chapter 4: bulk thermodynamics of a cloud-free and cloudy atmosphere thermodynamic diagrams brenda...
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Chapter 4:Bulk Thermodynamics of a
Cloud-Free and Cloudy Atmosphere
Chapter 4:Bulk Thermodynamics of a
Cloud-Free and Cloudy Atmosphere
THERMODYNAMIC DIAGRAMSTHERMODYNAMIC DIAGRAMS
Brenda Dolan
AT620
September 14, 2005
Brenda Dolan
AT620
September 14, 2005
Recall…Recall…
In the first part of the chapter, we derived various forms of the 1st Law of Thermodynamics under moist conditions, and for a cloud-free atmosphere– Most generally, 1st law for open thermodynamic multi-
phase system:
In the first part of the chapter, we derived various forms of the 1st Law of Thermodynamics under moist conditions, and for a cloud-free atmosphere– Most generally, 1st law for open thermodynamic multi-
phase system:
Vapor
ice Heat storage radiation dissipation
For a cloud-free atmosphere: For a cloud-free atmosphere:
Recall…Recall…
We also introduced thermodynamic variables which are conserved for adiabatic motions:– θ: Potential temperature
Conserved for dry, isentropic motions
– θl,i: Ice-liquid water potential temperature Conserved for wet adiabatic (liquid and ice
transformations) Reduces to θ in the absence of cloud or precip.
– θe: Equivalent potential temperature Useful in diagnostic studies as a tracer of air parcel
motions Conserved during moist and dry adiabatic processes θeiv is conservative over phase changes but not if
precipitation fluxes exist
We also introduced thermodynamic variables which are conserved for adiabatic motions:– θ: Potential temperature
Conserved for dry, isentropic motions
– θl,i: Ice-liquid water potential temperature Conserved for wet adiabatic (liquid and ice
transformations) Reduces to θ in the absence of cloud or precip.
– θe: Equivalent potential temperature Useful in diagnostic studies as a tracer of air parcel
motions Conserved during moist and dry adiabatic processes θeiv is conservative over phase changes but not if
precipitation fluxes exist
Recall…Recall…
Recall…Recall…
Wet-Bulb temperature, Tw
– Temperature that results from evaporating water at constant pressure from a wet bulb
Wet-bulb potential temperature, θw
– Determined graphically– Conserved during moist
and dry adiabatic processes
(as is θe)
Wet-Bulb temperature, Tw
– Temperature that results from evaporating water at constant pressure from a wet bulb
Wet-bulb potential temperature, θw
– Determined graphically– Conserved during moist
and dry adiabatic processes
(as is θe)
Recall…Recall…
Energy Variables– Dry static energy (s)
– Moist static energy (h)
In water-saturated environ:– Static energy (S)
– Mosit Static energy (H)
Energy Variables– Dry static energy (s)
– Moist static energy (h)
In water-saturated environ:– Static energy (S)
– Mosit Static energy (H)
4.6 Thermodynamic Diagrams4.6 Thermodynamic Diagrams
Provide graphical display of lines representing major kinds of processes to which air may be subject– Isobaric, isothermal, dry adiabatic and
pseudoadiabatic processes
Provide graphical display of lines representing major kinds of processes to which air may be subject– Isobaric, isothermal, dry adiabatic and
pseudoadiabatic processes
4.6 Thermodynamic Diagrams4.6 Thermodynamic Diagrams
Three desirable characteristics1. Area enclosed by lines representing any cyclic
process be proportional to the change in energy or the work done during the process(in fact, designation thermodynamic diagram is reserved for only those in which area is proportional to work or energy)
2. As many as possible of the fundamental lines be straight
3. The angle between the isotherms and the dry adiabats shall be as large as possible (90º)
- makes it easier to detect variations in slope
Three desirable characteristics1. Area enclosed by lines representing any cyclic
process be proportional to the change in energy or the work done during the process(in fact, designation thermodynamic diagram is reserved for only those in which area is proportional to work or energy)
2. As many as possible of the fundamental lines be straight
3. The angle between the isotherms and the dry adiabats shall be as large as possible (90º)
- makes it easier to detect variations in slope
Thermodynamic diagramsThermodynamic diagrams
Would like to use p-α, but the angle between isotherms and adiabats is too small
To preserve the area enclosed on one diagram to be equal to the area enclosed on another, we found:
Thus, if we specify the thermodynamic variable B, we can determine what A must be in order to have an equal-area transformation from α, -p to A,B
Would like to use p-α, but the angle between isotherms and adiabats is too small
To preserve the area enclosed on one diagram to be equal to the area enclosed on another, we found:
Thus, if we specify the thermodynamic variable B, we can determine what A must be in order to have an equal-area transformation from α, -p to A,B
TephigramTephigram
Let B=T Now determine A:
From eqn. 4.37:
Let B=T Now determine A:
From eqn. 4.37:∂A∂α
⎞⎠⎟B
=∂p
∂B⎞⎠⎟α
Plug in B=T: Plug in B=T: ∂A∂α
⎞⎠⎟T
=∂p
∂T⎞⎠⎟α
From ideal gas: pα=RT => p=RT/α From ideal gas: pα=RT => p=RT/α
∂A∂α
⎞⎠⎟T
=∂RT
α∂T
⎞
⎠
⎟⎟⎟α
TephigramTephigram
∂A∂α
⎞⎠⎟T
=R
α
∂T
∂T
∂A)T =∂α R
α
A =Rlnα + F(T )
Not real useful to plot… introduce Poisson’s equation to get potential temperature:
Not real useful to plot… introduce Poisson’s equation to get potential temperature:
T
θ=
p1000
⎛⎝⎜
⎞⎠⎟
k
=RT
1000α⎛⎝⎜
⎞⎠⎟
k
TephigramTephigram
Take the log and solve for lnα: Take the log and solve for lnα:
lnα =1k
lnθ −lnT[ ] + lnT + lnR−ln1000
R lnα =cp lnθ +G(T ) Now back to A: Now back to A:
A =cp lnθ + F(T ) +G(T )
Arbitrarily choose F(T)=-G(T), resulting in the following coordinates:
Arbitrarily choose F(T)=-G(T), resulting in the following coordinates:
A =cp lnθB=T (4.38)
TephigramTephigram
We also know cplnθ as entropy (ϕ), leading the developer of this diagram, Sir Napier Shaw, to call it the T- ϕ diagram, or tephigram
The equation for the isobars can be found by taking the logarithm of Poisson’s equation:
We also know cplnθ as entropy (ϕ), leading the developer of this diagram, Sir Napier Shaw, to call it the T- ϕ diagram, or tephigram
The equation for the isobars can be found by taking the logarithm of Poisson’s equation:
lnθ =lnT + const.
Properties of a TephigramProperties of a Tephigram
Isobars are logarithmic curves sloping upward to the right, decreasing in slope with increasing temperature (because of log vs. linear coordinates)
Pseudoadiabats are curved– Recall, a pseudoadiabat is a moist-adiabatic process in which
the liquid water that condenses is assumed to be removed as soon as it is formed, by idealized instantaneous precipitation.
Saturation mixing ratio lines are nearly straight Angle between isotherms and adiabats is exactly
90º– Thus, changes in slope are easily detected and compared
Isobars are logarithmic curves sloping upward to the right, decreasing in slope with increasing temperature (because of log vs. linear coordinates)
Pseudoadiabats are curved– Recall, a pseudoadiabat is a moist-adiabatic process in which
the liquid water that condenses is assumed to be removed as soon as it is formed, by idealized instantaneous precipitation.
Saturation mixing ratio lines are nearly straight Angle between isotherms and adiabats is exactly
90º– Thus, changes in slope are easily detected and compared
Tephigram-SummaryTephigram-Summary
Tephigram isCoordinates are:
1) Area proportional to energy2) Four sets of lines which are exactly or nearly straight
and only one set which is curved3) An isotherm-to-adiabat angle of 90º
Satisfies all three criteria nearly perfectly!
Especially useful in tropics
Tephigram isCoordinates are:
1) Area proportional to energy2) Four sets of lines which are exactly or nearly straight
and only one set which is curved3) An isotherm-to-adiabat angle of 90º
Satisfies all three criteria nearly perfectly!
Especially useful in tropics
A =cp lnθB=T
Tephigram--ExampleTephigram--Example
isobars
Dry adiabatic lapse rate
Saturation mixing ratio
isotherms
moist adiabatic lapse rate
Skew T-log p DiagramSkew T-log p Diagram
Introduced by Herlofson in 1947, to make the isotherm-adiabat angle more 90º
B=-lnp– Thus, equation 4.37 becomes
Introduced by Herlofson in 1947, to make the isotherm-adiabat angle more 90º
B=-lnp– Thus, equation 4.37 becomes
∂A∂α
⎞⎠⎟
ln p
= −1
R
∂p
∂ln p
⎛⎝⎜
⎞⎠⎟⎞
⎠⎟α
∂A∂α
⎞⎠⎟
ln p
= −p
R
Which reduces to: Which reduces to:
(4.39)
Skew T-log p DiagramSkew T-log p Diagram Integrating and holding lnp (and therefore p)
constant: Integrating and holding lnp (and therefore p)
constant:A =−
pαR
+ F(lnp) => A=−T + F(lnp)
A =T + K lnpB=−Rlnp
Choosing an arbitrary function, F(lnp)=-Klnp, where K is a constant we choose, and realizing that we are not concerned with the sign of the area (it only determines the direction of the cycle), we arrive at the coordinates for a Skew T-log p:
Choosing an arbitrary function, F(lnp)=-Klnp, where K is a constant we choose, and realizing that we are not concerned with the sign of the area (it only determines the direction of the cycle), we arrive at the coordinates for a Skew T-log p:
(4.41)
(4.40)
B is the ordinate and A is the abscissa B is the ordinate and A is the abscissa
Skew T-log p DiagramSkew T-log p Diagram
Isotherms have the equation: Isotherms have the equation:
A =const+ K lnp
A =const−KR
B
B =−RK
A+ const.
oror
oror
(4.42)
The equation for the dry adiabats is found by taking the log of Poisson’s equation while holding θ constant:
The equation for the dry adiabats is found by taking the log of Poisson’s equation while holding θ constant:
lnT =−Rcp
+ const.
Properties of a Skew T-log pProperties of a Skew T-log p
Isotherms are straight parallel lines whose slope depends on the value of K– If K is chosen to make the isotherm-adiabat angle close to 90º, then
the isotherms slope upward to the right at an angle of ~45º to the isobars
Adiabats are gently, but visibly, curved, since Rlnp is one of the coordinates but lnT is not. They run from lower right to the upper left of the diagram, concave upwards
Pseudoadiabats are distinctly curved– This is true of all diagrams which preserve the energy-area
proportionality
Saturation mixing ratio lines are essentially straight
Isotherms are straight parallel lines whose slope depends on the value of K– If K is chosen to make the isotherm-adiabat angle close to 90º, then
the isotherms slope upward to the right at an angle of ~45º to the isobars
Adiabats are gently, but visibly, curved, since Rlnp is one of the coordinates but lnT is not. They run from lower right to the upper left of the diagram, concave upwards
Pseudoadiabats are distinctly curved– This is true of all diagrams which preserve the energy-area
proportionality
Saturation mixing ratio lines are essentially straight
Skew T-log p--SummarySkew T-log p--Summary
Skew-T isCoordinates are:
1) Area proportional to energy2) Three sets of exactly or closely straight lines; one set
of gently curved lines, and only one set which is markedly curved
3) An isotherm-to-adiabat angle which varies with position on the diagram but is about 90º
Satisfies all three criteria!
Skew-T isCoordinates are:
1) Area proportional to energy2) Three sets of exactly or closely straight lines; one set
of gently curved lines, and only one set which is markedly curved
3) An isotherm-to-adiabat angle which varies with position on the diagram but is about 90º
Satisfies all three criteria!
A =T + K lnpB=−Rlnp
Skew T-log p--ExampleSkew T-log p--Exampleisobars
Saturation mixing ratio
isotherms
Dry adiabats (θ)
moist adiabatic lapse rate
Stüve DiagramStüve Diagram B=T A=pk
Pk is on the ordinate with p increasing downward, and T on a linear scale
PROPERTIES– Dry Adiabats are straight lines– Pseudoadiabats are curved– Saturation mixing ratio lines are essentially straight– Adiabat-isotherm angle is usually ~45º– NOT an equal-area transformation of α,-p, so the area
is not proportional to energy Not as widely used anymore (Also called pseudo-adiabatic)
B=T A=pk
Pk is on the ordinate with p increasing downward, and T on a linear scale
PROPERTIES– Dry Adiabats are straight lines– Pseudoadiabats are curved– Saturation mixing ratio lines are essentially straight– Adiabat-isotherm angle is usually ~45º– NOT an equal-area transformation of α,-p, so the area
is not proportional to energy Not as widely used anymore (Also called pseudo-adiabatic)
Stüve DiagramStüve Diagram
isotherms
isobars
Dry adiabatic lapse rate
moist adiabatic lapse rate
Saturation mixing ratio
Thermodynamic Diagram ChoiceThermodynamic Diagram Choice There are many thermodynamic diagrams (Aerogram,
Clapeyron, Emagram, pastagram, Thetagram…) Most meteorologists seem to have an aversion to diagrams
other than the one with which they are most familiar Not a great deal of difference between various diagrams
– Sometimes one diagram is preferred over another because of excellence of the printing, the skillful use of color, and minimization of eye strain.
– In other cases, the deciding factor may be the presence or absence of some auxiliary graphical calculating device (ie: for rapidly calculating the distance between pressure levels)
Over-all, it seems that the tephigram and the skew T-log p are superior to all others by a small margin, but it is likely many of the major diagrams will continue for many years.
There are many thermodynamic diagrams (Aerogram, Clapeyron, Emagram, pastagram, Thetagram…)
Most meteorologists seem to have an aversion to diagrams other than the one with which they are most familiar
Not a great deal of difference between various diagrams– Sometimes one diagram is preferred over another because of
excellence of the printing, the skillful use of color, and minimization of eye strain.
– In other cases, the deciding factor may be the presence or absence of some auxiliary graphical calculating device (ie: for rapidly calculating the distance between pressure levels)
Over-all, it seems that the tephigram and the skew T-log p are superior to all others by a small margin, but it is likely many of the major diagrams will continue for many years.