SO345: Atmospheric Thermodynamics

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SO345 Atmospheric Thermodynamics

• CHAPTER 21
• THE PARCEL METHOD

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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

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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.

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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.

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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.

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ABSOLUTE INSTABILITY

• Recall the discussion from the previous
  chapter on the atmospheres 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.
• ? gt Gd gt Gs --------gt atmosphere is absolutely
  unstable

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

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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 gt Gs gt ? --------gt atmosphere is absolutely
  stable

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

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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.

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CONDITIONAL INSTABILITY

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

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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 gt ? gt Gs --------gt atmosphere is conditionally
  unstable

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

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CONDITIONAL INSTABILITY

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

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

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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)

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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 parcels vertical velocity ---gt
  dw/dt is the parcels vertical acceleration.

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CONDITIONAL INSTABILITY

• From this we can get (after some equation
  smashing)
• (Eq 21.2)

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CONDITIONAL INSTABILITY

• Equation 21.2 is a standard differential equation
  with
• three solutions
• 1. If (? - G) lt 0, the solution for z (the parcel
  motion) is a sinusoidal function of time this is
  the stable case
• 2. If (? - G) gt 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 ----gt (d2z)/(dt)2 0
  ----gt displaced particle does not accelerate and
  this is the neutral case.

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CONDITIONAL INSTABILITY

• These solutions confirm our earlier qualitative
  results that the atmosphere is
• stable if ? lt G
• neutral if ? G and
• unstable if ? gt G.
• where G represents either Gd or Gs.
• (This derivation can be found in Appendix P).

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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.

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