Stability

C LIMATE AND A TMOSPHERIC T HERMODYNAMICS

3.10 Stability

1 2

1

m m

w w r

w r

 

 (43)

v d

w m

m (44)

is warmer than its surroundings rises, and the rise continues (unstable) until the parcel’s temperature equalizes with that of its surroundings. At this point it remains at the approached level (neutral) or becomes cooler so that it starts descending (stable).

In other words, to reverse a stable system to its original status, you only need to remove the driving force for causing the instability—the stressor.

However, for an unstable system, lack of the driving force is not sufficient to make the system stable again. In a system with neutral stability, removing the driving force will leave the system at its current, and not initial, state. A technique to determine stability is to compare the actual lapse rate with that of the standard one. If the real-time lapse rate is smaller than the standard- adiabatic lapse rate, the air parcel is stable; if they are equal, the air parcel is neutral; for lapse rates larger than the standard-adiabatic lapse rate, the air parcel is unstable. Another method to identify the stability level is to calculate the gradient of potential temperature with respect to the altitude:

positive values are associated with stable air while zero and negative values represent neutral and unstable conditions. Convective Available Potential Energy (CAPE)—the amount of energy an air parcel requires to be lifted vertically to a certain distance—is related to the positive buoyancy of the air parcel and is another indicator of atmospheric stability [81]. Figure 9 depicts the state of hydrostatic equilibrium (single-line arrows) as well as an accelerating cloud scenario (double-line arrows). Balance of forces is applicable to the arrows of similar line type (Figure 9).

FIGURE 9 Air parcel in hydrostatic equilibrium and acceler ation scenarios.

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Applying the Archimedes’ principle to the parcel of air shown in Figure 9 reveals that upward force imposed on the body from its surroundings and downward force imposed by gravity on the air parcel are to be equal for the case of a parcel in equilibrium with its environment. If the buoyancy force exerted on the air parcel equals its weight, hydrostatic equilibrium is in place and the air parcel does not move (single-line arrows). Equation (32), dP  gdz, is then transformed to equation (47). Average virtual temperature (T_v) may be represented by equation (48).

( 0)

0 ^v

g z z

P P eRT

 

  

 

 (47)

 



⁰ 0



LnP LnP v v

T d LnP T  LnP LnP





(48)

There are also cases in which the air parcel accelerates or decelerates—

given the balance of said forces (double-line arrows). This scenario requires that the upward force imposed on the body by its surroundings and the downward force imposed by gravity on the air parcel are not equal—the difference is the acceleration force (F₃ in Figure 9). The balance of forces for an isobaric process results in equation (49), in which acceleration is given as a function of the air parcel temperature (T) and its surrounding temperature (T_). Equation (49) shows that if the temperature of the air parcel is lower than that of its surroundings, the acceleration is negative, meaning that the air parcel accelerates downward (stable). The opposite is the case when the temperature of the air parcel is higher than that of its surroundings, in which case the acceleration is positive, meaning that the air parcel accelerates upward (unstable).

( )

S

a T T g

T

 

 (49)

Substituting the hydrostatic relation presented by equation (32) into the semi-adiabatic process presented by equation (33) results in the saturated-adiabatic lapse rate (G_w) for the semi-adiabatic process presented by equation (50).  is the ratio of the gas constant for the dry air to that of the water vapor (  R_d /R_v ).

2 2

1 1

s d S

s d P

Lw dT R T

dZ L w

R C T

 

  

 

   



 

 

G G



(50)

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Assuming that the air parcel in Figure 9 is homogeneous, meaning that its density and therefore volume do not vary spatially (i.e., horizontally or vertically), the condition that is possible in some hurricanes, you may calculate the lapse rate for the homogeneous atmosphere by combining the hydrostatic relation—equation (32)—and equation of state—equation (56)—P  RT—to obtain equation (51), in which the homogeneous gradi- ent of temperature or lapse rate is a function of the mass of the air parcel and gas constant. By substituting the numerical values in equation (51), the homogeneous lapse rate of 10.42 °C per 1,000 ft is obtained (34.2 °C per km). If the lapse rate is higher than the homogeneous atmospheric lapse rate, the conditions are absolutely unstable, which is associated with the formation of tornadoes or hurricanes.

mg g

dT

dZ R R (51)

It is possible to calculate the altitude for the homogeneous atmosphere by combining the hydrostatic relation—equation (32)—and equation of state—equation (56)—and solving for the altitude. As a result, equation (52) is obtained, in which the homogeneous height is the function of the standard atmosphere absolute temperature, mass of the air parcel, and universal molar gas constant. By substituting the numerical values in equation (52), the homogeneous atmospheric height 8,267 m (27,119 ft) is obtained.

0 0

RT RT

Hmg g (52)

If the lapse rate of the air parcel is between the ones of the saturated and dry adiabatic, the air parcel moving upward first faces resistance due to a larger lapse rate and then less resistance due to a smaller lapse rate. This scenario is called conditional stability.