On different approaches to calculating lightning electric fields

Three different approaches to the computation of lightning electric fields are compared. These approaches are the traditional dipole (Lorentz conditio n) technique and two versions of the monopole (continuity equation) techniq ue. The latter two techniques are based on two different formulations of th e continuity equation, one used by Thottappillil et al. [1997] and the othe r by Thomson [1999], the difference between the formulations being related to different treatments of retardation effects. The three approaches involv e the same expression for the vector potential but different expressions fo r the scalar potential. It is analytically shown that the three different e xpressions for the scalar potential are equivalent and satisfy the Lorentz condition. Further, the three approaches yield the same total fields and th e same Poynting vectors. However, expressions in the three approaches for t he individual electric field components in the time domain, traditionally i dentified by their distance dependence as electrostatic, induction, and rad iation terms, are different, suggesting that explicit distance dependence i s not an adequate identifier. It is shown that the so identified individual field components in the electric field equation in terms of charge density derived by Thottappillil et al. [1997] are equivalent to the corresponding field components in the traditional equation for electric field in terms o f current based on the dipole technique. However, the individual field comp onents in the electric field equation based on Thomson's [1999] approach ar e not equivalent to their counterparts in the traditional dipole technique equation. Further, in Thottappillil et al.'s [1997] technique and in the tr aditional dipole technique, the gradient of scalar potential contributes to all three electric field components, while in Thomson's [1999] technique i t contributes only to the electrostatic and induction components. Calculati ons of electric fields at different distances from the lightning channel sh ow that the differences between the corresponding field components identifi ed by their distance dependence in different techniques are considerable at close ranges but become negligible at far ranges.