# Turbulence Properties of Interplanetary Coronal Mass Ejections in the Inner
Heliosphere: Dependence on Proton Beta and Flux Rope Structure

S. W. Good Department of Physics, University of Helsinki, PO Box 64, FI-00014
Helsinki, Finland  O. K. Rantala Department of Physics, University of
Helsinki, PO Box 64, FI-00014 Helsinki, Finland  A.-S. M. Jylhä Department of
Physics, University of Helsinki, PO Box 64, FI-00014 Helsinki, Finland  C. H.
K. Chen Department of Physics and Astronomy, Queen Mary University of London,
London E1 4NS, UK  C. Möstl Austrian Space Weather Office, GeoSphere Austria,
Reininghausstrasse 3, 8020 Graz, Austria  E. K. J. Kilpua Department of
Physics, University of Helsinki, PO Box 64, FI-00014 Helsinki, Finland

###### Abstract

Interplanetary coronal mass ejections (ICMEs) have low proton beta across a
broad range of heliocentric distances and a magnetic flux rope structure at
large scales, making them a unique environment for studying solar wind
fluctuations. Power spectra of magnetic field fluctuations in 28 ICMEs
observed between 0.25 and 0.95 au by Solar Orbiter and Parker Solar Probe have
been examined. At large scales, the spectra were dominated by power contained
in the flux ropes. Subtraction of the background flux rope fields reduced the
mean spectral index from $-5/3$ to $-3/2$ at $kd_{i}\leq 10^{-3}$. Rope
subtraction also revealed shorter correlation lengths in the magnetic field.
The spectral index was typically near $-5/3$ and radially invariant in the
inertial range regardless of rope subtraction, and steepened to values
consistently below $-3$ with transition to kinetic scales. The high-frequency
break point terminating the inertial range evolved almost linearly with radial
distance and was closer in scale to the proton inertial length than the proton
gyroscale, as expected for plasma at low proton beta. Magnetic compressibility
at inertial scales did not grow with radial distance, in contrast to the solar
wind generally. In ICMEs, the distinctive spectral properties at injection
scales appear mostly determined by the global flux rope structure while
transition-kinetic properties are more influenced by the low proton beta; the
intervening inertial range appears independent of both ICME features,
indicative of a system-independent scaling of the turbulence.

Solar coronal mass ejections (310) – Interplanetary magnetic fields (824) –
Interplanetary turbulence (830) – Solar wind (1534)

##  1 Introduction

Alongside the continuous outflows of the fast and slow solar winds,
interplanetary coronal mass ejections (ICMEs; Kilpua et al., 2017) represent a
third, impulsive type of solar wind in the heliosphere. The plasma, magnetic
field, and compositional properties of ICMEs differ to those of the other
solar wind types in various significant ways (Zurbuchen & Richardson, 2006),
making ICMEs a distinctive environment for investigating a range of space
plasma phenomena.

A characteristic signature of ICME wind at 1 au is the dominance of magnetic
pressure over the ion component of the plasma pressure (i.e. $\beta_{i}\ll
1$), a result of strong magnetic fields combined with low proton temperatures.
Strong fields in ICMEs are often associated with the presence of large-scale,
nearly force-free magnetic flux ropes (Goldstein, 1983). An ICME is the
interplanetary manifestation of an erupted coronal flux rope, although ICMEs
are not always observed to have a flux rope structure in situ (e.g. Cane &
Richardson, 2003). ICMEs typically have field strengths in excess of the
ambient solar wind field across a wide range of heliocentric distances (e.g.
Wang et al., 2005). Low proton temperatures and the absence of a proton
temperature-velocity correlation in ICMEs have been attributed to rapid
expansion close to the Sun (e.g. Matthaeus et al., 2006) and to the dominance
of magnetic forces in the magnetically closed structure of an ICME flux rope
(Démoulin, 2009). When modeled with polytropes, ICMEs show a non-adiabatic
expansion with radial distance that suggests some local heating by turbulence
(Liu et al., 2006) as in the solar wind generally. The radial evolution of the
magnetic field strength, proton temperature, and density in ICMEs are such
that low $\beta_{i}$ is maintained within ICMEs from the Sun to 1 au and
beyond. This contrasts to the fast and slow winds, in which $\beta_{i}$
approaches unity well before reaching 1 au.

Like other types of solar wind, ICMEs contain fluctuations at all measurable
scales (e.g. Leamon et al., 1998) though at relatively small amplitudes
(Borovsky et al., 2019). The canonical power spectrum of fluctuations in the
interplanetary magnetic field comprises distinctive power-law ranges (e.g
Verscharen et al., 2019). These include a $k^{-1}$ wavenumber spectrum (‘$1/f$
range’ or ‘injection range’) of non-interacting fluctuations at spatial scales
exceeding the correlation length, thought to be an imprint from the wind’s
coronal sources. These large-scale fluctuations supply energy to a
magnetohydrodynamic (MHD) cascade with a $k^{-3/2}$ or $k^{-5/3}$ spectrum
between the correlation length and ion scales, the spectral indices in this
‘inertial range’ being in agreement with various theories of Alfvénic MHD
turbulence (Schekochihin, 2022). Depending on the precise definition of
compressibility used, typically 2-10% of total fluctuation power is contained
within compressive modes at MHD scales (Chen, 2016). At ion scales, spectra
with power-law indices significantly less than the $-7/3$ and $-8/3$ indices
predicted for kinetic Alfvén wave turbulence may be observed; this range,
which arises from a gradual transition between the MHD and kinetic regimes, is
more consistently observed near the Sun than at 1 au (Bowen et al., 2020, and
references therein). At still smaller scales, ion-kinetic effects become fully
developed and a $k^{-2.8}$ spectrum, most likely dominated by kinetic Alfvén
wave turbulence, is present.

In this Letter, we present spectral analysis of magnetic field fluctuations
within ICMEs in terms of the phenomenology described above. For this analysis,
ICMEs observed by the Solar Orbiter (SolO; Müller et al., 2020) and Parker
Solar Probe (PSP; Fox et al., 2016) spacecraft have been selected, allowing
the radial evolution of spectral properties in the inner heliosphere to be
probed. A focus of the study has been to consider how certain well-known
properties that distinguish ICME wind from other wind types – namely,
background fields with a flux rope geometry and low proton $\beta$ across a
wide range of heliocentric distances – may affect various spectral properties
relating to MHD turbulence. Properties analyzed include fluctuation power and
compressibility, spectral indices and break scales, and correlation lengths in
the magnetic field. More broadly, we consider whether ICMEs support turbulence
that is different in nature to the turbulence present in non-ICME wind
occupying a similar parameter regime (e.g. the near-Sun wind at low
$\beta_{i}$), and, if there is some difference, whether the ICME environment
could provide a new regime for understanding aspects of the turbulence.

##  2 Data and Methods

The Helio4Cast ICMECAT database (Möstl et al., 2017, 2020) provides a
regularly updated list of ICMEs observed by SolO and PSP. From the current
list, a total of 28 ICME intervals with good magnetic field and plasma data
coverage were identified and selected for analysis; details of these ICMEs are
listed in the Appendix. The ICMEs were observed at heliocentric distances
ranging from 0.25 to 0.95 au. All of the selected ICMEs had mean proton
$\beta$ less than 0.4 and at least some coherent rotation of the large-scale
magnetic field, signatures characteristic of magnetic clouds. The mean value
of proton $\beta$ across all events was 0.16. Only the proton contribution to
$\beta_{i}$ has been considered, and $\beta_{i}$ is hereafter equated with
proton $\beta$. Magnetic field data from the SolO/MAG instrument (Horbury et
al., 2020) and PSP/FIELDS instrument suite (Bale et al., 2016), and ion
moments from the SolO/SWA (Owen et al., 2020) and PSP/SWEAP-SPC instrument
suites (Kasper et al., 2016), have been used. The SolO/MAG data were at
$\sim$0.127 s resolution and PSP/FIELDS data at 0.007 to 0.438 s resolution
(typically 0.11 s), while data resolution from the plasma instruments varied
between $\sim$1 and 28 s. Power spectral densities (PSDs) of magnetic field
fluctuations in the ICME intervals have been determined using a multitapered
fast-Fourier transform with bandwidth product NW = 5/2.

##  3 Analysis

Figure 1: Smoothed, trace PSD of magnetic field fluctuations for a selection
of the ICMEs analyzed. Red points mark spectral break frequencies.

We begin this section by summarizing key features identified in the spectra,
with the analysis described in detail in the following subsections. Figure 1
shows the smoothed, mean-subtracted, trace PSD (equivalent to the total power
in fluctuations of the magnetic field vector) as a function of spacecraft-
frame frequency, $f_{sc}$, for a selection of the ICMEs analyzed. The spectra
are color-coded according to heliocentric distance, $r$. There is a general
trend toward lower fluctuation power with increasing $r$ at all frequencies. A
similar trend can be seen in an analogous figure with solar wind intervals by
Chen et al. (2020), but with PSD lower in the ICMEs (e.g. by a factor of
$\sim$5 at $f_{sc}=10^{-2}$ Hz) at most scales for a given $r$. The spectral
indices are typically close to $-5/3$ at $f_{sc}\lesssim 10^{-1}$ Hz in the
inertial range, with steepening of the spectral slopes characteristic of a
transition toward the kinetic range evident at higher frequencies (Section
3.1). Only irregular signatures of an $1/f$ range are present at low
frequencies; at $f_{sc}\lesssim 10^{-3}$ Hz, slopes are in some cases steeper
than $-5/3$ due to power in the flux rope fields (Sections 3.1 and 3.3). The
spectral breaks at the high-frequency end of the inertial range, marked with
the red points in Figure 1, show an almost linear tendency towards lower
frequencies with $r$ (Section 3.4).

###  3.1 Spectral Breaks and Slopes

Figure 2: Example spectra from two ICMEs. From top to bottom, the panels show
the trace PSD with power-law fits in the inertial and transition ranges,
magnetic compressibility spectrum, and spectral index of the trace PSD.
Vertical lines mark spacecraft-frame frequencies associated with the spectral
break point, $f_{b}$, the $k\rho_{i}=1$ scale, $f_{\rho_{i}}$, and the
$kd_{i}=1$ scale, $f_{d_{i}}$. Horizontal lines in the bottom panels mark
characteristic $\alpha$ values ($-1$, $-5/3$, and $-2.8$).

The top panels in Figure 2 show the trace PSD for two of the ICMEs, one
observed by PSP at 0.39 au and the other by SolO at 0.89 au. Solid black lines
represent power-law fits in the inertial and transition ranges. The fits have
been extrapolated until they intersect, with the intersection point giving the
spectral break frequency, $f_{b}$. The fits were applied to linear regions of
the spectra such that $f_{b}$ approximately coincides with the midpoint of the
frequency range where the spectral slope, $\alpha$, steepens in the transition
range. The bottom panels in Figure 2 show $\alpha$ values calculated using a
sliding window, with errors determined from the linear fit quality. For both
ICMEs, it can be seen that $\alpha$ mostly fluctuates between $-1$ and $-2$ at
$f_{sc}\leq 10^{-2}$ Hz, is approximately $-5/3$ at $10^{-2}$ Hz $\leq
f_{sc}\leq 10^{-1}$ Hz, and steepens to values below $-3$ at higher
frequencies.

Spacecraft-frame frequencies associated with the proton gyroscale, $\rho_{i}$,
and the proton inertial length, $d_{i}$, are marked in the figure. Applying
Taylor’s hypothesis, which gives wavenumber $k=2\pi f_{sc}/\langle v\rangle$
in the plasma frame, it may be shown that

$k\rho_{i}=\frac{2\pi\sqrt{2k_{B}\langle T_{i}\rangle m_{i}}}{e\langle
v\rangle\langle B\rangle}f_{sc}$ (1)

and

$kd_{i}=\frac{2\pi}{e\langle v\rangle}\sqrt{\frac{m_{i}}{\mu_{0}\langle
n\rangle}}f_{sc},$ (2)

where $T_{i}$, $m_{i}$, $e$, and $v$ are the proton temperature, mass, charge
and speed, respectively, and angle brackets denote interval time averages.
Thus spacecraft-frame frequencies $f_{\rho_{i}}$ and $f_{d_{i}}$ in Figure 2
correspond to the scales at which $k\rho_{i}=1$ and $kd_{i}=1$, respectively.
It can be seen that $f_{b}$ is closer to $f_{d_{i}}$ than to $f_{\rho_{i}}$ in
both of the examples shown.

Of the 28 ICME intervals analyzed, 16 had spectra without any significant non-
power-law features (e.g. spikes, humps, or noise-floor flattening) near the
spectral break. For these 16 spectra, $f_{b}$ could be accurately determined.
The top panel of Figure 3 shows the ratio of $f_{b}$ to $f_{\rho_{i}}$ and
$f_{d_{i}}$ for the 16 events, as a function of the mean $\beta_{i}$ value in
each ICME. Across the range of relatively low $\beta_{i}$ examined,
$f_{b}/f_{d_{i}}$ was nearer unity than $f_{b}/f_{\rho_{i}}$, but with
$f_{b}/f_{\rho_{i}}$ approaching unity as $\beta_{i}$ increased. This trend is
in agreement with the findings of Chen et al. (2014), who determined that
$d_{i}$ is systematically closer to the observed break scale at low
$\beta_{i}$ and $\rho_{i}$ is closer at high $\beta_{i}$.

In order to make comparisons in the same reference frame and in terms of more
physically meaningful units (e.g. Wicks et al., 2010a), all of the spectra are
henceforth considered as a function of $kd_{i}$ rather than $f_{sc}$, with
conversion from $f_{sc}$ given by Equation 2. The bottom panel of Figure 3
shows the sliding-window profile of $\alpha$ across the inertial and ion-
kinetic ranges ($10^{-4}<kd_{i}\leq 3$), averaged across all events. Standard
deviations are indicated by error bars. It can be seen that $\alpha$ is near
$-5/3$ at $kd_{i}\lesssim 0.2$ in the inertial range and steepens to
approximately $-3.5$ at $kd_{i}>1$ in the kinetic range, but with high
variability between events in the latter case. Variability is similarly large
at $kd_{i}\lesssim 3\times 10^{-3}$, where there is some indication of
$\alpha$ being closer to $-3/2$ than $-5/3$. However, single fits at
$kd_{i}\leq 10^{-3}$ that include points below the $kd_{i}\simeq 10^{-4}$
cutoff in Figure 3 give a mean $\alpha\simeq-5/3$. The nature of the spectrum
at low $kd_{i}$ is further considered in Section 3.3. We note also that none
of the trends identified in Figure 3 showed a significant dependence on $r$.
Radial evolution is considered explicitly in Section 3.4.

Figure 3: Ratios of $f_{b}$ to $f_{d_{i}}$ and $f_{\rho_{i}}$ as a function of
$\beta_{i}$ with five-point running average lines (top panel), smoothed
compressibility spectra (middle panel), and mean spectral index (bottom panel)
with characteristic values ($-1$, $-3/2$, $-5/3$, and $-2.8$) marked by
horizontal lines.

###  3.2 Compressibility Spectra

The middle panels of Figure 2 show the magnetic compressibility,
$C=(\delta|\textbf{{B}}|/|\delta\textbf{{B}}|)^{2}$, here obtained by dividing
the PSD of fluctuations in the magnetic field magnitude by the trace PSD (e.g.
Telloni et al., 2021; Zhao et al., 2021). Compressibility gives the fraction
of total fluctuation power in non-Alfvénic compressions as measured by
$\delta|\textbf{{B}}|$. The smoothed spectra of $C$ given by the darker
overlaying lines show the same trend in both ICMEs: a broad minimum centered
in the inertial range and rising values through the transition range, with the
rise beginning at $f_{sc}<f_{b}$. A gradual plateauing of the rise can be seen
between $f_{d_{i}}$ and $f_{\rho_{i}}$ with $C$ reaching $\sim$1/3 at the
highest frequencies, this value being consistent with variance isotropy
saturation (e.g. Matteini et al., 2020). The $C$ values in the inertial range
are similar to those typically seen in non-ICME wind in the inner heliosphere
(Chen et al., 2020), but with $\delta\textbf{{B}}$ and $\delta|\textbf{{B}}|$
power both at lower levels in the ICMEs.

Smoothed compressibility spectra for 15 events are shown in the middle panel
of Figure 3, color-coded to $\beta_{i}$. A moderate correlation between $C$
and $\beta_{i}$ that is qualitatively consistent with an expected relationship
(discussed in Section 4) can be seen in Figure 3 at inertial scales. The
spectra generally take sigmoid forms for $kd_{i}\gtrsim 0.1$, converging to
$C\sim 1/3$ at $kd_{i}\gtrsim 3$.

###  3.3 Correlation Lengths and Flux Rope Fields

Figure 4: Low-frequency analysis. The top-left panel show the magnetic field
timeseries for an example ICME, overlaid with dash-dotted lines indicating the
component mean values and solid lines showing a GH fit. Vertical lines mark
the ICME boundaries. The right panel shows trace PSDs of the mean-subtracted,
GH-subtracted, and modeled GH flux rope field. The bottom-left panel shows
$\lambda$, $\lambda_{M}$, and $\lambda_{GH}$ versus ICME radial width, $L$,
with five-point running averages shown by the smooth lines and $y=x$ by the
dashed line.

We now turn to the low-frequency limit of the inertial range, which is often
found at scales near the correlation length. Magnetic field correlation
lengths, $\lambda$, have been determined for 27 of the ICMEs using data at 1
min resolution. The ICME with the shortest duration (197 min) has been
excluded from this analysis. For each ICME, the autocorrelation of a 224-min
interval starting from the leading edge has been calculated, with lags,
$\tau$, ranging from 0 to 112 min. The interval length has been chosen to
equal two-thirds the duration of the shortest-duration ICME analyzed. An
autocorrelation curve for each field component $j=\\{R,T,N\\}$ has been
obtained and replotted as a function of spatial scale $l=\langle
v\rangle\tau$, where $\langle v\rangle$ is the mean proton speed in the ICME.
The curves have then been fitted with exponentials of the form
$e^{-l/\lambda_{j}}$, with the average of the three $\lambda_{j}$ values,
$\lambda$, taken as the overall value for the ICME interval. A similar
technique has been used by Wicks et al. (2010b).

The correlation length determined in this way is likely sensitive to the
spatial variation of the ICME’s flux rope structure. To reduce this
sensitivity, and thus to estimate the correlation length of the fluctuations
rather than the background structure, correlation lengths have also been
calculated with subtraction of the background field from the timeseries before
performing the autocorrelation. The background field has been estimated with
(i) the mean of each field component in the interval and (ii) with a Gold-
Hoyle (GH) fit to the ICME flux rope (Farrugia et al., 1999), giving
correlation lengths $\lambda_{M}$ and $\lambda_{GH}$, respectively. The axial,
poloidal, and radial components of the GH flux rope in cylindrical coordinates
are given by

$B_{z}(\rho)=B_{0}/(1+\Gamma^{2}\rho^{2})$ (3a) $B_{\phi}(\rho)=\Gamma\rho
B_{0}/(1+\Gamma^{2}\rho^{2})$ (3b) $B_{\rho}(\rho)=0$ (3c)

respectively, where fit-obtained constants $B_{0}$ and $\Gamma$ are the field
magnitude at the axis and the field-line twist per unit length, respectively,
and $\rho$ is the radial distance from the axis. Of the 27 ICMEs analyzed, 15
could be well fitted with the GH rope model. Key parameters of the 15 fits are
listed in the Appendix. Further details of the GH fitting procedure are
described by Kilpua et al. (2019) and Good et al. (2019).

The top-left panel of Figure 4 shows the magnetic field timeseries of an
example ICME in RTN coordinates, with horizontal lines corresponding to the
component mean values and smooth lines to the GH fit profile. The rotation of
the magnetic field at the scale of the ICME duration is well captured by the
GH fit, and is a better approximation of the ICME’s structure than the
component mean values. The bottom left panel in Figure 4 shows the trace PSD
for the same ICME with subtraction of the mean field (i.e. PSD calculated in
the usual way) and with subtraction of the GH fit. While the mean-subtracted
and GH-subtracted spectra overlap at $kd_{i}\gtrsim 10^{-4}$, it can be seen
that power at the lowest frequencies (equivalent to the largest spatial
scales) is reduced in the GH-subtracted spectrum, with a corresponding
reduction in the spectral index from $-1.36$ to $-1.21$ at $kd_{i}\leq
10^{-3}$. Across all ICMEs fitted with the GH model, the average spectral
index reduced from $-1.65$ in the mean-subtracted spectra to $-1.53$ in the
GH-subtracted spectra at $kd_{i}\leq 10^{-3}$. Also displayed in Figure 4 is
the PSD of the GH fit profile (gold line, right panel), which shows the power
contained within the low-frequency, large-amplitude rotation of the flux rope
field. The significantly enhanced power and steep slopes seen at low
frequencies in Figures 1 and 2 are due to such fields.

The bottom-left panel in Figure 4 shows the three ICME correlation lengths
versus ICME radial width, $L=\langle v\rangle T$, where $T$ is the ICME
duration. The mean values of $\lambda$, $\lambda_{M}$ and $\lambda_{GH}$ are
2.3, 1.8 and $1.4\times 10^{6}$ km, respectively, with $\lambda$ also being
larger than both $\lambda_{M}$ and $\lambda_{GH}$ in all ICMEs individually.
All three ICME correlation lengths tend to increase with $L$, and in all cases
are significantly less than $L$. Five-point running average lines indicate
some levelling-off in the increase in correlation lengths at large $L$,
consistent with a nonlinear trend.

###  3.4 Radial Evolution

Figure 5: Radial evolution of magnetic field parameters within the ICME
intervals, color-coded to $\beta_{i}$. From top to bottom, the panels show the
integrated fluctuation power, fluctuation amplitude normalized to the mean
field strength, compressibility, inertial-range spectral index, spectral break
frequency, and mean-subtracted correlation length. Red lines show power-law
fits as a function of $r$.

Figure 5 shows a range of parameters calculated in the ICME intervals as a
function of $r$, with points color-coded to $\beta_{i}$. Values in the top
four panels are averages in the inertial range at $10^{-2}\leq kd_{i}\leq
10^{-1}$. Power-law fits of the form $r^{\gamma_{r}}$ are displayed in red,
with uncertainties in $\gamma_{r}$ determined from the spread in points around
the fit lines. For some parameters, it can be seen qualitatively that at least
some of this spread is due to a dependence on $\beta_{i}$; dependencies on
$\beta_{i}$ in terms of $\beta_{i}^{\gamma_{\beta}}$ power-law fits have also
been estimated. Given that the $\beta_{i}$ dependencies were determined from a
relatively narrow range of low-$\beta_{i}$ values, they may be less accurate
than the estimated $r$ dependencies, or may only be valid at low $\beta_{i}$.
There is some indication of a weak growth in $\beta_{i}$ with $r$
($\beta_{i}\propto r^{0.38\pm 1.29}$), although there is no statistically
significant trend, with a scatter in $\beta_{i}$ values at all $r$.

The integrated power, $\delta B^{2}$, shows a strong $r$ dependence
($\gamma_{r}=-1.52$) and weak $\beta_{i}$ dependence ($\gamma_{\beta}=-0.22$),
while the fluctuation amplitude normalized to the mean field strength across
the interval, $\delta B/\langle B\rangle=\sqrt{\delta B^{2}}/\langle
B\rangle$, grows weakly with $r$ ($\gamma_{r}=0.50$) and $\beta_{i}$
($\gamma_{\beta}=0.26$). Magnetic compressibility, $C$, shows a significant
$\beta_{i}$ dependence ($\gamma_{\beta}=0.77$) and negligible $r$ dependence
($\gamma_{r}=-0.16$). The inertial range spectral index, $\alpha$, is
invariant with both $r$ and $\beta_{i}$ ($\gamma_{r}=-0.01$,
$\gamma_{\beta}=-0.004$). The $f_{b}\propto r^{-0.86}$ dependence is slightly
weaker than the $r^{-1.09}$ dependence found by Bruno & Trenchi (2014) in fast
wind and the $r^{-1.11}$ dependence found by Duan et al. (2020) in slow wind.
Similarly to the $f_{b}/f_{d_{i}}$ ratio (Figure 3), the value of $f_{b}$ is
only very weakly dependent on $\beta_{i}$ within the low-$\beta_{i}$ range
examined here ($\gamma_{\beta}=-0.19$). The growth with $r$ in the mean-
subtracted correlation length, $\lambda_{M}$ ($\gamma_{r}=0.81$), is weaker
than that of the ICME radial width, $L$ ($\gamma_{r}=1.52$), consistent with
the trend seen in the top-right panel of Figure 4. Ruiz et al. (2014) found an
$r^{-0.43}$ dependence for the correlation length in ecliptic solar wind, a
somewhat weaker dependence than our finding for ICMEs. However, the mean value
found by Ruiz. et al. in ICMEs at 1 au ($2.33\times 10^{6}$ km) is consistent
with our trend.

##  4 Discussion

At large scales ($kd_{i}\leq 10^{-3}$), removal of the flux rope fields rather
than the mean fields reduced the mean spectral slope to $-3/2$ from $-5/3$.
Subtraction of the flux rope field is a form of detrending or frequency
filtering that removes power in the spectrum associated with rotation of the
background field, a signature of the flux rope transit over the spacecraft.
Time-varying background fields in ICME intervals contrast with the
approximately constant, Parker-spiral background in the solar wind more
generally. It has been implicitly assumed that the flux rope field is a
passive background that does not participate in the turbulent cascade, with
the residual timeseries representing waves, turbulence, or structures not
related to the global flux rope. The shallower slopes found at large scales
with subtraction of the flux rope field may represent a distinct sub-region of
the inertial range or a roll-over toward a $1/f$ range. Sliding-window
estimates of the spectral index in some cases do show $\alpha\sim-1$ values at
large scales with or without flux rope subtraction, but only sporadically. The
properties of the fluctuations at these scales merit further investigation
given their potential role in supplying energy to the turbulent cascade at
smaller scales.

Correlation lengths of $\sim$4-hr intervals were longer in the total ICME
fields than in the residual fields (mean or flux-rope-subtracted). Longer
correlation lengths in the total field were likely due to the dominant
influence of the background field rotation, with strongly autocorrelated
variations at timescales of order $\sim$1 hr. ICMEs not displaying simple
rotations in the background field may also have ordered global structures with
relatively long correlation lengths. The correlation lengths in the
background-subtracted fields may be related to the outer scale of turbulence,
or, alternatively, to some typical mesoscale length of ICME substructure
(Lugaz et al., 2018). Taking the mean-subtracted correlation length,
$\lambda_{M}$, as the turbulent outer scale and the high-frequency break scale
calculated with Taylor’s hypothesis, $l_{b}$, as the inner scale, then the
span of the inertial range defined as $\lambda_{M}/l_{b}$ was approximately
invariant with $r$ and $\beta_{i}$, at a constant value of $2.0\pm 1.2\times
10^{4}$ in the ICMEs analyzed. Correlation lengths increased with radial
distance (as in the solar wind generally) and with the ICME radial width, $L$.
Treating $L$ as the system scale size, then the normalized correlation length,
$\lambda_{M}/L$, weakly fell with $r$, i.e. $L$ increased more strongly than
$\lambda_{M}$ with $r$.

At $kd_{i}\gtrsim 10^{-3}$, spectral indices in the ICMEs were the same in the
mean and flux-rope-subtracted fields. The presence at inertial scales of a
$-5/3$ index across a range of heliocentric distances suggests turbulent
states that are already well developed close to the Sun, that develop
independently of the specific energy injection processes occurring at larger
scales, and that may be related to the low cross helicity commonly found in
ICMEs (Good et al., 2020, 2022; Soljento et al., 2023). Low cross helicity and
an associated $-5/3$ index (Podesta & Borovsky, 2010) may in turn be general
features of solar wind originating from closed-field regions in the corona
(e.g. Borovsky et al., 2019). For example, solar wind near the heliospheric
current sheet (HCS) has also been observed to have a $-5/3$ index close to the
Sun (Chen et al., 2021), in contrast to the more typical $-3/2$ value seen
further from the HCS at the same distances (e.g. Chen et al., 2020); the
radial evolution from a $-3/2$ to $-5/3$ index seen in the solar wind away
from the HCS is likely caused by an evolution in cross helicity from highly
imbalanced to balanced values (McIntyre et al., 2023). Like most ICMEs, the
HCS forms at the closed-field streamer belt, and ICMEs often locally replace
the HCS in situ (Crooker et al., 1998): thus some similarity between inertial-
range spectral properties of near-HCS and ICME plasma might be expected.
Steepening of the spectral index to values significantly below $-2.8$ in the
transition range was observed consistently at all radial distances, unlike in
the solar wind more generally. This behavior may be related to the ICMEs’ low
$\beta_{i}$, a dependency recently highlighted by Matteini et al. (2020).

The approximate invariance of magnetic compressibility with $r$ is notable. It
has been suggested by Verscharen et al. (2017) that the small amount of
compressive fluctuation power in the solar wind is primarily due to the MHD
slow mode rather than the kinetic slow mode. By assuming that the majority
incompressible power is due to the MHD Alfvén mode, Chen et al. (2020) find
that the magnetic compressibility is given by
$C=(\epsilon^{2}\beta_{i}\gamma\sin^{4}\theta_{kB})/2$, where $\epsilon$ is
the Alfvén-to-slow-mode amplitude ratio, $\gamma$ is the adiabatic index, and
$\theta_{kB}$ is the slow-mode propagation angle relative to the mean field.
Assuming further that $\gamma$ and $\theta_{kB}\sim 90^{\circ}$ are constant
with $r$, the $C\propto r^{-0.16}$ and $\beta_{i}\propto r^{0.38}$
dependencies found in the ICMEs suggests that $\epsilon^{2}\propto r^{-0.54}$.
While this estimate carries a large uncertainty, it indicates that ICMEs are a
plasma environment in which the slow mode does not develop (or is even
suppressed) relative to the Alfvén mode with heliocentric distance, in
contrast to the solar wind more generally. The particularly low power in
compressive modes, which are a source of ion heating at inertial scales
(Schekochihin et al., 2019, and references therein), may partly contribute to
the low proton temperatures observed in ICMEs. A better understanding of
turbulent heating in ICMEs could provide more stringent constraints for
modeling their thermodynamic evolution with radial distance, and thus lead to
more accurate modeling of ICME expansion and propagation for space weather
prediction purposes.

##  5 Conclusion

We have analyzed power spectra of magnetic field fluctuations within 28 ICMEs
observed throughout the inner heliosphere by the PSP and SolO spacecraft. Low
$\beta_{i}$ across a wide range of heliocentric distances and a global
magnetic flux rope structure make ICMEs a distinctive space plasma
environment. At large spatial scales comparable to the correlation length
($kd_{i}\lesssim 10^{-4}$), a significant fraction of power was contained
within the background flux rope fields of the ICMEs. Subtraction of these
fields from the timeseries revealed shorter correlation lengths and shallower
spectral slopes that on average reduced from $-5/3$ to $-3/2$ at $kd_{i}\leq
10^{-3}$, with some sporadic signatures of a $-1$ slope also at these scales.
The mean value of the spectral slope deep in the inertial range was $-5/3$ and
did not show the $-3/2$ to $-5/3$ radial evolution observed in non-ICME wind
away from the HCS. The inertial range terminated at scales closer to the
proton inertial length than the proton gyroscale, consistent with previous
analysis of low-$\beta_{i}$ plasma. Steepening of the spectral slope to values
below $-3$ in the transition range was observed at all radial distances.
Magnetic compressibility in the inertial range held similar values to the non-
ICME wind in the inner heliosphere but did not grow with radial distance to 1
au, likely mirroring very weak radial variations in $\beta_{i}$ and the
Alfvén-to-slow-mode fraction. Scaling of the Alfvénic turbulence in the
inertial range appears to be independent of the global flux rope structure and
low $\beta_{i}$ that characterize ICMEs, suggestive of its universal, system-
independent nature.

This work was funded by an Academy of Finland Research Fellowship (grants
338486 and 346612; INERTUM). C.H.K.C. is supported by UKRI Future Leaders
Fellowship MR/W007657/1 and STFC Consolidated Grants ST/T00018X/1 and
ST/X000974/1. C.M. is funded by the European Union (ERC, HELIO4CAST,
101042188). E.K.J.K. acknowledges support from Academy of Finland Centre of
Excellence FORESAIL (grant 336807) and from the European Research Council
under the European Union’s Horizon 2020 research and innovation programme,
grant 724391 (SolMAG). Views and opinions expressed are those of the authors
only and do not necessarily reflect those of the European Union or the
European Research Council Executive Agency. Neither the European Union nor the
granting authority can be held responsible for them. We thank the Solar
Orbiter and Parker Solar Probe instrument teams for providing the data used in
this work. The start and end times of the ICMEs analyzed in this study are
listed in Table 1, and are taken from the Helio4Cast ICMECAT database
(https://helioforecast.space/icmecat). For those ICMEs that have been fitted
with the Gold-Hoyle flux rope model, key parameters of the fits are also
listed in the table. The fit parameters include: impact parameter of the
spacecraft with the flux rope, $p$ (‘0’ indicating an axis encounter, ‘1’ an
edge encounter); the angle between the rope axis and the R–T plane,
$\theta_{0}$; the angle between the projection of the rope axis onto the R–T
plane and the R direction, $\phi_{0}$; the magnetic field strength at the rope
axis, $B_{0}$; and the field-line twist per unit length, $T$.

Table 1: List of ICMEs Analyzed Event number | Start time (UT) | End time (UT) | Spacecraft |  $r$ (au) | $p$ |  $\theta_{0}$ (o) |  $\phi_{0}$ (o) |  $B_{0}$ (nT) |  $T$ (au-1)
---|---|---|---|---|---|---|---|---|---
1 | 2018-10-30 20:25 | 2018-10-31 08:19 | PSP | 0.26 | 0.17 | -11 | 153 | 56.0 | -5.2
2 | 2018-11-11 23:51 | 2018-11-12 05:59 | PSP | 0.25 | 0.21 | 18 | 79 | 100.6 | 7.3
3 | 2019-03-15 12:11 | 2019-03-15 17:49 | PSP | 0.55 | 0.49 | 10 | 106 | 36.0 | 3.6
4 | 2019-03-24 03:45 | 2019-03-24 17:38 | PSP | 0.39 | 0.58 | 31 | 119 | 36.7 | 3.0
5 | 2020-02-11 05:08 | 2020-02-11 11:35 | PSP | 0.42 | 0.02 | 10 | 89 | 43.6 | 4.6
6 | 2020-05-28 08:50 | 2020-05-28 14:59 | PSP | 0.35 | – | – | – | – | –
7 | 2020-06-23 07:07 | 2020-06-23 16:52 | PSP | 0.48 | 0.70 | 39 | 310 | 20.1 | -2.7
8 | 2020-06-25 15:59 | 2020-06-26 08:15 | PSP | 0.52 | – | – | – | – | –
9 | 2020-09-12 13:33 | 2020-09-12 19:35 | PSP | 0.47 | 0.56 | 32 | 322 | 23.7 | 7.3
10 | 2020-10-28 13:24 | 2020-10-28 16:41 | PSP | 0.70 | – | – | – | – | –
11 | 2021-02-11 16:17 | 2021-02-12 00:57 | PSP | 0.63 | 0.26 | 15 | 291 | 15.4 | 2.6
12 | 2021-02-12 11:17 | 2021-02-12 23:59 | PSP | 0.64 | – | – | – | – | –
13 | 2021-05-06 18:26 | 2021-05-07 15:07 | SolO | 0.91 | 0.48 | 18 | 250 | 13.6 | -3.1
14 | 2021-05-10 14:01 | 2021-05-11 11:39 | SolO | 0.92 | – | – | – | – | –
15 | 2021-05-27 20:14 | 2021-05-28 10:27 | SolO | 0.95 | 0.06 | 2 | 274 | 11.1 | 4.0
16 | 2021-05-30 14:18 | 2021-05-31 00:44 | PSP | 0.70 | – | – | – | – | –
17 | 2021-06-10 16:52 | 2021-06-11 08:53 | PSP | 0.77 | – | – | – | – | –
18 | 2021-06-12 09:20 | 2021-06-13 01:12 | PSP | 0.77 | – | – | – | – | –
19 | 2021-06-21 12:22 | 2021-06-22 01:29 | SolO | 0.93 | – | – | – | – | –
20 | 2021-06-22 22:28 | 2021-06-23 10:50 | SolO | 0.93 | – | – | – | – | –
21 | 2021-06-24 08:10 | 2021-06-25 06:05 | PSP | 0.78 | – | – | – | – | –
22 | 2021-07-11 10:25 | 2021-07-11 18:31 | PSP | 0.68 | – | – | – | – | –
23 | 2021-08-25 01:06 | 2021-08-25 10:43 | SolO | 0.63 | 0.04 | 5 | 272 | 16.2 | -2.3
24 | 2021-10-04 07:01 | 2021-10-04 12:59 | SolO | 0.65 | 0.18 | -7 | 285 | 11.6 | -6.6
25 | 2021-10-05 05:09 | 2021-10-05 11:00 | SolO | 0.66 | – | – | – | – | –
26 | 2021-10-15 09:54 | 2021-10-15 23:11 | SolO | 0.72 | 0.01 | -28 | 271 | 14.7 | -2.2
27 | 2021-11-04 01:25 | 2021-11-04 19:47 | SolO | 0.83 | 0.31 | 64 | 311 | 20.6 | -1.0
28 | 2022-01-25 14:02 | 2022-01-27 07:15 | SolO | 0.89 | 0.12 | -5 | 261 | 12.5 | -0.7

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Dimerization in honeycomb Na2RuO3 under pressure: a DFT study Dimerization in
honeycomb Na2RuO3 under pressure: a DFT study D. D. Gazizova, A. V. Ushakov,
S. V. Streltsov Gazizova, Ushakov, Streltsov 06 March 2018*

# Dimerization in honeycomb Na2RuO3 under pressure: a DFT study

D. D. Gazizova∗+ e-mail<EMAIL_ADDRESS>A. V. Ushakov+ S. V. Streltsov∗+
∗Ural Federal University, Mira Str. 19, 620002 Ekaterinburg, Russia

+Institute of Metal Physics, Russian Academy of Science, S. Kovalevskaya Str.
18, 620041 Ekaterinburg, Russia

###### Abstract

The structural properties of Na2RuO3 under pressure are studied using density
functional theory within the generalized gradient approximation (GGA). We
found that one may expect a structural transition at $\sim$ 3 GPa. This
structure at the high-pressure phase is exactly the same as the low-
temperature structure of Li2RuO3 (at ambient pressure) and is characterized by
the $P2_{1}/m$ space group. Ru ions form dimers in this phase and one may
expect strong modification of the electronic and magnetic properties in
Na2RuO3 at pressure higher than 3 GPa.

74.50.+r, 74.80.Fp

##  1 Introduction

Compounds with the honeycomb lattice are under intensive study in last two
decades, since many of them show intriguing and sometimes rather unexpected
physical properties. There are materials demonstrating the spin gap behaviour
as, e.g., (Na, Li)3Cu2SbO6 [1, 2, 3], $\alpha-$MoCl3 [4] and systems having
different types of the long range magnetic order in the low-temperature (LT)
region: zigzag antiferromagnet (AFM) as, e.g., Na2Co2TeO6 [5, 6], Neel AFM
(Li2MnO3 [7, 8] and possibly SrRu2O6 [9]), and even ferromagnetic honeycomb
planes, as in Ni3TeO6 [10]. Moreover, they can host an unusual spin-liquid
ground state as was proposed by Kitaev [11]. While initial candidate, Na2IrO3
[12], does not seem to be a physical realization of the Kitaev model, another
system, $\alpha-$RuCl3, is under a close investigation now [13, 14, 15].

Another very important class of materials having honeycomb lattice are the
systems, where this lattice turns out to be dimerized at the LT phase. One of
the examples is Li2RuO3. This is a layered material. In the low-temperature
(LT) phase two out of six Ru–Ru bonds in a hexagon dimerize, which results in
formation of the spin gap [16]. With increase of temperature Li2RuO3 exhibits
an unusual phase transition at $T_{d}\sim 540K$, which was initially thought
as a transition from a dimerized to uniform structure [17]. However, more
careful study using X-ray pair distribution function analysis shows that
dimers as rigid units survive even at $T>T_{d}$, while in the average this
system can be described as undimerized, uniform (C2/m space group) [18]. Thus,
this transition at $T_{d}$ was described as a transition to the so-called
valence-bond liquid phase, where these Ru–Ru dimers start to flow over the
honeycomb lattice [19].

Surprisingly, similar compound Na2RuO3 order magnetically below $T_{N}\sim 30$
K (AFM zigzag) without any sizeable structural distortions in honeycomb
lattice [20, 21]. In high-temperature phase Na2RuO3 crystallizes in the same
C2/m space group as Li2RuO3, but the volume of the unit cell in Na2RuO3 is
$\sim 5\%$ larger. Thus, one might expect that under the pressure Na2RuO3 may
start to dimerize and exhibits all unusual properties as its sister compound
Li2RuO3.

In the present work we study a possible formation of the dimerized crystal
structure in Na2RuO3 under pressure using first-principle calculations using
density functional theory (DFT) in the generalized gradient approximation
(GGA). We found, that at $\sim 3$ GPa dimerized phase becomes lower in energy
than uniform structure without any Ru–Ru dimers. Thus, we predict the phase
transition to dimerized structure in Na2RuO3 at $\sim 3$ GPa, which may result
in strong modification of the electronic and magnetic properties of Na2RuO3
related to destroy of the long range magnetic order in the LT phase and
formation of the spin gap.

Figure 1: Fig. 1. The crystal structure of Na2RuO3. Ru ions are shown as grey,
Na as yellow and O as red balls.

##  2 Calculation details

The band structure calculations of Na2RuO3 were mostly performed using the
Vienna Ab-initio Simulation Package (VASP) [22, 23]. We utilized the projector
augmented-wave (PAW) method [24] with the Perdew-Burke Ernzerhof (PBE) GGA
functional [25]. The energy cutoff was chosen to be $E_{cutoff}\sim 500$ eV
and the 4$\times$4$\times$4 Monkhorst-Pack grid of k-points was used in the
calculations. Analysis of the electronic structure was performed within the
TB-LMTO-ASA code, which is based on the linearized muffin-tin orbital method
[26].

In this work we considered two crystal structures: uniform and dimerized ones.
The uniform structure was taken from known experimental data [20]. The
dimerized phase of Na2RuO3 was obtained from corresponding Li2RuO3 structure
[16], where the Li ions were substituted by the Na ions and volume of the unit
cell was rescaled accordingly. For the crystal structure optimization we used
conjugate gradient algorithm.

##  3 Crystal structure

The crystal structure of Na2RuO3 consists of the magnetic honeycomb RuO2
layers in the $ab$ plane, which are separated by nonmagnetic sodiums (see Fig.
1). Other Na ions are in the hexagons formed by edge sharing RuO6 octahedra.
According to the chemical formula Ru ions have a valency $4+$ with four
electrons on the Ru $t_{2g}$ orbitals, which results in $S=1$.

At ambient pressure Na2RuO3 (non-dimerized lattice) crystallizes in the
monoclinic structure (space group $C$2/$m$). The lattice constants and atomic
coordinates were determined in Ref. [20]: $a$=5.3456(1) Å, $b$=9.2552(18) Å,
$c$=5.5504(1) Å, $\beta$=108.74(3)∘. The unit cell volume is 260.046 Å3. There
are two slightly different Ru-Ru bonds in the $ab$ plane: $a_{1}$ = 3.084 Å,
$a_{2}$ = 3.089 Å, so that $r=a_{2}/a_{1}\sim 1$.

We used the LT Li2RuO3 structure, which is described by the $P$21/$m$ space
group [16], to create dimerized Na2RuO3 structure. There are two nearly the
same Ru–Ru bonds in the hexagon $a_{1}$=3.045 Åand $a_{3}$=3.049 Åand one
short $a_{2}$=2.568 Å, so that the ratio between longest and shortest bonds is
$r\sim 1.2$ in Li2RuO3 at the LT phase. Shortest Ru-Ru bonds form the
herringbone structure. Schematic representations of both uniform and dimerized
phases is shown in Fig. 2.

Figure 2: Fig. 2. The Ru-Ru bonds in the uniform (left part) and dimerized
(right part) phases. The short Ru-Ru bonds are marked by the red dashed lines.

##  4 Calculation results

In this study we carried out the DFT calculations for non-dimerized
(experimental) and dimerized (artificial) structures for several unit cell
volumes. At each point the shape of the unit cell was fixed and positions of
the Na, Ru, and O ions were optimized. Analysis of the calculation results
shows that both structures survive in chosen volume range and do not transform
to any other structure.

The volume dependence of total energies, $E$, is presented in Fig. 3(a). One
can see that the uniform structure of Na2RuO3, which is experimentally
observed at ambient pressure, does have the lowest total energy and this
structure corresponds to the global minimum of $E(V)$ in the GGA. However, the
situation changes with decrease of the volume and the dimerized structure
turns out to be the lowest in energy for $V<266$ Å3. Intristingly, the ratio
between longest and shortest Ru–Ru bonds in dimerized case is
$r=a_{2}/a_{1}\sim 1.2$, very close to what we have in Li2RuO3.

Figure 3: Fig. 3. (a) Total energy dependence of Na2RuO3 of two structures
(with and without structural Ru-Ru dimers) for different volumes. (b) Pressure
dependence on the volume for these two structures.

There were debates whether the structural structural transition in Li2RuO3 is
of first or second order [16, 27], but recent differential thermal analysis
demonstrates a heat anomaly at transition temperature, which is in favour of
first order transition. In order to estimate critical pressure and volume jump
for similar transition in Na2RuO3 we first interpolated $E(V)$ (to have a
smooth function) and then recalculated the pressure dependence on volume as
$P=-\partial E/\partial V$, which is shown in Fig. 3(b). Applying external
pressure to undimerized phase of Na2RuO3 we reduce volume of the unit cell
down to critical value $V_{undim}$, when the total energies of uniform and
dimerized phases are the same. This point defines a critical pressure for
transition, $P_{c}\sim 3$ GPa, see Fig. 3(b). Further compression drives the
system to the dimerized phase. Corresponding jump in the unit cell volume at
the transition is $\delta V=V_{undim}-V_{dim}\sim$10 Å3. Similar transition in
Li2RuO3 (but caused by temperature) gives $\delta V\sim 8$Å3 [16].

One may expect that in Na2RuO3 there also (as in Li2RuO3) can develop a spin
gap in the dimerized phase, so that one might expect a decrease of magnetic
susceptibility at $P_{c}$, and the electronic structure of Na2RuO3 can also be
very different. One cannot estimate reliably the value of the spin gap using
the GGA calculations, but it is rather straightforward to study the electronic
structure of this phase. We chose for this different calculation scheme - so
called tight-binding linearized muffin-tin orbitals method, as realized in the
TB-LMTO-ASA code [26]. In contrast to VASP used for the volume optimization,
in TB-LMTO-ASA one can easily change a local coordinate system for any ion.
This helps in analysis of the electronic structure.

The partial density of state (DOS) plots for the Ru-$4d$ and O-$2p$ states for
Na2RuO3 in the dimerized phase at the transition volume ($\sim 267$ Å3 on the
unit cell), as calculated in the TB-LMTO-ASA, are presented in Fig. 4(b). One
may see that in the GGA approximation Na2RuO3 is a metal, which is due to the
absence of strong electronic correlations in this method [28]. The O-$2p$
bands are in a region from -6.5 eV to -1.8 eV, whereas the Ru-$t_{2g}$ states
are located mostly in the energy interval from -1.5 eV to 1 eV.

Figure 4:  Fig. 4. Partial densities of states (DOS) of the Ru-$4d$ and O-$2p$
shells and partial DOS, corresponding to the Ru $t_{2g}$ orbitals for the
uniform and dimerized Na2RuO3 structures at the transition volume. The local
coordinate system where $x$,$y$, and $z$ axes point to the ligands in RuO6
octahedron was chosen. The Fermi energy is in zero.

The electronic structure in the vicinity of the Fermi level in the dimerized
phase is shown in Fig. 4(c). We used the local coordinate system with
coordinate axes pointing to the oxygen ions. One may notice a strong bonding-
antibonding splitting for part of the Ru $t_{2g}$ states. These are the $xy$
orbitals, which look to each other in the common edge geometry [18, 29, 30].
This large bonding-antibonding splitting is due to a direct overlap between
the $xy$ orbitals on two different sites forming Ru-Ru dimers. The splittings
of the $zx/zy$ states are much smaller. One may compare corresponding
splittings in dimerized phases of Na2RuO3 and Li2RuO3. The bonding-antibonding
splitting between the $xy$ orbitals in Na2RuO3 is $\sim 2$ eV, i.e. on $\sim
0.6$ eV smaller than in Li2RuO3 [18]. The dimers are mainly stabilized by the
formation of these bonding orbitals and thus one may see that even at pressure
Ru-Ru dimers are much less rigid and stable in Na2RuO3 than in Li2RuO3. This
also suggests that one may expect a smaller spin gap at low temperatures in
the high-pressure phase of Na2RuO3 (smaller than in Li2RuO3).

In contrast to the dimerized case the width of the t2g band in the uniform
Na2RuO3 at the transition volume is much smaller (in $\sim 2$ times, see Fig.
4(c) and (d)) because of the absence of bonding-antibonding splitting. Also
the O-$2p$ and Ru-$4d$ states are separated by the energy gap about $1.6$ eV.

In the end one needs to comment on possible importance of the strong
electronic correlations in Na2RuO3. We have seen that the GGA overestimates
equilibrium volume of the uniform phase on $\sim$6% (experimental unit cell
volume is 260 Å3 whereas the minimum of the total energy in the GGA
corresponds to $\sim 276$ Å3; this is rather typical that the local density
approximation (LDA) underestimates, while the GGA overestimates equilibrium
volume [28]). An account of the strong electronic correlations, which are
typical for the transition metal ions in insulating oxides, may, possibly,
improve agreement between a theory and an experiment. However, the simplest
LDA+$U$-like methods [31] will not help in this situation. This method tends
to stabilize a solution with local spin moments forming a long range magnetic
order and breaks molecular-orbitals, which are a driving force of a
dimerization. As a result LDA+$U$ would always favour an uniform, non-
dimerized structure. More appropriate in this situation would be cluster
LDA+DMFT calculations [32], which take into account on equal footing both a
formation of molecular-orbitals and strong Coulomb correlations. However,
presently it is impossible to calculate forces and optimize the crystal
structure within the cluster LDA+DMFT approach for real materials.

##  5 Conclusion

To sum up, in the present paper we predict a structural phase transition to
dimerized phase in Na2RuO3 at pressure $\sim$ 3 GPa. One may expect that this
transition will be accompanied by strong changes in the magnetic and
electronic properties, related to the formation of the spin gap and strong
bonding-antibonding splitting. These changes might be detected by the
susceptibility measurements and by the X-ray and optical spectroscopy.
Interestingly similar transition (and at similar pressures $\sim$ 3 GPa) to
the dimerized phase was recently observed in $\alpha-$Li2IrO3 [33], which also
(as Na2RuO3) has undimerized lattice at normal conditions.

## Acknowledgment

The authors would like to thank I. Mazin, R. Valenti, and D. Khomskii for
numerous discussions of physical properties of 213 ruthenates. Present work
was supported by the project of the Ural branch of RAS 18-10-2-3, by the FASO
through research program “spin” AAAA-A18-118020290104-2, by Russian ministry
of education and science via contract 02.A03.21.0006 and by the Russian
foundation for basic research (RFBR) via grant RFBR 16-02-00451.

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