An energy cascade transfers kinetic energy among scales through nonlinear motion. A forward cascade carries energy toward smaller spatial scales, where kinematic viscosity can dissipate it. Conservation and a mean scale-space flux do not uniquely prescribe the geometry or locality of the transfer mechanism.
The quadratic nonlinear term of the Navier-Stokes equation couples three Fourier modes whose wave vectors close to a triangle. These spectral triad interactions transfer kinetic energy between wave numbers. A turbulence closure can model their triple correlations while retaining more scale information than a prescribed local power law.
At fixed outer velocity and integral scale of turbulence, the mean viscous dissipation per unit mass approaches a positive finite value as the Reynolds number becomes large. The dimensionless coefficient is then of order one. This is a phenomenological property of established three-dimensional turbulence, not a consequence of taking kinematic viscosity to zero in an arbitrary smooth flow.
Although the local viscous dissipation is , increasingly intense velocity gradients can compensate for vanishing kinematic viscosity. A Burgers vortex illustrates this mechanism: its radius shrinks while its excess integrated viscous dissipation remains finite. Such a local model does not by itself prove the zeroth law of turbulence for a whole flow.
The eddy turnover time estimates how long a velocity difference takes to distort a structure of size . In a local strong energy cascade, the transfer time is assumed comparable to it. A weak wave cascade can transfer energy much more slowly because a single encounter makes only a small fractional change.
A two-dimensional forward two-dimensional enstrophy cascade has the idealized spectrum . The energy integral above a wavenumber scales as its inverse square, whereas the cumulative enstrophy integral grows logarithmically. The spectrum used here omits possible further logarithmic corrections.
For a wide range and separation well below its outer scale, the second-order longitudinal structure function is controlled by . The low-wavenumber filter term gives , larger than the order- energy above the cutoff. This is a marginal dependence on larger-scale enstrophy.
The Townsend-Betchov cascade cartoon organizes inertial range vorticity into sheets by bi-axial strain, then uses sheet instability and roll-up to generate finer vortex tubes. Dissipative tube cores balance stretching with kinematic viscosity. Such differing mechanisms across scales challenge literal identical-step, memoryless cascade models without contradicting a mean energy flux.
Kolmogorov 1941 theory assumes locally universal small-scale velocity statistics set by mean viscous dissipation and kinematic viscosity at large Reynolds number. In its inertial range dimensional form, . Large-scale modulation and internal intermittency require qualifying this unconditional scaling.
In the inertial range, velocity increment statistics also become independent of kinematic viscosity. Combined with the Kolmogorov first similarity hypothesis, this gives by dimensional analysis. Signed integer moments and absolute noninteger moments must be distinguished; the signed third-order coefficient is negative in a forward energy cascade.
The second-order longitudinal structure function has the displayed form in the idealized inertial range of Kolmogorov 1941 theory. It predicts that the typical longitudinal velocity increment grows as the cube root of separation. Unlike the Kolmogorov four-fifths law, its exponent and coefficient are not exact consequences of the Navier-Stokes equation.
In the locally isotropic equilibrium range, the small-scale velocity increment statistics depend on the separation, mean viscous dissipation and kinematic viscosity, independently of large-scale forcing details. Dimensional analysis leaves the Kolmogorov microscales as normalization scales and gives the displayed second-order longitudinal structure function. Universality of is part of the hypothesis, not something proved by dimensional analysis.
The equilibrium range consists of scales below the energy-containing motions whose statistics can adjust rapidly to the local energy supply. It includes the inertial range and dissipative scales. Approximate equilibrium concerns mean transfer and viscous dissipation, not absence of intermittent spatial fluctuations.
The dissipation range contains scales at which kinematic viscosity appreciably affects velocity gradients and converts kinetic energy into heat. In a high-Reynolds-number three-dimensional cascade its characteristic length and time are the Kolmogorov microscales. Intense vortex cores can be intermittent within this range.
The Kolmogorov length, velocity and time are , and . With , their ratios to the integral scales of turbulence are powers of the Reynolds number. They describe a mean scale balance; intermittent dissipative structures can have other local scales.
The Kolmogorov length scale is the length among the Kolmogorov microscales. Kinematic viscosity and mean energy dissipation per unit mass give the displayed dimensional scale. The associated speed is , so the Reynolds number formed at this length is one.
The inertial range lies between the energy-containing and viscous scales. Direct forcing and viscous dissipation are small there, and nonlinear transfer carries an approximately constant mean energy flux in a three-dimensional forward cascade. Scale similarity and universality are additional statistical hypotheses.
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The concept of energy cascade is commonly used in the fields of fluid dynamics and turbulence. It refers to the process through which energy is transferred from larger scales of motion to smaller scales, eventually dissipating as heat due to viscous effects. The energy cascade typically occurs in turbulent flows where energy introduced at larger scales (like waves or large eddies) is broken down into progressively smaller scales of motion.