Numerical modelling is presented of the JET Scrape-off-Layer (SOL) time evolution with the fluid code EDGE2D. The code includes direct SOL ionization by the LH wave and effects of near LH grill limiters. In [1] and [2], a stationary state was modeled. In contrast, the original contribution here is that we also explore for the first time the SOL time evolution when switching on/off the LH and near grill gas puff. Further from the separatrix, where the LH parasitic dissipation and corresponding local SOL heating takes place in the code, the computed plasma density increases with time at the onset of the local LH wave ionization with a short characteristic time of less than about 10ms. Closer to the separatrix, the plasma density approaches the new equilibrium with a several times longer characteristic time. Evolution of the SOL neutral particles density, SOL flows, and of other SOL parameters was also explored. Naturally, the SOL plasma variations at LH switching on/off are larger with a larger near grill gas puff and also with larger variations of the LH power. The results are relevant to LH coupling and modulated LH power experiments. Similar short and longer characteristic times for the SOL plasma density evolution in locations magnetically connected and not-connected to the LH launcher are measured by frequency swept reflectometry on Tore Supra during LH power modulation.