A Stochastic Perturbed Parameterization Tendency ... - AMS Journals

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A Stochastic Perturbed Parameterization Tendency Scheme for Diffusion (SPPTD) and Its Application to an Idealized Supercell Simulation XIAOSHI QIAO Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, and Shenyang Central Meteorological Observatory, Shenyang, China

SHIZHANG WANG AND JINZHONG MIN Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China (Manuscript received 9 August 2016, in final form 11 February 2017) ABSTRACT Diffusion plays an important role in supercell simulations. A stochastically perturbed parameterization tendency scheme for diffusion (SPPTD) is developed to incorporate diffusive uncertainties in ensemble forecasts. This scheme follows the same procedure as the previously published stochastically perturbed parameterization tendencies (SPPT) scheme but uses a recursive filter to generate smooth perturbations. It also employs horizontal and vertical localization to retain the impact of perturbation in areas with strong shear. Three additional restrictions are added for the sake of integration stability; these restrictions determine the area and amplitude of the perturbation and the situation to suspend SPPTD. The performance of this scheme is examined by using an idealized supercell storm. The model errors are simulated using different resolutions in the truth run (1 km) and ensemble forecasts (2 km). The results indicate that the ensemble forecasts using SPPTD encompass the intensity and displacement of maximum updraft helicity in the truth run. This scheme yields better results than can be obtained using initial perturbations or larger computational mixing coefficients. The sensitivity of SPPTD to each of its parameters is also examined. The results indicate that the optimal horizontal and temporal scales for SPPTD are 40 km and 30 min, respectively. Moderately adjusting the spatiotemporal scale by 10 km or 10 min does not significantly change the SPPTD performance. In this case study, an ensemble size of 20 is sufficient. Perturbing the diffusion terms of all variables using the same approach does not provide additional benefits other than that of selected variables and thus requires further study.

1. Introduction Errors caused by the need to truncate model equations and parameterize subgrid-scale processes using physical parameterizations are defined as model errors. One approach to representing model errors is to employ several numerical models as well as several physical parameterization schemes (Stensrud et al. 2000; Eckel and Mass 2005; Clark et al. 2008, 2010; Hacker et al. 2011b; Wang et al. 2011). However, Hacker et al. (2011a) and Berner et al. (2011) pointed out that this approach has drawbacks, especially in that it requires the development of multiple

Corresponding author e-mail: Shizhang Wang, szwang@nuist. edu.cn

models and parameterizations. An alternative method is to perturb the physical parameterizations by stochastically perturbing either the inputs (Lin and Neelin 2000; Majda and Khouider 2002; Plant and Craig 2008; Bengtsson et al. 2013) or the outputs (Lin and Neelin 2003; Palmer et al. 2009, hereafter P09; Christensen et al. 2015, hereafter CH15) of deterministic physical schemes. These inputs and outputs are often referred to as the prescribed parameters and the tendencies of parameterization schemes, respectively. One stochastic perturbation technique is the stochastically perturbed parameterization tendencies (SPPT) scheme, which was first proposed by Buizza et al. (1999). In an early version of the SPPT scheme, the parameterization tendency was multiplied by random noises that were piecewise constant in space and time. In P09, a

DOI: 10.1175/MWR-D-16-0307.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

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revised SPPT scheme was proposed, introducing the use of a smooth pattern in space and time for random noises, as described by Berner et al. (2009), and replacing the multivariate uniform distribution with a univariate Gaussian distribution. Romine et al. (2014, hereafter R14) applied the SPPT technique (described in P09) to convective-permitting simulations with the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008) at a horizontal grid spacing of 15 km, successfully increasing the ensemble spread of precipitation forecasts. However, their results showed that this improvement in the spread was accompanied by an increase in forecast bias. This may have been due partly to the design of the SPPT scheme, which cannot distinguish differences between parameterization schemes. CH15 stated that it is therefore better to separately perturb individual physical parameterization schemes because the error characteristics of different parameterization schemes cannot be neglected. They then separately applied multiplicative noise to each parameterization scheme and were able to improve their forecasts. To date, most studies using the SPPT scheme on the mesoscale and the convective-permitting scale have focused on their ability to forecast conventional observed variables, such as wind components, temperature (Berner et al. 2011; Berner et al. 2015, hereafter BR15), and precipitation (R14, CH15). This paper represents the first study to investigate the impact of the SPPT scheme on supercell forecasts. Intensity and displacement forecasts are still well-known challenges for supercell simulations, especially for low-level mesocyclones. Many parameterization schemes, such as microphysical schemes (e.g., Snook and Xue 2008; Snook et al. 2011) and surface physics schemes (e.g., Schenkman et al. 2012), can affect forecasts of supercells and low-level mesocyclones. In fact, physical parameterization schemes related to diffusion, such as the subgrid-scale turbulence kinetic energy (TKE) scheme (e.g., Klemp and Wilhelmson 1978) and the computational mixing scheme [or the numerical filter in some studies such as Takemi and Rotunno (2003), hereafter TR03], can also influence the prediction of mesocyclones (Adlerman and Droegemeier 2002, hereafter AD02). AD02 studied cyclic mesocyclogenesis and demonstrated that increasing the computational mixing coefficient tends to slow the evolution process. Fiori et al. (2009) further indicated that turbulence schemes and computational mixing have clear impacts on predicting storm paths. Additionally, Langhans et al. (2012) discovered that diffusion not only impacts convective intensity and cloud distribution but also has an upscale impact on mesoscale dynamics. Meanwhile, many discussions about subgrid-scale turbulence and computational mixing have revealed that the optimal parameters for these parameterizations vary

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widely and that they depend on factors such as grid spacing and environmental conditions. TR03 stated that computational mixing can be avoided if a larger viscosity coefficient is used in the TKE scheme. However, later studies have argued that computational mixing is still necessary to maintain the stability of the simulation. Knievel et al. (2007) noted that implicit diffusion in the WRF Model is sometimes insufficient for damping noise; therefore, they introduced a sixth-order computational mixing scheme (Xue 2000) to the WRF Model. This diffusion scheme worked well in the experiments of Kusaka et al. (2005). Bryan (2005) also concluded that simply increasing the TKE viscosity coefficient may not be sufficient for all applications. Fiori et al. (2009) demonstrated that the optimal setting of the TKE and computational mixing schemes depends on the grid spacing. Early studies, such as that of Canuto and Cheng (1997), revealed the relationship between environmental conditions and subgrid-scale parameterizations. Bryan (2005) later concluded that the sensitivity of the simulation of squall lines to subgrid-scale schemes is more significant in low-shear conditions than it is in high-shear conditions. Because diffusion has significant impacts on supercell storms and there is currently no optimal coefficient for this parameterization, it is desired to modify the SPPT scheme to account for these factors. However, even the most recent convective-scale research (CH15) has not incorporated this stochastically perturbed diffusion term into the SPPT scheme. The impacts of incorporating this perturbed diffusion term into the model’s dynamic components were investigated by Koo and Hong (2014) using a T254L64 model for seasonal forecasts; they obtained reasonable results when the dynamic components were perturbed. These results indicate that it would be worthwhile to fully examine the effects of incorporating a stochastically perturbed diffusion term on the convective scale simulation. An alternative to the stochastically perturbed diffusion term is that of the stochastic kinetic energy backscatter scheme (SKEB; Shutts 2005; Berner et al. 2009; Tennant et al. 2011), which is concerned with kinetic energy loss and uses streamfunction forcing perturbations to provide substantial input of energy in subsynoptic scales of motion (P09). This backscatter scheme uses horizontally nondivergent streamfunction perturbations (Palmer et al. 2009), although the balance assumptions on which the SKEB scheme is based may not work with small-scale systems (Bouttier et al. 2012). As a result of these limitations, it is preferable to use the SPPT scheme for supercell cases. In the present work, we use the Advanced Regional Prediction System (ARPS; Xue et al. 2000; Xue et al. 2001) to test the performance of the stochastically perturbing diffusion term because this model has previously been

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successfully applied to many supercell simulations (Schenkman et al. 2011; Schenkman et al. 2012; Tanamachi et al. 2013; Xue et al. 2014). Because the SPPT scheme is not currently implemented in the ARPS model, we develop a simple SPPT version that only perturbs the diffusion term in ARPS based on both the TKE and computational mixing schemes. As a preliminary study of the SPPT scheme with perturbing subgrid-scale turbulence and computational mixing schemes, an idealized supercell storm case is used to examine its performance. This approach has been widely employed in studies focusing on diffusion (AD02; TR03; Potvin and Flora 2015; Verrelle et al. 2015). It is widely regarded as a good testbed for numerical experiments because the impacts of changes to the model configuration are relatively easy to recognize in the results (Fiori et al. 2009). The outline of this paper is as follows. In section 2, the SPPT algorithm is briefly reviewed, and the SPPT scheme for diffusion (SPPTD) in the ARPS model is introduced. Model configurations and experiment designs are described in section 3. The results of these experiments are investigated and discussed in section 4. Finally, a summary of this paper is provided in section 5.

2. Methodology a. The SPPT scheme In the SPPT scheme presented by P09, parameterization tendencies, including wind components u and y, temperature T, and the water vapor mixing ratio qy, are multiplied by perturbations. These variables are also used in the recent work of BR15 and CH15. According to CH15, the tendencies of these variables in the SPPT scheme can be represented as follows: N ›X 5 D 1 K 1 (1 1 r)å Pi , ›t i51

(1)

where X denotes the model variables listed above, D represents the tendencies of the dynamic component, K corresponds to the diffusion tendency, Pi represents the tendency of the ith parameterization scheme, and r denotes the random perturbation of the SPPT scheme. The total number of parameterization schemes involved in the SPPT scheme is N. In recent convective-permitting work, the random perturbation r(x, y, t) often has a barotropic vertical structure. The horizontal pattern of r(x, y, t) is then determined using either spherical harmonics (CH15) or 2D Fourier modes (BR15). For a given set of wavenumbers, the evolution of the random perturbation rl,k(x, y, t) is governed by a firstorder autoregressive (AR1) process (Berner et al. 2011): pffiffiffi rl,k (x, y, t 1 Dt) 5 (1 2 a)rl,k (x, y, t) 1 asl,k «(x, y, t), (2)

where l (k) is the wavenumber in the x (y) direction, a is the autoregressive parameter, sl,k is the amplitude of noise, and « is the Gaussian white noise. The decorrelation time is defined as Dt/a. The sl,k in Eq. (2) represents the function of the spatial decorrelation scale and is wavenumber dependent in the implementation of BR15 and CH15.

b. The SPPT scheme for diffusion The current implementation of the SPPTD scheme essentially utilizes the general SPPT concept with two major differences. One is that the diffusion term involves three wind components, temperature, water vapor and other hydrometeor variables, whereas the published scheme only perturbs u, y, T, and q. The second major difference lies in the generation of the random perturbation r. Because the random perturbations in the SPPT scheme are multiplied by physical tendencies, any method that can produce similar patterns should have an equivalent effect. Therefore, the recursive filter employed in the threedimensional variational data assimilation system (Gao et al. 2004) in the ARPS package is used to generate the horizontal pattern of random perturbations for the SPPTD scheme. This recursive filter is defined by the following equations: fi 5 bfi21 1 (1 2 b)ci ,

for

i 5 1, : : : , n

xi 5 bx i21 1 (1 2 b)fi ,

for

i 5 n, : : : , 1,

(3)

where ci is the initial value at grid point I, fi is the value after filtering from I through n, xi is the final value after one pass of the recursive filter, and b is the decorrelation coefficient that determines the scale of the pattern after filtering, as defined by b511E2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E(E 1 2), E 5 2Npass Ds2 /(4L2 ) .

(4)

Here, L represents the decorrelation scale, Ds is the grid spacing in any direction (e.g., Ds is Dx in the west–east direction and Dy in the south–north direction), and the number of passes of the recursive filter is defined as Npass. Larger Npass indicate more significant isotropic features after filtering; given that this pattern is only used for perturbations, a small Npass value should be sufficient. In the present work, the recursive filter is only applied to a twodimensional perturbation field. The barotropic vertical structure is then employed to create the similar threedimensional perturbation seen in other studies (e.g., R14). A horizontal localization procedure is applied to prevent SPPTD from influencing areas where the value of the diffusion term is small. For a given model level, the coefficient rh of the horizontal localization at grid point (i, j) is defined as follows:

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rh (i, j) 5 [K(i, j)/Kmax ]g ,

(5)

where K is the tendency of diffusion as described in the last section, Kmax is the maximum value of K at the given model level, and the coefficient g determines the scale of localization and is always greater than zero. This localization produces values of the coefficient rh ranging from 0.0 to 1.0, where a value of 1.0 is assigned to a grid point in which K is equal to Kmax, and a value of 0.0 is assigned to grid points in which K is equal to zero. The rate of reduction from 1.0 to 0.0 depends on the coefficient g. For instance, large g values result in high

reduction rates such that fewer grids have rh greater than 0.1, whereas small g values have the opposite impact, producing more grids with values of rh greater than 0.1. In this study, the coefficient g is empirically set to 4.0 for all SPPT experiments. Moreover, vertical localization is also employed to prevent numerical instability or spurious oscillation in the midto upper troposphere. The correlation function defined in Gaspari and Cohn (1999) is used to perform this vertical localization; this correlation function is widely applied in data assimilation using the ensemble Kalman filter (e.g., Snook et al. 2011, 2015) and is defined as follows:

8 1 1 5 5 > > 2 z5 1 z4 1 z3 2 z2 1 1 > > 2 8 3 > < 4 2 ry (k) 5 1 5 1 4 5 3 5 2 z 2 z 1 z 2 z 2 5z 1 4 2 > > > 12 2 8 3 3z > > : 0 where k is the model level, z 5 d/c, d is the distance between an arbitrary grid point (AGL) and a specified location, and c is the prescribed cutoff radius. In the present work, the specified location is set to 2 km AGL, and c is 3 km. With this configuration, the maximum impact of the SPPTD scheme occurs at 2 km AGL. This impact is reduced by half at the ground and reduced to zero at 5 km AGL. This empirical setting is based on the fact that strong rotation flows in supercells usually occur at mid- to low levels (e.g., Fig. 3 in AD02). Additionally, the value of rh 3 ry is set to zero if it is less than 0.1. The SPPTD procedure is now summarized by the following seven steps: (i) Gaussian white noise with a mean of zero and a standard deviation of one is generated. This perturbation has a barotropic vertical structure. (ii) Equations (3) and (4) (which define the recursive filter) are applied to this Gaussian noise with the prescribed scale coefficient. The smoothed perturbation x is obtained. (iii) Horizontal localization is performed using Eq. (5) at every grid, level by level, to obtain rh, which is then multiplied by x to yield the horizontally localized perturbation rhx. (iv) The vertical localization coefficient ry is calculated by applying Eq. (6) to all grids and is multiplied by rh x to produce the vertically and horizontally localized perturbation ryrhx. Moreover, « in Eq. (2) is replaced by this perturbation in the SPPTD scheme. (v) A prescribed standard deviation of perturbation s is multiplied by ryrhx.

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for

z 2 [0, 0:5)

for

z 2 [0:5, 1)

for

z 2 [1, ‘)

,

(6)

(vi) The smoothed and localized perturbation sryrhx is substituted into the AR1 process [Eq. (2)] to produce the perturbation r(x, y, t), of which the initial value, i.e., r(x, y, 0), equals zero. (vii) The perturbation r(x, y, t) is multiplied by the tendency of diffusion in terms of Eq. (1). This tendency is the sum of the subgrid-scale turbulence and computational mixing. In terms of P09, the random perturbation r follows a univariate distribution and is applied to all variables to keep the model state close to its attractor. To achieve numerical stability, three additional restrictions are introduced to the SPPTD scheme. First, if the distance between a grid point and the nearest precipitation point with reflectivity greater than 30 dBZ at the first model level above the ground is larger than 15 km, this grid point is excluded from the SPPTD procedure. Second, the SPPTD algorithm is suspended if the horizontal wind components, vertical velocity, or perturbed potential temperatures exceed 95 m s21, 80 m s21, or 70 K, respectively. Finally, there is a limit above which the absolute values of perturbation are reset to a prescribed value. These settings are empirical and are based on an ARPS model configuration wherein integration will be stopped if the wind speed exceeds 150 m s21; however, these settings must be carefully refined for more general applications. In the current study, the reflectivity constraint is performed every 5 min until the end of the forecast because the precipitation area moves with the convective system. This interval is set to 5 min for optimal

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computational costs and because variations in storms within 5 min are usually neglected as a result of 5-min radar volume data often being used to represent the instantaneous state of the storm in data assimilation studies (e.g., Gao et al. 2004; Xue et al. 2014; Snook et al. 2015). At every time step, thresholds that may trigger the suspension of the SPPTD procedure (such as the 95 m s21, 80 m s21, or 70-K thresholds mentioned above) are checked.

c. Metrics for forecast evaluation Given that this work focuses on a supercell storm and the low-level spiral updraft, updraft helicity (UH) is used to evaluate experiment results. As defined by Kain et al. (2008), UH is the integration of the vertical component of helicity over a given layer. It is defined as UH 5

ð z1

k V = 3 V dz,

(7)

z0

where k is the unit vector in the vertical direction, V is the wind vector, and z0 and z1 are the bottom and top of a specified layer, respectively. In this work, z0 and z1 are set to 0 and 3 km AGL, respectively, to assess the low-level spiral updraft. The method used by Fiori et al. (2009) is adopted here to track the UH evolution in the simulated supercell; however, the details of this method have been slightly modified for the purposes of the present work. For example, vertical velocity is replaced by the maximum UH (maxUH) within the domain, and maxUH values of less than 300 m2 s22 are excluded to focus on strong stages during the UH evolution. The performance of the SPPTD scheme is evaluated using the ensemble spread and the fractions skill score (FSS). The ensemble spread is the second moment of the ensemble (see section 3b for more details), which measures the width of the forecast probability distribution (Whitaker and Loughe 1998) and is computed using Eq. (16) in Fortin et al. (2014) for u, y, w, u, p, qy, reflectivity Z, and UH at grid points with Z greater than 5 dBZ in any member. For UH, the threshold is set to 50 m2 s22 to concentrate on the strong low-level spiral updraft. Given that the mean-square error of the ensemble mean may be inappropriate for evaluating small-scale details (Mittermaier and Roberts 2010) and that the horizontal UH scale in this study is relatively small, the prediction skill of UH is evaluated using the FSS proposed by Roberts and Lean (2008). For brevity, the FSS used in this study is the mean of the FSS values of all members. The spatial scale of the FSS for UH is 15 km because the mean displacement error of maxUH is less than 15 km in most experiments. To

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focus on the strongest stage of UH, thresholds of 300 and 400 m2 s22 are selected.

3. Experimental design The ARPS model used for idealized storm simulations is a three-dimensional, compressible, nonhydrostatic model. Most model settings are nearly identical to the typical parameters of idealized storms used by Xue et al. (2001) and Tong and Xue (2005). The common settings used in all experiments are as follows. The Lin (Lin et al. 1983) scheme is used for the parameterization of microphysical processes with default intercept parameters. The 1.5-order TKE scheme follows the subgrid-scale turbulence parameterization of Moeng (1984). Fourth-order computational mixing is selected and applied in both the horizontal and vertical directions; the coefficients of computational mixing are individually set for each experiment and will be discussed in detail later. Earlier studies (Xue 2000; Kusaka et al. 2005) have suggested that higher-order diffusion could be beneficial to these schemes, although the fourth-order scheme is often employed in recent research (e.g., Dawson et al. 2010; Van Weverberg et al. 2011) and works relatively well. Fourthorder advection is applied for momentum and scalars, whereas the positive-definite scheme is employed for scalars. Open conditions are used for the lateral boundaries, and the radiation condition is applied to the top boundary. The bottom boundary is flat, and no surface physics are applied; no friction or surface fluxes are involved. All simulations are run for 180 min.

a. Settings for control Because this is an idealized storm case, a reference experiment must be conducted as a truth run. To simulate model errors due to low resolution, the truth run must have a higher resolution than the other experiments. Verrelle et al. (2015) demonstrated that results obtained using 2- and 1-km grid spacings differ by more than that of 1 and 0.5 km, respectively. Potvin and Flora (2015) concluded that a resolution of 1 km is likely sufficient to simulate rapid variations of lowlevel rotation, which is important for predicting convective hazards. However, horizontal resolution of 2 km or greater is currently used in most operational centers (Potvin and Flora 2015). Based on these data, a grid spacing of 1 km is used for the truth run, and a grid spacing of 2 km is used for the ensemble runs. A time step of 4 s is used for both the truth and ensemble runs. This value was first tested for all 1-km runs and then directly applied to all 2-km cases. Although slightly larger time steps also worked at 2 km, a time step of 4 s is selected for stability.

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remaining experiments produce results similar to those observed by AD02, in which mesocyclone cycles are produced. An experiment with cfcm4 equal to 0.0004 s21 yields two mesocyclone cycles, an experiment with cfcm4 equal to 0.0008 s21 yields only one cycle, and experiments with cfcm4 between 0.001 and 0.0012 s21 produce three cycles (Fig. 1). Given that three cycles are found in the results of AD02 and that the same sounding is used in this study, the experiment with cfcm4 of 0.001 s21 is selected here as the truth run.

b. Settings for the ensemble forecasts

FIG. 1. The evolution of maxUH in the truth run (large dots) and other 1-km runs (using different computational mixing coefficients, small dots) from 90 through 180 min.

The domain of the truth run comprises 103 3 103 3 53 grid points. The vertical grid stretches from approximately 20 m above the ground to 700 m near the top. A horizontally homogeneous environment is generated through a modified real sounding obtained on 20 May 1977 from Del City, Oklahoma, which has been widely used in previous studies (Xue et al. 2001; AD02; Tong and Xue 2005; Potvin and Flora 2015). The simulated supercell storm is initialized using a thermal bubble with a maximum perturbed potential temperature of 6 K. This bubble is centered at x 5 60, y 5 40, and z 5 1.5 km, with horizontal and vertical diameters of 20 km and 3 km, respectively. According to previous studies, the nondimensional coefficient of the fourth-order computational mixing scheme [Eq. (3.13) in Klemp and Wilhelmson (1978)] with a horizontal resolution of 1 km can vary from 0.002 (Dawson et al. 2010) to 0.012 (TR03). As this coefficient increases, a given storm evolves more slowly and more steadily (AD02). However, TR03 found that larger coefficients do not necessarily reduce the intensity of convection; well-organized cells have also been observed under stronger diffusion conditions. Therefore, we performed a series of runs with nondimensional coefficients ranging from 0.0004 to 0.006 (corresponding to the parameter cfcm4 in ARPS ranging from 0.0001 to 0.0015 s21) to select the necessary parameter for the truth run. To obtain fair comparisons and prevent sampling issues, the finite-difference equation of UH involves data with grid intervals of 4Dx instead of 2Dx. Convective systems in the experiments with cfcm4 less than 0.0004 s21 maintain the form of the supercell for the first 100 min before turning into linear convective systems. Based on results of previous studies, these values are not desirable and are not used as truth. The

The ensemble forecasts use a grid spacing of 2 km and contain 55 3 55 3 53 grid points. The vertical grids used in ensemble forecasts are identical to those used in the truth run. The cfcm4 value that produces a diffusion term of equivalent magnitude (cf. the truth run) is selected as the reference value for the ensemble runs, with the exception of some sensitivity experiments. To compute this value, the states of the truth run at 90 and 121 min are interpolated to a small area (51 3 51 grids in the horizontal direction) with a horizontal resolution of 2 km. The centers of this small area and of the 1-km domain are coincident. After this interpolation, the diffusion terms with a grid spacing of 2 km are calculated using the same cfcm4 value. The results show that the magnitude of the diffusion term of the u component after interpolation (Figs. 2b,e) becomes about 3 times as large as the original value (Figs. 2a,b) at both 90 and 121 min. This is because the magnitude of the diffusion is proportional to the amplitude of the short wave, and the transformation of variables from a 1-km grid mesh to a 2-km mesh causes a relatively long wave in the 1-km grid mesh to become a relative short wave in the 2-km grid mesh. Given that the intensity of UH increases between 90 and 121 min, a large cfcm4 is not desired. Therefore, a cfcm4 value of 0.0003 s21 is selected as the reference because the magnitude of diffusion computed by this cfcm4 (Figs. 2c,f) value is comparable to that obtained in the truth run. In addition to experiments using SPPTD, ensemble forecasts using only initial perturbations are also performed to examine whether the issues on cyclic mesocyclogenesis (no mesocyclone cycle at 2-km spacing found by AD02) can be addressed without using stochastic perturbations. The parameters for conducting reference SPPTD experiments (SPref) are as follows. First, the coefficient of the decorrelation time is set to 2.22 3 1023, which is equivalent to a decorrelation time of 30 min. The coefficient of the horizontal scale is defined to be 40 km. These settings are based on the spatiotemporal scale of the supercell storm. Figure 3 displays an example pattern of perturbations using these spatiotemporal settings, and

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FIG. 2. The vertical vorticity (shaded) and the diffusion term of u (contours) at model level 7 (about 500 m AGL) for (a),(d) the truth run and (b),(c),(e),(f) its interpolation to horizontal grid spacing of 2 km at (a)–(c) 90 and (d)–(f) 120 min. The diffusion terms in (b) and (e) [(c) and (f)] are computed using cfcm4 of 0.001 (0.0003) s21.

Fig. 3b indicates that these perturbations are limited in area around the strong low-level rotation core. The standard deviation in the AR1 equation is 1.3, with a limiter of 1.6. Twenty members are used; no initial perturbation is applied. Errors in the initial conditions and model parameterization are well-known sources of forecast error;

therefore, many ensemble forecast systems represent both (Bowler et al. 2009; Xue et al. 2009; Wang et al. 2011; Romine et al. 2014; Berner et al. 2015). However, this approach makes it difficult to distinguish the impact of model error on the final results. To highlight the impact of the SPPTD scheme on the ensemble forecasts, initial

FIG. 3. (a) Example of the perturbation pattern at 120 min without horizontal and vertical localization, where the interval of contours is 0.5 and solid (dashed) contours correspond to positive (negative) values. (b) The vertical vorticity (shaded) and the pattern of horizontal and vertical localization (contours with interval of 0.2) in the y–z plane along the thick line A–A0 in (a).

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perturbations are excluded in the experiments using the SPPTD scheme. Only the diffusion terms of the wind components are perturbed in SPref. Additionally, limiter values greater than 1.0 may result in an increase in shortwave energy. Although these short waves are often a source of error, TR03 demonstrated that there are still some physically based short waves; thus, moderate increases in short-wave energy are supposedly acceptable. Koo and Hong (2014) multiplied the dynamic tendencies by perturbations ranging from 0 to 2.0 for seasonal forecasts and obtained reasonable results. Given that the diffusion term is assumed to be part of the dynamic component in the Koo and Hong study, multiplying large perturbations by dynamic tendencies is equal to multiplying these perturbations by each dynamic component, including the diffusion term. If any of these perturbed terms caused instability, the seasonal forecast would not have been completed successfully. Therefore, it is apparent that large perturbations of diffusion do not cause instability, although they cannot be too large. The parameters for the reference experiment (IPref) using only initial perturbations are as follows. The value of cfcm4 is set to 0.0003 s21, and the ensemble size is equal to that used in SPref. Initial perturbations are generated using Gaussian noise, with a mean of zero and different standard deviations (STDs) for each variable. The STDs for u, y, w, and u are 0.1 m s21 and 0.1 K, respectively. The entire domain except for the uppermost 10 levels is perturbed. Given that the truth environment and thermal bubble are used in ensemble runs, the initial conditions are only affected by resolution-based errors, for which small STD values are likely sufficient. The spatial scales of these perturbations are 5 km and 2 km in the horizontal and vertical dimensions, respectively. Several groups of sensitivity experiments are designed to comprehensively evaluate the performance of the SPPTD scheme. The first set of experiments compares the impacts of the SPPTD scheme to those of the ensemble runs using different magnitudes of the initial perturbations, including those of the SPref, IPref, IP2std, and IP5std experiments. The suffix Nstd denotes that the STD is N times as large as the reference value. The goal of this set of experiments is to determine whether the magnitude of the initial perturbations is sufficient. The goal of the second set of experiments is to compare the results using the SPPT scheme to those of the ensemble runs using reference initial perturbations but with larger cfcm4 values. This set includes the experiments SPref, IPref, IP4cmix, and IP5cmix, where the suffix Ncmix denotes that the value of cfcm4 is N 3 1024 s21. This set of experiments is used to investigate whether the reference cfcm4 value is reasonable and if perturbing parameters of computational mixing works better than

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other experiments. The third set of experiments contains those of SPref, IP5std, IP5std4cmix, and IP5std5cmix, in which the suffix notations of Nstd and Ncmix have the same meanings defined above. These experiments are performed based on the consideration that larger STD in the initial perturbations may result in stronger spurious oscillations that require larger diffusion coefficient to suppress. The sensitivities of the SPPTD scheme to its parameters are also examined. First, the impacts of the spatiotemporal scales are investigated. Sensitivity experiments include SP30h, SP50h, SP20t, and SP40t, where the suffix Nh defines a horizontal scale of approximately N km, whereas the suffix Nt defines a temporal decorrelation scale of approximately N min. Experiments SP10m, SP30m, and SP40m are performed to examine the impact of ensemble size, where the suffix of Nm represents N ensemble members. Because SPref only involves the diffusion terms for the wind components, it is desirable to investigate the performance of experiments perturbing all diffusion terms (SPall). It is also desirable to examine the experiment (SP1.0L) in which the limiter is equal to 1.0, which is widely adopted in SPPT studies. The configurations of all ensemble runs are summarized in Table 1.

4. Results a. Comparison of the SPPT scheme and initial perturbations First, the performances of SPref and the experiments using different magnitudes of the initial perturbations are compared. Figure 4 shows that the growth of the ensemble spread in the first 60 min in SPref is faster than for IPref for all variables except qy (Fig. 4f) and Z (Fig. 4g), which is associated with hydrometeors. This result is not unexpected because no perturbations are multiplied by the water-related variables. The ensemble spread for UH is much larger in SPref than for IPref after 90 min (Fig. 4h). However, with the exception of UH, the magnitudes of the ensemble spread in SPref are essentially comparable to those in IPref. When initial perturbations are doubled, the ensemble spread in IP2std is larger than for SPref for all variables except for UH throughout the entire simulation period. Further increasing the initial perturbations produces a much larger ensemble spread; however, the UH has the smallest ensemble spread after 40 min. The ensemble spread of UH being much smaller in IPref than in SPref may not be repeatable for experiments in other perturbation ranges not examined here. Figure 5a demonstrates that the FSS values of UH obtained using a threshold of 300 m2 s22 in SPref are

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QIAO ET AL. TABLE 1. Configurations of ensemble forecasts.

Expt

Horizontal scale (km)

Temporal scale (min)

Perturbed diffusion terms

Ensemble size

Limiter

STD of initial perturbations

Coef of computational mixing (s21)

SPref IPref IP2std IP5std IP4cmix IP5cmix IP5std4cmix IP5std5cmix SP20t SP40t SP30h SP50h SP10m SP30m SP40m SP1.0L SPall

40 — — — — — — — 40 40 30 50 40 40 40 40 40

30 — — — — — — — 20 40 30 30 30 30 30 30 30

u, y, w — — — — — — — u, y, w u, y, w u, y, w u, y, w u, y, w u, y, w u, y, w u, y, w All

20 — — — — — — — 20 20 20 20 10 30 40 20 20

1.6 — — — — — — — 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.0 1.6

— 0.1 0.2 0.5 0.1 0.1 0.5 0.5 — — — — — — — — —

0.0003 0.0003 0.0003 0.0003 0.0004 0.0005 0.0004 0.0005 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003

significantly higher than those obtained in other experiments. Throughout the duration of the first mesocyclone cycle, the FSS values in SPref often exceed 0.3, with a maximum value greater than 0.5. Given that this FSS represents the mean of the ensemble and that some members have an FSS value of zero when maxUH never exceeds 300 m2 s22, this result implies that some members do have high FSS values. Therefore, we examined the FSS of each member and discovered that some members have an FSS value of 1.0 (not shown). In contrast, IPref, IP2std, and IP5std yield much lower FSS values; the

highest measured FSS value, which is approximately 0.25, was produced by IP 2std at 110 min. FSS values of the second and third cycles are also higher in SPref. Although the measured FSS values are small when obtained using a threshold of 400 m2 s22 (Fig. 5b), the SPref experiment still produces the highest values among the conducted experiments, indicating that the SPPTD scheme has the ability to improve forecasts of strong low-level spiral updrafts with relatively coarse resolution. To further understand these results, it is necessary to examine the evolution of maxUH and its locations.

FIG. 4. The evolution of ensemble spread, averaged over grid points at which the reflectivity is greater than 5 dBZ, for (a) u, (b) y, (c) w, (d) perturbation potential temperature, (e) perturbation pressure, (f) perturbation qy, (g) reflectivity Z, and (h) UH for experiments SPref (dark solid), IPref (gray solid), IP2std (dark dots), IP5std (dark dashed), IP4cmix (gray dashed), and IP5cmix (gray dots). Units are shown in the plots.

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FIG. 5. The FSSs of UH with thresholds of (a) 300 and (b) 400 m2 s22 for experiments SPref (dark dots), IPref (gray dots), IP2std (circles), IP5std (rectangles), IP4cmix (triangles), and IP5cmix (crosses).

Figure 6a shows that, after 90 min, maxUH in SPref synchronously increases with the truth run and that the truth maxUH is encompassed by the ensemble forecasts. The maxUH value in the truth run reaches approximately 580 m2 s22 at 121 min; this state is essentially simulated in SPref. The evolution of maxUH in the second and third cycles is also well predicted in SPref. Throughout the simulation, the displacement error of maxUH in SPref is less than 5 km in the x direction (Fig. 6g) and less than 12 km in the y direction (Fig. 6m). Figure 6b shows the opposite results in that the maxUH value in IPref is much weaker than it is in the truth run, and that the number of members producing maxUH values exceeding the threshold of 300 m2 s22 is much smaller than in SPref, consistent with the lower FSS value in IPref.

However, the paths of maxUH in IPref are reasonable (Figs. 6h,n). With the magnitude of the initial perturbations being doubled, the maxUH in IP2std increases (Fig. 6c). However, the path forecast in IP2std is worse than in IPref; the displacement error is larger (Figs. 6i,o), especially in the y direction, where the maximum displacement error is greater than 15 km. Further increasing the magnitude of the initial perturbations does not improve the prediction of maxUH because the maxUH in IP5std (Fig. 6d) is smaller than in IP2std, especially within the first mesocyclone cycle. The error in the path forecast increases as the initial perturbation is increased, with the maximum value reaching almost 20 km in IP5std (Fig. 6p). To qualitatively evaluate the performance of the SPPTD scheme, the members producing the strongest

FIG. 6. (a)–(f) The evolution of maxUH, and the locations of maxUH in the (g)–(l) x and (m)–(r) y directions for experiments (a),(g),(m) SPref; (b),(h),(n) IPref; (c),(i),(o) IP2std; (d),(j),(p) IP5std; (e),(k),(q) IP4cmix; and (f),(l),(r) IP5cmix. The large dots represent the truth run and the small dots represent the ensemble forecasts.

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FIG. 7. The reflectivity (shaded) and UH (contours with interval of 50 m2 s22 set in boldface every two contours) at model level 7 for (a) the truth run, (b) member 19 in SPref, (c) member 7 in IPref, (d) member 3 in IP2std, and (e) member 7 in IP5std at 121 min.

UH values at approximately 121 min in each experiment are investigated. Storm patterns, especially the hook echo in member 19 of SPref (Fig. 7b), look similar to those obtained in the truth run (Fig. 7a). The UH intensity observed in SPref is also close to the intensity observed during the truth run. However, the hook echo is not clear in IPref (Fig. 7c) and is even weaker in IP2std (Fig. 7d). The displacement error in IP5std (Fig. 7e) is so large that the supercell cannot be shown in the same subdomain. It should be noted that supercell storms in the 2-km runs are larger than those obtained in the truth run; this phenomenon may be related to the resolution of the experiment and cannot be influenced by subgridscale perturbations. To further investigate the impact of the SPPTD scheme on supercell storm simulations, we compared the evolutions of maxUH and r(x, y, t) for good (e.g., members 7 and 20) and bad (e.g., members 5 and 14) members. The first mesocyclone cycle is presented here as an example. Given that the maximum maxUH value of this cycle obtained in the truth run is approximately 580 m2 s22, the corresponding maxUH value predicted by the ensemble members over 400 m2 s22 is considered to be good. Figure 8 shows that r(x, y, t) is essentially negative (positive) before the intension of maxUH in good (bad) members in SPref. With negative r(x, y, t)

values appearing before the intension of maxUH, the value of maxUH exceeds 400 m2 s22, even in cases when small positive values of r(x, y, t) appear during the intension of maxUH (i.e., in member 7). The maximum maxUH value obtained in this member is approximately 425 m2 s22. If negative r(x, y, t) values appears again during the intension, a much larger maxUH value can be obtained (i.e., a value of 527 m2 s22 is obtained in member 20). In contrast, if positive r(x, y, t) values appears before the intension of maxUH (as it does in members 5 and 14), the maximum maxUH value will fail to exceed 400 m2 s22 (as it only reaches approximately 313 and 325 m2 s22 in members 5 and 14, respectively), even in cases in which negative r(x, y, t) values appear during the intension of maxUH. These results imply that the intension of wind shear before the intension of lowlevel spiral updraft plays an important role. However, further investigation of these dynamic mechanisms is beyond the scope of this work and requires additional study. Additionally, these results demonstrate that the response time of this model to the SPPTD scheme is short enough that it is possible to apply the SPPTD scheme to storm-scale ensemble forecasts. Figure 4 shows that compared to IPref, the ensemble spreads for all variables decrease as cfcm4 increases. Moreover, the observed difference between IP4cmix

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FIG. 8. The normalized maxUH (solid) and domain-averaged perturbation r (dots) for two (a) good and (b) bad members in SPref. The domain-averaged perturbation is the mean of r(x, y, t) with the corresponding coefficient of localization (ry 3rh) greater than 0.1. The mN in the legend means that member N is shown. The maximum UHs of the first mesocyclone cycle predicted by ensemble members are listed in the plots. The distinction between the good and bad members is based on whether they predict maxUH over 400 m2 s22.

and IP5cmix is smaller than that observed between IPref and IP4cmix, which implies that further increasing the value of cfcm4 is not beneficial for representing forecast uncertainties. The FSS value of maxUH is also smaller in IP4cmix and IP5cmix (Fig. 5). In the third cycle, the FSS value in IP4cmix is higher than in SPref, which is likely because more of its members are producing maxUH values greater than 300 m2 s22 (Fig. 6e). However, most maxUH values in IP4cmix fail to exceed 400 m2 s22, a result that is well estimated in SPref. The ensemble forecasts of maxUH in IP4cmix and IP5cmix further confirm that simply increasing cfcm4 throughout the entire domain not only suppresses the intensity of the low-level spiral updraft but also reduces the ensemble spread for all variables. Given that the cfcm4 value in IP5std and the magnitudes of the initial perturbation in IP4cmix and IP5cmix may be small, we also performed experiments using large initial perturbations with large cfcm4 values. The

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ensemble spreads in IP5std4cmix and IP5std5cmix are smaller than those in IP5std, although they are still significantly larger than those in SPref (Figs. 9a–g). However, in regard to small-scale motions in which subgrid scales or unsolved activities play an important role, perturbing the diffusion term is more effective because the ensemble spread of UH in SPref is clearly larger than in IP5std4cmix and IP5std5cmix (Fig. 9h). In contrast to the results obtained for the ensemble spread, the ensemble forecasts of the maxUH intensity in IP5std4cmix and IP5std5cmix are reasonable. Figure 10a shows that the maxUH value in IP5std4cmix exceeds 500 m2 s22 in the first cycle and then essentially follows the same pattern as the truth run. The maxUH obtained in IP5std5cmix is smaller, although it still exceeds 400 m2 s22 (Fig. 10b). However, IP5std4cmix yields a significantly larger error of displacement, even in the first cycle, during which this error reaches approximately 12 km in the x direction and 6 km in the y direction. The corresponding values in SPref are approximately 2 and 4 km, respectively. Large displacement errors are also found in IP5std5cmix. In both the truth run and SPref, the right- and leftmoving cells are the only convective systems in the domain (Figs. 11a,b). However, many cells can be seen in the storm pattern of IP5std at 121 min (Fig. 11c), implying that the significant displacement error in IP5std is likely the result of interactions between cells. Given that the cells stimulated by perturbations are randomly scattered throughout the domain, strong supercells can therefore be randomly located in the domain. These spurious cells are gradually removed as the cfcm4 value increases (Figs. 11d,e). However, spurious cells in IP5std5cmix indicate that larger values of cfcm4 are needed to suppress spurious convection, although the intensity of the lowlevel spiral updraft will be further weakened by doing so. After systematically considering the number of spurious cells as well as the intensity and displacement of maxUH, it appears that the SPPTD scheme is preferable to combinations of large cfcm4 values and initial perturbations as a result of its ability to produce reasonable intensities, fewer spurious cells, and more accurate displacements.

b. Comparison of the SPPT parameters The SPPTD scheme features several parameters that need to be defined. Although the parameters described in section 3 are partially based on the knowledge of physical processes, these values may not be optimal. The sensitivity of the SPPTD scheme to its parameters also requires investigation to determine whether its performance is stable or sensitive to individual settings. For brevity, only the ensemble spreads of Z and UH are examined in this section.

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FIG. 9. As in Fig. 4, but for experiments SPref (dark), IP5std (dark dashed), IP5std4cmix (gray dashed), and IP5std5cmix (dots).

Figure 12a indicates that experiments using different spatiotemporal scales produce very small differences in both Z and UH. These differences are slightly larger in UH but are still essentially insignificant (Fig. 12b). These results indicate that increasing or reducing the spatial scale by 10 km or the temporal scale by 10 min does not significantly influence the performance of the SPPTD scheme within the context of the ensemble spread. However, Fig. 13 shows that there is a clear influence on the FSS of UH. With a threshold of 300 m2 s22, SP50h yields a higher score (approximately 0.4) during the first mesocyclone cycle than SP30h, which yields a slightly higher FSS value than SP20t and SP40t. On average, the FSS of UH with a threshold of 300 m2 s22 in the first cycle ranges from 0.2 to 0.4 in these four experiments, while the corresponding range in SPref is slightly larger and extends from 0.3 to 0.5. During the second cycle, the FSS values of maxUH in SP20t and SP40t are larger than those in SP30h and SP50h, although they are still smaller than that in SPref. This situation occurs again during the third cycle except that SP20t yields the highest FSS value. However, with a threshold of 400 m2 s22, SP40t yields the highest FSS of UH among all cycles (Fig. 13b). Figure 14a shows that the intensity of the low-level spiral updraft in SP20t is relatively weak in the first cycle but becomes stronger in the second and third cycles; it is also accompanied by changes in the displacement error that are comparable with those observed in SPref (Figs. 14e,i). In SP40t, the maxUH evolution essentially follows that of the truth run, with the maxUH value exceeding 500 m2 s22 at approximately 120 min (Fig. 14b). During the second and third cycles, several members in SP40t produce maxUH

values greater than 600 m2 s22, which are much higher than those produced in the truth run. The displacement error in SP40t is approximately comparable with that of SPref, although it is slightly larger in the y direction during the third cycle by approximately 4 km (Figs. 14f,j). Given that the third cycle occurs after 150 min, this slightly larger displacement error is likely acceptable to a certain extent. The ensemble forecasts of maxUH in SP30h are less than 450 m2 s22 in the first cycle, whereas they essentially match the values of the truth run in the second and third cycles (Fig. 14c). In SP50h, more members produce maxUH values greater than 300 m2 s22 at approximately 120 min, consistent with the higher FSS of UH obtained in this experiment relative to SP30h, SP20t, and SP40t (Fig. 14d). The relatively low FSS in SP50h in the second cycle is due to a temporal error in which the peak intensity of the ensemble forecasts occurs approximately 10 min earlier than it does in the truth run. Similar temporal errors can also be observed in the third cycle. The displacement errors in SP30h and SP50h (Figs. 14g,k,h,l) are comparable to those observed in SP20t and SP40t. Although using different spatiotemporal scales has a clear impact on the SPPTD scheme, this does not yield different performances based on the SPPT experiments compared with the results obtained in the experiments using only the initial perturbations. As previously stated in section 3, we are also concerned with the sensitivity of the SPPTD scheme to the ensemble size. Thus, we performed the SPref experiment with ensemble sizes of 10, 30, and 40 members. Figure 15 indicates that ensemble size does not have a significant impact on the ensemble spread of Z, although it does have a relatively large impact on UH. SP10m yields the largest ensemble spread of UH after 70 min,

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FIG. 10. As in Fig. 6, but for experiments (a),(c),(e) IP5std4cmix and (b),(d),(f) IP5std5cmix.

followed by SPref. The ensemble spreads of UH for SP30m and SP40m are slightly smaller than those of SPref. There is also only a small difference in the FSS value for UH between these experiments (Fig. 16), implying that an ensemble size of 20 is sufficient for the SPPTD scheme. The results presented in Fig. 15 also indicate that further decreasing the limiter yields smaller ensemble spreads for Z and UH in SP1.0L compared with SPref. Perturbing the diffusion terms for all available variables

significantly increases the ensemble spread for Z and dramatically decreases that of UH in SPall. This phenomenon is similar to that observed in IP5std (Figs. 4g,h). The FSS values of UH in SP1.0L and SPall are lower than those in SPref (Fig. 16), especially for the first cycle, in which both SP1.0L and SPall fail to produce maxUH values greater than 450 m2 s22 (Figs. 17a,b). In these two experiments, the spreads of maxUH in the second and third cycles are larger than in the first cycle, although they are accompanied by larger displacement errors. For

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FIG. 11. As in Fig. 7, but for (a) the truth run, and experiments (b) SPref, (c) IP5std, (d) IP5std4cmix, and (e) IP5std5cmix.

example, SPall achieves a maximum displacement error of 20 km. Examining the storm patterns in SPall (not shown) further indicates that cells in this experiment evolve into linear convective systems after 120 min, similar to the behavior observed in the runs with cfcm4 less than 0.003 s21. Therefore, at least in the present work, it appears that perturbing the wind components, temperature, and hydrometeors using the same approach

is not ideal for improving the performance of the ensemble forecasts.

5. Summary and conclusions The SPPT scheme has been successfully employed in convective-scale simulations. However, until now, none of these studies has assessed the ensemble forecasts of

FIG. 12. As in Fig. 4, but for experiments SPref (dark solid), SP20t (gray solid), SP40t (dots), SP30h (gray dashed), and SP50h (dark dashed).

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FIG. 13. As in Fig. 5, but for experiments SPref (dark dots), SP20t (gray dots), SP40t (circles), SP30h (crosses), and SP50h (asterisks).

supercell storms and their related mesocyclones. Simulations of these convective systems are sensitive to the parameterization of subgrid-scale turbulence and computational mixing, which have not yet been examined in the SPPT scheme. Therefore, the goal of this work is to investigate the impact of perturbing diffusion terms using the SPPT technique to simulate supercells and their related mesocyclones. Here, a SPPTD scheme, based on the ARPS model, is proposed. A recursive filter is used

to produce a smooth random field with a prespecified spatial scale. Horizontal and vertical localizations are added to this SPPTD scheme to limit the impact of perturbations within the area with intense variation of the gradient of variables and within low levels, where mesocyclones often occur. Several restrictions are applied to maintain the stability of the integration. First, perturbations are only applied in areas that are close to the convective system. Second, the SPPTD scheme is suspended

FIG. 14. As in Fig. 6, but for experiments (a),(e),(i) SP20t; (b),(f),(j) SP40t; (c),(g),(k) SP30h; and (d),(h),(l) SP50h.

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FIG. 15. As in Fig. 4, but for experiments SPref (dark solid), SP10m (gray solid), SP30m (dots), SP40m (gray dashed), SP10L (dark dots), and SPall (dark dashed).

when the wind components or temperature exceeds a preset threshold and only resumes when the thresholds are not exceeded. Finally, a limiter is applied to the perturbation fields to reset the absolute values of the perturbations to a prescribed value when the limiter is exceeded. To investigate the performance of the SPPTD scheme, several sets of experiments are performed, which are evaluated based on their resulting ensemble spreads and FSS values. First, we compare the SPPTD scheme to experiments performed using only initial perturbations with different standard deviations of perturbations. The results indicate that the ensemble spreads of model variables obtained using the SPPTD scheme throughout the simulation period are comparable to those obtained using small initial perturbations. In contrast, the SPPTD scheme produces a much larger ensemble spread of UH. The ensemble forecasts using the SPPTD scheme match the truth run more closely than the other experiments, especially for the intensity of the low-level spiral updraft. The

use of large initial perturbations may introduce many spurious convective cells and degrade the performance of ensemble forecasts, even in cases using large cfcm4 values. The SPPTD scheme also has a significant impact on lowlevel wind shear, which is likely conducive to the intension of the low-level spiral updraft (and vice versa). Further study is required to better understand the dynamic mechanisms of this relationship and to determine whether this phenomenon occurs in other supercell cases. Second, the sensitivities of the performance of the SPPTD scheme to its parameters are investigated. Optimal conditions for simulating a supercell storm include using a spatial scale of 40 km and a temporal scale of 30 or 40 min. In this case, 20 members are sufficient because the ensemble spread of UH decreases with increasing ensemble size and the FSS value for UH is higher in SPref than in other experiments using different ensemble sizes. Reducing the limiter to 1.0 produces smaller ensemble spreads for both Z and UH, resulting in lower FSS values

FIG. 16. As in Fig. 5, but for experiments SPref (dark dots), SP10m (crosses), SP30m (asterisks), SP40m (triangles), SP1.0L (gray rectangles), and SPall (circles).

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FIG. 17. As in Fig. 6, but for experiments (a),(c),(e) SP1.0L and (b),(d),(f) SPall.

of UH and smaller maxUH values during the simulation. These results imply that there may be cases in which physically based small eddies are weaker than those obtained during the truth run, increasing the energy of the short wave required for representing such errors or uncertainties. Therefore, using limiter values greater than 1.0 is acceptable. It is not recommended to perturb the wind components, temperature, and hydrometeors using the same approach because this produces large errors in storm structure and displacement. Further work is needed to ascertain the proper approach to perturbing the diffusion terms for temperature and hydrometeors. Given these results, several conclusions are drawn: (i) the SPPTD scheme can represent the uncertainties

of unsolved activities and improve ensemble forecasts of mesocyclones in supercells, especially for the intensity of the low-level spiral updraft; (ii) the intension of lowlevel (below 5 km AGL) wind shear is likely conducive to the enhancement of subsequent low-level spiral updraft, and vice versa; (iii) the SPPTD scheme is not very sensitive to its parameters; thus, moderately tuning the parameters of the SPPTD scheme will not significantly change the performance of this scheme; (iv) in this case, a moderate ensemble size of 20 members is sufficient for the SPPTD scheme to produce reasonable results; and (v) perturbing the diffusion terms for all variables using the same approach does not improve the performance of the ensemble forecasts.

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Although this work focuses on the impact of the SPPTD scheme on ensemble forecasts, the influence of cfcm4 on supercell simulations is also investigated (not shown). When a grid spacing of 500 m and a time step of 1 s are used, the simulated supercell is not very sensitive to variations in the cfcm4 value from 0.0015 to 0.0025 s21 because all runs produce two mesocyclone cycles with similar onset times. When using a grid spacing of 1 km with a time step of 4 s, the supercell storm cannot be well maintained with a cfcm4 value of less than 0.0004 s21. When cfcm4 is greater than 0.001 s21, both the simulated storm patterns and the maxUH evolutions are similar. However, further increasing the value of cfcm4 leads to instability. When using a grid spacing of 2 km with a time step of 4 s, the simulated storm again becomes less sensitive to cfcm4 because the storm patterns and maxUH evolutions are similar when cfcm4 is greater than 0.0004 s21. These results indicate that the influence of cfcm4 on the simulated storm depends on the horizontal resolution. Compared to a grid spacing of 1 km, subgrid activities (on a scale of less than 1 km) are better solved at higher resolutions (0.5 km); therefore, the model becomes less sensitive to cfcm4. This sensitivity to cfcm4 is also weakened when the scale of these activities is too small to be reasonably addressed by parameterizations with lower resolutions (2 km). This weaker sensitivity means that tuning the prescribed cfcm4 value for a grid spacing of 2 km does not improve simulations of cyclic mesocyclogenesis and maxUH intensity. This work represents a preliminary study of stochastic perturbing diffusion terms and presents potential beneficial applications of this scheme to convective-scale ensemble forecasts. Although several sensitivity experiments have been performed, many parameters in this scheme have not yet been analyzed for their sensitivity. Further work is also required to validate these results because this study only represents one case, which is relatively insufficient. Additionally, the idealized supercell studied here represents a case with strong vertical wind shear and large convective available potential energy (CAPE); cases with weaker shear or CAPE are not discussed in this paper. Therefore, more work is required before this scheme can be applied to real cases. Acknowledgments. This work is jointly sponsored by the National Natural Science Foundation of China (41505090, 41430427, and 41505089), the Major State Basic Research Development Program of China (973 Program: 2013CB 430102), the Startup Foundation for Introducing Talent of NUIST (2014R007), the Foundation of Beiji Ge (BJG201409), the Special Program for Applied Research on Super Computation of the NSFCGuangdong Joint Fund (the second phase), and a Project

Funded by the Priority Academic Program Development (PAPD) of the Jiangsu Higher Education Institutions.

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