A Stratification-Dependent, Enstrophy-Controlled Regime in Baroclinic Turbulence Experiments in the Laboratory

The circulation of the Earth’s atmosphere and those of many other planets is dominated by turbulent interactions in a baroclinically unstable, rotating, stratified flow. Even for the Earth, which has been well observed for many years, the energy spectrum and complex properties of the anisotropic and inhomogeneous turbulent cascades of energy and enstrophy remain poorly understood and difficult to model accurately. Here we measure geostrophic turbulence energised by baroclinic instability in a rotating, differentially heated fluid annulus in the laboratory, which is bounded by convectively-driven warm and cold flows at the outer and inner boundaries, respectively (see Fig. 1a). Horizontal velocity fields (Fig. 1b-c) are obtained via particle image velocimetry of neutrally buoyant particles suspended in the flow, while the temperature structure is sampled using a vertical array of thermocouples located in the middle of the channel. The horizontal kinetic energy spectra exhibit a wavenumber range at relatively large length scales which scales as k⁻³, where k denotes the horizontal wavenumber (see Fig. 1d-e). Moreover, the spectral amplitude is found to correlate with the square of the Brunt–Vaisala frequency N at the same heights as the velocity measurements. The observed turbulent state exhibits a net forward enstrophy cascade across all scales, along with bidirectional kinetic energy transfer, which is indicated by a reversal in the sign of the spectral energy flux. The change of sign of the kinetic energy cascade occurs at a scale proportional to the internal Rossby radius of deformation L_d. These findings highlight the role of baroclinic instability in shaping the distribution of energy across scales with implications for synoptic- and meso-scale turbulent flows in the atmospheres of the Earth and other terrestrial planet atmospheres and oceans.

FIG. 1. (a) Schematic plot of the convective tank. Snapshots of vorticity ζ for thermal Rossby number Ro_T = 5.41 (b) and RoT = 0.03 (c). On the scale bar, Lⁱ_d = 2.4 cm and Lⁱⁱ_d = 22.6 cm are the Rossby radius of deformation for (c) and (b), respectively. (d) Kinetic energy spectra, E(k), for various values of Ro_T. The arrow indicates the wave number k_p corresponding to the peak of E(k) when Ro_T = 0.03. Inset: radial profiles of temporal- and zonal-averaged azimuthal velocity, U_θ. (e) Kinetic energy spectra compensated by k⁻³ and normalised by N² versus L_Rk. The dashed line indicates the plateau segment for L_Rk ∈ [2, 10] and has a magnitude of ∼ 0.5. Data are for height h = 0.18 m.