The Analysis

Introduction3534035357_60d308f4da

 This theoretical analysis began as an attempt to write a program to simulate the wind gradient model of dynamic soaring including drag losses. It soon became clear that the wind gradient model did not work because we have to take into account the acceleration of the bird during the windward and leeward turns. The problem here is that any attempt to analyse turning flight would normally involve the whole 360 degree circle whilst it appears the albatrosses use only part of the circle - the crosswind parts. They only gain advantage from two segments of the circle, that is they fly crosswind plus and minus about 20 to 30 degrees. The other 240 odd degrees of the circle are useless to them. Also, successful dynamic soaring depends on a minimum wind velocity as well as a particular flight technique. So, proving a theory of dynamic soaring is probably impossible. The best we can achieve is an illustration of what an animal does given a certain wind.

 So the windward turn theory evolved. There are a large number of terms and variables and so I have not attempted to reduce the maths to a single equation. It was a bit like juggling with jellyfish. The albatross model is known to work, so its data were used as a starting point. The data used is that of an albatross in typical wind conditions as  published in Avian Flight by J J Videler 2010. Many of the nominated parameters are estimated by observation of film of albatross in flight.

 The object of this exercise is to see whether albatross flight can be modeled using the triangle of velocities, the equations of motion and the forces acting on the bird. The end result depends on not just the equations but also certain parameters such as the range of wind-angles and the wind speed. By accepting a limited range of wind-angles, typically crosswind plus and minus 20 to 30 degrees, we can calculate for each wind-angle, the effect of angle of bank and angle of pitch. The angle of bank will give a centripetal force which will act with the drag force to give a rate of turn and therefore a rate of change of headwind or tailwind component. They will also control the rate of tangential acceleration, which acts opposite to the rate of change of the head/tailwind component to control the airspeed.

 The motion of the bird is defined by airspeed, lift/drag ratio,  load factor and angles of pitch and roll combined with the wind speed and wind-angle. The roll and pitch profile is intended to give a flat windward turn and a leeward wing-over. Using a spread sheet, different angles of bank can be tested to see which produce an airspeed, ground-speed and height profile similar to those seen in nature. The important point being to test whether the result gives at least as much airspeed and height at the end as at the beginning.

Results

 The following diagram shows an example of the results of the calculations. See figure 8. The horizontal axis is crosswind distance in tens of metres, an overall distance of 430m and lasting about 20 seconds. The left side is the windward turn and the right side is the leeward turn, with the bird flying from left to right, reversing the direction of turn at about 340m. The model bird has a mass of 10kg, and a lift/drag ratio of 20. The program works with a starting point of airspeed 20m/sec and height 2m. The wind-velocity is a uniform 10m/s with no wind gradient or any vertical motion.

 A profile of angle of bank (10 degrees left in the windward turn and 80 degrees right in the leeward turn) and angle of pitch is then applied at 10m intervals. The program uses the triangle of velocities, the laws of motion and the forces acting on the bird,  to calculate the load factor, the centripetal and tangential accelerations, the rate of turn, airspeed and wind-angle (heading relative to the wind), ground speed and track, the rate of climb or descent and height.

 The first diagram shows variation of airspeed and ground-speed against distance flown. Starting at 20 m/s the airspeed increases and the ground-speed decreases during the windward turn from 0 to 340m. During the leeward turn  the airspeed decreases as the bird gains height in the wing-over and then increases slightly as the bird descends, leaving a slight gain of airspeed overall. The ground-speed decreases slightly as the bird climbs and then increases as it descends with a slight gain of ground-speed at the end.

Figure 8

Spreadsheet plot2

 In the middle diagram of height against distance, the height at the beginning is 2 metres. There is a slight loss and then a slightly greater gain of height in the windward turn. The leeward turn is then a wing-over rising to just over 10 m before descending to leave a slight gain of height overall. Notice that from about 300m to 350m the model albatross gains both height and airspeed as seen in the data from actual albatross GPS tracking. (See the Data section) The small kink in the line is the point at which the model albatross reverses direction. Bear in mind that, in straight flight with a lift/drag ratio of 20, at constant speed the model glider will descend from 2m to the surface in a distance of 40m.

 The lower diagram is the plan view of the  air-path (wind-angle), the motion of the bird within the air mass, and the ground track, the birds actual path over the ground, with the wind coming from the top. It shows the range of wind-angles, turning left during the windward turn from 120 through 90 to 60 degrees and turning right during the leeward turn from 60 through 90 to 120 degrees. The gap between the ends of the lines on the right is the distance lost downwind, about 200m, the effect of the wind. The birds are not drawn to scale but do illustrate the angle of drift.

 The result is an illustration of how it is possible for an albatross to dynamic soar in a uniform horizontal wind maintaining average airspeed and height. It requires the typical profile of roll and pitch angles which albatrosses are seen to perform. Any other profile results in a loss of airspeed or height.

 The viability of the theory depends on there being a single basic mechanism, controlled primarily by the angle of bank, which will enable the bird to maintain airspeed and height throughout the windward and leeward turns. That mechanism is the tendency of the airspeed to change due to two effects. Firstly, the changing headwind component caused by the changing wind-angle, that is the rate of turn. Secondly the opposite acceleration of ground-speed caused by the aerodynamic forces acting on the bird.

The rest of this page is an explanation of the maths used to produce these results.

Angle of bank in the Windward Turn

 So, as a starting point, what minimum angle of bank will provide a rate of turn which will enable the bird to just maintain airspeed, in the windward turn?

Angle of bank in level flight

 The angle of bank is measured from the vertical. The centripetal acceleration it provides is horizontal. See figure 9 in which a level turn is depicted where the vertical component of lift is equal to the weight.

AoB is Tan x Tan x = a / g                                          x is angle of bank

 a = v2 / r                                              a is centripetal acceleration m/s2

                                                             v is tangential air-speed m/s

                                                             r is radius of turn m

 Tan x = v2 / g . r

 r = v2 / g .Tan x                                   (1)

Rate of turn Ry

 The angle of bank x  gives the rate of turn Ry

 Ry = 360 . v / C                                  Ry is rate of turn deg/sec

                                                            C is circumference of circle radius r

 Ry = 360 . v / 2 . pi . r                                (2)

 Ry = 360 . v . g . Tan x / 2 . pi .v2             (1&2)

 Ry = 180 . g . Tan x / pi . v                      (3)

 So the rate of turn Ry is a function of tangential speed v and an angle of bank x assuming a level turn in which the vertical component of lift is equal to the weight.

Load factor Lf

 Load factor is the ratio of lift to aircraft weight. In straight and level flight, lift equals weight and Lf is one (1G). In a straight pitch-up the load-factor is increased by the increased angle of attack, so that in a loop for example the load factor might be three (3G). In a level turn the vertical component of lift must remain equal to weight and to achieve this the actual lift is increased by increasing the angle of attack, to correspond with the increasing angle of bank, so that the load factor is equal to 1/ cosine of the angle of bank. In a 30 degree banked level turn the load factor is 1.15G

 In a wing-over at any given angle of bank, the load-factor can be whatever we want but the vertical component of lift will not necessarily be equal to the weight. In this case we can introduce the load-factor to equation (3) by saying that tan x = sin x / cos x and Lf = 1/cos x therefore tan x = Lf . sin x Substituting in equation (3)

Rate of turn

 Ry = 180 . g . Lf . sin x / pi . v                    (4)

 Rw Rate of change of the headwind component H versus the wind angle y

 Rw is a function of trigonometry and simply depends on the aircraft heading relative to the wind, the wind-angle. Given air-velocity and wind-velocity there is a headwind component H corresponding to each wind angle y.

 H = W . cos y

 Taking a given interval between wind-angles, (say 1 degree), then in a turn between each pair of wind angles there is a corresponding change of headwind component. Therefore we get a rate of change of headwind component with respect to the wind-angle.

 The rate of change of head-wind component per degree of wind-angle

 Rw = (H1 - H2) / (y1 - y2)               (5)

 (m/ s per deg) where H1 and H2 are headwind components at points y1 and y2 separated by a small angle (y1 - y2)

RH rate of change of headwind component with time

Note that Rw , in units m/s per deg, does not depend on the rate of change of y with time. It simply depends on the heading during the turn. The time element is introduced by the aircraft rate of turn Ry deg per sec. Therefore the rate of change of the headwind component RH m / sec2

 RH =Rw. Ry                        (6)

 Looking at the units here we get:

 m / sec2 =  m / sec per deg x deg per sec

The Drag Force F1

 The drag force depends approximately on the lift divided by the lift/drag ratio Ld. The lift is equal to the weight (m.g) at 1G. In a turn it is then  multiplied by the load factor Lf which, in level flight, is inversely proportional to the cosine of the angle of bank x. Note that, for aircraft, the lift/drag ratio varies with the angle of attack which also changes the load factor but in birds the relationship is unknown, so I will assume constant L/D ratio over the small range of airspeeds and angles of attack used in dynamic soaring.

 F1 = m . g . Lf / Ld                                (7)

The Centripetal Force F2

 The centripetal force F2 in level flight (corresponding to acceleration a in figure 9) depends on the lift (weight) multiplied by the tan of the angle of bank or, in a wing-over, by the sine of the angle of bank times the load-factor

 F2 = m . g . Lf . sin x                         (8)

Now we can calculate the minimum angle of bank x, which will give constant airspeed at each wind-angle

For simplicity we will use F1  for the tangential load and Lf sinx for the centripetal acceleration.

Ry = (180 . g . Lf . sin x) / (pi . v)                       (4)

RH = Ry . Rw                                                     (6)

RH is a function of the rate of turn but is the same as the acceleration caused by aerodynamic force when the acceleration of airspeed is zero. It is therefore the accelaration term in F = m . a

RH = F1 / m                                                        (9)

RH = F1 / m  = m . g . Lf / Ld . m = Rw . 180 . g . Lf . sin x / pi . v                     (4) (6) (7) (9)

m.g . Lf / m . Ld = Rw 180 .  g . Lf . sin x / pi . v

1 / Ld   = Rw . 180 . sin x / pi . v

sin x = pi . v / (Ld . Rw . 180  )                       (10)

Rw = (H1 - H2) / (y1 - y2)                                (5)

 So, for each wind-angle, at one degree increments, there is a value of Rw (m/s per deg) which will give a rate of change of tail/headwind component which will make the airspeed increase. There is a drag load, represented by the lift/drag ratio Ld , which causes the airspeed to reduce. Together they give an angle of bank x  which gives the a rate of turn at which airspeed is constant.

 This calculation is repeated for each wind-angle in the windward turn to generate a profile of minimum angle of bank versus wind-angle. The result gives small angles of bank which correlates quite well with the small angles of bank observed in film of albatross. The nominated values of airspeed and wind-speed can then be modified to see, for example, what the minimum wind speed needs to be and what effect the airspeed /windspeed ratio has on the useable range of wind-angles. 

 For example:

Given Wind angle 80 to 79 deg,  bird airspeed 16.8m/s, K is 15.4m/s, Bird L/D 20, Wind 8m/s,   Drift 27.8 deg

Rw = H1 - H2 = (8 .cos 80) - (8 .cos 79)

Rw = 0.14 m/s per deg

Angle of bank x = 5.5 deg

 This is the minimum angle of bank needed to maintain height and airspeed at one point in the windward turn, at a wind angle of 80 degrees. In practice, the GPS data suggest that the airspeed increases during the windward turn with a gain of height before rolling into the leeward turn. To achieve this, the angle of bank is slightly greater than the minimum.

 Increasing the angle of bank in the windward turn increases the rate of turn which reduces the duration of the turn within the limited range of wind-angles and reduces the distance flown. Therefore to maximise the distance flown, the albatross will use the least angle of bank which will still allow it to maintain airspeed and height.

                                            Figure 10

components F1 F2 F3 windward turn2Force and acceleration components

 Now things get complicated. On the face of it there appears to be two horizontal force components: that is the drag force F1 and the centripetal force F2. In the windward turn it is easy to see how the drag force causes the airspeed and ground-speed to reduce, while the centripetal force provides a rate of turn which makes the head-wind component and therefore the airspeed to increase. However, in the leeward turn it is necessary for the ground speed to increase and it is less easy to see how the drag and centripetal forces make this possible.  The best way to explain this is to say that the drag force and the centripetal force are really components of a single horizontal force, the horizontal resultant F3 and this is  the starting point for our calculations.

Horizontal resultant and components

 Figure 10 is a plan view of a windward turn with the wind coming from the top of the diagram and the bird is flying left to right in a left turn.  The horizontal component of lift F2 is normal to the direction of the air velocity and combines with the drag F1 to create the horizontal resultant F3.

 We can calculate F3 using F1 and F2 and pythagoras which means that F3 is always positive.

F3 =sqrt (F12 + F22)

 Force F3 is not exactly in line with the ground velocity  and it can be seen that when the wind is a large proportion of the airspeed,  the drift angle d  is also large. In the windward turn a tangential force component   acts opposite to the  direction of the ground-velocity. Fgt = F3 cos e

 The centripetal component  provides the centripetal acceleration which creates the curved path relative to the ground. Fgc = F3 sin e

 The logical inference is that under acceleration, the aerodynamic forces cause acceleration of the ground velocity and then the ground acceleration causes acceleration of the air-velocity. This appears to be somewhat different to the situation when the aircraft is under equilibrium, when the ground-velocity is the vector sum of air-velocity and wind-velocity. In fact, the only difference is that under equilibrium the accelerations are all zero.

 (It should be noted that this process is the same as that which causes ground speed to change during upwind turns by a normal powered aircraft where thrust is equal to drag. F1 is then effectively zero and F3 equals F2. However F3 still produces values of Fgc and Fgt which causes the ground speed to reduce.)

 Angle b is the angle between F3 and the glider heading   sin b = F2 / F3

 Angle e is the angle between F3 and the ground track (the direction of the tangential acceleration) By inspection angle e = b - d

 The tangential acceleration of the ground-velocity is : Agt = F3 cos e / m

 The centripetal acceleration of the ground-velocity is Agc = F3 sin e / m

Sorting out the positives and negatives                                                                                                                                         Figure 11components Fgc Fgt leeward turn2

 Angles b, d and e are illustrated in figure 10 for the windward turn.

 By inspection in the windward turn, angle e = b - d     Angles b, d and e all are positive in the sense that they increase in a clockwise direction. Therefore

(+e)  = (+b) - (+d)

e = b - d

However in the calculations, b has a negative value in the windward turn, due to the angle of bank having a negative sign, being opposite to the angle of drift . Therefore

e = (-b) - (+d)   so that in the calculation e has a large negative value.

 In the leeward turn in figure 11, by inspection, e = b + d but angles b and e increase anti-clockwise, opposite to the windward turn, while d increases clockwise, the same as the windward turn therefore

-e = (-b) + (+d)

 e = b - d

 So that in the calculation, e is a small positive value and b is positive because F2 is positive because the angle of bank is on the same side as the angle of drift.

 Therefore in the calculations, using e = b - d gives the correct result in both windward and leeward turns.

 The sign of the drag force F1 is negative, opposite to the direction of flight. The value of centripetal force F2 can be negative or positive depending on the angle of bank which is measured left or right from the vertical and is made negative for a windward turn and positive for a leeward turn. The value of angle e  results in F3 being retarding in the windward turn and propulsive in the leeward turn. Using pythagoras means that F3 is always positive however angle e is negative in the windward turn and positive in the leeward turn this gives the correct sign to Fgt and Fgc

 This seems rather complicated but it is necessary treat all the forces equally and account for the acceleration of ground speed.

Components of airspeed and ground-speed

 There is a component of the ground speed K = G . cos d and a component of the wind-speed H = W . cos y  Both are parallel with the air-velocity which is the sum of the two components. See figure 12

Airspeed V= K + H.

The rate of change of airspeed is the sum of the rate of change of the headwind component H and the rate of change of the ground-speed component K.

dV/dt = dK/dt + dH/dt

Figure 12

Effect of changing wind angle 9

 In order for the airspeed to be constant, the ground-speed component K must reduce and the wind component H must increase at the same rate RH

In other words  when airspeed is constant, the rate of change of the headwind component RH is equal to dH/dt and also equal (and opposite) to dK/dt. But force F1 does not directly affect component H, it only affects component K by causing a tangential acceleration of the aircraft.

RH = -dK/dt = dH/dt

(Fear not, dear reader, we are using Leibnitz notation to express acceleration but we are not doing calculus!)

 (For comparison, in straight and level flight in a uniform wind, headwind component H is constant. Therefore, an unbalanced drag force F1 would cause the airspeed V to reduce but in reality it is only making component K reduce. Therefore we use component K in the acceleration calculations and not airspeed V)

Acceleration of component K: Aat Aac

The  tangential acceleration of the component K will be reduced by the cosine of the angle of drift

 Aat = Agt . cos d

        = Fgt . cos d / m

        = F3 . cos e . cos d/ m.

 The air centripetal acceleration is also reduced by the angle of drift

 Aac = Agc . cos d

         = Fgc . cos  d / m

         = F3 . sin e . cos d /m

 As mentioned before, the calculation gives e a large negative value in the windward turn which makes Aat a retarding acceleration and Aac a centripetal acceleration to windward.

 Referring to equation (4) the rate of turn is Ry = 180 . g . Lf . sin x / pi . v The term g . Lf sin x is the centripetal acceleration caused by the angle of bank in units m/s2  . In the spreadsheet this is substituted with centripetal acceleration Aac.

 Meanwhile the tangential acceleration of component K is given by Aat .

 The rate of change of component H is given by RH= Ry . Rw            (6)

The story so far

 We have started with a single horizontal resultant force F3 which causes tangential and centripetal accelerations of the bird, seen as accelerations of its ground-velocity. These then combine with the rate of change of the head-wind component to cause acceleration of the air-velocity.

The Leeward Turn

  Having  lost ground-speed in the windward turn, but having maintained airspeed and height, the albatross now has to recover that ground speed in the same range of wind angles and end up with at least the original airspeed and height. Figure 11 shows a plan view of the leeward turn with the wind coming from the top. Drag force F1 and centripetal force F2 combine vectorially to give the Horizontal Resultant F3. Force F3 then provides a propulsive force in the direction of the ground-tangential-velocity. Fgt = F3 . cos e This force produces an acceleration of the ground velocity Agt = F3 . cos e / m This  tangential acceleration of the ground-speed then appears as a tangential acceleration of component K but reduced again by the cosine of the drift angle

 Aat = F3 cos e . cos d/ m.

 The centripetal component of acceleration is

 Aac = Agc . cos d

         = Fgc . cos  d / m

         = F3 . sin e . cos d /m

 In the calculation e has a small positive value, Aat is positive and therefore propulsive. Aac is also positive and gives a turn to leeward

 However, whereas in the windward turn the increasing headwind component tends to increase airspeed, in the leeward turn the reducing headwind component tends to reduce airspeed. See figure 13. Therefore the propulsive effect of F3 needs to be substantial and  requires a large angle of bank but without an increase in drag.

Airspeed components in the Leeward turn

 In figure 13 can be seen a plan view of the leeward turn with the wind coming from the top. The airspeed V consists of a ground-speed component K and a wind-speed component H (which could be a headwind or a tailwind). As the wind angle y increases, the headwind component reduces and then becomes an increasing tailwind with the same effect. Acceleration of airspeed is the sum of the acceleration of the ground speed component K and the acceleration of the wind-speed component H. When these two accelerations are equal and opposite the airspeed is constant. (The albatross tracking GPS data suggest that airspeed reduces slightly in the leeward turn balanced by an increase of airspeed in the windward turn).

 The maths are the same as the windward turn except that in the leeward turn, the bank angle and the drift angle are on the same side, so that the horizontal resultant has a propulsive component in the direction of ground-velocity.

Figure 13

Leeward turn 10

 Although it seems as if drag should reduce airspeed, in fact the drag force only exists as a part of the horizontal resultant F3. When the angle of bank and the angle of drift are big enough and on the same side, the propulsive component makes the ground speed increase which in turn makes component K increase. Airspeed remains approximately constant because of the balance of accelerations; a combination of increasing ground-velocity and reducing headwind component, the ground acceleration allowing the bird to ‘keep up with the wind’ as it turns downwind. The centripetal component of the horizontal resultant provides the centripetal acceleration which creates the curved path relative to the ground. In the case of albatross dynamic soaring there is a very small drag force and a large horizontal resultant which is only possible in a low G low drag wing-over.

 (It should be noted that this process is the same as that which causes ground speed to increase during a downwind turn by a normal powered aircraft where thrust is equal to drag. Also, note that in a level turn, drag increases with angle of bank and any shortfall in the propulsive component of lift is made up by gravity with consequent loss of height)

 Component K is part of the airspeed. Therefore tangential acceleration  component Aat = F3 cos e . cos d/ m causes tangential acceleration of the air-speed, while the other component of the airspeed, the wind component H is reducing due to the bird turning downwind. As in the windward turn, under acceleration, the aerodynamic forces cause acceleration of the ground velocity and then the ground acceleration causes acceleration of the air-velocity.

 This boils down to the fact that we can use the same set of equations to describe the bird motion in both the windward and leeward turns. The only difference is that the angle of bank has a negative sign in the windward turn and a positive sign in the leeward turn.

Flat turns and wing-overs

 Whereas in the windward turn the vertical component of lift is equal to the weight of the bird and therefore the actual lift increases by the load factor; in the leeward turn, because the vertical component of lift is less than the weight of the bird, we can assume (guess!) that the load factor is one and therefore the actual lift is equal to the bird weight. The rate of change of the headwind component Rw versus the wind angle y is the same at any particular wind-angle in both turns although the rate of change of the headwind component with time RH depends on the rate of turn Ry and is therefore greater in the leeward turn.

 Rate of turn is proportional to load-factor but increasing the load-factor will increase the drag load. To avoid increasing the drag load, we keep the load-factor at one (1G) and the rate of turn is then only proportional to the sine of the angle of bank. For example, if we need to  triple the rate of turn at only 1G then the angle of bank increases as follows:

 Angle of bank in the windward turn 15deg

sine 15 = 0.25

0.25 times 3 = 0.75

 Angle of bank in the leeward turn: 

sine-1 0.75 = 48deg

 However at 48 degrees angle of bank and load factor of 1G the vertical component of lift will be cos 48 = 0.7G which is not enough to sustain level flight. So, in the leeward turn the rate of turn can be increased at a load-factor of one without increasing the drag loading but only by doing a wing-over and not a level turn.(A level turn at 48deg would require a load factor of 1.5G, which is relatively high-stress. 70 degrees angle bank would require 3G which is jet-fighter territory!)

How does a wing-over differ from a level turn?                                                                                         Figure 14

forces in wing-over 3 In level flight, when an aircraft banks, the lift force is tilted off the vertical and the horizontal component provides the centripetal acceleration causing the aircraft to turn. (See figure 14) The total lift force must be increased, normally by increasing the angle of attack, in order that the vertical component can balance the weight of the aircraft and prevent a loss of height. This increase in load-factor will increase the drag loading.

 In dynamic soaring the leeward turn is a wing-over in which total lift is not increased and remains approximately equal to weight. The vertical component of lift partly supports the weight whilst the horizontal component provides the centripetal acceleration which makes the bird turn. Drag is not shown. The flight path is partly ballistic and the bird climbs and descends but gains time to complete the turn before running out of height.

Angle of climb

 During the leeward turn the bird climbs and descends in a wing-over. An angle of climb will give an additional retarding force equal to the weight times the sine of the climb angle but in a climb the bird gains potential energy as it loses airspeed. However, the propulsive acceleration will reduce the airspeed losses and the height energy then converts to speed in the descent

Vertical Acceleration

 Whenever the vertical component of lift is less than the weight, the bird will accelerate downwards. Its rate of descent will then be the sum of of the effect of gravity and the effect of its angle of climb or descent. This limits the time and length of the leeward wing-over.

Roll reversals

 Between each turn there is a roll reversal with associated drag losses. The gain of height in the leeward turn will have to compensate for this. Also a little height can be gained by briefly increasing the angle of bank and rate of turn at the end of the windward turn. The penalty is a small loss of the distance flown in the windward turn.

Limits on the energy gained

 Doing a wing-over is simply a way of gaining time to gain ground-speed before running out of height. The amount of energy gained in this way will be limited by the amount of time the bird can spend in the wing-over in a steeply banked attitude (about 2 to 5 seconds) and this will depend on how much excess height or air-speed can be gained first.

 The amount of turn in the leeward turn has to be the same as that in the windward turn in order to maintain an approximately straight average course relative to the wind and a suitable range of wind-angles is found to be not less than about 50deg and not greater than about 130deg (probably less than this in nature). In effect, the bird is flying a crosswind heading +/- 20 to 30 deg. Even with this reduced range of wind-angles, the bird can still gain sufficient momentum because the drift angle and the propulsive force, is greatest in the middle part of the leeward turn.

The effect of the wind gradient

 As the bird turns it climbs and descends and airspeed converts to height and back to airspeed again. If there is a wind gradient, the wind will change with height and the effect of this will be to reduce the change of airspeed. During the climb the reducing airspeed will be offset by the increasing headwind. During the descent the increasing airspeed will be assisted by the decreasing tailwind. Thus the wind-gradient, by adding a little airspeed, may improve the efficiency of the wing-over by decreasing the loss of airspeed.  This, combined with the reduction of drag due to the ground-effect, may be the reasons why the albatross stays close to the surface

 However, it should be noted that the overall effect of the wind gradient is to reduce the average wind near the surface so that the wind-gradient will reduce the birds ability to dynamic soar. Also bear in mind that according to GPS data logging, dynamic soaring does not appear to involve 180 degree turns and therefore the effect of any wind gradient will be diminished by the minimum wind-angle achieved between windward and leeward turns.

 (For simplicity, and to make a point, I have left the wind gradient out of the calculations as the results seem to work without it)

Downwind Drift

 The bird can maintain height using dynamic soaring but can only do so at the expense of drifting downwind due to the consistently large drift angles. This is probably why albatross circumnavigate the Antarctic continent with the prevailing wind and fly downwind around the high pressure patterns of the North Pacific

 It is not possible to soar upwind using the windward turn theory. It is a mechanism for the bird to use wind energy to maintain average airspeed and height. The geometry of the windward and leeward turns means that there will always be a fairly large downwind drift angle and it only works when the wind-speed is a large proportion of the birds airspeed. On the other hand, the manoeuvre is inherently low-G and therefore requires minimum effort from the bird. If the albatross is only just maintaining average height, then it always has a large drift angle in both turns and will therefore always be drifting downwind. If the bird needs a wind speed equal to half its airspeed then its maximum angle of drift will be 30 degrees at the cross wind position and it never gets within about 60 degrees of an upwind heading. If the bird flies an air-distance of 290 m at 20m/s in 14 sec with a 15m/s wind, the distance lost downwind will be 210m. With an upwind ground speed of 5m/s that distance would take 42 secs. Thats a lot of flapping!

 It is not possible to do the dynamic soaring manoeuvre with 180 deg turns. The bird cannot extend the windward turn because the reducing wind-angle means that it cannot maintain airspeed and height. It cannot extend the leeward turn because again the reducing drift-angle means it loses the propulsive effect and it cannot gain any more ground momentum. The turns have to join up in terms of speed, height and wind-angle. Although it appears the bird could gain height slowly, it would lose the advantage of ground effect and also the ability to judge whether it is climbing or descending.

 Alternatively, if the bird can find some vertical component in the wind, for example on the upwind side of a swell or a ship, then it might be possible to make distance against the wind but this would not be dynamic soaring and of course a swell will normally be moving with the wind.

Upwind dynamic soaring is possible using a different technique and this is discussed on the Upwind Dynamic Soaring page

Conclusions

1: In a Windward Turn, airspeed is constant or increasing slightly because a  reducing ground-speed component is balanced by an increasing headwind/decreasing tailwind component. Bird momentum reduces and wind momentum increases.

2: In a Leeward Turn, aerodynamic forces provide an acceleration of ground speed. A component of this balances the decreasing headwind component. Bird momentum is gained and wind momentum is lost.

3: Provided that the bird gains at least the same momentum in the leeward turn than it loses in the windward turn then it can maintain or gain height.

4: The bird can maintain height in a uniform horizontal wind although the wind has lost speed after the bird has passed.

5: The wind gradient does not add energy to the manoeuvre but may improve the efficiency of the leeward turn by reducing airspeed losses. Ground-effect will reduce drag in the windward turn. These two effects plus the birds desire to make distance rather than height explain why it stays at low level.

6: Dynamic soaring probably works better over the oceans than over the land because the wind gradient is smaller and the wind is stronger and steadier

7: The Windward Turn Theory accurately models albatross flight manoeuvres and explains how they are able to maintain average airspeed and height over long distances. It helps to explain some aspects of albatross and petrel physiology such as tube nostrils and high aspect ratio wings and of behaviour such as the direction of large scale flight patterns and the shape of small scale repetitive turns. The relatively low G forces help to explain the albatrosses low in-flight metabolism

8: The theory generates a mathematical model which in turn produces a representation of albatross flight which is similar to that seen in nature.

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