A Survey on Thermal Updraft Models

Sreekanth Ganapathi Raju Eric Poquillon Emmanuel Rachelson

Abstract

This report presents a survey on the existing Ther-mal updraft models. These models are studied toperform autonomous soaring simulations that canhelp in improving aircraft endurance. This workcomprises of several existing approaches used insoaring studies to model updrafts as close as pos-sible to reality. The models surveyed here weredeveloped based on data measurements from re-gions of updrafts. Thermal updrafts in the formof plume rise called as Chimney Thermals, theseChimney thermals in the presence of light horizon-tal wind characterized by leaning and drifting andfinally completely detached Bubble Thermals arethe different kind of updrafts presented here. Toclose, a software tool is developed based on thesemodels. This tool can be used to create updraftsover a given region based on a selected model. Thetool simulates a wind field where updrafts will ran-domly form and vanish. Hence, it can be used insimulation studies to give the effect of updrafts onthe aircraft.

1 Introduction and Motivation

During the past few decades with major advance-ments in precision sensing, fast actuation and au-tonomous computing, the controllability and auto-maticity of modern aircrafts have improved to alarge extent. This has also increased the number ofboth civil and military applications of unmannedaerial vehicles (UAVs). However, as the complexityof their tasks is extending, there is a great demandfor seeking more possibilities to extend the rangeand flight duration of UAVs. A key difference be-tween UAVs and manned aircraft is that their en-durance is not limited by operator endurance. Apromising idea will be use of atmospheric energy inthe form of gusts and updrafts to extend aircraftendurance. Its use could significantly augment the

mission duration. With the growing demand forinnovation in areas of green technology, energy ex-traction from atmospheric sources will simultane-ously conserve fuel or electrical energy.

Autonomous soaring gliders have shown to begreatly efficient in conserving energy from the at-mosphere in the form of Thermal updrafts. Thanksto their very efficient aerodynamics with less dragthey can significantly improve range. Any au-tonomous soaring study can be characterized intothe atmospheric model consisting of the winds andthermal updrafts, the flight dynamics of the air-craft and the control algorithm governing the au-tomaticity of the flight. The first step is to modelthe atmosphere and the thermal as close to realityas possible. For this reason, there is a great inter-est and also a great importance to understand upwinds and thermals in the atmosphere. It is im-portant to look into the available work done in thefield of thermal updrafts.

Thermal updrafts are raising mass of air fromthe ground, caused due to temperature variationbetween the air close to the ground and the air inthe atmosphere. These updrafts are thermal con-vection currents fuelled by radiating heat from theground. Since thermal updrafts are very complexphenomenon there exist many approaches to calcu-late the updraft velocity and size of the updrafts.This bibliography survey report comprises of var-ious previous work done in the study of thermalupdrafts. The motivation behind this is to under-stand the existing research in this field and analyzetheir advantages and disadvantages. The modelsexplained here are purely empirical and are notbased on CFD or any other numerical computa-tions. These models are very simple and effectivein terms of computation time.

A software tool has also been developed alongwith this survey, which can be used along with anysoaring simulation studies. This tool is parametricin nature with input variables as position and sim-

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ulation time. The output variable will be a windvector at the given point and time. The tool canbe directly used as a blackbox to give the effect ofupdrafts on the aircraft. The data for all graphsand plots shown in this report are generated usingthis software tool.

This survey report and the software tool are de-veloped as a first step in performing autonomoussoaring study.

2 Thermal Updraft Models

During daytime when the ground gets heated updue to the sun’s radiation, the temperature of theground rises. This results in heat transfer betweenthe ground and the air close to it. This heat trans-fer is mainly conductive in nature. Due to differ-ences in the nature of the ground, different regionsget heated up differently resulting in areas calledhot spots. Usually this will be a parking lot closeto forest areas, farmland surrounded by trees etc.As the air in these areas have higher temperaturethan that of the surrounding air, convection cur-rent is set up causing this air to rise. The rising airexchanges heat with the surrounding atmosphere.This air rises over several hundred meters until itcools down and mixes with the surrounding atmo-sphere. The height until which this phenomenonhappens is called the convective mixing-layer thick-ness zi shown by the Figure 1.

The type of updraft shown in Figure 1 is alsocalled as chimney thermal and is predominant inthe absence of high horizontal winds. In the pres-ence of high horizontal winds the rising air mightget detached and start traveling along with thewind and this type of thermals are called Bubblethermals. More about them in section 2.9.

Various models can be found in the literatureto describe this phenomenon and the main modelfeatures of a thermal are its radius and the updraftfield. Both these variables as a function of altitudeare studied. The updraft field representing how thevertical airspeed varies with the position relative tothe thermal center, and its radius defining wherethe updraft speed is null or almost null, are thevariables of main concern.

Figure 1: Thermal updraft and Convective bound-ary layer thickness zi

2.1 Overview

Lenschow [11] in 1980 performed experiment bytaking humidity measurements over sea and for-mulated expressions for updraft velocity, diameterof the thermal and various other parameters as afunction of vertical height. His analytical expres-sions are cited by all the other authors as well andhence serve as a basis for general updraft study.The model expresses statistical mean and variancesof updraft velocity and diameter, for a given nor-malized thermal height based on measured data.His equations give variation of updraft velocity andthermal radius with altitude but assumes them tobe constant in the horizontal direction. In this re-port his equations are extended to 3D by imposinga given profile on the updraft velocity showing thevariation along the diameter of the thermal, hencein the horizontal direction.

M.J. Allen [1] puts forth a model based on bal-loon temperature readings and surface radiationmeasurements. The model is a mathematical rep-resentation of Chimney thermals in 3D. He devel-ops his equations based on Lenschow’s [11] workand based on flight test results from Konovalov [8].This model gives the variation of updraft velocitywith respect to height as well as with respect tothe radius of the thermal. He considers the ther-mal as trapezoidal in shape. Using Lenschow [11]equation for the average updraft velocity and then

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correlating this to the average velocity from thetrapezoidal, he develops equations for updraft ve-locity at each point. The model also comprises ofan environmental sink rate developed to conservemass.

Childress [4] is based on collected data by fly-ing at constant altitudes in and out of the thermalsusing an instrumented glider. A new phenomenondifferent from traditional plume rising thermals wasobserved using flight tests. The main differencebetween the conclusions of Childress and others isthat, this author puts forth an idea of thermals withstrong downdraft at the center of the thermal sur-rounded by maximum updraft regions. He says thiskind of thermals with a core downdraft is formedwhen there are disturbances in the environment.The author maps the flight test data collected toa mathematical model forming equations based onLenschow [11] and Allen [1].

Lawrance [10] Ph.D. thesis is about harvestingatmospheric energy for autonomous soaring usingpath planning. As the main idea is not on updraftmodeling Lawrance in his Ph.D. thesis only brieflytalks about a thermal model characterized by bub-ble formation. The author proposes detached ther-mal updrafts in the form of bubble leaving theground in case of high winds. The model definesa 3D toroidal updraft flow field and the equationsgiving the wind velocity field at each location insidethe bubble.

2.2 Lenschow’s model

The study done by D.H. Lenschow and P.L.Stephans is one of the very first models proposedgiving an analytical expression for the thermal up-draft velocity and thermal diameter as a func-tion of vertical distance. The equations proposedby Lenschow and Stephan, are used by all theother authors as a staring point in their own stud-ies. Hence this paper forms as a basis for an up-draft study. The equations in this paper are de-rived based on measurements taken using a NCARElectra aircraft flying through convective bound-ary layer. All the results and conclusions of thispaper are based on thermals formed over sea andnot over land. The experiment consisted of flyingthe aircraft at constant altitude for 30km and mea-suring humidity changes. If the humidity changeobserved was more than half the standard devia-

tion of humidity fluctuations, then the region wascharacterized as thermal.

The updraft properties derived are normalized tofit for any thermal irrespective of their height andsurface heating, using scaling parameters w* theconvective velocity scaling and, zi convective mix-ing layer thickness. Based on the measurements,expressions are formed for the normalized numberof thermals, diameter of the thermal and the up-draft velocity.

The normalized number of thermals at regionsclose to the ground, z < 0.2zi was predicted to bea function of -1/3 powers of normalized height. Forregions above this height the number of thermalsare nearly constant.

For z < 0.2zi,

N = 0.68(z

zi)

13

An expression is given for the diameter of the ther-mal as a function of 1/3 powers of normalizedheight,

d = zi × 0.16(z

zi)

13 (1− 0.25z

zi) (1)

The average updraft velocity at any given heightinside the thermal is given as a function of convec-tive velocity scale parameter and normalized heightby Equation 2.

wTw∗

= 1.0(z

zi)

13 (1− 1.1z

zi) (2)

In addiction to the above, an expression given byEquation 3 is also derived for temperature excessin the thermal.

θTθ∗

= 1.5(z

zi)

13 (1− 2z

zi) (3)

From the mean updraft velocity and the tem-perature excess formed in the thermal, quantitiessuch as the total contribution of heat flux of ther-mal, humidity excess, virtual temperature excessare also expressed in this paper. After expressingthe mean quantities, the author extends his studyby defining the variances of the calculated quanti-ties as the function of normalized height also. Thepaper concludes by deriving an equation for meanvertical velocity equation for a thermal includingthe pressure and edge effect terms.

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Hence this paper provides expressions for up-draft and downdraft velocities, diameter of ther-mal, temperature excess and humidity excess in thethermal and the heat flux a function of normalizedheight. The variances of these quantities as a func-tion of normalized height are also calculated. Asthis model comprises only on dependencies alongthe vertical direction it can be approximated as a1D model for the thermals.

2.3 3D modeling of Lenschow equa-tions

Lenschow model provides equations to calculate up-draft velocity and thermal diameter at any givenheight but does not talk about the variation ofthose quantities along the horizontal direction.

The most basic modeling extension of thermalsin 2D is Gaussian modeling of the vertical updraftvelocity. This is done by imposing a Gaussian onthe thermal with the maximum updraft velocity atthe center of the Gaussian and the variance ad-justed such that the vertical velocity at the end ofthe Gaussian is zero. This thermal profile has nodowndraft associated to the thermal. This modelcan be seen in many previous work associated toautonomous soaring and modeling of thermals forsuch studies [15, 2, 5]. In this report and in the vi-sualization model, this is done by assuming that theupdraft velocity proposed by Lenschow as the corethermal center velocity and Lenschow diameter asthe diameter of the Gaussian. Hence for a thermalwith centerline updraft velocity wcore and radiusR, the updraft velocity w at any given distance rfrom the center is given by Equation 4.

w = wcoree−( r

R)2 (4)

Figure 2 shows the plot of Equation 4. The ther-mal was considered to axis-symmetric and this wasextended to 3D in the visualization model and thiscan be seen in the Figure 3. As the Gaussian hasno downdraft associated to it, Gedeon [7] proposedan improvement by adding a downdraft at the endof the Gaussian. In this model the thermal velocityis maximum at the center and becomes zero at theend of the thermal radius also but outside the ther-mal radius there is a region of downdraft where thevertical velocity is negative. This thermal profile is

Figure 2: Variation of Updraft along the radiusafter imposing a Gaussian

Figure 3: 3D representation of the updraft calcu-lated by imposing a gaussian on Lenschow model

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Figure 4: Variation of Updraft along the radiusafter imposing Gedeon’s Profile

widely used in many static soaring studies associ-ated with [9, 12, 6]. Here the updraft velocity wat any given distance r from the center is given byEquation 5.

w = wcoree−( r

R)2 [1− (

r

R)2] (5)

The Figure 4 shows the 2D profile and Figure 5 theaxis-symmetric 3D extension of the Gedeon equa-tion.

The figures 6 and 7 show the variation of theLenshcow models for different altitudes. It can beseen that as height increases the updraft velocitydecreases but the effective radius of influence ofthe thermal increases. Though the effective radiusof the thermal is the same between Gaussian dis-tribution and Gedeon distribution, the decrease inupdraft is steep in Gedeon model and also there ispresence of downdraft at outer edge of the thermal.Hence for a location close to the thermal edge Gaus-sian imposed Lenschow model will produce a pos-itive updraft while, for the same location Gedeonimposed Lenschow model will give a negative up-draft.

Both the Gaussian and Gedeon approaches arepure empirical extensions of the lenschow modelsand the mass conservation of these approaches werenot tested and validated. Also, the lenschow modeldoes not give a smooth transition between the endof the convective layer zi and heights above it. The

Figure 5: 3D representation of the updraft calcu-lated by imposing Gedeon’s profile on Lenschowmodel

Figure 6: Variation of Updraft profile with Altitudefor Lenschow with Gaussian imposed

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Figure 7: Variation of Updraft profile with Altitudefor Lenschow with Gedeon imposed

updraft velocity is maximum near the ground andit does not reaches zero when z = zi. It reacheszero at approximately 0.9z* and then goes nega-tive until the end of the convective layer. If the netupdraft velocity outside the convective layer is con-sidered as zero then this model, if used to performsoaring studies, will create a discontinuity when thealtitude crosses the convective layer limit zi.

2.4 Allen’s model

M. J. Allen [1] collected data through surface radi-ation measurements and balloon-measured temper-ature over a year at Nevada, USA. The measure-ment consists of surface temperature, wind and ra-diation measurements using surface radiation sta-tion as well as temperature and humidity collectedfrom balloon for an entire year of 2002. Using thisdata the primary observed and calculated quanti-ties were, the convective velocity scale w* and theconvective mixing layer thickness zi. The convec-tive layer thickness zi is calculated by plotting mea-sured balloon temperature and dry adiabatic lapserate. Convective scale velocity w* is calculated us-ing equations of sensible heat from Stull [14]. Inthis paper only conditions with horizontal windsless than 12.87 [m/s] (25 knots) are considered, forthe rest w* is equated as zero. Equations fromLenschow [11] were used to calculate the averageupdraft velocity based on w* and zi. The equa-

Figure 8: Updraft velocity distribution as revolvedtrapezoid from [1]

tions used by Allen for the radius of the thermal hasdifferent coefficient compared to that of Lenschow[11].

Using the flight test results from Konovalov [8]this paper maps the relation between the radius ofthe thermal and updraft velocities. Thereby ex-tending the model to 2D. Konovalov [8] shows thatthe maximum updraft velocity is not a single pointbut instead can be felt over a region in the cen-ter. This means that inside a thermal there willbe a region with constant peak updraft velocitysurrounded by a region of decreasing velocity un-til zero outside the thermal. To satisfy this Allenassumed the thermal to be having a revolved trape-zoidal shape with peak updraft velocity at the cen-ter. The vertical velocity starts to decrease andreaches zero at the radius of the thermal. The re-volved trapezoidal is shown below in Figure 8.

The updraft outer radius given by Allen is givenby the Equation 6.

r2 = max(10, 0.102(z

zi)

13 (1− 0.25

z

zi))× zi (6)

The average velocity given by Lenschow in Equa-tion 2 is correlated with that of the area of the peakvelocity and the volume of the entire trapezoid. Us-ing that the peak velocity is given by the Equation7

wpeak = 3wr32 − r2

2r1

r32 − r3

1

(7)

As the height increases the thermal updraft ve-locity decreases and then turns into a downdraft.

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Figure 9: Toroidal Shape of the thermal from [1]

The toroid shape of the thermal showing this phe-nomenon is shown below in the Figure 9. Allen hasdeveloped equations for the downdraft developedas the height increases. The equations given by 8and 9 are used to calculate the downdraft velocity.

wl =

{−π6 sin(π × r

r2) for r1 < r < 2r2

0 otherwise(8)

wD =

{2.5wl(

zzi− 0.5) for 0.5 < z

zi< 0.9

0 otherwise(9)

To have a smooth transition between the peakupdraft velocity in the thermal to the surroundingair, Allen has imposed a bell shaped curve basedon shape constants k1−4 which depend on the ra-dius ratio r1/r2 given by the table 1. The result-ing profile for the updraft velocity for different bellshapes are shown in the Figure 10. To make thebell curves symmetric about r/r2=0 the constantk3 was made zero in all cases. Based on all theabove given characteristics the updraft velocity dis-tribution along the radial direction is given by 10

w = wpeak(1

1 + |k1 × rrr

+ k3|k3+ k4 ×

r

r2+ wD)

(10)

r1r2

k1 k2 k3 k4

0.14 1.5352 2.5826 -0.0113 0.00080.25 1.5265 3.6054 -0.0276 0.00050.36 1.4866 4.8354 -0.0320 0.00010.47 1.2042 7.7904 0.0848 0.00010.58 0.8816 13.972 0.3404 0.00010.69 0.7067 23.994 0.5689 0.00020.80 0.6189 42.797 0.7157 0.0001

Table 1: Shape constants for different radii ratio

Figure 10: Bell curves for different shape constants

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In a region of given area the maximum numberof updrafts is computed using the equations fromLenschow [11]. For a given domain of horizontalarea S, number of thermals N of height zi and ra-dius R is given by,

N =0.6S

ziR(11)

As updrafts are basically mass of air rising, to haveconservation of mass equal about of mass has to de-scend. This paper also puts forth the environmen-tal sink rate caused by a given number of thermalsin a particular terrain. The resulting downdraft ve-locity in areas where there is no thermals is givenby Equation 12.

we =−wNπr2

2swd

S −Nπr22

swd =wDwl

(12)

To have a smooth transition from the calculatedupdraft in the center to the environmental sink rateEquation 13 is used.

wtotal = w(1− wewpeak

) + we (13)

2.5 Discussion on Allen’s model

Figure 11 shows the updraft velocity for differentaltitudes. It can be seen that the updraft velocitydecreases with altitude and the radius of the ther-mal increases with height. The negative velocityoutside the Thermal is the environmental sink rate.Here while calculating the environmental sink rate,it was considered that one thermal is present in aregion of 300x300 [m2] area. Figure 13 shows the3D profile of the updraft velocity calculated usingallen model at an altitude of 800[m]. The environ-mental flow rate calculated by Allen is done by ap-plying mass conservation for every altitude i.e. if aslice is made at any given altitude then the amountof mass going upward in the form of the updraftis equal to the amount of air going down as downas downdraft. Because of this kind of modeling theenvironmental sink rate invariable becomes a func-tion of altitude. Figure 12 shows the variation ofenvironmental sink rate with respect to altitude.

Figure 11: Allen updraft velocity profile for variousaltitudes

Figure 12: Variation of environmental sink ratewith altitude

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Figure 13: 3D representation of Allen updraft ve-locity at z=800[m]

2.6 Allen’s model with ambient wind(Bencatel’s model)

In the presence of horizontal winds the shape andposition of the thermal updrafts may be disrupted.Thermal updrafts are formed over surfaces calledhotspots. These hotspots depend on the topogra-phy of the terrain and because of which, the ther-mals remain anchored to this location. In mostcases with horizontal winds, the thermal center onthe ground does not drift along with the wind. Thiscauses the thermal to drift along the direction ofthe blowing wind. In his report Allen says thathis model is valid for horizontal winds less than12.87[m/s] but he mentions no effect on winds be-low this value. To include the effect of winds intothe Allen model equations from Bencatel [3] is used.Bencatel [3] states that, when the thermal is an-chored to their hotspot or if the thermal driftingis slower than the wind speed, the thermals havetendency to lean along the wind. He has men-tioned equations using which the leaning can becalculated.

In our model we have included the equationsfrom Bencatel [3] to calculate the leaning of thethermal. The drift velocity of the thermal centernear the ground can be modeled either as zero or a

constant. The leaning of the thermal is character-ized as thermal centerline leaning towards the wind.The centerline drifting is a function of the updraftvelocity; if the updraft velocity is high the driftingis less. The following Equation 14 , 15 gives theposition of the centerline at any given height basedon wind velocity in the horizontal directionWx andWy.

xt(H) = xc +

∫ H

0

Wx(h)− u

wz(h)dh (14)

yt(H) = yc +

∫ H

0

Wy(h)− v

wz(h)dh (15)

Where, xt(H) and yt(H) are position of thermalcenterline at height H, xc and yc are the positionof thermal center on the ground, Wx(h) and Wy(h)are the ambient wind velocities in X and Y direc-tions, wz(h) is the average updraft velocity of thethermal as a function of height and u, v are the ve-locity with which the thermal center on the grounddrifts. The drift velocities u, v in most cases as-sumed to be zero. Figure 14 shows the leaning ofthermal center line along the direction of wind andFigure 15 shows the contour of the updraft withhorizontal winds.

In case of thermals formed over sea or dessertwith very even topography, this drift velocity canbe assumed to be equal to the wind velocity, inwhich case the thermal only drifts and does notlean.

2.7 Childress’ modelC.E. Childress, An Empirical Model of ThermalUpdraft, [4] is characterized by taking measure-ments from an instrument glider flying in and out ofthermals at different altitudes and comparing themto the traditional view of thermals as a single risingplume. From the data collected arguments were putforth saying that the thermal cannot be consideredas a single rising plume after a certain height but in-stead another phenomenon, expressed as convectivecells, is occurring. The paper builds mathematicalmodels based on work by Allen [1] and Lenschow& Stephens [11].

The convective cell phenomenon expressed hererefers to a cell that is formed when there is asym-

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Figure 14: Leaning of Thermal Centerline with hor-izontal Wind

Figure 15: Updraft Contour in the presence of Hor-izontal Wind

Figure 16: Measured Updraft velocity by Childressat z*=0.87 from [4]

metry or disturbances in the atmosphere. This phe-nomenon is prominent at heights above half theover all height of the Convective Boundary Layer(CBL). Above 50% CBL the thermal is no moresingle toroid shape with strong core section but in-stead two peaks with maximum updraft surround-ing a core with strong downdraft. This core col-lapse phenomenon is a conclusion based on thedata collected which is shown below. The quan-tity z*=current altitude(z)/height of CBL(zi) Fig-ure 16 shows one of the flight test data collected byChildress at z*=0.87 showing the updraft velocity.It can been seen here that a downdraft region issurrounded by regions of updraft.

With the data collected and based on equationsfrom Lenschow the author develops equations forthe diameter of the thermal and the diameter ofthe downdraft core as functions of vertical height.Based on measurement the author gives his vali-dation on Lenschow model. It is shown that thediameter calculated using lenschow equations arethree times smaller than the ones measured by theexperiment. To account for the increased diameterthe author used the same equations of lenschow butwith different constants. The author also adds anadditional term to account for the convective cellgrowth. The diameter of the thermal is given bythe Equation 16.

dT = zi × (0.43√z∗ × (1− 0.5z∗)) +

(z∗ − 0.6)(z∗)z

π(16)

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The paper also calculates the diameter of the coredowndraft. The equation for the core downdraftdiameter is given by 17.

d1 = 0.17dT + 0.5× (z∗ − 0.6)× dT (17)

The updraft velocity along the vertical and ra-dial directions is calculated based on the equationsused by Allen [1]. Same methodology as that ofAllen is adapted to find the peak velocity at thethermal center as a function of height. In addictionthe downdraft velocity at the convection cell coreis calculated using dimensionless decelerating termfrom Lenschow [11]. The core downdraft velocityas a function of dimensionless decelerating term wdand the peak updraft velocity wpeak is given by 18.

wD = wd(z∗ + 0.45)− 0.5wpeak (18)

The variation of vertical velocity along the radialdirection is not derived based on experimental databut it is done by mathematically imposing an equa-tion. The mathematical model imposed here can behence divided into three, one for z* less than 0.5,another for z* between 0.5 and 0.9 and the third tocharacterize the region where the thermal dies outz* greater than 0.9.

For z∗ < 0.5,

w(r) = wpeak cos(r

r2× π

2) (19)

For 0.5 < z∗ < 0.9,

w(r) =

{wD × cos( rr1 ×

π2 ) for 0 < r < r1

wpeak × sin( r−r1r2× 1.212π) for r1 < r < r2 + r1

(20)For z∗ > 0.9,

w(r) =

{ wD2 × cos( rr1 ×

π2 ) for 0 < r < r1

(1− z∗)wpeak × sin( r−r1r2× 1.212π) for r1 < r < r2 + r1

(21)A comparison of the Childress equation with theobserved data can be found in Childress [4].

2.8 Discussion on Childress’ model

Figure 17 shows the output of the Childress modelfor different altitudes. It can be seen that the ef-fective area of the thermal increases with altitudei.e. the thermal radius increases with altitude. In

Figure 17: Childress vertical velocity profile for var-ious altitudes

the domain where the thermal is considered as tra-ditional plume (region below z∗ < 0.5) as heightincreases the updraft velocity decreases and the ra-dius of the thermal increases. The variation in up-draft velocity is significantly higher compared toLenschow model. Also, the absolute peak updraftvelocity in higher in Childress model than Allen andlenschow models. In the region where the convec-tive cell phenomenon is observed. Both the updraftand the downdraft velocities decrease with altitude.The updraft smoothly reduces and become zero atz*=0.9 while the downdraft continue to decreaseuntil z∗ = 1. The radius of the downdraft core in-creases with respective to altitude. At which pointthere is still some downdraft. It should be notedthat there is a strong discontinuity at the centerline when z* is 0.5, 0.9 and 1. Figure 18 shows thevariation of outer radius and the inner radius of thethermal with respect to altitude. Figure 26 showsthe variation of centerline velocity with respect toaltitude.

2.9 Lawrence’s model

Lawrance [10] talks about thermals that are causedin places where there are high horizontal winds.This kind of thermal, due to the winds get detachedfrom the ground and start rising independently inthe form of a bubble, as shown in the Figure 20.They can be classified as bubble thermals and are

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Figure 18: Variation of Childress thermal radiuswith altitude

Figure 19: 3D representation of Childress updraftvelocity at z=850[m]

Figure 20: Formation of Bubble thermal

not continuous rising air mass like chimney ther-mals.

This kind of thermal is previously studied byYoung [16] and the average updraft velocity as thefunction of height is given by 22.

wTw∗ = 0.85(

z

zi)

13 (1.3− z

zi) (22)

The resulting updraft can be seen as a bubbledetaching from the earth surface having a doughnutshaped with updraft at the middle and downdraftalong the outer surface. The model defines a 3Dtoroidal updraft flow field given by the Equations23, 24, 25. The Figure 21 from the reference [10]shows the thermal in space.

The velocity of wind along the directions X, Y,Z inside the thermal is a function of the distancefrom the thermal bubble center. If x, y, z are thedistance from the bubble center then the velocityfield is given by,

wx =

wzzx

dh(dh−R)k2 if dh 6= 0, Rwcore2k2R (1 + cos( πzkR )) if dh = R

0 if dh = 0

(23)

wy =

wzzy

dh(dh−R)k2 if dh 6= 0, Rwcore2k2R (1 + cos( πzkR )) if dh = R

0 if dh = 0

(24)

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Figure 21: Shape of the Bubble thermal given by[10]

wz =

wcore if dh = 0

wcoreRπdh

sin(πdhR )1+cos( πzkR )

2 dh ∈ (0, 2R]

0 if |z| > kR(25)

Where, wcore is the centerline updraft velocity, R isthe radius of the thermal bubble, k =

∆zflow2R with

∆zflow is the vertical thickness of the bubble anddh is the horizontal distance from the centerlinegiven by dh =

√x2 + y2.

Figure 23 shows the wind vector for every pointinside the thermal bubble developed using Equa-tions 23, 24 and 25. For a thermal bubble posi-tioned at z=800[m] Figure 22 gives the updraft ve-locity distribution. The toroidal shape of the bub-ble thermal can be clearly observed. The radius ofthe thermal and the peak velocity are calculatedbased on Section 2.4.

3 Comparison of models

Figure 24 shows the comparison of updraft veloc-ity profile of all the thermal models at an altitudeof 600[m] for the same w* and zi. It can be seenthat though the convective velocity scale w* is the

Figure 22: Updraft Contour of a thermal bubblecentered at z=800[m]

Figure 23: Wind vector inside a Bubble thermal

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same, the updraft predicted by the models are verydifferent. The Updraft given by Childress model ishigher than all the other models, while that pre-dicted by Lenschow is the least. It can be arguedthat as Lenschow [11] is based on measurementsfrom updrafts formed over the sea, the predictedupdraft velocity is lower than the rest. Due to theenvironmental sink rate allen model gives a con-stant downdraft everywhere outside the thermal.The downdraft given by Gedeon [7] can be com-pared to the immediate downdraft around the ther-mal, formed due to the air being sucked near theground by the growing thermal. Lawrance bubblethermal with center position at 600[m] is modeledusing equations from Allen hence the peak updraftvelocity and the radius of the thermal are sameas that of Allen. But it can be clearly seen thatthe toroidal bubble of Lawrance prroduces moredowndraft compared to others. This is becausethe bubble is characterized by more circulating air.When seeing the updraft profile at z=900[m] showby the Figure 25 the differences in magnitude ofthe updraft between the different models still holdsi.e. Childress is higher than all the other modelsand lenschow updraft remains the lowest. But theshape of the updraft is different between Childressand Allen. The convective cell formation is promi-nent in Childress model while the excess downdraftformed outside the thermal in Allen is due to theshape of thermal shown in Figure 9. Figure 26shows the variation of centerline velocity with re-spect to height for the different thermal models andFigure 27 the variation of thermal outer radius withrespect to height. The Lawrance bubble thermal ismodeled to move up with wcore and radius R as thatof Allen. Hence, in both these cases the Lawarancebubble thermal, produced the same results as thatof Allen.

4 Thermal Life Cycle

When thermals are formed, each thermal under-goes a specific lifecycle. A thermal lifecycle consistsof creation, a resting or formation phase, a grow-ing phase, a matured phase, a fade off phase anddeath. The time taken to heat the air around theregions of hotspot until a natural convection cur-rent is formed can be termed as resting phase. Dur-ing this phase the location of the thermal is known

Figure 24: Comparison of Models at z=600[m]

Figure 25: Comparison of Models at z=900[m]

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Figure 26: Comparison of Center line updraft ve-locity

Figure 27: Comparison of Thermal Radius

Figure 28: Evolution of time coefficient

but as the thermal is not yet developed it has noupdraft. Once the thermal starts rising upward, thetime taken to achieve its maximum updraft veloc-ity can be termed as the growing phase. Once thethermal is matured it continues to have maximumupdraft velocity. This is usually during the datetime or when then sun is at the peak. Once the en-vironment cools down the thermal slowly fades outand dies. To model the thermal life cycle a mul-tiplication factor or the time coefficient given bythe following graph show in the Figure 28 is used.This thermal time coefficient varies from zero toone, zero representing no effect of the thermal andone representing the maximum effect of the ther-mal. The thermal time coefficient is multiplied tothe updraft velocity to get the effect of Thermallife cycle. If tSim is the simulation time and trestand tlife are the resting and lifetime of the thermalthen the thermal time coefficient is calculated usingEquation 26 from [13].

c =

1 if ‖τ‖ ≤ D,

12 [1 + cos(πTξ (|τ | −D))] if D < |τ | ≤ 1+ξ

2T ,

0 else(26)

Where,

τ = tsim − (trest +tlife

2) (27)

D =1− ξ2T

(28)

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T =1 + ξ

tlife(29)

The factor ξ is the parameter that defines the shapeof the function, randomly chosen for each thermal.

5 Windfield GeneratorThe computerized tool created along with this re-port is developed using C++ language. It is para-metric in nature, meaning that it operates on agiven set of parameters and on analytical equations.This tool can be used to include the effects of ran-domly formed thermals on a given domain. Theinput variables for the program are position andtime. The output will be wind vector at the givenpoint and time.

Inside the program a wind field is generated. Awind field is a given domain in space with dimen-sions specified. Inside the wind field, thermals willrandomly form and vanish. A scenario file is cre-ated or loaded from an existing file at the beginningof the simulation, which consists of dimensions ofthe domain, the minimum and maximum simula-tion time and the ambient wind in the horizontalaxis in the absence of thermal. Each thermal infor-mation consists of the location at which the ther-mal will be formed, the convective velocity scalingparameter w∗, time at which the thermal will beformed and the lifetime of the thermal. The effectof the thermals on the aircraft is based on the modelchosen. The output wind vector will be the super-position of the ambient wind and the effect of ther-mals. The effect of thermal once calculated basedon the chosen model is multiplied by the thermaltime coefficient to take into account the lifecycle ofthe thermal. The number of thermals at any giveninstant of time is decided based on the equationsgiven by Lenschow [11] and Allen [1] as given byEquation 11.

To show the working of the software tool the sce-nario file shown in Figure 29 in the form of a .txtfile is loaded to the program. Any line with # isread as a comment. But the order of the givendata should be respected as the program reads thedata line by line. After the general domain charac-teristics, the information regarding the thermal isadded continuously. This consists of the center po-sition of the thermal and the time characteristics.This is read till the end of the file. The program

Figure 29: Configuration file containing all the in-formation about the Windfield

reads this file and creates a wind field object. Thewind field object can also be created without load-ing the scenario file but just the domain parame-ters. Using the domain parameters the thermalswill be created automatically and randomly overthe domain by the program, respecting the Equa-tion 11 at any given time. Using functions definedwithin this wind field object, one can obtain thewind vector for a given point at given time.

An example code to include the software tooland showing the usage is shown below. The headerfile “WindField.hpp” contains the class “Windfield”,whose members and methods are used to createthe wind field with thermals, and which can beused to get the wind vector using the functioncalcwind(x,y,z,t). To create a windfield objectthe user can either use default parameters or pa-rameters from a file. In the code below the objectwindfld is created automatically using the defaultparameters for the domain and the object windfld2is created from the file "config.txt".

Windfield windfld;Windfield windfld2("config.txt");

double wind[3];double t=8.0, x=300.0,y=300.0,z=650.0;

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Figure 30: Windfield showing updraft at Z=650mat time "t" seconds

wind=windfld.calcwind(t,x,y,z);

Using the calcwind function the value of wind canbe calculated at any point and time. It was donefor the entire domain and the Figure 30, 31, 32below show the vertical velocity at every point inthe windfield. The updraft velocity was calculatedusing Allen [1] model. All the thermals have thesame characteristics and hence will result in thesame updraft velocity, for a given height. Here itis calculated at the height of 650m. However, at agiven time all thermals have same updraft velocitythis is because of the thermal time coefficient. Itcan be seen that some thermals are fully developedand some of them are still in the developing phasecharacterized by lower vertical velocity. The threepictures shown here are the wind field statuses attime t, t+2, t+4 seconds.

6 ConclusionThis survey puts forth the developments and re-search undergone in the area of thermal updraftmodeling. The data collected and the equationsused by Lenschow [11] can be applied to find thegiven number of thermals in any region along withthe detailed variation of updraft velocity and di-

Figure 31: Windfield showing updraft at Z=650mat time t+2 seconds

Figure 32: Windfield showing updraft at Z=650mat time t+4 seconds

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ameter for thermals with altitude. The imposedGaussian and Gedeon [7] approach help in giving a3D shape for vertical updraft velocity. Allen [1] hasgiven a detailed 3D model of the thermals, whichform the basis of Plume type revolved trapezoidalupdrafts also called as Chimney Thermals. Withthe environment sink rate, [1] can be directly usedto replicate thermal soaring studies without hori-zontal winds. The equations given by Bencatel [3]extend Allen [1] with horizontal winds and hencecan be used to replicate thermal leaning and drift-ing. In regions where there is frequent mergingof thermals, Childress [4] can be used to representdowndrafts surrounded by updrafts. The formationof convective cells over certain altitudes and datacollected showing the same by [4], is a new phe-nomenon different from the traditional plume typethermal and hence can serve as a starting pointfor new research. The thermal bubble phenomenonproposed by Lawrance [10] is a good approximationto model thermals formed over regions with highwinds and that are highly disrupted.

As it is clearly visible the sophistication and com-plexity in modeling thermals has increased chrono-logically. The naturally occurring updrafts are ex-tremely complex and depend on several other nat-ural, turbulent and unknown atmospheric phenom-ena. Hence it is impossible to create a mathemati-cal model taking into account all of them. However,the above references can provide a very good ap-proximation and can be used directly. Future workcan be done to come up with new theories and, the-ories based on the mentioned work to capture thereal world occurrences as close as possible.

References

[1] M. J. Allen. Updraft model for developmentof autonomous soaring uninhabited air vehi-cles. Technical report, NASA Dryden FlightResearch Center, 2006.

[2] K. Andersson and I. Kaminer. Stability ofa thermal centering controller. In AIAAGuidance, Navigation, and Control Confer-ence,(Chicago, Illinois), AIAA, 2009.

[3] R. Bencatel, J. T. de Sousa, and A. Gi-rard. Atmospheric flow field models applicable

for aircraft endurance extension. Progress inAerospace Sciences, 61:1–25, 2013.

[4] C. E. Childress. An empirical model of thermalupdrafts using data obtained from a mannedglider. Master’s thesis, University of Tennessee- Knoxville, 2010.

[5] D. J. Edwards. Implementation details andflight test results of an autonomous soaringcontroller. In AIAA Guidance, Navigation andControl Conference and Exhibit, pages 18–21,2008.

[6] C. Gao. Autonomous soaring and surveillancein wind fields with an unmanned aerial vehicle.PhD thesis, University of Toronto, 2015.

[7] J. Gedeon. Dynamic analysis of dolphin stylethermal cross-country flight. In Proceedings ofthe XIV OSTIV Congress, Organisation Sci-entifique et Technique Internationale du Vol àVoile, 1974.

[8] D. A. Konovalov. On the structure of ther-mals. Technical report, Main geophysical Ob-servatory, Leningrad, USSR, 1970.

[9] J. W. Langelaan. Gust energy extraction formini and micro uninhabited aerial vehicles.Journal of guidance, control, and dynamics,32(2):464–473, 2009.

[10] N. R. J. Lawrance. Autonomous Soaring Flightfor Unmanned Aerial Vehicle. PhD thesis,University of Sydney, 2011.

[11] D. H. Lenschow and P. L. Stephens. The roleof thermals in the convective boundary layer.Boundary-Layer Meteorology, 19(4):509–532,1980.

[12] Y. Qi and Y. J. Zhao. Energy-efficient tra-jectories of unmanned aerial vehicles flyingthrough thermals. Journal of Aerospace En-gineering, 18(2):84–92, 2005.

[13] E. Rachelson, M. Melo, and S. Rapp. Devel-opment of a q-learning algorithm for improvedautonomous soaring. Technical report, Insti-tut Supérieur de l’Aéronautique et de l’Espace,Toulouse, 2014.

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[14] R. B. Stull. An Introduction to BoundaryLayer Meteorology. Kluwer Academic Publish-ers, 1994.

[15] J. M. Wharington. Autonomous Control ofSoaring Aircraft by Reinforcement Learning.PhD thesis, Royal Melbourne Institute ofTechnology, 1998.

[16] G. S. Young. Turbulence structure of the con-vective boundary layer. part ii: Phoenix 78aircraft observations of thermals and their en-vironment. In Journal of the Atmospheric Sci-ences, 1988.

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