2.2.1 Activation

It is well known that the aerosol distribution and the activation process are of great importance for the life cycle of fog (e.g., Gultepe et al., 2007). The amount of activated aerosols determines the number concentration of droplets within the fog, which, in turn, has a significant influence on radiation through optical thickness as well as on sedimentation and consequently affects macroscopic properties of the fog, like, for instance, its vertical extent. For these reasons, a sophisticated treatment of the activation process is an essential prerequisite for the simulation of radiation fog. Several activation parameterizations for bulk microphysics models have been proposed in literature. In this work, three of these activation schemes were compared with each other in order to quantify their effect on the development of a radiation fog event. The schemes considered in this scope are the activation scheme of Twomey (1959), which was used, for example, by Bott and Trautmann (2002) to simulate radiation fog, the scheme of Cohard et al., 1998 (used by, for example, Stolaki et al., 2015; Mazoyer et al., 2017), and the one by Khvorostyanov and Curry (2006). The latter two represent an empirical and analytical extension of Twomey's scheme, respectively. Consequently, these parameterizations are frequently termed Twomey-type parameterizations that have the following form:

where N_CCN values are the number of activated cloud condensation nuclei (CCN), N₀ and k are parameters depending on the aerosol distribution, and s is the supersaturation. The three parameterizations considered in the present study are variations of Eq. (5) differing in mathematical complexity:

1. Twomey (1959). The power law expression (Eq. 5) is well known and has been used for decades to estimate the number of activated aerosols for a given air mass in dependence of the supersaturation. A weakness of this approach is that the parameters N₀ and k are usually assumed to be constant and are not directly linked to the microphysical properties. Furthermore, this relationship creates an unbounded number of CCN at high supersaturations.

2. Cohard et al. (1998). This extended Twomey's power law expression by using a more realistic four-parameter CCN activation spectrum as shaped by the physiochemical properties of the accumulation mode. Although an extension to the multi-modal representation of an aerosol spectrum would be possible, all relevant aerosols that are activated in typical supersaturations within clouds and especially fog are represented in the accumulation mode (Cohard et al., 1998; Stolaki et al., 2015). Following Cohard et al. (1998) and Cohard and Pinty (2000), the activated CCN number concentration is expressed by

   $$\begin{array}{}\text{(6)}& {\displaystyle }{N}_{\text{CCN}}\left(s\right)=C{s}^k\cdot F\left(\mathit{\mu },{\displaystyle \frac{k}{\mathrm{2}}},{\displaystyle \frac{k}{\mathrm{2}}}+\mathrm{1};\mathit{\beta }{s}^{\mathrm{2}}\right),\end{array}$$

   where C is proportional to the total number concentration of CCN that is activated when supersaturation s tends to infinity. Beside k, the parameters μ and β are adjustable shape parameters associated with the characteristics of the aerosol size spectrum such as the geometric mean radius and the geometric standard deviation as well as with chemical composition and solubility of the aerosols. Thus, in contrast to the original Twomey approach, the effect of physiochemical properties on the aerosol spectrum are taken into account.

3. Khvorostyanov and Curry (2006). This found an analytical solution to express the activation spectrum using Köhler theory. Therein, it is assumed that the dry aerosol spectrum follows a log-normal size distribution of aerosol f_d:

   $$\begin{array}{}\text{(7)}& {\displaystyle }{f}_{\text{d}}={\displaystyle \frac{\text{d}{N}_{\mathrm{a}}}{\text{d}{r}_{\mathrm{d}}}}={\displaystyle \frac{N_{\mathrm{t}}}{\sqrt{\mathrm{2}\mathit{\pi }}\ln {\mathit{\sigma }}_{\mathrm{d}}{r}_{\mathrm{d}}}}\exp \left[-{\displaystyle \frac{{\ln }^{\mathrm{2}}(r_{\mathrm{d}}/r_{\mathrm{d}\mathrm{0}})}{\mathrm{2}{\ln }^{\mathrm{2}}{\mathit{\sigma }}_{\mathrm{d}}}}\right].\end{array}$$

   Here, r_d is the dry aerosol radius, N_t is the total number of aerosols, σ_d is the dispersion of the dry aerosol spectrum and r_d0 is the mean radius of the dry particles. The number of activated CCN as a function of supersaturation s is then given by

   $$\begin{array}{}\text{(8)}& {\displaystyle }{N}_{\text{CCN}}\left(s\right)={\displaystyle \frac{N_{\mathrm{t}}}{\mathrm{2}}}[\mathrm{1}-\text{erf}(u\left)\right];u={\displaystyle \frac{\ln (s_{\mathrm{0}}/s)}{\sqrt{\mathrm{2}}\ln {\mathit{\sigma }}_{\mathrm{s}}}},\end{array}$$

   where erf is the Gaussian error function, and

   $$\begin{array}{}\text{(9)}& {\displaystyle }{s}_{\mathrm{0}}{\displaystyle }=r_{\mathrm{d}\mathrm{0}}^{-(\mathrm{1}+\mathit{\beta })}{\left({\displaystyle \frac{\mathrm{4}{A_{\text{K}}}^{\mathrm{3}}}{\mathrm{27}b}}\right)}^{\mathrm{1}/\mathrm{2}},{\mathit{\sigma }}_s={\mathit{\sigma }}_{\mathrm{d}}^{\mathrm{1}+\mathit{\beta }}.\end{array}$$

   In this case, A_K is the Kelvin parameter and b and β depend on the chemical composition and physical properties of the soluble part of the dry aerosol.

Since prognostic equations were neither considered for the aerosols nor for their sources and sinks, a fixed aerosol background concentration was prescribed by setting parameters N₀, C and N_t for the three activation schemes. The different nomenclature of the aerosol background concentration is based on the nomenclature used in the original literature.

The activation rate is then calculated as

where n_c is the number of previously activated aerosols that are assumed to be equal to the number of pre-existing droplets and Δt is the length of the model time step. Note that this method does not take into account reduction of CCN. However, this error can be neglected, since processes like aerosol washout and dry deposition are of minor importance for radiation fog. For all activation schemes it is assumed that every activated CCN becomes a droplet with an initial radius of 1 µm. This results in a change of liquid water, which is considered by the condensation scheme and is described in the next section. Furthermore, we performed a sensitivity study with initial radii of 0.5 to 2 µm, which showed that the choice of the initial radius had no impact on the results (not shown). This is consistent with the findings of Khairoutdinov and Kogan (2000) and Morrison and Grabowski (2007).

2.2.2 Condensation and supersaturation calculation

The representation of diffusional growth, evaporation and calculating the underlying supersaturation (which is the main driver for activation) is one of the fundamental tasks of cloud physics. Three different methods have been evaluated and widely discussed in the scientific community. Namely, these are the saturation adjustment scheme, the diagnostic scheme, where the supersaturation is diagnosed by the prognostic fields of temperature and water vapor, and a prognostic method for calculating the supersaturation following, for example, Clark (1973), Morrison and Grabowski (2007), and Lebo et al. (2012). Basically, the supersaturation is given by $s=q_{\text{v}}/q_{\text{s}}-\mathrm{1}$, while the absolute supersaturation (or water vapor surplus) is defined as $\mathit{\delta }=q_{\text{v}}-q_{\text{s}}$, where q_v is the water vapor mixing ratio and q_s is the saturation mixing ratio. In the following, these three methods are briefly reviewed.

1. Saturation adjustment. In many microphysical models, a saturation adjustment scheme is applied. The basic idea of this scheme is that all supersaturation is removed within one model time step and supersaturations are thus neglected. Saturation adjustment thus potentially leads to excessive condensation. Despite the many years of application of this scheme, its impact on microphysical processes is discussed controversially (e.g., Morrison and Grabowski, 2008; Thouron et al., 2012; Lebo et al., 2012). Saturation adjustment might hence especially be a source of error in fog simulations, where very small time steps are used due to small grid spacings, as already discussed. Using the saturation adjustment scheme, q_l represents a diagnostic value calculated by means of

   $$\begin{array}{}\text{(11)}& {\displaystyle }{q}_{\mathrm{l}}=max(\mathrm{0},q-q_{\text{r}}-q_{\text{s}}),\end{array}$$

   where q is the total water mixing ratio. The saturation mixing ratio, which is a function of temperature, is approximated in a first step by

   $$\begin{array}{}\text{(12)}& {\displaystyle }{q}_{\text{s}}\left(T_{\text{l}}\right)={\displaystyle \frac{R_{\text{d}}}{R_{\text{v}}}}{\displaystyle \frac{e_{\text{s}}\left(T_{\text{l}}\right)}{p-e_{\text{s}}\left(T_{\text{l}}\right)}},\end{array}$$

   where T_l is the liquid water temperature and p is pressure. R_d and R_v are the specific gas constants for dry air and water vapor, respectively. For the saturation vapor pressure (e_s) an empirical relationship of Bougeault (1981) is used. In a second step, q_s is corrected using a first-order Taylor series expansion of q_s:

   $$\begin{array}{}\text{(13)}& {\displaystyle }{q}_{\text{s}}\left(T\right)=q_{\text{s}}\left(T_{\text{l}}\right){\displaystyle \frac{\mathrm{1}+\mathit{\gamma }q}{\mathrm{1}+\mathit{\gamma }{q}_{\text{s}}\left(T_{\text{l}}\right)}},\end{array}$$

   with

   $$\begin{array}{}\text{(14)}& {\displaystyle }\mathit{\gamma }={\displaystyle \frac{L_{\text{v}}}{R_{\text{v}}{c}_{\text{p}}{T}_{\text{l}}^{\mathrm{2}}}},\end{array}$$

   where c_p is the specific heat of dry air at constant pressure and L_v is the latent heat of vaporization. As previously mentioned, in each model time step, all supersaturation is converted into liquid water or, in subsaturated regions, the liquid water is reduced until saturation. In order to use this scheme with aerosol activation parameterizations, it is necessary to estimate the supersaturation (see Eq. 5). This can be achieved for the activation scheme of Cohard et al. (1998) following, for example, Thouron et al. (2012), Mazoyer et al. (2017) and Zhang et al. (2014), directly translating into a droplet number concentration by

   $$\begin{array}{ll}\text{(15)}& {\displaystyle }& {\displaystyle }{s}^{k+\mathrm{2}}\cdot F\left(\mathit{\mu },k/\mathrm{2},k/\mathrm{2}+\mathrm{1},-\mathit{\beta }s\right)={\displaystyle }& {\displaystyle \frac{{\left({\mathit{\varphi }}_{\mathrm{1}}w+{\mathit{\varphi }}_{\mathrm{3}}\frac{\text{d}T}{\text{d}t}{|}_{\text{rad}}\right)}^{\frac{\mathrm{3}}{\mathrm{2}}}}{\mathrm{2}kC\mathit{\pi }{\mathit{\rho }}_l{\mathit{\varphi }}_{\mathrm{2}}B\left(\frac{k}{\mathrm{2}},\frac{\mathrm{3}}{\mathrm{2}}\right)}},\end{array}$$

   where ϕ₁, ϕ₂ and ϕ₃ are functions of temperature and pressure and are given in Cohard et al. (1998) and Zhang et al. (2014). w is the vertical velocity and B is the beta function.

2. Diagnostic supersaturation calculation. Supersaturation is calculated diagnostically from q_v and temperature T (from which q_s can be derived). However, since it is assumed that the supersaturation is kept constant during one model time step, the diagnostic approach requires a very small model time step of

   $$\begin{array}{}\text{(16)}& {\displaystyle }\mathrm{\Delta }t\le \mathrm{2}\mathit{\tau },\end{array}$$

   due to stability reasons (Árnason and Brown Jr., 1971). Here, τ is the supersaturation relaxation time which is approximated by

   $$\begin{array}{}\text{(17)}& {\displaystyle }\mathit{\tau }\approx (\mathrm{4}\mathit{\pi }D{n}_c\stackrel{\mathrm{‾}}{r}{)}^{-\mathrm{1}},\end{array}$$

   where $\stackrel{\mathrm{‾}}{r}$ is the average droplet radius, and D is the diffusivity of water vapor in air. Due to the low dynamic time step in the present study imposed by the Courant–Friedrichs–Lewy criterion (on the order of 0.1 s), however, the condensation time criterion is fulfilled, and no additional time-step decrease is needed. The rate of cloud water change due to condensation or evaporation is given by

   $$\begin{array}{}\text{(18)}& {\displaystyle }{\left({\displaystyle \frac{\partial q_{\mathrm{l}}}{\partial t}}\right)}_{\text{cond}}& {\displaystyle }={\displaystyle \frac{\mathrm{4}\mathit{\pi }G(T,p){\mathit{\rho }}_{\mathrm{w}}}{{\mathit{\rho }}_{\mathrm{a}}}}s\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}rf\left(r\right)\text{d}r\text{(19)}& {\displaystyle }& {\displaystyle }={\displaystyle \frac{\mathrm{4}\mathit{\pi }G(T,p){\mathit{\rho }}_{\mathrm{w}}}{{\mathit{\rho }}_{\mathrm{a}}}}s{r}_{\mathrm{c}},\end{array}$$

   where r_c is the integral radius and $G=\frac{\mathrm{1}}{F_{\mathrm{K}}+F_{\mathrm{D}}}$ included the thermal conduction and the diffusion of water vapor (Khairoutdinov and Kogan, 2000). The density ratio of liquid water and the solute is given by ρ_w∕ρ_a.

3. Prognostic supersaturation. The prognostic approach, which was first introduced by Clark (1973), includes an additional prognostic equation for the absolute supersaturation. Even though this requires solving one more prognostic equation, it mitigates the problem of spurious cloud-edge supersaturations and prevents inaccurate supersaturation caused by small errors in the advection of heat and moisture (Morrison and Grabowski, 2007; Grabowski and Morrison, 2008; Thouron et al., 2012).

   The temporal change of δ is given by

   $$\begin{array}{}\text{(20)}& {\displaystyle \frac{\partial \mathit{\delta }}{\partial t}}-{\displaystyle \frac{\mathrm{1}}{\mathit{\rho }}}\mathrm{\nabla }\cdot \left(u\mathit{\rho }\mathit{\delta }\right)=A-{\displaystyle \frac{\mathit{\delta }}{\mathit{\tau }}},\end{array}$$

   with A being described by

   $$\begin{array}{}\text{(21)}& {\displaystyle }A=-q_{\text{s}}{\displaystyle \frac{\mathit{\rho }gw}{p-e_{\text{s}}}}-{\displaystyle \frac{\text{d}{q}_{\text{s}}}{\text{d}T}}\cdot \left[{\displaystyle \frac{gw}{c_{\mathrm{p}}}}+{\left({\displaystyle \frac{\text{d}T}{\text{d}t}}\right)}_{\text{rad}}\right],\end{array}$$

   with g being gravitational acceleration. The supersaturation relaxation time is given in Eq. (17). The second term on the left-hand side of Eq. (20) describes the change of the absolute supersaturation due to advection, while the right-hand side considers changes of δ due to changes in pressure, adiabatic compression and expansion, and radiative effects (from left to right). By doing so, the predicted supersaturation is used for determining the number of activated droplets as well as the condensation and evaporation processes. Note that here the absolute supersaturation is taken, as using s would involve more terms and is more complex to solve (Morrison and Grabowski, 2007).

4.2.1 One-moment microphysics scheme: impact of supersaturation calculation on diffusional growth

In this section we discuss the error introduced by using saturation adjustment for simulating radiation fog with a one-moment scheme in a LES. For this, we compare three simulations with identical setups (cases SAT, DIA, and PRG), which differ only in the way supersaturation is calculated and consequently the amount of condensed or evaporated liquid water. To isolate this effect, activation is neglected in all cases and n_c is set to a constant value of 150 cm⁻³ (a typical value in fog layers). The effect on different supersaturations driving the diabatic process of activation is discussed in Sect. 4.2.2. As mentioned before the time step is roughly 0.1 s, which is more than 1 order of magnitude smaller than the allowed values of 2–5 s for assuming saturation adjustment (Thouron et al., 2012). The present case hence is an ideal environment evaluating the error introduced by using saturation adjustment and by keeping all other parameters fixed.

Figure 4 shows time series of the horizontally averaged saturation (supersaturation) for cases SAT, DIA and PRG at selected heights close to the surface. In all cases saturation occurs simultaneously around 01:20 UTC. In case SAT, relative humidity does not exceed 100 % due to its limitation by saturation adjustment, while in case DIA and PRG, average supersaturations of 0.05 % are reached at a height of 2 m, which corresponds to typical values within fog (Hammer et al., 2014; Mazoyer et al., 2019; Boutle et al., 2018).

For cases DIA and PRG, starting from 06:15 UTC (in 2 m height) and 07:15 UTC (in 20 m height), supersaturations are removed and the air becomes subsaturated (on average). This is in contrast with case SAT, where the saturation adjustment approach keeps the relative humidity at 100 % as long as liquid water is present (i.e., until the fog has dissipated). Around 06:00 UTC, which is shortly after sunrise, relative humidity drops rapidly in PRG and DIA as a direct consequence of direct solar heating of the surface and the near-surface air, preventing further supersaturation at these heights. While we cannot clearly identify the lifting of the fog in case DIA and PRG (due to the limited humidity range displayed), we note that for case SAT we can identify lifting times as a decrease of relative humidity around 08:45 UTC at 2 m height and around 09:10 UTC at 20 m height.

Besides this inherent difference in relative humidity, the general time marks (formation, lifting and dissipation, defined by Maronga and Bosveld, 2017) of the fog layer are identical for cases SAT, DIA and PRG.

Figure 5Time series of liquid water path (LWP) for cases using saturation adjustment, the diagnostic approach and a prognostic method for the diffusional growth.

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Figure 5 shows the liquid water path (LWP) for all cases. Differences in the LWP appear between 04:00 and 11:00 UTC and do not exceed 1 % (lower values for cases DIA and PRG), indicating that the choice of the condensation scheme does not affect the total water content of the simulated fog layer.

It can be summarized that, although the assumptions of saturation adjustment are not valid for the simulation of fog when using a very small time step, the mean liquid water content is not changed by more than 1 % and the general fog structure is not altered when using a one-moment microphysics and neglecting supersaturation. This is probably due to the very small supersaturation that is not strong enough to generate a significant change in the effective droplet radius and which could possibly lead to stronger sedimentation or higher radiative cooling rates.

Figure 6Time series of LWP for simulations using saturation adjustment (N2SAT in black), the diagnostic scheme (N2DIA in blue) and the prognostic method (N2PRG in red). All cases use the activation scheme of Cohard et al. (1998).

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Figure 7Profiles for liquid water mixing ratio (a) and droplet number concentration (b) at 04:00, 06:00 and 08:00 UTC.

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4.2.2 Two-moment microphysics scheme: impact of supersaturation calculation on CCN activation

Even though different methods for calculating supersaturation which interacts with the diffusional growth are not strong enough to generate any noteworthy differences by using a one-moment microphysics (considering a constant value for n_c), the impact of different methods modeling supersaturation on CCN activation by using a two-moment microphysics might be significant.

Table 1Overview of conducted simulations. The droplet number concentration n_c is only prescribed for simulations without activation scheme. In the simulations N1DIA–N3DIA, n_c is a prognostic quantity and is thus variable in time and space. The aerosol background concentration is abbreviated with N_a,tot and used to initialize the activation schemes. Note for the scheme after Cohard et al. (1998) a conversion to the parameter C must be applied, while for both other activation schemes this value is directly used to prescribe N₀ and N_t, respectively.

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Figure 6 shows the LWP for simulations applying the activation scheme of Cohard et al. (1998) in conjunction with the usage of saturation adjustment (N2SAT), the diagnostic scheme (N2DIA) and the prognostic scheme (N2PRG) for calculating supersaturations. It can be seen that the prognostic and diagnostic methods produce similar LWP values. However, for case N2SAT the LWP is nearly 70 % higher than for the other two cases. In Fig. 7 profiles of the liquid water mixing ratio (left) and droplet number concentration (right) are shown. From that figure it can be seen that in case of N2SAT, both the fog height as well as the liquid water mixing ratios within the layer are higher than in N2DIA and N2PRG, respectively. However, small differences in q_l can also be found between N2DIA and N2PRG (e.g., at 06:00 UTC in the second third of the fog layer). This is explained by slightly higher values for the number concentration in case of N2DIA than in N2PRG. However, both are at approximately 75 cm⁻³ at 06:00 UTC. In contrast, in simulation N2SAT, a number concentration of 120 to 150 cm⁻³ (at the top) is observed, which is about 60 %–100 % higher in comparison to N2DIA and N2PRG. These differences can be explained by the different methods for calculating the supersaturation, since activation is the main process altering the droplet number concentration. Therefore, we can implicitly derive from the droplet number concentration that the predicted and diagnosed supersaturations using the prognostic and diagnostic method are similar. These differences between N2SAT and N2DIA–N2PRG are, however, in good agreement with values reported for a stratocumulus case by Thouron et al. (2012). Their Fig. 2 shows that the number concentration of the diagnostic and prognostic method were also similar and the case with saturation adjustment overestimated the supersaturation and therefore the droplet number concentration. As the fog droplet number concentration has a crucial feedback on the overall LWP of the fog layer, the times of lifting and the time of its dissipation, the reported differences in n_c are significant regarding the accurate modeling and prediction of fog. The reason why the number concentration is such a critically parameter can be ascribed to their impact on sedimentation and radiative cooling, which is explained in more detail in Sect. 4.4.3.

Figure 8As in Fig. 6 but also for 2 m (dotted–dashed) and 4 m (dashed).

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In order to evaluate the possible effect of the grid spacing, in conjunction with different methods for calculating the supersaturation, on CCN activation, we repeated each of the cases N2SAT, N2DIA and N2PRG with two coarser grid spacings of 2 and 4 m. The general effect of the grid spacing on the temporal development and structure of radiation fog is discussed in detail in Maronga and Bosveld (2017). In this section, we will focus only on changes in LWP due to different supersaturation calculations at different spatial model resolutions. For isolating the effect of the grid spacing, all simulations with a coarser grid spacing were carried out with the same time step of 0.125 s, which corresponds to the average time step of the simulations at highest grid spacing of 1 m. In this way, effects of different time steps induced by different grid spacings could be eliminated.

Figure 8 shows the LWP for all grid sensitivity runs. First of all, note that for 1 m grid spacing, the results reflect the results shown in Fig. 6 and discussed above (i.e., significantly higher LWP for case N2SAT than for cases N2DIA and N2PRG). Moreover, Fig. 8 reveals that these results are somewhat sensitive to changes in the grid spacing. For all cases we observe a tendency towards higher LWP values with increasing grid spacing, at least for cases N2DIA and N2PRG. These difference are, however, not larger than 4 g m⁻² and are thus significantly smaller than the observed differences found between the different methods to calculate supersaturation. Note, however, that the relative change in LWP with grid spacing is higher for case N2DIA than for case N2PRG. Quantitatively speaking, in case of 1 m grid spacing the relative difference of the LWP is 2.1 % between N2DIA and N2PRG during the mature phase, while for the case with a grid spacing of 4 m it reaches 8.1 %. This might be explained by the fact that the diagnostic scheme is very sensitive to small errors (e.g., induced by the numerical advection) in the temperature and humidity fields (e.g., Morrison and Grabowski, 2008; Thouron et al., 2012). A coarser spatial resolution here can lead to larger error introduced by spurious supersaturation. We thus suppose that the increased differences (see Fig. 8) by larger grid spacings are induced by spurious supersaturation, which affect the CCN activation and hence influence the LWP of the fog layer.

Furthermore, we note that coarser grid spacings lead to a later fog formation time, which is in agreement with Maronga and Bosveld (2017) and can be ascribed to under-resolved turbulence near the surface at coarse grids.

In summary, we can thus conclude that the sensitivity to changes in the grid spacing is rather small, but it might imply differences in the LWP of the simulated fog layer of up to 4 g m⁻².

Figure 9Time series of LWP and n_c (as a horizontal and vertical average of the fog layer) for the reference and N1DIA–N3DIA case.

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4.4.1 Visibility

In Fig. 11 the simulated visibility is shown for the cases N1DIA–N3DIA in 2 m height together with the observed values at Cabauw (for illustration only). Visibility is calculated from the LES data following Gultepe et al. (2006) as

This visibility estimation (with n_c and q_l given in units of cm⁻³ and g m⁻³, respectively) thus significantly depends on the droplet number concentration and the liquid water content. Unlike in the first part of this paper, analyzing visibility estimations from the simulations might illuminate the capability of LES to predict visibility. Figure 11 reveals that visibility follows the same general temporal developed for all cases, with a rapid decrease at fog formation, deepening and dissipation, with minimum values at around 100 m (which is close to the observed values). We also see noteworthy differences, particularly shortly before 02:00 UTC (before fog deepening) at around 05:45 UTC (shortly after sunrise). For both time marks, cases N1DIA–N3DIA display sudden increases in visibility, due to an fast decrease in n_c in 2 m height, which are not reproduced by case SAT, as n_c is fixed value in this case. The sudden increase in visibility around 00:45 UTC in the observations is possibly related to this process. Also, the time marks of formation and dissipation vary. For cases N1DIA–N3DIA, the formation time is significantly advanced compared to case SAT, while dissipation time only shows a small tendency towards earlier times, at least for N1DIA and N3DIA. Case N2DIA displays a different behavior, with a later fog formation and higher visibility and accordingly earlier dissipation time. This is in line with the findings discussed above (i.e., a much weaker fog layer that, as a direct consequence, can dissipate much faster). Otherwise, all cases display almost identical visibility as soon as the fog has deepened.

Table 2Table of fog's life cycle time marks.

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4.4.2 Time marks of the fog life cycle

The effect of the different droplet concentration (induced by the usage of different activation schemes) on the time marks of the fog life cycle is summarized in Table 2. While N1DIA and N3DIA have similar time marks, N2DIA stands out and shows a delayed onset by 25 min, while the maximum liquid water mixing ratio is reached 45 min earlier than in the other cases. Also lifting and dissipation are affected and occurred 15 and 40 min (with respect to simulation N3DIA) earlier. This is due to a lesser absolute liquid water mixing ratio which evaporates faster by the incoming solar radiation. Therefore, it can be concluded that the use of different activation schemes (if they change the droplet number concentration) has an effect on the time marks on the life cycle as well as on the fog height and the amount of liquid water within the fog layer.

Figure 12Profiles (instantaneously and horizontally averaged) of liquid water mixing ratio at 04:00, 06:00 and 08:00 UTC, and profiles of liquid water budget terms at 06:00 UTC.

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4.4.3 Budgets of liquid water and droplet number concentration

In this section we will analyze the budgets of liquid water and droplet number concentration in physical terms. As in the preceding section, we will use the cases with different activation parameterizations, since they provide us a range of different CCN concentrations. Figure 12a shows the profiles of the liquid water mixing ratio at 04:00, 06:00 and 08:00 UTC, i.e at different times during the mature phase of the fog. A detailed analysis of budgets at other stages of the life cycle of the fog is beyond the scope of this paper. The maximum q_l in the fog layer is reached at approximately 06:00 UTC at a height of 60 m. Afterwards a further vertical growth of the fog can be observed, where no further increase in liquid water takes places as a result of larger vertical extent of the mixing layer and due to rising temperatures after sunrise. Moreover, Fig. 12b and c show the liquid water budget during the mature phase of the fog at 06:00 UTC, when the fog was fully developed. Almost all three cases show identical values for condensation rates in the lowest part of the fog layer, with values being in the same order as the evaporation rates so that the net gain in this region appears to be small (see Fig. 12b). However, the N2DIA case (with the lowest n_c) exhibits a generally lower absolute evaporation rate compared to both other cases, which can be attributed to the slightly higher mean values of the relative humidity (not shown) than in N1DIA and N3DIA. In the upper part of the fog layer, higher values of the condensation rate are observed (especially for N1DIA and N3DIA) with a concurrent decrease in evaporation rates, leading to differently strong deepening of the fog layer. At a height of approximately 80 m, a maximum of the evaporation rates can be observed, representing the presence of subsaturated regions at this height and the top of the fog. Larger differences can be observed in the sedimentation rates. First and foremost the sedimentation is proportional to the liquid water mixing ratio (see also Eq. 4). The strength of sedimentation also depends on the mean radius of the droplets, which increases with a decreasing number of activated drops. Here, a lower n_c for a given amount of liquid water leads to a higher mean radius, compared to a higher n_c where the same amount of water is distributed to more drops, decreasing the mean radius. Integrated over height, all three cases exhibit approximately the same sedimentation rates. Therefore, case N2DIA experiences the strongest loss of liquid water due to sedimentation (in relative terms). Moreover, Fig. 12c shows that sedimentation partially counteracts the gains caused by condensation at the upper edge of the fog. The net advection transports liquid water from the second third of the fog layer (position of the maximum) to higher levels. It can be summarized that all terms contribute significantly to the net change of the liquid water mixing ratio, illustrating that all microphysical processes deserve a proper modeling for radiation fog. In the mature phase, however, sedimentation plays a key role, showing the highest values for the individual tendencies. As a result liquid water is slowly and constantly removed from the fog layer. These findings are in good agreement with previous investigations by Bott (1991).

The sum of all tendencies, which is shown in Fig. 12d, is the height-dependent change of the liquid water. Also here it can be seen that in the lower 50 m the net tendency is negative, while in higher levels we observe a positive tendency so that the fog continues growing vertically while the liquid water content within the fog layer decreases.

Figure 13a additionally shows the profiles of n_c. We note that the profiles of the different cases differ quantitatively but not qualitatively. The stage of the fog can thus be identified in the profiles for all cases. At 04:00 UTC, the highest supersaturations occur close to the ground due to cooling of the surface and near-surface air, leading to high activation rates and therefore high n_c near the surface (not shown). At 06:00 UTC a well-mixed layer has developed that is driven by the radiative cooling from the fog top. While the turbulent mixing leads to a vertical well-mixed n_c, we note the maximum at the top, where the radiative cooling induces immense aerosol activation. This is further illustrated in the budget of the n_c in Fig. 13b and c, where instantaneous data at 06:00 UTC are shown. Here, we see clearly that aerosol activation at the top of the fog layer is the dominant process in the mature phase of the fog, while activation near the surface is comparably small. Evaporation of droplets, though small in magnitude, occurs only at the fog top, reflecting upward motions of foggy air penetrating the subsaturated air aloft where droplets then evaporate. Also, we see that both advection and sedimentation rates are much smaller than activation rates so that the net change in n_c is controlled by the activation near the fog top during the mature phase of the fog.

Figure 13Profiles (instantaneously and horizontally averaged) of n_c at 04:00, 06:00 and 08:00 UTC, and profiles of n_c budget terms at 06:00 UTC.

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