Femtosecond laser filamentation has drawn much attention from researchers recently， which contains fruitful physical processes， such as the self-focusing induced by the optical Kerr effect， plasma defocusing， self-phase modulation and so on. It has shown great potential for many applications in the atmospheric field， such as remote sensing， lightning control， rain making. When propagating in the atmosphere， the influence of the turbulence on the filament spatial distribution has to be taken into consideration. Many studies have been dedicated to the filamentation distance， filament survival rate， transverse wandering， multiple filaments formation induced by turbulence.

Interestingly， it has been demonstrated that the air turbulence can have opposite effects of either increasing or decreasing the filamentation probability and either decreasing or increasing the filament onset distance depending on the initial experiment conditions. However， few studies have been dedicated to explore the mechanism of this phenomenon based on numerical simulation method， which would be significant for understanding the mechanism of the interaction between turbulence and laser filamentation.In this paper， the propagation of a single laser filament was simulated based on the laser filament nonlinear propagation model and the turbulent phase screen model.

The filamentation probability as a function of initial laser beam power was investigated in the turbulent air with turbulence structure constant C_n^{\mathrm{2}} of 10^-13 cm^-2/3， 10^-12 cm^-2/3 and 10^-11 cm^-2/3. The corresponding minimum initial laser powers for the formation of the laser filament with the filamentation probability larger than 0 under the above three turbulent conditions are 0.96 P_cr，0.68 P_cr and 0.14 P_cr， respectively. Moreover， the filamentation probability increases with the turbulence intensity C_n^{\mathrm{2}} when the initial laser beam power is less than P_cr. Thus， the air turbulence can trigger the laser filamentation for an initial laser power below the filamentation threshold. In the next step， the effect of a lens as a focusing optics on the filamentation probability in the turbulent air with C_n^{\mathrm{2}} of 10^-12 cm^-2/3 has been studied. By using a lens with focal length of 5 m as the focusing optics， the laser filamentation probability is always larger than that without any focusing optics in the turbulent air when the initial laser power is less than the filamenting threshold. However， for a lens with focal length of 2 m， the laser filamentation probability is less than that without any focusing optics in the turbulent air. Thus， using a lens with a relative large focal length could promotes the filamentation probability effectively in the turbulent air. Further， the filament onset distances in the turbulent air with different turbulent intensities have been investigated. The initial laser power is chosen to be 1.4 P_cr. The filament onset distance is 354 cm in the calm air. When C_n^{\mathrm{2}} is 10^-14 cm^-2/3， the filament onset distance varies between 349 cm and 358 cm. The average filament onset distance is 353.3 cm. When C_n^{\mathrm{2}} is 10^-11 cm^-2/3， the filament onset distance varies between 107 cm and 290 cm. The average filament onset distance is 149.1 cm. The relationship between the average deviation of the filament onset position 〈\mathrm{\Delta }r〉 and \sqrt[]{C_n^{\mathrm{2}}} has been obtained. For 4×10^-13 cm^-2/3≤C_n^{\mathrm{2}}≤4×10^-12 cm^-2/3， the relationship between 〈\mathrm{\Delta }r〉 and \sqrt[]{C_n^{\mathrm{2}}} can be fitted linearly well.

The interaction mechanism between the air turbulence and the laser beam can be determined by the parameter Q， which is defined by the ratio of the coherent area in the beam and the power it contains over the critical laser power for self-focusing. When C_n^{\mathrm{2}} is larger than 2.25×10^-12 cm^-2/3， Q is larger than 1. Under this condition， the air turbulence is relative strong， leading to the modulation instability of the beam phase， which can promote the filament formation finally. Thus the filament onset distance is always shorter than that without any turbulence. When C_n^{\mathrm{2}} is less than 2.25×10^-12 cm^-2/3， Q is much larger than 1. Under this condition， the air turbulence is relative weak， which will act as a noise and suppress the whole beam self-focusing effect. The filamentation probability will decrease and the filament onset distance will be lengthened. This study will pave the way to understand the mechanism of the influence of air turbulence on the filamentation probability and filament onset distance and would be valuable for many applications of the laser filament in the atmosphere.