3. Initial and boundary conditions
There are two approaches to analysing the KH instability. One is to perturb a flow at a particular point in space and observe how the disturbance grows as it propagates downstream. The other is to perturb a flow everywhere and observe how the disturbance grows with time. The former is referred to as a spatial approach and has been adopted by, for example, Hardee & Norman (1988) and Hardee & Stone (1997). The latter is called the temporal approach and has been used by Bodo et al. (1994) and Rossi et al. (1997). Here we use the temporal approach.
In order to do this the initial conditions were set up as shown in Fig. 2. The slab jet occupies all of the grid within 1 jet radius of the `lower' edge. The boundary conditions at either end of the grid are set to periodic so that, effectively, we have a jet with infinite length. The boundary away from the jet axis is set to have gradient zero boundary conditions, while the jet axis boundary conditions are set to reflecting.
The grid cells are uniform in the direction parallel to the jet axis with a size of cm. Perpendicular to this direction the cell spacing obeys
where j is the cell index perpendicular to the jet axis and cm. The grid size was set to 400 200 cells, or about 48 jet radii. We `stretch' the cells perpendicular to the jet axis in order to avoid reflections from the upper boundary. The jet radius was set at 50 cells (i.e. cm) which is typical of YSO jets (e.g. Raga 1991).
The jet to ambient density ratio was set to 1 and the jet and ambient medium were set to be in pressure equilibrium also. Three different density regimes were used: 20 cm-3, 100 cm-3 and 300 cm-3. These densities, although perhaps rather low for YSO jets (e.g. Bacciotti et al. 1995), were selected to ensure that the cooling lengths behind shocks were resolved. The pressure was chosen so that the initial temperature on the grid was K. This temperature being set as the equilibrium temperature of the system (see Sect. 2.3).
The calculation was performed in the rest frame of the jet in order to reduce the effect of advection errors on the growth of the instability. The ambient medium was given an initial velocity of Mach 10 ( 112 km s-1). The jet material was given a small transverse velocity perturbation:
where is the number of perturbation wavelengths, c is the jet sound speed, R is the jet radius, and are approximately 0.21, 0.42, 0.52, 0.84, 1.05, 1.68, and 2.09. These values were chosen so that the grid length corresponded to an integral number of wavelengths of each perturbation. The sine term outside the summation gives the perturbation a profile across the jet radius such that it is zero both on the axis and at the edge of the jet.
It is well known (e.g. Bodo et al. 1994) that introducing a shear layer dampens the growth rates of all surface modes and body modes which have wavelengths shorter than the width of the layer. In simulations of adiabatic jets with high Mach numbers as the growth rates of the surface waves are usually much less than the growth rates of the body modes. However, in radiatively cooled jets analytic studies suggest that a surface mode exists which has growth rates higher than all other modes for a large range of wavelengths if the cooling function around is shallow enough (see Hardee & Stone 1997). Therefore it is important that we run simulations with different thicknesses for the shear layer in order to study the effect of this surface mode if it is present.
The jet itself is given a `top-hat' velocity profile modified by a term so that a shear layer exists initially between the jet and ambient medium. Two forms of this term have been tested. The first is identical to that of Rossi et al. (1997), i.e.
which gives a shear layer of approximately 20 cells. The second was chosen to be
This gives a shear layer of about 5 cells. This latter shear layer is wide enough to be well resolved by the code, and hence avoids numerical errors which lead to spurious waves being emitted from the jet boundary. The latter was chosen for the majority of simulations in the parameter space study.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998