Erik Lindborg

Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence?

The statistical features of turbulence can be studied either through spectral quantities, such as the kinetic energy spectrum, or through structure functions, which are statistical moments of the difference between velocities at two points separated by a variable distance. In this paper structure function relations for two-dimensional turbulence are derived and compared with calculations based on wind data from 5754 airplane flights, reported in the MOZAIC data set. For the third-order structure function two relations are derived, showing that this function is generally positive in the two-dimensional case, contrary to the three-dimensional case. In the energy inertial range the third-order structure function grows linearly with separation distance and in the enstrophy inertial range it grows cubically with separation distance. A Fourier analysis shows that the linear growth is a reflection of a constant negative spectral energy flux, and the cubic growth is a reflection of a constant positive spectral enstrophy flux. Various relations between second-order structure functions and spectral quantities are also derived. The measured second-order structure functions can be divided into two different types of terms, one of the form r 2/3 , giving a k -5/3 -range and another, including a logarithmic dependence, giving a k -3 -range in the energy spectrum. The structure functions agree better with the two-dimensional isotropic relation for larger separations than for smaller separations. The flatness factor is found to grow very fast for separations of the order of some kilometres. The third-order structure function is accurately measured in the interval [30, 300] km and is found to be positive. The average enstrophy flux is measured as Π ω ≃ 1.8 × 10 -13 s -3 and the constant in the k -3 -law is measured as K ≃ 0.19. It is argued that the k -3 -range can be explained by two-dimensional turbulence and can be interpreted as an enstrophy inertial range, while the k -5/3 -range can probably not be explained by two-dimensional turbulence and should not be interpreted as a two-dimensional energy inertial range.

The energy cascade in a strongly stratified fluid

A cascade hypothesis for a strongly stratified fluid is developed on the basis of the Boussinesq equations. According to this hypothesis, kinetic and potential energy are transferred from large to small scales in a highly anisotropic turbulent cascade. A relation for the ratio,

Scaling analysis and simulation of strongly stratified turbulent flows

l_{v}/l_{h}

Horizontal velocity structure functions in the upper troposphere and lower stratosphere: 1. Observations

, between the vertical and horizontal length scale is derived, showing how this ratio decreases with increased stratification. Similarity expressions are formulated for the horizontal and vertical spectra of kinetic and potential energy. A series of box simulations of the Boussinesq equations are carried out and a good agreement between the proposed hypothesis and the simulations is seen. The simulations with strongest stratification give horizontal kinetic and potential energy spectra of the form

Horizontal velocity structure functions in the upper troposphere and lower stratosphere: 2. Theoretical considerations

E_{K_{h}} \,{=}\, C_{1} \epsilon_{K}^{2/3} k_{h}^{-5/3}

Correction to the four-fifths law due to variations of the dissipation

and

The effect of rotation on the mesoscale energy cascade in the free atmosphere

E_{P_{h}} \,{=}\, C_{2} \epsilon_{P} k_{h}^{-5/3}/\epsilon_{K}^{1/3}

Stratified turbulence forced in rotational and divergent modes

, where

The kinetic energy spectrum of the two-dimensional enstrophy turbulence cascade

k_{h}

A note on Kolmogorov's third-order structure-function law, the local isotropy hypothesis and the pressure–velocity correlation

is the horizontal wavenumber,