Illustration of the two kinds of inertial modes propagating in a fluid with conical rotation profile Ω(θ) = Ω₀(θ) / Ω_ref = 1 + εsin²θ, for m = { 0,1,2,4 }from top left to bottom right. The eigenfrequency in the inertial frame ω_p is shown in x-axis and the differential rotation parameter ε is in y-axis. Blue: D modes that exist in the whole shell (ξ> 0 everywhere). White: DT modes that exhibit at least one turning surface inside the shell (the sign of ξ changes in the shell). Red: No inertial modes may exist in the shell (ξ< 0 everywhere). Orange: the modes in the region between the two orange lines feature a corotation layer () inside the shell. The meaning of the black solid, dotted, and dashed lines is explained in Sect. 3.2.2.