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2. The model

2.1. Turbulence, structure functions and power spectra

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).

  figure226
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 tex2html_wrap_inline1297. These have 3-D and 2-D Fourier Transforms given by tex2html_wrap_inline1299 and tex2html_wrap_inline1301, 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 tex2html_wrap_inline1307 and tex2html_wrap_inline1309. The autocorrelation functions for n(x,y,z) and tex2html_wrap_inline1313 are tex2html_wrap_inline1315 and tex2html_wrap_inline1317, 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 tex2html_wrap_inline1319 that gives the variance of the delay difference between two lines of sight separated by tex2html_wrap_inline1321, is proportional to tex2html_wrap_inline1323. 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 tex2html_wrap_inline1325. The focus of this analysis is the spatial power spectrum of the wavefront delay tex2html_wrap_inline1327. 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.

  figure253
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, tex2html_wrap_inline1333 are the corresponding spatial frequencies; tex2html_wrap_inline1335 and tex2html_wrap_inline1337 are a position and displacement in the 3-D space, respectively; tex2html_wrap_inline1339 and tex2html_wrap_inline1341 are their equivalents in the x-y plane; tex2html_wrap_inline1345 and tex2html_wrap_inline1347 are 3-D and 2-D spatial frequencies: tex2html_wrap_inline1349; tex2html_wrap_inline1351; tex2html_wrap_inline1353; tex2html_wrap_inline1355

2.2. From refractivity to interferometer phase

  figure279
Figure 3: The response of an interferometer with baseline tex2html_wrap_inline1357 to a wavefront delay tex2html_wrap_inline1359 moving at windspeed tex2html_wrap_inline1361. On the left, tex2html_wrap_inline1363 is mapped in stages onto the phase difference tex2html_wrap_inline1365 measured by the interferometer as a function of time. The corresponding power spectra at each stage are shown on the right

  figure289
Figure 4: a) Contour plot showing phase power as a function of spatial frequency tex2html_wrap_inline1367 for (tex2html_wrap_inline1369 m, tex2html_wrap_inline1371) i.e. wind blowing parallel to the baseline, as indicated by the symbol. The contours are evenly spaced in the quantity tex2html_wrap_inline1373, 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 tex2html_wrap_inline1375. 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 tex2html_wrap_inline1377 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 tex2html_wrap_inline1379 axes. The asymptotic gradients for the unbounded case are shown. The corresponding frequency scale is obtained by the relation tex2html_wrap_inline1381. All Log scales are to base 10

An interferometer, comprising a pair of antennas looking vertically up through the atmosphere separated by a baseline tex2html_wrap_inline1383, 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 tex2html_wrap_inline1385, 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. tex2html_wrap_inline1387) is very similar at each antenna and therefore gets canceled out to some extent. The power spectrum tex2html_wrap_inline1389 of this filtered signal is the product of the atmosphere's intrinsic power spectrum tex2html_wrap_inline1391 and the square of the filtering function:


 eqnarray302
An interferometer actually measures the difference in the phase of the two signals received: tex2html_wrap_inline1393 and tex2html_wrap_inline1395, where tex2html_wrap_inline1397 is the frequency of the radiation being observed.

Figures 4a & b show contour plots of the quantity tex2html_wrap_inline1399 as a function of tex2html_wrap_inline1401 and tex2html_wrap_inline1403 for the cases of (tex2html_wrap_inline1405 m, tex2html_wrap_inline1407) and (tex2html_wrap_inline1409, tex2html_wrap_inline1411 m), respectively, i.e. wind along the baseline and wind perpendicular to the baseline. The spatial frequencies tex2html_wrap_inline1413 and tex2html_wrap_inline1415 have units mtex2html_wrap_inline1417, such that tex2html_wrap_inline1419 for a disturbance with wavelength tex2html_wrap_inline1421 (note: tex2html_wrap_inline1423 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 tex2html_wrap_inline1425 the volume under the contours is proportional to the variance tex2html_wrap_inline1427, 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 (tex2html_wrap_inline1435) in both the x and y directions. It can be shown that the volume under the contours, that is the total variance in tex2html_wrap_inline1441, is proportional to tex2html_wrap_inline1443. 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). tex2html_wrap_inline1455 is sampled only along the x-axis, such that x=wt, where t is time: tex2html_wrap_inline1463. The power spectrum tex2html_wrap_inline1465 is derived by integrating tex2html_wrap_inline1467 over tex2html_wrap_inline1469. Here tex2html_wrap_inline1471 is the temporal frequency of phase variations in the output of the interferometer, not to be confused with the frequency of the observed radiation tex2html_wrap_inline1473. For example, a fluctuation in the refractivity with a spatial periodicity of 200 m in the x-direction (i.e. tex2html_wrap_inline1477 mtex2html_wrap_inline1479, tex2html_wrap_inline1481) gives rise to a measured phase fluctuation of period 40 s (tex2html_wrap_inline1483 Hz) if w=5 m stex2html_wrap_inline1487. tex2html_wrap_inline1489 is plotted against tex2html_wrap_inline1491 (tex2html_wrap_inline1493) in Figs. 4c and d. Plotting tex2html_wrap_inline1495 ensures that area under the curve is proportional to the variance of tex2html_wrap_inline1497, i.e. the curve represents a spectral density distribution.

The same models are plotted on tex2html_wrap_inline1499 axes in Figs. 4e and f, illustrating the broken power law dependence of tex2html_wrap_inline1501 on tex2html_wrap_inline1503: fluctuations much smaller than the baseline are uncorrelated between the two antennas and tex2html_wrap_inline1505; fluctuations much larger than the baseline give tex2html_wrap_inline1507.

2.3. The effect of baseline orientation and length

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 tex2html_wrap_inline1509 are quite sharply peaked around scales corresponding to tex2html_wrap_inline1511, compared to a softer peak centered on scales of tex2html_wrap_inline1513 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 tex2html_wrap_inline1515. 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 tex2html_wrap_inline1517 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 tex2html_wrap_inline1523; the vertical scale should also be increased by a factor tex2html_wrap_inline1525. 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.

2.4. The finite thickness of the turbulent layer

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 tex2html_wrap_inline1527 of 5 km, 2 km and 1 km. In Fig. 2 (click here), the expressions for tex2html_wrap_inline1529 and tex2html_wrap_inline1531 are now given by


 eqnarray343
The values of tex2html_wrap_inline1533 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 tex2html_wrap_inline1535 takes effect for tex2html_wrap_inline1537: the distribution of phase power becomes narrower as there is less power on large spatial scales (or lower temporal frequency) and the tex2html_wrap_inline1539 plots deviate from the gradient of tex2html_wrap_inline1541. 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 tex2html_wrap_inline1543.

  figure359
Figure 5: tex2html_wrap_inline1545 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 (tex2html_wrap_inline1547 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 tex2html_wrap_inline1551 flattens to an index of -5/3 for larger scales (Fig. 5 (click here)). There is further flattening when the baseline length is exceeded.

2.5. The effect of elevation

The results so far have dealt with antennas observing at an elevation of tex2html_wrap_inline1555. 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 tex2html_wrap_inline1563. The component of tex2html_wrap_inline1565 in the x-direction is the projected windspeed tex2html_wrap_inline1569. The z-axis has elevation tex2html_wrap_inline1573 and azimuth in the horizontal plane tex2html_wrap_inline1575 with respect to tex2html_wrap_inline1577. The baseline components tex2html_wrap_inline1579 and tex2html_wrap_inline1581 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
 eqnarray372
where A is proportional to the strength of the turbulence,
eqnarray382
and tex2html_wrap_inline1589 is the vertical component of each wavevector. Note that for tex2html_wrap_inline1591 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 tex2html_wrap_inline1593 in the same way as before:
equation386
with tex2html_wrap_inline1595.

  figure393
Figure 6: Spectral density plots for elevations of tex2html_wrap_inline1597, tex2html_wrap_inline1599, tex2html_wrap_inline1601 and tex2html_wrap_inline1603, in order of increasing phase power. The turbulent layer is 100 m thick, the baseline has tex2html_wrap_inline1605 and tex2html_wrap_inline1607 m, and the lines of sight are inclined along the wind direction (tex2html_wrap_inline1609 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 tex2html_wrap_inline1611, tex2html_wrap_inline1613, tex2html_wrap_inline1615, tex2html_wrap_inline1617 and tex2html_wrap_inline1619. It can be seen that the phase power on small spatial scales (high tex2html_wrap_inline1621) is proportional to tex2html_wrap_inline1623 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 tex2html_wrap_inline1625, the distance travelled through the turbulent layer tex2html_wrap_inline1627 and the intensity of the turbulence A.

2.6. Summary of the model

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.


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