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Syllabus: Physical MeteorologyMid –Term Examination

Part-I Thermodynamics

1.1.1 -Gas laws, ideal gas equation, concepts of internal energy and specific heats of gas. -First and second laws of thermodynamics.-Concepts of adiabatic process, potential temperature and entropy.-Dry adiabatic lapse rate, Moisture in the atmosphere, vapour pressure, saturation vapour pressure, relative humidity, mixing ratio, virtual temperature, dew point and wet-bulb temperature. Changes in saturation vapour with temperature-Moist adiabatic lapse rate-Statement of Normnd’s theorem, Equivalent potential temperature

1.1.2 -Geopotential. Pressure height Curve, Barometric altimetry, Standard atmosphere- T gram and its uses- Computation of height of pressure surface-Moisture varables-Study of stability by Parcel method

Post-Mid –Term Examination

Syllabus2.1.1- Clouds

- Fog and precipitation. Basic knowledge of their formationRadiation

2.2.2 -Kirchof’s law-Stefan –Boltzman Law-Wein displacement law-Planck’s law-Beer’s law and radiative equilibrium-Elementary ideas of absorption, emission and scattering of radiation in the atmosphere-Solar radiation, direct and Diffuse and their measurements-Solar constant-Albedo

2.2.3 -Terrestrial radiation-Green house effect-Simpson’s diagram-Heat balance of the earth and atmosphere-Minimum and Maximum temperature

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Thermodynamics: It is predominantly an empirical science based on axioms whose validity has been established experimentally. The treatment of problems in thermodynamics is done wholly based on such axioms or postulates. In thermodynamics, we infer the existence of certain fundamental laws from experimental evidence and then draw conclusions from these fundamental laws.

It is the study of the initial and final equilibrium state of a system which has been subjected to a specific energy process or transformation. A system means a specific sample of matter. Laboratory experiments have been shown that the equilibrium state of a system can be completely specified by a finite number of properties such as pressure, temperature and volume. These properties are known as variables of state or thermodynamic variables.Thermodynamics is based on three simple but empirically derived laws of nature, acquired from long experience. They are:

1. Ideal gas law (Equation of state of a perfect gas)

2. First law of thermodynamics3. Second law of thermodynamics

Gas laws1. Boyle’s law: States that if the temperature of fixed mass of gas is constant, the specific volume (α) of a gas is inversely proportional to its pressure (P).

i.e.1

P

P Const (1)

1 1 2 2P P (for two gases)

2. Charles Law: There are two ways of define it. The first one states that, for a fixed mass of gas at constant pressure, the specific volume (α) of the gas is directly proportional to its absolute temperature.

T

The second states that, for a fixed mass of gas held within a fixed specific volume the pressure of the gas is proportional to its absolute temperature (T).

P T

Therefore, the Charles two laws are given by:

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constT

and

Pconst

T (2)

or 1 2

1 2T T

(for two gases) and

P

T = constant or 1 2

1 2

P P

T T ( for two gases)

Ideal gas law (Equation of state of an ideal gas)

The relationship among the variables of state P and T or quantities derived from them is known as equation of state. All gases are found to follow approximately the same equation of sate over a wide range of conditions. This equation of state is referred to as the ideal (or perfect) gas equation. We have Boyle’s law for unit mass of gas as (1) & (2)respectively. On combining them we get,

P RT (3)

Or

1 1 2 2

1 2

P P

T T

(For any two gases)

Where, R is the proportionality constant called specific gas constant, which depends on particular gas under consideration. If m is the mass of a gas in kilograms one mole of the ideal gas equation may be written as

PV =m RT (4)

Where, V

m

Therefore, for one kilomole of gas it becomes i.e. when m =M

PV=MRT (5)

Where, M = molecular weight

1 Kilomole or 1 Kilogram molecular weight of a material is its molecular weight M expressed in Kilograms. For example, the molecular weight of water is 18.016; therefore one Kilogram of water is 18.016 Kg of water. According to Avogadro

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hypothesis, at equal pressures and temperatures, a kilogram molecular weight (called kilo mole) of two different gases will occupy the same volume (For an ideal gas at 273 0 K

and 1013.25 hPa this volume is 22.4 m3 . Therefore, the quantity MR

T is a constant for

all gases. Let, R* = MR, where, R* is called the universal gas constant, equal to 8314.3

deg

J

Kmole.

Then the new equation of state for one kilomole of gas is

PV = R* T (6)Therefore, for n kilomoles of gas the Equation (6) becomes,

PV = n R* T

If m kilogram is the mas of n kilomoles of gas i.e. n =m

M, we get

*mPV R T

M

Or*R

P TM

(7)

For most purpose, we may assume that atmospheric gases weather considered individually or as a mixture obeys the ideal gas equation exactly.

Ideal gas law for mixture of gases: Assumptions:

- Dry air is composed of mix. Of non-interacting gases- Obey ideal gas laws (KE is a function of temperature only and P.E ~0)- Obey Dalton’s law of partial pressure- Obey Avogadro hypothesis

Air is a mixture of gases which is observed to behave in nearly ideal fashion so long as condensation does not take place. Air obey Dalton’s law of partial pressures which states that, the total pressure exerted by a mixture of gases which do not interact chemically is equal to the sum of the partial pressures of the gases. The partial pressure of a gas is the pressure it would exert at the same temperature as the mixture if it alone occupied the volume that the mixture occupies. i.e for a mixture of K components

1 2

k

k ii

p p p P p (8)

Where, P – totl pressure and ip - partial pressure of the thi constituent of Molecular

weight iM . If each gas obeys separately the ideal gas law then

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* ii

i

R mp T

M V (9)

k

ii

p p =* i

i

mR T

V MOr

*

i

i

i

m

Mp R T

M (10)

Where, i

V

m

is the specific volume of the mixture

If we define an apparent molecular weight ___

M (mean molecular weight) as

___i

i

i

mM

m

M

(11)

i.e mass weighted harmonic mean

For dry air, ___

M = 28.97The specific gas constant for 1 kg of dry air is

*d

d

RR

M = 287

deg

J

kg

For dry air

dP R T (13)

Ideal gas for moist air:

Consider a volume V of moist air at temperature T and total pressure P which contains mass dm of dry air and mass vm of the water vapour.

The mean molecular weight ___

M of moist air is given by

___

1 1 d v

d v d v

m m

m m M MM

(14)

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dM and vM are the molecular weights of dry air and water vapor respectively.

On rearranging Equation (14) becomes

___

1 11d v d

d d v d v

m m M

M m m m MM

(15)

But, v

d

m

m = , is the mixing ratio

and v

d

M

M = =0.622, is a constant

Equation (15) can be written as

___

11 1

1dMM

(16)

Equation (3) shows that moist air has a lower apparent molecular weight than dry air.Therefore, the equation of state for moist air becomes

11

*1d

P R TM

i.e.1

*

1d

R TP

M

(17)

Equation (4) is exactly the same as the equation of state for dry air except for the factor in parenthesis, which is obviously a correction to apply to the specific gas constant for dry air to obtain the value of the gas constant for moist air containing an amount of vapor given by . However, this would mean a variable gas constant, which is awkward. Therefore, it is more convenient to retain the gas constant for dry air and apply the correction to temperature itself and define a new temperature.

1*

1T T

(18)

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This new temperature is called the virtual temperature. It is fictitious temperature that satisfies the equation of state of moist air. The virtual temperature is obviously the temperature that dry air would have if its pressure and specific volume were equal to those of a given sample of moist air.

Since water vapour is less dense than air, its presence always increases the specified volume. Hence, the virtual temperature is always higher than or equal to the observed temperature. However, even for very warm, moist air the virtual temperature exceeds the observed temperature by only a few degrees, never more than 70 K and usually less than 10 K.

Internal Energy: Internal energy of a body or a system is the sum of kinetic and potential energy of its molecules or atoms. Increases in internal kinetic energy in the form of molecular motions are manifested as increase in temperature of the body, while changes in potential energy of the molecules are caused by changes in their relative configurations. In general internal energy is a function of thermodynamic variables. For an ideal gas it is assumed that internal energy is function of temperature only, i.e. heat added at constant volume is applied only to increasing the random motion of the molecules i.e. vdu C dT . In real gases some of the energy may be used in overcoming

their inter molecular forces, but according to postulate, these do not exist in an ideal gas.Specific heat capacities of gas: Suppose a small quantity of heat is given to a unit mass of material and as a consequence its temperature increases from T to T+dT, without

change the phase. The ratio dhdT is called the specific heat capacity (C) of the material.

However, the specific heat capacity can have any number of values, depending on how the material changes its state as it receives the heat. For gas, there are mainly two types of specific heat capacities. vC and pC . If the specific volume of the gas is kept constant,

the specific heat capacity vC is defined as

vdhC dT

and Specific heat capacity at constant pressure is given as

pP

dhC dT

At constant pressure, a part of heat added is used to increase the temperature and another part is used to work for the expansion of the system against constant pressure. Whereas at constant volume no work is performed and all the heat added to the system is used to raise the temperature. Therefore, pC is greater than vC . The unit of specific heat

capacity is degJ

Kg .

First law of thermodynamics:

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This is the law of conservation of heat energy for a thermodynamic system. It is an empirical statement which tells that like other forms of energy, heat can neither be created nor destroyed. Let us suppose that a body of unit mass takes certain quantity of heat energy dh (measured in joules) which it can receive either thermal conduction or radiation. As a result the system (body) may do a certain amount of external work dw (also measured in joules). The excess of the energy supplied to the body over and above the external work done by the body is dh dw . Therefore, if there is no change in the macroscopic kinetic and potential energy of the body, it follows from the principle f conservation of energy that internal energy u of the body must increase by dh dw .i.e. du dh dw (19)

This is first law of thermodynamics, where du is the differential increase in the internal energy of the system. The work done dw by a unit mass of a system when its specific volume increases by small amount d is equal to the pressure of the substance multiplied by its increase in specific volume i.e dw pdPutting this in equation (19), we getdh du pd (20)Which is an alternative statement of first law of thermodynamics. For an isosteric process, d =0 i.e. is constant.

dh =du

i.e vdhC dT

= dudT

For an ideal gas, u is a function of temperature only.

v

duC

dT

or vdu C dT

change in internal energy is the heat transferred during an isosteric process. Putting this in Eq-20 we get

vdh C dT pd (21)

Relation between pC & vC :

On differentiating the ideal gas law, we getpd dP RdT

Or

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pd RdT dP (22) From (21) and (22) we have,

vdh C dT RdT dP

vdh C R dT dP (23)

For an isobaric process, dp=0, i.e. p is constant

vdh C R dT

i.e. p vC C R (24)

Where, pP

dhC

dT

Or

pdh C dT dP (25)

This form of the first law of thermodynamics is especially useful because the increment dT and dP are those most commonly measured in meteorology.

vC and pC for dry air are 717 degJ

Kg and 1004 degJ

Kg respectively and the

difference between them is 287 degJ

Kg , which is numerically equal to the specific gas

constant for dry air.

Adiabatic process: An adiabatic process is one in which a system changes its physical state (if it’s P, V or T change) without any heat being either added to it or withdrawn fro it. (i.e. heat transfer, dh=0)

Heat may be added to a parcel of air by many processes like radiation, friction, condensation of water vapour or turbulent transfer of heat.

From the first law of thermodynamics

dh du pd for an adiabatic process dh=0

du=-dwDuring an adiabatic expansion dw pd is positive ( i.e d = positive)

du= negative or internal energy decreases, i.e. the system is cooled during an adiabatic expansion.

Equation for a dry adiabatic process:

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For an adiabatic process, we have the first law of thermodynamics given as0 du pd (dh=0) (26)

From ideal gas law (3)RT

p

On putting this equation in equation (26) and rearranging

0p

dT R dp

T C p (27)

p

R

C = k =0.286

Or. kT const P (28)

This is known as Poisson’s equation and describes the adiabatic path in the state space of an ideal gas.

Consider an air parcel moving vertically between two levels. If vertical motion occurs on a time scale short compared to that for heat transfer, the process may be regarded as adiabatic. Under these conditions, the changes in temperature are related directly to the change in pressure through equation. Adiabatic motion turns out to be good approximation for many processes because the time scale for radiative transfer is of the order of a week and therefore long by comparison with the characteristics time scale of air displacements.

Concept of potential temperature:

In equation (28), the constant can assume different values, depending upon the initial pressure and temperature of the gas engaging in the adiabatic process. For example, if the initial pressure is 1000 hPa and initial temperature is the equation (28) can be written as

1000

kT P

(29)

Then the temperature is called potential temperature. This may be defined as the temperature that an air parcel could have if compressed /expanded adiabatically from a given state p and T to the pressure of 1000 mb. A function of state variable is a state variable. From Equation (29) , it follows that is invariant for an adiabatic process. Hence, the of an air parcel conserved for an adiabatic motion. Thus, the potential temperature can be used to label a parcel of air in order to facilitate its identification, as it moves about from day to day.

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Adiabat and isotherm: Consider a P diagram (Fig below), suppose that the initial state of a material is represented by a point A and that when it undergoes an isothermal transformation (compression) it moves along the line AB (isotherm). If the body underwent a similar change in specific volume but under adiabatic conditions, the transformation can be represented by a curve AC which is called an adiabat. The adiabat AC is steeper than isotherm AB. This is because during the adiabatic compression the internal energy increases (dh=0 and Pd is negative, From 1st law of thermodynamics du =positive) and therefore the temperature of the material rises. However, for the isothermal compression the temperature remains constant. Hence, C BT T and therefore,

C BP P .

Fig: P diagram (slope of isotherm and adiabatic process)

Reversible & Irreversible process: In all thermodynamic events, it is necessary to consider the changes that occur in the immediate environment of a system, as well as the alterations in the system under consideration.

A reversible process is one in which each state of a system is an equilibrium state. Thus the system is brought to pass through the process via an infinite number of balanced states which are infinitesimally different from each other. In such a case one can reverse the process and cause the system to return to its original state, the environment will then be found to have returned to its original state. Irreversible process proceeds in finite steps, if the system is restored to its original state by whatever means the surroundings will have suffered a change from their original conditions. Thus the term irreversible does not mean that a system cannot be returned to its original state but that the system plus its environment cannot be restored. Most of the natural processes are spontaneous and irreversible. Since, they move a system from non equilibrium state towards a condition of equilibrium.

Entropy ( ): The change in the entropy d of a system may be defined as

rev

dhd

T

(30)

Where, dh is the quantity of heat which is added added reversibly to a unit mass of the substance at temperature T. Entropy is a function only of the state using the above relation First law of the thermodynamics for a reversible transformation may be written asTd du pd (31)

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2

2 1

1 rev

dh

T

When, an irreversible process take place, some of the energy becomes unavailable for the production of work. The amount of unavailable energy is T , where, is the difference in entropy between the final and initial states of the system and its surroundings. Thus entropy is a measure of the unavailability of energy. Since work is an orderly process of motion, we can also say that entropy is a measure of disorder in systems.

Relationship between entropy and potential temperature:

For a dry adiabatic process, we have the Poisson’s equation (29) given by:

1000

kT P

By taking logarithms we get,

ln ln ln ln1000T k p k On differentiating and rearranging, it becomes

p

dT d R dp

T C p

p p

d dT dpC C R

T p

(32)

From stI law of thermodynamics (25)

pdh C dT dP

On dividing this equation by T and substituting RTp from equation (3) we get,

p

dh dT RdpC

T T p (33)

Right hand side of equation (32) and (33) are same

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p

dh dC

T

(34)

For a reversible process

rev

dhd

T

p

dd C

On integrating, we get

lnpC const (35)

For an adiabatic process is constant. From equation (35), also is constant, i.e. reversible adiabatic process also is an isentropic process.

Second law of thermodynamics

Through experience it is found that while work can always be transformed completely into heat, heat can not be transformed completely into work. The second law of thermodynamics is concerened with the maximum fraction of quantity of heat that can be converted into useful work. For any given system there is a theoretical limit to this conversion factor. The various statements of 2nd law of thermodynamics are given belw:

1. It is only by transferring heat from warmer to a colder body that heat can be converted into work in a cyclic process.2. Heat cannot itself (that is, without the performance of work by some external agency) pass from a cold to a warm body.3. In any reversible cycle the total change in entropy is zero.

0rev

dhd

T Ñ Ñ (36)

The above statements of the 2nd law of thermodynamics were done only with respect to ideal reversible processes in which a system moves through a series of equilibrium states. However, all natural processes are spontaneous and irreversible.The generalized statement of the second law can be given in terms of the following four postulates:

1. There exists a function of state for a body called the entropy (per unit mass) 2. may change either because the body comes into thermal contact with its environment ( ed ) or a result of internal changes within the body ( id ). The total

change in the entropy of a body d is given by

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e id d d (37)

3. The change ed is given by e

dhd

T (38)

Where, dh is the heat received by a unit mass of the body.

4. For reversible changes id =0 and for irreversible changes 0id f . Hence

using (37) and (38) for reversible changes,

erev

dh dhd

T T

(39)

And for irreversible changes, ed d because 0id (40)

By combining these two equations, i.e. (39) & (40), 2nd law of thermodynamics can be expressed as

rev

dh dhd

T T

Putting this in 1st law of thermodynamics we get,

Td du pd (41)

Where, equality sign applies to reversible transformations and its inequality sign to irreversible transformation.

Moisture Variables:

Water is an important constituent of atmosphere, and it occurs in the atmosphere in all three phases namely gas, liquid and solid. The amount of water vapour present in a certain quantity of air may be expressed in many different ways called moisture parameters. Some of the more important of these are considered below:

1. Vapour pressure (e): In the moist air, is the pressure exerted by water vapour when it alone occupied the volume2. Saturation Vapour pressure ( se ): Consider a small box containing air at

temperature T0 K and let the floor of the box be covered with pure water. If the air in the box is initially dry, the water will evaporate and the water vapour pressure in the air will increase. Eventually, an equilibrium state will be reached where the rate of evaporation of molecules from water will be equal to their rate of condensation on the water from moist air. When this condition is realized the air is said to be saturated with respect to the plane

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surface of pure water, and the pressure exerted by water vapour is called the saturation vapour pressure over a plane surface of pure water. Similarly, air, which is equilibrium with respect to a plane surface of ice, is said to be saturated with respect to ice and the pressure exerted by the water vapour is called the saturation vapour pressure over a plane ice surface ( sie ). The magnitudes of the saturation vapour pressures depend only on

temperature and they both increase rapidly with increasing temperature. It is found that

se is more than sie at all temperatures because water evaporate more rapidly than ice and

that the magnitude of s sie e reaches a peak value at about –12 0 C. It follows that if if

an ice particle is in water-saturated air it will grow due to deposition of water vapour upon it.3. Mixing –ratio ( ): is defined as the ratio of the mass vm of water vapour to the

mass dm of dry air.

i.e.

( )

v v v

d d

m e M e e

m e M p e p

p e

?

(42)

Mixing ratio is generally expressed in grams of water vapour per kilogram of air. In the atmosphere the magnitude of is typically a few gms per/kg in the middle latitudes but in the tropics it can reach a value of 20 g/kg. It should be noted that if neither condensation nor evaporation take place, the mixing ratio of an air parcel is a conservation quantity.

4. Saturation mixing ratio ( s ): is defined as the ratio of the mass vsm of water

vapour in a given volume of saturated air to the mass dm of the dry air.

i.e.

[ ]

vs vs vs s ss

d d s

s

m e M e e

m e M p e p

p e

?

(43)

i.e. at a given temperature the saturation mixing ratio is inversely proportional to the total pressure.

5. Relative humidity (R.H): is the ratio, expressed as a percentage, of the actual mixing ratio to the saturation-mixing ratio s at the same temperature and

pressure.

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exp se

p

s

L

C T

. 100

s

R H x

(43a)

6. Virtual temperature (T*): is the temperature which dry air would have to be in order to have the same density as a sample of moist air, assuming both have the same pressure. The equation for virtual temperature is

1 0.6* 1

1T T T

(44)

Thus T* is a function T &

7. Dew point temperature ( dT ): is defined as the temperature to which moist air

must be cooled at constant P and in order to saturate it with respect to plane surface of water. At dT the actual mixing ratio of the air becomes its saturation mixing ratio.

A meteorological process by which air can be brought to its dew point is radiational cooling of a layer of air near the ground during night. Indeed this is a primary process by which dew forms.

8. The wet bulb temperature ( wT ): is defined as the temperature to which air may

be cooled by evaporating water in to it at constant pressure until it is saturated with respect to a plane surface of pure water. e.g. An evaporating cloud droplet or raindrop will be at the wet bulb temperature.

The definition of the wet bulb temperature is rather similar to that of the dew point, but there is a distinct difference. If the unsaturated air approaching, the wet bulb has mixing ratio , the dew point temperature dT is the temperature to which air must be cooled at

constant pressure in order to become saturated. The air, which leaves the wet bulb, has a

mixing ratio ' , which saturate it at temperature T . If the air approaching the wet bulb

is unsaturated ' is greater than . Therefore, wT is greater than dT .

If T is the dry bulb temperature, then,

d wT T T (45)

Where, the equality sign apply only under saturated conditions.

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9. The lifting condensation level (LCL): is the level to which a parcel of moist air can be lifted adaibatically before it becomes saturated with respect to plane surface of water. During lifting the mixing ratio of the air and its potential temperature remain constant, but the saturation mixing ratio s decreases until it becomes equal to at the

LCL.

The Clausius –Clapeyron equation:

This is a relationship for the change in the saturated vapour pressure above a liquid with temperature. It is expressed as

2 1

sde L

dT T

(46)

Where, se = Saturated vapour pressure

L =Latent heat of vaporization 1 and 2 are specific volumes of liquid and vapor at temperature T respectively.

In case of water, 2 1 ? and the equation shows that se increases with the

temperature.

Adiabatic processes of saturated air: (Saturated –adiabatic and Pseudo adiabatic process):

It can be shown that till condensation occur process in saturated moist air is similar to that of dry air. However, as soon as condensation of the vapour occurs, large differences appear because the latent heat of condensation is released. The rate at which saturated air-cools as it expands adiabatically is smaller than the rate at which unsaturated air cools adiabatically because the part of the cooling is cancelled by the latent heat released. There are two extreme cases of adiabatic process of saturated air in the atmosphere.

When a parcel of air rises in the atmosphere, the temperature decreases with altitude at the dry adiabatic lapse rate until the air becomes saturated with water vapour. Further, lifting results in the condensation of liquid water (or deposition of ice), which releases latent heat. If all the condensation products remain in the rising parcel, the latent heat released by the phase change will warm the dry air, the water vapour and the condensation products. Consequently, the rate of decrease in the temperature of the rising parcel becomes less. Such a process is adiabatic in the sense that no heat is added from outside the air parcel, even though the latent heat released appears as sensible heat within a parcel. This process also is a reversible because later if it begins to sink, the compression will cause a rise in temperature. This warming will be converted into latent heat by evaporated water (or ice) into vapour. As a result the parcel return back to its original state. This is called saturated adiabatic process.

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If on the other hand, all the condensation products immediately fall out of the parcel of air, the process is irreversible. This is because in case of sinking motion later warming due to compression cannot be converted into latent heat due to the absence of condensation products (water or ice) in the parcel. Thus, the parcel will increase in temperature at the dry adiabatic lapse rate, which is different from the rate at which it cooled during expansion. Therefore, the parcel cannot return to its original state without alterations to its environment. Since the condensed water that falls out carries some heat with it, this process is not strictly adiabatic and accordingly is known as pseudo-adiabatic process.

The real atmospheric situation lies between two extremes, since some of the condensed water may drop out and some of it remains suspended as cloud particles in the air.

The dry adiabatic lapse rate (DALR): This is defined as the rate of change of temperature with height of a parcel of dry air, which is being raised or lowered adiabatically in the atmosphere. For such an air parcel its potential temperature is conserved ( i.e. =constant). It may be shown that

dp

dT gDALR

dz C (47)

Since an air parcel expands as it rises in the atmosphere, its temperature decreases with height so by definition ( )dDALR is a positive quantity. Substituting g =9.81 m/sec

and pC =1004, we get, d =9.8 deg/Km 10 deg/km

The saturated adiabatic lapse rate (SALR): This is the rate of change of temperature with height of a saturated air parcel, which is being raised or lowered in the atmosphere. It can be shown that

1

ds

s

p

dTSALR

ddz LC dT

(48)

Where, d is DALR, L = latent heat of condensation/evaporation

s = Saturated mixing ratio

Since, sddT

is always positive, from the above equation s < d i.e. Latent

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heat released partially compensate by cooling. Actual values of s range from about 4

deg/Km near the ground in warm humid air masses where sddT

is very large, to

typical values of 6-7 deg/Km in middle troposphere. For typical temperature near the tropopause s is only slightly less than d because moist capacity is so small that the

effect of condensation is negligible.

Equivalent potential temperature ( e ): The equivalent potential temperature is given

as

exp se

p

L

C T

(49)

e is defined as the potential temperature of a saturated air parcel when its saturation

mixing ratio, s become zero. A parcel may attain the temperature by the following

processes. From an initial saturated state T and P the air is expanded pseudo-adiabatically until all the vapour has condensed, released its latent heat, and fallen out. The air is than compressed dry adiabatically to the standard pressure of 100 hPa, when it will attain temperature e . The value of e is conserved during both dry and saturated adiabatic

process.

The concept of static stability: Here we will see the response of an air parcel in an atmosphere, which is at hydrostatic equilibrium, to the disturbing forces. When such a parcel of air is displaced by some impulse from its initial position, the atmosphere is said to be-stable at that level if the displaced parcel is forced back to its initial position, on the other hand if the parcel is subject to forces which push it further away we shall consider the atmosphere to be unstable at that level and if the displaced parcel is subjected to no net force, we shall consider the atmosphere to be in a state of neutral equilibrium at that level. Criteria for determining the stability of vertical moving air is important in understanding and predicting significant atmospheric phenomenon such as convection and turbulence .There are several methods f or the study of static stability of atmosphere such as parcel method ,slice method, entrainment theory and bubble theory etc .Parcel method of static stability analysis of atmosphere: Consider the vertical motionof an individual parcel of air with the following simplifying 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.

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Unsaturated air parcel: Consider an atmosphere in hydrostatic equilibrium with a certain lapse rate of virtual temperature (as measured by radiosonde) .Imagine a parcel of air at a level O which has the same temperature, pressure and density as its surroundings (i.e. both system and surrounding are at hydrostatic equilibrium). There will be no net vertical force on this parcel. Now suppose the parcel is raised to the level defined by points A & B. If the parcel remains unsaturated, it will expand and cool at the DALR. If

d , the parcel will be at lower virtual temperature ( AT ), then its environmental

temperature BT at this level. Since the parcel immediately adjust to the pressure of its

environment it is clear from the ideal equation that the colder parcel must be denser from than the warmer environmental air. Therefore, the parcel tends to return to its original level (O). Thus, the condition d represents the stable (Positive static stability) for

unsaturated air (In general, larger difference between d the greater the restoring

force for a given displacement and the greater the static stability). At the original level it will once more be in buoyant equilibrium with no net vertical force. However, it will have a certain downward momentum which will carry it beyond this point (original level). As it sinks, it will warm at the rate d and will find itself warmer and lighter

then its new environment, thus, there will be net upward force which will ultimately reverse its motion and send it upward. In this way in the stable case the displaced parcel continue to oscillate about its original position until viscosity robs the oscillation of its energy.

If d , a parcel unsaturated air displaced upward from O will find itself at A with a

temperature greater than that of its environment. Therefore, it will be less dense than the environmental air and will be subjected to a net upward force. In this case, the parcel will continue to move upward and will not return to its original location. Similarly, a parcel displaced downward will continue to move down. This condition represents unstable case (Negative static stability). Such unstable situations generally do not persist in the free atmosphere, since the instability is eliminated by strong vertical mixing as fast as it forms. The only exception is in the layer just above the ground under the condition of very strong heating from below.

R.K.Giri 21

Finally, the condition d corresponds to zero or neutral stability. When unsaturated

air parcels are displaced upwards or downwards, it will have the same pressure and density as its environment. Therefore, it will be subject to no net force in either direction. In summary, the stability criteria for an unsaturated air parcel displacements are as given below:

d

d

d

stable

Neutral

Unstable

For a saturated air parcel:

s

s

s

stable

Neutral

Unstable

Saturated air: If a parcel of air is saturated, its temperature will decrease with height at a saturated adiabatic lapse rate (SALR). If follows from arguments similar to those applied for unsaturated air if, is the actual lapse rate of temperature in the atmosphere saturated air will be stable, neutral or unstable with respect to vertical displacements, depending on conditions said above.

Conditional instability: Suppose of the atmosphere lies between s and d . An

initially unsaturated air parcel forced to ascend will be stable since d . But if the

impulse forcing the air upward last long enough the parcel will reach the LCL and become saturated. Then the parcel lapse rate immediately become less then that of environment and instability results, known as conditional instability. Thus, in this case where, d s , the parcel is stable with respect to unsaturated lifting process and

unstable with respect to saturated lifting process.

R.K.Giri 22

In summary, five states can be recognized with respect to parcel displacement in an atmosphere of lapse rate . The atmosphere is said to be:

s

s

s

d s

d

d

stable

Absolutly stable

Neutral

Conditionally unstable

Dry neutral

Absolutely unstable

Thermodynamic Diagrams: A thermodynamic diagram provides graphical display of the lines representing the major kinds of processes to which air may be subject, namely isobaric, isothermal, dry adiabatic and pseudoadiabatic processes. Thermodynamic diagrams which are used to study thermodynamic process which occur in the atmosphere are often referred to as aerological diagrams. Any thermodynamic diagram should have the following desirable characteristics:1. The area enclosed by the lines representing any cyclic process be proportional to the change in energy or the work done during the process.2. As many as possible of the fundamental lines should be straight.3. The angle between the isotherms and the dry adiabats shall be as large as possible.

The Tephigram (T gram):

The Tephigram is a thermodynamic diagram. This was adopted for use in meteorology for the first time by Sir Napier Shaw. Its coordinates are temperature (T) and entropy ( ). Hence its name is Tephigram. Since entropy ( ) is related with the potential temperature ( ) by the equation (35) as lnpC const , in addition to entropy ( )

potential temperature ( ) also has been labeled along Y axis.

In T gram, the vertical lines parallel to the axis are isotherms and horizontal lines, parallel to T axis represents to dry adiabats. The isobars on the T gram can be

obtained using the Poisson’s equation, i.e. 1000

p

RCT P

, by putting the various values

of T and computing for a constant pressure. Since one coordinate of this diagram is logarithmic ( lnpC ), but the other is a linear scale of T the isobars are logarithmic

curves which slope upward to the right and decrease in slope with increasing in temperature. However, in the normal range of meteorological conditions the isobars have only gentle curvature and are nearly straight. The pseudoadiabats are appreciably curved, but the saturation mixing ratio (isohygric) lines are nearly straight on Tephigram.

Uses of Tephigram:

R.K.Giri 23

1. To derive meteorological parameters such as T* , , e , R.H etc. at various

levels from the aerological ascents.2. To study the stability of atmospheric layers3. To compute height of a pressure level.4. To compute precipitable water in an atmospheric layers.

Normands Rule:

States that on a Tephigram the LCL of an air parcel is located at the intersection of the following lines.1. Dry adiabat through the point located by the temperature T, and pressure P of the air parcel.2. Pseudoadiabat through the point located by the wet bulb, temperature wT and

pressure P of the air parcel.3. Saturation mixing ratio lines (isohygric) through the point determined by the dew point, dT and pressure, P of the air parcel.

Standard atmosphere:

International standard atmosphere (ISA): It has been specified the average atmosphere (hypothetical) which is necessary for various purposes like testing of aircraft, evaluation of aircraft performance, calibration of altimeter etc. The most widely used is one defined by ICAO, known as ISA. Its specifications are:

Air is dry Temperature at mean sea level (MSL) 150 C Pressure at MSL 1013.25 hPa Density at MSL 1225 g/m3

Acceleration due to gravity 980.665 cm/sec2

Lapse rate up to 11 km 6.5 0C/km Temperature is assumed constant -56.5 0C from 11 km to 20 km From 20 km to 30 km there is a rise of temperature at the rate of 1 0C/km

with a temperature of -44.5 0C at 32 km.

Barometric altimetry:In pressure altimetry the atmospheric pressure is measured by aneroid barometer and elevation of that level is determined from the standard atmosphere. The possible source of errors at the time of observation can be MSL pressure and temperature. In mountain

R.K.Giri 24

waves an aircraft is subjected to rapid height fluctuations, due to up and down currents. The aneroid capsules of the altimeter are unable to respond correctly to these fluctuations. This is called Hysteresis effect. Altimeter also gives erroneous readings if the temperature differs from ISA. If colder than ISA, the readings are too high and vice versa. Hence altimeter readings are not reliable and deceptive when used for terrain clearance. Geo-potential: Defined as the potential energy imparted to a unit mass if it is lifted from sea level to some height z.

0

( )z

z gdz (J/kg) (50)

Geo-potential height: 0

gZg

Where, 0g is standard gravity at MSL. It is an

adjustment to geometric height with the variation of gravity with latitude and elevation.The influence of the acceleration of gravity on atmospheric properties varies with latitude and with altitude as the acceleration of gravity similarly varies with these parameters.One method for normalizing the variation of atmospheric properties with respect to the variation of the acceleration of gravity is to express the values of the atmospheric properties as a function of geopotential. The relationships between geopotential and geometric altitude are then used to apply the atmospheric data to the proper altitudes at any particular latitude. (Geo-potential meter = 0.98 dynamic meter)

Pressure height Curve: The hydrostatic balance can be written as:

pg

zp gp

z RT

(51)

Orz RT

p gp

Noting that,

lnP

p p

ln

z RTH

p g

Where, H is the vertical scale height. For an isothermal atmosphere ‘z’ varies as ln p or we can say that p varies exponentially with ‘z’.The height of the given pressure surface is dependent on the average temperature below that surface and the surface pressure sp . Thus integrating Eq-50 we have:

R.K.Giri 25

sp

p

T dpz p R

g p (52)

OrThickness of an atmospheric layer, sandwiched between two pressure surfaces, such as 1p and 2p .

1

2

2 1

p

p

T dpz z R

g p (53)

Fig: warm column of air expand, cold columns contract, leading to a tilt of pressure surfaces, a tilt which typically increases with height in the troposphere.

References:

1. S.L. Hess: (1952) An introduction to theoretical Meteorology, Holt, Rinehart and Winstin, Newyork

2. J.M Wallace and P.B. Hobbs: (1977) Atmospheric Science –An Introductory Survey, academic Press, New York

3. P.N. Sen: (2000) Atmospheric Thermodynamics –Lecture notes of the first SERC School on Cloud Physics and Atmospheric Electricity –Fundamental. IITM, Pune, India

4. D.K. Paul: (2000) Basic concepts in Meteorology – Lecture notes of the first SERC School on Cloud Physics and Atmospheric Electricity –Fundamental. IITM, Pune, India

5. G.J. Haltiner and F.L. Martin: (1957) Dynamical and Physical Meteorology, McGraw Hill Book Inc, New York

6. J.R. Holton: (1992) An Introduction to Dynamic Meteorology, Third Edition, academic Press, New York.

7. I.C. Joshi: (2009) Aviation Meteorology, second edition, Himalayan Books, Pvt Ltd.

8.

R.K.Giri 26

Appendices:

DALR

1000

p

RCT P

Taking log on both sides—

ln ln ln ln1000p

RT p

C

Differentiating with respect to z we get,

1

p

dT R dp

T dz PC dz

Hydrostatic Equation dp

gdz

1

p

dT Rg

T dz PC

Therefore,

p

dT RTg

dz PC

P RT

P RT

p

dT gCdz

Or DALR = d = p

gC 100 C/km

SALR: From 1st law of thermodynamics

pdh C dT dP

Hydrostatic equation dp

gdz

dP gdz

pdh C dT gdz

R.K.Giri 27

Now if the saturation mixing ratio ( s ), the quantity of heat dh released into or

absorbed from a unit mass of dry air due to condensation or evaporation of liquid water is

sLd , where L is the latent heat of condensation. Therefore,

s pLd C dT gdz If we neglect the small amounts of water vapour associated with unit mass of dry air, which are also warmed or cooled, by the absorption of latent heat, then pC is he specific

heat at constant pressure of dry air. Dividing both sides by pC dz we get,

s

p p

ddT L g

dz C dz C

s

p p

ddT L dT g

dz C dT dz C

But,

d = p

gC

SALR =

1

ds

s

p

dT

dz dLC dT

Equivalent Potential Temperature

pdh C dT dP Dividing by T, we get

pC dTdh Rdp

T T p

1000

p

RCT P

ln ln ln ln1000p

RT p

C

ln ln ln ln1000p

RT p

C

On differentiating,

p p

d dT dpC C R

T p

Or

R.K.Giri 28

p

dh dC

T

sdh Ld

s

p

dL d

C T

Or

s s

p p

d LLd

C T C T

;

s

p

L dd

C T

Integrating above equation

lns

p

Lconst

C T

0s

T

as e

lns

p e

L

C T

exp se

p

L

C T

e is the equivalent potential temperature 0s . Therefore, the air is expanded

pseudoadabatically until all vapours have been condensed, released its latent heat and fallen out. The air is then compressed dry adiabatically to the standard pressure of 1000 hPa when it attains the temperature e . If the air is compressed dry adiabatically to the

surface pressure level instead of the standard 1000 hPa the temperature is called Equivalent Temperature eT .

exp se

p

LT T

C T

We know that the potential temperature is conserved in adiabatic process. e , however

conserved both dry and saturated adiabatic process. The lines of constant e are known

as Pseudoadiabats.

If the Pseudoadiabat traced up to 1000 hPa isobar, the temperature of this intersection is called wet-bulb potential temperature ( w ). w is also conserved for dry and saturated

adiabatic process.

Convective or Potential Instability

R.K.Giri 29

When vertical motion affects the whole layer of air, instead of a small parcel of air, the lapse rate of the air undergoing a vertical displacement is changed so that the stratification may become unstable when it was stable to begin with. For a particular vertical distribution of humidity, forced ascent of stable layer causes it become unstable. The layer has a lapse rate s and so it is stable in the sense that it resists the motion of

he air parcel through it. Suppose, however that layer is lifted bodily viz ascent over high ground, in such a way that the same pressure depth is maintained. With the particular distribution of humidity the base of the layer becomes saturated on ascent sooner than does the top of he layer. The fraction of the ascent at DALR is therefore greater for the top of the layer than the base and the lapse rate within the layer steepens. Since each part of its ascents, after saturation, the lapse rate in the layer after the whole of it saturated exceeds s . This type of instability, where the layer was stable originally, but become

unstable owing to vertical lifting is called Potential instability or Convective instability.

Variations of saturation vapour pressure with temperature

2 1

sde L

dT T

(Clausius-Clapeyron equation)

In the atmosphere the specific volume of water vapour is more than specific volume of liquid water and during this transformation liquid to vapour we need certain amount of heat. Therefore keeping the latent heat of vaporization constant with temperature, we want to see the variation of saturation vapour pressure.

2

s vde L

dT T

2s ve R T

2v

s

R T

e

R.K.Giri 30

2s v s

v

de L e

dT R T

Integrating above equation assuming vL constant, we have,

00 02

s

s T

e T Ts v

e Ts v

de LdT

e R T

0 0 0

1 1 1 1ln s v v

s v v

e L L

e R T T R T T

00

1 1( ) exp v

s sv

Le T e

R T T

Where, 0se is known at 0T , at T=0 0K, 0se =611 Pa Lv = 2.50x106 J kg-1

( ) exps

Be T A

T

Where,

00

exp vs

v

LA e

R T

=2.56 x1011 Pa

v

v

LB

R =5.42 x 103 K

As the air temperature increases the amount of water vapor required for the air to become saturated increases at an exponential rate. Similarly, as air is cooled the amount of water vapor required for the air to become saturated decreases at an exponential rate.

Questions:1. From the Fig below what will be the value of dh in each cases??

R.K.Giri 31

2. In reference to first law of thermodynamics the quantity of dh dw remain ----------for all processes 3. In a cyclic process dh = -------.

R.K.Giri 32

4. In free expansion if the process is adiabatic then there is no work done by the system and surrounding and the change of the internal energy of the system is zero. (T/F)5. An accidently dropped egg in a cup is splatters but it can not forming whole egg by reverse process. This is an example of -------------process.6. Pseudoadiabatic process in the atmosphere is an --------------process.7. There is difference in entropy and energy as the former is not obeying the conservation law. The entropy of a closed system is always increases but energy remains constant. (T/F) 8. The second law of thermodynamics can be written in terms of entropy as 0d (T/F).9. What is the significance of temperature inversion?10. In equation (50) the integral of geopotenatial is exact, so it will depends on the ------taken.11. What is hypsometric equation??12. Calculate the thickness between the 1000 and 500 mb levels for an atmosphere with __

vT =250 K

13. Lower constant pressure heights correspond to ---- column temperature and near the poles in the atmosphere.14. For an adiabatic process potential temperature is ----------.15. Dew point temperature always indicates the --------------amount of water vapour in the air. 16. How wet bulb temperature differs from dew point temperature?17. The potential temperature of the air parcel continues to decrease as it is lifting above LCL. Comments?18. Saturated adiabatic process is -------- and --------------process. 19. Saturated adiabatic process and pseudoadaibatic processes are virtually identical. Comments ??20. In thermodynamic diagram like T gram the adiabats are also called isenropes (T/F)