A catheter comprises a flexible shaft having a length for accessing the renal artery. A balloon arrangement, dimensioned for deployment within the renal artery, is provided at the shaft's distal end and fluidly coupled to a lumen arrangement. A balloon body is configured to complete at least one revolution of the shaft's distal end to form a spiral configuration thereabout, and is secured to the shaft by one or more movable mounting arrangements. The balloon body supports an electrode arrangement for ablating perivascular renal nerves in a substantially spiral pattern. A cooling arrangement cools at least the electrode contact regions during ablation. The distal end of the shaft has a stiffness sufficient to oppose a force produced by the balloon body when inflated that tends to contort the shaft's distal end and to maintain a substantially constant apposition force between the electrodes and an inner wall of the renal artery.