## 1. Introduction

Surface visibility is crucial for aviation and land transportation safety. One of the reasons for a reduction in visibility is fogs—in particular, radiation fogs, which are a significant percentage of all fogs in the central region (Westcott 2004), northeastern region (Tardif 2004a), and southern region of the United States (Croft et al. 1997). The mechanism for radiation fog is very complex and has been extensively studied for over almost a century (e.g., Taylor 1917). Many comprehensive observational programs and in situ measurements since the 1970s (e.g., Pilié et al. 1975; Roach et al. 1976; Pinnick et al. 1978; Jiusto and Lala 1980; Gerber 1981; Meyer et al. 1986; Fitzjarrald and Lala 1989; Fuzzi et al. 1992) along with a series of numerical simulation studies (e.g., Brown and Roach 1976; Welch et al. 1986; Turton and Brown 1987; Zhou 1987; Musson-Genon 1987; Bott et al. 1990; Duynkerke 1991, 1999; Golding 1993; Bergot and Guedalia 1994; Von Glasow and Bott 1999; Nakanishi 2000; Pagowski et al. 2004) have greatly improved our knowledge of radiation fog. Nevertheless, progress in operational forecasting of fog is still slow although much effort has been devoted to it (e.g., Ballard et al.1991; Guedalia and Bergot 1994; Croft et al. 1997; Meyer and Rao 1999; Bergot et al. 2005; Müller et al. 2005; Zhou et al. 2007).

It is well accepted that radiation fog is a threshold phenomenon. In other words, it forms only after wind speeds cease or turbulence intensity drops below a certain level as the air is saturated. However, precise prediction of the onset and dissipation of radiation fog is often difficult. The reason could be that fog is too sensitive to the subtle balance among all processes (Turton and Brown 1987; Stull 1988; Meyer and Lala 1990). The balance mechanism—in particular, the turbulence threshold—is still not known and was paid little attention in the previous studies. This work attempts to examine this problem through an asymptotic analysis of the *steady state* or *equilibrium* of radiation fog. The advantage of the analytic approach is that its solution is explicit and the relationship among all of the responsible parameters can be directly and quantitatively assessed. The presence of a steady state or equilibrium in radiation fog has long been documented in the literature by Roach et al. (1976), Brown and Roach (1976), Jiusto (1981), Zhou (1987), and Musson-Genon (1987), among others. These detailed observations or numerical studies have demonstrated that the evolution of radiation fog from formation to dissipation happens in stages. During the mature period, after initial growth and before the dissipation phase, the fog bank usually stays steady or quasi steady in its depth, liquid water content (LWC), and droplet size distributions. The persistence of the steady stage implies that there exists a certain self-sustenance mechanism or a balance in its liquid water budget (Choularton et al. 1981). It is evident that the dissipation of radiation fog is the result of the breakdown of such a balance as one or more variables suddenly change—for instance, a fluctuation in turbulence associated with a variation in wind speeds or in cooling rate associated with nocturnal stratus clouds or sunrise. An examination of the equilibrium or persistence condition gives an alternative insight into radiation fog.

Forecasting fog is in essence the prediction of a mean visual range in foggy conditions. Because the visibility in situations with radiation fog can be related to its LWC magnitude, the starting point of this study begins from the LWC equation of radiation fog in an attempt to solve it at its steady state with an asymptotic method. The primary objective of this study is to estimate the mean feature of the LWC vertical structure and the liquid water budget inside radiation fog rather than to solve its more detailed characteristics using sophisticated physics. As we know, radiation fog forms, develops, and persists in a weak turbulence environment associated with a stable nocturnal surface layer. Under a weak turbulence condition, the LWC equation falls into the applicable scope of the *singular perturbation* technique (Van Dyke 1964). The singular perturbation, an asymptotic approach, was first suggested by Prandtl (1904) to solve the motion of an incompressible fluid passing an object and has since then been extended and widely applied in other scientific and engineering fields (Kelley 1994). For example, this method was explored by Blackadar and Tennekes (1968) to investigate the Ekman flow in the neutral barotropic planetary boundary layer. In the current study, the singular perturbation method was successfully applied in solving the LWC equation for steady radiation fog. Section 2 first examines the self-stabilization property of radiation fog and then presents the asymptotic analysis procedure to obtain an approximation solution for the LWC of steady fog. Section 3 discusses the asymptotic solution, from which the persistence conditions of fog and the turbulence threshold are discovered. Section 4 presents error evaluations of the asymptotic solution followed by verifications using observations of a fog event over Long Island, New York.

## 2. The solution of the LWC equation at steady state

### a. Self-stabilization property of the LWC equation

*K*closure for turbulence, the governing equation for the LWC of radiation fog can be expressed as the following partial differential equation (PDE):where

*W*is the LWC (g kg

^{−1}),

*K*is its turbulent exchange coefficient,

*G*is the droplet gravitational settling flux onto the ground,

*β*(

*p*,

*T*)

*C*(

_{o}*z*) is the condensation rate per unit mass due to cooling of the air,

*C*(

_{o}*z*) = −(∂

*T*/∂

*t*) is the total local cooling rate, hereinafter referred to as cooling rate. The slope

*β*(

*p*,

*T*) can be expressed using the Clausius–Clapeyron equation aswhere

*p*and

*T*are the air pressure and temperature, respectively,

*L*and

_{υ}*R*are the latent heat and the gas constant for vapor, respectively, and

_{υ}*e*is the saturation vapor pressure.

_{s}The turbulent exchange coefficient in radiation fog is small, with a typical range from 10^{−2} m^{2} s^{−1} in its early stage to 10^{−1}–10^{0} m^{2} s^{−1} in its mature stage (Lala et al. 1975; Roach et al. 1976; Pickering and Jiusto 1978). In confining this study to the layers close to the surface, *K* is hypothesized as a constant value without variation in height, allowing us to examine steady fog in an average turbulent environment. In real fogs, typical *K* is not a constant. The impact of this assumption will be discussed in the later sections. The cooling rate inside a fog is the residual of divergences of turbulent heat and radiative fluxes in the fog, depending on its developmental stages. In its early stage when the fog bank is still optically shallow, the total cooling is nearly uniform. In its mature stage, the maximum total cooling is located near the fog top as a result of strong radiative cooling of droplets. For purposes of demonstration, we first assume the cooling rate is vertically uniform and solve the LWC equation to show the overall solution procedure. Thereafter, a linearly distributed cooling rate is assumed.

*G*can be parameterized aswhere

*υ*is the average droplet terminal velocity, which depends on the droplet distribution spectrum and can be derived from the Stokes terminal velocity law. For simplicity,

_{t}*υ*is linearly related to LWC by the relationship

_{t}*υ*= −

_{t}*αW*, where

*α*is the gravitational settling parameter. For radiation fog,

*α*= 0.062, as suggested by Brown and Roach (1976). The impact of

*α*on LWC will be examined in section 3.

Assume that the depth of the saturated surface layer is *H* with two boundary conditions, *W* = 0 at the ground (*z* = 0) and at *H*. The lower boundary condition means that the ground is a water droplet–absorbing surface because of either settling or turbulent diffusion. The upper boundary layer condition means that fog cannot exceed the top of the saturation layer.

PDE (1) is a nonlinear equation and its solution stability is a major concern, but testing the solution stability of a nonlinear PDE would be cumbersome (Seydel 1988). Therefore, a numerical solution was employed to examine whether the steady solution(s) of PDE (1) exists and how long it takes to reach its equilibrium. PDE (1) was numerically integrated respectively with two different initial conditions, *W*(*0*) = 0.0 and *W*(*0*) = 0.4 g kg^{−1}. The results are compared in Figs. 1a,b, where *T* = 0 °C, *p* = 1000 hPa, *C _{o}* = 1.0 °C h

^{−1},

*K*= 0.01 m

^{2}s

^{−1}, and

*H*= 30 m, showing that the vertical distributions of LWC consistently approach a unique and steady solution in about 30 min for both initial conditions. The numerical test of PDE (1) reveals that indeed the LWC of radiation fog has a tendency of self-stabilization for a given set of environmental parameters (e.g.,

*K, C*, and

_{o}*H*).

On the other hand, if the depth of the saturated layer is lowered to 1 m with the other conditions held the same, fog will not form (not shown). Even for large initial LWC, fog cannot persist and disappears within a few minutes (Fig. 1c), implying that a persistence condition for a fog is also a necessary condition for fog formation, although it is not sufficient. In other words, if the air near the surface is saturated but the turbulence intensity is still large, fog cannot form. However, if *K* drops to 0.001 m^{2} s^{−1} in the same case, a shallow fog forms and takes only about 5 min to reach its steady state as shown in Fig. 1d. It appears that there probably exists a turbulence pivot for the persistence of radiation fog and that such a pivot depends on the depth of saturated air/fog. In the following sections, we try to confirm this conjecture.

### b. Singular perturbation solution

*W*/∂

*t*∼ 0, PDE (1) reduces to a second-order ordinary differential equation (ODE),where

*H*is the depth of the fog bank. For the relatively small

*K*in radiation fog, ODE (4) can be solved using the singular perturbation method. The idea of the singular perturbation approach is that in solving a boundary value problem with a small parameter, when a regular perturbation series may not match all of the boundary conditions, there probably exist one or more regions in which the solution varies rapidly. The procedure is first to assume a transition region on the ground and then respectively to solve the reduced equations (i.e.,

*K*→ 0) above (outer solution) and inside (inner solution) the transition region with different scales,

*z*for the outer solution and

*τ*=

*z*/

*K*for the inner solution. The last step is to match the inner and the outer solutions under certain matching constraint conditions so that the inner solution consistently and smoothly approaches the outer solution across the transition region. The detailed procedure can be found in Zhou (2006). Suppose the outer solution takes the form of a regular perturbation seriesBy substituting (5) into (4) and setting

*K*→ 0, we have the following reduced differential equation that only satisfies the upper boundary condition:The solution of (6) isThus, the outer solution (5) can be expressed aswhere

*O*(

*K*) is the truncation error of the termsIt is obvious that (8) does not satisfy the lower boundary condition

*W*(0) = 0.

*τ*=

*z*/

*K*in (4), we have the LWC equation in the new scale:Suppose the inner solution also has the form of a regular perturbation series

The matching constraint conditions for the layer correction are as follows: if and only if both *W*_{in}(*τ*, *K*) → 0 and *dW*_{in}(*τ*, *K*)/*dτ* → 0 as *τ* → ∞ (or *z* → ∞), then *W*(*z*, *K*) → *W*_{out}(*z*, *K*) consistently as it crosses the transition region from the lower boundary.

*η*

_{0}(0) = [(

*βC*)/

_{o}H*α*]

^{1/2}. The last two lines in (12) are matching constraint conditions for the reduced inner solution. The solution for (12) is (see the appendix)Then we have the inner solutionThe minus sign in (13) or (14) means that the inner solution is a removable effect of the LWC. The entire solution can then be uniquely obtained by combining (8) and (14) into (11) asOmitting the terms with a higher order of

*K*and substituting

*τ*=

*z*/

*K*, we obtain the final asymptotic solution for the liquid water content equation:Because the inner solution satisfies the matching requirement, it can be tested that as

*z*≫ 0 (16) will approach the first term, or the outer solution. On the other hand, if

*z*→ 0, (16) approaches zero, satisfying the lower boundary condition

*W*(0) = 0.

*C*, which must be positive (continuously decreasing in temperature) to produce the liquid water droplets, 2) the droplet gravitational settling parameter

_{o}*α*, which acts as a regulator to adjust the cooling-produced liquid water content in the first term, and 3) the turbulence term, represented by the inner solution in (16). The roles of turbulence and the inner solution will be discussed in detail in sections 3a and 3c, respectively. Because the exponential term in the inner solution is unitless, it appears that a scaling sublayer can be characterized asWith the definition of (17), (16) can be simplified as

*z*/

*δ*. The

*δ*can be thought of as a fog boundary layer (FBL). The roles of

*δ*will be discussed in later sections (especially section 3c). Further integrating over the entire fog layer with substitutions of

*x = z/H*and

*t*=

*z*/

*δ*, we have the layer-averaged liquid water content

*W*for the fog bank:Considering that

_{a}*δ*is much less than

*H*in most cases (see later sections), the second integration on the rhs of (19) can be estimated byThen we obtain

*C*(

_{o}*z*) =

*C*+ (

_{b}*C*−

_{t}*C*)

_{b}*z*/

*H*, where

*C*is the temperature variation at the bottom (>0 for cooling and <0 for warming) and

_{b}*C*is the cooling rate at the fog top (

_{t}*H*). Such an assumption may not precisely represent the real situations within fog layers. However, it is simple and captures the general feature of cooling rates in radiation fog. Using this linearly distributed cooling rate and repeating the above procedure, we have the solution for (4) (Zhou 2006):whereThe expression of the average LWC is very complex for (21) and will not be presented here. For a special case in which

*C*=

_{b}*C*=

_{t}*C*, however, (21) and (22) will reduce to (17) and (18), respectively. For another special case in which the slight warming is neglected, or

_{o}*C*= 0, (21) and (22) becomeThe layer-averaged LWC is

_{b}As we know, in a dense fog, the fog top is dominated by strong radiative cooling while the bottom is in a slightly warming regime. If one neglects the warming at the bottom, (23) can be approximately applied for dense fogs to replace (21).

## 3. Discussion

### a. Outer solution and inner solution

The asymptotic solution (18) or (21) has two components: the outer solution and the inner solution. The outer solution is controlled by cooling rate and droplet gravitational settling while the inner solution represents a negative contribution of turbulence to the liquid water content. The decreasing influence of turbulence with height is indicated by the dimensionless parameter *z*/*δ*. As *z* → *δ*, the term 2/(1 + *e*^{1}) → 0.54. That means that when *z* reaches the height of *δ* the influence of turbulence is decreased by one-half. As *z* further lifts away from *δ*, the influence of turbulence will decrease more rapidly. For a typical shallow radiation fog with a depth of 30 m and a cooling rate of 1.0 °C h^{−1}, the impacts of turbulence on the LWC distribution are illustrated in Fig. 2, in which turbulence has nothing to do with the outer solution but has a significant impact on the inner solution. Such a behavior of the inner solution demonstrates the dominant impact of turbulence on the lower parts of radiation fog. Only after the turbulence intensity increases substantially can its influence reach the upper regions of the fog. The asymptotic LWC distribution is the residual balance between cooling and turbulent diffusion, as shown by the combination of the outer and the inner solutions in Fig. 2b. The larger the turbulence intensity is, the smaller the LWC will be, reflecting the depletion effect of turbulence on the fog water. It can be expected that, if the turbulence intensity is strong enough, the liquid water in the fog layers will be completely exhausted. This is particularly true in the dissipation stage of radiation fog when turbulence intensity is substantially increased, as a result of either sunrise or increasing wind speeds.

Note that the input data used in this example are close to the radiation fog reported by Roach et al. (1976, their case A). In that particular fog event, a shallow fog (∼10 m) persisted for about 3 h before it rapidly developed into a 30-m mature fog. The mature fog bank lasted another 3 h before it dispersed after sunrise. The observed temperature was approximately 0°C, *K* was approximately 0.01 m^{2} s^{−1}, and the measured LWC was in a range between 0.08 and 0.22 g kg^{−1} at the lower levels. Figure 2b shows that the calculated LWC at the lower levels is around 0.1–0.2 g kg^{−1} (solid curve corresponding to *K* ∼ 0.01 m^{2} s^{−1}), which is close to the observation.

The average LWC (20) or (25) reveals another fact that the turbulence term has less impact on the liquid water budget in dense fogs than in shallow fogs because it is reduced by a factor of *H*^{−1}. For example, in comparing a 30-m shallow fog and a 100-m dense fog at 0°C and 0.01 m^{2} s^{−1}, it is seen that the ratio of the inner solution to the outer solution is 3% for the shallow fog and 0.25% for the dense fog. If turbulence is increased to 0.1 m^{2} s^{−1}, then the ratio increases to about 30% for the shallow fog but only 5% for the dense fog. Such a conclusion appears to agree with the observations reported by Fitzjarrald and Lala (1989) and the numerical study by Oliver et al. (1978). In Fitzjarrald and Lala’s observations, fogs were found to be classified as either surface layer fogs or boundary layer fogs. A surface layer fog is shallow and is characterized by the dominant role of turbulence, whereas a boundary layer fog is deeper and is primarily controlled by radiative cooling at the fog top. Oliver et al.’s numerical analysis shows that the role of radiation prevails over that of turbulence in a fog as it grows deeper than a few meters from the ground.

### b. Droplet gravitational settling

Droplet gravitational settling was not considered in early fog studies (e.g., Zdunkowski and Barr 1972). Its important role in the liquid water budget of radiation fog was first realized in the field observations by Roach et al. (1976) as well as in the numerical study by Brown and Roach (1976), who found that ignoring the droplet gravitational settling led to unrealistically large estimated LWC. They concluded that the observed LWC in radiation fog was only a small percentage of the total condensation and that most of the condensed liquid water had deposited onto the ground. Most current numerical fog models handle droplet gravitational settling by using a linear fitting between the terminal velocity and the LWC as expressed in (3). In this discussion, the linear relationship (3) is tested with different values of *α* to evaluate its effect on the vertical distribution of LWC.

The calculated LWC profiles and their layer averages for different values of *α* are presented in Fig. 3. The results show that the parameter *α* has a significant impact on both the LWC distribution and its average. If *α* is too small (e.g., *α* = 0.001, which represents an underestimation in the droplet gravitational settling), the LWC profile has a maximum LWC of 1.2 g kg^{−1} and an average LWC of 0.8 g kg^{−1}. Only a convective cloud can reach such a high LWC amount, which is unrealistic in comparison with the 0.1–0.2 g kg^{−1} LWC range observed in typical fogs. When *α* increases up to 0.03, the calculated LWC decreases sharply down to the realistic range for radiation fog. As *α* further increases, both the distribution profile and the LWC average vary little, indicating that the reasonable range for *α* is between 0.03 and 0.1. Such a range of *α* values supports the coefficient 0.062 for the linear fitting between LWC and terminal velocity of fog droplets that was suggested by Brown and Roach (1976).

### c. Fog boundary layer

The parameter *δ* is of particular interest to us for its role in the LWC distribution of radiation fog. In mathematical terms, it was defined to simplify the asymptotic solution. In physical terms, it represents the effective depth to which the influence of turbulence reaches. The formation of the FBL inside a fog is the consequence of interaction between two media, the fog bank and the ground, as the first medium “flows” over the second medium under the influences of cooling, turbulence, and gravity. The *δ* characterizes a liquid-phase sublayer adjacent to the ground, within which the turbulent dispersal of LWC is dominant and above which cooling prevails. The detailed numerical simulation of radiation fog by Bott et al. (1990) indicated that, in the mature stage of fog, droplets were generated near the fog top and then moved downward to the surface. The FBL near the ground inside a fog means that the condensed droplets of the upper parts of the fog will travel through the FBL before they settle down on the ground, except for a very shallow fog in which cooling may dominate both in and above the FBL. The formulation of the FBL in (17) or (22) is, in its expression, similar to the characteristic depth of the laminar layer *h _{ν}* = 5

*ν*/

*u*

_{*0}(Busch 1973), where

*ν*is the kinetic viscosity and

*u*

_{*0}is the surface friction velocity. In comparing this definition with (17) or (22), we may treat the denominator (

*αβC*)

_{o}H^{1/2}in (17) or [

*αβ*(

*C*+

_{b}*C*)

_{t}*H*/2]

^{1/2}in (22) as a scaling “velocity” for the fog droplets generated by cooling and deposition toward the ground by gravity (its unit is in velocity). Observing that the factor (

*βC*/

_{o}H*α*)

^{1/2}in (18) or [

*β*(

*C*+

_{b}*C*)

_{t}*H*/2

*α*]

^{1/2}in (21) is the maximum LWC, the scaling velocity characterizes the maximum deposition rate inside a fog. For a typical cooling rate of ∼1.0°C h

^{−1}, the scaling velocity ranges from 0.01 to 0.05 m s

^{−1}, depending on temperature and fog depth.

The magnitude of *δ* is a function of temperature, fog depth, and turbulence exchange coefficient. For a 30-m shallow fog with *C _{o}* = 1.0°C h

^{−1}and

*K*= 0.01–0.1 m

^{2}s

^{−1}, the

*δ*has a value of 0.3–3 m, or 1%–10% of the fog depth. Such a percentage of

*δ*is consistent with the depth of the gas-phase mixing layer suggested in observations and numerical simulations for shallow fogs (Roach et al. 1976; Musson-Genon 1987). A similar behavior can also be expected for deep fogs. For example, in a well-developed deep fog in which

*C*(negative value) or

_{b}*K*may increase substantially as a result of heating of the lower parts of the fog, (22) yields a very high FBL. For a 100-m fog with

*K*over 1.0 m

^{2}s

^{−1},

*C*= 1.0°C h

_{t}^{−1}, and

*C*= −0.5°C h

_{b}^{−1}, the FBL depth can reach as high as 70 m, which is in agreement with the depth of unstable layers observed in deep fogs (R. Brown 2006, personal communication). Therefore, although the FBL is defined from LWC, it reflects well the feature of the mixing layer near the surface observed in radiation fog. Such a similarity between the FBL and the surface mixing layer is of practical significance to the application of the asymptotic LWC formulation because an estimate of

*K*in a nocturnal boundary layer is not so reliable. If a relationship between the FBL and the surface mixing layer exists in radiation fog, the parameter

*δ*in the formulation can be substituted for by the depth of the surface mixing layer, which can be easily estimated from reliable temperature profiles.

### d. Persistence conditions and the critical turbulent coefficient

*ϕ*= exp[

*H*/(3

*δ*)]. The approximate solution for (27) is

*ϕ*> 1.62, or

*δ*/

*H*< 0.69. In other words, if the FBL depth reaches 70% of the entire fog bank, the shallow fog will become unstable and disperse. Therefore, the following inequality is a balance condition for a shallow fog:In a similar way, the following inequality can be derived for a deep fog:The approximate solution for (29) is

*ϕ*> 1.40, or

*δ*/

*H*< 1.0. Thus,

*δ*<

*H*is a persistence condition for deep fogs. It can be tested that as

*δ*→

*H*in Eq. (21) or (23)

*W*(

*z*) will approach zero at all levels.

*δ*< 0.69

*H*and

*δ*<

*H*into the definitions of

*δ*in (17) and (24), respectively, the following critical turbulent exchange coefficient

*K*for shallow and deep fogs can be obtained:

_{c}To keep the balance in a steady fog, the turbulence intensity in the fog must be less than the critical turbulent exchange coefficient *K _{c}*, which is more sensitive to the fog depth (

*H*

^{3/2}) than to the cooling rate (

*C*

^{1/2}

_{o}or

*C*

^{1/2}

_{t}).

The critical turbulent exchange coefficient in (30a) or (30b) defines the upper limit of turbulence intensity that a persisting fog can withstand. An initial ground fog usually forms below 10 m near the surface and remains stable for a long time (conditioning), during which the surface turbulence does not exceed the critical turbulent exchange coefficient *K _{c}*; otherwise, the ground fog will dissipate. Several factors may cause the turbulence intensity to exceed

*K*: 1) a reduction in cooling rate (

_{c}*K*decreases) due to sunrise, local clouds, or warm advection and 2) rising local wind speeds, which increase the surface mechanical turbulence (

_{c}*K*increases). On the contrary, an increase in cooling rate or cessation of winds will favor persistence of ground fog.

As indicated by (30), a deep fog (with large *H*) has a large *K _{c}*, implying that it is not easy for turbulence to disperse a deep fog because a strong turbulence intensity is required to break the balance. Various critical turbulent exchange coefficients for different

*H*and different cooling rates at different temperature are presented in Fig. 4. The parameter

*K*can be used as a threshold for diagnosing whether a fog persists or dissipates. For example, for a ground fog with

_{c}*H*∼ 1 m,

*T*= 10, and a cooling rate of ∼1°C h

^{−1},

*K*is about 4 × 10

_{c}^{−3}m

^{2}s

^{−1}(Fig. 4a). That is, if turbulence near the surface drops below 4 × 10

^{−3}m

^{2}s

^{−1}, a ground fog can persist; otherwise, the ground fog will soon disappear, which is consistent with the numerical experiment in the first section of this paper and also is in agreement with the general turbulence intensity range observed in typical shallow fogs. However, for a deep fog (e.g.,

*H*∼ 100 m),

*K*increases to 4 m

_{c}^{2}s

^{−1}(Fig. 4c). In other words, a 100-m-deep fog will persist except when the turbulence intensity exceeds 4 m

^{2}s

^{−1}. This explains why turbulence only disperses shallow fogs but not deep fogs as observed by Fitzjarrald and Lala (1989). The large-eddy simulation of radiation fog by Nakanishi (2000) revealed a sharp increase in the turbulent kinetic energy (TKE) inside a fog bank in its dissipation stage after sunrise. This result appears to be related to the breakdown of the balance condition as described by

*K*and

*K*because

_{c}*K*closely relates to TKE. A dramatic increase in TKE will lead to a significant rise in

*K*as well.

The role of the critical turbulent exchange coefficient in radiation fog can be more conveniently explained with an *H–K* state diagram (Fig. 5). The diagram is divided, according to the *K _{c}* curve, into two regimes:

*K*<

*K*(left of the

_{c}*K*curve), where it is possible for fog to exist, and

_{c}*K*>

*K*(right of the

_{c}*K*curve), where it is impossible for fog to be present. Line

_{c}*H*

_{AB}is a border that separates the upper unsaturated air from the lower saturated air. Because saturation of the air usually begins from the ground and expands upward as cooling continues, line

*H*

_{AB}represents an upward-moving saturation front, and the fog top is limited to below

*H*

_{AB}. That is, fog can only appear in the region where both

*K*<

*K*and RH = 100% are true. Point B is the intersection point between the

_{c}*K*curve and the saturation front line, and C is the intersection point between the

_{c}*K*curve and the turbulent exchange coefficient (e.g.,

_{c}*K*

_{1}). The steady state of a fog can be represented by a pair (

*H*,

*K*). If

*K*

_{1}is larger than

*K*

_{B}(

*K*

_{1}on the right side of point B) or

*H*

_{AB}is below point C, there is no fog even if the air is saturated. Only when

*H*

_{AB}lifts above point C or the turbulent exchange coefficient decreases to below

*K*

_{B}(

*K*

_{1}on the left side of point B) can a steady fog exist; however, its state (

*H*,

*K*) is restricted to the shaded triangle area enclosed by A, B, and C as shown in Fig. 5. It can be expected that the smaller

*K*

_{1}is, the lower the point C (

*H*) is and the thinner the fog depth can be. In other words, if turbulence does not cease completely, it is very difficult for a fog initially to appear (forms locally or moves over the scene from other places) and to persist on the ground. This implies that in a larger turbulent environment radiation fog is prone to appear initially at a higher level and then to diffuse downward to the ground, because it is impossible for a fog with a depth thinner than

_{C}*H*to persist in the region below point C.

_{C}After a fog appears, its persistence relies on the subsequent surface turbulence status. Turbulence in a nocturnal boundary layer is weak and fluctuates with time. The fluctuations in nocturnal turbulence are responsible, at least partially, for the periodic oscillations of fog layers (Jiusto and Lala 1980). The mechanism of how fluctuating turbulence affects a fog also can be explained by the *K–H* state diagram. For instance, if the turbulence intensity *K* in a fog fluctuates between *K*_{1} and *K*_{2} across *K*_{B} of point B as shown in Fig. 5, the fog will fluctuate with time, too. When *K* increases over *K*_{B}, the fog is not stable and disappears soon, whereas if *K* decreases below *K*_{B}, the steady fog resumes. Such a scenario for low values of *K _{c}* explains what was observed in shallow fog. If the saturation front

*H*

_{AB}steadily lifts to a higher level,

*K*

_{B}will increase as well and results in a larger shaded area in the diagram. The deep fog bank can consequently endure stronger turbulence and has less temporal variation in LWC. Deep and well-mixed fog is more like a stratocumulus cloud in which internal convection is reinforced—in particular in the lower regions of the fog. However, the fog-induced turbulence or convection is not able to disperse the fog itself, except that the turbulence balance inside the fog is broken by external reasons such as sunrise. If turbulence becomes adequately large but not large enough to exceed

*K*

_{B}, it has a positive impact on the generation of fog droplets without dispersing the fog layers, because adequate turbulence not only helps to broaden the droplet distribution spectrum in radiation fog (Gerber 1981) but also lifts the saturation level quickly (Welch et al. 1986; Zhou 1987). This also implies that if a fog appears aloft it is more likely to become a dense fog because it has a larger

*H*at the beginning. The behavior at the high end of

*K*explains the fact that growing, dense fogs are associated with strengthening turbulence without being dispersed like shallow fogs.

_{c}## 4. Error evaluation and verification

### a. Errors and uncertainties

Errors and uncertainties of the LWC formulation arise from the following reasons. First, it is an asymptotic solution with a truncation error of *O*(*K*). This means that the solution is more accurate for a small *K* than for a large *K*. The accuracy of the asymptotic solution for different *K* values can be evaluated by comparing the asymptotic solution with the steady numerical solutions of PDE (1) in different *K* values as presented in Fig. 6, which shows that the LWC profiles for the asymptotic and the numerical solutions are in close agreement, with a small positive bias of 10% for weak turbulence (Fig. 6a for shallow fog and Fig. 6d for deep fog) and a larger positive bias of 30% for strong turbulence (Fig. 6b for shallow fog and Fig. 6e for deep fog). However, if the turbulence intensity further increases, being close to *K _{c}*, both asymptotic and numerical LWC approach zero (Figs. 6c,f). This implies that

*K*is in good agreement with the value of turbulence pivot of PDE (1) for given conditions, which can be confirmed by comparison between

_{c}*K*and the turbulence pivots searched from the solution space of PDE (1) for different fog depths and temperatures as shown in Fig. 7.

_{c}Second, the asymptotic solution was derived under several hypotheses. It is not appropriate to apply the solution in rapidly growing stages of a fog, such as the formation or dissipation stages, or in a fog over very heterogeneous topography. The complex processes in multiple phases in ice fog (Girard and Blanchet 2001) were not considered in this study. Ice fog usually occurs at temperatures below −30°C (Thuman and Robinson 1954). Therefore, the formulation is not applicable in extreme cold environments. In addition to these limitations, the asymptotic formulation also requires a vertically uniform *K* as an input parameter. However, in radiation fog *K* varies with heights. To evaluate the impact of the uniform-*K* assumption on the asymptotic solution—in particular, in a deep fog—two numerical solutions of ODE (4) for a 100-m fog are compared in Fig. 8: one with *K* linearly increasing from zero at the surface to a maximum value of 1.0 m^{2} s^{−1} at 70 m and then linearly decreasing to zero at the fog top and another with a uniform value of 0.5 m^{2} s^{−1}, which represents the average of the first case. It can be observed that the two LWC profiles are not significantly different. Thus, the uniform *K* assumption is not, at least in terms of solution accuracy, a serious problem for applying the solution in deep fogs.

The accuracy of the asymptotic formulation also depends on where it is applied. If it is used in a field study, its accuracy relies on the quality of measurements. If it is applied in a numerical weather prediction (NWP) model, the modeled fog LWC is treated as a first guess and the asymptotic formulation can be used to diagnose fog conditions at grid points near the surface or to resolve/adjust the modeled LWC. In this case, the depth of the saturated surface layer is suggested as the fog depth in computation. For a shallow saturated surface layer at a grid point, the modeled cooling rate is reliable for use, because whether the model predicted light fog has relatively less impact on the modeled cooling rate and turbulence. If the saturated surface layer at the grid point is deep, however, the modeled cooling rate may have some uncertainties because the first guess from an NWP model may not be correct (Stoelinga and Warner 1999; Müller et al. 2005) to generate reliable cooling profiles. One suggestion is using the layer-averaged cooling rate over multiple levels within a deep fog to reduce the uncertainty. In this case, (18) instead of (21) or (23) is used. The sensitivity of the asymptotic LWC to various cooling rate profiles will be assessed in section 4b.

### b. Verification

In this section, verifications of the LWC in a deep radiation fog event, including its persistence and vertical distributions estimated from the asymptotic solution, with the observational data are presented. The fog episode occurred at midnight 10–11 October 2003 in east-central Long Island, and was described in detail by Tardif et al. (2004b). Weather conditions were favorable for radiation fog, with light winds and clear skies. Surface observational data were collected by the Automated Observing Surface Systems (ASOS) at major terminals and regional airports, and the data above the surface were recorded by a 90-m National Center for Atmospheric Research (NCAR) tower at Brookhaven National Laboratory. Both potential temperature and wind speeds measured at 10, 15, 20, 32, 45, 70, and 85 m (Figs. 9a,b) decreased steadily with time during the early evening before the onset of fog around 0400 UTC. The distributions of cooling rate before and after the fog formation (Fig. 9c) were calculated from the temperature profiles. After sunset (around 2220 UTC), the air near the surface began cooling at a rate of 1.5°C h^{−1} from 0000 to 0200 UTC and 0.8°C h^{−1} from 0200 to 0400 UTC. After formation of the fog, the layer’s maximum cooling separated from the surface because of warming of the ground. During the steady fog stage between 0400 and 0600 UTC, the cooling rate showed a maximum value of 1.75°C h^{−1} at 85 m and then decreased steadily downward to a 0.5°C h^{−1} heating rate near the surface. Because there was no direct LWC measurement in this fog event, it was approximately estimated with the formula of Kunkel (1984) from the visibility data observed from the ASOS at the 4- and 32-m levels, respectively. Deriving the LWC from the visibility measurements is an approximate estimation because the LWC of fog is also related to the droplet number concentration (Gultepe et al. 2006). The droplet number concentration was not available for this fog event; thus the usage of the more accurate formula was limited in this case. The variations of estimated LWC are presented in Fig. 9d, which shows the appearance of the fog earlier at 32 m than at 4 m, implying that the fog initially formed aloft and then thickened, growing upward and downward to the ground (Tardif et al. 2004b). It quickly became a stable dense fog by 0430 UTC with an LWC of 0.30 g kg^{−1} at 4 m and almost 0.5 g kg^{−1} at 32 m. The fog began to disperse after 0700 UTC when a low marine stratus cloud deck with a ceiling height of about 200 m moved over the fog region.

Because of uncertainties in computing turbulence intensity under stable conditions, three estimates of turbulent exchange coefficients were averaged using 1) the formulation of Businger et al. (1971), 2) the improved formulation of Duynkerke (1991), and 3) the level-2 formulation of Mellor and Yamada (1974). The calculated mean turbulent exchange coefficients at all levels (Fig. 10a) decreased with time prior to formation of the fog at 0400 UTC. Figure 10b shows a consistent temporal tendency among the three calculated turbulence intensities near the surface, where the variations among the three curves are highlighted by shading. All of the three calculated turbulence intensities dropped to their minimum values around 0400 UTC, at which time an inset with a 1-h window is centered to show the amplified variations in the three curves. It can be seen from the inset that the average value of the three turbulence intensities near 0400 UTC was about 0.1 m^{2} s^{−1}. The *K _{c}* around this time, computed with known temperature (13°C) and cooling rate (1.0°C h

^{−1}), was 0.05 m

^{2}s

^{−1}for

*H*= 4 m and 0.36 m

^{2}s

^{−1}for

*H*= 32 m, respectively. In other words,

*K*(4 m) <

_{c}*K*<

*K*(32 m). Thus, the fog could not form below the 4-m level as the saturation layer spread upward from the ground but could form somewhere else between 4 and 32 m.

_{c}Figure 10a shows that the magnitude of the turbulent exchange coefficient inside the fog was in a range of 0.05–0.1 m^{2} s^{−1} in its formation period and increased to a range of 0.3–2.0 m^{2} s^{−1} during its steady stage, in which the characteristic depth of the FBL was estimated to be 10 m—one-half of the temperature-defined surface mixing layer at 0600 UTC as indicated in Fig. 9a. Given that the maximum cooling rate was around 85 m, the fog top was assumed to be steady at 100 m. In response to longwave radiative heating from the emerging stratus cloud overhead, the turbulent exchange coefficient inside the fog by 0800 UTC increased substantially to 5.0 m^{2} s^{−1} at lower levels and to over 10 m^{2} s^{−1} at upper levels.

With *p* = 1000 hPa, *T* = 13°C, and *K* = 0.7 m^{2} s^{−1}, which represented a layer-averaged turbulence intensity in the period between 0400 and 0600 UTC, the vertical distribution of the LWC was computed with either formulation (21) or (23). In usual situations, however, the vertical profile of a cooling rate is not available. In this case, the formulation (16) may be an alternative, which requires a vertically uniform cooling rate. To compare the three vertical distribution formulations of LWC, three cooling rates were tested: 1) a vertically uniform rate of 1.0°C h^{−1}, which represents an average cooling rate over the entire fog bank; 2) a vertically linear distribution rate with 1.75°C h^{−1} cooling at the fog top and 0.5°C h^{−1} heating at the bottom; and 3) a vertically linear distribution rate with 1.75°C h^{−1} cooling at the fog top but zero heating at the bottom. Moreover, to assess the substitution of the FBL with a mixing layer in formulation (18), the fourth case with *δ* = 20 m was also tested. The calculated distribution profiles of the LWC for the four cases are compared in Fig. 11, showing a similarity in the vertical distribution patterns among the four LWC profiles and their average values with a slight difference in the maximum LWC locations. Considering that an error of 0.07 g kg^{−1} may be involved in Kunkel’s formula (Duynkerke 1991), the computed LWC generally agrees with the observed values at 4 and 32 m, respectively. The assessment indicates that the simplified deep fog formulation (23) is a good representation for the original formulation (21). If the vertical distribution of cooling rate for a deep fog is not available, an average cooling rate can be used in the computation. Moreover, substituting the FBL with the depth of mixing layer in the asymptotic formulation also obtains a reasonable result.

It can be observed from Fig. 10a that during the mature fog stage between 0400 and 0600 UTC, the turbulence intensity inside the fog steadily increased to as large as 2.0 m^{2} s^{−1}. Such a strong turbulence intensity did not disperse the fog bank, however, because the 100-m-deep fog could withstand a turbulence intensity of 4.0 m^{2} s^{−1}, as shown in Fig. 4c. The deep fog dispersed only after the cloud deck moved over the fog region when the turbulent exchange coefficient suddenly exceeded 4.0 m^{2} s^{−1}, the critical turbulence intensity for the fog bank. Thus, the potential temperature and cooling rate profiles at 0800 UTC show that the whole fog bank was in steady warming and that the surface unstable mixing layer had expanded from 20 m to the entire fog bank. It can be calculated that the turbulence intensity of 5.0–10.0 m^{2} s^{−1} could generate a 75–150-m FBL, which had reached the fog top and broken the persistence condition for the 100-m-deep fog.

## 5. Summary

At the steady state and under uniform turbulence assumption, the LWC equation for radiation fog was successfully solved using the singular perturbation along with the layer-correction matching technique. Despite simplifications without the sophisticated physics involved in the asymptotic analysis, the solution captures the balance relationship among cooling, gravitational settling, and turbulence when radiation fog is in its persistence stage, showing that the LWC ∼ *C*^{1/2} and *H*^{1/2}, where *C* is cooling rate and *H* is fog depth. The cooling produces the water content, which is distributed by the gravitational settling and depleted by turbulence. When turbulence is weak, its influence on fog is significant only near the surface, whereas as turbulence becomes strong its influence extends to higher levels. The influence height of turbulence can be characterized by a fog boundary layer, which is thin for a large cooling rate or weak turbulence and increases as cooling rate decreases or turbulence increases. Such an LWC-defined FBL from the asymptotic solution resembles the behavior of the mixing layer in mature fogs. The FBL also parameterizes the persistence condition for radiation fog. If the characteristic depth of FBL reaches the fog top, the condition is broken and the fog will disperse.

The critical turbulent exchange coefficient for steady radiation fog was identified from the asymptotic solution. Its value is proportional to *C*^{1/2} and *H*^{3/2}, which defines the strongest turbulence intensity a steady fog can endure. A weaker-than-critical turbulent intensity is a necessary condition for having a fog persist. Once turbulence exceeds its threshold, the fog will become unstable and disperse. This means that a deep fog can withstand a stronger turbulence intensity than a shallow fog does, and an initial fog forming aloft is more likely to become a dense fog because it is deep from the beginning. The roles of the turbulence threshold in radiation fog can be conceptually illustrated with a state diagram of fog depth versus turbulence intensity. Note that the concepts of FBL and critical turbulent exchange coefficient were first introduced for radiation fog in this study and were examined with only one observation. Their role and impact on radiation fog await further verification with more observational data or with numerical simulations.

The asymptotic LWC formulation was derived under the steady condition of fog; therefore it is only applicable in the mature fog stage and not in the fast-growing formation stage or in the dissipation stage. The accuracy of the formulation was accessed in terms of solution error and uncertainties in the input parameters. The asymptotic solution has a first-order approximation with respect to turbulence intensity; therefore it is more accurate in conditions with weak turbulence than with large turbulence. Although the asymptotic solution was derived under the uniform turbulence assumption, numerical tests have demonstrated that this is not a serious problem in terms of solution error in deep fogs.

The verification of the asymptotic LWC formulation was performed with a deep radiation fog event that was dispersed by a stratus cloud. The persistence conditions before and after the cloud moved overhead were investigated and examined with regards to cooling and turbulence intensity. The verification showed that the computed LWC magnitudes with either constant or linearly distributed cooling profiles generally agreed with the observed LWC. Although the verified fog event is not a common case in which sunrise is the major reason for fog dissipation, the asymptotic formulation is still applicable to a common case because heating and an increase in turbulence intensity inside fog layers are the major reasons for both the common case and the cloud-overhead case.

The three anonymous reviewers are appreciated for their valuable remarks and suggestions about this paper. The authors are also very grateful to Drs. Zavisa Janjic, Mike Ek, and Yucheng Song of NCEP/EMC for their EMC internal reviews of this work, to Dr. Rod Brown (retired) of the Met Office for his comments and suggestions, and to Dr. Stephen Lord, director of NCEP/EMC, for his support. We thank Dr. Robert Tardif of NCAR for his provision of the observational fog data. The site data of the New York 2003 fog event used in this study were collected from the equipment and facilities that were installed, operated, and funded by the FAA Aviation Weather Research Program. It is our pleasure to acknowledge Ms. Mary Hart of NCEP/EMC, who reviewed and edited early drafts of this paper.

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# APPENDIX

## The Inner Solution

*τ*=

*z*/

*K*and substituting (11) into (9), we have the following dual-scale (

*z*and

*τ*) equation:Because the outer solution

*W*

_{out}(

*z*,

*K*) is also satisfied by (4), (A1) must have the following formwhich reduces further to the following equation as

*K*→ 0:By substituting the asymptotic expansions for both the inner and the outer solutions given by (5) and (11) into (A3), the reduced inner equation becomesThe lower boundary condition for the inner solution isand (A4) can be conveniently transformed to (Zhou 2006)Integrating both terms in (A6) with respect to

*τ*, one obtainswhere

*C*

_{1}= 0 based on the matching constraint conditionsFurther integrating the second term in (A7), one obtainsThe general solution for (A8) iswhere

*C*

_{2}= −1 based on the lower boundary condition (A5). The solution for

*ξ*

_{0}is obtained by rearranging terms in (A9):