# Marine Stratus—A Boundary-Layer Model

## Abstract

**:**

## 1. Introduction

^{−1}). Sample profiles from Sable Island (WSA) are shown in Figure 2. In this case, we see that clouds would have been present from about 500–1250 m with a stable layer at the top. Note that, sadly, the Sable Island radiosonde program ended in 2019 [4].

_{s}) as the moisture-related variables. Lower boundary conditions appear to be based on mean surface values of total water mixing ratio and virtual potential temperature (OLW p 305), but it is not clear how liquid water droplets interact with the water surface. Our model will assume that they collide and coalesce so that the water surface is a sink for liquid water in the air (QL(0) = 0). Figure 3 of the OLW paper shows maximum values of QL at the surface in a warm surface fog situation (cold air advected over warm water). Our model would remove those surface droplets by coalescence with the underlying water surface.

## 2. A 1D PBL Model

_{pa}, starts as 1005 Jkg

^{−1}K

^{−1}but includes adjustments for temperature, and c

_{p}is for moist air accounting for mixing ratios and specific heats of water vapor and liquid water.

_{p}= (c

_{pa}(T) + Qc

_{pv}+ QLc

_{l})/M, in J kg

^{−1}K

^{−1}

_{s}is the gravitational settling velocity of the droplets that are formed. Turbulent fluxes are represented by

_{m}, K

_{h,}etc., could differ. In addition, we use an equation for turbulent kinetic energy per unit mass (TKE, E = 0.5 [<u

^{2}> + <v

^{2}> + <w

^{2}>]),

_{s}and P

_{b}are shear and buoyancy production terms, and ε is the rate of viscous dissipation, details in WT. The eddy diffusivities in this basic E-l, 1.5 order closure are

_{0}for all quantities and ϕ

_{m}is a function of z/L

_{o}where L

_{o}is the Obukhov length, based on local shear stress and heat flux values. In neutral stratification, L

_{o}is infinite and ϕ

_{m}= 1.

**U**= 0, $\Theta $ = T

_{surf}, the surface water temperature, Q = QSAT (T

_{surf}) and QL = 0. The surface can thus be a source of water vapor but is assumed to be a sink for cloud droplets as they collide and coalesce. Fluxes of momentum, heat, water vapor and liquid water evolve as a part of the solution and depend on the assumed roughness lengths. These can differ, z

_{0m}, z

_{0h}, etc., but are presently all set as z

_{0}= 0.001 m.

_{a}, and using mass absorption coefficients (k

_{a}, k

_{w}, k

_{sa}, k

_{sw}), ignoring backscattering, and with σ as the Stefan–Boltzmann constant, we can write the transfer equations for irradiance as,

_{a}is dry air density. Initially, we neglect clear air absorption (k

_{a}, k

_{sa}= 0), set the emissivity, ε = 1, and focus on absorption coefficients for cloud droplets (k

_{w}, k

_{sw}) with units of m

^{2}kg

^{−1}. A serious omission is the back scattering and multiple scattering of downwelling solar radiation (SFD) and the contribution to SFU. A more careful treatment of solar radiation is planned for future work.

^{−2}here) and must specify SFD. In realistic simulations, this will have a strong diurnal cycle, but in the test case considered in Section 3.3, we simply set SFD = 250 Wm

^{−2}and hold it constant.

_{1}, QL

_{1}, T

_{1}) predicted after a time step (Equations (1)–(7)) with Q

_{1}≠ QS

_{1}to an equilibrium state with Q

_{2}= QS

_{2}, where QS

_{i}= QS(T

_{i},P), the saturation mixing ratio. Note that no adjustment is needed (Case 1) if Q

_{1}< QS

_{1}and QL

_{1}= 0, but one is needed (Case 2) if Q

_{1}> QS

_{1}. In Case 4, we may find Q

_{1}< QS

_{1}and QL

_{1}> 0, and liquid water will evaporate, cooling the air. It may also be possible (Case 3), with Q

_{1}< QS

_{1}and QL

_{1}> 0, to evaporate all the droplets while Q

_{2}< QS

_{2}. Our approach would then predict QL

_{2}< 0, and adjustments are made to correct for that. The essential feature of the adjustment is that heat per unit mass of the mass of the material undergoing the adjustment is conserved, i.e.,

_{1}Q

_{1}+ Mc

_{p1}T

_{1}= L

_{2}Q

_{2}+ Mc

_{p2}T

_{2}.

_{p}

_{1}and c

_{p}

_{2}are the specific heats of the dry air plus water vapor and liquid water in states 1 and 2. M is the total mass (dry air plus water vapor and liquid water) per unit mass of dry air, as in Equation (4). This is a constant during the adjustment. In the cases shown, our desired saturation adjustments correspond to the points of intersection of the solid green and blue lines with the black line corresponding to QS(T). If we use the tangent QS line and set c

_{p}= c

_{p}

_{1}in the H = constant line, as in our initial estimates of state 2, we get the points of intersection of the dashed lines. Additional details are at https://www.yorku.ca/pat/AdjustJan2024.pdf (accessed on 7 May 2024).

## 3. Results

#### 3.1. Initial States

_{sur}

_{f}= 288 K. We assumed a geostrophic wind

**U**

_{g}= (20, 0) ms

^{−1}with z

_{0}= 0.001 m and Coriolis parameter, f = 10

^{−4}s

^{−1}. Initial TKE was set to a surface value based on the geostrophic drag law and we imposed an exponential decay with height exp(−z/2000 m). As initial conditions for these runs, we simply set

**U**=

**U**

_{g}at all levels. Initial values are not important—we are just seeking an equilibrium steady state. With neutral stratification, we get a typical Ekman spiral profile while Θ is constant and T decreases at the dry adiabatic lapse rate. Results are in Figure 4 and use 241 vertical levels between the surface and a model top at 3 km. The time step used is 2.5 s, and we impose a TKE (E) minimum of 0.00001 m

^{2}s

^{−2}.

#### 3.2. Adding Moisture

**U**, V, E, Θ stable layer “2K/km equilibrium” profiles discussed in the previous section as initial conditions and now adding moisture effects, and potential condensation. Results after 5 days are shown in Figure 6. Our surface boundary conditions will now include mixing ratios Q = QSAT (T

_{s}), QL = 0 while T

_{surf}is maintained, in this case, at 288 K. We can set initial profiles of Q and QL as we wish, but the extreme case is to have completely dry air with Q = 0 for all z > 0, and QL = 0 for z ≥ 0. Our air column may have been advected from an extremely dry desert out over an ocean.

_{s}, of 0.005 ms

^{−1}, appropriate to droplets of diameter near 13 μm. Gravitational settling over a day would lead to a descent of 432 m, so it can play a significant role and there will be some sensitivity to the value used.

_{s}, does, however, reduce QL at the cloud top, and by day 6, there is a smoother transition to clear air above the cloud (Figure 7). If we start with some moisture present, the cloud forms more quickly and initial cloud water mixing ratios are higher.

_{s}= −0.005 ms

^{−1}shown earlier, but as time moves on, the cloud top continues to rise, and peak QL values slowly increase. There is some sensitivity to the treatment of mixing in stable conditions at the top of the cloud.

#### 3.3. Adding Radiation

_{w}, k

_{sw}. Units will be m

^{2}kg

^{−1}. Stephens’s paper [18], (his Table 3), leads us to use k

_{w}= 80 m

^{2}kg

^{−1}for infrared irradiance. For solar radiation, we initially take k

_{sw}= 40 m

^{2}kg

^{−1}, but much deeper investigation is needed. Adjustment for solar angle with the time of day would be needed for SFD above, within the cloud, and for k

_{sw}

_{,}but for now, we will just set these as constant to illustrate potential solar heating effects. As discussed in Section 2 above, we also need to take into account backscattering within the cloud layer, find appropriate coefficients for that, and find a way to treat diurnal cycles.

^{−2,}and the surface irradiance, at z = 0 m, is the black body value, in this case at 288 K, 390 Wm

^{−2}. Selected results are in Figure 9, extending out to 60 h. Soon after that 60 h point, the model ran into computational problems, probably associated with the extremely strong thermal stability conditions that developed at the cloud top where radiational cooling had dropped the temperature down by about 7 K, as shown in Figure 9a. Results at the 60 h point show upwelling and downwelling long-wave radiation both equal to black body emissions at the cloud water temperature through most of the cloud layer (Figure 9c). Near the cloud base (~500 m), where QL is lower, there is still some unabsorbed radiation from the underlying water surface so that RFU > RFD. Clouds appear at around 30 h, as in the case with no radiation, after which the liquid water content of the cloud increases with time (Figure 9b) and has a maximum near the cloud top where radiational cooling is lowering temperatures, as illustrated in Figure 9a. In Figure 9c, one can see that just at the cloud top level, d(RFD − RFU)/dz will be negative, and, with no solar component, RFDIV is also negative and will cause the cloud top cooling.

^{−2}and hold the absorption coefficient, k

_{sw}= 40 m

^{2}kg

^{−1}, constant. In this case, a stable situation develops, and Figure 11 shows results after 5 days of development. The boundary layer cloud develops with a base of around 200 m and extends up to ~1100 m. The RH is 1.0 with Q = QSAT (Figure 11b) throughout the cloud layer, but the liquid water content is low and hardly visible in Figure 11b. With a time step of 2.5 s, results in the upper part of the cloud were a little noisy, so we reduced the time step to 0.25 s for Figure 11c (note also a different z scale) above. The maximum QL of around 0.03 g/kg can be compared to 0.5 g/kg in the case with no radiation and ~ 1 g/kg in the long-wave radiation case. At t = 5 days, the four irradiance components are shown in Figure 11d.

_{sw}

_{,}to 20 m

^{2}kg

^{−1}, the QL values are still low (max ~ 0.1 g/kg), and the depth increases steadily with time (~1650 m at t = 5 days), as with no solar radiation.

^{−3}range, according to Lowmann et al. [20], so g/kg values 0.04–0.5. Our modeled clouds are not precipitating but do have gravitational settling (0.005 ms

^{−1}). With no solar radiation, our model QL values (~1 g/kg) are rather high, while our first estimates for QL with solar impacts (~0.03 g/kg) are a little low. However, Isaac et al. [21], in their Figure 12, report groups of fog cases over the Grand Banks, at a height of 69 m, with LWC in ranges of 0.005–0.01 and 0.01–0.05 gm

^{−3}, and so our QL = 0.03g/kg, LWC = 0.025 gm

^{−3}, may be realistic while 1.0 g/kg is too high and we may need to assume larger droplet sizes and possibly allow rain to develop.

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Global cloud coverage, from https://earthobservatory.nasa.gov/images/85843/cloudy-earth (accessed on 1 May 2024).

**Figure 2.**Wind speed, temperature and humidity profiles on a cloudy day over Sable Island, 00Z, 3 July 2019, data from http://weather.uwyo.edu/upperair/sounding.html (accessed on 19 February 2024). Note Θ

_{e}, ThetaE, is the equivalent potential temperature with the latent heat of water vapor accounted for.

**Figure 3.**Two saturation adjustment cases (2 and 4) with initial temperature T

_{1}= 288 K plus an illustration of q

_{s}(T) and (dashed red line) the linear approximation q

_{sa}(T). The solid blue and green lines correspond to H = constant in the two cases. The corresponding dashed lines are first approximations with state 1 values of c

_{p}and L and QS varying linearly with T.

**Figure 4.**Equilibrium planetary boundary layer profiles with neutral stratification and dry air. Initial state had

**U**

_{g}= (20, 0) ms

^{−1}and Θ = 288K for all z. Also note z

_{0}= 0.001m and f = 10

^{−4}s

^{−1}. Equilibrium profiles are after integration for 5 days.

**Figure 5.**Equilibrium planetary boundary layer profiles with dry air and stable stratification aloft. Our initial state had

**U**= (20, 0) ms

_{g}^{−1}and Θ = 288 + 0.002 z K for all z(m). We use the same initial TKE, z

_{0}= 0.001 m and f = 10

^{−4}s

^{−1}, as in Figure 4. Surface temperature (T

_{s}) is maintained at 288 K.

**Figure 6.**Results after 5 days over a water surface. Thin lines define the initial conditions (equilibrium profiles from Figure 5) and o, +, x symbols are values after 120 h over a water surface with Q(z = 0) = QSAT (288K). Note that temperature and potential temperature are in K, Relative Humidity = 1 is 100% and that mixing ratios, Q, QL, QSAT are in kg/kg.

**Figure 7.**From the same case as in Figure 6, QL profiles every 6 h, every 12 h snd every 24 h. The first cloud appears after 30 h and a relatively steady profile after 120 h, with a maximum lower than the earlier peak.

**Figure 9.**Selected results with long-wave radiation, up to 60 h. (

**a**) Potential temperature (K), (

**b**) Liquid water mixing ratio, QL (kg/kg so 10

^{−3}g/kg), (

**c**) Radiative fluxes, long wave, infrared, Wm

^{−2}, (

**d**) Mixing ratios, Q, QSAT, QL (kg/kg).

**Figure 11.**Selected results with long- and short-wave radiation after 120 h. (

**a**) Temperature and potential temperature (K), (

**b**) mixing ratios (kg/kg) (

**c**) Liquid water mixing ratio(kg/kg), (

**d**) radiative fluxes (Wm

^{−2}).

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**MDPI and ACS Style**

Taylor, P.A.
Marine Stratus—A Boundary-Layer Model. *Atmosphere* **2024**, *15*, 585.
https://doi.org/10.3390/atmos15050585

**AMA Style**

Taylor PA.
Marine Stratus—A Boundary-Layer Model. *Atmosphere*. 2024; 15(5):585.
https://doi.org/10.3390/atmos15050585

**Chicago/Turabian Style**

Taylor, Peter A.
2024. "Marine Stratus—A Boundary-Layer Model" *Atmosphere* 15, no. 5: 585.
https://doi.org/10.3390/atmos15050585