text:
\section{Introduction}

Turbulent flows subject to background rotation are an important problem
in fluid mechanics as several systems are affected by rotation. Rotation
affects the Earth atmosphere and the oceans at large scales, it is crucial
in many engineering flows such as in turbomachinery, and is also
important in many astrophysical systems such as stellar convective
regions and gaseous planets atmospheres. As many applications arise in
nature and technology, it becomes important to understand
detailed properties of these flows.

In particular, a long-lasting problem in fluid mechanics is that of the
decay of turbulent fluctuations. The rate of energy decay in turbulent
flows is known to be sensitive to initial conditions, and no single
universal solution to the decay appears to exist to which all flows
converge asymptotically for infinite Reynolds number \cite{George1992}.
Considering isotropic and homogeneous turbulence at high Reynolds number,
there are two well known canonical cases as far as the energy spectrum
at large scales is concerned. Provided the cubic velocity correlation
tensor $\left<u_i({\bf x}) u_j({\bf x}) u_k({\bf x}+{\bf r})\right>$
decays sufficiently fast for large $r$, these cases correspond to
the so-called Saffman spectrum for which $E(k \rightarrow 0) \sim Lk^2$,
and to the so-called Batchelor spectrum where
$E(k \rightarrow 0) \sim Ik^4$. Which of these spectra is observed
depends on the initial conditions, and other non-canonical cases may arise
if the above condition on the decay of the correlation tensor does not
hold (see, e.g., \cite{Krogstad2010}). The quantities $L$ and $I$ are
approximately conserved integrals for fully developed, freely decaying
turbulence in each case ($I$ is often called the Loytsiansky
integral), and have been used to determine the decay rate of the total
energy (see \cite{Batchelor1956,Saffman1967,Saffman1967b,Ishida2006}
for examples). The quasi-conservation in each case is associated with
self-preserving solutions of the von K\'arm\'an-Howarth equations,
and can also be interpreted in terms of conservation of linear and
angular momentum. While constancy of $L$ is a consequence of linear
momentum conservation \cite{Saffman1967}, constancy of $I$ is a
consequence of the conservation of angular momentum
\cite{Landau1959,Davidson2009}.

The presence of background rotation breaks down isotropy as a preferred
direction arises along the axis of rotation (for detailed studies
of rotating turbulence, see \cite{Cambon1989,Cambon1997,Bellet2006}).
Flows subjected to rotation develop anisotropies which have been
shown to impact dramatically on the decay of energy, and there are
different and sometimes conflicting results in the literature
regarding the decay laws followed by the energy in rotating flows.
Moreover, theoretical arguments based on the isotropic von
K\'arm\'an-Howarth equations are not valid anymore, and must be
extended to include the effect of rotation. The integrals $I$ and $L$
become tensors, and for axisymmetric flows some of the diagonal
components of these tensors can be expected to replace $I$ and $L$
as approximately conserved quantities. Extensions of these arguments
to anisotropic cases such as conducting flows with an imposed
magnetic field, rotating, or stratified flows, have been recently
derived in Refs.~\cite{Davidson2009,Davidson2010,Okamoto2010}.

The development of anisotropies in rotating flows have been reported
both in experiments and simulations. Experiments of freely decaying
rotating turbulence show a reduction of the energy transfer, evidenced
by a decrease in the energy decay rate \cite{Bokhoven2009}. An
anisotropic energy flux with a trend towards
quasi-two-dimensionalization has been recently reported
\cite{Lamriben2011}, and is also evidenced, e.g., by an increase in the
correlation lengths along the axis of rotation
\cite{Baroud2003,Staplehurst2008}. Regarding the energy decay,
experimental results for non-rotating grid turbulence with integral
scales smaller than the size of the vessel \cite{Morize2006} show an
energy decay $E \sim t^{-1.1}$ compatible with the theoretical result
$E \sim t^{-6/5}$ expected for large scale initial conditions with
Saffman spectrum $\sim k^2$. In \cite{Moisy2011}, a transition
from a $E \sim t^{-6/5}$ decay to a decay law closer to
$\sim t^{-3/5}$ is reported once rotation starts affecting the flow
dynamics, together with a cyclone-anticyclone asymmetry evidenced by
positive skewness which grows with rotation rate. The decay for bounded
flows, in which initial integral scales are close to the size of the
vessel, seems to change from $E \sim t^{-2}$ to $\sim t^{-1}$ when rotation
is present \cite{Morize2006}.

Numerical simulations and models also show a slow down of the energy
decay for many different initial conditions
\cite{Jacquin1990,Yang2004,Bellet2006,Muller2007,Bokhoven2008,Teitelbaum2009},
together with the trend towards two-dimensionalization
\cite{Muller2007,Bourouiba2007}, and the higher cyclonic-over-anticyclonic
activity \cite{Teitelbaum2010,POF}. The first theoretical
study of the decay of rotating turbulence was reported in
\cite{Cambon1994}, and considered the decrease of the energy transfer
in the presence of rotation to explain the observed slow down in the
decay, and to predict decay rates for the energy. The exponents
reported were consistent with those obtained in simulations in
\cite{Jacquin1990}, and in the experiments in \cite{Morize2006}. Also,
a very close decay law $E \sim t^{-0.8}$ was found in
\cite{Bellet2006} for a wide range of Rossby numbers (from very small
to moderate values), when solving equations for inertial wave turbulence
and for an Eddy Damped Quasi-Normal Markovian model (EDQNM3).

Later, in \cite{Bourouiba2007} a distinction was introduced between
the energy contained in modes with $k_z=0$ ($E_{2D}$, corresponding
to two-dimensional, ``slow'' or ``vortical'' modes) and modes with
$k_z \neq 0$ ($E_{3D}$, corresponding to three-dimensional, ``fast'',
or ``wave'' modes). The authors carried a comprehensive study of
the transfer of energy from 3D to 2D modes, for a wide range of Rossby
numbers, and found a non-monotonic behavior for the decay of energy
for large, intermediate, and small Rossby numbers. They focused on the
intermediate Rossby range, where the maximal energy transfer between 3D
and 2D modes occurs (see also \cite{Bourouiba2011}). In this range,
they reported an initial growth of $E_{2D}$ as soon as the decay
begins, that eventually results in a cross-over with the
monotonously decaying $E_{3D}$. Reference \cite{Thiele2009}
also reported different behavior in the time evolution of $E_{2D}$ and
$E_{3D}$, but considered only the decay of the total energy
$E = E_{2D} + E_{3D}$ proposing a decay that depends on the amount of
background rotation $E \sim t^{-\gamma(\Omega)}$. The dependence of
$\gamma$ on the rotation rate was derived using phenomenological
arguments resulting $\gamma\sim \Omega^{-1}$.

This variety of solutions naturally led to parametric studies of the
initial condition space in numerical simulations. For non-helical
rotating flows with initial integral scale close to the size of the
domain \cite{Teitelbaum2009}, a decay $E \sim t^{-1}$ was found in
simulations, in agreement with the experimental results \cite{Morize2006}.
For flows with initial integral scale sufficiently smaller than the
domain size, the case of initial conditions with energy spectra
$\sim k^4$ was studied in detail in Refs.~\cite{Teitelbaum2010,POF},
where different decay rates were reported for the two-dimensional and
three-dimensional modes. While $E_{3D}$ was found to decay as in the
case of isotropic turbulence, a slower decay was found for $E_{2D}$ in
the intermediate-Rossby range as defined in \cite{Bourouiba2007}. The
decay of two-dimensional modes can be correctly explained if only a few
components of the Loytsiansky tensor are considered (here referred in
short as $I_{\perp}$ and $I_{\parallel}$ based on the
symmetries of the flow), and it was shown that $I_{\perp}$ remains
approximately constant during the decay \cite{POF}. Finally, it was also
found that the decay is influenced by the amount of total helicity
present in the initial conditions, and by the initial degree of
anisotropy in the flow.

Recently, the anisotropic von K\'arm\'an-Howarth equation was used to
consider the decay of a rotating turbulent flow with initial conditions
following a Saffman spectrum \cite{Davidson2010}. The approximately
conserved integral $L$ in the isotropic and homogeneous case, was
generalized in the axisymmetric case to two integrals
$L_{\perp} \propto L_{xx} = L_{yy}$ and
$L_{\parallel} = L_{zz}$. Assuming that both $L_{\perp}$ and $L_{\parallel}$
are simultaneously conserved (or $I_\perp$ and $I_{||}$ for
the case of Batchelor initial
spectrum), and using the empirical result that characteristic lengthscales
of the flow in the direction parallel to the rotation axis grow linearly
with time \cite{Davidson2010}, the author concludes that regardless of
the initial spectrum being $\sim k^2$ or $\sim k^4$, both cases should
decay with the same $E \sim t^{-1}$ law, thus implying that it is not
possible to set apart $\sim k^2$ from $\sim k^4$ turbulence by
measuring the energy decay exponent alone.

In this paper we analyze whether the decay of rotating flows is affected
by long-range correlations, studying the energy decay rate for flows with
$\sim k^2$ and $\sim k^4$ initial conditions. We first present a brief
theoretical discussion of conserved integral quantities for rotating
turbulence using a von K\'arm\'an-Howarth equation which includes a
Coriolis term due to rotation. We extend the previously derived results
for Saffman turbulence \cite{Davidson2010}, considering Batchelor
spectrum. The results corroborate the phenomenological
arguments used in \cite{POF} to explain the decay laws found numerically
for energy spectra $k^4$. In the second part of the paper we use
numerical simulations to analyze the decay of rotating Saffman and
Batchelor turbulence in periodic domains. We focus on the regime of
intermediate Rossby numbers (initial $Ro \approx 0.1$ \cite{Bourouiba2007}),
and therefore we will not attempt a study of the dependence of the decay
with the rotation rate. To the best of our knowledge, the free decay of
energy for $\sim k^2$ initial conditions has not yet been numerically
studied for rotating turbulence, and the resulting decay laws seem
to be relevant for experiments. We show that the decay of energy in
two-dimensional and three-dimensional modes follows distinct power
laws for the cases of Batchelor and Saffman turbulence. The decay
laws are consistent with the predictions derived from the anisotropic
von K\'arm\'an-Howarth equation, and with the approximate constancy of
either $L_\perp$ or $I_\perp$ as the flow decays.

\section{\label{sec:Karman}von K\'arm\'an-Howarth equations and
self-similar decay}

The dynamics of an incompressible fluid subjected to background
rotation is described by the Navier-Stokes equation with the addition of
the Coriolis acceleration,
\begin{equation}
\partial_t {\bf u} + \mbox{\boldmath $\omega$} \times
    {\bf u} + 2 \mbox{\boldmath $\Omega$} \times {\bf u}  =
    - \nabla {\cal P} + \nu \nabla^2 {\bf u} ,
\label{eq:momentum}
\end{equation}
together with the incompressibility condition,
\begin{equation}
\nabla \cdot {\bf u} = 0.
\label{eq:incompressible}
\end{equation}
Here, ${\bf u}$ is the velocity field, $\mbox{\boldmath $\omega$} =
\nabla \times {\bf u}$ is the vorticity, the centrifugal acceleration
is absorbed in the total pressure per unit of mass ${\cal P}$, and
$\nu$ is the kinematic viscosity. We assume uniform density and the
rotation axis in the $z$ direction so $\mbox{\boldmath $\Omega$} =
\Omega \hat{z}$, with $\Omega$ the rotation frequency.

We briefly present the derivation of the von K\'arm\'an-Howarth equation
including the Coriolis term in order to find invariant quantities during
the self-similar energy decay in rotating flows. To this end, we write
the Navier-Stokes equation in a rotating frame in index notation, and
evaluated at two points ${\bf x}$ and ${\bf x}'={\bf x}+{\bf r}$,
\begin{equation}
\frac{\partial u_i}{\partial t}=-\frac{\partial}{\partial x_k}(u_iu_k)-
    \frac{\partial p}{\partial x_i} + \nu \nabla^2u_i -
    2\epsilon_{imn}\Omega_mu_n ,
\label{eq:NSROT1}
\end{equation}
and
\begin{equation}
\frac{\partial u'_j}{\partial t}=-\frac{\partial}{\partial x'_k}(u'_ju'_k)-
    \frac{\partial p'}{\partial x'_j} + \nu \nabla'^2u'_j -
    2\epsilon_{jmn}\Omega_mu'_n ,
\label{eq:NSROT2}
\end{equation}
where ${\bf u}'={\bf u}({\bf x}')$, $\nabla'^2$ denotes
Laplacian with respect to the ${\bf x}'$ coordinate, and $\epsilon$ is
the Levi-Civita symbol. Multiplying Eq.~(\ref{eq:NSROT1}) by $u'_j$,
Eq.~(\ref{eq:NSROT2}) by $u_i$, summing and averaging, we get the
equation for the time evolution of the two-point velocity
correlation tensor,
\begin{eqnarray}
&&\frac{\partial}{\partial t} \left<u_iu'_j\right> =
    -\left( \frac{\partial}{\partial x_k}
    \left< u'_ju_iu_k \right> +\frac{\partial}{\partial x'_k}
    \left<u_iu'_ju'_k \right> \right)
    \nonumber \\
{} && - \left( \frac{\partial}{\partial x_i} \left< u'_jp \right> +
    \frac{\partial}{\partial x'_j} \left< u_ip' \right> \right) +
    \nu \left( \nabla^2 \left< u'_ju_i \right> + \right.
    \nonumber \\
{} && + \left. \nabla'^2 \left< u_iu'_j \right> \right)
    - 2\Omega_m \left<\epsilon_{imn}u_nu'_j+\epsilon_{jmn}u'_nu_i\right>.
\label{eq:KH}
\end{eqnarray}

If $\Omega = 0$, and assuming the cubic velocity correlation tensor
$\left<u_iu_ju'_k\right>$ and the pressure-velocity correlation
$\left<u_i p' \right>$ decay fast enough with $r$ when
$r \rightarrow \infty$, two possible integral conserved quantities can
be obtained from Eq.~(\ref{eq:KH}). When the terms on the r.h.s.~of the
equation go to zero as $\mathcal{O}(r^{-2})$, then it follows from the
time derivative on the l.h.s.~that
\begin{equation}
L={\int}\left<{\bf u\cdot u}'\right>d{\bf r}
\label{eq:L}
\end{equation}
is conserved. This is know as the Saffman integral \cite{Saffman1967}.
Multiplying
Eq.~(\ref{eq:KH}) by $r^2$, another possible invariant is obtained
when the terms on the l.h.s.~are $\mathcal{O}(r^{-4})$, namely
\begin{equation}
I=-{\int}r^2\left<{\bf u\cdot u}'\right>d{\bf r},
\end{equation}
which is known as the Loitsyanski integral and whether it actually
remains constant or not in decaying isotropic and homogeneous turbulence
has been a matter of debate \cite{Davidson2004}. As the energy
spectrum for small values of $k$ can be expanded as
\begin{equation}
E(k \rightarrow 0) \sim Lk^2+Ik^4+... \, ,
\label{eq:Elowk}
\end{equation}
the integrals $L$ or $I$ are then expected to be quasi-invariants during
the decay for initial large-scale energy spectra of the form $\sim k^2$
(Saffman spectrum) \cite{Saffman1967,Saffman1967b} or $\sim k^4$
(Batchelor spectrum) \cite{Batchelor1956} respectively. Using these
quasi-conserved integrals, the energy balance equation
$dE/dt \sim -E^{3/2}/l$ can be closed (as either $l^3u^2$ or $l^5u^2$ are
constant, with $l$ the energy-containing scale), and different decay laws
arise for each scaling of the initial spectra,
\begin{equation}
E(t) \sim \left\{ \begin{array}{ll}
 t^{-6/5} & \textrm{if $E(k,t=0) \sim k^2$}\\
 t^{-10/7} & \textrm{if $E(k,t=0) \sim k^4$}
\end{array} \right.
\label{eq:teodecay}
\end{equation}
These decay laws has been observed in numerical as well as in experimental
studies (see, e.g.,
\cite{Kolmogorov1941,Ishida2006,Saffman1967,Saffman1967b,Morize2006,POF,Moisy2011})

These isotropic integrals have been also assumed to remain constant during
the decay of rotating turbulence with $\sim k^2$ and $\sim k^4$ large-scale
energy spectra, to predict energy decay laws \cite{Cambon1994}. In the
presence of rotation, the energy per unit of time (flux) transferred
towards smaller scales is reduced as $\Omega$ is increased. A
phenomenological expression consistent with the observations is to assume
that the flux is reduced by the ratio of the turnover time
$\tau \sim l/E^{1/2}$ to the wave time $(2 \Omega)^{-1}$
\cite{Iroshnikov1963,Kraichnan1965,Muller2007,Mininni2009},
resulting in \cite{Cambon1994}
\begin{equation}
\frac{dE}{dt} \sim -\frac{E^2}{\Omega l^2}
\label{eq:balancerot}
\end{equation}
Further assuming $I$ or $L$ are conserved results in the following
decays,

\begin{equation}
E(t) \sim \left\{ \begin{array}{ll}
 t^{-3/5} & \textrm{if $E(k,t=0) \sim k^2$}\\
 t^{-5/7} & \textrm{if $E(k,t=0) \sim k^4$}
\end{array} \right.
\label{eq:teodecay2}
\end{equation}
Although these power laws are close to the behavior found in
experiments and simulations, as mentioned in the introduction a better
agreement with observations can be obtained when the arguments are
extended to consider the effect of anisotropy in the flow.

In the anisotropic case, and assuming absence of long-range correlations,
only the final term in Eq.~(\ref{eq:KH}) can contribute to the rate of change
of integrals of the type
\begin{equation}
I_{ijmn}={\int}r_mr_n\left<u_iu'_j\right>d{\bf r},
\label{Iijmn}
\end{equation}
for the Batchelor case, or
\begin{equation}
L_{ij}={\int}\left<u_iu'_j\right>d{\bf r}
\label{Lij}
\end{equation}
for Saffman turbulence. It is important to point out here that
it is not trivial that in this case long-range correlations should
vanish as in the isotropic case (specially for the pressure velocity
correlations, see \cite{Bellet2006} where these terms are computed
using the Fourier transform of Eq.~(\ref{eq:KH}) and the Poisson
equation for the pressure fluctuations). Thus, the validity of the
assumptions can only be verified {\it a posteriori} from the results
obtained from the numerical simulations.

Assuming long-range correlations vanish sufficiently fast at
large-scales, the problem is that of finding what components of
the tensors in Eqs.~(\ref{Iijmn}) and (\ref{Lij}) are still conserved
in the presence of rotation. We derive here the Batchelor case
($I_{ijmn}$) under the same assumptions for the decay of velocity
correlations with $r$ as in the isotropic case, multiplying
Eq.~(\ref{eq:KH}) by $r_m r_n$ and integrating to obtain
\begin{eqnarray}
&& \frac{\partial}{\partial t}{\int}r_mr_n\left<u_iu'_j\right>d{\bf r} =
    \nonumber \\
{} && -2 \Omega_l {\int} r_mr_n \left( \epsilon_{ilk}\left< u_ku'_j\right> +
    \epsilon_{jlk}\left<u'_ku_i\right> \right) d{\bf r} .
\end{eqnarray}
Let's assume $\Omega_l = \Omega_z \delta_{lz}$, so
\begin{eqnarray}
&& \frac{\partial}{\partial t}{\int}r_mr_n\left<u_iu'_j\right>d{\bf r} =
    \nonumber \\
{} && -2 \Omega_z {\int} r_mr_n \left(\epsilon_{izk}\left<u_ku'_j\right> +
    \epsilon _{jzk}\left<u'_{k}u_i\right> \right) d{\bf r} .
\end{eqnarray}
As the flow has axisymmetry, we are interested in the diagonal components
of the tensor. For $j=i$,
\begin{equation}
\Omega_z\epsilon_{izk}\left<u_ku'_i\right>=\Omega_z\left<u_xu'_y-u_yu'_x\right>,
\end{equation}
\begin{equation}
\Omega_z\epsilon_{jzk}\left<u'_{k}u_j\right>=\Omega_z\left<u'_xu_y-u'_yu_x\right>,
\end{equation}
where all terms with $i=z$ or $j=z$ vanish. For $m=n$, introducing
${\bf r_{\perp}} = (r_x,r_y,0)$ and ${\bf u}_{\perp}=(u_x,u_y,0)$ we get
\begin{eqnarray}
&& \frac{\partial}{\partial t}{\int}r_{\perp}^2\left<u_{\perp}u'_{\perp}\right>
    d{\bf r} =
    \nonumber \\
{}&& -2{\int}r_{\perp}^2\Omega_z[\left<u_xu'_y-u_yu'_x\right>-
    \left<u_xu'_y-u_yu'_x\right>]d{\bf r},
\end{eqnarray}
so the terms on the r.h.s.~of the equation cancel and
\begin{equation}
\frac{\partial}{\partial t}{\int}r_{\perp}^2\left<
    {\bf u}_{\perp}{\bf u}'_{\perp}\right>d{\bf r} = 0,
\end{equation}
or equivalently,
\begin{equation}
I_{\perp} = {\int}r_{\perp}^2\left<{\bf u}_{\perp}{\bf u}'_{\perp}\right>d{\bf r}
    = \textrm{constant} .
\label{eq:I2D}
\end{equation}

The case of Saffman turbulence was considered in \cite{Davidson2010}. The
relevant components of the tensor $L_{ij}$ for axisymmetric flows are
$L_{xx}=L_{yy}$ and $L_{zz}$. Similar arguments as the ones described
above lead, for the $xx$ and $yy$-components of the von K\'arm\'an-Howarth
equation, to the conservation of the integral
\begin{equation}
L_{\perp} = {\int}\left<{\bf u}_{\perp}{\bf u}'_{\perp}\right>d{\bf r} =
    \textrm{constant} .
\label{eq:K}
\end{equation}

Note that both $I_{\perp}$ and $L_{\perp}$ resemble integrals that arise in the
context of the decay of $2D$ turbulence \cite{Fox2009}. This is
to be expected as the spectrum of an axisymmetric flow can be expanded,
for the modes with $k_\parallel=k_z=0$, as
\begin{equation}
E(k_{\perp} \rightarrow 0,k_{\parallel}=0) \approx
    L_{\perp}k_{\perp}+I_{\perp}k^3_{\perp}+...
\label{eq:expansion2d}
\end{equation}
which is (except for differences in the dimensions) also the expansion
of a 2D energy spectra. The resemblance can be expected as rotating
flows tend to become quasi-2D, concentrating most of the energy in
the slow modes with $k_\parallel=0$.

For axisymmetric turbulence and a Batchelor initial spectrum $\sim k^4$
at large scales, Eq.~(\ref{eq:I2D}) leads to
\begin{equation}
I_{\perp} \sim l_{\perp}^4l_{\parallel}u_{\perp}^2 \approx\textrm{constant} ,
\label{eq:Iperp}
\end{equation}
where $l_\perp$ and $l_\parallel$ are characteristic (energy-containing)
scales in the perpendicular and parallel direction respectively. Assuming
2D and 3D modes are only weakly coupled in rotating turbulence, from
Eq.~(\ref{eq:balancerot}) we can write an equation for the decay of the
energy in 2D modes
\begin{equation}
\frac{dE_{2D}}{dt} \sim -\frac{E_{2D}^2}{\Omega I_{\perp}^2} ,
\label{eq:E2D}
\end{equation}
and from Eq.~(\ref{eq:Iperp})
\begin{equation}
\frac{dE_{2D}}{dt} \sim -\frac{E_{2D}^{5/2}l_{\parallel}^{1/2}}
    {I_{\perp}^{1/2}\Omega}.
\label{eq:balancerot2D}
\end{equation}
Assuming $E_{2D} \sim t^\gamma$ it follows that \cite{POF}
\begin{equation}
E_{2D} \sim t^{-2/3}
\end{equation}
if $l_\parallel$ remains constant (which is reasonable as 2D modes have
no dependence on the direction parallel to rotation, and therefore
$l_\parallel = l_0$ is the vertical size of the box).

For a $\sim k^2$ initial spectra,
\begin{equation}
L_{\perp} \sim l_{\perp}^2l_{\parallel}u_{\perp}^2 \approx \textrm{constant} ,
\end{equation}
and replacing in Eq.~(\ref{eq:E2D}) leads to
\begin{equation}
\frac{dE_{2D}}{dt} \sim -\frac{E_{2D}^3l_{\parallel}}{L_{\perp}\Omega} ,
\label{eq:balancerot2D2}
\end{equation}
and a decay
\begin{equation}
E_{2D} \sim t^{-1/2}.
\end{equation}

The results for the decay of energy
in 2D modes in rotating turbulence can be summarized as follows,
\begin{equation}
E_{2D}(t) \sim \left\{ \begin{array}{ll}
 t^{-1/2} & \textrm{if $E(k,t=0) \sim k^2$}\\
 t^{-2/3} & \textrm{if $E(k,t=0) \sim k^4$.}
\end{array} \right.
\label{eq:teodecayrot}
\end{equation}

It is worth mentioning that the decay of rotating turbulence with
$\sim k^2$ spectra is discussed in detail in \cite{Davidson2010}, where
the constancy of $L_{\perp}$ is obtained from the K\'arm\'an-Howarth equation.
In that case, the author also considers the von K\'arm\'an-Howarth
equation for the $zz$-component of the two-point correlation tensor, and
assuming cubic correlations decay fast enough also for that component,
obtains constancy of $L_\parallel \sim u_{\parallel}^2l_{\perp}^2l_{\parallel}$
as well as of $L_{\perp}$. The constancy of these two quantities, together with
the empirical result $l_{\parallel}=l_0(1+\kappa \Omega t)$ (where
$\kappa$ is a constant of order one \cite{Staplehurst2008,Jacquin1990})
leads to a decay $E_{2D}(t)\sim t^{-1}$. A similar argument for Batchelor
$\sim k^4$ turbulence leads to the same decay, which would result in an
impossibility to distinguish between $\sim k^2$ and $\sim k^4$ turbulence
by measuring the energy decay exponent alone.

Finally, it is interesting to point out that constancy of $I_{\perp}$ and
$L_{\perp}$ can be respectively associated with conservation of the
$z$-component of angular and linear momentum (see
\cite{Davidson2009,Davidson2010}).
In fact, the mean squared angular momentum $H_z$ averaged over a
cylinder-shaped volume $V$ is
\begin{equation}
\left<H_z^2\right> = 2\pi V l_\parallel I_{\perp},
\end{equation}
while the linear momentum $P_z$ is
\begin{equation}
\left<L_z^2\right> = 2\pi V l_\parallel L_{\perp}.
\end{equation}

\section{\label{sec:NumSim}Numerical Simulations}

In the following we resort to numerical simulations to verify the validity
of the assumptions and predictions discussed in the previous section.
As a well resolved initial spectrum $\sim k^2$ or $\sim k^4$ is needed
to observe constancy of $I$ or $L$ in simulations of isotropic and
homogeneous turbulence (see, e.g., \cite{Ishida2006}), we use
large-eddy simulations (LES) in periodic boxes to be able to have
initial energy containing wavenumber $k_0 \approx 40$ with large
Reynolds numbers at a reasonable computing cost.

Before proceeding, we must caution the reader on how the
simulations in periodic boxes should be interpreted. Flows in periodic
boxes are sometimes considered as describing a homogeneous flow in an
infinitely periodic domain. In this work, we will interpret instead
a flow in a periodic box as artificially confined. Indeed, as was
shown in \cite{POF}, when the integral scale of the flow is close to the
domain size, the eddies cannot become larger and the energy decay laws
obtained are those of confined flows. The confinement is artificial in
the sense that the flows have no Ekman layers, as the domain has no
rigid walls. Moreover, the confinement also results in a discrete set of
wavenumbers and selects a discrete set of inertial waves which are
normal modes of the domain, unlike homogeneous rotating flows in
infinite domains which have continuous wavenumbers (see, e.g.,
\cite{Smith2005,Bellet2006}). Note that due to the discretization of
wavenumbers in the former case, the existence and number of exact
resonances depend on the domain size \cite{Smith2005} (as this is
aggravated when boxes with large aspect ratio and lower resolution
in the vertical direction are used, we restrict our study to boxes with
aspect ratio of unity).

While these are unavoidable and inherent properties of the
simulations in periodic domains, the choice of a relatively large
initial energy containing wavenumber $k_0 \approx 40$
ensures that as the equations are advanced in time the energy
containing scale of the flow will remain smaller than the box size,
and that a large fraction of wavenumbers will be available at
large-scales for the system to evolve.

\subsection{Equations and model}

Equations (\ref{eq:momentum}) and (\ref{eq:incompressible}) are solved
numerically using a dynamical subgrid-scale spectral LES model of
rotating turbulence in which only the large scales are explicitly
resolved. The statistical effect of unresolved scales on the scales
larger than a cut-off wave number ($k_c=N/2$) are modeled with an
eddy-viscosity and eddy-noise that are obtained after solving in each
time step the eddy damped quasi-normal Markovian (EDQNM) equations for
the spectrum of the unresolved scales. Details of the LES can
be found in Ref.~\cite{Baerenzung2008}, and in
Refs.~\cite{Baerenzung2010,Baerenzung2011}
for its extension to the rotating case. A validation of
the LES against direct numerical simulations (DNS) for the case of
freely decaying rotating turbulence can be found in \cite{POF}.

The simulation domain is a 3D periodic box of length $2\pi$ with
spatial resolution of $256^3$ grid points (resulting in $k_c=128$). A
parallelized pseudo-spectral method without de-aliasing is used to
solve spatial derivatives, and an explicit second-order Runge-Kutta
method is used to evolve in time \cite{Gomez1}.

\subsection{Initial conditions}

In \cite{POF} it was observed that the energy decay is sensitive
to the initial anisotropy of the flow (in particular, to the initial
amount of energy in 2D modes). To enforce spectral and variance
isotropy of the initial conditions, we generate at $t=0$ a random
velocity field using the Craya-Herring decomposition
\cite{Craya1958,Herring1974} which associates to each wave vector
${\bf k}$ an orthonormal frame with axes dependant on ${\bf k}$ as
\begin{equation}
{\bf i}({\bf k}) = \frac{{\bf k} \times{\bf \alpha}}{|{\bf k} \times
{\bf \alpha}|},\qquad{\bf j}({\bf k}) = \frac{{\bf k}
    \times{\bf i}}{|{\bf k}|},
\qquad\frac{{\bf k}}{k};
\end{equation}
${\bf \alpha}$ is an arbitrary but fixed unit vector which we choose
to be parallel to the rotation axis, ${\bf \alpha} = (0,0,1)$. This
representation has been widely used to study turbulence (see, e.g.,
\cite{Lee1979,Cambon1985}). Random initial conditions are created through
a superposition of harmonic modes with random phases $\phi({\bf k}),
\phi'({\bf k})$ projected over the Craya reference system for each
${\bf k}$,
\begin{equation}
{\bf u}({\bf k})=e^{i\phi}{\bf i}+e^{i\phi'}{\bf j}.
\end{equation}
Note that due to the geometry of the Craya-Herring decomposition there
is no projection of ${\bf u}$ into ${\bf k}$ as the incompressibility
condition stands for ${\bf u}({\bf k}) \perp {\bf k}$.

As we are interested in the decay laws followed by flows with a particular
initial spectrum (either Batchelor's or Saffman's), we can further
control the shape of the initial isotropic spectrum by multiplying
each mode ${\bf u}({\bf k})$ in Fourier space with $0 < k \le k_0$ by
an amplitude $C k^\beta$, where C is a constant to get initial
r.m.s.~velocity $U\approx 1$, and $\beta$ is a parameter used to adjust the
power law in $E(k)$. For $k > k_0$, the spectrum is continued by an
exponential decay up to $k_c$ (the maximum wavenumber explicitly resolved
in the simulations).

\subsection{Spectra}

In order to characterize the simulations we consider isotropic as well as
anisotropic spectra. The latter is useful to analyze rotating flows in
which anisotropies grow with time.

In the simulations, the isotropic spectrum is defined by averaging in
Fourier space over spherical shells,
\begin{equation}
E(k,t) = \frac{1}{2} \sum_{k\le |{\bf k}|<k+1} {\bf u}^*({\bf k},t)
\cdot {\bf u}({\bf k},t) ,
\label{eq:isoenespec}
\end{equation}
where the star denotes complex conjugate. For the anisotropic spectrum
the components of ${\bf u}({\bf k})$ are integrated around the axis of
rotation to obtain a spectrum that depends only on $k_{\perp}$,
\begin{equation}
E(k_{\perp}) = \frac{1}{2} \sum_{k_{\perp} \le |{\bf k_{\perp}}|<k_{\perp}+1}
{\bf u}^*({\bf k},t)\cdot {\bf u}({\bf k},t) .
\label{eq:anienespec}
\end{equation}
From now on we will reference these two spectra simply as $E(k)$ and
$E(k_{\perp})$ respectively.

More detailed information on the energy spectral
distribution of anisotropic flows can be obtained from the
axisymmetric energy spectrum $e(k_{\parallel}, k_{\perp})$
(see, e.g., \cite{Cambon1985,Cambon1997,Mininni2012}).
This spectrum is obtained after integrating the three-dimensional
energy spectrum around the axis of rotation to obtain a spectrum
that depends only on $k_{\parallel}$ and $k_{\perp}$, and that relates to
$E(k_{\perp})$ as follows:
\begin{equation}
E(k_{\perp})=\sum_{k_{\parallel}}{e(k_\parallel,k_\perp)}.
\end{equation}

Finally, the spectrum of the initial conditions is such that the Saffman
or Batchelor power laws $\sim k^2$ or $\sim k^4$ are imposed in
$E(k)$. The relationship between the power law in $E(k)$ and of
that of $E(k_\perp)$ can be derived as follows. Equation
(\ref{eq:isoenespec}) in the continuous limit can be expressed as
\begin{equation}
E(k,t) = \frac{1}{2}\int \hat{\Phi}_{ii}(k,t)k^2d\varOmega_k,
\label{ec:Ephi}
\end{equation}
where $\hat{\Phi}_{ij}$ is the spectral tensor, i.e., the Fourier
transform of the second order velocity correlation function
$\Phi_{ij} \left<u_i({\bf x},t)u_j({\bf x}+{\bf r},t)\right>$, and where
$\varOmega_k$ is the solid angle in Fourier space. Per virtue of
Eq.~(\ref{ec:Ephi}), if $E(k,0) \sim k^{\sigma}$ then $\hat{\Phi}_{ii}
\sim k^{\sigma-2}$. For axisymmetric flows we can write the energy
spectrum $E(k_{\perp})$ integrating in complex space over cylindrical
shells,
\begin{equation}
E(k_{\perp},t)=\frac{1}{2}\int \hat{\Phi}_{ii}(k_{\perp},t)
    k_{\perp}d{\phi}_k dk_z.
\end{equation}
If $E(k,0)\sim k^{\sigma}$ then,
\begin{eqnarray}
E(k_{\perp},0)=\frac{1}{2}\int k^{\sigma-2}k_{\perp}d\phi_k dk_z =
    \nonumber \\
{} \frac{1}{2}\int(k^2_{\perp}+k^2_z)^{\frac{\sigma-2}{2}}k_{\perp} d\phi_k
    dk_z,
\label{eq:alto_paper}
\end{eqnarray}

When most of the energy is in the $k_z=0$ plane then, from
Eq.~(\ref{eq:alto_paper})
\begin{equation}
E(k_{\perp},0)\sim \frac{1}{l_\parallel}\int k_{\perp}^{\sigma-1}d\phi_k.
\label{eq:energiakperp}
\end{equation}

\begin{table*}
\caption{\label{table:runs}Parameters used in the simulations: number of
linear grid points $N$,kinematic viscosity $\nu$, rotation frequency
$\Omega$, Reynolds number $Re$, Taylor Reynolds number $Re_{\lambda}$,
Rossby number $Ro$, micro-Rossby number $Ro^{\omega}$, and a brief
description of the initial energy spectrum $E(k)$: the power law followed
at large-scales and the range of scales where this power law is satisfied.
Values of $Re$, $Re_{\lambda}$,$Ro$, and $Ro_{\lambda}$ are given at $t=0$,
and at $t=20$ between parentheses.}
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
Run&$N$&$\nu$&$\Omega$&$Re$&$Re_{\lambda}$&$Ro$&$Ro^\omega$& Initial $E(k)$ \\
\hline
A &$256$&$1.3\times 10^{-4}$& $0$ & $1580(330)$ & $780(100)$ &
  $\infty$($\infty$)
  & $\infty$($\infty$) & $k^{2}$ $(1 \leq k \leq 40)$\\
B &$256$&$1.3\times 10^{-4}$& $33$ & $1580(1090)$ & $780(520)$ &
  $0.15(0.004)$ & $0.9(0.024)$
  & $k^{2}$ $(1 \leq k \leq 40)$\\
C &$256$&$1.3\times 10^{-4}$& $0$ & $1280(150)$ & $730(60)$ &
  $\infty$($\infty$)
  & $\infty$($\infty$) & $k^{4}$ $(1 \leq k \leq 40)$\\
D &$256$&$1.3\times 10^{-4}$& $33$ & $1280(360)$ & $730(180)$ &
  $0.18(0.002)$ & $1.0(0.015)$
  & $k^{4}$ $(1 \leq k \leq 40)$
\label{tab:paratable}
\end{tabular}
\end{ruledtabular}
\end{table*}

Even in the case in which the energy is not concentrated in the
$k_z=0$ plane (as in the isotropic initial conditions), the dependence
of $E(k_\perp)$ will be $\sim k_\perp^{\sigma - 1}$ as it follows directly
from Eq.~(\ref{eq:alto_paper}). To summarize,
\begin{equation}
E(k) \sim k^{\sigma} \,\, \Rightarrow \,\, E(k_{\perp}) \sim k_{\perp}^{\sigma-1}.
\label{eq:Eisoani}
\end{equation}
It is important to note that this simple relation between
the isotropic $E(k)$ and the anisotropic $E(k_{\perp})$ spectra
only holds for the isotropic initial conditions. However, the
relation will be useful to understand the power laws observed
at large scales in the following section. As an example, when
analyzing rotating turbulence with an initial isotropic
spectrum $E(k)\sim k^2$, we will be facing the case
with initial $E(k_{\perp})\sim k_{\perp}$. Note the power
laws $E(k_{\perp})\sim k_{\perp}$ and $E(k_{\perp})\sim k_{\perp}^3$,
resulting respectively for Saffman and Batchelor isotropic initial
conditions, are the ones that arise in the expansion of the
axisymmetric energy spectrum for $k_\parallel=0$ and
$k_\perp \to 0$ in Eq.~(\ref{eq:expansion2d}).

\section{\label{sec:NumRes}Numerical Results}

We now present results stemming from four numerical simulations (see
table \ref{tab:paratable}). All runs were initialized using the random
(Craya-Herring) initial conditions described in the previous section,
with an initial isotropic spectrum $E(k) \sim k^2$ between $0 < k \le 40$
for runs A and B, and $E(k) \sim k^4$ between $0 < k \le 40$ for runs C and D.
Runs A and C correspond to isotropic and homogeneous turbulence (no
rotation), while runs B and D are rotating, resulting in homogeneous but
anisotropic flows as time evolves. The purpose of simulations $A$ and $C$
is to recover well known results for isotropic $\sim k^2$ and $\sim k^4$
turbulence, and to use them as a starting point to analyze the more
complex rotating cases. Note the decay of $\sim k^4$ turbulence
(corresponding to runs C and D) was studied in detail in
\cite{Teitelbaum2010}, and is considered here to compare with the
$\sim k^2$ case.

\begin{figure}
\includegraphics[width=8cm]{f_figure1.eps}
\caption{Evolution of the isotropic energy spectrum $E(k)$ for run A from
$t=0$ to $t=20$ with time increments $\Delta t=0.5$. Note how the initial
$E \sim k^2$ spectrum is preserved over time at the larges scales (smaller
wave numbers).}
\label{fig:espectrum_norot}
\end{figure}

Parameters of the runs listed in table \ref{tab:paratable} are defined
as follows: The Reynolds number is
\begin{equation}
Re = \frac{lU}{\nu}
\end{equation}
where $U$ is the r.m.s velocity and the energy-containing
scale $l$ is defined from the isotropic energy spectrum $E(k)$ as
\begin{equation}
l=2\pi \frac{\int E(k)k^{-1}dk} {\int E(k)dk}.
\end{equation}
The Reynolds number based on the Taylor scale $\lambda$ is
\begin{equation}
Re_\lambda = \frac{\lambda U}{\nu}
\end{equation}
where
\begin{equation}
\lambda = 2 \pi \left(\frac{\int E(k)dk}{\int E(k)k^2dk}\right)^{1/2}.
\end{equation}
The Rossby number is
\begin{equation}
Ro = \frac{U}{2 \Omega l},
\end{equation}
while the micro-Rossby number is defined as
\begin{equation}
Ro^\omega = \frac{\omega}{2 \Omega}.
\end{equation}
This number should be initially of order one for the scrambling effect of
waves not to completely damp the non-linear term in the Navier-Stokes
equation, which would result in just an exponential decay of
the energy \cite{Cambon1997}.

Time-dependent parameters in table \ref{tab:paratable} are given for all
runs at $t=0$, and also at $t=20$ (roughly ten turnover times after the
self-similar decay of energy starts).

\begin{figure}
\includegraphics[width=8cm]{f_figure2.eps}
\caption{Energy evolution for run A (solid) and C (dashed). Note the
decay laws with exponents $\approx -6/5$ and $\approx -10/7$ respectively,
after an initial transient of $\approx 10$ turn over times. Slopes
are shown as a reference. Inset: enstrophy decay for the same runs.
The decays approach $\approx -11/5$ and $\approx -17/7$ respectively after
$t\approx 10$.}
\label{fig:decay_norot}
\end{figure}

\subsection{Decay of non-rotating flows}

Figure \ref{fig:espectrum_norot} shows the evolution of the isotropic
energy spectrum for run A, at times ranging from $t=0$ to $t=20$. The
initial $\sim k^2$ law is approximately preserved over time at the
smallest wavenumbers, while higher wavenumbers decay due to the effect
of dissipation. This behavior is consistent with the quasi-conservation
of $L$ in Eq. (\ref{eq:Elowk}), for which a subsequent decay law
$E \sim t^{-6/5}$ is expected for the energy.

\begin{figure}
\includegraphics[width=8cm]{f_figure3.eps}
\caption{(a) Evolution of the integral $L$ normalized by its initial value
$L(0)$ for the isotropic run A (solid) and rotating run B (dashed).
(b) Evolution of the integral $I$ normalized by its initial value $I(0)$
for the isotropic run B (solid) and rotating run D (dashed).}
\label{fig:L_vs_time}
\end{figure}

Figure \ref{fig:decay_norot} shows the energy and enstrophy decay for
runs A and C. After a transient lasting for approximately $10$ turn-over
times, both runs decay with different power laws. In run A, the energy
decays close to $E \sim t^{-6/5}$ as predicted by Eq. (\ref{eq:teodecay}),
and as expected from the shape of the large-scale spectrum in Fig.
\ref{fig:espectrum_norot}. The enstrophy shows a faster decay close to
$Z \sim t^{-11/5}$, consistently with phenomenological arguments which
indicate that $E \sim t^{\alpha}$ leads to $Z \sim t^{\alpha -1}$ under
the assumption of isotropy, as
\begin{equation}
\frac{dE}{dt} = -2 \nu Z(t) .
\label{eq:Zdecay}
\end{equation}

The energy decay for run C is consistent with $E \sim t^{-10/7}$ as
expected by Eq. (\ref{eq:teodecay}), while the enstrophy decay is
close to $Z \sim t^{-17/7}$, also consistent with Eq.~(\ref{eq:Zdecay}).
The spectrum of run C maintains a $\sim k^4$ shape for large scales (not
shown). All these results are consistent with previous simulations and
theoretical results \cite{Kolmogorov1941,Ishida2006,POF}.

The prediction $E \sim t^{-6/5}$ for run A assumes $L$ remains constant
during the decay. To verify this, and to further study whether
$L$ also remains constant in the decay of rotating turbulence with
$\sim k^2$ spectrum, we show the time evolution of $L$ for runs A and B
in Figure \ref{fig:L_vs_time}(a). We estimated $L$ in two different ways:
A fit to the spectrum with a power law $\sim k^2$ was computed for the smallest
wavenumbers to obtain the multiplicative prefactor in Eq.~(\ref{eq:Elowk})
(proportional to $L$ if the spectrum is $\sim k^2$), as was done
in \cite{Ishida2006}. We also computed the two-point correlation function
from the energy spectrum using \cite{Davidson2004}
\begin{equation}
\left < {\bf u \cdot u'} \right > (r)=2\int_0^{\infty}
E(k)\frac{\sin (kr)-kr \cos (kr)}{(kr)^3}dk,
\end{equation}
and then used Eq.~(\ref{eq:L}) to compute $L$. Both methods give consistent
results and in the following we show results obtained with the last method.

\begin{figure}
\includegraphics[width=8cm]{f_figure4.eps}
\caption{Evolution of the isotropic energy spectrum $E(k)$ for run B
from $t=0$ to $t=20$ with time increments $\Delta t=0.5$. Note how the
initial $E \sim k^2$ spectrum for large scales (low wave numbers) is
preserved over time, but its overall amplitude increases.}
\label{fig:spectrum_rot_iso}
\end{figure}

In run A, $L$ decays to half its initial value between $t=0$ and
$t\approx 10$, but after this time it only changes slowly with time and
can be considered almost constant (note the energy between $t=10$
and $t=100$ changes by more than an order of magnitude). However, $L$ is far
from being constant in run B; the behavior of the rotating runs is
discussed in more detail in the next subsection.

Similar results were found for the evolution of $I$ in simulations C
and D, as shown in Fig.~\ref{fig:L_vs_time}(b). In run C, $I$ grows during
a short transient, but stays constant during the self-similar decay of
the energy.

While the behavior of $L$ and $I$ in the simulations is compatible with
the theoretical and phenomenological arguments discussed in
Sec.~\ref{sec:Karman}, the rapid growth of these quantities in the
runs with rotation seems to question the use of isotropic integrals
to derive decay laws for anisotropic flows.

\begin{figure}
\includegraphics[width=8cm]{f_figure5.eps}
\caption{Energy decay for runs B (solid) and D (dashed); $t^{-3/5}$
and $t^{-5/7}$ power laws are shown as a reference, corresponding to
phenomenological predictions for Saffman and Batchelor rotating
turbulence based on isotropic conserved quantities.}
\label{fig:Exy_rot_kz=0}
\end{figure}

\subsection{\label{sec:RotFlow}Decay of rotating flows}

The analysis for runs A and C was based on the isotropic energy
spectrum and on the quasi-invariance of the isotropic integrals
$L$ and $I$. However, the growth of $L$ ($I$) in run B (D) suggests
that different arguments should be used to predict their energy
decay. In this light, we now consider the anisotropic quantities
introduced in Sec.~\ref{sec:Karman}, $L_{\perp}$ and $I_{\perp}$, and
show that further analysis can be done by studying the evolution of
anisotropic spectra and splitting the energy decay in $2D$ and $3D$
modes. A similar method was used in \cite{POF} to study $\sim k^4$
turbulence.

The evolution of the isotropic spectrum for run B is shown in
Fig.~\ref{fig:spectrum_rot_iso}, for different times from $t=0$ to
$t=20$. As for run A, its initial $\sim k^2$ behavior for large scales
is preserved over time. However, its amplitude at large scales
increases with time, in agreement with the evolution of $L$.

\begin{figure}
\includegraphics[width=8cm]{f_figure6.eps}
\caption{Evolution of the axisymmetric energy spectrum $E(k_{\perp})$
for run B from $t=0$ to $t=20$ with time increments $\Delta t=0.5$.
The initial shape $E \sim k_{\perp}$ of spectrum at large scales (low
wavenumbers) is approximately preserved over time.}
\label{fig:spectrum_rot_ani}
\end{figure}

The time evolution of the total energy in run B shows that
the decay is close to, but steeper than, $E \sim t^{-3/5}$. In Figure
\ref{fig:Exy_rot_kz=0} we show the energy decay for this run (we also
show the $\sim k^4$ case for comparison, i.e., run D). The decay deviates
from the expected $\sim t^{-3/5}$ law derived via isotropic arguments (in
the same fashion, run D deviates from the expected $\sim t^{-5/7}$ law)
and strictly speaking, a power law is hard to identify. Indeed, as was
shown in Fig.~\ref{fig:L_vs_time}, both $L$ and $I$ grow fast in runs
B and D. As a result, arguments based on the quasi-conservation of
$L$ or $I$ should be expected to fail to predict the correct energy
decay. The reason for this failure can be associated with the fact that
rotating flows are not isotropic, and instead become axisymmetric with
a tendency towards two-dimensionalization.

From the arguments in Sec.~\ref{sec:Karman}, we expect only large-scale
correlations in the direction perpendicular to the axis of rotation
to be preserved, and either $L_\perp$ or $I_\perp$ to remain constant
depending on the initial conditions. Therefore, we analyze runs B and D
with the aid of the expansion in Eq.~(\ref{eq:expansion2d}) and of the
anisotropic energy spectrum $E(k_{\perp})$. Following Eqs.~(\ref{eq:E2D})
and (\ref{eq:teodecayrot}), we will also separate the energy into the
energy of slow modes $E_{2D}$, and the energy of fast modes $E_{3D}$.

\begin{figure}.
\includegraphics[width=8cm]{f_figure7.eps}
\caption{(a) Ratio $L_{\perp}(t)/L_{\perp}(0)$ as a function of time
for run B. (b) Evolution of $I_{\perp}(t)/I_{\perp}(0)$ in run D. Note
that in both cases, these magnitudes remain approximately constant after
a short initial transient.}
\label{fig:K_vs_time}
\end{figure}

In Fig.~\ref{fig:spectrum_rot_ani} we show the time evolution of
$E(k_{\perp})$ in run B. Per virtue of Eq.~(\ref{eq:Eisoani}), as the
isotropic spectrum is $E(k)\sim k^2$ for small wavenumbers, the
anisotropic spectrum is $E(k_{\perp})\sim k_{\perp}$. As for $E(k)$,
the shape of the spectrum is preserved during the decay, and
$E(k_{\perp})\sim k_{\perp}$ for small $k_{\perp}$ at all times. However,
unlike the isotropic spectrum, $E(k_{\perp})$ rapidly decreases its
amplitude during a short transient, and then the amplitude of the
spectrum at large scales stabilizes and varies only slowly with time.
Run D shows a similar behavior in $E(k)$ and $E(k_{\perp})$, but
following $E(k_{\perp})\sim k_{\perp}^3$ instead.

We saw in Sec.~\ref{sec:Karman} that constancy of $L_{\perp}$ ($I_{\perp}$)
may be expected for $E(k) \sim k^2$ ($\sim k^4$) initial spectra if
large-scale correlations decay fast enough, and that phenomenological
analysis leads then to $E_{2D}(t) \sim t^{-1/2}$ ($E_{2D} \sim t^{-2/3}$).
As for $L$, we estimate $L_{\perp}$ using two different methods:
By fitting the spectrum for small $k_\perp$, and by using Eq.~(\ref{eq:K}).
The two-point longitudinal correlation function for the axisymmetric
case is estimated using Bessel functions and the anisotropic perpendicular
spectrum (see, e.g., \cite{Davidson2004,Teitelbaum2010}),
\begin{equation}
\left < {\bf u_{\perp} \cdot u_{\perp}'} \right>
(r_{\perp})=2\int E(k_{\perp})J_0(k_{\perp}r_{\perp})dk_{\perp}.
\label{eq:corrperp}
\end{equation}
Both estimations give similar results and the curves discussed below
are obtained from Eqs.~(\ref{eq:K}) and (\ref{eq:corrperp}). The same
procedure was used to estimate $I_\perp$  from Eq.~(\ref{eq:I2D}).

In Fig.~\ref{fig:K_vs_time} we show the evolution of $L_{\perp}$ and
$I_{\perp}$ normalized by their values at $t=0$, for runs B and D
respectively. Although at early times their values decrease rapidly,
they remain afterwards approximately constant during the entire
simulation (note that in the same simulations and at the same times,
$L$ and $I$ increase by at least one order of magnitude).

\begin{figure}
\includegraphics[width=8cm]{f_figure8.eps}
\caption{(a) Time evolution of the energy $E_{3D}$ in modes with
$k_{\parallel} \neq 0$ (dashed), and of the energy $E_{2D}$ in modes with
$k_{\parallel}=0$, for run B. Power laws $t^{-6/5}$ and $t^{-1/2}$
are indicated as a reference, following phenomenological arguments
respectively for the decay of the 3D energy, and for the decay of the 2D
energy based on approximate constancy of $L_\perp$ in Saffman turbulence.
(b) Same for run D. The power laws $t^{-10/7}$ and $t^{-2/3}$ correspond
to the phenomenological predictions for Batchelor turbulence.}
\label{fig:energias2d3d}
\end{figure}

The evolution of the energy for run B shows interesting properties (see
Fig.~\ref{fig:energias2d3d}); $E_{2D}$ initially grows until
$t \approx 10$, when it reaches its maximum value and begins to decrease.
$E_{3D}$ decreases faster than $E_{2D}$ and at $t\approx 20$ both energies
are comparable. After that time, the system is dominated by the energy
in the slow modes. Starting at $t \approx 10$, both energies show a
decay compatible with power laws with different exponents. $E_{2D}(t)$
decays close to $\sim t^{-1/2}$, in agreement with
Eq.~(\ref{eq:teodecayrot}) and with the approximate constancy of
$L_\perp$,  while $E_{3D}$ decays close to $\sim t^{-6/5}$, the value
expected from Eq.~(\ref{eq:teodecay}) for the decay of three-dimensional
Saffman turbulence.

A distinct decay of $E_{2D}$ and $E_{3D}$ is also observed in run D
(corresponding to $\sim k^4$ turbulence, also shown in
Fig.~\ref{fig:energias2d3d}). The decay of the energy in slow modes
is compatible with $E_{2D}(t) \sim t^{-2/3}$, as expected for
Batchelor turbulence from Eq.~(\ref{eq:teodecayrot}) and with the
approximate constancy of $I_\perp$ shown in Fig.~\ref{fig:K_vs_time}.
The 3D energy shows a decay that is close to $E_{3D} \sim t^{-10/7}$.

\begin{figure}
\includegraphics[width=8cm]{f_figure9.eps}
\caption{$E_{2D}$ decay as a function of time for runs B (thin) and D
(thick). The expected decays based on constancy of $L_\perp$ and $I_\perp$
are indicated as a reference. The agreement in this case is much better
than when assuming constancy of the isotropic integrals $L$ and $I$ (see
Fig.~\ref{fig:Exy_rot_kz=0}).}
\label{fig:2Ddecays}
\end{figure}

The decay of $E_{2D}$ in runs B and D is compared in
Fig.~\ref{fig:2Ddecays}. The two different decay laws followed by the
energy in the two runs can be clearly identified. The result indicates
that the decay of rotating turbulence is affected by large scale
correlations, as different power laws can be observed for
$E(k) \sim k^2$ and $\sim k^4$ initial spectra.

The different power laws followed by $E_{2D}$ and $E_{3D}$ after
$t \approx 10$ in both runs point to a negligible interchange of energy
between slow and fast modes at late times, as required for an equation
like Eq.~(\ref{eq:E2D}) to hold. Runs B and D have initial values of
the Rossby number which correspond to the intermediate Rossby range
studied in \cite{Bourouiba2007}. In this range, maximal energy transfer
from 3D to 2D modes takes place at early times, with $E_{2D}$ in some
cases growing from energy in 3D modes as a result (see
Fig.~\ref{fig:energias2d3d} (a)). After this phase, the independent
decay of the two energies implies that these exchanges are small
compared with, e.g., the transfer of energy from vortical motions at
large scales to vortical motions at smaller scales in $E_{2D}$, and
the transfer from 3D modes to 3D modes in $E_{3D}$.

\begin{figure}
\includegraphics[width=8cm]{f_figure10.eps}
\caption{(a) Energy interchanged per unit of time $S$ between 2D and 3D
modes, (a) normalized by $|dE_{2D}/dt|$, and (b) normalized by $|dE_{3D}/dt|$,
for runs B (thin) and D (thick). Positive values of S represent flux
of energy from 2D to 3D modes, while negative values indicate flux
from 3D to 2D motions. At early times $S$ is negative, indicating
energy is transferred from 3D to 2D modes. After $t \approx 10$ in run
B, and $t \approx 10$ in run D, when the distinct power law decays are
observed for $E_{2D}(t)$ and $E_{3D}(t)$ (Fig.~\ref{fig:energias2d3d}),
$S$ becomes negligible.}
\label{fig:S}
\end{figure}

\subsection{Energy exchange between 2D and 3D modes}

A measure of the exchange of energy between 2D and 3D modes per unit
of time can be obtained from the flux of energy across planes in Fourier
space with normal $k_z$. The amount of energy per unit of time going
through any of such planes is given by
\begin{equation}
\Pi(k_\parallel) = - \iiint_{k_\parallel=0}^{k_\parallel} \hat{\bf u}^*_{\bf k} \cdot
    \widehat{\left({\bf u} \cdot \nabla {\bf u}\right)}_{\bf k} \,
    dk_x dk_y dk_z ,
\end{equation}
where the integrals in $k_x$ and $k_y$ run over their entire range
(see \cite{Mininni2011} for definitions of anisotropic fluxes and
spectral transfer functions), and the hat denotes Fourier transform.
The energy transferred from 2D modes to 3D modes per unit of time is
then
\begin{equation}
S = \Pi(k_\parallel=0) .
\end{equation}
When S is positive, it represents transfer of energy from 2D to 3D
motions (i.e., energy going from $k_\parallel=0$ towards larger values of
$k_\parallel$, resulting in an effective source of 3D energy coming from
2D loses), while when negative it represents transfer from 3D to 2D
motions (i.e., a source of 2D energy from 3D loses). For these sources
(or loses) of energy to be negligible, they must be small when compared
with the time derivatives $dE_{2D}/dt$ and $dE_{3D}/dt$. When they are,
the decay of $E_{2D}$ and of $E_{3D}$ can be considered independently.

\begin{figure}
\includegraphics[width=8.5cm]{f_figure13.eps}
\caption{Axisymmetric energy spectrum $e(k_{\parallel}, k_{\perp})/ \sin \theta$
at different times in run A. The circular contour levels indicate an isotropic
energy distribution.}
\label{fig:contourA}
\end{figure}

\begin{figure}
\includegraphics[width=8.5cm]{f_figure14.eps}
\caption{Axisymmetric energy spectrum $e(k_{\parallel}, k_{\perp})/ \sin \theta$
at different times in run B. For rotating flows, the spectral
distribution of energy becomes anisotropic with more energy near
the $k_{\parallel}= 0$ plane. The cross indicates the maximum of the
spectrum at the different times.}
\label{fig:contourB}
\end{figure}

Figure \ref{fig:S} shows the function $S$ normalized by $|dE_{2D}/dt|$
and $|dE_{3D}/dt|$ for runs B and D. In both runs $S$ is negative
before $t\approx 10$, indicating energy goes from 3D to 2D modes,
and explaining the growth of $E_{2D}$ at early times in run B. Note that
in run B, $|dE_{2D}/dt|$ becomes zero at $t\approx 5$, resulting in an
infinite value of the ratio $S/|dE_{2D}/dt|$ in Fig.~\ref{fig:S} (a). Later,
for $t \approx 10$ in run B, and for $t \approx 20$ in run C,
$S/|dE_{2D}/dt|$ and $S/|dE_{3D}/dt|$ become small, indicating the
interchange of energy between slow and fast modes becomes negligible
when compared with the energy decay rates, and justifying the use
of separate balance equations to study the decay of $E_{2D}$ and $E_{3D}$.
Indeed, to have $S \ll |dE_{2D}/dt|$ and $S \ll |dE_{3D}/dt|$ is enough
to consider equations for the evolution of $E_{2D}$ and $E_{3D}$ as was
done in Sec.~\ref{sec:Karman}, as is a more precise definition of
what was meant in that section by the condition of ``weakly coupled''
2D and 3D modes.

It is worth pointing out that the fact that $S/|dE_{2D}/dt|$ and
$S/|dE_{3D}/dt|$ are much smaller than unity does not imply that the 2D and
3D modes are completely decoupled, and only implies that almost no energy
is interchanged between slow and fast modes during the self-similar decay.
Triadic interactions, e.g., between two fast modes and one slow mode, can
still occur and be relevant, as long as the slow mode is only an intermediary
and does not receive from nor give net energy to the fast modes. Indeed, such
interactions between slow and fast modes that preserve the 2D and 3D energy
separatedly would be required to have $dE_{2D}/dt$ proportional to
$1/\Omega$ in Eq.~(\ref{eq:balancerot2D}), as in their absence the decay
of $E_{2D}$ should be that of 2D Navier-Stokes and independent of
$\Omega$ (note triadic interactions between three slow modes are
trivially resonant and independent of $\Omega$).

\begin{figure}
\includegraphics[width=8.5cm]{f_figure15.eps}
\caption{Axisymmetric energy spectrum $e(k_{\parallel}, k_{\perp})/ \sin \theta$
at different times in run D. As in run B, the spectral energy
distribution is anisotropic.  The cross indicates the maximum of the
spectrum at the different times.}
\label{fig:contourD}
\end{figure}

\subsection{Energy spectral distribution}

A deeper understanding of the development of anisotropy in
the flow cannot be obtained solely from studying the reduced
spectrum $E(k_\perp)$ and the isotropic spectrum $E(k)$. To
further investigate the energy spectral distribution, we present
contour plots of the axisymmetric energy spectrum
$e(k_{\parallel}, k_{\perp})$ for runs A, B, and D in
Figs.~\ref{fig:contourA}, \ref{fig:contourB}, and \ref{fig:contourD}
respectively. Note that in order to obtain circular contour levels
when the spectral distribution of the energy is isotropic, in these
figures the axisymmetric energy spectrum is divided by
$\sin \theta$, where $\theta = \arctan(k_{\parallel}/k_{\perp})$.

\begin{figure}
\includegraphics[width=8cm]{f_figure11.eps}
\caption{Axysimmetric energy spectrum $e(k_{\parallel}, k_{\perp})$ for
run B for different values of $k_{\parallel}$ at (a) $t=5$, and (b) $t=10$. A $k_{\perp}^{-3}$ slope is shown as a reference.}
\label{fig:kparaB}
\end{figure}

In the absence of rotation (Fig.~\ref{fig:contourA}) the spectrum
shows an isotropic energy distribution evidenced by circular contour levels
which maintain their shape as the flow decays. However, when rotation
is present, the spectral distribution of the energy becomes
anisotropic with more energy near the $k_{\parallel}= 0$ axis (see
Figs.~\ref{fig:contourB} and \ref{fig:contourD}). Already at $t=10$, the
maximum of the axisymmetric energy spectrum takes place in
the $k_{\parallel}= 0$ axis, in agreement with the preferential
transfer from 3D modes towards slow 2D modes observed at early
times in the flux in Fig.~\ref{fig:S} (as a matter of fact, the
maximum of the spectrum is already located in the $k_{\parallel}= 0$
axis for times as early as $t=5$).

After $t=10$, the exchange of energy between 2D and 3D modes
becomes negligible. As already observed in the time evolution of
$E_{2D}$ and $E_{3D}$, the energy in the 3D modes decays faster than
the energy in 2D modes. This results in an increase of the spectral
anisotropy as time evolves, with most of the energy near the
$k_{\parallel}= 0$ axis at late times. At the same time, the peak of
the spectrum slowly moves through the $k_{\parallel}= 0$ axis towards
smaller values of $k_\perp$, although even at late times the peak is
sufficiently far from $k_\perp=1$.

The shape of the spectrum near the $k_\parallel=0$ plane is of
interest for many theories of homogeneous rotating turbulence (see,
e.g., \cite{Cambon2004,Bellet2006}). It is important to note that here
the domain with finite size results in a discrete set of values for
$k_\parallel$ so the $k_\parallel \to 0$ limit cannot be studied,
unlike homogeneous flows in infinite domains for which
$k_\parallel$ is a continuum. Bearing in mind this limitation, we show
in Figs.~\ref{fig:kparaB} and \ref{fig:kparaD} the axisymmetric
energy spectrum $e(k_{\parallel}, k_{\perp})$ for fixed values of
$k_{\parallel}$ (from 0 to 5), for runs B and D at two different times
($t=5$ and $t=10$).

Figures \ref{fig:kparaB} and \ref{fig:kparaD} confirm that
most of the energy is contained in modes with $k_{\parallel}=0$
(2D modes), and that the wavenumber $k_\perp$ at which the peak
of the spectrum takes place slowly decreases with time (although
the wavenumber is larger than $k_\perp=1$). As an example, in run
B the peak of the spectrum for $k_{\parallel}=0$ is at
$k_{\perp} \approx 20$ at $t = 5$, at $k_{\perp} \approx 15$ at
$t=10$, and at $k_{\perp} \approx 8$ at $t = 35$. The spectrum
for modes with $k_{\parallel}=0$ also shows a positive slope for
wavenumbers smaller than the energy containing wavenumber,
and the slope seems to be approximately preserved as the system
evolves in time.

\begin{figure}
\includegraphics[width=8cm]{f_figure12.eps}
\caption{Axysimmetric energy spectra $e(k_{\parallel}, k_{\perp})$ for
run D for different values of $k_{\parallel}$, at (a) $t=5$, and (b)
$t=10$.}
\label{fig:kparaD}
\end{figure}

The spectra $e(k_{\parallel}, k_{\perp})$ for
$k_{\parallel} \geq 1$ are approximately flat for small wavenumbers (specially for
run B), and show a slope compatible with a $k_{\perp}^{-3}$ power law
(specially for early times, at $t=5$, and for the spectrum
corresponding to $k_{\parallel} = 1$). This is consistent with the
scaling $e(k_{\parallel}, k_{\perp}) \sim k_{\parallel}^{-1/2}k_{\perp}^{-3}$
valid for $k_\parallel/k_\perp \ll 1$
derived from inertial wave turbulence equations in
\cite{Cambon2004}, and also observed before in numerical
simulations in \cite{Thiele2009}.

\section{\label{sec:conclusions}Conclusions}
We have analyzed numerical simulations of freely decaying rotating
turbulence in periodic domains in the regime of intermediate Rossby number
($Ro \approx 0.1$), with initial energy spectra $\sim k^2$ and $\sim k^4$.
We first presented a brief theoretical discussion of conserved integral
quantities for rotating turbulence using a von K\'arm\'an-Howarth
equation which includes the Coriolis term due to rotation, extending
previously derived results for Saffman turbulence \cite{Davidson2010},
considering also the Batchelor spectrum. Assuming 2D and 3D modes are
only weakly coupled in rotating turbulence, constancy of the anisotropic
integral quantities $L_{\perp}$ or $I_{\perp}$ was used to derive
phenomenologically the energy decay rate expected for energy in 2D
modes.

In simulations without rotation, we recovered well known results for
isotropic $\sim k^2$ and $\sim k^4$ turbulence, and also verified the
approximate constancy of the isotropic integrals $L$ and $I$.
On the other hand, in the simulations with rotation a self-similar
decay was difficult to identify for the total energy, while $L$ and $I$
were found to grow rapidly during the entire decay. However, the decay of
energy in two-dimensional and three-dimensional modes followed distinct
power laws, in agreement with the phenomenological predictions and with
constancy of $L_\perp$ and $I_\perp$, which were also observed to remain
approximately constant in the simulations.

The separate power laws followed by $E_{2D}(t)$ and $E_{3D}(t)$ in
decaying rotating turbulence require the interchange of energy between
slow and fast modes to be negligible. We verified that this was the case by
studying the flux of energy across planes with normal $k_z$ in Fourier space.
The results showed that energy is initially transferred from 3D to 2D
modes, as expected for rotating turbulence at intermediate Rossby number
\cite{Bourouiba2007}, but that later the flux of energy between 3D and
2D modes becomes negligible. Contour levels of the axisymmetric
energy spectral distribution show that in presence of rotation the
energy accumulates rapidly near the $k_{\parallel}=0$ plane, and that
at later times the anisotropy increases as a result of the different
decay rates of the $E_{2D}$ and $E_{3D}$ components of the total
energy.

Altogether, the results show that rotating turbulence is indeed affected
by initial large scale correlations in the flow, as different decay laws
arise for $\sim k^2$ and $\sim k^4$ initial spectra.

\begin{acknowledgments}
The authors acknowledge support from UBACYT Grant No.~20020090200692, PICT
Grant No.~2007-02211, and PIP Grant No.~11220090100825.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Q: How can I simulate emission of a standard key sequence? I want to test the "copy" functionality of my Qt widget by simulating emission of QKeySequence::Copy, but QTest::keyClick doesn't accept QKeySequence (or QKeySequence::StandardKey):
void QTest::keyClick ( QWidget * widget, Qt::Key key, Qt::KeyboardModifiers modifier = Qt::NoModifier, int delay = -1 ) [static]
void QTest::keyClick ( QWidget * widget, char key, Qt::KeyboardModifiers modifier = Qt::NoModifier, int delay = -1 ) [static]

I could just hardcode copy as Ctrl+C:
QTest::keyClick(widget, Qt::Key_C, Qt::ControlModifier)

However that would be rather brittle and might fail when testing my widget in other environments, so I'd prefer to use QKeySequence::Copy if possible.
I know I can use QKeySequence::QKeySequence(StandardKey) to find the primary binding, but I can't see how to extract the key and modifier(s).  Alternatively, is there some other way to get the same result?

A: Worked it out: the operator[] returns a bitwise combination of a Qt::Key and the Qt::KeyboardModifiers flags type (this doesn't seem to be documented anywhere).  So the way to extract the key and modifiers is to mask with Qt::KeyboardModifierMask (also completely undocumented):
QKeySequence binding(QKeySequence::Copy);
for (uint i = 0; i < binding.count(); ++i) {
    uint key = binding[i];
    QTest::keyClick(widget, Qt::Key(key & ~Qt::KeyboardModifierMask),
        Qt::KeyboardModifiers(key & Qt::KeyboardModifierMask));
}

A: You could use the QKeySequence::keyBindings() method to get all available key sequences of the given SequenceKey and then extract the keys using the answer in this question.
Alternatively you could convert every QKeySequence to string using the toString function, split the string to a QStringList using + as a separator and then convert every part of the list to the corresponding Qt::Key using a helper function.
QList<QKeySequence> bindings = QKeySequence::keyBindings(QKeySequence::Copy);
for (int i=0; i<bindings.count(); i++)
{
    QStringList keysList = bindings[i].toString().split("+");
    QVector<int> keys = getKeys(keysList);
}

QVector<int> getKeys(const QStringList& list)
{
    QVector<int> keys;
    for (int i=0; i<list.count(); i++)
    {
        if (list[i] == "Ctrl")
            keys << Qt::Key_Control;
        else if (list[i] == "F11")
            keys << Qt::Key_F11;
        else if (list[i] == "Ins")
            keys << Qt::Key_Insert;
        // add remaining modifiers
        // single letters
        else
            keys << list[i][0].toAscii();
    }
    return keys;
}

A: I had the same problem and the accepted answer worked, however I discovered another simple way to do it (just want to add it here in case someone else is looking for a solution).
If using Qt 5.10 or later you can just call:
QTest::keySequence(widget, QKeySequence::Copy);

https://doc.qt.io/qt-5/qtest.html#keySequence-1

redpajama_set_name: RedPajamaStackExchange


text:
Q: Looking for eval function in any .NET language Do any .NET/CLR languages support an eval function and also allow calling into standard .NET code (e.g. calling into C# code)?  I know that neither F# nor C# support eval.  Are there any other options?
I am ideally looking for a solution compatible with .NET 3.5 (so I think this rules out Clojure-CLR), but I'm interested in any/all options.

A: An Eval Function for C# using JScript.NET (JavaScript)

A: Not sure if this helps, but as you said you're looking for any solution.
What about IronRuby?
public static class RubyEngineCreator
{
    private static ScriptEngine ironRubyEngine = null;
    private static ScriptEngine CreateEngine()
    {
        if (ironRubyEngine == null)
            ironRubyEngine = Ruby.CreateEngine();

        return ironRubyEngine;
    }

    public static dynamic GetRubyObject(string script)
    {
        return CreateEngine().CreateScriptSourceFromString(script).Execute();
    }
}

[TestClass]
public class UnitTest
{
    private T Eval<T>(string s)
    {
        return (T)RubyEngineCreator.GetRubyObject(s);
    }

    [TestMethod]
    public void Should_Return4_4()
    {
        var result = Eval<int>("2 + 2");
        Assert.AreEqual(4, result);
    }
}

Example taken from http://viniciusquaiato.com/blog/eval-em-c-com-ironruby/ (pt-BR)

A: JScript.NET has an eval function, see this answer

A: My eventual solution was to adopt IronPython.  JScript.NET technically does work, but it's clumbsy to use with Visual Studio 2008 and JScript.NET seems to be getting deprecated by Microsoft.  IronPython on the other hand has a fairly active community.


text:
Why Firstborns Are Such a Big Deal in the Torah
Bo, Exodus 10:1−13:16
D'Var Torah By: Rabbi Ana Bonnheim
The research abounds: birth order has an impact on development. The Internet teems with articles on expected personality traits for firstborn and later-born children and, in a rare moment of consensus, experts agree that birth order matters.1 It influences a child's need for attention, interest in interacting with adults versus peers, reactions to challenge and pressure, and relationship with parents. As the mother of two young sons (and a firstborn myself), I see evidence of this research daily and often wonder how birth order will affect my children's lives.
Birth order matters in Parashat Bo, too. Bo begins with the final four plagues, culminating in that infamous, horrifying last plague: makat b'chorot, the killing of the firstborn. God takes this concept to its extreme, condemning every single firstborn — whether human or animal — to perish. The Israelite firstborns were saved by placing lambs' blood on their doors.
But the emphasis on the firstborn doesn't end with the 10th plague. After the conclusion of the plagues, God commands Moses, "Consecrate to Me every first-born; human and beast, the first [male] issue of every womb among the Israelites is Mine" (Exodus 13:2). This statement lays the groundwork for a ritual called pidyon haben, in which parents must symbolically redeem their firstborn son from God's ownership via an offering of coins on the 31st day after birth.
While the Torah offers many examples of God's relationship with firstborn humans, animals, and even plants, in this commandment, God essentially places a claim on all Israelite firstborns, forever. While God does not require a sacrifice through death, this is still an exchange nonetheless. Is God really placing some kind of quid pro quo on the Israelites for saving their firstborns from the 10th plague?
God's covenant with the Israelites eschews human sacrifice, but by connecting the firstborn to God in a nonsacrificial way, a new category of religious expression is created. Now, the Israelites have a symbolic way of showing devotion to God and also of confirming that there is no joy taken from the plagues. The plagues culminate in Israel's redemption, and as a result, the Israelites redeem their firstborns as an eternal show of gratitude to God.
Liberal Jews have sometimes struggled with pidyon haben and with the concept of indebtedness to God. The idea of needing to redeem a new baby from God can seem like an ancient ritual, divorced from contemporary ideas of parenthood and newborn life. Additionally, Reform Judaism's emphasis on individual choice reflects that it's challenging to feel indebted to God: shouldn't we be worshipping God and performing rituals because we choose to, not because we're shouldering a never-ending payment on a millennia-old debt?
With all of this weight placed on the firstborn's shoulders, it also raises the question: What is the relationship to God of the other children who do not have to be bought back? Do they matter? I think about my own family: I love my sons differently but equally, something I couldn't conceive of before our younger child was born. The idea that one is more intrinsically holy or connected to God feels somehow like a betrayal of them.
So how, in the modern world, do we reinterpret the idea of firstborn symbolically versus literally? As a society, we no longer view the world in terms of rank by birth order: an older child doesn't receive any different dowry or inheritance than a younger sibling. Or to say it differently, birth order has been democratized.
Nevertheless, we still practice a form of Judaism that stems from God's commandment to symbolically consecrate firstborns. Of course, we no longer sacrifice lives to worship God, but we have built our communal practices of worship and ritual around this concept of expressing our gratitude to God.
Earlier in Exodus, God tells Moses that "Israel is My first-born son" (Exodus 4:22). This is another way of understanding the covenantal relationship between God and Israel. God loves and protects Israel as a firstborn, and in return Israel will forever belong to God. God expands the emphasis on the firstborn as a way to emphasize the developing peoplehood of Israel.
Today, we can use the concept of Israel as God's firstborn as a way to broaden the meaning of the firstborn's ties to God. Essentially, we are all symbolic firstborns, consecrating ourselves to God by actively choosing Judaism and a Jewish life, not because we are in debt to God or because we have to due to birth order or lineage, but because Jewish tradition offers a way to understand the world, our history, and our place in it with depth, meaning, joy, and compassion.
It is ironic that the killing of the Egyptian firstborns in the 10th plague is a primary event that sets up the covenantal relationship between the Israelites and God. Yet, it's also a reminder that our covenant came with a heavy price. We can choose that special firstborn relationship as a way to understand choosing Judaism, and yet we also must remember that our firstborn symbolic uniqueness is tempered with the memory of a legacy that includes that 10th plague.
1. Please see these sources:
E. Cole, Birth Order: An examination of its relationship with the Big Five personality theory and Trait Emotional Intelligence, doctoral thesis, UCL (University College London, 2014)
D.L. Paulhus, P.D. Trapnell, D. Chen, "Birth order effects on personality and achievement within families," Psychological Science, 1999
Rabbi Ana Bonnheim recently moved to Charlotte, NC with her husband and two young children. She served as the associate director, and director of year-round programs, at URJ Greene Family Camp in Texas for the past 8 years.
What It Means to Be the Firstborn
Daver Acher By: Rabbi Amanda Greene
This week's Torah portion forces us to consider the following: Is being the oldest child, the firstborn, really a reward? Or is it an obligation?
Indeed, as Rabbi Bonnheim suggests, symbolically, we all have the opportunity to be the firstborn as the Torah teaches, "Israel is My first-born son" (Exodus 4:22). But wasn't Esau the firstborn? What then is the meaning of the verse? Our ancient tradition offers the following response: "It refers to Jacob, their ancestor, who purchased the birthright in order that he might serve God" (Sh'mot Rabbah 5:7).
Being firstborn, then, is not at all about being born first. It's about a different kind of status, one of commitment to service and committing what we have to divine service. It's not only about service to God through worship, but also about service to our communities and to our world through our actions.
We learn about the firstborn's commitment to God in the very painful passage of the death of all Egypt's firstborn sons. This parallel cannot be ignored, and it calls us — through our traditional frame of the Exodus narrative — to be aware of injustice in our world, and to dedicate ourselves to repair. Since the time of our Exodus, to be a firstborn means to stand up, means to be a role model and a guide for moral compass. To be a firstborn is not merely a right, but an obligation.
"Israel is My first-born son" (Exodus 4:22). Being a firstborn is also not a choice. God has called each of us, all of us, the people of Israel, God's children. Therefore, we have the responsibility and the obligation to live up to that status. To live up to dedicating our hearts and our actions to what God expects: "To do justly and love mercy, and walk humbly with God" (Micah 6:8).
Rabbi Amanda Greene is the assistant rabbi at Chicago Sinai Congregation in Chicago, IL.
Bo, Exodus 10:1-13:16
The Torah: A Modern Commentary, pp. 448-471; Revised Edition, pp. 405–426;
The Torah: A Women's Commentary, pp. 355–378
Haftarah, Jeremiah 46:13−28
The Torah: A Modern Commentary, pp. 700−702; Revised Edition, pp. 427−429
When do we read Bo
January 20, 2024 / 10 Shevat 5784
February 1, 2025 / 3 Shevat 5785
January 24, 2026 / 6 Shevat 5786
Rabbi Ana Bonnheim
Rabbi Ana Bonnheim is the program director for Hebrew Union College-Jewish Institute of Religion's Founders' Fellowship. She lives in Charlotte, N.C.
Rabbi Amanda Greene
Bo Commentaries
Giving What's Due
A Sign on Your Hand, A Reminder Between Your Eyes: An Embodied Jewish Theology of Solidarity and Liberation
Reparations: Seeding a Better Future
Honoring the Innocent Victims of Conflict
Thinking Big and Failing Fast
On the Other Hand: Ten Minutes of Torah – Bo: What is Lost in Translation
English Translation of Bo

redpajama_set_name: RedPajamaCommonCrawl


Textual analysis essay that explores a relationship between Stranger in the Village by James Baldwin and Ceremony by Leslie Marmon Silko.
5-7 MLA-formatted pages, including in-text citations.
must ANALYZE the texts. Do NOT summarize.
The essay must be thesis-driven, with the purposeful organization.
I will not be needing the works cited page included in the essay. That I can do on my own.
The content of this assignment will have an analysis of Ceremony by Silko and Stranger in the Village by James Baldwin which is both exceptionally written pieces. To give a brief background of both; Ceremony is a story about Tayo, a Native American veteran of the World War II, while he struggles to live during a time when his people needed to fight between what the "whites" said is the authentic way of life, and what his culture believes. The story revolves around Tayo's hardship in finding the path meant for him, to other Native Americans that are fighting for their own. On the other hand, Stranger in the Village is about what the author went through in a village located in Switzerland. Baldwin says that he is black and due to his race, villagers see him as unique and fascinating. Based on his portrayal, it seems like these villagers have never laid eyes on a black man.
In the story of Ceremony, there is an exploration of gender roles of 4 women and how it is significant to the evolution of Tayo's persona and character. These 4 women she is surrounded with are her biological mother, Auntie, Grandma, and Ts'eh. Since Tayo has a mixture of different ancestries – half Native American and half white, he soon finds out he is naturally flawed in their culture, which would last until he is grown. However, the book begins by Tayo thinking about the war and recalls the way the Japanese look similar to him because they have the same complexion. One of the characters in the book named Rocky told him that they are ordered to the kill Japanese, but it confuses him why he needs to kill people that look the same and maybe share the same feelings as him. He remains there because he was told he has the privilege to fly planes and make the USA proud. Another memory that occurred upon him is when he clashed paths with the wanted white man and his own path. The thought of the Japanese being like him is startling, and he began putting the people's face he knew on the Japanese soldiers. As time goes by, he could not see why the Japanese had to be killed. When one of the Japanese soldiers killed Rocky, Tayo was shattered and shocked.
The story of Tayo during that time it could be compared to how some soldiers of today might feel when they are at war. However, even if Tayo had this confusion, he still saw his people as the ones who found knowledge. Regardless of what Tayo thought, they exemplified unity and harmony amongst themselves. In the entire book, the difficulties of Tayo are being reflected by every character in the story. They struggle during a drought, disconnect, and loss between the natural state of things and what they currently are. A lot of problems come from this, and these problems are healed by the rediscovery of their origin. In addition, they become part of ceremonies that are created to relate to the truths they lost.


text:
City of Toronto seeks input on vacant home tax intended to increase housing supply
By Noushin ZiafatiThe Canadian Press
Thu., Oct. 21, 2021timer3 min. read
Torontonians are being asked for their input on a new tax aimed at boosting housing supply amid the city's ongoing affordability crisis.
But some observers say they question just how much of an effect the tax, which kicks in next year, will have on Toronto's housing supply.
The vacant home tax is meant to encourage property owners to sell or rent out their unoccupied residential properties to increase Toronto's housing supply – or pay a tax to keep them vacant.
The city has asked residents to fill out a survey, which includes questions on how much they support the tax, whether they agree with proposed exemptions and how often the tax should be evaluated. Submissions are being accepted until Monday.
Deputy Mayor Ana Bailao, who is also the city's housing advocate, said the public consultation will allow the city to finalize some details of the tax, such as who can qualify for an exemption.
"The intent of the tax is really to motivate people who have the homes that are today sitting empty, and that are not fulfilling the role that I believe they should be fulfilling, which is actually being somebody's home ... to rent (them) or to sell (them)," she said.
Beginning in 2022, all property owners will be required to declare the status of their residential homes each year.
A unit is considered vacant if it's been unoccupied for more than six months during the previous calendar year and is not the principal residence of the owner.
People who declare their homes vacant will be required to pay a tax, which will be one per cent of their home's current assessed value. Those who don't could face a fine.
Since the tax is based on a residence's status from the year prior, owners of homes declared vacant in 2022 will have to pay the tax in 2023.
Some exemptions may apply, including for snowbirds, if the registered property owner dies or if the property is undergoing renovations.
The city estimates the tax will generate $55 million to $66 million in revenue annually, based on a one per cent vacancy rate and a one per cent tax.
Jason Mercer, chief market analyst for the Toronto Regional Real Estate Board, said "the jury's still out" on the tax.
"It will be important to track to see if we see more supply that can be attributed to the tax," he said.
"Hopefully, over time, it becomes a tax that essentially collects no revenue, because it will have been successful in prompting investors to bring their units online."
Murtaza Haider, a real estate management professor at Toronto's Ryerson University, said the city's motivation to implement the tax is "right and justified."
But, he questioned how much of an impact the tax will actually have.
"My estimate is that it will have some impact, but not much, primarily because not many homes are left vacant in a city that is seeing such high increases in rents and prices," said Haider.
"Why would somebody keep an asset like that vacant, and if somebody has kept a property vacant then they must have a good reason for it, for which they would apply an exemption status."
Haider said more homes need to be built to ultimately address the lack of housing supply and "rapid price escalation" seen in the market.
"What is needed is far more than what such a tax would deliver," he said. "What is needed is tens of thousands of more homes above and beyond what would have been constructed anyways."
Bailao said the city needs both — more newer builds and a vacant home tax to ensure "that all the units that we're building are being somebody's home."
She added that Toronto will review the impact of the tax and make adjustments if it is not having the effect the city wants to see.
Vancouver introduced an empty homes tax in 2017, making it the first major city in Canada to implement such a measure that was intended to crack down on foreign investment and property speculation. The city says the tax has cut down 25 per cent of vacant properties since its launch.
Ottawa is also introducing a vacant home tax in 2022, while other jurisdictions like Mississauga, Ont., are considering implementing similar taxes.
This report by The Canadian Press was first published Oct. 21, 2021.
This story was produced with the financial assistance of the Facebook and Canadian Press News Fellowship.


In an attempt to be more "missional" and attract new members, Pastor Dave of West Akron Mennonite planted five cleverly hidden geocaches in his church this past Sunday.
After seven schisms within the last two years, West Akron has struggled to keep the doors open, but Pastor Dave plans to hide new geocaches in different locations throughout the sanctuary to keep them coming.
The items inside Dave's geocaches have included an old hymnal, a copy of Martyr's Mirror, and a 6-pack of Diet Pepsi.
The West Akron Mennonite Church is located at N 40° 09.523 W 076° 11.869.


text:
import {
  AfterViewInit,
  ChangeDetectionStrategy,
  Component,
  ElementRef,
  OnDestroy,
  ViewChild,
  ViewEncapsulation,
} from '@angular/core';
import { Terminal } from 'xterm';
import { FitAddon } from 'xterm-addon-fit';
import {
  TerminalOutputService,
  TerminalOutputStreamer,
} from 'src/app/modules/shared/terminals/terminals.service';
import trackByIdentity from 'src/app/util/trackBy/trackByIdentity';
import { TerminalView } from 'src/app/modules/shared/models/content';
import { WebsocketService } from '../../../../../data/services/websocket/websocket.service';
import { AbstractViewComponent } from '../../abstract-view/abstract-view.component';

@Component({
  encapsulation: ViewEncapsulation.None,
  selector: 'app-terminal',
  styleUrls: ['./terminal.component.scss'],
  templateUrl: './terminal.component.html',
  changeDetection: ChangeDetectionStrategy.OnPush,
})
export class TerminalComponent
  extends AbstractViewComponent<TerminalView>
  implements OnDestroy, AfterViewInit
{
  @ViewChild('terminal', { static: true }) terminalDiv: ElementRef;

  selectedContainer = '';
  containers: string[] = [];

  private terminalStream: TerminalOutputStreamer;
  private term: Terminal;
  private fitAddon: FitAddon;

  trackByIdentity = trackByIdentity;

  constructor(
    private terminalService: TerminalOutputService,
    private wss: WebsocketService
  ) {
    super();
  }

  update() {
    if (this.v) {
      this.containers = this.v.config.containers;
    }
  }

  ngOnDestroy(): void {
    if (this.terminalStream) {
      this.terminalStream.scrollback.unsubscribe();
      this.terminalStream.line.unsubscribe();
      this.terminalStream.exitMessage.unsubscribe();
      this.terminalStream = null;
    }
  }

  ngAfterViewInit() {
    const logLevel = 'info';
    const terminal = this.v?.config.terminal;

    if (terminal) {
      const { active } = terminal;
      const disableStdin = !active;
      this.term = new Terminal({
        logLevel,
        disableStdin,
      });
      setTimeout(() => {
        this.initStream();
      });
      this.enableResize();
      this.term.onData(data => {
        if (active) {
          this.wss.sendMessage('action.octant.dev/sendTerminalCommand', {
            key: data,
          });
        }
      });
      this.fitAddon = new FitAddon();
      this.term.loadAddon(this.fitAddon);
      this.term.open(this.terminalDiv.nativeElement);
      this.term.focus();
      this.fitAddon.fit();
    }

    super.ngAfterViewInit();
  }

  enableResize() {
    let timeOut = null;
    const resizeDebounce = (e: { cols: number; rows: number }) => {
      const resize = () => {
        this.wss.sendMessage('action.octant.dev/sendTerminalResize', {
          rows: e.rows,
          cols: e.cols,
        });
        this.fitAddon.fit();
      };

      if (timeOut != null) {
        clearTimeout(timeOut);
      }
      timeOut = setTimeout(resize, 500);
    };
    this.term.onResize(resizeDebounce);
  }

  onContainerChange(containerSelection: string): void {
    this.terminalService.selectedContainer = containerSelection;
    this.selectedContainer = containerSelection;
    this.term.reset();
    this.initStream();
    this.term.focus();
    this.fitAddon.fit();
  }

  initStream() {
    const { namespace, podName, terminal } = this.v.config;
    const { container } = terminal;
    if (this.terminalService.selectedContainer) {
      this.selectedContainer = this.terminalService.selectedContainer;
    }
    if (namespace && podName && container) {
      if (
        this.terminalService.namespace === namespace &&
        this.terminalService.podName === podName &&
        this.selectedContainer
      ) {
        this.selectedContainer = this.terminalService.selectedContainer;
        this.terminalStream = this.terminalService.createStream(
          namespace,
          podName,
          this.selectedContainer
        );
      } else {
        this.terminalStream = this.terminalService.createStream(
          namespace,
          podName,
          container
        );
      }
      this.terminalStream.exitMessage.subscribe((exitMessage: string) => {
        if (exitMessage && exitMessage.length !== 0) {
          this.selectedContainer = undefined;
          this.terminalService.selectedContainer = this.selectedContainer;
        }
      });
      this.terminalStream.scrollback.subscribe((scrollback: string) => {
        if (scrollback && scrollback.length !== 0) {
          this.term.write(atob(scrollback).replace(/\n/g, '\n\r'));
        }
      });
      this.terminalStream.line.subscribe((line: string) => {
        if (line && line.length !== 0) {
          this.term.write(atob(line).replace(/\n/g, '\n\r'));
        }
      });
      this.terminalService.namespace = namespace;
      this.terminalService.podName = podName;
    }
  }

  onResize() {
    this.fitAddon.fit();
  }
}

redpajama_set_name: RedPajamaGithub


Founded by Master Chen Qing Yuan. 8 training venues are provided in Beijing. If you want to learn Tai Chi while are living, working, or travelling in Beijing, this school would be ideal for you.
A new simple and fashionable style of Tai Chi that created based on traditional Chen style Tai Chi; Founded by Jet Li and Jack Ma; located in Tongzhou (outskirts of Beijing) with an excellent study environment and training facilities.
Provide Tai Chi and Daoyin Qigong classes for those who come to Beijing for a short period of time. The classes take place at the famous Temple of Heaven, where you can approach the locals and experience the unique exercises from our specially selected masters.
The center is one of the research and promotion Chenjiagou Taijiquan health culture and practical fighting club, is founded by Mr. Shen Xiaojian, who is the thirteenth generations of outstanding heritage, come from the Taijiquan birthplace of Henan.


If you are in need of great birthday wishes for someone dear to you who is celebrating his or her 75th anniversary, you can start with the fine list we have on this page.
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It is truly joyful to know that the greatest and most caring father in the world is 75 years old today. May good health, joy and happiness never depart your world as you commemorate this Big Day.
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In you, I see a very knowledgeable man, a great counselor, a devoted teacher, and above all, my best role fried. May the divine favor of the Lord be richly present in your world. Have a joyful 75th birthday celebration!
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Congratulations, my dear Mother for achieving another exceptional milestone on this earth. Your life today is just a testament to how incredible a mother you are. May you enjoy a splendid 75th anniversary!
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Dear Mom, may excitement, good health and joy always flow in abundance in your world as you mark this extraordinary day. Happy 75th birthday!
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75? Burn the house down.
Happy 75th birthday, dear Granny! May the good Lord richly bless you for being an absolutely loving grandmother to my siblings and me.
Wishing you fine health and a heart that constantly brims with happiness! Happy 75th birthday, my sweet Grandmother.
As the days turn into years, you tend to grow much stronger and more beautiful. You are indeed an amazing grandmother. Wishing you a blissful 75th birthday!
It will be too little to use words to express how grateful I am to you for all the awesome and lovely things that you have furnished my world with. Wishing a fabulously beautiful 75th birthday to you, Grandma!
Congratulations, dearest Grandma! Excitement and pride fill my heart knowing that you are 7.5 decades old today. May you experience the purest forms of happiness and good health throughout your life.
Wishing the greatest and most fantastic grandmother on this planet a joyful 75th anniversary! Hope your day is as joyful as the joy you bring to this family.
Indeed, you are truly a special blessing from the Lord and this milestone ceremony in your life is the perfect time to show a lot of appreciation to you. Hoping that you have a truly happy 75th birthday!
For a fabulously elegant 75th birthday.
I don't believe that there are many grandfathers out there who are as astounding as you are. It will be inadequate to say thank you to such a wonderful grandpa like you. May all the best of the treasures of heaven be yours as you celebrate your 75th birthday today.
Happy birthday, Grandpa. Hoping your 75th birthday is as brilliant as you are! May you always have good health and a gigantic smile on your handsome face.
A million and one special thanks to the most amazing and incredible grandfather in the world! Having you in our midst is undoubtedly a great blessing from the Almighty God. Have a joy filled 75th anniversary, Grandpa.
My warmest wishes to you on your 75th birthday, Grandpa! May the blessings of the heavens forever keep you happy.
Happy 75th birthday, dear Grandfather! I am proud to be part of a generation led by someone who is as spectacular as you are. May the sunlight always shine in your universe.
Wishing you a glorious 75th birthday, sweet Grandfather! May this exciting new chapter God has blessed you with bring you true excitement, good health, love and happiness without end.


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