I3            $4\times 10^4$  0.20           5844                   1.3%
  M1                 1.0        0.651          11298                  1.8%

  : Characters of the numerical runs.[]{data-label="runs"}

$a_{\rm f}$, $F_{\rm b}$, $t$ and $\sigma_{\rm max}$ are the amplifying parameter, dimensionless input energy flux, dimensionless relaxation time and maximum deviation of average total energy flux from input flux, respectively.

For convenience, we make all the quantities dimensionless by scaling the variables so that $\rho_{\rm t}$, $p_{\rm t}$, $T_{\rm t}$ and $d$ all have the value one. The scalings, $\rho_{\rm scl}$, $p_{\rm
scl}$ and $T_{\rm scl}$ are given in Table \[dpm\]. $d/R$ represents the radial fraction of domain included in the computation, and the last column gives the dimensionless energy fluxes of the unmodified models. All the scalings of the other variables can deduced from dimensional analysis.

Results and Discussions
=======================

![Height distributions of average fluxes for Case I2 (top) and Case M1 (bottom). The enthalpy flux, kinetic flux, viscous flux, diffusive flux and total flux are represented by dotted lines, dashed lines, dash dot lines, dash dot dot lines and solid lines, respectively. Double-headed arrow indicates the overshooting zone, defined by negative convective flux. The vertical dashed lines represent the integral pressure scale heights counted from the upper boundary.[]{data-label="flx"}](./figures/mm10-flxs.eps "fig:"){width="8.2cm"} ![Height 