Optimal dynamic soaring consists of successive shallow arcs

Abstract

Albatrosses can travel a thousand kilometres daily over the oceans. They extract their propulsive energy from horizontal wind shears with a flight strategy called dynamic soaring. While thermal soaring, exploited by birds of prey and sports gliders, consists of simply remaining in updrafts, extracting energy from horizontal winds necessitates redistributing momentum across the wind shear layer, by means of an intricate and dynamic flight manoeuvre. Dynamic soaring has been described as a sequence of half-turns connecting upwind climbs and downwind dives through the surface shear layer. Here, we investigate the optimal (minimum-wind) flight trajectory, with a combined numerical and analytic methodology. We show that contrary to current thinking, but consistent with GPS recordings of albatrosses, when the shear layer is thin the optimal trajectory is composed of small-angle, large-radius arcs. Essentially, the albatross is a flying sailboat, sequentially acting as sail and keel, and is most efficient when remaining crosswind at all times. Our analysis constitutes a general framework for dynamic soaring and more broadly energy extraction in complex winds. It is geared to improve the characterization of pelagic birds flight dynamics and habitat, and could enable the development of a robotic albatross that could travel with a virtually infinite range.

Keywords: albatross; dynamic soaring; energy; soaring; trajectory optimization; wind.

Figure 1.

Wind profile. (a) Wind field behind waves. Colour-coding: wind intensity, experimental data adapted from [10]. (b) The logistic wind profile in this study captures adequately the wind field in separated regions, such as behind ocean waves. More generally, it constitutes a robust way to approximate a wide class of wind fields, based on two parameters: a typical wind speed inhomogeneity W₀ separated by a shear layer of typical length-scale δ. (c) Rayleigh's wind model is the limit of the logistic profile for δ → 0.

Figure 2.

The albatross' trajectory. (a) Recording of a flying albatross from [11] (top view). In crosswind flight the typical turn of the albatross is approximately 50°–70°. Dot-dashed yellow portions of the trajectory: the albatross is involved in a 60° turn within ±20°. Dashed red portions: the albatross is involved in a 60° turn within ±10°. Note that while in the ground frame the mean albatross travel has a downwind component, in the frame moving with the average wind it is nearly crosswind. (b) Rayleigh cycle describes the albatross' flight as a sequence of half-turns between the windy and slow regions. At each layer transition, there is an airspeed gain equal to the wind speed, which compensates inherent drag losses that are quadratic in airspeed. However, this trajectory is suboptimal for energy extraction. Instead, for thin shear layers, the optimal cycle (c) is composed of a succession of small-angle arcs. The flight portion in the wind layer is functionally analogous to the sail of a sailboat, while the portion in the slow layer is analogous to the keel of a sailboat (d).

Figure 3.

Wandering albatrosses #2 and #4 from Yonehara et al. [24], analysed in this study. The track of albatross #2 is over 1700-km-long, lasts for approximately 2 days and is made of up-, cross- and downwind flights in low and high winds, separated by active foraging and resting periods. The track of albatross #4 is a nearly uninterrupted, 650 km, 9 h, approximately crosswind flight performed in 8–15 m s⁻¹ winds. Note that some data are missing or dropped due to poor GPS quality.

Figure 4.

Minimum wind trajectories for three shear layer thicknesses (see the wind profiles in the plots' backgrounds and how they relate to shear layer thickness in figure 1). (a) The trajectories are constrained to fulfil the specific requirement that the heading increases by 360° over a cycle, hence their loitering appearance. (b) The heading is required to be periodic, hence their travelling appearance. For the three-dimensional trajectories the scale is common and is indicated on the bottom right corner: the trihedral is of length λ = 2m/(ρS) (24 m for an albatross). The scale bars on the top view insets are of length λ. a(ii), b(ii) are representative of the shear layer thickness experienced by albatrosses. The travelling trajectory requires less wind than the loitering one, with an increasing advantage for thinner shear layers. When δ/λ → 0, the travelling trajectory becomes two-dimensional and is composed of a sequence of vanishingly small arcs of finite curvature performed at nearly constant speed. The behaviour of the loitering trajectory is qualitatively different: for decreasing shear layer thicknesses, it quickly converges to a limit trajectory that remains significantly three-dimensional even for an infinitely thin shear layer.

Figure 5.

Sketch of the evolution of airspeed and air-relative heading angle over one dynamic soaring cycle, in the large glide ratio approximation. Following a glide phase in the boundary layer, the glider transitions into the wind layer, experiencing a shift of its air-related heading to port, as well as an airspeed boost. A glide phase in the wind layer ensues and is followed by a transition into the boundary layer which is associated to a shift of air-related heading to starboard and an airspeed boost. The cycle in this figure starts a quarter period earlier than in the right-hand side of figure 4. In the thin shear layer limit, airspeed has double periodicity and air-related heading has double anti-periodicity, such that the physical cycle may be divided into two equivalent sub-units.

Figure 6.

Minimum wind and turn amplitude of the travelling and loitering trajectories as a function of the shear layer thickness from our numerical model, for various glide ratios. Unless otherwise indicated, the maximum glide ratio is reached at c_L = 0.5. The model is compared with experimental (exp.) data of flying albatrosses from [11,24], and simulations (sim.) of dynamic soaring in a logarithmic wind field from [25,33]. (a) In the thin shear layer regime δ → 0, the wind required for the travelling trajectories converges to our two-dimensional model in equation (4.2). (b) Similarly, the turn amplitude decreases and the trajectories become straighter. The histogram insets represent the turning statistics of Sachs et al. [11], Yonehara et al. [24] albatross #4 and #2 from bottom to top. Yonehara's albatrosses are recorded over hundreds of kilometres. In crosswind, the recorded albatrosses typically turn by 50°–70° while in the recorded mixed-flight the typical turn amplitude is 80°. Error bars represent the median turn ±1 s.d.

Figure 7.

Characteristics of the minimum-wind cycle. Same legend as in figure 6. (a) Height separation between the lowest and highest point of the cycle. For thin shear layers the travelling trajectory is nearly two-dimensional. Note that the convergence rate is only about z∼δ^2/3. (b) Cycle duration. (c) Maximum airspeed attained during the cycle. (d) Crosswind travel during one cycle. The blue (respectively, purple) markers correspond to twice the length of the sail (respectively, keel) phase in figure 2a.