Evolution of global properties of powerful radio sources. II. Hydrodynamical simulations in a declining density atmosphere and source energetics

We present two-dimensional numerical hydrodynamical simulations of light, supersonic jets propagating in atmospheres that decline in density with increasing distance in several ways: isothermal King law atmospheres with the power-law exponent beta = 1 and beta = 0.75 and in an isobaric King law atmosphere with power-law exponent beta = 1. We explore the same very broad range of parameter space in Mach number M and density contrast as in Paper I in this series. We compare our results with those for the constant density and pressure atmosphere simulations and with the predictions of the self-similar models (Paper I). We also discuss the global energetics of the sources in these different environments. Our comparison of the constant and declining density results shows the following. The overall morphology of the jet, cocoon, and bow shock is similar. However, there are some differences that start to appear when the jet has propagated about twice the core radius. In a declining density atmosphere, (1) there is less structure in the cocoon as a result of turbulence and (2) at a given source size, the cocoon seems to be underexpanded relative to a constant density atmosphere for a broad range of Mach numbers similar to5-30 (outside this range the cocoon may actually be wider). In an isobaric atmosphere with density gradient the general appearance of the source is similar to that of an isothermal atmosphere although the source size increases much faster. The overall distribution of pressure in the shocked ambient gas (SAG) region and cocoon and its decline with distance from the jet head are similar in both types of atmospheres. However, in jets in declining density atmospheres, for case 1 (pressure dominated by relativistic electrons), the cocoon and SAG region remain much more overpressured with respect to the ambient medium than in jets in constant density atmospheres. The lateral expansion speeds of the bow shock and cocoon are similar in the constant and declining density atmospheres. In a constant density atmosphere the jet head velocity decelerates slowly with time, while in the declining density atmosphere the jet head accelerates with time with a rate given by exponent z(h) proportional to t(m) with 1 < m < 1.5. However, at high Mach number and/or high jet density, the acceleration decreases with time and the head velocity approaches a nearly constant velocity, consistent with the predictions of the self-similar models. In an isothermal atmosphere with a density gradient, the volume of the region enclosed by the bow shock tends to increase with source size in agreement with the predictions of the type III models, i.e., V proportional to t(3). The estimated volume of the cocoon tends to vary with time possibly as a result of the influence of Kelvin-Helmholtz instabilities on the contact discontinuity, but in general V proportional to t(2) - t(3). We find that for all environments the behavior of the source volume as a function of time is closer to the self-similar predictions than the behavior of the source size as a function of time. This suggests that the shape of the bow shock adjusts to keep the time dependence of the source volume roughly self-similar. We examine the dependence of the pressure on the source volume as a function of the source age for the sources in different environments. In the constant density atmosphere, the pressure evolution of cylindrical slices of the SAG region departs from that expected for adiabatic expansion both near the head and near the jet inlet. Near the jet head the departure is due to energy input to the SAG region from the jet head (i.e., there is effective source term), and near the jet inlet the departure is due to the source coming into pressure balance with the ambient medium. The average global pressures in the SAG region and cocoon vary more slowly with volume than predicted by the self-similar models. These results for the evolution of the pressure are similar for the constant density or isothermal King atmosphere, although for the King atmosphere there tends to be better agreement with the predictions of the self-similar models. In general, the energetics of the source are similar whether the source is propagating in an isobaric or isothermal atmosphere, with some caveats. For both the SAG region and the cocoon, the pressure declines more slowly with source size in the isobaric atmosphere than in the isothermal atmosphere with a density gradient. In addition, the sources in an isobaric atmosphere show somewhat larger departure from the predictions of the self-similar models than do the sources in either a uniform atmosphere or an isothermal atmosphere with declining density gradient. The behavior of the source and the energetics are similar in the beta = 1 and beta = 0.75 King model atmospheres. However, we do note that some of the global parameters are more in agreement with the self-similar models for beta = 1. The self-similar models assume that the source is highly overpressured with respect to the ambient medium and thus do not explicitly consider a functional form for the ambient pressure. However, we have found that the agreement of the simulations with the predictions of the self-similar model does depend on the degree to which the source is overpressured and thus on the assumed ambient pressure pro le for the source. The agreement between the self-similar models and the simulations tends to improve as the atmosphere declines more steeply, i.e., from an atmosphere with beta = 0 to beta = 0.75 and beta = 1.