge boosts the original wave to much greater amplitude.
In 6E the spacing between ridges has been increased half a wavelength and now the descending streamline meets rising ground; this acts to cancel out the wave.
Ridge height and wave amplitude. Provided that the width of the ridge fits the lee wavelength, the higher the ridge the greater is the wave amplitude. However, high mountains are often wide mountains; wide mountains produce their greatest effect on the longer lee waves. Since long wavelength is closely associated with strong winds the bigger the mountain the stronger are the winds needed to produce the best waves. In contrast small ridges may set off large amplitude waves with relatively low wind speeds.
Separation of airflow. So far it has been assumed that the lowest streamline follows the shape of the ridge. Experience shows that this is not true with rugged or sharp edged ridges. There the airflow often breaks away leaving an eddy filling the gap left on the lee side. This eddy can act like an extension to the hill producing a smooth shape which can be followed by the wave pattern. If the lower air is very hazy one may observe the haze top undulating much more smoothly than the ground below.
lmprovements on the simple models. Although Scorers first model was a severe distortion at the real atmosphere the predictions have often proved remarkably good. However forecasters needed a more realistic variation of temperature and winds aloft from which to calculate lee wave behaviour. Casswell (1962) published a graphic method of working out wavelength, vertical velocity and the height of the best lift. His method was based on a paper by Foldvik who used a two layer atmosphere but instead of fixing the "natural" wavelengths in each layer allowed them to vary smoothly, increasing upward in an exponential curve which could often be adjusted to give a close fit to the real atmos- phere.
There was still a defect; waves were being observed when there was an unstable layer with cumulus in the lower layers. Wallington published equations for a three layer model. The bottom layer was the convective layer with no static stability. The next two layers were essentially the same as Scorer's two layer model. Wallington had effectively jacked up the Scorer model by inserting a convective layer underneath. This immediately increased the time needed for calculation but gave much better results on the many days when the waves were above the thermal layer. The effect of the convective layer was to increase the lee wavelength and decrease its amplitude. Thus the wavelength should become longer during the morning as the ground warms up and the convection deepens, but decrease in the evening when cooling begins In practical tests Wallington's model seems to give more accurate values for the wavelength, but is unreliable for the amplitude because the cumulus clouds themselves may act as extensions to the ridges.
Multi-layered models. The simple models ignored the effect of the stratosphere and yet were able to produce useful results on many occasions. This was because on the successful days the upper winds were strong enough to produce a very long "natural" wavelength which reflected practically all the wave energy before it reached the stratosphere. The tiny amount of energy which escaped into the stratosphere had no significant effect on the waVe train below.
Multilayered models were developed as soon as sufficient computing
power became available. The vast amount of extra work involved could
only be handled by a computer. With so many layers available the
modellers could follow the real atmosphere very closely instead of having
to apply heavy smoothing to simplify the calculations.
Inclusion of the stratosphere showed that lee waves could develop on
many more occasions than predicted by the simple models. However
long trains of lee waves did not develop at low levels on these days. The
waves were strong near the mountains but decayed downstream.
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More elaborate models were also able to study the changes in the wave pattern with time. The first models assumed the wave had reached a steady state. Now it was possible to show that even if the initial conditions in the approaching air were kept constant the development of the wave could produce startling changes in the flow over and down- stream. Such changes might take several hours to develop but the end result was sometimes a flow pattern totally unlike the srnooth sinusoidal pattern given in simple text books of meterology. In extreme cascs the air on the lee side of the mountains exhibited a vertical jump of several thousand feet and extreme turbulence developed in the lower part of the stratosphere. Such a pattern might have been disbelieved if it had not been previously observed by a number uf research aircraft making a deliberate transit of the wave system.
The shape of lee waves. In many esmentary text books the streamlines of lee wave flow show a symmetrical pattern with the wave crests on each streamline placed vertically above the one beneath. This kind of pattern seems to be broadly correct when there is a lonq train of lee waves well downstream of the mountain. High powered radar studies of such waves shows they are associated with a stable wavelength and steady flow pattern.
Fig 7 shows a computed pattern which is very different. In this diagram the crests of each streamline have been joined by a dot-dash line to emphasise the tilt of the wave front. The dotdash line represents the phase line of particular waves. The waves are not symmetrical, they tilt forward at a marked angle to the vertical and in certain sections the streamlines are practically vertical ahead of the wave crest but have only a gentle slope on the downstream side. The spacing between the streamlines is a guide to the wind velocity; the closer the lines the stonger is the flow at that point.
This kind of pattern is particularly interesting to a soaring pilot
Because it shows that:
(a) It is necessary to push forward during the climb to stay in lift.:
(b) The horizontal wind may drop off to nearly zero where the streamlines
approach the vertical. Circling in wave lift may be possible for
several thousand feet.:
(c) The apparently negligible horizontal wind found during part of the
climb can change into a strong headwind during a cruise to the next
upwind wave.:
(d) If the climb in the primary wave seems to die away there may be
better lift at that level in the downstream wave.:
The shape of the streamlines are modified by the shape of the ridge.
Fig 8 shows some of the variations.:
A. Here the ridge drops away too sharply; separation of flow occurs
leaving a lee eddy to fill the break in the flow.:
B. This shows a ridge where the upslope is gentle but the downslope is
steep. This produces very steep streamlines with a forward tilted wave
front.:
C. This shows the opposite situation to case B. The ascent is steep but
the descent from the ridge crc?st is gradual. The result is a much flatter
wave.
A practical example of this effect can be observed when a NW wind blows across the Ochils which have very steep slopes near Dollar but more gradual slopes north of Portmoak.
Three-dimensional waves. Studies of three dimensional waves are chiefly confined to the patterns which develops when the air flows over an isolated peak, or a very short length of ridge. These patterns are best seen on satellite pictures. They show up when there is a layer of stratocumulus under an inversion which is not far above the mountain peak. These waves resemble the wake left behind by a ship travelling across the ocean. They occur under the same conditions as those which cause the bettcr known parallel wave bars. Both types of wave can be seen on the same satellite picture and the interaction between them can produce rather complicated patterns. Although such patterns are theoretically calculable no-one seems to have considered the possible results worth the enormous effort needed to compute all the effects. See Fig 9.
Alignment of wave clouds. When the wind does not blow at right angles to a ridge the wave clouds tend to be aligned parallel to the ridge rather than at right angles to the wind. The effect shows up best when a north-westerly flow across Scotland produces waves which lie roughly parallel to the Ochils near Portmoak, but turn to lie nearly north to south downwind of the Kintyre peninsula.
When wake waves are added to wave bars which do not lie across the wind one may see streaks in the cloud lying at right angles to the wave bars. The effect of transverse and longitudinal waves must be to produce great variations in the amplitude at different places. If the pattern remains steady the positions for the best rate of climb can be identified with particular land marks but on many days slow changes of the various waves result in the best lift seeming to ]ump about with little obvious reason.
Cumulus streets under waves. Lee waves often affect the patterns of cumulus at low level; under wave troughs the cu are damped down or dispersed, under wave crests the cumuli build higher. In the commonest cases the waves are parallel to the lines of cu. Recent observations have shown that cumulus streets (which are aligned almost exactly with the low level winds direcrtion) may have wave bars above them which are at right angles to the streets.
So far there is no theoretical model which explains the way the flow
changes from convective streets to wave bars across the alignment of
the streets.
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Casswell, S.A., 1966, "A simplification of maximum velocities in mountain lee waves," Met Magazine, Vol 95 pp68-80
Scorer, R.S., 1949, "Theory of waves in the lee of mountains," Quarterly Journal of the Met Society, Vol 75, pp41-56.