An ensemble of convective thermals is distinguished from the surface layer of penetrative turbulent convection over a heated horizontally uniform surface. For an isolated convective element, an integral model of a quasi-stationary spontaneous jet is proposed which admits an exact analytical solution. A simple statistical model is constructed for an ensemble of dynamically identical thermals. In this model, convective thermals ascend in a static environment, their dynamic parameters are described by the equations of an isolated quasi-stationary jet, and their diameters are stochastic. It is shown that the ensemble of thermals rising in a horizontally homogeneous environment forms surface-layer turbulent moments. The analytical relationships for higher turbulent moments of vertical velocity and temperature are compared with experimental data from the second to the forth order inclusive.
The long periods/phases (with a duration of more than 10-15 years) of increased and decreased water flow of rivers in the Volga river basin were identified. The annual and seasonal water flow (over snow-melt flood period, as well as summer–autumn and winter low-water seasons) of six representative rivers over observation periods more than a century in duration, starting from the 1870s–1890s up to 2016, was used. In addition to this, periods with the average runoff close to its normal value were also observed. The boundaries of contrast phases were determined using normalized cumulative deviation curves in combination with Student’s test of the statistical homogeneity of the data series. The duration of the phases varies from 10 to 96 years. The phases of lower runoff were generally longer than those of higher runoff (this is especially typical of the winter and summer–autumn low-water season). The identified contrast phases show a statistically significant difference between the annual and seasonal runoff. The analysis of the data series of alteration of phases with increased and decreased water flow in hydrological seasons of the year allowed the authors to identify three major types of their long-term dynamics within the Volga river basin.
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