Abstract
Assuming that the vertical turbulent heat flux vanishes at extremely stable conditions, one should expect its maximal absolute value to occur somewhere at moderate stability, between a neutral and extremely stable equilibrium. Consequently, in some situations duality of solutions may be encountered (e.g. two different values of temperature difference associated with the same values of heat flux and wind speed). A quantitative analysis of this feature with a local equilibrium Reynolds-stress model is presented. The fixed-wind / fixed-shear maximum has been identified both in the bulk and in single-point flux–gradient relationships (that is, in the vertical temperature gradient and wind-shear parameter domain). The value of the Richardson number corresponding to this maximum is derived from the model equations. To study the possible feedback in strongly stable conditions, weak and intense cooling scenarios have been simulated with a one-dimensional numerical, high-resolution atmospheric boundary-layer model. Despite the rapid cooling, flow decoupling at the surface has not been observed; instead, a stability-limited heat flux is maintained, with a gradual increase of the Richardson number towards the top of the turbulent layer, with some signs of oscillatory behaviour at intermediate heights. Vertical changes of wind shear and the Brunt–Väisälä frequency display a remarkably non-monotonic character, with some signs of a gradually developing instability.
Highlights
At extreme stability, the vertical downward heat flux is supposed to vanish due to the suppression of turbulence by buoyancy forces and in spite of the presence of a large vertical temperature gradient
If turbulence ceases at a critical value of the Richardson number, there should be a maximum of downward heat flux1 found at a certain value of this number and an identical value of the heat flux should be found on both sides of the maximum’s location
It is worth noting that the Richardson number increases with height, and that most of the data points appear to lie in the smaller temperature gradient subdomain
Summary
The vertical downward heat flux is supposed to vanish due to the suppression of turbulence by buoyancy forces and in spite of the presence of a large vertical temperature gradient. It is not certain that the heat flux vanishes in extremely stable conditions These reservations should be made, as many recent studies undermined the belief in the existence of the critical gradient Richardson number He conjectured that only one of these solutions, that associated with a larger momentum flux, was a hydrodynamically stable one and should be chosen While the latter statement encountered criticism (Arya 1972; Carson and Richards 1978), the existence of the maximum was corroborated by experimental evidence (Malhi 1995; Mahrt 1998; Basu et al 2006; Sorbjan 2006). Non-linear effects such as a possible duality of solutions remained beyond the scope of the study
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