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Restoration opportunity area and costsMangrove restoration programmes have a greater chance of being successful when implemented in areas where mangroves have previously grown15. These areas have either been subject to deforestation or degradation and may be under government management or private ownership. They are locations that have undergone forest conversion into other land uses, including aquaculture, crops or plantations and urban settlements. Land ownership status is an important factor to consider for determining the availability of land for mangrove restoration7. For example, a higher opportunity and priority would be given to unproductive aquaculture ponds located in the protected and production forest areas which are under government management or leasehold, rather than in areas with other land uses that may be under private ownership (Methods gives detailed forest land tenure classifications in Indonesia). Therefore, managing mangrove rehabilitation should consider factors that include land tenure status and land-cover type as well as biogeomorphology (for example, ensuring that the correct mangrove species are used in hydrologically suitable locations) across landscape scales.We calculated that ~193,367 ha of land may be feasible for implementation of mangrove rehabilitation programmes (Fig. 4). This conservative assessment suggests that the potential for restoration may be only 30% of the current mangrove rehabilitation area target (600,000 ha). Depending on the challenges and opportunities for each of the biogeomorphological categories of land use and the forest land status we considered (see Methods for detailed mapping methodology), we identified that 9% of the potential restorable area was categorized as being within the high opportunity scenario, 33% as medium and 58% as areas falling within the low opportunity scenario. Among these scenarios, ~75% of identified areas have non-protected forest status, implying a greater tenurial challenge to establishing a rehabilitation programme. We identified the five provinces that are among the top ranked of high potential for mangrove restoration in Indonesia, namely East Kalimantan (20% of national restoration potential area), North Kalimantan (20%), South Sumatra (12%), West Kalimantan (5%) and Riau provinces (5%) (Fig. 1c). All of these provinces, except South Sumatra, are among the areas already identified in the current mangrove rehabilitation programme by the BRGM as having high opportunity for rehabilitation4. At the subprovincial scale, we identified the top six regencies with restoration area opportunity >10,000 ha, namely Banyuasin, Bulungan, Tana Tidung, Paser, Berau and Nunukan (Supplementary Table 1). Mangroves across these regions were commonly deforested after 2010 and converted into aquaculture ponds despite being designated as protected forest areas (Supplementary Table 1).Fig. 4: The distribution of mangrove loss area (in hectares) between 2001 and 2020 in Indonesia.Also shown are mangrove loss proportions within different biogeomorphological typology, loss drivers (land-use types), forest land status and identified scenarios of restoration opportunity (low, medium and high).Full size imageConsidering that previous successful (85% survival rates) mangrove rehabilitation around the world has been achieved only at small landscape scales (10–400 ha) with costs varying between US$1,500 ha−1 and US$9,000 ha−1 (refs. 8,16), the large-scale mangrove rehabilitation ambition of Indonesia must be carefully planned. Rehabilitating ~200,000 ha of degraded mangroves will require between US$0.29 billion and US$1.74 billion. The 2021 annual government budget allocation for mangrove rehabilitation under BRGM alone is ~US$0.10 billion17, which is 66–94% lower than the estimated total required budget but with additional international investment18 there is potential for scalable mangrove rehabilitation success.Lessons learned from the past failuresIn Indonesia, unproductive aquaculture ponds have become targets for mangrove rehabilitation programmes (Supplementary Fig. 1). However, metrics of rehabilitation success in these settings reveal low survival rates of planted seedlings, highlighting an urgency to develop new strategies for mangrove rehabilitation and strategies to assess the effectiveness of ecosystem rehabilitation6. For example, a silviculture approach—nursery-based mangrove planting using Rhizophora species—has been adopted for mangrove restoration and management for a long time in Indonesia19. When seedlings are directly planted in unused ponds (Supplementary Fig. 1), dense monoculture plantations often form, which despite providing some ecosystem services (for example, carbon sequestration20) have limited biodiversity value21 and may be less resilient to stressors compared to a diverse assemblages of tree species22.Mangrove restoration projects have often suffered low success rates due to inadequate hydrological site assessments before revegetation23. For example, mangrove planting programmes initiated after the 2004 tsunami were focused on mono-species planting and on reporting the number of seedlings being planted in a given area24. These planting projects most often occurred on undisputed land, such as mudflats, which are inappropriate locations for long-term mangrove growth because of high inundation frequency, high water flow rates and hypersaline conditions that limit seedling establishment and survival24. Planting has also focused in mangrove areas where low canopy cover is observed. While some mangrove areas with low canopy cover may respond to plantings because they are degraded, many sites naturally support low canopy cover, reflecting suboptimal environmental conditions for growth of Rhizophora species, instead favouring growth of highly salt tolerant species such as Avicennia spp.24. Such failures in mangrove rehabilitation efforts, however, have been under-reported with more than 50% of rehabilitation studies not monitored over time (Supplementary Fig. 1).Alternative restoration approaches through repairing hydrology, including excavation and removal of pond walls and tidal gates, have also been introduced15, although this approach has been only practiced in Indonesia at limited scales, mostly in unused aquaculture ponds25. A comprehensive understanding of the opportunity for mangrove rehabilitation in Indonesia is largely unquantified. Additionally, with limited monitoring of mangrove rehabilitation projects, the effectiveness and functionality of mangrove rehabilitation in Indonesia remains largely unknown and therefore it remains challenging to assess rehabilitation effectiveness between approaches and locations in Indonesia. Yet such assessments provide important data to achieve the ambitious mangrove rehabilitation goals of Indonesia.Mangrove governance in IndonesiaMangrove conservation in Indonesia was formally adopted in 1990 (Extended Data Fig. 1 and Supplementary Table 2), when mangroves were designated as protected forests under Law 5/1990 and the Presidential Decree 32/1990. When the Asian tsunami hit Aceh province in 2004, the role of mangroves in wave attenuation and therefore minimizing disaster risks for coastal communities was recognized26. As a result, nearly 30,000 ha of damaged mangroves were rehabilitated to recover coastal resiliency through planting of nearly 24 million seedlings over 60 projects24. However, the success of these programmes was low due to a lack of planning, monitoring and critical supplemental actions24,27. Despite the failure of many mangrove rehabilitation projects post-tsunami, the implementation of the subsequent programmes have not fully adopted best-practice mangrove rehabilitation principles6,7,15,23. In 2007, similar approaches to mangrove rehabilitation and conservation were adopted at a larger, national scale under the Spatial Planning Law (Law 26/2007) and the Coastal Area and Small Islands Management Law (Law 27/2007).In 2012, the National Mangrove Management Strategy (STRANAS Mangrove) was first established and followed by the formalization of the National and Regional Mangrove Working Group whose task was to guide mangrove conservation and rehabilitation. Its main goal was to involve more stakeholders, including civil society organizations and subnational government bodies, in mangrove conservation and rehabilitation28. Until 2017, the technical regulation of strategy and performance indicators for mangrove management was implemented with targets set to rehabilitate 3.49 Mha of mangroves by 204529. In 2020, however, the Mangrove Working Group and its supporting regulations were abolished and the mangrove rehabilitation strategy was subsequently managed by BRGM4. This effectively removed the regional governments (subnational working groups) from decisions related to mangrove management and concentrated development of policy at the level of the national government. The new strategy includes a tenfold increase in the annual rehabilitation target (from 11,250 to ~120,000 ha yr−1) with an overall target of 600,000 ha to be achieved within a shorter timeline (2020–2024). Without clear planning and appropriate strategies, these ambitious targets may not be feasible. For example, the annual mangrove rehabilitation area reached between 2017 and 2020 was only 5,318 ha (50% of the target) despite 2.6 million seedlings being planted (Supplementary Table 3). Given the lessons from the previous mangrove rehabilitation and the emerging processes of mangrove governance, it is timely to set an achievable restoration framework with improved planning, evaluation and monitoring.Implication for international environmental agendasA successful mangrove rehabilitation programme can directly contribute to reducing poverty (SDG 1) and maintaining food security and livelihoods (SDG 2), thereby increasing the health and well-being of 74 million coastal people in Indonesia (see Supplementary Table 1 for total population of regions with restoration potential area >5 ha). Additionally, mangrove rehabilitation will directly contribute to other relevant SDGs, such as improving water quality (SDG 6), providing healthy coastal habitats for fish and other marine biodiversity (SDG 14), contributing to emissions reductions and improving coastal resilience from sea level rise (SDG 13) and sustainably managing and protecting terrestrial ecosystems (SDG 15). Mangrove rehabilitation contributions to SDG 1 and 2 are particularly relevant as the current rehabilitation programme is delivered as cash-for-works activities under the National Economic Recovery strategy (PEN) as part of the social welfare payments to alleviate economic impacts of the COVID-19 pandemic17. With the current annual mangrove rehabilitation budget of US$0.10 billion17, further implementation of scalable community-based mangrove restoration with technical support from subnational and non-government stakeholders could increase the benefits to local communities, if administered properly. Therefore, the large investments planned for coastal communities via a national mangrove restoration programme will not only contribute to the economy of coastal communities, potentially reducing poverty across 199 regencies but will also help in securing nearly 4% of the national greenhouse gas emissions reduction target from the land sector.Restoring 193,367 ha of mangroves in the next 5 years (2021–2025) may contribute to carbon sequestration of 22 ± 10 MtCO2e by 2030 (see Methods for detailed estimate calculation and assumptions). Moreover, stopping the current annual rates of mangrove loss of 7,436 ha yr−1 between 2021 and 2030 will reduce up to 58 ± 37 MtCO2e or 12% of the national land sector emissions reduction targets. Clearly, climate benefits from mangrove rehabilitation and conservation in Indonesia are substantial if rehabilitation and conservation can be implemented appropriately and large annual rehabilitation targets are achieved. Indonesia has submitted its updated Nationally Determined Contributions (NDCs) to the United Nations Framework Convention on Climate Change, within which integrated management and rehabilitation of mangroves is a component of the actions to enhance the resilience of coastal ecosystems30. Further ecological aquaculture practices such as silvofisheries which are commonly applied in Indonesia31,32 may provide promising potential for climate change mitigation through mangrove biomass enhancement. With the increased potential for international investment to support mangrove rehabilitation in Indonesia, there is an opportunity for Indonesia to take the lead and show the world how mangrove conservation and rehabilitation can contribute to multiple international environmental agendas.In the past three decades, the governance of mangrove conservation and rehabilitation in Indonesia has been highly variable in approach (Extended Data Fig. 1). The current approach is top-down4 which has risks and may be ineffective at achieving landscape-scale increases in mangrove extent, as was demonstrated post-tsunami24,29. This top-down approach set by national-level agencies, which are responsible for achieving rehabilitation targets, has limited involvement (or investment) by subnational governments. While we have identified key factors that determine land available for mangrove rehabilitation, the success of mangrove rehabilitation is not necessarily assured because of the limited involvement of subnational mangrove working groups. A current ‘one size fits all’ strategy of the national government may not be appropriate to achieve successful mangrove rehabilitation and thus more flexible, localized approaches may increase the likelihood of success. More

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Plant species and pollinatorsMedicago sativa L. (Fabaceae), also called alfalfa or lucerne, is a perennial legume with flowers arranged in a cluster or raceme. It is a self-compatible plant with fairly high outcrossing rate (5.3–30%)46, and it requires insect visits for seed production47. No plant material was collected for this study. Honey bees, Apis mellifera, and alfalfa leafcutting bees, Megachile rotundata, are used as managed pollinators in alfalfa seed-production fields in the USA while bumble bees are commonly used in alfalfa breeding47.Experimental design and pollinator observationsFive 11 m × 11 m patches of M. sativa plants were set up in an east–west linear arrangement at the West Madison Agricultural Research Station in Madison, Wisconsin, USA. Within each patch, we transplanted 169 young plants grown from seeds in the greenhouse, each placed 90 cm apart. These plants grew and, at flowering, a plant had an average of 30.65 ± 16.4 stems per plant, with 4.93 ± 3.41 racemes per stem, and 7.53 ± 2.44 open flowers per raceme.A honey bee hive was placed approximately 100 m from the patches and a bumble bee hive was set up at the center of the southern edge of the patches. For leafcutting bees, a 60 × 30 × 7.6 cm bee board was set up in each of two boxes placed 1/3 and 2/3 along the southern edge of the patches and a half gallon of bees was released at periodic intervals throughout the alfalfa flowering season.Over two consecutive summers, observers followed bees foraging in the alfalfa patches, marked each raceme visited in succession within a foraging bout with a numbered clip, and recorded the number of flowers visited per raceme. After a bee had left a patch, observers went back to the marked racemes and measured the distance and direction traveled between consecutive racemes. Directions were recorded as one of the cardinal directions: North (N), South (S), East (E) or West (W), or inter-cardinal directions: Northeast (NE), Southeast (SE), Northwest (NW) and Southwest (SW). The frequency distributions of distances and directions traveled between two successive racemes are presented for each bee species each year in Figs. 1 (distances) and 2 (directions). The low pollinator abundance permitted observers to follow individual bees foraging in a patch. Little interference among bee species was observed in the patches.Figure 1Frequency distributions for distances traveled between consecutive racemes (cm) for each bee species each year.Full size imageFigure 2Frequency distributions of directions traveled between consecutive racemes for each bee species each year.Full size imageModel for the distance traveled between consecutive racemesWe first determined whether a statistical model best described the distance traveled between consecutive racemes (Modeled Distance), and examined whether the model differed among bee species. We used mixed effect linear models (proc Mixed in SAS 9.3)48 to identify the model that best described the distance traveled by pollinators between consecutive racemes. The model included loge distance as a linear function of loge flower number and bee species as fixed effects. The distance traveled between consecutive racemes and the number of flowers visited per raceme were log transformed prior to analyses in order to improve the models’ residuals. In addition, we included patch and foraging bout as random effects in the model. A foraging bout includes the racemes visited in succession from the time a bee is spotted in a patch to the time it leaves that patch. We used foraging bout instead of individual bee as the random effect because bees were not individually marked in this study. Moreover, to take into consideration the potential correlation between successive observations within a foraging bout, we added clip to the model. Clip 1 represents the first and second racemes visited in the foraging bout; clip 2, the second and third, and so on. Clip was added to the model either as a random effect or as a repeated measure with an AR(1) structure. The combination of random clip and random foraging bout creates a model that is sometimes called the “compound symmetry” model. The AR(1) structure represents correlations that decline exponentially as the gap between measurements increases such that measurements closer together in time are more strongly correlated than measurements further apart. Because we expected bees to visit flowers at close proximity when resources are abundant, we chose this correlation structure as a good potential descriptor of the way distances might be correlated within foraging bouts. We started with a full model which included loge flower number, bee species, patch, foraging bout, and clip either as a random effect or as a repeated measure with an AR(1) structure. We then removed variables and compared models by inspecting AIC values and the p values for each term in the model. We considered both low AIC and statistically significant (p  More

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In this section, we introduce flapping flight dynamics and describe the bird model used in our computational framework. Furthermore, we describe how such a dynamical model is used in order to identify steady and level flapping flight regimes, study their stability, and assess their energetic performance.Equations of motion modelling flapping flightFlight dynamics is restricted to the longitudinal plane and thus the bird main body is captured as a rigid-body with three degrees of freedom, i.e. two in translation and one in rotation. This model preserves symmetry with respect to this plane, without any lateral force and moment. The aerodynamic model of the wing relies on the theory of quasi-steady lifting line23. Additionally, the present work does not account for the inertial forces due to the acceleration of the wing, and thus also neglecting the so-called inertial power. This inertial power was shown to be negligible in fast forward flight conditions, in comparison to the other contributions to actuation power24, and is thus systematically neglected in similar work10,11,25 since wing inertia is neglected.The body is thus modelled with a mass (m_b) and a rotational inertia (I_{yy}) about its center of mass. The equations of motion are expressed in the body frame (G(x’, y’, z’)) with unit vectors ((hat{textbf{e}}_{x’}, hat{textbf{e}}_{y’}, hat{textbf{e}}_{z’})), and an origin located at the center of mass, as pictured in Fig. 1a. The state space vector is thus$$begin{aligned} textbf{x} = {u, w, q, theta } end{aligned}$$where u and w are the body velocities along the (x’-) and (z’-)axis and (theta) and q are the pitch angle and its time derivative about the (y’-)axis, respectively. Consequently, the equations of motion read11,13,26$$begin{aligned} begin{aligned} dot{u}&= -qw – gsin theta + frac{1}{m_b}big ( {F_{x’}(textbf{x}(t), t)} \&quad + {F_{x’, t}(textbf{x}(t), t)} + D (textbf{x}(t), t) big )\ dot{w}&= qu + gcos theta +frac{1}{m_b} big ( F_{z’}(textbf{x}(t), t) + F_{z’, t}(textbf{x}(t), t) big ) \ dot{q}&=frac{1}{I_{yy}} big ( M_{y’}(textbf{x}(t), t) + M_{y’, t}(textbf{x}(t), t) big )\ dot{theta }&= q end{aligned} end{aligned}$$
(1)
Figure 1(a) Bird model for describing the flight dynamics in the longitudinal plane. The state variables are expressed with respect to the moving body-frame located at the flier’s center of mass (G(x’,z’)). These state variables are the component of forward flight velocity, u, the velocity component of local vertical velocity, w, the orientation of this body-centered moving frame with respect to the fixed frame, (theta) and its angular velocity, q. A second frame (O(x’_{w}, z’_{w})) is used to compute the position of the wing, relative to the body. The wings (dark gray) and the tail (red) are the surfaces of application of aerodynamic forces. (b) Top view of the bird model. The left wing emphasizes a cartoon model of the skeleton. The shoulder joint s connects the wing to the body via three rotational degrees of freedom (RDoF), the elbow joint e connects the arm with the forearm via one RDoF and the wrist joint w connects the forearm to the hand via two RDoF. Each feather is attached to a bone via two additional RDoF, except the most distal one ”1” which is rigidly aligned with the hand. The right wing further emphasizes the lifting line (red) which is computed as a function of the wing morphing. The aerodynamic forces generated on the wing are computed on the discretized elements (P_{i}). The tail is modelled as a triangular shape with fixed chord (c_{t}) and maximum width (b_{t}) that can be morphed as a function of its opening angle (beta). (c) Wing element i between two wing profiles, identifying a plane (Sigma) containing the lifting line (red). (d) Cross section of the wing element, containing the chord point (mathbf {P_i}) where the velocities are computed (Color figure online).Full size imageThe forcing terms in Eq. (1) are the aerodynamic forces and moments applied to the wing (namely (F_{x’}), (F_{z’}), and (M_{y’}) ) and to the tail ((F_{x’, t}), (F_{z’, t}), and (M_{y’, t})). The whole drag is captured by an extra force D that sums contributions due to the body (D_{b}), the skin friction of the wing (wing profile) (D_{p,w}), and the skin friction of the tail (tail profile) (D_{p,t}). These terms are described in detail in the next sections. Importantly, we accounted for the drag acting purely along (x’) direction, after proving that the projection of the drag forces along (z’)-axis is between two and three orders of magnitude smaller with respect to the aerodynamic forces produced by two other main lifting surfaces. This assumptions holds for the fast forward flight regime that are subject of our study, but such components of drag along (z’) axis should be accounted for other flight situations.Wing modelThe bird has two wings. Each wing is a rigid poly-articulated body, comprising the bird arm, forearm and hand, as pictured in Fig. 1b. Each segment is actuated by a joint to induce wing morphing. We refer to13,15 for a complete description of this wing kinematic model.Each joint is kinematically driven to follow a sinusoidal trajectory specified as:$$begin{aligned} q_{i}(t) = q_{0,i}(t) + A_{i} sin (omega t + phi _{0,i}) end{aligned}$$
(2)
with (omega = 2 pi f) and f being the flapping frequency which is identical for each joint, (q_{0,i}) being the mean angle over a period (or offset), (A_i) the amplitude, and (phi _{0,i}) the relative phase of joint i. A complete wingbeat cycle is therefore described through a set of 19 kinematic parameters, including the frequency f.We assume that the wing trajectory is rigidly constrained, and therefore we do not need to explicitly solve the wing dynamics. Under this assumption, the motion generation does not require the computation of joint torques. The model further embeds seven feathers of length (l_{ki}) in each wing. The feathers in the model have to be considered a representative sample of the real wing feathers. They thus have a limited biological relevance; their number is chosen so as to interpolate the planform satisfactorily and to smoothly capture the morphing generated by the bone movements. These feathers are attached to their respective wing bones via two rotational degrees of freedom allowing them to pitch and spread in the spanwise direction. These two degrees of freedom are again kinematically driven by relationships that depend on the angle between the wing segments13. This makes the feathers spreading and folding smoothly through the wingbeat cycle. In sum, the kinematic model of the wing yields the position of its bones and feathers at every time step. This provides a certain wing morphing from which the wing envelope (leading edge and trailing edge) can be computed (see Fig. 1b). From the wing envelope, the aerodynamic chord and the lifting line are computed. The lifting line is the line passing through the quarter of chord, which is itself defined as the segment connecting the leading edge to the trailing edge and orthogonal to the lifting line (Fig. 1b). This extraction algorithm is explained in detail in15.In order to calculate the aerodynamic forces, the angle of attack of the wing profile has to be evaluated. Each wing element defines a plane containing the lifting line and the aerodynamic chord as pictured in Fig. 1c. The orientation of the plane is identified by the orthogonal unit vectors ((hat{textbf{e}}_n, hat{textbf{e}}_t, hat{textbf{e}}_b)), where (hat{textbf{e}}_n) is the vector perpendicular to the plane and (hat{textbf{e}}_t) is the tangent to the lifting line. To compute the effective angle of attack, the velocity perceived by the wing profile is computed as the sum of the velocities due to the body and wing motion, and the velocity induced by the wake. The first contribution, (textbf{U}), accounts for$$begin{aligned} textbf{U} = textbf{U}_{infty } – textbf{v}_{kin} – textbf{v}_{q}end{aligned}$$where (textbf{U}_{infty } = u hat{textbf{e}}_{x’} + w hat{textbf{e}}_{z’}) is the actual flight velocity, (textbf{v}_{kin}) is the relative velocity of the wing due to its motion, and (textbf{v}_{q}) is the component induced by the angular velocity of the body q and calculated as$$begin{aligned} textbf{v}_{q} = qhat{textbf{e}}_{y’} wedge (textbf{P}_{i} – textbf{G})end{aligned}$$This velocity vector (textbf{U}) defines the angle (alpha), as pictured in Fig. 1d.The second contribution is due to the induced velocity field by the wake, i.e. the downwash velocity (w_{d}), and acting along the normal unit vector (-w_{d}hat{textbf{e}}_n). The resulting effective angle of attack, (alpha _{r}), is thus$$begin{aligned} alpha _{r} = alpha – frac{w_{d}}{|textbf{U}|}end{aligned}$$The downwash velocity (w_d) is computed according to the Biot-Savart law23, assuming the wake being shed backwards in the form of straight and infinitely long vortex filaments at each time step of the simulation13,15. This quasi-steady approximation is justified a posteriori by ensuring that our reduced frequency, inversely proportional to the unknown airspeed, never exceeds the value of 0.2, below which the effects of time-dependent wake shapes on wing circulation are negligible (e.g. see discussion in27). Once the downwash is evaluated, it is possible to evaluate the circulation, and consequently the aerodynamic force and moment acting at the element (P_i), i.e. (F_{x’, i}(textbf{x}(t), t), F_{z’, i}(textbf{x}(t), t), M_{y’, i}(textbf{x}(t), t)), as explained in detail in13. We use the thin airfoil theory for the estimation of the lift coefficient, with a slope of (2pi) that saturates at an effective angle of attack (alpha _{r}) of (pm 15^{circ }).Drag production by body and wingThe main body and the wings induce drag that should be accounted for in a model aiming at characterizing energetic performance. Body-induced drag is named parasitic because the body itself does not contribute to lift generation, and only induces skin friction and pressure drag around its envelope28. The total body drag is$$begin{aligned} D_{b} = frac{1}{2}rho C_{d, b} S_{b}|textbf{U}_{infty }|^{2} end{aligned}$$
(3)
where (rho) is the air density. We implemented the model described by Maybury28 to compute the body drag coefficient (C_{d, b}). This depends on the morphology of the bird and the Reynolds number Re according to$$begin{aligned} C_{d,b} = 66.6m_{b}^{-0.511}FR_{t}^{0.9015}S_{b}^{1.063}Re^{-0.197} end{aligned}$$
(4)
with (S_{b}) and (FR_{t}) are respectively the frontal area of the body and the fitness ratio of the bird, and both of them can be estimated from other allometric formulas i.e.28,29.$$begin{aligned} S_{b}= & {} 0.00813m_{b}^{2/3} end{aligned}$$
(5)
$$begin{aligned} FR_{t}= & {} 6.0799m_{b}^{0.1523} end{aligned}$$
(6)
The Reynolds number (Re = rho |textbf{U}_{infty }| overline{c} / mu) is calculated with the reference length of the mean aerodynamic chord (overline{c}), with (mu) being the dynamic viscosity. This model is found to be suitable for Reynolds number in the range (10^{4}-10^{5})28. Another source of drag is the profile drag due to friction between the air and the feathers on the wings. It is the sum of the profile drag at each section along the wingspan, i.e.$$begin{aligned} D_{p,w} = frac{1}{2} rho C_{d, pro} sum _{i=1}^{n} c_{i}|textbf{U}_{r,i}|^{2} ds_{i} end{aligned}$$
(7)
with (c_{i}) the chord length, (ds_{i}) the length of the lifting line element along the wingspan, and (textbf{U}_{r,i}) the velocity at the wing section i accounting for the body velocity, the kinematics velocity of the wing and the downwash velocity (Fig. 1c,d). We used a value of profile drag of (C_{d, pro} = 0.02) and this is assumed to be constant over the wingspan and throughout the flapping cycle30. In reality, due to the wing motion, this value should be gait dependent. However, the aforementioned assumption has been largely used in previous works31,32.Tail modelSince the span of the tail is of the same magnitude as its aerodynamic chord, here the lifting line approach cannot be used23. Therefore, the tail is modelled according to the slender delta wing theory, as a triangular planform33. Its morphology is illustrated in Fig. 1b and characterized by the opening angle (beta) and the chord (c_t). This latter parameter is kept constant, thus the tail span is controlled via (beta) from the trigonometrical relationship$$begin{aligned} b_{t} = 2c_{t}tan frac{beta }{2}end{aligned}$$The main limitation of this framework is the low range of angles of attack ((alpha _{tail} More