A new approach to the theory of the small-scale dissipative turbulence based on non-equilibrium thermodynamics, continuum-mechanics and the statistical physics is developed in the monograph. The classical expression for the macroscopic kinetic energy of a small fluid particle in classical non-equilibrium thermodynamics is generalized by taking into account the local velocity shear related with the rate of strain tensor. The closure relation for the three-dimensional isotropic homogeneous small-scale dissipative turbulence (in an incompressible viscous Newtonian fluid) is founded by combining the classical statistical approaches, the conception of the statistical ensemble and the classical formulation of the law of large numbers. The significance of the new approach is evaluated for laboratory and oceanic stratified turbulence. The practical application of the new approach is shown in physical oceanography.

For specialists in non-equilibrium thermodynamics, continuum mechanics, statistical physics, hydrodynamics, physical oceanography and for students and post-graduates of related specialities.

Foreword	6

1. Introduction	9

2. The Macroscopic Kinetic Energy of the Small Fluid Particle	19
2.1. The Concept of Internal Shear Kinetic Energy	19
2.2. The Macroscopic Kinetic Energy in Classical Non-equilibrium
Thermodynamics	21
2.3. The Macroscopic Non-equilibrium Kinetic Energies
of the Small Fluid Particle	23
2.4. Homogeneous Fluid Sphere or Cube τ in a Shear Vortical
Three-dimensional Flow	32

3. The Kinetic Energy of Wave-Turbulent Pulsation	35
3.1. The Turbulent Kinetic Energy in Classical Semiempirical
Theories	35
3.2. The Kinetic Energy of the Small-scale Velocity Pulsations	39

4. Freely Decaying Three-Dimensional Isotropic Homogeneous Small-Scale Turbulence	48
4.1. Three-dimensional Isotropic Homogeneous Turbulence
of the Energy-containing Length Scale αL_K	48
4.1.1. Freely Decaying Three-dimensional Grid-generated
Turbulence in Homogeneous Fluid at the Early Stage
of Decay	48
4.1.2. Stochastic Model of the Three-dimensional Isotropic
Homogeneous Turbulence of the Energy-containing Inner
Kolmogorov Length Scale L_K	50
4.1.3. The Hydrodynamic Foundation of the Evolution Equation
for the Three-dimensional Isotropic Homogeneous
Turbulence of the Energy-containing Inner Kolmogorov
Length Scale L_K	52
4.1.4. The Thermodynamic Foundation of the Evolution Equation
for the Three-dimensional Isotropic Homogeneous Turbulence
of the Energy-containing Length Scale αL_K	55
4.1.5. The Time Evolution of the Freely Decaying Three-dimensional
Isotropic Homogeneous Turbulence of the Energy-containing
Length Scale αL_K	61
4.2. Three-dimensional Isotropic Homogeneous Turbulence of the
Energy-containing Length Scale α x L_FK (the Fossil Kolmogorov
Length Scale)	66
4.3. Three-dimensional Isotropic Homogeneous Turbulence of the
Energy-containing Ozmidov Inertial-buoyant Length Scale L_O	73

5. The Overturning Condition of the Small Fluid Particle in an Ideal Stratified Fluid	77
5.1. Generalization of the Classical Shear Stability Condition for the
Small Fluid Particle τ of Finite Size in the Three-dimensional
Shear Stratified Flow	77
5.2. Two-dimensional Parallel Shear Flow of an Ideal Stratified Fluid	86
5.2.1. The Overturning Conduction for the Two-dimensional
Parallel Shear Flow of an Ideal Stratified Fluid	86
5.2.2. Stratified Fluid Sphere and Straight Circular Cylinder	88

6. Practical Significance of the Macroscopic Internal Shear Kinetic Energy of the Small Fluid Particle	90
6.1. Three-dimensional Isotropic Homogeneous Small-scale
Turbulence of the Energy-containing Length Scale l of Turbulent
Eddies	90
6.2. Critical Kinetic Energy Dissipation Rate in the Stratified
Incompressible Viscous Newtonian Fluid for the Three-dimensional
Isotropic Homogeneous Turbulence of the Energy-containing
Ozmidov Inertial-buoyant Length Scale L_O	94

7. Critical Kinetic Energy Dissipation Rates for Non-Equilibrium Thermodynamic Regimes of the Three-Dimensional Anisotropic Small-Scale Stratified Turbulence	99
7.1 Introduction	99
7.2. The Small-scale Dissipative Structures of Turbulence	106
7.2.1. Foundation of the Closure Relation for the
Three-dimensional Isotropic Homogeneous Small-scale Turbulence
of the Energy-containing Length Scale I in an Incompressible
Homogeneous Viscous Newtonian Fluid	106
7.2.2. The Relative Significance of the Macroscopic Internal
Rotational and Shear Kinetic Energies for the Small-scale
Dissipative Structures of Turbulence	110
7.3. Critical Kinetic Energy Dissipation Rate in an Incompressible
Viscous Newtonian Stratified Fluid for the Three-dimensional
Anisotropic Small-scale Turbulence	112
7.3.1. Dependence of the Critical Kinetic Energy Dissipation Rate
on the Coefficient of Local «Rigidity» R of the Local Fluid
Motion, the Coefficient of Local Anisotropy a of the
Turbulent Velocity Pulsations and on the Critical Size l_cr of
the Energy-containing Turbulent Eddies	112
7.3.2. Relation of the Coefficient of Local «Rigidity» R(a) and the
Coefficient of Local Thermodynamic Non-equilibrium n_e
for the Small-scale Turbulent Velocity Field	130
7.3.3. Behavior of the Coefficient of Local Thermodynamic
Non-equilibrium n_e for the Turbulence-wave Transition
Processes	134
7.4. The Overturning Condition for the Small Fluid Particle in an
Ideal Stratified Fluid and the Critical Gradient Richardson
Number, which Characterizes the Transition of Turbulent
Regimes to Wave Regimes of Fluid Motions	137

8. Summary of main results	145

9. Conclusion	156

10. Perspectives of the Non-Equilibrium Statistical Thermohydrodynamics of Turbulence	161

References	167

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