The paper deals with the problem of characterizing partially ordered sets which are totally order-disconnected compact (Priestley) spaces with respect to intrinsic topologies. So posets with property DINT which are Priestley spaces with respect to interval topologies are investigated, where a partially ordered set is said to have property DINT if every set closed in the lower topology is a directed intersection of finitely generated upper sets. Also posets that are Priestley spaces with respect to bi-Scott topologies are characterized, where the bi-Scott topology is defined to be the common refinement of the Scott topology and the Scott topology on the dual poset.