In order to provide a detailed analysis of RR Lyrae instability strip topology, an extensive grid of nonlinear, nonlocal, and time-dependent convective models of RR Lyrae stars has been computed at fixed mass (M = 0.65 M.) and chemical composition (Y = 0.299, Z = 0.001). Four series of envelope models at different luminosity levels [log (L/L.) = 1.81; log (L/L.) = 1.72; log (L/L.) = 1.61; log (L/L.) = 1.51] and on a large range of effective temperatures (5700 K < T(e) < 8000 K) have been investigated. The nonlinear modal stability has been evaluated at limiting amplitude for both the fundamental and the first overtone. The equations governing both dynamical and convective interactions have been integrated in time until the initial perturbations and the nonlinear fluctuations due to superposition of higher order modes settled down. The theoretical observables obtained by the present survey (radius, luminosity, velocity and temperature amplitudes, periods) describe the pulsation characteristics of the models at full amplitude, hence they can be properly compared with observations. A linear, nonadiabatic survey of the first three modes of RR Lyrae models has been also computed to supply the static structure of the envelope to the nonlinear stability analysis. Several numerical simulations have been performed to test both the numerical accuracy (boundary conditions, time-step size, zoning) and the adequacy of the physical assumption (efficiency of the turbulent regime, artificial viscosity dependence, convective structure initialization) adopted to describe the coupling between dynamical and convective fields. The structure of the instability strip shows several striking features concerning the width in temperature of the region where only the first overtone is unstable. Indeed, the fundamental blue edge, moving from higher to lower luminosity levels, becomes redder, in contrast to previous findings but in agreement with globular clusters observed data. Moreover, using an improved treatment of the convective transport, the first-overtone red edge has been directly evaluated and hence also the width of the ''either-or'' region (i.e., the region where both the fundamental and the first overtone are simultaneously unstable). The width of this region is generally narrower than previously estimated and is mildly dependent on luminosity levels (decreasing at lower luminosities). Moreover, it has been found that the periods of the nonlinear convective models are systematically smaller than the corresponding periods of both linear and nonlinear radiative models. The differences between linear and nonlinear results are smaller than 2% of the period, however. This effect has been explained as a consequence of the changes induced during the phases of maximum compression by the convective transport on the adiabatic exponent, and on the density inversion located in coincidence with the hydrogen ionization region. The difference between the nonlinear periods of the present survey and the periods derived adopting the analytical formulae deduced by van Albada & Baker (1971) is positively correlated with the luminosity level, but it remains smaller than 3% of the period. The pulsation characteristics of the nonlinear models are discussed in detail, and the properties of the dynamical and convective structure both for models located close to the blue and for models close to the red edges are also described. Particular emphasis has been put on the evaluation of the convective timescales connected with the helium and hydrogen unstable regions along a period. Generally the convective transport in the helium ionization region becomes efficient only close to the phases of maximum compression, whereas the convective regime associated with the hydrogen ionization region is efficient for the entire period, but during the phases of maximum compression the velocity of the convective elements decreases consistently (alternate efficiency mechanism). The dependence of the amplitudes on luminosity and temperature and the morphology of the luminosity and velocity curves inside the instability strip obtained in this investigation are in agreement with observed values. The appearance of secondary features such as the Bump and the Dip along a full cycle and their dependence on luminosity and temperature have been analyzed, since it has been found that they are strongly correlated with the efficiency of the convective transport, and hence a detailed comparison with observations could in principle provide sound clues about both the adequacy of the physical approximations adopted in nonlinear pulsation codes and the choice of free parameters. A new method has been proposed to evaluate the average temperature over a full cycle, and the main differences from other methods are briefly discussed. Qualitative observational comparisons and possible implications are also reviewed.
