Energy spectrum of buoyancy-driven turbulence

Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux ${\Pi }_u$, we demonstrate that, for stably stratified flows, the kinetic energy spectrum $E_u\left(k\right)\sim k^{-11/5}$, the potential energy spectrum $E_{\theta }\left(k\right)\sim k^{-7/5}$, and ${\Pi }_u\left(k\right)\sim k^{-4/5}$ are consistent with the Bolgiano-Obukhov scaling. This scaling arises due to the conversion of kinetic energy to the potential energy by buoyancy. For weaker buoyancy, this conversion is weak, hence $E_u\left(k\right)$ follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh-Bénard convection, we show that the energy supply rate by buoyancy is positive, which leads to an increasing ${\Pi }_u\left(k\right)$ with $k$, thus ruling out Bolgiano-Obukhov scaling for the convective turbulence. Our numerical results show that convective turbulence for unit Prandt number exhibits a constant ${\Pi }_u\left(k\right)$ and $E_u\left(k\right)\sim k^{-5/3}$ for a narrow band of wave numbers.

Figure 1

Schematic diagrams of energy flux ${\Pi }_u\left(k\right)$: (a) In a stably stratified flow, ${\Pi }_u\left(k\right)$ decreases with $k$ due to a negative energy supply rate $\mathrm{Re}\left(\langle u_z\left(k\right){\theta }^{*}\left(k\right)\rangle \right)$. (b) In Rayleigh-Bénard convection, $\mathrm{Re}\left(\langle u_z\left(k\right){\theta }^{*}\left(k\right)\rangle \right)>0$, hence ${\Pi }_u\left(k\right)$ first increases for $k<k_t$, where $F\left(k\right)>D\left(k\right)$, then ${\Pi }_u\left(k\right)\approx \mathrm{const}$ $k_t<k<k_d$, where $F\left(k\right)\approx D\left(k\right)$; ${\Pi }_u\left(k\right)$ decreases for $k>k_d$ where $F\left(k\right)<D\left(k\right)$.

Figure 2

For stably stratified simulation with $\mathrm{Pr}=1$ and $\mathrm{Ri}=0.01$, plots of (a) normalized KE and (b) potential energy spectra for BO and KO scaling. BO scaling fits with the data better than KO scaling.

Figure 3

For stably stratified simulation with $\mathrm{Pr}=1$ and $\mathrm{Ri}=0.01$, plots of KE flux ${\Pi }_u\left(k\right)$, normalized KE flux ${\Pi }_u\left(k\right)k^{4/5}$, and potential energy flux ${\Pi }_{\theta }\left(k\right)$.

Figure 4

For stably stratified simulation with $\mathrm{Pr}=1$ and $\mathrm{Ri}=0.01$, plots of $-F\left(k\right),D\left(k\right),[-F(k)+D(k\left)\right]$, $-d{\Pi }_u\left(k\right)/dk$, and $k^{-9/5}$ line to match with $-d{\Pi }_u\left(k\right)/dk$ in the small-$k$ regime.

Figure 5

For stably stratified simulation with $\mathrm{Pr}=1$, and (a) $\mathrm{Ri}=0.5$ and (b) $\mathrm{Ri}=4\times {10}^{-7}$, the plots of normalized KE spectra for BO scaling and KO scaling.

Figure 6

For stably stratified simulation with $\mathrm{Pr}=1$, plots of $-F\left(k\right),D\left(k\right),[-F(k)+D(k\left)\right]$, $-d{\Pi }_u\left(k\right)/dk$, and $k^{-9/5}$ line to match with $-d{\Pi }_u\left(k\right)/dk$ in the small-$k$ regime. We follow the same convention as Fig. 4. (a) For $\mathrm{Ri}=0.5$, the negative $F\left(k\right)$ is shown as the chained red curve, and positive $F\left(k\right)$ for large $k$ is shown as the solid red curve. (b) For $\mathrm{Ri}=4\times {10}^{-7}$, $F\left(k\right)$ is multiplied by ${10}^5$ to fit in the same range.

Figure 7

For stably stratified simulation with $\mathrm{Pr}=1$ and $\mathrm{Ri}=0.01$, the vertical variation of horizontally averaged mean temperature $\overline{T}\left(z\right)={\langle T(x,y,z)\rangle }_{xy}$.

Figure 8

For RBC simulation with $\mathrm{Pr}=1$ and $\mathrm{Ra}={10}^7$, (a) plots of normalized KE spectra for BO and KO scaling; KO scaling fits with the data better than BO scaling. (b) KE flux ${\Pi }_u\left(k\right)$ and entropy flux ${\Pi }_{\theta }\left(k\right)$. The shaded region shows the inertial range.

Figure 9

For RBC simulation with $\mathrm{Pr}=1$ and $\mathrm{Ra}={10}^7$, plots of ${\Pi }_u\left(k\right)$, $F\left(k\right)$, and $D\left(k\right)$ for $10\le k\le 50$.

Figure 10

For RBC simulation with $\mathrm{Pr}=1$ and $\mathrm{Ra}={10}^7$, plots of $F\left(k\right)$, $-D\left(k\right)$, $F\left(k\right)-D\left(k\right)$, and $d{\Pi }_u\left(k\right)/dk$. $d{\Pi }_u\left(k\right)/dk>0$ for $k<22$, but $d{\Pi }_u\left(k\right)/dk<0$ for $k>22$.

Figure 11

For RBC simulation with $\mathrm{Pr}=1$ and $\mathrm{Ra}={10}^7$, plots of the entropy spectrum that exhibits a dual branch. The upper branch matches with $k^{-2}$ quite well, while the lower part is fluctuating.