1 Introduction

Planetary boundary layer (PBL) is the lowermost portion of the atmosphere roughly constituting 1–3 km of the lower troposphere region and is characterized by friction and vertical mixing [1, 2]. Hydrometeorological process and dispersion of pollutants occurring within this PBL region are strongly dependent on the turbulent vertical mixing, and thus a good representation of vertical mixing in the planetary boundary layer (PBL) is important for modeling of meteorological and air quality phenomena. In numerical mesoscale modeling, it is vital to account for the subgrid-scale processes occurring within the confine of model grid for improving the model performance when the processes are not explicitly resolved in the model at coarser resolution. Since the scale of turbulent mixing is relatively smaller than the model resolution, the overall impact due to turbulent forcing on grid-scale variables of meteorology and air pollution is expressed through PBL parameterizations in numerical models. During the past few decades, various turbulent vertical mixing schemes for use in the PBL were developed and tested in 1D and 3D simulations in both meteorological and air quality modeling-related studies. These vertical mixing schemes can be local schemes or K-schemes [3,4,5,6], non-local vertical mixing schemes [7,8,9,10] or may be combination of local and non-local schemes [11]. Studies conducted by most of these schemes [12, 13] infer that modeling is sensitive to the vertical mixing scheme and appropriate apportion of flux between local and non-local component is crucial in representing mixing occurring within the PBL. The turbulence mixing can occur in a wide range of scale starting from subgrid scale to a scale as high as to the depth of the convective boundary layer. Performance of these PBL schemes in numerical weather prediction models varies with respect to the physics options, geography of the study region and time of the year [14,15,16,17]. Therefore, careful examination of these schemes for a chosen domain is crucial to not only weather prediction and research, but also for air quality studies and other environmental investigations [18]. Atmospheric boundary layer representation during convective condition of atmosphere has long been a quest area in numerical simulation of meteorological processes and air quality prediction [10]. The subgrid-scale processes, viz. mixing of heat, momentum and moisture, occurring in the atmospheric boundary layer primarily take place through convective and mechanical forcing of the earth surface due to differential heating and cooling during the daytime and nighttime, respectively. Sensitivity experiments with different local (e.g., K-theory in Stull [1]) and non-local PBL schemes show that complex phenomenon occurring within the boundary layer is likely to affect the vertical mixing within the PBL and evolution of PBL parameters which ultimately influence the air quality dispersion of the region [19,20,21]. Performance of near-surface PBL structure variation is more influenced by the surface layer formulation, whereas the upper profile of PBL is more influenced by mixing algorithms of parameterization schemes [22]. Results also indicate that there is a large variation in the mixing layer height estimated by the model using different combinations of surface and PBL schemes [23]. All the above studies show that appropriate representation of vertical mixing occurring within the confine of planetary boundary layer is thus one of the important components which needs our keen attention for meteorological and air quality modeling under different conditions of atmosphere and geography. Moreover, uncertainty associated with the PBL schemes remains one of the key sources of inaccuracy in model simulations [12, 24]. Inadequate parameterizations of physical processes or faulty parameter estimation cannot be justified by merely optimizing the initial and boundary condition [25]. Data assimilation techniques enable us to improve the accuracy of parameterizations in PBL schemes by estimating the most appropriate value of the model parameter. Parameter estimation using variational data assimilation method and ensemble Kalman filter are robust approaches to deal with model error associated with incorrect parameter value in PBL scheme [26,27,28]. It is also appropriate to say that parameterization of subgrid-scale meteorological process often ends up with range of a model parameter values which need to be optimized for the chosen domain of study. Therefore, parameters optimized for specific purpose within a PBL scheme are not essentially confined to the constant values assigned in the parameterization formulation under different spatial and temporal domains. Hence, it is also important to optimize the parameter values intended for parameter estimation over specific temporal and spatial domains. A numerical weather forecasting model like WRF-ARW offers several options of parameterization schemes for different physical processes. Routinely Planetary Boundary Layer (PBL) schemes are appropriately used to parameterize the vertical turbulent fluxes of heat, momentum and moisture within the planetary boundary layer as well as in free atmosphere. Within these PBL schemes, there are many less known hidden or implicit parameters which govern the mixing within the boundary layer. Each parameterization scheme contains several parameters which can vary over a range of values, and the choice of a value of parameter may depend on the geographical and meteorological conditions. One of such parameter in ACM2 [11, 24] scheme is ‘p’ which appears in the model equation to express the eddy exchange coefficient Kz in non-local closure scheme. This parameter is believed to govern the local mixing profile within the unstable portion of the boundary layer. It determines the maximum value of Kz and the height at which Kz will be maximum. Study carried out by Nielsen-Gammon et al. [25] also states that among all the parameters, ‘p’ has maximum impact on the vertical mixing in the boundary layer during the daytime. Thus, sensitivity study with a suitable choice of implicit or less known parameters in the boundary layer parameterization schemes is crucial for accurate simulation of extreme weather conditions, especially when the system is dominated by convective-scale eddies. However, it is also appropriate to admit that performance of different meteorological variables in numerical model also depends on combination of different physical processes like cumulus convection, cloud microphysics and definition of model initial state in the simulation of low temperature, strong wind and heavy precipitation [29]. In order to evaluate the dependency, sensitivity experiments need to be carried out to determine the most appropriate value of ‘p’ for the region of study. Thus, the objective of the present study is to accurately simulate the meteorological variables such as potential temperature and relative humidity by modifying these parameters in the boundary layer parameterization scheme suitable for the chosen region of study.

Subgrid-scale turbulent fluxes of heat, momentum and moisture within the planetary boundary layer are expressed in terms of mean quantities and their gradients with the help of turbulent closure method in the PBL scheme. A local closure scheme is more appropriate in a stable atmosphere than in an atmospheric condition where the turbulent fluxes are dominated by large eddies. Since an extreme weather event like western disturbance is characterized by large-scale advective and convective types of phenomena, the local closure schemes will produce insufficient mixing. Thus, in contrast, a non-local scheme will be more appropriate to use the profile of eddy diffusivity and incorporates the non-local effect of transport by the large eddies in the model. Besides, the counter-gradient fluxes are also taken into consideration by the non-local approach. It is also considered to be more robust numerically as stability oscillations do not affect it considerably [30].

This paper is organized as follows. Section 2 provides a description of the model, and Sect. 3 provides the configuration of the simulations. In Sect. 4, the results are presented, mainly focusing on the comparison of the simulation results with observations or analyses. Section 5 formulates the conclusions.

2 Model description

Weather Research and Forecasting (WRF) is a non-hydrostatic, fully compressible numerical weather prediction (NWP) model designed by National Center for Atmospheric Research (NCAR) in collaboration with National Centers for Environmental Prediction (NCEP) and several other institutes. It is a community model and has been developed for both the research and operational applications. The WRF Software Framework (WSF) contains the dynamic solvers, physics packages, programs for initialization, WRF-Var and WRF-Chem. WSF has two dynamic solvers: the Advanced Research WRF (ARW) solver and the Non-hydrostatic Mesoscale Model (NMM) solver. ARW has been developed primarily at NCAR and NMM at NCEP. In the present study, version 3.3 of ARW has been used. A brief description of the model configuration used for the present study is given in Table 1. ARW uses a terrain-following hydrostatic-pressure vertical coordinate, η. A detailed technical description of ARW is available in [31]. WRF model has been very widely used over a range of applicability and operability and has been tested effectively in forecasting extreme rainfall event [32, 33], intercomparison of winter and non-winter hail storms [34], sensitivity of physical schemes in simulating intense western disturbances [35] and meteorological data source for dispersion modeling [36].

Table 1 Overview of the ARW model configuration used for the present study
Full size table

3 Methodology

In the present study, nine cases of western disturbances (WD) that occurred during the winter seasons between 2005 and 2009 are chosen and simulated by using WRF-ARW model keeping the model horizontal resolution at 27 km. Figure 1 shows the model domain. Table 1 states the nine cases that have been investigated in the present study.

Fig. 1
figure 1

Model domain of study

Full size image

The NCEP FNL (Final) Operational Global Analysis data of 1° × 1° resolution have been used as model initial and lateral boundary conditions. The static geographical input data of resolution 30′′, 2, 5′ and 10′ that include MODIS, GWD, SSiB and VARSSO data have been downloaded from official Web site of WRF. In each case mentioned in Table 2, the model integration starts at 00:00 UTC of the previous day. As an example, in case 1, the weather condition of January 11, 2007 has been investigated. The model run starts with initial condition of 00:00 UTC of January 10, 2007. Model result obtained after 24 h model run has been used for analysis. Lateral boundary condition gets updated at an interval of 6 h.

Table 2 Dates of the WD cases simulated in the present study
Full size table

A coarse horizontal grid of size 27 km cannot resolve subgrid-level phenomena; thus, parameterization schemes are required to represent the subgrid-scale processes. In the present case, cumulus parameterization is necessary to take care of latent heat release on a realistic timescale in the convective column. The modified version of the Kain–Fritsch scheme [37] has been chosen as the cumulus parameterization scheme in the WRF-ARW model system. It uses a simple cloud model with moist updrafts and downdrafts, which include the effects of detrainment, entrainment and relatively simple microphysics. This modified version of the Kain–Fritsch scheme is different from the original KF scheme [38] in many ways which is briefly described in the technical note of ARW version 3.

Lin et al. [39] scheme has been chosen as cloud microphysics scheme for this study. It has been taken from Purdue cloud model. The Lin et al. scheme includes six classes of hydrometeors. These hydrometeors are water vapor, cloud water, rain, cloud ice, snow and graupel. It uses the bulk water microphysical parameterization technique to represent the precipitation fields which follow exponential size distribution functions. The detailed description of this scheme is available in Chen and Sun [40].

For the present investigation, the asymmetric convective model, version 2 (ACM2) scheme has been used as planetary boundary layer parameterization scheme to simulate various cases of western disturbances in WRF-ARW model. ACM2 can better represent the shape of the vertical profiles of model variables, especially the gradually decreasing gradient near the surface. ACM2 has been chosen for our experiments as comparatively lower bias was reported in the sensitivity experiments than other non-local PBL schemes [12]. This PBL parameterization scheme has been included in WRF-ARW model very recently. ACM2 is a non-local model and it considers the non-local fluxes explicitly through a transilient term [9]. This scheme contains several parameters which can vary over a range of values, and the choice of a value of a parameter may depend on the geographical and meteorological circumstances.

Local time derivative of potential temperature (θ) in the PBL scheme proposed by Deardorff [41], Holtslag and Boville [5] and others can be represented by

$$\frac{\partial \theta }{\partial t} = \frac{{\partial \left( {\overline{{w^{{\prime }} \theta^{{\prime }} }} } \right)}}{\partial x} = \frac{\partial }{\partial z}\left[ { - K_{\text{h}} \left( {\frac{\partial \theta }{\partial z} - \gamma_{\text{h}} } \right)} \right]$$
(1)

where Kh is the vertical eddy diffusion coefficient of heat, γh is the gradient adjustment term and \(\overline{{w^{{\prime }} \theta^{{\prime }} }}\) is the kinematic heat flux. Similar equation is applicable for humidity also. In asymmetrical convective model, ACM1 [9] mass fluxes are represented by rapid upward convective non-local transport from the lowest model layers directly to all other layers but gradual downward asymmetric layer-by-layer transport. Provision of local upward diffusion from one layer to the next layer is absent in ACM1 which is unrealistic. Pleim [11, 24] has described the formulation of ACM2 which combines the non-local scheme of ACM1 with an eddy diffusion scheme to represent turbulent transport of mass, moisture and heat within a convective boundary layer.

In ACM2 scheme, a weighting factor fconv controls the mixing due to local diffusion and non-local transport. fconv = 0 corresponds to fully local mixing and fconv = 1 corresponds to fully non-local mixing. For stable and neutral conditions, the portion of mixing due to non-local transport becomes zero and this scheme handles vertical mixing by pure local eddy diffusion component.

In ACM2, the PBL scaling form of vertical eddy diffusion coefficient Kz within the planetary boundary layer is represented as

$$K_{z} = k\left( {\frac{{u_{*} }}{\varphi }} \right)z\left( {1 - \frac{z}{h}} \right)^{p}$$
(2)

where k = the von Karman constant, u* = friction velocity, φ = similarity profile function, z = height above ground level and h = height of the top of PBL measured from the ground level.

From Eq. 2, it can be seen that there are many parameters that affect the vertical mixing in this scheme. Values of these parameters used in this scheme are kept hidden from WRF-ARW model users. Different roles played by these parameters and their plausible values are tabulated in Nielsen-Gammon et al. [25]. The detailed investigation of Nielsen-Gammon et al. [25] reveals that among the ten parameters, ‘p’ plays the most important role in governing the vertical mixing in the daytime. This parameter (p) determines the value of the local eddy vertical mixing coefficient within the convective PBL, with larger p leading to smaller vertical mixing.

The parameter ‘p’ used in Eq. (2) is also a hidden parameter, and by default the value of ‘p’ used in the model is 2. The prescribed plausible range of ‘p’ values lies between 1 and 3. It is clear from Eq. (2) that ‘p’ determines the height at which eddy diffusivity becomes maximum. For p = 1, eddy diffusivity becomes maximum at the middle of the boundary layer. This height decreases with the increase of ‘p’ value. Thus, eddy diffusivity is determined by the magnitude of ‘p’. A smaller value of ‘p’ refers to stronger mixing. Figure 2 shows the vertical profiles of normalized Kz for different values of ‘p’ with PBL height arbitrarily set at 2000 m. This shows that vertical eddy diffusivity Kz varies considerably with the ‘p’ values and hence the vertical mixing strength in daytime within the PBL is largely dependent on the p value.

Fig. 2
figure 2

Normalized profiles of Kz for different values of p

Full size image

Selection of most appropriate value of hidden parameter ‘p’ through sensitivity experiments is important as ‘p’ influences the vertical mixing mostly within the PBL. The present study aims at finding out the impact of the variation of p values on boundary layer during WDs and the most appropriate value of ‘p’ for simulation of western disturbance over the region of study. Two variables that are mostly affected by the variation of vertical mixing are potential temperature (θ) and relative humidity (RH). The resultant effect due to change of ‘p’ values will be studied on them. Since the vertical profile form of Kz is applicable in the daytime boundary layer, the behavior of potential temperature (θ) and relative humidity (RH) profile within daytime boundary layer will be discussed. Thus, model outputs are taken at 00, 06, 12 and 18 UTC and the θ and RH profiles at 06 UTC are expected to be influenced most by the variation of p values. In the present study, six values of ‘p’ between 1 and 3 were chosen. These values are 1.25, 1.50, 1.75, 2.0, 2.50 and 2.75.

4 Results and discussion

4.1 Impact of ‘p’ values on potential temperature profile

Figure 3a shows the observed vertical profiles of potential temperature and those simulated by the model at Delhi region at 00:00 UTC for six cases out of the nine due to paucity of data. Observed vertical profile of potential temperature is obtained from radiosonde observation taken at Safdarjung Airport, New Delhi, by India Meteorological Department (IMD). Model simulates stable boundary layer at 00:00 UTC in all the six cases. Simulated results are in good agreement with the observation except in case 3 (Fig. 3c). In this case, observed profile shows that the atmosphere is in neutral condition in the lowest 1 km. Here, the model is unable to predict the intense mixing at the lower part of the boundary layer due to the local neutral or unstable condition. In cases 2 and 6 (Fig. 3b, d), the model-simulated potential temperature profiles depart appreciably from the observed potential temperature profile. Since in most of the cases atmosphere is stable during early morning hours, the eddy diffusivities are determined by local schemes and hence different values of ‘p’ do not have much impact on the potential temperature profiles.

Fig. 3
figure 3

Potential temperature profile at 00:00 UTC at Delhi for six cases

Full size image

The model-simulated potential temperature profiles for all nine cases of 06:00 UTC at Delhi region have been compared with the analysis profiles in Fig. 4, as the observations are not available at this time of the day. It is to be noticed that at 06:00 UTC, in most of the cases the potential temperatures are different for different values of ‘p’ at the lower part of the boundary layer where atmosphere is either neutral or unstable. This is because of the fact that non-local scheme of eddy diffusivity determines the mixing in the boundary layer and different values of ‘p’ cause different amounts of mixing. As p varies from 1.25 to 2.75, variation of potential temperature becomes as large as 1° K.

Fig. 4
figure 4figure 4

Potential temperature profile at 06:00 UTC at Delhi for nine cases

Full size image

In majority of the cases, the potential temperature profile for p = 1.25 shows closer proximity to the analysis profile within the lowest kilometer of the atmosphere. This reveals that modified formulation of ACM2 with p = 1.25 is capable of producing adequate mixing of heat flux in the morning neutral condition of lower boundary of the PBL during severe weather phenomenon of WD.

4.2 Impact of ‘p’ values on relative humidity profile

Figure 5 shows the model-simulated and IMD-observed relative humidity profiles (obtained from radiosonde observation) over Delhi region at 00:00 UTC for six cases due to non-availability of observational data for other three cases. Unlike potential temperature, the model-simulated relative humidity shows appreciably larger departure from the observations.

Fig. 5
figure 5

Relative humidity profile at 00:00 UTC at Delhi for six cases

Full size image

Similarly, the model-simulated and analysis of relative humidity profiles at 06:00 UTC for all nine cases are shown in Fig. 6 over the same place. Here, in most of the cases, the difference between analysis of and model-simulated relative humidity is found significant in the order of ~ 10%. This large difference of simulated relative humidity in the order of ~ 10% implies that the modified formulation of ACM2 has minimal ability to produce sufficient mixing of relative humidity (moisture) especially in the morning neutral atmospheric condition during severe weather events. Thus, the model is not efficient in capturing the necessary condition of moisture flux which is a highly variable parameter across the wide spectrum of convection in the organized severe weather event. Analysis of the simulated results conveys that other physical processes such as the microphysical processes, cumulus convection process may play more important role to account for the forcing due to turbulence to successfully simulate the relative humidity involved at the lower boundary during neutral or unstable condition of the atmosphere.

Fig. 6
figure 6figure 6

Relative humidity profile at 06:00 UTC at Delhi for nine cases

Full size image

4.3 Root-mean-square errors of potential temperature and relative humidity

Figure 7 presents the root-mean-square errors (RMSEs) of potential temperature with respect to analysis up to a height of 1 km. It shows that except in cases 2 and 3, the root-mean-square error of potential temperature corresponding to ‘p’ = 1.25 is always less than that for other ‘p’ values. Figure 8 shows that the mean value of RMSE calculated over all the cases is the least and it has a value of 0.86° K while the mean of RMSE corresponding to the default value of p, i.e., ‘p’ = 2, is 1.04° K.

Fig. 7
figure 7figure 7

RMSE of model-simulated potential temperature (K) at 06:00 UTC with respect to analysis calculated up to 1 km height at Delhi for all cases

Full size image
Fig. 8
figure 8

Mean value of RMSE of potential temperature (K) at 06 UTC at Delhi

Full size image

However, like potential temperature, the root-mean-square error of relative humidity with respect to analysis at 06:00 UTC over Delhi is least for p = 1.25 in majority of the cases (see Fig. 9). However, this is highest in cases 2, 8 and 9. The mean value of RMSE calculated over all the cases for relative humidity is also least for p = 1.25. Figure 10 shows that the mean value of the domain-averaged RMSE value of relative humidity for p = 1.25 is slightly more than 6.68%. The mean RMSE of relative humidity for the default p value (2) is slightly less than 7%. Thus, it can be inferred that the simulated potential temperature and relative humidity profiles for p = 1.25 are closest to the respective analysis profiles over Delhi during 06:00 UTC. Thus, the modified formulation of ACM2 scheme by lowering the value of ‘p’ = 1.25 is sensitive to the turbulent transport of heat and moisture during convective boundary condition by modifying eddy diffusion. This modified formulation in the ACM2 scheme is perhaps more effective in partitioning of the local and non-local mixing components in the model at least for the domain under study. Therefore, it will be more appropriate to modify the default value of ‘p’ to 1.25 in Eq. 1. The default value p = 2 may be more applicable for a less convective environment. The lower value of p (= 1.25) suitable for the present study indicates that large eddy diffusivity values (and hence larger mixing) are required to represent the convective planetary boundary layer over this tropical region of study. It also indicates that the maximum value of Kz occurs at higher level within boundary layer for p = 1.25 than for the default value of p.

Fig. 9
figure 9figure 9

RMSE of model-simulated relative humidity (%) at 06:00 UTC with respect to analysis calculated up to 1 km height at Delhi for all cases

Full size image
Fig. 10
figure 10

Mean value of RMSE of relative humidity (%) at 06 UTC at Delhi

Full size image

5 Conclusion

In the above study, we have presented the results of numerical meteorological simulation for greater area of Northern India at a spatial resolution of 27 km. Sensitivity experiments have been carried out to assess whether WRF-ARW model embedded with ACM2 PBL scheme is capable of correctly reproducing observed meteorological quantities at different values of model hidden parameter ‘p’. To demonstrate this, model-simulated potential temperature and relative humidity profiles over Delhi at 06:00 UTC were compared with analysis data and RMSEs were computed. The comparison shows that p = 1.25 scores better than other values of ‘p’ for both potential temperature and relative humidity. As a result, we believe that our approach with ‘p’ value of 1.25 in the model code is more promising in defining the vertical profile of the PBL especially when one is interested to simulate extreme weather events like WD dominated by convective phenomenon at least for the period and domain under study.