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**Properties of the Atmosphere:**

The terrestrial atmosphere is classified by given properties: density, pressure, temperature, composition and motion. Conventional meteorology handles with such quantities from sea-level to height of approx 20 kilometers. In past, kites, balloons, and aircraft were used to access such regions. Though, as experimental methods enhances, probing of the atmosphere extended in height. Today, rockets and satellites are being used to take measurements at much higher altitudes.

**Vertical Diminution of Density with Height:**

Density of atmosphere reduces with height. This vertical decrease of density with height is one of the most outstanding characteristics of terrestrial atmosphere. In practice, it may be supposed that atmosphere is horizontally stratified. Obviously, gravity is the root cause of this stratification. Hydrostatic equation defines balance between the gravitational force at any point and pressure gradient dp/dz at that point. Quantitatively, hydrostatic equation may be written as

dp/dz = -ρg

This signifies that

dp = -ρgdz

Here p is pressure, ρ is mass density, g is the acceleration due to gravity and z is height. Consider an ideal gas. Equation of state for ideal gas is pV = nkT'

Here p is pressure, V is volume, n is number of molecules, T' is absolute temperature and k is Boltzmann constant. Though, state temperature T (in energy units) such that ' T = kT'. In this situation, equation of state reduces to pV = nT or P = (n/V)T or

p = NT

Here N = n/V is number density.

dp/p = -(ρg/NT)dz

dp/p = -(1/H)dz where 1/H = (ρg/NT)

Of course, by making H subject of the formula, we obtain:

H = (N/ρ)(T/g) or H = (1/m)(T/g)

Here 1/m = N/ρ

So that m (that is equivalent to ρ/N = ρV/n) is mean molecular mass at given height. Quantity H is known as scale height. By integrating equation from some reference height z = z_{0} (at pressure p = p_{0 }) to arbitrary height z (at pressure p ) we obtain:

_{p0}∫^{p}dp/p = -_{z0}∫^{z}dz/H

[log_{e}p]_{po}^{p} = - _{z0}∫^{z}dz/H

log_{e}(p/p_{0}) = - _{z0}∫^{z}dz/H

p = p_{0}exp(-_{z0}∫^{z}dz/H)

Given equation illustrates how pressure diminishes with height. At reference point z_{0}, pressure is p_{0} and the number density is N_{0} so that Equation yields

p_{0} = N_{0}T_{0}

At arbitrary height z, ideal gas equation is

p = NT

Thus, by taking ratio, you obtain p/p_{0} = NT/N_{0}T_{0}

So that p = p_{0}(NT/N_{0}T_{0}) or p_{0}exp(-_{z0}∫^{z}dz/H) = p_{0}(NT/N_{0}T_{0})

or N = (N_{0}T_{0}/T)exp(-_{z0}∫^{z}dz/H)

Here T_{0} is temperature at z_{0}. From equations it becomes clear that scale height H is very significant in determination of pressure and density distributions.

**The Isothermal Atmosphere:**

Isothermal atmosphere is one in which temperature stays constant. This is highly idealized case. In practice, it is not possible to find perfectly isothermal atmosphere. Though, in region of upper thermosphere, temperature profile illustrates very little variation with height. Possible clarification for this Isothermality is existence of high thermal conductivity that tends to smooth out any temperature gradient. This takes place above height of approx 300km. Atmosphere in this region tends to attain thermal equilibrium, so that finally temperature approaches constant value. At such heights, there is practically no mixing. In result, different gases tend to exist in separate portions of space. Different gases that make up atmosphere are distributed with partial pressure provided by constituent gases. Assume that scale height H_{i} of ith gas is provided by

Hi = T/m_{i}g

Consider the limited region of atmosphere in which gravity stays practically constant. By using equations expression can be written as:

p_{i} = p_{0i}exp(-_{z0}∫^{z}dz/H_{i}) that simplifies to

p_{i} = p_{oi}exp[-(z-z_{0}/H_{i})]

By extension, density distribution is found in the given way:

p_{i} = p_{oi}(N_{i}T/N_{0i}T)

Thus p_{i} = p_{0i}(N_{i}/N_{oi}). Also p_{i} = p_{oi}exp(-_{z0}∫^{z}dz/H_{i})

By solving equation we get

N_{i} = N_{oi}e^{-(z-z0)/Hi}

Above equation shows that scale height, in the isothermal atmosphere, is e-folding distance for pressure and density. Large scale height is related with a light gas. Therefore, it is practical to expect that light gas will predominate at adequately high altitudes.

**Adiabatic Atmosphere:**

In lower atmosphere, convective motions are establish when atmosphere is heated from below. Consequently, gases that make up atmosphere are thoroughly mixed so that average mass stays practically unchanged with height. In this region, thermal conductivity is very low. Due to slow conduction, gas expands and contracts in a way that is nearly adiabatic as it moves from one part to another.

Equation describing adiabatic procedure is

pV^{γ} = k, Where k is constant and γ is ratio of principal specific heat capacities. That is,

γ = c_{p}/c_{v}

Here c_{p} is specific heat capacity at constant pressure and c_{v} is specific heat capacity at constant volume. The volume V is ratio of mass m to density ρ. That is, V = m/ρ. Thus, adiabatic equation becomes p(m/ρ)^{γ} = k, hence p = k(ρ/m)^{γ}

Hence p = Aρ^{γ}

Here A is constant defined by

A = k/m^{γ}

Differentiating p with respect to ρ, you obtain

dp/dρ = Aγρ^{γ-1}

In order that change in pressure dp is provided by

dp = γAρ^{γ-1}dρ

By eliminating p from hydrostatic equation and adiabatic equation

These yields

γAρ^{γ-1}dρ = -ρgdz or γAρ^{γ-2}dρ = -gdz

Integrating expression given above, and simplifying result, we get:

γA _{ρ0}∫^{ρ}ρ^{γ-2}dρ = -g _{z0}∫^{z}dz

(γA/γ-1)(ρ^{γ-1 }- ρ_{0}^{γ-1}) = -g(z-z_{0})

ρ^{γ-1} - ρ_{0}^{γ-1 }= -(γ-1/γA)g(z-z_{0})

ρ =[ρ_{0}^{γ-1} - g(γ-1/γA)(z-z_{0})]^{1/1-γ}

Equation signifies that mass density ρ decreases continuously with height. Interesting result is attained when ρ is set equivalent to zero.

In that situation, [ρ_{0}^{γ-1} - (γ-1/γA)g(z-z_{0})]

z-z_{0} = z-z_{0}p_{0}/ρ_{0}g(γ-1)

Here p_{0} (≡ Aρ_{0}γ ) represents adiabatic equation at z = z_{0}. This result illustrates that density of adiabatic atmosphere vanishes at certain height given by Equations. Therefore, adiabatic atmosphere should have a finite height.

**Temperature Profile of Adiabatic Atmosphere:**

Derive the expression for temperature profile of adiabatic atmosphere in given way. Equation shows that p = NT and equation illustrates that p = Aρ^{γ}

From two equations, we get that

NT = Aρ^{γ} or T = (A/N)ρ^{γ}

But, mass density ρ is expressed as follows:

ρ = (Number of particles x Mass of individual particles)/Volume or ρ = nm/V

Therefore V = nm/ρ

By definition of number density N, the equation is

N = n/V therefore V = n/N

Therefore, equating two expressions for volume, we get

nm/ρ = n/N => m/ρ = 1/N Therefore N = ρ/m

Therefore, temperature profile T of adiabatic atmosphere is provided by:

T = [A/(ρ/m)]ρ^{γ }thus Amρ^{γ-1}

Using Equation, we get

T = Amρ_{0}^{γ-1 }- mg(γ-1)/γ(z-z_{0})

Therefore, temperature gradient becomes

dT/dz = -mg(γ-1)/γ

The temperature gradient is also called the lapse rate.

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