Stability of negative ionization fronts: Regularization by electric screening?

We recently have proposed that a reduced interfacial model for streamer propagation is able to explain spontaneous branching. Such models require regularization. In the present paper we investigate how transversal Fourier modes of a planar ionization front are regularized by the electric screening length. For a fixed value of the electric field ahead of the front we calculate the dispersion relation numerically. These results guide the derivation of analytical asymptotes for arbitrary fields: for small wave-vector k, the growth rate $s\left(k\right)\mathrm{}$ grows linearly with k, for large k, it saturates at some positive plateau value. We give a physical interpretation of these results.