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
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
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Heat |
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Some |
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other |
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Electric |
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Latent |
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forms of |
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energy |
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energy |
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energy |
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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 |
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e ; Тi Тe |
However, |
Pi Pe P |
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Suppose a parcel of air has received an amount |
Static condition |
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of heat dq |
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As result its inner energy will increase by dui |
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Parcel will expand making work against |
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dq |
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external pressure force, dwi. |
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dq dui dwi |
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If the air is dry (non-saturated), it can be regarded as an ideal gas |
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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
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dui cvdTi |
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dwi Pdvi |
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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
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dq cpdTi |
RcTi |
dP |
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dq с R dT v dP |
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p |
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v c |
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In case of an isobaric process dP=0, |
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c dTi |
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and hence, dq сv R |
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On the other hand, at the isobaric process |
dq cpdTi |
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cv Rc cp |
c |
718 J |
kg K |
cp |
1,4 |
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cv |
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Rc cp cv |
cp 1005 J kg K |
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R 287 J |
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Mayer equation |
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kg K |
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p |
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5 |
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Adiabatic process
Thermo dynamical process going on without energy income or
outflow to a body ( |
dq 0 )is called adiabatic process. |
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Pdvi cvdTi |
dP |
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RcTi P |
cpdTi |
dvi 0 dTi 0 |
dvi 0 dTi 0 |
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dP 0 dTi 0 |
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dP 0 dTi 0 |
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6
Ti
сp
Ti 0
For an adiabatic process the equation of the first law of thermodynamics can be also written in integral form.
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p |
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R T |
dP c |
dT |
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dTi |
dP |
c i |
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T |
Rc p |
Index “0” refers |
cp ln |
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Rc ln |
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T |
P |
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p0 |
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to initial state. |
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i0 |
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This equation is called equation of adiabatic processes in integral 
form (Poisson’s equation)
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Ti |
P |
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Rc |
C p |
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i0 |
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Rc |
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287 |
0,286 |
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Ti |
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0,286 |
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c |
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i0 |
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7
Dry adiabatic lapse rate
Statics’ eq-on |
State eq-on |
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dP |
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dT R T |
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dP g edz |
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cpdTi RcTi |
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P |
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gdz 0 |
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PR T |
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cp |
dTi |
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dT |
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gdz |
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dz |
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Te |
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a |
dTi |
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Ti |
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0,00980 / m 10 /100m |
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dz |
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T |
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a |
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e |
P e RcTe
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RcTe |
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dP |
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gdz |
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R T |
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0 |
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c e |
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m kg K |
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m s2 K |
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s2 J |
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s2m2 |
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0,0098 |
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cp |
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s2 |
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kg |
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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 |
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Isobaric |
1005 |
cpdT |
dT≠0 |
Isosteric |
718 |
cvdT |
dT≠0 |
Isothermal |
∞ |
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dT=0 |
10