1 Introduction
This study was initiated when it was found that predicting fog in areas offshore from Atlantic Canada using the National Center for Atmospheric Research and University Corporation for Atmospheric Research (NCAR/UCAR) Weather Research and Forecasting model (WRF-ARW) was generally satisfactory in terms of fog occurrence but gave high values of cloud water mixing ratio, leading to visibilities that were too low compared to observations. Other studies of marine fog had encountered similar problems (e.g. Chen et al., 2020). Koračin et al. (2014) had noted that, “From the many modeling studies of sea fog, essentially numerical experiments/simulations/forecasting that started in the immediate post WWII period, it becomes clear that deterministic forecasting of sea fog onset and its duration has generally been unsuccessful”. On land and over the sea, the formation and decay of fog in the atmospheric boundary layer is a complex issue involving many processes including cloud microphysics, longwave and solar radiation, turbulent boundary layer mixing, advection and surface interactions. Modelling of fog, in idealized one dimensional or single column models up to operational 3-D weather prediction and climate models, is a challenge which many have addressed over the years, as noted by Koračin (2017), Gultepe et al. (2017) and many others. Koračin et al. (2014) review marine fog processes and studies up to 2014, noting the importance of air–sea interactions. They discuss fog water deposition to vegetation extensively but not turbulent deposition to water surfaces, and it is missing from their Fig. 1 (and Fig. 9.1 in Koračin, 2017) which shows “the main processes governing the formation, evolution, and dissipation of marine fog”. Although fog could be caused by mixing two slightly subsaturated air parcels and causing saturation due to curvature of the saturated mixing ratio versus temperature line, most fog formation is initialized by cooling the lower parts of a column of moist, but unsaturated, air. This can arise because of longwave radiative heat loss from the underlying surface (radiation fog), vertical displacement of the air column as it travels over sloping terrain or horizontal advection over a cooler surface. Our focus is on the advection fog situation over ocean waters, a frequent occurrence over areas such as the Grand Banks and offshore areas of eastern Canada as the wind blows moist air from over the Gulf Stream towards the Labrador Current (Taylor 1917; Isaac et al., 2020).
1.1 Fog and the underlying surface
The focus in this paper is on the interactions of fog water droplets with the underlying water surface, how this is being modelled and how it could be improved in the widely used WRF model, and it will briefly suggest some field measurements to support this work. The basic hypothesis will be that, in addition to gravitational settling, turbulence will induce collisions between fog droplets and the water surface, and that most of these collisions will lead to coalescence, so that the water surface is a sink for water droplets. This can be represented in terms of a deposition velocity over and above the settling or terminal velocity associated with small cloud droplets falling through the air under gravity, which is predictable assuming Stokes law (see, for example, Rogers and Yau 1989). Different authors use different symbols (, , LWC, , etc.) and different measures (grams per kilogram, hereafter ; kilograms per cubic meter, hereafter ; etc.) of fog or cloud water content. We will use for mixing ratio ( or kilograms per kilogram, hereafter ) and , where is air density, as liquid water content ( or grams per cubic meter, hereafter ) unless discussing results from specific papers where, for clarity, it is sometimes useful to use their symbols. If there is an enhanced turbulent deposition to the water surface, one would then expect the cloud water mixing ratio () to approach zero at the surface and increase with height () above the surface. In a constant flux layer, this would lead to a logarithmic profile and allow the concept of a roughness length for cloud droplets, , although the profile can be modified to incorporate gravitational settling (Taylor, 2021). Not included is the possible creation of spray droplets by breaking waves in high wind speeds, and this may need consideration in high seas with strong winds.
There have been many studies on the collision and coalescence of raindrops and cloud droplets, and of droplets impacting hydrophobic surfaces, but relatively few concerning interactions between cloud or fog droplets and ocean surfaces. Over water, the combination of wind and waves will lead to impacts occurring at a range of speeds and incidence angles, and relatively little is known about the details of this important interaction. The paper by Hallett and Christensen (1984) and the reference to it by Isaac and Hallett (2005), although primarily focused on impacts at normal incidence, do, however, support our expectation that fog droplets interacting with the ocean surface are likely to coalesce eventually, even if they may bounce on initial impact if that occurs at a shallow angle. If fog droplets do collide with the underlying surface, whether it is the ocean, a lake, a water puddle on land or wet vegetation, one would expect coalescence and deposition of the fog droplets to the surface. Gravitational settling will play a role in this, but droplet impacts on the surface due to turbulence also need to be considered. As a result of deposition, there would be a reduction in the fog–cloud water mixing ratio (), maybe to zero, at the lower boundary, which would lead to a positive value for d/d and a downward flux of .
1.2 Aerosol and vegetation
If we broaden our view and consider aerosols in general, we find that significant work has been done in the same size range as fog droplets (1–50 ). Recent reviews by Emerson et al. (2020) and Farmer et al. (2021) make it very clear that dry deposition (i.e. not rainfall related) of aerosol particles, solid or liquid, is a key process for their removal, and that it is driven by turbulence and strongly dependent on particle size. For aerosol with gravitational settling and turbulent diffusion, both contribute to the overall deposition velocity. The aerosol studies include both water surfaces and vegetation. It is clear from Farmer et al. (2021; Fig. 3) that deposition velocity, , over water increases significantly with aerosol diameter between 1 and 50 , while this variation is somewhat less over other surfaces. Farmer et al.'s (2021) plots are not normalized by friction velocity or wind speed, which probably accounts for some of the variability in at fixed diameters.
There have been studies of fog deposition to vegetation and also to meshes designed to catch fog water (e.g. Sect. 3.4 of Gultepe et al., 2017). However, as far as we are aware, the models of fog droplet deposition to water surfaces have either been via gravitational settling alone, ignored or considered as a part of a turbulent total water (vapour, , plus liquid droplets) flux at the surface. Right at the surface, the flux of water vapour will rely on molecular transfer alone while the collision and coalescence of water droplets can be much more efficient and require separate treatment.
2 Boundary layer modelling
For aerosols and sometimes other quantities, weather prediction and other models tend to use deposition velocities () to relate fluxes to an underlying surface to concentrations at some level above the surface. From a boundary layer perspective, one often looks at the concentration profile and an eddy diffusivity. The simplest, and traditional, way to model the flux profile relationships of a quantity, , in neutrally stratified, turbulent boundary layer flow near rough walls is via an eddy viscosity/diffusivity, , where is the Kármán constant (0.4), and is the friction velocity. The roughness length, , is specific to the property (horizontal velocity, temperature, mixing ratio, ) under consideration and will vary considerably, depending on the physics of the final transfer process at the surface. The traditional way to determine is to consider an approximately constant flux layer near the surface, leading to a logarithmic profile, as follows:
1 where is the surface value. This will imply that at and is the empirical way in which can be determined. It is well known, see, for example, Garratt (1992) or Brutsaert (1982), that roughness lengths for momentum () and heat or water vapour () transfers differ because form drag on roughness elements is the major cause of momentum transfer, while molecular diffusivity at the surface is needed to effect heat transfer. As a result, , except maybe over aerodynamically smooth surfaces. We will propose the use of for cloud droplet collision and coalescence with the water surface. We have no measurement data to determine a value, which might well vary with droplet size and sea state, but we can use reported aerosol studies to provide some guidance. We do, however, expect that .
If the fog has continued for some time, one might expect that the relative humidity, RH, is equal to 100 % in the fog layer, with no significant condensation or evaporation. There will then be a near-steady state in the lower fog layers with constant downward flux (). This flux will be a combination of turbulent diffusion and gravitational settling (), where is the gravitational settling velocity, based on Stokes law. If, as we will assume, as , then turbulent transfer will dominate as the surface is approached, and logarithmic profiles should result.
In our model calculations, with an eddy diffusivity, , we do find % in the fog layers, typically up to around 100 m, and we see constant flux layers with near-logarithmic profiles through most of this height range, as in Fig. 4. Departures from logarithmic profiles could arise in part due to the effects of gravitational settling.
Marine fog in the areas under consideration often occurs in moderate and high wind conditions (Isaac et al., 2020). Relatively low heights ( m) are used as the lowest model level, and in that lowest, constant flux wall layer with neutral stratification, we can assume horizontal homogeneity, a constant downward flux of and a steady state. We can then seek the solution to the following: 2 where is a downward flux of cloud droplet liquid water mixing ratio, and is introduced as a mixing ratio scale. With at , the solution is as follows: If is small, then to first order in , Eq. (3) simply becomes the following: If this is used to relate to a deposition velocity, , and with , we would have the following: 5 where is the height above the surface where is measured. This logarithmic profile approximation could be fit to measured profiles to determine from observations. As with , this is a somewhat empirical approach. In the same way that the use of the concept is widely accepted without precise calculation of the form drag on roughness elements, we would hope that future experimental determination of would be a way to account for the effects of turbulent collision and coalescence of fog droplets with a water surface. For radiation fog in low wind speeds over land, stable air density stratification effects could be significant and can be accounted for with Monin–Obukhov similarity modifications to , if the Obukhov length () can be determined.
The expected values of terminal velocity, , for a droplet of diameter, , and density, , falling under gravity () through air of density, , and molecular viscosity, , should be considered. In reality, the fog droplet size distribution will be broad and often bimodal (see Isaac et al., 2020). The two peaks in some of Isaac et al.'s (2020) measured droplet size distributions are at diameters near 6 and 25 , with Stokes law terminal velocities () of 0.001 and 0.019 . These are clearly small compared to wind speed, but for the larger diameter, where the bulk of the liquid water content (LWC) is often measured, the terminal velocity corresponds to 67 and will represent a considerable removal rate in fog which may last several days. The key parameter in our constant flux with gravitational settling model is . In moderate winds over the ocean, one might expect values in the 0.1–0.5 range, and so the parameter, , will generally be in the range 0.006 to 0.46, while may be 5–10 at the lowest grid point, implying that gravitational settling can play a significant role, and that Eq. (3) may provide a more appropriate profile for the larger droplets. In principle, Eq. (3) should be used to refine any estimates from measurements. For typical friction velocities (0.1–0.5 ) and with the lowest model level at m with or 0.01 m, values would be in the range 0.005 to 0.04 , which is quite comparable with the gravitational settling velocities, so both will play a role in the modelling of deposition to the surface. A more detailed analysis is presented in a companion ACP discussion paper (Taylor, 2021).
Ideally, values for would be established from field measurements, but we are not aware of any height profiles of in fog over water, and for now, we will treat as a tuning parameter in our models. Over most land surfaces, the surface roughness length for momentum, , is considered independent of the Reynolds number, and we might hope that the same would apply for . Over water surfaces with ripples and waves as the roughness elements, life becomes more complicated, and can be wind speed dependent, governed by the Charnock–Ellison relationship
Henry Charnock always told me that Tom Ellison had suggested the dimensional analysis behind what is generally referred to as the Charnock relationship, so I refer to it in this way. – Peter A. Taylor
(Charnock, 1955), , where is referred to as Charnock's constant, with typical values in the 0.01–0.03 range and values in the 0.05 to 1.5 mm range. Establishing precise over water values for will prove at least as difficult as for , noting that it may also vary with droplet size, but it does provide a framework for representing this potentially important fog deposition process. 3 Past field and laboratory measurementsThere have been many field measurements in marine fog, including, notably, Geoffrey Ingram Taylor's (1917) work over the Grand Banks and, more recently, the C-Fog study reported by Fernando et al. (2021). As far as we are aware, none have provided the profile data from which we could make determinations.
Over land there are some multilevel measurements, indicating lower values near ground than above and also lower droplet numbers. Kunkel (1984) reports measurements of advection fog in July 1980 and July 1981, at two levels (5 and 30 m) on a tower “in the middle of a large, flat, open area” about 12 km inland from the Atlantic on Cape Cod. There is some variability, but his liquid water content values (; ) are always higher at 30 than at 5 m, and the ratios are generally between 2 and 3. There are some differences in droplet size between the levels, but they are relatively modest and less consistent. Ignoring stratification effects, assuming that a logarithmic profile is appropriate and that , then the ratios of 2 and 3 in correspond to values of 0.833 and 2.04 m. If were , say some fraction of (5 m), then the values would be higher. Pinnick et al. (1978) report measurements, from February 1976 above an inland site in Germany at multiple heights up to 180 m, with light scattering instruments carried aloft by a tethered balloon. Water content was calculated from particle size distributions, and from their photographs, the local land surface appears open and flat. Their sample profiles, in fog and haze, generally show increasing with height, and three of four cases shown are consistent with increases by factors of 2–3 between 5 and 30 m. Most of their results appear to be in radiation fog with light wind conditions. Klemm et al. (2005) report eddy covariance measurements of fog water fluxes to a spruce forest at Waldstein, in a mountainous area of Bavaria, Germany, and compare results with related model studies. They report that “turbulent exchange dominates over sedimentation at that site” and investigate relationships between liquid water content (LWC; ) and visibility. Their flux model is based on a deposition velocity, , with deposition to the canopy, , including both turbulent flux and gravitational settling. They note that some studies at the same location (Burkhard et al., 2002) report significant differences in downward flux at different levels (flux at 22 m can be 45 % less than at 35 m), perhaps illustrating the difficulty of making representative measurements close to the canopy top. Evaporation of fog droplets is also cited as a possible cause of these differences. It is perhaps also worth adding that fog water collectors (e.g. Schemenauer and Cereceda, 1991) can enhance the amount of fog water that is removed at ground level and provide an important source of clean water for some isolated communities. A removal efficiency of 20 % is estimated for a two-layer m polypropylene mesh.
Turning to aerosol studies, Farmer et al. (2021) provide an extensive list of laboratory and field studies of aerosol deposition to both land (grassland, forest, snow and ice) and water surfaces. Many provide values for aerosols in our size range. Deposition velocity measurements in wind tunnel studies in a short report by Schmel and Sutter (1974) are interesting but lack details of how the aerosol flux to the surface was determined. From their Fig. 3, we can estimate average deposition velocities for selected particle sizes and wind speeds. Unfortunately, it is not clear at what heights their wind speeds were measured, and their and values are somewhat suspect. If we assume that m, and that wind speeds in their tunnel were measured at a height of 0.1 m, then their average (7.2 ) and (0.44 ) values are reasonably consistent, and their value of 0.04 for 6 diameter aerosol would lead to m. For larger diameter aerosol (28 ), and m with the same wind assumptions, suggesting strong size effects, but we are wary of suggesting precise values.
Field data studies in the Farmer et al. (2021) list include studies on Lake Michigan by Caffrey et al. (1998) and Zufall et al. (1998), with deposition to surrogate surfaces, and a recent report by Qi et al. (2020) from the NW Pacific Ocean. These and other papers confirm the strong size dependence of deposition velocity, and acknowledge wind speed dependence, but are often concerned with long-term estimates of the deposition of chemical species to the ocean or lake rather than short-term events. One way in which wind speed plays a role is via wave breaking and broken water surfaces, a concept used in a model proposed by Williams (1982). This proposes that dry deposition of aerosol particles is considerably different between smooth and broken patches of the water surface, with a much higher resistance over the smooth areas.
To briefly summarize, we believe that there are observations to support the idea that the underlying land or water surface can be an effective sink for fog droplets and other, similar sized, aerosol. The deposition velocity will have a dependence on droplet size, especially over water, but there is a lack of reliable data, even over land, to calibrate our simple roughness-length-based approach to modelling the turbulent deposition of fog droplets. Our roughness length, , will have to remain as a tuning parameter until more extensive fog droplet profile and flux measurements can be made.
4 Model studies
As reported by Koračin (2017), there have been many studies aimed at understanding and/or predicting the occurrence of fog, and Kim and Yum (2012) also provide a review focused on marine fog. For our purposes, it is relevant to see how different model papers discuss the deposition of fog water to the surface and their surface boundary conditions on . The model of Brown and Roach (1976) focusses on radiation fog in relatively low wind speeds and provides an excellent summary of the key components needed to model fog formation and its life cycle, including radiation, turbulent diffusion and gravitational settling. They note that “liquid water (as well as water vapour) is also lost to the ground by turbulent diffusion and gravitational settling of droplets”, and their lower boundary conditions include for and , where is their liquid water mixing ratio. Brown and Roach (1976) assert that “, , , exchange coefficients for heat, water vapour and liquid water (w) respectively” are assumed equal in their model. In adiabatic conditions, they state but avoid a discussion of roughness length. Extrapolating their liquid water, , vs. profiles to would indicate a value, for liquid water, of slightly less than m. This is consistent with their use of the model of Zdunkowski and Barr (1972), who set cm. Zdunkowski and Barr's (1972) treatment of the conservation equation and lower boundary condition for , the total moisture content (vapour plus droplets), plus zero flux of to the surface, generally leads, inappropriately, to liquid water profiles with maxima at the surface. Barker (1977) developed a similar model for maritime boundary layer fog and also uses the same eddy diffusivity and roughness length for heat, water vapour and liquid water. He assumes (Barker, 1977; Eq. 19) that cloud liquid water concentration (his ) is zero at the water surface.
The COBEL and COBEL–ISBA (COuche Brouillard Eau Liquide – Interactions Soil Biosphere Atmosphere) 1-D models developed in France (Bergot 1993; Bergot and Guedalia 1994; Bergot et al., 2005) have been used successfully at Paris's Charles de Gaulle Airport. Bergot and Guedalia (1994; hereafter referred to as BG) provide details of dew and frost deposition to the underlying surface and note its importance. However, their dew flux is based on direct condensation of water vapour to the surface (BG; Eq. 22) as the inverse situation of evaporation. Their liquid water () diffuses and has a gravitational settling velocity (BG; Eqs. 17 and 18), but no surface condition is specified, and one assumes that the only flux to the surface is through gravitational settling. Few details are given on the surface boundary conditions in the latest journal publications, but contour plots, e.g. Fig. 13c from Bergot et al. (2005), generally show maxima at the surface. COBEL has also been coupled with WRF (Stolaki et al., 2012) and used to simulate advection–radiation fog conditions at Thessaloniki Airport. Ducongé et al. (2020) report on recent radiation fog modelling studies with Meso-NH downscaled from the Métèo-France operational model of AROME.
Bott and Trautmann (2002) proposed PAFOG as being “a new efficient model of radiation fog”, and it has been used by others, including, recently, and coupled to WRF, in a study by Kim et al. (2020). PAFOG is a 1-D model, developed as a more practical version of the more complete MIFOG model (Bott et al., 1990), which carries multiple aerosol and size bins for fog droplets. The MIFOG model includes dynamics and thermodynamics but focuses on interactions of radiation (solar and longwave) with fog droplets of varying size. The cloud droplets that evolve in the model have a bimodal size distribution which varies with time, with large droplets descending under gravity, and being removed at the surface, at a faster rate than the small ones. The dynamics include turbulent mixing via eddy diffusivities for momentum and heat. Water droplet number concentrations in each size bin are also subject to diffusion with the same diffusivity as heat. The diffusivities are given by Forkel et al. (1987). It appears that a common roughness length, m, is used for momentum, heat and water droplets. No boundary conditions are given in Bott et al. (1990), but from the results presented, it would appear that there is no turbulent flux to the surface – only deposition via gravitational settling in MIFOG. The same appears to be true with PAFOG, apart from possible removal of cloud water by vegetation as described by Siebert at al (1992a, b). PAFOG appears to give good results for 2 m visibility (Bott and Trautmann 2002, Fig. 1). Their Fig. 2 generally shows high values (0.2 and 0.3 ) extending almost down to the surface but with a sudden drop near in three of the four contour figures shown. There is similar near-surface behaviour of in Siebert's results, but it is not clear why. All of the above papers have a lack of detail on surface boundary conditions.
Shuttleworth (1977) and later Lovett (1984) were early modellers of fog deposition to vegetation, using resistance concepts (). Katata et al. (2008) later developed a land surface model (mod-SOLVEG), including fog and cloud water deposition on vegetation and on forests. The downward flux of cloud water is due to both turbulent mixing and gravitational settling (Katata, 2014), and Katata et al. (2008) successfully compare their model predictions with field measurements from a forest site near Waldstein in Germany. The turbulent fluxes use a vertical eddy diffusivity, , and multiple vegetation levels are involved. They claim that their model results compare well in comparison with Klemm et al.'s (2005) application of the Lovett (1984) model. Lovett (1984) points out that there can be “turbulent transfer of cloud droplets to the canopy”, and that, in windy conditions, “inertial impaction is the dominant mechanism”. These model papers all deal with forests, and Katata et al. (2011) describe the implementation of the ideas within WRF using the Mellor–Yamada–Nakanishi–Niino (MYNN) 2.5 planetary boundary layer scheme and WSM6 cloud microphysics. The central assumption is that, within what Katata et al. (2011) call org-WRF, fog water deposition to the surface can be represented as follows:
6 where is the wind vector at the lowest model level, and is air density. is a bulk transfer coefficient for height above the surface (specifically the lowest model level, although was later defined as the canopy height), and is a deposition velocity associated with turbulent diffusion but including gravitational settling. In what Katata et al. (2011) call fog-WRF, the deposition velocity is set to the following: 7 Here LAI is leaf area index (square meters per square meter; hereafter ), and here is canopy height (in meters), so that the coefficient (0.0164) has units of . Values given for in Katata et al. (2008) for both needleleaf and broadleaf trees are mostly in the range 0.02–0.04, with measured “over the canopy”. If the and measurement height was at 10 m, and m, then, from Eq. (5) and the log wind profile, , but with m, the result is , in the middle of Katata et al.'s (2008) range. In their large eddy simulation (LES) modelling, Mazoyer et al. (2017) follow Zhang et al. (2014) and set . A similar approach is being made by Salomé Antoine (personal communication, 2021; ICCP poster – Improvement of fog forecast at hectometric scales in AROME).
Recent papers by Wainwright and Richter (2021) and Richter et al. (2021) focus on marine fog, using a LES model, following on from the work of Maronga and Bosveld (2017) and Schwenkel and Maronga (2019, 2020) on LES studies of radiation fog. The marine fog models use Morrison et al.'s (2005) microphysics. The cloud water () and cloud droplet number () equations include turbulent diffusion and sedimentation, but there seems to be no enhanced deposition to the surface. Most results (e.g. Figs. 3a, 6, 10 and most of Fig. 11 from Wainwright and Richter, 2021) appear to show maxima at the surface, although Fig. 7 in Schwenkel and Maronga (2019) suggests a rapid drop in near the surface. There seems to be little discussion of deposition of fog droplets to the surface in most of these papers, although, for their Lagrangian simulations, Richter et al. (2021) note that “At the bottom of the domain, droplets that hit the water surface are removed from the simulation, and a new super-droplet is immediately introduced randomly in the domain according to the same procedure for initialization”. It is not clear what this does in terms of a flux to the surface, but their results (Fig. 3 in their paper) in a simulation of advection fog show number densities that are maximum at the fog top, around 30 m after 10 h, while and mean droplet radius are maximum near the ground.
None of the papers that we have found use the approach that we have adopted, although the resistance and deposition velocity ideas of Lovett (1984), Katata et al. (2008) and Mazoyer et al. (2017) are closely related. When roughness lengths are used, the values for always appear to be the same as for water vapour.
5 Operational numerical weather prediction (NWP) modelsFog forecasts have been a challenge for operational numerical weather prediction (NWP) models as indicated by many authors, including Wilkinson et al. (2013), who note the Gultepe et al. (2006) opinion that “most NWP models were unable to provide accurate visibility forecasts, unless they accounted for both liquid water content and droplet number”. We also note the following comment of Bergot et al. (2007): “Current NWP models poorly forecast the life cycle of fog, and improved NWP models are needed before improving the prediction of fog”.
Wilkinson et al. (2013) focus on the droplet number issue and, in a somewhat ad hoc fashion, the UK Met Office Unified Model (MetUM) at that time applied “a taper curve for cloud droplets near the surface”. This reduces droplet numbers between the surface and 150 m without changing liquid water concentration. Droplets are then larger, have higher settling velocities and so “the impact … is greatest closest to the surface, where they increase the amount of () removed from the lowest model levels”. Boutle et al. (2016, 2018) and Smith et al. (2021) have adjusted the MetUM taper parameters and obtained improved matches with visibility observations of fog, including the LANFLEX (Price et al., 2018) study. It seems to work as a “tuning parameter”, but the taper curve approach could be considered somewhat unphysical.
Yang et al. (2010) made an evaluation of the Canadian GEM-LAM model for marine fog off the eastern coast of Canada, with nesting down to 2.5 km, using both visibility reports and comparisons with observed measurements from the FRAM project (Gultepe et al., 2009). In total, three case studies are presented, with the overall conclusion that GEM-LAM forecasts at 2.5 km resolution underestimate and have a warm and dry mean bias at the lowest model level. This is opposite to our WRF studies which predict high values at low levels. An earlier evaluation by de la Fuente et al. (2007) had reported that “It has been shown that the current operational 15 km regional GEM forecast is insufficient for forecasting (sea) fog”. The GEM-HRDPS (Milbrandt et al., 2016) uses a MoisTKE treatment of the boundary layer, which is described in Belair et al. (2005). It works with the variable , where is the total cloud water content () which is mixed vertically, using an eddy diffusivity , as for heat. Assuming that surface transfers are of , this suggests no special treatment of cloud droplets over water surfaces. Milbrandt et al. (2016) indicate that the cloud microphysics then used in GEM-HRDPS were based on MY2, which is the two-moment bulk microphysics scheme described in Milbrandt and Yau (2005). That paper includes the following statement: “… because cloud droplets are assumed to have negligible terminal fall velocity”. Fall speeds were given for different hydrometeor categories but not for fog droplets. As discussed above, terminal velocities under gravitational settling are small (millimeters per second; hereafter ) and can probably be considered negligible in a convective cloud, but for long-lasting marine fog, they can play an important role. Currently, GEM-HRDPS uses predicted particle properties (P3) microphysics (Morrison and Milbrandt, 2015). This includes gravitational settling of cloud droplets, but there are subtle distinctions between explicit and implicit from the microphysics and the boundary layer treatments, and there appears to be no surface flux of , just a flux of .
Teixeira (1999) reported on European Centre for Medium-Range Weather Forecasts (ECMWF) successes in fog forecasting at that time with the Tiedtke (1993) cloud scheme forecasting liquid water content. The Musson-Genon (1987) surface boundary layer treatment treats diffusion of total water with a low surface roughness length but includes gravitational settling of liquid water. Teixeira's (1999) conclusions include the following statement: “The comparison between the simulated and the observed visibility shows that the onset of fog, the lowest values of visibility and the dissipation stage are properly simulated”. In terms of marine fog in the Grand Banks area, the reanalysis data showed that “The comparison between the model's fog climatology and the climatological data shows that the model is able to reproduce most of the major fog areas, particularly over the ocean”. The ECMWF (2020) model physics are documented at
In the USA, there are many different forecast models, but we will just consider the Rapid Refresh (RAP) and High-Resolution Rapid Refresh (HRRR) models based on WRF-ARW (Skamarock et al., 2021). These are run operationally, with 13 and 3 km resolution meshes by National Centers for Environmental Prediction (NCEP) and National Oceanic and Atmospheric Administration (NOAA) Earth System Research Laboratories (ESRL) Global Systems Laboratory. They use the same MYNN boundary layer and Thompson microphysics modules as in our marine fog simulations and, thus, may have similar limitations in depositing fog droplets over water. Going back to a statement in Zhou and Du (2010), “Although one hopes that the liquid water content (LWC) at the lowest model level can be explicitly used as fog, experience indicates that an LWC-only approach does not work well with the current NWP models due mainly to two reasons: one is the too coarse model spatial resolution and the other is a lack of sophisticated fog physics”. Things have changed since then, but the recent, somewhat improved, statement (including the qualifier, somewhat) on visibility performance by Alexander et al. (2020) can be noted.
6
Fog deposition treatment in the WRF model with module_bl_mynn and module_sf_fogdes
WRF versions 4.1.2 and 4.2.1 (
WRF SCM setup and tests
As a basic test of our treatment of deposition of fog droplets to a water surface and for comparisons against the regular WRF schemes, we use the single column version (SCM) of WRF (em scm xy), which is one of the ideal test cases described by Skamarock et al. (2021). In our applications of this SCM, we used several boundary layer and microphysics schemes and set up various vertical grids, with up to 201 levels and different lower and upper levels. Initial soundings have close to 100 % relative humidity in the lowest few 100 m, moderate wind speeds typical of the NW Atlantic and WRF-SCM was typically run for 36–84 h. To simplify the interpretation of the results, our SCM runs are without any solar or longwave radiation. Surface temperatures were cooled for several hours and then held steady. The main interest is to see the impact of fog deposition to the underlying water surface. Physics and dynamics components of the WRF name list input are listed in the Supplement. Turbulent deposition to the surface is represented via a deposition velocity, , multiplying the lowest level value at . This is set as follows: 8 where is the friction velocity, () is the Kármán constant, and is a roughness length specific to water droplets diffusing to a water surface and coalescing. In principle it could be dependent on sea state and droplet size. Our assumption is that (for fog–cloud droplets) should be significantly larger than for water vapour.
WRF-SCM was run using modules bl_mynn, for boundary layer turbulent transfers, and mp_thompson (with ), for cloud microphysics, to generate the results shown in Figs. 1–3. Since gravitational settling is represented within mp_thompson, the parameter grav_settling was set to 0 in bl_mynn (see Olson et al., 2019; Sect. 6.4). No radiation effects are included. Lack of longwave radiation will affect mixing at the top of the fog layer, but we will focus on lower boundary issues. In the results below, the initial sounding has a potential temperature of 300 K at the surface, increasing with height at a rate of 4 . The initial relative humidity was 100 % at the surface, dropping to 0 % at 6 km. The wind profile was established with a long, no cooling run and has a geostrophic wind components, (U, V), of . Sea surface temperature was cooled at a rate of 3 for 6 h and then held fixed. The lower boundary condition included a flux of water droplets to the surface computed with a deposition velocity determined by Eq. (8) above and using a range of values.
Figure 1
Contours of () generated by WRF SCM, with 6 h of surface cooling at 3 . (a) MYNN boundary layer, using the turbulence deposition scheme described with m, plus Thompson microphysics with gravitational settling. (b) Original MYNN module with gravitational settling only in Thompson microphysics. The full vertical domain is shown to indicate that no upper-level cloud formed in these cases (it did with other input). Times on the axis are in the format DD HHZ (day, hour in UCT(Z)), with small tick marks 4 h apart. The run start time was 15:00 Z.
[Figure omitted. See PDF]
Figure 1 shows contours of () as it varies with (; eta grid level) from the model calculations over 4 d starting, somewhat arbitrarily, at 00:00 Z on day 15 of a month (15:00 Z) so that cooling runs to 15:06 Z. Some height levels are marked to indicate the grid stretching in . These runs are for latitude 44 N (Sable Island), with 101 eta grid levels. The WRF model operates with a sigma-type vertical coordinate (), decreasing from 1 at the lower boundary to 0 at the upper boundary, where . It has a simple form over a flat surface. Details are in Skamarock et al. (2021). Our model grid points are not uniformly spaced in , and the spacing increases smoothly with increasing height (decreasing ). We set Pa to give a top boundary at about 12 km. The eta levels start at (the surface), decreasing to and at eta level 101 (our SCM model top). In full 3-D runs, we take Pa. The grid is staggered so that variables like , , , and , where is potential temperature, and is the water vapour mixing ratio, are at mid-levels, while the lower boundary () is at the base of the lowest grid cell. Our grid levels start with the center of the lowest cell (0) and increase upwards. In Fig. 1a, m, while Fig. 1b is for results with the original MYNN scheme with no surface deposition, except for gravitational settling in the Thompson microphysics. Fog forms as a result of the surface cooling and extends from the surface to around eta level 20, which corresponds to m. We were initially concerned by the wave-like features in the contour lines. These have a period of around 17 h and arise because of inertial oscillations (of period ) in the wind field, , as it adjusts to the cooling of the surface and changing turbulent momentum transfers. They decay slowly as the wind profile adjusts to the cooler surface. Values of are lower in Fig. 1a because of turbulent deposition to the surface. Figure 2 shows profiles with the MYNN boundary layer, at 16:00 Z, 24 h after the start of the model calculations and 18 h after the end of surface cooling. The additional turbulent deposition can play an important role in lowering levels in the boundary layer while, in this case, not having a significant impact above 100 m. The amount of the reduction depends on the value chosen for .
Figure 2
profiles, 24 h after the start of the integration and 18 h after the end of the surface cooling, by 18 K. Results are shown with the original MYNN (gravitational settling in Thompson microphysics only) and with a range of values (in meters). A time step, , of 60 s was used with 101 eta levels.
[Figure omitted. See PDF]
Figure 3
(a) Potential temperature () and (b) profiles corresponding to Fig. 2, including the initial profiles. Note that the deposition of cloud droplets has minimal impact, and all curves are overlain.
[Figure omitted. See PDF]
It is interesting to note that the removal of at the lower boundary has minimal impact on the predicted temperature and water vapour, , profiles (Fig. 3). It could, however, be important when fog starts to evaporate if the air temperature rises. Note that, in generating these results, we have not included radiation (shortwave or longwave) effects in order to focus on the impacts of turbulent deposition at the water surface. Radiation can play a significant role once fog has formed, and in particular, longwave radiational cooling at the fog top (Yang and Gao, 2020) can add to the cooling rate and can enhance turbulent mixing in the upper part of the fog layer. The center of the lowest grid layer is at 1.7 m. Noting the kinks in the profiles at the lowest level in profiles of , and , we investigated possible causes and plotted them on an expanded height scale (not shown). They arise because in WRF modules sf_mynn and sf_fogdes the fluxes to the surface are computed with deposition velocities involving , while the eddy diffusivities used to compute fluxes at the top of the first level and levels above are based on length scales proportional to without the addition. This will not be significant for , but with the lowest computational levels close to the surface this could be modified. This is an internal WRF issue, noted in comments within the module_bl_mynn code.
A further point from Fig. 3b is that, with our near-saturated initial profile and strong cooling, there is a significant reduction in , of order 10 , throughout the lowest 100 m. This will be converted to , but after 24 h, most will have been deposited to surface through both gravitational settling, as in the original curves in Fig. 2, or by a combination of gravitational settling and turbulent deposition to the water surface, as in the other cases shown in Fig. 2. In runs with gravitational settling turned off in the microphysics and no turbulent deposition, the values increase significantly, to around 6 , near the surface after 12 h. This is not shown for this case, but see the 3-D case in Fig. 4b, although then there is less cooling. Gravitation settling generally prevents very high values from occurring, but additional turbulence-induced deposition further limits them.
Figure 4
The 2-D fog plots at lowest model level on 1 July 2018, 18:00 Z, from WRF. Thompson microphysics with gravitational deposition (a) with m and no turbulent deposition (b) is related to Fig. 5a. The black dot shows the point on the Grand Banks to which the profiles in Fig. 5 correspond.
[Figure omitted. See PDF]
7 3-D test casesTurning to the 3-D WRF model, we have been running the model for North Atlantic simulations for summer 2018 on a domain extending from eastern Canada out beyond the Grand Banks and including Sable Island. A separate paper on comparisons with visibility measurements on Sable Island is in preparation, while some sample results are in Cheng et al. (2021). These 3-D runs have no additional surface cooling and are simply run as hindcasts of the actual situation with initial and boundary conditions taken from NCEP analyses. The sea surface temperatures are held fixed for daily 36 h runs, generally with a 12 h spin up. Note that the input initial and boundary fields had zero . They are run with , and in the vertical direction we have a straight sigma coordinate as follows:
9 with Pa. Runs were also made with hybrid , and results were almost identical. Solar and longwave radiation can use either Goddard or RRTMG (Rapid Radiative Transfer Model for Global models) scheme, and we used the MYNN PBL (planetary boundary layer) scheme with either the Thompson or the WSM6 microphysics options. For details of these options, see Skamarock et al. (2021). Figures 4 and 5 show sample results from 6 h after the start of a run with the full 3-D model, using Thompson microphysics and Goddard radiation and long- and shortwave.
Figure 5
Sample 3-D WRF output at a fixed location over the Grand Banks, with different values (given in meters) in turbulent deposition, (a) with and (b) without gravitational settling. Start time (day, month, year, and hour) was 7 January 2018 12:00 Z, and results are for 7 January 2018, 18:00 Z. Results are with the MYNN boundary layer and Thompson microphysics.
[Figure omitted. See PDF]
With 3-D WRF simulations, we initially look at plots and animations over our d02 domain (see Supplement) at the lowest model level. Figure 4 is an example of 2-D plots of at the same time as in Fig. 5, with and without turbulent deposition. The black dot identifies the Grand Banks location (GB) used in Fig. 5. The value of was 0.01 m. In additional runs (not shown) with no gravitational settling, the spatial fog patterns are similar, but in the extreme case with no turbulent deposition, the values are up to 0.8 in some areas, although it is only 0.4 at our GB location.
In Fig. 5, the profiles show a similar response to the SCM (Fig. 2) when turbulent deposition of cloud water to the surface is introduced. Figure 5a shows a normal run, with the Thompson microphysics module accounting for gravitational settling effects. MYNN has turbulent deposition to the surface but no gravitational settling (). In Fig. 5b, we removed gravitational settling from the Thompson microphysics scheme () and from MYNN. With no turbulent deposition to the surface, and in one special case with no gravitational settling either, there are higher values as expected. These 3-D runs used NCEP analyses as initial conditions, but the initial was set to zero everywhere. In fog, the analysis would give 100 % RH, and the model then generated within a few hours but without the strong temperature and drops that were simulated in our SCM tests. Gravitational settling (Fig. 5a) has reduced the peak values at around 100 and 900 m from the case with no settling, and the removed from those levels has settled and mixed downwards to increase the values near the ground.
Additional 3-D runs were made with the standard MYNN codes and the Katata scheme using modified deposition velocities in the other case. These matched our results obtained with a modified MYNN code. Also, in place of the Thompson microphysics scheme, we ran tests with WSM6 microphysics. In all cases, there was a large impact of turbulent surface deposition of in the lowest 100 m, even with very low values for . As an initial guide, we suggest using m or 0.001 m as a modest value which has a solid impact. We should also emphasize that gravitational settling also has an impact on values near the surface, and both processes need to be included in models.
8 Visibility considerationsModels can predict liquid water mixing ratios, but the critical forecast issue is visibility, which will depend on the number and size distribution of the fog droplets. In dense marine fog ( ), Isaac et al. (2020; Fig. 12) show that the size distribution of marine fog droplets is generally broad and frequently bimodal, raising concerns about all simple diagnostic schemes. Despite such concerns, models such as the one proposed by Isaac et al. (2020) assume that visibility is proportional to , where is the droplet number density (cubic meters; hereafter ). Some models include dynamic equations for , while others assume prescribed values, typically . If the size distribution was well known and universal, then this could work, but as Isaac et al. (2020) note, the size distribution in fog over the ocean can be bimodal, and the number density can vary widely. In conditions with , the number density reported by Isaac et al. (2020) over a site in the Grand Banks area varies between and . Medians were close to . Note, however, that these measurements were at a height of 69 m above the ocean surface, and if the water surface is a sink for cloud droplets, then one would expect lower values, and maybe a different size distribution, at the World Meteorological Organization (WMO) standard visibility measurement height of 2.5 m (WMO, 2020). Chen et al. (2020) note problems with visibility that is too low from their WRF calculations coupled to the Kunkel (1984) visibility equation ( with the extinction coefficient (in kilometers; hereafter ), , where (or LWC) is in ). The contrast threshold, was given as 0.02 by Kunkel (1984) but is set to 0.05, as recommended by the WMO (Boudala et al., 2012; Chen et al., 2020). In the NASA Global Systems Division (GSD) algorithm used in NCEP's Unified Post Processor version 2.2, the Kunkel (1984) result is used with for visibility reductions in clouds, plus additional effects of aerosol, rainfall and humidity. The relationship between visibility and LWC can vary in these models between a power of , through to if were proportional to LWC, but all show that too high a value of LWC or will lead to too much reduction in visibility. Running standard versions of WRF, one can compute visibilities with either the Isaac et al. (2020) equations or the GSD algorithm used in NCEP's Unified Post Processor version 2.2 (for details, see Lin et al., 2017). Both led to significantly lower values of visibility than were reported on Sable Island. Typical WRF values being of order one-tenth to one-fifth of the reported visibility, suggesting values that may be high by a factor between 5 and 30. Visibility–cloud water relationships are open to revision, with different values of and noting the scatter in Isaac et al.'s (2020) data, but there is a strong suggestion that WRF values of are too high without adding additional deposition.
Figure 6
Sample June 2018 GSD visibility hindcasts for Sable Island at 2 m, using MYNN boundary layer and WSM6 microphysics, with different values (in meters).
[Figure omitted. See PDF]
Figure 6 shows sample visibility time series computed from 3-D WRF output for the Sable Island location, vertically interpolated to m, for two 36 h periods in 2018 when fog was reported at Sable Island. We should, however, note that these computations were made with a 10 km horizontal mesh, and there was no island. In reality, the presence of a land surface can modify the temperature, up or down, leading to relative humidity, LWC and visibility adjustments as air travels in from the shoreline (see, for example, Cheng, 2021). In these cases, the fog occurred in daytime, and could be lower at the weather station than offshore. Original WRF runs with just gravitational settling show seriously limited visibility ( m) on some occasions when METAR visibility was closer to 1 km, while, with added turbulent deposition and a range of values, the optical range was a better match to the observations. These are sample cases and a more extensive comparison is planned.
9 ConclusionsIt has been known for many years that fog water can be deposited on vegetation, and this has been incorporated into some boundary layer fog models. It is also known that size aerosols can be removed from the atmosphere by turbulence at water, and other, surfaces (Farmer et al., 2021). It then seems surprising that, for marine fog, turbulence-induced cloud–fog droplet deposition to water surfaces has not been recognized by most modellers as a significant potential addition to the deposition associated with gravitational settling. Neglecting this can then lead to fog liquid water mixing ratios being too high and visibility forecasts being too low. This applies to specialized boundary layer models and to numerical weather prediction models. Many authors have noted the difficulties and complexity of modelling fog and accurately forecasting visibility. Making everything right will be extremely challenging, but, for marine fog, recognizing that a significant process is missing from many models could be a step in the right direction.
WRF-ARW is a major contribution to the atmospheric research endeavour, and the developers and maintainers of this huge, multifaceted, publicly available model deserve huge credit. As with anything of this size and complexity, developed and modified over many years by many individuals, it can be very hard for new users to trace through the source codes and understand just how they work. Some module codes are well documented and commented on; others are less so. Running the model is made relatively easy, and it is designed to be robust. We have done our best to understand some details and ensure that our modifications, briefly explained in the Supplement, do what we expect, but we make no guarantees.
Recent fog field programs, including LANFEX (Price et al., 2018) in the UK, SoFog 3-D (
Based on our modelling of marine fog with WRF, and reviews of the treatment of boundary layer fog in WRF and other models, it seems that a better understanding of fog droplet interaction with the ocean surface, and other surfaces, is needed. Laboratory studies might be possible, and numerical simulations may be done, but with some good in situ profile measurements through fog layers over land and water, one could start to better understand and parameterize this process. Any foggy location on land could work, but Sable Island would offer an ideal location for such a study in marine fog. It is a 43 km long, narrow (mostly km wide) sand bar in the Atlantic Ocean about 175 km offshore from Nova Scotia, Canada, and will be field site during Fatima in summer 2022. Sable Island has some vegetation, cranberry bushes and grass, wild horses and many seals and is now a national park. Observations (
Code availability
WRF codes used are readily available from
Data availability
Details of the models used are provided in the text and in the coding notes provided in the Supplement.
The supplement related to this article is available online at:
Author contributions
ZC, LC, PAT, YC, SA and WW were involved in aspects of the WRF code adaptation and model runs. PAT, GAI and TWB were primarily involved in reviewing background information and interpretation of the results. PAT prepared the original paper and its revision, with contributions from all co-authors.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
Financial support for this research, for which we are very grateful, has come primarily through a Canadian NSERC Collaborative Research and Development grant program (High-Resolution Modelling of Weather over the Grand Banks), with Wood Environment and Infrastructure Solutions, Inc. as the industrial partner. Initial support was also received through Peter A. Taylor's NSERC Discovery grant. We would like to thank Anton Beljaars, for providing guidance and many valuable comments, and Ayrton Zadra and Jason Milbrandt, for their help in tracking down details of Environment Canada's GEM model. Trevor VandenBoer pointed us to the aerosol work, and Joe Fernando allowed two of us to attend a C-Fog meeting in 2019, where we also had useful discussions with Will Perrie and Rachel Chang.
Financial support
This research has been supported by the Natural Sciences and Engineering Research Council of Canada and Wood Group Canada Inc (grant no. CRDPJ 543476-19).
Review statement
This paper was edited by Barbara Ervens and reviewed by Anton Beljaars and Thierry Bergot.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
© 2021. This work is published under https://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Abstract
There have been many studies of marine fog, some using Weather Research and Forecasting (WRF) and other models. Several model studies report overpredictions of near-surface liquid water content (
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Details
1 Centre for Research in Earth and Space Science, Lassonde School of Engineering, York University, Toronto, Ontario M3J 1P3, Canada
2 Centre for Research in Earth and Space Science, Lassonde School of Engineering, York University, Toronto, Ontario M3J 1P3, Canada; Weather Impacts Consulting Incorporated, 20 Pine Ridge Trail, Barrie, Ontario L4M 4Y8, Canada
3 Met-Ocean & Digital Environment Solutions, 133 Crosbie Road, St. John's, NL A1B 4A5, Canada