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.