The simple model developed here will be used to interpret the data from the phase monitor presented in the next section. It is intended to emphasize the relationships between different quantities, rather than to be mathematically rigorous.
The inhomogenous distribution of water vapor in the atmosphere is the result of a turbulent velocity field acting on large scale concentrations of water vapor. Turbulence is injected into the atmosphere on large scales by processes such as convection, the passage of air past obstacles and wind shear, and cascades down to smaller scales where it is eventually dissipated by viscous friction. Between the outer scale of injection and the inner scale of dissipation--known as the inertial range--it is a good approximation to say that kinetic energy is conserved, and simple dimensional arguments predict that for 3-dimensional, isotropic turbulence, the power spectrum is described by a power law with an index of -11/3. This is the Kolmogorov Power Spectrum (Tatarskii 1961, 1971).
Figure 1: A plane wavefront from a distant point source is distorted by variations in the refractivity n as it passes through the atmosphere
Figure 1 (click here) shows how a plane wavefront from a distant point source is distorted as it passes through an atmosphere containing variations in the refractivity n(x,y,z) (= refractive index -1). Figure 2 (click here) illustrates the relationships between important quantities.
The refractivity field n(x,y,z) can be integrated along the line of sight (z-axis) to give the wavefront delay . These have 3-D and 2-D Fourier Transforms given by
and
, respectively, where q denotes a spatial frequency.
The Fourier Transforms are implicitly performed over a finite volume containing the scales of interest in the x-y-z domain, ensuring that the integrals remain finite.
The corresponding power spectra are
and
. The autocorrelation functions for n(x,y,z) and
are
and
, which in turn are related to the 3-D and 2-D structure functions, respectively, of the refractivity field. For the case of fully three-dimensional Kolmogorov turbulence, the 2-D structure function
that gives the variance of the delay difference between two lines of sight separated by
, is proportional to
.
The details of the calculations can be found in the literature (e.g. Tatarskii 1961, 1971; Thompson et al. 1986). An implicit assumption is that the wavefront delay at a given location depends only on the refractivity field along the line of sight; this is the geometrical optics approximation and is
discussed in Sect. 4.
The emphasis in the past has been on the 2-D structure function . The focus of this analysis is the spatial power spectrum of the wavefront delay
.
In the next section the layer of turbulence is considered to be of
effectively infinite thickness, so that the turbulence is isotropic;
subsequent sections deal with layers of finite thickness.
Figure 2: Relationships
between the refractivity field n(x,y,z),
wavefront delay, structure functions and power spectra. Power
law indices, where given, are appropriate for a very thick layer
of Kolmogorov turbulence. Variables: (x,y,z) are spatial
coordinates, are the corresponding spatial
frequencies;
and
are a
position
and displacement in the 3-D space, respectively;
and
are their equivalents in the x-y plane;
and
are 3-D and 2-D
spatial frequencies:
;
;
;
Figure 3: The response of an interferometer with baseline to a wavefront delay
moving at windspeed
. On the left,
is mapped in stages onto the phase difference
measured by the interferometer as a function of time. The corresponding power spectra at each stage are shown on the right
Figure 4: a) Contour plot showing phase power as a function of
spatial frequency for (
m,
) i.e. wind
blowing parallel to the baseline, as indicated by the symbol.
The contours are evenly spaced in the quantity
,
so that volume under the contours is proportional to phase power.
b) Contour plot for wind blowing perpendicular to baseline.
The corresponding spatial scales are also shown.
c & d) Result of integrating the above functions over
.
Area under the curves is proportional to phase power. The thick lines
are for an unbounded turbulent region; the thin lines show the change
in shape as the thickness of the turbulent layer
is reduced to
5 km, 2 km and 1 km, and have been normalised to have the same
power on small scales. e & f) Plots of the same curves on
axes.
The asymptotic gradients for the unbounded case are shown. The
corresponding frequency scale is obtained by the
relation
. All Log scales are to base 10
An interferometer, comprising a pair of antennas looking vertically up through the atmosphere separated by a baseline , is sensitive to the difference in the delay between the two signals received. This response is illustrated schematically in
Fig. 3 (click here), where the two circles represent positive and negative delta functions at the location of the antennas. The Fourier Transform of this response is given by
, so that the
interferometer acts as a spatial filter: the excess delay introduced by a fluctuation that is much larger than the baseline (i.e.
) is very similar at each antenna and therefore gets canceled out to some extent. The power spectrum
of this filtered signal is the product of the atmosphere's intrinsic power spectrum
and the square of the filtering function:
An interferometer actually measures the difference in the phase of the two signals received: and
, where
is the frequency of the radiation being observed.
Figures 4a & b show contour plots of the quantity as a function of
and
for the cases of (
m,
) and (
,
m), respectively, i.e. wind along the baseline and wind perpendicular to the baseline. The spatial frequencies
and
have units m
, such that
for a disturbance with
wavelength
(note:
is not the wavelength of the
radiation here). The logarithmic axes (base 10) are necessary to cover the large range of scales and by plotting equally spaced contours of
the volume under the contours is proportional to the variance
, i.e. the plot shows the contributions to the variance from different spatial frequencies.
Although these plots are for d=100 m, the response for a longer (shorter) baseline is simply obtained by shifting the contours down and to the left (up and to the right). For example, for d=200 m shift by -0.3 (
) in both the x and y directions. It can be shown that the volume under the contours, that is the total variance in
, is proportional to
. This is the 2-D phase structure function evaluated for a separation d (Fig. 2 (click here)).
The pattern of turbulence is blown at windspeed w over the interferometer. Here it is assumed that w is uniform in speed and direction over the volume containing the turbulence, and that the pattern of turbulence is essentially fixed over the time interval needed for the pattern to blow through a line of sight. This is the assumption of frozen turbulence and is addressed further in Sect. 5. It is convenient to consider the turbulence as fixed and the antennas as moving at speed w in the x-direction, as shown in Fig. 3 (click here). is sampled only along the x-axis, such that x=wt, where t is time:
. The power spectrum
is derived by integrating
over
.
Here
is the temporal frequency of phase variations in the output of the interferometer, not to be confused with the frequency of the observed radiation
. For example, a fluctuation in the refractivity with a spatial periodicity of 200 m in the x-direction (i.e.
m
,
) gives rise to a measured phase fluctuation of period 40 s (
Hz) if w=5 m s
.
is plotted against
(
) in Figs. 4c and d. Plotting
ensures that area under the curve is proportional to the variance of
, i.e. the curve represents a spectral density distribution.
The same models are plotted on
axes in Figs. 4e and f, illustrating the broken power law dependence of
on
:
fluctuations much smaller than the baseline are uncorrelated between the two antennas and
; fluctuations much larger than the baseline give
.
There are clear differences between the case shown in Figs. 4a, c and e where the
wind blows along the baseline, and that shown in b, d and f where they are perpendicular. In the former case the contributions to the variance of are quite sharply peaked around scales corresponding to
, compared to a softer peak centered on scales of
for the latter. The break in the power law is also more evident in Fig. 4e than in f, and occurs at a spatial frequency given by
. When the wind is neither parallel nor perpendicular to the baseline, the shapes of the curves are intermediate between the two extremes shown. The nulls in Figs. 4c and e are a direct result of the interferometer's
response--the
of
Eq. (2 (click here))--and are suppressed when the wind is not blowing along the baseline.
The effect of the size of the individual antennas has been ignored up to this point; fluctuations much smaller than the effective aperture are smeared out, but since there is very little power on scales less than d, this approximation is justified.
The curves plotted are for 100 m baselines. For a baseline length d, shift the the curves
in c and d to the left by ; the vertical scale should also be increased by a factor
.
The phase power is the same on small scales where there is no correlation between the two lines of sight and increases on large scales for the longer baselines.
Until this point it has been assumed that the turbulent region is effectively infinite in all directions. The three other
curves in Figs. 4c, d, e and f illustrate the effect on the phase power spectrum of an atmosphere with vertical thickness of 5 km, 2 km and 1 km. In Fig. 2 (click here), the expressions for
and
are now given by
The values of are calculated by numerical integration. The curves have been normalized so that the power on small spatial scales is the same, to emphasize the differences in the shapes of the distributions.
The finite value of
takes effect for
: the distribution of phase power becomes narrower as there is less power on large spatial scales (or lower temporal frequency) and the
plots deviate from the gradient of
. The exact behavior depends on the orientation of the wind direction with respect to the baseline. The curves shown are for a 100 m baseline, but the shape of the curve is the same for a given value of
.
Figure 5: plot to show how phase power
changes as a function of size scale when the baseline
is much longer than the thickness of the turbulent layer (
in this case)
When the baseline is much longer than the thickness of the turbulent layer, the power law index of -8/3 on scales smaller than flattens to an index of -5/3 for larger scales (Fig. 5 (click here)). There is further flattening when the baseline length is exceeded.
The results so far have dealt with antennas observing at an elevation of . The effect of the elevation of the line of sight through the turbulent layer is complicated, and requires further elaboration of the model.
The coordinate system is now defined with the z-axis along the line of sight
and the x-axis such that the x-z plane contains the wind vector . The component of
in the x-direction is the projected windspeed
. The z-axis has elevation
and azimuth in the horizontal plane
with respect to
. The baseline components
and
are in the x- and y-directions, respectively, perpendicular to the line of sight; together they give the projected baseline.
The power spectrum of the wavefront delay is now given by
where A is proportional to the strength of the turbulence,
and is the vertical component of each wavevector. Note that for
Eq. (5 (click here)) reduces to Eq. (4 (click here)). The temporal power spectrum of the phase fluctuations at the output of the correlator is related to
in the same way as
before:
with .
Figure 6: Spectral density plots for elevations of ,
,
and
, in order of increasing
phase power. The turbulent layer is 100 m thick, the baseline
has
and
m, and the lines of sight are inclined
along the wind direction (
in Eqs. (6) and (7)). The
elevations correspond to airmasses of 1.0, 1.4, 2.0 and 3.9
Figure 6 (click here) shows examples of how the power spectrum obtained from a given turbulent layer is a function of the elevation. The curves are calculated by numerical integration of
Eq. (5 (click here)). The shape is a function of the relative values of ,
,
,
and
. It can be seen that the phase power on small spatial scales (high
) is proportional to
as would be expected. It is also possible to show that the total phase power integrated over all timescales depends only on the length of the projected baseline
, the distance travelled through the turbulent layer
and the intensity of the turbulence A.
A model has been developed to interpret the power spectrum of atmospheric turbulence measured by an interferometer. The distribution of phase power has a strong dependence on the orientation of the baseline with respect to the wind direction. The effects of the thickness of the turbulent layer and the elevation of the line of sight have also been demonstrated. The assumptions are that the geometrical optics approximation is valid over the scales of interest and that the turbulent field can be regarded as "frozen" Kolmogorov turbulence.