Satellite observations are obtained from both the Tropical Rainfall Measuring Mission (Huffman et al., 2007) version 3B42 (TRMM 3B42, NASA, 2017) and the Global Precipitation Measurement (GPM, Skofronick-Jackson et al., 2017; NASA, 2018) mission data (Level-3). TRMM 3B42 has a high temporal coverage (1998–2014) which facilitates a climatological overview of the diurnal cycle over Lake Malawi. However, the spatial resolution of TRMM 3B42 (0.25∘) prohibits its use in the analysis of the spatial characteristics of the diurnal cycle over the lake and its shores. This difficulty is overcome by using GPM, the successor to TRMM 3B42, which has a higher spatial resolution of 0.1∘. This facilitates a more detailed study of spatial variations in the diurnal cycle of precipitation. The temporal resolution of the original GPM Level-3 data is every 30 min, which is averaged to hourly rainfall in this study.

The Advanced Research version of the Weather Research and Forecasting (hereafter referred to as WRF, Skamarock et al., 2008) model version 3.9.1 is used to investigate the diurnal variations around Lake Malawi. The domains used in all simulations are shown in Fig. 1a. The outer domain covers south-eastern Africa, -20.74902 to -2.958107∘ S and 23.3115 to 44.0885∘ E with 15 km grid spacings (171×117 grids), and the inner domain is centred on Lake Malawi, -15.87943 to -8.219772∘ S and 32.22042 to 37.06839∘ E with 3 km grid spacing (155×250 grids), respectively (Fig. 2a). Both domains have 56 vertical layers. The outer domain is forced laterally with 6-hourly ERA-Interim (Dee et al., 2011; ECMWF, 2018) data which have a grid spacing of 0.75∘ and at the lower boundary by the daily optimum interpolated sea surface temperature (OISST, Reynolds et al., 2007), which has a grid spacing of 0.25∘. The inner domain is forced laterally by the outer domain of WRF (the outer domain of WRF does not interact with the inner domain).

The following physical schemes are used in our WRF simulations: the WRF Single-moment (WSM) six-class scheme for microphysics (Hong and Lim, 2006) and the Yonsei University parameterization for the planetary boundary layer (PBL; Hong et al., 2006). The longwave and shortwave radiative forcings are parameterized by the Rapid Radiative Transfer Model (Mlawer et al., 1997) schemes. The Betts–Miller–Janjíc (Janjíc, 1994) scheme is used for parameterizing convective processes in the outer domain only; cumulus parameterization is switched off in the convection-permitting inner domain. A study of the sensitivity of precipitation in this region to the convective schemes used in the outer domain showed that simulations using the Betts–Miller–Janjíc scheme reproduced the observed precipitation over land better than simulations using the Kain–Fritsch (Kain, 2004) scheme (not shown). Therefore, the Betts–Miller–Janjíc scheme is chosen for the outer domain in this study with no cumulus scheme used in the inner, high-resolution domain. Over the land and lake grids that are based on MODIS land-use data, the NOAH land surface model consisting of four layers (Chen and Dudhia, 2001a, b) and the nine-layer lake model (Xu et al., 2016) are implemented and air–land/lake interactions are active in the simulations.

With the model configurations above, a control experiment is initialized on 1 January 2014 at 00:00 UTC of ERA-Interim for the atmosphere and land surface and integrated until 1 April 2015 (referred to as WRF-CTL hereafter). This run will complement the observations to gain insights into the diurnal variations around Lake Malawi. In a second experiment, the grid boxes over Lake Malawi are converted from water to land grid boxes (Fig. 1b and c). This facilitates an exploration of the role of Lake Malawi in the local diurnal variations (called WRF-NOLM in the rest of the paper). Due to this conversion, some land surface properties are modified in WRF-NOLM: the land-use index of the converted grids is set to be savanna, which is the most dominant land-use category in the inner domain of WRF experiments. The soil type of the converted grids is also replaced with sandy clay loam, which is the majority soil type for the savanna grids in the inner domain. Additionally, the surface albedo over Lake Malawi grids is set to a value of albedo averaged over the savanna grids in the inner domain. Finally, the soil moisture and temperature of the converted grids are initialized by the averaged values of the savanna grids. These modifications are done only in the inner domain to avoid any modulations in larger-scale meteorological and hydrological quantities associated with the absence of Lake Malawi. All settings of the outer domain of WRF-NOLM are exactly the same as those of WRF-CTL.

We analyse the hourly output of the 5 months from November in 2014 to March in 2015; that is, the first 10 months are designated as a spin-up period for initializing the land surface following the methodologies of Cosgrove et al. (2002) and Chen et al. (2007). In particular, in WRF-NOLM, the soil moisture and temperature are initialized with an averaged value, which is to a large extent artificial. Therefore, a long spin-up period is employed for initializing the land surface.

Harmonic analysis has been widely used to quantify the main characteristics of the diurnal cycle (e.g. Yang and Slingo, 2001; Diro et al., 2012; Mooney et al., 2017). One particular advantage of harmonic analysis is the estimation of the explained variance (%) of a specific frequency and its phase from a time series. This study follows Mooney et al. (2017) by fitting the following function to the NDJFM-averaged hourly data: 1R(t)≅a24cos⁡2π(t-ϕ24)24+a12cos⁡2π(t-ϕ12)12, where R(t) is the hourly variation of total rainfall and a24 (a12) and ϕ24 (ϕ12) are the amplitude and phase of the diurnal and semi-diurnal cycles, respectively.

The empirical orthogonal function (EOF) analysis is additionally used to capture the features of the diurnal cycle around Lake Malawi following previous studies (e.g. Kikuchi and Wang, 2008; Teo et al., 2011).

The EOF analysis is used to identify the dominant spatio-temporal patterns. For the diurnal cycle, it is known that the first mode represents a stationary dipole pattern between coastal land and ocean, while the second mode identifies a propagation pattern from land to sea (the EOF patterns and principle component scores between the first and second modes are out of phase by approximately π/4, e.g. Kikuchi and Wang, 2008; Teo et al., 2011). Employing these statistical methodologies, we will explore the details of the observed and modelled diurnal cycle around Lake Malawi in Sects. 3.2 and 3.3. The EOF analysis is adopted into the diurnal deviation components with 2A′(t)=A(t)-A‾, where A is a variable and t is time (hourly). The overbar and prime denote the daily-mean and daily-deviated components, respectively.

Firstly, we take an overview of the climatological diurnal cycle of precipitation around Lake Malawi using TRMM 3B42, which has good temporal coverage but relatively coarse resolution (temporarily and spatially). Figure 2 illustrates the 3-hourly precipitation obtained by TRMM 3B42 for NDJFM-mean climatology. Between 00:00–03:00 and 06:00–09:00 UTC (02:00–05:00 to 08:00–11:00 LST), the precipitation over Lake Malawi is enhanced and the precipitation over the surrounding land area becomes weaker. At 09:00–12:00 UTC, the precipitation is suppressed over the entire area. Later, from 12:00 to 15:00 LST, precipitation is activated over the land surrounding Lake Malawi. The land precipitation intensifies widely at 15:00–18:00 UTC, while rainfall over Lake Malawi is negligible. From 18:00–21:00 to 21:00–00:00 UTC, the land precipitation is gradually reduced and precipitation over Lake Malawi commences. That is, around Lake Malawi there is a well-organized diurnal variation in precipitation. Interestingly, the magnitude of land and lake precipitation is almost identical (0.9 mm h-1).

In this subsection, the more detailed nature of the diurnal cycle, which is indicated in the preceding subsection, is investigated with a finer-resolution satellite product and numerical simulation for a case study of November to March in 2014/15. Figure 3a–e show monthly-mean rainfall for GPM from November to March. In November, the daily rainfall around Lake Malawi is low compared to the other months. There is little rainfall over the southern part of Lake Malawi, but there is some intense rainfall over the northern part of the lake. Rainfall becomes more intense in December, particularly over the centre of Lake Malawi. Precipitation peaks in January and is very intense in the entire domain, with rainfall over Lake Malawi reaching ∼22 mm h-1. From February to March, the precipitation over land decreases while the lake precipitation over the lake remains strong, especially in the central area (around 18 mm h-1). The precipitation over Lake Malawi is not distributed homogeneously, but it seems that there is a dependency on location: the precipitation is intense in the central part of the lake in December to March; in particular, the precipitation spreads broadly around the centre of the lake. On the northern and southern edges of the lake, there are also moderate peaks of the precipitation in February and March. These distributions might be determined by several factors (for example, lake surface temperature), which is a highly complex process and beyond the scope of this study. Figure 3f–j show that WRF-CTL can capture the seasonal march of larger-scale precipitation. However, the land precipitation tends to be overestimated, in particular from January to February. This overestimation might be due to the high topography (higher than 2300 m) around Lake Malawi (see Sect. 4). WRF-CTL successfully reproduces the intense lake precipitation from November to March.

Figure 4 shows the key characteristics of the diurnal cycle of precipitation obtained by harmonics analysis (see Sect. 2.3) for NDJFM-mean hourly data of GPM and WRF-CTL. Over Lake Malawi, the GPM-observed sub-daily variations are dominated by the diurnal cycle as shown in Fig. 4a (about 70 %–80 % of explained variance). Other dominant diurnal cycles are seen along the coast of Lake Malawi and to the north-east of Lake Malawi, around 10∘ S and 35–36∘ E, with a similar explained variance. In WRF-CTL the dominant diurnal variations are captured well over Lake Malawi, with 60 %–70 % of the explained variance in Fig. 4e. Although the strength of the diurnal signal over the coastal region tends to be underestimated to some extent, the terrestrial diurnal cycle is well represented in WRF-CTL in terms of the explained variance.

The largest amplitudes of the diurnal cycle (a24 in Eq. 1) are observed over Lake Malawi (up to 0.5 mm h-1) and its coastal region (Fig. 4b). Over land, the amplitude is relatively large to the north-east of the lake (0.2–0.3 mm), where the diurnal cycle dominates the sub-daily variations (Fig. 4a). This distribution of amplitude is fairly well simulated by WRF-CTL in Fig. 4f. However, over Lake Malawi, the amplitude is lower than observed, while the amplitude over land to the north-east is too large (0.5 mm). This is consistent with the overestimated monthly-mean precipitation in Fig. 3. The observed phase of the diurnal cycle (ϕ24 in Eq. 1) shows a clear contrast over the lake and land in Fig. 4c; the maximum peak of the precipitation is at 02:00–03:00 UTC over the lake and surrounding coastal area and at 13:00–14:00 UTC over the land north of the lake where terrestrial precipitation is relatively large (Fig. 4a and b). This result is consistent with the climatological overview in the previous subsection. The timing of the WRF-simulated diurnal cycle in Fig. 4g agrees reasonably with that of the observations. Over the lake, the peak time is slightly late, especially in the south (at 03:00–05:00 UTC) compared to the observations, and the land precipitation is maximized at 13:00–14:00 UTC to the north of the lake. However, over the central–eastern coastal region, the timing of the rainfall is incorrectly simulated.

In Fig. 4d and h, the explained variance of the semi-diurnal cycle is given for GPM and WRF-CTL. Neither product shows a clear semi-diurnal cycle around Lake Malawi, although there are some spots with a relatively high variance of 40 %–50 %. These results suggest that the sub-daily variations in rainfall are mainly associated with the diurnal cycle over and around the lake, while the semi-diurnal cycle is almost negligible.

Figure 4i–l show the characteristics of the diurnal cycle of precipitation calculated by the harmonic analysis (Eq. 1) for WRF-NOLM. Compared to WRF-CTL (Fig. 4e), the explained variance of the diurnal cycle is almost identical around Lake Malawi, in particular to the north-east of the lake. Over the lake, the variance of the diurnal cycle is reduced remarkably in the southern part of the lake, which drops down to 20 %–30 % in Fig. 5i (50 %–60 % in WRF-CTL, shown in Fig. 4e). To the north of the lake, the diurnal cycle persists despite the absence of the lake. However, the amplitude of the diurnal cycle shrinks over the entire lake in Fig. 4j. Most notably, the reduction is largest in the central part and the northern part of the lake (a decrease from 0.5 to 0.1–0.2 mm h-1) even though the variance of the diurnal cycle is still comparable to the WRF-CTL case. Over land, the diurnal amplitude is largely unchanged when Lake Malawi is removed; this is most evident overland to the north-east of the lake. The phase of the diurnal cycle is also modified over the lake. Its peak is slightly earlier (around 02:00–03:00 UTC) than WRF-CTL (comparison between Fig. 4g and k). On the southern shore of the lake (where the diurnal cycle almost disappears), the phase is noisy with respect to WRF-CTL. The component of the semi-diurnal cycle is almost identical to that in WRF-CTL and the semi-diurnal cycle is not of importance in the sub-daily variations (Fig. 4l).

The dominant spatio-temporal pattern of variation is provided through the EOF analysis in Fig. 5. The EOF first mode of GPM shows a clear contrast between the land and lake (Fig. 5a). The amplitude is larger over the lake than over the land, suggesting that the lake rainfall is more intense than the land rainfall. The coastal land rainfall synchronizes with the lake rainfall on both the eastern and western shores. This mode explains 53.69 % of the total variance and its principal component (PC) score (Fig. 5h) shows a distinct diurnal cycle. The peak of rainfall over land is between 12:00 and 17:00 UTC and that of the lake rainfall over the lake is between 23:00 and 03:00 UTC. This seesaw pattern of daytime rainfall over land and nighttime rainfall over the lake is quite similar to the pattern described by sea–land contrast in the tropics (e.g. Teo et al., 2011; Bhatt et al., 2016). The EOF second mode has 15.77 % of the total variance and its spatial pattern and PC score do not indicate a propagation mode from land to lake (not shown). The PC score seems a semi-diurnal cycle, and the spatial pattern is quite spotty and appears to be unrelated to Lake Malawi. Its amplitude is considerably smaller than that of the first mode.

WRF-CTL represents well the sharp contrasting spatial pattern between the land and lake in Fig. 5b as an EOF first mode (the explained variance is 41.51 %). However, as shown in Figs. 3 and 4, the amplitude of the land precipitation is overestimated and coastal terrestrial rainfall synchronizing with the lake rainfall does not spread widely compared to the observation, although there is some coastal land precipitation occurring simultaneously with the lake precipitation. While the PC score of the first mode is roughly consistent with that of observation (Fig. 5h and i), the phase is somewhat shifted: the peak of the nighttime rainfall is around 03:00–07:00 UTC (later than the observation) and that of daytime is around 12:00–14:00 UTC, which is slightly earlier than the observation. In particular, the earlier simulated peak in the daytime precipitation is a common issue in regional climate modelling (e.g. Nikulin et al. 2012; Pohl et al., 2014; Mooney et al., 2016, 2017; Koseki et al., 2018). Similar to the GPM observations, WRF-CTL does not show any clear propagation mode by the second mode and the large variation is limited in some small areas (its variance is 18.36 %), although the PC score of the second mode is lagged by approximately π/4 (not shown).

The modelled surface zonal wind shows an interesting distribution by the EOF first mode in Fig. 5c: the lake shore is encompassed by the narrow bands of the negative and positive daily anomalies of surface zonal wind (77.88 % of the total variance), and those bands spread over Lake Malawi. Combined with its PC score it can be interpreted that the outgoing flow from the lake is maximized between 09:00 and 13:00 UTC (Fig. 5i) and the incoming flow into the lake is dominant between 21:00 and 03:00 UTC. This diurnally varying circulation is consistent with a well-characterized lake–land breeze (e.g. Keen and Lyons, 1978; Crosman and Horel, 2010). The PC score of the surface zonal wind leads that of the precipitation by approximately 3 h. The surface meridional wind also shows a remarkable pattern by the EOF first mode (61.46 % of the total variance) in Fig. 5d: with a macroscopic view, there is a dipole mode of positive in the north and negative in the south of Lake Malawi. Combining it with the PC score (Fig. 5i), there is an outgoing/incoming flow of meridional surface wind during daytime/nighttime, respectively. The EOF first mode of meridional wind varies approximately with the zonal wind as shown in Fig. 5i.

The EOF first mode also shows substantial changes in the diurnal cycle in WRF-NOLM as shown in Fig. 5e–g; the dipole pattern between the lake and surrounding terrestrial area almost disappears in the EOF first mode and the dominant variability is only over the land in Fig. 5e. The variance is still 35.60 % and the amplitude over the land is almost identical to that of WRF-CTL in Fig. 5b. While the harmonic analysis estimates the diurnal cycle independently at each grid cell, the EOF analysis calculates the most explainable variability in all the selected grids and, therefore, the amplitude at one grid would be affected by that at other grids. That is, in Fig. 5e, the variabilities at the lake grids are much smaller than those at land grids, which is consistent with the reduced amplitude of diurnal variation over the lake in Fig. 4j. The PC score indicates that the EOF first mode is a diurnal cycle in Fig. 5j with some modification in its peak time. Whereas the EOF first modes of surface zonal winds have the two narrow bands along the lake shore in WRF-NOLM (74.53 % of the total variance), their spreads over the lake are largely diminished on both sides of the lake shore with respect to that in WRF-CTL (Fig. 5c and f). The magnitudes of the WRF-CTL (Fig. 5i) PC scores are similar to those for WRF-NOLM (Fig. 5j) and the maximum and minimum of the PC scores for both WRF-CTL and WRF-NOLM occur during the day and the night, respectively. Similarly, the variability in surface meridional wind is also reduced over the lake as shown in Fig. 5g. However, there is still some evidence of a dipole pattern between the northern and southern parts of Lake Malawi, as shown in WRF-CTL (Fig. 5d). However, the maximum of PC scores for the meridional wind occurs slightly earlier in the WRF-NOLM (Fig. 5j) simulation compared to WRF-CTL (Fig. 5i).

As witnessed by the harmonic and EOF analyses above (Figs. 4–5), Lake Malawi plays a crucial role in the generation and/or amplification of the diurnal cycle of precipitation. At certain times in the day, the lake's role can be clearer than at other times (Fig. 6). During 00:00–03:00 UTC, the nocturnal precipitation occurs over Lake Malawi in WRF-CTL (Fig. 6a), but this lake-anchored precipitation is extensively reduced in WRF-NOLM (Fig. 6b). Its influence is remarkable over the entire lake, in particular over the northern and central parts of the lake (Fig. 6c). This indicates the importance of Lake Malawi for rainfall over the lake (as concluded by Diallo et al., 2018). Conversely, the surrounding area of the lake experiences a modest reduction in precipitation in the presence of the lake during midnight to early morning. During daytime when the precipitation peak is closely tied to the maximum in local solar heating (11:00–14:00 UTC), precipitation is more dominant over the surrounding area of the lake than over the lake in WRF-CTL (Fig. 6d). While precipitation over the lake is quite small, there is some increase in the precipitation over the southern part of the lake in WRF-NOLM (Fig. 6e). In contrast to the nocturnal precipitation, daytime precipitation is amplified over the southern part of the lake, although its response is relatively weaker than that in the nighttime (Fig. 6c and f).

Figure 7 presents the surface horizontal wind and its divergence anomalies from the daily mean at nighttime and daytime, estimated by Eq. (2). In WRF-CTL, the incoming flow from the shore toward the lake is detected and the strong convergence forms over the lake shown in Fig. 7a. These lakeward flows are land breeze circulations and penetrate deeply into the lake, as shown by the EOF analysis (Fig. 5c and d). The intense nocturnal rainfall (as in Fig. 6a) can be attributed to this strong convergence over the lake. In WRF-NOLM, the land breezes are extensively weakened and, as a result, the convergence over the lake shrinks considerably (Fig. 7b). The difference shows clearly that the intensification in the land breeze and convergence is due to Lake Malawi (Fig. 7c). While the daily-residual component of the surface wind can be seen not only around the lake, but also in the region (Fig. 7a and b), the influence of the lake on the wind seems to be limited around and over the lake. During daytime, on the other hand, the outgoing flows and thus lake breezes are organized well from the lake outward, and this flow is highly divergent over the lake in WRF-CTL (Fig. 7d). This outgoing circulation can also be seen in WRF-NOLM (Fig. 7e), but its magnitude is considerably reduced and the flow-forming divergence is also reduced. The difference during daytime is almost a mirror image of that during nighttime and it shows that Lake Malawi plays an important role in the diurnal variations of local wind circulations. The lake surface seems to create a heat contrast favouring the lake–land breeze circulation in nighttime and daytime: the surface temperature over the lake is higher in WRF-CTL (25.7 ∘C) than in WRF-NOLM (24.8 ∘C) during nighttime and lower in WRF-CTL (26.8 ∘C) during the daytime than in WRF-NOLM (32.8 ∘C). This behaviour in the surface temperature can create favourable conditions for more convergence (divergence) and, consequently, the precipitation over the lake is enhanced (suppressed) effectively.

The preceding subsections have shown that Lake Malawi radically drives the diurnal cycle in precipitation and local circulations. Since the moisture flux, Uq (here, U is horizontal wind vector and q is specific humidity at the surface), due to the lake–land breeze circulations can be highly related to precipitation, we quantify the surface moisture flux and its diurnal variation. Note that 10 and 2 m data are used for horizontal wind and specific humidity in this study. The moisture flux can be subdivided into four components as follows: Uq(t)=U‾+U′(t)⋅q‾+q′(t)=U‾q‾+U‾q′(t)+U′(t)q‾+U′(t)q′(t), where the overbar and prime denote daily mean and daily deviation as Eq. (2). U is the surface wind vector and q is the surface-specific humidity. The horizontal divergence of moisture flux is calculated as 3∇⋅Uq(t)=∇⋅U‾q‾︸A+∇⋅U‾q′(t)︸B+∇⋅U′(t)q‾︸C+∇⋅U′(t)q′(t)︸D. The term A is the moisture flux divergence/convergence due to daily-mean wind and humidity, which do not have diurnal variation, but its relevance is more to the moisture flux associated with the Indian Winter Monsoon over this region. The term B reflects the influence associated with the diurnal variation in the heat flux and the background wind. The term C indicates the contribution due to the lake–land breeze and the daily-mean humidity to the moisture flux divergence/convergence. The final term is attributed to the diurnal variations in local breeze and humidity. Since the term A does not contain any temporal change, only the three terms of B, C, and D are averaged over Lake Malawi and the surrounding area as shown in Fig. 8a.

During nighttime, the moisture flux converges over Lake Malawi and diverges over the surrounding area mainly by the lake–land breeze circulation and background humidity in WRF-CTL (term C in Fig. 8b and c). The daily-mean (background) latent heat flux averaged over the lake grids is 155.2572 and 56.9174 W m-2 for WRF-CTL and WRF-NOLM, respectively, and the lake surface is an important water source of the local precipitation, depending on the wind conditions and other characteristics (e.g. topography). The intense moisture flux convergence is responsible for the nocturnal precipitation as shown in Fig. 8c. Other terms in Eq. (3) do not substantially contribute to the moisture flux divergence/convergence. In WRF-NOLM, the diurnally varying breeze and background humidity also contributes to the moisture flux convergence/divergence, but its magnitude is much smaller than that in WRF-CTL in Fig. 8b and c. Consequently, the precipitation over the lake area is reduced without Lake Malawi. As shown in Fig. 6c, the precipitation surrounding the lake is somewhat enhanced in WRF-NOLM during nighttime (although the response of the rainfall is noisy and weak, the consistency with the response of the moisture flux is reasonable). During daytime, the lake–land breeze and background humidity are still the main driver of the moisture flux divergence/convergence over the lake and surrounding area in Fig. 8d and e. Without the lake, the divergence over the lake and convergence over the lake shore are weakened, which is consistent with the enhanced (reduced) daytime rainfall surrounding (over) the lake in WRF_CTL in Fig. 6f. The term C is mainly contributed by the zonal component, ∂(uq)/∂x, which is about 70 % to 80 % of the total divergence/convergence (not shown).

In both cases of nighttime and daytime, the other terms of B and D in Eq. (3) do not contribute to the diurnal changes in moisture flux divergence/convergence. That is, the land–lake breeze and the enriched background water vapour due to Lake Malawi mainly drive the diurnal variations in surface moisture flux and, consequently, the precipitation around Lake Malawi.