Errors caused by draft fraction in cumulus parameterization

GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L17802, doi:10.1029/2009gl039100, 2009 Errors caused by draft fraction in cumulus parameterization Akihiko Murata 1 Received 24 May 2009; revised 16 July 2009; accepted 3 August 2009; published 1 September 2009. [1] The assumption that the draft fraction is negligible in cumulus parameterization can be a major source of error in the parameterized tendencies of thermodynamic quantities. This study estimates the errors arising from this assumption based on an analysis of high-resolution cloud-resolving model output. The numerical results reveal a negative error associated with the assumption, indicating that the magnitude of the mean subgrid-tendency of moist static energy is underestimated when the assumption is employed. The magnitude of the error is proportional to the draft fraction, and is independent of horizontal resolution. The results are confirmed by theoretical analyses, which support a linear relationship between error magnitude and draft fraction, and confirm that the error magnitude is independent of horizontal resolution. The resulting underestimates of moist static energy tendency should not be neglected in the case that the horizontal resolution is sufficiently fine to enable the frequent occurrence of large draft fractions. Citation: Murata, A. (2009), Errors caused by draft fraction in cumulus parameterization, Geophys. Res. Lett., 36, L17802, doi:10.1029/2009gl039100. 1. Introduction [2] Convective clouds are often assumed to occupy only a small fraction of the grid domain of a numerical model, that is, the fraction of the area covered by convective clouds, referred to as the draft fraction, is assumed to be negligible. When this assumption is employed, the effects of the draft fraction on the parameterized tendencies of thermodynamic quantities (e.g., temperature and specific humidity) are also neglected. The assumption is valid when the grid domain is much larger than the horizontal scale of cumulus, but it is questionable at finer resolutions. Nevertheless, cumulus parameterizations that employ this assumption have been used in numerical models with resolutions finer than the horizontal scale of cumulus; consequently, it is necessary to estimate the magnitude of the errors arising from the assumption. [3] Several studies have investigated issues related to the draft fraction in cumulus parameterizations. Kurihara [2002] formulated a cumulus parameterization including the draft fraction. In this approach, the conditions of cloud and clear areas within a grid domain are split into an explicit component and the deviation from this component, which distinguishes cloud and clear areas. Gerard and Geleyn [2005] proposed a sophisticated cumulus scheme that included the effects of the draft fraction; a prognostic scheme was used to determine the 1 Typhoon Research Department, Meteorological Research Institute, Tsukuba, Japan. Copyright 2009 by the American Geophysical Union. 0094-8276/09/2009GL039100 draft fraction. The explicit calculation of the draft fraction allowed the authors to take into consideration the effects of the draft fraction on the tendency of static energy and specific humidity due to convection. [4] Cloud-resolving models (CRMs) are useful in evaluating and developing cumulus parameterization schemes. Moncrieff et al. [1997] reported that a CRM was able to determine the collective effects of subgrid-scale processes on the large-scale field and that bulk properties were readily calculated from CRM outputs. CRMs have recently been used to evaluate and develop physically-based cumulus parameterizations [e.g., Liu et al., 2001; Kuang and Bretherton, 2006]. [5] It is recognized that it is necessary to take into account the draft fraction in cumulus parameterizations to improve the accuracy of calculations of the heat budget. However, it appears that no previous study has estimated the error arising from neglecting the draft fraction in cumulus parameterizations. This study estimates the effect of the draft fraction in cumulus parameterizations on the magnitude of errors in moist static energy tendency, using data obtained from a high-resolution cloud-resolving numerical simulation. To the best of my knowledge, this is the first study to examine these errors. 2. Numerical Analysis [6] For a numerical model, the Japan Meteorological Agency Nonhydrostatic Model (JMANHM) [Saito et al., 2006] is employed, which has fully compressible Euler equations and employs a semi-implicit time integration scheme. The model includes bulk cloud microphysics developed by Ikawa et al. [1991]. The scheme predicts the mixing ratios of six water species (water vapor, cloud water, rain, cloud ice, snow, and graupel) and the number concentrations of cloud ice, snow, and graupel. The size distributions of the water substances are assumed to be inverse exponential for rain, snow, and graupel, and monodisperse for cloud water and cloud ice. The treatment is based on work by Lin et al. [1983], Murakami [1990], and Murakami et al. [1994]. A box-lagrangian rain-drop scheme [Kato, 1995] is incorporated for calculating the fallout of rain and graupel. [7] JMANHM with a horizontal grid spacing of 200 m is used as a CRM for numerical simulations of cumulus convection that occurred in a tropical region of the western North Pacific (around 7.5 N, 169.5 E) on 28 August 2004. A grid-nesting strategy is adopted for the initial and lateral boundary conditions. The vertical coordinate of JMANHM is terrain-following and contains 76 levels. The vertical grid increment is 40 m at the surface, gradually increasing to 1480 m at the highest model level (29 km). The depth of the Rayleigh friction layer is 10 km. A type of Arakawa Schubert cumulus scheme [Murata and Ueno, 2005; L17802 1of6

Figure 1. CFFD of relative error e plotted against draft fraction a u + a d for horizontal grid spacings of (a) 40, (b) 20, (c) 10, and (d) 5 km. Arakawa and Schubert, 1974] is included in the outermost JMANHM in coupling with a bulk cloud microphysical scheme. [8] In the analyzing the results obtained from cloudresolving simulations, a new method was developed that consists of two parts, for extracting areas of a convective core and its surroundings (i.e., the non-core cumulus area). The method employed in detecting the core is similar to that proposed by Xu [1995]. [9] For detecting convective cores, the present method is based on the horizontal distribution of the maximum cloud updraft strength, w x (x, y), in x and y (2-dimensional; horizontal) coordinates, below the melting level, as in the work by Xu [1995]. A grid column of the convective core satisfies one of the two following conditions: (1) w x (x, y) > 2 w xa (x, y), where w xa (x, y) is the average of w x (x, y) over the surrounding 24 grid columns, or (2) w x (x, y) >w xth, where w xth =3.0ms 1. [10] For detecting the surrounding area, grid points in each vertical level are considered. If a grid point adjacent to a cumulus grid point (i.e., a grid point in the core or surroundings of a cumulus area) satisfies the following condition, it is assumed to be included in the same cumulus area: (3) q c (x, y, z) +q i (x, y, z) >q th, where q c (x, y, z) and q i (x, y, z) are the mixing ratios of cloud water and cloud ice, respectively, in x, y, and z (3-dimensional) coordinates, and where q th = 0.1 g kg 1. It should be noted that the core grid points that do not satisfy condition (3) are excluded from the cumulus area. [11] Grid points within the cumulus area, which consists of the core and the surrounding grid points mentioned above, are classified into two groups (cumulus updraft and cumulus downdraft) according to vertical velocity, w(x, y, z). The area of cumulus updraft (downdraft) is defined as the area consisting of the grid points that satisfy w >0ms 1 (w <0ms 1 ). The remaining area (i.e., neither the area of cumulus updraft nor the area of cumulus downdraft) is referred to as the environmental area. It should be noted that these three areas (i.e., updraft u, downdraft d, and environment e) are vertically variable (dependent on height). [12] Next, an area equivalent to a coarse grid (e.g., a square with sides of 20 km in length) is set. Using the criteria mentioned above, the area, which in this example consists of 100 100 grid points with a grid spacing of 200 m, is divided into the three sub-areas (i.e., u, d, and e). The occupancy rate of each sub-area (a u (z), a d (z), and a e (z), respectively) is calculated against the total area (i.e., the square with sides of 20 km in this example), where a denotes the fractional cloud coverage. Sub-area-averaged quantities of moist static energy (i.e., h u (z), h d (z), and h e (z)) and vertical velocity (i.e., w u (z), w d (z), and w e (z)) are also calculated. All these variables are vertically variable. 3. Results [13] An equation is derived to obtain the mean subgridscale tendency of moist static energy. Following Siebesma and Cuijpers [1995] and Siebesma and Holtslag [1996], the equation can be expressed as follows: @rw0 h 0 ¼ @M u h u h T 0 ; h d h @M d @M e h e h ð1þ M ¼ raðw wþ; ð2þ 2of6

Figure 2. As in Figure 1, but for another case. where r is average density; z is height; and M is mass flux defined in equation (2) Overbars in the equation indicate the spatial horizontal average over the area equivalent to the coarse grid described above. Primes denote deviations from the horizontal average. The first, second, and third terms on the right-hand side of equation (1) describe the contribution of average organized cumulus updrafts, cumulus downdrafts, and compensating subsidence in the environment, respectively. The term that describes the correlated fluctuations with respect to the cumulus updraft area is neglected; similar terms for the cumulus downdraft area and the environmental area are also omitted. This approximation was proposed by Tiedtke [1989]. [14] We now consider the condition of a negligible draft fraction, which is represented as In this case, a u þ a d ffi 0 or a e ffi 1: ð3þ h e ffi h: It should be noted that the updraft and downdraft fraction are considered together because, in cumulus parameterization, downdraft areas are included in clouds. [15] Applying this condition to equation (1) results in @rw0 h 0 ffi @M u h u h h d h @M d T 1 : The third term on the right-hand side of equation (1) vanishes in this case. If the sign of the term is changed from negative to positive, the term represents an error in the mean subgrid-tendency of moist static energy associated with the assumption of negligible draft fraction. ð4þ ð5þ [16] The relative error e is calculated as follows: e T 1 T 0 T 0 ¼ @M e h e h, @rw 0 h 0 ¼ @M e h e h : @rw 0 h 0 The results are shown in a diagram of relative error e plotted against draft fraction a c (=a u + a d ) for the areas of 40-, 20-, 10-, and 5-km square grids during a 5-h period at 30-min resolution (Figure 1). The square area is a proxy for a coarse grid domain where a cumulus parameterization is applied. The diagram, referred to as contoured frequency by fraction diagram (CFFD), is similar to the contoured frequency by altitude diagram (CFAD) introduced by Yuter and Houze [1995] but uses draft fraction instead of altitude. The ordinate of the CFFD is draft fraction, and the abscissa is the relative error. Histograms as a function of the relative error are constructed for each draft fraction using a bin size for relative error of 0.05. The histograms are then normalized to the number of data points in each draft fraction and are arranged in sequence along the axis of draft fraction (the bin size of the draft fraction is 0.01). Therefore, the shading in the CFFD represents the rate of points for each draft fraction, and summarizes the characteristics of the histograms for all draft fractions. [17] Figure 1 shows that the relative error is generally negative, demonstrating that the magnitude of the mean subgrid-tendency of moist static energy is underestimated when the draft fraction is not considered. The magnitude of the relative error increases with increasing draft fraction, and appears to be proportional to the draft fraction although the data show some scatter. The maximum magnitude of the relative error increases with increasing horizontal resolution, as the maximum draft fraction also increases with increasing horizontal resolution. However, the relationship ð6þ 3of6

Figure 3. CFFD of d, defined by equation (7), plotted against draft fraction a u + a d for horizontal grid spacings of (a) 40, (b) 20, (c) 10, and (d) 5 km. between the relative error and the draft fraction appears to be independent of the horizontal resolution; that is, the constants of proportionality for the different resolutions are nearly equal. [18] To assess the reliability of the obtained results, another simulation was conducted of cumulus convection that occurred in a tropical region of the western North Pacific (around 10 N, 140.5 E) on 27 June 2002. The CFFD for this case is shown in Figure 2. Figure 2, as well as Figure 1, demonstrates that the magnitude of the relative error increases with increasing draft fraction and appears to be proportional to the draft fraction. The results also reveal that the maximum magnitude of the relative error increases with increasing horizontal resolution. 4. Discussion [19] To examine the relationship between e and draft fraction a c (=a u + a d ), another quantity d is defined as follows: d M e h e h : ð7þ rw 0 h 0 This quantity is similar to e, but does not contain the vertical derivative. [20] The CFFD in Figure 3 shows the relationship between d and a c for the areas of 40-, 20-, 10-, and 5-km square grids. d is proportional to a c ; this relationship is clearer than that in Figure 1. The constant of proportionality appears to be independent of the horizontal resolution, similar to the results obtained for the relative error. [21] The proportionality between d and a c can be explained within a theoretical framework. For simplicity, a c is not divided into a u and a d. The fractional area coverage obeys the equation a c þ a e ¼ 1: The equations for moist static energy and mass flux can be expressed as follows: a c h c þ a e h e ¼ h; ð8þ ð9þ M c þ M e ¼ ra c ðw c wþþra e ðw e wþ ¼ 0: ð10þ In the case of the two areas (cumulus area and environmental area), equation (7) gives d ¼ M e h e h M e h e h ¼ : ð11þ rw 0 h 0 M c h c h þ Me h e h The cumulus-area-averaged moist static energy h c eliminated between equations (9) and (11), giving a c M e h e h d ¼ a e M c h h e þ ac M e h e h a c M e h e h ¼ ¼ a c ¼a c a e M e h e h þ ac M e h e h a e þ a c is ð12þ where equation (10) is applied for converting M c to M e in the denominator. Equation (12) shows that d is proportional to a c and that the constant of proportionality is 1, consistent with the results displayed in Figure 3. [22] Some data in Figure 3 do not support a linear relationship between the relative error and the draft fraction, 4of6

Figure 4. CFFD of D, defined by equation (13), plotted against downdraft fraction a d for horizontal grid spacings of (a) 40, (b) 20, (c) 10, and (d) 5 km. particularly in smaller-area grids (i.e., the areas of 10- and 5-km square grids). This finding arises because the contributions of the updraft and downdraft fractions are not distinguished in the theoretical framework. [23] The relationship in equation (12) becomes increasingly non-linear with increasing downdraft fraction a d. Figure 4 shows a CFFD for D with the downdraft fraction, instead of the draft fraction (i.e., updraft plus downdraft), as the ordinate, where D is defined as D ¼ d þ a c ð13þ Figure 4 shows that the area of maximum frequency shifts to larger D with increasing a d. The results demonstrate that departures from the linear relationship between d and a c are significant when the downdraft fraction is nonnegligible. [24] We now consider the proportionality between e and a c within the theoretical framework. From equations (11) and (12), we have M e h e h ¼ ac rw 0 h 0 : ð14þ Applying equation (14) to equation (6) gives If the condition e ¼ @a crw 0 h 0 @rw 0 h 0 ¼a c rw 0 h 0 @a c @rw 0 h 0 : @a c @rw0 h 0 rw 0 h 0 a c ð15þ ð16þ is satisfied, equation (15) becomes e ffia c : ð17þ [25] This particular derivation, equation (17), allows us to interpret the physical situation as one in which e is proportional to a c (constant of proportionality = 1) if the change in a c (i.e., the total area of cumulus ensembles, not each cumulus area) is small enough to satisfy equation (16). A comparison of Figures 1 and 3 suggests that equation (16) is generally satisfied although not for all data. It is found, from comparison of Figure 2 and its counterpart (not shown), that equation (16) is also satisfied in another dataset. 5. Concluding Remarks [26] This study investigated the errors induced by the assumption that the draft fraction is negligible in cumulus parameterizations using data obtained from high-resolution cloud-resolving simulations. The relative errors in the mean subgrid-tendency of moist static energy were estimated. The numerical results show that the relative error in moist static energy associated with the assumption of small draft fraction is generally negative. The magnitude of the relative error increases with increasing draft fraction, and is approximately proportional to the draft fraction. The results demonstrate that the magnitude of the mean subgrid-tendency of moist static energy is underestimated when the assumption is employed, particularly in the case of a large draft fraction. [27] The magnitude of relative errors generally increases with decreasing horizontal grid spacing. This finding is understandable for two reasons. First, the relationships (i.e., constants of proportionality) between the relative errors and draft fraction are independent of the horizontal 5of6

resolution. Second, the draft fraction tends to increase with increasing horizontal resolution. The rate of grids where the draft fraction is more than 0.5 is 5% at the 5-km horizontal resolution in the first case. Because the rate decreases at the more coarse grids, an answer for the question from which resolution the errors are not negligible appears to be 5 km in this case. [28] The relationship between the relative error and draft fraction is considered within a theoretical framework. The derived equations describe a relationship that is consistent with the results obtained from the data analysis: the equations demonstrate that the relative error is proportional to the draft fraction under a condition (i.e., equation (16)) and is independent of horizontal grid spacing. This finding should be tested in fully-fledged weather prediction models and climate models in future work. [29] Acknowledgments. The author would like to thank R. Sakai for providing the initial data for the numerical simulations and T. Kato and W. Mashiko for the use of their nesting tools. References Arakawa, A., and W. H. Schubert (1974), Interaction of a cumulus cloud ensemble with the large-scale environment, Part I, J. Atmos. Sci., 31, 674 701, doi:10.1175/1520-0469(1974)031<0674:ioacce>2.0.co;2. Gerard, L., and J.-F. Geleyn (2005), Evolution of a subgrid deep convection parametrization in a limited-area model with increasing resolution, Q. J. R. Meteorol. Soc., 131, 2293 2312, doi:10.1256/qj.04.72. Ikawa, M., H. Mizuno, T. Matsuo, M. Murakami, Y. Yamada, and K. Saito (1991), Numerical modeling of the convective snow cloud over the Sea of Japan Precipitation mechanism and sensitivity to ice crystal nucleation rates, J. Meteorol. Soc. Jpn., 69, 641 667. Kato, T. (1995), A box-lagrangian rain-drop scheme, J. Meteorol. Soc. Jpn., 73, 241 245. Kuang, Z., and C. S. Bretherton (2006), A mass-flux scheme view of a high-resolution simulation of a transition from shallow to deep cumulus convection, J. Atmos. Sci., 63, 1895 1909, doi:10.1175/jas3723.1. Kurihara, Y. (2002), A new approach to the cumulus parameterization, paper presented at 25th Conference on Hurricanes and Tropical Meteorology, Am. Meteorol. Soc., San Diego, Calif. Lin, Y. H., R. D. Farley, and H. D. Orville (1983), Bulk parameterization of the snow field in a cloud model, J. Clim. Appl. Meteorol., 22, 1065 1092, doi:10.1175/1520-0450(1983)022<1065:bpotsf>2.0.co;2. Liu,C.,M.W.Moncrieff,andW.W.Grabowski(2001),Hierarchical modelling of tropical convective systems using explicit and parametrized approaches, Q. J. R. Meteorol. Soc., 127, 493 515, doi:10.1002/ qj.49712757213. Moncrieff, M. W., S. K. Krueger, D. Gregory, J.-L. Redelsperger, and W.-K. Tao (1997), GEWEX Cloud System Study (GCSS) working group 4: Precipitating convective cloud systems, Bull. Am. Meteorol. Soc., 78, 831 845, doi:10.1175/1520-0477(1997)078<0831:gcssgw>2.0.co;2. Murakami, M. (1990), Numerical modeling of dynamical and microphysical evolution of an isolated convective cloud The 19 July 1981 CCOPE cloud, J. Meteorol. Soc. Jpn., 68, 107 128. Murakami, M., T. L. Clark, and W. D. Hall (1994), Numerical simulation of convective snow clouds over the Sea of Japan: Two-dimensional simulation of mixed layer development and convective snow cloud formation, J. Meteorol. Soc. Jpn., 72, 43 62. Murata, A., and M. Ueno (2005), The vertical profile of entrainment rate simulated by a cloud-resolving model and application to a cumulus parameterization, J. Meteorol. Soc. Jpn., 83, 745 770, doi:10.2151/ jmsj.83.745. Saito, K., et al. (2006), The operational JMA nonhydrostatic mesoscale model, Mon. Weather Rev., 134, 1266 1298, doi:10.1175/mwr3120.1. Siebesma, A. P., and J. W. M. Cuijpers (1995), Evaluation of parametric assumptions for shallow cumulus convection, J. Atmos. Sci., 52, 650 666, doi:10.1175/1520-0469(1995)052<0650:eopafs>2.0.co;2. Siebesma, A. P., and A. A. M. Holtslag (1996), Model impacts of entrainment and detrainment rates in shallow cumulus convection, J. Atmos. Sci., 53, 2354 2364, doi:10.1175/1520-0469(1996)053<2354: MIOEAD>2.0.CO;2. Tiedtke, M. (1989), A comprehensive mass flux scheme for cumulus parameterization in large-scale models, Mon. Weather Rev., 117, 1779 1800, doi:10.1175/1520-0493(1989)117<1779:acmfsf>2.0.co;2. Xu, K.-M. (1995), Partitioning mass, heat, and moisture budgets of explicitly simulated cumulus ensembles into convective and stratiform components, J. Atmos. Sci., 52, 1 23. Yuter, S. E., and R. A. Houze Jr. (1995), Three-dimensional kinematic and microphysical evolution of Florida cumulonimbus, Part II: Frequency distributions of vertical velocity, reflectivity, and differential reflectivity, Mon. Weather Rev., 123, 1941 1963, doi:10.1175/1520-0493(1995)123<1941:tdkame>2.0.co;2. A. Murata, Typhoon Research Department, Meteorological Research Institute, Nagamine 1-1, Tsukuba, Ibaraki 305-0052, Japan. (amurata@ mri-jma.go.jp) 6of6