Editorial Type:
Article
Article Type:
Research Article

Closed-Form Analytic Solution of Cloud Dissipation for a Mixed-Layer Model

Bengu Ozge Akyurek Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California

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Jan Kleissl Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California

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Print Publication:
01 Aug 2017
DOI:
https://doi.org/10.1175/JAS-D-16-0303.1
Page(s):
2525–2556
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Abstract

Stratocumulus clouds play an important role in climate cooling and are hard to predict using global climate and weather forecast models. Thus, previous studies in the literature use observations and numerical simulation tools, such as large-eddy simulation (LES), to solve the governing equations for the evolution of stratocumulus clouds. In contrast to the previous works, this work provides an analytic closed-form solution to the cloud thickness evolution of stratocumulus clouds in a mixed-layer model framework. With a focus on application over coastal lands, the diurnal cycle of cloud thickness and whether or not clouds dissipate are of particular interest. An analytic solution enables the sensitivity analysis of implicitly interdependent variables and extrema analysis of cloud variables that are hard to achieve using numerical solutions. In this work, the sensitivity of inversion height, cloud-base height, and cloud thickness with respect to initial and boundary conditions, such as Bowen ratio, subsidence, surface temperature, and initial inversion height, are studied. A critical initial cloud thickness value that can be dissipated pre- and postsunrise is provided. Furthermore, an extrema analysis is provided to obtain the minima and maxima of the inversion height and cloud thickness within 24 h. The proposed solution is validated against LES results under the same initial and boundary conditions.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Bengu Ozge Akyurek, bakyurek@ucsd.edu

Abstract

Stratocumulus clouds play an important role in climate cooling and are hard to predict using global climate and weather forecast models. Thus, previous studies in the literature use observations and numerical simulation tools, such as large-eddy simulation (LES), to solve the governing equations for the evolution of stratocumulus clouds. In contrast to the previous works, this work provides an analytic closed-form solution to the cloud thickness evolution of stratocumulus clouds in a mixed-layer model framework. With a focus on application over coastal lands, the diurnal cycle of cloud thickness and whether or not clouds dissipate are of particular interest. An analytic solution enables the sensitivity analysis of implicitly interdependent variables and extrema analysis of cloud variables that are hard to achieve using numerical solutions. In this work, the sensitivity of inversion height, cloud-base height, and cloud thickness with respect to initial and boundary conditions, such as Bowen ratio, subsidence, surface temperature, and initial inversion height, are studied. A critical initial cloud thickness value that can be dissipated pre- and postsunrise is provided. Furthermore, an extrema analysis is provided to obtain the minima and maxima of the inversion height and cloud thickness within 24 h. The proposed solution is validated against LES results under the same initial and boundary conditions.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Bengu Ozge Akyurek, bakyurek@ucsd.edu
Keywords: Convective clouds; Marine boundary layer; Mixed layer; Stratiform clouds; Surface fluxes; Single column models

1. Introduction

Stratocumulus clouds (Sc) cover 21% of Earth’s surface on average annually and have a relatively high albedo, resulting in a cooling contribution to climate (Klein and Hartmann 1993; Eastman and Warren 2014). Sc also impact photovoltaic (PV) generation output in coastal areas, such as Southern California (Jamaly et al. 2013). Sc are prevalent over the ocean and the coastline but less so inland, yet there are also studies focusing on continental Sc (e.g., Kollias and Albrecht 2000). Their global abundance and the increase in coastal populations make it important to accurately model and forecast their behavior. However, global forecast models fail to accurately represent and forecast Sc (Bony 2005).

Sc generally form under a strong inversion layer and the resulting boundary layer (BL) is spatially homogeneous and well mixed both day and night because of buoyant turbulence forcing from longwave cooling at the cloud top (Lilly 1968; Bretherton and Wyant 1997). In observational studies, it has been shown that Sc can also form during the day under decoupled conditions, especially for deeper BLs and stronger winds, temperature and moisture gradients, yet they are less prevalent than the well-mixed cases (Serpetzoglou et al. 2008; Rémillard et al. 2012). Mixed-layer models (MLMs) are therefore an appropriate tool and have been widely applied to Sc since the groundbreaking work of Lilly (1968). Many studies improved physical model components, such as entrainment (Stevens 2002; Fang et al. 2014; Caldwell et al. 2005), radiation (Larson et al. 2007; Duynkerke 1999), surface fluxes (Bretherton and Wyant 1997), and advection (Seager et al. 1995). MLMs are typically numerically integrated, validated against other numerical simulations, such as large-eddy simulation (LES), and applied to study specific cases or sensitivities in the Sc-topped BL [Stevens et al. (2005) and references within]. Numerical integration is required because the MLM integro-differential equations are often very complex with multiple feedback loops (Ghonima et al. 2016). However, to understand interdependencies between variables or sensitivity of the system to a parameter, multiple case studies and simulations need to be performed. Even then, hidden interdependencies or feedback effects may not be discovered using trial-and-error methods.

There are also studies that use analytic models to understand the underlying behaviors (van der Dussen et al. 2014; Duynkerke et al. 2004; Stevens 2002). However, these studies focus on the modeling of the physical phenomena with better analytic equations, rather than solving the time evolution of cloud variables analytically.

In this work, we build up a physical MLM with radiation, buoyancy flux, and surface schemes and use mathematical approximations to obtain a closed-form analytic solution to inversion height, cloud-base height, and ultimately cloud dissipation. The advantage of an analytic solution is that the dependencies and sensitivities are observable directly from equations. For example, and related to our application of solar forecasting over coastal lands, the dependence of cloud thickness on Bowen ratio can be directly inferred, given the initial conditions of the system. The temporal evolution of the system can be described without numerical approximations, and steady-state conditions or attraction points can be detected.

We provide sensitivity and extrema analysis for inversion height, cloud-base height, and cloud thickness, to infer how they depend on the initial and boundary conditions and understand when their minima and maxima occur during the diurnal cycle.

This paper is structured as follows. Section 2 provides a background on the models that constitute the system of equations. In section 3 the analytic solution is derived. In section 4, closed-form analytic solution is verified against LES and is shown to closely follow the numerical results. Section 5 contains detailed analysis on the sensitivity of inversion height, cloud-base height, and cloud thickness evolution in time with respect to the system parameters and initial conditions and the timing of their extrema during the diurnal cycle.

The radiation and surface models used in this work are similar to those of Ghonima et al. (2016), who use numerical time stepping to solve a similar single-column mixed-layer model. Even though the authors show that the MLM results are close to a more complex simulation method (LES), the underlying connections and interdependencies between the cloud variables and the initial conditions are not analyzed. Such an analysis using numerical solution techniques is impractical because of the vast number of variables in the solution space (as shown in Fig. 2 later), motivating our analytic solution to this problem.

2. Background

In this section, we define the models that approximate the physical processes. Consider a well-mixed single vertical column with a single cloud layer bounded by its base height and the inversion height . An illustration is shown in Fig. 1. We assume constant air density and constant values for the jumps at the inversion layer for total water mixing ratio and liquid potential temperature (Lilly 1968).

Fig. 1.
Fig. 1.

Simulation domain. The system is modeled as a single well-mixed air column. The stratocumulus cloud layer (gray) is bounded by the temperature inversion and cloud base. The liquid water content linearly increases with height in the cloud layer, and the resulting liquid water path is shown. The surface boundary conditions of LHF and SHF are also shown.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

The cloud thickness h is the primary parameter of interest, and its tendency can be defined as follows:
e1
We use the inversion tendency definition from Caldwell et al. (2005) and Duynkerke et al. (2004), where the inversion height changes with the entrainment parameter and the subsidence . Subsidence is further approximated by a divergence (Caldwell et al. 2005):
e2
The cloud-base height tendency expression (Ghonima et al. 2015) depends on the conserved variables of liquid potential temperature and total moisture :
e3
where and represent the gas constants for dry air and water vapor, respectively, is the latent heat of evaporation, is the specific heat, g is the gravitational acceleration, is the temperature at the cloud base, and is the Exner function evaluated at the cloud base. In the following sections, the inversion height and cloud-base height tendencies are derived based on the budget equations for heat and moisture.

a. Budget equations for conserved moisture and temperature variables

The MLM budget conservation equations are given for the liquid potential temperature and the total moisture as follows (Lilly 1968):
e4
The large-scale advection values of total moisture and liquid potential temperature are assumed to be zero throughout this work. While advection effects are important for the MBL over coastal lands, the advection terms complicate the integration of the equations and are left for future study. Here, and represent the average liquid potential temperature flux and average total moisture flux, respectively; represents the net radiation flux. Because of the well-mixed assumption, both conserved variables can be assumed to be independent of height. This forces the right-hand side of the equations to be also independent of height, resulting in a linear height dependency for the partial derivatives. Representing the partial derivatives as E and W, respectively, we can derive the full expressions using the boundary conditions at z = 0 and z = zi, as given in Bretherton and Wyant (1997):
e5
e6
The boundary conditions at the surface and inversion height are obtained as follows:
e7
e8
The final expressions for and tendencies are obtained as follows:
e9
e10

b. Radiation model

In this section we derive equations for the components of the net radiation flux and their attenuation through the cloud layer. Net radiation flux is decomposed into net longwave and net shortwave components:
e11

1) Liquid water path and optical depth

Both radiation terms are attenuated by an optical depth term designated as τ. This term depends on the total columnar liquid water content. We assume that the liquid water mixing ratio within the cloud increases linearly with height proportional to a constant , which can be calculated from thermodynamics or observations:
e12
The liquid water path (LWP) then becomes
e13
The optical depth τ is defined with respect to the optical depth at the cloud top, which is assumed to be zero, while is the optical depth at and below the cloud base:
e14
where is the density of water and is the effective droplet radius.

2) Longwave radiation

For the longwave radiation, we utilize the model in Larson et al. (2007), which assumes isothermal blackbody radiation and single scattering. The net radiative longwave flux is defined as
e15
where represents the optical depth scale for longwave radiation. The coefficients L and M are obtained by solving the second-order radiation differential equation in Goody (1995):
e16
e17
The coefficients are defined as follows:
e18
e19
e20
e21
Here, designates the single scattering albedo, and is the asymmetry factor. The , , and terms are blackbody radiation arising from , , and :
e22
where and designate the effective radiation temperatures of the cloud base and ground surface, respectively, and is the effective downwelling longwave radiative temperature of the column above the cloud top.

3) Shortwave radiation

We utilize the delta-Eddington approximation in Duynkerke et al. (2004) and Shettle and Weinman (1970) as a shortwave radiation model. Using the Eddington approximation, the diffuse radiance can be divided into a linear combination of a term independent of the solar zenith angle and a term dependent on solar zenith angle, yielding the analytic solution for the net shortwave radiation flux as follows:
e23
where designates the single scattering albedo for shortwave radiation, is the asymmetry factor, , A is the surface albedo, and k is the optical depth scale for shortwave radiation. Note that the incoming downward shortwave radiation is different from the net shortwave radiation at the cloud top as the net radiation includes radiation reflected from clouds and/or the ground surface. The coefficients are as follows:
e24
e25
e26
e27
e28
e29

c. Boundary conditions

To close the system of budget equations, the boundary conditions at the ground surface and inversion are needed. Entrainment at the top can be expressed as a function of the virtual potential temperature flux through a convective velocity scale defined as in Turton and Nicholls (1987), Caldwell et al. (2005), and Bretherton et al. (1999):
e30
where is a tuning parameter, is a reference virtual potential temperature, and Ri is the Richardson number. Combining the velocity scale equations and the Richardson number, we obtain
e31
Finally, we need the surface boundary conditions to close the system of equations. Surface fluxes of heat and water are connected to the net surface radiation through surface flux efficiency and the Bowen ratio β as follows (Ghonima et al. 2016):
e32
e33
A value of is applied in all simulations, while Bowen ratio is also constant for a particular simulation but will vary from simulation to simulation to investigate effects of soil moisture content.

Interdependencies of atmospheric variables are abundant as illustrated in Fig. 2 through an automatically generated dependency graph.

Fig. 2.
Fig. 2.

Dependency graph for all variables in the system. An arrow between two variables indicates a dependency, where the source of the arrow depends on the end of it (e.g., cloud thickness depends on inversion height).

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

3. Analytic closed-form solution

a. Inversion height tendency

The objective of this section is to obtain a closed-form solution for Eq. (2). This requires the entrainment velocity in Eq. (31), which depends on the virtual potential temperature flux . The virtual potential temperature flux depends on the surface heat fluxes as follows (Bretherton and Wyant 1997):
e34
From the surface to the cloud-base height, the coefficients and are used. For the cloud layer, spanning from the cloud-base height to the inversion height, and (Bretherton and Wyant 1997).
We start by scaling Eqs. (9) and (10) by and , respectively, and summing them:
e35
The left-hand side can also be expressed using Eq. (4):
e36
Equations (35) and (36) are equal to each other. We use the fact that the left side of both Eq. (35) and Eq. (36) are independent of z as a result of the well-mixed assumption to take the integral of both equations from z = 0 to an arbitrary z. Leaving the virtual potential temperature flux on the left side of the equation, the expression becomes
e37
Utilizing the same scaling operation as in Eq. (35) for the surface flux definitions from Eqs. (33) and (32),
e38
Substituting Eq. (38) into Eq. (37), we obtain
e39
Next, we integrate over the boundary layer depth to obtain the entrainment velocity in Eq. (31):
e40
Combining all terms on the left side, we obtain
e41
Substituting this result in Eq. (2), we obtain the inversion height tendency:
e42
where ζ coefficients are employed to simplify the equation. This is a nonlinear differential equation. Each net radiation term depends on the cloud thickness through the optical depth term. Furthermore, the columnar integral of the net radiation is called a Dawson function and is not an analytic function. Thus, an analytic solution requires approximations, as explained in section 3c.

b. Cloud thickness tendency

In addition to the inversion height tendency, the cloud thickness tendency requires the cloud-base height tendency. The solution strategy is to manipulate Eq. (3) into simpler variables analogous to the derivation of the inversion height tendency. The total moisture and liquid potential temperature tendencies appear in Eq. (3), but their tendencies given in Eqs. (9) and (10) depend on . Since inversion height is a complex expression itself, it would be difficult to solve the tendencies in their current form. To simplify the inversion height dependency, we multiply both differential equations by and add and , respectively, so that the resulting expressions are the derivatives of the product of the conserved variables with the inversion height:
eq1
This manipulation simplifies the right side of the differential equation by eliminating the inversion height term. Only its tendency remains. Using the inversion tendency [Eq. (2)] and the surface fluxes [Eqs. (32) and (33)], we obtain the following:
e43
e44
Note that we again need the net radiation expressions as in the inversion height expression to solve these differential equations. Finally, we use the cloud thickness tendency in Ghonima et al. (2015) to obtain the cloud thickness:
e45

c. Approximation of net radiation flux term

The net radiation flux appears in three forms: 1) surface , 2) inversion height , and 3) columnar average . We start with the approximations for the net longwave expressions at z = 0 and zi based on Eq. (15). Then we continue with the net shortwave expressions at z = 0 and zi. Finally, we approximate the columnar integral of net radiation as a linear combination of the net radiation at z = 0 and zi:
e46
e47
We simplify these expressions by neglecting higher-order (<−1) exponential optical depth terms as follows:
e48
e49
Even though this simplification is not required for the analytic solution, it simplifies the sensitivity analysis in section 5, and the error is less than 1%. Specifics for the error estimation are provided in appendix B(b).
To permit integration of net shortwave radiation into the cloud tendency expressions, we need to simplify solar zenith angle–dependent terms, since solar zenith angle changes with time in a sinusoidal shape, and complex nonlinear dependencies on such as in Eq. (29) or the third exponential in Eq. (23) are difficult to integrate. We use the following approximations for and in Eq. (29) and Eq. (24), with less than 2% and 1% error, respectively [see appendix B(c)]:
e50
To approximate the net shortwave radiation at the inversion height, we use its mathematical bounds for clear sky and infinite depth :
e51
The following approximation assumes an exponential dependence of net shortwave radiation on optical depth between these limits. The error of approximation is less than 6% [see appendix B(c)]:
e52
The net shortwave radiation at the surface is approximated in terms of the value at the inversion height scaled by a factor of attenuation depending on the optical depth, with an error of less than 7% [see appendix B(c)]:
e53
The columnar integral of net (shortwave and longwave) radiation flux can be approximated by a linear combination of net radiation values at the surface and inversion height with an error of 6% [see appendix B(d)]:
e54

d. Inversion height solution

Using the simplified, integrable approximations for the net radiation terms, a closed-form solution for the inversion height in Eq. (42) can be obtained. The columnar integral expression in Eq. (54) is employed to write Eq. (42) as a combination of net radiation terms at the surface and inversion height.
e55
e56

The solution strategy is to find all time-dependent variables inside the net radiation expressions and then solve the differential equation. For net longwave [Eq. (15)] the blackbody radiations are time dependent, and for net shortwave [Eqs. (52) and (53)] the solar zenith angle is time dependent. Furthermore, both radiation terms depend on the optical depth exponentially, and optical depth depends on the square of the cloud thickness given in Eq. (14). We use two approximations, which are further discussed in the following paragraphs: 1) Surface, cloud-base, and are constant over a 24-h period. As a result, the blackbody radiation terms are constants. 2) The change in cloud thickness h is negligible compared to the radiation length scales. This means that the effect of change in optical depth can be ignored only for radiation terms resulting in constant exponential optical depth terms. The actual cloud thickness solution is not a constant, and the actual time-dependent expression is presented in section 3e.

The first approximation can be supported as follows: 1) The model is only valid in overcast conditions. In overcast conditions, the daily range in surface temperature compared to the actual temperature is small, where the root-mean-square error (RMSE) of the constant temperature assumption is about 6%. 2) Surface and cloud-base temperatures follow similar diurnal patterns decreasing the error of the difference of blackbody radiation differences in Eqs. (16) and. (17). The RMSE of a constant blackbody difference assumption is about 4%. 3) The change in surface and cloud-base temperatures is largest near solar noon because of the peak in net shortwave irradiance at small solar zenith angle. However, at noon the net longwave radiation is only ~10% of the net shortwave radiation, and therefore the longwave balance does not contribute significantly to the overall net radiation. In conclusion, it is justifiable to approximate the differences in blackbody radiations as constant. To further reduce the error, rather than using the initial temperatures at midnight, the mean temperatures of the previous day are used.

For the second approximation, we need to investigate the exponential optical depth terms for net longwave and net shortwave expressions separately. Using the optical depth expression in Eq. (14), the exponent of the shortwave radiation can be written in the form: , where , and for k taken from Shettle and Weinman (1970), from Larson et al. (2007), and for in the range . The cloud thickness has to change on the order of to cause a significant change in the value of the exponent. The same notation for longwave yields the exponent in the form of with ; , resulting in an even smaller exponent value than the shortwave. For a cloud thickness of 250 m, the RMSE of keeping the exponential optical depth term constant with respect to a varying numerical optical depth solution is ~7%, as demonstrated in appendix B(e). The appendix also provides comparisons of daily model runs for constant and variable optical thickness under different Bowen ratios and values. The constant optical thickness results follow the variable optical thickness results, but differences increase for greater and smaller Bowen ratios. Large results in smaller scales, causing larger deviation from the constant optical thickness assumption, whereas smaller Bowen ratios delay cloud dissipation, resulting in the accumulation of errors over longer time horizons.

Using both approximations, the only time-dependent terms are the solar zenith angle terms and , and the inversion height tendency equation simplifies into
e57
The solution of differential equations of type is
e58
Assuming that , , and are the solutions of
e59
we can write the inversion height as follows:
e60
We use the solar zenith angle definition of , where , , lat is the local latitude, dec is the declination, and H is 12 h. We solve for the functions , , and using Eq. (58). The equations for a single day are given below (note that the general forms for multiple days are more complex and provided in appendix C.):
e61
e62
e63
The unit of these functions is seconds as a result of the time integration. Using these functions, the inversion height can be calculated for any t without numerical integrations that would be required in mixed-layer models. The functional forms as plotted in Fig. 3 directly reveal the following. At nighttime, when , and follow the same exponential trend as as in with additional oscillatory terms; therefore, and decrease during the night; also follows a negative exponential trend as a result of the negative sign of D. This means that the combined effect of all three functions causes the inversion height to change exponentially, and the exponent is the subsidence divergence parameter D. The value of D is hard to determine and difficult to measure; it typically assumes values on the order of . During the day, and increase, dominate over , and behave like a sigmoid function. The signs and magnitudes of the coefficients for the u functions also determine the trends for the cloud-base height as will be shown in section 5.
Fig. 3.
Fig. 3.

Time evolutions of the , , and functions that constitute the inversion height solution in Eq. (60). For subsidence, is used. Solar zenith angle is calculated for a latitude of 32.85°N and a Julian day of 196.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

Since the initial condition is scaled by , the analytic solution also shows that the e-folding time for the effect of the initial condition on the inversion height to approach zero is 1/D ~ 3 days. This means that the initial inversion height has a negligible effect on the solution in ~3 days. Furthermore, since all u functions have the same exponential trend of , converges within 5% of steady state in approximately 1/3D ~ 9 days. Once reaches the steady-state solution, the inversion height oscillates with sinusoids of periods 12 and 24 h. However, in practice this finding is largely irrelevant, as the synoptic meteorological conditions induce change over shorter time scales, rendering the mixed-layer model results not applicable.

e. Cloud thickness solution

To obtain the final cloud thickness expression, the cloud-base height expression is subtracted from the inversion height expression. In Eq. (3) only and tendencies vary in time, as the other terms are either constant or assumed constant because of the assumption of constant effective radiative temperature. We integrate Eq. (3) to obtain
e64
Assuming the change in to be small compared to its initial value, we use to linearize the expression and denote the coefficients of the time-varying terms as and :
e65
e66
In Eqs. (43) and (44) the and differentials are of the same functional form as the inversion height tendency. Thus, we manipulate the cloud-base height expressions to obtain the same format so that the total moisture and liquid potential temperature results can be substituted directly. To achieve this, we multiply Eq. (65) by and Eq. (66) by and sum them up to obtain the following:
e67
Scaling Eq. (65) by and subtracting it from Eq. (67) yields
e68
e69
The and differentials can be substituted from Eq. (43) and Eq. (44):
e70
where . Aggregating all constant coefficients in and , we obtain
e71
e72
Equation (70) depends only on the radiation terms, which already had been derived for the inversion height expression:
e73
where the constants are combined into , , and for convenience. Solving for the cloud-base height, we obtain
e74

And, finally, the cloud thickness is obtained from .

4. Validation against LES

We verify our solution against an LES, specifically the University of California, Los Angeles (UCLA)-LES (Stevens et al. 2005) on a 100 × 100 grid with 193 vertical levels. The horizontal resolution is 25 m, and the vertical resolution is 5 m, resulting in a domain of 2.5 km × 2.5 km × 960 m. The LES land surface model is a constant–Bowen ratio model that converts the incoming net radiation into sensible heat flux (SHF) and latent heat flux (LHF) according to Eqs. (32) and. (33). Initial conditions are from the Cloud Feedbacks Model Intercomparison Project (CFMIP)/Global Atmospheric System Studies (GASS) Intercomparison of Large-Eddy and Single-Column Models (CGILS) s12 case from Zhang et al. (2012), and initial profiles of and are shown in Fig. 4. The initial inversion height is 677 m, the initial cloud thickness is 238 m, and LWP is 72.4 g m−2. LES is initialized at 0300 LST. The results for the first hour of integrations are considered spinup time and not shown. The LES is run for 23 more hours, with samples taken every 20 s and averaged over 10 min. The LES inversion height, cloud-base height, inversion jumps for total moisture and liquid potential temperature, total moisture at the surface, and the effective radiative temperatures at the surface and the cloud base at 0400 LST serve as initial conditions for the analytic model. The value of is obtained from the LES longwave flux; the constant value of the exponential optical depth is calculated from the LES shortwave flux; and the subsidence divergence D is extracted from the LES using Eq. (2).

Fig. 4.
Fig. 4.

Initial profiles of liquid potential temperature and total water mixing ratio used for LES are taken from CGILS s12 data (thick solid line). Throughout this work, the initial profiles have been slightly modified to study various conditions, such as total moisture jump sensitivity in Fig. 6 (dotted and thick dashed lines) and initial inversion height sensitivity in Fig. 10 (thin solid and dashed lines).

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

The validation consisted of two sets of sensitivity experiments: 1) varying Bowen ratio and 2) varying jump at the inversion. Bowen ratio sensitivity results in Fig. 5 show agreement in the inversion height and cloud thickness time series and cloud dissipation time; the inversion height RMSE compared to LES is less than 1.5%; and the cloud thickness RMSE is less than 9%.

Fig. 5.
Fig. 5.

Bowen ratio sensitivity comparison between the analytic solution (dotted) and LESs (solid lines). Simulations are shown until the cloud dissipated for the largest Bowen ratio (i.e., until 0800 LST), because the analytic model is not valid in clear conditions. Inversion height is plotted for (a) and (b) . Cloud thickness h is plotted for (c) and (d) . Cloud thickness was also calculated using LCL for the and cases (dashed). The vertical dashed lines mark sunrise.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

At this time, the cloud-base height can also be compared against the lifting condensation level (LCL)—the level where the moisture in air is expected to saturate based on surface temperature and relative humidity (Bolton 1980). The LCL results for and in Fig. 5 agree with our cloud thickness formulation. The small difference is due to the approximate nature of the LCL formulation. We use the current formulation (Ghonima et al. 2015) for the rest of this paper, since it is integrated with the simulated MLM profiles. In contrast, the LCL formulation depends on near-surface temperature and relative humidity, which would require additional equations to obtain the closed-form results.

Both inversion height [Eq. (56)] and cloud-base height [Eq. (73)] were shown to depend on the inversion jump, including the total moisture jump . Furthermore, the inversion jump also affects entrainment and the turbulent fluxes through the boundary conditions [Eqs. (7) and (8)]. Even though multiple interdependent variables depend on , we are able to infer how affects the cloud thickness through our analytic solution. A detailed sensitivity analysis is presented in section 5, where the analytic solution suggests that the inversion height decreases and cloud thickness increases with smaller magnitude inversion jumps. For the validation, LES were run for Bowen ratios of 0.3 and 1, and the jump was varied by ∓0.5 g kg−1 (moister and drier air in the free troposphere), while keeping the boundary layer value at 9.43 g kg−1. Figure 6 shows that the analytic solution closely follows LES results in both trend and dissipation times. The inversion height RMSE compared to LES is again less than 1.5%, and the cloud thickness RMSE is less than 5%. The cloud dissipates only for , and the time of dissipation differs only by 5 min.

Fig. 6.
Fig. 6.

Sensitivity of (top) inversion height and (bottom) cloud thickness to total moisture jump at the inversion and initial cloud-base height, computed by the analytic solution (dashed) and the LES (solid lines). Results for Bowen ratios of (left) 0.3 and (right) 1 are shown. The original total moisture jump is gray, the reduced case is marked with a triangle, and the increased case is marked with a circle.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

5. Sensitivity analysis

a. Inversion height sensitivity

In section 3, we found that the inversion height tendency is a linear combination of three functions: u1, u2, and u3. The common property of these functions is that they generally increase exponentially, and the exponent is the subsidence divergence D. The evolution of the inversion height in time then depends on the coefficients of these functions given in Eq. (57), where the coefficients were kept in their compact forms to emphasize the linear combination of the three functions. Now we write out these coefficients and analyze their dependence on the initial and boundary conditions:
e75
e76
e77
e78
where, for convenience, we defined and , and remember that the exponential optical depth value was assumed to be constant in section 3d.
The turbulent flux coefficients in Eq. (34), , , , , and the convective surface efficiency of in Eqs. (32) and. (33) are obtained from Ghonima et al. (2016). The value of in Eq. (30) is from Turton and Nicholls (1987). Constants related to longwave radiation are from Larson et al. (2007) and shortwave radiation from Duynkerke et al. (2004). The coefficients become
e79
e80
e81
where aggregates the inversion jumps and has been defined for notational convenience. Units are as follows: is in K, and are in W−1 s−1 m3, and , , and are in m s−1.
Furthermore, is a scalar multiple of , so we can combine the and functions into a new function: . Combining the coefficients, the inversion height expression becomes
e82

The functions and are always positive. Thus, their combined tendency in time depends on the sign and magnitude of their coefficients. Defined in Eq. (81), is the common denominator of all coefficients, and its only negative term is the inversion jump in total moisture. However, given the strong temperature inversions for the stratocumulus-topped marine boundary layer, the total moisture jump would have to be unrealistically large to create a negative sign for . For example, if , the jump in total moisture would have to be to reverse the sign, but typical values of in the boundary layer are only . Thus, let us assume that .

For the optical depth exponentials and , a thinner cloud in the range [0, 200]-m thickness and a thicker cloud in the interval [200, 400] m are analyzed. For the thinner cloud, the optical depth variables are calculated as and and for the thicker cloud and . For the thin-cloud case, is positive for all Bowen ratios. For , there is a balance between the cloud-base minus and surface minus cloud-base blackbody radiation differences, slightly weighted toward the latter. The effect of the Bowen ratio is small as a result of its coefficient being small relative to the rest of the terms. A low radiative temperature for the cloud base favors positive , whereas high surface temperatures or favor negative . Using the standard atmospheric lapse rate of −6.5 K km−1 and assuming that effective radiative temperature equals air temperature, is always negative for the thin-cloud case. A negative means that the inversion height increases proportionally with the cloud-base temperature and inversely proportionally with the surface temperature. This sounds counterintuitive at first, as a large cloud-base temperature would lead to a higher upwelling longwave radiation and thus faster cooling. However, for a thin cloud with low optical depth, a large proportion of the downwelling longwave radiation from the cloud top reaches the surface and contributes to the sensible heat flux. This leads to a temperature increase in the boundary layer, increasing the turbulent fluxes and entrainment, which results in increased inversion height.

For the thick-cloud case, is positive for all Bowen ratios. The sign of depends on both the Bowen ratio and the radiative temperature balance. However, only the sign of is negative for all Bowen ratios; thus, the inversion height is inversely proportional to . The change in the direction of the effect for a thicker cloud emerges since the net longwave radiation is attenuated through the cloud’s high optical thickness, and only a negligible fraction reaches the surface.

To infer the combined effect of the oscillating terms in Eq. (82), we need the numerical values of and . For , Julian day of 196 and latitude 32.85°N, in magnitude on average. For typical effective radiative temperatures, it is physically impossible for the weighted summation to be negative. For example, for thin clouds, if , would have to be more than 560 K to cause a negative trend. Increasing Bowen ratio increases . Since is the dominant term, the combined trend increases with Bowen ratio. To show this, we fix , , , and D and vary the Bowen ratio and , as shown in Fig. 7.

Fig. 7.
Fig. 7.

Inversion height trend with respect to different Bowen ratios and Tsky for (top) the thin-cloud case and (bottom) the thick-cloud case . Other variables are fixed at , , , and . Simulations with (solid) and (dashed) are shown. Simulations for Bowen ratios between 0.1 and 5 are shown as thin gray lines for the case with with 0.2 increments. The dotted line is an admittedly unrealistic parameter choice to show a case where decreases. For this and all future graphs, a latitude equal to 32.85°N and Julian day of 196 were used for solar zenith angle calculations.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

Before sunrise such that the results represent only and all lines for both thin and thick clouds show a downward slope since the negative term of is dominant. This comes from the fact that includes only net longwave radiation terms. During the night, the net longwave radiation causes the boundary layer to cool, decreasing the inversion height. For the thin-cloud case, throughout the day, higher are associated with larger inversion height since the cloud’s optical thickness is small enough to admit net longwave radiation to the surface, which is converted into sensible heat flux and warms up the boundary layer. For the thick case, we see exactly the opposite, where higher leads to lower inversion heights. A large optical thickness attenuates the radiation before it reaches the land surface, which results in a cooler mixing layer and reduces surface turbulent fluxes.

Larger Bowen ratio causes to increase by a factor of , as is illustrated by the spacing between the gray lines of constant Bowen ratios for . Decreased moisture content in the soil associated with larger Bowen ratio increases the sensible heat flux, and the warming increases the inversion height. Since the ratio of radiation flux converted into turbulent fluxes is fixed through , the rates of the increases in the sensible heat flux and inversion height slow with increasing Bowen ratio, as reflected in the closer line spacing. Finally, the trend of the inversion height is also affected inversely by . A larger jump in potential temperature results in a smaller change in inversion height, whereas a larger jump in the magnitude of total water mixing ratio causes, in contrast, a greater change. This arises mainly from the fact that the turbulent fluxes are bounded by the negative of the inversion jumps at the inversion layer, as presented in Eqs. (7) and (8).

b. Cloud-base height sensitivity

For the sensitivity analysis of the cloud-base height from Eq. (74), it is enlightening to analyze as—similar to —its functional form is a linear combination of the three u functions in Eq. (73). The coefficients of , , and are
e83
e84
e85
with
e86
The values of and from Eq. (66) are calculated using as −211 590 m and 125 m K−1, respectively. The coefficients become
e87
e88
Units are as follows: and are in W−1 s−1 m4 and , , and are in m2 s−1. As for the inversion height [Eq. (82)], and are combined into :
e89
As in the inversion height analysis in section 5a, we consider two cases: thin clouds with and and thick clouds with and .
As with the coefficient of inversion height , for there is a balance between the surface minus cloud-base and cloud-base minus radiation differences. Using a lapse rate for a standard atmosphere, is negative for any Bowen ratio. The equation for is very similar to , except that for Bowen ratios for the thin-cloud case and for the thick-cloud case, changes sign and becomes positive. The combined trend depends on the and functions. Since and is much greater than , is the dominant term in the equality. Therefore the sign of changes with the sign of during daytime. To show this, similar to the inversion height analysis, the sensitivity of β and is shown in Fig. 8. The results for the daytime reflect sign and magnitude variation in with Bowen ratio. As expected, cloud-base height starts to increase during daytime at a Bowen ratio of 0.47, and the cloud-base height increases with increasing Bowen ratio. The sensitivity to is small because of the dominance of . Note that is only an intermediate expression that allows understanding cloud-base height trends, but it does not have a physical meaning; instead, Eq. (74) is considered now as follows:
e90
Using the values from the sensitivity analysis for and , and neglecting the terms, as is the dominant term during daytime:
e91
For the thin-cloud case, cloud-base height changes direction for , whereas, for the thick-cloud case, the direction change occurs for . The cloud-base height for different Bowen ratios is plotted in Fig. 9. This result shows that the cloud-base height trend changes direction depending on the Bowen ratio. Only a single is shown as the effect of is negligible. Increasing Bowen ratio causes a decrease in the latent heat flux and an increase in the sensible heat flux. The resulting drying and heating of the boundary layer increases the cloud-base height more than the inversion height. The cloud then dissipates faster with increasing Bowen ratios. The effect of the Bowen ratio decreases with increasing cloud optical thickness, as more radiation is absorbed or reflected within the cloud, resulting in smaller surface turbulent fluxes.
Fig. 8.
Fig. 8.

The (proxy for cloud-base height) time series for different Bowen ratios and . Cases for (solid) and (dashed) are shown. Other variables are fixed at , , , and .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

Fig. 9.
Fig. 9.

Cloud-base height time series for different Bowen ratios for (top) the thin-cloud case and (bottom) the thick-cloud case . The kink in the line occurs when the cloud-base height reaches the inversion height. The thin gray lines represent Bowen ratios from 0.1 to 5 with 0.2 increments. Other variables are fixed at , , , , , , and .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

Furthermore, note that the cloud-base height converges to a steady state:
e92
As shown earlier in this section, is positive, and changes from negative to positive with higher Bowen ratios. Therefore, larger Bowen ratios lead to larger steady-state cloud-base height.

c. Cloud thickness sensitivity

Using inversion height and cloud-base height trends, we can directly infer the cloud thickness sensitivity. In this section, we study the maximum initial cloud thickness that can be dissipated 1) before sunrise and 2) before sunset, or whether the cloud dissipates within 24 h at all.

1) Cloud thickness evolution

Figure 10 shows the thickness evolution of a cloud with 200-m initial thickness (top) and the resulting surface shortwave radiative fluxes, which are especially important from the solar forecasting aspect (bottom). The expected dissipation times using Eqs. (94) and. (96), later presented in this section, are tabulated in Table 1 for the cases in Fig. 10. The initial conditions used for the cases are also shown in Fig. 4.

Fig. 10.
Fig. 10.

(top) Cloud thickness evolution and (bottom) resulting surface shortwave radiative fluxes for different Bowen ratios, initial inversion heights, and subsidence values. High initial inversion height cases of are represented by dashed lines, cases of are represented by solid lines, and high subsidence cases are represented by thick lines. , , , , , and .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

Table 1.

Projected critical cloud thickness values for the cases in Fig. 10.

Table 1.

The dashed lines compare the effect of the Bowen ratio under normal subsidence for an initial inversion height of 1500 m. Under these conditions, Eq. (94) predicts that the cloud does not dissipate before sunrise, but Eq. (96) predicts that the cloud dissipates during the day if . As expected in the figure only does not dissipate. The lines with markers compare the same Bowen ratio scenarios for a lower initial inversion height of 500 m. Under these conditions, the cloud dissipates at about the same time as for the initial inversion height of 1500 m for and . Finally, the thick solid lines compare the effect of a strong subsidence for different initial inversion heights. As expected, stronger subsidence decreases cloud thickness. For strong subsidence , Eq. (94) predicts that, for , the cloud dissipates before sunrise, and Eq. (96) predicts that and dissipates during the day. The results validate the analytically derived conditions.

2) Dissipation before sunrise

The expression for dissipation at will be derived to determine the critical initial cloud thickness . In order for the cloud to dissipate, the initial cloud thickness must be less than . Before sunrise, and is negligible compared to such that
e93
Since cloud thickness either monotonically increases or decreases during the night, the critical cloud thickness would dissipate exactly at sunrise. We manipulate Eq. (93) to obtain the maximum allowable initial cloud thickness for the dissipation condition to be satisfied:
e94
We infer the following points from this condition. 1) Deeper boundary layers can dissipate thicker clouds. This comes from the fact that the contribution of the initial inversion height decreases in time through subsidence [Eq. (82)], whereas the initial cloud-base height is not multiplied by a subsidence term in Eq. (90). For example, if surface, cloud-base, and cloud-top radiative temperatures were the same such that the net longwave radiation and related coefficients are zero, the inversion height would still decrease in time as a result of subsidence, whereas cloud-base height would stay constant, as shown in Eq. (90). Thus, a larger inversion height subsides faster, resulting in more dissipation. The physical mechanism behind this is a faster subsidence rate as a result of a high inversion height. A faster subsidence rate results in a faster decrease in the cloud thickness. 2) Stronger subsidence dissipates thicker clouds. This is expected because of the faster decrease in the inversion height. The physical process is the same as the previous item. As the subsidence divergence increases, the subsidence rate of the cloud top also increases, resulting in a thinner cloud. 3) The cloud-base analysis showed that is proportional to the . Thus, a higher increases the maximum “dissipatable” cloud thickness before sunrise. However, a 1-K increase in only leads to approximately a 7.5-m increase in ; thus, has a smaller effect compared to the initial inversion height. The maximum dissipatable cloud thickness before sunrise for various , subsidence values, and initial inversion heights is presented in Fig. 11.
Fig. 11.
Fig. 11.

Maximum cloud thickness that can be dissipated by sunrise for different , initial inversion heights, and subsidence values. For normal subsidence values of , only a very thin cloud dissipates for the cases of zi(0) = 1000 and 1500 m. A zero result means that the cloud will not dissipate before sunrise for the given conditions. . Other variables are fixed at , , , and .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

3) Dissipation after sunrise

The second dissipation option materializes through a closing of the gap between inversion height and cloud-base height during the day because of a faster increase in cloud-base height. Previously, we observed that the dominant daytime term is . Dropping the terms, the cloud thickness expression can be written as follows:
e95
The maximum dissipatable or critical initial cloud thickness at sunset is obtained as follows:
e96
Using , , and , the critical thickness is obtained as follows:
e97
For the thin-cloud case, the critical-thickness expression becomes
e98
We infer the following points based on this condition: 1) The dominant term is the negative Bowen ratio dependent term in the numerator of Eq. (98). The value of increases with increasing Bowen ratio. However, the dependence on the Bowen ratio weakens as , consistent with section 5a. When the Bowen ratio increases, the positive feedback on the inversion height is weaker compared to the positive feedback on the cloud-base height, and the combined effect is an increase in . Since the net radiation flux that is converted into turbulent fluxes is constant, the sensitivity to Bowen ratio decreases for high Bowen ratios. 2) The dominant term changes sign with Bowen ratio, making dissipation impossible for small Bowen ratios and possible for larger Bowen ratios. Therefore, there is a region of the parameter space without dissipation before sunset. The Bowen ratio threshold that causes dissipation before sunset is inversely proportional to the initial inversion height. 3) Larger initial inversion heights enhance dissipation (first quadratic term). As explained in the previous section for dissipation before sunrise, the term that contains the initial inversion height decreases exponentially with subsidence, whereas the term with the initial cloud-base height persists in time. 4) Larger potential temperature inversion jumps and smaller magnitudes of total moisture inversion jumps enable dissipation, as they have been shown in section 5a to limit inversion height growth [Eq. (96)]. 5) Stronger subsidence enables dissipation, resulting directly from the decrease in inversion height. We plot the maximum cloud thickness that can be dissipated during the day for various Bowen ratios, subsidence values, and initial inversion heights in Fig. 12. Combining both night and day results, stronger subsidence, larger inversion height, and higher Bowen ratio enable dissipation and result in higher values.
Fig. 12.
Fig. 12.

Maximum cloud thickness that can be dissipated by sunset for different Bowen ratios, initial inversion heights, and subsidence values. A zero result means that the cloud will not dissipate before sunset for the given conditions. Horizontal lines result when the dissipatable cloud thickness reaches the initial inversion height. Parameters are , , , , , , , and .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

d. Extrema analysis

One of the advantages of an analytic solution is the ability to analyze derivatives for extrema determination. Extrema may be of interest, for example, in solar forecasting, where the thickest cloud conditions determine the maximum required amount of back-up generation. We performed extrema analysis on inversion height and cloud thickness to find out where their minima and maxima occur. To find the extrema points, we take the first and second derivative of the inversion height Eq. (82):
e99
e100
The extrema points are obtained by solving the following equation:
e101

The terms of the equality are plotted in Fig. 13. The extrema are close to sunrise and sunset. A greater initial inversion height leads to extrema moving toward midday. Furthermore, since , the effect of Bowen ratio and longwave and shortwave radiation terms is small compared to the initial inversion height.

Fig. 13.
Fig. 13.

Evaluation of the terms in Eq. (101) to find the inversion height extrema points.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

The second derivative determines whether these points are maxima or minima. We know that , , and . So the sign is determined by the sign of the second derivative of . The sign is positive until midday as the cosine of the solar zenith angle is increasing, and it is negative after midday. This means that the first extremum after sunrise is a minimum, and the second extrema before sunset is a maximum. This is an expected result, as during nighttime longwave cooling decreases the inversion height. A minimum occurs when after sunrise net shortwave radiation counteracts longwave cooling and eventually becomes dominant to increase . Similarly, in the afternoon, net shortwave radiation results in an increase in inversion height until longwave cooling dominates closer to sunset.

We continue with the cloud thickness expression. The cloud-base height was [Eq. (90)]
e102
The cloud thickness is obtained by subtracting cloud-base height in Eq. (102) from :
e103
The derivatives are
e104
e105
The cloud thickness derivative contains the inversion height derivative. We expand the inversion height expression from Eq. (82) for the first derivative as follows:
e106
Using the fact that , , and , the expression becomes
e107
Equation (107) states that the cloud thickness extrema points exist when the derivative of the inversion height is equal to the right-hand side (RHS) of the expression. We utilize the sensitivity results presented previously in this section for all coefficients to assess the extrema of cloud thickness. During nighttime for , the RHS is positive. Therefore, no extremum is present before sunrise, as the derivative of the inversion height was shown to be negative.

During daytime for large Bowen ratios that lead to the dissipation of the cloud before sunset, the RHS has a small negative value close to zero because of the quadratic term in the denominator, and . This means that the extrema, if they exist, are close to the extrema of the inversion height—right after sunrise and right before sunset—since the inversion height extrema are when the inversion height derivative is zero. Inversion height is increasing during the day, except between sunrise and the inversion height minimum and between the inversion height maximum and sunset. The extremum for cloud thickness must occur during these two intervals when the inversion height decreases and the RHS is negative.

When smaller Bowen ratios lead to persistence of the cloud, the RHS changes sign during the day from negative to positive. Since the initial sign of RHS is negative, the first extremum between sunrise and the minimum inversion height still exists; however, the other extremum shifts to the interval between the minimum inversion height and maximum inversion height, where the inversion height derivative is positive and matches the sign of the RHS.

We check for cloud thickness minima and maxima conditions for the inversion height extrema points. The sign of the following expression determines the extrema condition:
eq2
The sign depends on the initial inversion height. The sign is the opposite of the second derivative of the inversion height for small initial inversion heights and the same for large initial inversion heights. Therefore, for shallow boundary layers, the morning cloud thickness extremum is a maximum and occurs between sunrise and the minimum inversion height, and the afternoon extremum is a minimum. For higher boundary layers, the morning cloud thickness extremum is a minimum, and the afternoon extremum is a maximum. However, since larger inversion heights were shown to increase in section 5c, the afternoon maximum may not be observed, as the cloud may already have dissipated before the extremum depending on the Bowen ratio. Two examples are shown in Fig. 14, where and the only difference is the initial inversion height. As expected, the minimum and maximum switch intervals between the two examples.
Fig. 14.
Fig. 14.

Evaluation of the cloud thickness and its extrema values. Both cases have an initial cloud thickness of 200 m, but with zi(0) = (top) 500 and (bottom) 1500 m. The remaining parameters are the same (i.e., , , , , and ).

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

Combining this extrema result with the inversion height extrema, we have three scenarios for dissipation: 1) Cloud dissipation occurs before the minimum inversion height, and then no cloud thickness maximum occurs as, for example, for the high subsidence and cases in Fig. 10. 2) For larger initial inversion height, dissipation occurs after sunrise. 3) For small initial inversion height, dissipation occurs after sunrise and before sunset, with a maximum after sunrise depending on the Bowen ratio. Since the extrema analysis can only give the extrema of the cloud thickness and not the values at those points, it is possible that the cloud may dissipate before the minimum.

6. Conclusions

We have provided an analytic closed-form solution to the cloud thickness evolution of stratocumulus clouds in a mixed-layer model framework with a focus on application over coastal lands. This solution enabled sensitivity studies for inversion height, cloud-base height, and cloud thickness. While the parameter space was not explored exhaustively, for the typical base case chosen here, the following parameters influenced cloud thickness: Bowen ratio, subsidence, and initial inversion height. Critical initial cloud thicknesses, which can be dissipated pre- and postsunrise, were derived. Furthermore, we provided extrema analyses for inversion height and cloud thickness expressions to show when these variables reach their maximum and minimum values. Cloud dissipation can occur presunrise, but this situation is unlikely in practice, as such adverse conditions would likely have prevented cloud formation in the first place. If cloud does not dissipate presunrise, then a morning maximum and afternoon minimum in cloud thickness is observed. For large initial inversion heights, this observation is reversed as a morning minimum for cloud thickness. If this minimum is associated with a cloud thickness of zero, then the cloud deck breaks up during the day. If the minimum is associated with a cloud thickness greater than zero, then clouds are guaranteed to be maintained throughout the day.

The work in this paper will be used as the fundamental building block for future research on physical effect on cloud lifetime. In the present analysis that does not consider advection, clouds are sustained only for unrealistically small Bowen ratios. Even though our solution provided a good match against LES results, the models and assumptions that were required to solve the equations limit its application compared to the variable meteorological conditions in the real world. Examples include soil moisture change, precipitation, wind profiles, advection, and decoupling. Future work will include large-scale advection effects to analyze more realistic scenarios over coastal lands. We plan to create a multicolumn structure, where the columns are coupled through large-scale advection. The addition of moisture and cooling from the ocean is expected to increase the sustenance of the clouds over the coast, creating more realistic dissipation times. Furthermore, our current model does not consider the decoupling process. Even though decoupling occurs less frequently than the well-mixed conditions, multilayer clouds can form in deep boundary layers that can result in the vertical column deviating from well-mixed conditions. Decoupling can occur under stronger winds and stronger temperature and moisture gradients. We plan to extend our current model to study multiple cloud layers in a single column to observe the effects of decoupling on cloud dissipation.

Acknowledgments

The authors thank Dr. Mohamed Ghonima and Dr. Joel Norris for helpful discussion and comments for this work.

APPENDIX A

Nomenclature

Optical depth scale for longwave radiation

Surface turbulent efficiency

Entrainment tuning parameter

Cloud blackbody radiation

β

Bowen ratio

Surface blackbody radiation

Downwelling blackbody radiation above the cloud top

Specific heat constant

Total water vapor mixing ratio jump at the inversion

Liquid potential temperature jump at the inversion

Virtual potential temperature jump at the inversion

Net longwave radiation flux

Net radiation flux

Net shortwave radiation flux

Asymmetry factor for longwave radiation

Asymmetry factor for shortwave radiation

Latent heat of evaporation

Cosine of the solar zenith angle

Single-scattering albedo for longwave radiation

Single-scattering albedo for shortwave radiation

Liquid water mixing ratio

Total water vapor mixing ratio

Horizontal advection of water vapor mixing ratio

Total water vapor mixing ratio at the inversion

Gas constant for dry air

Effective droplet radius

Density of air

Density of water

Ri

Richardson number

Gas constant for moist air

Optical depth of the cloud

Cloud-base temperature

Effective cloud temperature

Liquid potential temperature

Horizontal advection of liquid potential temperature

Liquid potential temperature at the inversion

Virtual potential temperature

Virtual potential temperature reference

Surface temperature

Effective downwelling radiative temperature above the cloud top

Horizontal wind speed

Entrainment velocity

Mean turbulent flux for total water vapor mixing ratio

Subsidence velocity

Mean turbulent flux for liquid potential temperature

Mean turbulent flux for virtual potential temperature

Cloud-base height

Inversion height

D

Subsidence divergence

g

Gravitational acceleration

t

Time

APPENDIX B

Error Calculations

a. Error calculation methods and metrics

We use the RMSE definition as follows:
eb1
where, represents the ith point generated by our model, whereas is the ith point associated with a reference (usually the ground truth). The percentage error is defined by normalizing the RMSE by the mean reference value:
eb2
This percentage error model is only used for positive-valued variables.

Throughout this section, errors are assessed by comparing the results of our approximations with their original forms. For longwave calculations, we use the parameters from Larson et al. (2007), and for shortwave calculations we use the parameters from Duynkerke (1999). The errors are calculated numerically over a range of parameter values and then averaged. For inversion height, the interval of with 50-m resolution is used, whereas, for the cloud thickness, is used. Since optical thickness depends on , we use the interval with a resolution of . Longwave radiation depends on the surface, cloud-base, and cloud-top radiative temperatures. The standard atmosphere adiabatic lapse rate of −6.5 K m−1 allows for calculating the cloud temperatures and from the surface temperature. We use the interval with 1-K resolution. For solar zenith angle calculations, we use daytime with 100-s resolution.

b. Longwave error calculations for the approximations in Eqs. (46) and (47)

We set and for all longwave calculations. We performed more than 35 million experiments, where we calculated the percentage error of our approximation in Eqs. (49) and (48) with respect to the original formulation in Eqs. (47) and (46). The maximum RMSE observed is 0.53, and the maximum percentage error is 0.05%, while the mean percentage error is 0.03%. The maximum error is observed for Eq. (47), , , and .

c. Shortwave error calculations for the approximations in Eqs. (53) and (52)

We set and for all shortwave calculations. We calculate and [Eq. (50)] and compare against Eqs. (29) and. (24), respectively, to obtain the error performance. We performed more than 32 million experiments. The resulting mean percentage error is 2%. The percentage error of our approximation at the inversion height in Eq. (52) is 6%, and the maximum RMSE observed is 43 W m−2. The maximum error is observed for the case with , , and . For the shortwave approximation at the surface in Eq. (53), the percentage error is 7%, and the maximum RMSE is 44 W m−2. The maximum error is observed for the case with , , and .

d. Net radiation error calculations for the approximations in Eq. (54)

We use the same configurations as in sections B(b) and B(c). The mean percentage error of the columnar integral linear approximation in Eq. (54) is 6%, and the RMSE is 41 W m−2.

e. Constant assumption validations

The first assumption states that the surface, cloud-base, and variations are small compared to the actual temperature. Assuming a 30-K sinusoidal variation during the day from 265 to 295 K and back to 265 K, the RMSE of assuming a fixed temperature is only 4.5K, corresponding to less than a 2% error. The errors are amplified to 6% in the blackbody radiation calculation because of the fourth-order temperature dependence. The second assumption states that similar trends in temperature will decrease the effective error since the equations depend on the difference of the blackbody radiations. To verify this claim, we create a second temperature time series at a height of 1 km. Under the standard atmosphere assumption, the lapse rate is −6.5 K km−1 so the second temperature time series therefore varies sinusoidally from 256.5 to 286.5 K instead. The error of the difference of blackbody radiation drops to 5%. The third assumption states that the net shortwave radiation is greater than the net longwave radiation in the cloud layer during the day. Using the assumptions in the previous example, the average ratio of net shortwave to net longwave during the day is 8.7.

The constant-optical-depth assumption with calculated once using the initial thickness and then set constant is validated against a model run with a variable (real) optical depth that is solved iteratively at every minute. Different optical depth variations were created through two scenarios with different Bowen ratios of 0.2 and 1. Furthermore, since the optical depth depends on , we analyzed two scenarios with (Fig. B1, top) and (Fig. B1, bottom). The results in Fig. B1 show that the iterative and constant solutions are close in all Bowen ratio cases. In the case of , the distances between the solutions increase relative to the case. The main reason is that the LWP and the cloud optical depth are 5 times higher, resulting in the optical thickness scale that is 5 times smaller. The difference is largest for , since the cloud does not dissipate within 24 h, and the error accumulates over a longer time.

Fig. B1.
Fig. B1.

The constant-optical-depth solution (solid) follows the iterative-optical-depth solution (dashed) closely, showing that our constant optical depth assumption is valid. for the normal case, and for the dense case. The rest of the simulation parameters are , , , , , and .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-16-0303.1

APPENDIX C

Derivation of , , and Functions

We start the solution from :
ec1
ec2
We continue with . This function involves the solar zenith angle and can be written in a general form as . The solution is as follows:
ec3
Note that the expression in the maximum is a periodic expression. The nonzero region within a day spans from (sunrise) to (sunset). If t is greater than 1 day, then the solar-zenith-angle expression will be repeated. The general solution for a time t on following days is as follows:
ec4
ec5
We start with the solution of the integral with a general bound:
ec6
ec7
ec8
ec9
ec10
Using this result, we construct :
ec11

The resulting equation has three components: a constant, an oscillatory component with a periodicity of 24 h, and an exponentially decreasing component, which has subsidence as its exponent. As in the previous component, this means that the exponential term will vanish after roughly 10 days.

We continue with . We now deal with the square of the solar zenith angle. Taking the square of the expression, we obtain a very similar expression as before:
ec12
We again start with the solution of the integral with a general bound for :
ec13
ec14
ec15
ec16
Using the respective x values, we obtain the following:
ec17
This is similar to the previous result and results in three different components: a constant, an oscillatory, and an exponential component with subsidence as its exponent.

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  • Duynkerke, P. G., 1999: Turbulence, radiation and fog in Dutch stable boundary layers. Bound.-Layer Meteor., 90, 447–477, doi:10.1023/A:1026441904734.

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  • Duynkerke, P. G., and Coauthors, 2004: Observations and numerical simulations of the diurnal cycle of the EUROCS stratocumulus case. Quart. J. Roy. Meteor. Soc., 130, 3269–3296, doi:10.1256/qj.03.139.

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  • Eastman, R., and S. G. Warren, 2014: Diurnal cycles of cumulus, cumulonimbus, stratus, stratocumulus, and fog from surface observations over land and ocean. J. Climate, 27, 2386–2404, doi:10.1175/JCLI-D-13-00352.1.

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  • Fang, M., B. A. Albrecht, V. P. Ghate, and P. Kollias, 2014: Turbulence in continental stratocumulus, part I: External forcings and turbulence structures. Bound.-Layer Meteor., 150, 341–360, doi:10.1007/s10546-013-9873-3.

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  • Ghonima, M. S., T. Heus, J. R. Norris, and J. Kleissl, 2016: Factors controlling stratocumulus cloud lifetime over coastal land. J. Atmos. Sci., 73, 2961–2983, doi:10.1175/JAS-D-15-0228.1.

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  • Duynkerke, P. G., 1999: Turbulence, radiation and fog in Dutch stable boundary layers. Bound.-Layer Meteor., 90, 447–477, doi:10.1023/A:1026441904734.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Duynkerke, P. G., and Coauthors, 2004: Observations and numerical simulations of the diurnal cycle of the EUROCS stratocumulus case. Quart. J. Roy. Meteor. Soc., 130, 3269–3296, doi:10.1256/qj.03.139.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Eastman, R., and S. G. Warren, 2014: Diurnal cycles of cumulus, cumulonimbus, stratus, stratocumulus, and fog from surface observations over land and ocean. J. Climate, 27, 2386–2404, doi:10.1175/JCLI-D-13-00352.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Fang, M., B. A. Albrecht, V. P. Ghate, and P. Kollias, 2014: Turbulence in continental stratocumulus, part I: External forcings and turbulence structures. Bound.-Layer Meteor., 150, 341–360, doi:10.1007/s10546-013-9873-3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ghonima, M. S., J. R. Norris, T. Heus, and J. Kleissl, 2015: Reconciling and validating the cloud thickness and liquid water path tendencies proposed by R. Wood and J. J. van der Dussen et al. J. Atmos. Sci., 72, 2033–2040, doi:10.1175/JAS-D-14-0287.1.

    • Crossref
    • Export Citation

  • Ghonima, M. S., T. Heus, J. R. Norris, and J. Kleissl, 2016: Factors controlling stratocumulus cloud lifetime over coastal land. J. Atmos. Sci., 73, 2961–2983, doi:10.1175/JAS-D-15-0228.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Goody, R., 1995: Principles of Atmospheric Physics and Chemistry. Oxford University Press, 324 pp.

  • Jamaly, M., J. L. Bosch, and J. Kleissl, 2013: Aggregate ramp rates of distributed photovoltaic systems in San Diego county. IEEE Trans. Sustainable Energy, 4, 519–526, doi:10.1109/TSTE.2012.2201966.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Klein, S. A., and D. L. Hartmann, 1993: The seasonal cycle of low stratiform clouds. J. Climate, 6, 1587–1606, doi:10.1175/1520-0442(1993)006<1587:TSCOLS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation