1 Introduction

Convective parameterization is an important part of numerical weather prediction and climate models. There are many cumulus parameterization schemes, such as Kuo scheme (Kuo 1965), moist convection adjustment scheme (Manabe et al. 1965; Betts 1986), mass flux scheme (Arakawa and Schubert 1974; Kain and Fritsch 1990; Zhang and McFarlane 1995). So far, one of the most complex and classic achievements on convective parameterization is the work published by Arakawa and Schubert (1974). The Zhang–McFarlane (ZM) deep convective scheme utilizes some of the basic concepts set forth by it to construct a bulk representation of the effects of an ensemble of cumulus clouds. In the Community Atmospheric Model version 5 (CAM5) of the National Center for Atmospheric Research (NCAR), the process of deep convection is parameterized by the ZM deep convective scheme. Due to the characteristics and differences of various cumulus parameterization schemes in several aspects, the application of convective parameterization has become one of the important sources of model biases and uncertainties. The optimization of cumulus convection schemes also plays an important role in the bias correction of precipitation. How to optimize parameters to obtain better simulations has always been a hot topic.

Reducing precipitation bias has been one of the subjects of intensive research in the climate modeling community. Jia (2019) found that the global climate models (GCMs) of the Coupled Model Intercomparison Project Phase 5 (CMIP5) generally overestimate the mean monthly precipitation across annual cycle over the Tibetan Plateau, on average by 48.2 mm, especially in spring and summer. Evaluating the performance of 23 CMIP6 models, Zhu and Yang (2020) found that despite substantial model developments in recent years, the systematic model biases (overestimated precipitation) over the Tibetan Plateau still exist, even worse in some cases. Similar problems also impair precipitation simulation in other steep mountain regions, such as the Andes Cordillera in South America. The double intertropical convergence zone (ITCZ) bias is one of the most outstanding errors in all previous generations of climate models, which may reduce the reliability of future climate predictions based on these models. Tian and Dong (2020) found that the double ITCZ bias with a big inter-model spread persists in all CMIP models and remains a serious problem in the latest CMIP6 models. However, the bias is slightly reduced in the CMIP6 models when compared to the CMIP3 or CMIP5 models based on several precipitation bias indices (Tian and Dong 2020).

Some parameters in cumulus parameterization schemes are constant. Models generally overestimate the frequency of light precipitation and underestimate the intensity of heavy precipitation (Sun et al. 2006). To reduce uncertainties in these parameters, Yang et al. (2013) investigated the sensitivity of precipitation to the key parameters in CAM5 and ZM scheme. Qian et al. (2015) identified that six parameters related to cloud physics and convection have the greatest influences on the global mean precipitation; three of them are related to the deep convection scheme: evaporation efficiency of precipitation (ke), maximum cloud downdraft mass flux fraction (alfa), time scale for consumption rate of convective available potential energy (τ), which are the primary contributors to the total variance of the phase and amplitude of precipitation diurnal cycle over land (Qian et al. 2015). Yang et al. (2013) found that the simulated convective precipitation is most sensitive to the parameters of τ, parcel fractional mass entrainment rate, and maximum downdraft mass flux fraction. Some studies improved the Kain and Fritsch (1993) scheme in the regional model to realize the dynamically computed parameters (Bullock et al. 2015; Zheng et al. 2016). In some studies, using the global model, machine learning methods were used to optimize the parameters of the deep convection trigger function (Zhang et al. 2018), but a dynamic computation of the parameters in the global model has not been realized.

The characteristic adjustment time scale τ is one of key parameters in a convective parameterization scheme. It indicates the hypothetical time required for deep moist convective overturning to adjust thermodynamic profiles to a quasi-equilibrium or neutral state (Bullock et al. 2015). Through convection, the convective available potential energy (CAPE) decreases at an exponential rate. The parameter τ was originally used in the mesoscale model (Fritsch and Chappell 1980), and represents the time of convection activity at the model grid. Fritsch and Chappell (1980) focused on the idea that moist convection stabilizes the local environment by replacing unstable air at low levels with less unstable air from above, through the action of convective down-drafts. They linked the convective adjustment time scale to the advective time scale of clouds. This concept was used in the Kain and Fritsch (1993) parameterization. A large number of studies have shown that various simulations are sensitive to the specific value of τ (e.g., Alapaty et al. 1994a, b; Emanuel et al. 1994; Lin et al. 2000). Therefore, the value of τ is usually determined on the basis of repeated experiments. τ is set to a constant value in many regional and global models, with a few exceptions (e.g., the European Center for Medium-Range Weather Forecasts model; Bechtold et al. 2008). In many general circulation models, τ ranges from one to several hours (e.g., Zhang and McFarlane 1995; Collins et al. 2006; Wilcox and Donner 2007). Take the CAM5 as an example, this parameter is usually set to the global unified value: 3600 s. According to the early test of typical cloud lifetime, the variation range of τ is limited to 1800–3600 s (Bullock et al. 2015). Studies showed that there are some uncertainties in the effect of this parameter. For example, aqua-planet simulations revealed that when τ increases, rate of deep convective precipitation decreases; and this leads to an accumulation of convective instability in the atmosphere (Mishra and Srinivasan 2010). On the other hand, the enhancement of shallow convective precipitation and large-scale precipitation limit the accumulation of convective instability in the atmosphere.

While Done et al. (2006) found that changing the adjusted time scale from minutes to one day resulted in all subgrid-scale precipitation generated by convective parameterization becoming grid-scale precipitation. Liu et al. (2007) studied the response of climate to various constant τ values using a community atmosphere model; and they found that any single value of this parameter would not produce the best results for all aspects of climate. Tannahill et al. (2010) concluded that τ was one of the most influential parameters among all tunable parameters in global climate simulations. The results obtained from these and many other climate simulation studies showed that using fixed τ value is problematic. This highlights the necessity of allowing the value of τ to change in time and space.

Some geographical and atmospheric factors contribute to the development of deep convection, such as CAPE, latent heat flux, and relative humidity in the middle troposphere (Shinoda and Uyeda 2002). A close relationship between CAPE and convective events was confirmed by the studies of CAPE and convective events (Adams and Souza 2009; Suhas and Zhang 2014). CAPE theoretically reflects the intensity of vertical motion due to the work of environmental buoyancy, and can reflect the overall structural characteristics of the atmosphere. In Zhang–McFarlane scheme, when its value is greater than 70 J/kg, deep convective precipitation will be triggered in the numerical model. Therefore, the dynamic computation of characteristic adjustment time scale by CAPE in this study may effectively improve precipitation simulation.

Therefore, the main purpose of this paper is to improve the ZM deep convection scheme, by realizing the dynamic calculation of characteristic adjustment time scale parameter, to reduce precipitation deviation of the model. We use CESM simulation to test whether the modified scheme can improve precipitation simulation. The rest of the paper is organized as follows. In Sect. 2, we describe the model and data used. The precipitation fields simulated by original and dynamic τ are analyzed and compared in Sect. 3. In Sect. 4, the major points are summarized, and the implications of these results are discussed.

2 Deep convection scheme and experimental design

2.1 ZM deep convection scheme

The ZM scheme has been used in the NCAR CAM since 1995 (Zhang and McFarlane 1995). In this scheme, moist convection occurs only when there is CAPE for which parcel ascent from the sub-cloud layer acts to destroy the CAPE at an exponential rate using a specified adjustment time scale τ (Neale et al. 2010). CAPE theoretically reflects the intensity of vertical motion due to the work of environmental buoyancy, and can reflect the overall structural characteristics of the atmosphere. The closure of the ZM scheme is:

$${M}_{b}=\frac{CAPE-{CAPE}_{0}}{\tau F}$$
(1)

where \({M}_{b}\) represents cloud bottom mass flux, \({CAPE}_{0}\) represents the CAPE threshold (70 J/kg) that triggers deep convection, τ represents a time scale of CAPE value exceeding \({CAPE}_{0}\) for convection consumption, and F is the CAPE consumption rate per unit cloud base mass flux. Adjustment time scale τ is the time when CAPE decreases to stabilize the atmosphere, which was first proposed by Fritsch and Chappell (1980). In the default configuration of many models, τ is specified as a constant.

When CAPE becomes larger, the convection will develop stronger, and the reaction time required for precipitation (the adjustment time scale τ) will be shorter. Also, when τ becomes shorter, the cloud base mass flux will be larger. The cloud base mass flux is positively correlated with precipitation (Zhang and McFarlane 1995). Based on these, we use CAPE to modulate τ, and realize a dynamic computation of the parameter, which aims to improve precipitation simulation.

2.2 Dynamic τ formulation

Zheng et al. (2016) modified the Kain–Fritsch scheme in the regional model, in which τ is inverse ratio with cloud depth-averaged vertical velocity (its calculation is related to entrained CAPE). Xie and Zhang (2000) proposed that τ be prolonged when the large-scale dynamic forcing is positive but the intensity is very weak. This is consistent with the conclusion of Wang and Randall (1996); they pointed out that τ is related to the change rate of CAPE caused by large-scale processes (including horizontal and vertical advection, radiation transfer), and that a larger CAPE change rate corresponds to a smaller value of τ, and vice versa. Based on these ideas and the characteristics of the ZM scheme itself, an inverse relationship exists between vertical velocity (w) and τ.

$${\uptau }\propto \frac{1}{w}$$
(2)

Vertical velocity is not an available parameter in the ZM scheme. CAPE is a significant parameter in the ZM scheme. Holton (1992) indicated that CAPE corresponds to the maximum vertical kinetic energy for a parcel of unit mass \({\text{W}}_{\text{max}}\) at EL (Equilibrium level). \({\text{W}}_{\text{max}}\) is a theoretical value, and the actual value does not exceed 50% of the theoretical maximum value. Therefore, we can write the maximum value of vertical velocity as follows:

$$\text{CAPE}=\frac{{\text{W}}_{\text{max}}^{2} }{2}$$
(3)
$$\text{W}=\frac{{\text{W}}_{\text{max}}}{2}=\frac{\sqrt{2\text{*CAPE}}}{2}=\sqrt{\frac{\text{CAPE}}{2}}$$
(4)

Based on the inverse relationship between vertical velocity (w) and \({\uptau }\) on convection mentioned above, when CAPE > 70 J/kg, the new function of \({\uptau }\) is proposed as follows:

$${\uptau }\left(\text{CAPE}\right)=\frac{\text{AW}}=\frac{\text{A}}{\sqrt{\frac{\text{CAPE}}{2}}}$$
(5)
$$\text{A}={{\uptau }}_{0}{W}_{0}$$
(6)

where A is a constant, \({\text{W}}_{0}=\sqrt{\frac{{\text{CAPE}}_{0}}{2}}\) (m/s), and \({\text{CAPE}}_{0}=70\) J/kg. When the deep convection is triggered (\(\text{CAPE}={\text{CAPE}}_{0}=70\) J/kg), the value of τ equals its maximum value \({ {\uptau }}_{0}\) = 3600 s (the default value of τ in CAM5). Constant A is calculated according to formula (6). In order to easily compare D_Tau with the original ZM scheme, the calculation method of dynamic τ is constructed in formula (5), meanwhile, the default empirical values of τ and trigger of deep convection in the original ZM scheme are used. In formula (6), \({\text{W}}_{0}=\sqrt{\frac{{\text{CAPE}}_{0}}{2}}\). The default τ value (\({ {\uptau }}_{0}=\) 3600 s) and trigger of deep convection (\({\text{CAPE}}_{0}=\) 70 J/kg) are adopted from the original ZM scheme (Zhang and McFarlane 1995) in CAM5. The differences of simulated precipitation are compared when τ using dynamic calculated value and original constant value.

According to formula (5), when the instantaneous CAPE reaches 10,000 J/kg, τ is about 300 s, which is a reasonable time scale for a severe deep convection.

2.3 Experimental design

Since the ZM scheme was proposed in 1995, different versions of the ZM scheme have been developed. At present, the latest version is the ZM scheme used in CAM6. The main differences between CAM6 and CAM5 is that the Cloud Layers Unified by Binormals (CLUBB) scheme (Golaz et al. 2002; Bogenschutz et al. 2013) replaces the earlier schemes for boundary layer turbulence, shallow convection and cloud macrophysics. There is a little change in the usage of τ in the ZM scheme between CAM6 and CAM5. In fact, it is also possible to modify τ in CAM6. In this study, we intended to analyze changes in convective precipitation, including both deep and shallow convection. Therefore, we turned off the CLUBB scheme in CAM6 and performed sensitivity experiments using CAM6. The experimental results of the CAM6 were similar to those of the CAM5. In order to compare with previous researches (e.g., Cui et al. 2021; Zhang and McFarlane 1995), we adopted CAM5 in this paper.

We use CAM5.3 and the experimental case F_2000_CAM5. In the control run (CTL), the deep convection scheme uses the ZM scheme (Zhang and McFarlane 1995); the shallow convection scheme uses the University of Washington (UW) shallow convection scheme (Park and Bretherton 2009); the boundary layer scheme is the diagnostic Turbulent Kinetic Energy (TKE) scheme (Bretherton and Park 2009); the scheme of cloud microphysics process used is described in Gettelman et al. 2008a, 2010a); the Rapid Radiative Transfer Method for GCMs (RRTMG) scheme (Iacono et al. 2008; Mlawer et al. 1997) is used as the radiation scheme; the model resolution is about 1° (0.9° × 1.25°), and 30 hybrid sigma-pressure levels with a time step of 1800 s. The experiment with the modified ZM scheme (Exp D_Tau) runs for six years, and the results of years 2–6 are analyzed. The monthly mean precipitation (1°, daily, 1997–2014) from the Global Precipitation Climatology Project (Huffman et al. 2001) is used to evaluate simulated precipitation. The National Centers for Environmental Prediction (NCEP)/NCAR reanalysis (Kalnay et al. 1996) is used to evaluate the Pacific Walker circulation. Student’s t test method is used to do statistical significant testing.

3 Results

3.1 Temporal and spatial features of dynamic τ

Through analyzing the spatial distribution of CAPE in CTL, we found that CAPE values are relatively large in some areas. Table 1 shows the mean CAPE in CTL averaged between 45° S and 45° N in JJA and DJF are 76.85 and 72.18 J/kg, respectively. In comparison, the regionally averaged CAPE in some regions are relatively large; e.g., 121.35 J/kg south of the Tibetan Plateau (70°–110° W, 25°–30° N) in JJA, 152.53 J/kg north of South America (50°–70° W, 0°–30° S) in DJF, 139.22 J/kg in Northern Hemisphere eastern equatorial Pacific Ocean (140°–80° W, 0°–10° N) in JJA, and 139.22 J/kg in the central Indo-Pacific Warm Pool (90°–150° E, 10° S–20° N) in JJA.

Table 1 Mean CAPE of different regions in CTL (units: J/kg)
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Figure 1 shows the pattern between τ and CAPE in Exp D_Tau. Unlike the original ZM scheme, τ changes over time in Exp D_Tau. When the instantaneous CAPE is 70 J/kg, τ equals 3600 s, which happens to be the default value in the original ZM scheme. When the instantaneous CAPE reaches 10,000 J/kg, the τ value is about 300 s, which is a reasonable value for extremely strong convection. The spatial distribution of τ in Exp D_Tau is shown in Fig. 2. The seasonal mean τ is calculated from the timesteps that deep convection occurs. When there is no convection or when the atmosphere is stable, τ is undefined. Compared to the default value of 3600 s in CTL, the τ value in Exp D_Tau significantly decreases in areas such as south of the Tibetan Plateau, north of South America, Northern Hemisphere eastern equatorial Pacific Ocean, the central Indo-Pacific Warm Pool, and southern Africa. Because these regions also correspond to large CAPE regions, after we use CAPE to modulate τ, τ becomes smaller in these regions. According to the statistical analysis results of CAPE (Fig. 3), compared to the plain, the CAPE value on the windward slope of high-altitude areas is larger, and the spatial variability of CAPE is also large, such as on the south side of the Tibetan Plateau and the east side of the Andes Mountains. In the improvement scheme, τ and CAPE are power functions, and their relationships are displayed in formulas (5) and (6). τ is sensitive to changes in CAPE. Therefore, dynamic τ also changes significantly in high-altitude regions. The areas with relatively large CAPE values and spatiotemporal variability are mainly located over the windward slopes. The use of dynamic τ leads to significant change of precipitation in these areas. Analysis of precipitation results is described next.

Fig. 1
figure 1

The pattern between τ and CAPE in Exp D_Tau

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Fig. 2
figure 2

Mean τ (shading; units: J/kg) of D_Tau in JJA (a) and DJF (b)

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Fig. 3
figure 3

Annual mean CAPE of D_Tau (a), CTL (b) and difference between D_Tau and CTL (c) (shading; units: J/kg). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line in c denotes the topography of 3000 m

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3.2 Improved precipitation and Walker circulation over tropical oceans

The overall performance of Exp D_Tau in simulating precipitation is evaluated by comparing with that of CTL. Figure 4a shows that the annual mean precipitation amount over the ITCZ, northern Australia, New Guinea, and some other islands around New Guinea is overestimated in CTL. The overall performance of D_Tau in simulating precipitation is evaluated with respect to that of CTL (Fig. 4a). Differences between CTL and D_Tau in terms of annual mean precipitation amount (Fig. 4b) indicate that D_Tau improves the positive biases in most of the above mentioned areas. Significant improvement can be found in New Guinea and eastern tropical Pacific (Fig. 4). By analyzing the change of large-scale precipitation and convective precipitation over the tropical oceans (Fig. 5), we found that large-scale precipitation decreases in this region (Fig. 5a), while deep convective precipitation mainly increases (Fig. 5c). Convective precipitation consists of deep convective precipitation and shallow convective precipitation. Deep convective precipitation is generated by deep convective processes, which occurs at altitudes above 600 hPa in the ZM scheme. Deep convective precipitation is calculated by using Zhang-McFarlane Scheme (Zhang and McFarlane 1995), and shallow convective precipitation is calculated by using University of Washington Shallow Convection Scheme (Park and Bretherton 2009).

Fig. 4
figure 4

Differences in annual mean precipitation (shading; units: mm/day) between CTL and GPCP (a), and between D_Tau and CTL (b). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line denotes the topography of 3000 m

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Fig. 5
figure 5

Differences in annual mean large-scale precipitation (shading; units: mm/day) between D_Tau and CTL (a), convective precipitation (b), and deep convective precipitation (c). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line denotes the topography of 3000 m

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Figure 6a presents differences in summer (JJA) mean precipitation amount between CTL and GPCP. It can be seen that the mean precipitation amount over the ITCZ, New Guinea, and some other islands around New Guinea is overestimated in CTL. Differences between CTL and D_Tau of summer mean precipitation amount show that D_Tau improves the biases around New Guinea, the central Indo-Pacific Warm Pool and eastern ITCZ (Fig. 6b). The improvement by D_Tau in winter (DJF) is also obvious. Compared with CTL, D_Tau reduces the biases of winter precipitation amount over the New Guinea, and Northern Hemisphere ITCZ (Fig. 6d).

Fig. 6
figure 6

Differences in summer (JJA) mean precipitation (shading; units: mm/day) between CTL and GPCP (a), and between D_Tau and CTL (b). c, d Are the same as a and b, respectively, but for winter (DJF). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line denotes the topography of 3000 m

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The Pacific Walker circulation is an important tropical atmospheric circulation system, which has a significant impact on climate anomalies in the tropics and many other regions of the world. The analysis results show that dynamic τ has a significant impact on the proportion of deep convection and large-scale precipitation in the equatorial Pacific, which is the region of the Pacific Walker circulation. The changes in vertical motion caused by dynamic τ and its interaction with the Pacific Walker circulation affect the vertical distribution of water vapor in the ascending and descending areas of the Pacific Walker circulation.

Using the zonal mass stream function over the equatorial Pacific (5° S–5° N) to indicate the intensity of the Pacific Walker circulation, we can see that the position and intensity of the strong center in CTL and D_Tau are similar (Fig. 7b, c). However, when comparing the negative centers on the west side of the strong positive center of the Walker circulation, CTL has two negative centers on both sides of 150° E, and D_Tau shows a westward shift negative center, which is closer to the results of the NCEP data. In addition, compared with CTL, D_Tau increases the intensity of the ascending on the west side of the Walker circulation and that of the descending on the east side.

Fig. 7
figure 7

Climatological zonal-vertical profile of Pacific Walker circulation of NCEP (a), CTL (b), and D_TAU (c). Shading and contour are annual mean zonal mass stream function (units: 1011 kg/s) along the equatorial Pacific (5° S–5° N). Vector is the composite of pressure velocity (\(\omega \times -100;\) units: Pa/s) and zonal divergent wind (units: m/s)

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Figure 8 shows the profile of annual mean specific humidity and vertical velocity difference between D_Tau and CTL in ascending region (150° E–180°, 5° S–5° N) and descending region (90°–120° W, 5° S–5° N) of the Pacific Walker circulation. It can be seen from Fig. 8a that the specific humidity at 600–800 hPa in the ascending region significantly increases in D_Tau, and the specific humidity below 400 hPa in the descending region significantly reduces. The dynamic τ significantly increases the upward vertical velocity below 400 hPa in the updraft region and the downward vertical velocity below 300 hPa in the descending region (Fig. 8b). Overall, specific humidity and vertical velocity differences of the ascending region in the middle and lower layers show opposite changes with those of the descending region. The increase of specific humidity in the middle layer and the increase of upward vertical velocity may result in an increase in the mean precipitation amount in the updraft region, while the decrease of specific humidity and the increase of downward vertical velocity may result in a decrease in the mean precipitation amount in the descending region.

Fig. 8
figure 8

Profile of annual mean specific humidity difference (units: g/kg) between D_Tau and CTL in updraft (dashed line) and downdraft region (solid line) of Pacific Walker circulation (a), vertical velocity (\(\omega \times -100;\) units: Pa/s) (b)

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When the lower atmosphere has conditional instability, there will be both convective scale updraft and corresponding downdraft. On the one hand, a parcel with convective available potential energy moves upward with buoyancy until it reaches a neutral buoyancy height. During this process, the vertical transport of convection redistributes water vapor and energy, and rolls out warm and moist air to affect environmental temperature and humidity. On the other hand, the compensatory downdraft generated by the environment fills the air quality gap caused by convective rise in the lower layers, causing large-scale environmental warming and drying, and then tending to an overall equilibrium state. This concept is the theoretical foundation of mass flux schemes and the most classic achievement related to convective parameterization (Arakawa and Schubert 1974; Kuo 1974; Manabe 1969; Zhang and McFarlane 1995). According to the seasonal mean τ and CAPE (Figs. 2, 3), it can be seen that the largest CAPE is located in the Pacific Walker circulation region (around 150° E) between 5° S and 5° N, corresponding to a sharp decrease in τ. It means that in this area, the parcels will achieve vertical water vapor and energy transport in a short period of time, rapidly increasing environmental temperature and humidity, and also enhancing the vertically ascending movement in the Pacific Walker circulation region. At the same time, the compensatory descending generated by the environment on the east side of Pacific Walker circulation region also plays a certain role in enhancing the Pacific Walker circulation. Therefore, the changes of dynamic τ can affect vertical motion, environmental temperature, water vapor, and the Pacific Walker circulation.

3.3 Improved precipitation simulation over steep terrain areas

Besides the tropical oceans, significant improvement can be found in some other regions. Because the modification has its greatest effect during summer when convection is prevalent, we mainly focus on the summer of each year. In summer, large rainfall amounts are found surrounding the southern periphery of the Tibetan Plateau in the GPCP data. With a horizontal resolution as high as 0.9° × 1.25° of the model, the topographical forcing process benefits from the resolved detail, and the model can depict the fine structure of orographic precipitation modulation. Both CTL and D_Tau reproduce the precipitation belt to the south of the Tibetan Plateau, with comparable spatial width of the GPCP (figures not shown).

To present the biases of the model simulations, we show differences in summer precipitation over Tibetan Plateau between CTL and GPCP over Tibetan Plateau in Fig. 9a. The precipitation amount of CTL has large positive bias compared to the GPCP, while the positive bias of D_Tau is significantly smaller. The most striking feature of the bias distribution is its close relation with topography. Comparing with the orographic height shown by red thin line (3000 m), the positive biases are located in highlands. Positive precipitation bias dominating the Tibetan Plateau has been a systematic error in many climate models for a long time (Yu et al. 2015). The largest overestimate of precipitation in this area is found along the southern edge of the Tibetan Plateau (Fig. 9a).

Fig. 9
figure 9

Differences in summer (JJA) mean precipitation (shading; units: mm/day) over the Tibetan Plateau between CTL and GPCP (a), between D_Tau and GPCP (b), and between D_Tau and CTL (c). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line denotes the topography of 3000 m

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Note that the excessive amount of precipitation near 3000 m (the red thin line denotes the topography of 3000 m) along the southern and eastern edge of the Tibetan Plateau is reduced in D_Tau (Fig. 9c). Next, we analyze changes in convective and large-scale precipitation separately. As shown in Fig. 10, the change in large-scale precipitation (Fig. 10a) presents similar spatial pattern with the decrease of mean precipitation around 3000 m in Fig. 9c. Deep convective precipitation increases around 3000 m and below 3000 m along the southern edge of the Tibetan Plateau (Fig. 10c).

Fig. 10
figure 10

Differences in summer (JJA) large-scale precipitation (shading; units: mm/day) over the Tibetan Plateau between D_Tau and CTL (a), convective precipitation (b), and deep convective precipitation (c). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line denotes the topography of 3000 m

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In the GPCP data, large rainfall is found northern South America, especially around the area of the Amazon Plain. Although both CTL and D_Tau reproduce the precipitation over northern South America, with comparable spatial width to the GPCP; large positive bias exists along the eastern edge of the Andes Mountains (figures not shown). However, compared to the precipitation amount bias between CTL and GPCP, the bias in D_Tau is smaller. Figure 11a shows the biases of precipitation simulated by CTL in South America during the austral summer. CTL overestimates the precipitation over the Andes Mountains, and underestimates the precipitation in some low-altitude regions on the northeastern plain. Clearly, D_Tau reduces both positive and negative biases (Fig. 11c). The negative biases in the northern plain of South America and the positive biases around the eastern edge of the Andes Mountains are improved. According to changes in convective and large-scale precipitation (Fig. 12), the changes of deep convective precipitation contribute to the increase of mean precipitation amount over northern South America. And the large-scale precipitation contributes to the decrease of mean precipitation amount around the Andes Mountains.

Fig. 11
figure 11

Differences in austral summer (DJF) mean precipitation (shading; units: mm/day) over South America between CTL and GPCP (a), between D_Tau and GPCP (b), and between D_Tau and CTL (c). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line denotes the topography of 3000 m

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Fig. 12
figure 12

Differences in austral summer (DJF) large-scale precipitation amount (shading; units: mm/day) over South America between D_Tau and CTL (a), convective precipitation (b), and deep convective precipitation (c). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line denotes the topography of 3000 m

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The root mean square errors (RMSEs) in different regions are shown in Table 2. Comparing with the RMSE between CTL and GPCP, the RMSE between D_Tau and GPCP is smaller in many regions, including the Tibetan Plateau (70°–110° W, 25°–40° N), northern South America (50°–70° W, 0°–30° S), Northern Hemisphere equatorial eastern Pacific Ocean (140°–80° W, 0°–10° N) and central Indo-Pacific Warm Pool (90°–150° E, 10° S–20° N). After the dynamic τ is adopted, the RMSE of mean precipitation amount in JJA over the Tibetan Plateau is reduced by 12.56%. And the results in Northern Hemisphere equatorial eastern Pacific Ocean (20.15%) and central Indo-Pacific Warm Pool (16.40%) are also impressive in JJA.

Table 2 RMSE of mean precipitation amount in different regions (units: mm/day). The percentage shows the reduction of RMSE between CTL and D_Tau
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As stated above, the areas with relatively large CAPE value and spatiotemporal variability are mainly located over the windward slopes. The use of the dynamic τ modulated by CAPE results in the increase of deep convective precipitation and the significant decrease of large-scale precipitation over high-altitude regions of the windward slopes. Overall, exaggerated precipitation in the highlands of CTL is reduced by using dynamic τ.

3.4 Improved simulations of intensity and frequency of deep convection precipitation

In this paper, the intensity of precipitation is defined as the average precipitation rate of all precipitation events with a specific intensity. The intensity of deep convective precipitation and annual mean frequency of deep convective precipitation amount are shown in Figs. 13 and 14, respectively. For the problem that models generally overestimate precipitation frequency and underestimates precipitation intensity mentioned in Sect. 1, the use of dynamic τ improves the precipitation frequency of deep convection and the simulation of precipitation intensity. Through the analysis of intensity of deep convective precipitation, we find that compared with CTL, D_Tau significantly increases the intensity of deep convective precipitation in most regions by at least 1 mm/day (Fig. 13); some regions show increase exceeding 3 mm/day. As can be seen from Fig. 14, compared with CTL, D_Tau increases the frequency of large deep convective precipitation (more than 20 mm/day) in the equatorial Pacific (Fig. 14a). The maximum increase of the frequency of deep convective precipitation exceeds 4%, almost reaching 50% of CTL. D_Tau also reduces the frequency of deep convective precipitation with precipitation less than 20 mm/day in many regions (Fig. 14c).

Fig. 13
figure 13

Differences in the annual mean intensity of deep convective precipitation amount (shading; units: mm/day) between D_Tau and CTL. The shading areas indicate that the differences are statistically significant at 0.05 level

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Fig. 14
figure 14

Annual mean frequency (shading; units: %) of deep convective precipitation amount exceeding 20 mm/day in CTL (a), difference between D_Tau and CTL (b). c, d Are the same as a and b, respectively, but for no more than 20 mm/day. The shading areas indicate that the differences are statistically significant at 0.05 level

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Fig. 15
figure 15

Climatological zonal-vertical profile of differences in temperature (units: K/day) (a), specific humidity (units: g/kg) (b), and vertical velocity (\(\omega \times -100;\) units: Pa/s) (c) averaged along − 5° S ~ 5° N between D_Tau and CTL

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3.5 Mechanisms behind the impact of dynamic τ on deep convective precipitation

According to the spatial distribution of deep convection precipitation in the Pacific Walker circulation region (− 5° S ~ 5° N) (Fig. 5) and the climatological zonal-vertical profile of differences in temperature, specific humidity and vertical velocity averaged along − 5° S ~ 5° N between D_Tau and CTL (Fig. 14), while the deep convection precipitation over this region increases, the feedback effect of convection-induced vertical heating on the large-scale circulation field (Arakawa 2004) is reflected in the significant increase of temperature above 850 hPa and the decrease in low-level temperature corresponding to the large-value region of deep convection. The enhancement of ascending motion in the ascending region (near 150° E) and the increase in water vapor at 400–850 hPa lead to abundant water vapor in this region. The large-scale precipitation has not been suppressed. However, the water vapor below 400 hPa in the descending area (west from 90° W) has significantly decreased and the ambient temperature has increased, which is not conducive to the occurrence of large-scale precipitation, resulting in decreases in large-scale precipitation in the descending area. This reflects the interaction between convection and large-scale circulation, and results in decreases or little changes in large-scale precipitation in some regions, that is, deep convective precipitation increases and large-scale precipitation decreases in D_Tau (Fig. 15).

In addition, the numerous mountains on the island of New Guinea act as a “thin wall”. Due to the abundant water and heat conditions near the equator, a small amount of uplift disturbance can produce heavy precipitation. The existence of the “thin wall” forces the wind carrying water vapor to rise after encountering a barrier. However, due to the insufficient thickness of the mountain body of the “thin wall”, the precipitation in this region may be a combination of two precipitation modes, both of which are generated by the uplift of airflow completely passing through the mountain terrain and the backflow of airflow on the leeward slope. According to spatial distribution of difference of deep convection and large-scale precipitation (Fig. 4), the increase in deep convection precipitation is caused by circulation around the New Guinea Island and nearby regions, and the decrease in large-scale precipitation is caused by topographic uplift. From the vertical velocity profile Fig. 14, it can be found that the ascending motion near 135° E weakened, which may be due to the compensatory effect of the enhanced ascending airflow on its eastern side. This is the reason that using the new scheme of dynamic τ will lead to significant changes in deep convection and large-scale precipitation near the island of New Guinea.

We investigate relationships among τ, CAPE, and deep convective precipitation to explore the mechanisms behind the impact of dynamic τ on precipitation simulation. The annual mean CAPE for CTL and D_Tau is calculated based on results of CTL and D_Tau for 2–6 years. CAPE is only calculated when deep convection occurs. By analyzing annual mean CAPE values of CTL and D_Tau between 45° S–45° N, we find that CAPE of the Tibetan Plateau in the model is 0 J/kg (Fig. 3), and the CAPE value in D_Tau is smaller in most regions (Fig. 3c). The mean CAPE in 45° S–45° N are, respectively, 61.07 and 74.47 J/kg in D_Tau and CTL, while the maximum values are 233.84 and 359.75 J/kg, respectively. Compared with CTL, the dynamic τ significantly reduces the annual mean CAPE of the Northern Hemisphere ITCZ by more than 40 J/kg (Fig. 3). Note that the CAPE of D_Tau is reduced by more than 120 J/kg at the coast of the eastern equatorial Pacific Ocean. Although the CAPE values of some regions in D_Tau decrease, it does not mean that the precipitation amount in these regions decreases. In fact, the change of CAPE means that the frequency and intensity of deep convective precipitation have changed. According to the seasonal mean τ and CAPE (Figs. 2, 3), it can be seen that in areas with large CAPE, τ decreases sharply. When deep convective precipitation occurs, CAPE is consumed significantly in a short period of time. As a result, the intensity of deep convection increases, the frequency of light deep convective precipitation decreases and the frequency of large deep convective precipitation increases. Therefore, the dynamic τ can affect the intensity and frequency of precipitation.

To explore the impact of dynamic τ on deep convective precipitation, we use a scatter plot of τ and deep convective precipitation amount difference between D_Tau and CTL by using the hourly outputs in year 2. The hourly data used to calculate the fitting curve of deep convective precipitation differences and τ differences between D_Tau and CTL has a relatively large amount for 2–6 years, so only one year is selected for calculation. In order to avoid the impact of the initial field, the data of the first year is not used. The differences between the second and the sixth year are not significant, so the data of the second year is used in analysis. And we use the least square method to fit the quadratic function curve. Figure 16a shows the original data points (hourly data of the second year) and the fitting curve. The upper x-axis represents the CAPE value calculated by the formula of dynamic τ, which corresponds to the difference of τ between D_Tau and CTL. Because the curve is very close to y = 0, the lowest point of the fitting curve and the intersection with y = 0 are not clear. Therefore, the enlargement of fitting curve in the red box of Fig. 16a without original data points is shown in Fig. 16b.

Fig. 16
figure 16

The fitting curve and the original data points (hourly data of the second year) of deep convective precipitation difference and τ difference between D_Tau and CTL (a), the enlargement of fitting curve in the red box of a, b. The upper x-axis represents the CAPE value in D_Tau (R2 = 0.4051, RMSE = 5.46)

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According to the fitting curve in Fig. 16b, when CAPE value is greater than 114.42 J/kg, the difference of deep convective precipitation is greater than 0 mm/day, and the difference of deep convective precipitation amount is greater when CAPE value in D_Tau gets larger. When CAPE value is less than 114.42 J/kg, the difference of deep convective precipitation amount is less than 0 mm/day. When CAPE value equals 82.80 J/kg, the difference of deep convective precipitation amount arrives at the minimum point around − 2 mm/day. In addition, the pattern of deep convective precipitation difference and CAPE value can also be approximated as a logarithmic function, that is, when CAPE value in D_Tau is greater, the difference of τ is greater (τ in D_Tau is smaller), the increase of the difference in deep convective precipitation amount between D_Tau and CTL is more dramatic.

As stated above, τ in Exp D_Tau is calculated by CAPE value, namely, the dynamic τ is affected by CAPE, resulting in the difference between dynamic τ and the default τ in the original scheme. Compared D_Tau and CTL, the change in CAPE has a non-monotonic effect on the change of deep convective precipitation. Therefore, CAPE = 114.72 J/kg can be used as the critical point to further analyze different effects in deep convective precipitation caused by the dynamic τ.

Figure 17a shows the spatial distribution of the annual mean CAPE value simulated in D_Tau, and the value less than 70 and more than 114 J/kg is shaded based on the results of the fitting curve. Comparing Fig. 17a with Fig. 17b, which shows the difference of the annual mean deep convective precipitation amount between D_Tau and CTL, we can see that the area with CAPE value greater than 114 J/kg corresponds to the positive center of the difference in deep convective precipitation amount. Specifically, the annual mean CAPE value is more than 114 J/kg in central Africa, the tropical Indian Ocean, and the areas extending from the islands near New Guinea along the Northern Hemisphere ITCZ to the equatorial eastern Pacific and northern South America, and the differences of deep convective precipitation amount in these areas are also beyond 0.5 mm/day. In the area with CAPE value of 70–114 J/kg, there is a negative center of deep convective precipitation difference, especially in the eastern equatorial Pacific region. Analyzing the CAPE output data of CTL in the first year indicates that the maximum values of CAPE over the Tibetan Plateau and Andes are at 1647.6 and 6975.6 J/kg, respectively.

Fig. 17
figure 17

Annual mean CAPE of D_Tau (shading; units: J/kg) (a), difference of annual mean deep convective precipitation amount between D_Tau and CTL (shading; units: mm/day) (b). The shading areas indicate that the differences are statistically significant at 0.05 level. The red thin line in b denotes the topography of 3000 m

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Dynamic τ has achieved improvement in some regions with steep terrain and low latitude areas where deep convection is prevalent, but its effect is not significant in India and some high latitude areas. From formula (5) of dynamic τ, it can be seen that the changes in τ mainly occur in areas with large CAPE or high variability of CAPE, thereby affecting the precipitation over these areas. According to the fitting curve shown in Fig. 16, the relationship between the difference in τ and the difference in deep convective precipitation is not a simple linearity, but a power function. If the CAPE ranges from 70 to 114 J/kg, the difference in precipitation decreases. It leads to more significant effect of the modified scheme in areas with larger CAPE, especially in areas with an annual mean CAPE exceeding 114 J/kg, where deep convective precipitation increases significantly, such as steep terrain areas and low latitude areas where deep convection is prevalent. Therefore, the dynamic τ scheme is not effective in areas where deep convection rarely occurs or in some inland plain areas, such as India.

In summary, although the CAPE value of most regions in D_Tau decreases, it does not mean that the deep convective precipitation amount in these regions decreases. This is because the change in CAPE has a non-monotonic effect on the change of deep convective precipitation. Dynamic τ in D_Tau is calculated by CAPE value, so it is affected by CAPE, resulting in the difference between dynamic τ and the default constant τ, that is, the time of duration of deep convection varies with time and space in D_Tau, resulting in changes in the intensity and frequency of deep convection precipitation. The impact of dynamic τ on deep convection processes can also cause changes in CAPE, that is, CAPE decreases in most regions. There is a critical point in D_Tau where the CAPE is equal to 114.72 J/kg. The regions with an annual mean CAPE value greater than 114 J/kg in D_Tau correspond to the positive centers of deep convective precipitation differences between D_Tau and CTL, while in regions with CAPE values ranging from 70 to 114 J/kg, there are negative centers of strong convective precipitation differences.

4 Conclusions and discussion

In this study, a new formulation for the dynamic convective adjustment time scale τ has been developed, and the performance of dynamic τ in the ZM deep convection scheme was tested and evaluated. We are motivated to realize the dynamic computation of the original constant parameter, and to improve the simulation of the general atmosphere circulation model on precipitation. Dynamic τ in D_Tau has effectively improved the precipitation deviation in most areas, including some steep terrain areas (such as Tibetan Plateau, Andes), New Guinea and its surrounding islands, especially in the tropical Pacific. Dynamic τ in D_Tau significantly reduces the positive deviation of CTL precipitation simulation in the Northern Hemisphere eastern equatorial Pacific Ocean. Dynamic τ in D_Tau also improves the Walker circulation by strengthening the western updraft and eastern downdraft.

The improvement of precipitation mainly comes from the change of precipitation properties. The change of τ value affects the reaction time of deep convective precipitation, thus the frequency of heavy precipitation, and the proportions of deep convective precipitation, large-scale precipitation, and shallow convective precipitation. In the region with improvement of mean precipitation bias, dynamic τ in D_Tau mainly increases deep convective precipitation, and reduces large-scale and shallow convective precipitation amounts. Dynamic τ in D_Tau increases the frequency of large deep convective precipitation (more than 20 mm/day) in the equatorial Pacific and reduces the frequency of deep convective precipitation with precipitation less than 20 mm/day. At the same time, dynamic τ reduces CAPE in most regions.

The impact of dynamic τ on the precipitation of deep convection significantly corresponds to the spatial distribution of CAPE value in the model. Exp D_Tau has the most significant improvement over the tropical oceans, probably because the CAPE value over the tropical oceans is larger than that over the other regions. When CAPE is used to modulate τ, the τ over the ocean changes obviously, which has a great impact on the precipitation in this region. Also, the large CAPE and its spatiotemporal variability over steep terrain regions may be the reason for the improvement over the Tibetan Plateau and Andes. The areas with relatively large CAPE values and spatiotemporal variability are mainly located over windward slopes. The use of dynamic τ modulated by CAPE leads to an increase in deep convective precipitation and a significant decrease in large-scale precipitation over windward slopes of the highlands. Overall, exaggerated precipitation in the highlands is reduced in the model. Although τ value in D_Tau is always smaller than the default in the CTL, deep convective precipitation does not increase globally, and there is a decrease of deep convective precipitation in some regions. The reasons for some regions where deep convective precipitation does not increase may be as follows: the calculated value of τ is not small enough because the value of CAPE in this region is understated by the model; in CAM5, the contribution of τ in this region is small, and the change of τ is not able to completely affect the precipitation in this region; and the relationship between the variation of deep convective precipitation and CAPE (or the variation of τ) is not monotonous. Through experiments at different resolutions (results are not shown), it was noticeable that the effect of dynamic τ became better when spatial resolution was higher. Also, through comparative analysis of inter-annual and inter-decadal precipitation results of dynamic τ, we found that the improvement by dynamic τ was still effective (figures not shown).

We also analyzed the diurnal cycle of deep convection precipitation based on our experimental results. It was found that using dynamic τ did not significantly change the peak time or phase of the simulated daily cycle of deep convection precipitation, but enhanced the intensity or amplitude of the simulated daily cycle of deep convection precipitation in the afternoon of local time (Figure not shown). The reason probably is that the focus of our work is to import vertical velocity (Equilibrium level maximum vertical velocity) into the ZM scheme, which is a power function of CAPE, in order to improve precipitation intensity. Cui et al. (2021) indicated that the improvement of diurnal precipitation cycle mainly relies on the modification of convective triggering mechanisms and thresholds. The results of diurnal cycle of deep convection precipitation indicate that dynamic τ mainly enhances the precipitation intensity in the afternoon of local time, which is consistent with the conclusion that dynamic τ improves the simulation of precipitation by increasing the intensity of deep convective precipitation.

Different from the original ZM scheme, the modified scheme uses dynamic parameter τ. Dynamic calculation of characteristic adjustment time scale parameters can help improve precipitation simulation. However, dynamic τ is not effective in some regions, which may be related to the small CAPE value in these areas. The constant used in the function of modified scheme is based on the default τ value in CAM5. More experiments are needed to test whether it is the most suitable value for this scheme. In addition, whether the scheme can be applied to other models with higher resolution on the premise of ensuring stability, so as to improve the simulation of extreme precipitation also needs to be verified. Bayr et al. (2020) pointed out that biases in the mean state of the rising branch of the Pacific Walker circulation plays an important role for the underestimated El Niño-Southern Oscillation (ENSO) atmospheric feedbacks. The impact of the ZM scheme on the Walker circulation may also have impact on the ENSO. The modified ZM scheme can be applied in coupled ocean-atmosphere models in future research, which may lead to more new findings.