Relationship between spectral truncation and grid point resolution - linear, quadratic and cubic grids

The relationship between the spectral resolution, governed by the truncation number T, and the grid resolution depends on the number of points on the grid at the equator, 4N, at which the shortest wavelength field is represented:

linear grid:  each wavelength is represented by 2 grid points → 4N = 2

\simeq

quadratic grid: each wavelength is represented by 3 grid points → 4N = 3(T_Q + 1)

cubic grid: each wavelength is represented by 4 grid points → 4N = 4(T_C + 1)

Until the implementation of IFS cycle 18r5 on 1 April 1998, the IFS used a quadratic grid. The introduction of the two-time level semi-Lagrangian numerical scheme at IFS cycle 18r5 made possible the use of a linear Gaussian grid reflected by the T_L notation.  The linear grid has been used since then, up to and including IFS cycle 41r1. At IFS cycle 42r1, the cubic grid is used and is indicated by the T_C notation.