Thermodynamics of the atmosphere

It is widely used in the thermodynamics of the atmosphere.
The first law of thermodynamics holds: the total amount of energy remains constant or is conserved. It is not possible for the energy to be destroyed or to arise. It can only be transformed from one form to another. For instance:

The first law of thermodynamics as applied to the atmosphere

Radiant energy

Heat

Electric energy

Latent energy

Some other forms of energy

This statement can be expressed in form of energy equation

it, can be expressed by three parameters.

Pressure Pi

Density ρi

Temperature Ti

The same parameters for environment we’ll denote as Pe, ρe, Te respectively.

In general case

However,

Static condition

Suppose a parcel of air has received an amount of heat dq

dq

As result its inner energy will increase by dui

Parcel will expand making work against external pressure force, dwi.

If the air is dry (non-saturated), it can be regarded as an ideal gas

is a theoretical gas composed of a set of randomly-moving, non-interacting point particles is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state

be measured. Therefore, this formula must be transformed to include those parameters which are measured.

In case of an isobaric process dP=0, and hence,

On the other hand, at the isobaric process

Mayer equation

to a body ( )is called adiabatic process.

thermodynamics can be also written in integral form.

Index “0” refers to initial state.

This equation is called equation of adiabatic processes in integral form (Poisson’s equation)

γa=-dTi/dz can be integrated and brought in the following form.

This equation is known as approximated equation of a dry adiabat.
The dry adiabat is also called curve of the state of a dry air parcel

polytropic processes.
The process is called polytropic in the case as a heat influx to an air parcel is proportional to the parcel temperature variation.

Polytropic processes

potential temperature

1. It remains unchanged at adiabatic displacement (up or down)

potential temperature remains unchanged, while its molecular temperature changes.
Any variation of the potential temperature means that the air gains or looses energy.

available potential energy)

Total energy of a particle of a unit mass

Potential temperature variation is uniquely determined by the total energy variation of the air particle.

At adiabatic displacement, the total energy of the air particle remains unchanged.

and other meteorological parameters distribution with height is known as stratification of the atmosphere.

Case 1

The atmosphere is unstable (unstable stratification)

Case 2

The atmosphere is stable (stable stratification)

Case 3

Neutral state

is in the neutral state

Our reasoning and conclusion are valid assuming the process goes without any exchange with environment. In the real atmosphere at leastways some exchange takes place. However, in spite this exchange, conclusion on stability conditions and corresponding vertical displacement are valid.

Potential temperature variation at different types of the stratification

Case 1. Unstable stratification

temperature decreases with height, it means that the atmosphere is unstable

Case 2. Stable stratification

Potential temperature increases with height in the stable atmosphere or if potential temperature increases with height, it means that the atmosphere is stable

From the similar reasoning it goes without saying that at the neutral condition potential temperature does not vary with height.

case of very dry air, instead of cloud formation, overturning of layer occurs. Cold air moves down, and the overheated air ascents up in convective flows.

This process results in formation of eddies similar to tornado, but these vortexes are very small. In our latitudes, this process leads to gusty winds and squalls.

of energy dq.

Adiabatic processes in moist but non-saturated air

S denotes a fraction of water vapor

and

are specific heat capacities of the dry air and water vapor respectively.

Equation of state for moist, non-saturated air