Constant Flux Layers with Gravitational Settling: with links to aerosols, fog and deposition velocities over water

Constant Flux Layers with Gravitational Settling: with links to aerosols, fog and deposition velocities over water. Peter A. Taylor1 1Centre for Research in Earth and Space Science, York University, Toronto, M3J 1P3, Canada Correspondence to: Peter A. Taylor (pat@yorku.ca) 5 231 Oct3 July 2021 Abstract. Turbulent boundary layer concepts of constant flux layers and surface roughness lengths are extended to include aerosols and the effects of gravitational settling. Interactions between aerosols and the Earth's surface are represented via a roughness length for aerosol which will generally be different from the roughness lengths for momentum, heat or water vapor. Gravitational settling willThese impact vertical profiles and the surface deposition 10 of aerosols, including fog droplets. , especially over water. Simple profile solutions are possible in neutral and stably stratified atmospheric surface boundary layers. These profiles can be used to predict deposition velocities and to illustrate the dependence of deposition velocity on reference height, friction velocity and gravitational settling velocity. 15


Introduction
Within the turbulent atmospheric "surface layer", typically 0 < z < ~50 m, it is helpful to look at idealized situations 20 where fluxes of momentum, heat or other quantities are considered independent of height, z, above a surface which is a source or sink of the quantity being diffused by the turbulence. Garratt (1992, Chapter 3) or Munn (1966, Chapter 9) discuss this "constant flux layer" concept and, for momentum, the paper by Calder (1939), discussing earlier work by Prandtl, Sutton and Ertel, is an early recognition of the utility of this idealized concept. Monin-Obukhov Similarity Theory (MOST) is based on constant flux layer situations in steady state, horizontally 25 homogeneous, turbulent atmospheric boundary layers and leads to suitably scaled, dimensionless velocity and other profiles being dependent on z/L where z is height above the surface and L is the Obukhov length (defined below).
With no sources or sinks of momentum or heat within these constant flux layers one can use dimensional analysis to establish the form of the profiles while observational data or hypotheses are needed to establish the detailed profile forms. Munn (1966, Chapter 9), Garratt (1992, section 3.3) or Kaimal and Finnigan (1994) explain Monin-Obukhov 30 similarity while Monin and Obukhov (1954) is a translation of the original Russian work. The simplest case is with neutral stratification (1/L = 0) where dimensional analysis can be used to infer that the velocity shear, dU/dz is simply proportional to u⁎/z where the shear stress, assumed constant with height, is ρu⁎ 2 , with ρ as air density.
Integration of this relationship leads to U(z) = (u⁎/k) ln(z/z0m), with the roughness length for momentum, z0m, being defined as the height at which a measured profile has U = 0 when plotted on a U vs ln z graph, and where k is the Karman constant with a generally accepted value of 0.4.

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Noting that z0m values are generally small compared to measurement heights, and after a z0m value has been established for the underlying surface, it is mathematically convenient to modify the relationship to U = (u⁎/k) ln((z+z0m)/z0m), 45 so that we have U = 0 on z = 0. In eddy viscosity terms (u⁎ 2 = Km dU/dz) this corresponds to Km = ku⁎(z+z0m).
In situations with constant, or near constant fluxes of heat (H) or water vapour, similar, near logarithmic,

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MOST profiles and eddy diffusivities can be established, based on measured profiles, involving z/L where the Obukhov length, L = -ρcpθu⁎ 3 /(kgH) in which cp is the specific heat of air at constant pressure, g is acceleration due to gravity and θ is the potential temperature. Application of Buckingham's pi theorem, assuming steady state, horizontally homogeneous conditions, with a constant (positive upwards) heat flux, (H/ρcp = -u⁎θ⁎) leads to 55 (kz/θ⁎) dθ/dz = ΦH(z/L) where ΦH(z/L), referred to as a dimensionless temperature gradient. This needs to be established experimentally but should approach one when z/L → 0. In the limit for small z, or large |L|, we again get a logarithmic profile after integration but a complication arises over what we define as surface temperature, or surface water vapour mixing 60 ratio. Integration of Eq (4) and a similar equation for water vapour leads to For potential temperature and water vapour profiles thatese can involve additional "scalar" roughness lengths, z0h and z0v. Much has been written about roughness lengths and ratios between z0m and z0h, including Chapter 5 of Brutsaert (1982) and Chapter 4 of Garratt (1992). For momentum transfers, pressure differences and form drag on roughness elements, sand grains, blades of grass, bushes, trees, buildings and water waves can provide most of the drag on the surface. E and, except over water, z0m is considered as a Reynolds number independent surface property. Water waves are wind speed dependent and z0m needs to take this into account. For heat and water vapour the final transfers from air to the surface involve molecular diffusion and, as a result, values of z0h, z0v are generally lower than z0m.
For aerosol particles orand droplet concentrations we can will introduce an additional roughness length, z0c, on the basis that their interactions with the surface will again be different from momentum and from other 70 quantitiescalars. Aerosol type, density and size, as well as u⁎, may also cause variability in z0c. As was necessary with the established roughness lengths for momentum and heat, field measurements over a variety of surfaces will be needed to establish appropriate values. As a first approach, for fog droplets and other aerosol particles deposited to water, and other, surfaces we assume Qc → 0 as z → 0 and, as a trial value, will generally use z0c = 0.01 m for illustration. This is somewhat larger than values typically assumed for water vapour or heat. The main innovation in 75 this short communication will be to combine the effects of turbulent transfer towards an underlying surface with gravitational settling (Vg). This is done in a similar way to that proposed by Venkatram and Pleim (1999) and differs from the additive deposition velocity format used by Zhang et al (2001) and Slinn (1982). The parameter, S = Vg/ku⁎ plays a key role..

A simple model with added gravitational settling
We will consider situations where there is aerosol present with a concentration or mass mixing ratio, Qc. For simplicity it is assumed to consist of uniform particles with a constant gravitational settling velocity, Vg, and is at a density low enough to have no impact on the density of the combined air plus+ aerosol mixture. We assume no mass 85 exchange between the aerosol and the surrounding air, which may be a concern for fog droplets which require an additional assumption that the air is always at 100% relative humidity.
If we have a net upward or downward flux of aerosol we need to discuss the source. If we are considering sand or dust being picked up from the surface by wind then upward diffusion will be countered by downward 90 gravitational settling, while if the source of the aerosol is above our constant flux layer then the turbulent fluxes and gravitational settling combine. This could be the case with long range transport of aerosol in air blowing out over a rural area, a lake or the ocean. AnOur other example couldwill be fog droplets, formed at the top of a fog layer and being deposited at the underlying surface (Taylor et al, 2021).

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In a horizontally homogeneous, steady state situation, and with a simply specified eddy diffusivity (Eq (3) but with z0m replaced by z0c) and neutral stratification we just need to consider vertical turbulent transfers and gravitational settling where Vg represents the gravitational settling velocity. One could then model the constant downward flux of aerosol, FQc, as 100 VgQc + Kqc ku⁎(z + z0c) dQc/dz = FQc = u⁎qc⁎., where Vg represents the gravitational settling velocity, Csanady (1973) proposed this approach and Venkatram and Pleim (1999) obtained essentially the same solution as we will find below. They commented, in 1999, "... why not 105 use a formulation that is consistent with the mass conservation equation (Eq. 5)." More recently Giardina and Buffa (2018) raise the same issue. Note that Vg is generally proportional to d 2 , where d is the diameter, via Stokes law for small (d <60 μm) spherical particles (Rogers and Yau,1976, p125), and u⁎ is the friction velocity. We introduce qc⁎ as a mixing ratio scale via this constant flux definition. The eddy diffusivity Kqc is assumed to be where z0c is a roughness length for the aerosol with the assumption that Qc = Qcsurf at z = 0.

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The upward flux case with a surface source of aerosol is interesting in the sense that there will only be a steady, horizontally homogeneous, state when the net flux is zero, i.e, upward turbulent transfer is balanced by gravitational settling. Xiao and Taylor (2002), in an aside from relation to a blowing snow study, show, by solving Eq.(5) with FQc = 0, show that this leads to the classic power law solution (e.g, Prandtl, 1952), which in the current context is or Profiles of suspended sediment, and velocity, in water currents can be treated in a similar way but there is an 125 interesting twist if the density of the sediment and water mix is sufficient to modify the turbulent mixing through stable stratification. Taylor and Dyer (1977) rediscovered an interesting result due to Barenblatt (1953) showing that a modified solution allowing for stratification effects on the eddy diffusivity could be obtained. Observations were sometimes misinterpreted as power laws with a modified value of k (Graf, 1971, p180).

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For the case of downward flux case to the lower boundary in the atmospheric surface layer it is easiest if we assume Qcsurf = 0, which may be most relevant over water but is also often assumed for dry deposition of particles (Seinfeld and Pandis, 1998, p960). Material starts from a source above the constant flux layer and travels downwards due to both turbulent mixing and gravitational settling. Assuming constant values for z0c, u⁎ and Vg one can then solve the first order differential equationODE, Eq (54), by integrating factor techniques. Multiplying Eq.

Dry deposition velocities
For aerosol dry deposition (i.e. not involving rain or snow -wet deposition) to any surfaces the traditional way to parametrize the process is with a deposition velocity, Vdep. Then the flux to the surface is represented as, 155 In a numerical model the reference height zref is often the lowest grid level. If gravitational settling is the main cause of FQc, we would expect little change in Qc with height, but if turbulent transfer is dominant then the choice of zref could be important.

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Dry deposition can involve many aspects and is often modelled in terms of a series of resistances. The deposition velocity used generally includes the effects of both gravitational settling and turbulent collisions of particles with vegetation or the ground, or water surface. The expression used for deposition velocity by Zhang et al (2001), and others, is where Vg is the gravitational settling velocity and the resistances to deposition are aerodynamic (Ra) and surface (Rs).
The aerodynamic resistance is given as where z0 is a roughness length, presumed to be z0m and ψH is a stability function from MOST. It is applied with zref >> z0 and so one can use ζref = ln ((zref + z0)/z0). In neutral stratification ψH = 0 and for deposition to a water surface it is reasonable to set Rs = 0, unless it could be used to differentiate between z0m and z0c. We can then write the 175 relationship as From our CFLGS profile (Eq 8) we can derive an alternative expression for deposition velocity, or if we set Rs = ln(z0m/z0c)/(ku⁎). The Zhang et al (2001) and z0c approaches differ in detail between those limits and an illustration is given in Section 4, Fig 3. The ζref → 0 limit is similar in both approaches with Rs = 0 since then Vdep → ∞ as z → 0 and the aerodynamic resistance goes to 0.
There is little discussion of the variation of Vdep with zref in the literature, most of the focus being on variation with light winds over land it could be lower. The parameter, S will thus generally be in the range 0.0 to 0.3 in marine situationsover water but could be unlimited in light winds with low u⁎ over land. With high values of S gravitational settling will be the dominant process except very close to the surface.
At low values of S gravitational settling will have littleow impact and the Qc profiles will be are approximately 225 logarithmic.
To illustrate this Fig. 1 shows Qc constant flux profiles with linear and log vertical axes and a range of S values.
We have scaled Qc with a value at 50m. The main unknown is the value of z0c. Here we use our first guess value (z0c = 0.01m) indicating relatively efficient capture of water droplets, or other aerosol, by the water surface. These calculations are for uniform sized aerosol particles or droplets. Note that with high S (=Vg/ku⁎) values, maybe 230 occurring with low u⁎ and minimal turbulence, the limiting case would be constant Qc down to z = 0 and a discontinuity to Qc = 0 at the surface. Calculations with S = 1 and 5 (not shown) confirm this. The essential point from Fig. 1 is that, if there is gravitational settling involved then the profiles will depart from the simple logarithmic profiles that one might expect in a neutrally stratified near-surface atmospheric boundary layer. Note that these profiles depend on z0c but not directly on z0m, except via u⁎.

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For aerosol dry deposition to any surface a traditional way to parametrize the process is with a deposition velocity, Vdep, based on a Qc measurement at zref, and simply defined via,

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In a constant flux layer, Vdep(zref), shown in Fig. 2, is simply proportional to the inverse of Qc(zref) provided that FQc is constant between the surface and zref. The dependence of Vdep on the reference height, zref, for Qc is seldom acknowledged in papers reporting measured Vdep values, or in the review by Farmer et al (2021). The height, zref, is often not discussed and hard to find, e.g. in Sehmel and Sutter (1974). In addition, there is a strong dependence on u⁎ and any value of Vdep will depend on zref, u⁎ and Vg as well as the nature of the underlying surface, which we have

Vdep/ku⁎ = S/(1 -e -Sξ ) ≈ S/(1 -exp(-Sku⁎(Ra + Rs)))
One way to look at the relative importance of gravitational settling for these uniform size droplets is to consider the 270 relative contributions to the total downward flux of water droplets (u⁎qc⁎). The gravitational contribution is simply VgQc while the turbulent diffusion contribution is, 275 The ratios of turbulent transfer (TT)/total flux and gravitational settling (GS)/total flux then become

TT = e -Sζ and GS = 1 -e -Sζ
Noting that ζ = ln ((z+z0c)/z0c) we can see that these ratios depend on both z0c, through the z(ζ) relationship, and S 280 and will vary with z. Fig. 2 illustrates this. It is important to note that Fig. 2 is based on our relatively low estimate for z0c, (0.01 m). If we increase it to z0c = 0.1 m then turbulent fluxes become more important. We can see that the TT ratio is formally 1 at the surface, where Qc = 0 so there is no gravitational component. For very large ζ the TT term would decay to 0 but this would be well above the constant flux layer approximation. At 50 m the value will depend on S and z0c.

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OnAnothere way to look at the relative importance of gravitational settling for these uniform size droplets is to consider the relative contributions to the total downward flux of water droplets aerosol (u⁎qc⁎). The gravitational contribution is simply VgQc while the turbulent diffusion contribution is,

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The ratios of turbulent transfer (TT)/total flux and gravitational settling (GS)/total flux then become

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Noting that ζ = ln ((z+z0c)/z0c) we can see that these ratios depend on both z0c, through the z(ζ) relationship, and S and will vary with z. Fig. 32 illustrates this. It is important to note that Fig. 32 is based on our relatively low estimate for z0c, ( = 0.01 m). If we increase it to z0c = 0.1 m then turbulent fluxes become more important (Fig 2c). We can see that the TT ratio is formally 1 at the surface, where Qc = 0 so there is no gravitational component. For very large ζ 325 the TT term would decay to 0 but this would be well above the constant flux layer approximation. At 50 m the value will depend on S and z0c. We can also use Equations (12) and (13)

Stable Stratification Case
For fog applications, Oover land, radiation fog often occurs at low wind speeds with stable stratification. Advection fog when warm, moist air is advected over a colder surface is another case with stable stratification. For constant flux 365 boundary layers in these circumstances MOST has, for velocity, Km = k(z+z0m)/ ΦM(z/L) and

[(z+z0c) S exp(Sβz/L)Qc] /dz = (qc⁎/k)(1+β(z+z0c)/L) (z+z0c) S-1 exp(Sβz/L)
and we need to integrate the RHS. To do this it is convenient to let β(z+z0c)/L = x and the integral that we need is of After some guidance and a few trials one can see that d/dx{x S exp(Sx)} = (Sx S-1 + Sx S )exp(Sx) and the integral required and we can then plot the ratio Qc(z)/Qc(ztop) as in Fig. 4. For S = 0, with no gravitational settling, the profile will be 395 essentially the same as the velocity profile in Eq. (18)(A1) above, i.e.

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In addition to z0c and S the key parameter is the Obukhov length, L = -ρcpu⁎ 3 θ/(kgH), (>0). Neutral stratification corresponds to L → ∞ while stable stratification relationships (H < 0, L > 0) are generally limited to 0 < z/L < 1. If we are concerned with height ranges up to 10 or 20m then L = 10m would be considered as a very low value maybe with u⁎ ≈ 0.13 ms -1 and H ≈ -20 Wm -2 as possible values. Figure 4 shows Qc(z)/Qc (20m)  it is customary to ignore that difference.

Conclusions and Suggestions
surface through gravitational settling and be diffused towards the surface by turbulence and on contact they can coalesce with an underlying water surface. Taylor et al (2021) apply these ideas to fog modelling with the WRF model. During reviews of that work, and an earlier version of the current paper, it became clear that some reviewers were reluctant to accept that turbulence could cause fog droplets to collide and coalesce with an underlying water surface, and even more reluctant to see this as a constant flux layer situation. Fog droplets are perhaps a special case 450 but the CFLGSin that there could be fluctuations in relative humidity allowing transfers between water droplets and water vapour, and variations of droplet size. It can still be argued that our conceptual model of fog droplets and cloud liquid water being generated near the top of a fog layer, perhaps as a result of radiative cooling is useful.
concept is equally applicable to aerosol particles or droplets in general, Once created the droplets can travel downward via both gravitational settling and turbulent diffusion towards a sink at the water surface. If the relative 455 humidity is at 100% throughout this descent it seems reasonable assume a constant flux layer.
The same constant flux layer concept can apply in the case of other aerosols, provided that they are inert and without sources or sinks in the air. Desert dusts, various pollutants or micro-plastic fragments being blown out over lakes or the sea from sources on land may beare examples. Here we could anticipate a situation with initial mixing through 460 a relatively deep atmospheric layer over land with minimal deposition being advected over an aerosol capturing water surface so that one could envisage a situation over the water with a constant downward flux of aerosol due to gravitational settling plus turbulent diffusion in a low level constant flux layer.
One implication of the CFLGS model is that simply adding gravitational settling (Vg) to a deposition velocity (Vdep) based on aerodynamic and surface resistances may overestimate the combined effects. If we use the CFLGS 465 model it can indicate reductions of order 20%. These are small compared to the uncertainties based on deposition velocity measurements but may well be worth considering.
In considering aerosol the recent review of dry deposition by Farmer et al (2021) and the widely used scheme of Zhang et al (2001) clearly show us that deposition velocity frequently exceeds gravitational settling velocity, 470 especially over water. This seems to be readily accepted in the atmospheric chemistry community with models developed such as Eqs (10-12) above, and also for fog deposition to vegetation (Katata, 2014). One can use these ideas in modelling work, adapting the approach of Katata et al (2010Katata et al ( , 2011 for radiation fog over forests. This is the approach adopted in Taylor et al (2021) to deal with marine advection fog over the ocean. A critical unknown parameter in this work is the deposition velocity relating Qc at the lowest model level to the downward flux to the surface due to turbulent transfer. As in the analysis above, one can use a roughness length for cloud droplets, z0c, as a tuning parameter when suitable Qc profile measurements are available.
The bottom line is that this removal process needs to be taken account of in modelling and forecasting fog occurrence and development and we need to know more about it. Fog is an intermittent phenomenon so setting up 2000) included 30-m masts and LANFEX (Price et al, 2018) used 50-m masts but the profile measurements did not include fog water, Qc, or visibility. In-situ vertical profiles of Qc were also missing in field programs like FRAM (Gultepe et al, 2009) and C-Fog (Fernando et al, 2021). C-Fog instrumentation at various sites included 10-m and 15-m masts and also a Radiometrics microwave radiometer for Qc profile measurements. These may well report 485 interesting measurements but better vertical resolution is desirable. There were Qc measurements at two or more levels in earlier field measurements reported by Pinnick et al (1978) and Kunkel (1984) showing increases with height. More such measurements are needed with multiple measurement levels and measuring droplet size distributions, Qc or LWC values and ideally Qc fluxes, along with wind, turbulence, temperature and humidity profiles plus surface pressure and fluxes of momentum, heat and water vapour. Visibility measurements at multiple 490 levels, 4 component radiation and air, aerosol and fog chemistry measurements could also play an important role in fog. From the modelling perspective we need values for z0c, which will depend on surface type and, on droplet diameter and on wind speed or friction velocity. Assuming that the lower layers, say 10-30 m of a deep fog layer, are in a relatively steady, constant flux layer situation then the CFLGS profiles developed above could provide a framework for analysis of fogs and the improvement of fog models.