estimated value based on standard model (Melia & Falcke 2001). At low luminosities (less than a few percent of the Edington luminosity), black holes can accrete via advection dominated accretion flows (ADAFs) (Ichimaru 1977, Narayan & Yi 1994, Kato, Fukue & Mineshige 2008 and Yuan & Narayan 2014 for review). In such a flow, radiative losses are small compare to viscously heating because of low particle density of accreting flow at low accretion rate. Consequently, most of the energy released via viscosity is stored as entropy and transport inward with accretion. ADAFs are optically thin, geometrically thick and hot (compare the virial temperature of the gas in the flow) and radiate mostly in X-ray band (see Narayan et al. 1996). In the past decades the ADAFs models have captured great attentions and rapid progress has been made.

At the same time as ADAFs model was introduced, it was realized that they are likely to be unstable against convection in the radial direction. Because of low radiative efficiency in hot accretion flow, since the gas is heated but hardly cools, the entropy increases with decreasing radius. Hot accretion flows are therefore potentially unstable to convection. However, according some numerical simulations (e.g. 