Small-scale structure of radiation fog: a large-eddy simulation study

Authors


Abstract

In order to study the small-scale structure of radition fog, large-eddy simulations (LESs) of a fog case are analysed. The LESs were performed at very high resolution –2 m in the horizontal and 1 m in the vertical. Despite uncertainties in the measurements, particularly for advection, the main characteristics of the fog layer were well captured by the model. Radiation fog forms in statically stable stratification near the ground. During the formation phase, small stripes occur in the middle of the fog layer, associated with a significant burst in the turbulent kinetic energy (TKE). During the development phase, the dynamics of the fog layer change significantly. The maximum of the variance moves to the top of the fog layer where horizontal rolls appear clearly. These eddies have their centre near the mean top of the fog layer and have a depth corresponding to about one third of the fog layer height. This leads to a maximum of TKE at the top of fog, and to very strong scatter on the liquid water content. The energy is clearly produced at a length-scale corresponding to the fog height. The turbulence is 3D homogeneous inside the fog layer, while it is better characterized by 2D turbulence near the top. During the dissipation phase, the radiative heating of the surface increases the convective structure of the fog. The dissipation of fog at ground level takes a long time (about 2 h), even if the soil conditions are homogeneous. The top of the stratus layer is homogeneous, while the spread of the base height reaches a value typical of the cloud thickness. Copyright © 2012 Royal Meteorological Society

1. Introduction

The life cycle of a fog layer, from formation to dissipation, clearly has a strong impact on personal safety and the economy. Gultepe et al. (2007) estimated that the financial and human cost related to fog events is comparable to that of tornadoes and storms. Adverse ceiling and visibility conditions contribute to 35% of the weather-related accidents in the civil aviation sector and cause 168 fatal casualties per year on average (Herzegh et al., 2004).

In consequence, there is an increasing demand for accurate fog forecasts. While current numerical weather prediction models are able to forecast situations that are favourable to fog events, these NWP models are usually unable to forecast the exact location and life cycle of a fog layer (formation, vertical development and dissipation), e.g. Roquelaure and Bergot (2009), Van der Velde et al. (2010), Zhou et al. (2012).

This study focuses on radiation fogs, which are prevalent in continental areas. Radiation fog is mainly formed during the night above continental surfaces under high pressure systems. The radiative cooling of the surface can lead to saturated conditions and hence to the formation of cloud droplets. Fog formation results from the condensation of water vapour into liquid droplets, reducing the horizontal visibility near the surface. Fog formation and evolution are directly related to thermodynamic, turbulent, radiative and microphysical processes as well as surface conditions. However, our precise understanding of fog physics remains partial, due to the small scales of the processes at work in a fog layer.

The mechanisms of fog formation, development and dissipation are very complex and are very difficult to study. Many efforts have been made since the 1970s to improve our knowledge of fog. Several comprehensive observational programmes associated with in situ measurements have been carried out in Europe and North America: Roach et al. (1976) at Cardington (UK); Meyer and Lala (1986) for Fog82 at Albany (New York); Guedalia and Bergot (1994) for Lille88 and Lille91 (France); Duynkerke (1999) at Cabauw (Netherlands); Fuzzi et al. (1992, 1998) at the Po Valley (Italy). Based on these observations, numerical simulation studies have improved our knowledge of fog physics. Worthy of note are Brown and Roach (1976), Musson-Genon (1987), Bott et al. (1990), Duynkerke (1991), Bergot and Guedalia (1994) and Pagowski et al. (2004), among others.

However, these studies have improved the quality of the operational forecasting of fog only slowly. It is now well demonstrated that the vertical resolution is one of the key points in fog forecasting (e.g. Tardif, 2007). Following this idea, an alternative for improving fog forecasting is based on column (1D) models. One-dimensional models have been successfully used to forecast fog in various places (e.g. Bergot et al., 2005; Clark, 2006; Terradellas and Cano, 2007; Stolaki et al., 2012). Intercomparison experiments among different operational 1D models have been done by Bergot et al. (2007) and Terradellas and Bergot (2007), and have proven that 1D models can simulate the major features of the fog life cycle. However, a high sensitivity to the physical parametrizations has been found revealing significant differences in the nocturnal cooling near the ground. These works show that the simulation of a fog layer is very sensitive to the parametrization of turbulence, and that these parametrizations are particularly deficient in the case of stable stratification.

The understanding of the physical mechanisms involved in a fog layer is currently a key point if fog forecasting is to be improved. The goal of this study is not to directly improve an operational forecasting model, but to focus purely on understanding the physical mechanisms of a fog layer. Here, large-eddy simulation (LES) modelling at very high resolution (1 m vertically and 2.5 m horizontally) is used to capture the main characteristics of the development of the fog layer. The quality of these kind of simulations depends on the vertical and horizontal resolution as discussed in Beare and MacVean (2004) who showed that the simulations converge at 2 m resolution. It should be noted that, to our knowledge, this is the first time that an LES study of radiation fog has been performed at such high resolution, to resolve the turbulence processes as much as possible, and thus to explicitly simulate the processes involved in the fog life cycle. The objective of this work is to study the mechanisms associated with the radiation fog during its whole life cycle. The life cycle of radiation fog can be decomposed into three steps according to the behaviour of the turbulent kinetic energy (TKE): the onset phase, the development phase and the dissipation phase (e.g. Nakanishi, 2000). The formation stage is characterized by low intensity of turbulence and a strong inversion near the ground. The development phase is characterized by an increase in TKE and the appearance of a mixed fog layer. Finally, during the dissipation stage, the unstable fog layer is intensified by surface heating (by solar radiation) or by an increase in wind intensity. To study the transition between these three phases better, this LES study will be separated into two different periods. The first one concerns the formation and the beginning of the development phase. The second one concerns the mature fog and its dissipation.

In section 2, the case studied and the model used will be presented. Section 3 focuses on the first period of the night concerning the formation of the fog layer and the beginning of its development. The influence of large-scale tendencies will be explored in section 4. In section 5, the mature phase of the fog layer and the dissipation will be studied. Finally, conclusions and perspectives are drawn in section 6.

2. Descriptions

2.1. Description of the case studied

The case studied here occurred during the night of 18 to 19 February 2007. The observations were made near Paris, France, at the Site Instrumental de Recherche en Télédétection Atmosphérique (SIRTA; Bergot et al., 2008; Haeffelin et al., 2010).

On 18 February, following two days of unsettled weather with light precipitation, a weak ridge developed over France associated with high pressure centred over England. There was a light wind from the west and the sky was cloudless during daytime. The maximum temperature was about 16 °C (Figure 1(b)), and the maximum visibility was about 30 km at 1500 UTC. At sunset, at about 1700 UTC, the wind shifted to the east and strengthened to reach about 3.5 m s−1 at 10 m (Figure 1(c)). The cooling near the surface, mainly due to radiative processes, reached about −3 K h−1 (Figure 1(b)). At 1900 UTC the vertical stratification in the surface layer was about 0.13 K m−1. A cloud passing briefly at around 1900 UTC appeared on the downwelling long-wave radiative flux (Figure 1(a)). This cloud temporarily reduced the stability of the surface layer (Figure 1(b)). Afterwards, the wind speed at 10 m was about 1 m s−1 (Figure 1(c)). Between 1900 and 2300 UTC, the visibility decreased progressively. At 2240 UTC, fog droplets appeared near the surface, and the horizontal visibility became less than 600 m. Unfortunately, the droplet concentration was not a routinely observed variable, and so it is not possible to know the microphysical properties. Over the next 30 min, the surface temperature increased, and the stable surface layer was destroyed. The downwelling long-wave radiative flux suddenly increased by 65 W m−2 (Figure 1(a)). This increase is typical for fog formation, e.g. Vehil et al. (1889). A well-mixed, saturated fog layer developed rapidly from the surface. Oscillations could be observed on the downwelling radiative flux during the fog event, and particularly during the mature fog phase. This flux was directly related to the optical thickness of the fog layer, and the oscillations were certainly a consequence of local variation in the fog height.

Figure 1.

Time series of observations of (a) the downwelling long-wave radiative flux, (b) temperature at 1 m (bold line), at 2 m (dotted line), at 5 m (dash-dotted line), at 10 m (dashed line) and at 30 m (solid line) and (c) wind speed at 10 m. Vertical profiles of (d) observed temperature, (e) humidity and (f) wind speed at 2100 UTC (bold line), 0300 UTC (dashed line), 0600 UTC (dotted line) and 1200 UTC (dash-dotted line).

Figures 1(d, e, f) illustrate the vertical profiles of temperature, relative humidity and wind speed from sonde data at different times during the life cycle of the fog layer. It is very difficult to precisely define the fog top from the 100% threshold on relative humidity profiles because of measurement uncertainties. By comparing the inversion on temperature profiles and 100% threshold on relative humidity profiles, it can be seen that the fog height was more than 200 m during the mature phase. However, significant variation can be observed during the night in both the 100% relative humidity threshold and the inversion height in the temperature profiles. The base of the inversion layer on the temperature profiles was at approximatly 120 m at 0000 UTC, 175 m at 0300 UTC and 125 m at 0600 UTC. The difference in the inversion is clearly illustrated by comparing temperature profiles at 0300 and 0600 UTC on Figure 1(d). A comparison between temperature profiles at the top of the boundary layer (around 1000 m) also illustrates advection processes. For example, advection leads to a cooling of about 2 K at 1000 m between 0300 and 0600 UTC.

The wind speed profiles are plotted in Figure 1(f), from which significant variation of the wind during the night can be observed. At the beginning of the night (between 1800 and 0000 UTC), the wind at 500 m was about 5 m s−1. At 2100 UTC, the wind showed two nocturnal jets at 75 m and at 300 m. During the fog event (between 0000 and 0600 UTC), the wind was weak inside the fog layer, but strong variation can be seen near the top of the fog layer, with, for example, a wind speed of about 8 m s−1 at 300 m at 0300 UTC. During the dissipation phase, the wind speed at the top of the fog or low cloud layer also showed a strong jet of about 12 m s−1. The wind at 1500 m was moderate during the night, between 8 and 10 m s−1.

2.2. Model description

The non-hydrostatic anelastic research model Meso-NH (Lafore et al., 1998) designed to simulate atmospheric motion over a broad range of scales, from mesoscale to turbulent eddies, was used here (http://mesonh.aero.obs-mip.fr gives more details).

The configuration chosen for LES modelling of the life cycle of a fog layer used an anelastic system of equations and a 3D turbulent scheme with a prognostic TKE (Cuxart et al., 2000) and a Deardorff mixing length (Deardorff, 1980). The lateral conditions were cyclic.

The radiative transfer was computed by solving long-wave (LW) and short-wave (SW) radiative transfers separately using the European Centre for Medium-Range Weather Forecasts (ECMWF) operational radiation code (Morcrette, 1991).

For the microphysics, the model includes a one-moment bulk microphysical scheme. The liquid cloud water is calculated prognostically, but condensation and evaporation are assumed to be instantaneous. The droplet sedimentation is computed by considering Stokes' Law for the cloud droplet sedimentation velocity and assuming that the cloud droplet size distribution follows a generalized gamma function:

equation image(1)

where

equation image(2)

Γ is the gamma function, D is the cloud droplet diameter, N is the number of cloud droplets, rc is the cloud liquid water content (LWC), and ρw and ρa are the densities of liquid and atmosphere rerspectively. α and ν are parameters allowing distribution fitting. These parameters were adjusted using typical fog droplet spectra (Pawlowska and Brenguier, 2000; Remy, 2006; Rangognio, 2009): α = 3, ν = 1, and N = 300 cm−3. No ice was considered, given the temperature of the fog layer.

The atmospheric model was coupled with the ISBA surface scheme (Interaction between Soil Biosphere and Atmosphere; Noilhan and Planton, 1989; Noilhan and Mahfouf, 1996). This scheme simulates the exchanges of energy and water between the land surface (soil, vegetation and snow) and the atmosphere above it. It uses five prognostic equations for the deep temperature, the deep soil water content, the surface temperature, the surface soil water content and the water interception storage by vegetation (http://www.cnrm.meteo.fr/surfex/ gives more details).

The horizontal resolution varied from 2.5 to 20 m depending on the life cycle of the fog layer. The horizontal domain size was 200 × 200 grid points, and consequently varied from 500 m × 500 m at the beginning of the fog to 4000 m × 4000 m during the dissipation phase. The domain was extended because the turbulent structures in the fog layer grew progressively during the life cycle of the fog layer. 135 vertical levels were used between the soil and the top of the boundary layer (1500 m). The vertical resolution was 1 m for the first 50 points nearest the ground and increased slighly up to the top of the model. For the mature phase of the fog, the 1 m resolution grid was shifted to the top of the fog layer and the vertical grid was stretched below and above the inversion. The time step varied from 0.1 s during the formation phase to 0.5 s during the dissipation phase. The radiation code was called every 60 s to reduce the computational cost.

3. Formation and development of the fog layer

3.1. Simulation set-up

The model was initialized with the sonde data before the fog formation at 2100 UTC. The vertical profile of temperature, relative humidity and wind speed are illustrated in Figure 1(d–f). In order to generate turbulence, white noise of 0.5 K was applied in the first 50 m. Following the profiles from soundings, a geostrophic wind of 5 m s−1 was prescribed up to 500 m and accelerated to reach 8 m s−1 at 1000 m. The runs finished at 0300 UTC.

As shown in Bergot and Guedalia (1994), the values near the ground should be precisely initialized. The temperature and moisture of the ground were initialized from ground measurements. For this simulation, the soil temperature and soil moisture were estimated as the average of soil measurements, obtaining a surface temperature of 281 K and a normalized soil moisture of 98%.

3.2. Reference simulation

Figure 2(a) shows the time–height cross-section of the simulated mean LWC. The fog started to form near the ground at about 2245 UTC, close to the observed time of fog formation. In the first 30 min, the LWC was small and the fog layer was shallow. The depth of the fog layer increased monotonically with time until 0100 UTC. Afterwards, the fog grew more rapidly, and the vertical gradient at the top of the fog seemed to reduce. At 0300 UTC, the fog height was about 140 m, which is close to the observed fog height. Initially, the LWC had a maximum near the ground, but when the fog became optically thick, this maximum rose close to the fog top. Note, however, that the fog vertical development between 2300 and 0100 UTC was smaller than the observed one.

Figure 2.

Time–height cross-section of (a) the simulated mean liquid water content (with contour intervals 0.05 g kg−1) and (b) the simulated turbulent kinetic energy (intervals 0.05 m2s−2) for the reference simulation.

As a consequence of the underestimated modelled fog thickness, the modelled temperature profile showed a lower mixed layer than the observed one (30 m for the simulation against 90 m for the observation at 0000 UTC). At 500 m, the modelled and observed temperatures were close. The evolution of the ground temperature was also correctly modelled (not shown). A comparison of the modelled and observed wind speed shows that the wind intensities were close above 150 m. The difference between the simulated and observed fog vertical development could be mainly due to inaccuracy in the large-scale forcing, as will be discussed in section 4. However, the main characteristics of the fog layer are well modelled by our LES and this simulation illustrates the mechanism involved in the life cycle of a fog layer.

Figure 2(b) shows the time–height cross-section of the simulated mean TKE. A burst of TKE can be observed at 2310 UTC. This burst corresponds to the destruction of the ground inversion, when the fog became dense, and concerns a small atmospheric layer about 15 m in height. Afterwards, the TKE stayed relatively constant inside the fog layer. During this period, the fog underwent significant changes in its thermodynamic properties. At 0100 UTC, the TKE inside the fog layer increased, which is clearly correlated with the vertical development of the fog layer. As discussed by Nakanishi (2000), this corresponds to the beginning of development stage of the fog layer. At this time, the turbulence became greater and the dynamics of the fog layer started to change significantly. This point will be studied in section 5 concerning the mature phase of the fog.

To estimate the fog optical thickness, Figure 3(a) shows the evolution of the mean modelled downwelling long-wave radiative flux compared to the observations at ground level. Before the appearance of the fog layer, the value of the modelled downwelling longwave flux is very close to the observed one. The increase in the downwelling radiative flux occurred at the appearance of the fog layer and appeared simultaneously in the model and the observation. The optical thickness of the growing fog layer, causing the strong radiative cooling at the top of the fog layer, is one of the key factors in the life cycle of the fog. Figure 3(a) clearly shows that the modelled flux is smaller than the observed one during the formation phase of the fog, until 0100 UTC. The observed downwelling radiative flux increased by about 65 W m−2 between 2245 and 2315 UTC while the modelled flux increased by about 65 W m−2 between 2245 and 0100 UTC. Consequently, the modelled optical thickness of the fog was smaller than that observed up to 0100 UTC.

Figure 3.

Time series of (a) the downwelling long-wave radiative flux at the ground, and (b) the turbulent kinetic energy at 10 m: observations (bold line), and reference simulation (thin line).

Figure 3(b) shows the evolution of the TKE at 10 m for the model and the observations. Before the fog formation, the modelled TKE was smaller than the observed one. This is likely due to the heterogeneous ground conditions or to nocturnal wind jets not being correctly modelled. The hypothesis of a geostrophic wind in forcing data seems to be too simple to correctly simulate the complexity of the real wind and consequently to precisely reproduce the small-scale variation of TKE (Porson et al., 2011). However, between 2215 and 2245 UTC just before the fog appeared, the TKE was very small in both model and observations. Near 2300 UTC, a sudden increase in the TKE near the surface appeared in both observations and model. It can be noted that this burst in TKE was underestimated by the model. The burst seems to be related to the fog formation, and the lower vertical development of the fog layer during the formation phase could explain the difference. After 0100 UTC, the modelled and observed TKE were close.

Figure 4(a) shows the instantaneous horizontal fluctuations for the cloud water content at 2310 UTC, the time of the TKE burst, in a horizontal plane at z = 5 m. Small stripes occur near the middle of the growing fog layer and they are aligned almost perpendicular to the mean wind direction. The modelled wavelength is about 40 m, which corresponds to 2–3 times the fog layer depth. These patterns disappear near the top of the fog layer. The existence of these stripes is clearly associated with a significant burst in TKE. To better illustrate these stripes, Figure 4(b) shows a vertical cross-section of the LWC along the x-axis. The stripes can be clearly seen to be localized in the first half of the fog layer, and produce strong variation of the LWC in the lower half of the fog layer. Above, the fog layer is homogeneous and the fog top height shows no significant variation over the different grid points. The stripes are strongly correlated in time and space with the TKE burst.

Figure 4.

(a) Horizontal cross-section at 2310 UTC and 5 m height of the liquid water content fluctuation (equation image, contours) and wind arrows (the mean wind speed at 5 m is about 1.6 m s−1). Axes show grid points along x and y. (b) Vertical cross-section of the liquid water content along the x-axis, with contour intervals 0.01 g kg−1.

The vertical shear of the horizontal wind is provided to examine the cause of the stripes and TKE burst. A large vertical shear, of about 200 m s−1km−1, appears near 2310 UTC inside a small layer between 2 and 10 m (Figure 5(a)). This kind of large wind shear inside a fog layer was also observed by Uematsu et al. (2005). The profile of the Richardson number, Ri, is plotted Figure 5(b). The Ri is less than the critical Richardson number (Ric = 0.25) in the first 10 m. The small Ri inside the fog layer is clearly caused by extremely large wind shear. The value of Ri < Ric and the strong wind shear allow us to conclude that the type of structures is induced by Kelvin–Helmholtz (KH) instability inside the fog layer. This kind of KH instability was observed previously by Uematsu et al. (2005, 2007) and also modelled by Nakanishi (2000). The effect of these structures seems very important during the formation phase of the fog layer. These shear-induced KH stripes may influence the drop-size distribution inside the fog, and consequently modify the microphysical properties of the fog layer. Mixing by KH waves clearly influences the dynamical conditions near the ground surface and allows the fog to develop vertically. The turbulence characteristics of these stripes will be studied in the next section.

Figure 5.

(a) Time–height cross-section of the simulated vertical shear of the horizontal wind, with contour intervals 10 m s−1km−1 and (b) vertical profile of the Richardson number at 2310 UTC.

3.3. Energy spectra

As previously shown, turbulence plays a critical role in the first phase of the life cycle of the fog. In order to better understand the variability of the turbulence field in the first hours of the fog, one can look at the spectra of the variance of potential temperature (Ricard et al., 2011). Figure 6 shows these spectra at the moment of the burst of KE at 2310 UTC (Figure 6(a)) and at 0100 UTC (Figure 6(b)). At the beginning of the fog, the maximum variance is clearly located in the middle of the fog layer. This maximum is reached for wavelengths between two and seven times the depth of the fog layer. The maximum clearly corresponds to the KH waves shown previously. At 0100 UTC, when KE suddenly increases inside the fog layer, the shape of the spectra changes. The maximum of variance is now located at the top of the fog layer, and the variance is relatively weak inside the fog layer. The maximum variance is reached for wavelengths between one and three times the depth of the fog layer.

Figure 6.

Distribution of the potential temperature variance according to the wavelength and height at (a) 2310 UTC and (b) 0100 UTC. The wavelength and the height are normalized by the height of the fog layer, Hfog.

Detailed spectra at 0100 UTC are plotted inside the fog layer and at the top of the fog in Figure 7(a, b). Inside the fog layer, energy is clearly produced at a length-scale corresponding to the fog height. Afterwards the downscale energy cascade occurs at a rate of −5/3, which is typical of the Kolmogorov hypothesis. This implies a 3D homogeneous turbulence inside the fog layer with an isotropic cascade of energy to dissipation. Since slightly stratified 3D homogeneous turbulence is highly dissipative, it must be sustained by some forcing, like large-scale forcing or radiative cooling near the top of the fog layer.

Figure 7.

Distribution of the potential temperature variance at 0100 UTC according to the wavelength at (a) Z/Hfog = 0.75 and (b) Z/Hfog = 1. The wavelength is normalized by the height of the fog layer, Hfog. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

At the top of the fog layer, the downscale energy cascade occurs at a rate of −3. The Kolmogorov hypothesis no longer seems valid, and the turbulent eddies are heterogeneous. It seems that waves (section 5) produce transverse rolls. These waves supply energy inside the fog layer, presumably through local breaking events. The turbulent eddies originating from both buoyancy and shear cause the fog layer to expand vertically.

4. Advection sensitivity

Radiosonde observations (Figure 1) clearly show that advection processes occurred during the night. The goal of this section is to evaluate their influence on the fog and turbulence properties. Moisture advection was introduced to test the impact of advection on the results presented above. It should be noted that, in order to simplify this sensitivity study, we focus on moisture advection only. Model sensitivity to temperature advection was mentioned in Van der Velde et al. (2010), and will be investigated in future work. The moistening rate was constant with height up to 200 m, and decreased linearly to zero between 200 and 250 m. The moistening started at 2100 UTC and ended at 0000 UTC. Two experiments were conducted. In the first, named advq1, a moisture advection of 3×10−5 g kg−1s−1 was introduced and, in the second, named advq2, the moisture advection was increased to 6×10−5 g kg−1s−1. These moistening rates were chosen approximately, but they improved the performance of some modelled fog characteristics against observations.

The time–height cross-sections of the simulated mean LWC for the advq1 and advq2 experiments are plotted in Figure 8(a, b). Compared to the reference run (Figure 2(a)), there is a significant improvement for the fog top height. However, the fog forms slightly too soon in both advq1 and advq2. At 0000 UTC, the fog top reaches 80 m for advq1 and 130 m for advq2 as against 30 m for the reference simulation (and about 90 m for the observations). However, the main properties of the fog layer are not modified. During the formation phase, the LWC is small and the level of maximum LWC is close to the ground. During the mature phase, the level of maximum LWC is no longer near the ground and moves to the upper level of the fog layer.

Figure 8.

As Figure 2(a), but for the sensitivity simulation with large-scale advection: (a) run advq1 and (b) run advq2.

Figure 9(a) shows the profiles of mean simulated potential temperature at 0000 UTC against the observations. The reference simulation shows a very shallow well-mixed layer of about 25 m near the ground while the observed one is more developed and reaches about 100 m. The advq1 and advq2 simulations are closer to the observation, even though the inversion is slightly too low in advq1 and slightly too high in advq2. In both simulations, a well-mixed fog layer is developed. As a consequence of the lower modelled fog layer in the reference, the modelled temperature near the ground is lower than the observed one by about 1.5 K. For simulations with advection, the modelled and observed temperatures of the fog layer are close. The difference between the simulated temperatures is relatively small – about 0.4 K – but the difference between the modelled fog layer heights is about 25 m, which is about one quarter of the mean fog height.

Figure 9.

(a) Vertical profiles of the potential temperature at 0000 UTC. (b) Time series of downwelling long-wave radiative flux at the ground for observation (bold line), the sensitivity simulation advq1 (thin solid line), the sensitivity simulation advq2 (dash-dotted line), and the reference simulation (dashed line). (c) is as Figure 2(b), but for the sensitivity simulation advq1.

To evaluate the change in the optical thickness of the fog layer, observed and modelled downwelling long-wave radiative fluxes are plotted in Figure 9(b). The increase in the radiative flux occurs simultaneously with the vertical development of the fog layer. The increase in the optical thickness occurs earlier in the simulation with advection; the fog appears about 20 min earlier in advq2 than in the reference simulation. However, the increase in radiative flux is more sudden in the simulation with advection, corresponding to a more sudden increase of the fog layer. After 0100 UTC, the difference between observation and model has about the same value as the measurement uncertainty (5 W m−2). These simulations show that the rapid development of the fog layer between 2300 and 0100 UTC could be a consequence of local or large-scale advection.

It would be interesting to know the influence of advection on the TKE burst during the formation phase. To illustrate this point, Figure 9(c) shows the time–height cross-section of the simulated mean TKE for the advq1 simulation. The TKE burst is larger for this simulation, and reaches a maximum closer to observation values, about 0.35 m2s−2. The vertical extension of the TKE burst is also larger, about 40 m, and corresponds to the vertical development of the fog layer. As for the reference simulation, this TKE burst is a consequence of shear-induced KW waves.

Several advection mechanisms are possible. For example, observations show variations in wind direction at 10 m, favouring advection from other directions around the site. This study suggests that local heterogeneities play an important role in the formation and vertical development of the fog layer. Further work is needed to better understand the impact of these local heterogeneities. To summarize, the analysis in sections 3 and 4 shows that key characteristics of the fog layer during the formation phase are well captured by the model. These are the burst in TKE, a well-mixed saturated layer, and KH waves in the middle of the fog layer.

5. Mature and dissipation phases of the fog layer

5.1. Simulation set-up

The model was initialized with the sonde data at 0300 UTC. The data were corrected to initialize the saturated fog layer with a LWC of 0.2 g kg−1. In order to generate the turbulence, white noise of 0.5 K was applied inside the fog layer. The runs finished at 1500 UTC.

The horizontal resolution was 20 m and, consequently, the horizontal domain size (200 × 200 grid points) was 4000 m × 4000 m. 135 vertical levels were used between the ground and the top of the boundary layer (1500 m). The vertical resolution was 1 m close to the fog top and the vertical grid was stretched below and above. The time step was 0.5 s and the radiation code was called every 60 s to reduce the computational cost.

5.2. Reference simulation

The base and top of the fog layer are plotted in Figure 10. During the whole simulation time, the evolution of the mean top of the fog/cloud layer is more or less linear, at a rate of about 30 m h−1. During the mature phase of the fog (before 0800 UTC), the variability of the top of the cloud layer is substantial, more than one third of the height of the fog layer. For example, at 0600 UTC, the spread of the top of the fog layer is about 125 m while the mean top is about 250 m. Regarding the mean LWC, the fog lifts at the ground at about 0800 UTC, which is close to the observation, but the dissipation at ground level, considering the different simulated points of the horizontal domain, varies over about 2 h, which is a long time given the fact that the soil conditions are homogeneous. During these 2 h, the fog persists in some points of the domain (4 km × 4 km), while at other points the fog has evolved into low stratus. This demonstrates that, even with homogeneous ground conditions, fog takes a long time to dissipate due to internal processes inside the fog layer, and it is very difficult to forecast the dissipation time precisely. During the dissipation phase (after 0800 UTC), the spread of the top of the cloudy layer decreases and reaches less than 30 m after 1200 UTC. However, the spread of the cloud base reaches about 350 m at 1500 UTC, which roughly corresponds to the depth of the cloud layer. During the dissipation phase, the variability is concentrated at the base of the stratus layer. This is clearly a consequence of the spread of the mixed layer above the stratus and shows that the turbulence in the mixed layer above the cloud plays a significant role in the fog dissipation. These differences between the cloud top and base also show that cloud-base thermodynamic properties are uncorrelated with cloud-top properties.

Figure 10.

Time series of (a) the base and top of the fog/cloud layer and (b) the liquid water path. Mean values are shown as solid lines, minimum and maximum values by the grey area, and standard deviation by the vertical bars.

Figure 10(b) shows the liquid water path (LWP) in the fog layer. The LWP grows during the night to reach about 0.1 g m−2 at 0800 UTC, which corresponds to the mean dissipation time at ground level. During the mature phase, the spread of the LWP is relatively large and has values comparable to the mean LWP. During the dissipation phase, the mean LWP decreases slightly, but stays close to 0.1 g m−2. The main characteristic of the dissipation phase is the significant increase in the spread of the LWP, which reaches 0.2 g m−2 at 1500 UTC, for a mean value of about 0.1 g m−2. Figure 10(a) shows that this spread is mainly a consequence of the spread of the base of the cloudy layer.

To illustrate the characteristics of the fog layer at the end of the mature phase, just before the dissipation phase, Figure 11 shows the mean profiles of LWC, potential temperature, wind speed and vertical shear of the mean horizontal wind at 0500 UTC. The mean value of the LWC (Figure 11(a)) varies from 0.05 g kg−1 at the ground to 0.25 g kg−1 at the top of the fog layer. These values are consistent with the values observed during various field experiments. Classically, the values are a maximum near the top of the fog layer. It is interesting to note that the scatter on the LWC is very strong. Near the ground, the spread is about 0.05 g kg−1 and the mean LWC is about 0.05 g kg−1. The spread increases in the middle of the fog layer and reaches about 0.2 g kg−1. In the upper third of the fog, the fog layer is patchy, showing points without LWC and points with maximum LWC, about 0.45 g kg−1. This shows that the LWC varies strongly during the mature phase of the fog. This is mainly due to waves at the fog top, as shown later.

Figure 11.

Vertical profiles of (a) the liquid water content, (b) potential temperature, (c) wind speed and (d) vertical shear of mean horizontal wind, all at 0500 UTC. The mean value is plotted as a solid line and the minimum and maximum values as dashed lines.

Concerning the temperature profiles (Figure 11(b)), the spread of temperature is small in the lower half of the fog layer (less than 0.2 K). However, the height of the inversion varies significantly among the different grid points, showing a variability that reaches about one third of the height of the fog layer. The fluctuation of the wind speed (Figure 11(c)) is strong throughout the fog layer, between 2 and 5 m s−1. A jet of about 1 m s−1 can also be seen at the top of the fog layer. It produces a large vertical shear of the horizontal wind at the top of the fog layer (about 70 m s−1km−1, Figure 11(d)). This has a clear impact on the TKE, which is maximum at the top of the fog layer.

Figure 12 shows the time–height cross-section of the simulated TKE. The TKE inside the fog layer is about 0.2 m2s−2, and is relatively homogeneous during the mature phase, until 0800 UTC. This value is consistent with observations. The TKE is relatively homogeneous inside the lower half of the fog layer. However, the TKE exhibits a burst towards the top of the growing fog layer, with a maximum value of 0.8 m2s−2. This burst in TKE affects a vertical area of about one third of the fog height, and is strongly correlated with the wind shear at the fog top. During the dissipation phase, the maximum of TKE is near the ground, due to intensified buoyancy production initiated by surface heat flux. The growth of TKE near the ground reaches about 0.5 m2s−2 which is consistent with observations.

Figure 12.

As Figure 2(b), but for the mature and dissipation phases.

In order to better understand the scatter of the LWC, and the burst in TKE at the top of the fog layer, Figure 13 shows a cross-section of the LWC fluctuation superimposed on the wind field at 0500 UTC at two levels: at the mean top of the fog (230 m) and inside the fog layer (75 m). Waves are clearly seen at the top of the fog (Figure 13(a)), but they gradually become unclear inside the fog layer. The length-scale is about 1.5 km, and the waves are aligned at about 45° to the mean wind. Inside the fog layer (Figure 13(b)), the fog patterns seem more organized into cells, which seem to be distorted by the horizontal wind. However, we again find a small signature of the organized fog-top rolls. Inside the fog layer, the Richardson number Ri is computed from the wet equivalent potential temperature (Duynkerke, 1987). At the top of the fog layer, the Richardson number Ri ∼ 0.22 is close to the critical value (0.25). The waves at the top of the fog layer seems to be a consequence of shear-induced KH instability. Moderate wind at the top of the fog layer, as in this case, could be favourable for the formation of a large shear and consequently for the appearence of waves. Nakanishi (2000) also reported this kind of structure inside the fog layer. However, the characteristics of the waves produced in this simulation differ slightly from those of Nakanishi; in particular, the wavelength is larger (1.5 km) than those in the Nakanishi study (175 m).

Figure 13.

Horizontal cross-section at 0500 UTC of the liquid water content fluctuation (equation image, contours) and wind arrows (a) at 230 m and (b) at 75 m. (c) shows a vertical cross-section of the simulated liquid water content along the plane y = 200 − x. Contour intervals are (a, c) 0.05 g kg−1, and (b) 0.01 g kg−1.

The wave at the top of the fog layer is clearly illustrated in Figure 13(c). The eddies have their centre near to the mean fog top, and their depth is about 75 m. The LWC is larger in the ridge of the fog-top rolls and smaller in the troughs. This implies that the LWC is larger in updraught regions, and that the instantaneous vertical velocity plays a role in the condensation process. These rolls in the fog top have a significant impact on the spread of the fog top, but seem to not accelerate the vertical fog growth (Figure 10(a)). In spite of waves of significant amplitude, the fog layer has difficulty in penetrating the stable layer beyond the inversion. This stable layer is relatively dry, and thus fog droplets penetrating the layer evaporate rapidly.

The spread of LWP, mainly due to the waves at the top of the fog layer, has a strong impact on the radiative fluxes. To illustrate this point, Table 1 shows the values of the sensible heat fluxes, latent heat fluxes, net radiative fluxes, net radiative fluxes in the long-wave and short-wave and the ground heat flux. The short-wave radiative flux at the ground increases gradually until 1200/1300 UTC and reaches about 75 W m−2. However, the spread is substantial, about 100 W m−2. This spread is clearly a consequence of the spread of the LWP. The long-wave radiative flux at the ground is more homogeneous, with small spread, about 5 W m−2. Consequently, the net radiative flux is a maximum (about 60 W m−2) between 1200 and 1300 UTC but is heterogeneous with a spread of about 100 W m−2. However, the temperature below the cloudy layer increases slowly, at a rate of about 0.4 K h−1 and is relatively homogeneous, with a spread smaller than 0.5 K. The energy brought by the short-wave radiation is mainly used to develop the mixed layer below the cloud. This can explain the strong spread of the cloud base during the afternoon (Figure 10).

Table 1. Fluxes (W m−2) at the ground obtained from the simulations: sensible heat flux (H), latent heat flux (LE), net radiative flux (Rn), net radiative flux for long wave and short wave (RnLW and RnSW respectively), and flux into the ground (G).
Time (UTC)H LE RnRnLWRnSWG
  1. Upper values show mean values and lower values the (minimum/maximum) values over the simulation domain.

070010.281.36−3.77−4.050.29−15.41
 (6.26/17.15)(0.86/2.30)(−4.51/−3.31)(−4.82/−3.68)(0.18/0.51)(−23.04/−10.83)
080013.243.9310.31−5.1615.48−6.86
 (7.90/21.49)(2.56/6.50)(6.09/22.66)(−6.31/−4.49)(11.22/28.91)(−16.62/4.41)
090018.349.5525.10−7.4232.54−2.78
 (10.52/31.88)(5.60/14.91)(16.39/40.00)(−8.80/−6.56)(23.26/47.99)(−19.73/18.37)
100023.2417.2140.81−9.7650.580.35
 (13.20/40.22)(10.77/25.98)(25.67/69.66)(−11.26/−8.61)(35.14/80.07)(−26.96/31.43)
110027.2824.7453.87−11.7165.591.85
 (16.23/45.99)(16.99/38.13)(35.42/100.80)(−13.70/−10.15)(46.24/113.90)(−38.76/60.05)
120029.2731.4761.00−13.0774.080.26
 (16.00/49.42)(19.18/46.82)(39.53/119.20)(−15.45/−11.42)(52.98/134.30)(−43.99/65.60)
130027.0634.3359.02−13.4072.44−2.37
 (15.62/45.64)(23.08/53.45)(32.75/132.50)(−15.57/−11.64)(45.94/147.20)(−45.45/76.70)
140020.8333.6449.16−12.9762.13−5.32
 (11.87/37.67)(22.96/51.12)(25.13/135.70)(−15.81/−10.73)(37.22/150.80)(−44.93/86.95)
150012.1229.0932.90−12.0144.91−8.31
 (6.44/22.72)(17.46/47.20)(10.85/122.60)(−25.89/−9.59)(21.14/148.20)(−42.26/85.37)

6. Conclusions

A case of fog associated with a moderate wind (around 8 m s−1 at 1000 m) is analysed in this study. The fog event occurred during a cloudless night, forming at 2240 UTC and dissipating at 0800 UTC at ground level. Observations show significant variation during the second part of the night in long-wave downwelling radiative flux, wind profiles and height of the mixed fog layer. The fog layer developed rapidly and reached more than 200 m.

In order to study the small three-dimensional structures of the fog layer, a simulation with LES modelling at 1 m vertical resolution and 2.5 m horizontal resolution was performed. The model was run with an interactive land-surface scheme, which accurately reproduced the effect of fog on the ground, particularly during the dissipation phase. Despite numerous observations, uncertainties remain, principally in horizontal advection, surface heterogeneities and microphysical properties. Given these uncertainties, the model is able to reproduce the main characteristics of the fog layer, such as the scatter of fog properties (e.g. fog height, LWC).

During the formation phase, a burst in TKE appeared suddenly in the surface layer. This burst was confirmed by observations and corresponded to the formation of a dense fog layer. At this time, the simulation clearly showed stripes in the middle of the fog layer with a wavelength corresponding to two to three times the height of the fog layer. At this time, the middle of the fog layer was characterized by a large wind shear and a small Richardson number. Consequently, these structures seems induced by KH instability inside the fog layer. The end of the formation stage was characterized by an increase in the TKE inside the fog layer. The maximum of temperature variance was clearly located in the middle of the fog layer during the formation phase. This maximum moved to the top of the fog layer at the onset of the development stage. At this time, the maximum of variance was reached for wavelengths between one and three times the depth of the fog layer. The energy was clearly produced at a length-scale corresponding to the fog height. The turbulence inside the fog layer was 3D homogeneous, with an energy cascade at a rate of −5/3 characteristic of the Kolmogorov hypothesis. At the top of the fog layer, the turbulence was more characteristic of 2D turbulence, with an energy cascade at a rate of −3.

During the mature phase of the fog, waves were clearly found at the top of the fog layer with a horizontal scale of about 1.5 km. These waves were aligned at about 45° with the mean wind. Given a Richardson number close to the critical value and a large wind shear, these waves seems to be a consequence of shear-induced KH instability. Inside the fog layer, the structures seemed more organized, in cells distorted by the horizontal wind. The moderate wind of the case studied (about 8 m s−1 at 1000 m) could be favourable to the development of wind shear sufficiently strong to trigger KH instability. These waves led to a burst in TKE (at about 0.8 m2s−2) near the top of the fog layer. The variability of the fog top height was substantial, about one third of the fog layer height. The LWC varied strongly inside the fog layer, and the fluctuation of the wind speed was considerable inside the fog layer, between 2 and 5 m s−1. However, the temperature was fairly homogeneous inside the fog layer (scatter less than 0.2 K).

During the dissipation stage, the radiative heating of the surface increased the convective structure of the fog layer. This phase was characterized by an increase in the scatter of the LWP. This led to a strong spread of the ground fluxes (except for the long-wave radiative flux). The dissipation of fog at ground level took a long time, about 2 h. It should be emphasized here that the ground conditions were homogeneous and this range of dissipation time is due only to dynamical processes of the fog layer. During the dissipation stage, the height of the cloud top was relatively homogeneous while the spread of the cloud base reached values typical of the low-cloud thickness. The energy brought by the solar radiation was mainly used to develop the mixed layer below the low cloud.

The results presented here show that various organized structures occur in a fog layer. Particularly, this study shows that inhomogeneities within fog are not only a consequence of surface heterogeneities but can also be a consequence of internal processes inside the fog layer, such as waves at different scales. New observations would be helpful to improve our knowledge of fog physics. It would be especially interesting to improve the capture of fog heterogeneities, both horizontal and vertical. Recently developed techniques, such as lidar, could help in the organization of this kind of field experiment.

Acknowledgements

I would like to address many thanks to the late Laurent Gomes for all the work he did during and after the ParisFog experiment, and for many friendly discussions on fog. The support team of Meso-NH is acknowledged for their help during this study. I thank Valery Masson from CNRM-GAME for fruitful discussions and for his valuable comments on an earlier draft of this paper. Finally, I would like to acknowledge the reviewers for helpful comments on the first version of this manuscript.

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