Additional Clearance over Obstacles to Determine Minimum Flight Altitude in Mountainous Terrain

Abstract

The International Civil Aviation Organization (ICAO) specifies that in the design phase of instrument flight procedures, an additional clearance may be added to an obstacle when flights are over mountainous terrain. This clearance increase can be up to 100 per cent of the minimum obstacle clearance (MOC). Airspace and instrument flight procedure designers usually face the problem of determining what value should be applied, since setting the maximum value of 100% often implies operational penalties, but there are no standardized criteria to determine lower values. The ICAO PANS-OPS indicates that the additional clearance over obstacles in mountainous areas is caused by two effects, both related to orography and wind speed. The first effect is due to the altimeter indication error. The second one is related to the loss of altitude when an aircraft is exposed to turbulence produced by mountain waves. This paper presents a methodology for determining the additional clearance to be applied over obstacles when, in the flight procedure design phase, the overflight of mountainous terrain is expected. Through this methodology, results have been achieved for the proposal of an appropriate additional clearance. The development of graphs and tables allows us to identify which additional value should be considered in each case.

1. Introduction

In the design phase of instrument flight procedures, among other aspects, the minimum flight altitude of each segment should be stated. For this purpose, the designer applies a minimum obstacle clearance (MOC) over the obstacles. When mountainous areas are foreseen in the procedure planning and design, the ICAO establishes the addition of a clearance to compensate for the effects of strong winds. Nevertheless, the ICAO does not determine the precise value to be added, specifying only that it may be up to 100 per cent of the MOC. Since this value is not clearly defined, designers usually apply this maximum of 100% directly, which in many cases can lead to operational constraints. These operational constraints are due to the fact that, by applying a higher MOC, an aircraft will fly at higher altitudes, leading to pronounced descents in the approach and steep ascents in take-off, or resulting in longer distances to comply with the slopes, which translates in any case into a lower flight efficiency. This lower efficiency involves a higher fuel consumption, with its economic and environmental consequences, as well as airspace limitations that may lead to the inability to implement the procedure or make it unnecessarily complex. Furthermore, the ICAO specifies that only the necessary airspace should be used, so any other use of airspace exceeding the need would be inadequate. In addition, raising flight altitudes unnecessarily could also result in unstable approaches, which increases the runway excursion probability during landing, which is an unacceptable consequence. Hence, the need to establish a way to calculate the additional margin for mountainous terrain has prompted the analysis presented in this paper, in order to avoid applying an excessive clearance while maintaining safety.

Currently, the early detection of turbulence remains a challenge of which numerous studies have been carried out. Some research tries to identify turbulence areas from the eddy dissipation rate (EDR) or other turbulence indicators, as well as using detection and simulation programmes [1,2,3,4,5,6]. Additionally, much research has been based on the observation of these turbulences and on detecting the conditions under which they occur [7,8,9,10,11].

Turbulences caused by strong winds are very common on mountainous terrain due to the formation of mountain waves that are generated when the wind speed reaches a certain value above 15–20 kt and, at the same time, has a normal direction to the orographic system. Under these conditions, vertical upward and downward waves are generated by pressure differences between hills and valleys, with the winds on the leeward side being particularly severe.

These effects have been studied since the beginning of aviation because of the large number of accidents and incidents that have occurred throughout history as a result of the loss of aircraft control [12,13,14]. Due to the importance of mountain waves in flights over these areas, there are many articles about mountain wave formation and the types of waves that exist, as well as their characteristics, parameters, and consequences [15,16,17,18].

When an aircraft is flying towards its destination airport, at some point, it will begin to descend and, in many cases, may fly over mountainous areas at minimum flight altitudes, since the objective is to descend to land. Thus, a flight procedure designer faces the challenge of keeping an adequate clearance over obstacles while ensuring that this margin is not excessive, with the consideration that the aircraft should descend with appropriate gradients and flight regimes to maintain safety. Therefore, a methodology for calculating this additional clearance over mountainous terrains must be provided.

When the aircraft is over a mountainous terrain, it usually encounters winds with a certain intensity, which can produce two effects on the aircraft: erroneous information from the barometric altimeter and, on the other hand, a loss of altitude due to the effect of turbulence.

An altimeter error due to wind speed is indicated by the International Civil Aviation Organization (ICAO) [19]. Furthermore, certain articles analyse how turbulence also affects radio altimeter height measurements and errors in pressure altitudes [20,21].

The turbulence effect on an aircraft is more complicated to determine, either because of the ambiguity of turbulence and the lack of knowledge about it or because of the difficulty measuring such effects. However, several parameters can be useful when studying the effects of turbulence in an aircraft. Garman K et al. [22] establish a relationship between vertical wind and induced lift. Gao Z et al. [23] show in their paper a method for estimating an aircraft’s vertical acceleration and eddy dissipation rate in turbulent flights. Deskos G et al. [24] evaluate low-altitude atmospheric turbulence patterns for monitoring aircraft aeroelasticity. Elisov N et al. [25] study the influence of turbulence on the aircraft’s aerodynamic properties. Meanwhile, Gabriela S et al. [26] analyse the effects of atmospheric turbulence on aircraft dynamics, and Wang H et al. [27] estimate the severity of vertical displacements of aircrafts in turbulent flights.

Moreover, a large number of projects attempt to perform flight and turbulence simulations for the observation of their effects using collected data and estimations [28,29,30,31,32]. In addition, articles have been found with instructions on what procedures should be followed in the presence of severe turbulence [33,34,35].

However, no project, article, or study has been found about the correlation between the turbulence intensity and vertical displacement in aircraft altitude, which normally occur in these conditions, and its calculation is necessary to safely and efficiently set the additional clearance for mountainous areas. Therefore, this paper is mainly based on research into the effects of turbulence on an aircraft’s altitude. From all the information gathered and the collection of parameters affecting an aircraft’s attitude, a methodology was developed which provides a numerical result of this vertical displacement depending on initial conditions that must be defined.

Finally, an aircraft’s loss of altitude due to turbulence was obtained as a function of the wind speed, which combined with the altimeter error determines the additional clearance in mountainous terrains that should be added according to the flight segment.

2. Materials and Methods

This section explains the methods used for the calculation of the fundamental errors to which the additional clearance in mountainous terrains is due. Firstly, a barometric altimeter error calculation caused by strong winds is explained. Later, the following subsection describes how to obtain the turbulence error, which is the loss of aircraft altitude due to the effects of strong winds on the aircraft’s stability. These two errors have been called the altimeter error clearance and turbulence clearance, respectively, meaning that they both constitute the additional clearance for mountainous areas.

2.1. Altimeter Error Clearance

The ICAO PANS-OPS Volume III [19] states that “The combination of strong winds and moutainous terrain can cause local changes in atmospheric pressure due to the Bernoulli effect. This occurs particularly when the wind is across mountain crests or ridges. It is not possible to make an exact calculation, but theoretical studies (CFD Norway, Report 109.1989) have indicated altimeter errors (…)”. These altimeter errors are shown in the table below.

From the values in

Table 1. Altimeter error as a function of wind speed (PANS-OPS, ICAO).

Wind Speed [kt]	Altimeter Error [ft]
20	53
40	201
60	455
80	812

, a function can be defined to obtain the altimeter error for any wind speed in order to estimate an altimeter error clearance for each phase of flight, according to its wind speed, as proposed by the ICAO.

In Figure 1, the relation between the altimeter error in feet and the wind speed in knots is graphically represented, as is the trend line that best fits the function, obtaining an equation that relates these values.

Several functions were tested in MATLAB R2023b to find the one that best fit the values in Table 1, finding that the best one was a degree 3 polynomial function, resulting in the expression defined as follows:

$$\mathrm{F}\left(\mathrm{x}\right)=-6.25·{10}^{-5}{ \mathrm{x}}^3+0.14 {\mathrm{x}}^2-0.825 \mathrm{x}+14$$

(1)

where the value “x” is the known wind speed in knots, and “F(x)” would be its corresponding altimeter error in feet. The goodness of fit obtained from this function is as follows:

• SSE: 8.1284·10⁻²⁷;
• R-square: 1;
• Adj R-square: 1.

In this way, the altimeter error can be determined for any wind speed value. This altimeter error is proposed to be added to the turbulence clearance, and it can be obtained by using the graph or the table below or by applying the formula provided, using in both cases the wind speed specified by the ICAO for each phase of flight.

Finally, some values for the altimeter error [ft] as a function of wind speed [kt], from 10 kt to 80 kt, are shown in

Table 2. Altimeter error according to wind speed.

Wind Speed [kt]	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80
Altimeter Error [ft]	20	33	53	80	114	154	201	255	315	382	455	535	621	713	812

.

2.2. Turbulence Clearance

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.

This methodology was developed from the research on the loss of aircraft altitude due to vertical gusts in mountain waves and on how this wind speed affects the turbulence experienced by the aircraft. Thus, information was extracted from scientific references; aeronautical and meteorological books; research articles; and official websites belonging to international organizations such as the International Civil Aviation Organization (ICAO) [36], government agencies such as the National Weather Service [37] or the National Oceanic and Atmospheric Administration (NOAA), and aviation administrations such as the Federal Aviation Administration (FAA) [38]. Therefore, it is assumed that vertical displacements can be numerically reflected in the load factor variation, flight speed variation, and/or in the aircraft acceleration.

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, ${\mathrm{U}}_{\mathrm{e}}$, 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, ${\mathrm{U}}_{\mathrm{de}}$, 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. Turbulence and mountain wave intensity.

	Light	Moderate	Severe
Wind perpendicular to the mountain range at 700 hPa, ${\mathrm{V}}_{\mathrm{norm}}$	15–25 kt	25–45 kt	>45 kt
Vertical gust of mountain waves, ${\mathrm{V}}_{\mathrm{gust}}$	9–27 kt	27–53 kt	53–107 kt
Load factor variation, ∆n	0.05–0.2 g	0.2–0.5 g	0.5–1.5 g
Effective gust velocity, ${\mathrm{U}}_{\mathrm{e}}$	3–12 kt	12–21 kt	21–30 kt

, 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]:

$$∆\alpha ={\tan }^{-1}\left(\frac{{\mathrm{V}}_{\mathrm{gust}}}{{\mathrm{V}}_{\infty }}\right)$$

(2)

where the following applies:

• $∆\alpha$ is the angle of attack’s variation;
• ${\mathrm{V}}_{\mathrm{gust}}$ is the vertical velocity of the mountain wave or vertical gust velocity;
• ${\mathrm{V}}_{\infty }$ 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]

$$\Delta {\mathrm{C}}_{\mathrm{L}}={\mathrm{C}}_{\mathrm{L}}∆\alpha$$

(3)

where ${\mathrm{C}}_{\mathrm{L}}$ is the lift coefficient as a function of the angle of attack, obtained according to the following expression [44]:

$${\mathrm{C}}_{\mathrm{L}}={\mathrm{C}}_{{\mathrm{L}}_{\alpha }}(\alpha - {\alpha }_{\left({\mathrm{C}}_{\mathrm{L}}=0\right)})$$

(4)

where the following applies:

• $\alpha$ is the angle of attack at which the aircraft operates.
• ${\alpha }_{\left({\mathrm{C}}_{\mathrm{L}}=0\right)}$ is the angle of attack for which the lift coefficient is zero and, therefore, the lift is zero.
• ${\mathrm{C}}_{{\mathrm{L}}_{\alpha }}$ is the slope of the lift curve, referred to as the angle of attack, and is calculated as follows:

$${\mathrm{C}}_{{\mathrm{L}}_{\alpha }}=\frac{{\mathrm{C}}_{\mathrm{L}}-0}{{\alpha }_{\left({\mathrm{C}}_{\mathrm{L}}\right)} - {\alpha }_{({\mathrm{C}}_{\mathrm{L}}=0)}}$$

(5)

• ${\alpha }_{\left({\mathrm{C}}_{\mathrm{L}}\right)}$ is the angle of attack corresponding to a certain lift coefficient, ${\mathrm{C}}_{\mathrm{L}}$.

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]:

$$∆\mathrm{L}=∆{\mathrm{C}}_{\mathrm{L}}\frac{1}{2}\mathsf{\rho }{\mathrm{V}}_{\infty }^2\mathrm{S}$$

(6)

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, $∆\mathrm{n}$, which is due to the wind, is obtained using the following expression [44]:

$$∆\mathrm{n}=\frac{∆\mathrm{L}}{\mathrm{W}}=\frac{{\mathrm{C}}_{\mathrm{L}}{\tan }^{-1}\left(\frac{{\mathrm{V}}_{\mathrm{gust}}}{{\mathrm{V}}_{\infty }}\right)\mathsf{\rho }{\mathrm{V}}_{\infty }^2\mathrm{S}}{2\mathrm{W}}$$

(7)

where the following applies:

• ${\mathrm{C}}_{\mathrm{L}}$ is the lift coefficient, obtained as a function of the angle of attack;
• ${\mathrm{V}}_{\mathrm{gust}}$ is the vertical velocity of the mountain wave or vertical gust velocity;
• ${\mathrm{V}}_{\infty }$ is the true airspeed (TAS), expressed in the same units as the vertical gust velocity;
• $\mathsf{\rho }$ 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:

$$\mathrm{T}={\mathrm{T}}_1+{\mathrm{T}}_{\mathrm{h}}(\mathrm{h}-{\mathrm{h}}_1)$$

(8)

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;
• ${\mathrm{h}}_1$ is the altitude where the atmospheric layer in which the aircraft is located starts;
• ${\mathrm{T}}_1$ is the temperature corresponding to the lowest altitude of each atmospheric layer;
• ${\mathrm{T}}_{\mathrm{h}}$ is the temperature gradient, defined as the quotient of the temperature variation by the difference of their corresponding altitudes:

$${\mathrm{T}}_{\mathrm{h}}\equiv \frac{\mathrm{dT}}{\mathrm{dh}}=\frac{\mathrm{T}-{\mathrm{T}}_1}{\mathrm{h}-{\mathrm{h}}_1}$$

(9)

${\mathrm{T}}_1$, ${\mathsf{\rho }}_1$, and ${\mathrm{h}}_1$ vary depending on the atmospheric layer, being in the Troposphere ${\mathrm{T}}_1=288.15 \mathrm{K}$, ${\mathrm{h}}_1=0 \mathrm{m}$, and ${\mathsf{\rho }}_1=1.225 \mathrm{kg}/{\mathrm{m}}^3$. Likewise, the air density at a given altitude is obtained from the following equation:

$$\frac{\mathsf{\rho }}{{\mathsf{\rho }}_1}={\left(\frac{\mathrm{T}}{{\mathrm{T}}_1}\right)}^{-\left[\frac{\mathrm{g}}{{\mathrm{T}}_{\mathrm{h}}\mathrm{R}}+1\right]}$$

(10)

where the following applies:

• ${\mathsf{\rho }}_1$ 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:

$$\mathrm{g}={\mathrm{g}}_0{\left(\frac{{\mathrm{R}}_{\mathrm{T}}}{{\mathrm{R}}_{\mathrm{T}}+{\mathrm{z}}_{\mathrm{g}}}\right)}^2$$

(11)

where ${\mathrm{R}}_{\mathrm{T}}$ is the Earth’s radius, ${\mathrm{z}}_{\mathrm{g}}$ is the flight altitude, and ${\mathrm{g}}_0$ is the gravity at sea level (${\mathrm{g}}_0=9.80665 \mathrm{m}/{\mathrm{s}}^2$).

On the other hand, the wind speed (${\mathrm{V}}_{\mathrm{wind}}$) 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:

$$\mathrm{ICAO} \mathrm{standard} \mathrm{wind} \left[\mathrm{kt}\right]=\frac{2\mathrm{h}}{1000}+47$$

(12)

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 (${\mathrm{V}}_{\mathrm{norm}}$) [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:

$$\mathrm{h}=\frac{\mathrm{T}-{\mathrm{T}}_0}{{\mathrm{T}}_{\mathrm{h}}}$$

(13)

Following the expression, first, it is necessary to calculate the temperature, T, at that pressure, using the formula below [45]:

$$\mathrm{p}={\mathrm{p}}_0{\left(\frac{\mathrm{T}}{{\mathrm{T}}_0}\right)}^{-\frac{\mathrm{g}}{{\mathrm{T}}_{\mathrm{h}}·\mathrm{R}}} ➔ \mathrm{T}={\mathrm{T}}_0{\left(\frac{\mathrm{p}}{{\mathrm{p}}_0}\right)}^{\frac{{\mathrm{T}}_{\mathrm{h}}·\mathrm{R}}{\mathrm{g}}}$$

(14)

where the following applies:

• p is the atmospheric pressure at altitude h, expressed in the same units as ${\mathrm{p}}_0$. In this case, it is 700 hPa.
• ${\mathrm{p}}_0$ corresponds to sea level pressure (1013.25 hPa).
• ${\mathrm{T}}_0$ is the sea level temperature, expressed in °K (288.15 K).
• ${\mathrm{T}}_{\mathrm{h}}$ 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]:

$$\mathrm{v}={\mathrm{v}}_0{\left(\frac{\mathrm{h}}{{\mathrm{h}}_0}\right)}^{\alpha }$$

(15)

where the following applies:

• ${\mathrm{h}}_0$ is the reference altitude. In this study, this would be equivalent to the flight altitude.
• ${\mathrm{v}}_0$ is the wind speed at the reference altitude. In this case, it corresponds to the wind speed at the flight segment (${\mathrm{V}}_{\mathrm{wind}}$).
• 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 (${\mathrm{V}}_{\mathrm{norm}}$) at 700 hPa.
• α is the power law index or Hellman exponent, which is obtained using the expression shown below [47]:

$$\alpha =\frac{0.37-0.088\ln \left({\mathrm{v}}_0\right)}{1-0.088\ln \left(\frac{{\mathrm{h}}_0}{10}\right)}$$

(16)

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. Aircraft approach categories (PANS-OPS, ICAO). Speed in knots.

Aircraft Category	Vat	Range of Speeds for Initial Approach	Range of Final Approach Speeds	Maximum Speeds for Visual Manoeuvring (Circling)	Maximum Speeds for Missed Approach
					Intermediate	Final
A	<91	90/150	70/100	100	100	110
B	91/120	120/180	85/130	135	130	150
C	121/140	160/240	115/160	180	160	240
D	141/165	185/250	130/185	205	185	265
E	166/210	185/250	155/230	240	230	275

), 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]:

$$\mathrm{TAS}=\mathrm{IAS}·171233· \frac{{(288+\mathrm{VAR}-0.00198 · \mathrm{H})}^{0.5}}{{(288-0.00198 · \mathrm{H})}^{2.628}}$$

(17)

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. Wing area and weights according to aircraft category.

Category	Reference Aircraft	S [m2]	MTOW [kg]	W [kg]
A	C172	16.17	1157	1040
B	ATR 72–600	61	23,000	20,000
C	A320	122.6	78,000	56,400
D	A340-600	439.4	380,000	226,100

, 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 (${\mathrm{V}}_{\mathrm{norm}}$) 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 (${\mathrm{V}}_{\mathrm{norm}}$) to the vertical gust (${\mathrm{V}}_{\mathrm{gust}}$), as the values indicated in Table 3, which are shown again in

Table 6. Wind speed perpendicular to the mountain range vs. vertical gust velocity.

	Light	Moderate	Severe
Wind perpendicular to the mountain range at 700 hPa, ${\mathrm{V}}_{\mathrm{norm}}$	15–25 kt	25–45 kt	>45 kt
Vertical gust of mountain waves, ${\mathrm{V}}_{\mathrm{gust}}$	9–27 kt	27–53 kt	53–107 kt

.

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:

$${\mathrm{V}}_{\mathrm{gust}}=1.8 {\mathrm{V}}_{\mathrm{norm}}-18$$

(18)

Vertical gust velocity for moderate-intensity mountain waves:

$${\mathrm{V}}_{\mathrm{gust}}=1.3 {\mathrm{V}}_{\mathrm{norm}}-5.5$$

(19)

Vertical gust velocity for severe-intensity mountain waves:

$${\mathrm{V}}_{\mathrm{gust}}=2.16 {\mathrm{V}}_{\mathrm{norm}}-44.2$$

(20)

After obtaining the vertical gust (${\mathrm{V}}_{\mathrm{gust}}$), as well as the true airspeed ($\mathrm{TAS}={\mathrm{V}}_{\infty }$), lift coefficient (${\mathrm{C}}_{\mathrm{L}}$), air density ($\mathsf{\rho })$, 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 (${\mathrm{C}}_{\mathrm{L}}$), 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 (${\mathrm{U}}_{\mathrm{e}}$) 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. Load factor variation vs. effective gust velocity.

	Light	Moderate	Severe
Load factor variation, ∆n	0.05–0.2 g	0.2–0.5 g	0.5–1.5 g
Effective gust velocity, ${\mathrm{U}}_{\mathrm{e}}$	3–12 kt	12–21 kt	21–30 kt

.

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 (${\mathrm{U}}_{\mathrm{e}}$) was obtained for each type of turbulence intensity, as shown below:

Turbulence experienced by the aircraft as light:

$${\mathrm{U}}_{\mathrm{e}}=60 ∆\mathrm{n}+5·{10}^{-15}$$

(21)

Turbulence experienced by the aircraft as moderate:

$${\mathrm{U}}_{\mathrm{e}}=30 ∆\mathrm{n}+6$$

(22)

Turbulence experienced by the aircraft as severe:

$${\mathrm{U}}_{\mathrm{e}}=9 ∆\mathrm{n}+16.5$$

(23)

The effective gust velocity should be adjusted to obtain the vertical speed variation of the aircraft, resulting in the derived gust velocity (${\mathrm{U}}_{\mathrm{de}}$), 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]:

$$\mathrm{a}=\frac{\mathrm{V}-{\mathrm{V}}_0}{\mathrm{t}-{\mathrm{t}}_0}$$

(24)

This equation establishes the following:

• t as the final time, i.e., the instant when a vertical gust ends;
• ${\mathrm{t}}_0$ as the initial time, i.e., the instant at which a vertical gust begins;
• $(\mathrm{t}-{\mathrm{t}}_0)$ as the duration time of the vertical gust affecting the aircraft;
• $(\mathrm{V}-{\mathrm{V}}_0)$ as the vertical speed variation experienced by the aircraft due to the turbulence intensity, which corresponds to the derived gust velocity (${\mathrm{U}}_{\mathrm{de}}$);
• $a$ 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 ($∆\mathrm{n}$) [43]:

$$\mathrm{a}=-(\mathrm{g}+∆\mathrm{n} \mathrm{g})=-\mathrm{g} (1+∆\mathrm{n})$$

(25)

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:

$$-\mathrm{g} \left(1+∆\mathrm{n}\right)=\frac{-{\mathrm{U}}_{\mathrm{de}}}{\mathrm{t}}\to \mathrm{t}=\frac{{\mathrm{U}}_{\mathrm{de}}}{\mathrm{g} \left(1+∆\mathrm{n}\right)}$$

(26)

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:

$$\Delta \mathrm{h}={\mathrm{U}}_{\mathrm{de}} (\mathrm{t}-{\mathrm{t}}_0)$$

(27)

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 ($\mathrm{h}\left[\mathrm{ft}\right]=\left(\mathrm{WindSpeed}\left[\mathrm{kt}\right]-47\right)·1000/2$).

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.

3. Additional Clearance over Mountainous Terrain

The Figure 7 shows a diagram summarising the calculation process to obtain the additional clearance in a mountainous terrain, with the estimates and expressions used.

The following graph shows the results obtained from the altimeter error and vertical displacement due to turbulence for the different categories. As both parameters depend on the wind speed, it was possible to develop functions depending only on wind speed. Since aircraft altitude losses vary according to the category and angle of attack, twelve lines were plotted, three per category (A, B, C, D), referring to each angle of attack (AOA = 2; AOA = 4; AOA = 6), in order to show the variation in the additional margin between categories and the angle of attack involved in the phase of flight. Thus, the altimeter error obtained from each wind speed was added to the vertical displacement, resulting in the following graph.

As Figure 8 shows, the variation in the angle of attack does not greatly affect the final value to be considered as an additional margin, since its impact just affects the component of turbulence and not the component of altimeter error. The altimeter error is the most significant factor in determining the additional clearance and remains consistent regardless of the category or angle of attack, so the differences observed in Figure 8 correspond to the turbulence clearance.

From 15 to 30 kt of wind speed, there is a greater difference between category A’s values and those of the rest of the categories due to the fact that the load factor variation in category A’s aircraft is more affected by these wind speeds, while the other categories are not as susceptible to those intensities of turbulence. This results in higher vertical displacements for category A’s aircraft and, therefore, in greater turbulence clearance. In other words, lower-category aircrafts are more susceptible to the effects of turbulence, requiring a greater additional margin over mountainous terrain.

Likewise, it can also be noticed from Figure 8 that the discrepancies between the additional margin values begin to remain almost constant above a 40 kt wind intensity, where the maximum difference is in the order of 60 ft and occurs between categories A and D, since they present the greatest disparity in size and weight and, therefore, in vertical displacement due to mountain wave turbulence.

The results displayed in Figure 8 are presented in the following three tables [

Table 8. Additional clearance for mountainous terrain for a 2° angle of attack.

Wind Speed [kt]	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80
Cat. A [ft]	20	63	105	143	182	223	272	327	389	457	530	610	697	789	889
Cat. B, C, and D [ft]	20	37	61	92	130	175	227	286	349	418	492	572	659	752	852

,

Table 9. Additional clearance for mountainous terrain for a 4° angle of attack.

Wind Speed [kt]	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80
Cat. A [ft]	20	74	118	150	186	228	278	335	397	465	539	619	706	799	898
Cat. B, C, and D [ft]	20	41	70	104	143	188	241	301	366	436	510	591	678	772	873

and

Table 10. Additional clearance for mountainous terrain for a 6° angle of attack.

Wind Speed [kt]	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80
Cat. A [ft]	20	79	124	154	190	234	284	342	405	474	548	628	715	808	908
Cat. B, C, and D [ft]	20	47	78	113	153	200	254	316	381	449	522	603	689	781	880

] for 2°, 4°, and 6° angles of attack, where the additional clearance for mountainous areas is indicated as a function of the wind speed, constituted by the altimeter error and the loss of aircraft altitude. As observed in the figure, the additional clearance of categories B, C, and D are practically the same, so they have been combined in the same row, since the maximum difference in their values is between categories B and D and is less than 3 m (9 ft), which is negligible compared to the altimeter error. Therefore, the values shown in the row correspond to category B, as they are the most conservative.

In summary, the clearance to be applied over a mountainous terrain should be the sum of the nominal MOC, the clearance due to the barometric altimeter error, and the clearance related to the loss of altitude caused by turbulence effects.

4. Discussion and Validation

The validation of the results obtained in this paper could not be performed with real flights, so we decided to validate them by comparison with research results published by reference organizations. In this sense, documents published by Airbus and NASA were found. Additionally, pilot and expert statements [56] were gathered, which also agree with the values achieved by the proposed methodology, as shown in the following figure.

The figure shows that the maximum values of aircraft altitude loss due to turbulence can reach 30 m (100 feet), categorising this type of turbulence as severe. This value coincides with those obtained in the proposed method, whose results were shown in the graphs above according to the categories and angles of attack [Figure 3, Figure 4, Figure 5 and Figure 6].

Depending on the angle of attack, the turbulence intensity experienced by the aircraft changes, so higher values of vertical displacement can be achieved as the angle of attack in-creases. For example, wind intensities associated with moderate turbulences (25–45 kt) result in maximum vertical displacements of 6 m (20 ft), following the values shown in Figure 9. However, for that same turbulence intensity, maximum vertical displacements of 26 m (87 ft) are achieved in the graph developed for category A [Figure 3] and around 18 m (60 ft) for the other categories operating at maximum angles of attack (6°) [Figure 4, Figure 5 and Figure 6]. These values are properly shown in Appendix A [

Table A1. Vertical displacement according to wind speed for 2° angle of attack.

Wind Speed [kt]	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80
Cat. A [ft]	0	30	52	63	68	69	71	72	74	75	75	75	76	76	77
Cat. B [ft]	0	4	8	12	16	21	26	31	34	36	37	37	38	39	40
Cat. C [ft]	0	3	6	10	13	18	23	28	32	33	34	34	35	36	37
Cat. D [ft]	0	3	5	8	11	15	21	26	30	31	32	32	33	34	35

,

Table A2. Vertical displacement according to wind speed for 4° angle of attack.

Wind Speed [kt]	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80
Cat. A [ft]	0	41	65	70	72	74	77	80	82	83	84	84	85	86	86
Cat. B [ft]	0	8	17	24	29	34	40	46	51	54	55	56	57	59	61
Cat. C [ft]	0	7	13	20	26	31	36	42	47	49	50	51	52	54	56
Cat. D [ft]	0	6	12	18	24	29	34	39	43	46	47	48	49	50	52

and

Table A3. Vertical displacement according to wind speed for 6° angle of attack.

Wind Speed [kt]	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80
Cat. A [ft]	0	46	71	74	76	80	83	87	90	92	93	93	94	95	96
Cat. B [ft]	0	14	25	33	39	46	53	61	66	67	67	68	68	68	68
Cat. C [ft]	0	12	22	29	35	41	48	56	63	66	67	67	67	67	68
Cat. D [ft]	0	10	20	27	33	38	45	52	59	62	64	65	66	67	67

]. Meanwhile, for lower angles of attack (2°), the altitude loss for categories C and D under wind intensities of 25–45 kt ranges between 3 and 8 m (8–28 ft). It is important to note that the data shown in Figure 9 correspond to categories C and D.

In any case, the results obtained and those found coincide in that the maximum value of the aircraft altitude loss is around 30 m (100 ft). Thus, assuming 6° as a high angle of attack, the maximum altitude loss for a category A aircraft is around 29 m (96 ft) according to the proposed methodology, while for categories B, C, and D, the maximum vertical displacement achieved could be up to 21 m (68 ft).

Referring to the document provided by Airbus [57], a figure presents the vertical wind speed variations as a function of time during a turbulence event that they categorize as excessive, and whose data have been obtained using flight recorder data. Analysing the 30 kt downdrafts and identifying that their duration time is approximately 2 s, it is determined that the aircraft’s altitude loss would correspond to about 30 m, which is in agreement with the maximum vertical displacement value determined in this paper.

On the other hand, based on the data gathered in NASA Technical Note D-5573 [32], which consists of an analytical study about the effects of severe turbulence on a subsonic jet aircraft, the maximum normal accelerations experienced by the aircraft were found. The following table includes these accelerations, which were calculated by a high-speed digital computer during the flight after taking eighteen different samples of two hundred-second periods. Each of these samples are listed in

Table 11. Range of variation and root mean square of normal and lateral accelerations.

Sample	Maximum Normal Acceleration, ${\mathbf{a}}_{\mathbf{n}}$ [g units]	Root Mean Square, ${\mathbf{a}}_{\mathbf{n}}$
1	−1.00 to 4.75	0.891
2	0.20 to 3.00	0.723
3	−0.50 to 1.60	0.310
4	0.00 to 2.00	0.249
5	−0.20 to 1.80	0.293
6	0.57 to 1.10	0.065
7	−0.20 to 1.90	0.298
8	−0.90 to 2.50	0.606
9	−0.20 to 1.90	0.296
10	−0.20 to 1.80	0.297
11	−0.20 to 1.90	0.298
12	−0.20 to 1.80	0.291
13	−0.20 to 1.90	0.296
14	−0.40 to 2.40	0.379
15	−0.20 to 4.00	0.779
16	−0.60 to 2.30	0.382
17	−0.20 to 1.90	0.304
18	−0.50 to 3.60	0.722

. The second column of this table indicates the maximum normal acceleration reached due to turbulence in each of the samples, and the third column represents the root mean square of the normal acceleration, obtained from all the values recorded in each sample.

The data shown in Table 11 were obtained through a figure shown in the reference document [32], and each sample is represented in graphs as a function of time, from which these values of maximum accelerations are obtained. By analysing these plots, it is possible to obtain the duration time of each vertical gust and the normal acceleration variation given in that time, which is related to the load factor variation.

Then, for a better understanding of how the following

Table 12. Vertical displacement as a function of normal acceleration variation and time.

Sample	1		2		3		4		5		6		7		8		9
${\mathbf{a}}_{\mathbf{n}}$ [g]	−1	4.75	0.2	3	−0.5	1.6	0	2	−0.2	1.8	0.57	1.1	−0.2	1.9	−0.9	2.5	−0.2	1.9
Δ${\mathbf{a}}_{\mathbf{n}}$ [g]	−2	3.75	−0.8	2	−1.5	0.6	−1	1	−1.2	0.8	−0.43	0.1	−1.2	0.9	−1.9	1.5	−1.2	0.9
t [s]	1	1	2	1	1	1	1	1.8	1.5	2	1	1	1.6	1.8	1.3	1.5	1.6	1.8
Δh [m]	−20	37	−31	20	−15	6	−10	32	−26	31	−4	1	−30	29	−31	33	−30	29
Δh [ft]	−66	121	−101	65	−49	20	−33	105	−85	102	−13	3	−98	95	−102	108	−99	95

data were obtained, the graph corresponding to “Sample 1” is represented in Figure 10, where the maximum normal acceleration values measured are −1 g and 4.75 g, with “g” being the Earth gravity (g ≈ 9.81 m/s²). Therefore, knowing that the aircraft is operating with a load factor of 1.00 g units, the maximum normal acceleration variations in this case are −2 g and 3.75 g (${\mathrm{a}}_{\mathrm{n}}-1\mathrm{g}$). On the other hand, the duration time is obtained approximately by analysing the abscissa axis of the graphs, where time is expressed in seconds and is measured from the time when the normal acceleration begins to increase or decrease until it reaches its maximum value.

Figure 10 shows that the maximum normal acceleration values reached in Sample 1 are 4.75 g and −1.00 g, and that the time in which these values are achieved is barely one second. This procedure was repeated with each of the reference document figures, obtaining the normal acceleration variations and times shown in Table 12.

The load factors given in Figure 10 and Table 11 usually range between −1 and 2.5 (orange lines), which coincide with the maximum values for clean configuration specified by EASA for the flight manoeuvring envelope in the CS-25 certification document [58]. This range for the load factor of an aircraft corresponds to the theoretical limit within which there would be no structural damage that would prevent aircraft operation, although such damage should be reviewed after landing. If these limit values (−1, 2.5) are exceeded, as in “Sample 1”, the structural integrity would be severely threatened, and it must be evaluated through an exhaustive inspection and its corresponding maintenance specified by the manufacturer. Moreover, Airbus specifies that, after the occurrence of severe loads and overload, they must be notified of the event and receive the event data for analysis; they also indicate the steps to follow when experiencing any type of load [57].

Once “time” and “maximum acceleration variation” are estimated, the maximum vertical displacements experienced by the aircraft are calculated following Expressions (24) and (27), with the difference that, in this case, the time and acceleration variation are known parameters, while the unknown is the variation in the aircraft’s vertical speed. In this sense, based on the data provided by NASA for a turbulence event that is considered severe, the proposed methodology is applied in order to verify whether the aircraft altitude losses calculated for real values are consistent with those presented by Airbus, pilots, and experts, as well as those determined in this paper.

$$\mathrm{a}=\frac{\mathrm{V}-{\mathrm{V}}_0}{\mathrm{t}-{\mathrm{t}}_0}➔\left(\mathrm{V}-{\mathrm{V}}_0\right)=\mathrm{a}·\mathrm{t} ➔ \Delta \mathrm{h}=\left(\mathrm{V}-{\mathrm{V}}_0\right)·\mathrm{t}=\Delta {\mathrm{a}}_{\mathrm{n}}·{\mathrm{t}}^2$$

(28)

Applying this expression to each maximum value of normal acceleration variation, maximum vertical displacements (Δh) were calculated for the values in Table 11. The following table shows the first nine samples, as the results obtained for the rest of the samples have the same order of magnitude.

Negative values in the table refer to a loss of altitude, while positive values refer to aircraft vertical displacements in the opposite direction to the force of gravity.

As mentioned above, values higher than 2.5 g and lower than −1 g are not usual, being considered under extreme events, and thus, Samples “1” (4.75 g) and “2” (3 g) must be reported to the aircraft manufacturer for a structural integrity inspection.

The following graph represents the impact of the turbulence duration time on an aircraft’s vertical displacement as a function of different normal acceleration variations, according to the proposed methodology [Equation (28)].

Observing Table 12 and Figure 11, it can be concluded that the exposure time to turbulence is also a very influential factor in the vertical displacement of an aircraft. For example, comparing Samples “5” and “7” [Table 12], which have the same normal acceleration variation of −1.2 g, with a difference of one tenth of a second, Sample “7” presents four meters more in the aircraft height loss. Likewise, Figure 11 shows that as the normal acceleration variation increases, the influence of time on the aircraft’s vertical displacement becomes greater, following an exponential function. Thus, a turbulence of 1.8 s duration with a normal acceleration variation of 1 g presents a difference of 10 m compared to one of 2 s, while a turbulence of 3 g variation differs by more than 20 m.

According to the values of time and normal acceleration provided by NASA [Table 12], which were measured during in-flight turbulence events, it is observed that for larger values of normal acceleration variations, the duration times of turbulence gusts are shorter, resulting in more abrupt altitude losses but with the same order of magnitude. Thus, the estimated maximum altitude losses remain around 30 m, as shown in Table 12, since these values are consistent with those obtained in this paper and also with those provided by Airbus and experts.

Finally, it should be noted that the obtained additional clearance values for mountainous terrains prove the need to apply an additional margin of 100% of the MOC on certain occasions, especially in phases of flight at high altitudes where wind speeds are stronger, whereas in phases of flight within the Terminal Control Area, which are at lower altitudes, the application of a 100% margin would penalize the procedures and could even make their design unfeasible when a lower clearance also guarantees their safety.

5. Conclusions

The aim of this research was to obtain a unified and established criteria for the determination of the additional clearance over obstacles in mountainous areas. This clearance is constituted by the barometric altimeter error due to strong winds and by the loss of aircraft altitude caused by turbulences due to the presence of mountain waves. The following conclusions were reached:

• A loss of altitude due to mountain wave turbulence depends not only on the wind intensity, but also on aircraft characteristics such as weight, wing area, angle of attack, and flight speed.
• The lower aircraft categories (A and B) are more susceptible to turbulence than the higher categories (C and D), resulting in greater vertical displacements.
• Aircrafts of the same category assume lower vertical displacements the higher the wing loading is, meaning a greater the weight or smaller the wing area. Therefore, an identical aircraft will be less affected by turbulence at high weights.
• An aircraft flying under the same conditions of weight, airspeed, wind, and altitude experiences greater vertical displacements due to turbulence effects when operating at higher angles of attack.
• The maximum vertical displacement due to mountain wave turbulence for a category A aircraft is approximately 30 m (100 ft), while for categories B, C, and D, it is around 20 m (67 ft).
• The altitude loss due to turbulence is less significant than the altimeter error. This difference increases the higher the wind speed is, which means that at strong winds, the altimeter error margin is the main one, since the turbulence clearance is almost negligible.
• The value obtained by adding the altimeter error and the aircraft altitude loss due to turbulence is proposed as the additional clearance to be applied in mountainous terrains.