Thermodynamics of the

atmosphere

The atmosphere is a medium where transition of energy from one form to another constantly occurs.

The branch of meteorology dealing with the general regularities of energy transformation and variation of the state of the atmosphere under influence of heat influxes is called

THERMODYNAMICS OF THE ATMOSPHERE

1

Radiant energy

The first law of thermodynamics as applied to the atmosphere

This law is also known as the law of energy conservation. 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:

This statement can be expressed in form of energy equation

	Heat
							Some
							other
Electric
					Latent		forms of
energy					energy		energy

2

Energy equation

State of the atmosphere, as well as any parcel of 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	i	e ; Тi Тe	However,	Pi Pe P
Suppose a parcel of air has received an amount				Static condition
of heat dq		As result its inner energy will increase by dui
		Parcel will expand making work against
dq		external pressure force, dwi.
					dq dui dwi
If the air is dry (non-saturated), it can be regarded as an ideal gas						3

Ideal Gas

•An ideal gas 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

4

dui cvdTi		dwi Pdvi
Specific heat capacity at the constant volume		Volume increment

dq cvdTi Pdvi

The volume can not be measured. Therefore, this formula must be transformed to include those parameters which are measured.

Pvi RcTi Pdvi vidP RcdTi Pdvi RcdTi vidP

dq cpdTi

RcTi

dP

dq с R dT v dP

p

v c

i

i

In case of an isobaric process dP=0,

c dTi

and hence, dq сv R

On the other hand, at the isobaric process

dq cpdTi

cv Rc cp

c

718 J

kg K

cp

1,4

cv

v

Rc cp cv

cp 1005 J kg K

c

c

R 287 J

Mayer equation

kg K

p

v

c

5

Adiabatic process

Thermo dynamical process going on without energy income or

outflow to a body (	dq 0 )is called adiabatic process.
Pdvi cvdTi	dP
	RcTi P	cpdTi

	dvi 0 dTi 0	dvi 0 dTi 0
	dP 0 dTi 0
		dP 0 dTi 0

6

Ti

сp

Ti 0

For an adiabatic process the equation of the first law of thermodynamics can be also written in integral form.

		p		R T	dP c		dT
	dTi		dP	c i	P	p		i	T		P
	T	Rc p		Index “0” refers				cp ln	i	Rc ln
									T		P
	i	p0		to initial state.
									i0		0

This equation is called equation of adiabatic processes in integral

form (Poisson’s equation)

	Ti	P			Rc	C p
T			P
	i0		0

Rc			287	0,286		Ti		P		0,286
c		1005
					T			P
	p
						i0		0

7

Dry adiabatic lapse rate

Statics’ eq-on

State eq-on

dP

c

dT R T

0

dP g edz

p

i

c

i

p

cpdTi RcTi

P

gdz 0

PR T

c

e

T

cp

dTi

Ti

g

c

dT

i

gdz

0

dz

Te

T

p

i

e

a

dTi

Ti

0,00980 / m 10 /100m

dz

T

a

e

P e RcTe

e

P

RcTe

dP

P

gdz

R T

0

c e

m kg K

m s2 K

K

s2 J

s2m2

m

g

a

0,0098

cp

J

m

2

s2

8

kg

Regarding dry adiabatic lapse rate as a constant value, the expression γa=-dTi/dz can be integrated and brought in the following form.

Ti Ti0 a z z0 Ti0 0,01 z z0

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

9

Polytropic processes

Along with adiabatic processes there are more general ones

known as 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.

dq cdT

Name of process	C	dq	dT
Adiabatic	0	0	-
Isobaric	1005	cpdT	dT≠0
Isosteric	718	cvdT	dT≠0
Isothermal	∞	-	dT=0

10