The integral constitutive relation for high frequency waves propagating in toroidal axisymmetric plasmas, obtained by formal integration of the linearized Vlasov equation, is simplified assuming the range of spatial dispersion to be small compared to the linear dimensions of the plasma. We propose to call this the `quasi-local approximation'. A (formally infinite) system of purely differential wave equations is obtained, which should be a good approximation under conditions similar to those which would justify an Eikonal Ansatz for the form of the wave fields. This system is valid to all orders in the Larmor radius, and, in the presence of a poloidal static magnetic field, predicts a different plasma response to each poloidal Fourier component of the h.f. field. Compared to ray tracing based on the Eikonal approximation, these wave equations have the advantage of allowing to take into account periodicity, boundary conditions, and toroidicity-induced coupling between poloidal Fourier modes.

As an example, the quasi-local wave equations are used to model propagation and absorption of the compressional wave at frequencies higher than the ion cyclotron frequency in the high-β plasma of the National Spherical Tokamak Experiment in Princeton, USA. Because of the low magnetic field and the tight aspect ratio of this device, large Larmor radius effects and toroidicity play an important role in these experiments. This example, therefore, illustrates well the importance of taking into account these effects, and, in particular, the different response of the plasma to each poloidal Fourier mode.