Aspect ratio dependence of charge transport in turbulent electroconvection

We present measurements of the normalized charge transport or Nusselt number $\mathrm{N}\mathrm{u}$ as a function of the aspect ratio $\mathrm{\Gamma }$ for turbulent convection in an electrically driven film. In analogy with turbulent Rayleigh-B{'e}nard convection, we develop the relevant theoretical framework in which we discuss the local-power-law-scaling of $\mathrm{N}\mathrm{u}$ with a dimensionless electrical forcing parameter $\mathcal{R}$. For these experiments where $10^4 \lesssim {\cal R} \lesssim 2 \times 10^5$wefindthat${\rm Nu} \sim {\rm F}(\Gamma){\cal R}^\gamma$with$\gamma$either$ = 0.26 (\pm 0.02)$or$0.20 (\pm 0.03)$, in excellent agreement with the theoretical predictions of $\gamma =1\mathrm{/}4$ and 1/5. Our measurements of the aspect ratio-dependence of $\mathrm{N}\mathrm{u}$ for $0.3 \leq \Gamma \leq 17$comparesfavorablywiththefunction${\rm F}(\Gamma)$ from the scaling theory.