According to Wikipedia The Atmosphere of:
Earth is a layer of gases surrounding the planet Earth that is retained by Earth’s gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention greenhouse effect, and reducing temperature extremes between day and night.
Atmospheric stratification describes the structure of the atmosphere, dividing it into distinct layers, each with specific characteristics such as temperature or composition. The atmosphere has a mass of about $5.0*10^18$ kg, three quarters of which is within about 11 km (6.8 mi; 36,000 ft) of the surface. The atmosphere becomes thinner and thinner with increasing altitude, with no definite boundary between the atmosphere and outer space. An altitude of 120 km (75 mi) is where atmospheric effects become noticeable during atmospheric reentry of spacecraft. The Kármán line, at 100 km (62 mi), also is often regarded as the boundary between atmosphere and outer space.
Air is the name given to atmosphere used in breathing and photosynthesis. Dry air contains roughly (by volume) 78.09% nitrogen, 20.95% oxygen, 0.93% argon, 0.039% carbon dioxide, and small amount s of other gases. Air also contains a variable amount of water vapor, on average around 1%. While air content and atmospheric pressure varies at different layers, air suitable for the survival of terrestrial plants and terrestrial animals is currently known only to be found in Earth’s troposphere and artificial atmospheres.
Just to show the wide range of scales we show this diagram from the book Atmospheric Chemistry and Physics by Seinfeld, Pandis:
The atmosphere itself presents a range of spatial scales in its motions that spans eight orders of magnitude (Figure above). The scales of motion in the atmosphere vary from tiny eddies of a centimeter or less in size to huge airmass movements of continental dimensions.
From wikipedia:
Free oxygen did not exist until about 1.7 billion years ago and this can be seen with the development of the red beds and the end of the banded iron formations. This signifies a shift from a reducing atmosphere to an oxidising atmosphere. O2 showed major ups and downs until reaching a steady state of more than 15%.[19] The following time span was the Phanerozoic eon, during which oxygen-breathing metazoan life forms began to appear.
So far, only terminology and definitions… To be expanded
(Below, we use the convention $\tau=2\pi$)
The flux (density) is the energy per unit time flowing through a surface. The spectral flux $F(\lambda)$ is the differential flux with respect to the wavelength. The radiant intensity/radiance $L(\Omega)$ $[W \cdot m^{-2} \cdot sr^{-1}]$ is the amount of flux associated to a specific direction $\Omega$.
Usually one is interested in the outgoing or incoming flux, with respect to a surface. E.g. the outgoing flux $F^u$ can be obtained by integrating the radiance over the upper hemisphere of a surface, taking into account the angle between the surface normal and the outgoing direction of the radiance:
The spectral emissivity is the ratio of the spectral emission (or spectral radiance) of a real grey object to that of an ideal black-body,
$\epsilon(\lambda,T) = \frac{L(\lambda)}{B(\lambda;T)},$
where $B(\lambda;T)$ is the Planck function. $\epsilon(\lambda,T)$ lies in the interval $[0,1]$. Kirchhoff’s law states that the spectral emissivity is equal to the absorptivity (the ratio between the absorbed radiation and Planck function) and holds if the object is at thermodynamic equilibrium. The brightness temperature of a gray body is the equivalent black-body temperature that corresponds to the same spectral radiance $L(\lambda)$:
A greybody has constant $\epsilon$.
The Beer-Lambert law expresses the variation of the spectral radiance $L(\lambda)$ along its path of propagation ($s$) in an atmospheric medium:
The volume extinction coefficient $\beta_e$ reflects that radiation is attenuated when it propagates through the atmosphere. It can be converted into a mass extinction coefficient $k_e$ by $\beta_e=\rho k_e$ (density $\rho$). The extinction coefficient can be partitioned into absorption and scattering contributions $\beta_e=\beta_a+\beta_s$. The single scattering albedo is the ratio of scattering to absorption plus scattering $\omega = \beta_s/\beta_e$ and should not be confused with the albedo introduced below.
The optical path $\tau_{(OP)}$ is the integral of the volume extinction coefficient along a path. The transmittance is the exponential of the optical path. Optical depth/optical thickness is the optical path when the integration path is taken along the vertical axis of the atmosphere.
The bidirectional reflectance distribution function BRDF expresses the angular relationship between incoming and outgoing radiation. It is defined as:
Here $L^u$ outgoing (reflected) radiation with outgoing angle $\Omega$ and $L_d$ incoming radiation with direction of incidence $\Omega_0$. The BRDF is an important quantity in remote sensing. For Lambertian reflection the angular distribution of the reflected radiation is uniform. In the case of specular reflection the relected and incident direction are uniquily related.
The albedo or bi-hemispherical reflectance assesses the overall reflective characteristics of a surface and is the ratio of the reflected radiance to the incident radiance:
According to Wikipedia it is:
Static atmospheric models describe how the ideal gas properties (namely: pressure, temperature, density, and molecular weight) of an atmosphere change, primarily as a function of altitude. For example, the US Standard Atmosphere is essentially a table of values for air temperature $T$, pressure $p$, and mass density $\rho$, as a function of altitude $z$ above sea level.Other static atmospheric models may have other outputs, or depend on inputs besides altitude.
$R=8.314 J/K mole$ is the gas constant. H is the scale height, which Wikipedia explains:
A scale height is a term often used in scientific contexts for a distance over which a quantity decreases by a factor of $e$ (the base of natural logarithms). It is usually denoted by the capital letter H. For planetary atmospheres, it is the vertical distance upwards, over which the pressure of the atmosphere decreases by a factor of $e$. The scale height remains constant for a particular temperature.
According to Wikipedia:
An atmospheric model is a mathematical model constructed around the full set of primitive dynamical equations which govern atmospheric motions. It can supplement these equations with parameterizations for turbulent diffusion, radiation, moist processes (clouds and precipitation), heat exchange, soil, vegetation, surface water, the kinematic effects of terrain, and convection. Most atmospheric models are numerical, i.e. they discretize equations of motion. They can predict microscale phenomena such as tornadoes and boundary layer eddies, sub-microscale turbulent flow over buildings, as well as synoptic and global flows. The horizontal domain of a model is either global, covering the entire Earth, or regional (limited-area), covering only part of the Earth. The different types of models run are thermotropic, barotropic, hydrostatic, and nonhydrostatic. Some of the model types make assumptions about the atmosphere which lengthens the time steps used and increases computational speed. Forecasts are computed using mathematical equations for the physics and dynamics of the atmosphere. These equations are nonlinear and are impossible to solve exactly. Therefore, numerical methods obtain approximate solutions. Different models use different solution methods. Global models often use spectral methods for the horizontal dimensions and finite-difference methods for the vertical dimension, while regional models usually use finite-difference methods in all three dimensions. For specific locations, model output statistics use climate information, output from numerical weather prediction, and current surface weather observations to develop statistical relationships which account for model bias and resolution issues.
Its governed by these continuous equations:
Climate modeling like Global circulation models- GCM and limited area forecasting for e.g. hurricanes, cyclones anf air quality.
Atmosphere, Wikipedia
Atmospheric models, ibid.
Atmospheric model, ibid.
Primitive equations, ibid
J. Seinfeld S. Pandis,2006 Atmospheric Chemistry and Physics 2e