Relevance of the specific heat of Kelvin modes \\ in superfluid %turbulence experiments at very low temperatures
Uwe R. Fischer
% DON'T CHANGE THIS LINE It is pointed out that %at low temperatures of order mK, the specific heat of helical vortex line excitations, in low temperature superfluid turbulence experiments carried out in helium II, can be of the same order as the specific heat of the phononic quasiparticles. The ratio of Kelvin mode and phonon specific heats scales with $L_0 \, T^{-5/2}$, where $L_0$ represents the smoothed line length per volume within the vortex tangle, such that the contribution of the vortex mode specific heat should %in principle be %measurable % observable for %sufficiently %large, yet experimentally feasible %vortex line densities $L_0 = 10^6 \cdots 10^8 \, {cm}^{-2}$, and at temperatures which are of order $1\cdots 10$ mK.
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\twocolumn[\hsize\textwidth\columnwidth\hsize\csname @twocolumnfalse\endcsname
%\title{Relevance of the specific heat of Kelvin modes \\ in superfluid 
%turbulence experiments at very low temperatures }
\title{%Relevance of the 
Specific heat of the Kelvin modes %energy %specific heat 
%of the vortex tangle 
in  %ultralow 
low temperature superfluid 
turbulence} %experiments}
%\title{Relevance of the 
%thermal energy of the Kelvin oscillation modes in \\ low temperature superfluid 
%turbulence experiments}
%\title{%Relevance of the  
%Low temperature specific heat of Kelvin oscillation modes \\ in superfluid 
%turbulence experiments}
% with excitation of internal oscillation degrees
%of freedom along their axes} 
%concurrent excitation \\ of ring modes}
\author{Uwe R. Fischer} 
%\author{Uwe R. Fischer and Nils Schopohl}
\address{%Lehrstuhl f\"ur Theoretische Festk\"orperphysik \\ 
Eberhard-Karls-Universit\"at T\"ubingen, 
Institut f\"ur Theoretische Physik \\
Auf der Morgenstelle 14, D-72076 T\"ubingen, Germany}


\date{\today}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}                % DON'T CHANGE THIS LINE
It is pointed out that %at low temperatures of order mK,  
the specific heat of helical vortex line excitations,  
in low temperature superfluid turbulence experiments 
carried out in helium II, can be of the same order as the specific heat 
of the phononic quasiparticles. 
The ratio of Kelvin mode and phonon specific heats scales 
with $L_0 \, T^{-5/2}$, where $L_0$ represents the smoothed 
line length per volume within the
vortex tangle, such that the contribution of the vortex mode specific heat
should %in principle 
be %measurable %
observable for 
%sufficiently %large, yet experimentally feasible 
%vortex line densities 
$L_0 = 10^6 \cdots 10^8 \, {\rm cm}^{-2}$,  
and at temperatures which are of order $1\cdots 10$ mK.
\end{abstract}
%\begin{abstract}                % DON'T CHANGE THIS LINE
%It is pointed out that at temperatures of order mK,  
%the specific heat of vortex line excitations in superfluid 
%turbulence experiments is, for sufficiently large 
%vortex line densities in the tangle, 
%of the same order than that of the phononic quasiparticles. 
%%The ratio of vortex mode and phonon specific heat scales 
%%as $L_0/T^{5/2}$.
%\end{abstract}
\

%\pacs{PACS numbers: 66.35.+a, 67.40.Vs} %, 11.27.+d}

\

] \narrowtext



%\section{Comparison of vortex and phonon specific heats}
The problem of superfluid turbulence has seen a renewed  
upsurge of interest, largely due to the fact that turbulence
in superfluids at temperatures of order one Kelvin 
\cite{smith93,maurer,stalp}, has proved to be similar 
to turbulence in normal fluids \cite{frischbook}. 
Of particular interest should be the range of  
lower temperatures, of order a few mK, where the normal fluid density 
is negligible, and a particularly interesting behaviour of 
superfluid turbulence in a very pure form should be 
observable \cite{vinen2000,hendry2000}.  
Since the pioneering work of Vinen \cite{vinen57}, 
the dynamics of superfluid turbulence, especially its generation and 
decay, is to a large extent %as yet 
still not understood. 
More recent efforts in this direction, investigating, {\it e.g.}, 
the energy spectrum and reconnection events %probabilities 
in the tangle, can
be found in \cite{noreabidbrachet97,tsubotaPRB00}. An interesting thermodynamic
approach to directly observable effects of dissipation in low temperature 
superfluid turbulence has been proposed by Samuels and Barenghi 
\cite{samuelsbarenghi98}.  
The effect to be observed, according to this proposal, 
consists in the conversion of (incompressional) kinetic energy 
of the vortex tangle into compressional energy, which then heats the sample. 
In that work, it has been assumed that the specific heat of the turbulent 
helium II sample is given solely by phononic excitations.  
Here, it will be shown that, at temperatures of the 
order a few mK, the total specific heat of line excitations 
in the vortex tangle %with core diameter $\xi_c$ 
%for a total line length encountered in experiments with large line densities, 
is of the same order as the %than the % total 
specific heat of phonons, integrated over the volume of the tangle. 
This will be true for large, yet experimentally feasible, line densities 
in the tangle. Though vortex lines fill out a very small fraction of the 
volume of the turbulent helium sample, the contribution of their oscillations 
to the specific heat should, as will now be shown, 
 be discernible from the phonon specific heat at %these 
low temperatures.  
%The probability of thermal excitation of 
%waves on the vortex line, which are helical Kelvin waves for wavelengths
%large compared with the vortex core size is hence for large, yet feasible
%line densities comparable to the thermal excitation of phonons. 
%Any small temperature increase in the vortex tangle (due to dissipation), 
%at absolute temperatures of the mK order, 
%%reducing line length   
%%due to the radiation of phonons by oscillating lines,  
%pumps to a significant extent energy 
%into the line oscillations inside the vortex tangle. 
   
%At any given temperature
%of this order, the thermal energy stored in Kelvin modes exceeds the thermal
%energy stored in phonons.  
%This has, in particular,  
%implications for the ``vortex heating''
%process proposed by Samuels and Barenghi \cite{samuelsbarenghi98},
%which consists in the conversion of (incompressional) kinetic energy 
%of the vortex tangle into compressional energy. 
%, which then heats the sample. 

   

The total specific heat of the helium II sample is assumed to be composed 
of the specific heat of the elementary {nontopological} 
excitations which constitute the heat bath, living in the three dimensions
of the sample, and the specific heat of the excitations 
propagating along the topological line vortices, 
which live in one spatial dimension. 
Below temperatures of a fraction of one Kelvin, only phonons 
contribute to the specific heat of the nontopological 
excitations, the contribution of the gapful excitations (rotons) 
%largely 
being exponentially suppressed. 
The specific heat of sound excitations in a volume $\cal V$ is 
%\cite{samuelsbarenghi98} 
\begin{eqnarray}
\frac{C_{\rm ph}}{k_B} & = & 
\frac{2\pi^2}{15} {\cal V} \left( \frac{k_B T}{\hbar c_s}\right)^3
%\nonumber\\
%=\frac{2\pi^2}{15} \left( \frac{k_B T}{\hbar c_s/d}\right)^3
%& = &\frac{2\pi^2}{15} \left( \frac{T}{T_0}\right)^3
%\nonumber\\
%& = & 
%%\frac{2\pi^2}{15}\left( \frac{T}{0.7638 \cdot 10^{-7}\,{\rm K}}\right)^3
%%\left( \frac{\tilde L_0}{\tilde c_s}\right)^3
%2.95 \left( \frac{T}{10^{-7}\,{\rm K}}\right)^3
%\left( \frac{\tilde d}{\tilde c_s}\right)^3
\,,  \label{Cph}
\end{eqnarray} 
where $c_s$ is the speed of sound. 
%where $\tilde c_s = c_s[10^4$\,cm/sec] and 
%$\tilde L_0 = L_0$[cm] are in units of typical experimental values and and 
%where the volume ${\cal V}=l^3$ is assumed to be a cube of side length 
%$l$. 
%The temperature $T_0 = \hbar c_s/(k_B l)$ characterizes the crossover to the
%dominance of the $T^3$ law in a given volume. 
The specific heat of the oscillation modes of an isolated 
vortex ring of radius $r_0$ 
%, for temperatures fulfilling $k_B T\ll 0.01 \hbar\omega_c$,  
may generally be written in the form 
\begin{eqnarray}
\frac{C_{\rm L}}{k_B} 
%& \simeq & \frac{2\pi r_0}{\xi_c}\left( 
%\frac{2.22}{\sqrt{\ln n^*_{c}}}\,\sqrt{\frac{k_B T}{\hbar\omega_c}}
%+ 23.27\,  \frac{k_B T}{\hbar\omega_c}
%\right)\nonumber\\
\label{CL}
& = & \frac{2\pi r_0}{\xi_c}\, f \left(\frac{k_B T}{\hbar\omega_c} \right) 
\,. 
%\nonumber\\
\end{eqnarray}
The nondimensional function $f$ depends on the ratio of the 
temperature and the `cyclotron' energy of the vortex core,   
$k_B T/\hbar\omega_c$, %which  
%The (small) ratio $k_B T/\hbar\omega_c$ 
characterizing the separation 
between thermal and core energy physics \cite{BDVPhysFluids85}.
In helium II, $\hbar\omega_c \sim 1 \cdots 10$ K, so that the %small 
parameter
$k_B T / \hbar\omega_c \sim 10^{-3}\cdots 10^{-4}$ for temperatures in the
mK range.     
The cyclotron frequency of the core is defined by  
$%\begin{equation}
\omega_c={\Gamma}/({\pi (\xi_c/2)^2})\,, 
$ %\end{equation}
with $\xi_c$ the core diameter and $\Gamma = h /m$ the quantum 
of velocity circulation. For a line of arbitrary global shape, $2\pi r_0$ 
in (\ref{CL}) is replaced by the smoothed 
arc length of vortex line, {\it i.e.}, the length of line 
with the Kelvin disturbances subtracted \cite{vinen2000}. 
This undisturbed, smoothed  length of line plays the role of ``volume'' 
in the specific heat of the Kelvin modes. 
The smoothed line density of vortices %\cite{vinen2000} 
is designated $L_0=L_0[{\rm cm}^{-2}]$, 
so that we may assign $2\pi r_0 = L_0 {\cal V}$, as an (average)  
filament length entering the specific heat (\ref{CL}). %=2\pi r_0 /l^3$. 
The line density $L_0$, around which there are Kelvin 
fluctuations, is to be 
distinguished  from the full line density $L$,  
which derives from the total (incompressible) 
kinetic energy, with the contribution of the Kelvin waves included.
They differ by a factor logarithmically dependent on the product 
of the largest Kelvin wave number excited and the 
average vortex element 
distance $l=L_0^{-1/2}$ (the smallest possible 
Kelvin wave number is $l^{-1}$) \cite{vinen2000}, so that $L_0$ can be 
about an order of magnitude less than $L$.  

The function $f$ in (\ref{CL}) depends on the spectrum of waves on the 
vortex line.  
%Assuming for a simple estimation that the vortices are arranged in the 
%form of an array 
For wavelengths much larger than the core diameter, 
a single ring vortex has the Kelvin spectrum 
\begin{eqnarray}
\omega_{\rm K} & = &
%\frac{\Gamma n^2}{4\pi r_0^2}\ln 
%\left[8\sqrt e\, \frac{r_0}{\xi_c}\, \frac 1n\right]\equiv 
%\gamma_1  
\frac{\Gamma n^2}{4\pi r_0^2} \ln [{n_c^*}/{n}]\,, \label{Kelvin}
%\qquad \left(1<n \ll n_c/2\right)\,, 
\end{eqnarray}
%\nonumber\\
where $n$ is the number of waves on the circumference (the mode number),
so that the wave number %has magnitude 
$k=n/r_0$.  
%and the parameters 
%$\gamma_1$ %=\gamma_1(\frac{r_0}{\xi_c}),\,
%and $n_c^*$ %=n_c^*(\frac{r_0}{\xi_c})$ 
%are expressed by 
%\begin{equation}  
%$\gamma_1 = {\Gamma }/({4\pi r_0^2})$\,, 
%= \omega_c \left({\xi_c}/{4r_0}\right)^2$\,, 
%and $n_c^*$ is 
The ultraviolet cutoff mode number %$n_c^*$ 
is for a vortex ring  $n_c^* = O(2\pi r_0/\xi_c)$; 
%$n_c^* = O(10r_0/\xi_c)$; 
for a filament of more general shape, the cutoff wave number
$k_c=O(\xi_c^{-1})$, and $n_c^*$ is of order the total length 
of a filament, divided by $\xi_c$. 

If vortex filaments approach each 
other to within a distance corresponding to the inverse Kelvin wave vector 
under consideration, so that $kl\lesssim 1$, 
the Kelvin spectrum will be changed.  
In the expression (\ref{Kelvincontr}) 
for the specific heat of the oscillations  
presented below, we will %henceforth 
assume that vortices remain, at least on average, 
sufficiently well separated to retain the validity of the Kelvin 
spectrum (\ref{Kelvin}) for most of the 
%thermodynamically relevant 
excitations on the filaments.  
Deviations from that spectrum 
%from the Kelvin dependence 
are then too small to be thermodynamically relevant for
the specific heat %contribution 
of the oscillations, and in particular its temperature dependence. 
Conversely, if there are any measurable deviations from the temperature 
dependence of the specific heat of Kelvin waves,  
%manifesting  themselves in the specific heat, 
we will have an indicator 
that the Kelvin spectrum has changed due to the presence of a 
very dense vortex tangle.  
Strictly spoken, these conclusions can %obviously 
only claim validity if the vortex system is in thermodynamic 
equilibrium with itself and the surrounding fluid.
This is the case for a vortex array, generated by a constant rotation rate
$\Omega$ of the superfluid, with resulting 
densities $L_0 = 2\Omega/\Gamma= 2\cdot 10^3 {\rm cm}^{-2}{\rm sec}\,
\,\Omega$. 
The rotation rates to achieve the line densities of order
$L_0 = 10^6 \cdots 10^8 $ cm$^{-2}$, discussed below, are thus in the range 
$10^3 \cdots 10^5 $ rad/sec, {\it i.e.}, of an order of magnitude 
reached in centrifuges only, so that this appears difficult to realize
in experimental low temperature practice.  
In general, the turbulent vortex tangle will not be in a
(local) thermodynamic equilibrium state. 
We may, however, expect that the contribution 
of line oscillations in the tangle to the internal energy 
will still be proportional to the line density $L_0$. 
Furthermore, any deviation from the predicted 
dependence of the line oscillation contribution on temperature (for
line densities which are not too high, so that the Kelvin spectrum 
remains valid),  
will give a measure of how {\em far}
the turbulent tangle is away from thermodynamic equilibrium.

The one-dimensional density of states for the Kelvin modes, within 
logarithmic accuracy, may be written in the form  
\begin{equation}
N_{\rm K}(E)\simeq \frac{\sqrt{\pi}\,r_0}{\sqrt{\hbar \Gamma \ln[n_c^*]}}\, E^{-1/2}
\,.
\end{equation}  
%(4\hbar \gamma_1 \ln[n_c^*]\, E)^{-1/2}
For rings large compared  to $\xi_c$, 
in the continuum approximation of closely spaced excitation 
frequencies, this expression for the density of states 
then enables the calculation of the specific heat 
of the Kelvin modes according to 
\begin{equation}
C_{\rm L}= L_0{\cal V} \frac{\partial}{\partial T}\left( 
\int_0^\infty\!\! dE\, N_{\rm K}(E)\, \frac {E}{\exp[E/k_B T]-1}\right).
\end{equation}
There results the following expression for the 
dimensionless function in (\ref{CL})
%the specific heat of the helical vortex line excitations becomes 
\begin{equation}  
%\frac{C_{\rm L}}{k_B}= \frac{2\pi r_0}{\xi_c}
%\left( 
f_{\rm K}= \frac{3\zeta(\frac32)}{4\sqrt{\pi \ln[n^*_{c}]}}
\,\sqrt{\frac{k_B T}{\hbar\omega_c}}\,.
\end{equation}
%It is easily checked that for the 
%Kelvin spectrum (\ref{Kelvin}), 
More explicitly, 
we may write for the specific heat of the Kelvin modes 
\begin{eqnarray}
\frac{C_{\rm L}}{k_B} 
%& \simeq & 
%%2.22\pi^{3/2} r_0 \sqrt{\frac{k_B T}{\hbar \Gamma \ln [n_c^*]}}
%%12.36
%12.4 \, r_0 \,\sqrt{\frac{k_B T}{\hbar \Gamma \ln [n_c^*]}}\nonumber\\
& \simeq & 
{\cal V} L_0\, \frac{3\zeta(\frac32)}{8}\,
\sqrt{\frac{k_B T}{\hbar \Gamma \ln [n_c^*]}}
\nonumber\\
& = & 
%3.5 %
3.54
\cdot 10^{11}\,{\cal V}[{\rm cm}^3] \,L_0[10^6 {\rm cm}^{-2}] \,  
\sqrt{\frac{T[{\rm mK}]}{\ln [n_c^*]}}\,,
%\qquad\qquad \mbox{(Kelvin modes)}. 
\label{Kelvincontr}
\end{eqnarray} 
the second line giving the absolute magnitude of the specific heat
expressed in terms 
of the temperatures, line densities, and sample dimensions of 
a typical experiment in $^4\!$He.
There is no dependence on $\xi_c$ (save for the very weak one contained in 
$\ln [n_c^*]$), 
as it should be for the large wavelength %Kelvin 
oscillation modes of a vortex line.  
%Recently, Fischer and Schopohl \cite{0008215} have derived 
%the existence of a branch of large mode number waves, linear in $n$, with 
%wavelengths about one order of magnitude above the core diameter. 
%This branch, for temperatures $k_B T\ll 0.01 \hbar\omega_c$,  
%gives a contribution to $f= 23.3\, %23.27 
%k_B T /\hbar\omega_c$, and a dependence
%of $C_{\rm L}$ proportional to the area $2\pi r_0\xi_c$. 
%At temperatures of order mK,
%this term in the specific heat resulting  
%from the large wavenumber end of the spectrum will be subdominant 
%compared to the Kelvin contribution (\ref{Kelvincontr}), 
%and be neglected in what follows. 
\vspace*{2em}
\begin{center}
\begin{figure}[b]
\psfrag{T[mK]}{$T$\,[mK]}
\psfrag{C/sqrtT}{\large
$\frac {C/k_B}{{\cal V}[\mu {\rm m}^3]\sqrt{T[{\rm mK}]}}$}
%{$C/(k_B{\cal V}[\mu {\rm m}^3]\sqrt{T[{\rm mK}]})$}
\psfrag{L1}{$10^6$\,cm$^{-2}$}
\psfrag{L2}{$10^7$\,cm$^{-2}$}
\psfrag{L3}{${\displaystyle\frac{L_0}{\sqrt{\ln[n_c^*]}}}=2.5\cdot 10^7$\,cm$^{-2}$}
{\includegraphics[width=0.48\textwidth]{RevFig1.eps}}
\vspace*{1em}
\caption{\label{C} Low temperature asymptotics of the total specific heat 
per volume ${\cal V}={\cal V}
[\mu {\rm m }^3]$, divided by $\sqrt T=\sqrt{T[{\rm mK}]}$,   
%as a function of $T$, % =T$[mK] 
for three different values of %smoothed line densities 
$L_0/\sqrt{\ln[n_c^*]}$ (the speed of sound $c_s= 240$ m/sec).
%The quantity ${\tilde L}_0\equiv \sqrt{\pi/\ln [n_c^*]}\,L_0$. 
}
\end{figure}\vspace*{0.5em}
\end{center}
The ratio of the specific heats (\ref{CL}) and (\ref{Cph}) is now given by  
\begin{eqnarray}
\frac{C_{\rm L}}{C_{\rm ph}}& = &
\frac{15\left(\hbar c_s\right)^3 L_0}
{4\pi ^2 \xi_c \left( k_B T\right)^3}\,\, f
\left(\frac{k_B T}{\hbar\omega_c}\right)\nonumber\\
%\nonumber\\& = &  \frac{(d^3 L_0/ \xi_c)}
%{2.95 \left( \frac{T}{10^{-7}\,{\rm K}}\right)^3
%\left( \frac{\tilde d}{\tilde c_s}\right)^3
%}\,\, f \left(\frac{k_B T}{\hbar\omega_c} \right)
& = & 
%\frac{15}{2 \pi^2}\sqrt{\frac{\pi}{4\Gamma}} 
%\frac{15}{4 \pi^{3/2}}\frac{2.22}{\sqrt{\ln [n^*_{c}]}}
%\frac1{\sqrt{\Gamma}} 
%\left(\frac{\hbar}{k_B T}\right)^{5/2} c_s^3 L_0\,.
%\frac{1.49}{\sqrt{\ln[n^*_{c}]}}
\frac{0.74}{\sqrt{\ln [n^*_{c}]}}
\frac1{\sqrt{\Gamma}} 
\left(\frac{\hbar}{k_B T}\right)^{5/2} c_s^3 L_0\,.
\label{ratioExplicit}
%& = & 33.86 \left(\frac{c_s[10^4\,{\rm cm/sec}]}{T[{\rm mK}]}\right)^3
%\frac{L_0[10^6 {\rm cm}^{-2}]}{\xi_c [\mbox{\AA}]}
%\, f\,.  
\end{eqnarray}  
We may  express this ratio in units of relevance 
for turbulence experiments in superfluid $^4\!$He:  
%\cite{smith93}, 
%in which typical vortex line densities 
%are of order $L_0\simeq 10^5 \cdots 10^7 $ cm$^{-2}$: % \cite{smith93}: 
\begin{eqnarray}
\frac{C_{\rm L}}{C_{\rm ph}}& = &
%33.86 \left(\frac{c_s[10^4\,{\rm cm/sec}]}{T[{\rm mK}]}\right)^3
%\frac{L[10^6 {\rm cm}^{-2}]}{\xi_c [\mbox{\AA}]}
%\frac{1.67}{\sqrt{\ln[n_c^*]}} \,\frac{c^3_s[2.4\cdot 10^4\,{\rm cm/sec}]}{
\frac{1.67}{\sqrt{\ln[n_c^*]}} \,\frac{c^3_s[2.4\cdot 10^4\,{\rm cm/sec}]}{
T^{5/2}[{\rm mK}]}\,
L_0[10^6 {\rm cm}^{-2}]\,. \label{numbers}
\end{eqnarray} 
%where the cutoff distance $b= L_0^{-1/2}$ ($b= 10^{-3}$\,cm for 
%$L_0=10^6 {\rm cm}^{-2}$ and $\ln[b/\xi_c]\sim O(10)$). 
The contribution of line oscillations to the specific heat thus becomes 
noticeable, using the indicated line density of $L_0=10^6\,{\rm cm}^{-2}$,  
for temperatures which are of the order mK. 
The total specific heat $C=C_{\rm ph}+ C_{\rm L}$ per volume $\cal V$, 
divided by $\sqrt{T}$, is displayed in Fig. \ref{C}. 
The extrapolation of $C/\sqrt T$  
to $T=0$ yields the smoothed line density, up to the very weak 
(square root of) logarithmic 
cutoff dependence on the largest possible 
number of waves on the filaments (the ultraviolet cutoff mode number), 
{\it i.e.}, it gives the quantity  $L_0/\sqrt{\ln[n_c^*]}$. 

In higher temperature superfluid turbulence experiments, 
values of $L\sim 10^8\, {\rm cm}^{-2}$ have been reported \cite{stalp}. 
The number for $L_0$ put into the 
estimates (\ref{Kelvincontr}) and (\ref{numbers}) 
above is thus rather conservative.  
The ultimate limit of achievable line densities in superfluid turbulence, and
in particular the {\em absolute} 
line densities which are reached in turbulence 
experiments at mK temperatures, 
are as yet not clear \cite{hendry2000}. 
Because the conventional second sound 
technique to detect vortices \cite{vinen57}, 
fails at these temperatures, one has to look for
different means to measure the vortex density. One conceivable possibility, 
using a plot like that shown in Fig. \ref{C}, is to measure the specific heat
of the developed turbulent state in the mK range,  divide by $\sqrt T$, 
and extract the (smoothed)  
vortex density from an extra\-polation of the resulting asymptote 
to absolute zero. It should of course be mentioned %at this point 
that an experimental determination of specific heats as small as 
those estimated in equation (\ref{Kelvincontr})
%, for realistic line densities, 
is a less than trivial affair
(though not entirely unrealistic).  
However, even if a direct determination 
of the specific heat of line oscillations $C_{\rm L}$ will prove to be
difficult, this quantity is %certainly 
of relevance for the (thermo-)dynamical
behaviour of the superfluid at the lowest temperatures. 

%This constitutes, in particular, an independent 
% means to measure the smoothed vortex density $L_0$, as compared to the 
%conventional  measurement of  the full vortex 
%density $L$. %, which is  related to the total kinetic energy density.  
%The estimate (\ref{numbers}) above is thus rather conservative.
In the present context, it is also of interest to note 
that in the numerical simulation work of 
Nore {\it et al.} \cite{noreabidbrachet97}, quite large line densities 
of order $L_0 \sim 10^{10}\cdots 10^{11} {\rm cm}^{-2}$ 
have been assumed. 
%(the resulting average line spacing 
%$l=L_0^{-1/2}\simeq 10^{-5}\, {\rm cm}\cdots 10^{-6}\, {\rm cm}$ is 
%only two to three orders of magnitude above $\xi_c$).  
The resulting crossover temperatures, for which 
$C_{\rm ph} = C_{\rm L}$, are consequently 
in these simulations already of order 100 mK. However, 
as already alluded to above, it is likely that 
for very large %st 
densities, modifications of the Kelvin spectrum,  
by mutual induction of the vortex filaments, 
have to be taken into account.

  

According to (\ref{ratioExplicit}), the ratio of the contributions of 
Kelvin line oscillations and phonons to 
the specific heat scales with $L_0\,T^{-5/2}$, 
so that simultaneously relatively %very 
low temperatures and high line densities are
necessary to observe the influence of the specific heat of the Kelvin modes
upon the thermodynamical behaviour of the turbulent superfluid. 
From equation (\ref{numbers}), we can %however 
conclude that the required values are not unreasonably large, and that 
the contribution of the Kelvin %mode  
oscillations in the vortex tangle 
%to the total specific heat of the helium sample 
should indeed be 
observable. % influence. 



The author %has been provided 
acknowledges financial support %by a {\it Forschungsstipendium} of 
by the DFG (FI 690/1-1). 



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%: {\em Local investigation of superfluid turbulence}, 
Europhys. Lett. {\bf 43}, 29-34 (1998).
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%: {\em Kolmogorov Equation in a Fully Developed Turbulence Experiment}, 
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%: {\em Decay of Grid Turbulence in a Finite Channel}, 
Phys. Rev. Lett. {\bf 82}, 4831-4834 (1999). 
\bibitem{frischbook} U. %riel 
Frisch, {\em Turbulence: The legacy of A.N. 
Kolmogorov} (Cambridge University Press, Cambridge, %England, 
1995). 
\bibitem{vinen2000} W. F. Vinen, %: {\em Classical character of turbulence 
%in a quantum liquid}, 
Phys. Rev. B {\bf 61}, 1410-1420 (2000).
\bibitem{hendry2000} S. I. Davis, P. C. Hendry, and P. V. E. McClintock, 
%: {\em Decay of quantized vorticity in superfluid $^4\!$He at mK temperatures}, 
Physica B {\bf 280}, 43-44 (2000).
\bibitem{vinen57} W. F. Vinen, 
%{\em Mutual friction in a heat current 
%in liquid helium II; III. Theory of the mutual friction}, 
%Proc. R. Soc. (London) A {\bf 242}, 493-515 (1957).
Proc. Roy. Soc. (London) A {\bf 242}, 493-515 (1957).
\bibitem{noreabidbrachet97} C. Nore, M. Abid, and  M. E. Brachet,  
%{\em Kolmogorov Turbulence in Low-Temperature Superflows}, 
Phys. Rev. Lett. {\bf 78}, 3896-3899 (1997); C. Nore, M. Abid, and  M. E. Brachet, 
%{\em Decaying Kolmogorov turbulence in a model of superflow}, 
Phys. Fluids {\bf 9}, 2644-2669 (1997). 
\bibitem{tsubotaPRB00} M. Tsubota, T. Araki, and S. K. Nemirovski\v\i,  
%Makoto Tsubota, Tsunehiko Araki, Sergey K. Nemirovski\v\i: 
%{\em Dynamics of vortex tangle without mutual friction in superfluid $^4\!$He}, 
Phys. Rev. B {\bf 62}, 11751-11762 (2000).  
\bibitem{samuelsbarenghi98} D. C. Samuels and C. F. Barenghi, 
%: {\em Vortex Heating in Superfluid Helium at Low Temperatures}, 
Phys. Rev. Lett. {\bf 81}, 4381-4383 (1998).
\bibitem{BDVPhysFluids85}  
Compare the present discussion of the thermodynamics of excitations 
on vortices to the one contained in
C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, 
%: {\em Thermal excitation of waves on quantized vortices}, 
Phys. Fluids {\bf 28}, 498-504 (1985).
%, contains a discussion of the effect 
%of thermal excitation of vortex waves on the free energy of the superfluid   
%at more elevated temperatures. 
%\bibitem{pismennepo} L. M. Pismen, A. A. Nepomnyashchy: {\em Stability of vortex 
%rings in a model of superflow}, Physica D {\bf 69}, 163-171 (1993).
%\bibitem{0008215} U. R. Fischer and  N. Schopohl, 
%{\em Low Energy Short Wavelength Modes on Vortex Rings}, 
%submitted to PRL 15.8.2000, 
%Los Alamos preprint 
%cond-mat/0008215. 

%\bibitem{KoplikLevine96} Joel Koplik, Herbert Levine: {\em Scattering
%of Superfluid Vortex Rings}, Phys. Rev. Lett. {\bf 76}, 4745 (1996)
\end{references}
\end{document}




\bibitem{svistunov} B. V. Svistunov: {\em Superfluid turbulence in the
low-temperature limit}, Phys. Rev. B {\bf 52}, 3647-3653 (1995)





%the heating process of the vortex tangle in superfluid $^4\!$He 
%turbulence experiments 
%carried out in superfluid $^4\!$He 
%does occur by excitation of the one-dimensional 
%filament oscillation modes. % in the vortex tangle.
At these temperatures, 
%the macroscopic motion of the vortex tangle, 
any temperature change  by some dissipation process
does not predominantly increase the energy of the phonon system,  
%(increase the compressible energy),   
but rather excites internal oscillations of the vortex lines in the tangle.  
These internal oscillations themselves radiate phonons and heat 
the thermal bath, so that 
the process is self-amplifying 
until a temperature is reached, above which the phononic specific heat 
dominates.   




This result does not invalidate the conclusion 
that ultimately dissipation of the turbulent kinetic energy 
is due to the radiation of phonons by Kelvin waves 
\cite{vinen2000}, but leads to a different interpretation for  
their existence, at least up to temperatures above which the 
phonon specific heat becomes dominant. 




,   
the thermal excitation energy of the Kelvin modes alone, converted into 
energy of phonons, can give a measurable contribution to the heating 
of the sample, %for typical line densities $L$,  
if the starting temperature is in the range of 10$^{-2}$K. 
The contribution of the competing reconnection processes exciting 
the Kelvin disturbances on the filament \cite{svistunov,tsubotaPRB00} 
should be distinguishable from the contribution of thermal excitations
if the heating for different initial temperatures is measured. 
%, because excitations are already generated thermally to a sufficient extent.
% to be the primary cause of vortex sound radiation. 





%These temperatures 
%are consistent with the requirement $k_BT \ll 0.01 \hbar\omega_c$ 
%for the first line of (\ref{CL}) to hold.   







%$b\simeq 10^{-3}$ cm. 
%Further, then, $a\simeq 10^{-5} \cdots 10^{-4}$ cm, 
%so that 








The line density $L$  
decays in a superfluid turbulence experiment after the source of vortex
excitation has been turned off, 
%and hence $a$ increases and $T_1$ decreases, 
so that at some instant $C_{\rm ph} = C_{\rm L}$,  
and there is a crossover in the characteristic temperature dependence
of the total specific heat. 
%,so that the phonon specific heat part begins to dominate. 





A physically transparent measure for a crossover temperature can be given as
follows. 
First, we define the distance 
\begin{equation}
a\equiv \left(\frac{\xi_c}L\right)^{1/3}= 
\left({b^2}{\xi_c}\right)^{1/3}\,,
\end{equation} 
such that $(l/a)^3$ gives a measure of the total length of
vortex in units of $\xi_c$ in the volume $\cal V$ for given $L$, {\it i.e.}
the prefactor of $f$ in the specific heat (\ref{CL}).    
%The ratio of the specific heats may then further be rewritten to read 
%by the value of the function $f$ 
%and the parameter $T/T_1= k_B T /(\hbar c_s /a)$: 
%\begin{eqnarray}
%\frac{C_{\rm L}}{C_{\rm ph}} & = & 
%%\frac{15}{2\pi ^2}\,\,\frac{f \left(\frac{k_B T}{\hbar\omega_c} \right)}
%%{\left( \frac{k_B T}{\hbar  c_s /a}\right)^3}= 
%\frac{15}{2\pi ^2}\,\frac{f \left(\frac{k_B T}{\hbar\omega_c} \right)}
%{\left( {T}/{T_1}\right)^3} \,,
%\end{eqnarray}
We may then define a crossover temperature 
\begin{eqnarray}
T_1 & = & \frac{\hbar}{k_B} \frac{c_s}{ a } %\nonumber\\
=  7.6%38
\, {\rm mK} \,\, \frac{c_s[10^4\,{\rm cm/sec}]}{a[10^{-5}\, {\rm cm}]}\,.
\end{eqnarray}
below which effects of the specific heat of line oscillations should 
be discernible. The frequency $c_s/a$ 
corresponding to the temperature $T_1$ is the inverse time within which 
a phonon travels the distance $a$. 










A criterion for the validity of the Kelvin spectrum may be derived as 
follows. The part of the 
self-induced velocity of the vortex of radius $r_1$, which is  
due to the Kelvin modes, is of order $\Gamma /(4\pi r_1) n^2 u_n/r_1 $, where
$u$ is the amplitude of the Kelvin wave of mode number $n$. 
The dipole velocity field of another vortex at the place of the first 
(oscillating) vortex, with radius $r_2$, is of order
$\Gamma \pi r_2^2 /l^3$. The Kelvin spectrum will still hold within a tangle 
if the self-induced Kelvin field is much larger than the dipole field, so that
the condition 
\begin{equation}
\left( \frac l{2\pi r_2}\right)^2 \frac{l}{r_1} \frac{n^2 u_n}{r_1}\gg 1
\end{equation} 
obtains. The first two factors both being large for a sufficiently dilute 
tangle, this condition will be fulfilled, because a simple estimate shows





The specific heat %at constant volume 
then evaluates from
\begin{eqnarray*}
C_L & = & \frac{\partial}{\partial T}\left(
2\int_{0}^{\infty}
dE\,\frac{1}{2\sqrt{\gamma_1\ln n^*_c}}
\frac{\sqrt{E}}{e^{\beta E}-1}\right)\\
&=&\frac{1}{\beta ^{\frac{3}{2}}}
\frac{1}{\sqrt{\gamma _{1}\ln n_{c}}}\int_{0}^{\infty }dx\,\frac{\sqrt{x}}
{e^{x}-1}\\
&=&T^{\frac{3}{2}}\cdot 
\frac{1}{\sqrt{\gamma _{1}\ln n_{c}}}\frac{1}{2}\sqrt{\pi }\zeta \left( 
\frac{3}{2}\right) 
\end{eqnarray*}






%, in particular its temperature dependence.   
%$n_c^*= 8\sqrt e\, {r_0}/{\xi_c}\,$. 
cond-mat0102020.extracted_bib
\bibitem{smith93} M. R. Smith, R. J. Donnelly, N. Goldenfeld, and 
W. F. Vinen, %: {\em Decay of Vorticity in Homogeneous Turbulence}, 
Phys. Rev. Lett. {\bf 71}, 2583-2586 (1993). 

\bibitem{maurer} J. Maurer and P. Tabeling, 
%: {\em Local investigation of superfluid turbulence}, 
Europhys. Lett. {\bf 43}, 29-34 (1998).
%
\bibitem{moisy} F. Moisy, P. Tabeling, and H. Willaime, 
%: {\em Kolmogorov Equation in a Fully Developed Turbulence Experiment}, 
%Phys. Rev. Lett. {\bf 82}, 3994-3997 (1999).

\bibitem{stalp} S. R. Stalp, L. Skrbek, and R. J. Donnelly, 
%: {\em Decay of Grid Turbulence in a Finite Channel}, 
Phys. Rev. Lett. {\bf 82}, 4831-4834 (1999). 

\bibitem{frischbook} U. %riel 
Frisch, {\em Turbulence: The legacy of A.N. 
Kolmogorov} (Cambridge University Press, Cambridge, %England, 
1995). 

\bibitem{vinen2000} W. F. Vinen, %: {\em Classical character of turbulence 
%in a quantum liquid}, 
Phys. Rev. B {\bf 61}, 1410-1420 (2000).

\bibitem{hendry2000} S. I. Davis, P. C. Hendry, and P. V. E. McClintock, 
%: {\em Decay of quantized vorticity in superfluid $^4\!$He at mK temperatures}, 
Physica B {\bf 280}, 43-44 (2000).

\bibitem{vinen57} W. F. Vinen, 
%{\em Mutual friction in a heat current 
%in liquid helium II; III. Theory of the mutual friction}, 
%Proc. R. Soc. (London) A {\bf 242}, 493-515 (1957).
Proc. Roy. Soc. (London) A {\bf 242}, 493-515 (1957).

\bibitem{noreabidbrachet97} C. Nore, M. Abid, and  M. E. Brachet,  
%{\em Kolmogorov Turbulence in Low-Temperature Superflows}, 
Phys. Rev. Lett. {\bf 78}, 3896-3899 (1997); C. Nore, M. Abid, and  M. E. Brachet, 
%{\em Decaying Kolmogorov turbulence in a model of superflow}, 
Phys. Fluids {\bf 9}, 2644-2669 (1997). 

\bibitem{tsubotaPRB00} M. Tsubota, T. Araki, and S. K. Nemirovski\v\i,  
%Makoto Tsubota, Tsunehiko Araki, Sergey K. Nemirovski\v\i: 
%{\em Dynamics of vortex tangle without mutual friction in superfluid $^4\!$He}, 
Phys. Rev. B {\bf 62}, 11751-11762 (2000).  

\bibitem{samuelsbarenghi98} D. C. Samuels and C. F. Barenghi, 
%: {\em Vortex Heating in Superfluid Helium at Low Temperatures}, 
Phys. Rev. Lett. {\bf 81}, 4381-4383 (1998).

\bibitem{BDVPhysFluids85}  
Compare the present discussion of the thermodynamics of excitations 
on vortices to the one contained in
C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, 
%: {\em Thermal excitation of waves on quantized vortices}, 
Phys. Fluids {\bf 28}, 498-504 (1985).
%, contains a discussion of the effect 
%of thermal excitation of vortex waves on the free energy of the superfluid   
%at more elevated temperatures. 
%
\bibitem{pismennepo} L. M. Pismen, A. A. Nepomnyashchy: {\em Stability of vortex 
%rings in a model of superflow}, Physica D {\bf 69}, 163-171 (1993).
%
\bibitem{0008215} U. R. Fischer and  N. Schopohl, 
%{\em Low Energy Short Wavelength Modes on Vortex Rings}, 
%submitted to PRL 15.8.2000, 
%Los Alamos preprint 
%cond-mat/0008215. 

%
\bibitem{KoplikLevine96} Joel Koplik, Herbert Levine: {\em Scattering
%of Superfluid Vortex Rings}, Phys. Rev. Lett. {\bf 76}, 4745 (1996)

\bibitem{svistunov} B. V. Svistunov: {\em Superfluid turbulence in the
low-temperature limit}, Phys. Rev. B {\bf 52}, 3647-3653 (1995)





%the heating process of the vortex tangle in superfluid $^4\!$He 
%turbulence experiments 
%carried out in superfluid $^4\!$He 
%does occur by excitation of the one-dimensional 
%filament oscillation modes. % in the vortex tangle.
At these temperatures, 
%the macroscopic motion of the vortex tangle, 
any temperature change  by some dissipation process
does not predominantly increase the energy of the phonon system,  
%(increase the compressible energy),   
but rather excites internal oscillations of the vortex lines in the tangle.  
These internal oscillations themselves radiate phonons and heat 
the thermal bath, so that 
the process is self-amplifying 
until a temperature is reached, above which the phononic specific heat 
dominates.   




This result does not invalidate the conclusion 
that ultimately dissipation of the turbulent kinetic energy 
is due to the radiation of phonons by Kelvin waves 
\cite{vinen2000}, but leads to a different interpretation for  
their existence, at least up to temperatures above which the 
phonon specific heat becomes dominant. 




,   
the thermal excitation energy of the Kelvin modes alone, converted into 
energy of phonons, can give a measurable contribution to the heating 
of the sample, %for typical line densities $L$,  
if the starting temperature is in the range of 10$^{-2}$K. 
The contribution of the competing reconnection processes exciting 
the Kelvin disturbances on the filament \cite{svistunov,tsubotaPRB00} 
should be distinguishable from the contribution of thermal excitations
if the heating for different initial temperatures is measured. 
%, because excitations are already generated thermally to a sufficient extent.
% to be the primary cause of vortex sound radiation. 





%These temperatures 
%are consistent with the requirement $k_BT \ll 0.01 \hbar\omega_c$ 
%for the first line of (\ref{CL}) to hold.   







%$b\simeq 10^{-3}$ cm. 
%Further, then, $a\simeq 10^{-5} \cdots 10^{-4}$ cm, 
%so that 








The line density $L$  
decays in a superfluid turbulence experiment after the source of vortex
excitation has been turned off, 
%and hence $a$ increases and $T_1$ decreases, 
so that at some instant $C_{\rm ph} = C_{\rm L}$,  
and there is a crossover in the characteristic temperature dependence
of the total specific heat. 
%,so that the phonon specific heat part begins to dominate. 





A physically transparent measure for a crossover temperature can be given as
follows. 
First, we define the distance 
\begin{equation}
a\equiv \left(\frac{\xi_c}L\right)^{1/3}= 
\left({b^2}{\xi_c}\right)^{1/3}\,,
