Analytical solutions for the propagation of heat pulses with temperature gradient length-dependent diffusion coefficient

Abstract

An original method is presented to solve the linearized heat transport equation for a general class of χ models, of the form χ = χ0TμG(∇T/T), where χ0 is a constant and G an arbitrary function which depends only on ∇T/T. These solutions allow one to address the propagation of heat and cold pulses in a tokamak plasma. By looking for solutions of the problem as functions of ∇T/T instead of the usual radial coordinates, the linearized transport equation is reformulated into a much simpler second-order differential equation. In this new form, it may be solved directly, if an analytical solution of the problem exists. The calculations are carried out both in slab and cylindrical geometry, and can be generalized to other geometries. For slab geometry, the exact solutions of the linearized transport equation are found in the case G = (|∇T|/T)α. An approximate solution is given in the case with critical gradient length G = (|∇T|/T)α(|∇T|/T−κ)β, valid for strong profile stiffness (∇T/T≈κ). In cylindrical geometry, the Wentzel–Kramers–Brillouin solutions are found in the case G = (|∇T|/T)α. These solutions are validated by comparison with numerical simulations. Their dependence on the modulation frequency and the model parameters is investigated. Finally, a method is proposed to identify the model parameters (e.g. χ0, α and μ) which best fit given temperature modulation data, as an original application of these analytical calculations to experimental transport studies.