In ground based telescopes, this diffraction-limited resolution is
never reached. Ground-based telescopes must look through miles
of the Earth's
atmosphere. Turbulent fluctuations
in the wind velocities in the upper atmosphere mix layers of differing
temperatures, densities and water vapor content. Thus the index of refraction
of each level of the atmosphere fluctuates. The wavefront incident on
the telescope has spatial and temporal variations in phase and amplitude due
to these fluctuations in the index of refraction along the optical path.
Across the diameter of large telescopes (4 to 8 meters), phase errors are
a few 
m and dominate the degradation of spatial resolution.
The spatial resolution of the telescope is degraded to an order of magnitude
greater than the diffraction limit and the central intensity of a point
source image is degraded by two orders of magnitude. (Hubin & Noethe, 1993).
Turbulence is a non-linear process, and so the equations governing it are non-linear. In 1941, Kolmogorov attempted to model turbulence in the atmosphere using a statistical approach. The Kolomogorov theory of turbulence is based on the assumption that wind velocity fluctuations are approximately locally homogeneous and isotropic random fields for scales less than the largest wind eddies. (Beland, 1993; Beckers 1993).
How do phase variations due to atmospheric turbulence affect the intensity
of the incident electro-magnetic wave? Assuming the incoming electric field
portion of the electro-magnetic wave E is proportional to 
with 
the angular
frequency and t time, the wave equation
reduces to
where k is the wavenumber for propagation in free space, n is the index of refraction and the dispersion
relation 
holds. (Note that the wavenumber for propagation through the
atmosphere is 
.)
This equation cannot be solved exactly, so we approximate the solution using
a perturbation expansion of n and E. This assumes that for the area of the sky
we are interested in, the index of refraction undergoes slow, small perturbations due to the
atmospheric turbulence.
Let 
where 
is the incident electric field and 
is the scattered
field. 
where 
.
Dropping the second order 
term,
Putting in the perturbation to E
Dropping the small 
term and using
equation 22 becomes
This can be solved in terms of using Green's function for the wave equation, giving
integrating over the scattering volume between source and telescope.
(Beland, 1993).
However, this solution is only valid for very weak perturbations of E and
the integral over the scattering volume between the source and telescope is
not very convenient since the effects of phase variations are not obvious.
A less limiting method is the Rytov method of smooth
perturbations. Here E is written as
where 
and 
, A is the
amplitude and S is the phase of E. Lutomirski and Yura, using an extended
Huygens-Fresnel theory find
here integrating over a surface.
The intensity of the incoming wavefront will tell us how the phase fluctuations
degrade the resolution.
The mutual coherence function (MCF) in the focal plane of the telescope is given
by 
and is equal to 
where 
is the transverse coherence length. A spatially coherent wavefront will
have its spatial coherence degraded to after passing through turbulence.
Most astronomical applications translate this spatial
mutual coherence function to the modulation transfer function (MTF) in the spatial
frequency domain. The modulation transfer function 
where f is the focal length, 
is the spatial frequency (cycles/length),
is the wavelength of light, and 
is the atmospheric coherence length
or Fried parameter. is about 10 cm
for a vertical path for visible wavelength and characterizes the effect of turbulence on the
incoming wavefront. Since optical turbulence blurs out point source
functions, there is a cutoff spatial frequency 
and 
gives the limit of resolution due to optical turbulence.
(Beland, 1993; Beckers, 1993.)
The diffraction limit of resolution for 4m telescope at
500 nm is given by equation 17
Assuming 
, the angular resolution due to atmospheric
turbulence R is
So turbulence in this case degrades the resolution by a factor of 100!