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3.2 Richardson numberWhich radiosonde measurements are turbulent must first be established before the structure constant can be calculated, or risk including nonturbulent measurements in a theory that is specifically suited for turbulence only. To accomplish this the Richardson number is calculated per measurement. Simply put the Richardson number is a measure of how turbulent an atmospheric layer is. A stability criterion for the spontaneous growth of smallscale waves in a stably stratified atmosphere with vertical wind shear, it yields the ratio between the work done against gravity by the vertical motions in the waves to the kinetic energy available in the shear flow.
where g is the gravitational acceleration, T_{v} the virtual temperature, γ_{a} is the adiabatic rate of decrease of temperature = 0.0098 K/m. With z the height and ΔU, ΔV the components of the wind. The smaller the value of the Richardson number, the less stable the flow is in terms of shear instability. The most commonly used value for the start of shearinduced turbulence is between 0.15 and 0.5, usually set at Ri_{cr} = 0.25. However, once turbulence is established within a shear layer, it should be sustained as long as Ri < 1.0, Wallace [1977]. The impact of using either 0.25 or 0.5 for the critical Richardson number was evaluated and there are no significant differences between the two values. 3.3 Potential Refractive Index GradientThe potential refractive index 'vertical' gradient, M, is needed to compute the refractive index structure constant. This is not a 'full' gradient, meaning it does not comprise all derivatives to variables the refractive index is dependent of. This is because the only relevant gradient, or variation of the refractive index, is the one due to turbulence alone. To that end, the refractive index variation must be inspected in terms of conservative additives. Now, with the expression for the potential temperature
where γ_{a} the adiabatic lapse rate of temperature and z the height. And the expression for specific humidity
with e is the partial water vapour pressure and p the pressure and ε = 0.622 the ratio of gas constants for dry air to that for water vapour, for an upwelling parcel of air moving from height z_{1} to z_{2} due to turbulent mixing
where the potential refractive index gradient, M, can be written as
This yields, Tatarskii [1961]
A more generalised approach for M used by Warnock & VanZandt [1985] has the form:
This formulation yields betterbehaved results and will be used here. Equation 10 tends to overestimate the potential refractive index. With these expressions C_{n}^{2} can be calculated using Tatarskii [1971]:
where a^{2} is a dimensionless constant between 1.5 and 3.5, but most commonly used at a value of 2.8, Monin and Yaglom [1971]. A = K/(K_{m}(1Ri)) is a numerical constant generally considered equal to unity. L_{0} is the outer scale of turbulence, which has been set equal to the resolution of the radiosonde data. Using equation 12 the refractive index structure constant can be calculated for every radiosonde measurement considered as turbulent given a subcritical Richardson number. All necessary differentials needed to calculate the Richardson number are determined by the values from two consecutive height measurements. If a measurement is considered stable, a C_{n}^{2} value of 10^{21} m^{2/3} is assigned. This value was chosen to reflect, in a simplified manner, the sensitivity of C_{n}^{2} radar measurements and also, for convenience, since for all the results we use logarithmic scales. Layers with sustained turbulence, where the critical Richardson number may be 1, are in this approach considered to be stable. This will lead to an underestimation of the statistics of C_{n}^{2} . As radiosonde measurements do not have values for the same heights, performing statistics on the dataset requires mapping of the C_{n}^{2} to fixed heights with 10 meter spacing. This makes it possible to arrange all data in a histogram per location, making it readily apparent at which values the various percentiles are and how they compare to each other. Moreover probability distribution functions (pdf) may easily be derived for each height as crosssections of this histogram. Note that all stable measurements get binned to the lower excess class of the histogram of the refractive index structure constant. 