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Convective Instability
Figure 2.4:
Convection.

If water is heated from the bottom and temperature difference between the top
and the bottom exceeds a limit, convection is driven.
Water heated from the bottom climbes and cool water on the top descends.
This transfers the thermal energy from the bottom to the top.
In this section, we describe the condition in which convection is driven.
Consider a hydrostatic balanced atmosphere in which the hydrostatic
balance equation is satisfied:

(2.78) 
where we assumed
in equation (2.2)
and the gravity is working downwards in direction ()
The pressure and density of the atmosphere are and .
We consider a gas element (hatched region in Fig.2.4),
whose density and are equal to those of the
atmosphere and as

(2.79) 
Further, we assume this gas element to move from to
adiabatically, that is,

(2.80) 
Pressure balance is required between the pressures of the gas element
at ,
,
and the atmosphere .
If the density of the gas element at ,
,
is smaller than that of the atmosphere,
the gravity force in equation (2.78) is weaker than
the pressure force and the element keeps climbing.
Thus the condition for the convective instability is written as

(2.81) 
Using

(2.82) 
we can rewrite equation (2.81) into
the relation in the variables of the atmosphere (variables without *) as

(2.83) 
or

(2.84) 
This means the specific entropy
decreases upwardly. Thus, if we consider the adiabatic process, the atmosphere
in which a specific entropy decreases upwardly is unstable for the convection.
Problem
Obtain equation (2.83) from equation
(2.81) using equation
(2.82).
Next: Super and Subsonic Flow
Up: Physical Background
Previous: Rotating Disk
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Kohji Tomisaka
20070708