The variation of the GPS dropwindsonde drag coefficient and its impact on the wind retrieval from dropwindsonde measurements

Theor Appl ClimatolDOI 10.1007/s00704-012-0792-x

ORIGINAL PAPER

The variation of the GPS dropwindsonde drag coefficientand its impact on the wind retrieval from dropwindsondemeasurements

Sunwei Li Craig Miller

Received: 2 March 2012 / Accepted: 29 October 2012 Springer-Verlag Wien 2012

Abstract The wind finding equations currently used toretrieve horizontal winds from Global Positioning Systemdropwindsonde wind measurements are derived based ona point object model in which the drag coefficient of thedropwindsonde is assumed to be a constant. The windtunnel tests performed as part of this study showed, how-ever, that the dropwindsonde aerodynamic coefficients varyappreciably with angles of attack. To investigate the impactof this finding, the dropwindsonde motion in a pseudo-stochastic wind field has been simulated using a motionmodel more sophisticated than the point object model. Theresults showed that, although the constant drag coefficientassumption is not supported by the wind tunnel test results,the wind finding equations still correctly calculate both themean and the turbulence intensity profiles. In addition, arevised method to calculate the vertical wind was proposedbased on the derivation of the improved motion model,which enhanced the accuracy of vertical wind estimates byincluding the real-time dropwindsonde drag coefficient andthe dropwindsonde vertical acceleration into calculation.

1 Introduction

Measurements taken by the Global Positioning System(GPS) dropwindsonde, which has been introduced by Hockand Franklin (1999), provide one of the few reliable datasources about the wind characteristics in the hurricaneboundary layer (HBL). For example, the behavior of the seasurface drag coefficient under high wind speed conditions

S. Li () C. MillerBoundary Layer Wind Tunnel Lab., University of WesternOntario, London, ON, N6G 5B9, Canadae-mail: [email protected]

has been found by analyzing the mean wind profiles com-puted by compositing GPS dropwindsonde measurements(Powell et al. 2003). As the airsea interactions under highwind speed conditions are more accurately modelled withthe help of this finding, the boundary layer parametrization,which is a critical part in numerical simulations of tropi-cal cyclones, has been improved (Nolan et al. 2009a, b).Moreover, the wind field reconstructed based on GPS drop-windsonde (or dropsonde for simplicity) observations wasused as a validation criterion for modelling studies (Kepert2006a, b). In addition to advancing our understanding oftropical cyclone meteorology, the dropsonde measurementsyielded a mean wind profile model for engineering applica-tions (Vickery et al. 2009).

As described by Hock and Franklin (1999), the drop-sonde horizontal motions are used to derive the hori-zontal wind measurements. It should be pointed out thatthe measurement taken by the dropsonde is in neitherthe conventional Eulerian framework, which requires thatthe measurement is taken at a fixed point, nor a perfectLagrangian framework, which requires the dropsonde fol-lows the motion of air parcels. As a result, a thoroughunderstanding of the dropsonde motion in the wind fieldis important to interpret the horizontal wind measurementtaken by the dropsonde. However, the dropsonde aerody-namic parameters, which significantly influences its motion,have not been investigated in a controlled environment yet.In addition, Hock and Franklin (1999) provided the windfinding equations, which have been used extensively toretrieve the horizontal winds, based on a point object modelin which the drag coefficient is assumed to be constantregardless of the angle of attack. This assumption lacks anyexperimental or theoretical basis since the angle of attackobviously influences the aerodynamic forces experiencedby the dropsonde due to the fact that the shape of dropsonde

S. Li, C. Miller

(including the parachute) is strongly anisotropic. Since thewind finding equations are important in reducing the lageffect of the dropsondes in a sheared wind field (Hock andFranklin 1999), it is necessary to evaluate the reliability ofthe point object model, which should include an evaluationof the constant drag coefficient assumption.

In order to check the constant drag coefficient assump-tion, wind tunnel tests were conducted as a part of this studyto measure the dropsonde aerodynamics and their variationwith angles of attack. The test results not only produce areliable estimate of the dropsonde drag coefficient in a con-trolled environment but also quantify the variation of thedrag coefficient with angles of attack.

Based on the wind tunnel test results, dropsonde motionsin a pseudo-stochastic wind field were simulated using amotion model taking into the consideration the variationof the drag coefficient with angles of attack. Accordingto Cockrell (1987), the parachute and its payload can bemodelled separately to take into account their aerodynamicparameters difference. Although Cockrell gave severaleven more advanced modelling approaches to account forthe variation of the parachute drag coefficient with anglesof attack, a simple separation model is adopted in thisstudy since the parachute in the dropsonde system has amass far less than the dropsonde body which means thatthe parachute is much more responsive to wind directionchanges. The simple separation model treats the dropsondebody (the payload to the parachute) as a rigid body with2 translational degrees of freedom and 1 rotational degreeof freedom moving in an imaginary plane. The parachute,on the other hand, is modelled as an external decelerationforce whose direction is opposite to the dropsonde relativemovement. From the simulation results, the wind findingequations are validated in a more reliable and strict way. Inaddition, adding the rotation into the model allows the drop-sonde body orientation to be explicitly solved, and thereforeallows the variation of the dropsonde aerodynamics withangles of attack.

The vertical turbulent wind measurement is importantsince it is necessary to calculate turbulent fluxes, and cor-rectly modelling the vertical variation of the turbulent fluxesis critical in the hurricane intensity prediction since the heatfluxes, especially the latent heat fluxes, provide the energyrequired for the hurricane intensification while the momen-tum fluxes transport surface drags upwards to reduce thehurricane wind strength (French et al. 2007; Drennan et al.2007). Direct observations of the vertical turbulent windwere, however, rather rare. Although French et al. (2007),Drennan et al. (2007) and Zhang et al. (2009) revealedimportant information about the turbulent fluxes within theHBL, they did not provide the vertical variation of turbu-lent fluxes in the entire HBL due to their instrumentationlimitations. The dropsonde, on the other hand, provides an

unprecedented opportunity to measure the vertical variationof turbulent fluxes in the entire HBL (from sea surface upto a height of 10,000 m). However, the conventional wayto calculate the vertical wind from raw dropsonde measure-ments, as the measured dropsonde falling speed subtractedfrom the theoretical value, is crude, and the accuracy ofthis method has yet been evaluated. The alternative motionmodel described above, which takes into account the varia-tion of the drag coefficient with angles of attack, leads to atheoretically better vertical wind estimation method. Basedon the dropsonde motion simulation results, the improve-ment of this new estimation method was evaluated in acomparison with the crude calculation since the true ver-tical wind was known from the pseudo-stochastic windfield.

The organization of this paper is as follows. Section 2covers the technical details and the results of the windtunnel study. Section 3 describes the alternative motionmodel and the dropsonde motion simulation procedures.The discussions on the simulation results, including the val-idation of the wind finding equations and the evaluationof the new vertical turbulent wind estimation method, aregiven in Section 4. Finally, the conclusions are presented inSection 5.

2 Wind tunnel test

As discussed in Section 1, the aerodynamics of the drop-sonde body and of the parachute significantly influence thedropsonde motion in its fall. In order to gain a better under-standing of the dropsonde motion, and then a more theo-retically sound wind retrieval method, a wind tunnel studyto measure the dropsonde aerodynamics and their variationwith angles of attack was conducted as a part of this study.This section covers the wind tunnel test which includes adescription of the test configuration and a discussion on thetest results.

2.1 Test configuration

The wind tunnel test was conducted in the BLWTL-2wind tunnel in the Boundary Layer Wind Tunnel Labora-tory of the University of Western Ontario. The dropsondemodel was placed and tested in the high-speed section ofBLWTL-2, which has the dimensions 39 m3.4 m2.5 m(length width height), and the maximum wind speedallowed is 28 ms1.

The dropsonde model tested was a full scale, geometricalreplica of the real dropsonde body, which roughly is a cylin-der with a diameter of 6.86 cm and a height of 40.89 cm.The parachute used in this test is real from the dropsondesystem, which is a unique square-cone parachute of the

The impact of the variation of drag coefficients on the dropsonde wind retrieval

Fig. 1 Setting of the dropsonde wind tunnel test

National Center for Atmospheric Research. A more compre-hensive technical description of the dropsonde system andthe parachute can be found on the dropsonde website.1

The model was installed close to the flow inlet. Theforces in two perpendicular directions and the torque weremeasured by the wind tunnel balances. A steel bar was usedto support the model between two balances. The contribu-tion of the bar to the measured aerodynamic forces was thensubtracted based on the forces measured in a separate test inwhich only the bar was installed. The general setting of thewind tunnel test is shown in Fig. 1.

The angle of attack is defined as the angle betweenthe model orientation and the horizontal flow, as shown inFig. 2. The range of the angles of attack tested is from75 to 75 increasing at a 5 step, while the wind speedused ranges from 6.1 to 21.3 ms1 at an increment of3.05 ms1. Although the wind speeds utilized in the windtunnel test are less than the hurricane wind speeds foundin a real dropsonde fall, the velocity influencing the drop-sonde aerodynamics is the relative wind velocity calculatedby subtracting the dropsonde motion from the surround-ing wind, and the relative wind velocity can be reasonablyassumed to fall into the range of the tested wind speeds.Moreover, as will be seen, the dropsonde aerodynamicsare Reynolds number independent when the relative windvelocity exceeds some critical value. This Reynolds num-ber independence suggests that the aerodynamics measuredunder low wind speed conditions are still valid under highwind speed conditions. As for the flow configuration, thetest was conducted in a smooth flow, where the flow wasonly confined by tunnel walls, and in a turbulent flowwhere the flow passed a grid shield to generate the small

1www.eol.ucar.edu/isf/facilities/dropsonde/gpsDropsonde.html

Fig. 2 Sketch showing the definition of the angle of attack and theorientation of the drag and lift

scale-turbulence. As the grid cell dimension is approxi-mately 0.3 m 0.2 m, the largest spatial scale of generatedturbulence is in the order of 0.1 m. In addition, the turbu-lence intensity, which is defined as the standard deviationof the free stream wind velocity divided by the mean freestream wind velocity, was measured for two test settings.The variation of the turbulence intensity with the free streamwind velocity is shown in Fig. 3. Using the small-scaleturbulence test setting, we investigate the influence of thesmall-scale turbulence on the aerodynamics of the drop-sonde. For the sake of simplification, the angle of attacktested in the small-scale turbulence only ranges from 15to 75, and the symmetric behavior is assumed.

To identify the parachutes contribution, the test is con-ducted for both the whole system including the parachuteand for the dropsonde body alone. The aerodynamics of theparachute itself were then estimated by subtracting the testresults of the dropsonde body from the results of the wholesystem.

0

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1

1.2

1.4

4 6 8 10 12 14 16 18 20 22

Turb

ulen

ce in

tens

ity (%

)

Wind Speed (m/s)

smoothturbulent

Fig. 3 Variation of turbulence intensity with free stream wind velocityfor two wind tunnel test settings, i.e., smooth condition and small-scaleturbulence condition

S. Li, C. Miller

2.2 Test results and discussion

The most useful results from the wind tunnel test are thedrag, lift, and torque coefficients of the whole dropsondesystem and of the parachute. Since the horizontal flow inthe wind tunnel test represents the relative wind in a realdropsonde fall, the drag and lift definition shown in Fig. 2is equivalent to defining the drag and the lift according tothe direction of the relative wind velocity vector. The vari-ation of the drag, lift, and torque coefficients with anglesof attack, for both the whole system and for the dropsondebody alone, is shown in Fig. 4. These coefficients are cal-culated by normalizing the measured drag, lift, and torquewith the free stream wind speed, the dropsonde sectionalarea, and the air density, i.e.,

CD = D12AV

2r

CL = L12AV

2r

CT = T12AlV

2r

(1)

In expression (1), C represents the aerodynamic coefficientin which the subscript gives the coefficient category (D rep-resents the drag, L represents the lift, and T represents thetorque). Vr represents the relative wind velocity which isthe free stream wind speed in the wind tunnel test and A isthe dropsonde sectional area. Following the EDITSONDEsource code (J. L. Franklin 2010, personal communication),this parameter takes the value of 0.0676 m2 to make thewind tunnel measurements comparable to the drag coeffi-cient implied by Hock and Franklin (1999). In addition, is the air density, which takes the value of 1.235 kgm3 inthe coefficient calculation, and l is the half of the dropsondebody length (20.45 cm).

The aerodynamic behaviors of both the dropsonde bodyand the whole system are roughly symmetric. Although theparachute weight force violates the symmetry for the wholesystem, its influence becomes negligible under high windspeed conditions (as revealed in Fig. 5a). There is a slightasymmetry seen in Fig. 4a, which can be explained by thewake effect. In detail, when the angle of attack is nega-tive, the parachute is strongly influenced by the wake ofthe dropsonde body since gravity makes the parachute stayin a position lower than the horizontal plane. In contrast,when the angle of attack is positive, the parachute is lessaffected by the wake. Since this asymmetry is minimal andis not seen in the variation of the lift coefficient, whichimplies that its influence is limited, no explicit correctionconcerning this asymmetry was implemented.

Figure 5a, b shows the drag coefficient of the dropsondebody and of the whole system under different testing wind

0

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Dra

g Co

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ient

Angle of Attack (degree)(a) Drag Coefficient

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ue C

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Angle of Attack (degree)

Torque Coefficient

(c) Torque CoefficientFig. 4 Variation of the dropsonde drag, lift, and torque coefficientswith angles of attack, shown in a, b, and c, respectively. For coeffi-cients of the whole system, the solid line shows results in the smoothflow and plus signs show results in the turbulent flow; for coefficientof the dropsonde body, the dashed line shows results in the smoothflow and dark circles show results in turbulent flow. The thicker lineindicates the drag coefficient implied by Hock and Franklin (1999)

speeds. In both cases, the curves collapse when the windspeed exceeds 10 ms1. In addition to demonstrating thatthe influence of the parachute weight force is minimal whenthe relative wind velocity is high, this also substantiatesthat the drag coefficients of both the dropsonde body andthe parachute are Reynolds number independent. Similarbehavior can be seen in Figs. 6 and 7 for the lift and thetorque coefficients.


0.45

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6m/s12m/s18m/s21m/s

(a) Whole system

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(b) Dropsonde bodyFig. 5 Variation of the drag coefficient with angles of attack under dif-ferent testing wind speeds, the wind speed is indicated by the legend.a shows the variation for the whole system and b shows the variationfor only the dropsonde body

The variation of the aerodynamics with angles of attackis appreciable as seen in the figures. The standard devia-tion of the drag coefficient of the whole dropsonde systemin the smooth flow is 0.099 or 15.3 % in a relative sense.Thus, assuming the dropsonde drag coefficient is constantregardless of the angle of attack is not supported by thewind tunnel test results. As regard to the drag coefficientvalue, the one calculated implicitly using the terminal drop-sonde falling rate Hock and Franklin (1999) is also shownin Fig. 4a. On average, the drag coefficient measured in thistest is close to the value implied by the terminal dropsondefalling rate, but the difference becomes considerable whenthe angle of attack is in the range of (30, 30), larger than60, or less than 60.

As shown in Fig. 4, the small-scale turbulence introducedby the grid shield installed in front of the dropsonde modelhas little impact on the aerodynamics measured, except forthe drag and lift coefficients of the whole dropsonde sys-tem. This can be explained by the random vibration of theparachute. When the parachute randomly vibrates in thesmall-scale turbulence, with a magnitude larger than thatfound in the smooth flow, the parachutes added mass

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Lift

Coef

ficie

nt


6m/s12m/s18m/s21m/s

(a) Whole system

-0.2

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ft Co

effic

ient


6m/s12m/s18m/s21m/s

(b) Dropsonde bodyFig. 6 Variation of the lift coefficient with angles of attack under dif-ferent testing wind speeds, the wind speed is indicated by the legend.a shows the variation for the whole system and b shows the variationfor only the dropsonde body

increases. According to Cockrell (1987), the added mass,which can be roughly defined as the mass of the air stronglyperturbed by the parachute, acts as the additional massassociated with the parachute. Therefore, the increase ofthe parachutes added mass produces a higher apparentparachute weight force, which in turn leads to a negativecontribution to the lift force measured by the balances sincethe lift is defined positive when it points upwards. As aresult, the lift coefficient measured in the small-scale tur-bulence is smaller than that measured in the smooth flow(shown in Fig. 4b). Meanwhile, the random nature of thevibration also makes the parachute drag turn into the liftwhen the parachute is not in the exact horizontal position,which explains that the drag coefficient measured in theturbulence is constantly lower than that measured in thesmooth flow (shown in Fig. 4a). The added lifts, however,are then cancelled out in the averaging process to calculatethe lift coefficient since it is evenly distributed across 0. Thisexplains that the lift coefficient measured in the small-scaleturbulence is not larger than that measured in the smoothflow (shown in Fig. 4b). In conclusion, the small-scale tur-bulence hardly influences the dropsonde aerodynamics, and

S. Li, C. Miller

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6m/s12m/s18m/s21m/s

(a) Whole system

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6m/s12m/s18m/s21m/s

(b) Dropsonde bodyFig. 7 Variation of the torque coefficient with angles of attack underdifferent testing wind speed, the wind speed is indicated by the legend.a shows the variation for the whole system and b shows the variationfor only the dropsonde body

therefore the aerodynamics measured in the smooth flow areutilized to describe the dropsonde motion in the followingdiscussions.

As shown in Fig. 8a, b, the parachute drag is nearly con-stant regardless of the angle of attack while the parachutelift coefficient fluctuates around 0. The torque coefficient ofthe parachute (shown in Fig. 8c) ensures that the dropsondebody keeps the right vertical position, since it has a signopposite to the angle of attack. In other words, the parachuteprovides a recovery torque to reduce the angle of attack backto 0.

In summary, the mean of the drag coefficients of thewhole dropsonde system (including the parachute) in thesmooth flow is 0.65, while the maximum, 0.80, takes placeat an angle of attack of 70 and the minimum, 0.51, takesplace at an angle of attack of 0. In the mean time, thecoefficient derived implicitly from the terminal dropsondefalling rate (Hock and Franklin 1999) is 0.61, as shown inFig. 4a. The averaged lift coefficient is 0.0011 and the aver-aged torque coefficient is 0.0033. The drag coefficient forthe parachute alone, which can be modelled as a constant, is0.48.

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Drag CoefficientHF

(a) Drag Coefficient

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(b) Lift Coefficient

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Torque Coefficient

(c) Torque CoefficientFig. 8 Variation of the drag, lift, and torque coefficients of theparachute with angles of attack, shown in a, b, and c, respectively. HFrefers the drag coefficient implied by Hock and Franklin (1999)

3 Motion simulation with an alternative model

Adopting the aerodynamics gathered in the wind tunnel test,it is possible to analyze the dropsonde motion characteristicsin a more sophisticated way which separates the parachutefrom the dropsonde body. In this model, the variation ofdropsonde body aerodynamics with angles of attack can beincluded since the dropsonde body orientation is one modelvariable. This section covers the derivation of this alterna-tive motion model and the details of the dropsonde motionsimulation based on the alternative motion model.


3.1 The alternative motion model of the dropsonde

Implicitly adopted by Hock and Franklin (1999), the pointobject model, in which the dropsonde drag coefficient isassumed to be a constant regardless of the angle of attack,helped deriving the wind finding equations. According toHock and Franklin (1999), the wind finding equations areimportant in reducing the dropsonde lag effect to calculatethe correct mean wind profile in a sheared wind field. Con-sidering the importance of the wind finding equations, itis necessary to further validate the wind finding equationssince the wind tunnel test results showed that the dropsondedrag coefficient apparently varies with angles of attack.

As introduced in Section 1, an alternative motion modelwhich separates the parachute from the dropsonde body isexpected to be more appropriate to describe the dropsondemotion since the angle of attack, and then the variation ofdropsonde aerodynamics with angles of attack, is includedin the model. According to Cockrell (1987), the parachuteand its payload are supposed to be modelled as two rigidbodies in this alternative motion model. Since the mass ofthe parachute is far less than that of the dropsonde body,the parachute is more responsive to wind direction changes.As a result, the influence of the parachute can be simpli-fied as an external deceleration force. Moreover, Cockrell(1987) described both 2 degrees of freedom motion (the pla-nar translation) and 3 degrees of freedom motion (addingthe rotation to the planar translation). To explicitly modelthe angle of attack, the 3 degrees of freedom motion modelwas used. In summary, the 3 degrees of freedom motionmodel, with the parachute modelled as an external decel-eration force, is adopted to simulate the dropsonde motionin the pseudo-stochastic wind field. This motion model isreferred as the alternative motion model in the followingdiscussions.

In the alternative motion model, the drag of the drop-sonde body needs to be calculated based on the angle ofattack as,

FDx = 12ACD()M(u x)

FDy = 12ACD()M(v y)

FDz = 12ACD()M(w z) + mg (2)where M is the magnitude of the relative wind velocityvector, i.e., (u x, v y, w z), and can be expressed as

M =

(u x)2 + (v y)2 + (w z)2. (3)In expressions (2) and (3), the drag coefficient CD() ismade up of two parts: one is provided by the dropsondebody (CDbody()) which depends on the angle of attack,, and the other is a constant provided by the parachute

(CDparachute). FD represents the drag force and the subscriptindicates the force direction. A is the area of dropsondebody cross section. The vector (x, y, z) shows the drop-sonde velocity at the position (x, y, z) while the vector(u, v,w) shows the wind velocity. The upward direction isdefined as positive in the following equations, and thereforethe gravity g should take the value of 9.8 ms2.

In contrast to the point object model, the alternativemotion model allows an explicit expression of the lift, whichshows

FLx = 12ACLbody()M(c(v y) b(w z))

FLy = 12ACLbody()M(c(u x) + a(w z))

FLz = 12ACLbody()M(b(u x) a(v z)) + mg. (4)

All symbols have the same meanings as in expression (2).Different from the drag coefficient CD, the lift coefficientCL contains only the contribution made by the dropsondebody. q = (a, b, c) is a parametric vector with the unit mag-nitude, i.e., a2 +b2 +c2 = 1. This parametric vector is usedto determine the lift direction. While the total aerodynamicforce could be decomposed into three spatial components,the lift force calculated using the lift coefficient measuredin the wind tunnel tests is a combined force whose compo-nents are perpendicular to the relative wind velocity vectorin two orthogonal directions if the force is described in thecoordinate system defined by and follows the change of therelative wind velocity vector. As a result, the aerodynamicforce experienced by the dropsonde body needs to only bedecomposed into a drag and a lift in our simulation, andthe parametric vector q, which shows the direction of thecombined lift force, is calculated as the cross product of therelative wind velocity vector, m = (u x, v y, z z), andthe dropsonde body orientation vector, p = (px, py, pz),i.e.,

q = m p= (u x, v y, w z) (px, py, pz)= (pz(v y) py(w z), px(w z)

pz(u x), py(u x) px(v y)). (5)

In a component fashion, the parametric vector q shows,

a = pz(v y) py(w z)b = px(w z) pz(u x)c = py(u x) px(v y). (6)

Considering that the rigid body motion of the dropsondebody is driven by a combination of the aerodynamic forcesand the weight force as ma = FD + FL (the weight force is

S. Li, C. Miller

implicitly included in the expression of FD), the governingequation of the dropsonde motion reads

mx = 12AM(CD()(u x)

+CLbody()[c(v y) b(w z)])my = 1

2AM(CD()(v y)

+CLbody()[c(u x) + a(w z)])mz = 1

2AM(CD()(w z)

+CLbody()[b(u x) a(v z)]) + mg(7)

In addition, since the alternative dropsonde motion modelincludes 1 rotational degree of freedom, the dropsonde bodyorientation should be calculated based on the torque as

I = T () (8)where I is the moment of inertia of the dropsonde bodyfor rotating around its center, is the angle between thedropsonde body orientation and a reference direction, and shows the angular acceleration. One uncertainty in Eq. (8)is the moment of inertia value, I . Since the exact calcula-tion of I is extremely difficult, a sensitivity analysis of theI value was performed. In the sensitivity analysis, a singledropsonde drop was simulated based on different moment ofinertia values, including I1, which was calculated assumingthe mass is evenly distributed in the cylinder defined by theouter geometry of the dropsonde body; I2, which was calcu-lated assuming the mass is concentrated in a slender cylinderwith a diameter only 10 % of the value found in a real drop-sonde body; and I3, which was calculated assuming themass is concentrated in a tube with an outer diameter thesame as the dropsonde body cylinder and a thickness 10 %of that diameter. Figure 9 shows the dropsonde profiles sim-ulated using the I1 value and the difference between usingthe I1 value and using the I2 and I3 values. It is found thatthe influence of using different moment of inertia values isnegligible, since most differences shown in Fig. 9b are in therange of (1.5 ms1, 1.5 ms1), which is relatively smallcompared with the simulated dropsonde velocity (approx-imately 40 ms1). Therefore, the I1 value is used in oursimulations.

In summary, Eqs. (7) and (8) govern the 3 degrees of free-dom motion of the dropsonde system, when the parachute ismodelled as an external deceleration force. These equationsformulate the alternative dropsonde motion model.

3.2 Dropsonde motion simulation

Based on the alternative motion model described above,dropsonde motions in a pseudo-stochastic wind field weresimulated to further evaluate the wind finding equations

(a) I1 Profile

(b) Difference ProfileFig. 9 A single dropsonde pseudo profile simulated using I1, shownin a, and the differences between pseudo dropsonde profiles simulatedusing I1 and I2 (labelled as I1 I2) plus the difference gener-ated by using I1 and I3 (labelled as I1 I3), shown in b. Thecalculations of I1, I2, and I3 can be found in the text. In addition,I1 = 0.00498 kgm2, I2 = 0.00488 kgm2, and I3 = 0.00506 kgm2

introduced by Hock and Franklin (1999) and to investigatethe vertical wind retrieval from raw dropsonde measure-ments. In the simulation, Eq. (7) was solved numerically togive the dropsonde translations in two spatial directions, andEq. (8) was solved for the dropsonde body rotation whichwas then used to calculate the angle of attack.

One precondition of the dropsonde motion simulationis the wind velocities that appear in the equation, i.e.,(u, v,w). To provide these wind velocities, a pseudo-stochastic wind field was generated numerically. FollowingSolari et al. (2007) and Carassale et al. (2007), the properorthogonal decomposition (POD) technique was utilizedto simulate the target pseudo-stochastic field based on a


spectrum matrix, which describes all second-order spectralstatistics of the target pseudo-stochastic field. According toCarassale et al. (2007), the spectrum matrix consists of N N elements for a target pseudo-stochastic filed containingN discrete points, which shows

S() =

S11 SiN...

. . ....

SN1 SNN

(9)

Every element in matrix Eq. (9) is also a 3 3 small matrixrepresenting either the spectral statistics for a single point(Sii for point i) or the spectral cohesions for two givenpoints (Sij for points i and j ). The matrix Sii is calculatedas,

Sii =

suu(), suv(), suw()

svu(), svv(), svw()

swu(), swv(), sww()

(10)

In expression (10), suu, svv, and sww are the wind spectra forwind components u, v, and w and suv, suw, and svw are theco-spectra. It should be noted that suv = svu, suw = swu andsvw = swv . Based on the point spectral statistics calculatedfollowing formula (10), the spectral cohesion matrix Sij iscalculated as

Sij = Sii Sjj i,j () (11)in which the spatial cohesion coefficient function iscalculated based on the distance between points i andj . In formulating the spectrum matrix, the wind spec-tra (suu, svv, sww) were calculated following Von Karman(1948), which is also known as Karman spectrum, andthe wind co-spectra (suv, suw, svw) were calculated basedon the point cohesion coefficient model given by Solari andPiccardo (2001). Since the Karman spectrum requires thestandard deviation of the wind velocity series at the discreteheights in the pseudo-stochastic wind field, the turbulenceintensity model given by the Engineering Science Data Unit(ESDU) (1993) was utilized to calculate the standard devi-ation of wind velocities in the pseudo-stochastic wind field.A simple exponential decay model was used to calculate thespatial cohesion coefficient i,j (Davenport 1967), and thedecay rates (dimensionless model parameters, see the studyof Davenport (1967) for details) are specified as 10 for theu component, 6.5 for the v component, and 3 for the wcomponent.

In the POD simulation, decomposing the spectrummatrix produced a matrix containing the magnitudes ofthe periodical variations constituting the target pseudo-stochastic field in the frequency domain, and summing theseperiodical variations yielded the target pseudo-stochasticfield. In detail, the elements in the decomposed matrixwas first multiplied by a random process with an unit

magnitude tuned by a random phase. The resulting ran-dom processes were then transformed from the frequencydomain to the time domain by discrete Fourier transfor-mations. Finally, the transformed results, which are alsorandom processes, were added together to produce the targetpseudo-stochastic wind field. As regard to the configura-tion of the target pseudo-stochastic wind field, the variationof the wind velocity in the longitudinal and lateral direc-tions was neglected, which means that the wind velocityonly varied with height and time. When Taylors hypothesisis invoked, the wind velocity time histories are transferredinto the longitudinal wind variations. In the target pseudo-stochastic wind field, 120 discrete points in the verticaldirection showed the wind velocity vertical variation and65,536 discrete steps were used to generate the wind veloc-ity time histories. Since the height step and the time stepwere 5 m and 0.1 s, respectively, the total height is 600 mwhile the time length is 1.8 h. To evaluate the performanceof the wind finding equations in sheared flows, the targetpseudo-stochastic field was superimposed onto a logarith-mic mean wind profile to finally provide the wind velocityfield ((u, v,w) appeared in the governing Eqs. (7) and (8).The profiles of mean wind velocities and turbulence inten-sities (calculated by normalizing the wind velocity standarddeviation by the mean wind velocity at the same verti-cal level) of the pseudo-stochastic wind field are shown inFig. 10, in which the target is calculated using the ESDUmodel.

Other parameters required in solving Eqs. (7) and (8)were the dropsonde aerodynamics, such as CD(). Thewind tunnel test results presented in Section 2 were usedto give these parameters. As discussed in Section 2, thedropsonde aerodynamics vary with angles of attack appre-ciably (see Fig. 4a), and therefore the angle of attack needsto be calculated to find the real-time dropsonde aerody-namics. In the simulation, the angle of attack was thedifference between the relative wind velocity vector, i.e.,(u x, v y, w z), and the dropsonde body positionvector, i.e., (px, py, pz). While the relative wind velocityvector was calculated directly based on the solved drop-sonde motion (x, y, z), the dropsonde body position vectorwas calculated by integrating the solved dropsonde angularacceleration ().

Under the condition that both the wind velocities and thereal-time dropsonde aerodynamics were found, the drop-sonde translations and rotations in the pseudo-stochasticwind field were simulated by numerically solving Eqs.(7) and (8). More specifically, we used the fourth-orderRungeKurta method with an integration time step of 0.05 sto solve the governing equations which produced boththe accelerations and velocities of the dropsonde body.The dropsonde positions and orientations were then calcu-lated by integrating the directly solved dropsonde motions.

S. Li, C. Miller

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Mean Wind Velocity (m/s)(a) Mean Wind Profile

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Turbulence Intensity (%)(b) Turbulence Intensity Profile

Fig. 10 The mean wind velocity, shown in a, and the turbulenceintensity profile, shown in b, of the pseudo-stochastic wind field

Afterwards, a single pseudo dropsonde profile was gener-ated by sampling the numerically solved dropsonde posi-tions, velocities, and accelerations at a frequency of 2 Hzand compositing a considerable number of pseudo drop-sonde profiles yielded the composited mean wind andcomposited turbulent statistics profile of the pseudo-stochastic wind field. Since it is an important in derivingwind statistics from dropsonde measurements, the compo-sition process is elaborated as follows. Firstly, differentpseudo dropsonde profiles were simulated by assuming thatdropsondes were released at different moments within thetime length of the pseudo-stochastic wind field (1.8 h).Secondly, these different pseudo dropsonde profiles weredivided into segments according to the heights of measure-ments points in an individual pseudo dropsonde profile. Indetail, as 100 height bins were formulated successively inthe vertical direction, measurement points with heights inthe range of one height bin were grouped in one segment,and measurement points in the same height bin from dif-ferent pseudo dropsonde profiles were averaged using aweighted-average method to calculate the desired statisticat the center of the height bin. Finally, the statistic pro-file of the pseudo-stochastic wind field was formulated bylining up the statistics calculated for different height bins.In the weighted-average process, the weight was assigned

Fig. 11 A sketch illustrating the composition process to calculate themean and the turbulent wind profile

according to the distance from the measurement point to thebin center, and it linearly varies from 0 (at the bin bound-ary) to 1 (at the bin center). As 1,000 pseudo dropsondeprofiles were generated in the dropsonde motion simula-tion and each height bin spanned 6 m vertically, around1,000 (920 1,103) measurement points were averagedfor each height bin given that the pseudo dropsonde ver-tical sampling interval is 6 m. The composition wasutilized to calculate the profiles of the mean wind, the tur-bulent wind, and the turbulent flux. In calculating the meanwind profile, raw pseudo dropsonde profiles were compos-ited, and then the same composition process was appliedto these turbulent components, which were calculated asraw pseudo dropsonde measurement minus the mean pro-file derived previously. While the calculations of the meanand turbulent wind profiles are illustrated in Fig. 11, thecalculation of the turbulent flux is detailed in the follow-ing subsection focusing the vertical wind retrieval (Section4.3). Since the statistics profiles can be reliably calculatedfrom the pseudo-stochastic wind field, the statistics calcu-lated by compositing pseudo dropsonde profiles were thencompared to the true statistics calculated directly based onthe pseudo-stochastic wind field.

4 Simulation results and discussion

To thoroughly discuss the dropsonde motion simulationresults, this section is structured as follows. Firstly, a the-oretical analysis is presented to illustrate the improvementof using the alternative model to describe the dropsondemotion on both the horizontal and vertical wind retrievals.Secondly, a discussion is presented on the mean wind pro-file calculated by compositing pseudo dropsonde profilesin a context of showing the effects of using the wind find-ing equations. Finally, a comparison is conducted betweenthe conventional way and the revised way of calculating thevertical turbulent wind.


4.1 Theoretical analysis

To show the effect of using the alternative motion model,the velocity profiles from a single dropsonde drop simu-lated based on different models (the point object model andthe alternative model) are presented in Fig. 12. As indi-cated by Fig. 12b, the difference of the dropsonde profilessimulated based on different models can go up to 4 ms1,or nearly 10 % considering that the dropsonde velocity isapproximately 45 ms1. When compared to the differenceshown in Fig. 9b, the relatively large difference indicatesimprovements of using the alternative motion model.

The governing equations of the dropsonde translationadopted in the alternative dropsonde motion model (Eq. (7))

(a) Simple Profile

(b) Difference ProfileFig. 12 A pseudo dropsonde profile simulated based on the simplemodel (point object model), shown in a, and the difference betweenthe pseudo dropsonde profiles simulated based on the simple and thealternative motion models, shown in b

can be simplified under two conditions: the neglect of thelift and the linearization of the term

M =

(u x)2 + (v y)2 + (w z)2. (12)If the horizontal velocity differences are significantly lessthan the vertical velocity difference, i.e., |u x|

S. Li, C. Miller

In addition to validating the use of the wind findingequations, the alternative motion model also reveals a moresophisticated way to calculate the vertical wind. The cal-culation is also derived based on Eq. (7) (the equation forthe vertical motion) under the same linearization assump-tion. The equation governing the vertical dropsonde motioncould reduce to

mz = 12A|w z|(w z)CD() + mg (18)

under the linearization assumption. From Eq. (18), thevertical wind can be solved as

w =

2m(z g)ACD()

+ z (19)

under the assumption that the vertical wind w is alwayslarger than the falling rate z (around 12 m/s near thesurface). When compared to the conventional way to calcu-late the vertical wind using the measured dropsonde fallingrate subtracted from the theoretical value, which can beexpressed as

w = zcons + z (20)where the theoretical dropsonde falling speed is

zcons =

2mgACD

. (21)

Equation (19) takes two additional factors into considera-tion: the dropsonde vertical acceleration and the variation ofthe drag coefficient with angles of attack.

4.2 Validation of the wind finding equations

In addition to the theoretical validation presented in Section4.1, the validity of the wind finding equations was eval-uated based on the dropsonde motion simulations. Morespecifically, the composited mean and composited turbulentwind profiles were compared to the true mean and turbu-lent wind profiles calculated based on the wind velocitydata in the pseudo-stochastic wind field . Since the drop-sonde motion simulations were conducted based on boththe simple motion model (point object model), in whichthe dropsonde drag coefficient is assumed to be constantregardless of the angle of attack, and the alternative motionmodel derived in Section 3.1, the impact of the variationof dropsonde drag coefficient can be investigated throughinvestigating the improvements of using the wind findingequations in two simulations (based on the simple motionmodel and the alternative motion model). In the simulationbased on the simple model, the dropsonde drag coefficientimplied by Hock and Franklin (1999) (0.61) is utilized.

Figure 13 shows a comparison of the mean wind profilederived based on the two different motion models. The sim-ilar improvements in Fig. 13a, b substantiates that the windfinding equations are still valid to calculate the correctmean wind profile in sheared flows even when the drop-sonde drag coefficient variations are included in the drop-sonde motion model. As in Fig. 13, similar improvementsare also seen in Fig. 14 which compares the turbulenceintensity profiles. In order to be more clear, the process tocalculate the turbulence intensity profile is described as fol-lows. Firstly, the mean wind profile calculated previouslywas subtracted from 1,000 individual pseudo dropsondeprofiles to find the turbulent components of the pseudodropsonde measurements. Secondly, a weighted-averageprocess same as in calculating the mean wind profile wasutilized to calculate the mean of the squared turbulent com-ponents found in the previous step. Finally, the square rootsof the averaging results, corresponding to different heightbins, were divided by the mean wind velocity at the samevertical level to yield the turbulence intensity profile.

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TrueRaw

Find1

(a) Simple Model

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TrueRaw

Find1Find2

(b) Alternative ModelFig. 13 Comparison of the mean wind profiles, Raw refers to the com-position results calculated based on the raw pseudo dropsonde profiles,Find1 refers to the calculation results of the wind finding equations,Find2 refers to the calculation results of Eq. (15), and True refers to thestatistics calculated directly from the pseudo-stochastic wind field. ashows the results from the point-object model and b shows the resultsfrom the alternative model


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(a) Simple Model

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TrueRaw

Find1Find2

(b) Alternative ModelFig. 14 Comparison of the turbulence intensity profiles, Raw refers tothe composition results calculated based on the raw pseudo dropsondeprofiles, Find1 refers to the calculation results of the wind findingequations, Find2 refers to the calculation results of Eq. (15), and Truerefers to the statistics calculated directly from the pseudo-stochasticwind field. a shows the results from the point-object model and bshows the results from the alternative model

In conclusion, the similar improvements of using thewind finding equation to calculate the statistics profile basedon different dropsonde motion models (shown in Figs. 13and 14) support that the wind finding equations are stillvalid to retrieve both the mean and the turbulence intensityprofiles of the wind field even if the variation of the dragcoefficient with angles of attack is not negligible.

From the derivation of the wind finding equations basedon the alternative motion model, an additional correction isrequired to account for the influence of the dropsonde ver-tical acceleration (shown in Eq. (15)). The effect of usingthe term z g, instead of g, in the wind finding equa-tions is illustrated in Figs. 13b and 14b. It is found thatthe improvement is insignificant (< 0.1 ms1 for the meanwind profile and < 0.5 % for the turbulence intensity pro-file). Moreover, since the dropsonde vertical acceleration isnot directly reported but must be calculated by differenti-ating the dropsonde falling rate, the improvement resultedfrom this additional correction could be easily cancelled outby extra errors introduced in the finite difference process.Furthermore, since the dropsonde vertical acceleration is

0

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0.15

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0.25

-10 -8 -6 -4 -2 0 2 4 6 8 10

Dis

tribu

tion

Vertical Acceleration (ms-2)Fig. 15 Distribution of the dropsonde vertical acceleration over arange [10 ms2, 10 ms2]; boxes show the probability density

relatively small when compared to the gravitational acceler-ation g, its influence is expected to be minimal in the termz g. As evidence of this statement, the majority of thedropsonde vertical accelerations solved directly from sim-ulations concentrate in the range of (2 ms2, 2 ms2)(shown in Fig. 15). Therefore, this additional correc-tion is not recommended in processing actual dropsondemeasurements.

It can be seen from Fig. 14b that the pseudo dropsondeprofiles produced by the alternative motion model, afterdynamically corrected using the wind finding equations,overestimate the turbulence intensity for both the origi-nal equation introduced by Hock and Franklin (1999) andfor Eq. (15) with the additional correction. This can beexplained by the alternative model nature which includesthe variation of the aerodynamics with angles of attack.From Eqs. (7) and (8), it can be seen that the influenceof the wind velocities (u, v,w) is not only on the aero-dynamic forces but also on the angle of attack, which inturn influences the dropsonde aerodynamics calculated byinterpolating the wind tunnel test results. Therefore, thedropsonde motion described by the alternative model ismore sensitive to horizontal wind fluctuations than thatdescribed by the simple model. This effect is illustrated inFig. 14, as the turbulence intensity calculated by composit-ing raw dropsonde measurement is higher in Fig. 14b than inFig. 13a. Moreover, the dynamic correction term in the windfinding equations (xz/g and yz/g) duplicates the influenceof the dropsonde aerodynamics variations since both thehorizontal dropsonde velocity x, y and the dropsonde fallingrate z implicitly reflect the variations. Therefore, the useof the wind finding equations overestimates the turbulenceintensity. However, since a 5-s low-pass filter is commonlyutilized in processing the actual dropsonde measurements,which produces a length filter scale of 60 m 150 m giventhat the dropsonde falling rate ranges from 12 ms1 (nearthe surface) to about 30 ms1 (above 100 mb), the overes-timation of the turbulence intensity shown in Fig. 14 needs

S. Li, C. Miller

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TrueRaw

Find1Find2

Fig. 16 Comparison of the turbulence intensity profiles calculatedby compositing pseudo dropsonde profiles low-pass filtered by a 5-sspline smooth filter. Raw, Find1, Find2, and True have the samemeanings as in Fig. 14b

to be further investigated. By applying a spline-smooth fil-ter (as described by Franklin et al. (1987)) to the individualpseudo dropsonde profiles simulated based on the alter-native motion model, the influence of low-pass filters isillustrated in Fig. 16 which compares the turbulence inten-sity profiles derived based on two different wind findingequations. In the filter design, a spatial filter scale of 60 mis used since the dropsonde falling rate in our simulationis approximately 12 ms1 given that the air density usedin simulations is close to the air density value near ground.In addition, the second derivatives of the fitted spline areminimized which indicates k = 2 when using the notationspecified in Franklin et al. (1987). When comparing Fig. 16to Fig. 14b, it can be seen that the overestimation of turbu-lent wind is reduced by the low-pass filer (or more preciselya 5-s spline-smooth filter) and the calculation results of boththe wind finding equation and Eq. (15) are close to the trueturbulent wind profile calculated directly from the pseudo-stochastic wind field. Although it seems that the 5-s filtereffectively corrects the overestimation produced by usingthe wind finding equations (as shown in Fig. 14b), we didnot conduct a comprehensive investigation on the low-passfilter by comparing the impacts of different low-pass filterswith different cut-off time scales. Therefore, Fig. 16 shouldbe interpreted with cautions.

4.3 Calculation of the vertical wind

Although the vertical turbulent wind estimation is criticalin calculating turbulent fluxes, the dropsonde vertical tur-bulent wind measurement is not thoroughly investigated.Presently, only a crude estimation is available which isderived under the assumption that the theoretical drop-sonde falling speed varies only with the air density. Morespecifically, the vertical wind is calculated by subtractingthe measured dropsonde falling speed from the theoretical

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Error Velocity (m/s)

conventionaleqn(19)

Fig. 17 Comparison of the mean estimation error profiles. conven-tional refers to the error resulted from the conventional calculation; Eq.(19) refers to the error resulted from the calculation results of Eq. (19)

value. When compared to this crude calculation, Eq. (19)provides a theoretically better vertical wind estimate.

If the estimation error is defined as the absolute valueof the difference between a single vertical wind measure-ment in any pseudo dropsonde profile (estimated usingboth the conventional method and Eq. (19) and the cor-responding true vertical wind (calculated by interpolatingfrom the pseudo-stochastic wind field), the mean estimationerror profile can be calculated using the same compositionapproach as in the calculation of the mean wind profile, andthen the improvement of using Eq. (19) can be quantified.Figure 17 shows the comparison of the mean estimationerror profiles. Obviously, Eq. (19) yielded more accuratevertical wind estimates for all the heights in the pseudo-stochastic wind field. More specifically, Eq. (19) improvedthe accuracy in estimating the vertical wind by nearly 70 %(the error difference divided by the error resulted from theconventional calculation). In addition, the turbulent momen-tum fluxes were calculated based on two vertical windestimates. In extending the turbulent wind profile calcula-tion, the turbulent flux profile can be calculated as follows.Using the turbulent components of pseudo horizontal andvertical wind measurements (u and w) found followingthe methodology detailed in the previous subsection, theturbulent wind products uw from different pseudo drop-sonde profiles within the same height bin were averagedusing the same weighted-average methodology to calcu-late the turbulent momentum flux at the center of theheight bin. The turbulent flux profile was then formulatedby lining up the averaged turbulent wind products uwfor all height bins. As in the error comparison, the turbu-lent flux profile comparison shown in Fig. 18 substantiatesthat Eq. (19) indirectly gives a more accurate estimate ofthe turbulent momentum flux. Since the turbulent momen-tum flux calculation is crucial in numerical simulationsof tropical cyclones as it determines the boundary layer


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Flux Difference (m2s-2)

conventionaleqn(19)

Fig. 18 Comparison of the difference between turbulent flux profilescalculated using different vertical wind estimates and that calculateddirectly from the pseudo-stochastic wind field. conventional and Eq.(19) have the same meanings as in Fig. 17

parametrization used in numerical weather prediction sys-tems (Nolan et al. 2009a, b), Eq. (19) is important in fullyutilizing dropsonde measurements to give a more reliableboundary layer turbulence model.

It should be noted that the improvement of using Eq. (19)to retrieve the vertical winds depends on the accuracy of twoestimates: the dropsonde vertical acceleration and the real-time dropsonde aerodynamics. While the dropsonde verticalacceleration can be calculated by differentiating the drop-sonde falling rate, there is no way to find the real-time drop-sonde aerodynamics presently due to the limitation of thecurrent dropsonde instrumentation. As a result, the improve-ment resulted from an accurate dropsonde drag coefficientis not possible at current stage. In contrast, the vertical windestimate can be improved by including the dropsonde ver-tical acceleration through Eq. (19). As a result, the errorin the vertical wind estimation has two sources: the ran-dom error in measured dropsonde falling rate and the errorintroduced by calculating the dropsonde vertical acceler-ation through a finite difference approach. In some case,the directly measured dropsonde falling rate is consideredunreliable and the height measurements are differentiatedto calculate the dropsonde falling rate (J. L. Franklin 2010,personal communication). If the accuracy of dropsonde ver-tical motion measurements is assumed to be similar tohorizontal motion measurements, the random measurementerror is in the order of 0.5 2 ms1 according to Hock andFranklin (1999). The error introduced by the finite differ-ence approach was investigated. The first-order backwardfinite difference, which is used widely in processing theactual dropsonde measurement, was conducted to calculatethe dropsonde falling rate from the pseudo height mea-surements and the dropsonde vertical acceleration from thepseudo falling rate measurements. Using the differentiatedand the true dropsonde motions (the true value is directlysolved in the dropsonde motion simulation), different ver-tical wind estimates were made using the conventional

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)

Error Velocity (m/s)

conventionaleqn(19)true Vztrue az

Fig. 19 Comparison of the mean error profiles. conventional refersto the error profile resulted from the conventional calculation, eqn(19)refers to the calculation results of Eq. (19) using differentiated drop-sonde falling rate and vertical acceleration, true Vz is similar to eqn(19)but uses the true dropsonde falling rate, and true az is similar toEq. (19) but uses both the true dropsonde falling rate and verticalacceleration

method and Eq. (19). To eliminate the influence of the dragcoefficient shown in Eq. (19), the constant value impliedby Hock and Franklin (1999) (0.61) was utilized to calcu-late the vertical winds, i.e., CD() 0.61. Following themethodology shown in Fig. 17, the mean estimation errorprofiles were calculated and compared in Fig. 19. Obviousin this mean estimation error profile comparison, the mostsignificant improvement was resulted from the use of thetrue vertical dropsonde acceleration which is directly solvedin the simulation. This trend is also seen in Fig. 20 for thecomparison of the turbulent momentum flux profiles. Thissuggests that the improvement of using Eq. (19) to retrievethe vertical winds from raw dropsonde measurements relieson the accuracy the dropsonde vertical acceleration calcu-lation. However, even if the vertical acceleration is onlycrudely calculated by differentiating the dropsonde fallingrate, Eq. (19) still improves the estimates of both the verticalwind and the turbulent momentum flux. Since Fig. 17 shows

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conventionaleqn(19)true Vztrue az

Fig. 20 Comparison of the difference between the turbulent fluxprofiles calculated using different vertical wind estimates and that cal-culated directly from the pseudo-stochastic wind field; conventional,Eq. (19), true Vz, and true az have the same meanings as in Fig. 19

S. Li, C. Miller

the error of using Eq. (19) when all required informationis available (including the real-time dropsonde aerodynam-ics), the improvement of using an accurate drag coefficientcan be roughly estimated by comparing Figs. 17 to 19. It isclear that an accurate drag coefficient further improves thevertical wind estimate. More specifically, the error level isreduced from about 1.1 ms1 (shown as true az in Fig.19) to 0.3 ms1 (shown as Eq. (19) in Fig. 17).

5 Conclusion

The point object model is conventionally used to describethe dropsonde motion in the wind field, in which the drop-sonde drag coefficient can only be assumed as a constantregardless of the angle of attack, and the wind finding equa-tions introduced by Hock and Franklin (1999) are derivedbased on the point object model. Since the wind findingequations are important to reduce the dropsonde lag effect tocalculate the correct mean wind profile in sheared flows, itis necessary to validate the point object model by investigat-ing the variation of dropsonde drag coefficient with anglesof attack. To do that, a series of wind tunnel tests are con-ducted as a part of this study to examine the dropsondeaerodynamics. The test results showed that, although thedrag coefficient of the parachute alone can be modelled asa constant, the variation of the drag coefficient of the wholedropsonde system (including the parachute) is appreciable.

Using the dropsonde aerodynamics gathered in the windtunnel tests, numerical simulations of dropsonde motions ina pseudo-stochastic wind field based on an alternative drop-sonde motion model were conducted. When compared tothe point object model, this alternative model showed thefollowing three improvements:

The variation of aerodynamics with angles of attackis explicitly modelled which allows a more realisticdescription of aerodynamics forces experienced by thedropsonde.

The lift force is added into the model. The rotation is included in the model, and therefore the

orientation of the dropsonde body is solved.

Based on the simulation results, individual pseudo drop-sonde profiles were generated and composited to calcu-late the statistics profiles of the pseudo-stochastic windfield. Since the dropsonde motion simulations were con-ducted based on both the simple point object model andthe alternative model, the impact of the dropsonde dragcoefficient variation was investigated through a compari-son approach. Through comparing the mean and turbulentwind profiles calculated in two simulations (based on twodifferent motion models), it has been found that the windfinding equations are valid for computing both the mean

and the turbulence intensity structure of the wind field, eventhough they were derived based on a simple point objectmodel in which the dropsonde drag coefficient is a con-stant regardless of the angle of attack. This is theoreticallyunderstandable since the variation of the drag coefficientis reflected by the variation of the measured dropsondefalling rate. In discussing the validity of the wind find-ing equations theoretically, it has been found that the windfinding equations can be additionally improved by includ-ing the dropsonde vertical acceleration. However, since thisimprovement is insignificant and will introduce extra errorsin the finite difference process to calculate the dropsondevertical acceleration, it is not recommended in processingthe actual dropsonde measurements.

Based on the alternative motion model, a new approachto retrieve vertical winds from raw dropsonde measurementsis proposed. When compared to the conventional verticalwind retrieval method (subtracting the measured dropsondefalling speed from the theoretical value), this approach takestwo additional factors into account: the variation of the dragcoefficient with angles of attack and the dropsonde verticalacceleration. While there is no way to find out the real-time angle of attack currently (therefore the improvementresulted from using an accurate dropsonde drag coeffi-cient is impossible), the vertical wind retrieval can still beimproved by incorporating the dropsonde vertical acceler-ation through Eq. (19). As the vertical wind estimate iscritical in turbulent flux calculation, the use of Eq. (19) alsoimproves the turbulent momentum flux estimation.

Acknowledgments This work was made possible by the facilitiesof the Shared Hierarchical Academic Research Computing Network(SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada.

References

Carassale L, Solari G, Tubino F (2007) Proper orthogonal decompo-sition in wind engineering. Part II: theoretical aspects and someapplications. Wind Struct 10:177208

Cockrell DJ (1987) The aerodynamics of parachutes. Tech. rep., TheUniversity of Leicester, UK, available from the library of the WindTunnel Lab. The University of Western Ontario, London, Canada

Davenport AG (1967) The dependence of wind loads on meteorologi-cal parameters. In: International research seminar on wind effectson buildings and structures, National Research Council of Canada,Ottawa, ON, Canada, pp 1982

Drennan WM, Zhang J, French JR, McCormick C, Black PG (2007)Turbulent fluxes in the hurricane boundary layer. Part II: latentheat flux. J Atmos Sci 64:11031115

Engineering Science Data Unit (ESDU) (1993) Characteristics ofatmospheric turbulence near the ground. Part ii: single point datafor strong winds (neutral atmosphere). Tech. rep., EngineeringScience Data Unit 85020

Franklin JL, Ooyama KV, Lord SJ (1987) Two improvements inomega wind finding techniques. J Atmos Ocean Technol 4:214219


French JR, Drennan WM, Zhang J, Black PG (2007) Turbulent fluxesin the hurricane boundary layer. Part I: momentum flux. J AtmosSci 64:10891102

Hock TF, Franklin JL (1999) The NCAR GSP dropwindsonde. BullAmer Meteor Soc 80:407420

Kepert JD (2006a) Observed boundary layer wind structure and bal-ance in the hurricane core. Part I: Hurricane Georges. J Atmos Sci63:21692193

Kepert JD (2006b) Observed boundary layer wind structure and bal-ance in the hurricane core. Part II: Hurricane Mitch. J Atmos Sci63:21942211

Nolan DS, Stern DP, Zhang J (2009a) Evaluation of planetary bound-ary layer parameterizations in tropical cyclones by comparisonof in situ observations and high-resolution simulations of Hurri-cane Isabel (2003). Part II: inner-core boundary layer and eyewallstructure. Mon Weather Rev 137:36753698

Nolan DS, Zhang J, Stern DP (2009b) Evaluation of planetary bound-ary layer parameterizations in tropical cyclones by comparison

of in situ observations and high-resolution simulations of Hurri-cane Isabel (2003). Part I: initialization, maximum winds, and theouter-core boundary layer. Mon Weather Rev 137:36513674

Powell MD, Vickery PJ, Reinhold TA (2003) Reduced drag coefficientfor high wind speed in tropical cyclone. Nature 422:279283

Solari G, Piccardo G (2001) Probabilistic 3-D turbulence modeling forgust buffeting of structures. Probab Eng Mech 16:7386

Solari G, Carassale L, Tubino F (2007) Proper orthogonal decom-position in wind engineering. Part I: a state-of-the-art and someprospects. Wind Struct 10:153176

Vickery PJ, Wadhera D, Powell MD, Chen Y (2009) A hurricaneboundary layer and wind field model for use in engineeringapplication. J Appl Meteor Climatol 48:381405

Von Karman T (1948) Progress in the statistical theory of turbulence.Proc Natl Acad Sci USA 34:530539

Zhang J, Drennan WM, Black PG, French JR (2009) Turbulence struc-ture of the hurricane boundary layer between the outer rainbands.J Atmos Sci 66:24552467

The impact of the variation of drag coefficients on the dropsonde wind retrievalAbstractIntroductionWind tunnel testTest configurationTest results and discussion

Motion simulation with an alternative modelThe alternative motion model of the dropsondeDropsonde motion simulation

Simulation results and discussionTheoretical analysisValidation of the wind finding equationsCalculation of the vertical wind

ConclusionAcknowledgmentsReferences

Documents

The variation of the GPS dropwindsonde drag coefficient and its impact on the wind retrieval from dropwindsonde measurements