# What is the lapse rate in a simplified model of cloud formation?

• Saitama
In summary, the pressure at a height ##z## is given by: $$p=p_0\left(\frac{T_0-\Gamma z}{T_0}\right)^{\alpha}$$ where ##T_0## is the pressure and temperature at sea level, ##\gamma## is the lapse rate, and ##m_a## is the relevant molar mass.
Saitama

## Homework Statement

Consider a simplified model of cloud formation. Hot air in contact with the earth’s surface contains water vapor. This air rises convectively till the water vapor content reaches its saturation pressure. When this happens, the water vapor starts condensing and droplets are formed. We shall estimate the height at which this happens. We assume that the atmosphere consists of the diatomic gases oxygen and nitrogen in the mass proportion 21:79 respectively. We further assume that the atmosphere is an ideal gas, g the acceleration due to gravity is constant and air processes are adiabatic. Under these assumptions one can show that the pressure is given by
$$p=p_0\left(\frac{T_0-\Gamma z}{T_0}\right)^{\alpha}$$
Here ##p_0## and ##T_0## is the pressure and temperature respectively at sea level ##(z = 0)##, ##\Gamma## is the lapse rate (magnitude of the change in temperature ##T## with height ##z## above the earth’s surface, i.e. ##\Gamma > 0##).

(a) Obtain an expression for the lapse rate ##\Gamma## in terms of ##\gamma, R, g## and ##m_a##. Here ##\gamma## is the ratio of specific heat at constant pressure to specific heat at constant volume; ##R##, the gas constant; and ##m_a##, the relevant molar mass.

(There are few more parts to this problem, they seem to be related to (a), I will post them when I solve a) ).

## The Attempt at a Solution

Consider a layer at height ##z##, cross section ##A## and thickness ##dz##. Let ##p## and ##p+dp## be the pressure on two sides as shown in figure. The mass of the air in this layer is ##A\rho \,dz## where ##\rho## is the density of air at height ##z##. Equating the forces, I get
$$\frac{dp}{dz}=-\rho g$$
Now I am not sure what to do after this. From the ideal gas, I have ##\rho=(pm_a)/(RT)## but how do I introduce ##\gamma##?

Is my approach even correct?

Any help is appreciated. Thanks!

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The key assumption in deriving the lapse rate is that when air rises, it expands, does work, and cools down.

Hi voko!

voko said:
The key assumption in deriving the lapse rate is that when air rises, it expands, does work, and cools down.

How do you conclude this from the problem statement? And how do I make the equations? Do I have to use ##PV^{\gamma}=\text{constant}##?

The statement is a bit obscure in this respect, but the following is consistent with what I said: "This air rises convectively"; "air processes are adiabatic".

Your equation is a good starting point.

voko said:
The statement is a bit obscure in this respect, but the following is consistent with what I said: "This air rises convectively"; "air processes are adiabatic".

Your equation is a good starting point.

Let the initial pressure be ##P## and volume occupied be ##Az##. When the gas expands by height ##dz##, I have:
$$P(Az)^{\gamma}=(P+dP)(A(z+dz))^{\gamma} \Rightarrow P=(P+dP)\left(1+\frac{dz}{z}\right)^{\gamma} \approx (P+dP)\left(1+\gamma\frac{dz}{z}\right)$$
$$\Rightarrow \frac{dP}{dz}=-\gamma \frac{P}{z}$$

Is this correct?

No, that is not correct. You assumed that ##A## was constant, but there is no reason to assume that. Besides, you obtained the relationship between ##p## and ##z## in #1; it is good. What you need here is a relationship between ##T## and ##p##.

How is the temperature related to the pressure in the adiabatic reversible expansion of an ideal gas?

voko said:
No, that is not correct. You assumed that ##A## was constant, but there is no reason to assume that. Besides, you obtained the relationship between ##p## and ##z## in #1; it is good. What you need here is a relationship between ##T## and ##p##.

Chestermiller said:
How is the temperature related to the pressure in the adiabatic reversible expansion of an ideal gas?

$$PT^{\gamma/(1-\gamma)}=\text{constant}$$

But how to make use of this?

You have two equations: one linking ##p## and ##z##, another linking ##p## and ##T##. You need to find ##\Gamma(z) = - T'(z) ##.

voko said:
You have two equations: one linking ##p## and ##z##, another linking ##p## and ##T##. You need to find ##\Gamma(z) = - T'(z) ##.

I differentiate the equation relating p and T with respect to z.
$$T^{\gamma/(1-\gamma)}\,\frac{dp}{dz}+ \frac{\gamma p}{1-\gamma}\frac{T^{\gamma/(1-\gamma)}}{T}\frac{dT}{dz}=0$$
$$\Rightarrow \frac{dp}{dz}=-\frac{\gamma}{1-\gamma} \frac{p}{T}\frac{dT}{dz}$$
Also,
$$\frac{dp}{dz}=-\frac{pm_a g}{RT}$$
Equating both, I get
$$\frac{dT}{dz}=\Gamma=\frac{m_ag}{R}\frac{1-\gamma}{\gamma}$$
But this is not correct, I have got a sign error somewhere. :(

Look again at #9.

voko said:
Look again at #9.

Why a negative sign?

Pranav-Arora said:
Why a negative sign?

The sign is correct. The temperature decreases with altitude in the atmosphere. When a parcel of air rises, the pressure decreases, and the parcel expands adiabatically against the surrounding parcels, doing work. So you get expansion cooling.

1 person
Pranav-Arora said:
Why a negative sign?

Look how the lapse rate is described in your #1.

1 person
Chestermiller said:
The sign is correct. The temperature decreases with altitude in the atmosphere. When a parcel of air rises, the pressure decreases, and the parcel expands adiabatically against the surrounding parcels, doing work. So you get expansion cooling.

voko said:
Look how the lapse rate is described in your #1.

Ah yes, thanks a lot both of you, the other parts have been solved too.

## 1. What is the cloud formation problem?

The cloud formation problem refers to the scientific mystery of how clouds are formed in the Earth's atmosphere. While it is a well-known phenomenon, the exact processes and mechanisms behind cloud formation are still not fully understood by scientists.

## 2. What are the main factors that contribute to cloud formation?

The main factors that contribute to cloud formation include atmospheric temperature, humidity, and air pressure. These factors affect the amount of water vapor in the air and how it condenses to form clouds. Other factors such as air currents and the presence of particles in the air can also play a role.

## 3. How do clouds form differently in different regions of the world?

Cloud formation can vary depending on the region due to differences in climate, topography, and other environmental factors. For example, in areas with high mountains, clouds may form at higher altitudes due to the cooling effect of the mountains on the air. In coastal regions, sea breezes can bring in moisture that contributes to cloud formation.

## 4. Can human activities affect cloud formation?

Yes, human activities such as pollution and deforestation can have an impact on cloud formation. Pollution can affect the size and properties of cloud particles, while deforestation can alter the amount of water vapor in the air and the local climate, both of which can affect cloud formation. These human activities can also have indirect impacts on cloud formation by contributing to climate change.

## 5. Why is understanding cloud formation important?

Understanding cloud formation is crucial for predicting weather patterns and climate change. Clouds play a significant role in regulating Earth's temperature by reflecting sunlight and trapping heat. Changes in cloud formation and properties can have a significant impact on the Earth's climate, making it essential for scientists to continue studying and unraveling the cloud formation problem.

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