We investigate the properties of r -mode and inertial mode of slowly rotating , non-isentropic , Newtonian stars , by taking account of the effects of the Coriolis force and the centrifugal force .
The Coriolis force is the dominant restoring force for both r -mode and inertial mode , which are also called rotational mode in this paper .
For the velocity field produced by the oscillation modes the r -mode has the dominant toroidal component over the spheroidal component , while the inertial mode has the comparable toroidal and spheroidal components .
In non-isentropic stars the specific entropy of the fluid depends on the radial distance from the center , and the interior structure is in general divided into two kinds of layers of fluid stratification that is stable or unstable against convection .
Because of the non-isentropic structure , low frequency oscillations of the star are affected by the buoyant force , which has no effects on oscillations of isentropic stars .
In this paper we employ simple polytropic models with the polytropic index n = 1 as the background neutron star models for the modal analysis .
For the non-isentropic models we consider only two cases , that is , the models with the stable fluid stratification in the whole interior and the models that are fully convective .
For simplicity we call these two kinds of models “ radiative ” and “ convective ” models in this paper .
For both cases , we assume the deviation of the models from isentropic structure is small .
Examining the dissipation timescales due to the gravitational radiation and several viscous processes for the polytropic neutron star models , we find that the gravitational radiation driven instability of the nodeless r -modes associated with l ^ { \prime } = |m| remains strong even in the non-isentropic models , where l ^ { \prime } and m are the indices of the spherical harmonic function representing the angular dependence of the eigenfunction .
Calculating the rotational modes of the radiative models as functions of the angular rotation frequency \Omega , we find that the inertial modes are strongly modified by the buoyant force at small \Omega , where the buoyant force as a dominant restoring force becomes comparable with or stronger than the Coriolis force .
Because of this property we obtain the mode sequences in which the inertial modes at large \Omega are identified as g -modes or the r -modes with l ^ { \prime } = |m| at small \Omega .
We also note that as \Omega increases from \Omega = 0 the retrograde g -modes become retrograde inertial modes , which are unstable against the gravitational radiation reaction .