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helicity_forcing_jz.tex
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\documentclass[pre,preprint,showpacs]{revtex4}
% \documentclass{article}
\usepackage{graphicx}
\begin{document}
\title{forcing}
\author{Mark A. Taylor}
\subsection{Deterministic forcing \label{forcing_det}}
We make use of a determinisitc low wave number forcing, modeled after
the schemes described in \cite{Kerr94,SVBSCC96,OvePop98}, where the
energy in a few low wave numbers is relaxed back to a target spectrum.
Here we instead relax the flow at each time step towards a configuratin
with maximum helicity, as in \cite{Kurien04}, which is based on
that first used in \cite{PolSht89}
We force only those modes in the first wave number shell,
$\tilde {\bf u}_{\bf k}$ for $0.5 \le |{\bf k}| < 1.5$,
where $\tilde {\bf u}_{\bf k}$ is the $k$th Fourier coefficient of
${\bf u}$.
We define the energy in this shell in
the usual way:
\begin{equation}
E(1) = \sum_{0.5 \le |{\bf k}| < 1.5} \frac12 | \tilde {\bf u}_{\bf k} | ^2.
\end{equation}
Let ${\bf u'}$ denote the truncated ${\bf u}$,
\begin{equation}
\tilde {\bf u'}_{\bf k} = \tilde {\bf u}_{\bf k}, \quad 0.5 \le |{\bf k}| <1.5,
\qquad {\bf u'}_{\bf k}=0 \quad \text{otherwise}.
\end{equation}
The flow will be relaxed back to a field
${\bf v}$, chosen so that
$| \tilde {\bf v}_{\bf k}| = |\tilde {\bf u}'_{\bf k}| $
and the phases of $\tilde {\bf v}_{\bf k}$ are chosen so that $\bf v$ has
maximum helicity, using the procedure from \cite{PolSht89}.
Note that $< {\bf u}' \cdot {\bf u}' > = < {\bf v} \cdot {\bf v} > = 2 E(1) $.
The simplest form of this forcing term would be
\begin{equation}
{\bf f} = \tau_1 ( {\bf v} - {\bf u'} )
\end{equation}
and $\tau_1$ is chosen at each timestep in order to obtain a constant
energy injection rate. This forcing function has
the drawback that as ${\bf u}$ approaches maximal helicity,
${\bf v} - {\bf u'}$ will approach zero, forcing
$\tau_1$ to approach infinity, creating a very short
relaxation time scale which requires a very small time step
to resolve. We overcome this drawback
by introducing an addition term,
\begin{equation}
{\bf f} = \tau_1 ( {\bf v} - {\bf u'} ) + \tau_2 {\bf u'}.
\end{equation}
With this form, we can choose $\tau_1 = 1/\Delta t$,
the largest numerically stable
value, and then take $\tau_2$ to insure a constant energy injection rate
$< {\bf u} \cdot {\bf f} > $.
Note that
\begin{equation}
< {\bf u} \cdot {\bf f} > = \tau_1 < {\bf u'} \cdot {\bf v} >
- 2 \tau_1 E(1) + 2 \tau_2 E(1)
\end{equation}
so that $\tau_2$ is given by
\begin{equation}
2 \tau_2 E(1) = < {\bf u} \cdot {\bf f} > +
\tau_1 \left( 2 E(1) - < {\bf u'} \cdot {\bf v} > \right ).
\end{equation}
Using the inequality
\begin{equation}
< {\bf u'} \cdot {\bf v} > \le \left( < {\bf u'} \cdot {\bf u'} >
< {\bf v} \cdot {\bf v} > \right)^{1/2} = 2 E(1),
\end{equation}
we can see that $\tau_2>0$ and thus the forcing will always be the sum
of a term $\tau_1 ( {\bf v} - {\bf u'} )$,
which relaxes the solution towards a maximal helicity configuration,
and a term $ \tau_2 {\bf u'}$,
which increases the energy while preserving the phases. This later
term is most active only when the forced modes of ${\bf u}$ have
large helicity.
\begin{thebibliography}{99}
\bibitem{SVBSCC96} K.R. Sreenivasan, S.I. Vainshtein, R. Bhiladvala,
I. SanGil, S. Chen and N. Cao, Phys. Rev. Lett., {\bf 77} pp. 1488--1491
(1996).
\bibitem{OvePop98} M. R. Overholt and S. B. Pope,
Comp. Fluids, {\bf 27} pp. 11--28 (1998).
\bibitem{Kerr94} N. P. Sullivan, S. Mahalingam, R. M. Kerr,
Phys. Fluids, {\bf 6} pp. 1612--1614 (1994).
\bibitem{Kurien04} S. Kurien, M. A. Taylor, T. Matsumoto,
{\it Cascade time scales for energy and helicity in homogeneous
isotropic turbulence,} Phys. Rev. E, {\bf 69} (2004)
\bibitem{PolSht89} W. Polifke and L. Shtilman,
{\it The dynamics of helical decaying turbulence},
Phys. Fluids A, {\bf 1} pp. 20252033 (1989)
\end{thebibliography}
\end{document}