NOTES AND CORRESPONDENCE. Parameterization of Wind Gustiness for the Computation of Ocean Surface Fluxes at Different Spatial Scales

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1 15 NOTES AND CORRESPONDENCE Parameterization of Wind Gustiness for the Computation of Ocean Surface Fluxes at Different Spatial Scales XUBIN ZENG Institute of Atmospheric Physics, The University of Arizona, Tucson, Arizona QIANG ZHANG Cold and Arid Regions Environmental and Engineering Research Institute, Lanzhou, China D. JOHNSON AND W.-K. TAO Mesoscale Atmospheric Process Branch, NASA Goddard Space Flight Center, Greenbelt, Maryland 16 February 001 and 0 February 00 ABSTRACT Analysis of the Goddard cloud-ensemble (GCE) model output forced by observational data over the tropical western Pacific and eastern tropical North Atlantic has shown that ocean surface latent and sensible heat fluxes averaged in a typical global-model grid box are reproduced well using bulk algorithms with grid-box-average scalar wind speed but could be significantly underestimated under weak wind conditions using average vector wind speed. This is consistent with previous observational and modeling studies. The difference between scalar and vector wind speeds represents the subgrid wind variability (or wind gustiness) that is contributed by boundary layer large eddies, convective precipitation, and cloudiness. Based on the GCE data analysis for a case over the tropical western Pacific, a simple parameterization for wind gustiness has been developed that considers the above three factors. This scheme is found to fit well the GCE data for two other cases over the tropical western Pacific and eastern tropical North Atlantic. Its fit is also much better than that of the traditional approach that considers the contribution to wind gustiness by boundary layer large eddies alone. A simple formulation has also been developed to account for the dependence of the authors parameterization on spatial scales (or model grid size). Together, the preliminary parameterization and formulation can be easily implemented into weather and climate models with various horizontal resolutions. 1. Introduction The atmosphere and ocean interact through the exchange of surface fluxes of heat, freshwater, and momentum. Systematic biases in ocean surface fluxes have been found in atmospheric general circulation models (Gleckler et al. 1995), and become even more severe in ocean atmosphere coupled climate models so that flux adjustments (e.g., Roberts et al. 1997) have been applied in some of the coupled models. Surface fluxes are computed using bulk methods in numerical models and for data analyses. The sensitivity of surface fluxes to the choice of algorithms (particularly under low wind conditions) has been widely recognized (e.g., Webster and Corresponding author address: Xubin Zeng, Dept. of Atmospheric Sciences, The University of Arizona, P.O. Box 10081, Tucson, AZ xubin@atmo.arizona.edu Lukas 199; Miller et al. 199). The sensitivity of cloudresolving models to surface flux algorithms has also been reported (Wang et al. 1996). Various bulk algorithms have been developed in the past several decades (e.g., Fairall et al. 1996; Zeng et al. 1998, and references therein), but most of them deal with fair weather conditions. On the other hand, a number of studies have shown the significant modulation of surface fluxes by precipitation convection using observational data over the eastern tropical North Atlantic (Gaynor and Ropelewski 1979; Barnes and Garstang 198; Johnson and Nicholls 1983) and over the western Pacific warm-pool region (Ledvina et al. 1993; Young et al. 1995; Saxen and Rutledge 1998). For instance, using the Tropical Ocean and Global Atmosphere Tropical Atmosphere Ocean Array (TOGA TAO) buoy data over the tropical Pacific, Esbensen and McPhaden (1996) found that the magnitude of the mesoscale enhancement of monthly averaged surface heat fluxes is 00 American Meteorological Society

2 16 MONTHLY WEATHER REVIEW VOLUME 130 TABLE 1. Correlation coefficient r, bias, and mean absolute deviation (Mad) between the wind gustiness values computed using our parameterization [i.e., (7), (9), and (10)] and those averaged over the GCE domain [i.e., from (6)] for three cases: COARE1 and COARE (over the western Pacific warm-pool region for and 19 7 Dec 199, respectively), and GATE (over the eastern tropical North Atlantic for 1 8 Sep 1974). The values in brackets represent the corresponding results if (7) alone is used (i.e., only the contribution to gustiness from boundary layer large eddies is considered, as used in the traditional approach). The gusty, vector, and scalar wind speeds (U g,u, and U, respectively) averaged over the GCE domain and over the modeling integration period are also shown. All variables are in meters per second except for the nondimensional correlation coefficient. COARE1 COARE GATE r Bias Mad U g U U 0.68 (0.71) 0.66 (0.58) 0.76 (0.84) 0.9 ( 1.6) 0.10 ( 1.) 0.10 ( 1.) 0.47 (1.6) 0.46 (1.) 0.41 (1.) about 10% or less of the total, but can reach 30% during occasional periods with weak and variable winds over precipitation zones. Using a cloud-resolving model, Jabouille et al. (1996) obtained similar results. In order to consider the above mesoscale enhancement in surface flux parameterization, Redelsperger et al. (000) reported their preliminary parameterization of mesoscale gustiness as a function of convective precipitation rate or cloud mass flux. Williams (001) also developed a simple theoretical model for moist convective gustiness based on convective rainfall rate and ambient thermodynamic structure for the lower atmosphere. The purpose of this work is to develop a comprehensive parameterization scheme of wind gustiness that is caused by atmospheric boundary layer large eddies (under unstable conditions), convective precipitation, and convective cloudiness at different spatial scales. This scheme can then be used in bulk algorithms (e.g., Fairall et al. 1996; Zeng et al. 1998) to compute ocean surface fluxes in weather and climate models.. Results a. Cloud-resolving model Our new scheme is developed primarily by analyzing the Goddard cloud-ensemble (GCE) model output forced by observational data over different climate regimes. GCE is a finescale cloud-resolving model (Tao and Simpson 1993; Simpson and Tao 1993). Further improvements in recent years include the implementation of the four-class, multiple-moment microphysics scheme (Ferrier 1994), the solar and infrared radiative transfer scheme (Chou et al. 1998), and the sophisticated seven-layer soil/vegetation land process model (Lynn et al. 1998). Turbulence parameterization in the atmospheric boundary layer includes the effects of both dry and moist processes on the generation of subgrid-scale kinetic energy. The Coupled Ocean Atmosphere Response Experiment (COARE) bulk algorithm (Fairall et al. 1996) is used to compute ocean surface fluxes. GCE is used here to understand and parameterize the mesoscale effect of convective precipitation and clouds on surface fluxes, similar to its use for developing and testing cumulus parameterizations. We have analyzed data from two-dimensional (D) GCE simulations with 51 horizontal grid points with a grid spacing of 1 km for three cases: the fast-moving, slow-moving, and nonorganized convection over the eastern tropical North Atlantic (1 8 September 1974) (denoted as the GATE case) and the squall lines and convective systems prior to and during westerly wind bursts over the western Pacific warm-pool region on and 19 7 December 199, denoted as COARE1 and COARE cases, respectively. The overall model setup is discussed in Johnson et al. (001, manuscript submitted to J. Atmos. Sci.), and the model performance for the COARE1 and COARE cases is discussed in Tao et al. (000). Table 1 shows that the average vector wind speed at a height of 37 m (i.e., the first model level above the surface) is lowest for the GATE case and highest for the COARE case. b. Parameterization of wind gustiness due to boundary layer large eddies, convective precipitation, and cloudiness For a model grid box (e.g., approximately.8.8 for a T4 global spectral model), average ocean surface fluxes are computed from (e.g., Zeng et al. 1998) LH alcu(q e h s q a), (1) SH ccu( ), () a p h s a CUU, and (3) a d s acu, d (4) where the overbars denote averaging over a model grid box; LH and SH are latent and sensible heat fluxes, respectively; s and are scalar and vector mean wind stresses, respectively; C h and C d are the turbulent exchange coefficients for heat (or moisture) and momentum, respectively; a is air density; L e is latent heat of vaporization; c p is specific heat of air; q a and a are the near-surface air humidity and potential temperature, respectively; and s and q s are surface potential temperature and saturated specific humidity, respectively. The near-surface vector (U ) and scalar (U) mean wind speeds are 1/ U (u ), and (5) 1/ 1/ U (u ) (U U g ), (6)

3 17 FIG. 1. Time evolution of GCE domain-averaged (a) latent heat flux LH, (b) sensible heat flux SH, and (c) vector wind stress over the western Pacific warm-pool region from 19 to 7 Dec 199 (the COARE case), denoted by solid lines. Fluxes computed from (1) (3) directly (i.e., using domain-averaged scalar wind) and those using domain-averaged vector wind [i.e., replacing U by U in (1) (3)] are denoted by dotted and dashed lines, respectively. where u and are the two horizontal components of near-surface wind vector, and U g is the wind gustiness. An atmospheric model would provide u and, and hence U, but it does not provide the scalar wind U so that the wind gustiness U g needs to be parameterized. Using the GCE domain (of 514 km) to represent a global-model grid box, the overbars in (1) (6) then refer to averages over all GCE grids (at 1 km) except for the scalar mean wind U. Because boundary layer large eddies are largely unresolved by GCE with a grid size of 1 km, GCE includes the gustiness parameterization of Fairall et al. (1996), which is similar to (7) (to be discussed later), in the computation of scalar mean wind at each 1-km grid box. Therefore, the mean scalar wind U in (6) over the GCE domain is computed as the average of scalar wind speed at each GCE grid box. Figure 1 compares surface fluxes computed using three methods for the COARE case: the horizontally averaged fluxes from GCE flux data, fluxes computed using (1) (3) directly (i.e., using horizontally averaged scalar wind), and fluxes computed using (1) (3) with vector wind speed (i.e., replacing U by U ). It is seen that surface latent and sensible heat fluxes (LH and SH) computed using horizontally averaged scalar wind in (1) (3) are much closer to fluxes averaged over the GCE domain than those using horizontally averaged vector wind. For instance, the mean differences between the horizontally averaged LH values from GCE and those computed using the mean scalar and vector wind speeds are 0.4 and 10.5 W m, respectively, while the maximum differences are 4.9 and 43.9 W m, respectively. This implies that, if the scalar mean wind speed U is used in (1) and (), the nonlinear effects due to correlation between, for example, the mean wind speed and the surface air humidity difference would be relatively small. In other words, if the gustiness parameter U g in (6) can be properly parameterized, (1) and () would give reasonable grid-box heat fluxes in global models. Therefore, only the parameterization of U g is emphasized here, just as in Redelsperger et al. (000). As

4 18 MONTHLY WEATHER REVIEW VOLUME 130 FIG.. (a) Wind gustiness U g from (6) and (b) mesoscale gustiness U gm from (9) vs precipitation rate using the GCE data for the COARE case. compared with the horizontally averaged vector wind, the use of scalar mean wind also improves the computation of vector mean wind stress (Fig. 1c). In fact, the ratio of average fluxes computed using the scalar versus vector wind speed is nearly the same (i.e., 1.08 for LH and SH, and 1.09 for ). However, the averaged computed using the scalar wind is still significantly lower than that from the GCE flux data (Fig. 1c). In contrast, the difference between the scalar wind stress in (4) from the GCE flux data and that computed using the scalar wind would be relatively small (not shown). These results are consistent with previous observational and cloud-resolving modeling studies (e.g., Esbensen and McPhaden 1996; Jabouille et al. 1996; Redelsperger et al. 000). To illustrate and understand the above results, we can consider the averaging of fluxes over two grid boxes with one horizontal component of wind vector being u 1 and u (with u 1 0 and u 0). For simplicity, we assume that 1 0 and that all variables except U and U in (1) (4) are nearly the same over the two grid boxes. Then the average of fluxes over the two boxes would be proportional to (u 1 u )/ for LH, ( u1 u )/ for vector mean wind stress (denoted as o ), and ( u1 u )/ for scalar mean wind stress (denoted as so ). Using the scalar mean wind speed U (u 1 u )/, we would obtain the same LH as above. However, computed using U would be proportional to u1 u u1 u u1 u UU, 4 or 50% of o. In contrast, s computed using U would be proportional to UU (u1 u ) /4, which is smaller than so because (u1 u ) u1 u (u1 u ) but larger than 50% of so because

5 19 (u1 u ) u1 u u1u For most global models, the wind gustiness under unstable conditions is parameterized as 1/3 g U w* w z, (7) gb i where w * is the convective velocity scale, g is acceleration due to gravity, z i is the convective boundary layer height, is virtual potential temperature, and w is the surface buoyancy flux. The value of z i is taken as 1000 m and is taken as unity in (7) for easy implementation in global models (e.g., Zeng et al. 1998; Beljaars 1995), and Redelsperger et al. (000) gave additional discussion on the value of. This accounts for the contribution of large eddies in the convective boundary layer to surface fluxes, and the wind gustiness is denoted as U gb. Note that the buoyancy flux in (7) depends on the scalar mean wind [see (1) and ()], and (1) (7) need to be solved iteratively. When convective precipitation occurs, Redelsperger et al. (000) parameterized the wind gustiness as Ugr log(1 67R 0.48R ) (8) to fit their cloud-resolving model output. Equation (8) accounts for the mesoscale contribution of convective precipitation (through the occurrence of downdrafts and updrafts) to surface fluxes, and the wind gustiness is denoted as U gr. Here, R is precipitation rate in centimeters per day, and U gr is in meters per second in (8). As the precipitation rate approaches zero, the wind gustiness computed from (8) approaches zero rather than a nonzero gustiness from (7), which was used by Redelsperger et al. (000) under fair weather conditions only. To maintain a smooth transition between rain versus no rain, here we parameterize the total (i.e., boundary layer and mesoscale) wind gustiness as Ugt (U gb) (U gm), (9) where U gm represents the mesoscale gustiness due to convective precipitation and cloudiness. Figure shows the horizontally averaged U g computed in (6) (i.e., based on the definition of U g ) from the scalar and vector mean wind speeds using the GCE data as a function of precipitation rate. Obviously, U g never goes to zero [as implied by (8)]. The mesoscale gustiness U gm can be computed by taking U g as the total gustiness in (9) and using the GCE data to compute U gb in (7). Figure also demonstrates that U gm varies between 0.3 and 1.7 m s 1 for very small precipitation rate, which has to be (at least partially) associated with cumulus cloudiness. Just as with convective precipitation, convective cloudiness could be associated with mesoscale gustiness through two possible mechanisms: 1) updrafts associated with nonprecipitating convective clouds could generate (albeit weak) local circulations and ) both convective cloudiness and mesoscale wind variability take a long time to decay after convective precipitation ends. Because the impact of clouds on grid-box averaged variables is already considered in a numerical model, it is reasonable to assume that the impact of cloudiness on U gm is symmetric to cloud fraction f c of 0.5. For instance, stratus clouds with 100% cover would not affect gustiness here. Therefore we parameterize U gm in (9) as 1/ 1/3 Ugm min[3, max(.4r, 1.8 f c )], (10) where precipitation rate R is in millimeters per hour and f c min( f c,1 f c ) is the cloud fraction that is symmetric to 0.5. Based on the above formulation, the maximum U gm due to cloudiness alone is 1.4 m s 1 for f c 0.5, which is equivalent to the U gm value at R 0.35 mm h 1. Therefore, when R is smaller than 0.35 mm h 1, the impact of cloudiness could become very important. The coefficients and exponents in (10) are chosen to fit the GCE data for the COARE case and are then tested for the COARE1 and GATE cases. Further work is still needed to determine whether (10) [or (8)] is applicable over other regions. Figure 3 compares the total wind gustiness U gt computed using our parameterization [i.e., (7), (9), and (10)] with that from GCE directly [i.e., through (6)] for the COARE case. The correlation between parameterized and GCE wind gustiness values is 0.66, the bias (i.e., the mean difference) is 0.10 m s 1, and the mean absolute deviation is 0.46 m s 1. To assess the improvement of our parameterization over the traditional approach (e.g., Fairall et al. 1996; Zeng et al. 1998) that considers the wind gustiness due to boundary layer large eddies alone, Fig. 3 also compares U gb using (7) alone with GCE values. Apparently, results using our parameterization in Fig. 3a are much better than those using the traditional approach in Fig. 3b. For instance, the maximum U gb from (7) is only 1. m s 1 in Fig. 3b. Table 1 shows that the correlation coefficient using (7) is smaller than that from our parameterization for the COARE case, and both the bias and mean absolute deviation using the traditional approach are much larger in magnitude than the values using our parameterization. Table 1 also shows that, for all three cases, using (7) alone would more than double the mean absolute deviation as compared with our parameterization. Although the bias from our parameterization is between 0.1 and 0.3 m s 1 for all cases, it is more than 1.1 m s 1 using (7) alone. For the GATE case, the mean vector wind U for the 7-day period is very small (1.0 m s 1 ) so that the mean wind gustiness U g (1.9 m s 1 ) is even larger than U, illustrating the substantial role of U g in computing the scalar wind speed. For the COARE1 case, the mean U g is about 59% of U and results in 0.8 m s 1 (or 16% of the U value) in (U U ). For the COARE case, the mean U g is larger than that in the GATE case, but it results in only 0.5 m s 1 (or 9.4%

6 130 MONTHLY WEATHER REVIEW VOLUME 130 FIG. 3. Comparison of wind gustiness computed (a) using our parameterization [i.e., (7), (9), and (10)] (denoted as U gt ) and (b) from (7) alone (denoted as U gb ) with that from the GCE data directly [i.e., through (6), denoted as U g ] for the COARE case. of the U value) in (U U ). The above value of 9.4% is also consistent with the 7.6% relative difference in latent heat flux using scalar versus vector wind speeds in Fig. 1. The implication of the above results is that wind gustiness becomes very important only for weak large-scale flow. Here we can also quantitatively estimate the condition under which wind gustiness becomes unimportant. Because the maximum value of U gm is3ms 1 in Eq. (10), the maximum value of U gt in our parameterization is about 3.4 m s 1 (assuming a maximum U gb value of 1.5 m s 1 ). If this maximum U gt value results in less than 10% in the difference between scalar and vector wind speeds relative to the scalar wind, Eq. (6) would yield 1/ 1 U /(U 3.4 ) 0.1. The solution is U 7ms 1. In other words, when U is greater than 7ms 1, the effect of wind gustiness is significantly reduced. If U gt is less than 3.4 m s 1, the corresponding U value when wind gustiness becomes unimportant is also lower (than 7 m s 1 ). We have also compared our parameterization with that of Redelsperger et al. (000) [i.e., Eq. (8) for precipitation rate greater than zero]. For the COARE1 and COARE cases, it is not surprising that results are very similar, because they also used their cloud-resolving model output over the same region for the same period for the development of their parameterization. For the GATE case, our results are slightly better in correlation, bias, and mean absolute deviation. Because our parameterization considers the impact of cloudiness, which could be important only when R is lower than 0.35 mm h 1 (as discussed earlier), it is probably more meaningful to compare results for 0 R 0.35 mm h 1. Then both the mean absolute deviation and bias using (8) more than double those using our parameterization for each of the three cases.

7 131 U gm values are normalized by their maximum value (taken as an average of U gm values for L km). These results for all three cases are given in Fig. 4. Overall, the normalized U gm value increases more rapidly with L when L is relatively small. Based on Fig. 4, a simple scale-dependent function can be developed: 0.37 F(L) (L/400) (11) with L in kilometers and under the restriction of 0 F(L) 1. Then (10) multiplied by (11) gives our parameterization of U m that depends on convective precipitation rate, convective cloudiness, and horizontal scales. Figure 4 also shows that the scale dependence is different for the three cases. To assess the statistical significance of these differences, the two-sample t-test statistic (Wilks 1995) is computed. It is found that the differences between any two cases are not statistically significant at the 90% level. FIG. 4. Mesoscale wind gustiness U gm from GCE data averaged for the whole integration period as a function of horizontal scale L. Results for each case have been normalized by the average of U m values for L km. Results for the COARE1, COARE, and GATE cases are denoted by the solid, dotted, and dashed lines, respectively, while our scale-dependent formulation [i.e., (11)] is represented by the plus signs. c. Dependence of the parameterization on spatial scales Our parameterization was developed for the GCE domain of about 500 km. An obvious issue is the scale dependence of our scheme or other schemes (e.g., Williams 001; Redelsperger et al. 000; Emanuel and Zivkovic-Rothman 1999; Zulauf and Krueger 1997). The dependence of sea surface fluxes on horizontal scales has been demonstrated before (e.g., Sun et al. 1996). Mahrt and Sun (1995) and Vickers and Esbensen (1998) have also proposed empirical formulations to compute wind gustiness as a function of horizontal scales directly (without explicitly considering precipitation rate or cloudiness). There are two approaches to address this issue: one is to develop a different set of coefficients in (10) for each horizontal scale (or model grid size); the other is to multiply our parameterization by a horizontal-scale-dependent function. The latter approach is adopted here for its simplicity. Furthermore, because U gb represents the contribution to wind gustiness of boundary layer large eddies with a horizontal scale of about km, it is reasonable to consider the dependence of U gm alone on horizontal scales. For each case, we use the GCE data to compute U gm averaged over horizontal scales from 10 to 510 km. Then for each horizontal scale L, U gm is averaged for the whole integration period. Furthermore, these scale-dependent 3. Conclusions The Goddard cloud-ensemble (GCE) model output forced by observational data over the western tropical Pacific and the eastern tropical North Atlantic is used here to understand and parameterize the mesoscale effect of convective precipitation and cloudiness on surface fluxes at various spatial scales, similar to its use for developing and testing cumulus parameterizations. Consistent with previous observational and modeling studies, ocean surface fluxes are found to have significant spatial variability (particularly under weak wind conditions) within a typical global model grid box. Most of this variability is explained by the subgrid directional variability in the near-surface wind field, or wind gustiness defined as the difference between average scalar and vector wind speeds in (6). Because the vector wind speed is used in weather and climate models, a parameterization of the wind gustiness is needed. Here we have developed such a parameterization that considers the contributions to wind gustiness from boundary layer large eddies, convective precipitation, and convective cloudiness. The parameterized gustiness can be as large as about 3.4 m s 1 and hence is very important under weak wind conditions. Its importance is significantly reduced under strong wind conditions (e.g., greater than 7ms 1 ). This scheme is found to fit the GCE data much better than the traditional approach (as used by almost all weather and climate models) does that considers the impact of boundary layer large eddies alone. When compared with the precipitation-dependent scheme of Redelsperger et al. (000) that was developed based on the analysis of cloud-resolving model output over the tropical western Pacific, it is not surprising that the two schemes have an overall similar performance. However, since the main difference between these two schemes is our consideration of cloudiness (in addition to precipitation) that is most important for precipitation rate less than 0.35 mm h 1, it is more meaningful to

8 13 MONTHLY WEATHER REVIEW VOLUME 130 compare the two schemes for 0 R 0.35 mm h 1. Then our scheme has much smaller biases and mean absolute deviations. Because of the abundance of drizzles (with some being convective drizzle rainfall) in climate models (e.g., Chen et al. 1996), the consideration of convective cloudiness in wind gustiness parameterization becomes very important. Whereas boundary layer large eddies have an inherent spatial scale of about km, our GCE data analysis has shown that the impact of convective precipitation and cloudiness on wind gustiness depends on the spatial scales, in agreement with previous studies. Accordingly, a simple formulation has also been developed to account for the scale dependence of our parameterization. When both the parameterization and the simple formulation are taken together, our scheme for wind gustiness can be written as Ug (U gb) [F(L)U gm], (1) where U gm and F(L) are given in (10) and (11), respectively, and U gb is given in (7) under unstable stratification conditions. For stable conditions, U gb can be replaced by a small value (e.g., 0.1 m s 1 ). Equation (1) can be easily implemented into weather and climate models with various horizontal resolutions. Our parameterization presented here and those of Redelsperger et al. (000) and Zulauf and Krueger (1997) were all developed based on two-dimensional cloud-resolving models. At least for fair weather conditions, wind gustiness derived from one-dimensional data is found in Vickers and Esbensen (1998) to be smaller than that from two-dimensional data in the horizontal direction. Therefore, analysis of the output from three-dimensional cloud-resolving models in the future could result in slightly different coefficients in Eqs. (10) and (11). Even though the output of a cloud-resolving model forced by observational data may be appropriate for our purpose, our parameterization or other parameterizations (e.g., Redelsperger et al. 000) have to be further evaluated and possibly validated using observational data directly over various regions. Furthermore, because these schemes and that of Williams (001) were developed over the tropical oceans where both the precipitation and cloudiness are primarily due to cumulus convection, their applicability to midlatitude and high-latitude oceans remains to be studied. Other processes for generating wind gustiness, for example, evaporatively driven downdrafts with little or no precipitation (i.e., microbursts), may also need to be considered. Acknowledgments. 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