expected. The result (67) can be compared with the result of the LRA in Kaneda
et al. (1999),
$G^{(L,T=0)}=1-0.530C_{K}\epsilon^{2/3}k^{4/3}\varepsilon^{2}+O(\varepsilon^{3})$.
The numerical constant in (67), namely $(2\sqrt{3}/35)\pi\simeq 0.311$, is
smaller than that of the LRA result.

Next, let us consider the Lagrangian response function with the ordering
$t_{\ell}\geq t_{m}$, which is the same ordering considered in our numerical
study in section 3. Specifically, setting
$t_{\ell}=t_{m}+\varepsilon~{}(\varepsilon\geq 0)$, the short-time expansion
is

$\displaystyle
G^{(L,T)}_{jj}({\bm{k}},-{\bm{k}},t_{m}+\varepsilon|t_{m}+\varepsilon,t_{m})$
$\displaystyle=$
$\displaystyle\frac{1}{4\tilde{\sigma}^{2}(k)T}\langle\hat{v}_{j}({\bm{k}},t_{m}+\varepsilon|t_{m}+\varepsilon)\hat{\zeta}_{j}(-{\bm{k}},t_{m}+\varepsilon|t_{m})\rangle$
(68) $\displaystyle=$
$\displaystyle\frac{1}{4\tilde{\sigma}^{2}(k)T}\langle\hat{u}_{j}({\bm{k}},t_{m}+\varepsilon)\hat{\zeta}_{j}(-{\bm{k}},t_{m}+\varepsilon|t_{m})\rangle$
$\displaystyle=$
$\displaystyle\tilde{b}_{0}+\tilde{b}_{1}\varepsilon+\tilde{b}_{2}\varepsilon^{2}+\ldots.$

The factor $4$ in the denominator is due to the incompressibility as we will
see. We wish to write the coefficients $\tilde{b}_{0},\tilde{b}_{1}$ and
$\tilde{b}_{2}$ in terms of the Eulerian velocity modes.

To evaluate the velocity-noise correlation, we need the labelling-time
evolution of the random forcing, which is described by the passive vector
equation. Hence its Duhamel solution is

$\displaystyle\hat{\zeta}_{j}({\bm{k}},t_{m}+\varepsilon|t_{m})=\hat{\zeta}_{j}({\bm{k}},t_{m}|t_{m})-{\mathrm{i}}k_{a}\int_{t_{m}}^{t_{m}+\varepsilon}\sum_{\begin{subarray}{c}{\bm{p}},{\bm{q}}\\\
{\bm{p}}+{\bm{q}}+{\bm{k}}={\bm{0}}\end{subarray}}\hat{u}_{a}(-{\bm{p}},s)\hat{\zeta}_{j}(-{\bm{q}},s|t_{m}){\mathrm{d}}s.$
(69)

Similarly, we use the formal solution of the Eulerian velocity to the equation
(63),

$\displaystyle\hat{u}_{j}({\bm{k}},s)=\hat{v}_{j}({\bm{k}},s|s)$
$\displaystyle=\hat{u}_{j}({\bm{k}},t_{m})+\int_{t_{m}}^{s}ds_{1}\big{[}\Lambda_{j}({\bm{k}},s_{1})$
$\displaystyle\quad+(2\pi)^{3}P_{ja}({\bm{k}})\sum_{{\bm{p}}}\hat{\zeta}_{a}({\bm{p}},s|s_{1})\hat{\psi}({\bm{k}},s_{1}|-{\bm{p}},s)\big{]}.$
(70)

Notice that we set the labelling time appearing on the right hand side of (70)
to the same time on the left hand side. Here again we ignore the viscous term
and the large-scale forcing. We also need the formal solution of the labelling
time evolution of the position function

$\displaystyle\hat{\psi}({\bm{k}}^{\prime},t_{m}|{\bm{k}},t_{m}+\varepsilon)=\hat{\psi}({\bm{k}}^{\prime},t_{m}|{\bm{k}},t_{m})-{\mathrm{i}}k_{a}\int_{t_{m}}^{t_{m}+\varepsilon}ds\sum_{\begin{subarray}{c}{\bm{p}},{\bm{q}}\\\
{\bm{p}}+{\bm{q}}+{\bm{k}}={\bm{0}}\end{subarray}}\hat{u}_{a}(-{\bm{p}},s)\hat{\psi}({\bm{k}}^{\prime},t_{m}|-{\bm{q}},s).$
(71)

Using (69)–(71) successively, the velocity-noise correlation in (68) can be
calculated as

$\displaystyle\langle\hat{v}_{j}({\bm{k}},t_{m}+\varepsilon|t_{m}+\varepsilon)\hat{\zeta}^{*}_{j}({\bm{k}},t_{m}+\varepsilon|t_{m})\rangle$
$\displaystyle=4\sigma^{2}(k)T-\sigma^{2}(k)T\varepsilon^{2}\bigg{[}6k_{b}k_{d}\sum_{{\bm{p}}}\langle\hat{u}^{*}_{b}({\bm{p}},t_{m})\hat{u}_{d}({\bm{p}},t_{m})\rangle$
$\displaystyle+k_{b}\langle\hat{u}_{j}({\bm{k}}),t_{m})\hat{u}^{*}_{c}({\bm{k}},t_{m})\rangle\sum_{{\bm{p}}}P_{bcd}(-{\bm{p}})P_{dj}({\bm{k}}+{\bm{p}})$
$\displaystyle+P_{jab}({\bm{k}})\sum_{{\bm{p}}}\left\\{P_{bcj}({\bm{p}}+{\bm{k}})-2k_{c}P_{bj}({\bm{p}}+{\bm{k}})\right\\}$
$\displaystyle\qquad\times\langle\hat{u}^{*}_{a}({\bm{p}},t_{m})\hat{u}_{c}({\bm{p}},t_{m})\rangle\bigg{]}$
$\displaystyle+O(\varepsilon^{3}).$ (72)

Now we again assume that the dominant part is the diagonal components of the
second-order velocity correlations. This leads to the expansion of the
response function

$\displaystyle
G^{(L,T)}_{jj}({\bm{k}},-{\bm{k}},t_{m}+\varepsilon|t_{m}+\varepsilon,t_{m})\simeq
1-\frac{\varepsilon^{2}}{6}$ $\displaystyle\bigg{[}4k^{2}\langle E\rangle$
$\displaystyle+\frac{1}{2}\langle|\hat{{\bm{u}}}({\bm{k}},t_{m})|^{2}\rangle\sum_{{\bm{p}}}\frac{\\{{\bm{k}}\cdot({\bm{k}}+{\bm{p}})\\}\\{{\bm{p}}\cdot({\bm{p}}+2{\bm{k}})\\}}{|{\bm{k}}+{\bm{p}}|^{2}}$
$\displaystyle-\sum_{{\bm{p}}}({\bm{p}}\cdot{\bm{k}})\langle|\hat{{\bm{u}}}({\bm{p}},t_{m})|^{2}\rangle\bigg{]}+O(\varepsilon^{3}).$
(73)

The last term in the $\varepsilon^{2}$ coefficient vanishes. The second term
can be estimated by approximating it with the integral

$\displaystyle\sum_{{\bm{p}}}\frac{\\{{\bm{k}}\cdot({\bm{k}}+{\bm{p}})\\}\\{{\bm{p}}\cdot({\bm{p}}+2{\bm{k}})\\}}{|{\bm{k}}+{\bm{p}}|^{2}}$
$\displaystyle\simeq\left(\frac{L}{2\pi}\right)^{3}\int
d{\bm{p}}\frac{(k^{2}+{\bm{k}}\cdot{\bm{p}})(p^{2}+2{\bm{k}}\cdot{\bm{p}})}{|{\bm{k}}+{\bm{p}}|^{2}}$
$\displaystyle=2\pi
k\int_{k_{0}}^{k_{1}}dp~{}p^{3}\int_{0}^{\pi}d\theta~{}\sin\theta\frac{(k+p\cos\theta)(p+2k\cos\theta)}{k^{2}+p^{2}+2kp\cos\theta}$
$\displaystyle=\frac{2}{3}\pi k^{2}k_{1}^{3}+2\pi k^{3}k_{1}+\mbox{(subleading
terms)}.$ (74)

Here we have dependence on the high-wavenumber cut-off $k_{1}~{}(\gg k)$.
Therefore, the linear response function is written as

$\displaystyle
G^{(L,T)}_{jj}({\bm{k}},-{\bm{k}},t_{m}+\varepsilon|t_{m}+\varepsilon,t_{m})$
$\displaystyle\sim$ $\displaystyle
1-\frac{\varepsilon^{2}}{6}\left[4k^{2}\langle
E\rangle+\frac{1}{6}C_{K}\epsilon^{2/3}k^{-5/3}k_{1}^{3}+\ldots\right]+O(\varepsilon^{3}).$

In this case, the dominant time scale of the linear response function is
$\epsilon^{-1/3}k^{5/6}k_{1}^{-3/2}=\epsilon^{-1/3}k_{1}^{-2/3}(k/k_{1})^{5/6}$,
which is an unexpected result. So far, we do not have a clear interpretation
of this time scale. Nevertheless, this manifests non-locality, which can be
ascribed to the position function. In fact, the $k_{1}$ dependence is coming
from the correlation between the velocity and the advection term in (69).
Although the wavenumber $k$ which we are now probing is much less than
$k_{1}$, the time scale determined by $k_{1}$ (the highest active wavenumber
of the velocity) is a reminiscent of the viscous-convective-range picture of
the passive scalar transport at high Schmidt numbers, see, e.g., Davidson
(2004).

Here we notice that the scaling behaviour obtained above cannot be compared
with our DNS result described in section 3, since our simulation did not have
sufficient scale separation, $k_{0}\ll k\ll k_{1}$, between the beginning and
the ending wavenumbers of the inertial range. Nevertheless, we comment on the
short-time behaviour of the Lagrangian response function observed in DNS shown
in section 3. As shown in figure 7, the Lagrangian response functions for
large wavenumbers ($k_{\eta}/16$ and $k_{\eta}/32$) at short times are so flat
that the parabolic decrease does not fit well. If we plot
$1-G^{L}(k,t_{\ell}-t_{m})$ as a function of $t_{\ell}-t_{m}$ in the log-log
coordinates, the short-time part is almost flat for $k_{\eta}/32$ and is close
to $(t_{\ell}-t_{m})^{0.50}$ for $k_{\eta}/16$. We speculate that these
apparent scaling behaviours are caused by the large-scale forcing (2) and by
some competition between the $\varepsilon$ and $\varepsilon^{2}$ terms of the
short-time Taylor expansion.

##  Appendix E Probability density functions of Eulerian and Lagrangian
velocity Fourier modes

In this section with our numerical data, we show that probability density
functions (PDFs) of the velocity Fourier modes are self-similar and close to
Gaussian. In particular, this holds not only for the Eulerian velocity, but
also for the Lagrangian history velocity. For the Eulerian velocity, we
consider both cases with and without the random noise ${\bm{\xi}}({\bm{k}},t)$
in (10). The setting of the noise is the same as in section 2.3, namely
$\sigma(k)=k^{-1}$ and $T=10^{-6}$. For the Lagrangian velocity, we consider
only the case without the noise.

To calculate the PDF of the Eulerian velocity modes, we consider a shell in
the wavenumber space, $k\leq|{\bm{k}}|<k+\Delta k$ with $\Delta k=1$, as we do
in calculating the energy spectrum. Within this shell characterised with $k$,
we calculate the PDF of the real part of the $\varphi$-component of the
Fourier mode, ${\rm Re}[\hat{u}_{\varphi}({\bm{k}},t)]$. We take 10 snapshots
in the statistically steady state, starting from different initial conditions.
The real parts are then standardised to have zero mean and unit standard
deviation. The resultant PDFs shown in figure 13 for five different wavenumber
shells indicate that they are self-similar and close to Gaussian. This is in
stark contrast to behaviour of PDFs for the Eulerian velocity increments in
the physical space, which are not self-similar nor Gaussian, see, e.g., Frisch
(1996).

To estimate quantitatively how close they are to Gaussian, we calculate the
skewness and kurtosis, which are defined respectively as the 3rd and 4th
moments of the standardised variables. The results listed in table 1 are
indeed around those of the Gaussian distribution, although the smallest and
largest wavenumber cases in table 1 have somewhat larger kurtosis. Similar
results are obtained for the imaginary parts of the $\varphi$-components and
both parts of the $\theta$-components of the Eulerian velocity modes (figure
not shown).

Figure 13:  Probability density functions of the real parts of the Eulerian velocity Fourier modes, ${\rm Re}[\hat{u}_{\varphi}({\bm{k}},t)]$, for $R_{\lambda}=210~{}(k_{\eta}=160)$. Top: without the random noise, or equivalently $T=0$. Bottom: with the random noise. shell wavenumber | skewness | kurtosis
---|---|---
| (without noise) | (with noise) | (without noise) | (with noise)
$k_{\eta}/32$ | $-1.09\times 10^{-1}$ | $-5.70\times 10^{-2}$ | $3.19$ | $3.16$
$k_{\eta}/16$ |  00$1.75\times 10^{-2}$ |  00$4.30\times 10^{-2}$ | $3.08$ | $3.00$
$k_{\eta}/8$ | $-7.89\times 10^{-3}$ |  00$1.06\times 10^{-2}$ | $3.05$ | $3.02$
$k_{\eta}/4$ | $-1.42\times 10^{-2}$ | $-9.03\times 10^{-3}$ | $3.04$ | $3.03$
$k_{\eta}/2$ |  00$2.50\times 10^{-3}$ |  00$3.06\times 10^{-4}$ | $3.05$ | $3.05$
$k_{\eta}$ |  00$9.80\times 10^{-4}$ | $-3.98\times 10^{-3}$ | $3.11$ | $3.05$
Table 1: Skewness and kurtosis of the real part of the Eulerian velocity
Fourier mode, ${\rm Re}[\hat{u}_{\varphi}({\bm{k}},t)]$, for
$R_{\lambda}=210\hphantom{0}(k_{\eta}=160)$ with and without the random noise.
For Gaussian random variables, the skewness is zero and the kurtosis is $3$.

For the Lagrangian history velocity, we use the same method to calculate PDFs
as in the Eulerian case. However, it should be noticed that the Lagrangian
history velocity, $\hat{{\bm{v}}}({\bm{k}},t_{\ell}|t_{m})~{}(t_{\ell}\geq
t_{m})$, is not statistically steady and that it is not solenoidal in general,
${\bm{k}}\cdot\hat{{\bm{v}}}({\bm{k}},t_{\ell}|t_{m})\neq 0$. Hence the
decomposition of the Lagrangian mode has three components as

$\hat{{\bm{v}}}({\bm{k}},t_{\ell}|t_{m})=\hat{v}_{\varphi}({\bm{k}},t_{\ell}|t_{m}){\bm{e}}_{\varphi}+\hat{v}_{\theta}({\bm{k}},t_{\ell}|t_{m}){\bm{e}}_{\theta}+\hat{v}_{c}({\bm{k}},t_{\ell}|t_{m})\hat{{\bm{k}}}$
(76)

in contrast to the Eulerian mode given in (4) with two components. Here
$\hat{{\bm{k}}}={\bm{k}}/|{\bm{k}}|$ and $\hat{v}_{c}$ is the compressible
component which is zero at $t_{\ell}=t_{m}$.

We calculate PDFs of the real and imaginary parts of the three components at
11 different instances,
$t_{\ell}=t_{m}+0.05\tau_{to},~{}t_{m}+0.1\tau_{to},~{}t_{m}+0.2\tau_{to},\ldots,~{}t_{m}+\tau_{to}$.
We use the same 10 snapshots (as we used in the Eulerian case) as the initial
Lagrangian velocity fields ($t_{\ell}=t_{m}$) for the passive vector equations
(18). In figure 14, we plot PDFs of the real parts of the compressible
components at two particular instances $t_{\ell}=t_{m}+0.05\tau_{to}$ and
$t_{m}+0.50\tau_{to}$. We choose this component since it may behave
differently from the solenoidal Eulerian modes, $\hat{u}_{\varphi}$ and
$\hat{u}_{\theta}$. Although the compressible components are zero at
$t_{\ell}=t_{m}$, they quickly develop and their PDFs become self-similar and
close to Gaussian at $t_{\ell}=t_{m}+0.05\tau_{to}$ as shown in figure 14 (how
they develop to Gaussian from zero is beyond the scope of this paper). We
observe as small differences between the two times shown in figure 14 that the
earlier time PDFs have less developed tails and less fluctuations. The PDFs of
the imaginary parts of the compressible component behave in a similar manner.
Both real and imaginary parts of the $\varphi$\- and $\theta$-components at
both times are also similar to those at $t_{\ell}=t_{m}+0.50\tau_{to}$ shown
in figure 14. To observe how close the PDFs are to Gaussian, we list skewness
and kurtosis of ${\rm Re}[\hat{v}_{c}({\bm{k}},t_{\ell}|t_{m})]$ in table 2.
Similar results are obtained for another part of the same components and both
parts of the other components at other instances. In conclusion, we observe
that the PDFs of the Eulerian and Lagrangian history velocity Fourier modes
are close to Gaussian in the inertial and dissipation ranges. Thus we have
verified, for $R_{\lambda}=210$, the same Gaussianity of the Eulerian velocity
modes as have been numerically found for $R_{\lambda}=80$ by Brun & Pumir
(2001).

Figure 14:  Probability density functions of the real parts of the Lagrangian velocity Fourier modes, ${\rm Re}[\hat{v}_{c}({\bm{k}},t_{\ell}|t_{m})]$, for $R_{\lambda}=210~{}(k_{\eta}=160)$. Top: $t_{\ell}=t_{m}+0.05\tau_{to}$. Bottom: $t_{\ell}=t_{m}+0.50\tau_{to}$. shell wavenumber | skewness | kurtosis
---|---|---
| ($t_{\ell}=t_{m}+0.05\tau_{to}$) | ($t_{m}+0.5\tau_{to}$) | ($t_{m}+0.05\tau_{to}$) | ($t_{m}+0.50\tau_{m}$)
$k_{\eta}/32$ |  00$3.77\times 10^{-2}$ | $-2.92\times 10^{-2}$ | $3.08$ | $3.05$
$k_{\eta}/16$ |  00$3.64\times 10^{-2}$ | $-7.28\times 10^{-3}$ | $3.07$ | $3.13$
$k_{\eta}/8$ |  00$7.60\times 10^{-3}$ |  00$2.57\times 10^{-2}$ | $3.04$ | $3.07$
$k_{\eta}/4$ | $-1.14\times 10^{-2}$ | $-8.68\times 10^{-3}$ | $3.02$ | $3.01$
$k_{\eta}/2$ |  00$2.82\times 10^{-3}$ | $-1.03\times 10^{-2}$ | $3.05$ | $3.02$
$k_{\eta}$ | $-1.35\times 10^{-3}$ | $-1.85\times 10^{-3}$ | $3.16$ | $3.04$
Table 2: Skewness and kurtosis of the real part of the Lagrangian velocity
Fourier mode, ${\rm Re}[\hat{v}_{c}({\bm{k}},t_{\ell}|t_{m})]$ at
$t_{\ell}=t_{m}+0.05\tau_{to}$ and $t_{m}+0.50\tau_{to}$ for
$R_{\lambda}=210\hphantom{0}(k_{\eta}=160)$.

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