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/m15-flxs.eps "fig:"){width="8.2cm"}

![Height distribution of superadiabatic gradient for Case I1 – I3 (top) and Case M1 (bottom). In the top panel, solid line: I1; dotted line: I2; dashed line: I3. Double-headed arrow indicates the overshooting zone. The vertical dashed lines represent the integral pressure scale heights counted from the upper boundary. The very large superadiabatic gradients near the upper boundary are mainly caused by the solid boundary conditions. In the middle region of convection zone, the small superadiabatic gradient could be affected by the numerical errors. (see text for details)[]{data-label="sags"}](./figures/sags-im.eps "fig:"){width="8.2cm"} ![Height distribution of superadiabatic gradient for Case I1 – I3 (top) and Case M1 (bottom). In the top panel, solid line: I1; dotted line: I2; dashed line: I3. Double-headed arrow indicates the overshooting zone. The vertical dashed lines represent the integral pressure 