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Appendix F Background and Assumptions: Wind Data Interpolation

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This document outlines the background and assumptions related to wind data interpolation (WDI) within urban environments, focusing on the estimation of mean wind speed profiles. It discusses the importance of microscale roughness length, displacement height, and local friction velocity in determining wind profiles. By applying adapted mathematical models, the WDI enables a more flexible approach to calculating wind speed amplification in urban settings. It highlights the need for future research to further refine the understanding of how urban canopies affect wind speed, pointing out variability in the literature regarding these impacts.

Appendix F. Background and Assumptions Relating to Wind Data Interpolation The mean wind speed profile is comprised of a local scale mean wind speed portion, between the top of the surface layer [zSL] and the top of the UCL [zH], a transitional portion, and a microscale portion below the UCL. The applicable equations, by sublayer within the SL, are as follows: ISL: Logarithmic Profile , where (F.1) z0 is microscale roughness length, d is displacement height, and u* is the local maximum friction velocity u*(z*). The microscale roughness length and displacement height are provided by the Urban Parameterisation table based on urban sub-region category. This equation is restricted to heights [z] in the range where z > z*. RSL: Transitional Profile (MacDonald, 2000), where (F.2) A and B are calculated parameters (see Appendix D - 2), H is the mean building height, u* is the friction velocity, and u(Z2H) is the mean wind speed at z2H ~ 2H. The Urban Parameterisation table provides A, B, and H, dependent on urban sub-region category. Ideally the friction velocity profile should be used to produce the height-dependent friction velocity values below the top of the RSL. The use of the maximum local friction velocity, calculated by the WDE, produces a slight overestimation of the mean wind speed within this portion of the profile. On comparison to general practice, which extrapolates the logarithmic profile through this region, this slight over estimation produces a comparable mean wind speed within the UCL and is thus considered acceptable for the sake of simplifying the calculation. Whether the profile over or under estimates the mean wind speed, in relation to the profile created through extrapolation of the log profile beyond its intended range, depends on the morphology of the underlying terrain. Figure F.1, below, illustrates a case where there is a slight overestimation by the transitional profile. Equation F.2 is first used in the form of , where (F.3) is the mean wind speed at the top of the RSL, as estimated by the logarithmic profile in the first section. With u(z2H) thus determined, the original form of the equation, Equation F.2, is used to generate the mean wind speed profile from z2H < z < z*. UCL: exponential profile , where (F.4) a is the attenuation coefficient, calculated for each urban sub-region and tabulated within the Urban Parameterisation table. In the region where zH < z < z2H the attenuation coefficient is considered to be a function of height (i.e., at z = zH: a is as calculated using Equation D.7 (Appendix D -2 ), at z = z2H: a is as calculated using Equation D.6 (Appendix D – 2), and between these two heights . Below the top of the UCL (i.e., where z = zH) the UWEP DSS holds the attenuation coefficient constant. This adaptation of the exponential profile equation is a first attempt to account for the fact that lf is actually a function of height, reaching its maximum value at ground level, in heterogeneous urban sub-regions. The range of calculated attenuation coefficients produced plots, illustrated below, which compared favourably to those published by MacDonald (2000). Figure F.1. Nondimensionalised wind speed as a function of height within the UCL, by attenuation coefficient. Figure F.2 depicts the mean wind speed profile as developed through this multi-layer approach, compared to the traditional logarithmic profile extrapolation. By establishing the complete profile, as opposed to merely calculating individual mean wind speeds at specific heights, the WDI provides the Wind Amplifier sub-module with the flexibility to determine the appropriate reference wind speed for calculating the amplified wind speed at various reference heights. The Wind Amplification sub-module primarily selects the suitable reference wind speed from the exponential UCL profile, which represents the microscale mean wind speed. Figure F.2. Mean wind speed profile generated by the WDI. The profile segments are as follows: solid black - Logarithmic, green square - Transitional, and blue dashed - Exponential. zSL is the height of the top of the SL, z* is the height of the RSL, and zH is the height of the UCL. Further research is required to determine at what height the urban canopy starts to affect the mean wind speed. As discussed in other Appendices and in the Literature Review Chapter, there are differing views and theories on this matter. The selection of z = 2H, producing conservative results on comparison against the values generated by the transitional profile equation, may be either too high or too low. The modification to the exponential portion of the mean wind speed profile was deemed necessary given that the transitional profile does not contain parameters that can readily be correlated to height. It is proposed that the mixing length profile as opposed to the mixing length be used to calculate the parameters of the transitional profile so as to make this portion, as opposed to the exponential portion, of the profile a function of height. References MacDonald, R. W. (2000). Modelling the mean velocity profile in the urban canopy layer. Boundary-Layer Meteorology, 97(1), pp. 25-45. .4