Global sonde datasets do not support a mesoscale transition in the turbulent energy cascade

TL;DR

This study tests whether atmospheric turbulence obeys a scale-dependent transition among regimes or a single anisotropic cascade as proposed by Lovejoy–Schertzer. By computing structure functions from dropsonde and radiosonde data, the authors find vertical Hurst exponents around $H_v\approx0.6$ and horizontal exponents near $H_h\approx0.4$ over broad scale ranges, incompatible with gravity waves, 3D isotropic turbulence, or a large-scale enstrophy cascade. A joint 2D structure-function fit yields $H_h\approx0.37$ and $H_v\approx0.63$, closely matching Lovejoy–Schertzer predictions ($H_h=1/3$, $H_v=3/5$), with vertical smoothing identified as a plausible source of small deviations. The results argue for a cohesive, anisotropic turbulent cascade governing atmospheric dynamics across scales, though further multi-directional measurements and sub-200 m insight are needed to fully confirm Lovejoy–Schertzer scaling and quantify smoothing effects.

Abstract

Conceptual and theoretical models describing the dynamics of the atmosphere often assume a hierarchy of dynamic regimes, each operating over some limited range of spatial scales. The largest scales are presumed to be governed by quasi-two-dimensional geostrophic turbulence, mesoscale dynamics by gravity waves, and the smallest scales by 3D isotropic turbulence. In theory, this hierarchy should be observable as clear scale breaks in turbulent kinetic energy spectra as one physical mechanism transitions to the next. Here, we show that this view is not supported by global dropsonde and radiosonde datasets of horizontal winds. Instead, the structure function for horizontal wind calculated for vertical separations between 200 m and 8 km has a Hurst exponent of $H_v \approx 0.6$, which is inconsistent with either gravity waves ($H_v = 1$) or 3D turbulence ($H_v = 1/3$). For horizontal separations between 200 km and 1800 km, the Hurst exponent is $H_h \approx 0.4$, which is inconsistent with quasi-geostrophic dynamics ($H_h = 1$). We argue that sonde observations are most consistent with a lesser known "Lovejoy-Schertzer" model for stratified turbulence where, at all scales, the dynamics of the atmosphere obey a single anisotropic turbulent cascade with $H_v=3/5$ and $H_h =1/3$. While separation scales smaller than 200 m are not explored here due to measurement limitations, the analysis nonetheless supports a single cohesive theoretical framework for describing atmospheric dynamics, one that might substitute for the more traditional hierarchy of mechanisms that depends on spatial scale.

Figures (10)

• Figure 1: Simulations of a synthetic stochastic process with varying $H$lovejoy2010FIF, representing example wind profiles that might be observed in hypothetical atmospheres where the theories represented by Eqns. \ref{['eq:kolmogorov law']}-\ref{['eq:GW spectrum']} apply. The profiles are generated with the same random seed but have varying degrees of "smoothness" as specified by the value of $H$.
• Figure 2: Plot of the Lovejoy-Schertzer turbulent structure function (Eqn. \ref{['eq:2D structure function']}), representing the average sizes, shapes, and strengths of turbulent circulations. The function is shown in nondimensional form, i.e. $\phi = \varepsilon=1$, and using the theoretical values $H_h=1/3$ and $H_v=3/5$. Note that the empirical structure functions plotted in Sect. \ref{['sec:results']} consider absolute values for $\Delta x$ and $\Delta z$, which correspond to the first quadrant of this plot.
• Figure 3: Illustration of how dropsonde measurements of horizontal wind fluctuations $\Delta v(\Delta z)$ are effectively smoothed by sonde inertia. Wind fluctuations that occur over a smaller spatial scale than the sonde's adjustment scale $\Delta z_\text{min}$ cannot be reliably measured.
• Figure 4: Vertical structure functions for the IGRA radiosonde dataset (pink solid), the NOAA hurricane dropsonde dataset (black dashed) and the ACTIVATE dropsonde dataset (blue dot-dashed). Structure functions and Hurst exponents (Eqn. \ref{['eq:general structure function']}) are calculated for the lowest 8km of the troposphere, which is the layer measured by all three datasets.
• Figure 5: Hurst exponents and 95% confidence (shaded) calculated for structure functions as shown in Fig. \ref{['fig:8km vertical spectrum']} but evaluated within stacked layers $2\,\mathrm{km}$ thick.