## 1 Introduction  
Ion transport is relevant in numerous biological and engineering systems [1]. Selective ion transport is the key process in desalination of salt solutions by means of electrodialysis as well as in electrochemical energy conversion systems, such as redox flow batteries or reverse electrodialysis [2; 3; 4; 5]. In the above-mentioned processes, ion-exchange membranes ensure ionic continuity and selective transport of ions between different compartments. Ion-exchange membranes are polymeric films containing fixed charges in their structure, which make them selective for mobile ions with the opposite charge (counter-ions) and impermeable for mobile ions with the same charge (co-ions). Therefore, two main types of ion-exchange membranes exist: anion- and cation-exchange membranes. Ion transport through ion-exchange membranes has been widely studied using electrolytes composed of monovalent salts, since desalination of brackish water was one the first applications of electrodialysis membranes [6]. While transport of monovalent salt solutions across charged membranes has been widely investigated, phenomena involved during the transport of multivalent ions are not as widely understood. Treatment of industrial effluents from mining sites or metal finishing industries, recovery of valuable metals from electronic waste or transport of multivalent ions in redox flow batteries, to name a few, are some of the applications where transport of multivalent ions is involved [7; 8; 9]. Three are the main properties that are desired in ion-exchange membranes: long-term stability, high selectivity for the counter-ions of interest and high ionic conductivities. Membranes having these properties ensure high faradaic efficiencies and low energy consumption. Nonetheless, selectivity of ion-exchange membranes implicitly entails the development of concentration polarization. When a constant current density is imposed within an electrodialysis cell, ionic current is mainly transported by counter-ions in the membrane phase, whereas both counter- and co-ions contribute to ionic current through diffusion boundary layers (DBLs). Consequently, concentration profiles are developed at the membrane-neighboring solution layers. This phenomenon, known as concentration polarization, is intensified as current density is increased. Ultimately, when the current is increased up to the point that the concentration at the membrane depleting surface approaches zero, the resistance of the membrane system increases substantially. The current density at which this phenomenon occurs is conventionally known as the limiting current density (\(\mathrm{i}_{\mathrm{lim}}\)). After surpassing \(\mathrm{i}_{\mathrm{lim}}\), the well-known plateau region of current-voltage curves develops. Electrodialysis systems are usually operated below the \(\mathrm{i}_{\mathrm{lim}}\), in order not to increase energy costs. Contrariwise to pressure-driven membrane processes, flux of ions through the membranes can be further increased beyond the \(\mathrm{i}_{\mathrm{lim}}\) via activation of additional ion transport mechanisms. Among them, electroconvection, water dissociation and natural convection are the most common. Electroconvection is the formation of vortices under strong polarization conditions and an extended space charge layer at the depleting side of the membrane. The created instabilities result in mixing of the DBL and an increased supply of ions from the bulk solution towards the membrane [10]. Opposite to the dissociation of water, in the case of electroconvection, the ions transported through the membrane are the target ones. For this reason, several researchers have investigated different strategies to potentiate electroconvection at low driving forces, that is to reduce the length of the plateau region in order to shift the operation point above the \(\mathrm{i}_{\mathrm{lim}}\) without implying very high increases in energy losses.