Free energy and spatial periodicity of generalized cnoidal ion holes

The free energy density of an ion hole excited coherent state plasma is calculated. The underlying ion trapping nonlinearity is shown to support expressions which appear one order earlier in the expansion scheme compared with conventional, linearly based derivations. This energy argument hence provides a further hint for the necessity of a full nonlinear Vlasov equation in describing coherent wave phenomena found ubiquitously in space, lab and fusion plasmas. The most prominent holes, the privileged cnoidal ion holes, being characterized by regularly trapped ions, can achieve states of lower energy than the ones of the unperturbed plasma and are hence attractors in the nonlinear dynamical evolution being approached in long term runs. Such negative energy holes can also be found by an extended description of holes that admits singularities of the trapped ion distribution fit, namely a moderate slope singularity at the separatrix of fit. A stronger singularity in form of a jump of fit at the boundary of the trapped ion region, on the other hand, results in a positive energy expression and is already for this reason less probable. As a biproduct, a novel expression is obtained for the wave length of a cnoidal ion hole in terms of the other relevant parameters.