  Retrograde waves with frequencies much lower than the rotation frequency
become trapped in the solar radiative interior. The eigenfunctions of the
compressible, nonadiabatic, Rossby-like modes ($\epsilon$-mechanism and
radiative losses taken into account) are obtained by an asymptotic method
assuming a very small latitudinal gradient of rotation, without an arbitrary
choice of other free parameters. An integral dispersion relation for the
complex eigenfrequencies is derived as a solution of the boundary value
problem. The discovered resonant cavity modes (called R-modes) are
fundamentally different from the known r-modes: their frequencies are functions
of the solar interior structure, and the reason for their existence is not
related to geometrical effects. The most unstable R-modes are those with
periods of 1--3 yr, 18--30 yr, and 1500--20000 yrs; these three separate period
ranges are known from solar and geophysical data. The growing times of those
modes which are unstable with respect to the $\epsilon$-mechanism are $\approx
10^2, 10^3,$ and $10^5$ years, respectively. The amplitudes of the R-modes are
growing towards the center of the Sun. We discuss some prospects to develop the
theory of R-modes as a driver of the dynamics in the convective zone which
could explain, e.g., observed short-term fluctuations of rotation, a control of
the solar magnetic cycle, and abrupt changes of terrestrial climate in the
past.
