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We complement the analytical work with direct numerical simulations, and show that $\\mathrm{Nu} \\sim \\mathrm{Ra}^{\\gamma}$ with $\\gamma \\approx (0.30-0.32)$, $\\mathrm{Pe} \\sim \\mathrm{Ra}^{\\eta}$ with $\\eta \\approx (0.57-0.61)$, and $\\theta_\\mathrm{rms} \\sim \\mathrm{const}$. 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