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*Layers of the Atmosphere:*

The atmosphere of the earth may be categorized in terms of:

(i) Temperature

(ii) Ionization

(iii) Magnetic field, and

(iv) Radio wave propagation.

In terms of temperature, atmosphere may be categorized in four distinct layers, that is:

(i) The troposphere

(ii) The stratosphere

(iii) The mesosphere, and

(iv) The thermosphere.

In terms of magnetic field the atmosphere may be categorized into two layers, that is:

(i) The dynamosphere, and

(ii) The magnetosphere.

In terms of radio wave propagation, terrestrial atmosphere may be classified categorized into two:

(i) troposphere, and

(ii) ionosphere.

Ionosphere is dispersive medium in which propagation is frequency dependents. Troposphere is a non-dispersive medium specifically largely neutral.

*Governing Equations:*

**Equation of State:**

Charles' law defines that volume of fixed mass of gas is directly proportional to temperature, given that pressure remains constant. Therefore, at constant-pressure, all gases expand by the constant amount (equal to 1/273 of volume at 0^{o}C) for every 1^{o}C rise in temperature. This is called as Charles' constant-pressure law. Conversely, if volume is remaining constant, all gases experience increase in pressure (equal to 1/273 of their pressure at 0^{o}C) for every 1^{o}C rise in temperature. This is called as Charles' constant-volume law. Therefore, Charles' law defines that ratio of volume to temperature of fixed mass of gas is constant i.e. constant, V/T = constant if the pressure remains constant.

Boyles' law defines that volume of fixed mass gas is inversely proportional to pressure, if the temperature remains constant. Therefore, at constant temperature product of pressure and volume remains constant. This signifies that

pV = constant

In light of the laws, equation of state for ideal gas may be devises as follows:

pV = nkT'

Here p is pressure V is volume, n is number of molecules, T' is absolute temperature and k is Boltzmann constant. Equation of state reduces to pV = nT or p = (n/V)T

By defining number density N as number of molecules per unit volume:

N = n/V

Therefore, Equations may be written as:

p = NT,

Here N is number density.

**Equation of Hydrostatic Equilibrium:**

Equation of hydrostatic equilibrium expresses balance between gravitational force at any point and pressure gradient at that point. Therefore, hydrostatic equation may be defined as

dp/dz = -Ρg

Here p is pressure, Ρ is mass density, g is acceleration because of gravity and z is height. Equation signifies that change in pressure dp may be defined as

dp = -Ρgdz

Equations imply that

dp/P = -Ρgdz/NT or dp/p = -dz/H

Here scale height H is provided by

H = NT/Ρg

Scale height H may be expressed in terms of mean molecular mass m at any given height. The total mass M of molecules is provided by M = ΡV,

nm = ΡV

As

M = nm

Equation signifies that mean molecular mass m is provided by

m = Ρ(V/n)

Equation shows that mean molecular mass m is provided by

m = Ρ/N

Equation may now be rewritten as

H = (N/Ρ)(T/g) or H = (1/m)(T/g)

Thus, scale height H may be written as,

H = T/mg,

Here m is mean molecular mass at given height.

By integrating Equation from some reference height z = z_{0} (at pressure p = p_{0} to arbitrary height z (at a pressure p), you get following:

_{P0}∫^{P}dP/P = - _{z0}∫^{z}dz/H

ln p - lnp_{0} =- _{z0}∫^{z}dz/H

ln(p/p_{0}) = - _{z0}∫^{z}dz/H

Thus p = p_{0}exp(-_{z0}∫^{z}dz/H)

Equation illustrates how pressure p differs with height z. It illustrates that pressure decreases exponentially with height. Clearly, a large scale height is related with low-pressure regions of atmosphere. Equation may also be utilized to find distribution of number density with height. You will recall that

p = NT

So that , at some reference point where P = P_{0} and T = T_{0} number density N_{0} is provided by expression

P_{0} = N_{0}T_{0}

But equation illustrates that pressure p is ptovided by

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

Equation shows that number density N diminishes exponentially with height also. It signifies that a large scale height is related with light gas. In the isothermal atmosphere temperature stays constant and scale height becomes constant as well. In consequence, equation reduces to

P = p_{0}exp[-(z-z0)/H]

Hre p_{0} is pressure at some reference height z_{0}. Also, in isothermal atmosphere, temperature remains constant so that T = T_{0} In this case, equation reduces to

N = N_{0}exp[-(z-z0)/H]

Here N is number density at some reference height z_{0}.

**Snell's Law:**

Snell's law defines that ratio of sine of angle of incidence to the sine of angle of refraction is a constant. That constant is a measure of refractive index. When the radio signal passes through atmosphere, its direction changes because of refraction. Consider neutral atmosphere which is horizontally stratified into m layers. If ionospheric refraction is neglected then total change in direction of radio signal passing through the atmosphere can be created by repeatedly applying Snell's law to every layer of atmosphere. In this case, Snell's law is provided by

n_{1}sinψ_{1} = sinψ_{m}

**Water Vapor:**

Troposphere has both water vapor and dry air. Amount of water vapor differs with time and location. On the other hand, composition of dry air doesn't differ considerably with height. To understand water vapor one needs to have good knowledge of some vapor-related quantities like:

(i) Mixing ratio

(ii) Partial pressure of vapor, and

**(iii) Relative humidity.**

**Mixing Ratio:**

Mixture of dry air and water vapor is known as moist air. Mixing ratio provides a measure of amount of moisture in air. Mixing ratio w is stated as ratio of mass of water vapor to mass of dry air. Therefore,

w = m_{v}/m_{d} = (m_{v}/V)(V/m_{d}) = (m_{v}/V)÷(m_{d}/V) = Ρ_{v}/Ρ_{d}

Here Ρ is density of water vapor and Ρ_{d} is density of dry air.

Equation of state holds for fixed mass of gas. It also holds for mixture of gases. Now, equation of state

pV = RT can be written in terms of specific volume and specific gas constant for i^{th }gas as follows:

pV = m_{i}R_{i}T

Here R_{i} is specific gas constant and m_{i} is mean molecular mass of i^{th} gas. Of course, specific gas constant R_{i} = R/m_{i}, where R is universal gas constant. Equation can also be re-written in terms of specific volume as follows:

pV = m_{i}R_{i}T or p = (m_{i}/V)R_{i}T

Or p = Ρ_{i}R_{i}T Here Ρi is density of ith gas stated by Ρ_{i} = m_{i}/V. Therefore, pressure p may be written as

p = Ρ_{i}R_{i}T

The specific volume v_{i} is mass per unit volume i.e. v_{i} = V/m_{i} = 1/Ρ_{i}

If this new version of th equation of state is applied to water vapor, you get

e = Ρ_{v}R_{v}T

Here e is partial pressure of water vapor. If same version of equation of state is applied to dry air, you get

P_{d} = p-e = Ρ_{d}R_{d}T

Where p is total pressure of moist air and d R is specific gas constant of dry air. Mixing ratio may thus be stated in light of equations:

Ρ_{v} = e/R_{v}T

Ρ_{d} = P-e/R_{d}T

But, the mixing ratio

w = Ρ_{v}/Ρ_{d }or w = e/R_{v}T ÷ p-e/R_{d}T

w = e/R_{v}T x R_{d}T/p-e

w = eR_{d}/(R_{v}(p-e)) Or w = ε(e/p-e)

Where ε is ratio of specific gas constant for dry air to specific gas constant for vapor (i.e. ε = R_{d}/R_{v})

*Saturation Region:*

**Partial Pressure of Saturated Air:**

In closed system, equilibrium will be recognized when number of water molecules passing from liquid phase to vapor phase is equal to number of water molecules passing from vapor to liquid phase. Any vapor which meets requirement is known as saturated vapor. Mixture of vapor and air under equilibrium conditions is known as a saturated air. When saturated air comes in contact with the unsaturated air, diffusion starts to happen. Molecules of water move from areas of higher concentration of water vapor to areas of lower concentration of water vapor. Partial pressure of the saturated water vapor depends on temperature. Warm air can have large amounts of water vapor. When warm air is cooled, surplus of the water vapor (over and above saturation value) at the new temperature condenses to form water. Condensation procedure leads to energy release. Energy released per unit mass is equal to latent heat of vaporization. Two more latent heats exist: latent heat of fusion and latent heat of sublimation. Latent heat of fusion is energy needed to change unit mass of ice to liquid water at same temperature. Latent heat of sublimation is sum of latent heat of fusion and latent heat of vaporization. By extension of Equation saturation mixing ratio w_{sat} may be expressed as

w_{sat} =ε(e_{sat}/p-e_{sat})

If total pressure p is considerably higher than saturated partial pressure e of water vapor then saturation mixing ratio becomes w_{sat} ≈ ε(e/p)

Relative humidity r_{h} is ratio of mixing ratio to saturation mixing ratio. Therefore, if p is large compared with both e and e_{sat} relative humidity r_{h} is provided by:

r_{h} = ε(e/P)÷ε(e_{sat}/P)

=ε(e/P)x(Pεe_{sat})

e/e_{sat}

Therefore

r_{h} = w/w_{sat}≈e/e_{sat}

It is frequently essential to define relative humidity in percentage. In this situation two expressions for relative humidity become:

r_{h} = 100w/w_{sat}% ≈ 100e%/e_{sat}

Here r_{h} is relative humidity and p is large compared with both e and e_{sat}.

**Propagation Delay:**

Condition of troposphere has the deep effect on radio wave propagation. Radio signal suffers delay by neutral atmosphere. Total delay relies on refractivity along the path of radio signal. Refractivity itself relies on primary parameters such as temperature and pressure.

Fermat's principle is basic physical law governing radio wave propagation. Fermat's principle illustrates that light and other electromagnetic waves will follow path between two points that involves least travel time. The optical (or electromagnetic) distance S between source and receiver as:

S = ∫cdt where c is speed of light and t is time. The electromagnetic path s as:

S = ∫vdt

Here v is propagation speed given by v = ds/dt

Refractive index n of medium is ratio of the speed of light in free space to speed of light in medium i.e.

n = c/v or v = c/n

Where c is speed of light in vacuum and v is speed of light in medium. Excess path length D caused by troposphere is provided by:

D = [∫n(s)ds - ∫ds] + [∫sds - ∫ldl]

Excess path length D is also known as delay. It is estimated in slant direction caused by troposphere.

**Refractivity N: **

Refractive index n is always small number (in the neighborhood of 1). It is frequently more suitable to work with parameter with large spread. For that reason, refractivity N is frequently utilized. In relation to refractive index n, refractivity N is stated by expression

N = (n-1)x10^{6}

Plainly, range of values of refractivity can be much more manageable than corresponding range of values of refractive index. For example, in ionosphere values of refractivity N ranges from zero to approx 300, while values of refractive index n stays clustered in neighborhood of unity. Neutral atmosphere has both dry air and water vapor. For this reason, refractivity N can be split in dry air component and water vapor component. Therefore, we may write

N = N_{d} + N_{v}

Here N_{d} refractivity of is dry air and N_{v} is refractivity of water vapor. This is a wise decision as dry air content stays almost constant in time while water vapor content differs widely in time and space.

For frequencies up to 20GHz, refractivity may be written, as function of temperature and partial pressure, in the given manner:

N_{d} = (k_{1}(Pd/T)Z_{d}^{-1}) and Nv = (k_{2}(e/T) + k_{3}(e/T^{2}))Z_{v }^{-1}

Here Z_{λ} is compressibility factor of dry air, Z_{v} is compressibility factor for water vapor and k_{i}(i=1,2,3) are dimensional constants. Unit of k_{1} and k_{2} is Kelvin per millibar; (Kmbar-1). Unit of k_{3} is Kelvin squared per millibar (K_{2} mbar^{-1}). For the ideal gas, compressibility factor is equal to 1. Any departure of Z from unity accounts for non-ideal behaviour of affected gas. Luckily, compressibility factor for both dry air and water vapor is roughly equal to 1, so that Z_{d} ≈ Z_{v} ≈ 1

Therefore, dry air and water vapor may be treated as ideal gases. In this situation, equations reduce to

N_{d} = k_{1}(P_{d}/T) and N_{v} = k_{2}(e/T) + k_{3}(e/T^{2}) respectively.

Above troposphere, there is practically no more water vapor. Therefore, at high altitudes contribution of wet refractivity to total refractivity becomes small. This is so due to wet refractivity drops much faster than dry refractivity as height increases.

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