Thermal convection in rotating spherical shells: Temperature-dependent internal heat generation using the example of triple-$\alpha$ burning in neutron stars

We present an extensive study of Boussinesq thermal convection including a temperature-dependent internal heating source, based on numerical three-dimensional simulations. The temperature dependence mimics triple-$\alpha$ nuclear reactions and the fluid geometry is a rotating spherical shell. These are key ingredients for the study of convective accreting neutron star oceans. A dimensionless parameter ${\mathrm{Ra}}_n$, measuring the relevance of nuclear heating, is defined. We explore how flow characteristics change with increasing ${\mathrm{Ra}}_n$ and give an astrophysical motivation. The onset of convection is investigated with respect to this parameter and periodic, quasiperiodic, chaotic flows with coherent structures, and fully turbulent flows are exhibited as ${\mathrm{Ra}}_n$ is varied. Several regime transitions are identified and compared with previous results on differentially heated convection. Finally, we explore (tentatively) the potential applicability of our results to the evolution of thermonuclear bursts in accreting neutron star oceans.

Figure 1

(a) Time series of the volume-averaged kinetic energy $K$ at $\mathrm{Ra}=6.2\times {10}^3$ and ${\mathrm{Ra}}_n={10}^{18}$. The initial condition corresponds to $\mathrm{Ra}=6.2\times {10}^3$ and ${\mathrm{Ra}}_n=0$. From a purely differentially heated convective flow and after a long transient $\left(t>20\right)$ the burning flow attractor is reached. (b) Conductive temperature $T$ versus radial coordinate $r$. The bottom curve corresponds to ${\mathrm{Ra}}_n=0$ with $T=T_c$ given by Eq. (10) (differentially heated conductive state). From bottom to top, the remaining curves correspond to ${\mathrm{Ra}}_n={10}^{11},{10}^{12},{10}^{13}$, all with $\mathrm{Ra}={10}^3$.

Figure 2

In the first row, the left three plots are the spherical, equatorial, and meridional cross sections of the contour plots of the nonaxisymmetric component of the temperature $T$ at $\mathrm{Ra}=6.2\times {10}^3$ and ${\mathrm{Ra}}_n={10}^{17}$; the right three plots are the same as the left, but for $\mathrm{Ra}=6.2\times {10}^3$ and ${\mathrm{Ra}}_n=0$. The second to fifth rows show $v_r,v_{\theta },v_{\phi }$, and ${\mathbf{v}}^2/2$, from top to bottom. All meridional cross sections are selected at a relative maximum. The spherical cross sections of $v_{\theta },v_{\phi }$, and ${\mathbf{v}}^2/2$ are taken at the outer boundary. All of them cut a relative maximum except for $v_{\phi }$ and ${\mathrm{Ra}}_n=0$, where the maximum is close to the inner boundary. The spherical cross sections for $T$ and $v_r$ pass through the maximum, which is located inside the shell.

Figure 3

Time series of the kinetic energies $K,K_a^T,K_{\text{na}}^T,K_a^P$, and $K_{\text{na}}^P$ represented by black solid, blue dashed, light blue dotted, red dash-dotted, and light red dash–double-dotted lines, respectively. The parameters are $\mathrm{Ra}=6.2\times {10}^3$ and (a) ${\mathrm{Ra}}_n=0$, (b) ${\mathrm{Ra}}_n={10}^{17}$, (c) ${\mathrm{Ra}}_n={10}^{19}$, and (c) ${\mathrm{Ra}}_n={10}^{20}$.

Figure 4

Time series of the temperature at $r=r_i+0.5d,\phi =0$, and three different colatitudes $\theta =3\pi /8,\theta =\pi /4$, and $\theta =\pi /8$, represented by black solid, blue dashed, and light blue dotted lines, respectively. The parameters are $\mathrm{Ra}=6.2\times {10}^3$ and (a) ${\mathrm{Ra}}_n=0$, (b) ${\mathrm{Ra}}_n={10}^{17}$, (c) ${\mathrm{Ra}}_n={10}^{19}$, and (c) ${\mathrm{Ra}}_n={10}^{20}$.

Figure 5

Physical properties versus $\sqrt{{\mathrm{Ra}}_n}$: (a) time-averaged global (volume-averaged) kinetic energies $K(•),K_a(\circ )$, and $K_{\text{na}}(*)$, with fitting lines $\overline{K}=8\times {10}^{-8}{\mathrm{Ra}}_n^{1/2}$ and $\overline{K}=5\times {10}^1{\mathrm{Ra}}_n^{1/8}$; (b) ratios of the time-averaged equatorially symmetric over global kinetic energies $K^s/K(•),K_a^s/K_a(\circ )$, and $K_{\text{na}}^s/K_{\text{na}}(*)$; (c) time-averaged rms curl of forces integrals $\mathbf{\nabla }\times {\mathcal{F}}_I(•$, inertial), $\mathbf{\nabla }\times {\mathcal{F}}_V(\circ$, viscous), and $\mathbf{\nabla }\times {\mathcal{F}}_C(*$, Coriolis); and (d) time-averaged temperatures $\overline{T_1},\overline{T_2}$, and $\overline{T_3}$ taken at $\phi =0,\theta =3\pi /8$, and $r=r_1=r_i+0.15d(*),r=r_2=r_i+0.5d(\circ )$, and $r=r_3=r_i+0.85d(•)$, respectively (where $d=r_o-r_i)$, with the fitting line corresponding to $\overline{T_2}=(1.95\pm 0.15){\mathrm{Ra}}_n^{0.223\pm 0.001}$.

Figure 6

Time-averaged mean zonal flow (azimuthally averaged $v_{\phi })$ versus $\sqrt{{\mathrm{Ra}}_n}$ at colatitude (a) $\theta =\pi /8$, (b) $\theta =\pi /4$, and (c) $\theta =3\pi /8$. The symbols indicate the radial position: $*,r=r_1=r_i+0.15d$; $\circ ,r=r_2=r_i+0.5d$; and $•,r=r_3=r_i+0.85d$ (where $r_i=1.5$ and $r_o=2)$.

Figure 7

Instantaneous contour plots at ${\mathrm{Ra}}_n={10}^{20},{10}^{21},{10}^{22},{10}^{23},{10}^{26},{10}^{27}$ (from top row to bottom row). The left three plots are the spherical, equatorial, and meridional cross sections of the temperature $T$. The right three plots are the same cross sections for $v_{\varphi }$. Meridional cross sections and the spherical cross section of $T$ are taken at a relative maximum. The spherical cross section of $v_{\varphi }$ is at the outer surface.

Figure 8

Instantaneous contour plots at ${\mathrm{Ra}}_n={10}^{20},{10}^{21},{10}^{22},{10}^{23},{10}^{26},{10}^{27}$ (from left to right) showing the meridional cross sections of the kinetic energy density ${\mathbf{v}}^2/2$ (first row) and the mean zonal flow (azimuthally averaged $v_{\phi })$ (second row). Cross sections of ${\mathbf{v}}^2/2$ pass through a maximum.

Figure 9

Instantaneous contour plots at $\mathrm{Ra}=6.2\times {10}^3$ and ${\mathrm{Ra}}_n={10}^{19},{10}^{20},{10}^{21},{10}^{22}$ (from left to right and from top to bottom group of plots). Each group of three plots consists of the spherical, equatorial, and meridional cross sections of the axial vorticity $w_z$. Spherical cross sections are taken at $r=r_i+0.95d,r=r_i+0.8d,r=r_i+0.3d$, and $r=r_i+0.5d$ for ${\mathrm{Ra}}_n={10}^{19},{10}^{20},{10}^{21},{10}^{22}$, respectively. Meridional cross sections are taken where $w_z$ has a relative maximum.