The interaction of radio frequency (RF) waves with edge turbulence has resurfaced after the results obtained on many tokamaks showing that edge turbulence decreases when the ion cyclotron frequency heating (ICRH) is switched on. Using the lower hybrid (LH) waves to drive current into tokamak plasmas, this issue presented contradicting results with some tokamaks (FTU & HT-7) showing a net decrease, similar to the ICRH results, and others (Tore Supra) did not. In this article, these apparent discrepancies among tokamaks and RF wave frequencies are removed. It is found that turbulence large-scale structures in the scrape-off layer decrease at high enough plasma currents (Ip) on the Tore Supra tokamak. We distinguish three regimes: At low Ip's, no modification is detected with statistical properties of turbulence similar to ohmic plasmas even with PLH reaching 4.8 MW. At moderate plasma currents, turbulence properties are modified only at a high LH power. At high plasma currents, turbulent large scales are reduced to values smaller than 1 cm, and this is accompanied by a net decrease in the level of turbulence of about 30% even with a moderate LH power.
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