Comments on “A New Second-Order Turbulence Closure Scheme for The

3478 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 56

NOTES AND CORRESPONDENCE

Comments on ``A New Second-Order Turbulence Closure Scheme for the Planetary Boundary Layer''*

D. V. MIRONOV Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany

V. M . G RYANIK Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany, and A. M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, Russia

V. N . L YKOSSOV Institute for Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

S. S. ZILITINKEVICH Institute for Hydrophysics, GKSS Research Centre, Geesthacht, Germany

11 November 1997 and 30 March 1998

1. Introduction tional downgradient diffusion term and the term pro- portional to the gradient of mean potential temperature, Abdella and McFarlane (1997, henceforth AM97) 2 have proposed a second-order turbulence closure their expression for wЈ ␪Ј contains a nongradient ``advection'' term. That term is identical to the expression scheme for the planetary boundary layer. The scheme 2 contains a prognostic equation for the turbulence kinetic for wЈ ␪Ј obtained by AM97 [Eq. (3) below]. energy and algebraic expressions for the other second- In this note we show that the parameterization for the 2 order moments. The expressions for the potential tem- ¯ux of potential temperature variance, wЈ␪Ј , proposed perature ¯ux and the temperature variance incorporate by AM97 is inconsistent with the physical requirements nonlocal eddy diffusivity and countergradient terms. of symmetry. We then develop a parameterization that These expressions are derived through the use of an possesses necessary physical properties and verify it advanced parameterization of the third-order moments against large-eddy simulation (LES) and observational based on convective mass-¯ux arguments. Remarkably, data. the AM97 parameterizations for the ¯ux of potential temperature ¯ux, wЈ 2␪Ј, and the ¯ux of potential tem- 2. Abdella and McFarlane's parameterizations for perature variance, wЈ␪Ј 2 , do not have a traditional down- w؅ 2␪؅ and w؅␪؅ 2 gradient diffusion form. Simultaneously, Zilitinkevich et al. (1997, henceforth ZGLM; see also Zilitinkevich AM97 used the convective mass-¯ux concept (e.g., et al. 1999) proposed a ``turbulent advection ϩ diffusion Randal et al. 1992) to derive expressions for the ¯ux of 2 parameterization'' for wЈ ␪Ј. Apart from the conven- potential temperature ¯ux and the ¯ux of potential temperature variance. They considered an idealized convective circulation composed of rising branches (up- * Alfred Wegener Institute for Polar and Marine Research Contri- drafts) covering fractional area a, and sinking branches bution Number 1332. (downdrafts) covering fractional area (1 Ϫ a). The area mean X of a generic variable X is given by Corresponding author address: Dr. Dmitrii Mironov, Deutscher Wetterdienst, Abteilung Meteorologische Analyse und Modellierung, X ϭ aXu ϩ (1 Ϫ a)Xd, (1) Referat FE14, Postfach 10 04 65, Frankfurter Str. 135, D-63067, Offenbach am Main, Germany. where Xu and Xd are the mean values of X within the E-mail: [email protected] updrafts and downdrafts, respectively. The vertical tur-

᭧ 1999 American Meteorological Society

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FIG. 1. Vertical velocity skewness (left) and potential temperature skewness (right) vs dimensionless height z/h, where h is the CBL depth. Dotted curves represent LES data. Crosses are data from the water tank experiment of Deardorff and Willis (1985), ®lled circles are the wind tunnel data of Fedorovich et al. (1996), and open circles are data from measurements in the atmospheric CBL taken from Sorbjan (1991). bulent ¯ux of X due to convective circulation is given formulated in terms of probabilities of the upward and by downward motions rather than in terms of fractional areas covered by these motions), and (iii) the third- wЈXЈϭa(w Ϫ w )(X Ϫ X ) uu order-moment budget equations. ZGLM emphasized that Eq. (3) satis®es basic phys- ϩ (1 Ϫ a)(wddϪ w )(X Ϫ X ) ical requirements, namely, the requirements of sym- ϭ a(1 Ϫ a)(wududϪ w )(X Ϫ X ), (2) metry. A close look at Eq. (4) suggests that it does not satisfy these requirements. Indeed, if wЈ is replaced with where wu and wd are the mean values of the vertical velocity within the updrafts and downdrafts, respec- ϪwЈ, the lhs of Eq. (4) changes its sign, while the rhs of Eq. (4) does not. The invariance with respect to the tively. In a similar way, the variance of X, the ¯ux of → X, and the ¯ux of variance of X are expressed through transformation ␪Ј Ϫ␪Ј is also violated. a, Xu, Xd, wu, and wd. Combining the expressions for these second- and third-order moments, then replacing 3. Proposed parameterization for w؅␪؅2 X with potential temperature ␪, AM97 obtained the following parameterizations for the ¯ux of potential tem- We observe that if the temperature skewness S␪ ϵ perature ¯ux and the ¯ux of potential temperature var- ␪Ј3 /(␪Ј 2 )3/2 is used instead of the vertical velocity skew- iance: ness Sw on the rhs of Eq. (4), it would acquire the form that possesses all necessary physical properties. In the wЈ2␪ЈϭS (wЈ 2) 1/2 wЈ␪Ј, (3) w framework of the mass-¯ux model used by AM97, the 2 2 1/2 wЈ␪ЈϭSw(␪Ј ) wЈ␪Ј. (4) expression of S␪ in terms of the fractional area covered by updrafts, a, has exactly the same form as the second Here, Sw is the vertical velocity skewness, equality on the rhs of Eq. (5). This does not mean, wЈ3 1 Ϫ 2a however, that the two quantities can be used inter- Sw ϵϭ , (5) changeably since their physical nature is different. The (wЈ2)[ 3/2a(1 Ϫ a)] 1/2 fact that S␪ is numerically equal to Sw stems from the where the second equality on the rhs of Eq. (5) follows simpli®ed character of the mass-¯ux model. It disagrees from the mass-¯ux arguments. with the empirical evidence. The LES (the data used are As mentioned above, the rhs of Eq. (3) coincides with described below), water tank, wind tunnel, and atmo- the advection part of the ZGLM turbulent advection ϩ spheric data presented in Fig. 1 all show that vertical 2 diffusion parameterization for wЈ ␪Ј. Three sets of ar- pro®les of S␪ and Sw are not similar over most of the guments were used to derive that advection part, namely, convective boundary layer (CBL). The temperature (i) the tensor nature of the third-moment considered, (ii) skewness is larger than the vertical velocity skewness a bimodal bottom-up/top-down turbulence model (that in the lower and middle parts of the CBL. is similar to the mass-¯ux model used by AM97 but is In order to illustrate how the difference in magnitude

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between S␪ and Sw can be accounted for in the framework of the model of mid-CBL convective circulation, we introduce modi®cations into the earlier model of ZGLM and AM97. Following ZGLM, we formulate the modi®ed model in terms of probabilities rather than in terms of fractional areas. Without the loss of generality, the mean values of potential temperature and vertical velocity can be set to zero. We assume that convective circulation in mid-CBL involves warm updrafts, where

¯uctuations of potential temperature ␪w and vertical velocity wu occur with the probability Pwu; and cold downdrafts, where ¯uctuations ␪c and wd occur with the probability Pcd. We introduce cold updrafts, where vertical velocity ¯uctuations are wu and potential temperature ¯uctuations are ␪c, and the probability of these to occur is Pcu. The following equations specify the parameters Pwu, Pcd, Pcu, wu, wd, ␪w, and ␪c:

Pwuϩ P cdϩ P cu ϭ 1,

wuwu(P ϩ P cu) ϩ wP dcdϭ 0,

␪wP wuϩ ␪ c(P cdϩ P cu) ϭ 0, (6)

222 wuwu(P ϩ P cu) ϩ wP dcdϭ wЈ , ␪ 22P ϩ ␪ (P ϩ P ) ϭ ␪Ј 2, (7) FIG. 2. Vertical pro®les of the ¯ux of potential temperature variance w wu c cd cu made dimensionless with the Deardorff velocity, w ϭ (,␤wЈ␪Ј h)1/3 * s 3323/2and temperature, ␪ ϭ wЈ␪sЈ /w , scales (where ␤ ϭ g/T0 is the buoy- wuwu(P ϩ P cu) ϩ wP dcdwϭ S (wЈ ), * * ancy parameter, g is the acceleration due to gravity, T0 is the reference value of the absolute temperature, andwЈ␪Ј is the surface potential ␪ 33P ϩ ␪ (P ϩ P ) ϭ S (␪Ј 23/2) . (8) s w wu c cd cu ␪ temperature ¯ux). Dotted curve represents LES data, solid curve is The above idealization of mid-CBL convective circu- computed from Eq. (9) with C␪ ϭ 1, dashed curve is computed from the AM97 parameterization [Eq. (4)], heavy dotted curve is the Len- lation implies that positive ¯uctuations of potential tem- schow et al. (1980) ®t to the AMTEX data, and heavy dashed curve perature, ␪w, are more localized than positive ¯uctua- is the surface-layer free-convection similarity prediction (after Wyn- tions of vertical velocity, wu. As the skewness is indic- gaard et al. 1971). ative of this localization, our modi®ed model gives S␪ Ϫ1/2 ϭ (1 Ϫ 2Pwu)[Pwu(1 Ϫ Pwu)] , which exceeds Sw ϭ Ϫ1/2 [1 Ϫ 2(Pwu ϩ Pcu)][(Pwu ϩ Pcu)(1 Ϫ Pwu Ϫ Pcu)] . inequality to the turbulence moments on the rhs and the Obviously, with Pcu ϭ 0 the earlier model of ZGLM lhs of Eq. (3) and then to express the forth-order mo- and AM97 is recovered. ments through the second-order moments using the qua- It can be easily veri®ed that Eq. (3) for wЈ2␪Ј is si-normal approximation. The same procedure applied consistent with Eqs. (6)±(8) and thus remains in force. to Eq. (9) yields C␪ Յ 1. The derivations are straight- Instead of Eq. (4), the following parameterization for forward. We do not present them here and refer to ap- the ¯ux of potential temperature variance holds: pendix A of ZGLM for details.

2 2 1/2 wЈ␪Ј ϭ C␪S␪(␪Ј ) wЈ␪Ј, (9) 4. Veri®cation of the proposed parameterization where C␪ is a dimensionless constant. The earlier ZGLM and AM97 model and the modi®ed model of mid-CBL In order to verify the proposed parameterization of convective circulation suggest the value of C␪ ϭ 1. We the ¯ux of the temperature variance [Eq. (9)], we use allow this constant to differ from 1 to have some free- large-eddy simulation data generated with the LES mod- dom in ®tting empirical data. We keep in mind that Eq. el of Moeng (1984). The data are from the simulation (9) is not an identity but an approximation based on an of the surface-heating-driven CBL with strong temper- idealized view of convective circulation. It should be ature inversion at its top and no mean wind. A detailed emphasized that the value of C␪ ϭ 1 is the upper limit description of the simulation is given in ZGLM where set by the requirements of realizability. ZGLM consid- it is referred to as ``case I.'' ered realizability conditions for the advection part [Eq. In Fig. 2, the ¯ux of potential temperature variance (3) with a dimensionless coef®cient on the rhs similar computed from Eq. (9) with the maximum allowable to C␪ on the rhs of Eq. (9)] of their advection ϩ diffusion value of C␪ ϭ 1 is compared with LES data. As seen parameterization for the ¯ux of potential temperature from the ®gure, the proposed parameterization ®ts LES ¯ux, wЈ 2␪Ј. The procedure was to apply the Schwartz data well in the CBL interior. The two curves prove to

Unauthenticated | Downloaded 09/23/21 10:52 AM UTC 1OCTOBER 1999 NOTES AND CORRESPONDENCE 3481 be similar in shape in the inversion layer as well, though parameterization are that it does not have a traditional the proposed parameterization underestimates the ¯ux downgradient form and that it includes the potential of temperature variance there. It is not unexpected, how- temperature skewness. Likewise, the nongradient part ever, as the concept of the mid-CBL convective circu- of the parameterization for the ¯ux of potential tem- lation that stands behind Eq. (9) is hardly applicable to perature ¯ux [Eq. (3)] includes the vertical velocity the stably strati®ed inversion layer. An account should skewness. The physical meaning of both parameteri- be taken of the effects of gradients of mean temperature zations is illustrated by the model of mid-CBL convec- and temperature variance. tive circulation [Eqs. (6)±(8)]. 2 1/2 The AM97 parameterization [Eq. (4)] shown for com- The combination wa ϵ Sw(wЈ ) was termed by parison underestimates wЈ␪Ј 2 over the CBL interior. In ZGLM ``large-eddy skewed-turbulence velocity.'' By 2 1/2 order to obtain a satisfactory ®t to LES data (see Fig. analogy, the combination ␪a ϵ S␪(␪Ј ) may be termed 16 of AM97), AM97 introduced an arbitrary scaling ``large-eddy skewed-turbulence temperature ¯uctua- factor into the rhs of Eq. (4). Notice further that the tion.'' The term re¯ects the most important feature that AM97 and the LES curves are different in shape in the Eq. (9) accounts for: the skewed nature of convective inversion layer. turbulence. Heavy dots in Fig. 2 represent the Lenschow et al. (1980) ®t to data from the Air-Mass Transformation Acknowledgments. This work was supported by the Experiment (AMTEX) observations in the baroclinic German Coordinating Of®ce of the World Ocean Cir- 2 2 3 culation Experiment (Grant 03F0157A); the German atmospheric CBL, wЈ␪Ј /w**␪ ϭ 3.1(1 Ϫ z/h) . Al- though the CBL was affected by the mean wind shear, Federal Ministry for Education, Research and Technol- the atmospheric curve appears to be fairly close to the ogy (Grant 03F08GUS); the Russian Fund for Funda- free-convection LES curve and to the proposed param- mental Investigations (Grant 95-05-14172); Project IN- eterization. TAS 96-1682; EC Contract JOR3-CT95-0008 of the The shape of the wЈ␪Ј 2 pro®le in the surface layer JOULE Programme (through the National Observatory merits consideration. Both dotted and solid curves in of Athens, Greece); and EU Project SFINCS-EC Con- Fig. 2, computed using LES data on turbulence mo- tract ENV4-CT97 0573. ments, tend to zero as the underlying surface is ap- proached. This contradicts the well-established surface REFERENCES 2 layer free-convection similarity prediction, wЈ␪Ј / Abdella, K., and N. McFarlane, 1997: A new second-order turbulence 2 1.06(z/h)Ϫ1/3, shown by the heavy dashed curve w**␪ ϭ closure scheme for the planetary boundary layer. J. Atmos. Sci., in Fig. 2. The empirical coef®cient 1.06 is taken from 54, 1850±1867. Wyngaard et al. (1971). The disagreement re¯ects the Deardorff, J. W., and G. E. Willis, 1985: Further results from a lab- oratory model of the convective planetary boundary layer. de®ciency of LES in the surface layer. The subgrid clo- Bound.-Layer Meteor., 32, 205±236. sure model used in the LES code provides subgrid-scale Fedorovich, E., R. Kaiser, M. Rau, and E. Plate, 1996: Wind tunnel contributions only to the second-order turbulence mo- study of turbulent ¯ow structure in the convective boundary layer ments. The third- and higher-order moments are com- capped by a temperature inversion. J. Atmos. Sci., 53, 1273± puted from the resolved scale ®elds. Since the small- 1289. Lenschow, D. H., J. C. Wyngaard, and W. T. 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