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Stability - I

Equilibrium and stability:

In dynamics, ‘equilibrium’ means – no change in motion. So, if a moving body is in equilibrium, it cannot have any acceleration. Either it is at rest or it moves at uniform velocity.

To understand ‘stability’ we can consider a ball at rest ( in equilibrium, obviously). We want to know, how stable its equilibrium is. Let us consider the ball in three different situations as shown in the diagram.

In case-A, the ball is kept on the top of an inverted bowl. In case-B, the ball is inside the bowl and in case-C, the ball is placed on a plane surface.

Three different things are observed in these situations:

• If we now displace the ball slightly, in the case A, the ball displaces ifself further. So, it is said to be in unstable equilibrium.
• In case-B, the ball, when displaced (slightly), goes back to its previous position and so, this is the case of stable equilibrium.
• In case-C, the ball, when displaced, neither can goes back to the previous position, nor does it go away from that location. So, we call it to be in neutral equilibrium .

Stability in (static) atmosphere:

We may observe an air parcel in the atmosphere. At a certain position z = z0,it is part of the atmosphere and its density is ρ0 . So, it’s weight (the force due to gravity) per unit volume is ρ0g. This air-parcel is now  displaced to z = z1, where the density of the surrounding is ρ1.

At this point, we should renew our knowledge about buoyancy. Buoyancy is the force that is applied by a fluid, on a body that is immersed in it. This buoynat force develops due to the fluid-pressure on  the body. The fluid-pressure on the top(downward) of the body is always less than that on the lower-face (upward). The reason is simple, - the top side is always at a smaller depth than the bottom. Due to this difference in pressure, the buoyant force acts in the upward direction, trying to counterbalance the force of gravity.

The force of buoyancy, on a body in a fluid, is equal to the weight of the fluid that is displaced by the body itself.

So, if the parcel we are observing doesn’t mix with its surrounding, then the net force F on it, per unit volume, is:

F  =  o – ρ1)g.

We define stability S as,

S = (ρo- ρ1)/(zo – z1)  when,(z – z1)->

or, S = -(dρ / dz)

Comparing with the previous examples for explaining stability, we can say that the parcel is :

• Stable          if  S > 0, or positive.
• Unstable        if  S < 0, or negative.
• Neutral         if  s = 0

# A better atmospheric-model:

Stability in a polytropic atmosphere:

This model is more realistic because it fits the atmospheric properties better.

Assumptions: the process is adiabatic, which ensures that the parcel under consideration does not exchange heat with its  surroundings.

It can be derived that for a polytropic atmosphere :

ρo / ρ1     =  (po – p1)1/n

where, n is some index. Now, the stability factor S can be shown to  be depending on the relative value of n as compared to γ, the ratio of two specific heat of the fluid (here, air). So,

• If  n < γ    Stable    as  S > 0, or position.
• If  n > γ    Unstable  as  S<0, or negative.
• If  n = γ    Neutral   as  S = 0.

Tapas bhattacharya
Web-project : Phys-645, Fall-2007, UAF
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