Tilts of Atmospheric Radar-Scattering Structures Measured by Long-Term Windprofiler Radar Studies

Abstract

Month-long and seasonally persistent apparent tilts in atmospheric radar scatterers have been measured with a network of six windprofiler radars over periods of two or more years. The method used employs cross-correlations between vertical winds and horizontal winds measured using the radars. It is shown that large-scale apparent tilts that persisted for many weeks and months were not uncommon at many sites, with typical tilts varying from horizontal to ~3–4° from horizontal. The azimuthal and zenithal alignment of the tilts depend on local orography as well as local seasonal atmospheric conditions. It is demonstrated that these apparent tilts are not, in general, true large-scale phenomena, but rather are a manifestation of coordinated motions within turbulent and quasi-specular radar-scattering structures at scales between a few metres and tens of metres, with these structures themselves being defined by larger-scale and longer-term physical processes. Windshear combined with breaking gravity waves seems to be a particularly effective mechanism for producing these tilts, although other possibilities are also discussed. Implications for the interpretation of the nature of turbulent eddies, the accuracy of vertical wind measurements, and the nature of layering and scattering in the real atmosphere, are discussed. A method which allows for accurate measurements of the mean off-horizontal alignment of anisotropic scatterers and turbulent eddies is introduced.

1. Introduction

From as early as the 1970s, atmospheric radar scattering from well-defined layers of refractivity index in the atmosphere have been observed using windprofiler radars (e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13]. More recently, such observations have been further summarized by [14] (among others). Full explanation of these results is still incomplete, since the scattering demonstrates some peculiar characteristics, including strong aspect sensitivity. Interestingly, different models even propose seemingly opposite background atmospheric conditions, from highly stable regions to extremely turbulent regimes.

Figure 1 shows sketches of some representative models which have been presented in the literature, in order to give an idea of the variation in explanations.

Figure 1a shows entrainment, in which sheets are drawn horizontally into each other and form alternating layers of refractivity, e.g., [10]; this model requires quite still air and little to no turbulence. Refractive index edges between adjacent layers must be quite sharp, being completed in less than half a radar wavelength (i.e., <2–3 m thick). In contrast, Figure 1b shows extreme turbulence with sharp edges, as proposed by [15]; this model is now generally considered as unrealistic, but has been retained here for historical reasons. Figure 1c shows the possibility of small-scale waves called “viscosity” waves, aligned quasi-horizontally, with a vertical wavelength equal to half the radar wavelength, acting as radio wave reflectors [16,17]. These require non-turbulent conditions. Figure 1d–f show more sophisticated turbulence models than Figure 1b. In particular, they simulate radio-scattering eddies as ellipsoids, with eddies at the edges of the layer having the largest axial ratios. Elongated horizontal structures with rapid height varying vertical refractive indexes can also exist at the edges, as illustrated by undulating horizontal broken lines (e.g., [18] and references therein). Both the eddies and the striated structures can back-reflect radio waves, although more experimental evidence for such structures is needed. Figure 1d shows the case that the eddies are aligned with the edges of the layer, while Figure 1e shows the possibility that the eddies are aligned at an angle with the layer. Figure 1f shows the case that the eddies are aligned quasi-randomly. Figure 1e,f are considered more appropriate for turbulent scatter than Figure 1b. In Figure 1a,c, the radar signal that is returned to the ground is considered to behave like reflection of light from a rough mirror. In the cases in Figure 1e,f, the process is not considered to be a reflection, but rather a “scattering” process (e.g., [19]). For radar signals incident perpendicular to the long axis of the ellipse, strong signals are scattered perpendicular to that axis, but the signal is also scattered at angles which are not perpendicular, albeit with diminishing energy as one moves further from perpendicularity. Only ellipsoids with a width of order of one half of the radar wavelength produce significant scatter; other ellipsoids at other scales exist, but are not drawn. More details on the relation between the so-called “scatter anisotropy parameter” and the axial ratio of the ellipsoids can be found in figure 7.18 in [20]. The radar-signal is especially sensitive to these eddies: if the mean layer and the eddies have different orientations, then the scattering eddies will be the main defining feature, unless other quasi-horizontal extended strata (like the broken undulating lines) also co-exist nearby.

Such layers have also been seen by other methods. Reference [21] has observed similar structures by optical methods in clouds, and these have thicknesses of ~7–30 m with relatively sharp edges. In situ observations using kites, radiosondes, high-altitude balloons, helicopters, drones, and other instruments also reveal similar structures, e.g., [7,8,9,10,14,22], among others.

However, most studies have been relatively short term, covering a few hours or perhaps a few days. Our purpose in this paper will be to make studies over much longer time-scales: monthly, seasonal, and over multiple years. The above shorter-term discussions will nevertheless prove important in the interpretation of our results.

As a final point in this introduction, it is of value to determine typical isobaric slopes that might be expected in “normal” atmospheres. Recognizing that the pressure in an isothermal atmosphere is given by P = P₀ exp{−z/H}, where H is the scale height, then dP = P₀/H exp{−z/H}dz, so δz = H/P₀ exp{z/H}δP. Hence, at ground level, for H ≈ 8500 m, a change in pressure of 1 hPa corresponds to approximately 8.5 m in height. At 10 km altitude, the temperature is lower, so the scale height is 6.5 km, but the exp{z/H} term needs to be included, so 1 hPa corresponds to ~30 m of height.

As an example, if a low-pressure system is at 980 hPa and a high-pressure system is at 1020 hPa, and these are separated by 500 km, the mean slope between systems is tan⁻¹{(1020 − 980) × 8.5/(500 × 10³)} = 0.04°. If we consider a frontal system, and suppose that the change in pressure is 20 hPa across a distance of 50 km, the mean slope is 0.2°. At 10 km in height, 1 hPa corresponds to a 30 m displacement in height, so these angles will be ~30/8.5 times higher, so up to 0.6° in an extreme frontal system.

Another useful example comes in the form of sea and lake-breezes. Reference [23] estimates a pressure gradient of 1 hPa per 50 km as typical of a sea breeze, which would give a slope of ~0.01°. A more extreme but nevertheless realistic case would be a situation in which the change in pressure might typically be 4 hpa across a horizontal traverse of 50 km, giving an isobaric slope of tan⁻¹{4.0 × 8.5/(50 × 10³)} = 0.04°. At the top of a sea breeze circulation (say at 4 km altitude, where the temperature is ~270 K, so that H~7.6 km and exp {z/H} = exp(4000/7600) ≈ 1.7), the pressure changes by 1 hPa across ~13 m in height, so the angle of an isobar could be ~13/8.5 times more, or ~0.06°.

These values will prove useful later as limits for our data. We will measure monthly averages of the slopes, so it is probably fair to say that slopes of more than 0.2° might be considered unrealistic if they were to be explained as simply slanted isobars. Slopes larger than 0.5 degrees would definitely need alternate explanations.

2. Methodology

Testing of the short-term behaviour of atmospheric layers by radar methods usually involves beam steering methods, spatial interferometry, or spatial correlation techniques using multiple antennas (e.g., see [24,25,26], among others; in [20], chapter 9, further details can be found). These procedures are well suited for studies using time-scales ranging from hours to a few days. However, for this current longer-time-scale study, we adopt a different technique. Correlations between horizontal winds and vertical winds were used. Studies were performed on time-scales of one month for each month of the year at multiple sites; hourly averages of both horizontal and vertical winds were used. Horizontal winds were calculated using radial velocities determined with 4 off-vertical beams, and vertical winds were calculated using radial velocities recorded with a nominally vertical beam. Using four off-vertical beams for the horizontal wind determinations helps null out the impact of potential effects of biases in zenithal pointing directions (see the next section). Correlations between the hourly vertical wind and the horizontal wind component were determined for all azimuthal directions from 0 to 350°, and then the direction associated with the largest statistically significant correlation coefficient determines the azimuthal direction of any layer tilt which might be present. Once the optimum azimuthal direction has been found, the zenithal layer tilt is found by calculating a least-squares fit between the vertical winds and the horizontal wind components along the azimuthal direction of the layer tilt. Early attempts at this procedure were presented by [27], but this new study substantially expands the scope of that work.

3. Instrumentation

3.1. Radar Details

Data from six radars in Canada were used. Five of these formed part of the O-QNet, a network of windprofilers in Ontario and Quebec in Canada. The O-QNet radars that were used were near Walsingham, Harrow, Negro Creek, Wilberforce, and Montreal (McGill Macdonald campus). All of the radars operated at frequencies between 40 and 55 MHz and had peak pulse powers of 32–40 kW. More details about radar locations, configurations, and specifics can be found in Appendix A and [28]. For the purposes of this paper, the main points are that (i) two-way half-power beam-half-widths were typically from 1.6° to 1.9°, with the exception of Negro Creek, which had a value of 2.3°, and (ii) pulse-length resolutions were 500 m above 2.6 km altitudes and 250 m below. Duty cycles were typically 5%, and the antenna array one-way gain was ~25 dB. Off-vertical beams, which are used to determine horizontal winds, were generally 10.9° off vertical.

Data analysis of the raw data followed the procedure outlined in [29] for the CLOVAR radar. This includes some extra features not normally applied on other radars, but which were applied for all radars used in this study. In particular, data lengths from ~30 to 35 s were recorded on each beam, and extensive Fourier and spectral analyses were applied. In that paper, figures 5, 6, and 7 demonstrate the methods used in real-time to recognize and remove spurious signals. The removal of ground echoes and low-frequency oscillations is demonstrated in figure 7 in [29]. This capability is unique to the O-QNet radars and is of particular value to our experiments in this paper, in which we need to have reliable vertical winds; all ground echo signatures within ±0.03 Hz of zero frequency were automatically notched out, but broader spectra close to 0 Hz could still be retained to some degree (with the centre notched out), often with sufficient information to allow for some spectral fitting. As a rule, however, radial velocities less than ~0.1 m/s in magnitude (with frequency offsets of 0.03 Hz) were ignored, or at least were treated cautiously.

3.2. The Vertical Beam

A very important aspect of this analysis is the degree to which the main radar beam is truly vertical. Even a perfectly flat antenna array might have unknown tilts due to refractive index structures below the ground, variations in soil composition, and also possibly due to errors in the length of the feeder cables connecting to the antennas. So care is needed here.

Before we begin this discussion, it is emphasized that, in regard to array construction, all surveying pertaining to height for the O-QNet radars was conducted using a laser leveller. Critical measurements were made at night-time when the laser beams were clearly visible over distances of ~100 m. Measurements of the horizontal plane of the array were also taken from multiple redundant positions. An error of 30 cm in height across the array of width ~80 m corresponds to an error of 0.1°, and laser levelling produces an error of less than 2–5 cm over that distance; so, from a surveying perspective, our radars are mechanically flat to better than 0.02°. All feeder cables were cut to exact multiples of one wavelength (±3 mm), using impedance meters to verify that resonance had been achieved. Accuracies better than 0.5 cm per cable can be assured. Nevertheless, caution is always required, and so we now discuss other procedures by which the degree of net tilt can be evaluated.

One such test is to consider opposite pairs of tilted beams. A tilt in the radar’s vertical beam will also lead to tilts in the off-vertical beams; if the nominally vertical beam in fact has a tilt of 0.2 degrees, say, this will result in opposite pairs of beams having differing tilts. For such a 0.2° tilt, if the nominal off-vertical tilt is 10.9°, then opposite vertical beams along the line of “vertical-beam-tilt” will appear as 11.1° and 10.7° when the off-vertical beams are employed. When averaged across both beams, the “horizontal winds” will be quite accurate, as one beam underestimates and the other overestimates. Nevertheless, in such a case, a detailed comparison of individual measurements on opposing beams will show an offset between the two beams. This, therefore, offers a way to check on the vertical nature of the array. (It should be noted that even if the array tilt was 0.5° (an extreme case), so that one off-vertical beam was at 10.4° and the other at 11.4°, the actual altitude of scatterers at 10 km range in these two off-vertical cases would be 9.802 km and 9.836 km, which differ by only 34 m, so the sampled data would still be in the same range gate. Hence, there are no altitude biases in this scenario).

The vertical accuracy of the array was checked by comparing radial measurements on oppositely directed off-vertical beams. As an example, figure 11 in [29] was used. This figure shows radial velocities on nominal east and west beams for the Clovar radar. A regression calculation of radial velocities between beams, taken at similar times (to within 10 min of each other), shows regression slopes of g_0x = 0.9500 ± 0.054 and g_0y = 1.006 ± 0.058 for regression of the “west” beam on the “east” beam and the converse, respectively. The offsets of the best-fit lines were 1.54 ± 0.87 and −0.672 ± 0.953, respectively.

Taking this one step further and applying the dual regression procedure presented by [30], which is designed for cases in which both variables have errors, shows a most likely regression slope of 0.97 ± 0.06. This corresponds to a possible tilt of 0.2 ± 0.25 degrees. The possibility that the true tilt is zero is not excluded by these results, but even a tilt of 0.2° will prove acceptable for our studies (as will be established in more detail later). The Clovar radar was the prototype of all of the O-QNet radars, so the tilt angles presented here are representative of all subsequent O-QNet radars. [The Clovar radar was not used in this study because it was a developmental system; it ran from 1994 until 2008, but usually in a mixed-mode configuration, alternating between tropospheric mode, mesospheric mode, and meteor-radar mode, so no months were used exclusively for tropospheric studies. It was also low power (6 kW peak compared to 32–40 kW for the O-QNet radar). Furthermore, the Clovar radar was turned off in 2008 due to a large metal building that was built within 10 m of its edge, drowning the radar signal and contaminating the data. Since the O-QNet radars offered higher power and continuous data, and our data were concentrated on the period from 2010 to 2012, the Clovar radar was excluded from further contributions to this study]. Finally, it is worth noting that the above analysis assumes that the “horizontal wind” was indeed truly horizontal; we will return to this point later.

It is therefore concluded that the mechanical alignments of the arrays used in this study were in general better than 0.2°. Further commentary will ensue later in the Section 5.

3.3. Correlation Analysis

The key point in this analysis is that we do not assume that the nominally vertical beam on average receives maximum scatter from directly overhead, but rather allow for the direction of the received signal to be more general. Denoting the horizontal hourly averaged wind speed as U, assuming that this wind blew from a direction Ψ clockwise from geographic north, assuming that the direction of maximum scatter received on the vertical beam on average was at a zenithal angle of θ from overhead, and assuming that the maximum scatter on the vertical beam occurred from a direction ϕ₀ clockwise from true north, it would be expected that the radar would, on average, record a mean radial velocity on the nominally vertical beam of

W_u = U sinθ cos(Ψ + 180° − ϕ₀)

(1)

where, as usual, a positive value for W_u means a motion away from the radar. Angles are considered to be in degrees. If ϕ₀ corresponds with the direction toward which the mean wind blows, then ϕ₀ = Ψ + 180°, and the cosine term is unity. The directions θ and ϕ₀ may be non-zero for a variety of reasons, e.g., the scattering entities and layers may not be horizontal (see Figure 1), or the nominally vertical beam might not be truly vertical. These (and other) possibilities will be considered in due course; for the moment, it is assumed that θ and ϕ₀ may be non-zero.

The radial wind measured using the vertical beam of the radar (as determined from frequency offsets in the raw spectrum determined when the nominally vertical beam is used, e.g., [29]) will be referred to as the “nominal vertical wind” (w); this terminology is used in order to recognize that w may not be the true vertical wind and may suffer biases due to scattering layer tilts and even modest tilts of the radar beam itself. The horizontal winds (U and Ψ) were determined from radial velocity measurements on the four off-vertical radar beams, and are assumed to be accurate. The radar-derived horizontal winds have been shown to be reliable by extensive comparisons with the NCEP North American Regional Reanalysis [31]. Throughout the analysis, we used hourly averages of w and W_u. Representative wind data rates are also presented in [31]. Hourly data were generally available continuously from ~400 m to ~7 km altitude, with 90% acceptance from 7 to 9 km and ~70% acceptance from 9 to 11 km. Data above 11 km were available ~50% of the time up to ~14 km altitude.

Next, a value for θ was chosen, and ϕ₀ was systematically varied from 0° to 350° in steps of 10°. We will refer to this initial guess of θ as θ_i. Normally, a value of θ_i = 1° was chosen, but other choices may be selected at times. When using θ_i, W_u is referred to as W_ui. For each new ϕ₀, W_ui was determined for every hour of the month, and then the Pearson cross-correlation coefficient between the measured nominal hourly “vertical wind” (w) and W_ui was determined for each recorded altitude. Our primary purpose here was to find the value of ϕ₀ for which the Pearson coefficient maximized in magnitude. This was done independently at each recorded altitude. The choice of θ_i does not affect the correlation coefficient, and can be chosen at the discretion of the user. Determination of the true value of θ will be described shortly.

A typical result of this correlation analysis is shown in Figure 2. More detailed discussion will be left to the next section, but the following points are evident. First, high levels of correlation were found, especially above ~6 km. Equally notably, the heights around 3 km showed a very weak correlation in this case. It is emphasized that, while the cases of high correlation catch the eye, the cases of low correlation are equally important. These do not mean poor or missing data; they indicate cases where the scattering layers were truly horizontal, and the horizontal winds and vertical winds were uncorrelated. In fact, to some extent, they represent the “ideal” situation of a truly flat atmosphere.

The correlation coefficients are clearly rotationally anti-symmetric, with strong correlations and strong anti-correlations occurring 180° apart (reds and blues). This is expected from Equation (1). Physically, this simply means that by viewing the layers with an azimuthal addition of 180°, the regression slopes of the layers change from positive to negative (or conversely). It is also important to note that, in Figure 2, the correlations (both magnitudes and positions of maximum ϕ₀) are strongly height-dependent. (Note that within this paper, several uses of the word “slope” exist. The word is used to refer to slopes in the atmosphere associated with layer tilts, but is also used in regard to slopes of regression fitting, as in this paragraph).

The next step was to find the correct value for θ for each height. To do this, the regression plot of w vs. W_ui was re-determined, but only for W_ui values determined using the value of ϕ₀ at the optimum cross-correlation coefficient. The offset of the fit at W_ui = 0 was also recorded, as it is to some extent a measure of the “true” mean vertical velocity. Afterall, the atmosphere as a whole does not rise and fall significantly on scales of months; so, in the main the average vertical velocity should be zero, meaning that the offset in our graphs at W_ui = 0 should be close to zero. However, even that needs to be stated cautiously, as instrumental/geophysical biases still exist. These arise for reasons related to selectivity due to the effects of natural atmospheric stability biases, and they are discussed in Section 5.3 later. However, the mean offset at W_ui = 0 is generally quite small and will not affect our subsequent analysis; so henceforth, our main concentration will be on the fitted slopes.

Two examples of such fits are shown in Figure 3, with Figure 3a showing the case of a relatively good correlation and Figure 3b showing a case of poor correlation.

Adapting Equation (1) for cases where the θ_i is user-defined, we write

w = s_i W_ui = s_i U sinθ_i cos(Ψ + 180° − ϕ₀)

(2a)

where we have included the best-fit regression slope as s_i. In general, s_i will not be unity, since θ_i was user-defined (for purposes of performing the cross-correlation). If the true angle sought is denoted as θ, then there is no need for a “regression slope” term, since the condition we seek is w = W_u, or a “slope” of unity. In this case,

w = U sinθ cos(Ψ + 180° − ϕ₀).

(2b)

Comparing Equations (2a) and (2b), it is seen that sinθ = s_i sinθ_i. Thus, once the best-fit line has been produced for our chosen θ_i, the value for θ (which is the tilt angle we seek) can be deduced through the following relation:

sin(θ) = (measured slope deduced using θ_i) × sin(θ_i).

(3a)

or for small angles,

θ ≈ (measured slope deduced using θ_i) × θ_i

(3b)

In this last equation, the angles may be in either radians or degrees; if we choose θ_i = 1°, then the ideal angle θ is just equal to the “regression slope of best fit for θ_i = 1”, expressed in degrees. Note that here we use two separate applications of the word “slope”, which we distinguish as “regression slope” and “tilt slopes”. When we use θ_i = 1, the “regression slope” and the geophysical tilt slopes of the atmospheric layers become numerically equal, but this only occurs if θ_i = 1. However, the regression slope and the geophysical (tilt) slope are conceptually very different. If we use a different value for θ_i (e.g., 2°), then the regression slope needs to be converted to the tilt-slope by either Equation (3a) or (3b).

Before proceeding to look at the results, it is worth commenting on the rotational anti-symmetry shown by the red and blue colourings demonstrated in Figure 2. Red (positive correlations) correspond to cases in which, along the direction ϕ₀, the vertical velocity increases whenever the mean horizontal wind component along the direction ϕ₀ increases. Blue (negative correlations) mean that, along the direction ϕ₀, the vertical velocity decreases as the mean horizontal wind component along the direction ϕ₀ increases. Each provides the same information in a slightly different manner. We choose to accept the red cases (positive correlation), although the blue cases carry the same information, albeit in a slightly different way.

4. Results

4.1. Correlations

The value of ϕ₀ at which the correlation coefficient maximizes is called ϕ_M (M means the maximum value of the cross-correlation coefficient). ϕ_M varies as a function of month and height. The value of ϕ_M at each height and for each month was subsequently found, and we then recorded the value of ϕ_M, the value of the correlation coefficient in that direction (ρ_M), the regression-slope of the best-fit line (s_B), and the offset of the best-fit line at W_ui = 0. The offset was not further used, but the reasons that the offset exists are discussed later in Section 5.3. The value s_B was then used to determine θ, as described above. The optimum value of θ will be referred to as θ_B, where B stands for “Best -fit” (see Figure 3a as an example). The offset of the least-squares fit (demonstrated in Figure 3) at W_u = 0 was also recorded. The layer-tilt (also called the layer-slope) is usually expressed in degrees, and represents the tilt of a layer from horizontal. Graphs of these three parameters were then produced; examples are shown in Figure 4.

While the different months are colour-coded, the main purpose of the graphs is to show typical values, variability, and spreads in values. Of particular note are the facts that (a) the correlations were generally >0.2, with only 50 points out of 325 (15%) having correlations <0.2 (and only 20 out of 325 (6%) having a correlation <0.15), (b) the azimuthal directions were not random, but were mainly concentrated in a range between approximately −30° and 130° for Harrow in 2010, and (c) the layer slopes (tilts) were generally between 0.4° and 3° in this case.

Again, it is emphasized that a correlation coefficient of around zero does not mean in any sense “bad data”; indeed, the cases with ρ_M = 0 actually represent the “ideal” cases when the layers were close to horizontal. As seen, this seems true for only ~20% of occasions. The other 80% correspond to situations with non-horizontal layering. The reasons for this will be discussed further in Section 5.

Of significant note is the fact that tilts as high as 2.5° and even 3° occur. These are much larger than the estimates demonstrated at the end of Section 1, so it is clear that the simple hypothesis that these tilts are due to large-scale tilts in the isobars is not at all appropriate. For now, we will concentrate on reporting the measurements; the explanation of the reasons for the tilts will be left to Section 5, although we will hint here that Figure 1 will be key to these discussions.

The raw data shown in Figure 4 are not ideal for analysis purposes, and some averaging is needed before any trends stand out. Different averaging schemes were tested, including “height-averaging” and cross-month averaging. In the end, the best scheme for interpretation proved to be a three-point running mean vertically and a simultaneous three-point running mean over successive months. In cases where there are months with no data, forbidding calculation of running means across successive months, a five-point running mean with height was used, as it gives reasonable smoothing, but retains sufficient height-structure. Three-point running means with monthly averages mean that only the data at 3-month steps are truly independent, so the results can, at best, be considered to be “seasonal” results.

A sample plot of the correlation coefficients smoothed with this 3 × 3-point running average is shown in Figure 5 for the Negro Creek Site. This figure shows that with the 3 × 3 running average, the graph appears less noisy and trends like the increasing mean correlation as a function of height are now clearly visible. In this case, months range only from February to November, since, for example, “February” is an average over January, February, and March. Hence, an average for January is not possible without also using December 2009. Likewise, December 2010 is missing.

In later discussions, we will, in fact, create running means which cross over successive years, but this is not necessary in this illustrative example. This graph is quite representative of all sites, with the average values of ρ_M increasing with height and typical largest correlations reaching ~0.4–0.6.

No further analysis of the correlation coefficients will be presented; the primary purpose of their calculation was to identify layering alignment (when layers existed), and so allow for the determination of ϕ_M and, thence, θ_B. Our primary focus will be on the tilt angle θ_B.

Because the procedures and results presented here-in are quite new, we chose to be very cautious in the data quality accepted. Hourly average data required at least 5 horizontal wind measurements per hour at any height, and at least 3 measurements of w per hour at each height. Furthermore, over 90% of the hours in each month had to have acceptable data, so while most radars had data in all months of the year, there were frequent months which were excluded from our particular analyses due to insufficient data. In general we used the years of 2009 and 2010, though in some cases data from 2011 and 2012 were available.

4.2. Azimuthal Alignment

Figure 6 shows some representative azimuthal angles ϕ_M. Data in Figure 6 were determined as a “composite year”, taking the year (either 2009 or 2010) with the largest number of available months as the “anchor-year” and filling in months missing from that year with the corresponding month from the alternate year. It will be noted that, for the sites of Negro Creek, Walsingham, and Harrow, the azimuthal alignment of the tilted layer was between 30° and 130°, and so was roughly aligned with the direction of the prevailing winds at these sites. We did not investigate the relationship between the tilt alignment and the prevailing wind in detail here, as our main purpose is to establish the existence of these tilted layers. Detailed studies of the relationships between the tilt alignment and the prevailing wind directions could be a topic of future study.

Generally, most months at all sites showed consistency between consecutive months. Behaviour is modestly well defined. ANOVA (ANalysis Of VAriance) tests were performed and showed that all sites behaved independently, and hence were not simply some form of “noise”. The ANOVA tests will be discussed in more detail later.

In regard to Figure 6, apparently anomalous profiles do appear, particularly in April at Eureka and in February at Negro Creek. Unusual events are a natural part of any weather history. In the case of Eureka, no data were available in the preceding or following months (March and May), so sanity checks were not available. It is most likely that there was a sustained event of strong winds from the south for a significant fraction of April in this year, rather than the more common prevailing winds from the west, which may have resulted in the re-alignment of the tilted layers for a significant portion of the month.

4.3. Zenithal Tilts

We now turn to study the tilt-angles. Figure 7 shows five-point averages of θ_B over height for the available months of 2010, 2011, and 2012 for Walsingham. No averaging as a function of month were used here; only smoothing with height was used.

Some points are obvious here. First, the winter-time values (black) tend to be smaller, and January and February in both 2010 and 2011 were similar. Second, the largest values are evident in March and May in 2010, in June and July in 2011, and in August and September in 2012 (no summer months were evaluated for this site in 2010). October and November, where available, were generally in the middle. There is thus modest evidence of seasonal effects. Hence, in order to reduce statistical scatter, some averaging over successive months seemed appropriate.

Consequently, running averages across successive months were employed. Specifically, we used the following process. First, for each site, all years analyzed were used; the number of years were not the same in all cases. Walsingham had data in 2009, 2010, 2011, and 2012. McGill only had data in 2009 and 2010. But all sites had at least 2 years of data. Then, values of θ_B were averaged across all years for each month at each site. Some years had missing months; only available years were used in the averaging process. A month of data formed across several years of data in this way is referred to as a “composite month”, and a collection of all such composite months is referred to as a “composite year”. By doing this, most composite years included at least 10 months, and, in many cases, all 12 months were covered. The next step was to create both three-point and five-point running means averaged over height. Finally, three-month running means were created from the three-point height-based running means for all uninterrupted sequences of composite months. However, for cases where a composite month did not have a neighbour on one or both sides, a running mean across months was not used, but, rather, the five-point running means as a function of height were used as a substitute. As an example for a sequence of composite months comprising March, June, July, August, September, and October, with no data in April, May, or November, three-point height averages coupled with monthly three-point running means (referred to as a “3 × 3 running mean”) were used for July, August, and September, while five-point height-only running means were used for March, June, and October. Use of a five-point mean was chosen rather than, for example, a seven-point mean, as seven-point averages and even larger averages degraded the height resolution too much. Finally, December and January were considered to be “sequential”, so that a running mean for January meant averaging over December, January, and February, and a running mean for December was formed by averaging over November, December, and January.

Results of this averaging process are shown compactly in Figure 8 for all sites. ANOVA tests verified the independence of each site, as will be discussed later. Behaviours are clearly different at all sites. For example, Negro Creek showed very little seasonal dependence, while Harrow showed a very strong seasonal dependence above 7 km, with the smallest values of θ_B in Winter, moderate values in Spring, and then the largest values (up to 2°) in Summer, with modest values appearing again in Autumn. On the other hand, Wilberforce showed greatest variability below 4 km, again with a clear annual cycle, with the maximum being in Summer. Walsingham showed the greatest variability to be between 4 and 8 km altitudes.

Eureka shows June to be distinct from the other months of the year, but it must be noted that this profile came from only 1 month of 1 year, so it is not subject to the same level of averaging as some other months. As discussed earlier, occasional anomalous data should come as no surprise in weather-related studies.

4.4. Statistical Details

Figure 6, Figure 7 and Figure 8 are presented without error bars, so it makes sense to briefly look at the possible errors, especially for the azimuthal angles ϕ_M and the tilts θ_B. To do this, running “3 × 3” standard deviations were created. These were formed similarly to the process used to create Figure 6, but for running standard deviations rather than running averages, using mainly 2009 and 2010 data independently. Thus, the errors deduced would be upper limits as they involve less smoothing than, say, the procedures used for Figure 8.

Mean standard deviations of the azimuthal alignment (ϕ_M) over the 2 years were then calculated, and the results are as follows: Eureka 22.2°; McGill 24.2°; Wilberforce 22.9°; Negro Creek 29.8°; Walsingham 33.6°; and Harrow 27.6°. These numbers include any natural geophysical variability, and so are larger than random errors alone. They are standard deviations for a single month and height. Using the Central Limit Theorem, and recognizing the smoothing applied in producing the profiles, the standard errors for the mean of any profile will be given by the errors shown above (~20–30°) divided by √24 (24 points in a profile) or ~5× smaller. Typical errors for the mean were, therefore, about 5–6° for each profile, so individual profiles that differ consistently across all heights by more than ~5–6° are statistically significant. A more thorough treatment using ANOVAs will be discussed shortly. Greater detail is given in [32]).

In regard to tilts, a similar treatment gives standard deviations of θ_B as follows: Eureka 0.17°; McGill 0.15°; Wilberforce 0.12°; Negro Creek 0.11°; Walsingham 0.22°; and Harrow 0.19°. As above, standard errors for full individual monthly profiles in Figure 8 were ~5× smaller, so ~0.04°. Hence, in Figure 8, the differences between Spring and Summer below 5 km altitude at Wilberforce, the differences between Spring and Summer at 3–9 km at Walsingham, and the seasonal cycle at Harrow above 7 km all have high geophysical significance. In contrast, the differences between the different composite months at Negro Creek in Figure 8 are likely not significant. Indeed, Negro Creek gives a good opportunity to obtain a visual estimate of the errors. At the 8 km altitude, where variability between profiles is least (and hence is least likely to be affected by geophysical variability), the values of θ_B range from 0.7 to 0.85 i.e., ±0.075. There are 12 profiles, so the error for the mean is 0.075/√(12), or 0.02°. Although only a visual estimate, this is not too different from our estimate of 0.04° above; thus, a non-geophysical error for the mean in any profile of ~0.04° is reasonable. Additional fluctuations due to geophysical effects occur on top of this. Overall, these tilts are unquestionably real, and not some sort of noise artefact.

We now turn to ANOVA studies, which involves the F-factor. The F value calculates the ratio of (i) the variance obtained with all of the data being treated as one large block, irrespective of classification (group), vs. (ii) the mean of the variances of selectively classified sub-groups (referred to here simply as “groups”).

The fundamental idea is as follows. First, the user defines what a “group” constitutes. In our case, a sensible choice is to regard every site as a different group. Then, the variance of each group is calculated and these are averaged across the groups. The mean value for each site is removed, of course, during the determination of each variance. Then, we consider all of the data as one giant collective and calculate its variance. The larger dataset indirectly incorporates all of the means of the separate sites, so if the means at each site differ, this will increase the total combined variance relative to the typical individual variance for the separate sites. Hence, if the means all differ, the variance found using all of the data will exceed the typical variance for any individual site. So the ANOVA essentially tests the null hypothesis that all means at all sites are equal. If all of them are equal, it might be true that the “tilts” and “azimuths” determined were due to some common aspect of the radar design or analysis technique and so might not be geophysical. This will be revealed by values of F around unity. If, on the other hand, the mean values at each site are different, then F will be much larger than unity, and will indicate that the sites are geophysically different. The p-value gives the probability that the data at all sites were due to a process in which all means are equal (e.g., noise or some radar-related quirk). A Python program was used for our calculations, specifically Python version 3.7 with scipy:stats.f-oneway.

First, an ANOVA was performed on all of the tilt-angles for all six sites separately for 2009, and then again for 2010. For 2009, F = 66.24 and p = 6.33 × 10⁻⁶⁰ while for 2010, F = 63.03 and p = 1.41 × 10⁻⁶⁰. So the probability that the data at all sites all originated from the same underlying dataset was exceedingly small, strongly supporting the proposal that the tilts were truly geophysical in origin.

As a second test, since Walsingham was the only site that had 4 years of data, these data were also treated with an ANOVA analysis. The four groups were considered to be the four separate years. The goal of this procedure was to investigate if the data from different years were broadly similar. We would expect some year-to-year differences, but generally similar behaviour (see Figure 7). The results were F = 3.45 with a p-value of 0.019, or approximately 2%. More thorough treatments are presented in [32], but a p-value of 2% is reasonable; it suggests some level of similarity between the different years, but not exact agreement, which is consistent with natural year-to-year variability.

5. Discussion

Various other trends are evident in Figure 6, Figure 7 and Figure 8; these will not be fully discussed here, but will be the foci of later studies. The key points in this paper are as follows: (i) non-horizontal tilt-measurements are real; (ii) there are clear site-to-site differences, as verified by ANOVA tests; and (iii) these tilts are too steep to be explained by simple isobaric slopes. A primary remaining objective is to explain the origin of these tilts.

Before proceeding with a discussion of the mechanisms, it is worth once more contemplating the nature of the vertical beam. If the beam was indeed tilted off-vertical, then one can imagine specific scenarios in which the layer coincidentally forms perpendicular to the beam, so the layer appears “horizontal” as far as the radar is concerned. However, one can also envisage many more scenarios where the combined effects of the beam-direction and the layer orientation makes the calculation of the true layer orientation worse than it should be. In fact, it is to be expected that any error correction will be azimuthally dependent (for example, if the tilt is perpendicular to ϕ₀, there will be no bias) and that the net effect will be an integrated effect across all azimuthal angles available in the data. In general, it is to be expected that cases in which the beam’s direction and the layer tilt partially cancel out are the least common cases. While it may be possible to develop a detailed model of correction for beam tilts, it is clear that, in the worst case, the effect of the beam and the layer will add, and when averaged over multiple azimuths, will be less than additive but will still increase relative to the true value. Considering the worst-case scenario to be statistically additive between the layer and beam tilts, then the tilts shown in Figure 7 and Figure 8 should all exceed the actual vertical beam tilt. Therefore, the lowest values seen in Figure 7 and Figure 8 would be upper limits on the tilt of the vertical beam. Occurrences of a “layer-tilt” of 0.2° are moderately common in Figure 7 and Figure 8, and tilts of 0.2° to 0.4° are very common. Hence, we may conclude that all vertical radar beams are offset from vertical by less than 0.4° and very likely less than 0.2°. Other studies, also reported in [32], looked for instrumental biases related to the azimuthal angles, and no biases were found. These further demonstrated that tilts in the main beam were of little consequence in our study (see table 6.1 in [32]); so, henceforth we assume that all of the tilts that were measured were atmospheric in origin.

In addition, it is important to recall (again) that values of the maximum correlation coefficient ρ_M of less than 0.15 occurred for layers which were close to horizontal, and, in many ways, such occasions apply to the ideal situation of a perfectly horizontal layer; but the fact remains that such “ideal” situations are less common than might be naively expected. We will argue shortly that this anticipation of “flatness” may itself be a premature expectation.

Next, possible atmospheric causes of these tilts need to be considered. The discussion of these causes will be broken into subsections.

5.1. Linear (Un-Saturated) Gravity Waves

In this section, the possibility that linear gravity waves could reflect radio-waves at a detectable level is discussed. We do not include saturating or breaking waves here—these may be important but are discussed in a later section.

Recall from the Introduction of this paper that radio-waves are only reflected effectively from reflecting structures if the structures have significant Fourier components at the Bragg scale. This requires that either (i) the structures have sharp edges and that these edges are less than half of the radar wavelength in width, or that (ii) the scatterers are in fact sinusoidal with a wavelength along the beam equal to the Bragg wavelength, or that (iii) the scatterers are isolated entities with widths of the order of half of the radar wavelength (like the ellipsoids in Figure 1). Viscosity waves were discussed as an example of case (ii) in the Introduction of this paper.

It has been proposed by [33] that gravity waves might be able to satisfy this criterion, so we need to briefly address that possibility here. The proposal by [33] was somewhat speculative, and, in the end, the authors concluded that, even if it could happen, the Bragg scales of the gravity waves could not be less than 20 m or so. The Bragg scales for our radars are all in the range 3 to 4 m, so this is not a feasible explanation for our results. Furthermore, [16] analyzed the effects of viscosity on the gravity-wave equations and showed that this model was quite infeasible at our scale.

So the only way in which unsaturated gravity waves could cause significant reflections is through reflections from entities embedded inside the waves that are moved around by the waves. In the following paragraphs, we assume that pre-existing “tracers” cause the radio reflections and that the waves simply perturb them. It should also be noted that we use hourly averaged data, so the smallest waves that could be detected in the vertical motions will have period of 2 h and more; shorter period motions will be significantly dampened in our data. A discussion of the possibility that perturbations in linear gravity waves can lead to the correlations demonstrated in this paper now ensues.

Several points need to be noted. First, gravity waves have polarization relations between the horizontal and vertical components of the wave (e.g., [34,35] and [20], equation (11.18)), so that horizontal and vertical wind components are in phase or in anti-phase. However, this effect is an additional effect on top of the mean wind; our studies have looked at correlations between the vertical velocities and the total mean horizontal winds, which is not the same thing. Furthermore, while the vertical and horizontal winds are well understood when they are measured at the same point in space, the radar measures horizontal winds using the off-vertical beams and the vertical wind on the vertical beam, and so each are measured at different times and different points in space. Therefore, any phase relationship depends on the wavelength and period of the wave. Even further, the ratio of vertical to horizontal velocity components becomes smaller as the period increases, and as discussed above, we used hourly averages, so any effects at periods less than 2 h (when vertical velocities are more dominant) are washed out. In addition, there may be many waves present at any one time, with different wavelengths, periods, and directions of travel. All in all, the effect of polarization relations of gravity waves will be diminished.

There is, however, one type of gravity wave for which the polarization relations between the vertical and horizontal velocities might have relevance for our studies. Lee waves, generated by flow over mountains, can produce steady-state conditions overhead of a radar close to the mountain (e.g., [36], or [20], figure 12.13). If a single wave were dominant, it is possible that it may lead to fixed tilts over the radar. But, over the course of a month, multiple waves may be generated, with different horizontal wavelengths, and these would produce varying layer-slopes in the scattering layers. If the mean wind changes direction, then the situation will also change; for some wind alignments, there will be no lee waves over the radar. Such a scenario involving lee-waves also requires a nearby mountain; most of our radars do not have nearby mountains. So, lee-waves cannot explain our results generally, although are likely to be relevant in some special circumstances.

In an interesting paper, [37] has tried to simulate the anisotropy in power as a function of angle seen by the MU radar in Japan. They assumed linear gravity waves, and their Figure 1 shows a schematic of radio-waves being reflected from a sinusoidal surface. They are careful to say that the “corrugated surface” presented in that figure is not due to gravity waves directly, but is caused by the “effects of gravity waves”; in other words, they are referring to “tracers” which are being orientated by linear gravity waves. Despite some success in replicating their own observations, they also recognized errors. However, their results did not offer anything that could explain the correlations shown in Figure 2 of this paper; all of our comments above still apply. They did comment that as the waves saturate, and if horizontal velocities are included, then the waves take on a saw-tooth structure, which may be important—but that refers to non-linear waves, which will be discussed later.

Thus, unsaturated linear gravity waves cannot explain our long-term correlations. Other types of waves, like larger-scale planetary waves, have far gentler sloping isobaric tilts, and will produce angles comparable to those smaller slopes discussed at the end of the Introduction of this paper; so these waves cannot explain our results either.

Nevertheless, we have not finished with gravity waves yet, and we will return to a discussion of saturated and breaking waves, as well as Kelvin-Helmholtz billows, in Section 5.5.

5.2. Viscosity Waves

Viscosity waves are mentioned above, and it is of value to consider them here. Reference [16] discusses them in some detail, as does [38], although different applications were employed. Reference [38] uses turbulent viscosity coefficients, while [16] uses molecular kinematic viscosity. We do not need to discuss the waves studied by [38], as they have vertical wavelengths of the order of tens of metres. The vertical wavelengths are equal to 4√π√(νT), where ν is the molecular kinematic viscosity and T is the wave period, and, when using the molecular kinematic viscosity, [16] shows that, if T = 5 min and ν = 2 × 10⁻⁴ m² s⁻¹, the vertical wavelength is 1.7 m, and, for a period of 1 h, the vertical wavelength is 6.0 m, so a Bragg wavelength of 3 m (as needed for our radars) is in the right ball-park. These values were applied at 20 km in height, which is above our height regime, but near the ground, the kinematic viscosity is around ν = 1.5 × 10⁻⁵ m² s⁻¹, so a wave period of 3.3 h gives a wavelength of 3 m at ground level.

So viscosity waves of the right scales do exist, but if they are to play a role in explaining our results, then there should be a reason why they lead to the strong correlation of “nominally vertical” winds with the mean wind. At this time, such a explanation is not obvious. Presumably, the waves would have tilted wavefronts, but how these are organized is not known. But perhaps the main reason that these are less likely to be important for our data is their altitude. Reference [16] discusses the lower stratosphere because, in that region, the air can be quite calm, and turbulence can be spatially and temporally intermittent there [39,40]. Patches of non-turbulent air can therefore occur there, allowing for molecular diffusion to be dominant at those times. However, in the troposphere, the air is almost always turbulent to some degree, so the chance for molecular processes to dominate are limited. Viscosity waves are subsequently more likely to be important in the stratosphere, and they may play a role above ~9–10 km in Figure 7 and Figure 8, but are probably not so important at lower altitudes.

5.3. Stability Conditions

Now we turn to a different class of possible explanations. This class refers to systematic biases in the radar’s detection capability. Reference [41] has proposed and demonstrated that there is a slight tendency for vertically directed radars to measure stronger and more frequent echoes from scatterers embedded in more stable regions of the atmosphere. Downward motions of gravity waves tend to produce greater stability, so there is thus a slight tendency for hourly averages to be downward because, within that hour, more measurements of upward velocities will be rejected than downward ones. Such biases could appear in our data and have indeed been discussed earlier in regard to offsets in our fitting procedures (see Figure 3a and Section 4.1, paragraph 1). However, there is no obvious reason why such biases should change as a function of mean wind speed, so this explanation cannot help us understand the reason(s) for non-zero slopes in regression fits like those seen in Figure 3a.

Another issue related to stability is entrainment, as shown in Figure 1a. This is a possibility, and could produce tilted structures. But entrainment is usually associated with stable conditions and fairly still air, and we need a mechanism that not only applies to a significant fraction of all months, but also produces layers that always (or at least dominantly) tilt in one direction relative to the mean flow. Observations of such entrainment seem rare, e.g., [10], and reasons for such thin entrained layers are still uncertain. They could be some form of fossil turbulence, but fossil turbulence will mix, diffuse, and weaken any small-scale gradients, so sharp near-discontinuities with scales comparable to the Bragg scale of the radar will be erased rather than enhanced.

So it seems that these stability issues, while important, cannot explain our correlations with the mean wind.

5.4. Small-Scale Reflectors and Scatterers

In Section 5.1 and Section 5.2, there was little mention of the lengths of the reflecting/scattering “tracers”. It is easy to believe that they might perhaps be hundreds of metres, or even kilometres, long, perhaps like Figure 1a. But there is no reason that this should be so. To begin this section, let us return to Figure 1e. This figure proposes that, within a patch of turbulence, ellipsoidal eddies form and that they have systematic tilts. In intense turbulence, there can be some questions about whether such eddies even exist, but they are frequently used in theoretical discussions of turbulence. To look at it another way, all atmospheric refractivity structures can be Fourier transformed, and radar back-scatter occurs from the Fourier components of length λ/2, aligned perpendicularly to the direction of travel of the radar wave. But we could equally mathematically decompose the entire refractivity structure into a collection of ellipsoids of different widths, lengths, and refractive indices, and base our modelling on the scatter from these ellipsoids (similarly to [20], figure 7.18).

In order to broaden the scope, let us also assume that these ellipsoidal eddies could even be tilted refractive index structures analogous to “shards” of partially reflecting glass. If the shards are, say, ten wavelengths long, they can be thought of as little reflecting mirrors. If the length of the reflectors is only a few wavelengths, or even less, then it needs to be recalled that the back-scattered radiation will not be purely perpendicular, but will spread out in a manner proportional to the diffraction pattern of a hole (of the same size and shape of the shard) in an opaque screen (Babinet’s principle). We will be able to distinguish between the different models using an “anisotropy parameter”, which we will represent as θ_a, and which will be defined shortly.

In fact, in an important development, [42] has shown that for cases of semi-transparent media (as here), the use of three-dimensional ellipsoidal structures with weak variations in refractive index relative to the background is equivalent to using flat weakly scattering quasi-reflectors (the “shards of glass”), possibly with “wrinkles”. The latter approach has commonly been used by [3]. We will adopt this model of ellipsoidal refractive-index structures for our discussions. Figure 1d assumes that the ellipsoids are aligned with their long axis horizontal, and even Figure 1f assumes that the ellipsoids are, on average, horizontal. But is this reasonable? After all, the development of dynamic turbulence relies on the Richardson number being less than ~0.25, so it relies on the wind-shear dominating over stability. So if the effect of stability is reduced, then why should the mean alignment of the eddies have to be exactly horizontal? Figure 1e becomes a better representation. Even better would be Figure 1e with some random fluctuations super-imposed (like Figure 1f). Indeed, the average tilt of the eddies might be partially defined by the wind shear, or some ratio of wind shear to buoyancy, or even by the Richardson number itself.

For now, we will consider the model shown in Figure 1e as a basis, with some allowance for quasi-random fluctuation about the mean tilt like in Figure 1f. As noted, we also allow for the tilted scatterers/reflectors to take various forms. As a further example, [43] has proposed that tilted structures associated with Kelvin-Helmholtz Instabilities (KHI) could exist, and our following discussions will also cover that possibility. However, one advantage of the ellipsoid approach is that the ratios of the major to minor axis lengths can be found [44,45], and this gives a good physical parameter for specifying the degree of anisotropy.

The question now is the following: how will such a model affect the radar measurements? To answer this, we need to turn to Figure 9. This figure shows a stream of tilted ellipsoids blowing through the beam. It is assumed that the wind is purely horizontal. In this case, we have used downward sloping eddies (mean alignment is shown by the sloping broken lines). This tilt direction is the opposite to Figure 1e (downward here compared to upward there), but the concept remains. The separate “eddies” drawn in the figure can be considered as all co-existing, or can be considered as a single eddy at various points in time.

In regard to Figure 9, it is worth noting that a single eddy (or any scatterer) travelling at 30 m/s for a period of 10 s will travel 300 m in that time. At 5 km altitude, this corresponds to an angular change of tan⁻¹(0.3/5) = 3.5°. So such a scatterer has time to cross from one side of the beam to the other in 10 s. Hence, even a single scatterer will travel from C to D in Figure 9 during a typical data-acquisition time. If longer acquisition samples of, say, 20–30 s are used, the same is still true: the eddy will exist at different points within the beam during a typical acquisition, and the positions will cover a substantial portion of the beam width. Any spectrum formed using data-lengths of~10–30 s will encompass locations of the eddy (or eddies) covering a relatively large fraction of the beam.

Now consider scatter from the ellipsoids in Figure 9. Pulses from the transmitter are transmitted radially and will follow trajectories represented by T_A, T_B, and T_C as examples. Each strikes a different eddy (or, if there is only one eddy, the radar paths strike that eddy at different positions as time passes). Due to the alignment of eddy “A”, it reflects most of its energy back to the receiver. But eddies B and C reflect energy off-axis (as shown by the broken arrows) and so the signal received at the radar will be diminished. (While we speak of “reflection”, in reality, the process should be treated as a scattering process, but this simple colloquial discussion is adequate here).

From Figure 9, scatter will be strongest from eddy “A”, and the more anisotropic the ellipsoids, the more dominant will be the effect of eddy “A”.

Hence, despite the fact that the eddy moves uniformly through the beam, scatter will be dominated by eddy “A” due to its alignment. The radial component of the velocity at this point will be positive (away from the antenna array), and so the Fourier spectrum of the time-series will be dominated by positive frequencies. Hence, the radar analysis will produce a positive velocity, even though there is no vertical motion. Furthermore, the radial velocity “measured” will be proportional to the horizontal wind.

At the current point in time, this model seems to best explain our data, as it gives a natural way for some of the horizontal motions to be converted to apparent vertical motions. Turbulence is a frequent occurrence in the atmosphere, so the radar effects produced by this mechanism can easily persist over the course of months and years, as seen in our results. Not only does the model match our data, but our data allow us to quantify the mean eddy-tilting in patches of turbulence to an accuracy (in principle) of 0.2° and better—a new capability not previously available, giving new insights into the nature of atmospheric turbulence.

Even if turbulence is not involved, tilted quasi-reflectors offer the same simple means of explaining the linear relationship between horizontal and vertical speeds; if a mirror is tilted at an angle θ_∗ to the horizontal orientation and moves horizontally a distance δx in time δt at speed U_x, then the perpendicular reflection point of the mirror for the vector T_A will move δxsinθ_∗ radially, so the apparent radial velocity is δx/δt sinθ_∗ = U_xsinθ_∗. So the idea of tilted, elongated scatterers/ellipsoids/quasi-reflectors existing with slight tilts is the simplest model to explain the observed correlations. Reference [44] considered the mathematical treatment of scatter from ellipsoids, and [45] extended that discussion to different types of scatterers and showed how the ratio of major to minor axis lengths of ellipsoids can be deduced from radar measurements. But if such tilted scatterers are to explain our observations, it still needs to be determined how they are produced, and why they all have similar tilts.

The above discussions have been somewhat qualitative, but we now note that the “tilt angle” we measure will be moderated by the beam itself (see Figure 9), and the angles shown in Figure 8 could even be slightly larger in reality. Figure 10 will now be employed to obtain an estimate for the typical corrections required for the values of θ_B shown in Figure 7 and Figure 8.

Figure 10a shows an adaptation of figure 7.4 from [20], which was used to determine a relationship between the radial velocity measured with the tilted off-vertical beam of a radar compared to the true horizontal velocity component along the beam direction. It is shown in [45] as well as [20] (and various other references cited within [20]) that the radial velocity measured is that which would be determined as if the radar beam was aligned at the angle θ_eff shown in Figure 10a rather than the true angle of the beam θ_T. The angles θ_eff and θ_T are related by

sin(θ_eff) = sinθ_T [1 + sin²θ₀/sin²θ_a)]⁻¹

(4)

This is equation (7.18) from [20], where θ₀ and θa are determined as follows. First, the polar diagram of the two-way main beam of the array is assumed to have a form proportional to exp{−sin²θ/sin²θ₀}. Then, θ₀ represents a radar-related “beam-width” parameter. As discussed in [20], this means that the two-way half-power half-width of the beam θ_1/2 relates to θ₀ as θ₀ = θ_1/2/√(ln2). Our values of θ₀ thus vary between 2.0° (Walsingham and McGill), 2.3° (Harrow, Wilberforce, and Eureka) and 2.8° (Negro Creek); see Appendix A.

In regard to θ_a, the back-scattered power is assumed, for simplicity, to have a Gaussian distribution of the form exp{−sin²θ/sin²θ_a}, where θ is the angle of back-scatter and θ = 0 corresponds to direct back-scatter (perpendicular to the major axis of the ellipse). Note that [20] used the symbol θ_s instead of θ_a; we have used θ_a here, as “θ_s” has been used for other purposes in this paper. In this case, consider the subscript “a” to mean “anisotropic”, so that θ_a represents the “anisotropy” parameter discussed a few paragraphs above, and can be used to describe either our ellipsoids or the “shards of glass” analogy previously discussed.

Now turn to Figure 10b. This shows the situation under current consideration. The ellipsoids (or scatterers generally) are pointing slightly downward in the direction of the mean wind, as observed in our data. The radar beam is set to vertical. The wind is shown to come from the right rather than the left, for reasons that will become clear shortly, but the key point is that the eddies slope down and forward in the direction of the mean wind.

This figure is now modified by rotating it clockwise so that the ellipses become horizontal, producing Figure 10c. The axis A′ in this figure is parallel to the minor axis of the ellipse, and the vertical axis of the atmosphere (i.e., the component parallel to the force of gravity) is now sloping (labelled “vertical”). The figure now looks like Figure 10a. Hence, whatever mathematics was applied to Figure 10a in [20] can now also be applied here, but in the new coordinate system. So we may write

sin(θ′_eff) = sinθ_P [1 + (sin²θ₀/sin²θ_a)]⁻¹

(5)

However, θ′_eff is the angle relative to A′, which is, in turn, parallel to the minor axes of the ellipses. We are interested in the angle relative to true (gravitational) vertical, which is labelled θ_s in the figure, and is conceptually the same angle as θ_B displayed in Figure 7 and Figure 8 (i.e., it is the tilt of the major axes of the ellipses relative to horizontal). Clearly, θ′_eff = θ_p − θ_s. So, if we assume small angles, it is relatively easy to produce the expression

θ_s = θ_P [1 − (1 + (θ₀²/θ_a²))⁻¹],

(6)

which can be rewritten as

θ_P = θ_s (θ₀² + θ_a²)/θ₀².

(7)

Hence, the true slope of the scatterers θ_P can be found from the measured values in Figure 7 and Figure 8, with θ_s = θ_B, provided that θ₀ and θ_a are known. It will be seen that in this coordinate system the horizontal wind points up and to the left, but this has no impact on the determination of the effective beam-pointing direction, and the nominal vertical velocity can still be found as the dot-product of the wind vector and a unit vector parallel to the effective beam.

The point needs to be made that this calculation is only a first-order approximation. Figure 10a–c are not drawn to scale; the angles in the figure are more extreme than in real life. In Figure 10a, the angle θ_T is realistically in the order of 10 to 15 degrees. But in Figure 10c, the angle θ_P is much smaller than is drawn; the actual values are of the order of 1° to 4°, or even less. But in any region of space, there are likely to be multiple ellipsoids, each with different values of θ_a. For a near-vertical beam, as in Figure 10b,c, the dominant scatterers will have small θ_a values, but at larger beam-tilts away from the vertical orientation (as in Figure 10a), the more specular scatterers will not be visible due to their low values of θ_a. Thus, for the large beam-tilt case, concentration can be focused on the values with larger θ_a. In this case, the scatterers do not vary much across the beam. But in Figure 10b,c, the situation is better represented by Figure 9, where the paths to the scatterers (shown by the vectors T_A, T_B, and T_C) have different angles of attack on their respective ellipsoids. A more thorough treatment is then required which recognizes the different weightings relevant for ellipsoids A, B, and C. We will not do that here.

Nevertheless, Equation (7) gives a good first-order approximation for correcting the tilt angle. As an example, representative values of θ_a can be found from [20], figures 7.19, and 7.20a. The figure 7.19 in [20] suggests a value for θ_a of the order of 4° for a vertical beam in the presence of strong anisotropy, while figure 7.20a in [20] suggests values ranging from 3° to 6° for a very anisotropic tropopause. Reference [16] presents evidence for values of θ_a as small as 1°. If we assume a typical value for θ₀ for the radars of 2.5°, and use a value for θ_a of 3°, then from Equation (7), θ_P/θ_s ≈ 2.5. This may be slightly larger than a full theory would predict, but is suitable as a first guess; so the slopes in Figure 7 and Figure 8 should be of the order of 2–3 × larger than those shown in Figure 7 and Figure 8.

It is worth considering two other special cases. First, the case that θ_a = 0 corresponds to a mirror-like reflector at T_A in Figure 9, and the term (θ₀² + θ_a²)/θ₀² in Equation (7) gives 1.0, so θ_P = θ_s, which makes sense. Second, if θ_a is very large, then the scatterers are close to isotropic. Strictly, Equation (7) only applies for small angles, but it is still of interest to look at application for the isotropic case. Then, (θ₀² + θ_a²)/θ₀² is also large—it could even be infinite—but, at the same time, isotropic scatter corresponds to the zero slope in Figure 3a, so θ_s(θ₀² + θ_a²)/θ₀² is zero multiplied by a large number (possibly infinity) and so could be considered to be undefined. In some ways, this is to be expected, since it corresponds to a zero correlation (a condition that we have already determined should occur for isotropic scatter), and hence an unknown slope. So that makes sense too.

We will leave Figure 7 and Figure 8 as they are, so they basically show the effective pointing angles of the beams, and readers should recognize that the true tilt angles of the scatterers from horizontal are somewhat greater, with the correction being given by Equation (7) or an equivalent expression. To obtain estimates of the length to depth ratios of the ellipsoids (ratio of major to minor axes), the reader can consult [45] or [20] or Equation (7).

5.5. Saturated Waves, Breaking Waves, and Kelvin–Helmholtz Braids

Reference [43] has looked at the possibility that tilted quasi-flat reflecting undulations can be produced by breaking gravity waves.

In the proposal by [43], the jet presents a resonant region in which a gravity wave amplifies, saturates, and finally breaks via Kelvin–Helmholtz Instability (KHI), producing billows in the process; see Figure 11a. KHI is a form of dynamical instability. The billows are tilted, and it is assumed that, within the billows, there are tilted quasi-horizontal structures which are capable of reflecting radio waves. It is proposed that these billows have systematic slopes that might be the scattering layers responsible for our measured tilts. Sloping anisotropic scatters, like those discussed in Section 5.4, seem likely again.

In Section 5.1, the work of [37] was noted, which tried to simulate the distribution of radar aspect-sensitivity using linear gravity waves, but it could not explain our particular results. The possibility that the waves start to saturate and then produce a sawtooth pattern was noted, but since it is a non-linear effect, it was left until now to discuss. A sawtooth wave surface modulating the scatterers has the potential to explain asymmetry in the anisotropy. It would be expected, however, that the asymmetry would be a function of the direction of the phase velocity of the waves, rather than of the direction of flow of the mean wind. This does not help explain our results, unless an explanation can be found that shows that some sort of filtering has taken place that means that the phase velocity directions are correlated with the mean wind. No such mechanism comes to mind, but it might be a topic for future studies. For now, we turn to further considerations of the wind shear.

5.6. The Importance of Wind Shear

The theory presented by [43] utilizes a wind jet, but it is possible that similar processes can occur in a more general wind profile which includes a wind shear with increasing altitude. Such additional cases are shown in Figure 11b. In Figure 11b, it is envisaged that there exists a wind profile which increases with height, while a spectrum of gravity waves is produced independently lower down. Waves from below therefore move upward as well as horizontally to both the left and right. Waves moving to the right will rise into ever-increasing mean winds until eventually the point is reached where the phase speed of the wave equals the background wind speed. This is called a “critical level”, at which point the wave may break and produce turbulence. On the other hand, waves moving to the left will grow in amplitude, and the point may be reached where the amplitude equals the intrinsic wave speed, giving rise to convective breakdown (e.g., see [20], equations (11.27)–(11.30)), and also [46]). Convective breakdown commonly leads to wave breakdown and, therefore, to turbulence. The structures produced may have a preferential tilt bias as they break (as in Figure 11a), which in turn leads to anisotropic eddies within the turbulence having a tilt bias.

Reference [47] has produced arguments that suggest that the degree of anisotropy of an eddy depends on the background wind shear du/dz. Although they did not discuss the mean orientation of the anisotropic eddies, it seems likely that du/dz might also affect the tilt of these entities. For example, if an anisotropic eddy forms pointing downward in the direction of the mean wind, then, in the presence of an increasing vertical wind shear, the tail of the eddy, being higher up and in a region of increased wind speed, will move forward faster than the head, resulting in a rolling motion that steepens the slope of the eddy. If, on the other hand, the eddy was formed with a slight upward tilt parallel to the wind, then the front will move ahead faster than the tail, and the eddy may be stretched and become flatter. So, in both cases, there is a tendency to “roll” the eddy clockwise. It seems likely, then, that the background wind shear plays a role in producing a downward slope to the eddies. This process is indicated in Figure 12.

Very clear evidence of this possibility comes from [48]. This paper presents contours of power as a function of angle very close to vertical, using 64 closely spaced near-vertical beams with the MU radar. The results clearly show that the maximum power was frequently not quite vertically overhead, but offset. Indeed, that seemed to be true more often than not. The results are very similar to Figure 10c, recognizing that true vertical is indicated by the sloping axis labelled “vertical”. Reference [48] then entered into an extensive discussion about the reasons for this, and concluded that the reason was due to wind-shear rotating the more anisotropic eddies off-vertical. The results of [48] are also qualitatively consistent with Figure 9.

Reference [48] also presents the following quote from [49] to support their hypothesis of the importance of wind shear: “gravity-wave breaking, being asymmetrical according to the direction of propagation of the waves relative to the mean shear, will modify an existing wave-field of internal waves, and possibly leave a wave-field with directional asymmetry”.

At this stage, the results are interesting, but somewhat speculative. Therefore, it is important to quantitatively ascertain the impact of shears. Figure 12 helps us do that.

Figure 12 shows that the change in angular tilt γ of an anisotropic eddy due to a wind shear du/dz during a time δt is

δγ = (l_z/l_h)(du/dz) δt

(8)

where the variables are defined in Figure 12. The higher wind speed at greater altitude pushes the upper tail of the eddy forward relative to the lower front end. It is of value to calculate a typical value for δγ. First, an eddy lifetime is needed. Taking a typical energy dissipation rate ε at the edge of the turbulent layer (where turbulent dissipation is weakest) as ε = 10⁻⁴ Wkg⁻¹ (e.g., see [20], figure 11.30), and using ε ≈ L²T⁻³, and supposing a typical value for l_h of 10 m (the eddy thickness along its minor axis will be of the order of the radar Bragg scale, ~3 m) gives an eddy lifetime of around (100/10⁻⁴)^0.333 ≈ 100 s. Assuming γ₀ = 1°, then l_z/l_h ≈ 0.02, and taking a strong but not excessive local wind-shear of 20 ms⁻¹ km⁻¹ (0.02 s⁻¹) gives δγ = 0.02 × 0.02 × 100 = 0.04 radians, or 2.3°. So an initial tilt angle of 1° is changed to 3.3° during the lifetime of the anisotropic eddy. This is a significant change, and is compatible with our typical tilt angles. So, depending on the stage of the development of the eddy and its initial tilt on formation, it could have any tilt from 1 to 3.5 or more degrees. The turbulent layer will have a mixture of such eddies at various temporal stages of formation and decay.

The discussions in [48] are therefore extremely supportive of our own data, although the techniques used in their work and ours are entirely different. We consider rotating of the eddies within turbulence via wind shear to be a very strong candidate to explain our observations.

5.7. Further Matters

Finally, two additional points need to be made. First, it should be noted that many previous authors have used vertical velocities measured with a vertical beam to study short-term gravity wave activity. As long as the mean wind is relatively constant during the study, or has an identifiable behaviour, then the effects presented in this paper can be successfully removed, and the remaining short-term fluctuations can still be used for gravity wave studies. However, studies of the long-term behaviour of the mean vertical wind must include corrections for the effects presented herein.

As a second point of interest, it is interesting to compare the results here to the results in [28]. That paper looked at the distribution of regions in the atmosphere where gravity waves dominated, compared to regions where large-scale 2D geostrophic turbulence was more prevalent. Both Walsingham and Harrow appear to have significant gravity wave production, especially in Summer, presumably due to lake breezes. Both sites show more gravity wave activity than the other sites used in that study, and both show significant wave production in the lowest 2 km. Comparing these results to Figure 8 in this paper, it is seen that our tilts become steeper at >6 km in Summer at Harrow and at 6–9 km at Walsingham. These results would be consistent with gravity wave production near the ground, free growth up to ~6 km altitude, and then breaking of the waves to produce strong turbulence and subsequent secondary gravity-wave growth at 6 km and upward. The occurrence of somewhat large layer tilts seen in Figure 8 of this paper at similar heights to the active gravity-wave growth at 6 km may be further support for our model. These results thus lend support to our discussions in Section 5.5 and Section 5.6.

Multiple explanations have been presented above that might explain the scatterers and quasi-reflectors that have been discussed. Some were rejected, but some seem feasible. There is no reason why there has to be only one answer. Just as gravity waves can have multiple sources (convection, flow over mountains, volcanoes, wind shear, resonant cavities, etc.), so the causes of quasi-specular reflectors can also have multiple sources. Viscosity waves and entrainment require still air and high stability, and may be more common at higher altitudes, like in the stratosphere. Turbulence requires instability and shear, so this requires a different environment, and yet anisotropy can still result. It is acceptable that there are multiple possible sources, but it does seem that wind-shear-driven realignment of anisotropic eddies toward off-vertical during breaking gravity-waves is a strong candidate to explain our observations.

6. Conclusions

Extensive investigations with six radars over a time-frame of up to 4 years show significant correlation between long-term vertical and horizontal winds as measured by VHF radars. An interpretation of these data leads to the conclusion that scatter on the vertical radar beam is biased due to mean tilts in the scattering eddies and reflectors in the range from 1 to 3 or 4 degrees (after correcting Figure 7 and Figure 8 with Equation (7)), and, at times, more. Wind-shear coupled with breaking gravity waves seems to be important in producing this bias, as is the Kelvin-Helmholtz breakdown of gravity waves. Extensive statistical tests confirm that these measurements are not artefacts, and significant variations are evident due to variations in season and local atmospheric conditions. This conclusion does not preclude other processes like intrusions and viscosity waves contributing to the tilting, and these latter processes may be especially important in the stratosphere, where turbulence-free zones often exist. The analysis method discussed in this paper has the unique potential to enable measurements of mean eddy and scatterer tilts in cases where turbulence is involved.