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This article was automatically translated from the original Turkish version.

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Thermal Air Current (Thermal)

Thermal air currents (thermals) are a natural phenomenon that enables birds, gliders, and even unmanned aerial vehicles (UAVs) to remain airborne and cover long distances without additional engine power. While soaring flight conserves energy, achieving very long distances solely through soaring is a significant challenge for both birds and humans; the solution lies in utilizing thermal air currents—columns of warm air rising faster than the descent rate of birds or gliders, heated by the Sun in the atmosphere. Thanks to thermal updrafts, glider pilots and raptors can stay aloft for hours without flapping their wings or using engine power, converting the altitude gained into forward distance. This phenomenon plays a critical role in natural ecosystems—for example, along migratory bird routes—and is also of great importance in sports aviation and UAV technology for enhancing efficiency.


A thermal air current is a localized column of warm air rising in the atmosphere due to the Sun heating the Earth’s surface. These localized convective currents experience an upward lift force because they are warmer and therefore less dense than the surrounding air, typically reaching vertical speeds of about 1–10 m/s. Thermals can reach diameters of several hundred meters depending on the terrain and atmospheric conditions; as they rise, they mix with surrounding air, expand, and gradually weaken. If the rising air contains sufficient moisture, water vapor condenses at a certain altitude, forming cumulus clouds; indeed, cumulus clouds are commonly observed at the upper limits of strong thermal currents. Thermal air currents, also known in aviation circles as “thermals,” can occur in any season, but the most intense and powerful examples are found especially during sunny summer months.

Formation Mechanism and Theoretical Foundations

The formation mechanism of thermals is based on the differential heating of different regions of the Earth’s surface. Air above an area intensely heated by the Sun warms more than its surroundings, expands, and begins to rise. This column of warm air continues to ascend to a certain altitude as long as a temperature and density difference persists between it and the surrounding air; over time, it mixes with cooler surrounding air, cools, and spreads out, eventually dispersing horizontally at the thermal’s peak and concluding with downward movements. This cycle, repeating throughout the day in unstable atmospheric layers, continuously generates new thermal cells.


Terrain and atmospheric conditions play a decisive role in thermal formation. For instance, cities, dry soil, or rocky terrain absorb more solar radiation and generate stronger thermals, whereas water surfaces or dense vegetation produce weaker thermals because a significant portion of the energy is consumed by evaporation. Sudden temperature differences at boundary zones such as forest clearings or hill slopes can also trigger thermal uplift. A rising thermal continues upward until it reaches a height determined by the atmospheric temperature stratification; if there is no strong inversion layer above, it may rise to altitudes of 3000–5000 meters, but in most cases, it flattens and stops due to the inversion layer.


The mass of warm air within a thermal experiences a net upward lift force because it is less dense than the surrounding air. The magnitude of this lifting force depends on the temperature difference between the rising air and its environment; as the air mass ascends, ambient pressure decreases, causing the mass to expand and cool. Consequently, the vertical velocity profile of the thermal diminishes with altitude and eventually reaches zero. In the real atmosphere, the strength and diameter of thermals do not remain constant with height; typically, an airflow that begins narrow and strong near the ground widens and weakens as it rises. Therefore, strong, large thermals usually develop over land during midday, while weak and small thermals are observable in the morning or evening. Thermal formation is a significant convective process that also triggers vertical mixing in the atmosphere and cloud development.


Use by Humans and Animals

Birds

Many large bird species (eagles, hawks, storks, vultures, etc.) skillfully use thermal air currents to fly for extended periods without flapping their wings. Predatory birds, in particular, spiral upward within thermals until reaching a suitable altitude, then lock their wings and glide, diving rapidly toward their prey. Some birds can cover hundreds of kilometers in a single day using only thermal currents. Migratory species also make effective use of this natural energy source; for example, certain migratory birds such as the white stork can travel thousands of kilometers along their migration routes by continuously following thermals, thereby avoiding wing flapping. This reduces energy expenditure during migration and helps birds avoid excessive fatigue. Flight patterns vary among bird species depending on their physical characteristics. Birds with heavier bodies and higher wing loading must perform wider turns to ascend within thermals. In contrast, lighter birds with broader wings can maneuver more easily within narrow thermals and exploit these areas more efficiently.


Another notable observation is that the flight behaviors birds exhibit when using thermals resemble the flight strategies employed by experienced glider pilots. Observations have shown that birds appear to intuitively predict the location and strength of the next thermal and adjust their flight speed accordingly. This behavior parallels some flight theories developed by humans. Consequently, it can be said that there is a striking overlap between instinctive natural movements and engineering-based approaches; both are direct consequences of aerodynamic principles.


Hang Glider and Glider Pilots

Hang glider, paraglider, and glider pilots engage in thermal hunting under suitable meteorological conditions to perform soaring flights. A pilot entering a thermal ascends in spiral turns, much like a large bird, gaining altitude; once sufficient height is reached, the pilot transitions to straight flight to locate the next thermal. Repeating this cycle enables covering distance without engine power. Experienced glider pilots can predict the presence of thermals from ground clues—for example, observing a flock of rising birds or a dust devil—and use an instrument called a variometer to measure instantaneous climb or sink rates while airborne, attempting to locate the thermal’s center.


The MacCready Speed-to-Fly theory, widely used among glider pilots, determines the optimal cruise speed based on the expected thermal climb rate. In practice, pilots following this strategy during transitions between thermals can reduce total flight time and enhance performance. This method is frequently employed in competitive gliding; flight data from successful pilots demonstrate that they adjust their speeds in accordance with these calculations.


Natural observations reveal that pilots rely not only on instruments but also on attention to nature when locating thermals. In particular, the climbing behavior of birds often serves as a guide for pilots. In some cases, birds have even been observed heading toward a thermal discovered by a glider and sharing the same air column. This mutual interaction is a compelling example of how nature and human technology can operate in harmony.


Unmanned Aerial Vehicles

The ability to harness thermal air has emerged in recent years as an innovative approach to extend flight duration and range in unmanned aerial vehicles (UAVs). Especially fixed-wing, glider-type UAVs, much like small gliders, can detect rising columns of warm air and ascend within them by performing circular maneuvers, thereby conserving battery energy. As a result, the flight duration and mission scope of these vehicles, which have limited energy capacity, are significantly enhanced. Systems that effectively utilize thermal currents have been observed to achieve far more efficient performance compared to conventional manual flight methods.


Different control approaches have been developed for UAV thermal utilization. One such approach, rule-based systems, analyzes sensor data from the vehicle to infer entry into a thermal based on signals such as altitude gain or airspeed changes. In this case, the vehicle automatically initiates a turn and begins a spiral ascent at a calculated radius, orienting itself toward the center of the warm air column. These algorithms operate within predefined rules, aiming to keep the vehicle effectively within the thermal.


More advanced systems are based on artificial intelligence approaches. In these methods, UAVs acquire the ability to analyze environmental conditions and adapt their flight strategies accordingly. In particular, using a machine learning technique known as reinforcement learning, UAVs are trained in simulation environments and, through these experiences, develop the skill to detect and remain within thermal currents during actual flights. Such systems draw inspiration from the flight behaviors of birds, integrating this natural mechanism—perfected over thousands of years—into engineering solutions. All these developments not only enhance the efficiency of UAV technology but also demonstrate how successful flight strategies from nature can be imitated through technology. The potential offered by thermal flight for designing energy-efficient, long-range, and environmentally friendly aerial vehicles serves as an inspiring example for modern engineering.

Detection Methods

Detecting thermal currents is the first prerequisite for their efficient utilization. In sports aviation, pilots observe various indicators of thermal presence: for example, puffy cumulus clouds forming in the sky typically mark the top of a strong thermal; the formation of a dust devil (small-scale dust whirlwind) or vertical rising smoke indicates thermal activity; additionally, seeing a group of birds rapidly ascending in circles signals the proximity of a strong thermal. Glider pilots continuously monitor their variometer while airborne; a positive climb rate indicates that the aircraft has entered an ascending air current within a thermal. Pilots often sense, as they enter a thermal, that one wing rises before the other, allowing them to determine the thermal’s location and begin turning in that direction. Since the central region, known as the thermal core, provides the strongest lift, pilots must turn as tightly as possible around the core to achieve maximum climb rate. During this process, experienced pilots combine variometer readings with atmospheric and terrain indicators to master the skill of centering the thermal.



In autonomous flying systems, algorithms based on vehicle sensor data are applied for thermal detection. For instance, if an abrupt increase in barometric altitude is detected by a UAV, it is assumed to have entered a thermal, prompting the vehicle to automatically switch to a search/climb mode. In this mode, the flight control system begins tracing a circular path around the thermal’s center, calculating the required turn radius to keep the vehicle within the region of strongest lift. If the vehicle continues to climb, it maintains the turn; if climbing stops or descent begins, it concludes it has lost the thermal and re-enters thermal search mode. Modern glider-type UAV autopilots further enhance thermal detection by integrating wind sensors, accelerometers, and GPS data.

Applications and Significance

The utilization of thermal air currents has broad applications and significant contributions in both natural ecosystems and human activities. Ecologically, thermals are a decisive factor in the migration strategies and daily movements of large birds. This enables birds to cover vast distances while conserving energy and surveying wide areas in search of food. For example, migratory birds such as storks and pelicans that ascend via thermals follow routes over land, avoiding oceanic paths, because strong thermals do not form over large water bodies, making continuous long flights over water challenging. In sports aviation and recreational flying, thermal currents form the fundamental basis for motorless aircraft (gliders, hang gliders, paragliders, etc.). Thanks to thermals, glider pilots can remain airborne for hours, break international distance records, and complete competition courses using only atmospheric energy. This provides aviation enthusiasts with an interactive flight experience with nature while also offering a silent and environmentally friendly platform for purposes such as surveillance, exploration, or photography without requiring powered flight.


From an engineering and industrial perspective, thermal utilization holds great potential for UAVs and autonomous aerial vehicles. A UAV with battery-limited flight range can significantly extend its airborne duration by encountering thermals along its route, thereby expanding its operational coverage for tasks such as search and rescue, agricultural spraying, or aerial mapping. Thermal-assisted flight, which can be performed without fossil fuels or additional power sources, also contributes to sustainable aviation approaches. Moreover, studying thermal flight serves as an inspiring subject for biomimetic and artificial intelligence research. Understanding the mastery of birds in thermals aids in developing automatic pilot systems for autonomous gliders. Indeed, the phenomenon of thermal ascent is regarded in the fields of autonomous motion control and reinforcement learning as an “ideal model problem” because, despite its dynamic and unpredictable constraints, it can be modeled with relatively few control parameters. Finally, thermals play a key role in vertical transport and cloud formation processes in the atmosphere. Therefore, research on thermal air currents contributes significantly to scientific fields such as weather forecasting and climate modeling, enhancing the understanding of atmospheric convection.

Mathematical Modeling

Various mathematical models have been developed to understand and optimize thermal flight. For example, the ideal turn radius formula, derived from the balance between the required lift force and the bird’s weight, and the need to counteract centrifugal force during a steady circular turn within a thermal, is as follows:


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  • <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mord mathnormal">s</span><span class="mord mathnormal">in</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4539em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1089em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight">co</span><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> ; from this; <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6289em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1089em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">in</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">μ</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight">co</span><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>


  • Substituting the equations; <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6289em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">W</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ρ</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1089em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">in</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">μ</span><span class="mclose mtight">)</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1433em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">co</span><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>


where r is the turn radius, W is the weight, S is the wing area, W/S is the wing loading, ρ is the air density, g is the standard gravitational acceleration, µ is the bank angle, γ is the glide angle, and CL is the lift coefficient. This expression quantitatively reveals how characteristics such as wing area and weight affect thermal turning performance. It predicts that a lighter or broader-winged (i.e., lower wing loading) object can turn with a smaller radius within the same thermal, whereas a heavier or smaller-winged object will have a larger turn radius. Indeed, data from different birds and gliders confirm that those with higher wing loading can only turn in larger circles, while lighter, broad-winged ones can fit within narrow thermals. Analytical models are also used to represent the vertical velocity profile of thermals. Generally, it is assumed that the upward velocity within a thermal decreases with distance from the center. A simple but common approach assumes a Gaussian distribution of velocity around the thermal core. For example, the vertical air velocity within a thermal, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>, can be modeled as a function of its horizontal position:


<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.7256em;vertical-align:-0.6166em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">e</span><span class="mord mathnormal">x</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1089em;"><span style="top:-2.6264em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8051em;"><span style="top:-2.1528em;margin-left:-0.0077em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">h</span></span></span></span><span style="top:-2.8448em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3472em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6166em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>


where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the maximum vertical velocity within the thermal core (m/s), <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are the thermal core coordinates, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> represents the thermal’s characteristic radius. This model reflects that the lifting effect is highest at the thermal’s center and decreases exponentially toward its edges. In the real atmosphere, thermal structure can vary with altitude, and wind/turbulence effects may cause the core position to fluctuate; however, the above idealized formulas serve as important guides for both pilot flight planning and autonomous system control algorithms. For example, MacCready’s famous speed-to-fly theory enables a glider pilot to determine the optimal cruise speed based on the expected climb rate in the next thermal, allowing the glider to minimize total flight time during transitions between thermals. Similar principles are applied instinctively by wild birds, as demonstrated by comparisons between bird and glider data.


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AuthorEmre KarapınarDecember 4, 2025 at 1:21 PM

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Contents

  • Formation Mechanism and Theoretical Foundations

  • Use by Humans and Animals

    • Birds

    • Hang Glider and Glider Pilots

    • Unmanned Aerial Vehicles

  • Detection Methods

  • Applications and Significance

  • Mathematical Modeling

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