# Convection of Moist Saturated Air: Analytical Study

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## Abstract

**:**

## 1. Introduction

## 2. Basic Equations of Moist Saturated Air Convection

_{i}is the moist air parcel density; ρ

_{e}the air parcel surrounding atmosphere density; and g the free fall acceleration. We accept the parameters of the surrounding atmosphere as an undisturbed state. Hence, the pressure may be written as: $p=\overline{p}+p\text{'}$, here p′ is the pressure disturbance relative to the statics state.

_{0}is the air thermal expansion coefficient; T

_{0}= 273 K; ΔT(z) = T

_{i}(z)–T

_{e}(z); ΔT(z) the overheat function; T

_{i}, T

_{e}the air parcel internal and external temperatures; s the water vapor mass fraction (also called specific humidity or moisture content); Δs=s

_{i}–s

_{e}the supersaturation function; β ≡ M

_{d}/M

_{v}–1 = 0.608; M

_{d}= 29 g/mol the dry air molar mass; M

_{v}= 18 g/mol the water vapor molar mass. In other words, we assume that the density depends on temperature and vapor mass fraction and not depends on pressure [33]. Here we do not consider the water loading.

_{ec}the surrounding air temperature at the condensation level (z

_{c}). The moist saturated air temperature gradient is determined by [34]

_{a}is the dry-adiabatic temperature gradient; L the specific heat of condensation; c

_{p}the specific heat capacity at constant pressure; s

_{m}the saturated vapor mass fraction. The quantity ds

_{m}/dz appearing in Equation (6) is a complicated function of temperature and pressure [34]; this is why Equation (6) is usually analyzed numerically. To obtain an analytical solution, we should propose an adequate parametrization for the quantity ds

_{m}/dz. We specify the saturated vapor mass fraction change with altitude parametrically by expanding in a Taylor series:

_{mc}).

_{mac}is the moist-adiabatic temperature gradient at the condensation level [34] (this is the known function of temperature and pressure).

_{p}≈ 3·10

^{–7}°C/m

^{2}.

_{ic}is the rising air parcel temperature at the condensation level. The function T

_{i}(z) graph presents the ’state curve’; the family of such curves is displayed on aerological diagrams. The table values of air temperature obtained from the state curve demonstrates the satisfactory agreement with the values calculated by Equation (9) within the convection layer of interest z

_{w}– z

_{c}≤ 5 km. Here z

_{w}is the convection level. Taking into account Equations (5) and (9) the overheat function will be:

_{c}T is the overheat function at the condensation level; Δγ

_{mac}= γ – γ

_{mac}is the difference of the ambient air temperature gradient and the rising air parcel temperature gradient at the condensation level. The quantity Δγ

_{mac}determines the angle between the state curve and the stratification curve at the condensation level on the aerological diagram.

_{ec}is the water vapor mass fraction in the surrounding atmosphere at the condensation level; b the vapor mass fraction gradient. Then for the supersaturation function we get

_{c}s = s

_{mc}– s

_{ec}is the supersaturation at the condensation level.

## 3. Solution of the Moist Saturated Air Convection Equations

**Figure 1.**Stream function at b = 10

^{–6}m

^{–1}, Δ

_{c}T = 0.5 °C, Δ

_{c}s = 0, Δγ

_{mac}= 5 × 10

^{–4}°C/m, ε = 3 × 10

^{–7}°C/m

^{2}, k = 10

^{–3}m

^{–1}, ${\text{d}{s}_{\text{m}}/\text{d}z|}_{z={z}_{\text{c}}}=-1.6\times {10}^{-6}{\text{m}}^{-1}$ .

**Figure 2.**Convection cell vertical size vs. water vapor mass fraction gradient. The atmospheric parameters are the same as in Figure 1.

## 4. Analysis and Discussion of the Obtained Solution

_{c}T and the oversaturation Δ

_{c}s are equal to zero at the condensation level), then for the convection level and for the maximal velocities level we have

_{max}– z

_{c}into Equation (44), we get for the ascending moist saturated air maximal velocity the equation:

^{2}kx = 1. From here we have

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Zakinyan, R.; Zakinyan, A.; Ryzhkov, R.; Avanesyan, K.
Convection of Moist Saturated Air: Analytical Study. *Atmosphere* **2016**, *7*, 8.
https://doi.org/10.3390/atmos7010008

**AMA Style**

Zakinyan R, Zakinyan A, Ryzhkov R, Avanesyan K.
Convection of Moist Saturated Air: Analytical Study. *Atmosphere*. 2016; 7(1):8.
https://doi.org/10.3390/atmos7010008

**Chicago/Turabian Style**

Zakinyan, Robert, Arthur Zakinyan, Roman Ryzhkov, and Kristina Avanesyan.
2016. "Convection of Moist Saturated Air: Analytical Study" *Atmosphere* 7, no. 1: 8.
https://doi.org/10.3390/atmos7010008