JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, D17101, doi: /2011jd017179, 2012

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jd017179, 2012 Computation of clear-air radar backscatter from numerical simulations of turbulence: 3. Off-zenith measurements and biases throughout the lifecycle of a Kelvin-Helmholtz instability David C. Fritts, 1 Kam Wan, 1 Patricia M. Franke, 2 and Tom Lund 3 Received 16 November 2011; revised 21 June 2012; accepted 3 July 2012; published 5 September [1] Previous papers by Franke et al. (2011) and Fritts et al. (2011) described the computation of radar backscatter power and vertical velocities from numerical simulations of turbulence arising due to Kelvin-Helmholtz (KH) shear instability. Comparisons of backscatter power and inferred velocities with the distributions of turbulence and the true velocities revealed biases in the identification of active or intense turbulence and in the inferred Doppler spectrum and vertical velocities throughout the flow evolution. This paper extends these analyses to off-zenith viewing angles typical of multiple-beam MF, HF, and VHF radars. These reveal similar biases in the identification of turbulence occurrence, Doppler spectra, and inferred radial velocities, with additional sensitivity to the off-zenith angle relative to the mean shear across the turbulence layer. Radial velocities are typically underestimated during turbulence generation and breakdown of the KH billows, except where turbulence refractive index gradients are strong. Doppler spectra are biased toward regions retaining strong refractive index gradients, implying strong aspect sensitivity at later stages in the evolution. Persistent tilted structures at late stages of the evolution contribute to radial velocity measurement biases that also are functions of off-zenith angle and time. Citation: Fritts, D. C., K. Wan, P. M. Franke, and T. Lund (2012), Computation of clear-air radar backscatter from numerical simulations of turbulence: 3. Off-zenith measurements and biases throughout the lifecycle of a Kelvin-Helmholtz instability, J. Geophys. Res., 117,, doi: /2011jd Introduction [2] Franke et al. [2011] and Fritts et al. [2011] (hereafter Franke11 and Fritts11) previously described the motivations for numerical studies of radar backscatter accompanying turbulence and small-scale refractive index fluctuations due to Kelvin- Helmholtz (KH) instability (or KHI) and the implications of these dynamics for radar measurements employing vertical beams. KH instability was chosen as the underlying dynamics in these initial studies because of its ubiquitous occurrence extending from the stable boundary layer into the thermosphere [Witt, 1962; Browning, 1971; Gossard et al., 1971; Røyrvik, 1983; Fritts and Rastogi, 1985; Eaton et al., 1995; Chilson et al., 1997; Blumen et al., 2001; Hecht, 2004; Kelley et al., 2005; Lehmacher et al., 2007; Luce et al., 2007, 2008; Woodman et al., 2007]. KH instability is dynamically significant because it contributes vertical mixing of heat, momentum, and 1 Boulder Division, GATS, Inc., Boulder, Colorado, USA. 2 Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA. 3 Colorado Research Associates Division, NorthWest Research Associates, Boulder, Colorado, USA. Corresponding author: D. C. Fritts, GATS, Inc., Boulder Division, 3360 Mitchell Ln., Boulder, CO 80301, USA. (dave@gats-inc.com) American Geophysical Union. All Rights Reserved /12/2011JD constituents, it limits the maximum shears that can occur and the amplitudes of wave motions contributing these shears, it poses flight risks for aircraft in extreme cases, it impacts laser propagation and astronomical applications, and it creates much of the persistent small-scale turbulence, and refractive index variations, that enable measurements of atmospheric flows with various radars. As such, it has received considerable research attention in the past, and it remains an area of active research interest. [3] In the stable boundary layer (SBL), KH billow wavelengths leading to turbulence vary from 1 m to hundreds of meters, with these limits imposed by kinematic viscosity and the maximum depth of SBL shear layers, respectively. Minimum KH billow wavelengths increase inversely with density as r 1/2 with increasing altitude, with maximum scales dictated by large-scale mean and inertia-gravity wave shear depths, and which may achieve scales of 10s of km in the stratosphere and above [Fritts and Alexander, 2003]. KH billow evolution and turbulence generation depends on various parameters, most critically the Richardson number, Ri = N 2 /U z 2 (where N is the buoyancy frequency and U z is the mean wind shear) which measures the tendency for the available kinetic energy in wind shear to overcome the stabilizing influences of stratification, and the Reynolds number, Re, which measures the stabilizing influence of kinematic viscosity. Values of Ri near zero favor strong instability, deep billows (1/2 the horizontal wavelength), and strong 1of22

2 turbulence, whereas values of Ri nearer 1/4 favor weak instability, shallow billows, very different instability and turbulence evolutions, and weak turbulence [Thorpe, 1973; Werne et al., 2005]. KH instability for which Re = Uh/n (where U is half the velocity difference of the shear layer, h is half the initial shear depth, and n is kinematic viscosity) is less than a few hundred tends to remain laminar for any Ri. But values of Re 1000 or greater, especially at lower Ri, favor rapid instability growth and strong three-dimensional (3D) turbulence [Klaassen and Peltier, 1985, 1991; Thorpe, 1987; Fritts et al., 1996, 2003; Smyth, 1999; Werne and Fritts, 1999a, 2001]. Further discussion of KH instability dynamics is provided in the reviews by Thorpe [1987], Fritts and Alexander [2003], and Peltier and Caulfield [2003] and references therein. [4] A potential for biases in radar wind measurements due to correlations of refractive index fluctuations with the underlying dynamics has been recognized for several decades [Kudeki et al., 1993; Muschinski, 1996, 2004; Muschinski et al., 1999; Gibson-Wilde et al., 2000; Tatarskii and Muschinski, 2001; Worthington et al., 2001]. Specific cases include mean vertical motions that differ from expectations in both sign and magnitude [Balsley and Riddle, 1984; Nastrom et al., 1985; Fritts and Yuan, 1989; Meek and Manson, 1989; Rüster and Reid, 1990; Wang and Fritts, 1990; Fritts et al., 1990; Fukao et al., 1991]. Nastrom and VanZandt [1994] noted a correlation between upward motions and enhanced high-frequency variances in tropospheric wind profiler data. They suggested that a negative bias in mean vertical velocities resulted from a correlation between higher radar backscatter power with the higher refractive index fluctuations accompanying the downward (more stable) phases of gravity waves (GWs) modulating turbulence intensities. Hoppe et al. [1990] and Hoppe and Fritts [1995a, 1995b] assessed the correlations between backscatter power, vertical velocity, and spectral width using the EISCAT 224 MHz radar under polar mesosphere summer echoes (PMSE) conditions and found high power to be well correlated, in general, with narrow spectral widths and downward motions in the more stable phases of the GW motions. [5] Measurement biases accompanying radar observations of KH instability were first suggested by Muschinski [1996], who argued that the systematic slopes of layered features (essentially anisotropic and/or inhomogeneous turbulence fields) accompanying KH evolution in mean shear could lead to errant vertical velocity estimates. Other biases were suggested by observations of the familiar Kelvin s cat s eye structures in radar backscatter power, which clearly delineate the KH billows but appear to fail to capture the stronger turbulence often occurring within the billow cores [Fritts and Rastogi, 1985; Eaton et al., 1995; Woodman et al., 2007; Lehmacher et al., 2007; Fritts et al., 2003], and by more recent numerical studies [Gibson-Wilde et al., 2000; Hill and Mitton, 1998; Hill et al., 1999]. [6] Turbulence anisotropy and inhomogeneity are expected to arise due to stratification and shear [Uberoi, 1957; Townsend, 1959; Bolgiano, 1959, 1962; Lumley, 1964] (also see early reviews by Champagne [1978], Mestayer [1982], Browne et al. [1987], Hunt et al. [1991], Sreenivasan [1991], and Van Atta [1991]). More recent studies further quantified our understanding of departures due to shear and/ or stratification for various geophysical flows [Durbin and Speziale, 1991; Werne and Fritts, 1999a, 1999b, 2000, 2001; Smyth and Moum, 2000; Pettersson-Reif et al., 2002; Fritts et al., 2003; Pettersson-Reif and Andreassen, 2003; Wroblewski et al., 2003; Ruggiero et al., 2005; Werne et al., 2005, Fritts et al., 2009a, 2009b]. [7] The biases suggested by various previous studies to accompany radar measurements of KH instability dynamics were examined in greater detail for zenith measurements in our first studies in this series by Franke11 and Fritts11. These studies employed a large-eddy simulation (LES) description of KH instability dynamics and turbulence arising in an environment having Ri = 0.05 and Re = 10,000 that was benchmarked against a direct numerical simulation (DNS) of the same flow with Re = 2,500 to ensure that the LES adequately described the secondary instability transition to turbulence within the primary KH billows. Radar backscatter computations employed the Born approximation for radar backscatter developed for isotropic turbulence by Tatarskii [1961, 1971] and extended to more general anisotropic turbulence and Fresnel backscatter by Doviak and Zrnic [1984], Muschinski et al. [1999], and Muschinski [2004]. Results of these two studies included (1) backscatter power predictions that agreed reasonably with observations of such events and their temporal evolutions, (2) a lack of significant backscatter power in well-mixed billow cores, suggesting possibly weak turbulence where in fact it may be strongest (as observed and anticipated in our earlier studies), (3) maximum backscatter power in the billow exteriors, where refractive index fluctuations are large but turbulence is weak, (4) underestimated vertical velocities at early times, with the smallest biases in the KH billow exteriors where turbulence and refractive index variations are both significant, and (5) significant inferred vertical velocities where true vertical velocities are near zero at late stages of restratification, especially in the edge regions of the turbulence layer. [8] Our purposes in this paper are to extend the analyses by Franke11 and Fritts11 to off-zenith radar measurements. For easy comparison with those results, we employ the same simulated KH instability and turbulence fields and the same methodologies for defining the sampling volumes and backscatter spectra, but now for volumes that are tilted by their respective off-zenith beam angles. Here, however, we present results only for the assumed 10 MHz radar having a 15-m Bragg scale, a 45-m range resolution, and narrow beam width, as those are generally representative of the results for the 3 MHz radar and coarser range resolution also considered by Fritts11. The description of the KH instability simulation, the assumed radar sampling, and the method for estimating radar backscatter are summarized in section 2. Section 3 presents the results of our study, including backscatter power, measured Doppler velocities and biases, and measured and true (i.e., assuming weighting dependent only on the beam parameters) Doppler spectra, for representative off-zenith beam angles. A discussion of these results and their implications is provided in section 4, and our conclusions are presented in section Dynamical Environment and Radar Assumptions and Backscatter Computation 2.1. KH Instability Simulation [9] Following Franke11, we assume radar backscatter from the life cycle of a KH instability provided by an LES description of the evolution for an initial Richardson number 2of22

3 Ri = N 2 /U z 2 = 0.05, a Reynolds number Re = Uh/n = 10,000, and a Prandtl number Pr = n/k = 1. The LES simulation (employing a dynamical sub-grid-scale model) was benchmarked against a direct numerical simulation (DNS) of the incompressible, Boussinesq Navier-Stokes equations at Re = 2,500 to ensure that the LES properly described the secondary instability dynamics leading to turbulence. Prior to turbulence generation, the LES model describes KHI and transitional dynamics as a DNS. Once turbulence arises, the LES sub-grid-scale model is activated and accounts for inertial-range structure and energy dissipation thereafter. The initial horizontal mean wind was given by U(z)= U tanh (z/h), with U =5ms 1 and h = 150 m, such that U z = U/h = s 1, the KH instability had a horizontal wavelength L = h, and the billow and turbulence layer approached a typical depth D 6 h 900 m. The quantities n and k are kinematic viscosity and thermal diffusivity, respectively, and Pr differed from that of the atmosphere, 0.7, in order to achieve uniform resolution requirements in the velocity and potential temperature fields. These choices yielded a time scale h/u, a buoyancy frequency N = Ri 1/2 / s 1, and a buoyancy period T b =2p/N =28h/U = 843 s. The LES was performed in a computational domain of dimensions (12.566, 4.2, 25)h, and employed spatial resolution with (720, 240, 1440) spectral components and a 2/3rd spectral truncation to eliminate aliasing [Fritts et al., 2009a]. This yielded a spatial resolution of Dz 2.6 m to ensure sufficient resolution of structures at the assumed radar Bragg scale of 15 m for our radar backscatter assessments. To ensure resolved radar phase variations and Doppler velocities, we chose a sampling interval of 1.5 s and 64 samples per spectral computation. This implied a sampling interval of 96 s or 0.11 T b. [10] We note, however, that the LES KHI simulation is representative of the same dynamics for the same nondimensional parameters at larger and smaller spatial scales. For example, these results can be thought of as describing a KH billow of depth 3 km occurring 10 km higher in the atmosphere or a KH billow of depth 100 m occurring 20 km lower in the atmosphere. [11] The KH instability and turbulence evolution (reproduced from Fritts11) is shown in Figure 1 with streamwisevertical cross sections of potential temperature, vertical velocity, and vorticity magnitude at times of t = 37, 54, 84, 129, 256, 312, and 364 (in units of h/u, which we will use in our discussion below) spanning 11 T b. These illustrate the KH evolution from the late two-dimensional (2D) flow through the turbulence transition, billow mixing and breakdown, and formation and restratification of the turbulence layer at late times. Our evaluation of radar backscatter for off-zenith beams will employ data at these times. [12] Secondary instabilities accompanying this KH instability evolution are of two types: (1) smaller-scale quasi-2d KH instabilities arising primarily in the strongly stratified braids between adjacent billows (accounting for the strong initial modulations of the outer-most sharp potential temperature gradients seen in Figure 1 at t = 54) and (2) streamwise-aligned instabilities (having spanwise wave numbers) arising along the 2D potential temperature and radial velocity fields immediately downstream of the billow inflow regions that are strongly modulated in the radial direction. These secondary KH and streamwise-aligned instabilities are shown in Figure 2 with images of vorticity magnitude in a streamwise-vertical cross section of the primary KH billow at t = 50, 52, and 54 (top three panels) and in subsets of the spanwise-vertical and streamwise-spanwise cross sections (lower left and right, respectively) at the upper edge and the right end of the KH billow at t = 52 and 54. The former are smaller-scale versions of the primary KH instability that become possible as the primary KH instability Re increases, but having local values of Re describing the secondary instability that preclude their becoming turbulent. The latter are the same secondary instabilities (though at smaller spatial scales) that drive the transition to turbulence at lower Re [Werne and Fritts, 1999a; Werne et al., 2005], suggesting that the LES for Re = 10,000 described above does indeed describe the transitional instability dynamics adequately for our purposes here. [13] Shown in Figures 3 and 4 are (1) expanded 2D views of the perturbation potential temperature and vorticity fields at t = 256 to 364 and (2) horizontal velocity and potential temperature through the billow center and between adjacent billows (solid and dashed, respectively), and Ri profiles at the billow center, throughout the evolution for reference in the discussion of measurement accuracies and biases below Radar Assumptions [14] For consistency with our previous results (Franke11, Fritts11), and to utilize the KH instability LES results described above with maximum spatial resolution, we assume: (1) a Doppler radar having a frequency of 10 MHz and a Bragg scale of 15 m; (2) a vertical beam having a sin x/x shape with sidelobes suppressed and a main lobe full width of 180 m at mesosphere and lower thermosphere (MLT) altitudes; (3) a radar baud length yielding a Gaussian full-width, half-maximum (FWHM) pulse width of 90 m (and an altitude resolution of 45 m); (4) the radar samples successive spatial positions within the KH billow, separated by the full beam width in the streamwise direction, simultaneously in order to achieve the optimal definition of the KH billow structure and evolution in space and time; (5) there is no background noise; and (6) the moments of the Doppler spectrum can be assessed following Franke11 (see below). [15] As noted in the discussion in section 2.1, the assumed range resolution and beam width would be correspondingly larger if we assumed a larger KH billow depth. For example, a KH billow depth of 3 km would imply a range resolution and beam width of 150 m and 500 m, respectively, in close agreement with the current capabilities of the Jicamarca VHF radar. [16] Backscatter computations were performed for each time throughout the KH evolution displayed in Figure 1, from the late 2D flow through the turbulence transition, billow mixing and breakdown, and formation and restratification of the turbulence layer, in order to evaluate measurement accuracies and biases at all stages of the evolution. Doppler spectra, from which backscatter power and Doppler velocities were obtained, were determined from 64 LES turbulence volumes spaced by 1.5 s (spanning 0.11 T b or 3.2 h/u) for each of the times displayed in Figure Radar Backscatter Computation [17] Our computation of radar backscatter employs the Born approximation to solve the Helmholtz equation. Index 3of22

4 Figure 1. KH LES evolution shown with 2D streamwise-vertical cross sections of (left) potential temperature, (middle) vertical velocity, and (right) vorticity magnitude at the times shown at upper left. Red/ orange and blue are positive and negative values in Figure 1 (left) and Figure 1 (middle), respectively. of refraction perturbations are assumed to respond to the incident field by acting as current sources in the scattering volume with the backscattered signal being the sum of all responses to the distributed sources [Tatarskii, 1961; Ishimaru, 1978]. This includes both Bragg scattering from the smaller-scale structures and Fresnel scattering from coherent layered structures, but neglects multiple scattering effects. A unique feature of our approach is to compute the integral in the time domain. This avoids making assumptions about the distribution of the different scales inside the scattering volume and handles scenarios where the radar pulse spans volumes with very different spatial distributions [Nickisch and Franke, 1998]. In all other respects, our procedure employs standard radar data processing methods described by Doviak and Zrnic [1984]. For additional details on our specific approach, the interested reader is referred to Franke Radar Backscatter Results 3.1. Backscatter Power Distributions [18] Computed backscatter power plots for off-zenith beam angles of 0, 7.5, and 15 (negative and positive relative to the flow above the shear layer to the right) are shown in Figure 5 for the 7 times displayed in Figure 1. Each panel shows sampling volumes of 45-m depth and 180 m width, with 20 volumes spanning the KH billow or turbulence layer depth and 10 beam widths spanning the KH horizontal wavelength. Sampling volumes having power less than 1% of the maximum in that beam at that time are 4of22

5 Figure 2. KH secondary instabilities viewed in a streamwise-vertical plane at t = 50, 52, and 54 (top three panels), (bottom left) in a spanwise-vertical plane at the top edge of the KH billow at t = 52 and 54, and (bottom right) in a streamwise-spanwise plane at the right end of the billow at the midpoint at t = 52 and 54. Only the streamwise-aligned secondary instabilities (bottom set of panels) contribute to turbulence generation for the flow parameters considered here. assumed to be inadequate for valid measurements (shaded gray). As discussed by Fritts11 for the vertical beam, the dominant backscatter power, following initial specular returns at the earliest time, is seen to occur in the billow or turbulence layer edge regions except at the latest time, where significant power is also seen in the center of the restratifying turbulence layer. [19] Backscatter power at the earliest times (t = 37 and 54) is very localized and relatively strong. Prior to generation of turbulence, it occurs only where potential temperature gradients are large and aligned nearly normal to the radar beams, enabling specular backscatter. For a vertical beam, this occurs only at the highest and lowest excursions of the KH billow at t = 37 (see the first row of Figure 1 and the second solid profile in Figure 4a). For off-zenith beams, the peak responses occur in the billow edge regions having alignments nearly orthogonal to the radar beam or in the strongly stratified braids between adjacent billows. Backscatter remains specular and confined to the billow edge regions (and between adjacent billows) at t = 54 because secondary instabilities, while now having large amplitudes (see Figure 2), have not yet led to well-developed and nearly isotropic 3D turbulence. Indeed, the strongest backscatter occurs between adjacent billows at this time because gradients are very sharp, coherent in the spanwise direction, and exhibit a range of slopes. Large-amplitude instabilities in the billow exterior also enable specular backscatter at multiple beam angles, thus accounting for the ability of all five beam angles to estimate velocities around much of the billow exterior. Influences of the larger-scale potential temperature gradients remain, however, as the sampling volumes having power above the threshold are more frequent where the edge regions are nearly normal to the beam at the various beam angles. [20] Radar backscatter power distributions at off-zenith beam angles during the turbulent billow phase of the KH evolution (at t = 84 and 129) resemble those for the vertical beam fairly closely, as turbulence containing refractive index fluctuations is the dominant source of backscatter during these times. As for the vertical beam, backscatter power is maximum in the outer portions of the billow at both times. At t = 84, this is because turbulence has not yet penetrated into the billow center. At t = 129, the radial distribution of turbulence intensity has reversed, with the strongest turbulence now in the billow core. This causes the eradication of strong refractive index gradients in the core, sharp refractive index variations in the edge regions instead, and a larger absence of significant backscatter within the billow than at t = 84 having backscatter power <1% of the maximum (see the very weak potential temperature gradients in the billow center in the 4th and 5th profiles in Figure 4b). [21] Despite the largely turbulent backscatter character at these times, there remain biases in the locations of peak power. These occur more frequently at the upper and upperright regions and the lower and lower-left regions of the billow exterior for all off-zenith beam angles. There is also a tendency for these regions of higher power to rotate clockwise around the core for positive beam angles and counterclockwise for negative beam angles. These suggest systematic tilts of the larger-scale potential temperature gradients that are nevertheless enhancing power in these regions, as seen in Figure 1. [22] While the KH billow is fully turbulent at t = 129, it is still highly coherent at large spatial scales, as can be seen from the strongly rotating core in the vertical velocity cross section shown in the middle column of Figure 1 and in the horizontal velocity profiles in Figure 4. This changes rapidly, however, as billow turbulence shears horizontally and 5of22

6 Figure 3. As in Figure 1, but for (left) perturbation potential temperature and (right) vorticity magnitude at t = 256, 312, and 364. achieves a more horizontally homogeneous distribution of velocities at much smaller spatial scales thereafter. The consequences of this evolution of the turbulence layer for backscatter power are several. Strong stratification and weaker turbulence in the edge regions enable layering and enhanced backscatter power (see the left and right columns of Figure 1 and Figure 4b) at t = 256 and thereafter). Both the enhanced stratification and weaker turbulence contribute to suppression of vertical motions in the edge regions, while enabling specular contributions to emerge (see below). Finally, persistent tilted layers are created by the background wind shear that evolve toward shallower slopes and smaller vertical scales, and which imply strong aspect sensitivity (see the right column of Figure 1 and below). [23] Evidence for evolving aspect sensitivity and specular backscatter at times beyond t = 129 are seen clearly in Figure 5. The turbulence layer edge regions dominate the power profiles at t = 256 and 312 because of strong refractive index fluctuations accompanying weaker, but persistent, turbulence, especially at negative beam angles. The power is also significantly stronger at negative than at positive beam angles at these times. These characteristics suggest that coherent tilted structures primarily in the edge regions, but also in the layer interior (see Figure 3 at these times), contribute preferential specular backscatter in these regions. [24] As the edge regions restratify and turbulence at the Bragg scale subsides (t = 312, see Figures 1 and 4), backscatter power in the interior (where turbulence remains more vigorous, but refractive index fluctuations are smaller) becomes competitive with that in the edge regions, again with larger power at negative than at positive beam angles. By t = 364, turbulence has decayed sufficiently in the edge regions that power maxima are now in the layer interior for all beam angles, again with higher power for negative than for positive beam angles. [25] Finally, we note that the vertical beam power distributions at late times differ somewhat from those discussed by Fritts11, which exhibited more distinct power maxima in the edge regions and layer center at late times. This is due to a more extended integration over the Gaussian pulse in altitude than employed by Fritts11. The two methods yield essentially identical velocity estimates, however. [26] The tendencies for stronger power at more negative beam angles from t = 256 to 364 clearly arise from the sloping turbulence and fine structure layers due the sustained large-scale shears (see Figures 3 and 4), which cause these layers to be more nearly normal to the negative beam angles, respectively. This bias continues to shift toward smaller beam angles with time because of the continuing shearing by the large-scale flow. The positive beam angles at these times exhibit significantly smaller power in the edge regions and in the layer interiors. This is direct evidence of specular backscatter because the turbulence structures and refractive index variations are the same for each radar beam. Referring to Figure 3 at t = 364, we see that the edge regions are now nearly horizontal because of their stronger shearing, suggesting similar influences at smaller positive and negative beam angles, but the interior layering still exhibits predominant slopes of 15 or larger, thus favoring greater backscatter at larger negative beam angles. [27] We summarize the aspect sensitivity discussed above in two ways. Figure 6 (top) shows vertical profiles of the mean backscatter power for each of the four beam angles at t = 84, 129, 256, 312, and 364 from left to right (with blue, 6of22

7 Figure 4. (a) Horizontal velocity, (b) potential temperature, and (c) Richardson number profiles (showing 0.5 < Ri < 2) throughout the KH instability and turbulence layer evolution shown in Figure 1. Times are t = 0, 37, 54, 84, 129, 256, 312, and 364. Solid profiles are at the billow center, dashed profiles are between adjacent billows, and Ri profiles are averaged over the central 10% of the horizontal domain in each case. green, orange, and red lines representing beam angles of 15, 7.5, 7.5, and 15, respectively). Each profile was obtained using a sliding 90-m FWHM Gaussian weighting in altitude (yielding 45-m resolution), the sinx/x beam width, and averaging over all beam positions. Figure 6 (bottom) shows horizontal mean backscatter power averaged over three range gates centered at the upper and lower turbulence layer edges (red and blue lines, respectively) and in the layer center (green line) for beam angles ranging from 20 to 20 in 2 increments at t = 256, 312, and 364 (left to right). These results further quantify the points made in the discussion above. Specifically, backscatter power exhibits the following characteristics throughout the KH evolution: (1) maxima in the billow or turbulent layer edge regions for all beam angles until late times in the turbulence layer restratification; (2) maxima at the larger negative beam angles throughout the billow and turbulence layer, with larger differences at the latest times; (3) a shift of edge region maxima to smaller off-zenith angles after t = 256; and (4) surprisingly symmetric and sharp maxima centered at 10 beam angles in the layer edge regions at t = 364. [28] Referring to Figures 1 and 3, we see that this evolution in aspect sensitivity accompanies a shearing and tilting of potential temperature gradients at large and small scales that is initially more coherent horizontally in the layer edge regions, but which extends into the layer interior at later times. At each stage of this evolution, the layers defining the refractive index variations achieve shallower slopes in the edge regions because of the higher shears and weaker turbulence than occur in the layer center. This aspect sensitivity has significant implications for radial velocity measurements and biases that will be discussed further below Radial Velocity Distributions [29] Measured radial velocities for each time, beam angle, and sampling volume for which backscatter power is judged to be sufficient (>1% of the maximum at that beam angle at that time) are shown in Figure 7 in the same format as in Figure 5. True radial velocities for these beam angles evaluated with the same spatial weighting are shown in the same format in Figure 8 for comparison. [30] Radial velocity estimates are available at only a few locations at t = 37 (see Figure 7) because only specular 7of22

8 Figure 5. Backscatter power for beam angles of 15, 7.5, 0, 7.5 and 15 (relative to the mean wind to the right above the shear layer) at the times displayed in Figure 1. backscatter occurs at this time and only at the upper and lower edges of the KH billow. There is also a tendency for velocity estimates to occur only where the billow edge is approximately normal to the beam direction, and hence to yield small velocity estimates. The exception is the upper right panel (beam angle of 15 ), where sufficient power enables velocity estimates extending from the normal beam positions (at upper right and lower left) backward (counterclockwise) toward the regions of air inflow into the billow. In all cases, measured velocities are much smaller than true velocities at this time (note the different color scales in Figures 7 and 8). Together with the limited spatial coverage, this suggests that radars are unable to define KH billow structure at early stages in their evolution. [31] Velocities at t = 54 extend further around the billow than at t = 37 because emerging secondary instability structures distort the edge regions significantly and provide many additional sites where locally normal conditions allow stronger specular backscatter. This enables velocity estimates that are now quite accurate (compare the second rows of Figures 7 and 8), despite the specular character of backscatter at this time. Because gradients are weak and there is 8of22

9 Figure 6. (top) Vertical profiles of horizontally averaged power at (left to right) t = 84, 129, 256, 312, and 364 for the 15 (blue), 7.5 (green), 7.5 (orange), and 15 (red) beams. (bottom) Backscatter power averaged horizontally and over the three altitudes centered on the peak power at the turbulence layer edge regions and layer center as functions of off-zenith angle in 2 increments at (left to right) t = 256, 312, and 364. no turbulence in the billow interior, however, velocity estimates are confined to the billow edge regions. Positive and negative measured radial velocities rotate around the billow with varying beam angle, consistent with the true velocities. [32] At t = 84, the outer edges of the billow have become fully turbulent, and turbulence has also extended nearly to the billow core (see Figures 1 and 4). Thus, there is sufficient backscatter power to enable measured radial velocities except in the billow center. Radar backscatter is now due largely or entirely to turbulence, as all coherent structures having large potential temperature gradients have experienced instability and turbulence generation (see the left and right columns of Figure 1). Measured radial velocities exhibit close agreement with the true velocities for most sampling volumes and all beam angles (see third rows of Figures 7 and 8). There are, nevertheless, small differences throughout the billow, with measured velocities appearing to be somewhat smaller than true velocities at most locations. Clearly, however, radar measurements at this stage in the KH evolution, where turbulence is well developed, but has not yet fully mixed the billow interior, yield a close approximation to the true billow velocity field. [33] Measured power and radial velocity distributions at t = 129 yield very much less sensitivity to the billow structure than seen at t = 84. Referring to Figure 1, we see that this is because turbulence is now strongest in the billow core and has largely eradicated the potential temperature gradients necessary for radar backscatter (see the largest vorticity magnitudes at right and the lack of significant potential temperature gradients at left in the billow core in Figure 1). This causes significant turbulence backscatter only in the billow edge regions to which the potential temperature gradients have been mixed. In the edge regions, however, where sufficient turbulence and potential temperature gradients exist together, measured velocities are now systematically smaller than true velocities (see the fourth rows of Figures 7 and 8), in contrast to the very good agreement seen at t = 84. These comparisons suggest that radar measurements during the decaying phase of a KH billow are likely to correctly describe the billow structure for a wide range of beam angles, but also to underestimate its velocity field and other quantities (such as momentum flux) derived from this field. [34] Following billow collapse, motions become increasingly uniform in the horizontal, but the measured radial 9of22

10 Figure 7. As in Figure 5, but for measured radial velocities. velocities exhibit significant, and beam-angle dependent, differences from true velocities. True radial velocities are composed largely of projected (largely horizontal) velocities to the right and left in the upper and lower portions of the turbulence layer, respectively, with maximum magnitudes of 1.4 ms 1 at 15. Referring to the bottom three rows in the first columns of Figures 7 and 8, we see that measured velocities at a 15 beam angle are reasonable approximations to the true velocities over much of the turbulence layer, but underestimate true velocities near the layer edge regions. Measured velocities at 7.5 at these same times are likewise reasonable approximations to the true velocities, but also underestimate true velocities in the edge regions (also see additional discussion below). [35] In contrast, measured velocities in the 7.5 and 15 beams are in qualitative agreement with true velocities at t = 256, but are smaller by 50%. Thereafter, they depart increasingly from true velocities, suggesting both a layering of measured quasi-horizontal velocities with increasing vertical structure with time and horizontal structure in the layer interior and edge regions, neither of which are observed (see Figure 8). Both effects contribute to increasing departures from real velocities accompanying restratification of the turbulence layer. 10 of 22

11 Figure 8. As in Figure 5, but for true radial velocities. [36] Measured vertical velocities are seen to be reasonably accurate at t = 256, but to exhibit systematic departures from true velocities in the turbulence layer edge regions, as discussed previously by Fritts11. At both times, measured downward velocities at the upper edge and upward velocities at the lower edge represent measurement biases that appear to increase with time as the turbulence layer restratifies. [37] Collectively, these results suggest that there are significant biases in radial velocity measurements accompanying KH instability evolution that vary with time, the location within the KH billow or subsequent turbulence layer, and the beam angle at which these dynamics are measured. To examine these differences more quantitatively, we display in Figure 9 the differences between the true and measured velocities for each beam angle and time in the same format used for these velocities individually in Figures 7 and 8. [38] Velocity differences displayed in Figure 9 indicate that measured velocities yield accurate estimates in specific regions of the billow or subsequent turbulence layer at times ranging from t 84 to 364. Where measured velocities depart from true velocities, they prove to be underestimates at virtually every stage of the KH instability evolution for the beam angles considered. The exceptions occur at late 11 of 22

12 Figure 9. As in Figure 5, but for true minus measured radial velocities. times in the turbulence layer edge regions, where they occasionally exceed nonzero true velocities or are significantly different from true velocities that are near zero. Measurements arising from specular backscatter exhibit large biases prior to significant small-scale instability structures preceding turbulence (see Figure 9 at t = 37), but yield qualitative agreement (but underestimated velocities) as these structures induce a wide range of refractive index gradient orientations within a sampling volume (t = 54). As turbulence evolves and becomes quasi-isotropic (t = 84 and 129), measurement accuracies increase, and KH structures are well defined by the zenith and off-zenith velocity distributions, with measured velocities varying from % of true values. At t = 84, where turbulence is strong throughout most of the KH billow and the edge region, velocity measurement biases appear to be comparable at positive and negative beam angles. As turbulence mixes the billow interior and shears horizontally (t = 129 and 256), potential temperature gradients are destroyed in the billow interior, turbulence in the billow and layer edge regions is decaying in the presence of increasing potential temperature gradients, and velocity measurement biases are much more apparent at positive than at negative beam angles. This trend continues to later times (t = 312 and 364), where 12 of 22

13 Figure 10. Radial velocities at t = 129 for beam positions 1, 3, 5, 7, and 9 of the 10 shown in Figure 7 for beam angles (a) 15, (b) +15, (c) 7.5, and (d) 7.5. Yellow profiles show the true velocities, red profiles show the measured velocities, and white and green profiles show the minimum and maximum true velocities, respectively, within the sampling volume corresponding to the true velocity at each altitude. The spacing between adjacent profiles corresponds to a radial velocity of 10 ms 1 in each panel. measurement biases are quite small for negative bean angles, except in the turbulence layer edge regions, but are comparable to, or even exceed, true velocities in the edge regions at positive beam angles. These velocity measurement biases are clearly related to the influences of tilted layers on specular backscatter and will be discussed further below Radial Velocity Profiles [39] We now compare profiles of measured velocities with the distributions of true velocities within idealized sampling volumes having 45-m range resolution and 180-m width. These velocities are displayed in Figures 10 to 13 for beam positions 1, 3, 5, 7, and 9 of the 10 shown in Figure 7 for the 7.5 and 15 beam angles at t = 129, 256, 312, and 364. Figure 14 displays the same comparisons for vertical velocities at the same times. As above, gray regions indicate power <1% of the maximum in that profile. [40] Consistent with our discussion above, measured velocities (red profiles) at t = 129, for which there is a coherent and largely turbulent KH billow structure, are seen to be relatively accurate approximations to the true velocities (yellow profiles) in the billow edge regions where backscatter power is significant. More accurate velocity estimates are seen where the beam angles are more nearly aligned with the larger-scale potential temperature gradients (upper left and lower right quadrants in Figures 10a and 10c, upper right and lower left quadrants in Figures 10b and 10d and between adjacent billows where turbulence now extends over increasing altitudes (the 7.5 beam measurements at lower left and profiles 1 and 5 in the other panels are particularly good examples). Velocity underestimates occur preferentially in the opposite quadrants and in the billow interior where the beam angle is more nearly normal to, than along, the larger-scale potential temperature gradients. These are most apparent in profiles 2 and 4 in Figures 10a, 10b, and 10d, where systematic, and sometimes significant (30 100%), departures are seen. Velocity estimates are between the minimum and maximum true velocities (shown by the white and green lines, respectively) at this time. [41] Measured velocities are also relatively accurate following billow collapse (see Figure 11 at t = 256) as the turbulence layer begins to restratify, especially for the negative beam angles that are more nearly aligned with the larger-scale potential temperature gradients. Departures from true velocities are typically within 20% at negative beam angles, with slight underestimates more typical for the 15 beam and most overestimates occurring in the 7.5 beam. Greater velocity errors occur at beam angles of 7.5 and 15, with the majority of these being significant underestimates (up to 100%) of true velocities in the turbulence layer edge 13 of 22

14 Figure 11. As in Figure 10, but at t = 256. The spacing between adjacent profiles corresponds to a radial velocity of (a and b) 5 ms 1 and (c and d) 2.5 ms 1. regions, as noted in the discussion of Figures 7, 8, and 9 above. Given the aspect sensitivity of these measured velocity accuracies, it is clear that the tilted structures arising in the edge regions at this time have significant influences on the velocity measurements. We also note that velocity estimates are relatively accurate in the interior of the turbulence layer, where turbulence remains strong, despite the very weak potential temperature gradients and backscatter power occurring at this time. [42] As the turbulence layer restratifies further (at t = 312), backscatter power increases and measured velocities for negative beam angles remain nearly as good approximations to true velocities as seen at t = 256. Measured velocities are largely underestimates in the layer edge regions at a beam angle of 15, with somewhat larger departures than seen in Figure 11 at t = 256. Departures are likewise larger at 7.5, but are largely overestimates in the upper edge region and underestimates in the lower edge region at t = 312. Measurement errors are significantly larger at t = 312 than at t = 256 at both positive beam angles. Indeed, only two of the measured velocity profiles at the upper edge region at a beam angle of 15 appear at all similar to true velocities, while measured velocities in the edge regions at a 7.5 beam angle appear to be more nearly anti-correlated with true velocities. Measured velocities are nevertheless relatively accurate in the turbulence layer interior in all beam directions at this time, as seen at t = 256 in Figure 11. Thus, aspect sensitivity appears to be having an increasing influence on radar measurements of radial velocities as turbulence decays and restratification advances, but with these influences largely confined to the layer edge regions. [43] Measured velocity accuracies degrade further in the final stages of turbulence layer restratification (see Figure 13 at t = 364). Backscatter power exceeding 1% of the maximum in each beam has become intermittent throughout the turbulence layer, but particularly in the edge regions, at this time. As seen at t = 256 and 312, velocity estimates remain relatively accurate in the layer interior (where velocity shears are already small), but they are accurate over greater depths for negative beam angles and over decreasing depths at 7.5 and 15, respectively. Weak negative and positive measured velocities are seen to occur in the upper and lower edge regions at negative beam angles, with smaller amplitudes (but larger relative amplitudes) seen at 7.5 than at 15. Seen at positive beam angles is a weak layered structure in the vertical having negative velocities at the upper edge and positive velocities at the lower edge of the turbulence layer, with opposite velocities in the layer interior at smaller distances from the layer center. None of these profiles can be regarded as reasonable approximations to the true velocity profiles, however. Instead, they are relatively insensitive to the large-scale shear flow and appear to be at least as strongly influenced by specular reflections from tilted layers as by bulk motions at this time. [44] The evolution of measured vertical velocities from t = 129 to 364 is illustrated in Figure 14 in the same format 14 of 22

15 Figure 12. As in Figure 10, but at t = 312. The spacing between adjacent profiles corresponds to a radial velocity of (a and b) 3 ms 1 and (c and d) 1.5 ms 1. employed for Figures 10 to 13. Measured vertical velocity errors are to a large extent consistent with those seen at the smaller 7.5 beam angles at each time. They bear a close resemblance to the off-zenith errors seen to occur at t = 129 in Figure 10 (compare to Figure 14a). Biases are small at t = 256 (Figure 14b), but are slightly negative and positive at the upper and lower edge regions, respectively. Biases at t = 312 (Figure 14c) are roughly intermediate between those seen in the 7.5 and 7.5 beams in Figure 12, with overall biases toward negative and positive velocities at the upper and lower edges, respectively. Finally, biases at t = 364 (Figure 14d) more closely resemble those noted in Figure 13 in the 7.5 beam, which suggest negative and positive biases at the upper and lower edges, respectively. The biases at the upper and lower layer edges are also seen to become larger with time, achieving maximum apparent vertical motions of 0.3 to 0.5 ms 1 at t = Discussion [45] In our discussion of velocity measurements, accuracies, and apparent biases above, we noted that when backscatter arises primarily due to turbulence, measured velocities tend to be reasonable approximations to the true velocities. Close agreement occurs even where backscatter power is quite weak when the radar beam is aligned approximately normal to the refractive index gradients. In contrast, we found that when backscatter appears to be largely specular in nature, very significant biases in measured velocities may arise, depending on the orientation of the structures accounting for the specular backscatter. This occurs preferentially where refractive index gradients are relatively strong. [46] In reality, radar backscatter occurs via a continuum of reflections due to processes ranging from fully developed quasi-isotropic turbulence having quasi-random refractive index fluctuations to specular reflections from sharp, coherent gradients in fully laminar flows. For simplicity, however, we assume that backscatter power arises due to purely turbulent and/or purely specular processes such that we can write the power and inferred velocity as and P tot ¼ P turb þ P spec ð1þ v tot ¼ P turb v turb þ P spec v spec =Ptot ð2þ with subscripts tot, turb, and spec denoting the total, turbulence, and specular power and velocity estimates, respectively, and with each velocity expressed as v =(u, v, w). Our discussion below will assume one or the other of these limits in isolation, for simplicity, though our radar assessments also surely include both turbulence and specular 15 of 22

16 Figure 13. As in Figure 12, but at t = 364. contributions at all times following initial turbulence generation. We will also drop the subscripts where the meaning is clear Turbulence Backscatter [47] For backscatter due to quasi-isotropic and homogeneous turbulence (P spec = 0), we anticipate a radial wind measurement given by v r ¼ usinq þ wcosq where u and w are the mean horizontal and vertical velocities of the air parcel being sampled at a beam angle q from zenith in the direction of the horizontal velocity, assuming that all portions of the sampling volume are weighted equally. If weighting is not uniform, however, due to correlations of the local turbulence velocities and backscatter power [e.g., Nastrom and VanZandt, 1994; Hoppe and Fritts, 1995a; Muschinski, 1996, 2004; Gibson-Wilde et al., 2000; Tatarskii and Muschinski, 2001; Worthington et al., 2001], we expect the portion of the volume contributing the majority of backscatter power to contribute most to the estimated velocity, with apparent velocities given by Z v app ¼ ð1=pvþ V PðÞvr r ðþdr; with P = R V P(r) dr and dr = dxdydz, that may differ dramatically from true velocities. ð3þ ð4þ [48] Our assessments of radar measurements above demonstrate both the accuracy of these measurements when the assumptions underlying equation (3) are satisfied and the occurrence of significant measurement biases when the backscatter power and velocity fields exhibit strong correlations and equation (4) provides a more accurate characterization of apparent velocities. Specifically, turbulence backscatter without strong correlations between power and radial velocities accounts for (1) the close agreement between measured and true velocities in the billow edge regions at t = 84 and 129 (where potential temperature gradients are approximately aligned with the radar beam) and (2) the accurate velocity measurements in the turbulence layer interior following billow collapse. [49] By comparison, negative correlations of power and velocity magnitudes clearly contribute to the reasonable, but underestimated, velocities in the billow edge regions and interior at t = 129 (where the radar beam and the large-scale gradients are misaligned). Weaker or misaligned gradients in the billow interior allow the higher backscatter power (and smaller radial velocity magnitudes) nearer the edge regions to contribute preferentially to the velocity spectrum because of the Gaussian range weighting and the relatively broad beam width, thus reducing the velocity magnitude estimate. In the turbulence layer interior following billow collapse, power varies weakly in altitude and the velocity shear is fairly uniform, suggesting that even broad Gaussian weighting functions and large beam widths will provide relatively accurate velocity estimates. Thus, turbulence 16 of 22

17 Figure 14. As in Figure 10, but for vertical velocities at (a) t = 129, (b) t = 256, (c) t = 312, and (d) t = 364. The spacing between adjacent profiles corresponds to a vertical velocity of 10, 2, 1.5, and 1ms 1 in Figures 14a, 14b, 14c, and 14d, respectively. backscatter largely accounts for the relative accuracies of these measurements (with P spec 0), but correlations of P turb (r) and v(r) nevertheless contribute to measurement biases Specular Backscatter [50] Radial velocities are more complicated and potentially exhibit greater biases if Bragg-scale fine structure that contributes significantly to backscatter power exhibits phase variations determined by advecting tilted structures. Such structures appear to be playing significant roles in the turbulence layer edges regions following billow breakdown, given the beam-angle dependent biases noted above and the vertical beam measurements discussed by Fritts11. In these cases, we expect that apparent phase motions will depend on both the off-zenith beam angle and the angle of the tilted structures relative to the horizontal. Figure 15 shows the general geometry considered. Assuming that Bragg-scale phase variations accompany only the motions of tilted structures (P turb = 0), the radial velocity is given by with v r ¼ u rot tanðq þ fþþw rot ð5þ u rot ¼ u cosq w sinq ð6þ Figure 15. Schematic of true and apparent radar velocities that influence radar measurements when tilted layers are responsible for apparent phase motions. Measurement biases due to specular backscatter arise only from advection of tilted surfaces by u rot. 17 of 22