the problem of the second wind turbine – a note on a ... · 80 f. gans et al.: the problem of the...

Earth Syst. Dynam., 3, 79–86, 2012www.earth-syst-dynam.net/3/79/2012/doi:10.5194/esd-3-79-2012© Author(s) 2012. CC Attribution 3.0 License.

Earth SystemDynamics

The problem of the second wind turbine – a note on a common butflawed wind power estimation method

F. Gans, L. M. Miller, and A. Kleidon

Biospheric Theory and Modeling Group, Max Planck Institute for Biogeochemistry, Hans-Knoell-Strasse 10,07745 Jena, Germany

Correspondence to:F. Gans ([email protected])

Received: 29 April 2010 – Published in Earth Syst. Dynam. Discuss.: 8 June 2010Revised: 19 March 2012 – Accepted: 11 May 2012 – Published: 8 June 2012

Abstract. Several recent wind power estimates suggest thatthis renewable energy resource can meet all of the currentand future global energy demand with little impact on theatmosphere. These estimates are calculated using observedwind speeds in combination with specifications of wind tur-bine size and density to quantify the extractable wind power.However, this approach neglects the effects of momentumextraction by the turbines on the atmospheric flow that wouldhave effects outside the turbine wake. Here we show with asimple momentum balance model of the atmospheric bound-ary layer that this common methodology to derive windpower potentials requires unrealistically high increases inthe generation of kinetic energy by the atmosphere. This in-crease by an order of magnitude is needed to ensure mo-mentum conservation in the atmospheric boundary layer. Inthe context of this simple model, we then compare the ef-fect of three different assumptions regarding the boundaryconditions at the top of the boundary layer, with prescribedhub height velocity, momentum transport, or kinetic energytransfer into the boundary layer. We then use simulationswith an atmospheric general circulation model that explic-itly simulate generation of kinetic energy with momentumconservation. These simulations show that the assumption ofprescribed momentum import into the atmospheric boundarylayer yields the most realistic behavior of the simple model,while the assumption of prescribed hub height velocity canclearly be disregarded. We also show that the assumptionsyield similar estimates for extracted wind power when lessthan 10 % of the kinetic energy flux in the boundary layeris extracted by the turbines. We conclude that the commonmethod significantly overestimates wind power potentials byan order of magnitude in the limit of high wind power ex-

traction. Ultimately, environmental constraints set the upperlimit on wind power potential at larger scales rather than de-tailed engineering specifications of wind turbine design andplacement.

1 Introduction

Several recent studies have quantified large-scale or globalwind power availability by extrapolating kinetic energy avail-ability from measured wind speeds (Archer and Jacobson,2005; Sta. Maria and Jacobson, 2009; Lu et al., 2009; Liu etal., 2008; Leithead, 2007).

Although the exact methodologies may differ, these stud-ies use a similar procedure. Hereafter, we will collec-tively refer to these studies as the “common method”. Thismethodology can be generally described as follows:

1. Observations of wind velocities,v, are derived for alarge spatial area from a collection of surface stations(e.g. Archer and Jacobson, 2005), satellite measure-ments (e.g.Liu et al., 2008), or reanalysis data (e.g.Luet al., 2009).

2. Technical specifications for a turbine are then used tospecify the rotor height, rotor-swept area, and velocitydependent power characteristics.

3. Generated power resulting from the extracted ki-netic energy is calculated for a single turbine byPex = 1

2 ρ Cf Arotorv3, whereρ is the air density,Cf is

the capacity factor dependent on the selected turbine,andArotor is the rotor-swept area.

Published by Copernicus Publications on behalf of the European Geosciences Union.

80 F. Gans et al.: The problem of the second wind turbine

4. The area is then populated with wind turbines at a den-sity limited by engineering constraints related to turbinewake turbulence, under the basic assumption that windturbines do not influence the wind field outside theirspecific influential wake volume (e.g.Sta. Maria andJacobson, 2009).

These studies collectively result in high wind power avail-abilities and exclude effects on larger scale atmospheric flow,since they assume kinetic energy consumption is limited tothe turbine wake, but does not affect the large-scale flow.

In stark contrast, studies byKeith et al.(2004) andWangand Prinn(2010) use global climate models which includeconsumptive effects and induced feedbacks from large-scalewind power extraction. These two studies also show signifi-cant climatic impacts of wind power removal on the climatesystem. Additionally,Miller et al. (2011) included these con-sumptive effects and estimated the maximum global land-based wind power availability from the atmospheric bound-ary layer (ABL) to be 18–68 TW, which is much lowerthan previous estimates and unavoidably associated withsignificant climatic differences.

The discrepancy in results between the two differentmethodologies is significant and must be resolved. Advo-cates of the common method usually argue that imprecisionsin general circulation models (GCMs) and the associated rep-resentation of wind turbines therein cause the differences inclimatic effects, and therefore the results should be disre-garded (e.g. see open discussion onMiller et al., 2011). In ouropinion the assumption of an unaltered wind velocity out-side the turbine wake is the main reason for this difference,because it neglects the effects of turbine-related momentumremoval from the atmospheric boundary layer.

The aim of this paper is to investigate the effect of not ac-counting for kinetic energy and momentum removal from theatmospheric flow during wind power extraction on differentmagnitudes of power extraction.

We will present a simple model which is based on the bal-ance of momentum and kinetic energy fluxes in the atmo-spheric boundary layer. We then compare the results of thissimple model with the GCM simulation results ofMiller etal. (2011).

2 Momentum and kinetic energy extraction from theatmospheric boundary layer

Wind turbines generate electrical power by removing kineticenergy from the atmospheric flow. The instantaneous rateof kinetic energy extraction from a moving fluid can be ex-pressed as

P = F · v (1)

wherev is the velocity of the fluid, and the forceF is the rateof momentum extraction from the flow. Wind power is there-

fore proportional to the extracted momentum and the velocityat which this momentum is extracted. In general, the highestpotential of extracting wind power is realized when there isa high velocity as well as a continuous source of momentumthat can be utilized.

Equation (1) forms the basis for deriving the wind powerformula used in the common methodology. The momentumdensity of a fluid moving at a velocity isρ v. The momen-tum flowing through the cross-section of a single turbineequals the momentum density multiplied by the flow ve-locity, Jp =ρ v · v, whereJp is the horizontal momentumtransport in the fluid. The assumption that the drag fromturbines,Fturbine, is proportional toJp (i.e. Fturbine∝ ρ v2)leads to the commonly known wind power extraction equa-tionPex∝ Aρ v3, whereA is the wind-swept area of the windturbine blades.

This equation is likely to give the impression that windpower availability is determined by measured velocities andair densities only, but its derivation shows that the flux ofmomentum is equally important. Wind turbines depend onthe extraction of momentum, and in the absence of a con-tinual source of momentum, wind turbines would eventuallystop the air from moving. In order to quantify the large-scale extractability of wind energy, it is therefore essentialto quantify and model the sources of momentum and kineticenergy, thereby investigating their influence on wind powerextractability.

We next consider the case in which we account for themomentum balance of the fluid from which kinetic energyis extracted. In the limit case of a completely developedwind farm, where all the surface area is equally covered withequidistant wind turbines, lateral sources of momentum canbe neglected and only vertical fluxes of momentum add tothe overall momentum balance of the system. In the absenceof external forces, momentum can not be generated or dis-sipated within the boundary layer – it can only be trans-ferred from regions with higher velocities to regions withlower velocities. In the atmospheric boundary layer, momen-tum is transferred from higher altitude layers with higherwind velocities to lower levels with lower wind velocitiesand eventually to the surface, wherev = 0.

We define two reference heights in the atmospheric bound-ary layer (see Fig. 1). One is the top of the boundary layer,defined here as 2 km above the surface. The second layer ofinterest is the hub-height, which we take to be 80 m aboveground.

2.1 Fundamental balance equations

The balance equations for momentum and kinetic energy atthe two reference heights are the sum of all vertical fluxesthat enter and leave the layer. For simplicity, we assume thesebalances are in a steady state.

Earth Syst. Dynam., 3, 79–86, 2012 www.earth-syst-dynam.net/3/79/2012/

F. Gans et al.: The problem of the second wind turbine 81

2.1.1 Momentum balances at hub-height

The balance of momentum fluxes at hub-height is written as:

dphub

dt= 0 = Ftransfer− Fsurface− Fturbine (2)

wherephub is the momentum per unit surface area at hub-height,Ftransfer is the vertical transport of momentum fromabove,Fsurface is the momentum transferred to the surfaceby frictional dissipation andFturbine is the momentum ex-tracted by wind turbines. From this momentum balance, wecan derive that an increased momentum extraction by windturbines,Fturbine, must be balanced either by (a) decreaseddissipation at the surface,Fsurface, or (b) increased momen-tum influx from higher layers,Ftransfer. In common parame-terizations of surface momentum transfer, the rate of transferis a monotonic function of the wind velocity,v. This meansthat we expect a reduction in wind velocities at hub-heightwith increased values ofFturbine, which in return will reduceextractable wind power associated with this momentum ex-traction. However, if a reduction in surface dissipation is themain source of momentum for wind turbines, then the ex-tracted wind power is limited by the rate of momentum trans-fer to the surface,Fsurface. Under this assumption the kineticenergy dissipation associated with surface momentum trans-fer gives a first-order estimate of maximum power extraction.

The potential change in momentum transfer from higheraltitudes with increased wind turbine extraction cannot beestimated ad hoc. Assuming that the hub-height velocity de-creases, one would expect an increase in momentum trans-fer from above because there is a higher vertical veloc-ity gradient. However, to fully understand this effect, onemust know how the velocity reduction at hub-height affectswind velocities in higher layers. To do so, we define anothermodel height at the top of the ABL. The momentum balanceequation at the top is:

dptop

dt= 0 = Facc − Ftransfer (3)

whereFacc is the accelerating force or, equivalently, the mo-mentum flux into the top of the atmopsheric boundary layerper unit area.

Note that so far, we based our equations and reasoningonly on physical conservation laws. No assumptions havebeen made about how the fluxes are parameterized. In orderto calculate some example fluxes of momentum and kineticenergy and to understand how wind velocities change dueto turbine momentum extraction, we will now formulate amodel based on parametrizations of momentum fluxes.

2.1.2 Flux parameterizations

In the next step, we parametrize the exchange fluxes of mo-mentum between the two model layers. We use standard at-mospheric boundary layer theory (Stull, 1988) to formulate

these fluxes depending on the hub-height velocity,vhub, andthe top-of-ABL velocity,vtop. The transport of momentumbetween the top-layer and hub-height layer is expressed as:

Ftransfer= ρ Ctransfer·(vtop − vhub

)· vtop (4)

whereCtransferis a dimensionless momentum transfer coeffi-cient between the two layers which can be derived for a givensurface roughness.

The surface friction is parameterized as:

Fsurface= ρ v2hubCDN (5)

where CDN is the surface drag coefficient for hub-heightwind velocities. Assuming neutral stability conditions, thesurface drag coefficient for wind speeds atz = 80 m hub-height can be calculated using

CDN = k2(

ln

(z

z0

))−2

= 0.0013 (6)

wherek ≈ 0.4 is the von-Karman parameter and we use aroughness length typical for grasslands,z0 = 1 mm (Stull,1988).

2.1.3 Kinetic energy balance

In addition to the momentum fluxes, we are interested in thevertical flux of kinetic energy. Analogous to the momentumbalance, one can formulate balance equations for the verticalfluxes of kinetic energy at each model layer, which is

dKEtop

dt= 0 = Jin − Jtransfer− Dtransfer. (7)

Here,Jin =Facc · vtop is the kinetic energy imported into thetop of the ABL, Jtransfer=Ftransfer· vhub is the kinetic en-ergy transferred to hub-height, andDtransferis the energy lostby turbulent dissipation between the two layers. The kineticenergy balance equation at hub-height is:

dKEhub

dt= 0 = Jtransfer− Dsurface− Pex (8)

whereDsurface=Fsurface· vhub is the kinetic energy dissipatedat the surface andPex is the kinetic energy extracted by windturbines.

2.2 Turbine parameterization

For the parametrization of the wind turbines, we use the stan-dard wind power equation and the turbine specifications ofthe Tjaereborg 2 MW wind turbine with a rotor diameterof D = 61 m (Sta. Maria and Jacobson, 2009). We assumethat the wind farm is very large, so we can neglect energyand momentum influxes from the sides of the wind farm. Inthis setup each turbine extracts a certain amount of momen-tum and kinetic energy from the atmosphere, given by theequations:

Fex =1

2ρ v2

hubCf Arotor/Aturbine (9)

www.earth-syst-dynam.net/3/79/2012/ Earth Syst. Dynam., 3, 79–86, 2012


and

Pex =1

2ρ v3

hubCf Arotor/Aturbine (10)

whereCf ≈ 0.56 is the capacity factor,ρ is the air density,Arotor is the rotor-swept area andAturbine is the ground sur-face area that a single wind turbine uses. This parameter de-scribes the turbine spacing in the wind farm.

What is not specified in the model is howFacc andJin re-spond to kinetic energy extraction by wind turbines. In orderto derive such a relationship, one would have to model thedynamics of the higher atmosphere, which is not within thescope of this simple model but is investigated later in Sect.3.Here, we will investigate three different scenarios of how thetop of the ABL fluxes might respond to changes in surfacemomentum extraction: (1) a prescribed hub-height velocitycorresponding to the common method, (2) a prescribed mo-mentum flux,Facc, and (3) a fixed kinetic energy flux,Jin.

2.2.1 Assumption (1): prescribed hub-height velocity

In the first scenario, we assume that the wind turbine in-stallations do not have an effect on the bulk wind velocity,vhub, outside the wake volume, which is equivalent to the as-sumptions made by the common method. This assumptiomdoes not necessarily ensure that all kinetic energy and mo-mentum fluxes are balanced. In order to ensure a constanthub-height wind velocity while maintaining the balance ofmomentum fluxes, the momentum import into the top of theABL must increase with increased turbine installations tocompensate for the increased momentum extraction by windturbines. This would lead to the following relationship be-tween turbine momentum extraction and total momentumimport:

Facc = Facc,0+ Fex (11)

whereFacc,0 is the momentum flux without turbine installa-tions andFex is the momentum extracted by wind turbines.Using this relationship all the other fluxes and velocities canbe determined.

2.2.2 Assumption (2): prescribed import of momentum

In a second scenario, we will assume that the momentum in-put into the top of the ABL is constant and is not affected bywind turbine installations.

Facc = Facc,0. (12)

Under this assumption the momentum flux into the hub-height level is fixed. This means that increased wind tur-bine momentum extraction must be balanced by a decreasein turbulent surface dissipation, which is only possible whenthe bulk wind velocity decreases. We will therefore expect alower wind power extractability compared to assumption (1).

2.2.3 Assumption (3): prescribed import of kineticenergy

Our third scenario assumes that the kinetic energy flux perunit time is fixed when altering the momentum extraction atthe surface. This assumption implies that the momentum fluxat the top of the ABL is:

Facc = Jin,0/vtop (13)

whereJin,0 is the fixed flux of kinetic energy into the bound-ary layer. Since the total kinetic energy import is fixed, wewould expect that power extraction of turbines should sat-urate with increased turbine densities. Hence, this assump-tion should also lead to lower estimates of wind power ex-tractability than assumption (1).

2.3 Results

In our setup, we assume a hub-height wind velocity ofvhub= 10 m s−1 in the absence of wind turbines, which re-sults in an accelerating force ofFacc= 0.168 Ws m−3 and awind velocity at the top of the ABL ofvtop = 12.8 m s−1.The kinetic energy fluxes for this scenario in the absenceof wind turbines (Fturbine= 0) are: Jin = 2.15 W m−2 andJtransfer= 1.68 W m−2, which is the total input of kinetic en-ergy into the ABL and hub-height level. The dissipation ratesareDtransfer= 0.47 W m−2 during the downward transport be-tween top and hub-height, andDsurface= 1.68 W m−2 belowhub-height.

The fluxes of momentum and kinetic energy and the veloc-ities at the model heights were calculated for different windturbine densities under the three assumptions. Figure3 showsthe response of the energy and momentum fluxes to changesin wind turbine density. Under assumption (1) the extractedpower is a linear function of the number of wind turbinesand increases accordingly. The import of kinetic energy andmomentum also increase with increasing turbine density soas to offset extraction by the wind turbines, while hub-heightvelocities outside the wake volume remain constant.

The results of assumptions (2) and (3) resemble each other,both showing a saturating maximum in extracted wind poweronce a certain density of turbines is reached. Wind veloci-ties decrease with increasing turbine numbers, while kineticenergy and momentum input remain in the same order ofmagnitude.

For low turbine densities at less than 0.1 turbines perkm2, all assumptions lead to very similar results. For higherdensities assumption (1) significantly differs from the othertwo assumptions by predicting much higher values of windpower extractability, kinetic energy and momentum influxfrom higher altitudes, and constant hub-height velocities atthe wind turbine interface. The divergence of the differ-ent assumptions is apparent where the turbine power ex-traction is≈10 % of the total kinetic energy import to theboundary layer.



Ftransfer

Facc

80 m

2000 m

d

Vtop

Vhub

Fturbine

Fsurface

Fig. 1.Schematic illustration of the vertical fluxes of horizontal mo-mentum in the atmospheric boundary layer. Momentum is importedfrom the upper atmosphere at the rateFacc and diffuses to lowerlayers at the rateFtransfer, which is driven by the velocity gradient.Wind turbines extract some of this momentum atvhub to generateelectrical energy (Fturbine).

We then compared these estimates for the case of aturbine-specific land areaAturbine= 10D · 3D, whereD isthe diameter of the wind turbine, as suggested bySta. Mariaand Jacobson(2009). According to assumption (1), thedownward momentum flux would need to increase 10-foldto 0.92 Ws m−3 and the power input into the ABL would be17.8 W m−2, which is also an increase by roughly an orderof magnitude. Using assumptions (2) and (3) leads to rela-tively low changes in energy and momentum import at thetop of the ABL. A comparison of the energy fluxes impliedby the different assumptions at this specified turbine densityis shown in Fig.2

The results of assumption (1) pose the question of howthe atmospheric circulation would be able to increase powergeneration by an order of magnitude. Because an explanationof the processes that would lead to such an increase has notbeen given yet, it seems unlikely that this drastic change isactually possible.

3 Comparison to climate model simulations

3.1 Climate model description

In order to evaluate which assumption is most reasonable andwhether the atmosphere can generate much more power, weneed to resort to a GCM that explicitly represents momen-tum balance constraints by being based on the Navier-Stokesequations of fluid dynamics. We use the climate model simu-lations ofMiller et al. (2011) and compare their results withthe ones derived by the 3 different scenarios. We are partic-ularly interested in how the boundary layer kinetic energyimport responds to increased momentum extraction at the

Prescribed energy fluxPrescribed momentum fluxPrescribed hub-height velocity

extra

cted

pow

er (W

/m2 )

0

2

4

Turbine density km20.01 0.1 1 10

(a)


ABL

kinet

ic en

ergy

impo

rt (W

/m2 )

0

2

4

Turbine density per km20.01 0.1 1 10

(b)


ABL

mom

entu

m im

port

(Ws/

m3 )

0

0.2

0.4


(c)

Prescribed energy fluxPrescribed momentum fluxPrescribed hub-height velocityhu

b-he

ight

win

d ve

locit

y (m

/s)

6

8

10


(d)

Fig. 2.Comparison of the simple atmospheric boundary layer modelresults under 3 different assumptions represented by different col-ors. The graph shows the turbine power extraction (W m−2), kineticenergy import at the top (W m−2), momentum import at the top(Ws m−3) and wind velocity at hub-height (m s−1).



2000 m

d

Pex =

23.221.1 m/s

10 m/s

12.2

11.0

1.7Jsurface

Jtransfer

Jdiss

Jin

9.3

Vtop

Vhub

80 m

Prescribed hub-height velocity

2000 m

d

Pex =

1.48.3 m/s

0.7

0.7

0.1Jsurface

Jtransfer

Jdiss

Jin

0.6

Vtop

Vhub

80 m

3.9 m/s

Prescribed momentum import

2000 m

d

Pex =

2.29.6 m/s

1.1

1.1

0.2Jsurface

Jtransfer

Jdiss

Jin

0.9

Vtop

Vhub

80 m

4.5 m/s

Prescribed kinetic energy import

Fig. 3. Comparison of the kinetic energy fluxes for the 3 differentassumptions using a turbine spacing ofAturbine= 10D · 3D, whereD = 61 m is the rotor diameter. Unless otherwise noted, all quanti-ties have the units W m−2.

surface. We can therefore determine if the increase in powergeneration predicted by the common method is supported byGCM results.

In these simulations, wind turbines are represented as anadditional frictional force in the atmospheric boundary layer.Details of the study are described inMiller et al. (2011). Thewind turbines, represented by the additional frictional param-eterCex, cause a frictional force on the atmospheric boundarylayer, Ffric =ρ Cexv

2. The conversion efficiency from wind

power extracted from the atmosphere to mechanical powerof the turbine is estimated to be≈0.6 using the Betz limit.Extracted wind power per unit area is then

Pex = 0.6 · F · v = 0.6ρ Cexv3. (14)

To represent the common method, wind velocities weretaken from a control simulation without any additional fric-tion. Based on these velocities, the power output of a singleturbine at each time step was calculated using Eq. (9). Thetotal power extraction per unit surface area is then

Ptot =1

AturbinePex =

1

2ρ

N

ALandCf Arotorv

3 (15)

where 1Aturbine

is the turbine density on land. ComparingEqs. (14) and (15) lets us derive the relationship between thefriction parameterCex and the turbine density:

1

Aturbine=

1.2Cex

Cf Arotor. (16)

For example, a turbine density of 1 turbine per km2 wouldcorrespond to a friction parameterCex = 0.0013. The kineticenergy import into the boundary layer of the GCM is diag-nosed using the energy balance Eq. (8), resulting in

Jin = DBL + Pex (17)

whereDBL is the rate of kinetic energy dissipation withinthe boundary layer andPex is the turbine associated powerextraction.

3.2 Results

The calculated power extraction as well as the influence ofwind turbine density on boundary layer wind velocities andboundary layer power input is shown in Fig.4. For low tur-bine densities, the common method and the GCM results leadto very similar results. As the turbine density increases, theestimates begin to diverge. During the process of kinetic en-ergy extraction, wind velocities decrease and therefore ki-netic energy extractability is reduced. The onset of the di-vergence takes place when≈10 % of the kinetic energy dis-sipated in the boundary layer is extracted by the turbines,consistent with the results of the simple model.

The GCM simulations do not show any evidence that theatmosphere would respond to kinetic energy extraction bysimply generating more kinetic energy, which is predicted byassumption (1) in our simple model. Figure4 shows the im-plied increase in kinetic energy import assuming that kineticenergy is extracted and wind velocities are assumed to be un-affected outside the turbine wake. However, the existence ofsuch a generation mechanism is not supported by the GCMresults. Instead, the total boundary layer kinetic energy im-port decreases in response to kinetic energy extraction and istherefore most similar to the results of assumption (2) withprescribed momentum import.



(a)

(b)

(c)

Fig. 4. Sensitivity of wind power extraction and consequences forboundary layer flow from a set of climate model simulations inwhich all land regions are populated by wind turbines of differ-ent density. Shown are(a) mean power extraction by wind turbines,(b) mean boundary layer kinetic energy import, and(c) mean globalwind velocities, and mean land wind velocities. The estimates of thecommon method are depicted by red lines and as modeled by theGCM are shown as black lines. Note the divergence of the two linesstarting at≈2 turbines per km2, or a power extraction of 10 % ofthe total kinetic energy import into the boundary layer.

4 Discussion

This simple illustration as well as more complex GCMsimulations show that accounting for the balance of mo-mentum and kinetic energy is critical to the quantificationof large-scale extractability of wind power. Here, we will

discuss thermodynamic limits on wind energy generationrates within the atmosphere and link them with predictionsmade by GCM simulations. We then discuss how these pre-dictions meet the currently observed state of the atmosphere.

Theory: Using well-established energetics of the atmo-sphere, the global generation rate of kinetic energy is esti-mated to be approximately 900 TW (Lorenz, 1955; Peixotoand Oort, 1992), about half of which is imported intothe boundary layer. Less known thermodynamic derivationsshow that this rate of kinetic energy generation is at a max-imum value given the present-day radiative forcing gradi-ents, as demonstrated by simple theoretical considerations(Lorenz, 1960), box models (Paltridge, 1978; Lorenz et al.,2001) and general circulation model simulations (Kleidon etal., 2003, 2006). This therefore means that an increase intotal atmospheric power generation due to wind turbine in-stallations as formulated in assumption (1) should not be ex-pected. Thermodynamic considerations therefore support theuse of prescribed kinetic energy import or prescribed mo-mentum import, but rule out the assumption of a prescribedhub-height velocity, because it implies a significant increasein kinetic energy generation.

GCM results: This rather theoretical view is supported bythe results of our GCM. In this more complex model, ki-netic energy generation does not increase due to additionalwind turbine installations. Instead, the kinetic energy flux tothe boundary layer decreases in response to very high windturbine densities (see Fig.4) and therefore contradicts theassumption of unaffected hub-height wind velocities. EveryGCM is based on equations ensuring the conservation of mo-mentum, which is the main reason for the noted divergencein large-scale wind power estimates. So although the specificquantities of extractable wind power and related wind turbinedensity will certainly depend on specifications of the chosenGCM and its parameterization of wind turbines, the qualita-tive behavior should be reproducible with different models.

Observations: Advocates of the common method rou-tinely point out that this approach is based on observa-tions, therefore making it superior to the theoretical mod-eling approaches presented here. In response, we want tostress that the present-day installation scale of wind farmsover the global surface is still quite small. The results wehave shown here reflect that for these small installation den-sities, the wind power estimates from the various scenar-ios are quite similar. Thus, observational evidence actuallyreinforces none of the scenarios. This fact makes it evenmore critical then to use a thermodynamically consistentestimation method supported by fundamental physics. Thathas been completed here, highlighting the resulting errorwhen one applies a linear function derived from small scalewind farm installations to global-scale estimates of windpower potential. Considering the role of atmospheric mo-mentum and kinetic energy transfer in estimating large-scalewind power is essential for reliable estimates on wind powerextractability.



5 Conclusions

The ability of wind turbines of any size, quantity, or spa-tial arrangement to appropriate atmospheric kinetic energy islimited, first by the generation rate of kinetic energy in theatmosphere, and secondly, by the ability of these turbines toappropriate wind power from the atmospheric flow. The com-mon method assumes that the bulk velocity is not affected bythe wind power extraction, thereby overestimating availablewind power. Taken further, applying this “common method”for large-scale wind power estimates makes the resulting es-timate dependent on the turbine characteristics (swept-area,hub height, torque-to-electricity conversion efficiency, spa-tial arrangement), rather than the generation rate of motionwithin the atmosphere. While initially counter-intuitive, herewe have illustrated why it is the generation rate of kinetic en-ergy in the atmosphere and the associated momentum fluxes,and not a more detailed understanding of wind velocitiesglobally, that ultimately constrains wind power estimates torealizable limits.

It should also be clear that we are presently well belowthese limits to extraction. In 2008, the wind power produc-tion was 0.034 TW, or 0.2 % of the 2008 global energy de-mand of 16.5 TW (Arvizu et al., 2011). If wind farms con-tinue to increase in scale, our results suggest that physicallimitations imposed by the atmosphere, estimated here usingbasic physical principles, will decrease and ultimately limitthe power production of successive wind turbines. While itremains unclear how much energy the human population willdemand in the future, studying these fundamental limits ofrenewable energy will ultimately present possible pathwaysto a renewable future. At a bare minimum though, to ensurethat these estimates are relevant and scientifically defensible,careful attention must be paid to ensure that basic physicalprinciples are being obeyed.

Acknowledgement.The authors thank Ryan Pavlick andNathaniel Virgo for their comments which improved thismanuscript.

The reviewers (Valerio Lucarini, Klaus Hasselmann, Daniel Kirk-Davidoff and two anonymous referees) also aided in clarifyingseveral concepts and conclusions that appear in the final versionof this manuscript. This work has been partially supported by theHelmholtz research alliance “Planetary Evolution and Life”.

Edited by: H. Held

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http://dx.doi.org/10.1029/2004JD005462

http://dx.doi.org/10.1029/2003GL018363

http://dx.doi.org/10.1029/2005GL025373

http://dx.doi.org/10.1029/2008GL034172

http://dx.doi.org/10.1073/pnas.0904101106

http://dx.doi.org/10.5194/esd-2-1-2011

http://dx.doi.org/10.5194/acp-10-2053-2010

the problem of the second wind turbine – a note on a ... · 80 f. gans et al.: the problem of the...

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