Comments on “A New Second-Order Turbulence Closure Scheme for The
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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 w9 u9 contains a nongradient ``ad- vection'' 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 w9 u9 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, w9u9 , 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, w9 2u9, and the ¯ux of potential tem- 2. Abdella and McFarlane's parameterizations for perature variance, w9u9 2 , do not have a traditional down- w9 2u9 and w9u9 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 1 diffusion Randal et al. 1992) to derive expressions for the ¯ux of 2 parameterization'' for w9 u9. Apart from the conven- potential temperature ¯ux and the ¯ux of potential tem- perature variance. They considered an idealized con- vective 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 2 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 5 aXu 1 (1 2 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- q 1999 American Meteorological Society Unauthenticated | Downloaded 09/23/21 10:52 AM UTC 1OCTOBER 1999 NOTES AND CORRESPONDENCE 3479 FIG. 1. Vertical velocity skewness (left) and potential temperature skewness (right) vs dimen- sionless 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- w9X95a(w 2 w )(X 2 X ) uu order-moment budget equations. ZGLM emphasized that Eq. (3) satis®es basic phys- 1 (1 2 a)(wdd2 w )(X 2 X ) ical requirements, namely, the requirements of sym- 5 a(1 2 a)(wudud2 w )(X 2 X ), (2) metry. A close look at Eq. (4) suggests that it does not satisfy these requirements. Indeed, if w9 is replaced with where wu and wd are the mean values of the vertical velocity within the updrafts and downdrafts, respec- 2w9, 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 u9 2u9 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 w9u92 X with potential temperature u, AM97 obtained the fol- lowing parameterizations for the ¯ux of potential tem- We observe that if the temperature skewness Su [ perature ¯ux and the ¯ux of potential temperature var- u93 /(u9 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 w92u95S (w9 2) 1/2 w9u9, (3) w framework of the mass-¯ux model used by AM97, the 2 2 1/2 w9u95Sw(u9 ) w9u9. (4) expression of Su 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, w93 1 2 2a however, that the two quantities can be used inter- Sw [5 , (5) changeably since their physical nature is different. The (w92)[ 3/2a(1 2 a)] 1/2 fact that Su 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 1 spheric data presented in Fig. 1 all show that vertical 2 diffusion parameterization for w9 u9. Three sets of ar- pro®les of Su 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 Unauthenticated | Downloaded 09/23/21 10:52 AM UTC 3480 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 56 between Su and Sw can be accounted for in the frame- work 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 uw and vertical ve- locity wu occur with the probability Pwu; and cold down- drafts, where ¯uctuations uc and wd occur with the prob- ability Pcd. We introduce cold updrafts, where vertical velocity ¯uctuations are wu and potential temperature ¯uctuations are uc, and the probability of these to occur is Pcu. The following equations specify the parameters Pwu, Pcd, Pcu, wu, wd, uw, and uc: Pwu1 P cd1 P cu 5 1, wuwu(P 1 P cu) 1 wP dcd5 0, uwP wu1 u c(P cd1 P cu) 5 0, (6) 222 wuwu(P 1 P cu) 1 wP dcd5 w9 , u 22P 1 u (P 1 P ) 5 u9 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 5 (,bw9u9 h)1/3 * s 3323/2and temperature, u 5 w9us9 /w , scales (where b 5 g/T0 is the buoy- wuwu(P 1 P cu) 1 wP dcdw5 S (w9 ), * * ancy parameter, g is the acceleration due to gravity, T0 is the reference value of the absolute temperature, andw9u9 is the surface potential u 33P 1 u (P 1 P ) 5 S (u9 23/2) . (8) s w wu c cd cu u temperature ¯ux). Dotted curve represents LES data, solid curve is The above idealization of mid-CBL convective circu- computed from Eq. (9) with Cu 5 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, uw, 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 Su 21/2 5 (1 2 2Pwu)[Pwu(1 2 Pwu)] , which exceeds Sw 5 21/2 [1 2 2(Pwu 1 Pcu)][(Pwu 1 Pcu)(1 2 Pwu 2 Pcu)] . inequality to the turbulence moments on the rhs and the Obviously, with Pcu 5 0 the earlier model of ZGLM lhs of Eq. (3) and then to express the forth-order mo- and AM97 is recovered.