The Role of Wind Gusts in the Near-Ground Atmosphere

Transcription

1 The Role of Wind Gsts in the Near-Grond Atmosphere D. Keith Wilson ARL-TR-9 December Approved for pblic release; distribtion nlimited.

2 The findings in this report are not to be constred as an official Department of the Army position nless so designated by other athorized docments. Citation of manfactrer s or trade names does not constitte an official endorsement or approval of the se thereof. Destroy this report when it is no longer needed. Do not retrn it to the originator.

3 Army Research Laboratory Adelphi, MD ARL-TR-9 December The Role of Wind Gsts in the Near-Grond Atmosphere D. Keith Wilson Comptational and Information Sciences Directorate Approved for pblic release; distribtion nlimited.

4 Abstract Gsts from bondary-layer-scale convective eddies modify the strctre of the atmospheric srface layer by intermittently intensifying or diminishing the local wind speed. A simple model for the effects of these gsts is proposed, based on the following two assmptions: 1) the wind gsts have an isotropic Gassian probability distribtion with standard deviation proportional to w the convective velocity scale) and ) the srfacelayer wind and temperatre profiles attain local eqilibrim with the wind gsts. The minimm friction velocity predicted by the model has the same dependence on srface roghness predicted by Schmann s earlier slab model for convective bondary layers. However, the crrent model also applies to sitations where the mean wind is nonzero. It predicts the breakdown of global Monin-Obkhov similarity for the srface-layer wind shear and temperatre gradient in highly convective conditions U r /w approximately 1 or smaller, where U r is the mean wind speed at the top of the srface layer). Also in contrast to existing similarity theories, the horizontal wind variance exhibits a significant dependence on height and srface roghness near the grond, even for moderate convection. The temperatre variance is nearly naffected by the gsts, becase of its weak dependence on the local wind speed in convective conditions. ii

5 Contents 1 Introdction 1 Modeling Srface-Layer Gsts 3.1 Concept Statistical Distribtion Nondimensionalization Generalized Similarity Theory Effect of Gsts on Atmospheric Statistics Friction Velocity Mean Profiles TemperatreVariance Vertical Velocity Variance Horizontal Velocity Variance Smmary 35 Acknowledgments 36 Appendices 37 A Reference Height and Gstiness Coefficient 37 B Moments of Friction Velocity and Wind Direction 39 References 45 Distribtion 49 Report Docmentation Page 51 iii

6 Figres 1 Vertical profiles of wind and temperatre measred at differerent hypothetical stations, separated in either space or time.. 4 Local wind speed as a fnction of local friction velocity Global and average local friction velocity as a fnction of vectoraverage wind speed at reference height, U r Local average friction velocity as a fnction of roghness length for vanishing mean wind Nondimensional wind gradient zκ/ )d U /dz) as a fnction of ζ = z/l Nondimensional temperatre gradient zκ /P t Q s )d T /dz) as a fnction of ζ = z/l Ratio of friction velocity determined by fitting measrements at 1-m height to actal vale determined from local-eqilibrim model Relative error in %) between wind speed profile in localeqilibrim model and a profile determined by fitting and T to measrements at 1-m height... 9 Same as figre 8, except for temperatre profile... 1 Temperatre variance predictions for U r /w = Ratio of vertical velocity variance σ w) predicted by localeqilibrim model to global similarity prediction, for a smooth srface z /z i =1 5 ) Same as figre 11, bt for a rogh srface z /z i =1 3 ) Comparison of approximate formla, eqation 77), to localeqilibrim model prediction Ratio of along-wind velocity variance σ ) predicted by localeqilibrim model to global similarity formla.45 w +5.8, for a smooth srface z /z i =1 5 ) Same as figre 14, bt for a rogh srface z /z i =1 3 ) Ratio of along-wind to crosswind velocity variance σ /σ v )in local-eqilibrim model, as a fnction of height, for a smooth srface z /z i =1 5 ) Ratio of along-wind to crosswind velocity variance σ /σ v )in local-eqilibrim model, smooth srface case z /z i =1 5 ) Local eqilibrim predictions for along-wind velocity variance sing several methods of calclation B-1 Statistical moments L and L cos θ as a fnction of normalized wind speed U r iv

7 B- Same as figre B-1, except for statistical moments L cos θ and L cos θ v

8 1. Introdction One of the primary goals of micrometeorology is to develop statistical descriptions for the srface-layer near-grond) strctre of the atmosphere. These statistical descriptions have many valable applications, sch as forecasting the trblent transport of biological and chemical agents and calclating the propagation of acostic and electromagnetic waves. The most sccessfl model for srface-layer statistics is the trblence similarity theory developed by Monin and Obkhov 1954), which hypothesizes that the statistics depend on jst for parameters: the friction velocity, the srface temperatre flx Q s, the boyancy coefficient b = g/t s where g is gravitational acceleration and T s, the srface temperatre), and the height from the grond z. Note that no parameters in the Monin-Obkhov similarity theory are related to the large-scale strctre of the atmospheric bondary layer. The theory assmes that srface-layer processes are not significantly affected by the large bondary-layer-scale eddies also called thermals or plmes) characteristic of convective conditions. Primarily dring the 196s and 197s, semiempirical eqations for the vertical profiles of mean wind and temperatre were developed on the basis of Monin-Obkhov similarity. These eqations fit the vertical profiles qite well and have since come into widespread se. Other statistics, sch as the temperatre and vertical velocity, have also been fond to agree with the theory. Nonetheless, experiments have clearly demonstrated that not all near-grond statistics obey Monin-Obkhov similarity. Most significantly, the theory does not work well for the variance and spectrm of the horizontal wind components, which have been fond to depend on z i, the height of the bondary-layer capping temperatre inversion Kaimal et al, 1976; Caghey and Palmer, 1979; Khanna and Brasser, 1997). Frthermore, Beljaars 1994) and Grachev et al 1998) demonstrated that parameterizations for srface flxes and drag coefficients in convective conditions agree better with observations when the wind speed is replaced by a scalar average proportional to the mixed-layer velocity scale, w =Q s z i g/t s ) 1/3, which is representative of the large eddies. Potential limitations of Monin-Obkhov similarity in convective conditions were actally anticipated decades ago, apparently first by Bsinger The bondary layer is defined as the layer of air directly above the Earth s srface in which the effects of the srface friction, heating, and cooling) are felt directly on time scales less than a day Garratt, 199, p. 1). It is typically 5 to 3 m deep dring the daytime. The srface layer can be thoght of as the lowermost tenth of the atmospheric bondary layer. Convective conditions here means that bondary-layer trblence is being prodced primarily by boyant instabilities, as opposed to wind shear. The boyant instabilities sally dominate when there is solar heating of the grond. 1

9 1973a,b). The main prposes of this report are to systematically address the behavior of srface-layer statistics in convective conditions and to provide a qantitative assessment of the breakdown of Monin-Obkhov similarity in these conditions. The framework I adopt is an extension of concepts developed by several previos athors, inclding Schmann 1988), Sykes et al 1993), Beljaars 1994), and Grachev et al 1997, 1998). The nderlying principle is that large-scale convective eddies circlations), which span the depth of the bondary layer, affect near-srface transfer processes by indcing a flctating local srface stress. Grachev et al 1997) aptly call this the convection-indced stress regime to differentiate it from the sitation where stress is indced primarily by a steady wind. To qote those athors: Close to the srface, a large-scale convective eddy has an instantaneos local wind profile which rapidly achieves a steady state. The local strctre of sch a profile shold be essentially the same as in the sal mean wind profile Kraichnan, 196; Bsinger, 1973a). As a reslt, free convection can be treated as a particlar case of forced convection Sykes et al, 1993; Beljaars, 1994). Therefore it is reasonable to apply the Monin-Obkhov similarity theory locally to statistics in the near-srface region of the large-scale convective eddies, even thogh application to global ensemble) average statistics may be tenos. In section, I show how the premise of local eqilibrim, when combined with a probability distribtion for the wind gsts indced by the largescale eddies, leads to a simple and workable methodology for calclating srface-layer statistics. This model is applied to the calclation of mean vertical profiles and variances in section 3. Two appendices follow the body of the report: appendix A discsses the vale of nmerical constants sed in the model, and appendix B provides approximate analytical reslts for large-scale eddy effects on atmospheric statistics.

10 . Modeling Srface-Layer Gsts.1 Concept In the conventional decomposition of trblence first proposed by Reynolds, fields sch as wind and temperatre are written as the sm of mean and flctating parts. For the wind velocity component aligned with the mean flow direction, the Reynolds decomposition is U = U +, 1) where U is the instantaneos wind speed, the flctation, and angle brackets indicate the ensemble mean. If we hypothesize that the trblence attains a local eqilibrim with the large-scale convective eddies, an appropriate alternative decomposition wold be U = U L +, ) where the sbscript L on the angle brackets indicates a local mean vale. The meaning of local here is admittedly imprecise. The general idea is that a spatial filter is applied in the horizontal plane, eliminating motions with scales smaller than those of the large-scale convective eddies. A filter having a ctoff in the range.1z i to.3z i wold serve the desired prpose. A filter in time cold serve in place of the spatial one. Since is zero by hypothesis, if we take the global average ensemble mean) of eqation ), it follows that the global average of locally averaged qantities is simply the global average: U = U L. 3) Throghot the rest of this report I refer to the variations in wind velocity reslting from large-scale convective eddies simply as gsts. This terminology is borrowed from previos athors, inclding Godfrey and Beljaars 1991) and Grachev et al 1998). Colloqially, gsts can refer to variations in the wind velocity other than those originating from large-scale convective eddies, sch as activity associated with a thnderstorm or a frontal passage. Figre 1 illstrates the concept that local vertical profiles of wind and temperatre obtain eqilibrim with gsts, as opposed to the actal mean The concept of applying a filter to separate large-scale motions from smaller ones is discssed in more detail by Tong et al 1998). The prpose of the filter in this report is to separate the large-scale convective eddies from other smaller scale motions. The prpose of Tong et al was somewhat different: to mimic the resolvable-/sbgrid-scale partition in large-eddy simlation. 3

11 Figre 1. Vertical profiles of wind and temperatre measred at differerent hypothetical stations 1 4), separated in either space or time. Profiles are averaged with local filter. Upper: Profiles for a flow having no significant gstiness. Lower: Profiles for a flow having prononced gstiness. Local temperatre profiles are in eqilibrim with gsts. Arrows represent horizontal wind velocity. Gray scale represents temperatre, with light shading being warmer air.) 1) ) 3) 4) 1) ) 3) 4) wind, U. Shown in the figre are the vertical profiles of wind and temperatre, after applying the local filter L, at different measrement locations. For the hypothetical sitation illstrated in the pper part of the figre, the flow strctre at scales greater than the filter ctoff is negligible, so all the observed local profiles are nearly identical. The wind speed increases roghly logarithmically with height, and a layer of warm air lies next to the grond. The local profiles at each station are in eqilibrim with the mean wind speed. For the sitation illstrated in the lower part of the figre, gsting is significant, and the local wind and temperatre profiles come into eqilibrim with the gsts. At station 1), a locally strong gst is mixing warm air pward from the grond. At station ), little deviation of U L from U occrs so that the wind and temperatre profiles are the same as in the pper figre. At station 3), motion from a large-scale eddy is in the opposite direction of the prevailing wind, hence decreasing the local wind speed. As a reslt, trblent mixing is weakened, and most of the warm air remains near the grond. If the motion from a large-scale eddy is strong enogh, the wind may actally shift direction from the prevailing one, as shown for station 4). Strictly speaking, neither of the sitations illstrated in figre 1 pertains to the cstomary application of Monin-Obkhov similarity theory; that theory applies to the ensemble flow characteristics. Even thogh the wind may vary locally becase of individal gsts, the flow is not reqired to obtain a local eqilibrim with them. Local states of the flow are jst samples of the global ensemble, and the similarity theory is intended only for the global ensemble. However, one might anticipate that Monin-Obkhov similarity 4

12 . Statistical Distribtion cannot be applied satisfactorily to a global ensemble nless the local flow samples are all nearly the same as the global average, as in the pper part of figre 1. Otherwise, the premise that large- z i -) scale eddies do not significantly affect the near-grond flow is violated. What jstification does one have, thogh, for applying Monin-Obkhov similarity to locally averaged flow fields? A simple spportive argment follows from the relative time scales of the large-scale convective eddies and the smaller srface-layer ones. The convective eddies have a time scale of z i /w. Taking z i 1 m and w m/s as representative implies an approximate time scale of abot 8 min. The srface-layer eddies have a time scale of z/ ; with z 1 m and.3 m/s, z/ is abot 3 s. Therefore the time scale of the convective eddies shold be one to two orders of magnitde larger than that of the srface-layer eddies. Given this disparity, one might reasonably expect the srface-layer eddies to obtain a rogh eqilibrim with the gsts. So far, I have discssed only large-scale variations in wind velocity. Of corse other qantities, sch as the srface heat flx or temperatre, cold also vary locally. These variations cold even occr over niform terrain, as a reslt of clods blocking incoming solar radiation, and other processes. In this report I chose to neglect sch possible variations for simplicity, althogh these cold be treated in a manner analogos to the wind gsts. This section bilds a general framework for calclating modifications to srface-layer statistics cased by the gsts large-scale convective eddies). The basic idea is to specify a probability density fnction for the windvelocity components at some reference height. Local application of Monin- Obkhov similarity then allows calclation of other statistics. To begin, we follow Godfrey and Beljaars 1991) and Grachev et al 1998) by defining a horizontal gst velocity G the contribtion of the large-scale convective eddies to the local wind, denoted W G by Godfrey and Beljaars and Ũ by Grachev et al) throgh the relation U L z) =U z) + G z). 4) Boldface type in this report indicates a vector qantity; the same symbol withot bolding represents the magnitde of the vector.) Taking the sqared magnitde of both sides of eqation 4) reslts in U L z) =[U z) + G x z)] + G y z), 5) where G x z) and G y z) are the along-wind and crosswind components of the gst velocity, respectively. Averaging both sides, while keeping in mind that G z) =, reslts in U L z) = U z) + G z), 6) 5

13 where G z) = G x z) + G y z). Godfrey and Beljaars and Grachev et al set G z r ) = β w, 7) where β is an empirical factor, called the gstiness coefficient. In this report, I set β =.95 and z r =.1 z i for reasons discssed in appendix A. For brevity, I sally drop the explicit fnctional dependence of qantities sch as U z) and G z) on z when they are to be evalated at z = z r. I also abbreviate the local wind velocity U z r ) L as U L, and U z r ) as U r. Provided that the gsts have random orientation, eqation 7) implies that the variances G x and G y at the reference height mst individally be β w /. To complete a probabilistic model, I will make the assmption that G x and G y are independent, Gassian random variables. The approximate Gassian natre of trblent flctations is well established Batchelor, 1953, pp ), whereas the independence of G x and G y follows from the isotropic natre of the gsts. The joint probability density fnction pdf) for G x and G y is therefore ) p G G x,g y )= 1 πβ w exp G x + G y β w. 8) f U r,q s,z,z,z i,g/t s ) = Given the pdf for the gsts, srface-layer statistics can be calclated from the general formla f U r,q s,z,z,z i,g/t s,g x,g y ) p G G x,g y ) dg x dg y, 9) where f U r,q s,z,z,z i,g/t s,g x,g y ) is a fnction of interest, and f U r,q s,z,z,z i,...) its ensemble mean. As an example, consider the mean wind speed at the reference height, U L. Since we have U L U r,g x,g y )= U r + G x ) + G y, 1) U L = 1 ) πβ w U r + G x ) + G y exp G x + G y β w dg x dg y. 11) Soltion of this type of integral is discssed in appendix B. The reslt is [ U L π ) = βw e U r/ 1+U U ) r r I + U ri U )] r 1, 1) where U r = U r /βw, and the I n s are modified Bessel fnctions of the first kind. In the limit U r, one can show U L πβw /. 6

14 Of corse, to be sefl, this method mst be extended to qantities in addition to U L. We wold like to calclate, for example, mean profiles and variances in the presence of gsts. This is where the assmption of local eqilibrim enters into the problem. The basic strategy is to se local eqilibrim to determine a local friction velocity, L, as a fnction of U r,q s,z,z,z i,g/t s,g x,g y ). Then, given an eqation for the qantity of interest in terms of the friction velocity, we can apply eqation 9) to determine its statistics. The basic Monin-Obkhov similarity eqations for the vertical gradients of mean wind and temperatre are the starting point for formlating sch a soltion. These are e.g., Panofsky and Dtton, 1984; Kader and Yaglom, 199) d U dz d T dz = z ) κz φ M L = P tt κz φ H z L, and 13) ). 14) Here T = Q s / is the srface-layer temperatre scale, κ.4 is the von Kármán constant, and P t.9 is the trblent Prandtl nmber. The fnctions φ M and φ H are hypothesized to have a niversal dependence on the nondimensional height ζ = z/l, where L = 3 T s / κgq s ) is called the Obkhov length. The form I se in this report for the fnctions φ M and φ H is Carl et al, 1973) φ M,H ζ) =1 a M,H ζ) 1/3, 15) where a M and a H are empirical constants. The vertical profiles, fond by integrating eqations 13) and 14), are U z) = κ T z) = T z h )+ T P t κ [ ln z z ) z ) ] Ψ M +Ψ M, and 16) z L L [ ln z z ) zh ) ] Ψ H +Ψ H. 17) z h L L The qantity z is the aerodynamic roghness length the height at which the velocity vanishes), and z h is a reference height where the temperatre is measred. The fnctions Ψ M and Ψ H are defined by Ψ M,H ξ) ξ 1 φ M,H x) x dx. 18) Performing this integration, one finds Grachev et al, 1998) Ψ M,H ξ) = 3 ln 1+φ 1 M,H + φ M,H 3 3 arctan 1+φ 1 M,H 3 + π 3. 19) Throghot this report, I se temperatre implicitly to mean the virtal potential temperatre. The virtal potential temperatre incldes corrections for the effect of water vapor and air boyancy Stll, 1988). 7

15 The basic premise of the local eqilibrim theory is that these eqations apply locally to the gsts. Hence we replace them with U z) L = e x cos θ + e y sin θ) L κ T z) L = T z h ) P tq s κ L [ ln z z Ψ M z [ ln z z h Ψ H z L L L L ) +Ψ H zh L L ) +Ψ M z )] L L )], and ), 1) where θ is the local wind direction, L is the local friction velocity indced by the gsts, L L = 3 L T s/ κgq s ), and e x and e y are the nit vectors in the x- andy-directions. Orienting the coordinate system so that the mean wind direction is along the x-axis, one has immediately with application of the constraint U = U L ) z L cos θh M, z ) = κ U z), and ) L L L L z L sin θh M, z ) =, 3) L L L L where h M,H x, y) =ln x y Ψ M,H x)+ψ M,H y). 4) Taking the magnitde of eqation ) reslts in U L = L κ h zr M, z ) L L L L. 5) Eqating this reslt to eqation 1) yields U r + G x ) + G y = L κ h zr M, z ) L L L L. 6) This eqation can be solved, in principle, for L as a fnction of U r, Q s, z, z i, g/t s, G x, and G y. However becase h M in eqation 5) is a complicated nonlinear fnction of L L and L L is in trn proportional to 3 L,it is not possible to analytically solve eqation 5) for L. Fortnately the eqation can be solved nmerically for L withot difficlty. The simplest method is to calclate a look-p table of U L vales for a range of niformly distribted L vales. Then, given a new vale of U L, the corresponding vale of L can be fond by interpolating in the table. Figre shows the dependence of U L /w on L /w for several vales of z /z i. The motivation for normalizing the velocities by w and then plotting with z /z i as a parameter are discssed in sect..3.) When temperatre statistics are calclated, it is necessary to specify z h and T z h ) in eqation 1). This is actally very problematic in the context of the local eqilibrim model. The most reasonable approach may initially seem to be setting z h and T z h ) to constant global) vales, with T z h ) 8

16 Figre. Local wind speed as a fnction of local friction velocity. For crves are shown, corresponding to different vales of ratio z /z i. For z i = 1 m, z /z i =1 5 is representative of a grassy srface, whereas 1 4 is representative of matre crops or brsh.) Local wind speed, U L z r )/w * z /z i = 1 6 z /z i = 1 5 z /z i = 1 4 z /z i = Local friction velocity, * L/w *.3 Nondimensionalization being the srface temperatre and z h the thermal roghness length of the srface. The problem with this approach is that the local flctations in L and therefore T L ) then create extremely large temperatre flctations at z = z r. These flctations are associated with trblence length scales above the ctoff for the local filter; that is, they come from the convective, z i -scale eddies whose velocity flctations are specified by eqation 8). In actality, observations sggest that the temperatre variance σt does not depend on the large eddies, even well above the srface layer. Therefore this approach is not physically realistic. The root of the difficlty is that z h and T z h ) vary in response to the gsts. As discssed by Panofsky and Dtton 1984, p 147), z h shold be order k h /, where k h is the moleclar coefficient of heat condction. In the context of local eqilibrim, then, z h is order k h / L. When the wind slackens, the layer adjacent to the grond thickens where moleclar heat condction dominates. A strong gst extends trblent mixing closer to the srface, casing the moleclar layer to shrink. To circmvent the problem of specifying the dynamic response of z h and T z h ) to gsts, I will assme that the variance of the locally averaged temperatre flctations, T G where TG = T L T ), is zero at z = z r,asis consistent with observations. Then we can simply apply eqation 1) with z h = z r and with T z r ) fixed to the global vale. Reslts derived from this approach mst be viewed as speclative; fortnately only a few of the calclations specifically the temperatre gradient and variance) in this report depend on it. Sppose we apply the methodology in the previos section to determine statistics in the atmospheric srface layer. Sch statistics wold generally 9

17 depend on the parameters U r, Q s, g/t s, z, z, and z i. Other parameters, sch as L and w, can be written as combinations of this set.) It wold be rather clmsy, thogh, to systematically stdy the dependence of the statistics on all five of these parameters. Therefore we might ask whether the parameter dependence can be simplified in some fashion. In this section I show that it is possible to redce the dependence to jst three basic ratios involving the five original parameters, specifically U r /w, z/z i,andz /z i. The z/z i dependence drops ot if z is not part of the original parameter set. As will become apparent in section 3, most wind and temperatre statistics of interest are proportional to ensemble averages involving powers of the local friction velocity, along with cosines and sines of the local wind direction. Therefore we are faced with the general problem of converting a fnction whose explicit dependence is on L, cos θ, andsin θ, to one with dependence on U r, Q s, etc, so that eqation 9) can be applied. To start, consider eqation 5) evalated at z = z r and recast in the following form: U L = 1 L h M κz rw 3 w κ w z i 3, κz w 3 ) L z i 3. 7) L This version ses the relationship L L = z i /κ) L /w ) 3. Since the vale.1 z i is sed for z r, the right-hand side is a fnction entirely of L /w and z /z i. If I assme that h M is an invertible fnction, then some fnction b mst exist sch that L w UL = b, z ) w z i = b Ur + G ) x + w w Frthermore, from basic trigonometric argments, Gy w ), z. 8) z i cos θ = sin θ = U r + G x, and 9) U r + G x ) + G y G y U r + G x ) + G y. 3) With h M given by eqations 19) and 4), analytical determination of the fnction b is impossible. The important point here, thogh, is the existence of the fnction b. Now, sppose we wish to know the average vale of some fnction f L /w,θ). Using eqations 8), 9), and 3) to determine the dependence of f on U r /w, G x /w, G y /w, and z /z i reslts in the probability integral ) L f,θ = f b w arctan Gy U r + G x ), z, Ur + G ) x Gy + w w w z i )] p G G x,g y ) dg x dg y. 31) 1

18 Transforming the variables of integration to the dimensionless ratios G x = G x /w and G y = G y /w, and assming that the joint pdf depends only on these ratios as does the Gassian pdf in the previos section, eq 8)), I have ) ) L f,θ = f b Ur + G w w x + G y, z, z i )] Gy arctan p G G U r + G x,g y) dg x dg y. 3) x The fnction f now depends entirely on U r /w, z /z i, G x,andg y. The dependence of f on G x and G y is of corse eliminated by the integration, leaving ) L Ur f,θ = g, z ), 33) w w z i where g is the reslt of the integration. This theorem demonstrates that statistics involving L /w and/or θ depend only on U r /w and z /z i, althogh in practice it may be impossible to actally obtain an analytical expression for the fnction g U r /w,z /z i ). An interesting special case of the preceding theorem that applies to the average friction velocity in the limit of vanishing mean wind was proposed by Bsinger 1973b). He sggested that L w = g B zi z ), 34) where g B is an nknown fnction. The validity of this scaling relationship was later spported by Schmann s 1988) analysis based on a simplified blk model for the atmospheric srface layer. Eqation 33) implies the same scaling relationship, since the dependence on U r /w disappears in the limit of vanishing U r. In this report, the procedre for deriving this relationship was mch different from Schmann s, however, being based instead on the statistical natre of the gsts. Frthermore, it is now clear that any statistic of L and/or θ depends solely on z /z i when the mean wind is zero. Several valable corollaries of eqation 33) can be easily derived. First, let s consider statistics of qantities that additionally depend on z/l L.Since z/l L z/z i )w / L ) 3 and the ratio w / L has already been acconted for, a dependence on z/z i is introdced: f L w, z L L,θ ) = g Ur, z, z ) w z i z i. 35) A local dependence on z /L L wold not add new dependencies to the reslt, since z /L L z /z i )w / L ) 3. Next consider statistics that depend on T L = Q s / L, as is sally the case for fnctions involving temperatre. Defining a mixed-layer temperatre scale as θ = Q s /w, we can 11

19 write T L /θ = w / L. Again, no new dependencies are added to the reslt. Hence we have, most generally, L f, T L z,, z ) Ur,θ = g, z, z ). 36) w θ L L L L w z i z i An important application of eqation 35) made later in this report is to statistics of the height-dependent gst profiles G x z) and G y z). From eqations ) and 4), we find G x z) w G y z) w.4 Generalized Similarity Theory = 1 L κ z cos θh M, κz w 3 w L L z i 3 L z, κz w 3 ) L L z i 3 L = 1 L sin θh M κ w ) U z) w, and 37). 38) The terms in eqations 37) and 38) involving the fnction h M depend on L /w, z/l L, and z /z i. Therefore, eqation 35) shows that statistics of these terms depend on U r /w, z/z i, and z /z i. Taking the global average of eqation ) yields U z) = 1 L z cos θh M, κz w 3 ) w κ w L L z i 3. 39) L We see that this term also depends on U r /w, z/z i, and z /z i. Therefore Gx z) f, G ) y z) Ur = g, z, z ). 4) w w w z i z i Althogh it is not immediately obvios, the nondimensionalization theorems in the previos section imply modifications to the Monin-Obkhov and other similarity theories for the srface layer. Before discssing these modifications, I will review several of the common similarity theories: Monin-Obkhov similarity Monin and Obkhov, 1954): The basic physical parameters hypothesized to be significant are, Q s, b = g/t s, and z. This parameter set possesses only three fndamental physical dimensions: length, time, and temperatre. According to Bckingham s π theorem see, for example, Stll, 1988), we shold therefore select jst three of these parameters as key. The three that are conventionally selected are, Q s, and z. Qantities scaled made dimensionless) by these parameters mst be fnctions of a single dimensionless ratio. This dimensionless ratio is taken to be z/l, where L = 3 T s / κgq s ) is the Obkhov length. The mins sign and constant κ in L are also matters of convention.) Local free-convection similarity Wyngaard et al, 1971): The basic physical parameters are the same as Monin-Obkhov similarity, except that is dropped from the set. The remaining three parameters incorporate the 1

20 three physical dimensions of length, time, and temperatre. As no dimensionless combinations can be formed from these parameters, qantities scaled by them mst be constant. For example, qantities with dimensions of velocity, when divided by the derived scale f =Q s zg/t s ) 1/3,are constant. Mixed-layer similarity Deardorff, 197): This scheme is the same as freeconvection similarity, except that z i replaces z. The velocity scale w = Q s z i g/t s ) 1/3 therefore replaces f. All three of the preceding similarity theories have been applied to srface-layer statistics. Monin-Obkhov similarity is widely applied to the mean profiles for both mechanical and convective trblence. Local freeconvection similarity is applied to the temperatre and vertical velocity variance in convective conditions. Mixed-layer similarity is applied to the horizontal velocity variance in convective conditions. Sppose now that the parameters from all three of these schemes are combined, reslting in the five-parameter set, Q s, b, z, and z i. Three of these parameters or combinations thereof) can be taken as key. Qantities scaled by the three key parameters mst depend on two dimensionless ratios. Nmeros combinations of key parameters and dimensionless ratios are possible. To generalize Monin-Obkhov similarity, we wold logically take, Q s, and z as the key parameters, and the two dimensionless ratios as z/l and z i /L. Alternatively, we cold formlate a generalized mixed-layer scheme with Q s, b, and z i as the key parameters. For the dimensionless ratios, the only restriction is that and z mst be niqely recoverable. So we may take these ratios to be /w and z/z i, for example. The roghness length z is not inclded in any of the three similarity theories described earlier. Omission of z can be jstified on the basis that the strctre of a high Reynolds nmber flow shold be independent of roghness at heights z z Tennekes and Lmley, 197, p 146). However, this reasoning overlooks possible interactions between large-scale eddies and the srface, a phenomenon that plays a central role in the model described in this report. Therefore z shold generally be inclded in the parameter set. Forming a six-parameter set by adding z to the five discssed in the preceding paragraph, we conclde that scaled qantities mst depend on three dimensionless ratios. For the generalized mixed-layer scheme, these can be taken to be /w, z/z i, and z /z i. The nondimensionalization theorems in section.3 are closely analogos to this generalized roghness-inclsive mixed-layer scheme. The main difference is that mean wind speed U r appears in place of. Sch a scaling cold have been arged by choosing U r, Q s, b, z, z, and z i as the initial six-parameter set, as long as the reference height does not introdce a new scale i.e., the reference height mst be proportional to either z or z i ). The reason for preferring U r over in the present model is that it simplifies the formlation of the pdf for the wind velocity at the reference height. We now see that the nondimensionalization theorems are consistent with, and 13

21 14 cold have been anticipated on the basis of, a generalized mixed-layer similarity theory that incorporates U r, z, and z, in addition to the standard parameters Q s, z i, and b. The model in section. provides a framework for calclating the similarity fnctions.

22 3. Effect of Gsts on Atmospheric Statistics 3.1 Friction Velocity In this section, I apply the framework for calclating gst effects on atmospheric statistics. The calclations involve the nmerical integration of eqation 9) with p G G x,g y ) given by eqation 8), and L determined from eqation 5). The nmerical integrations were done on a sqare grid defined by 1w G x,g y ) 1w. The grid had 1 evenly spaced points in both the x- and y-directions. The calclations reqire only a few seconds each on a desktop personal compter. Let s begin by applying the methodology developed in section. to the friction velocity. The familiar global version of the friction velocity is defined throgh the eqation = w. 41) In this eqation, is the flctation in the horizontal wind component aligned with the mean wind and w is the flctation in the vertical component. Similarly, the local friction velocity is defined as L = w L, 4) where the prime indicates the coordinate system aligned with the local wind direction θ. By elementary trigonometry, = cos θ v sin θ, 43) where v is the horizontal crosswind component of the wind. Mltiplying throgh by w and taking the local average, one finds w L = w L cos θ + v w sin θ. 44) L When I sbstitte eqation 4) and set v w L =a reslt of local symmetry in the trblence statistics), this becomes w L = L cos θ. 45) Now, by taking the global average and recalling that the global average of a local average eqals the cstomary global average see sect..1), one obtains the reslt = L cos θ. 46) We see that the global friction velocity is not the simple average of the local one. The changing wind direction plays a role throgh the factor cos θ. Ifθ 15

23 is randomly distribted in all directions, the cos θ factor has zero mean, and vanishes. The global friction velocity is compared to the averaged local vale, L, in figre 3. For wind speeds U r larger than w, the global and averaged local vales are nearly the same. As the wind speed approaches zero, however, the global vale vanishes, whereas the local average becomes constant. This asymptote is the minimm friction velocity recognized by Bsinger 1973a,b), Schmann 1988), and others. Also shown on the figre are approximate reslts for L based on the average local wind speed at the reference height, U L, as given by eqation 1). I fond these reslts by solving eqation 5) for L, except that I replaced the local wind speed by U L. The reslting vale of L will be designated as,eff in the remainder of this report; that is, U L = ),eff κ h zr z M,, 47) L eff where L eff =,eff T s / κgq s ).Thefigre demonstrates that,eff is a very good approximation to the actal vale of L for all reasonable combinations of U r /w and z /z i. This observation lends spport to the analyses of Godfrey and Beljaars 1991) and Grachev et al 1998) who se an effective vale for based on the mean wind speed. Let s analyze now in more detail the vanishing mean wind speed, sing approximate forms for φ M ζ) that will allow s to obtain an analytical soltion. Simple approximations for φ M ζ) are available for the two limiting cases of netral and free-convection stability. It may seem obvios, at first glance, that we shold se the free-convection stability limit, namely φ M ζ) a M ζ) 1/3 Carl et al, 1973; Kader and Yaglom, 199). However, we shold keep in mind that even when the mean wind vanishes, there L eff Figre 3. Global solid lines) and average local dashed lines) friction velocity as a fnction of vector-average wind speed at reference height, U r. Dotted lines are an approximation for average local friction velocity based on mean local wind speed, U L. Normalized friction velocity, * /w * z /z i = 1 3 z /z i = z /z i = Normalized wind speed, 16

24 is still local shear from the gsts. Sykes et al 1993) actally based their analysis on locally netral profiles. So, to be carefl, we shold calclate the netral and free-convection stability limits for the local wind profiles separately and then compare the reslts to nmerical calclations made with the fll Monin-Obkhov similarity eqation for φ M ζ). For the netral limit, φ M ζ) =1, and one has app B, eq B-6)) L w = πβκ lnz r /z ). 48) Replacing the constants with their nmerical vales reslts in ) L =.34 ln 1 zi. 49) w 1z For the free convection limit app B, eq B-34)), [ L = Γ5/4) β1/ κ /3 a 1/6 ) 1/3 ) ] 1/3 1/ M zi zi. 5) w 3 z z r Replacing the constants with their nmerical vales, [ ) ] L 1/3 1/ zi = ) w z Generally, z i is mch larger than z, and we may neglect the second term in the sqare brackets, leaving L w =.4 zi z ) 1/6. 5) This reslt is very similar to an approximation derived by Schmann 1988) his eq 37)), the only difference being the nmerical coefficient, which Schmann determined as.5. It is reassring that the approach in this report, based on a statistical model for the gsts, yields nearly the same reslts as Schmann s model, which is based on an entirely different approach involving blk approximations. Figre 4 compares a nmerical evalation of the average friction velocity, based on local application of the fll Monin-Obkhov eqation 15) for φ M ζ), to the two approximate analytical soltions, eqations 49) and 51). Also shown is,eff from eqation 47), which is observed to be an excellent proxy for L throghot the range of roghnesses. The approximation that the profiles nder the gsts locally obey free-convection similarity prodces better agreement with the fll local Monin-Obkhov evalation than does netral similarity. For nrealistically small roghness lengths not shown on the figre), the netral similarity crve does eventally become the better approximation. We conclde that in conditions of vanishing wind speed, the local profiles more closely resemble their freeconvection forms than their netral ones. 17

25 Figre 4. Local average friction velocity as a fnction of roghness length for vanishing mean wind. Three crves are based on different methods for calclating local wind profiles nder large eddies. Average local friction velocity, * L /w * Local Monin-Obkhov similarity *,eff/w * Local free-convection similarity Local netral similarity Normalized roghness length, z /z i 3. Mean Profiles Becase Monin-Obkhov similarity is widely sed to model the mean vertical profiles of wind and temperatre, an important practical matter is whether the local-eqilibrim model implies any significant, systematic violations of global Monin-Obkhov similarity. More precisely, when the mean gradients are normalized by and T as in eqations 13) and 14), are they still niversal fnctions of z/l? We might frthermore ask, even if the global application of Monin-Obkhov similarity is tenos, can empirically fitted vales for and T be devised that still prodce good agreement with atmospheric data? Starting with the first qestion, let s write down eqations for the mean wind and temperatre gradients in the context of local eqilibrim. Applying eqations 13) and 14) locally and then taking the global average, yields d U dz d T dz = 1 κz = P tq s κz L cos θφ M z 1 L φ H z L L L L ) ), and 53). 54) Next, when we nondimensionalize by mltiplying with κz/ and κz /P t Q s, we have ) κz d U z = L cos θφ M L cos θ 1/, and 55) dz L L κz ) d T z = 1 L P t Q s dz φ H L cos θ 1/. 56) L L Plots of the nondimensional wind and temperatre gradients calclated from these eqations are shown in figres 5 and 6. For different vales 18

26 Figre 5. Nondimensional wind gradient zκ/ )d U /dz) as a fnction of ζ = z/l. Nondimensional wind gradient =.3 = 3 = 1 = Nondimensional height, z/l Figre 6. Nondimensional temperatre gradient zκ /P t Q s ) d T /dz) as a fnction of ζ = z/l. Nondimensional temperatre gradient =.3 = 1 = 3 = Nondimensional height, z/l of the ratio U r /w are shown:.3, 1, 3, and 1. For a srface roghness z /z i =1 4, these for vales correspond to z i /L = κ w / ) 3 = 1, 19, 6, and 1.6. The first vale is extremely convective and probably rare for the atmosphere. The second represents a fair smmer day over land. For the global application of Monin-Obkhov similarity to be valid, the crves in figres 5 and 6 can depend only on z/l; that is, for all vales of U r /w, they mst collapse onto a single crve. Clearly this does not happen. Sbstantial deviations from global Monin-Obkhov similarity are fond when U r /w 1 for the wind gradient and U r /w.3 for the temperatre gradient. The dependence of the nondimensional gradients on U r /w 19

27 may explain some of the scatter in measrements sch as those made by Bsinger et al 1971) and by Högström 1988). The data shold be reexamined for dependence on the inversion height. Besides U r /w, the local-eqilibrim theory also predicts a dependence of the nondimensional gradients on z /z i. By performing nmerical evalations, I fond this effect to be qite insignificant, being most discernable when U r /w 1. Next we trn to the qestion of whether we can still fit eqations 16) and 17) to srface-layer data with good reslts, even when there is a breakdown in the global application of Monin-Obkhov similarity. This amonts to selecting vales for and T, possibly different from their actal vales, to fit the profiles at some set of sensor heights. Then we assess whether these fitted vales prodce good agreement with the actal overall profiles. To gain some insight into how the fitted vales for and T can differ from their actal vales, consider the eqations for the ensemble mean profiles derived from the local-eqilibrim theory. The wind profile follows directly from eqation ): U z) = 1 z L cos θh M, z ). 57) κ L L L L The mean temperatre profile follows by averaging eqation 1), with T z) = T z) L : T z) = T z h )+ P t κ z T L h H, z ) h L L L L. 58) Becase h M and h H are complicated fnctions of L, it is not possible to analytically derive a general eqation for effective vales of and T. For the idealized sitation where netrally stable conditions prevail locally, h M and h H are logarithmic, and the two preceding eqations become U z) = 1 κ ln z z L cos θ, and 59) T z) = T z h ) P tq s κ ln z z h 1 L. 6) Since the mean profile in netral conditions, according to the globaleqilibrim theory, is U z) = /κ)lnz/z ), the first of these eqations implies simply that L cos θ plays the role of an effective vale for. If the gsts are relatively strong and randomly oriented, L cos θ can be mch smaller than L. Similarly, Q s 1 L plays the role of an effective temperatre scale in place of T = Q s / in the global theory. Trning now to the other limiting approximation, where free convection prevails locally, one can show app B) that U z) L cos θ. Since the global-eqilibrim model predicts U z), and = L cos θ eq 46)), the effective friction velocity in free convection is the same as the

28 actal friction velocity. In regards to the temperatre profile, it becomes independent of in free convection see, for example Kader and Yaglom, 199)). Therefore in this extreme, there is no local variability in the temperatre profiles, and T = Q s / can be sed as an effective temperatre scale. To elcidate these ideas, let s consider an example atmospheric bondary layer with w = 3 m/s, z =1 5 m, and z i = 1 m Q s =.81 Ks/m). The wind speed U r is varied. For each vale of U r, fitted vales of and T are determined by matching eqations 13) and 14) to the wind speed and temperatre at 1-m height. Figre 7 shows the reslting ratio of the fitted vale of,fit ) to the actal vale. For large U r /w, the fitted friction velocity is the same as the actal vale. As U r /w becomes less than abot 1,,fit becomes less than the actal vale, as wold be anticipated from the preceding discssion. Srprisingly, as U r /w, the ratio approaches a constant vale smaller than one. Based on the preceeding discssion, the asymptote shold eqal one. The reason behind this apparent discrepency is that the free-convection limit is never trly attained. The gsts prevent the trblence from being in sch a state locally. Figres 8 and 9 illstrate agreement between wind and temperatre profiles from the local-eqilibrim model to those based on fitted vales for and T. Plotted are the ratios [U fit z) U z) ] / U z) and [T fit z) T z) ] / T z) for z = m and z = 1 m, where U fit z) and T fit z) are the profiles calclated by fitting and T to measrements at z = 1 m. Also shown are [U glob z) U z) ] / U z) and [T glob z) T z) ] / T z) for z = m and z = 1 m, where U glob z) and T glob z) are the profiles calclated from the global-eqilibrim model. The profiles calclated from the fitted vales are always in good agreement relative error less than abot 4%) with the local-eqilibrim model, even Figre 7. Ratio of friction velocity determined by fitting measrements at 1-m height to actal vale determined from local-eqilibrim model. Fitted friction velocity, *,fit/ * Mean wind speed, 1

29 Figre 8. Relative error in %) between wind speed profile in local-eqilibrim model andaprofile determined by fitting and T to measrements at 1-m height. Errors are shown for - and 1-m heights. Also shown are relative errors for global Monin-Obkhov similarity based on actal vales of and T. Wind speed discrepancy %) Global fit, z = m Global similarity, z = m Global similarity, z = 1 m Global fit, z = 1 m Mean wind speed, Figre 9. Same as figre 8, except for temperatre profile. 1.8 Global fit, z = m Temperatre discrepancy %) Global similarity, z = m Global fit, z = 1 m Global similarity, z = 1 m Mean wind speed, for very small U r /w. The global-eqilibrim profiles for the wind speed are in poor agreement with the local-eqilibrim model, althogh agreement for the temperatre profiles is qite good. The reason for this behavior was mentioned earlier: the local temperatre profiles are independent of in highly convective conditions, and therefore the global and local models match in this case. In interpreting the reslts for the wind profiles, one shold keep in mind that vales for U r /w < 1 are nsal for the atmosphere. Even for U r /w = 1, thogh, the global-eqilibrim model can

30 3.3 Temperatre Variance be off by more than 1 percent. Application of global similarity theory to the wind profiles in highly convective atmospheric conditions is therefore very tenos, althogh when fitted vales for and T are sed in place of the actal ones, the theory can be made to work fairly well. In this section, I explore the implications of the local-eqilibrim model for the temperatre variance. The temperatre variance is defined as σt z) = [T z) T z) ]. 61) Previos stdies, based on a global similarity approach, have fond the srface-layer temperatre variance in netral conditions to be abot 4.T e.g., Stll s 1988) eqation 9.5.3k). In freely convective conditions, the previos stdies sggest σt =.9T z/l) /3 Kaimal et al, 1976; Caghey and Palmer, 1979). The following eqation smoothly interpolates between these two limits: σt [ = L) z ] /3 1. 6) T Note that the temperatre flctations are entirely small scale ; that is, since neither w nor z i appear in the previos eqation, the large-scale convective eddies have not previosly been observed to affect the temperatre variance. In the context of local eqilibrim, the large convective gsts can modify the temperatre variance in two ways. First, they can directly vary the temperatre throgh local adjstments to the vertical profile. Second, they can indirectly affect the temperatre variance by locally modlating σt throgh changes in L. Let s decompose the temperatre as follows: T = T + T G + t, 63) where T G is the change indced directly by the gsts and t represents the small-scale flctations obeying eqation 6), except with local scales replacing the global ones. The variance is therefore σt = T G + t) = TG +TG t + t. 64) Let s decompose t as t = σ T,L ξ, 65) where ξ is a random variable having zero mean and nit variance. The factor σ T,L scales the variance locally in response to the gsts. Having factored ot σ T,L, I can now consider the random variable ξ a process independent from the gsts. As a reslt, the direct and indirect contribtions separate from each other: σ T T G +TG σ T,L ξ + σ T,L ξ = T G + σ T,L. 66) 3