Numerical modelling of the evolution of the boundary layer during a radiation fog event

Despite the impact it has on human activity, particularly transport, accurate forecasting of fog remains a major challenge for numerical weather prediction models. The complex interaction between various physical processes, many of which are parametrised and highly sensitive to small changes, is one of the key reasons for poor fog forecasts. One challenge for numerical models is predicting the structure of the boundary layer, which often undergoes a transition from statically stable to weakly unstable during the life cycle of a fog event. The recent local and non‐local fog experiment (LANFEX) has provided a new comprehensive and detailed observational dataset of fog events. Here, a case study has been used as the basis for an investigation of the effect of the humidity of the residual layer and wind speed on the stability of the boundary layer during a fog event. We find a very high sensitivity in the timing of the stability transition during the fog event; for example, a +3% relative humidity perturbation results in a delay of almost 3h, while a 0.45ms−1 10m wind speed perturbation results in a delay of more than 8h.

by fog leads to huge disruptions for air, sea and land transport, leading to financial and human losses comparable to losses from tornadoes or severe tropical storms (Gultepe et al., 2007). The accurate forecasting of fog is essential to efforts to reduce these disruptions, and this need is becoming ever greater as dependencies on transport, particularly aviation, increase. Although there is an abundance of studies on fog, it is still a phenomenon which is not fully understood due to the complex interaction between a myriad of physical processes and its variability in time and space.
Different types of fog are commonly categorised by the mechanisms responsi-

Introduction
The impact of fog is often understated; however, the reduction in visibility caused Numerical modelling of the evolution of the boundary layer during a radiation fog event tion of turbulence occurs, allowing radiative cooling to result in the formation of fog, while Rodhe (1962) and Duynkerke (1999) alternatively propose that turbulent mixing of near-saturated eddies of different temperatures results in saturation and the formation of fog.
After initial formation the fog can deepen, initially within the surface inversion. This stage of the fog life-cycle will be referred to as 'shallow stable radiation fog' (see Figure  1(b)). As this shallow stable radiation fog develops, it eventually becomes optically thick fog, usually defined as being opaque to thermal radiation (in the 8-12μm range) and may be detected at the surface by an increase in downwelling longwave radiation. After a fog becomes optically thick, the fog top becomes the primary location of radiative cooling. Consequent fog top cooling -which produces negatively buoyant air at the fog top that sinks and eventually reaches the surface -combined with a halt in surface cooling erodes the stability of the boundary layer, evolving it to a saturated adiabatic temperature profile. A fog which has undergone this transition in boundary layer stability is referred to as a 'deep adiabatic radiation fog' (see Figure 1(c)). The transition from a shallow stable radiation fog to a deep adiabatic radiation fog takes on average 2h from when the fog becomes optically thick, and it has been found to occur in approximately 50% of radiation fog cases in central England (Price, 2011). Deep adiabatic radiation fogs tend to be longer lived and can persist throughout the day, unlike shallow stable radiation fogs, which tend to dissipate after sunrise (Price, 2011). In addi-tion to the lifespan difference between shallow and deep fog, another important reason to understand the deepening process is that many boundary layer parametrisation schemes, such as that of Lock et al. (2000), rely on the model producing the correct boundary layer types in order to calculate the turbulent transport of heat and moisture. This highlights the importance that understanding and accurately simulating the evolution of the boundary layer has on accurate forecasts of fog.
The aim of this study is to investigate the influence of both wind speed and humidity of the residual layer -the portion of the nocturnal atmospheric boundary layer which is well mixed and above the surface inversion (labelled in Figure 1) -on the timing of the evolution and eventual dissipation of radiation fog. However, there are also other factors which are important, such as aerosol-fog interactions  and advection (Guedalia and Bergot, 1994;Porson et al., 2011), but investigating these processes is beyond the scope of this article.

Method
To investigate the influence that the wind speed and humidity of the residual layer have on the development of fog, we have carried out experiments using a singlecolumn atmospheric model. Single-column models have the benefit of being computationally cheap, containing the equivalent of a single grid point from a 3D model, and are easier to focus on particular questions through control of their large-scale forcing.
The single-column version of the Met Office's Unified Model (MetUM) has been used for these experiments and will hereafter be referred to as the SCM. It has been configured with 140 vertical levels, with 18 below 250m and the first model level at 1m for wind and 2m for temperature and humidity. The MetUM contains various physical parametrisations for sub-grid scale processes, including radiation (Edwards and Slingo, 1996), vertical mixing in the boundary layer (Lock et al., 2000), a large-scale cloud parametrisation (Smith, 1990), and a cloud microphysics parametrisation (Wilson and Ballard, 1999) which incorporates the updated cloud droplet number profile discussed by . The MetUM is coupled to the Joint UK Land Environment Simulator (JULES; Best et al., 2011). JULES contains information about the properties of the land surface, including soil moisture and temperature, albedo and surface roughness. Fog can be diagnosed using the model's visibility scheme, which uses a single monodisperse dry aerosol concentration that is hydrated using a Köhler curve (Clark et al., 2008). Given significant moisture, the scheme forms fog with the size and number of particles used to calculate an extinction coefficient which is then used in Koschmieder's Law (Koschmieder, 1924) to calculate visibility.
A case from the recent LANFEX (the local and non-local fog experiment; Price et al., 2018) field campaign was selected as a basis for the SCM experiments. The case chosen was the first intensive observation period of LANFEX (IOP1), which was based at the Met Office research unit site at Cardington, Bedfordshire, UK. IOP1 was a case of prolonged shallow stable radiation fog which formed at 1745 UTC on 24 November 2014. Figure 1. Schematic showing the structure of a typical radiation fog during its life cycle from pre-fog to a deep adiabatic radiation fog. The red arrows represent the region of the largest emission of longwave radiation, the curved blue arrow represents turbulent mixing, U is the wind speed, T is the temperature and RH is the relative humidity.
It remained a shallow stable radiation fog for 10h and only became a deep adiabatic radiation fog an hour before dissipation at 0815 UTC on 25 November 2014. This case is ideal for simulation using an SCM as there was no significant advection and the synoptic conditions remained approximately stationary over the night of the case study. Figure 2 shows the surface analysis chart at 1800 UTC on 24 November 2014, shortly after the fog was observed at Cardington. Overnight on 23 November 2014 an area of high pressure developed over the UK. This remained the situation throughout 24 November 2014, providing ideal conditions for fog to form. The area of high pressure moved on 25 November 2014, with higher winds and cloud cover preventing the reformation of fog during the following night.
A wide array of measurements were taken during IOP1, and in-depth details are presented in Price et al. (2018). Instruments included a cloud droplet probe and turbulence probe flown from a tethered balloon, a 50m tower -which houses instruments that take high frequency measurements of temperature, humidity and wind -surface radiation instruments, sub-surface instruments and radiosondes.
The SCM was initialised, prior to fog formation, with radiosonde data from a launch at 1700 UTC on 24 November 2014, and with observed soil properties. The initial winds have an approximate log-profile, with winds of 1.15ms −1 at 10m, 2.25ms −1 at 100m and approximately 4ms −1 in the boundary layer above 200m. Note that the SCM was run with a wind relaxation forcing which relaxed the model winds towards the observed winds from the five subsequent radiosonde launches during the night. The 10m wind remained below 1.75ms −1 overnight. This setup formed the control simulation of the SCM.
Two sets of sensitivity experiments were run -perturbation of the wind and perturbation of the humidity of the residual layer. To perform the wind perturbation experiments, the initial winds and wind forcing were perturbed by 1, 2 and 3ms −1 at 100m and above (with a linear interpolation of the perturbation to zero at the surface which results in a ~0.15, ~0.3 and ~0.45ms −1 perturbation, respectively, at 10m). The wind perturbations were selected to be within the observed range of variability at Cardington. The aim of these experiments was to assess the influence of wind speed and wind shear changes on the development of fogs and the consequent timing of the stability transition, rather than attempting find a critical speed at which fog will not form. The maximum perturbation was therefore restricted to 3ms −1 .
The humidity perturbation experiments were performed by perturbing the relative humidity (RH) by ±3% above the stable surface layer at 42m at 1700 UTC. This falls within the 2-5% RH uncertainty of the radiosonde relative humidity measurements during the LANFEX campaign . Finally, a combination of both the humidity and wind perturbations was performed to assess the interaction between the two. Figure 3 shows the temporal evolution of the simulated liquid water content and the observed fog top height (derived from a cloud droplet probe attached to a tethered balloon which was used to conduct profiles during IOP1). The control simulation produces fog from 1750 UTC. The fog develops vertically within a stable boundary layer until 2130 UTC, when a thin mixed layer at the base of the fog forms. By 2220 UTC the fog in the control simulation is of a similar depth to the observed fog (42m compared with the observed 50m). However, the observed fog is within a stable boundary layer without the thin mixed layer at its base. The fog continues to develop in the control simulation, and by 0330 UTC the fog is 100m deep -an over-estimate compared with observations. The control simulation continues to overestimate fog depth until the final observed liquid water profile at 0750 UTC. The observed fog appears to lift into stratus cloud from 0815 UTC on 25 November 2014, whereas the control simulation produces fog which persists throughout the day. The overproduction of fog here is consistent with other cases for which the SCM has been run and is the focus of continuing work. Table 1 shows the formation time (i.e. when the 1.5m visibility drops below 1km), the timing of the stability transition (i.e. when the boundary layer first becomes unstable) and the dissipation time (i.e. when the 1.5m visibility returns to being above 1km) for the wind perturbation experiments and observations. The perturbations have no effect on the formation time, as this only occurs 50min after the model is initialised, and this is not enough time for the perturbations to have a significant impact on the boundary layer temperature and humidity. However, they make a significant difference to the time at which the boundary layer becomes unstable: 2130 UTC in the control simulation, compared with 0545 UTC in the 3ms −1 wind speed experiment (with this transition occurring at 2300 UTC and 0200 UTC in the 1 and 2ms −1 perturbation respectively). This result is the converse of that for clear stable boundary layers, where the stability of the boundary layer erodes in higher winds. Also, in the highest wind speed case the fog dissipates at 1100 UTC, in contrast to the persistent fog in the control simulation.

Results
The mechanism responsible for the difference in the stability transition is investigated further here for the SCM simulations. Figure 4 shows the difference in the evolution of the lowest 250m of the atmosphere between the wind perturbation experiments. At 1800 UTC, only 1h after initialisation, all the simulations have a fog layer with a depth of 10m and a maximum liquid water content of 0.075gkg −1 , large enough for the diagnosed visibility to be less than 1km. There is greater specific humidity above the fog layer than in it, indicating that cooling is responsible for its formation rather than any changes in specific humidity. There is a significant difference in the profile of total water content between the experiments, but little difference in the temperature profile. The total specific humidity difference is due to higher wind shear in the higher wind speed experiments, leading to a greater turbulent kinetic energy and greater turbulent moisture flux transport ( Figure 5). The greater vertical mixing (with the perturbed winds) increases the moisture flux divergence in   the lowest 100m, resulting in a turbulent transport of moisture out of this layer, thus drying it. The impact of this specific humidity difference on the fog layer can be seen in the 2100 UTC profiles, for which the control experiment produces a fog layer of approximately 50m, whereas the +3ms −1 run has a shallower fog of about 30m. By 0000 UTC the influence the fog has on boundary layer stability is becoming more evident, with the boundary layer in the control and +1ms −1 runs becoming unstable near the surface. By 0300 UTC this difference is clear in all four simulations, with the control simulation having a deeper unstable boundary layer with a fog depth just below 100m, while the highest wind speed case has a fog layer of 25m within a stable boundary layer. Note that the increase in negative moisture flux at the lowest model level in the higher wind speed runs indicates an increase in dew formation and a greater removal of moisture from the atmosphere ( Figure 5). After sunrise all four simulations produce rather similar fog layers (i.e. liquid water contents) but with very different total specific humidity and potential temperature structures ( Figure 6). The control simulation has the coolest boundary layer, at approximately 271K, and the lowest total specific humidity, at approximately 3.7gkg −1 at 1000 UTC. The boundary layer structure differences are created by the differing wind forcings and consequent fog developments. In the morning the highest wind forcing run has a larger relative humidity above the fog layer as it is cooler and contains more moisture (i.e. it has a higher specific humidity), which enhances growth. The warmer boundary layer in the +3ms −1 simulation results in the earlier dissipation of the fog; although this occurs in a very thin layer near the surface at 1200 UTC, it is sufficient to increase the surface visibility to over 1km.
The results from the humidity perturbation experiments are summarised in Table 2. Like the wind perturbation experiments, there is no difference in the fog formation time. The residual layer humidity does have an effect on the stability transition: a 1.5h difference is observed between the −3% RH perturbation and +3% perturbation, with the transition occurring at 2230 UTC and 2100 UTC, respectively. All three simulations maintain the fog during the day on 25 November 2014. Figure 7 shows the difference in the evolution of the lowest 250m of the atmosphere between the humidity perturbation experiments. There is no difference in the fog depth and liquid water content until the fog reaches the height of the relative humidity perturbation (42m). After the fog reaches this height it develops more rapidly as there is more moisture available, so by 2100 UTC the +3% perturbation is deeper with a greater liquid water content (as less fog top cooling is necessary for saturation to be reached). The difference in the fog structure also effects the temperature, causing a deeper stable layer with higher near-surface temperatures in the +3% perturbation run. By midnight the fog in the −3% run has a lower liquid water content than the control and a shallower unstable layer. By 0300 UTC the +3% perturbation has the deepest unstable layer, and the −3% perturbation has the shallowest.
The fog depth, liquid water content and specific humidity profiles below 42m are the same in the simulations prior to the fog reaching the height of the perturbations, which indicates that there has been little or no mixing between the residual layer and the stable surface layer prior to the fog developing above 42m. The very small turbulence kinetic energy (TKE) in these simulations restricts moisture transport.
To examine humidity sensitivity with slightly stronger wind forcing, the humidity perturbation experiments were re-run with the highest wind forcing (the +3ms −1 run) to investigate the influence that the humidity of the residual layer has on the timing of the stability transition in more turbulent conditions. Table 3 summarises the formation, stability transition and dissipation times of the humidity perturbation for the +3ms −1 wind forcing experiments. The relative humidity perturbation with the greater wind forcing has a greater impact on the timing of the stability transition than in the control wind case; the difference in the timing of the transition between the −3% and +3% relative humidity runs is now 5h, compared with just 1.5h with the control wind forcing. However, note that in the −3% run the stability transition occurs after sunrise, and the transition is therefore influenced by insolation. The difference between the dissipation times is less than the difference in the stability transition: a 2h difference between the −3% and +3% (with +3ms −1 wind forcing) runs.
The inclusion of the +3ms −1 wind forcing leads to the production of fog of different depths before the fog reaches the height of the perturbation (Figure 8). At 0300 UTC all three simulati ons produce a shallow stable fog, with the +3% RH perturbation simulation producing a slightly thicker, deeper fog. By 0700 UTC the difference in the fog depth between the three simulations is greater: the +3% simulation is three times deeper (with a peak liquid water content of 0.288gkg −1 ) than the −3% simulation (which has a peak liquid water content of 0.085gkg −1 ).
The dissipation in all three runs occurs with the fog layer lifting into stratus cloud, which can be seen in the −3% run at 1000 UTC and occurs later in the other simulations. The fog top continues to develop in the morning from continued radiative cooling and entrainment. The thinner fog in the −3% simulation allows more solar radiation to reach the surface and, despite a lower incoming longwave radiation from the thin- Persists to next day  ner fog, the net radiation is greater at the surface, leading to a warmer surface. The combination of this increased warming and the slightly lower specific humidity in the boundary layer (from the entrainment of the dryer air caused by the perturbation) results in an earlier dissipation in the −3% RH run. The difference between the simulations with higher wind forcing shows that humidity of the residual layer can play an important role in the life cycle of fog in more turbulent conditions, when mixing between the fog layer and the residual layer is enhanced.

Conclusions
The processes dictating the evolution of a shallow stable radiation fog into a deep adiabatic radiation fog are not well understood but are critical for improved prediction of fog. This work has used a set of single-column model experiments to investigate the effects that humidity and wind-driven mixing in the residual layer can have on the development of shallow stable radiation fog, including the transition to a deep adiabatic radiation fog.
Using a series of perturbed wind forcing experiments, it has been shown that turbulence has an important role in modifying the vertical profiles of humidity and temperature. In slightly stronger wind speed conditions, the modification of the vertical profile of humidity has the consequence of slowing the rate of vertical development of the fog during the shallow stable phase, thus prolonging the observed shallow stable radiation fog phase and eventually leading to an earlier dissipation time.
The humidity of the residual layer has an important role in the timing of the stability transition. However, it has a more significant role during modestly higher wind speed conditions due to additional mixing between the residual layer and the surface layer. This additional mixing modifies the surface layer moisture and resultant liquid water content, altering the rate of development of the fog layer in its shallow stable phase, thus affecting the transition timing.
In summary, subtle changes to wind speed and consequently turbulence, as well the humidity of the residual layer, significantly affect the evolution of shallow stable radiation fog -in particular on the timing of the transition to a deep adiabatic radiation fog. In order to forecast a fog event accurately, it therefore is necessary to accurately predict wind speed to within ~0.5ms −1 at 10m and relative humidity to within 3%, as these factors can change the timing of the transition to a deep adiabatic radiation fog by 8h 15min and 2h 40min respectively. The implementation of ensemble forecasts, using suitable wind and humidity perturbations to represent the uncertainty in a fog forecast, may be useful as a mitigating factor for this sensitivity. Also, as the perturbations Figure 7. Simulated profiles of liquid water content (LWC; gkg −1 ), potential temperature (θ; K), total water content (q tot ; gkg −1 ) and relative humidity (RH; %) in columns from left to right at (a) 2100 UTC and (b) 0000 UTC for the relative humidity perturbation experiments with the control wind forcing. Figure 8. Simulated profiles of liquid water content (LWC; gkg −1 ), potential temperature (θ; K), total water content (q tot ; gkg −1 ) and relative humidity (RH; %) in columns from left to right at (a) 0300 UTC, (b) 0700 UTC and (c) 1000 UTC for the relative humidity perturbation experiments with the +3ms −1 wind forcing. used in this study are similar to the errors in the observations used in data assimilation, the study highlights the importance of accurate and representative observations for accurate fog forecasts.