We present a comprehensive study and full classification of the stationary solutions in Leith's model of turbulence with a generalised viscosity. Three typical types of boundary value problems are considered: Problems 1 and 2 with a finite positive value of the spectrum at the left (right) and zero at the right (left) boundaries of a wave number range, and Problem 3 with finite positive values of the spectrum at both boundaries. Settings of these problems and analysis of existence of their

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... ions are based on a phase-space analysis of orbits of the underlying dynamical system. One of the two fixed points of the underlying dynamical system is found to correspond to a 'sharp front' where the energy flux and the spectrum vanish at the same wave number. The other fixed point corresponds to the only exact power-law solution-the socalled dissipative scaling solution. The roles of the Kolmogorov, dissipative and thermodynamic scaling, as well as of sharp front solutions, are discussed. Q1 Keywords: Leith model of turbulence, stationary solutions, solvability of boundary value problems, bottleneck phenomenon ( P Q , 2 11 2 2 3 where = ( ) c 24 11 2 3 and P and Q are arbitrary constants. For Q=0, this gives the pure Kolmogorov cascade solution, whereas for P=0 this is a pure thermodynamic spectrum. For J. Phys. A: Math. Theor. 00 (2016) 000000 V N Grebenev et al