We perform simulations using the Weather Research and Forecasting model coupled with chemistry (WRF-Chem) version 4.4.1 . WRF-Chem is coupled with a budget calculation tool WRFlux which provides flux computations and online time averaging. The use of WRFlux is further discussed in Sect. .

The model domain dimensions are 200 km in the y direction, 40 km in the x direction and the model top is at the height of 12 km. The horizontal grid spacing is 200 m and the simulations are run with a 2 s time step. Sensitivity of the model results and overall conclusions to the horizontal grid spacing was tested with an additional simulation run with a horizontal grid spacing of 100 m and a time step of 1 s (see Appendix for more details). The 50% reduction in the horizontal grid spacing results in a negligible change in the valley-volume-averaged fields or the general flow structure. The domain has 100 model levels with the lowest model level at 15 m above the surface and the model level spacing is less than 100 m below the height of 3 km (Supplement Fig. S1a). The domain has a 5 km deep w-Rayleigh damping layer at the top. The lateral boundaries are symmetric in y direction (along-valley) and periodic in x direction (cross-valley). The lateral boundary conditions in WRF-model are described in Chapter 6 in . The idealised valley topography shown in Fig.  is defined by Eq. () given in Appendix . The valley is south-north oriented, 100 km long, 20 km wide (ridge-top to ridge-top distance), 2 km deep (ridge-top height) and the cross-valley shape is defined by a cosine function. In the along-valley direction, the other half of the model domain (y<0 km) is flat (i.e. is a plain), also 100 km long, except the gradual rise of the ridges near the valley entrance (0 km>y≥-12 km).

The simulations are run in LES-mode (large-eddy simulation) with a three-dimensional 1.5-order TKE (turbulent kinetic energy) closure, third order Runge-Kutta time-integration, fifth order horizontal advection, and third order vertical advection of both momentum and scalars. No boundary layer scheme is used in the simulations. Surface layer physics are parameterised using Janjic Eta similarity scheme . Land surface physics are parameterised using the unified Noah land surface model which has 4 soil layers. Microphysics are parameterised using the Purdue-Lin scheme , however no clouds are formed during the simulations. Shortwave and longwave radiation is parameterised using the RRTMG scheme . Terrain features affecting the radiation (slope angle and topography shadowing) are considered and the radiation is updated every 60 s. The latitude of the domain is set at 28 degrees North, and the simulation is initialized for the equinox, when daylight is 12 h.

Two simulations are performed, one with aerosol-meteorology feedback and one without. These simulations are referred to as AER and CTRL, respectively. Both simulations are 24 h long starting at 06:00 local time in the morning. Henceforth, all times are local time. The simulations start from the same meteorological state, which is the result of a 24 h spin-up run without the aerosol-meteorology feedback. Case CTRL continues from the spin-up without any changes. Case AER continues from the spin-up, but now with the aerosol-meteorology feedback being switched on. The chemistry setup of case AER is described in the following paragraphs. The spin-up simulation is initialised with a stable atmosphere using a temperature of 295 K and a pressure of 1000 hPa at the altitude of 0 m, a constant Brunt-Väisälä Frequency of 0.11 s-1 (Fig. S1b). The initial wind components and relative humidity are set to zero throughout the domain. The land use category and vegetation type are defined as evergreen needle-leaf forest uniform across the whole domain. Soil moisture is initialised as 0.1 m3m-3 in all 4 soil layers. This causes a minimal moisture flux, leading to a surface latent heat flux at maximum of 10 Wm-2 (Fig. S2c). The simulation is relatively dry on purpose to avoid cloud formation, and in fact the surface sensible heat flux is one or two order of magnitude stronger than the latent heat flux during the simulation (Fig. S2b–c).

The chemistry is described by the GOCART (Goddard Chemistry Aerosol Radiation and Transport) simple aerosol scheme (chem_opt=300 in WRF-Chem). As we focus on black carbon (BC), all the other species than hydrophilic BC are set to zero in the model initialisation. During the 24 h spin-up run, BC is transported by developing winds and 3 % of the initialised total BC mass is removed by dry deposition at the surface. During the subsequent 24 h of simulation AER, an additional 1.5 % is removed by deposition at the surface. Neither BC nor any other compound are emitted or produced by secondary processes during the simulation. The aerosol optical properties are computed using the exact shell approximation method, which assumes a shell surrounding the BC core and incorporates a full Mie calculation . The particle size distribution of BC in this aerosol scheme is 25 % of modal mass in Aitken mode and 75 % in accumulation mode. The aerosol processes (dry deposition and radiation) are updated every 60 s. As already said in the Sect. , we only include aged hydrophilic BC.

The 24 h-long spin-up run is initialised with an idealised BC profile with high concentrations near the surface which decrease rapidly with height (Fig. f). By initializing most of the BC mass near the surface, we ensure its transport and redistribution will primarily occur within the boundary layer during the simulation, reflecting its typical real-life location . Consequently, the actual AER simulation starts after a 24 h spin-up with a BC field that has been adjusted to a realistic boundary layer structure (Fig. b, g), rather than with the artificial idealised vertical shape of the initial state (Fig. a, f). The BC concentrations in AER are up to 17 µgm-3 in the boundary layer after the spin-up (Fig. g) and the BC column mass is around 2 mgm-2 above the plain and the valley centre (not shown). For reference, reported a daily average of 10.2 µgm-3 of surface BC concentration during 2013–2015 in Kathmandu, Nepal, located in the Himalayan foothills. reported daytime surface BC concentrations between 8 to 20 µgm-3 during September and November 2012 in La-Paz, Bolivia, located in the Andes. reported mean BC concentrations of 8–10 µgm-3 throughout the mixed layer during February 2010 in Milan, Italy, located in the Po valley. In Beijing, China, the surface BC concentrations reach 20 µgm-3 and beyond during haze pollution events . estimated the BC column mass in Beijing using ground-based remote sensing and reported an annual variation between 2.7 and 7.3 mgm-2 in 2009–2010.

Budget computation and online time-averaging is handled by the WRFlux-tool v1.4.1 which is a fork of the default WRF v4.4.1 repository . All the model output shown in this article is based on the hourly averages from WRFlux. The analysis focuses on the valley volume, which refers to the air volume defined by y≥0km and z≤2km, shown by the dashed lines in Fig. b. As the lateral boundaries of the model domain are periodic in the x-direction, an identical parallel valley is located across the model x-boundaries (|x|>10 km) in addition to the valley in the middle of the domain (|x|≤10km). The analysis covers the whole domain width in the x-direction, meaning also the two valley halves at |x|>10 km are included in the spatially averaged or integrated values.

Along-valley momentum and heat budgets are retrieved using the post-processing tool of WRFlux to compute the tendency of each budget term from the hourly averaged WRFlux output. The tendency budget terms include transport of heat and momentum decomposed into resolved mean advection (referred to as ADV in this study), resolved turbulent (TRB) and sub-grid-scale (SGS) diffusion components using Reynolds decomposition . The momentum budget includes a tendency from the pressure gradient forcing (PGF) which is retrieved from the Runge-Kutta and acoustic step modules in WRF-Chem. The heat budget includes tendencies from the radiation scheme separately for shortwave (RSW) and longwave (RLW) components. The components ADV, TRB, and SGS are output in the x, y and z-directions. In this study, the y-component of the resolved transport and SGS diffusion is referred to as the along-valley component. The sum of the x and z-components is referred to as the cross-valley component. The along-valley momentum tendency is given as 1∂v∂t=∑iFv,i=ADVv+TRBv+SGSv+PGFv where v is the y-component of the wind vector. The heat tendency is given as 2∂θ∂t=∑iFθ,i=ADVθ+TRBθ+SGSθ+RSWθ+RLWθ where θ is the potential temperature. Fθ,i and Fv,i are the individual terms on the right-hand-side of the heat and along-valley momentum budgets, respectively. Finally, the post-processed tendency terms are mass-weighted over the valley volume to give the along-valley momentum and heat budget by 31M∫Vρ∂S∂tdV=∑i1M∫VρFS,idV, where S is either v or θ, M is the total mass of air in the valley volume V and ρ is the dry air density. The volume integral of Eq. () is approximated by a mass-weighted sum of the contribution of all grid boxes in the valley, taking into account the grid box volume.

Figure shows vertical profiles of potential temperature, BC concentration, and potential temperature tendency from the shortwave radiation scheme, subsequently referred to as the heating rate. The hourly averages are spatially averaged along the centre line of the domain (x=0 km) above the valley (y>0 km, solid lines) and above the plain (y<0,km, dashed lines).

The simulations start at 06:00 from the same initial state as shown in Fig. b. The initial state has a stable layer below 1 km and above that there is a less stable layer up to 2 km formed by the nocturnal cooling during the spin-up. Above these stable layers, there is a neutral residual layer that extends up to 3.5 km which was formed during the daytime of the spin-up. In CTRL by midday, a convective surface-based layer up to 0.5 km has developed but above that the stable layer below 2 km still remains (Fig. c). In the afternoon at 14:00–15:00, a well developed convective boundary layer reaches up to about 3.5 km above the plain (dashed lines in Fig. d) and 4 km above the valley (solid lines). Case AER develops in a very similar manner to case CTRL with 1–2 K colder potential temperatures below 0.5 km and slightly higher potential temperatures (less than 0.5 K margin) above 0.5 km at midday (Fig. c). This means the stability below 2 km is stronger in case AER. In the afternoon, case AER has a well developed convective boundary layer up to a similar height as in CTRL with around 1 K higher potential temperatures above 1 km. The convective boundary layer within and above the valley (solid lines) is around 2 K warmer than above the plain (dashed lines) in both simulations. At 17:00–18:00 a surface-based inversion develops in the valley in both simulations, whereas over the plain the near-surface stability is near-neutral (Fig. e).

The initial BC profile in case AER has two distinct layers with concentrations around 17 µgm-3 below 2 km (Fig. g) in the stable layer and concentrations around 8 µgm-3 between 2 and 3.5 km. BC concentration decreases rapidly with height above the inversion top at 3.5 km. During the night of the spin-up, BC has accumulated more in the near-surface layer than at higher altitudes (Fig. g). This is a result of the nocturnal winds in the valley flowing down the slopes and the surface-based inversion that develop during the night (Fig. b), as a similar vertical distribution of BC is found above the plain as well (Fig. g). Through the course of the day, the vertical gradient of BC weakens (Fig. h–j) as the convective mixing and vertical transport by the valley and slope wind circulation takes place (Sect. ). The well-mixed layer above the valley is deeper and, therefore, BC is mixed in a deeper layer than above the plain (Fig. i–j).

The absorption of shortwave radiation by BC in case AER causes a heating rate of up to 0.3 Kh-1 at midday (Fig. h). The heating rate is on a similar scale to what is reported in previous studies focusing on BC absorption with similar BC concentrations e.g.,. In case AER, the shortwave heating rate does not increase linearly with the BC concentration (Fig. h–j). This is due to less of the incoming solar radiation reaching the lower levels (see Fig. S2a) as absorption by BC occurs above. At midday with the clearly higher BC concentrations below 2 km, the shortwave radiative heating is strongest there (Fig. h). In the afternoon when the vertical gradient of BC is weaker, the shortwave heating rate is near equal throughout the well-mixed layer even though the BC concentrations near the surface are still higher than aloft (Fig. i–j). The direct radiative effect of BC causes warming rates up to 0.3 Kh-1 due to shortwave absorption. However, BC causes a differential heating effect within the boundary layer leading to both cooling and warming at different altitudes. This is seen when the potential temperatures of cases CTRL and AER are compared in Fig. c–e. The cooling due to BC below 0.5 km at midday is at most around 2 K above both the valley and the plain (Fig. c). In the afternoon in the valley, BC causes warming above 1 km and cooling below 1 km. The simultaneous cooling near the surface and warming of the upper boundary layer means BC makes the boundary layer more stable. Above the plain, the cooling effect near the surface is also seen at midday but in the afternoon the entire boundary layer above the plain is warmed with increasing strength towards the inversion top (Fig. i–j). The warming above 1 km is around 1 K with increasing strength towards the inversion top in the afternoon at 15:00 and 18:00 (Fig. d–e). Although the BC concentrations are higher at lower levels in the beginning of the day, the effect from BC on the potential temperature profile resembles the dome effect (Sect. ), in which the stability in the boundary layer is strengthened due to the BC interaction with radiation e.g.,.

Figure shows vertical cross-sections of the v-component of the wind speed (shaded) and potential temperature (black lines) along the valley centre line x=0 km and across y=25 km averaged at 14:00–15:00 for the two simulations. Panels b and d also show the cross-valley wind in x-z-plane as gray vectors. In addition, panels e and f show the BC concentration, and panels g and h show the potential temperature tendency from the shortwave radiation scheme for the case AER.

Daytime plain-to-valley and namely up-valley winds (red colors in Fig. a–d) form in both simulations as a result of the temperature difference between the valley atmosphere and the air above the plain below the ridge height of 2 km. The warmer temperatures in the valley compared to over the plain, also seen in the vertical profiles in Fig. d, is evident in Fig. a, c. Above the up-valley wind layer outside the valley forms the return flow in the down-valley direction (blue colors in Fig. a–d). The up-valley winds reach their maximum strength near the valley entrance and decay towards the end of the valley. Qualitatively the spatial structure of the up-valley winds and potential temperature do not differ drastically between the cases CTRL (Fig. a–b) and AER (Fig. c–d). However, the up-valley winds are stronger in the up-valley jet (0 km <y<25 km) in case AER at the shown time 14:00–15:00 and they extend slightly further into the valley. The two cases develop a surface-based convective boundary layer of similar depth but towards the inversion top the case AER is around 1 K warmer than case CTRL (Fig. c, e), which was also seen in Fig. d–e.

In the cross-valley direction, up-slope winds form near the heated slopes (Fig. b, d). The up-slope winds from both sides converge at the ridge top (|x|=10 km) resulting in an updraft above the ridge. The cross-valley circulation below the inversion top at 4 km includes also the return circulation away from the slopes and subsidence above the valley centre. Due to the diurnal cycle of solar radiation, there is an asymmetry within the valley as the eastward facing slope receives solar radiation earlier in the day than the westward facing slope. In the afternoon the westward facing slope (shown in Fig. b, d) has stronger winds than the eastward facing slope. The up-slope winds flow in a deeper layer in case CTRL (Fig. b) than in AER (Fig. d). Based on theoretical models and real-life observations of up-slope winds , reduced surface heating and increased stability would both make the up-slope winds weaker and shallower which is in line with Fig. b, d. In case AER, the reduction in surface heating stems from the absorption of shortwave radiation by BC within and above the valley. In addition, the heating due to shortwave absorption above the valley increases the stability. Two vertically stacked cross-valley circulation cells separated by an inversion layer (at around z=1.5 km) are seen in case AER (Fig. d) but not in case CTRL (Fig. b). The formation of such a stacked circulation is favored by increased stability .

Spatial variation of BC in AER is highly influenced by the valley circulation (Fig. e–f). The plain-to-valley winds transport BC towards the valley (Fig. c, e). From the bottom of the valley centre BC is carried up the slopes and out of the valley by the cross-valley circulation (Fig. d, f). The lateral and vertical transport of BC away from the valley centre is seen as a drop in BC concentration along the valley axis (Fig. e) when entering from the plain into the valley. Over the plain below 2 km there is less variation in BC in the y-direction. Above the valley the BC layer is deeper than over the plain. Once BC is transported above the valley, the return winds above the ridge height carry some of it towards the plain (Fig. c, e). In addition, the convective mixing above the plain causes vertical transport of BC. The subsidence over the centre of the valley by the cross-valley circulation brings cleaner air (i.e. air with lower BC concentration) from above into the valley atmosphere, seen around x=0 km between a height of 1.5 to 2.0 km (Fig. f).

Spatial variation in the shortwave heating rate (Fig. g–h) is strongly correlated with the BC concentration (Fig. e–f). At 14:00–15:00, the strongest shortwave heating occurs where BC concentrations are largest, i.e., below 2 km above the plain, above the ridges, and in the down-valley wind return flow. Similarly, as seen in the averaged vertical profiles (Fig. h–j), stronger heating is found at higher altitudes (Fig. g–h). The reason is, that due to absorption aloft, less solar radiation reaches the lower levels which leads to weaker heating rate although the BC concentrations are nearly the same.

Figure shows the temporal evolution of the average v-component of the wind in the valley volume, the u-component of the wind averaged over the lowest 300 m above ground level (a.g.l.) within the valley, and the potential temperature difference between the valley volume and the plain below 2 km. The 2 km reference height above the plain in the potential temperature difference calculation is used to match the ridge height of the valley volume. Positive v-component of the wind refers to up-valley wind. Positive (negative) u-component of the wind on the east (west) slope refers to up-slope wind. Hourly averages are plotted at the end time of each hour and the volume averages are calculated by weighting the individual grid-point values by both density and grid-box volume.

The simulations start at 06:00 with down-valley and down-slope winds that were formed during the night of the spin-up run (Fig. S3). In both simulations, the cross-valley winds in the lowest 300 m above the slopes turn up-slope during the first hour on the west slope (solid line in Fig. b) and by 10:00 on the east slope (dashed line). The up-slope winds peak in strength around 13:00 on the west slope and around 16:00 on the east slope. The cross-valley winds turn down-slope on the west-slope around 17:00 and on the east-slope by 19:00. The time lag between the slopes is caused by the diurnal cycle of solar radiation which causes the west slope (east facing) being sun-lit first in the morning and in turn the east slope (west facing) is sun-lit later in the afternoon. The up-slope winds peak in strength between 1.8 and 2.7 ms-1 with some variation between the simulations and the slopes. The up-slope winds are stronger in case CTRL on both slopes. Given the deeper flow layer, the stronger up-slope winds in CTRL would lead to even larger difference in the up-slope mass flux (Fig. S4). The up-slope wind maxima are stronger on the east slope in both simulations.

In both simulations, the along-valley wind turns up-valley around 09:00 and reaches the peak strength of around 5 ms-1 in the evening (Fig. a). The up-valley winds peak at 19:00 in case AER and at 20:00 in case CTRL. Until midday both cases have similar up-valley wind speeds but between 12:00 and 18:00 case AER has stronger up-valley winds. After the up-valley wind speeds peak at 4.7 ms-1 in AER at 19:00, the case CTRL has stronger up-valley winds with also a higher maximum strength at 5.2 ms-1. The average along-valley wind speed between 12:00 and 20:00 is 2.3 ms-1 in case CTRL and 2.6 ms-1 in case AER. The along-valley winds turn down-valley at 23:00 in case AER and at 01:00 in case CTRL.

As noted in Sect.  and , in both simulations the valley atmosphere is warmer than the air above the plain below the ridge height during the day. This is seen from the very beginning of the simulations and the maximum potential temperature difference of approximately 1.6 K is found for both simulations around noon (Fig. c). After that, in the case AER the valley-plain temperature difference slowly decays while in the case CTRL it stays near constant until 16:00. Slightly after the up-valley winds peak in strength, the potential temperature difference turns negative, hence the air above the plain becomes warmer than in the valley volume and a few hours later the along-valley winds turn down-valley (Fig. a) in both simulations.

An important remark to make is the contradiction between the up-valley wind speeds (Fig. a) and the valley-plain potential temperature differences (Fig. c). In the afternoon, when the case AER has stronger up-valley winds, the valley-plain potential temperature difference is in fact weaker than in the case CTRL, which is opposite to what the classic valley wind theory states (Sect. ). According to this theory, the valley-plain potential temperature difference that causes the pressure gradient force between the valley and the plain should be the main driver of the plain-to-valley and up-valley winds. In the following Sect. , we attempt to explain why the case AER has stronger up-valley winds even with a weaker valley-plain potential temperature gradient.

Figure shows the temporal evolution of the along-valley momentum budget for the valley volume. For a detailed description of the budget analysis refer to Sect. . In the left-hand-side panels (a–c), the four terms in the along-valley momentum budget are plotted (see Eq. ): the pressure-gradient force (PGF), mean advection (ADV), resolved turbulence (TRB), and sub-grid-scale diffusion (SGS). On the right-hand-side panels (d–f) ADV, TRB and SGS are decomposed into the along-valley (y) and cross-valley (x+z) components. Panels (c) and (f) show the difference between the cases AER and CTRL.

The along-valley momentum tendency (black dotted) and the sum of the budget terms (black dashed) show good agreement in both simulations (Fig. a–b). This comparison ensures that the momentum budget closes and thus is computed correctly. In case CTRL the net tendency reaches a maximum in the afternoon at 17:00, turns negative around the hour of the peak up-valley winds (Sect. ) at 20:00 and reaches a minimum during the evening transition at 21:00 (Fig. a). performed an along-valley momentum analysis for idealised valley simulations. The temporal evolution of the net momentum tendency and the budget terms in their simulation called “periodic” are similar to what is shown for case CTRL in Fig. a. The individual terms of both their and our simulation have similar magnitudes. Detailed comparison between case CTRL and the simulation in is not performed as there are major differences in the simulation setup, such as the numerical model, horizontal grid spacing, domain latitude, and valley geometry.

In case AER the net tendency reaches the maximum, turns negative and reaches the minimum one to two hours earlier compared to CTRL (Fig. a–b). Overall, the diurnal evolution of the net tendency is similar between the simulations but there are some notable differences in the magnitude, especially from 09:00 until 18:00. Between 10:00 and 15:00 case AER has a stronger positive tendency of the along-valley momentum (Fig. a–b) which is in line with the stronger up-valley winds (Sect. ). In contrast, the peak strength in the momentum tendency in case CTRL is higher (Fig. a–b), which is again in line with the stronger peak up-valley wind speeds (Sect. ).

In both simulations the PGF has a similar diurnal cycle (Fig. a–b) as the valley-plain potential temperature difference (Sect. ), driving the daytime up-valley winds and night-time down-valley winds. The temporal evolution of the PGF is similar between the cases CTRL and AER but with a larger daytime magnitude in CTRL (Fig. c) which is in line with the stronger valley-plain temperature difference in CTRL discussed in Sect. . TRB and SGS reduce the along-valley momentum during the day and have negligible values during the night (Fig. a–b). The along-valley component of TRB and SGS (dashed lines in Fig. d, e) are negligible low throughout the simulation. TRB and SGS include, for example, tendencies such as the surface drag but also tendency due to turbulent exchange between the valley volume and the surroundings at the valley entrance and at the ridge-top height. TRB shows very similar values for both simulations (Fig. c). Along-valley momentum is reduced more by SGS in case AER which can be explained by the higher up-valley wind speeds hence stronger surface drag (z-component included in the cross-valley, Fig. d–f).

During the day, the largest difference in the along-valley momentum between the cases AER and CTRL results from the mean advective tendency (ADV, Fig. c). In AER, ADV stays positive from 11:00 until the end of the simulation (Fig. b) where as in CTRL ADV has negative values in the afternoon from 13:00–16:00. During daytime, positive ADV would refer to advection of air with high along-valley momentum at the valley entrance by the plain-to-valley winds (Sect. ), which is seen as positive values of the ADV along-valley component (dashed line in Fig. d, e). Interestingly, in CTRL the daytime cross-valley circulation reduces the along-valley momentum (negative values, dotted line in Fig. d) as the up-slope winds carry air with high along-valley momentum (i.e. up-valley winds) vertically out of the valley volume at the ridge height. This is also seen in the cross-section in Fig. b as there are up-valley winds outside the valley volume above the ridge height which may be interpreted as a direct consequence of a vertical transport of along-valley momentum. In addition, the return flow and subsidence above the valley centre bring air with low or down-valley momentum into the valley volume from above which reduces the up-valley momentum. However, in the case AER the export of up-valley winds/import of down-valley winds by the cross-valley circulation is weak or even absent between 10:00 and 15:00 (Fig. e). In case AER, up-valley winds are seen above the ridge in Fig. d at 15:00, but much weaker than in the case CTRL. The transport by TRB and SGS may also contribute in the export of along-valley momentum, but the most striking difference between the simulations are found in the mean advective part (Fig. c, f).

Figure shows the temporal evolution of the heat budget for the valley volume (see Eq. ). The heat budget involves a resolved mean advective term (ADV), a resolved turbulent term (TRB), a sub-grid-scale diffusion term (SGS) which involves the surface sensible heat flux, and the heating tendencies due to shortwave (RSW) and longwave radiation (RLW).

Similar to the momentum budget, the sum of the budget terms (black dashed) and the independently computed heat tendency (black dotted) show a good agreement throughout the simulations (Fig. a, b), indicating that the budget is closed. In both simulations, the warming is strongest at midday and the valley volume starts cooling two hours before sunset around 16:00. The strongest cooling occurs around 18:00 during the evening transition and decays with time during the night. In case CTRL, the net heat tendency in the valley volume and the individual heat-budget terms exhibit a diurnal-cycle shape and magnitude similar to those reported by , who performed a comparable heat-budget analysis for their idealised valley simulation, “periodic”.

The mean daytime valley circulation acts to cool the valley volume, seen as negative ADV in Fig. a–b. This includes import of cold air at the valley entrance by the plain-to-valley winds (ADV along in Fig. d, e), and export of warm air at the ridge-top height by the up-slope winds (ADV cross in Fig. d, e). During the evening transition, the net tendency turns negative before sunset as the heating from the solar radiation weakens and the prevailing daytime valley circulation cools the valley volume (Fig. a, b).

Sub-grid-scale diffusion (SGS) shows a clear diurnal cycle with a peak at noon and a constant negative tendency during the night (Fig. a, b). In case AER there is heating due to shortwave radiation (RSW) during the day with a heating rate less than 0.2 Kh-1 in the valley volume. RSW has the strongest values at 10:00–11:00. This is probably due to BC ventilation out of the valley volume by the daytime cross-valley circulation (Sect. ). Earlier in the day, more BC is located in the valley volume compared to above the valley (Fig. h–j). As the BC is transported out of the valley, the heating due to BC absorption occurs more and more outside the valley. This reduces the shortwave radiation aloft, hence, there is both less shortwave radiation and less BC mass in the valley volume.

When comparing the two simulations in Fig. c, f, the reduction in SGS due to the BC absorption is stronger than the positive heat tendency due to RSW in the valley volume. The surface heating (SGS) is reduced as the absorption by BC reduces the incoming solar radiation reaching the ground. However, the absorption occurs also above the valley (Sect. –), not only within the valley volume for which the heat budget is computed. This leads to a reduction in the shortwave radiation reaching the top of the valley volume in case AER when compared to the case CTRL (Fig. S2a). The unequal heat input from shortwave radiation leads to asymmetry between the reduction of heating by SGS and the increase of heating by RSW. However, the difference in TRB seems to even out the asymmetry between SGS and RSW as the net difference between the simulations follows very closely the ADV term (Fig. c). All of the TRB comes from the cross-valley component (Fig. f) which would mean a weaker export of heat at the valley top in AER in the afternoon. The temporal evolution of ADV in the along and cross-valley directions look qualitatively very similar between the simulations (Fig. d, e) with the one to two hour time lag in the evening transition. The stronger negative values in the along-valley component of ADV in case AER in the afternoon might be caused by the stronger plain-to-valley winds, however, the weaker valley-plain potential temperature difference would compensate this. Throughout the day, the export of heat by the cross-valley component of ADV is stronger in case CTRL than in case AER. This is likely caused by the reduced surface heating and increased stability due to the shortwave absorption by BC leading to weaker up-slope winds in case AER (Sect. –).