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Aims. A first-order asymptotic representation of higher-order non-radial g + -modes in spherically symmetric stars with a convective core is constructed from the full fourth-order system of governing equations. Stars are considered that, besides their convective core, also contain a radiative envelope, or both an intermediate radiative zone and a convective envelope. At the same time, the earlier asymptotic theory of Willems et al. (1997, A&A, 318, 99) relative to stars consisting of a convective core and a radiative envelope is made more transparent. Methods. As in the asymptotic theory of Smeyers (2006, A&A, 451, 223) for low-degree, higher-order p -modes, two-variable expansion procedures and boundary-layer theory are applied to the fourth-order system of differential equations established by Pekeris (1938, ApJ, 88, 189). Results. Eigenfrequency equations are derived in terms of the radial order  n of the g + -mode. The first $n - 1$ nodes of the radial component of the Lagrangian displacement coincide with the $n - 1$ nodes of the divergence of the Lagrangian displacement, and are situated in the radiative envelope or in the intermediate radiative zone according to the type of star considered. The radial displacement displays an  n th node near the surface. In stars containing an intermediate radiative zone and a convective envelope, the n th node is situated in the envelope. Conclusions. As well as for higher-order p -modes of spherically symmetric stars, the divergence of the Lagrangian displacement plays a basic role in the development of the asymptotic theory.

The asymptotic representation of higher-order g$\mathsf{^+}$-modes in stars with a convective core