June 26, 2023, Revision: 1.102

1 Decaying vortical and irrotational initial conditions

ũ_{i}(k) = [(1 − q)δ_{ij} − (1 − 2q)k̂_{i}k̂_{j}] S̃_{j}(k). (1)

where S̃_{j}(k) is given by Eq. (24) of ?.

Figure 1: The black lines are for q = 1 and the colored lines are for q = 0. The upper (black or red) lines are for ω_{rms}, while the lower (black or blue)

lines are for (∇·u)_{rms}. pcomp_vortexgen_p4096a3

For moderate Mach numbers, the ratio between (∇ · u)_{rms} and ω_{rms} remains nearly constant during the evolution.

figure 2

In decaying turbulence, vortical and irrotational turbulent flows are seen to have similar effects on GW production. This is probably because the flow

Figure 2: GW production EEGW_vs_EEKM

Figure 3: GW production with rescaled stress by factor (t − 1)^{n}. pcomp_stress

is rapidly decaying and there is no vigorous time dependence anymore. One could artificially upscale the turbulence by some factor. One could also make the viscosity ν smaller, or perhaps even the sound speed smaller.

Figure 4: Kinetic, stress and mean GW spectra. The black and red curves in kinetic and stress spectra show the spectra at initial and final time of simulation, respectively. Note the presence of the bump in all irrotational runs. It is absent in all vortical runs. It could be caused by the bottleneck effect.

Figure 5: kω diagrams of ln |T̃_{+}(k, ω)| for the ir-

rotational run calc_ko_K512irrok6c_rep1_cont.

The white solid line indicates the diagonal, ω = ck with c = 1, the dashed-dotted line indicates the dispersion relation of sound waves, ω = c_{s}k with c = 1/\sqrt{3} ≈ 0.58.

figure 4 shows that in the irrotational case, the spectrum shows a sharp rise just before the maximum. This could be related to the oscillation in the stress, but this is not clear. It is also strange that the oscillations in E_{K}(k) do not automatically imply oscillations in the stress.

Table 1:

irro  vort
u_{rms} 0.184 0.184
ω_{rms}  —
(∇ · u)_{rms} 8.8 × 10^{−6}
ν  0.066
k_{1}  0.00154
k_{p}  0.007
β + 1  3
β − α + 1  −3

Figure 6: kω diagrams for a vortical run. calc_ko_

K512sig0_k6_ramp1a_cont

2 Rescaled stress

The goal is to assess the importane of GW production from the turbulence relative to the kick produced by switching on the GW solver and from the increase of the mean stress. With a rescaled stress, we have the possibility of putting more emphasis on the subsequent turbulence.

In figure 3, we plot the rms value of the unprojected stress, T_{rms}, as well as T_{rms}/t (which enters in the GW equation), and E_{GW}. As an example, the red line shows that the stress declines like t^{−1.2}, and therefore T_{rms}/t ∝ t^{−1.2}, and that then E_{GW} ∝ t^{+0.4}, so it still increases. There are other such examples.

As an example, the red line in figure 3 shows that the stress declines like The red straight likes show the scalings t^{−1.2}, T_{rms}/t ∝ t^{−1.2}, and E_{GW} ∝ t^{+0.4}. The dashed red line is without phase information.

3 kω diagrams

As in ?, we consider vortical plane wave forcing with k_{f} = 200π in a domain with k_{1} = 100, irrotational plane wave forcing, also with k_{f} = 200π in the same domain, and the bubble-like forcing, again Figure 7: kω diagrams for irrotational run with

blob-like forcing. calc_ko_K512irro_k6_ramp1c_
rep

Figure 8: kω diagrams for irrotational run with blob-like forcing. calc_ko_K512irro_k6_

dtf0p05e

Figure 9: Comparison of spectra for vortical (red), irrotational plane-wave (blue), and irrotational blob-like forcing (green) runs. pspec_comp_ko

with k_{f} = 200π (thus using R = 2/200π ≈ 0.0032), which differs from the plane wave forcing in that it is causal. Plane wave forcing, by contrast, violates causality, because it produces a forcing function that has an infinite correlation length.

The results are shown in figures 5 and 7. In figure 7, we see a marked trough along the line ω = c_{s}k with c = 1/\sqrt{3} ≈ 0.58. This is actually also seen in figure 5. For vortical turbulence, no such trough is seen; see figure 6. The emergence of a trough is unexpected, because it means that the power is reduced, but for resonant frequencies, the power is again large. On the other hand, for k = 100 and 200, there is actually indeed enhanced power, but it comes from the lower patch in the kω diagram.

4 Wave propagating in energy spectra

When starting a simulation with an initial condition that is not a statistically relaxed solution of the dynamical equations, we see the emergence of travelling waves in the energy spectrum. A quantitative description was given in ? for self-gravitating turbulence. To emulate a relaxed state from an artificial initial condition that is not a solution of the dynamical equations, we could try the following: generate ũ(k) using a standard procedure with a given spectrum, then compute ρ̃(k) from it as fol-