Rapidly rotating internally heated convection: bounds on long-time averages

Abstract

Convection on geophysical and astrophysical scales is subject to rapid rotation and strong heating from within the domain. In studying the long-time behaviour of the solutions for such a system, energy identities fail to capture the effects of rotation because the Coriolis force does no work, and rapid rotation can be prohibitive for direct numerical simulations. Instead, we derive an asymptotically reduced model for rapidly rotating convection driven by uniform internal heating between isothermal stress-free boundaries in a plane periodic layer. The main contribution is the proof of bounds on the mean temperature, and the mean vertical convective heat transport, in terms of the Rayleigh and Ekman numbers, in the limit of infinite Prandtl number. The first quantity represents the mixing of the flow, and the second the asymmetry in heat leaving the bottom and top boundaries due to convection, and unlike Rayleigh-Bénard convection, the two are not a priori related. We employ alternative estimation techniques to those used in previous studies (Grooms \& Whitehead, 2014 \textit{Nonlinearity}, 28, 29) and identify two distinct scaling behaviours for both quantities. Finally, our bounds are optimised, within the methodology, and provide a rigorous constraint for future studies of rotation-dominated internally heated convection.

Figures (7)

• Figure 1: Left: a schematic diagram of rotating internally heated convection between two isothermal parallel plates with stress-free boundary conditions. The uniform heating source is denoted by $S$, and $H$ is the separation between the two plates in the direction of gravity, $g$. Horizontal periodicities are $L_x H$ and $L_y H$. The convection is subject to global rotation $\Omega$. Without loss of generality, the boundary temperature is set to $T_b = 0$. Circles indicate flow structures with a characteristic vertical scale $H$ and horizontal scale $E^{1/3}H$. Right: The same flow structures shown in dimensionless coordinates. The vertical and horizontal length scales are $H$ and $E^{1/3}H$ respectively, with $E\ll 1$.
• Figure 2: A sketch of the background profile \ref{['eq_phi']} featuring quadratic boundary layers.
• Figure 3: Lower bounds on $\langle \Theta \rangle_{V,\,t}$ from \ref{['eq_bound_Theta']} (purple solid line) and \ref{['eq_bound_Theta_2']} (red solid line), compared with the uniform upper bound of $\tfrac{1}{12}$ (blue solid line) and the numerical maximum (black dots) of \ref{['eq_Theta_B']} subject to \ref{['eq_est_SC']} and the condition $\delta \leq \frac{1}{2}$. Triangle markers denote the ${\mathcal{R}}$ above which \ref{['eq_bound_Theta']} and \ref{['eq_bound_Theta_2']} are valid, with the red triangle corresponding to a lower bound on the nonlinear onset of convection, and the star indicates the linear onset of convection. Dotted and dashed lines respectively indicate regions where the bounds are invalid and valid but suboptimal.
• Figure 4: A sketch of the background field \ref{['eq_phi_wtheta']} with a quadratic boundary layer and linear bulk.
• Figure 5: Upper bounds on $\langle w\theta \rangle_{V,\,t}$ from \ref{['eq_wtheta_bound']} (purple solid line) and \ref{['eq_wtheta_bound2']} (red solid line), compared with the uniform bounds of $0$ and $\frac{1}{2}$ (blue solid lines), along with the numerical minimum (black dots) of \ref{['eq_wtheta_B']} subject to the condition \ref{['eq_est_wtheta_SC']} and $\delta \leq 1$. The triangle markers denote the ${\mathcal{R}}$ above which \ref{['eq_wtheta_bound', 'eq_wtheta_bound2']} are valid, and the star indicates the linear onset of convection. Dotted and dashed lines respectively indicate regions where the bounds are invalid and valid but suboptimal.

Theorems & Definitions (30)

• Theorem 1: Lower bound on mean temperature
• Theorem 2: Upper bound on mean heat flux
• Proposition 1: bound on $\langle\Theta\rangle_{V,\,t}$
• proof
• Remark 1
• Lemma 1: Sufficient conditions for $Q_{\bm{k}}\geq 0$, large $k$
• proof
• Lemma 2: Sufficient conditions for $Q_{\bm{k}}\geq 0$, small $k$
• proof
• Lemma 3: Sufficient conditions for spectral constraint