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4. Discussion

The model provides a good framework for understanding and interpreting the data. In the datasets examined so far there are no obvious discrepancies with the model predictions. It would be straightforward to extend the model to different power-law relationships for the turbulence, but in the absence of obvious problems with the Kolmogorov model, and the lack of physical basis for another power law, this is not considered necessary at this stage. The data show that it is possible to have significant phase power on timescales as short as 10 s and as long as an hour.

4.1. Turbulence and water vapor

A turbulent velocity field can be generated in the atmosphere by a number of different processes, e.g. (1) convective activity from heating of the ground, (2) the passage of air past an obstacle, (3) instability at the interface of two layers with different wind vectors, and (4) the large scale motions associated with weather systems.

Each energy injection mechanism has a characteristic range of scales over which turbulence is generated. The Kolmogorov law assumes that the turbulence is in statistical equilibrium, with a constant energy input to replenish the energy dissipated on small scales. If turbulence is generated in a particular location (e.g. on the lee of a mountain) then at short distances downwind the turbulence will be lacking power on short scales since there has not been enough time for the energy to cascade down. Conversely, further downwind the large scale motions will have decayed without being replenished. Each case would be apparent as a deviation from the Kolmogorov behavior, the first as a decrement on short scales, the second as a decrement on large scales. Beyond the outer scale of the dominant mechanism there should also be a marked reduction in turbulent power. There is no clear evidence for any of these effects in the data sets studied so far. In Fig. 7d, for example, there is no large deviation from the model at tex2html_wrap_inline1719 s; with a windspeed of 6.5 m stex2html_wrap_inline1721 this implies a lower limit to the outer scale of tex2html_wrap_inline172310 km. It is interesting to note that studies of the atmosphere at optical wavelengths, where the dominant contribution to phase fluctuations is the variation of the refractive index with temperature, suggest an outer scale size of order 5 m (e.g. Treuhaft et al. 1995; Coulman & Vernin 1991). This is clearly not the case at radio wavelengths.

A mountain-top site is likely to be more complicated than a flat location. For example, experience with interferometry at submillimeter wavelengths on Mauna Kea has shown that there are times when there is substantial phase variation on 1-second timescales; similar periods are also apparent in data from the Plateau de Bure in France (Bremer 1995).

The presence of a turbulent velocity field is not enough in itself to generate inhomogeneity in the distribution of water vapor. There must also be some initial density contrast in the distribution of water vapor within the turbulent zone; a uniform distribution cannot be mixed, and will not give rise to variations in the refractivity under the influence of turbulence. The total power of the phase fluctuations therefore depends on the density contrast in the water vapor that would be present in the turbulent region in the absence of turbulence. For example, convection mixes water-rich air from low altitude with drier regions higher up. A higher column density of water vapor does not directly imply stronger phase fluctuations. This may explain the weak dependence of radio seeing with altitude found by Masson (1993).

The model may be used to estimate the fractional contribution to the water vapor column along a single line of sight from the varying component. For typical conditions in the Owens Valley (5-minute rms phase at 12 GHz of tex2html_wrap_inline1725, and 5 mm of precipitable water vapor) the varying component constitutes tex2html_wrap_inline1727% of the total water vapor column.

4.2. Frozen turbulence

It is possible to estimate the validity of the assumption of frozen turbulence and to show the effect that non-frozen flows will have on the temporal power spectrum.

Consider an element of turbulence with size-scale l. The lifetime of this feature is of order tex2html_wrap_inline1731 (see Tatarskii 1961, Chap. 2), where tex2html_wrap_inline1733 is the turbulent velocity of the element. Therefore the timescale over which the feature retains a coherent identity is tex2html_wrap_inline1735. In this time the feature is blown a distance tex2html_wrap_inline1737, where w is the windspeed. If tex2html_wrap_inline1741, i.e. tex2html_wrap_inline1743 then the feature passes through a given line of sight relatively unchanged, and the assumption of frozen turbulence is a good approximation for this scale.

From simple dimensional arguments (Tatarskii 1961, Chap. 2), tex2html_wrap_inline1745, so that the velocity of the turbulent motion is highest on the largest scales, and this is where the frozen turbulence assumption will break down first. The wind is usually the result of motions on the scale of hundreds of kilometers. It is reasonable to assume that when turbulence is produced by the wind blowing past an obstacle or by the wind shear between layers, the turbulent motions do not have speeds exceeding w. If the turbulence is dominated by convection and the systematic "background" windspeed is very low, then it is possible that the velocity of the turbulence exceeds the windspeed and large structures evolve faster than the blow-by time. In this case there will be an apparent deficit of phase power on long timescales. Frozen turbulence should therefore be a good approximation whenever the systematic windspeed exceeds the speed of convective motions.

4.3. The geometrical optics approximation

This approximation, as noted in Sect. 2, assumes that the wavefront delay having passed through an inhomogeneous medium is given by the integral of the refractivity variations along the line of sight. The effects of diffraction are ignored. Tatarskii (1961, Chap. 6) shows that diffraction becomes important on size scales l for which tex2html_wrap_inline1751, where tex2html_wrap_inline1753 is the observing wavelength and h is the distance between the inhomogeneity and the observer.

The phase monitor observes at a wavelength of 25 mm, so that the approximation is valid only for scales exceeding tex2html_wrap_inline1757 5 m. Reference to Fig. 4 shows that there is actually very little phase power from scales less than 5 m. Diffraction results in some of this power being spread to larger spatial scales, so in this case the approximation has very little impact on the power spectrum. It will become more important for observations at lower frequencies and shorter baselines.

4.4. Turbulence in the Owens Valley

The Owens Valley (Fig. 9 (click here)) runs North-South, is approximately 8 km wide, and the floor is at an elevation of 1200 m. To the West, the mountains of the Sierra Nevada rise abruptly to over 4000 m; to the East the White Mountain range rises to tex2html_wrap_inline1759 m. The wind direction at the observatory on the valley floor is almost always North-South, but the prevailing wind direction for elevations exceeding 4000 m is approximately East-West. The obvious sources of turbulence are convective activity from the valley floor, eddies generated by the passage of air over the Sierra Nevada, and shearing between the volume of air in the valley and the air moving over the mountains. An analysis of the rms phase measured by the phase monitor over 5-minute intervals over a period of several months shows clearly that the phase tends to be worst during the middle of the afternoon. This suggests that convective activity inside the valley, blown in a North-South direction perpendicular to the baseline of the phase monitor, is a dominant contributor. This is consistent with the findings of the previous section. Turbulence generated by the fast air blowing over the Sierra Nevada from West to East is the likely cause of the high frequency component in Figs. 8f and h. This is shown schematically in Fig. 9 (click here).

  figure468
Figure 9: Schematic illustration of the Owens Valley. Convective turbulence in the valley is blown North-South. High-altitude turbulence is generated by the Sierra Nevada mountains and is blown predominantly to the West. The phase monitor is located on the valley floor with an East-West baseline

4.5. Extrapolating phase monitor data

The main aim of this paper is to demonstrate that a simple model with few assumptions can explain the basic features of the phase monitor data taken in the Owens Valley. This model can then be used to extrapolate the phase monitor data to different baselines and elevations to assess the impact of atmospheric fluctuations on astronomical data measured with interferometers.

The model can be fitted to a measured temporal power spectrum to estimate the wind speed (horizontal shift of the curve) and direction (shape of the curve for the high frequencies), turbulent intensity (vertical scaling) and the thickness of the turbulent layer (width of the curve). An example is given in Fig. 8 (click here). With these quantities the model can then be used to predict the power spectrum for an arbitrary baseline, elevation and observing frequency.

4.6. Advantages of the temporal power spectrum approach

 

The primary motivation for investigating the temporal power spectrum here is to use the data to evaluate the effect of phase calibration and correction schemes using water vapor radiometry. A knowledge of the timescales of fluctuations is clearly needed to do this, and the traditional structure function provides no such information. The spectral density distribution is also much more sensitive to the presence of multiple components of turbulence (e.g. Fig. 7h) which can be distinguished on the basis of windspeed.

The measured spectral density distributions also show that there can be substantial phase power on timescales longer than 1000 s for a baseline of only 100 m. When measuring the structure function, it is vital that the phase is monitored over a sufficiently long period of time. The longer the baseline, the longer the time required. Figure 4d has power on spatial scales 100 times the baseline length and the necessary sampling period can be prohibitive, particularly if the windspeed is low. If the period is too short, the power law index of the structure function will be underestimated.

A discussion of phase calibration procedures based on the temporal power spectrum of phase fluctuations, and the implications for water vapor radiometer schemes that will attempt to correct the fluctuations, is the subject of a companion paper (Lay 1997). It will also be instructive to study a much larger sample of data to look for cases where the model is inadequate, and to investigate diurnal variations.


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