This study analyzes eddy correlation data from a 34 m tower in FLOSSII (Fluxes over Snow Surfaces; Mahrt and Vickers, 2006) instrumented from 20 November 2002 to 2 April 2003 by the National Center for Atmospheric Research (http://www.atd.ucar.edu/rtf/projects/FLOSS). The tower site is located over a locally flat grass surface south of Walden, Colorado, USA (40.8°N, 106.3°W) in the Arapaho National Wildlife Refuge. The grass is often covered by a thin snow layer during the field programme. This study is based on data collected at 1, 2, 5, 10, 15, 20 and 30 m with Campbell CSAT3 sonic anemometers. We also briefly analyze data collected at 2 m above brush approximately 3.3 km northnortheast of the tower site, and 2 m above a snow-covered dry lake bed, approximately 1.5 km northwest of the tower site.
The North Park Basin lies between two large mountain ranges oriented in the north–south direction with elevations of 1000–1500 m above the floor of the basin. The basin is approximately 50 km from south to north and 30 km from west to east. The FLOSSII tower is located at the south end of a shallow sub-basin about 4 km across. The sub-basin is defined by hills on all sides except for an outflow region towards the northeast to the larger main basin. The wind direction in FLOSSII shows a strong preference for south–southwest flow with a secondary maximum of northerly flow. The predominance of southerly flow is related to the prevailing synoptic pressure gradient and the north–south oriented mountain ranges on the east and west edges of the basin. The local slope at the tower site is weak depending on direction and distance from the tower. On average, the terrain slopes upwards toward the south with a magnitude of about 1%.
The turbulent fluctuations are computed as deviations from an average computed over a variable averaging width; the averaging width is chosen independently for each 1 h record based on the record cospectra for heat, as in Mahrt and Vickers (2006). This OGIVE approach (Friehe et al., 1991) defines the turbulence as the smaller-scale part of the cospectra where the heat flux is systematically gradient (positive diffusivity). Once the perturbation quantities are computed, covariances are averaged over 1 min records. We will emphasize the 2 m data where fluxes are thought to be normally a reasonable approximation to the surface flux. For the most stable cases, the 1 m flux appears to suffer some flux loss due to path-length averaging. More detailed structure of the microfront will be examined only briefly in terms of fast response data. Otherwise, this study analyzes 1 min averages within the nocturnal period 2000–0600 local standard time for the entire four-month winter season.
We define a local potential temperature, θ,
where δz is the height above the surface, and the 0.01 constant has units K m−1. The sonic anemometer data is used for the temperature at each level. Based on strong-wind cases when well-mixed conditions are expected, the temperature at several of the levels was adjusted by ± 0.1 K.
The vertical motion shows a strong dependence on wind direction, particularly at higher levels on the tower. The attack angles for the 1 min flow are often small (mainly horizontal flow) so that small errors in the inability to correct for sonic misalignment and flow distortion could lead to serious errors in the vertical velocity. Acevedo and Mahrt (2010) found that the tilt coordinate rotation significantly influenced the computed mesoscale fluxes. In the current study, the tilt rotation did not significantly influence the composited within-sample structure of the vertical motion field, but did influence the mean vertical motion. We do not apply a tilt coordinate rotation to the 1 min winds used in this study, but do remove the sample mean vertical motion. In the analysis below, we use information on vertical velocity mainly at the top tower level, 30 m, where the horizontal structure of the vertical motion is best defined.
2.1. Detection of events
Solitary events will be detected in terms of changes of a detector variable, ϕ, such as temperature or one of the wind components across a sampling window of τF data points. To identify changes with time, we compute the similarity of the local structure with a detector function, H, such that in discrete form
where ti is the relative time within the sampling window, equal to t − to, and where to is the beginning of the sampling window. To select sharp changes with time, we assign the detector function to be the Haar function, which for a given sampling window can be written as
where the colon signifies sequentially incrementing through the data points. Application of the Haar function in Eq. (2) computes the difference between the means of ϕ for two halves of the sampling window, which we write equivalently as
and so forth. A time series of δt(ϕ) is computed by moving the sampling window of width τF through the data. To generate a time series of δt(ϕ), the sampling window is shifted 1 min between calculations throughout each of the 12 h nocturnal periods.
2.3. Variance analysis
To examine the relative importance of the solitary structures, we now construct a variance analysis. We first remove the sample means for the variance analysis to simplify the ensuing algebra, such that
where the square brackets indicate averaging over the sample window. The variance captured by the Haar transform of ϕ is then computed by first decomposing ϕ* at each point within the sampling window as
where is the deviation of the half-window means from the sample mean and projected onto all of the points within the sampling window. ϕ′ is the deviation of the point value of ϕ* from the half-window means and represents the more complex deviation of the parent motion from the Haar function and also includes smaller-scale minute-to-minute variations.
Since the Haar function corresponds to simple unweighted averaging for each half sampling window, the total within-sample variance is simply
where the first term on the right-hand side is the residual variance due to structure not explained by the Haar function (half-window means). The second term is the variance due to the half-window means. The fraction of variance explained by the Haar transform for each sample is then
which will be used to assess the relative importance of the structures similar to the simple Haar function.
We will also analyze the composited structure by averaging over all of the selected samples and assess the importance of the composited within-window structure compared to the between-sample variability. Towards this goal, we define the difference of ϕ*(ti) for a given sample from the average of ϕ*(ti) over all of the samples, such that for the jth sample
where the operator < ·> averages over all of the J samples for a given point within the sampling window. Had we not removed the sample means, the computed between-sample variance might be dominated by differences between nights, for example between very cold nights and mild nights.
The variance between samples at point ti, , is subsequently averaged over the sampling window, to obtain a total between-sample variance
The within-window variance of the composited flow (<ϕ(ti)>) is defined as
If the within-window variance of the composited structure, VARc, is small compared to the between-window variance, VARb, then the structure of the composited flow is of questionable significance. That is, the variance explained by the composited structure is less than the variability between the samples. Thus, a measure of the relative importance of the structure of the composited flow can be defined as
This ratio can be converted to a version of the relative standard error with respect to the significance of the variation within the composited structure relative to the between-sample variation
where J is the number of samples. Such tests indicate that the composited structure is highly significant for at least some of the variables, as discussed below. However, the significant skewness of for some of the variables, and the unlikely existence of an ensemble average, both compromise the interpretation of Eq. (15) as a formal version of the standard error. That is, the structures contributing to the composite arise from different situations (analogous to different populations). Nonetheless, Eq. (15) remains a relative measure of the significance of the structure for different variables.
While the composited structure for the variables not used in the selection process could potentially be dissimilar to all of the individual structures, the collection of samples of solitary events in this study generally includes a substantial number of cases that qualitatively resemble the composited structure. At the same time, the samples also include numerous other signatures quite different from the composited structure.