atmosphere

Article
Parameterization of Radiation Fog-Top Height and
Methods Evaluation in Tianjin
Tingting Ju 1,2 , Bingui Wu 2,3, *,† , Hongsheng Zhang 4 and Jingle Liu 2
1 Navigation College, Dalian Maritime University, Dalian 116026, China; bnutingting@163.com
2 Tianjin Key Laboratory for Oceanic Meteorology, Tianjin Meteorological Bureau, Tianjin 300074, China;
liujinglexxx@163.com
3 Laboratory of Straits Meteorology, Xiamen Meteorology Bureau, Xiamen 361012, China
4 Laboratory for Climate and Ocean-Atmosphere Studies, Department of Atmospheric and Oceanic Sciences,
School of Physics, Peking University, Beijing 100871, China; hsdq@pku.edu.cn
* Correspondence: tjwbgtjwbg@126.com; Tel.: +86-022-2335-2889
† Current address: Tianjin Meteorological Bureau, No.100 Weather Station Road Hexi District,
Tianjin 300074, China.




Received: 25 March 2020; Accepted: 5 May 2020; Published: 8 May 2020


Abstract: Different methods have been developed to estimate the fog-top height of radiation fog and
evaluated using the measurements obtained from a 255-m meteorological tower located in Tianjin
in 2016. Different indicators of turbulence intensity, friction velocity (u* ), turbulence kinetic energy
(TKE), and variance of vertical velocity (σw 2 ) were used to estimate the fog-top height, respectively.
Positive correlations between the fog-top height and u* , TKE, and σw 2 were observed, with empirical
0.68
 0.51
parameterization schemes H = 583.35 × u1.12∗ , H = 205.4 × (TKE) , and H = 420.10 × σ2w being
obtained. Among them, σw is the most appropriate indicators of turbulence intensity to estimate
2

the fog-top height. Compared with sensible flux and condensation rate, the new form of convective
velocity scale (w* ) was the most appropriate indicator of buoyancy induced by radiative cooling,
and the relationship H = 328.33 × w1.34 ∗ was obtained. σw 2 and with w* , which represents the
intensity of turbulence and buoyancy, were used to estimate the fog-top height. The relationship
H = 396.26 × (σw + 0.1 × w* ) − 16 was obtained, which can be used to accurately estimate the fog-top
height. Moreover, the temperature convergence (TC) method was used to estimate the fog-top height;
however, the results strongly rely on the threshold value.

Keywords: fog-top height; turbulence; radiative cooling; parameterization; surface measurements

1. Introduction
Fog is a boundary layer weather phenomenon composed of suspended water droplets or ice
crystals, which reduces horizontal visibility to less than 1 km [1]. The low visibility due to fog
events is a hazard to navigation, motor-vehicle accidents [2], plane delays [3], air crashes, and public
health [4,5], and the impact of fog has significantly increased because of increasing air, marine, and road
transportation [6,7]. Therefore, more attention needs to be paid on fog.
Previous studies have shown that fog and low stratus significantly influence the earth’s radiation
balance [8,9]. Fog-top height (fog thickness) is considered to be very useful information for aircraft
maneuvers, especially important for aircrafts having to land in foggy conditions. Moreover, there is
no doubt about the importance of an accurate information of fog thickness for data assimilation of
Numerical Weather Prediction (NWP) models, due to the significant impact of this parameter on the
radiation budget close to the surface [10]. The estimation of observed fog-top height is important and
useful for validation of model simulations [11], since comparisons between observed and simulated

Atmosphere 2020, 11, 480; doi:10.3390/atmos11050480 www.mdpi.com/journal/atmosphere

Atmosphere 2020, 11, 480 2 of 18

fog thickness cannot be performed in many cases due to no supplementary observational evidence of
fog-top height. Moreover, the height of the fog top at the end of the mature phase is useful information
for estimating the beginning and duration of the fog dissipation phase [12,13], with the thicker the
fog layer, the more time is needed for fog dissipation. Therefore, it is also crucial to improve the
nowcasting of fog dissipation [14]. However, many studies cannot provide information about observed
fog-top height due to a lack of vertical measurements, and the height of fog top is usually difficult to
determinate, since the fog-layer top cannot be observed from the surface.
The vertical extent of fog (fog thickness) ranges between a few meters and several hundred
meters. Previous studies showed that turbulent mixing and radiative cooling are thought to be
responsible for the increase in the depth of the fog. Numerical results showed that the fog-top height
depends on the intensity of turbulence; the stronger the turbulence, the higher the top of the fog.
In addition, some researchers have pointed out that the depth of fog is mainly due to the intensity of
the inversion layer and that the inversion layer plays an important role in suppressing the vertical
extension of this fog event; that is, the stronger the inversion, the lower the height of the fog-top.
Moreover, both the concentration and size of aerosol particles in the polluted fog can affect urban
fog thickness [15]. Therefore, fog-top height is closely related to turbulence and the upper inversion.
However, the quantitative relationships between fog-top height and intensity of turbulence and
inversion are ambiguous in the present study [16].
Many instruments and methods have been developed to determine the height of fog; satellite
data, ground remote sensing instruments, and atmospheric sounding are usually used to provide
approximations of fog-top height. In many case, temperature and humidity data from atmospheric
sounding are used to estimate fog thickness [17,18]. The relative humidity profile is the most common
and simple method to estimate the height of fog top; however, it is still uncertain to precisely define the
fog top using the relative humidity threshold because of measurement uncertainties [19]. Cai et al. [20]
defined the height above with a relative humidity that was smaller than 80%, while Sorli et al. [21]
set the threshold of relative humidity to 95%, and 100% was used in Huang et al. [22]. Moreover,
a mixing ratio of cloud water has also been used to determinate the height of fog top, with a threshold
value 0.05 g kg−1 [23]. By considering the inversion intensity of temperature profiles and the 100%
threshold on relative humidity profiles, the fog height can be estimated relatively accurately [24].
Moreover, the height of the fog top can be determined using virtual temperature profiles [11]. However,
to determine fog-top height accurately, the measure points under and above the fog top should be dense.
Therefore, there is discrepancy between the height of fog top estimated by atmospheric sounding and
the real value, due to the vertical resolution of atmospheric sounding data. Moreover, ground remote
sensing instruments are used to study radiation fog [25]; as an active sensor, it has advantages of high
vertical resolution and detection sensitivity, however, the results are not satisfactory. A method of
fog detection has been proposed by Wu et al. [26] and this method seems to be more reliable than
the passive satellite measurements; however, it fails to identify dense fog layers because of the high
signal attenuation.
Data and products from satellite have been widely used to detect fog or low clouds in foggy
conditions [27]. Most applications retrieve fog-top height by comparing the satellite-derived infrared
(IR) (infrared) temperature with temperature of soundings nearby. The premise of the scheme is that
fog is related to an uplifted temperature inversion with inversion base height delimiting the fog top;
therefore, the inversion base height is the fog top height [28]. Yi et al. [29] presented a novel method
to retrieve low stratus/fog top heights using the infrared (IR) water vapor and split-window bands
from geostationary (GEO) systems. However, the equations to retrieve inversion and thickness under
foggy conditions were taken from Liu and Key [30], which were originally developed for clear sky
situation. Thus, in future work, the adjustment of the new scheme should be developed to improve the
adaptation of the original scheme. Despite its merit, the study also showed that a proper discrimination
between low stratus and fog from satellite data remain challenging. A discrimination between fog at
the ground and other low stratus situations from satellite data requires information on cloud vertical
Atmosphere 2020, 11, 480 3 of 18

geometry to establish whether the cloud touches the ground. Cermak and Bendix [31] introduced
a technique that allows for the discrimination between low stratus and (ground) fog on the basis
of geostationary satellite imagery. The cloud base height is derived using a sub-adiabatic model of
cloud microphysics. In this model, the cloud base is varied until model liquid water path (LWP)
matches that retrieved from satellite data. The performance of this technique was shown to be good in
comparison with METeorological Aerodrome Reports (METAR) data comprising 1030 satellite scenes.
The main challenge in model development is to accurately quantify the deviation from the adiabatic
profile. However, there are still misclassifications between low stratus and fog in some situations.
In conclusion, it is very different to differentiate between fog and low clouds using satellite data [29,32].
As Marchand et al. [28] stressed in their review, fog-top height retrievals are a hitherto not yet
a solved problem. In addition, satellite data, ground remote sensing instruments, and atmospheric
sounding are expensive and not always available. Román-Cascón et al. [33] proposed two methods
based on the surface data, and line correlations were observed between the height of fog top and
surface friction velocity, and surface sensible flux. However, the results showed a relatively large bias
and their methods were thus not appropriate to estimate the height of shallow fog.
In this study, different methods for estimation of fog-top height were evaluated and validated
using the profile and atmospheric turbulence data obtained from a 255-m meteorological tower located
in Tianjin. Moreover, two new estimated methods using the surface measurements were developed
and evaluated. In Section 2, the information of the observation site, the instruments and data set are
described. Moreover, several important parameters used in the study are calculated. In Section 3,
different methods are used to estimate the fog thickness and compare with the observation results.
A few new estimated methods based on turbulence intensity and radiative cooling are developed and
validated using radiation fog events. In Section 4, the conclusions are summarized.

2. Data and Methods

2.1. Experiment Site and Data

Tianjin is located in the eastern of North China Plain (NCP), with rather flat terrain. Furthermore,
it is the largest coastal city in North China, with the Bohai Sea to the east. The results of Quan et al. [34]
have shown that the mean occurrence number of fog events over the NCP is 11.4 ± 6.9 days yr−1 .
Long-term fog trends are observed at 13 weather stations in Tianjin from 1980 to 2010, showing that
the total amount of fog has increased since 1980 [35]. The data used in this study were obtained from a
meteorological tower at a height of 255 m, located at the atmospheric boundary layer Meteorological
Observation Station in the south of Tianjin city (39.08◦ N, 117.21◦ E). There are no tall buildings around
the tower in a radius of 50 m, and heights of the surrounding buildings in the radius of 300 m are within
30 m. Therefore, the observation site can represent the homogeneous urban underlying surface [36].
The data and instruments used in this study are presented in Table 1. The measurements
of meteorological parameters (wind speed, wind direction, air temperature, etc.) were recorded
automatically and continuously at a 10-min interval. The 10 min data were then converted to a 30 min
moving average. The sampling frequency of atmospheric turbulence data is 10 Hz, and a preprocessing
was performed using Eddy Pro software (Advanced 4.2.1, LI-COR Biosciences, Inc. LI-COR, Lincoln,
NE, USA). The preprocessing includes spike removal [37], double coordinate rotation [38], and trend
removal. In addition, the block time average method was used in the trend removal process [39],
with an average time interval of 30 min. The results of Eddy Covariance (EC) depend on wind speed,
stability parameter, and friction velocity; strict quality control was performed on the atmospheric
turbulence data, for detailed criteria, please refer to Ye et al. [40] and Ren et al. [41]. Moreover, it should
be noted that due to the limitation of the maximum height of the observation tower and because the
fog-top heights of radiation fog were typically below 200 m [42], fog-top heights that were higher than
250 m are out of the scope of this paper.
Atmosphere 2020, 11, 480 4 of 18

Table 1. List of instruments used in this study.

Instrument Mounting Height Measurements Sampling Interval Accuracy

Cup and vane
Wind speed
anemometer 15 levels a 20 s 0.1 m s−1
Wind direction
(Changchun, China)
Temperature and
T: ±0.2 ◦ C
relative humidity Temperature
15 levels a 20 s RH: ±2% (0–90%)
probe (HMP45C, Relative humidity
±5% (90–100%)
CAMPBELL, USA)
Three-dimensional (3D)
Sonic anemometer- u, v: < ± 0.04 m s−1
wind components
thermometer (CSAT3, 40, 120, 20 m 0.1 s w: < ± 0.02 m s−1
Sonic virtual
CAMPBELL, USA) Tθ : 0.01 ◦ C
temperature
CO2 /H2 O Analyzer H2 O: 0.0047
Water vapor density
(LI-7500, 40, 120, 220 m 0.1 s mmol/mol
CO2 concentration
LI-COR, USA) CO2 : 0.11 ppm
Visibility sensor
± 10% (5 m–10 km)
(Model 6000, 2m Visibility 1 min
± 20% (10 m–20 km)
Belfort, USA)
Net radiometer (CNR4, Downward/upward
Short-wave: ± 5%
Kipp & Zonen, 40, 120, 220 m short-wave and <18 s
long-wave: ± 10%
Netherlands) long-wave radiation
a 15 vertical levels: 5,10, 20, 30, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, and 250 m.

2.2. Calculation Methods

Turbulence kinetic energy (TKE) and friction velocity (u* ) can be calculated according to
Equations (1) and (2), respectively:

1  02 
TKE = u + v02 + w02 (1)
2

2 1/4
 
2
u∗ = u0 w0 + v0 w0 (2)

where u0 , v0 , and w0 are fluctuating values respect to average values of the two horizontal wind speed
u and v and vertical wind speed w (m s−1 ).
The turbulent exchange coefficient Km (m2 s−1 ) can be calculated according to Equation (3):
κu∗ z
Km =   (3)
ϕm Lz

where κ is the von Karman’s constant and has a value of 0.4; z is the reference level and the observational
height (m); L is the Obukhov length (m); z/L represents the stability factor; ϕm is a dimensionless wind
shear [43,44].
The sensible heat (Hc , w m−2 ) is calculated as follows:

Hc = ρcp w0 T0 = −ρcp u∗ T∗ (4)

where ρ is the air density (kg m−3 ), cp the specific heat of air (J kg−1 K−1 ), T0 is the fluctuation of air
temperature (K), and T* is temperature scale (K).
The parameter critical turbulent exchange coefficient (Kc) (m2 s−1 ) can be calculated according to
Equations (5) and (6), for a shallow fog and a deep fog, respectively:

Kc = 1.38[αβ(p, T )C0 ]1/2 H3/2 (5)

Kc = 1.41[αβ(p, T )Ct ]1/2 H3/2 (6)

Atmosphere 2020, 11, 480 5 of 18

where α = 0.062 is the gravitational settling parameter for radiation fog [45], β(p,T)C0 (z) is the
condensation rate per unit mass due to cooling of the air, and C0 (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 as:
622Lv es (T )
β(p, T ) = (7)
Rv T2 p
where Lv is the latent heat of vaporization of water (J kg−1 ), es represents the saturation vapor pressure
(h pa), and Rv the gas constant for vapor (J kg−1 K−1 ). Ct is the cooling rate at the fog top (K h−1 ), H is
the depth of the fog bank (m). Moreover, the parameter can be used as a threshold; when Km < Kc ,
the fog can persist at a fog mature phase [46].
The convective velocity scale (w∗ ) represents the scaling velocity for the convective boundary
layer, and can be calculated according to Equation (8) as follows:
!1/3 !1/3
gzi   gzi
w∗ = w0 θ 0 = − u∗ θ∗ (8)
s
θ θ

where zi is the convective boundary layer height and can be estimated using the revised Carson scheme,
by setting the initial boundary layer height to 150 m according to the potential temperature profile.
The term (g/θ)(w0 θ0 ) represents the buoyancy heat flux.
The liner correlation between fog thickness and u* is obtained in Román-Cascón et al. [33] and the
relationship is:
H = 1369 × u∗ − 28 (9)

3. Results and Discussions

There are no unified fog criteria in previous literature, and different criteria have been used
to identify fog events [47–49]. In this study, the criteria of fog were as follows [40]: (1) RH ≥ 90%,
(2) precipitation less than 1 mm per hour to eliminate the effect of precipitation (precipitation fog is out
of our scope), and (3) the 10-min visibility ≤1 km. Moreover, we only focus on radiation fog in this
study, which is the primary and most important inland fog form. Therefore, the ‘fog’ in this study
refers to radiation fog, unless otherwise stated. All radiation fog events observed in 2016 were selected
to develop new parameterizations of the fog-top height and evaluate different methods. The selected
radiation fog events all formed in a nocturnal boundary layer and mainly due to radiative cooling
at the surface. The wind speed was always low, and the durations of fog were between 7 and 18 h,
which is similar to the statistical result in Liu et al. [50]. In this study, the fog-top height was estimated
using the relative humidity profile (not shown here), and the fog thickness was determined as the
height above where the value of RH was smaller than 90%, and under where the value of RH was
larger than 90%.

3.1. Fog-Top Height Estimations Using Turbulence Intensity

Previous studies [19] have shown that the whole fog life cycle is linked to the turbulence intensity.
Though the role of turbulence in the formation of radiation fog remains controversial, turbulence
within the fog layer is crucial for the exchange of material and energy, important for the development
of fog. Therefore, turbulence mixing is considered to be the main mechanism that influences the
fog-top height, and different described parameters related to turbulence intensity are used to estimate
the fog-top height. In this study, friction velocity (u* ), TKE, and variance of vertical velocity (σw 2 ) were
used to estimate the fog-top height, and the results were evaluated. The relationships between the
fog-top (H) height and u* , TKE, and σw 2 are shown in Figure 1a,b,c, respectively. A positive correlation
between H and u∗ is shown in Figure 1a, and the relationship

H =583.35 × u1.12
∗ (10)
Atmosphere 2020, 11, 480 6 of 18

can be obtained, with a coefficient of determination of R2 = 0.41. The result confirmed the hypothesis
that the turbulence intensity is crucial in the life cycle of fog and is closely linked to the fog-top height.
Although there is a low turbulence intensity during the fog formation phase, turbulence mixing leads
above air to be saturated conditions at the higher levels, and hence, to the development of fog bank, while
strong turbulence leads to the fog collapse [12]. Therefore, the range of u* in the scatterplot (Figure 1a)
can be interpreted as the minimum and maximum turbulence intensity required for fog formation and
dissipation, which are approximately 0.037 and 0.281 m s−1 , respectively. The minimum turbulence
intensity was slightly larger than that in Román-Cascón et al. [33], which were 0.025 and 0.030 m s−1
for different sites. The larger values of turbulence in this study were due to the sonic instrument
installation height being 40 m in this study, which was 3 m or 1.5 m in Román-Cascón et al. [33].
However, the result verified that there was a threshold value of turbulence that influenced the
formation of fog [51]. The maximum turbulence intensity in this study was much larger than those in
Román-Cascón et al. [33]; this discrepancy was due to the sonic instrument installation height and
the thickness of fog, with the thicker the fog, the stronger the turbulence the fog can endure [46].
The estimations with H = 1369 × u∗ − 28 (Equation (9)) obtained in Román-Cascón et al. [33] and similar
linear correlation H = 502 × u∗ − 4 obtained in this study are also shown in Figure 1a. The results
show there are obvious overestimations with H = 1369 × u∗ − 28, compared with the observations in
Tianjin. There were few discrepancies between the estimations with H = 502 × u∗ − 4 and Equation (10),
though the coefficient of determination of estimations with Equation (10) was a little higher. Therefore,
Equation (10) obtained in this study seemed to be more appropriate to estimate the fog-top height
in Tianjin, compared with Equation (9). The main reason of the discrepancies between the result
in this study and Román-Cascón et al. [33] was that the relationship between fog-top height and u*
in Román-Cascón et al. [33] was obtained by fitting seven or four mean values of u* associated to
each discrete value of fog thickness, while the relationship in this study was obtained from a large
amount of original values. In addition, the vertical resolution in Román-Cascón et al. [33] was low,
seven layers under 200 m or four layers under 100 m, which led to largely misestimate the actual
fog-top height. The vertical resolution in this study was higher, 15 layers under 250 m and eight
layers under 100 m, which is favorable for the accurate estimation of fog-top height. Therefore,
the relationship (Equation (10)) in this study was more appropriate to estimate the fog-top height,
which represents the effect of turbulence on fog-top height and can show the temporal variations
in the fog life cycle. However, there are some underestimations in large u* and overestimations for
weak turbulence, which may be due to the effect of other meteorological parameters and large-scale
atmospheric background fields.
Similar to the result of u* , the estimations of fog-top height with TKE are shown in Figure 1b,
and the relationship
H = 205.43 × (TKE)0.68 (11)

with a coefficient of determination of R2 = 0.44 was obtained, which indicates that TKE can be used to
estimate the fog-top height. Compared with the estimations of the fog-top height with u* and TKE,
the estimations of the fog-top height with σw 2 seem to be more agreed with the observations (Figure 1c).
An obvious correlation between H and σw 2 is shown in Figure 1c, and the relationship
 0.51
H = 420.10 × σ2w (12)

can be obtained, with a coefficient of determination of R2 = 0.53. The results suggest that the variance of
vertical velocity (σw 2 ) is a more appropriate turbulence intensity indicator than the first two parameters,
which can be used to estimate the fog-top height. Moreover, the minimum value of σw 2 is 0.0031 m2 s−2
in this study, that is, in the threshold range for fog formation (0.002–0.005 m2 s−2 ) obtained in Price [51],
can prove the existence of a threshold value of turbulence for fog formation. Though there are some
biases between estimations with turbulence and observations, the results in this section suggest the
possibility of using turbulence to estimate the fog-top height.
 Atmosphere 2020, 11, x FOR PEER REVIEW 7 of 19

Though there are some biases between estimations with turbulence and observations, the results in
this section
Atmosphere 2020, 11,suggest
480 the possibility of using turbulence to estimate the fog-top height. 7 of 18

Figure 1. 1. (a)
Figure Scatterplots
(a) Scatterplots of observations
of observations of height
of fog-top fog-top height
against againstvelocity.
the friction the friction velocity.
The bottom
The bottom
solid linesolid linethe
denotes denotes
relationship H = 583.35 × u1.12
the relationship *H = 583.35
, the × u1.12
bottom , theline
∗dashed bottom dashed
denotes line denotes
the relationship
H = 502 × 𝑢∗ –H4,=
the relationship and the upper
502×u ∗ − 4, solid line upper
and the denotessolid
the relationship
line denotesH = the × 𝑢∗ – 28. (b)H
1369relationship Scatterplots
= 1369×uof∗ − 28.
0.68
observationsof
(b) Scatterplots of observations
fog-top height against TKE.height
of fog-top The solid line denotes
against TKE.the
Therelationship
solid lineHdenotes
= 205.43 the
× TKE .
relationship
(c) Scatterplots of
0.68observations of fog-top height against the variance of vertical velocity (σw2). The
H = 205.43 × (TKE) . (c) Scatterplots of observations of fog-top height against the variance of vertical
0.64
solid line denotes the relationship H = 568.18 × σ2w .  0.64
velocity (σw 2 ). The solid line denotes the relationship H = 568.18 × σ2w .

3.2. Fog-Top Height Estimations using Radiative Cooling

Radiative cooling and heating impact the liquid water balance of fog, and therefore, play an
important role in the evolution of fog. There are two main radiative processes that affect the evolution of
fog: one is the longwave radiative cooling at the fog top, which produces liquid water by condensation;
the other one is the heating of the ground by absorption of solar radiation, which can cause a sensible
Atmosphere 2020, 11, 480 8 of 18

heat transfer to the fog, causing the fog dissipation [45]. Therefore, radiation fog often occurs at
night, and dissipates a few hours after sunrise due to the solar radiation and turbulence. Therefore,
condensation rate related to radiative cooling should to be closely related to fog-top height. However,
the aim of this study was to estimate fog-top height only using the surface measurements, and previous
result confirmed that an average or surface cooling rate can be used to replace the cooling at the fog
top [46]. The condensation rate near the surface was used to estimate the fog-top height in this study;
the result is shown in Figure 2a. The result shows that the fog-top height seems to be less related to
the condensation rate near the surface, though fog optical depth is closely related to the condensation
rate. Moreover, the relationship between the difference of fog-top height and condensation rate is
shown in Figure 2b. The results show that the fog-top height increases (difference of fog-top height is
larger than zero) when the condensation rate is larger than zero, while fog-top height decreases when
condensation rate is smaller than zero. However, there are some cases where those are opposite to
the conclusion that radiative cooling supports the development of fog. The main reason for this may
be radiative heating and the difference of condensation rate between surface and fog top. Moreover,
the result indicates that turbulence mixing seems to be more important for the development of fog-top
Atmosphere 2020, 11, x FOR PEER REVIEW 9 of 19
height, though the condensation rate is crucial to the formation of fog and fog optical depth.

Figure 2. (a) 2.
Figure Scatterplots of observations
(a) Scatterplots of fog-top
of observations height against
of fog-top the condensation
height against rate. (b)rate.
the condensation Scatterplots
(b)
Scatterplotsof
of observations of the
observations
differenceofof
thefog-top
difference of fog-top
height height
against against the condensation
the condensation rate. rate.

Previous
Previous studies
studies show
show thatthe
that thefog-top
fog-top height
height is
is influenced
influencedbybythethebuoyancy
buoyancygenerated by by
generated
radiative cooling at the fog top, other than the turbulence. Therefore, sensible flux was
radiative cooling at the fog top, other than the turbulence. Therefore, sensible flux was chosen aschosen as the
scalingparameter
the scaling parameter to
to represent
represent thetheintensity
intensityofofbuoyancy
buoyancy and used
and usedto estimate the fog-top
to estimate height.
the fog-top height.
Compared with the estimations with turbulence intensity, the relationship between the fog-top
height and sensible flux is complicated, and the scatter points are discrete, especially when the values
of sensible flux are near zero (not shown here). It is obvious that the relationship between the fog-top
height and sensible flux can be divided into two phases: the positive and negative sensible flux.
Therefore, the relationship between the fog-top heights and absolute values of sensible flux is
developed, and Equation (13) is obtained, with a coefficient of determination of R2 = 0.40 (Figure 3a):
Atmosphere 2020, 11, 480 9 of 18

Compared with the estimations with turbulence intensity, the relationship between the fog-top height
and sensible flux is complicated, and the scatter points are discrete, especially when the values of
sensible flux are near zero (not shown here). It is obvious that the relationship between the fog-top
height and sensible flux can be divided into two phases: the positive and negative sensible flux.
Therefore, the relationship between the fog-top heights and absolute values of sensible flux is developed,
and Equation (13) is obtained, with a coefficient of determination of R2 = 0.40 (Figure 3a):

H = 29.49 × |Hc|0.45 (13)

Though the result indicates that the absolute values of sensible flux can be as a potential indicator
of the fog-top height, there are some discrepancies between estimations and observations, especially for
low sensible flux. The results are similar to the results in Román-Cascón et al. [33], with significantly
dispersed distribution and larger error bars. Therefore, sensible flux may be not the appropriate
indicator
Atmosphere to represent
2020, 11, x FORthe
PEERbuoyancy
REVIEW and other described parameters should be sought out to represent 10 of 19
the buoyancy induced by radiative cooling.
Oncefog
Once fogoccur,
occur,fogfog stratification
stratification underwent
underwent a transition
a transition from thermally
from thermally stable tostable
weaklytounstable,
weakly
unstable,
and the fogand the can
layer fog be
layer can betodeemed
deemed to be well-mixed.
be well-mixed. To represent
To represent the of
the intensity intensity
buoyancyof buoyancy
induced
induced
by by cooling
radiative radiativeatcooling
the fog at the
top, fog top,
a new forma of
newtheform of the velocity
convective convective velocity
scale scale
(w∗ ) was (w* ) was
introduced.
introduced. It should to be noted that new w in this study is calculated according
It should to be noted that new w∗ in this study is* calculated according to Equation (8) with the absolute to Equation (8)
with the
value absolute
of sensible value
flux. Theofnewsensible flux. Thethe
w∗ represents new w* represents
buoyancy the buoyancy
and neglects and neglects
the sign, which the sign,
means whether
which
the means
sensible whether
flux the sensible
is negative flux it
or positive, is contributes
negative or to positive, it contributes
the mixing to the
of fog layer andmixing
dry air,ofand
fogleads
layer
and
to thedry air, and leads
development to the
of fog development of fog layer.
layer.
The relationships
The relationships between
between the the HH and
and new
new w w** are
are shown
shown in in Figure
Figure 3b3b and
and aapositive
positivecorrelation
correlation
between H and 𝑤 is observed, with the
between H and w∗ ∗is observed, with the relationship relationship
H = 328.33 × w1.34 (14)
H =328.33 × w1.34
∗
* (14)
being obtained, with a coefficient of determination of R2 = 0.44. The results confirm that buoyancy
being
inducedobtained, with cooling
by radiative a coefficient of determination
is a contributor of R2 = 0.44. with
to fog development, The results confirm
the mixing that buoyancy
between a fog layer
induced
and dry air. There are some bias between estimations and observations, which is because thefog
by radiative cooling is a contributor to fog development, with the mixing between a foglayer
bank
and
neardry
theair. Thereisare
surface notsome biaswell-mixed
always between estimations and observations,
fog; therefore, which is cooling
the effect of radiative becauseat
thethefog bank
fog top
cannot be represented by the new 𝑤∗ . However, compared with sensible flux, the new 𝑤∗ is a more
near the surface is not always well-mixed fog; therefore, the effect of radiative cooling at the fog top
cannot be represented
appropriate indicator of the new winduced
bybuoyancy ∗ . However, compared
by radiative with at
cooling sensible
the fogflux,
top, the w∗ be
newcan
which is aused
more to
appropriate
calculate theindicator of buoyancy induced by radiative cooling at the fog top, which can be used to
fog-top height.
calculate the fog-top height.

Figure 3. Cont.
Atmosphere 2020, 11, 480 10 of 18

Figure
Figure 3. (a)Scatterplots
3. (a) Scatterplotsofofobservations
observationsofof
fogfog thickness
thickness against
against the the absolute
absolute values
values of sensible
of sensible flux.
0.45
flux. The solid line denotes the relationship H = 29.49×|Hc| 0.45 . (b) Scatterplots of observations of fog
The solid line denotes the relationship H = 29.49 × |Hc| . (b) Scatterplots of observations of fog
thickness against the convective velocity scale. The solid line denotes the relationship H =328.33 × w1.34 1.34 .
thickness against the convective velocity scale. The solid line denotes the relationship H = 328.33 × w*∗ .
3.3. Fog-Top Height Estimations Using Turbulence and Radiative Cooling
BasedAtmosphere
on the budget of liquid
2020, 11, x FOR water content (LWC), the asymptotic LWC distribution11was
PEER REVIEW of 19 obtained,

which is the result of the comprehensive effect among radiative cooling, droplet gravitational settling,
3.3. Fog-Top Height Estimations Using Turbulence and Radiative Cooling
and turbulence mixing in the liquid water budget of radiation fog. Based on the relationship between
Based on the budget of liquid water content (LWC), the asymptotic LWC distribution was
turbulenceobtained,
exchange coefficient and the fog thickness (Equation (6)), a new estimation method was
which is the result of the comprehensive effect among radiative cooling, droplet
developed:gravitational settling, and turbulence mixing in the liquid water budget of radiation fog. Based on
the relationship between turbulence exchange coefficient ! 13 fog thickness (Equation (6)), a new
2 and the
km
estimation method was developed: Hmod = a (15)
β(p, T )C1 t
k2m 3
where ‘a’ represents the coefficient constant Hmod = a the result is shown in Figure 4. The(15)
and relationship
β p,T Ct
 2  1
km ‘a’3 represents the coefficient constant and the result is shown in Figure 4. The2relationship H =
H = 1.45 βwhere
(p,T )C
+1 35 was obtained, with a coefficient of determination of R = 0.26, and the scatter
t 2
km 3
points were extremely
1.45
β p,T Ct
+ discrete. Obvious
35 was obtained, with aunderestimations were of
coefficient of determination observed; thus,
R2 = 0.26, and this method
the scatter points was not
appropriate to estimate
were extremelythe fog thickness.
discrete. The large biaswere
Obvious underestimations between the estimations
observed; and observations
thus, this method was not may
be becauseappropriate
Equationto(15) estimate
was the fog thickness.
obtained from Thethelarge bias between
balance conditionthe estimations
at the fog and observations
mature phase, which is
may be because Equation (15) was obtained from the balance condition at the fog mature phase,
not suitable for estimations of the whole fog life cycle. In conclusion, there are two main causes of
which is not suitable for estimations of the whole fog life cycle. In conclusion, there are two main
the poor estimations,
causes of the poor one estimations,
is that theoneasymptotic
is that the formulation is only suitable
asymptotic formulation for the
is only suitable for beginning
the of
dissipationbeginning
phase, of and the other cause is that the relationship between condensation
dissipation phase, and the other cause is that the relationship between condensation rate and the
rate and
fog-top height the fog-top height
is complicated is complicated
and influencedand byinfluenced by other
other factors, factors,
such as such as radiative
radiative heating
heating and droplet
and droplet gravitational settling.
gravitational settling.

Figure 4. Scatterplots
Figure 4. Scatterplots of observations
of observations of of
fogfogthickness
thickness against
againstthethe
estimations with Equation
estimations (15).
with Equation (15).
Previous studies have shown that the development of fog can be attributed to turbulence mixing
and radiative cooling at the fog top. In brief, the life cycle of fog depends mainly on the balance
between radiative cooling and turbulence. Moreover, the results in Section 3.1 and 3.2 suggest that
σw2 and 𝑤∗ , with the same dimension, are the most appropriate parameters to represent the intensity
of turbulence and buoyancy induced by radiative cooling, which are closely related to the fog-top
Atmosphere 2020, 11, 480 11 of 18

Previous studies have shown that the development of fog can be attributed to turbulence mixing
and radiative cooling at the fog top. In brief, the life cycle of fog depends mainly on the balance
between radiative cooling and turbulence. Moreover, the results in Sections 3.1 and 3.2 suggest that
σw 2 and w∗ , with the same dimension, are the most appropriate parameters to represent the intensity
of turbulence and buoyancy induced by radiative cooling, which are closely related to the fog-top
height. Therefore, the fog-top height is estimated using the comprehensive parameters that include the
two physics parameters. The relationship

H = 396.26×(σw + 0.1 × w∗ ) − 16 (16)

can be obtained, with a coefficient of determination of R2 = 0.55. Some improvements can be observed
(Figure 5), which confirms the previous conclusion that the fog life cycle is closely related to the
turbulence intensity and radiative cooling. In conclusion, Equation (16) can be used to quantitatively
Atmosphere 2020, 11, x FOR PEER REVIEW 12 of 19
estimate the fog-top height, using only the surface measurements. However, the estimation with
Equation (16) cannot
can be parameters estimate the
to estimate the fog-top
fog-top height.
height perfectly, the parameters
More scaling scaling parameters
represent σwthe
2 and w also
turbulence
∗
can be parameters to estimate the fog-top height. More scaling parameters represent the
and radiation and different combinations of these scaling parameters should be found to estimate the turbulence
and radiation
fog-top height. and different more
Moreover, combinations
radiationoffog
these scaling
events are parameters
required to should
be usedbetofound to estimate
estimate the
the fog-top
fog-top height. Moreover,
height and verify the results. more radiation fog events are required to be used to estimate the fog-top
height and verify the results.

Figure 5.
Figure 5. Scatterplots
Scatterplotsofofobservations
observationsofoffog-top
fog-top height
height against
against thethe variance
variance of vertical
of vertical velocity
velocity and and
the
the new
new w*. The
w* . The solid
solid lineline denotes
denotes the the
relationship H =H396.26
relationship = 396.26
×(σw + w0.1
× (σ + 0.1
× w×* ) w
−*)16.
– 16.

3.4.
3.4. Fog-Top
Fog-Top Height
Height Estimation
Estimation through
throughTemperature
TemperatureConvergence
Convergence(TC
(TCMethod)
Method)
The
The radiative
radiativecooling
coolingnear
nearthethesurface cancan
surface leadlead
to saturated air conditions
to saturated at theat
air conditions ground, and hence,
the ground, and
to
hence, to the formation of fog droplets, and it may promote condensation in a supersaturatedlayer
the formation of fog droplets, and it may promote condensation in a supersaturated surface surfaceof
sufficient depth, leading
layer of sufficient depth,toleading
fog formation. Once the fog
to fog formation. firstthe
Once forms
fog at theforms
first ground,at the
the fog stratification
ground, the fog
undergoes
stratification undergoes a transition from thermally stable to weakly unstable [52]. Thewithin
a transition from thermally stable to weakly unstable [52]. The turbulent mixing the
turbulent
fog causes the homogenization of the layer, and the mixing causes the convergence
mixing within the fog causes the homogenization of the layer, and the mixing causes the convergence of temperatures,
which leads the temperature
of temperatures, which leadsofthe different heightsof
temperature todifferent
be a constant
heights[11].toPrevious studies[11].
be a constant have shown
Previous
that the fog thickness can be estimated through the vertical temperature profile
studies have shown that the fog thickness can be estimated through the vertical temperature profile [18,53]. Considering
the fog toConsidering
[53,18]. be present atthea certain
fog toheight, when at
be present theadifference between
certain height, the potential
when temperature
the difference between (θz )the
at
the height and surface level (θs ) is less than a threshold value (θ ):
potential temperature (θz) at the height and surface level (θs) is less than a threshold value (θT):
T

|4θ| | z<| <θθ

|△θ|==|θ|θs s−– θ-zθ TT (17)
(17)
The fog-top height is estimated to be the maximum height under where this criterion (in
Equation (17)) is always satisfied. With the use of potential temperature, the height-related influence
can be avoided. The determination of threshold value is crucial to estimate the fog thickness
accurately. The uncertainty between two level temperatures is 0.4 K, because the instrument-related
uncertainty of temperature measurements is 0.2 K. Small differences in potential temperatures are
allowed between different heights, and this strategy has been used in Price [52] and Román-Cascón
Atmosphere 2020, 11, 480 12 of 18

The fog-top height is estimated to be the maximum height under where this criterion
(in Equation (17)) is always satisfied. With the use of potential temperature, the height-related influence
can be avoided. The determination of threshold value is crucial to estimate the fog thickness accurately.
The uncertainty between two level temperatures is 0.4 K, because the instrument-related uncertainty of
temperature measurements is 0.2 K. Small differences in potential temperatures are allowed between
different heights, and this strategy has been used in Price [52] and Román-Cascón et al. [33], who used
threshold value settings of 0.8 K and 1.2 K, respectively. In this study, different threshold values were
used in the estimations, and the threshold value was set to 1.2 K based on the estimated results of the
fog thickness. The results of the fog thickness estimation versus the observations are shown in Figure 6a.
The estimation results are consistent with the observations, with a coefficient of determination of
R2 = 0.61, which indicates that the TC method based on the differences in potential temperature is
suitable for estimating the fog thickness, though there are some underestimations. Similar with the
Atmosphere 2020, 11, x FOR PEER REVIEW 13 of 19
theory mentioned in Section 3.1, though the fog stratification is weakly unstable during the fog life
cycle, atmospheric boundary structure may be different during the development phase and dissipation
during the two phases. Ultimately, the threshold value during fog development phase and
phase. Therefore, to estimate the fog thickness accurately, different threshold values were used, and the
dissipation phase was set to 1.2 K and 1 K, respectively. The relationship between the fog thickness
least square method was used to determine the θT during the two phases. Ultimately, the threshold
estimated by TC method and the observation are divided into two phases: the development phase
value during fog development phase and dissipation phase was set to 1.2 K and 1 K, respectively.
and the dissipation phase; the results are shown in Figure 6b. There was only a little improvement
The relationship between the fog thickness estimated by TC method and the observation are divided
with the segmentation of threshold values, with a coefficient of determination of R2 = 0.64; thus, the
into two phases: the development phase and the dissipation phase; the results are shown in Figure 6b.
same threshold value can be used for the whole fog life cycle. The TC method was based on the theory
There was only a little improvement with the segmentation of threshold values, with a coefficient of
that turbulent mixing within fog causes the temperature convergence, which results in the same value
determination of R2 = 0.64; thus, the same threshold value can be used for the whole fog life cycle.
of different heights. However, the physics mechanisms during fog development and dissipation
The TC method was based on the theory that turbulent mixing within fog causes the temperature
phase are different, with radiative cooling of the fog top and turbulence mixing leading to the increase
convergence, which results in the same value of different heights. However, the physics mechanisms
of fog top, while the fog dissipation is the comprehensive effect of turbulence, radiative cooling of
during fog development and dissipation phase are different, with radiative cooling of the fog top and
fog top, and entrainment of dry air. There may be a little difference in the degree of temperature
turbulence mixing leading to the increase of fog top, while the fog dissipation is the comprehensive
convergence during the two phases; however, this is not observed in this study, which suggests more
effect of turbulence, radiative cooling of fog top, and entrainment of dry air. There may be a little
fog events are required to improve the estimations.
difference in the degree of temperature convergence during the two phases; however, this is not
observed in this study, which suggests more fog events are required to improve the estimations.

Figure6.6.Scatterplots
Figure Scatterplotsofofobservations
observationsofoffog
fogthickness
thicknessagainst
againstthe
theestimation
estimationofoffog
fogthickness
thicknessbyby
temperature
temperature convergence (TC) method using in Equation (17), with (a) threshold value is setistoset
convergence (TC) method using in Equation (17), with (a) threshold value 1.2toK,
1.2
(b) threshold value during fog development phase and dissipation phase is set to 1.2 K andand
K, (b) threshold value during fog development phase and dissipation phase is set to 1.2 K 1 K,
1 respectively.
K, respectively.

The virtual potential temperature seems to be more appropriate, because this parameter includes
The virtual potential temperature seems to be more appropriate, because this parameter includes
the effect of water vapor and liquid water content on temperature, which is more reasonable to
the effect of water vapor and liquid water content on temperature, which is more reasonable to
describe the vertical distribution of temperature. Therefore, similar to in Equation (17), the difference
describe the vertical distribution of temperature. Therefore, similar to in Equation (17), the difference
between the virtual potential temperature at the height and surface level is used to determinate
between the virtual potential temperature at the height and surface level is used to determinate the
the fog top height. However, non-substantial differences are observed between using the potential
fog top height. However, non-substantial differences are observed between using the potential
temperature and virtual potential temperature (not shown here), which is consistent with the results of
temperature and virtual potential temperature (not shown here), which is consistent with the results
of Román-Cascón et al. [33]. Moreover, previous studies also show that although some improvements
have been made by using different thresholds for certain fog thickness estimations, it is always at the
expense of a worsening of the results for other fog thickness; therefore, the determination of the
threshold is still an intractable problem. Moreover, temperature convergence does not occur for
Atmosphere 2020, 11, 480 13 of 18

Román-Cascón et al. [33]. Moreover, previous studies also show that although some improvements
have been made by using different thresholds for certain fog thickness estimations, it is always at
the expense of a worsening of the results for other fog thickness; therefore, the determination of the
threshold is still an intractable problem. Moreover, temperature convergence does not occur for shallow
fog (not shown here), which is linked to strong thermal inversions and weak turbulence; therefore,
the TC method is not valid. The result suggests that although the TC method may be inadequate for
estimating the fog thickness of shallow fogs, it is still an appropriate method to estimate the deepen
fog thickness larger than 100 m, which is hazardous for transport.

3.5. Comparisons of Fog-Top Height Estimations with Different Methods

All parameterization schemes developed in this study and corresponding coefficients of
determination are shown in Table 2. The results confirm that σw 2 and w∗ , with the same dimension,
are the most appropriate indicators to represent the intensity of turbulence and buoyancy induced by
radiative cooling. Therefore, the combination of σw 2 and w∗ (Equation (16)) is the best parameterization
scheme developed in this study that can be used to estimate the fog-top height only using the surface
measurement. Moreover, the TC method can be used to accurately estimate the fog-top height using
the vertical profile data.

Table 2. All parameterization schemes developed in this study and corresponding coefficients
of determination.

Physical Mechanism Parameterization Scheme Coefficient of Determination (R2 )

H =583.35 × u1.12
∗ 0.41
Turbulent mixing H = 205.43 ×(TKE)0.68 0.44
 0.51
H = 420.10 × σ2w 0.53
H = 29.49×|Hc|0.45 0.40
Radiative cooling
H =328.33 × w1.34 ∗ 0.44
 2
1
km 3
H = 1.45 β(p,T)C + 35 0.26
Turbulent mixing and radiative cooling t
H = 396.26 × (σw + 0.1 × w* ) − 16 0.55
Temperature convergence |4θ| = |θs −θz | < 1 0.61

The fog-top height estimations of a thick fog and shallow fog with H = 1369 ×u* − 28 obtained in
Román-Cascón et al. [33] (Equation (9)), H =583.35 × u1.12∗ (Equation (10)), H = 396.26 × (σw + 0.1 × w* )
− 16 (Equation (16)) developed in this study, and the TC method (in Equation (17)) were compared and
are shown in Figure 7. The results show that there were obvious overestimations with the relationship
(Equation (9)) obtained in Román-Cascón et al. [33], for both thick and shallow fog, while the other
three estimation methods all perform better in thick fog than in shallow fog. The results in Section 3.4
has pointed out that the TC method is not suitable for estimating the fog-top height of shallow fog, in
which temperature convergence does not occur. Comparing the estimations in shallow fog with only u*
(Equation (10)) and the combination of u* and w* (Equation (16)), the estimations with the combination
of u* and w* are closer to the observations, since the new parametrization scheme (Equation (16)) in
this study considers the effect of radiative cooling and turbulence. Moreover, the new parametrization
scheme (Equation (16)) can approximately simulate the temporal variation of fog-top height in the
fog life cycle, though there are some underestimations. For thick fog, the TC method performed best,
especially during fog formation and development phase. However, the TC method is expensive and
not available in most areas. Moreover, to determine fog-top height with the TC method accurately,
the measure points under and above the fog top should be dense. Therefore, the TC method can only be
applied in a few sites, and the results strongly rely on the set of threshold value and vertical resolution
of profile data. Comparing the estimations in thick fog with only u* and the combination of u* and w* ,
there are better agreements between the estimations of the new parametrization scheme (Equation (16))
and observations, especially during the fog development phase. The new parametrization scheme
Atmosphere 2020, 11, 480 14 of 18

can also simulate the temporal variation of fog-top height during the fog dissipation phase, though
there are some underestimations. In conclusion, if only surface measurements are used to estimate the
fog-top height in Tianjin, the new parametrization scheme (Equation (16)) developed by combining
the effect of turbulence and radiative cooling in this study is more suitable compared with other
methods. However, the parametrization schemes, except the TC method, cannot used to simulate the
fog-top height during fog formation phase, during which the effect of turbulence on fog is ambiguous.
Therefore, more
Atmosphere 2020, 11,effort
x FOR should be made to improve our understandings of the effect of turbulence
PEER REVIEW on19
15 of
fog life cycle.

Figure 7. Observations of fog-top height, fog-top height estimations with H = 1369 × u* − 28

Figure 7.(9)),
(Equation Observations u1.12
H =583.35 ×of ∗ fog-top height,
(Equation (10)), H = 396.26
fog-top height w + 0.1 × wwith
estimations
× (σ * ) − 16H=1369 × u(16)),
(Equation * –28

(Equation (9)),
and the TC H = (in
method 583.35 × u1.12
Equation * (Equation
(17)) during a (10)),
shallowH = 396.26×
and thick (σ
fog w + 0.1
events ×
in w )
2018.
* – 16 (Equation (16)),
and the TC method (in Equation (17)) during a shallow and thick fog events in 2018.
4. Conclusions
4. Conclusions
In this study, different estimation methods for the fog-top height were developed and evaluated
using all radiation fog events obtained from a 255-m meteorological tower located in Tianjin in 2016.
In this study, different estimation methods for the fog-top height were developed and evaluated
The height of fog top was estimated using the relative humidity profile, and the threshold value of RH
using all radiation fog events obtained from a 255-m meteorological tower located in Tianjin in 2016.
is 90%.
The height of fog top was estimated using the relative humidity profile, and the threshold value of
Different indicators of turbulence intensity, friction velocity (u* ), TKE, and variance of vertical
RH is 90%. 2
velocity (σw ) were used to estimate the fog-top height, respectively, and the results were evaluated.
Different indicators of turbulence intensity, friction velocity (u*), TKE, and variance of vertical
The positive 2correlations between H and u∗ and TKE were obtained, with empirical parameterization
velocity (σw ) were used to estimate the fog-top height, respectively, and the results were evaluated.
scheme H =583.35 × u1.12 and H = 205.43 ×(TKE)0.68 . Compared with u∗ and TKE, the variance of
The positive correlations ∗ between H and 𝑢 ∗ and TKE were obtained, with empirical
vertical velocity (σw 2 ) was the most appropriate turbulence intensity indicator for estimating the
parameterization scheme H = 583.35 × u1.12 * and H = 205.43
 0.51 × TKE 0.68 . Compared with 𝑢∗ and TKE,
fog-top height,ofwith
the variance vertical velocity (σw2H
the relationship = 420.10
) was × σw appropriate
the most
2 and a higher coefficient
turbulence of determination
intensity indicator for
being obtained. The results confirm that turbulence
estimating the fog-top height, with the relationship H = 420.10 × σw intensity is crucial
2 0.51to the life cycle of fog and
and a higher coefficient of
isdetermination
closely linkedbeing
to theobtained.
fog-top height. Moreover,
The results confirmthe thatresult confirmed
turbulence the hypothesis
intensity is crucial tothat
thethere are
life cycle
threshold values of turbulence that can influence the fog formation and dissipation.
of fog and is closely linked to the fog-top height. Moreover, the result confirmed the hypothesis that
Radiative
there cooling
are threshold valuesimpacts the liquid
of turbulence thatwater balance ofthe
can influence fog
fogand therefore
formation andplays an important
dissipation.
role inRadiative
the evolution
coolingofimpacts
fog. Different
the liquidindicators
water balanceof radiative cooling,
of fog and surface
therefore playscondensation
an importantrate, role
and sensible flux were used to estimate the fog-top height. The result
in the evolution of fog. Different indicators of radiative cooling, surface condensation in this study show that rate,there
and
issensible
no obvious
flux relationship
were used tobetweenestimatesurface
the fog-topcondensation
height. The rateresult
and radiative
in this study cooling,
showwhich indicates
that there is no
that the condensation
obvious rate near surface
relationship between the surface is not suitable
condensation rate to
andestimate
radiative thecooling,
fog-top which
height,indicates
though itthatis
positively related to fog optical depth. The estimation results with sensible
the condensation rate near the surface is not suitable to estimate the fog-top height, though it is flux, which was chosen as
the described
positively parameter
related to fogrepresenting
optical depth. theThe buoyancy generated
estimation resultsby radiative
with sensible cooling at the fog
flux, which wastop, are
chosen
discrete, with large biases. Consider the effect of buoyancy on vertical mixing,
as the described parameter representing the buoyancy generated by radiative cooling at the fog top, the absolute values of
0.45
the sensible flux were used to associate with the fog-top height, and relationship
are discrete, with large biases. Consider the effect of buoyancy on vertical mixing, the absolute values H = 29.49×|Hc|
was 2 = 0.40), which indicates the effect
of obtained.
the sensibleHowever,
flux the coefficient
were used of todetermination
associate with was low
the (Rfog-top height, and relationship
0.45
H = 29.49 × |Hc| was obtained. However, the coefficient of determination was low (R2 = 0.40),
which indicates the effect of radiative cooling at the fog top on fog-top height cannot be represented
by sensible flux. To represent the buoyancy generated by radiative cooling at the fog top, a new form
of convective velocity scale (w* ), which was calculated with the absolute value of the sensible flux,
was introduced and used to estimate the fog-top height. An obvious positive correlation between H
Atmosphere 2020, 11, 480 15 of 18

of radiative cooling at the fog top on fog-top height cannot be represented by sensible flux. To represent
the buoyancy generated by radiative cooling at the fog top, a new form of convective velocity scale (w∗ ),
which was calculated with the absolute value of the sensible flux, was introduced and used to estimate
the fog-top height. An obvious positive correlation between H and new w∗ was observed, and the
relationship H =328.33 × w1.34 ∗ was obtained, with a higher coefficient of determination. The results
show that the new w∗ is more appropriate to represent the effect of radiative cooling at the fog top and
can be used to calculate the fog-top height.
To comprehensively consider the effect of turbulence and radiative cooling, different combinations
of different parameters were used to estimate the fog-top height. Based on the asymptotic LWC
distribution at the fog mature phase, which was the result of comprehensive effect among radiative
cooling, droplet gravitational settling, and turbulence mixing in the liquid water budget of radiation fog,
a new method (Equation (16)) was presented to estimate the fog thickness. However, the estimations
were extremely discrete, which indicates that this method is not appropriate to estimate the fog-top
height in the whole fog life cycle. σw 2 and w∗ , which are the most appropriate indicators of the
intensity of turbulence and buoyancy induced by radiative cooling, and are used to estimate the fog-top
height. The relationship H = 396.26 ×(σw + 0.1 ×w* ) − 16 was obtained, with a higher coefficient of
determination, which can be used to accurately estimate the fog-top height.
The fog thickness was also estimated through the vertical temperature profile, based on the
theory that the fog layer is homogeneous, and the temperature at different heights within the fog is
approximately constant, due to the turbulent mixing within fog. The fog-top height was estimated
based on Equation (17) and threshold value was set to 1.2 K. The results of fog thickness estimation
using the TC method were consistent with the observations, with a coefficient of determination of R2
= 0.61. Moreover, there was a little improvement with the segmentation of threshold values, with
a coefficient of determination of R2 = 0.64. The virtual potential temperature was also introduced
to determine the fog-top height, and few differences were observed between using the potential
temperature and virtual potential temperature. In addition, the determination of threshold value was
still an intractable problem and the TC method is not valid for shallow fog due to strong thermal
inversions and weak turbulence. In brief, the TC method based on the differences in the potential
temperature is suitable for estimating the fog thickness of deep fog.
This study compared the estimations of fog-top height with H = 1369 u* − 28 obtained in
Román-Cascón et al. [33], (Equation (9)), H =583.35 × u1.12 ∗ (Equation (10)), H = 396.26 × (σw + 0.1 ×
w* ) − 16 (Equation (16)) developed in this study, and the TC method (in Equation (17)) in a thick fog
and shallow fog, respectively. The results showed that TC method performs best in thick fog, though it
is not valid in shallow fog, which is not well-mixed. However, the TC method is expensive and not
available in most areas, and the results strongly rely on the set of threshold values, which are different
in different sites. Moreover, to determine fog-top height accurately, the measure points under and
above the fog top should be dense. Therefore, the TC method can only be applied in a few sites, and the
results strongly rely on the set of threshold and vertical resolution of profile data. There are better
agreements between the estimations of the new parameterization scheme (Equation (16)) developed
in this study and observations in both thick and thin fog, compared with the estimations of fog-top
height with relationship (Equation (9)) in Román-Cascón et al. [33] and with only u* (Equation (10)).
The results confirmed that the new parametrization scheme (Equation (16)) developed in this study,
considering the comprehensive effect of turbulence and radiative cooling, can be widely used to
estimate fog-top height only using the surface measurements.
It is not possible to state definitive conclusions after the analysis of several radiation fog cases at
one site; nevertheless, the main factors, turbulence intensity and radiative cooling, that influence the
fog-top height were determined. However, due to the limitation of observation height and vertical
resolution, more research is required to seek for more appropriate and common parameters that can be
used to estimate the fog-top height.
Atmosphere 2020, 11, 480 16 of 18

Author Contributions: T.J.: Conceptualization, Methodology, Formal analysis, Writing—Original Draft.

B.W.: Investigation, Writing—review and editing, Supervision. H.Z.: Conceptualization, Methodology. J.L.:
Data curation. All authors have read and agreed to the published version of the manuscript.
Funding: This research was jointly funded by the National Natural Science Foundation of China (41675018,
41675135, 41705045), the Natural Science Foundation of Tianjin (17JCYBJC23400), and the Bohai Rim Regional
Fund (QYXM201801).
Conflicts of Interest: The authors declare no conflict of interest.

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