## 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

*h*is the primary parameter of interest, and its tendency can be defined as follows:

*g*is the gravitational acceleration,

### a. Budget equations for conserved moisture and temperature variables

*E*and

*W*, respectively, we can derive the full expressions using the boundary conditions at

*z*= 0 and

*z*=

*z*

_{i}, as given in Bretherton and Wyant (1997):

### b. Radiation model

#### 1) Liquid water path and optical depth

*τ*. This term depends on the total columnar liquid water content. We assume that the liquid water mixing ratio

*τ*is defined with respect to the optical depth at the cloud top, which is assumed to be zero, while

#### 2) Longwave radiation

*L*and

*M*are obtained by solving the second-order radiation differential equation in Goody (1995):

#### 3) Shortwave radiation

*A*is the surface albedo, and

*k*is the optical depth scale for shortwave radiation. Note that the incoming downward shortwave radiation

### c. Boundary conditions

*β*as follows (Ghonima et al. 2016):

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

## 3. Analytic closed-form solution

### a. Inversion height tendency

*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

*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

### c. Approximation of net radiation flux term

*z*= 0 and

*z*

_{i}based on Eq. (15). Then we continue with the net shortwave expressions at

*z*= 0 and

*z*

_{i}. Finally, we approximate the columnar integral of net radiation as a linear combination of the net radiation at

*z*= 0 and

*z*

_{i}:

### d. Inversion height solution

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 *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

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 *k* taken from Shettle and Weinman (1970),

*H*is 12 h. We solve for the functions

*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

*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

*u*functions also determine the trends for the cloud-base height as will be shown in section 5.

Since the initial condition *e*-folding time for the effect of the initial condition *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 *D* ~ 9 days. Once

### e. Cloud thickness solution

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 ^{−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 *D* is extracted from the LES using Eq. (2).

The validation consisted of two sets of sensitivity experiments: 1) varying Bowen ratio and 2) varying

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

Both inversion height [Eq. (56)] and cloud-base height [Eq. (73)] were shown to depend on the inversion jump, including the total moisture jump ^{−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

## 5. Sensitivity analysis

### a. Inversion height sensitivity

*u*

_{1},

*u*

_{2}, and

*u*

_{3}. 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:

^{−1}s

^{−1}m

^{3}, and

^{−1}.

The functions

For the optical depth exponentials ^{−1} and assuming that effective radiative temperature equals air temperature,

For the thick-cloud case,

To infer the combined effect of the oscillating terms in Eq. (82), we need the numerical values of *D* and vary the Bowen ratio and

Before sunrise

Larger Bowen ratio causes

### b. Cloud-base height sensitivity

*u*functions in Eq. (73). The coefficients of

^{−1}, respectively. The coefficients become

^{−1}s

^{−1}m

^{4}and

^{2}s

^{−1}. As for the inversion height [Eq. (82)],

*β*and

### 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.

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

#### 2) Dissipation before sunrise

#### 3) Dissipation after sunrise

### d. Extrema analysis

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 second derivative determines whether these points are maxima or minima. We know that

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,

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.

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

## 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

*i*th point generated by our model, whereas

*i*th point associated with a reference (usually the ground truth). The percentage error is defined by normalizing the RMSE by the mean reference value:

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 ^{−1} allows for calculating the cloud temperatures and

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

We set

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

We set ^{−2}. The maximum error is observed for the case with ^{−2}. The maximum error is observed for the case with

#### 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 ^{−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

## APPENDIX C

### Derivation of , , and Functions

*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:

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.

*x*values, we obtain the following:

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