Electrical charge transfer modeling (lightning) in cloud and its implementation in a one-dimensional prognostic cloud model

One of the most dazzling events in the atmosphere is lightning. During updrafts in the life cycle of cumulus clouds, collision of graupels and ice crystals in the presence of liquid water results in vertical separation of electrical charges and lightning. There are four types of lightening depending on the location of discharge.
The first type is cloud-to-ground lightning or fork lightning that happens due to electrical discharge between the cloud base and the negatively charged earth. Where the regions with opposite electrical charge within a cloud are connected, intra-cloud lightning occurs. The third one is lightning between clouds with opposite electric charge namely cloud-to-cloud lightening or sheet or heat lightening and the last one is known as cloud-to-air lightening.
In spite of the fact that lightning is considered as a part of severe weather systems, but it is hard to be predicted in short-term prediction. A thunderstorm could contain several tens of Coulombs of charge. The negative charge region has a temperature of -5 to -10⁰C while the positive charge region is located 2-3km upper than the negative charge location. In the gravity separation theory (principle of this research) some microphysical processes lead to charge separation. Negative charges are carried by heavier particles (cloud droplets, ice crystals and ions). Therefore during precipitation, negative charges accumulate in the lower levels while positive charges moved upwards by updrafts within the cloud.
In this paper, it is shown that the charge transfer due to interaction of charged particles (collision of graupels and ice crystals) is the most important ionization process. Two processes of non-inductive and inductive ionization are almost considered. In non-inductive ionization process, collision of hydrometeors results in charge separation.  Whilst in the inductive one the existence of an external field induces polarization, and then charge separation occurs. However, non-inductive process that happens because of collision between the graupels and ice crystals in the presence of liquid water has the most significant role in the charge transfer (Sanders et al. 1991, Miller et al. 2001). In this work, the non-inductive ionization mechanism is applied.  
In this research outputs of a one-dimensional cloud model were used (including vertical velocity, mixing ratios of graupels and ice, liquid water content, terminal velocity of graupels and temperature) to simulate the charge transfer intracloud (at microphysics scale). The vertical one-dimensional cloud model (Explicit Time-dependent Model (ETM)) is based on Chen and Sun (2002) equations (Gharaylou et al., 2009). The cloud in this model is considered as a cylindrical column of air with a constant radius. Non-hydrostatic pressure is assumed within the cloud column while the environment is in hydrostatic equilibrium. In the cloud model, the microphysical processes such as evaporation/sublimation, deposition/condensation, melting, freezing, aggregation, accretion and Bergeron process, entrainment and detrainment and lateral and vertical eddy mixing effects have been considered. In the ETM model convection is initiated using a potential temperature perturbation based on Chen and Sun (2004) relation.
To proceed, an idealized sounding was used as the input data. This profile consists of temperature, relative humidity and ambient pressure. Surface temperature and relative humidity are equal to 298 K and 94.5%. Its temperature profile was determined according to dry adiabatic lapse rate below 1 km, saturated adiabatic lapse rate from 1 to 10 km and isothermal for upper levels. The relative humidity increased linearly below 1 km and afterwards decreased with a rate of 5%. The vertical velocity initialized based on Ogura and Takahashi (1971) relations.
In this research, the mean charge transferred per collision of graupels and ice crystals was simulated using parametric equations suggested by Sanders et al (1991). The simulation has been done for 70 minutes with 1 second time step. The vertical resolution was typically 250 meter up to 15 km above ground level. The studied parameters consist of relative vertical velocity of graupels, mixing ratios of graupels and ice, electric filed and mean charge transfer per collision.
The results showed that lightning happened between 35-50 minutes inward simulation. It is worth to note that simultaneous presence of graupels and ice crystals guarantees the initiation of electric field in the cloud. Once the electric field intensity (positive or negative) exceeded the threshold electric field defined by Marshall et al. (1995), lightning occurred. Decrease of mixing ratios of graupels and ice crystals leads to weakened electric field in the upper levels. Precipitation also results in electric fields form at the lower altitudes. The negative sign of electric field can be inferred from the negative charge transfer in collisions.