%$Id: notes.tex,v 1.102 2022/04/05 19:25:44 brandenb Exp $
%\documentclass{article}
\documentclass[twocolumn]{article}
\setlength{\textwidth}{160mm}
\setlength{\oddsidemargin}{-0mm}
\setlength{\textheight}{220mm}
\setlength{\topmargin}{-8mm}
\usepackage{graphicx,natbib,bm,url,color}
\graphicspath{{./figure1/}}
%\graphicspath{{./fig/}{./png/}}
\thispagestyle{empty}
\input macros
\def\red{\textcolor{red}}
\def\blue{\textcolor{blue}}
\title{}
\author{}
\date{\today,~ $ $Revision: 1.102 $ $}
\begin{document}
\maketitle

\newcommand{\jani}[1]{\textcolor{blue}{#1}}

\section{Decaying vortical and irrotational initial conditions}

%\begin{enumerate}\itemsep-0.2mm
%\item
%\end{enumerate}

\begin{equation}
\tilde{u}_i(\kk)=\left[(1-q)\delta_{ij}-(1-2q)\hat{k}_i\hat{k}_j\right]\tilde{S}_j(\kk).
\end{equation}
where $\tilde{S}_j(\kk)$ is given by Eq.~(24) of \cite{Bra+17}.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_vortexgen_p4096a3}
\end{center}\caption[]{
The black lines are for $q=1$ and the colored lines are for $q=0$.
The upper (black or red) lines are for $\orms$, while
the lower (black or blue) lines are for $(\nab\cdot\uu)_{\rm rms}$.
\url{pcomp_vortexgen_p4096a3}
}\label{pcomp_vortexgen_p4096a3}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{EEGW_vs_EEKM}
\end{center}\caption[]{
GW production
\url{EEGW_vs_EEKM}
}\label{pcomp_vortexgen_p4096a3}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_stress}
\end{center}\caption[]{
GW production with rescaled stress by factor $(t-1)^n$.
\url{pcomp_stress}
}\label{pcomp_stress}\end{figure}

For moderate Mach numbers, the ratio between $(\nab\cdot\uu)_{\rm rms}$
and $\omega_{\rm rms}$ remains nearly constant during the evolution.

\Fig{pcomp_vortexgen_p4096a3}

\begin{table}[b!]\caption{
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
                 & irro  &  vort  \\
\hline
$\urms$          & 0.184 & 0.184  \\
$\orms$          &  ---  \\
$\divu2m$        & $8.8\times10^{-6}$  \\
$\nu$            & 0.066  \\
$k_1$            & 0.00154  \\
$\kp$            & 0.007  \\
$\beta+1$        & 3  \\
$\beta-\alpha+1$ & $-3$ \\
\hline
\label{Ttimescale}\end{tabular}}\end{table}

In decaying turbulence, vortical and irrotational turbulent flows are
seen to have similar effects on GW production.
This is probably because the flow 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 $\nu$ smaller, or perhaps even
the sound speed smaller.

\begin{figure*}\begin{center}
\includegraphics[scale=0.4]{plotgw1}
\includegraphics[scale=0.4]{plotkin1}
\includegraphics[scale=0.4]{plotstr1}
\includegraphics[scale=0.4]{plotgw2}
\includegraphics[scale=0.4]{plotkin2}
\includegraphics[scale=0.4]{plotstr2}
\includegraphics[scale=0.4]{plotgw3}
\includegraphics[scale=0.4]{plotkin3}
\includegraphics[scale=0.4]{plotstr3}
\end{center}\caption[]{
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.
}\label{9panels}\end{figure*}

\Fig{9panels} 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 $\EK(k)$ do not
automatically imply oscillations in the stress.


\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{calc_ko_K512irrok6c_rep1_cont}
\end{center}\caption[]{
$k\omega$ diagrams of $\ln|\tilde{T}_+(\kk,\omega)|$ for the
irrotational run \url{calc_ko_K512irrok6c_rep1_cont}.
The white solid line indicates the diagonal, $\omega=ck$ with $c=1$,
the dashed-dotted line indicates the dispersion relation of sound waves,
$\omega=\cs k$ with $c=1/\sqrt{3}\approx0.58$.
}\label{calc_ko_K512irrok6c_rep1_cont}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{calc_ko_K512sig0_k6_ramp1a_cont}
\end{center}\caption[]{
$k\omega$ diagrams for a vortical run.
\url{calc_ko_K512sig0_k6_ramp1a_cont}
}\label{calc_ko_K512sig0_k6_ramp1a_cont}\end{figure}

\section{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 \Fig{pcomp_stress}, we plot the rms value of the {\em unprojected} stress,
$T_{\rm rms}$, as well as $T_{\rm rms}/t$ (which enters in the GW equation),
and $\EEGW$.
As an example, the red line shows that the stress declines like $t^{-1.2}$,
and therefore $T_{\rm rms}/t\propto t^{-1.2}$, and that then
$\EEGW\propto t^{+0.4}$, so it still increases.
There are other such examples.

As an example, the red line in \Fig{pcomp_stress}
shows that the stress declines like 
The red straight likes show the scalings $t^{-1.2}$,
$T_{\rm rms}/t\propto t^{-1.2}$, and $\EEGW\propto t^{+0.4}$.
The dashed red line is without phase information.

\section{$k\omega$ diagrams}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{calc_ko_K512irro_k6_ramp1c_rep}
\end{center}\caption[]{
$k\omega$ diagrams for irrotational run with blob-like forcing.
\url{calc_ko_K512irro_k6_ramp1c_rep}
}\label{calc_ko_K512irro_k6_ramp1c_rep}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{calc_ko_K512irro_k6_dtf0p05e}
\end{center}\caption[]{
$k\omega$ diagrams for irrotational run with blob-like forcing.
\url{calc_ko_K512irro_k6_dtf0p05e}
}\label{calc_ko_K512irro_k6_dtf0p05e}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pspec_comp_ko}
\end{center}\caption[]{
Comparison of spectra for vortical (red), irrotational plane-wave (blue),
and irrotational blob-like forcing (green) runs.
\url{pspec_comp_ko}
}\label{pspec_comp_ko}\end{figure}

As in \cite{Bra+17}, we consider vortical plane wave forcing with
$\kf=200\pi$ in a domain with $k_1=100$, irrotational plane wave forcing,
also with $\kf=200\pi$ in the same domain, and the bubble-like forcing,
again with $\kf=200\pi$ (thus using $R=2/200\pi\approx0.0032$), 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 \Figs{calc_ko_K512irrok6c_rep1_cont}
{calc_ko_K512irro_k6_ramp1c_rep}.
In \Fig{calc_ko_K512irro_k6_ramp1c_rep}, we see a marked trough
along the line $\omega=\cs k$ with $c=1/\sqrt{3}\approx0.58$.
This is actually also seen in \Fig{calc_ko_K512irrok6c_rep1_cont}.
For vortical turbulence, no such trough is seen; see
\Fig{calc_ko_K512sig0_k6_ramp1a_cont}.
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
$k\omega$ diagram.

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_ko}
\end{center}\caption[]{
Time series for vortical (red), irrotational plane-wave (blue),
and irrotational blob-like forcing (green) runs.
\url{pcomp_ko}
}\label{pcomp_ko}\end{figure}

\section{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 \cite{BN22} 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 $\tilde{\uu}(\kk)$ using a standard procedure with a given
spectrum, then compute $\tilde{\rho}(\kk)$ from it as follows.
We have
\begin{equation}
\frac{\DD}{\DD t}\ln\rho=-\nab\cdot\uu
\end{equation}
or, in Fourier space, we have
$-\ii\omega\,\widetilde{\ln\rho}=-\ii\kk\cdot\tilde{\uu}$, and
since $\omega=\cs k$ we obtain
\begin{equation}
\widetilde{\ln\rho}=\hat{\kk}\cdot\tilde{\uu}/\cs.
\end{equation}
This field could be outputted as an auxiliary field from the same routine.

\jani{Jani: I tried initializing the density using this equation but
when inverse transforming back to real space $\ln \rho$ is purely
imaginary. Like we talked in a meeting at the start of November, the
obtained real space array might need to be multiplied by a factor of $2
\pi i$ to take into account also the inverse transform in time. However,
taking only the imaginary part or multiplying by such a factor either
leads to no reduction in the oscillations or ends up blowing up the
density at early times.}

\section{Acoustic waves}
Question:
\begin{itemize}
    \item What determines the height of the linear part in GW spectrum?
    \item why the spectrum is broader around the peak in numerical simulation compared to the analytical one?
    \item Is the $k^9$ part in the GW spectrum also present for large Mach number?
    \end{itemize}
    Results:
    \begin{itemize}
    \item The spectrum after including the expansion is different than the one without expansion (may not have sharp drop feature around the peak). Another point, the spectrum gets saturated quickly.
    \item Oscillations at large $k$ are an artifact of large time step.
    \item In steady state, $k \dot{h} \sim k^2 h \sim T$. That implies if the spectrum of $T$ is white noise, the spectrumof $\dot{h} \sim k^0$
\end{itemize}
\begin{figure*}\begin{center}
\includegraphics[scale=0.8]{fig/spectrum_wo_exp.eps}
\end{center}\caption[]{GW spectrum without expansion at different times.
The amplitude of spectrum in between $k=1$ to $10$ is unchanged. However it continues increasing at $k=30$ with $\propto t$. The transition to $k^3$ propagate with $\propto t$ in the horizontal direction. }\label{9panels}\end{figure*}
\begin{figure*}\begin{center}
\includegraphics[scale=1]{fig/comparison.eps}
\end{center}\caption[]{Left: kinetic spectra, Right: GW spectra. Red curves shows the analytical spectra and blue curves represents numerical spectra. In simulations, we get more broader peak. Oscillations at large $k$ are an artifact of large time step.
}\label{9panels}\end{figure*}

\begin{figure*}\begin{center}
\includegraphics[scale=0.6]{fig/w_wo_exp_same_time.eps}
\includegraphics[scale=0.6]{fig/w_wo_exp_final_time.eps}
\end{center}\caption[]{Comparison between the GW spectra with and without expansion. Red: without expansion, Black dotted: with expansion. Legend represents the time of the spectra.
}\label{9panels}\end{figure*}

\begin{figure*}\begin{center}
\includegraphics[scale=0.4]{fig/SpecAll.png}
\end{center}\caption[]{Numerical integration with expansion.
}\label{9panels}\end{figure*}

\begin{figure*}\begin{center}
\includegraphics[scale=0.4]{fig/SpecAllNoExp.png}
\end{center}\caption[]{Numerical integration without expansion.
}\label{9panels}\end{figure*}

\bibliographystyle{unsrt}
\bibliography{ref}
\end{document}
