The turbulence clearance is based on the calculation of the altitude loss experienced by an aircraft due to the effects of turbulence caused by a mountain wave formation, which involves strong winds. For this purpose, a methodology was developed, from which this altitude variation is calculated as a function of all the parameters that influence an aircraft’s performance, such as wind speed, flight altitude, aircraft category, angle of attack, and flight speed.
2.2.1. Background
Turbulence is classified as light, moderate, severe, and extreme depending on the existing wind intensity and changes in the aircraft’s speed, in the normal acceleration variation, or in the load factor. Nevertheless, throughout this study, atmospheric turbulence classified as extreme turbulence is omitted, as aircraft operations over this type of turbulence are not possible for safety reasons and the fact that these turbulences occur in extreme cases, such as severe thunderstorms or cumulonimbus formation.
Then, the mountain wave intensity can be related to the wind component perpendicular to the mountain range; for example, if the mountain is oriented from east to west, the normal wind component would be either from north or south. For mountain waves to form, the normal wind speed must be higher than a value between 15 and 25 kt, depending on the type of mountain. Once this value is reached, the intensity of the wave depends on the wind speed at the mountain peak and the pressure difference between the peak and the leeward valley. Thus, the wave intensity can be classified according to the wind that is perpendicular to the mountain range at an altitude equivalent to 700 hPa pressure, resulting in the following classification [
39,
40]:
Light mountain waves are considered to occur when the normal wind speed is less than 25 kt.
Moderate mountain waves occur when the normal wind speed varies between 25 and 45 kt.
Severe mountain waves are formed with winds greater than 45 kt.
These mountain waves generate vertical upward and downward movements, also known as the vertical velocity of mountain waves. In this way, each type of wave corresponds to a range of vertical velocities [
39]:
Light-intensity waves have upward/downward movements with vertical velocities from 900 to 2700 ft/min (9–27 kt).
Moderate waves have vertical velocities from 2700 to 5400 ft/min (27–53 kt).
Severe waves range in vertical air currents from 5400 to 10,800 ft/min (53–107 kt).
On the other hand, turbulence generated by mountain waves causes variations in the aircraft load factor. Depending on the turbulence intensity, these perceived variations are as follows [
39,
40,
41,
42]:
For light turbulence, the load factor may fluctuate between 0.05 and 0.2 g.
In moderate turbulence, the load factor varies from 0.2 to 0.5 g.
In severe turbulence, the load factor variation ranges from 0.5 to 1.5 g.
Another concept to emphasize is the effective gust velocity,
, which establishes the buffeting intensity as a function of the overloads acting on the aircraft, depending not only on the magnitude of the gust but also on the aircraft characteristics [
39]. Each turbulence intensity (light, moderate, severe) has a range of effective gust velocities associated with it [
42]:
Light turbulence is considered to occur when the effective gust velocity is between 5 and 19 ft/s (3–12 kt).
Moderate turbulence gives rise to an effective gust velocity between 20 and 35 ft/s (12–21 kt).
Severe turbulence reaches up to 49 ft/s (30 kt).
The derived gust velocity,
, is established as the true variation in the aircraft’s vertical speed due to the wind gusts produced by the upward and downward movements of the mountain waves. The derived gust velocity is defined assuming sinusoidal variations in the effective gust velocity and is approximately 60% greater than the effective gust velocity [
39].
Table 3, whose data have been compiled from the information mentioned throughout this section, shows the characteristic values for each type of mountain wave according to its turbulence intensity [
39,
40,
41,
42,
43].
The normal acceleration experienced by the aircraft also varies as a function of the turbulence intensity and is directly related to the load factor variation, being the product of the load factor and gravity.
Then, the load factor is defined as the ratio of the lift force to the aircraft’s weight and is used as an indicator of stress on the aircraft structure due to the action of any force that diverts the flight from a straight-line direction. It is usually measured in “g”, gravitational acceleration, which is a unit of force that is equal to the force exerted by gravity and indicates the force to which a body is subjected when it is accelerated.
To calculate the load factor variation due to a given gust velocity, first, the magnitude of the change in the angle of attack due to this vertical gust velocity is defined, giving the following expression [
44]:
where the following applies:
is the angle of attack’s variation;
is the vertical velocity of the mountain wave or vertical gust velocity;
is the true airspeed (TAS), expressed in the same units as the vertical gust velocity.
Therefore, the change in the lift coefficient after the effect of the vertical gust would be [
44]
where
is the lift coefficient as a function of the angle of attack, obtained according to the following expression [
44]:
where the following applies:
is the angle of attack at which the aircraft operates.
is the angle of attack for which the lift coefficient is zero and, therefore, the lift is zero.
is the slope of the lift curve, referred to as the angle of attack, and is calculated as follows:
It should be noted that the angle of attack varies depending on the phase of flight, that is, whether the aircraft is performing departure, arrival, holding, approach, or en-route procedures. Therefore, the phase of flight also affects the lift coefficient and, thus, the vertical displacement of the aircraft.
Following this, the lift variation due to the vertical gust effect is defined as follows [
44]:
where L is the lift, a force produced by the dynamic effect of the air acting on the airfoil, and acts perpendicularly to the flight path through the centre of the lift and perpendicularly to the lateral axis. Finally, the load factor variation,
, which is due to the wind, is obtained using the following expression [
44]:
where the following applies:
is the lift coefficient, obtained as a function of the angle of attack;
is the vertical velocity of the mountain wave or vertical gust velocity;
is the true airspeed (TAS), expressed in the same units as the vertical gust velocity;
is the air density at flight altitude in kg/m3;
S is the wing area, measured in m2;
W is the aircraft weight in Newton.
Throughout this methodology, the air density and air temperature are obtained as a function of the flight altitude, since their variation influences the aircraft’s performance and, therefore, its vertical displacements due to turbulence. For this purpose, the standardized expressions based on the International Standard Atmosphere (ISA) model have been used [
44,
45]. Thus, the temperature at a given altitude is calculated using the ISA model expression shown below:
where the following applies:
T is the temperature in degrees Kelvin at a given altitude (h);
h corresponds to the altitude at which the temperature should be found;
is the altitude where the atmospheric layer in which the aircraft is located starts;
is the temperature corresponding to the lowest altitude of each atmospheric layer;
is the temperature gradient, defined as the quotient of the temperature variation by the difference of their corresponding altitudes:
,
, and
vary depending on the atmospheric layer, being in the Troposphere
,
, and
. Likewise, the air density at a given altitude is obtained from the following equation:
where the following applies:
is the air density of the minimum atmospheric layer altitude;
R is the ideal gas constant (287.053 J/Kg K);
g is the Earth gravity at that altitude in m/s2, which is calculated as follows:
where
is the Earth’s radius,
is the flight altitude, and
is the gravity at sea level (
).
On the other hand, the wind speed (
) is introduced in knots; and in the phases of flight where the “ICAO standard wind” should be applied [
46], the wind speed is calculated as a function of the flight altitude following the ICAO PANS-OPS expression indicated below:
where h is the flight altitude in feet.
For the rest of the cases, the wind speed values for obtaining the loss of altitude due to turbulence will be those specified in the PANS-OPS for the construction of protection areas, in accordance with the flight segment.
By having the flight altitude and wind speed, it is possible to obtain the wind speed equivalent to an altitude corresponding to 700 hPa of pressure by means of a mathematical relation. This conversion is needed, since the classification of mountain waves (light, moderate, severe) is developed according to the wind speed that is perpendicular to the mountain at that pressure altitude (
) [
39], as shown in
Table 3. That is why it is necessary to have an altitude associated with a wind speed in order to calculate the vertical displacement of an aircraft flying over a mountainous area.
Below, the steps developed to obtain the wind speed that is perpendicular to the mountain range at a 700 hPa pressure are defined. First, the altitude corresponding to a 700 hPa pressure according to the ISA model was calculated from the following equations:
Following the expression, first, it is necessary to calculate the temperature, T, at that pressure, using the formula below [
45]:
where the following applies:
p is the atmospheric pressure at altitude h, expressed in the same units as . In this case, it is 700 hPa.
corresponds to sea level pressure (1013.25 hPa).
is the sea level temperature, expressed in °K (288.15 K).
is the temperature gradient of the troposphere (−0.0065 °K/m).
R is the ideal gas constant (287.053 J/Kg/K).
g is the gravity at sea level (−9.80665 m/s2).
Secondly, once the altitude at a 700 hPa pressure has been calculated by replacing the indicated values (3010 m), Hellman’s exponential law or power law is used to calculate the perpendicular wind speed. Hellman’s exponential law was proposed in 1915 and is useful for estimating the wind speed at a given altitude based on data measured at another given altitude and is obtained as follows [
47,
48,
49]:
where the following applies:
is the reference altitude. In this study, this would be equivalent to the flight altitude.
is the wind speed at the reference altitude. In this case, it corresponds to the wind speed at the flight segment ().
h is equivalent to the altitude found above, relative to a 700 hPa pressure according to the ISA model, which is 3010 m.
v is the wind speed to be estimated, that is, the wind speed that is perpendicular to the ridge () at 700 hPa.
α is the power law index or Hellman exponent, which is obtained using the expression shown below [
47]:
Thus, following the expressions, the relative wind speed at a 700 hPa pressure is obtained for each case, which could cause mountain waves if its intensity exceeds 15 kt.
2.2.2. Methods
After having defined the theoretical basis of the turbulence effect on aircrafts due to the presence of mountain waves, this section specifies the assumptions made and the relationships and equations developed to calculate the vertical displacement that an aircraft would experience under these conditions.
In the PANS-OPS, the ICAO defines the different aircraft categories according to their indicated airspeeds (IAS) [
50]. From these categories, an aircraft model was chosen as a reference to define the wing area (S) and the aircraft’s weight (W), since these characteristics affect the turbulence perception and differ per category. Moreover, each category is associated with the maximum indicated airspeed (IAS) corresponding to the speed range according to the ICAO PANS-OPS categorization by approach speed (
Table 4), since the maximum airspeed gives the largest vertical displacements.
From the indicated airspeed and flight altitude, the true airspeed (TAS) is obtained using the following expression, provided by the PANS-OPS, for a non-international system unit [
46]:
where the following applies:
IAS is the indicated airspeed in knots [kt];
VAR is the temperature variation in ISA in °C. By default, ISA+15 is used, and thus, VAR takes the value 15;
H is the altitude in feet [ft].
Two tables are shown below; the first one corresponds to the table provided by the ICAO PANS-OPS, where the different aircraft categories are listed according to their indicated airspeeds (IASs) above the runway threshold (Vat) [
50]. The second one shows the aircrafts that have been taken as a reference in each category in order to define the wing area and the weights assumed, which will be explained later.
In
Table 5, the maximum take-off weight (MTOW) and wing area (S) were obtained from the corresponding aircraft reference manual [
51,
52,
53,
54]. “W” was defined as the approximate minimum weight at which the aircraft could operate in order to obtain the maximum vertical displacement achievable, since the lower the weight is, the greater the altitude loss from turbulence is. Therefore, the MTOW is not used, because the vertical displacements associated with this weight would be the smallest achieved.
For categories A and B, the weight used in this methodology was calculated taking into account the Operational Empty Weight (OEW) and estimating a minimum payload and fuel load. This results in an excess altitude loss, ensuring the vertical protection of the aircraft in case of any turbulence.
In contrast, for categories C and D, their weights (W) were obtained by analysing the landing weights of a sample of 110 and 155 real flights, respectively, which would be equivalent to the lowest weights with which the reference aircraft usually operates. Once the weights for each aircraft category had been collected, a probability density function was obtaining, so that a low landing weight was identified. Since, as mentioned above, the largest vertical displacements occur when the aircraft is operating at low weights (keeping all other variables constant), the weights used for categories C and D were taken as the value from which 90% of the landing weights are found following the probability density function. This means that the weight taken is a value so that the probability of not being lower than it is less than or equal to 0.1 (10%), being a reliable reference as the lowest weight at which the aircraft model may operate.
Figure 2 shows the histogram and probability density function obtained from analysing the landing weights of more than one hundred A320 flights. From the available sample, the histogram was fitted to a Weibull distribution. And from the probability density function, it can be said that with a probability of 0.1 (10%), the weight of the aircraft is 56,400 kg or less.
This same methodology was followed for the A340-600 (Category D), giving a weight of 226,100 kg.
It follows that the wind that is perpendicular to the mountain range at 700 hPa () identifies the intensity of the mountain wave and, therefore, the updrafts and downdrafts that will affect the aircraft. For this reason, it is necessary to convert the wind speed at a given flight altitude to its equivalent at the altitude corresponding to 700 hPa, as mentioned above.
To determine the vertical gust velocity according to a mountain wave, an expression was developed for each mountain wave intensity (light, moderate, severe) in order to relate the wind speed at 700 hPa (
) to the vertical gust (
), as the values indicated in
Table 3, which are shown again in
Table 6.
The values from the table were used to obtain equations relating to these data. First, an attempt was made to find a common equation for all the intensities. However, the trend line did not fit accurately in some parts of the function. Therefore, the functions were divided by the turbulence intensity, resulting in the following three linear equations:
Vertical gust velocity for light-intensity mountain waves:
Vertical gust velocity for moderate-intensity mountain waves:
Vertical gust velocity for severe-intensity mountain waves:
After obtaining the vertical gust (), as well as the true airspeed (), lift coefficient (), air density (, and aircraft characteristics such as the weight (W) and wing area (S), the load factor variation may be calculated (7).
The load factor variation indicates the turbulence intensity experienced by the aircraft, since depending on the weight (W), wing area (S), and lift coefficient (), it will be more or less affected at the same wind speed and altitude. The lift coefficient depends directly on the angle of attack, and hence, it is defined at the beginning.
In order to correlate the load factor variation with the buffeting experienced by the aircraft according to the type of turbulence perceived, the load factor variation and the effective gust velocity (
) were related. Following the same method as for the calculation of the vertical gust velocity, the values shown in
Table 3 were analysed, and a trend line with its corresponding equation was found to obtain an approximation of the effective gust velocity as a function of the load factor variation. The data used are shown again in
Table 7.
Nevertheless, a sufficiently accurate approximation was not obtained, so it was also decided to divide the function according to the turbulence intensities (light, moderate, or severe) for a more accurate approximation. In this way, the effective gust velocity () was obtained for each type of turbulence intensity, as shown below:
Turbulence experienced by the aircraft as light:
Turbulence experienced by the aircraft as moderate:
Turbulence experienced by the aircraft as severe:
The effective gust velocity should be adjusted to obtain the vertical speed variation of the aircraft, resulting in the derived gust velocity (
), which is 60% higher than the effective gust velocity [
39].
The next step is to calculate the time during which the aircraft is exposed to turbulence. For this, the equation of linear motion is applied [
55]:
This equation establishes the following:
t as the final time, i.e., the instant when a vertical gust ends;
as the initial time, i.e., the instant at which a vertical gust begins;
as the duration time of the vertical gust affecting the aircraft;
as the vertical speed variation experienced by the aircraft due to the turbulence intensity, which corresponds to the derived gust velocity ();
as the vertical acceleration experienced by the aircraft, being the sum of gravity and the vertical acceleration caused by the turbulence, which results from multiplying the gravity (g) and load factor variation (
) [
43]:
Substituting all these values in the previous equation and solving with time (t) as the unknown, the period of time in which the aircraft is affected by vertical air currents can be obtained:
Finally, once the turbulence duration time and derived gust velocity have been obtained, the aircraft’s vertical displacement caused by turbulence, Δh, can be calculated:
The turbulence clearance to be applied in mountainous terrains is proposed to correspond to that loss of altitude that the aircraft may experience when flying through mountain waves originated by the wind that is perpendicular to the mountain range.
For its application, the vertical displacement values due to turbulence may be determined by two methods, analytical or graphical. The analytical method would be the application of the previous expressions, and the graphical one would be taking the values shown below. These graphs are based on altitude losses associated with a defined wind speed and flight altitude, which is required to determine the true airspeed (TAS), among other parameters. Thus, from a 53 kt wind speed or higher, the expression of the “ICAO standard wind” (12) is applied to obtain the flight altitude ().
Following the equation, 47 kt corresponds to 0 ft altitude, making no sense, and a flight altitude of 2000 ft is associated with a wind speed of 51 kt, being an excessive wind speed for that altitude. For this reason, and since to calculate the true airspeed and, thus, the vertical displacement, it is necessary to define a flight altitude, 3000 ft has been kept constant for the graphs from 10 kt to 53 kt, varying the values of wind speed for the same altitude, without applying the “ICAO standard wind” expression. Moreover, the “ICAO standard wind” starts to apply for initial approach procedures, so a flight altitude of 3000 ft may be a generalized approximation of these altitude procedures to avoid having a huge number of graphs. Additionally, it should be noted that results obtained at altitudes above or below 3000 ft show no significant variations, where lower altitudes result in lower vertical displacements due to the reduction in the true airspeed and, thus, resulting in a smaller load factor variation.
Due to the influence of the aircraft category (A, B, C, D) and angle of attack on the turbulence effects, four graphs were developed [
Figure 3,
Figure 4,
Figure 5 and
Figure 6], one per category, and each of them has three series according to the angle of attack involved. The angles of attack (AOAs) used are 2°, 4°, and 6° in order to analyse how much the vertical displacements vary depending between them. We decided to assign these values, since aircrafts normally perform in en-route procedures with maximum angles of attack of 4°, while in arrivals, the angles of attack vary between −2° and 2°, and in take-offs, values up to 6° can be reached.
Therefore, the following graphs show the maximum altitude losses for each category as a function of the angles of attack indicated. In these graphs, the wind speed is related to the vertical displacement. Thus, if it is desired to obtain the turbulence clearance for a flight segment over a certain mountainous area, it is sufficient to take the value of the smallest category operating in that segment, since the smaller the category is, the greater the experienced loss of altitude is.
Furthermore, the 30 kt wind speed associated with the design of turns in the standard instrument departure (SID) and the final missed approach (MA) has been indicated with an arrow in the figures. To obtain the vertical displacement that would correspond to these phases of flight, it is sufficient to access the graph at 30 kt and identify the associated altitude loss as a function of the angle of attack. Here, for SID procedures, a maximum angle of attack of 6° can be estimated, resulting in a vertical displacement of around 35 ft (11 m) for category C or D. However, if 4° is assumed to be the angle of attack, the maximum altitude loss due to turbulence is reduced to 25 ft (8 m) in these same categories.
Comparing the graphs, it can be observed that the lower categories (A and B) are more susceptible to the effect of turbulence than the higher ones (C and D), meaning greater vertical displacements. This is due to the fact that wind loading (W/S) influences the aircraft’s stability, among other factors. With regards to the angle of attack (AOA), aircrafts are more affected by turbulence at higher angles of attack, since their stability decreases. Thus, minor disturbances in the air can result in significant lift variations, as well as increased structural stresses, which translate into greater load factor variations.
Additionally, three tables are included in
Appendix A, one for each significant angle of attack (AOA = 2; AOA = 4; AOA = 6), according to the values shown in the graphs as a function of the aircraft category.