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58 3 Basics of Geophysical Fluid Dynamicswhere 〈ψ〉 is a mean value and ψ ′ are fluctuations This approach, proposed byOsborne Reynolds (1895), is the key ingredient in the theory of turbulence.3.15.3 What Causes Turbulence?There are two principle causes of turbulence in a fluid One source of turbulence areshear flow . Shear fl w is a fl w that displays speed variations perpendicular to itsmovement direction. Figure 3.27 gives examples of shear fl ws. Owing to frictionaleffects, strong shear fl ws establish in the vicinity of flui boundaries. The othercause of turbulence is convective mixing that occurs in case of unstable verticaldensity stratification A stable density stratification on the other hand, can weakenor even suppress turbulence.Fig. 3.27 Examples of shear fl ws3.15.4 The Richardson NumberThe transition between laminar and turbulent fl w in a stratified vertical shear fl wcan be characterised by means of a nondimensional number – the Richardson number.This number compares turbulent energy production/dissipation associated withvertical density stratificatio and turbulent energy production associated with verticalshear of a mean horizontal fl w. For a mean fl w 〈u〉 running into the x-direction,for example, this number is given by:Ri =N 2(∂ 〈u〉 /∂z) 2 (3.59)where N 2 is the stability frequency of the ambient (mean) density field Theory andexperimental studies suggest that turbulence is created whenever Ri falls below athreshold value of around 1/4 = 0.25. Richardson (1920) was the firs to derive thisinstability criterion.