SO345: Atmospheric Thermodynamics

1. SO345: Atmospheric Thermodynamics CHAPTER 21: THE PARCEL METHOD

2. THE PARCEL METHOD Realistically, when an air sample (or air parcel) moves upward, it should leave a void of space which would then be filled by the surrounding air of the environment (a compensating motion by the environment). An air parcel also has no real barriers around it to prevent any mixing with the surrounding air

3. THE PARCEL METHOD Now, after stating the above realities of a vertically moving air parcel, we will at this point make two basic simplifying assumptions in this first approach to determining the hydrostatic stability of an atmosphere.

4. THE PARCEL METHOD As the parcel moves vertically, there will be no compensating motion by the environment, and 2. There will be no mixing of environmental air with the air parcel. These are obviously very simplistic assumptions which may even be thought of as fairly unrealistic; we will however start with this approach, called the �Parcel Method� in order to obtain a beginning simple result.

5. THE PARCEL METHOD Additional assumptions in this method are: 3. The �environment� is hydrostatic; (the environment has no vertical accelerations; the parcel, however, may experience vertical acceleration); 4. Dynamic equilibrium exists between the parcel and its environment (at any level, the parcel pressure equals the environment pressure); 5. Motion is frictionless.

6. ABSOLUTE INSTABILITY Recall the discussion from the previous chapter on the atmosphere=s stability with respect to Gd and with respect to Gs. Since the value of Gd is always greater than Gs, an environmental lapse rate (?) which is greater than Gd would also be greater than Gs. So as can be seen in Figure 21.1, an atmosphere with this large lapse rate is unstable with respect to both Gd and Gs, or in other words, �absolutely unstable.� ? > Gd > Gs --------> atmosphere is absolutely unstable

7. Absolutely Unstable Atmosphere Fig 21.1 ? of an absolutely unstable atmosphere.

8. ABSOLUTE STABILITY Accordingly, since Gs is always less than Gd, an environmental lapse rate (?) which is less than Gs would also be greater than Gd. So an atmosphere with this lapse rate is stable with respect to both Gs and Gd B or in other words, �absolutely stable� (Figure 21.2). Gd > Gs > ? --------> atmosphere is absolutely stable

9. Absolutely Stable Atmosphere Fig 21.2 ? of an absolutely stable atmosphere.

10. CONDITIONAL INSTABILITY What happens when the value of ? falls between the values of Gd and Gs? Let us look at this situation a little closer. Remember that a moist unsaturated air parcel will ascend at the dry adiabatic rate till saturation is reached, then will continue to ascend at the saturated adiabatic rate from then on.

11. CONDITIONAL INSTABILITY For an environmental lapse rate with the condition, Gd > ? > Gs, the air parcel will start out cooling at the dry adiabatic rate which is greater than the environmental rate (Gd > ?). With the parcel cooler and denser than the environment at these lower levels, we start with a stable atmosphere.

12. CONDITIONAL INSTABILITY Once the parcel reaches saturation, it switches from the dry adiabatic to the saturated adiabatic lapse rate. Eventually the air parcel becomes warmer than the environment, so the atmosphere turns unstable at some higher elevation. Because the atmosphere starts out stable, then becomes unstable, this condition is termed conditional instability. Figure 21.3 illustrates this situation. Gd > ? > Gs --------> atmosphere is conditionally unstable

13. Conditionally Unstable Atmosphere Fig 21.3 ? for a conditionally unstable atmosphere.

14. CONDITIONAL INSTABILITY This qualitative discussion results in the different stability situations summarized below and illustrated in Figure 21.4. ? > Gd > Gs --------> absolutely unstable atmosphere Gd > ? > Gs --------> conditionally unstable atmosphere Gd > Gs > ? --------> absolutely stable atmosphere

15. Atmospheric Stability situations Fig 21.4 Graphical summary of atmospheric stability situations.

16. CONDITIONAL INSTABILITY A more mathematically rigorous discussion of this method is presented by starting with the assumption that the environment is at rest and in hydrostatic equilibrium (assumption #3). Using the convention that environmental parameters are designated by �primed� quantities, the mechanical state of the �environment� is described by the hydrostatic equation: (Eq 17.2)

17. CONDITIONAL INSTABILITY or rearranged as: (Eq 17.2a) The air parcel moves through the environment and may experience a net force causing a vertical acceleration described by: (Eq 21.1) where: w is the parcel=s vertical velocity ---> dw/dt is the parcel=s vertical acceleration.

18. CONDITIONAL INSTABILITY From this we can get (after some equation smashing): (Eq 21.2)

19. CONDITIONAL INSTABILITY Equation 21.2 is a standard differential equation with three solutions: 1. If (? - G) < 0, the solution for z (the parcel motion) is a sinusoidal function of time; this is the stable case; 2. If (? - G) > 0, the solution for z is in terms of exponentials of time (the displacement increases indefinitely; this is the unstable case; 3. If (? - G) = 0, z = 0 ----> (d2z)/(dt)2 = 0 ----> displaced particle does not accelerate; and this is the neutral case.

20. CONDITIONAL INSTABILITY These solutions confirm our earlier qualitative results that the atmosphere is: stable if ? < G; neutral if ? = G; and unstable if ? > G. where G represents either Gd or Gs. (This derivation can be found in Appendix P).

21. ATMOSPHERIC STABILITY FROM A PLOTTED SOUNDING Once a temperature sounding is plotted on a thermodynamic diagram, the stability of the atmosphere at different layers can easily be determined by comparing the different lapse rates along the environmental temperature profile (use the principles of Figure 21.4 as your guide). Figure 21.5 illustrates the process.

22. Fig 21.5 Determining atmospheric stability from a temperature sounding.