ASL410 T-Phi Diagram

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The Parcel Method

In this simple approach we shall consider thevertical motions of an individual parcel of air

with the simplified assumptions

1. No compensating motions occur in the environment

as the parcel moves

2. The parcel does not mix with its environment and

so retains its identity

Neither of these two assumptions is completely justifiable



The stability criteria in this simple approach, canbe obtained without the formal use of

mathematics. Consider an atmospheric inhydrostatic equilibrium with a certain lapse rate of virtual temperature, . Imagine a parcel of airwhich initially has the same temperature,

pressure and density as its surroundings. Therewill be no net vertical force on this parcel since itis in hydrostatic equilibrium. Such an element of air is floating. Now suppose the is given a smallupward displacement by some external agency. If the parcel remains unsaturated it will expand andcool at dry adiabatic rate

d = g/Cp



i. If the environment lapse rate is less than the dryadiabatic lapse rate is

< d

the parcel will be at lower virtual temperature than itsnew surroundings. Since the pressure n the parcel veryquickly becomes equal to the pressure in theenvironment, it follows that the parcel will be more

dense than its surroundings and will not be buoyant, butwill sink back to the original level. These conditionsrepresents the stable case.

In fact in the stable case, the displaced parcel continuesto oscillate about its original position until viscosity robs

the oscillation of its energy. While sinking, the parcel willaccelerate downward because of the part gravity whichis not exactly cancelled by the vertical gradient of pressure.



As it sinks it will warm at the rate d > and will finditself warmer and lighter than its new environment.Thus there will be net upward force which will

ultimately reverse its motion and send it upward.This type of upward and downward motions willform the oscillations under the stable case.

ii.In this case d < , a parcel displaced upward will find

itself with a temperature greater than that of the newenvironment. Consequently it will be lighter than thesurroundings and will be subject to a net upwardforce. In this case the parcel will continue to move

upward and will not return to its original location.Similarly, a parcel displaced downloads under theunstable case will continue to move down.



iii. Finally, when d = , a parcel displaced upward ordownward will have the same temperature and densityas its surroundings. Consequently, it will be subjected to

no net force in either direction i.e. it will attain neutralequilibrium.

In summary the stability criteria for unsaturatedparcel displacements are,

d > ; stable

d = ; neutrald < ; unstable

Thus the dry adiabatic lapse rate is dividing linebetween mechanical stability and instability for dryair. The homogenous atmosphere, which has a lapserate for greater than d, is highly unstable and isfound only in shallow layers near the ground whereviscous and turbulent effects are dominant.



When the parcel of air is saturated, its lapse rate

is not dry adiabatic d but moist adiabatic s. The

same stability criteria apply in the saturated caseexcept that one must compare the environmental

lapse rate to the moist adiabatic value instead of

the dry adiabatic value. Since the moist adiabatic

lapse rate is smaller, it is easier to obtain

instability for saturated air than for unsaturated

air



Conditional Instability Suppose the environment lapse rate lies between

the moist and dry adiabatic values. An initiallyunsaturated parcel forced it ascend will be stable

since d > . But if the impulse forcing the air

upward lasts long enough the parcel will reach thelifting condensation level and become saturated.

Then the parcel lapse rate lapse rate immediately

becomes less than that of the environment and

instability results, known as conditional instability.

In this case when the environmental lapse rate lies

between d and s, a parcel is stable w.r.t. saturated

lifting process.



In the unsaturated case till d > it is stable. With

saturation one has to consider s which is less

than d and hence s < and hence it becomesunstable.

The adjacent tephigram may be used to explain

conditional instability. A parcel forced to ascenddry adiabatically from 1000mb at first is colder

than environment and is subject to negative or

downward buoyancy

Lifting condensation level (saturation occurs)

R Level of free convection

Shaded Area Latent instability





It reaches the LCL at 900mb and ascends moist

adiabatically thereafter. The parcel curve intersects the

environ curve at 800 mb, thus defining an area which liescompletely to the left of the sounding curve. This is

called the negative area since it is proportional to the

energy which must be supplied to the parcel in order to

lift it this high. Above 800mb the parcel is warmer than the sounding

and is subject to positive or upward buoyancy. From 800

to 500mb there is positive area to the right of the

sounding. This signifies positive area to the right of thesounding. This signifies positive area since it is

proportional to the energy which becomes available to

the parcel from the environ.



In summary five states can be recognize w.r.t. parcel displacement

in an atmosphere of lapse . The atmosphere is said to be

1. Absolutely stable if < s

2. Saturated neutral if = s

3. Conditional unstable if s < < d

4. Dry neutral if = d

5. Absolutely unstable if > d

The conditions that must e fulfilled before the element

may become unstable w.r.t. its surrounding are

1. A sufficient amount of moisture for the moving parcel to

become saturated soon.2. A mechanically produced lifting of sufficient strength to

overcome the stability forces at the lower levels and to carry

upwards.



Thermodynamic Diagrams The primary function of a thermodynamic diagram is

to provide a graphical display of the linesrepresenting the major kinds of process to which air

may be subjected, namely isobaric, isothermal, dry

adiabatic and pseudo adiabatic. The characteristic

properties of the thermodynamic diagram are:

i. Since energy changes are of primary importance, the

first desirable characteristic of such a diagram is that the

area enclosed by the lines representing any cyclic

process be proportional to the change in energy or thework done during the process. This is such an important

property that the designation thermodynamic diagram is

often reserve for those in which the area is proportional

to energy or work.



ii. The second desired characteristic is that as many as

possible of the fundamental lines be straight. The

more a diagram satisfies this criterion the easier itwill be to use.

iii. The angle between the isotherms and the dry

adiabats shall be as large as possible. When sounding

of the upper atmosphere are plotted.



For an element of specific work we write

dw = pd

And hence in order to satisfy the first criterion one should have p and ascoordinate.

However, the angle between the isotherms and adiabats of an , -p diagram is

quit small and hence does not satisfy the third criterion.

We must seek a means of setting up other suitable diagrams in which thecoordinates are too functions of thermodynamic variables, subject to the

restriction that the area enclosed by any cycle in the new diagram shall be

equal to area enclosed by the same cycle on a , -p diagram. Such a diagram

is called an equal area transformation of the , -p diagram.

We may then examine these new diagrams to see how well they satisfy the

other two criteria.



Consider two variables A and B. Let each be a function of one or more

thermodynamic variables. Since a thermodynamic variable is determined

by the sate of a system it suffices to know and p for a parcel in order to

determine A and B.



Thus each point on an , -p diagram corresponds to a point on an A, B

diagram and any closed cycle on one is a closed cycle (perhaps of different

shape) on the other. We shall require that the area enclosed on one diagram be

equal to the area enclosed on the other. This ensures that an A, B plot will be

a thermodynamic diagram.

Thus,

´ ´!

dBAd pfor any given cyclic process.

´ ! (1)0AdBd pThus

For the above integral to be zero, the integrand must be an exactdifferential-for example

pd + AdB = ds (2)

where S = f(, B)



From calculus,

)3(, dB B

S

d

S

BdS B EEEE ¹ º

¸

©ª

¨

x

x

¹ º

¸

©ª

¨

x

x

!

Comparing (2) and (3), sufficient conditions for an equal area transformation

are:

EE ¹ º

¸

©ª

¨

x

x

!¹ º

¸

©ª

¨

x

x

! B

S

A

S

p Band

Differentiating first w.r.t. B and the second w.r.t. ,

¹¹ º

¸

©©ª

¨

xx

x!

¹ º

¸

©ª

¨

x

x

¹¹ º

¸

©©ª

¨

xx

x!

¹ º

¸

©ª

¨

x

x

B

S A

B

S

B

p

B EEEE

22

and

)4(.,.EE¹ º

¸©ª

¨xx

!¹ º

¸©ª

¨xx

B

p Aei

B

is the sufficient condition for the area

to be equal.



The Tephigram

This diagram may be developed by letting B=T as for the emagram. Thus

A = R ln +F(T)

But this time instead of substituting from the equation of state let us

introduce potential temp from Poisson¶s eqn:k k

RT pT

¹¹ º

¸

©©ª

¨

!¹ º

¸

©ª

¨

! EU 10001000

Taking logarithms

ln T ± ln = k (ln T+ln R ±ln 1000 ±ln )

or, ln =(1/k) (ln -ln T) + ln T + ln R ±ln 1000

Or, R ln = C p ln +G(T)

where G(T) (1-C p) ln T +ln R ±ln 1000

A = C p

ln + F(T) +G(T)



Let us choose the arbitrary function

F(T) = -G(T)

So the coordinate become

A = C p ln

B = T

Since C p ln is equal to the entropy, apart from an additive constant,Sir Napier show, who introduced diagram, called it the T- diagram or

tephigram for short.

The equation of the isobars on the tephigram may be obtained by

taking the logarithm of Poission¶s equation. For constant values of p,

l n = l n T +Const



Since one coordinate of its diagram is ln but the other is a linear scale of

T, the isobars are logarithmic curves which decrease in slope withincreasing temp. In rather restricted range of meteorological conditions the

isobars have only gentle curvature and are nearly straight. It is possible to

rotate the diagram clockwise so that the isobars are essentially horizontal

with pressure decreasing upward as it does in the atmosphere. However this

is not absolutely necessary.The pseudoadiabats are appreciably curved, but the saturation mixing ratio

lines are nearly straight on a tephigram.



But the very nature of the diagram, the angle between isotherms and adiabats

is exactly 90o. This angle is double that of emagram and hence is the greatest

advantage of the tephigram.

Thus in summary, the tephigram has

(1) Area energy

(2) 4 sets of lines exactly straight or nearly straight and only 1 set quit

curved.

(3) Isotherm-adiabat angle large.

This diagram is used indely.

T- gram continued

= T(1000/p)(R/C p)

Taking natural logarithm and then differentiating,

l n = l n T + (R/C p) (ln 1000 - ln p)

C p d (ln)=C p d(ln T) ±R d (ln p)=C p (dT / T)±R (dp / p)=C p (dT/T)±(/T) dp



Multiplying with µT¶, C p T d (ln ) = C p dT ± dp = dH

From law of thermodynamics

H = C p dT - dp

=C p T d(ln T) - p d(lnp)

=C p T d(ln T) - R T d(lnp)

=C p T d(ln )For finite process,

Uln2

1

2

1d T C dH p´´ !

This relation suggests co-ordinates of x = C p

ln and y = T as in Tephi

diagram so as to have are representation.

Going in to the concept of entropy,

(H / T) =C p d(ln ) = C p d(lnT) ±R d(lnp)

The r.h.s is an exact differential.



For an ideal gas d = H / T

So the entropy = C p lnT ±R lnp +Constant

= C p ln +Constant

increases or decrease as heat is absorbed or removed.

is constant for dry adiabatic process.

Also a process where is constant is called the isentropic process.The constant of integration may be determined for any arbitary reference

level. Normally,

= c at p = 1022 mb and T =200oK

The relation = C p ln + Constant shows that a diagram with coordinatesT, C p ln has in effect a linear entropy scale.

Also 12

2

1

2

1JJJ !! ´´ T d T dH

Where is constructed in such a way that the areas A and B are equal.T





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ASL410 T-Phi Diagram