Quantitative investigation of physical properties of mantle plumes in three-dimensional numerical models

Numerical calculations have been carried out to investigate the physical properties of mantle plumes in highly viscous thermal convection depending on the Rayleigh number $(Ra)$⁠. The Boussinesq approximation was applied in a three-dimensional Cartesian domain filled with isoviscous, purely bottom-heated fluid with infinite Prandtl number. In order to monitor the dynamical behavior of plumes an automatic plume detecting routine was developed based on the temperature between the plume and its surroundings. It was established that as the convection becomes more vigorous with increasing Rayleigh number the average cross-sectional area of an individual plume decreases (⁠$∼Ra−2∕3$⁠), the vertical velocity in plumes increases $(∼Ra2∕3)$⁠, while the average temperature in plumes is independent of $Ra$⁠. It means that the volume and the heat transport in an individual plume is independent of the Rayleigh number. The number of plumes forming in the box increases $(∼Ra1∕3)$ which is in accordance with the scale analysis using the energy balance and the conservation of momentum. Furthermore, the Rayleigh number influences the temporal behavior of the average surface heat flow [Nusselt number, $Nu0(t)$] and the heat advected by plumes $[Twp(t)]$⁠. The characteristic frequencies of $Nu0(t)$ and $Twp(t)$ increase by $∼Ra2∕3$ in agreement with the rate of increase of the vertical velocity in plumes. The characteristic frequencies of $Nu0(t)$ and $Twp(t)$ are between the frequency corresponding to the time necessary for a plume to rise from the bottom to the top of the layer and the frequency of a whole convective cycle. The time series of $Twp(t)$ contain larger amplitudes and higher frequencies than $Nu0(t)$⁠. It was assumed that the heat in the top thermal boundary layer (TBL) propagates by conduction and using $Twp(t)$ as an input at the bottom of the top TBL the amplitude and the frequency of the heat flow series on the surface was calculated. It corresponds very well to the amplitude and the frequency of the observed $Nu0(t)$⁠. The correlation analysis between the time series of the surface Nusselt number and the heat advected by hot plumes showed that the time delay between the time series is equal to the time of the heat propagation by conduction through the TBL. The correlation between time series $Twp(t)$ at different depths demonstrated well that the main heat transfer mechanism in plumes is advection.