Radiative flux and forcing parameterization error in aerosol-free clear skies

The radiation pressure applied by an electromagnetic wave on a perfectly absorbing surface equals the energy density of the wave. The wave's momentum also gets transferred to the surface when an electromagnetic wave is entirely absorbed by it. The rate at which momentum is transmitted to an absorbing surface perpendicular to the propagation direction equals the force on the surface.
The average value of the rate of momentum transfer divided by the absorbing area represents the average force...

The average temperature of Earth is the subject of much current discussion. Earth is in radiative contact with both the Sun and dark space; it receives almost all its energy from the radiation of the Sun and reflects some of it into outer space. Dark space is very cold, about 3 K, so Earth radiates energy into it. For instance, heat transfer occurs from soil and grasses, the rate of which can be so rapid that frost can occur on clear summer evenings, even in warm latitudes.
The average...

During leveling, the Earth's curvature and atmospheric refraction introduce deviations in the line of sight from a true horizontal reference. When the line of sight is leveled, it remains perpendicular to the plumb line only at a single point. Beyond this, it deviates due to the Earth’s curvature, represented by the correction C. For a sight distance D, the deviation can be derived using the relationship:This relationship shows that the deviation increases quadratically with distance. Over a...

Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by

The rate of heat transfer by emitted radiation is described by the Stefan-Boltzmann law of radiation:

Change in atmospheric pressure with height is particularly interesting. The decrease in atmospheric pressure with increasing altitude is due to the decreasing gravitational force per unit area as we move away from the surface of the earth.
Assuming the air temperature is constant at a given altitude and that the ideal gas law of thermodynamics describes the atmosphere to a good approximation, one can find the variation of atmospheric pressure with height.
Let p(y) be the atmospheric pressure at...