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 = z0 (at pressure p = p0 ) to arbitrary height z (at pressure p ) we obtain:
p0∫pdp/p = -z0∫zdz/H
[logep]pop = - z0∫zdz/H
loge(p/p0) = - z0∫zdz/H
p = p0exp(-z0∫zdz/H)
Given equation illustrates how pressure diminishes with height. At reference point z0, pressure is p0 and the number density is N0 so that Equation yields
p0 = N0T0
At arbitrary height z, ideal gas equation is
Thus, by taking ratio, you obtain p/p0 = NT/N0T0
So that p = p0(NT/N0T0) or p0exp(-z0∫zdz/H) = p0(NT/N0T0)
or N = (N0T0/T)exp(-z0∫zdz/H)
Here T0 is temperature at z0. 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 Hi of ith gas is provided by
Hi = T/mig
Consider the limited region of atmosphere in which gravity stays practically constant. By using equations expression can be written as:
pi = p0iexp(-z0∫zdz/Hi) that simplifies to
pi = poiexp[-(z-z0/Hi)]
By extension, density distribution is found in the given way:
pi = poi(NiT/N0iT)
Thus pi = p0i(Ni/Noi). Also pi = poiexp(-z0∫zdz/Hi)
By solving equation we get
Ni = Noie-(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.
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
pVγ = k, Where k is constant and γ is ratio of principal specific heat capacities. That is,
γ = cp/cv
Here cp is specific heat capacity at constant pressure and cv 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 = Aργ
Here A is constant defined by
A = k/mγ
Differentiating p with respect to ρ, you obtain
dp/dρ = Aγργ-1
In order that change in pressure dp is provided by
dp = γAργ-1dρ
By eliminating p from hydrostatic equation and adiabatic equation
γAργ-1dρ = -ρgdz or γAργ-2dρ = -gdz
Integrating expression given above, and simplifying result, we get:
γA ρ0∫ρργ-2dρ = -g z0∫zdz
(γA/γ-1)(ργ-1 - ρ0γ-1) = -g(z-z0)
ργ-1 - ρ0γ-1 = -(γ-1/γA)g(z-z0)
ρ =[ρ0γ-1 - g(γ-1/γA)(z-z0)]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, [ρ0γ-1 - (γ-1/γA)g(z-z0)]
z-z0 = z-z0p0/ρ0g(γ-1)
Here p0 (≡ Aρ0γ ) represents adiabatic equation at z = z0. 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 = Aργ
From two equations, we get that
NT = Aργ or T = (A/N)ργ
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)]ργ thus Amργ-1
Using Equation, we get
T = Amρ0γ-1 - mg(γ-1)/γ(z-z0)
Therefore, temperature gradient becomes
dT/dz = -mg(γ-1)/γ
The temperature gradient is also called the lapse rate.
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