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\title[Generation of Gravitational Waves]
{\bf Gravitational Wave Spectra from Primordial 
Magnetohydrodynamic Turbulence}
\subtitle{\bf Nordita Astrophysics Seminar}
\author[Alberto Roper Pol]{
Alberto Roper Pol \\
\textit{Research Advisor: }Axel Brandenburg
}
\institute[University of Colorado] 
% Your institution as it will appear on the bottom of every slide, may be shorthand to save space
{
University of Colorado at Boulder \\
\medskip
Laboratory for Atmospheric and Space Physics (LASP)
}
\date{ August 22, 2018}


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\frametitle{Overview} % Table of contents slide, comment this block out to remove it
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\section{Introduction and Motivation} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
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\begin{frame}
\frametitle{Introduction and Motivation}
\begin{itemize}
\item \visible<1-> { Generation of cosmological 
gravitational waves (GWs) during the electroweak 
phase transition (EWPT)
\item Source of information from earlier times than 
recombination epoch (Cosmic Microwave Background)}
\item \visible<2-> { MHD sources of GWs: 
	\begin{itemize}
	\item Kinetic turbulence from phase transition bubbles nucleation
	\item Magnetic turbulence from primordial magnetic fields
	\end{itemize} }
\item \visible<3-> {Possibility of GWs detection with:
	\begin{itemize}
	\item Space mission LISA
	\item Polarization of CMB
	\item Pulsar Timing Arrays (PTA)
	\end{itemize}	}
\item \visible<5-> {Numerical simulations using {\sc Pencil Code} to solve:
	\begin{itemize}
	\item Relativistic MHD equations
	\item Gravitational waves equation
	\end{itemize}	}
\end{itemize}
\end{frame}

\section{Gravitational waves}

\begin{frame}
\frametitle{Gravitational waves equation}
\begin{block}{\bf GWs equation for an expanding 
flat Universe}
\begin{itemize}
\item Assumptions: isotropic and homogeneous Universe
\item Friedmann--Lema\^itre--Robertson--Walker (FLRW) 
metric $\eta_{ij}$
\item Tensor-mode perturbations above the FLRW model:
$$g_{ij} = \eta_{ij} + 
a^2 h_{ij}^{\rm phys}$$
\item GWs equation is\footnotemark
\vspace{-4mm}
$$ \left(\partial_t^2  - c^2 \nabla^2 \right) 
h_{ij} = \frac{16 \pi G}{a c^2} T_{ij}^{\rm TT} $$
\vspace{-4mm}
\begin{itemize}
\item $h_{ij}$ are rescaled $h_{ij} = a 
h_{ij}^{\rm phys}$
\item Comoving spatial coordinates $\nabla$ and 
conformal time $t$
\item Comoving stress-energy tensor components 
$T_{ij}$
\item Radiation-dominated epoch such that 
$a'' = 0$ 
\end{itemize}
\end{itemize}
\end{block}
\footnotetext{L. P. Grishchuk, {\em 
Sov. Phys. JETP}, 40, 409-415 (1974)}
\end{frame}

\begin{frame}
\frametitle{Gravitational waves equation}
\begin{block}{Properties}
\begin{itemize}
\item Tensor-mode perturbations are gauge invariant
\item $h_{ij}$ has only two degrees of freedom: $h^+$, $h^{\times}$
\item The metric tensor is traceless and transverse (TT gauge)
\end{itemize}
\end{block}
\begin{block}{Contributions to the stress-energy tensor}
\begin{columns}[c]
\column{.5\textwidth}
\begin{itemize}
\item From fluid motions \\
$ T_{ij} = \left( p/c^2 + \rho \right) \gamma^2 u_i u_j + p \delta_{ij} $
\end{itemize}
\column{.5\textwidth}
\begin{itemize}
\item From magnetic fields: \\
$ T_{ij} = - B_i B_j + \delta_{ij} B^2/2 $
\end{itemize}
\end{columns}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Traceless and transverse projection}
\begin{itemize}
\item For computational reasons, we solve GWs equation 
sourced by the stress-energy tensor\footnote{A. 
Roper Pol {\em et al.}, arXiv:1807.05479 (2018)}
\red{$$ \left( \partial_t^2  - c^2 \nabla^2 \right) 
h_{ij} = \frac{16 \pi G}{a c^2} T_{ij} $$}
\item Projection of $h_{ij}^{\rm TT}$ only at 
specific times, not at every time-step
\item Projection requires Fourier transform
$\tilde{h}_{ij}$ (computationally expensive)
\item TT projection:
\red{
$$ \tilde{h}^{\rm TT}_{ij} = \left( P_{il} P_{jm} -
\frac{1}{2} P_{ij} P_{lm} \right) \tilde{h}_{lm} $$}
where $P_{ij} = \delta_{ij} - \hat{k}_i \hat{k}_j$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Gravitational waves energy density}
{\large \bf GWs energy density:}
\small
$$ \mathcal{E}_{\text{GW}}(t) = 
\int_0^\infty E_{\rm GW} (k, t) {\rm \, d} k = 
\frac{c^2}{16 \pi G} \int_0^\infty \int_{4\pi} 
\left( \left| \tilde{\dot{h}}^+ \right|^2 + 
\left|\tilde{\dot{h}}^\times \right|^2 \right)
k^2 {\rm \, d} \Omega_k {\rm \, d} k$$
{\large \bf Antisymmetric GWs energy density:}
\small
$$ \mathcal{H}_{\text{GW}} (t) = \int_0^\infty 
H_{\rm GW} (k, t) {\rm \, d} k = \frac{c^2}{16 \pi G} 
\int_0^\infty \int_{4 \pi} 2 {\rm Im} \left(
\tilde{\dot{h}}^+ \tilde{\dot{h}}^\times \right) 
k^2 {\rm \, d} \Omega_k {\rm \, d} k$$
{\large \bf Polarization degree: }
$$ \red{\mathcal{P}(k) = \frac{H_{\rm GW}(k)}
{E_{\rm GW}(k)}} \in [-1, 1]$$
\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
\begin{block}{Initial conditions}
\begin{itemize}
\item Fully helical stochastic magnetic field
\item Batchelor spectrum, i.e., $E_{\rm M}(t_\ast)
\propto k^4$ for small $k$
\item Total energy density at $t_\ast$ is 
$\sim $ 10\% of the radiation energy 
density\footnotemark
\item Different positions of the spectral peak 
$k_{\rm M}$, normalized with $k_H = 1/(cH)$ 
\end{itemize}
\end{block}
\begin{block}{Numerical parameters}
\begin{itemize}
\item 1152$^3$ mesh gridpoints
\item 1152 processors
\item Duration of runs is $\sim$\ 1 -- 5 days
\end{itemize}
\end{block}
\footnotetext{limit from the Big Bang Nucleosynthesis 
bound}
\end{frame}

% UNCOMMENT FOR GWs RESULTS
\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
{\bf Negative initial helicity} \\
\begin{center}
\includegraphics[width = \textwidth]{GW_results/M1152e3} 
\vspace{-4mm}
\tiny $$ k_{\text{M}}/k_H = 6 $$
\end{center}
\blue{Blue dots indicate negative helicity spectrum 
(magnetic fields) and negative antisymmetric 
spectrum (GW)}
\end{frame}
\begin{frame}
\frametitle{Numerical results for decaying
MHD turbulence}
{\bf Positive initial helicity} \\
\begin{center}
\includegraphics[width = 0.33\textwidth]{GW_results/M1152c3} 
\includegraphics[width = 0.33\textwidth]{GW_results/M1152g3} 
\vspace{-4mm}
\begin{columns}[t]
\column{.33\textwidth}
\hspace{-5mm}
\tiny $$ k_{\text{M}}/k_H = 300$$
\column{.33\textwidth}
\hspace{5mm}
\tiny $$ k_{\text{M}}/k_H = 60$$
\end{columns}
\vspace{2mm}
\includegraphics[width = 0.33\textwidth]{GW_results/M1152f3} 
\tiny $$ k_{\text{M}}/k_H = 2$$
\end{center}
\vspace{3mm}
\red{Red dots indicate positive helicity spectrum 
(magnetic field) and positive antisymmetric 
spectrum (GW)}
\end{frame}

\begin{frame}
\frametitle{Numerical accuracy\footnote{A. Roper Pol
{\em et al.}, arXiv:1807.05479 (2018)}}
\begin{columns}[t]
\column{.6\textwidth}
\includegraphics[width=1.05\textwidth]{accuracy}
\column{.4\textwidth}
\vspace{-60mm}
\begin{itemize}
\item CFL condition is not enough for GW solution 
to be numerically accurate
\item $c \delta t/\delta x \sim 0.05 \ll 1$
\item Higher resolution is required
\item Hydromagnetic turbulence does not seemed to 
be affected
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
Box results for positive initial helicity:
\begin{center}
\includegraphics[width = 0.32\textwidth]{GW_results/M1152c3_hhT} 
\includegraphics[width = 0.32\textwidth]{GW_results/M1152g3_hhT} 
\includegraphics[width = 0.32\textwidth]{GW_results/M1152f3_hhT} \\
\includegraphics[width = 0.32\textwidth]{GW_results/M1152c3_hhX}
\includegraphics[width = 0.32\textwidth]{GW_results/M1152g3_hhX}
\includegraphics[width = 0.32\textwidth]{GW_results/M1152f3_hhX}
\vspace{-4mm}
\end{center}
\begin{columns}[c]
\column{.32\textwidth} \hspace{3mm} \tiny $k_{\text{M}}/k_H = 300$
\column{.32\textwidth} \hspace{3mm} \tiny $k_{\text{M}}/k_H = 60$
\column{.32\textwidth} \hspace{3mm} \tiny $k_{\text{M}}/k_H = 2$
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Comments}
\begin{block}{Polarization degree}
\begin{itemize}
\item Helical magnetic fields induce circularly 
polarized GWs
\item Helicity and GWs polarization have same sign
\item Agreement with analytical 
prediction\footnotemark
\item Illustrated with a 1D Beltrami field example
\end{itemize}
\end{block}
\begin{block}{Box simulations}
\begin{itemize}
\item For smaller $k_{\rm M}$, $h^+$ and 
$h^\times$ field structures become smoother
\item $h^+$ and $h^\times$ appear shifted, which 
indicates presence of helicity
\end{itemize}
\end{block}
\footnotetext{
T. Kahniashvili {\em et al., Phys. Rev. Lett.,} 95
(15):151301 (2005)}
\end{frame}

\begin{frame}
\frametitle{1D Beltrami field}
\begin{itemize} 
\item Magnetic field: $\pmb{B}(x) = B_0 \left(0,  \sin k_0 x, \cos k_0 x \right)$,
such that $\pmb{B} \cdot \left( \pmb{\nabla} \times \pmb{B} \right) = \text{sgn} \left(k_0\right)$
\item Stress-energy tensor: 
$ T_{ij}  = \frac{1}{2}B_0^2 \left( \begin{array}{c c c} 
0 & 0 & 0 \\ 0 & - \cos 2k_0 x & \sin 2k_0 x \\ 0 & \sin 2k_0 x & \cos 2k_0 x
\end{array} \right) $ \\
\vspace{2mm}
which is already TT with two independent modes:
$$ T^+ = -\frac{1}{2} B_0^2 \cos 2k_0 x, \quad T^{\times} = -\frac{1}{2} B_0^2 \sin 2k_0 x $$
\item Tensor-mode perturbations: 
{\small
$$ h^+(x, t) = -\frac{2 \pi G}{ c^4 k_0^2} B_0^2 \cos 2k_0x (1 - \cos 2k_0c t) $$
$$h^{\times}(x, t) = - \frac{2 \pi G}{ c^4 k_0^2} B_0^2 \sin 2k_0x (1 - \cos 2k_0c t) $$ } 
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{1D Beltrami field}
\begin{itemize}
\item GWs energy density:
$$ E_{\rm GW} (k, t) = \frac{\pi G}{c^4 k_0^2}
\mathcal{E}_{\text{M}}^2 \left[ 
\delta (k - 2k_0) + \delta(k + 2k_0) \right] $$ 
$$ H_{\rm GW} (k, t) = \frac{\pi G}{c^4 k_0^2}
\mathcal{E}_{\text{M}}^2 \left[ 
\delta (k - 2k_0) - \delta(k + 2k_0) \right] $$ 
\item Polarization degree:
$$\mathcal{P}(2k_0) = {\rm sgn} \left( k_0 \right) $$
\item We see that polarization degree is $\pm 1$, 
with the same sign as the sign of the magnetic 
helicity
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Numerical results for hydrodynamic 
forced turbulence}
\begin{block}{Physical conditions}
\begin{itemize}
\item Forced hydrodynamic term from phase transition 
bubble nucleation
\item Two cases considered:
\begin{itemize}
\item Rotational turbulence, e.g., fully helical 
kinetic velocity flow
\item Acoustic turbulence: kinetic flow is defined 
from a potential function and hence, ${\bf \nabla} 
\times {\bf u} = {\bf 0}$
\end{itemize}
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Numerical results for hydrodynamic 
forced turbulence}
{\bf Rotational turbulence}  \\
\begin{center}
\includegraphics[width = 0.6\textwidth]{GW_results/RT1152b} \\
\end{center}
{\bf Acoustic turbulence}  \\
\begin{center}
\includegraphics[width = 0.6\textwidth]{GW_results/AT1152c}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Numerical results for hydrodynamic forced turbulence}
\begin{block}{Conclusions}
\begin{itemize}
\item Much stronger signal found for acoustic 
turbulence than for rotational turbulence
\item Result in agreement with 
literature\footnotemark
\item In both cases, GWs polarization is zero
\end{itemize}
\end{block}
\footnotetext{M. Hindmarsh {\em et al., Phys. Rev. D,} 
92(12):123009 (2015)}
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
\begin{columns}[c]
\column{0.5\textwidth}
\begin{block}{LISA}
\begin{itemize}
\item Laser Interferometer Space Antenna (LISA) 
is a space based GWs detector
\item LISA is planned for 2034
\item LISA was approved in 2017 as one of the main 
research missions of ESA
\item LISA is composed by three spacecrafts in a 
distance of $\sim $2M km
\end{itemize}
\end{block}
\column{0.5\textwidth}
\begin{figure}[H]
\includegraphics[width=\textwidth]{LISA-waves}
\caption{Artist's impression of LISA from Wikipedia}
\end{figure}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
{\footnotesize
\begin{itemize}
\item LISA sensitivity is usually expressed as 
$h_0^2 \Omega_{\text{GW}}$
\item $\Omega_{\rm GW}$ is the ratio of GWs energy 
density to critical energy density
\item Critical energy density is
$$ \mathcal{E}_{\rm crit} = \frac{3 H_0^2 c^2 }
{ 8 \pi G} $$
\item Current Hubble parameter is usually expressed as
$$ H_0 = 100 \, h_0 \  {\rm km \, s}^{-1} 
{\rm Mpc}^{-1} $$
where $h_0$ represents the uncertainties in the 
actual value of $H_0$
\item We consider two different LISA configurations 
\footnote{C. Caprini {\rm et al., JCAP,} 2016(04):
001001 (2016)}
\begin{itemize}
\item 4-link configuration with $2 \times 10^9$ m 
arm length after 5 years of duration
\item 6-link configuration with $6 \times 10^9$ m 
arm length after 5 years of duration
\end{itemize}
\end{itemize}}
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
{\bf Decaying MHD turbulence with initial + 
fully helical magnetic field: }
\begin{center}
\includegraphics[width = 0.32\textwidth]
{GW_results/M1152c3_LISA} 
\includegraphics[width = 0.32\textwidth]
{GW_results/M1152g3_LISA} 
\includegraphics[width = 0.32\textwidth]
{GW_results/M1152f3_LISA} 
\end{center}
\vspace{-8mm}
\begin{columns}[t]
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 300$$
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 60$$
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 2$$
\end{columns}
\vspace{3mm}
{\bf Forcing hydrodynamic rotational and acoustic turbulence: }
\vspace{-2mm}
\begin{center}
\includegraphics[width = 0.49\textwidth]{GW_results/RT1152b_LISA} 
\includegraphics[width = 0.49\textwidth]{GW_results/AT1152c_LISA} 
\end{center}
\vspace{-6mm}
\begin{columns}[t]
\column{.5\textwidth}
\centering \tiny Rotational turbulence
\column{.5\textwidth}
\centering \tiny Acoustic turbulence
\end{columns}
\end{frame}


\begin{frame}
\begin{center}
\Huge{The End \\ Thank You! \\}
\includegraphics[width=0.5\textwidth]{question}
\end{center}
\end{frame}
%
%%------------------------------------------------
%
\section*{Evidence of primordial magnetic fields}

\begin{frame}
\frametitle{Evidence of primordial magnetic fields}

There are different evidence that indicate the presence of a large scale 
coherent magnetic field

\begin{block}{Fermi blazar observations\footnotemark}
\begin{itemize}
\item Gamma rays from blazars interact with 
extragalactic background light
\item Generation of electron - positron beam
\item Observed power removal from gamma-ray beam
\end{itemize}
\end{block}
\footnotetext{A. M. Taylor, I. Vovk, and A. Neronov, 
{\em Astronomy \& Astrophysics,} 529:A144 (2011)}
\end{frame}

\begin{frame}
\frametitle{Evidence of primordial magnetic fields}
\visible<1-> {
\begin{block}{Solution}
\begin{itemize}
\item Large scale (intergalactic) magnetic fields could deviate the electron-positron from beam in opposite directions
\item Recombination does not happen leading to lose of energy
\item Strength $\sim 10^{-16}$ G, scale $\sim 100 $ kpc
\end{itemize}
\end{block}
}
\visible<2> {
\begin{block}{Origin}
Intergalactic magnetic fields could have been originated from: 
\begin{itemize}
\item Astrophysical or
\item \alert{Cosmological } seed fields
\end{itemize}
subsequently amplified during structure formation
\end{block}
}
\end{frame}

\begin{frame}
\frametitle{Evidence of primordial magnetic fields}
\begin{block}{Helicity}
\begin{itemize}
\item Magnetic helicity is observed in present astrophysical objects
\item Fractional magnetic helicity is required in cosmological seed fields
\item Primordial helical magnetic fields require a
cosmological first order phase transition:
	\begin{itemize}
	\item \alert{Electroweak phase transition 
	(EWPT),} $t \sim 10^{-12}$ s, $\sim 250$ GeV
	\item Quantum chromodynamics (QCD) phase 
	transtion, $t \sim 10^{-6}$ s
	\end{itemize}
\end{itemize}
\end{block}
\begin{definition}[Magnetic Helicity]
$ \mathcal{H} = \left< \pmb{B} \cdot \left( \pmb{\nabla} \times \right)^{-1} \pmb{B} \right>  = \left< \pmb{A} \cdot \pmb{B} \right> $
\end{definition}
\end{frame}

%------------------------------------------------
\section*{Magnetohydrodynamic turbulence}
%------------------------------------------------

\begin{frame}
\frametitle{EWPT plasma}
\begin{block}{Characterization}
\begin{itemize}
\item Quark-gluon plasma 
\item  Charge-neutral, electrically conducting fluid 
\item  Relativistic magnetohydrodynamic (MHD) 
equations
\item  Ultrarelativistic equation of state
\end{itemize}
\end{block}
\begin{block}{Ultrarelativistic equation of state}
 $ p = \rho c_s^2, \quad c_s^2 = c^2/3$
\end{block}
\end{frame}


\begin{frame}
\frametitle{MHD equations}
Relativistic MHD equations are reduced 
to\footnote{A. Brandenburg, K. Enqvist, and P. 
Olesen, {\em Phys. Rev. D}, 
54(2):1291–1300, 1996.}
\begin{block}{Continuity equation}
$\frac{\partial \ln \rho}{\partial t} = - \frac{4}{3} \left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right) + \frac{1}{\rho} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta \pmb{J}^2 \right) $
\end{block}
\begin{block}{Momentum equation}
\small $ \frac{D \pmb{u}}{Dt} = \frac{4}{3} \left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right) - \frac{\pmb{u}}{p} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta J^2 \right] - \frac{1}{4} \pmb{\nabla} \ln \rho + \frac{3}{4 \rho} \pmb{J} \times \pmb{B} + \frac{2}{\rho} \pmb{\nabla} \cdot \left( \rho \nu \pmb{S} \right) $
\end{block}
\begin{block}{Induction equation}
$ \frac{\partial \pmb{B}}{\partial t} = \pmb{\nabla} \times \left( \pmb{u} \times \pmb{B} - \eta \pmb{J} \right) $ 
\end{block}
for a flat expanding universe with comoving magnitudes
\end{frame}

\begin{frame}
\frametitle{MHD energy density}
\small
\begin{columns}[t]
\column{.5\textwidth}
\blue{\textbf{Kinetic energy density: }
$$ \mathcal{E}_{\text{K}}(t) = \frac{1}{2} 
\left< \rho \pmb{u}^2 \right> = 
\int E_{\text{K}}(k, t) {\rm \, d}k $$ \\
\textbf{Kinetic helicity: }
$$ \mathcal{H}_{\text{K}}(t) = \rho 
\left< \pmb{\omega} \cdot \pmb{u} \right> 
= \int H_{\text{K}} (k, t) {\rm \, d}k $$ \\
\textbf{Kinetic correlation length: }
$$ \xi_{\text{K}}(t) =  \frac{\int {\rm \, d}k \ 
k^{-1} E_{\text{K}}}{\mathcal{E}_{\text{K}}} $$}

\column{.5\textwidth}
\red{\textbf{Magnetic energy density: }
$$ \mathcal{E}_{\text{M}}(t) = \frac{1}{2 } \left< 
\pmb{B}^2 \right> = \int E_{\text{M}}(k, t) 
{\rm \, d}k $$
\textbf{Magnetic helicity: }
$$ \mathcal{H}_{\text{M}}(t) = \left< \pmb{B} \cdot 
\pmb{A} \right> = \int H_{\text{M}}(k, t) 
{\rm \, d} k $$
\textbf{Magnetic correlation length: }
$$ \xi_{\text{M}}(t) =  \frac{\int {\rm \, d} 
k \ k^{-1} E_{\text{M}}}{\mathcal{E}_{\text{M}}} $$}
\end{columns}
\vspace{2mm}
\textbf{Realizability condition: }
$$ \frac{1}{2} k \left| H_{\rm M} \right|  
\leq E_{\rm M} \Rightarrow \green{
\xi_{\rm M} \geq \frac{{\cal H}_{\rm M}}{2 
{\cal E}_{\rm M}}}$$
\let\thefootnote\relax\footnotetext{
Lorentz-Heaviside units are used}
\normalsize
\end{frame}

\begin{frame}
\frametitle{Initial conditions}
\begin{block}{Fractional helicity 
$\sigma_{\rm M}$ and peak frequency $k_{\rm M}$}
\begin{columns}
\small
\column{.5\textwidth} 
$$ \varepsilon_{\text{M}} = \frac{k H_{\text{M}}
(k, t)}{2 E_{\text{M}}(k, t)} = 
\frac{2 \sigma_{\text{M}}}
{1 + \sigma_{\text{M}}^2}$$
\column{.5\textwidth}
\includegraphics[width=0.6\textwidth]{scheme}
\end{columns}
\normalsize
\end{block}
\begin{block}{Ratio of energy densities}
\begin{center}
$Q_* = B_*\left/\left(\sqrt{\rho_*} u_*
\right)\right., \quad $ where $t_\ast$ 
indicates the time of generation
\end{center}
Three cases:
\begin{itemize}
\item $Q_* \gg 1$:  Magnetically dominant 
turbulence $\left( E_{\rm M} \propto 
k^{\alpha_{\rm M}} \right.$, $\left. 
\alpha_{\rm M} \geq 4 \right)$ 
\item $Q_* \sim 1$: Equipartition case
\item $Q_* \ll 1$: Magnetically subdominant or 
kinetically dominant turbulence
$\left( E_{\text{K}} \propto k^{\alpha_{\text{K}}} 
\right.$, $\left. \alpha_{\text{K}} \geq 2 \right)$ 
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Spectral parameters}
\begin{block}{Parameters $\alpha_i$, $\beta_i$, $q_i$, $p_i$}
$$E_i(k, t) \propto k^{\alpha_i}$$
$$E_i(k, t) = \xi_i^{-\beta_i} \phi_i \left( k \xi_i \right)$$
$$\xi_i (t) \propto t^{q_i}$$
$$\mathcal{E}_i(t) \propto t^{-p_i}$$
\end{block}
\begin{block}{$pq$ diagram}
Since ${\cal E}_i = \int E_i {\rm \, d} k$, we have 
the relation
$$p_i = (1 + \beta_i) q_i$$
Straight lines for constant $\beta$
\end{block}
\end{frame}

\begin{frame}
\begin{block}{Inverse cascade condition}
$$ s_i = \frac{\partial \ln E_i}{\partial \ln t} = 
\left( \alpha_i - \beta_i \right) q_i > 0 $$
\end{block}
\begin{center}
\includegraphics[width=0.6\textwidth]{inverse_cascade}
\end{center}
\end{frame}
%
%% UNCOMMENT TO SHOW RESULTS
\begin{frame}
\frametitle{Numerical simulations}
{\large \textbf{Case 1.} Initial non-helical 
($\sigma_{\text{M}} = 0$) Batcehlor spectrum 
($\alpha_i = 4$) \\}
\centering
{\tiny $ Q_* = 10$ } 
\includegraphics[width=.8\textwidth]{MHD_results/A} \\
{\tiny $ Q_* = 0.1$ } 
\includegraphics[width=.8\textwidth]{MHD_results/C}
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
{\large \textbf{Case 2.} Initial helical 
($\sigma_{\text{M}} = 0.3$) white noise spectrum 
($\alpha_i = 2$) \\}
\begin{center}
{\tiny $ Q_* \rightarrow \infty$ } \includegraphics[width=1\textwidth]{MHD_results/G} \\
\end{center}
{\large \bf Realizabilty condition:}
$$\frac{1}{2} k |H_{\rm M}| \leq E_{\rm M} \Rightarrow
\green{\xi_{\rm M} \geq \frac{{\cal H}_{\rm M}}
{2 {\cal E}_{\rm M}} = \xi_{\rm M}^{\rm min}}$$
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
{\normalsize \textbf{Case 3.} Initial fully helical 
($\sigma_{\text{K}} = 1$) Batchelor spectrum 
($\alpha_{\text{K}} = 4$) \\}
\vspace{2mm}
\centering
{\tiny $\sigma_{\rm M} = 0$}
\includegraphics[width=.9\textwidth]{MHD_results/I} \\
{\tiny $\sigma_{\rm M} = -1$}
\includegraphics[width=.9\textwidth]{MHD_results/J}
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
\begin{block}{Conclusions}
\begin{itemize}
\item Inverse cascade can be predicted with the 
condition $s_i = \left( \alpha_i - \beta_i \right) 
q_i > 0$
\item With initial helicity, conservation of total 
helicity dominates ($\beta = 0$)
\item With no initial helicity, $pq$ diagram tends 
to $\beta = 2$ and conservation of Saffman integral 
dominates
\item With kinetically dominant turbulence, 
$pq$ diagram tends to $\beta = 4$ and conservation
of Loitsiansky integral dominates
\end{itemize}
\end{block}
\begin{itemize}
\tiny
\item Total helicity 
$$ \left< \pmb{\omega} \cdot \pmb{u} \right> + 
\left< \pmb{J} \cdot \pmb{B} \right> $$
\item Saffman integral
$$ \mathcal{S} = \int \left< \pmb{u}(\pmb{x}, t) \cdot 
\pmb{u}(\pmb{x} + \pmb{r}, t) \right> d\pmb{r} $$
\item Loitsianski integral
$$ \mathcal{L} = \int r^2 \left< \pmb{u}(\pmb{x}, t) 
\cdot \pmb{u}(\pmb{x} + \pmb{r}, t) \right> 
d\pmb{r} $$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Evolution of magnetic strength and correlation length\footnote{A. Brandenburg {\em et 
al. Phys. Rev. D,} 96, 123528 (2017)}}
\includegraphics[width = .9\textwidth]{MHD_results/B_corrl}
\end{frame}

%----------------------------------------------------------------------------------------

\end{document}

