Transition to inverse cascade in turbulent rotating convection in absence of the large-scale vortex

TL;DR

The study addresses how strong rotation induces an inverse energy cascade in convective turbulence and how the interaction with a large-scale vortex (LSV) can cause a discontinuous, hysteretic transition. By introducing an adapted hypoviscosity that dissipates the inverse flux on the horizontal manifold, the authors suppress the LSV in rotating Rayleigh-Bénard convection at $Ek=10^{-4}$ and $Pr=1$, exploring $Ra$ across regimes with varying scale separation. They find that suppressing the LSV yields a continuous transition to the inverse cascade and a more local energy transfer, while the presence of the LSV yields nonlocal upscale transfer and stronger inverse flux; the LSV also modestly reduces heat transport by $3$–$8\\%$. These results imply that the observed hysteresis in previous studies stems from LSV–turbulence interactions, providing insight into quasi-2D turbulence and informing how large-scale structures modulate small-scale transport in rotating convection.

Abstract

Turbulent convection under strong rotation can develop an inverse cascade of kinetic energy from smaller to larger scales. In the absence of an effective dissipation mechanism at the large scales, this leads to the pile-up of kinetic energy at the largest available scale, yielding a system-wide large-scale vortex (LSV). Earlier works have shown that the transition into this state is abrupt and discontinuous. Here, we study the transition to the inverse cascade at Ekman number $\textrm{Ek}=10^{-4}$ and using stress-free boundary conditions, in the case where the inverse energy flux is dissipated before it reaches the system scale, suppressing the LSV formation. We demonstrate how this can be achieved in direct numerical simulations by using an adapted form of hypoviscosity on the horizontal manifold. We find that in the absence of the LSV, the transition to the inverse cascade becomes continuous. This shows that it is the interaction between the LSV and the background turbulence that is responsible for the earlier observed discontinuity. We furthermore show that the inverse cascade in absence of the LSV has a more local signature compared to the case with LSV.

Figures (8)

• Figure 1: (a) Instantaneous kinetic energy spectra $E(k)$ for the cases with $\textrm{Ra}=8\times10^6$ with and without hypoviscosity and for different horizontal domain sizes (see table \ref{['tab:input']}). (b,c) The corresponding snapshots of kinetic energy $\tfrac{1}{2}|\bm{u}|^2$ for the cases with $L/H=2.235$ at the mid-plane $z=H/2$.
• Figure 2: Instantaneous snapshots of the temperature fluctuation field $T-\tilde{T}(z)$, where $\tilde{T}(z)$ is the horizontally and temporally averaged temperature profile, for the cases with $\textrm{Ra}=8\times10^6$ and $L/H=2.235$ without hypoviscosity (a) and with hypoviscosity (b).
• Figure 3: The total inverse flux of kinetic energy $\varepsilon_\textrm{inv}$ normalized by the total energy flux $\varepsilon = (\textrm{Nu}-1)/\sqrt{\textrm{Pr}\textrm{Ra}}$ as a function of $\textrm{Ra}$ for $\textrm{Ek}=10^{-4}$ and $\textrm{Pr}=1$ for the different series of runs with and without hypoviscosity. Panel (b) is a zoom of panel (a) that focuses on the cases with hypoviscosity, indicated by the dashed box in (a). For the runs without hypoviscosity, black diamonds indicate cases that show LSV formation. The figure reveals that while the transition to the inverse cascade without hypoviscosity is discontinuous and hysteretic (lower hysteretic branch is depicted by the dotted line), the transition for the case with hypoviscosity is continuous.
• Figure 4: Time-averaged kinetic energy transport maps from 3D (a,c) and 2D (b,d) modes $Q$ to 2D modes $K$, i.e. $T_\mathrm{3D}(K, Q)$ and $T_\mathrm{2D}(K, Q)$, respectively, for the cases with $\textrm{Ra}=8\times10^6$ and $L/H=2.235$ without hypoviscosity (a,b) and with hypoviscosity (c,d). Panels (a-d) also show the respective sums over the donating scales $Q$, obtained as $\mathcal{T}_\mathrm{3D}(K)\equiv \sum_Q T_\mathrm{3D}(K,Q)$ and $\mathcal{T}_\mathrm{2D}(K) \equiv \sum_Q T_\mathrm{2D}(K,Q)$ (blue lines). Panel (e) depicts the total 2D energy flux $\varPi_\textrm{2D}(K) \equiv -\sum_{K'<K} \mathcal{T}_\textrm{2D}(K)$, normalized by the total energy flux $\varepsilon$.
• Figure 5: (a) The average Nusselt number $\textrm{Nu}$ as a function of $\textrm{Ra}$ for $\textrm{Ek}=10^{-4}$ and $\textrm{Pr}=1$ for the series of runs with and without hypoviscosity and $L/H=2.235$. (b) The same data compensated by the $\textrm{Nu}$ of the runs with hypoviscosity. For the runs without hypoviscosity, black diamonds denote cases that show LSV formation. This indicates that the LSV lowers the $\textrm{Nu}$ by around 3-8%.