Wind Energy

24
Scaling Turbulent Atmospheric Stratification:
A Turbulence/Wave Wind Model
S. Lovejoy and D. Schertzer
Summary. Twenty years ago, it was proposed that atmospheric dynamics are
scaling and anisotropic over a wide range of scales characterized by an elliptical
dimension Ds = 23/9, shortly thereafter we proposed the continuous cascade
“fractionally integrated flux” (FIF) model and somewhat later, causal space–time
extensions with Dst =29/9. Although the FIF model is more physically satisfying
and has been strikingly empirically confirmed by recent lidar measurements (finding
Ds =2.55±0.02, Dst =3.21±0.05) in classical form, its structures are too localized,
it displays no wave-like phenomenology.
We show how to extend the FIF model to account for more realistic wavelike
structures. This is achieved by using both localized and unlocalized space–time
scale functions. We display numerical simulations which demonstrate the requisite
(anisotropic, multifractal) statistical properties as well as wave-like phenomenologies.
24.1 Introduction
According to a growing body of analysis and theory (e.g., [1–5] and references
therein) the 1D horizontal sections of the atmosphere follow Kolmogorov laws:
∆ν = ε 1/3 ∆x 1/3 and (assuming no overall advection/wind): ∆ν = ε 1/2 ∆t 1/2 ,
where ε is the energy flux. However, 1D vertical sections follow the Bolgiano–
Obukov law: ∆ν = φ 1/5 ∆z 3/5 , φ the buoyancy variance flux. The generalization
to arbitrary space–time displacements ∆R = (∆r, ∆t), ∆r =
(∆x, ∆y, ∆z), and multifractal statistics is the 23/9D spatial model, 29/9D
space–time model. The anisotropic extension of the Kolmogorov law for the
velocity (v) and of the Corrsin–Obukov law for passive scalar density (ρ)
satisfy
∆v (∆R) =ε 1/3
[∆R] [[∆R]]1/3 ; ∆ρ (∆R) =χ 1/2
[∆R] ε−1/6
[∆R] [[∆R]]1/3 , (24.1)
where ε and χ are the (multifractal) energy and passive scalar variance fluxes;
the subscripts indicate the scale at which they are averaged. The key idea
of this model is that the physical scale function [[∆R]] replaces the usual