††footnotetext: $\dagger$ These authors have contributed to this work
equally.††footnotetext: ✉ Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Laboratory observation of plasmoid-dominated magnetic reconnection in hybrid
collisional-collisionless regime

Z. H. Zhao† Center for Applied Physics and Technology, HEDPS, and SKLNPT,
School of Physics, Peking University, Beijing 100871, China  H. H. An†
Shanghai Institute of Laser Plasma, CAEP, Shanghai 201800, China  Y. Xie
Center for Applied Physics and Technology, HEDPS, and SKLNPT, School of
Physics, Peking University, Beijing 100871, China  Z. Lei  Center for Applied
Physics and Technology, HEDPS, and SKLNPT, School of Physics, Peking
University, Beijing 100871, China  W. P. Yao  Center for Applied Physics and
Technology, HEDPS, and SKLNPT, School of Physics, Peking University, Beijing
100871, China  W. Q. Yuan  Center for Applied Physics and Technology, HEDPS,
and SKLNPT, School of Physics, Peking University, Beijing 100871, China  J.
Xiong  Shanghai Institute of Laser Plasma, CAEP, Shanghai 201800, China  C.
Wang  Shanghai Institute of Laser Plasma, CAEP, Shanghai 201800, China  J. J.
Ye  Shanghai Institute of Laser Plasma, CAEP, Shanghai 201800, China  Z. Y.
Xie  Shanghai Institute of Laser Plasma, CAEP, Shanghai 201800, China  Z. H.
Fang  Shanghai Institute of Laser Plasma, CAEP, Shanghai 201800, China  A. L.
Lei  Shanghai Institute of Laser Plasma, CAEP, Shanghai 201800, China  W. B.
Pei  Shanghai Institute of Laser Plasma, CAEP, Shanghai 201800, China  X. T.
He  Institute of Applied Physics and Computational Mathematics, Beijing
100094, China  W. M. Zhou${}^{\textrm{{\char 0\relax}}}$ Science and
Technology on Plasma Physics Laboratory, Research Center of Laser Fusion,
China Academy of Engineering Physics (CAEP), Mianyang 621900, China  W.
Wang${}^{\textrm{{\char 0\relax}}}$ Shanghai Institute of Laser Plasma, CAEP,
Shanghai 201800, China  S. P. Zhu${}^{\textrm{{\char 0\relax}}}$ Institute of
Applied Physics and Computational Mathematics, Beijing 100094, China  B.
Qiao${}^{\textrm{{\char 0\relax}}}$ Center for Applied Physics and Technology,
HEDPS, and SKLNPT, School of Physics, Peking University, Beijing 100871, China

###### Abstract

Magnetic reconnection, breaking and reorganization of magnetic field topology,
is a fundamental process for rapid release of magnetic energy into plasma
particles that occurs pervasively throughout the universe. In most natural
circumstances, the plasma properties on either side of the reconnection layer
are asymmetric, in particular for the collision rates that are associated with
a combination of density and temperature and critically determine the
reconnection mechanism. To date, all laboratory experiments on magnetic
reconnections have been limited to purely collisional or collisionless
regimes. Here, we report a well-designed experimental investigation on
asymmetric magnetic reconnections in a novel hybrid collisional-collisionless
regime by interactions between laser-ablated Cu and CH plasmas. We show that
the growth rate of the tearing instability in such a hybrid regime is still
extremely large, resulting in rapid formation of multiple plasmoids, lower
than that in the purely collisionless regime but much higher than the
collisional case. In addition, we, for the first time, directly observe the
topology evolutions of the whole process of plasmoid-dominated magnetic
reconnections by using highly-resolved proton radiography.

Magnetic reconnection is a physical process occurring nearly anywhere there’s
plasma, in which the magnetic topology is rearranged and magnetic energy is
converted to kinetic energy, thermal energy, and particle acceleration
Zweibel2009 ; Yamada2010 . As plasma makes up the stars and ninety-nine
percent of the visible universe, magnetic reconnection is ubiquitous and plays
a key role in many energetic events throughout the whole universe such as
solar flares Schrijver1998 , gamma-ray-burst (GRB)Piran2004 and so on.
However, due to the vastly different plasma properties and spatiotemporal
scales, it is hard to give a universal model for describing the mechanism of
magnetic reconnection.

A key element of magnetic reconnection is the reconnection electric field $\bf
E_{rec}$, which plays a pivotal role in both energy conversion and production
of energetic particles by doing work on particles. The normalized reconnection
electric field is also employed to represent the reconnection rate.
Theoretical and simulation studies have shown that $\bf E_{rec}$ depends
heavily on the plasma propertiesBiskap1997 , in particular, the collision rate
as a combination of plasma density and temperature. In the strongly
collisional regime, such as magnetic reconnections in solar/stellar
photosphere and chromosphereShibata2007 , $\bf E_{rec}$ is induced by the
plasma resistivity as described by the Sweet-Parker model Sweet1958 ;
Parker1963 , which, however, leads to a rather slow reconnection with low
reconnection rate. By contrast, in the weakly collisional or collisionless
regime, such as those occurring in magnetopauseCassak2016 and
magnetotailPetrukovich2016 , $\bf E_{rec}$ is contributed mainly by the off-
diagonal (nongyrotropic) component of the electron pressure tensor Hesse2001 ;
Hesse2011 , resulting in a fast reconnection that is independent of the plasma
collision rate.

However, for most natural circumstances in the universe, a large class of
magnetic reconnection lies in a hybrid regime where the reconnection plasmas
have asymmetric collision rates with one side in a strongly-collisional state
and the other in a weakly-collisional or collisionless state. One of the most
typical scenarios for such hybrid collisional-collisionless magnetic
reconnection occurs in the solar/stellar atmosphereLi2016 ; Yang2019 , see
Figure 1(a), when the cool (electron temperature $T_{e}\sim 10^{4}$ K), dense
(plasma density $n_{e}\sim 10^{11}$ cm-3) filaments erupting from the
chromosphere collide with the hot ($T_{e}\sim 10^{6}$ K), tenuous ($n_{e}\sim
10^{8}$ cm-3) loops in the corona. As we know, the collisional mean free path
$\lambda_{\rm ei}\propto T_{e}^{2}n_{e}^{-1}$, therefore, the plasma in the
filaments is highly collisional while that in the coronal loops is
collisionless. Such hybrid magnetic reconnection may also occur when the
magnetic fields in the dense accretion disk collide with those in the tenuous
interstellar medium.

Fig 1: Hybrid collisional-collisionless magnetic reconnection configurations
and experimental set-up. (a) Magnetic reconnection in the hybrid regime
between the collisional solar filaments and the nearby collisionless coronal
loops. The original observation data is given by the Solar Dynamics
Observatory (SDO), and the picture, observed by Atmospheric Imaging Assembly
(AIA) at 171 $\rm\mathring{A}$ ($\sim 0.9$ MK), is re-edited from Li _et al._
Li2016 . (b) Setup for the experiment on magnetic reconnection in the hybrid
collisional-collisionless regime, which is achieved by irradiation of Cu
(yellow) and CH ($\rm C_{1}H_{1}$, green) foil targets respectively with long
ns laser pulses synchronously. Proton radiography is set up along the face-on
($-z$-axis) direction for probing the magnetic field topology changes during
the reconnection process, where the high-quality proton beam is generated from
a tantalum foil target driven by the relativistic ps laser pulse.

So far, most laboratory experiments Nilson2006 ; Li2007 ; Zhong2010 ;
Fiksel2014 of magnetic reconnections are focused on the purely collisional
and/or collisionless regimes. The reconnection experiment Rosenberg2015a ;
Rosenberg2015b by laser-driven colliding plasmas with asymmetric flow
velocities arising from the delay between laser drives are carried out, where,
however, almost no impacts on the reconnection dynamics have been observed
because the reconnection plasmas are both still purely collisionless. In
addition, to date, the evolutions of magnetic topologies for the whole process
of the plasmoid-dominated magnetic reconnectionsShibata2001 ; Shibata2016
including the growth of tearing instabilitiesUzdensky2016 and the formation of
multiple plasmoidsSamtaney2009 ; Uzdensky2010 have never been directly
observed, where only indirect measurements through the interferometry Dong2012
; Hare2017 have been given.

Here, we report result of the first experiment on asymmetric magnetic
reconnections in the novel hybrid collisional-collisionless regime by
colliding of laser-ablated high-Z Cu and low-Z CH plasmas. It shows that the
growth rate of the tearing instability in such a hybrid regime is still
extremely large, resulting in rapid formation of multiple plasmoids, lower
than that in the purely collisionless regime but much higher than the
collisional case (where only single X-point forms). Using the temporally and
spatially highly-resolved proton radiography, we provide, for the first time,
the direct measurement of magnetic topology evolutions for the whole process
of such plasmoid-dominated magnetic reconnection, so that the specific
reconnection dynamics including growth of the tearing instability and
formation of multiple plasmoids are well characterized. The experimental
results are well reproduced and explained by self-consistently combining the
radiation-magnetohydrodynamic (RMHD) and particle-in-cell (PIC) simulations as
well as the proton radiography iterative inversion algorithm. Physically, in
the hybrid magnetic reconnection, the reconnection electric field show much
distinct feature, which is large and grows fast at the collisionless plasma
side induced by the non-gyrotropic component of the electron pressure tensor,
whereas smaller and more slowly at the collisional side induced by only the
resistivity.

## Results

### Experimental setup

The hybrid magnetic reconnection experiments are carried out on the ShenGuang
II Upgrade (SG-II-U) laser facility that has 8 ns pulses and 1 relativistic ps
pulse. Figure 1(b) shows a diagram of the experimental setup, where the hybrid
collisional-collisionless reconnection is achieved by irradiation of Cu and CH
foil targets respectively with long ns laser pulses synchronously. The ablated
low-Z CH plasma is in the collisionless state, while the high-Z Cu plasma is
in the collisional state, due to their different ionization charge states.
These two plasmas expand and collide with each other, forming the magnetic
reconnection in the hybrid regime, because both of them advect the self-
generated Biermann magnetic fields together, as also shown in 1(b) (see the
density maps and field lines). In order to balance the aspect ratio of the
current sheet $L/\delta$ and the relative velocity of the plasma bubbles, the
distance between the two focal spots is appropriately set to be 1.8 mm. The
temporally and spatially highly-resolved proton radiography is set up along
the face-on ($-z$-axis) direction for probing the magnetic field topology
changes during the reconnection process, where the high-quality proton beam is
generated by target normal sheath acceleration from a tantalum foil target
driven by the relativistic ps laser pulse. The static radiography images in
Supplementary Fig. blueS1 clearly show that the proton beam is of high quality
with almost uniform distributions, where the cut-off energy exceeds 30 MeV.
For protons with energy of typically 13.5 MeV, we estimate the time for them
to pass through the reconnection region is about 20 ps, far less than the
characteristic plasma evolution time (generally $\sim$ ns), so that the
transient radiography can be guaranteed.

Fig 2:  Topologies of self-generated Biermann magnetic fields in laser-driven
expanding CH plasma bubbles. (a) and (b) are respectively the face-on proton
radiography images at $t=0.7$ ns and $1.0$ ns for CH plasma expansion, where
the higher the grayscale represents the higher the proton doses. (c) and (d)
are the strength of the reconstructed path-integrated magnetic fields $\psi$
(in units of $\rm T\cdot mm$), corresponding to the dashed-box region in (a)
and (b) respectively.

### Features of self-generated Biermann magnetic fields

To have an intuitive understanding of the self-generated Biermann magnetic
fields in laser-driven expanding plasmas, we firstly take proton radiography
for a single CH plasma bubble. The radiography images are shown in Figs. 2(a)
and 2(b) at time $t=0.7$ and $1.0$ ns, respectively, which correspond to the
radiography protons of $13.5$ and $7.7$ MeV. We see that the protons are
deflected into the inside of the plasma bubble and a clear low-dose ring
structure is formed on the periphery of the plasma bubble, which can be
regarded as the direct evidence for generation of a toroidal, clockwise
Biermann magnetic field. Note that, if it is a radial electric field, the
protons should be deflected outside the plasma bubble Petrasso2009 . Further,
comparing 2(a) and 2(b), we can also estimate the plasma expansion velocity
parallel to the target surface is about 700 km/sGao2015 , as about 2.0
$c_{s}$, where $c_{s}\equiv(\gamma\bar{Z}T_{e}/m_{i})^{1/2}$ is the ion sound
speed. Similar radiography images of laser-ablated Cu plasmas are also
obtained, by which we estimate that its expansion velocity is on the same
order due to the similar charge-to-mass ratio $\bar{Z}/A$. These self-
generated Biermann magnetic field topologies are also verified by our three-
dimensional (3D) synchronous proton radiography experiment Zhao2021 .

Fig 3: Experimental results of hybrid collisional-collisionless magnetic
reconnection. (a)-(d) are the proton radiography results of magnetic field
topologies at time $t=0.3$, $1.0$, $1.5$ and $1.8$ ns respectively, where the
upper half is for collisional Cu plasma and the lower is for collisionless CH.
The higher the grayscale represents the higher the proton doses, and all
images are rescaled to the realistic length through dividing by the
radiography geometric magnification $M\sim 6.0$, see Methods. (f)-(h) are the
reconstructed path-integrated magnetic fields $\psi$, corresponding to the
dashed-box region in (b)-(d) respectively, where the color map in (f)
represents $B_{y}$ component and those in (g) and (h) represent $B_{x}$
components. The contour lines in (f)-(h) represent the corresponding vector
potential $A_{z}$. (e) shows the profiles of $\psi$ in (g) (blue line) and (h)
(red line) along y-axis.

To extract more quantitative information from the radiographys, we use the
inverse field-reconstruction code “PROBLEM” Bott2017 to recover the path-
integrated magnetic fields $\psi$, see Methods. Reconstructed $\psi$ for the
dashed-box region in Figs. 2(a) and 2(b) are shown in 2(c) and 2(d)
respectively, which further confirms the existences of toroidal magnetic
fields in expanding plasmas (see red dashed curves). The value of $\psi$ is
about 4.0 $\rm T\cdot mm$, from which we can estimate the average field
strength about $>10$ T. This is in good consistence with the estimation under
paraxial approximation, where the path-integrated magnetic field $\psi$ is
estimated Li2007 ; Rosenberg2015a ; Rosenberg2015b as

$\psi({\rm T\cdot mm})\equiv\int_{0}^{d}\textbf{B}dz\approx
0.1445\frac{w({\rm\mu m})\sqrt{E_{p}({\rm MeV})}}{L({\rm mm})},$ (1)

where $w$ and $L$ are respectively the width of the low-dose ring and the
distance from the plasma to the RCF stack, and $E_{p}$ is the proton energy.
Assuming $w\sim 500\mu$m, $E_{p}\sim 13.5$ MeV and $L\sim 50$ mm for CH plasma
in 2(a), the magnetic field is roughly estimated as $\psi\approx 5.3$ $\rm
T\cdot mm$, and the field in 2(b) at $t=1.0$ ns is about $7.6$ $\rm T\cdot
mm$.

### Hybrid Collisional-Collisionless Magnetic Reconnection: Experimental
Results

The proton radiography results of the magnetic field topologies at various
times in Figs. 3(a)-3(d) provide, for the first time, direct picture of the
whole process for the hybrid collisional-collisionless magnetic reconnection.
At time $t=0.3$ ns [3(a)], we see that the toroidal Biermann magnetic field
still has not sufficiently developed and is not prominent, where some small-
scale jet-like proton accumulations exist possibly due to the filamentary
magnetic field arising from escaping of hot electronsGraziani2017 . At time
$1.0$ ns [3(b)], two low-dose rings (marked by the blue dashed lines) form at
respectively collisionless CH and collisional Cu plasmas, which indicate the
toroidal Biermann magnetic fields form, similar to those in Figs. 2(a) and
2(b). We also see that the two magnetic fields already start to touch each
other, consistent with the above estimations of their expansion velocities.
Further, we see that more protons accumulate inside the Cu plasma bubbles than
that of CH, indicating a stronger self-generated magnetic field, which is
attributed to a steeper temperature gradient $\nabla T_{e}$ of Cu plasmas, see
Supplementary Fig. blueS2. Also, the sharper proton accumulation radiography
in CH plasmas may benefit from the uniform drive laser focal spot in our
experiment, rather than the Gaussian one in Cu plasmas.

Fig 4: Kinetic 2D3V PIC simulation results of the hybrid magnetic
reconnection. Here the initial conditions are taken from the RMHD simulation
at $z=200\ \mu$m, and at $t=0.8$ ns [see Supplementary Fig. blueS2]. (a)-(d)
are distributions of the magnetic field strengths $|B|$ (in blue color) at
time $t=$ 1.0, 1.2, 1.5, 1.8 ns respectively during the magnetic reconnection.
The zoomed images show the current densities J (in blue-red color) and vector
potential $A_{z}$ (contours) of the reconnection region, corresponding to the
dashed box regions in (a)-(d) respectively. (e)-(h) show the corresponding
synthetic proton radiography images based on the magnetic fields [(a)-(d)]
obtained from PIC simulations (more details see Methods), where the higher the
grayscale represents the higher the proton doses. The radiography geometric
magnification $A$ and proton energy $E_{p}$ are consistent with the realistic
experimental radiography setup. In order to match the extracted path-
integrated magnetic fields $\psi$, the thickness of the fields is taken as
$d\sim 100\ \mu$m.

At later time $t=1.5$ ns [3(c)], both Cu and CH plasmas continue to flow in,
and as a result, the anti-parallel magnetic fields are strongly squeezed in
their colliding region forming a reconnection current layer. Correspondingly,
the proton radiography image [3(c)] shows that two low-dose rings merge into a
long and narrow ribbon with length $\Delta x_{\rm exp}\sim 1.9$ mm and width
$\Delta y_{\rm exp}\sim 0.3$ mm. Almost all protons are deflected out of this
narrow ribbon region indicating that rather strong magnetic field exist
locally, and the Y-shaped opening ends at both side of the ribbon region imply
that the magnetic fields in Cu and CH plasmas begin to separate Tubman2021 .
Finally and more importantly at time $t=1.8$ ns [3(d)], we see that the
uniform low-dose ribbon is disturbed, and several high-dose fine filaments
(marked by the red circles) appear, which means no proton deflection occurs
there. In other words, it means that the magnetic fields are dissipated at
these fine filamentary region, which is the key evidence for occurrence of the
hybrid magnetic reconnection.

To more accurately reflect the change of the magnetic field topology, the
reconstructed path-integrated magnetic field $\psi$ and the corresponding
vector potential $A_{z}$ (see field lines) are shown in Figs. 3(f)-(h), see
Methods. They clearly show that the two toroidal magnetic fields [see 3(f)]
are piled up together, forming a narrow reconnection layer dominated by only
the $B_{x}$ component [see 3(g)]. As shown in Fig. 3(e) for the profile of
$\psi$ along y-axis, we see that the asymmetry of the plasma properties
results in the asymmetric magnetic flux (blue curve), which is stronger in Cu
($y>0$) than CH ($y<0$). And at later time, the current sheet on the side of
the Cu plasma is obviously widened than that in CH plasmas (red curve), which
may be the result of the magnetic diffusion caused by the strong collision
effect. Using the Ampere’s law $\textbf{J}=\nabla\times\textbf{B}$, it is
estimated that the width of the current sheet (peak to peak) is about
$2\delta_{\rm exp}\sim 150\ \mu$m, which is much smaller than the length
$2L\sim 1.9$ mm. For such a long and narrow current sheet, the tearing
instability occurs and develops rapidly, resulting in formation of multiple
plasmoids Uzdensky2016 ; Samtaney2009 ; Uzdensky2010 , which are shown clearly
in 3(h) by the red contours, also corresponding to the high-dose filaments in
the radiography image 3(d). Further, we see that there are still lots of un-
reconnected magnetic fluxes on the outside of the fragmented current sheet,
which may be due to the inefficient plasmoid ejection (because of high
$\beta$) prevents further inflow of upstream magnetic fields Dong2012 ;
Hare2017 . In view of the above radiography results, we conclude that, we
have, for the first time, directly observed the growth of the tearing
instability and formation of multiple plasmoids as well as the whole
reconnection dynamics in the plasmoid-dominated magnetic reconnection.

### Hybrid Collisional-Collisionless Magnetic Reconnection: Simulation Results

Table 1:  Plasma parameters obtained from RMHD and following self-consistent
PIC simulations. (a) The expanding Cu and CH plasma parameters at $t=0.9$ ns
obtained from 3D RMHD simulations, which are used as the initial condition for
the following PIC simulations after self-similar scaling. (b) The plasma
parameters inside the compressed reconnection layer (current sheet) of
respectively Cu and CH sides at $t=1.3$ ns obtained from PIC simulations,
which are exactly the plasma states for magnetic reconnection. The listed
plasma parameters include electron density $n_{e}$, electron temperature
$T_{e}$, ion charge state $\bar{Z}$, magnetic field $B$, plasma flow velocity
$v$ and Alfvén speed $v_{A}$, the ratio of thermal pressure to magnetic
pressure $\beta$, the ratio of ram pressure to magnetic pressure $\beta_{\rm
ram}$, Reynolds number $R_{e}$, magnetic Reynolds number $R_{m}$ and Peclet
number $P_{e}$, electron mean free path $\lambda_{ei}$, Lundquist number $S$,
current sheet width of the Sweet-Parker model $\delta_{\rm sp}$, ion skin
depth $d_{i}$ and ion cyclotron radius $\rho_{ci}$. (a) plasma parameters
obtained from 3D RMHD simulations
---
para | $n_{\rm e}({\rm 10^{18}cm^{-3}})$ | $T_{\rm e}({\rm keV})$ | $\bar{Z}$ | $B({\rm T})$ | $v({\rm km/s})$ | $\beta$ | $\beta_{\rm ram}$ | $R_{\rm e}$ | $R_{\rm m}$ | $P_{\rm e}$
Cu111Plasma states as initial condition for PIC simulations. The plasma scale is about 1.0 mm and the Coulomb logarithm $\rm ln\Lambda$ is about 10. | $2.5\pm 1.0$ | $1.5\pm 0.2$ | $20\pm 2$ | $40\pm 10$ | $800\pm 100$ | $1.0\pm 0.2$ | $6.7\pm 1.0$ | $8000\pm 2000$ | $3000\pm 500$ | $0.03\pm 0.02$
CH111Plasma states as initial condition for PIC simulations. The plasma scale is about 1.0 mm and the Coulomb logarithm $\rm ln\Lambda$ is about 10. | $8.0\pm 3.0$ | $1.1\pm 0.1$ | $3.5$ | $25\pm 10$ | $1000\pm 100$ | $5.7\pm 1.0$ | $50\pm 10$ | $120\pm 30$ | $12000\pm 3000$ | $0.05\pm 0.03$
(b) plasma parameters in the reconnection layer obtained from PIC simulations
para | $n_{\rm e}({\rm 10^{18}cm^{-3}})$ | $T_{\rm e}({\rm keV})$ | $B({\rm T})$ | $v_{\rm A}({\rm km/s})$ | $\lambda_{\rm ei}({\rm\mu m})$ | $\beta$ | $S$ | $\delta_{\rm sp}({\rm\mu m})$ | $d_{\rm i}({\rm\mu m})$ | $\rho_{\rm ci}({\rm\mu m})$
Cu222Plasma states for hybrid magnetic reconnection. The half length of the current sheet is estimated to be $L\sim 1.0$ mm. | $30\pm 15$ | $2.5\pm 0.5$ | $100\pm 20$ | $200\pm 50$ | $100\pm 40$ | $4.0\pm 2.0$ | $1500\pm 500$ | $25\pm 10$ | $60\pm 10$ | $20\pm 5$
CH222Plasma states for hybrid magnetic reconnection. The half length of the current sheet is estimated to be $L\sim 1.0$ mm. | $40\pm 20$ | $2.5\pm 0.5$ | $50\pm 10$ | $130\pm 30$ | $500\pm 200$ | $16\pm 5.0$ | $5000\pm 1500$ | $15\pm 5$ | $50\pm 10$ | $70\pm 20$

The above experimental results are reproduced and explained by a self-
consistent combination of RMHD and PIC simulations as well as the proton
radiography iterative inversion algorithm. As mentioned, magnetic reconnection
is a rapid process of global magnetic topological self-organization triggered
by local reconnection points, which depends critically on the plasma states.
Before two plasma interactions, the laser-ablated plasma expansion dynamics is
majorly governed by RMHD. Therefore, we firstly run the RMHD simulations with
the “FLASH” code Fryxell2000 to obtain the basic plasma parameters as the
initial condition for the following PIC simulations that can describe the
interaction and reconnection dynamics of two plasmas. The expanding Cu and CH
plasma parameters obtained from RMHD simulations are summarized in Table 1(a),
where the electron density $n_{e}$, temperature $T_{e}$, flow velocity $v$ and
magnetic field strength $B$ are all taken for those at the periphery of each
plasma bubble. All of them are comparable to the experimental results shown in
the above Fig. 2. More details can also be seen from the Methods and
Supplementary Fig. blueS2. The differences in the equation of state (EOS)
Heltemes2012 and opacityMacfarlane1989 of the Cu and CH lead to the asymmetric
plasma states on both sides. For both plasmas, the thermal pressure to
magnetic pressure ratio $\beta>1$ and the ram pressure to magnetic pressure
ratio $\beta_{\rm ram}\gg 1$ indicate that reconnection is strongly
drivenRosenberg2015a ; Rosenberg2015b .

For the following plasma colliding and reconnection process, due to low
densities and high temperatures of plasmas at their interaction region, the
electron mean free path $\lambda_{\rm ei}$ comparable to the plasma scale $L$,
therefore, the kinetic effects start to play key roles and kinetic PIC
simulations are required Fox2011 ; Totorica2016 ; Xu2016 . In order to connect
the PIC simulation with the RMHD simulation in a self-consistent manner, a
self-similar transformation is proposed and applied here. Just like the self-
similarity principle of the magnetohydrodynamic equations Remington1999 ;
Ryutov2000 ; Ryutov2001 , we choose a set of free parameters to spatially
scale down the plasma parameters obtained from the above RMHD simulation to
the relative small scale that is practical for kinetic PIC simulation, where
the plasma properties including $\beta$, $\beta_{\rm ram}$, the Mach number
and the Alfvénic Mach number, etc. are all kept conserved between RMHD and
PIC. The Coulomb collisionsNanbu1998 are also well included by making
$L/\lambda_{ei}$ conserved. Through this novel methodology, see more details
in Methods, we carry out the whole simulation in a self-consistent manner.

Fig 5: Contribution terms of the reconnection electric field $\mathbf{E}_{\bf
z}$ in the hybrid magnetic reconnection. (a) and (b) are respectively the
contributions from the off-diagonal (non-gyrotropic) component of the electron
pressure tensor $E_{z,p_{e}}\sim-\nabla_{y}P_{yz}/(en_{e})$ (using the
colormap scale from black to red colors) and the resistivity $E_{z,\eta}=\eta
J_{z}$ at $t=1.4$ ns (using the colormap scale from purple to green colors),
where, the same as the setup in experiment, the upper half part is collisional
Cu plasma and the lower half part is collisionless CH plasma. (c) shows their
contributions varying with time from $t=1.2$ to $1.6$ ns for respectively Cu
and CH plasmas during reconnection. The normalized reconnection electric field
$E_{z}$ (normalized to local Alfvén speed $v_{A,loc}$ and magnetic field
$B_{loc}$) for respectively the cases of purely collisionless CH-CH (green
star), hybrid Cu-CH (green circle) and purely collisional Cu-Cu (green
diamond) reconnections are also marked in (c), where all the values are chosen
at the time when the changes of their magnetic field topologies are most
significant.

The fully-kinetic PIC simulations are performed in two-dimensional (2D) xy
plane with the code “EPOCH” Arber2015 . It has been identified that the 3D
geometric effect has little effect on the reconnection rate in such a
strongly-driven regime Rosenberg2015b . Topologies of magnetic field
evolutions at various times in the simulations are shown in Figs. 4(a) to 4(d)
from $t=1.0$ to $1.8$ ns. Based on these field topologies, we also carry out
the simulation for the synthetic proton radiography process, see Methods. The
synthetic proton radiography images corresponding to 4(a) to 4(d) are shown
respectively in Figs. 4(e) to 4(h). We see that the Cu and CH plasma bubbles
start to contact with each other at $t=1.0$ ns [4(a)], in consistence with the
experimental result [see 3(b)]. The corresponding synthetic radiography image
in 4(e) also show similar feature as that in the above 3(b), where the slight
difference may be due to lack of a driving source. At $t=1.2$ ns [4(b)],
similar as the experimental results, we see that the magnetic fields are
compressed and amplified in the colliding region, forming a long and narrow
current sheet. Due to the plasma pressure asymmetry, we also see the current
sheet drifts slowly towards the Cu plasma side.

Subsequently at time $t=1.5$ ns, we see clearly from 4(c) that the current
sheet breaks and multiple plasmoids form due to the tearing instability
Uzdensky2016 ; Samtaney2009 ; Uzdensky2010 , and magnetic field energy
dissipates at several local points (X-points), marked by the red crosses.
Furthermore, we see all the plasmoids protrude into the CH plasma side and are
more pronounced in the CH plasma side, which further verifying that the
asymmetry of the tearing instability and magnetic reconnection. This feature
is quite similar to the astronomical observations of the reconnection between
solar filaments and coronal loops Li2016 . Again, both the X-points and
plasmoids can be accurately reflected by the synthetic proton radiography, as
shown in 4(g), where several high-dose fine filaments appear (marked by the
red circles), agreeing well with the experimental results in Fig. 3(c). Note
that the length and width of the low-dose ribbon in 4(g) is about $\Delta
x_{\rm sim}\sim 1.7$ mm and $\Delta y_{\rm sim}\sim 0.25$ mm, which are also
both consistent with the experimental results. At much later time $t=1.8$ ns
[4(d)], we see that the magnetic fields, current sheet and plasmoids are
further continuously compressed, and the distance between these plasmoids
gradually increases due to stretch and ejection Dong2012 ; Hare2017 ($v\sim
10^{5}\ {\rm m/s}$). These plasmoids and X-points become more prominent in
radiographic images, see 4(h) and also 3(d). The number of plasmoids and
X-points does not change with time, implying that the merging of magnetic
islands and secondary tearing instability Shibata2001 ; Shibata2016 ;
Uzdensky2016 ; Samtaney2009 ; Uzdensky2010 both do not occur here. The
phenomenological consistency between the simulation and the experimental
results verifies that the tearing instability and plasmoid-dominated magnetic
reconnection indeed take place in our experiment.

To reveal the inherent mechanism of the hybrid collisional-collisionless
magnetic reconnection, the main parameters of both Cu and CH plasmas inside
the reconnection layer obtained from PIC simulations are summarized in Table
1(b), which are exactly the plasma states for magnetic reconnection. We see
that since both Cu and CH plasmas are highly compressed, their densities and
temperatures significantly increase, and the magnetic field strengths are
amplified to more than 2 times than their initial values, see also
Supplementary Fig. blueS3. The average width of the reconnection layer
(current sheet) is about $2\delta_{\rm sim}\sim 100\ \mu$m, which is similar
with the reconstructed experimental result $2\delta_{\rm exp}\sim 150\ \mu$m.
The Lundquist number $S\equiv Lv_{A}/\eta\gg 1$ in both plasmas, especially in
the CH plasma where $S\sim 0.5\times 10^{4}$, verifying that the tearing
instability can easily occur and develop. The half width of the Sweet-Parker
current sheet $\delta_{\rm sp}\equiv L/\sqrt{S}$ of Cu (about $0.4d_{i}$) and
CH (about $0.3d_{i}$) plasmas are both smaller than the ion skin depth $d_{i}$
or ion cyclotron radius $\rho_{ci}$, which means the two-fluid effect
Tubman2021 ; Rosenberg2015b needs to be considered. More importantly, we see
that, due to the very different ion charge states $\bar{Z}$ of Cu
($\bar{Z}\approx 20$) and CH ($\bar{Z}\approx 3.5$), the plasma on the Cu side
is in a strongly-collisional state ($\lambda_{\rm ei}\sim 100\ \mu$m), while
the plasma on the CH side is collisionless ($\lambda_{\rm ei}\sim 500\ \mu$m).
That is, an asymmetric magnetic reconnection in the hybrid collisional-
collisionless regime is formed.

From both the above experimental and simulation results [comparing Figs. 3(c)
and 3(d), or comparing 4(c) and 4(d)], we estimate that the characteristic
growth time of the tearing instability for the hybrid reconnection is rather
fast as about $\tau_{\rm TI}=0.5$ ns, at the same order of the Alfvén transit
time across the current sheet $\tau_{A}\equiv\delta/v_{A}$. Such fast tearing
instability and reconnection dynamics cannot be explained by only the two-
fluid effect. This can be proved from two aspects. One the one hand, as known,
the maximum growth rate of the spontaneous tearing instability from the
hydrodynamic perspective is about $\gamma_{\rm
max}\sim(v_{A}/\delta)(d_{e}/d_{i})^{0.7}(d_{i}/\delta)^{1.5}$ Fitzpatrick2004
; Loureiro2017 . Substituting the reconnection plasma parameter in Table 1(b)
into it and assuming $\delta\sim 50\ \mu$m, we estimate that $\gamma_{\rm
Cu}\sim(2.5\pm 1.0)\times 10^{8}\ {\rm s}^{-1}$ and $\gamma_{\rm
CH}\sim(1.5\pm 0.5)\times 10^{8}\ {\rm s}^{-1}$, which corresponds to the
characteristic growth time of the tearing instability as both larger than $10$
ns, too slow to explain the experimental and simulation results. On the other
hand, considering only the two-fluid effect, the predicted most unstable mode
is $k_{\rm
max}\sim(1/\delta)(d_{e}/d_{i})^{0.3}(d_{i}/\delta)^{0.5}$Fitzpatrick2004 ;
Loureiro2017 , which means that the most confident number of X-points in
reconnection is $M=Lk_{\rm max}/\pi\sim 2$, also smaller than 4 observed in
our experiment [see 3(c)].

Therefore, to explain the fast and effective tearing instability observed in
our experiment, we analyze the various contributions of the dissipative
reconnection electric field $E_{z}$, which after normalization is defined as
the reconnection rate. The two main contributions, the non-gyrotropic
component of the electron pressure tensor
$E_{z,p_{e}}\sim-\nabla_{y}P_{yz}/(en_{e})$ and the resistive component
$E_{z,\eta}=\eta J_{z}$, are shown in Fig. 5(a) and 5(b) respectively at time
$t=1.4$ ns. Their contributions varying with time from $t=1.2$ to $1.6$ ns for
respectively Cu and CH plasmas during reconnection are also shown in Fig.
5(c). We see clearly that the non-gyrotropic electron pressure tensor always
dominates [the lower half part in 5(a) and the blue square symbol in 5(c)] in
the collisionless CH plasma side, while the resistive electric field dominates
[the upper half part in 5(b) and the red triagnle symbol in 5(c)] in the
collisional Cu plasma side. And as a whole, for the hybrid reconnection, the
collisionless non-gyrotropic electron pressure tensor contributes (about
$75\%$) most of the reconnection electric field, while the resistivity only
contributes $\sim 25\%$. In other words, it is just the collisionless non-
gyrotropic electron pressure that results in the fast tearing instability and
reconnection in our experiment Hosseinpour2014 . The total reconnection
electric field is about $E_{z}\sim 6\times 10^{6}\ \rm{V/m}$, on the order of
$0.3\pm 0.1\ v_{A,loc}B_{loc}$ after normalization [marked by the green circle
in 5(c)], i.e., a fast reconnection rate Fox2011 ; Xu2016 .

To further demonstrate the dynamics of our hybrid reconnection, we also carry
out PIC simulations for the cases of the purely collisionless CH-CH and purely
collisional Cu-Cu magnetic reconnection, where the drive laser and other
parameters are also the same. The results are shown in Supplementary Fig.
blueS4. We see that in the purely collisionless reconnection case, the tearing
instability grows even more quickly, resulting in formation of more number of
multiple X-points, where the normalized reconnection electric field is
$E_{z}\sim 0.5\pm 0.05\ v_{A,loc}B_{loc}$ [marked by the green star in 5(c)],
a little higher than the hybrid case. By contrast, in the purely collisional
Cu-Cu reconnection case, the resistive electric field dominates, a Sweet-
Parker-like current sheet with single X-point is formed, where $E_{z}\sim
0.16\pm 0.03\ v_{A,loc}B_{loc}$ [marked by the green diamond in 5(c)] is much
lower. These further confirms the occurrence of a hybrid collisional-
collisionless magnetic reconnection in our experiment.

## Discussion

In summary, we have carried out the first experiment on an unexplored new
regime of magnetic reconnection, hybrid collisional-collisionless regime, by
colliding of laser-ablated high-Z Cu and low-Z CH plasmas. Using highly-
resolved proton radiography, we have, for the first time, directly observed
the topology changes for the whole process of the plasmoid-dominated magnetic
reconnection. We find that the growth rate of the tearing instability in such
a hybrid regime is still extremely large, resulting in rapid formation of
multiple plasmoids, lower than the purely collisionless case but much higher
than the collisional case. With the self-consistent simulations, we show that,
in this hybrid magnetic reconnection, the reconnection electric field show
much distinct feature, which is large and grows fast at the collisionless
plasma side induced by the off-diagonal (non-gyrotropic) component of the
electron pressure tensor, whereas smaller and more slowly at the collisional
side induced by only the resistivity.

The hybrid collisional-collisionless magnetic reconnection discussed here
occurs widely in astrophysics. For example, for the reconnection between the
erupting collisional filaments and the collisionless coronal loops in solar
and stellar atmosphere, the astronomical observations reveal that most of the
plasmoid formation and plasma heating occurred on the side of the
collisionless coronal loops Li2016 , in consistence with our results. Besides,
the possible reconnections occurring between the dense accretion disk and the
tenuous interstellar medium may also be in such hybrid regime. Our
experimental results show that the plasmoid formation is a universal feature
of magnetic reconnection, even in this hybrid regime. We also show that the
specific magnetic reconnection mechanism depends critically on the plasma
collision rates, which significantly affect the growth of the tearing
instability and further the global reconnection rate, eventually determining
the energy conversion efficiency from fields to plasmas, such as heating the
solar corona Schrijver1998 . Moreover, the lower plasma $\beta$, higher
Lundquist number $S$ and ubiquitous guiding fields in these astrophysical
environments may promote the plasmoid ejection, thereby help to establish the
positive feedback for global fast reconnection Shibata2001 ; Shibata2016 .

## Methods

### Experimental setup

The hybrid collisional-collision magnetic reconnection experiments presented
in this work are carried out on ShenGuang II upgrade (SG-II-U) laser facility
at the Joint Laboratory on High Power Laser and Physics, Shanghai Institute of
Optics and Fine Mechanics, Chinese Academy of Sciences. The laser facility
have 8 nanosecond (ns) pulses and 1 picosecond (ps) petawatt pulse. The main
target consists of two foils with the upper half part as Cu foil of density
$\rho=8.92$ g/cm3 and the lower half part as plastic CH foil of density
$\rho=1.02$ g/cm3. Both foil targets have thickness of 30 $\mu$m and width of
1.0 mm, and parallel to each other, and there is a 0.5 mm gap between the Cu
and CH targets, in order to eliminate the scattering effect of protons by the
solid targets.

The two foil targets are ablated with two $3\omega$ long laser pulses (green)
that have energy about $800\pm 100$J in a $1\pm 0.1$ ns square temporal
profile, with either a 200 $\mu$m FWHM Gaussian or a 450 $\mu$m diameter
uniform focal spot. The corresponding average laser intensities are about
$2.5\times 10^{15}$ W/cm2 and $4.0\times 10^{14}$ W/cm2 respectively. The
incident angle between the laser beam and the target normal is about 45∘, and
the distance between the two focal spots is about 1.8 mm. The proton
radiography is set up along the face-on ($-z$-axis) direction, by irradiating
the backlighter 20-$\mu$m-thick Ta foil (red) target with a relativistic ps
pulse. The ps pulse has energy of $350\pm 50$ J, wavelength of 1.053 $\mu$m,
duration of $1.0\pm 0.1$ ps and is focused by a 800 mm focal-length, $f/2.5$
off-axis parabolic mirror to an intensity about $3.0\times 10^{19}$ W/cm2,
with focal spot diameter $\sim 40\mu$m. The cut-off energy of the high-quality
laminar proton beam exceeds 30 MeV, and the deflected protons are finally
deposited on the HD-V2 radiochromic film (RCF) stack, whose response to proton
dose has been well calibrated with a good linear relationship between proton
dose and optical density (OD). The distance between the backlighter foil (Ta)
and the main foils (Cu and CH) is $h\sim 10$ mm, and that between the main
foils and the RCF stack is $L\sim 50$ mm, which result in the geometric
magnification factor of the proton radiography system as about
$M\equiv(h+L)/h\sim 6.0$.

### Path-integrated field reconstruction

The extraction of the path-integrated magnetic field $\psi$ from proton
radiographs in this work is done by using the algorithm and program “PROBLEM”
Bott2017 , where a parabolic Monge-Ampère equation is solved. Note that the
formation of the proton radiographys can be understanding as following.

First, the propagation of laser-produced proton beam initially can be regarded
as a uniform expansion from a point source (laser-driven target normal sheath
acceleration region) into the probing HED plasma region. Under the paraxial
approximation, $l_{p}/r_{i}\ll 1$, when the proton beam arrives at the probing
region, it can be regarded as a two-dimensional near-planar sheet, where
$l_{p}$ and $r_{i}$ are respectively the plasma lateral length and the
distance from the proton source to the plasma, in our experiments,
$l_{p}\approx 1$ mm and $r_{i}=h\approx 10$ mm.

Afterwards, the proton beam is deflected by the Lorentz force arising from
self-generated magnetic fields in HED plasmas, and the lateral deflection
velocity can be expressed as

$\textbf{v}(\textbf{x}_{\perp
0})\approx\frac{e}{m_{p}c}\hat{\textbf{z}}\times\int_{0}^{l_{z}}\mathrm{d}s\textbf{B}(\textbf{x}(s)),$
(2)

where $e$ is the proton charge, $m_{p}$ is the proton mass, $\textbf{x}_{\perp
0}$ and $\textbf{x}(s)$ are respectively the initial perpendicular position
and the proton trajectory. The outgoing proton beam moves approximately in a
straight line at a constant velocity, and the lateral deflection velocity
$\textbf{v}(\textbf{x}_{\perp 0})$ is amplified as

$\textbf{x}_{\perp}^{(s)}(\textbf{x}_{\perp
0})\approx\frac{r_{i}+r_{s}}{r_{i}}\textbf{x}_{\perp
0}+\frac{\textbf{v}(\textbf{x}_{\perp 0})}{v_{p}}r_{s},$ (3)

where $v_{p}$ is the proton velocity, $r_{s}$ is the distance from the plasma
to the RCF screen, $r_{s}=L\approx 50$ mm in experiments, and
$\textbf{x}_{\perp}^{(s)}$ is the proton position on the screen.

Eventually, the deflection effect by the magnetic field makes the initially
uniform proton beam redistribute on the RCF screen as

$\Psi(\textbf{x}_{\perp}^{(s)}(\textbf{x}_{\perp
0}))=\frac{1}{|\mathrm{Det}(\nabla_{\perp
0}[\textbf{x}_{\perp}^{(s)}(\textbf{x}_{\perp 0})])|}\Psi_{0},$ (4)

where $\Psi$ and $\Psi_{0}$ are respectively the redistributed proton flux on
the RCF screen and the initial proton flux (generally uniform), and
$\nabla_{\perp 0}$ is the gradient operator of the initial plasma coordinates.

Therefore, the path-integrated magnetic field can be obtained by solving the
Monge–Ampère equation (4), and

$\psi\equiv\int_{0}^{l_{z}}\mathrm{d}s\textbf{B}(\textbf{x}(s))=-\frac{m_{p}c}{e}\hat{\textbf{z}}\times\textbf{v}(\textbf{x}_{\perp
0}).$ (5)

Due to the divergence-free of magnetic field $\nabla\cdot\textbf{B}=0$, then
the deflection velocity is curl-free $\nabla\times\textbf{v}=0$. The
deflection velocity can be viewed as the gradient of a potential function
$\textbf{v}\equiv\nabla\phi$, and the corresponding magnetic field potential
function is expressed as

$\mathcal{A}=-\frac{m_{p}c}{e}\phi,$ (6)

under the quasi-two-dimensional approximation, this is equivalent to the
magnetic vector potential $A_{z}$, and $A_{z}\sim\mathcal{A}/l_{z}$.

This inversion problem is well-defined, the geometric distance and proton
energy are determined in the experiments, the redistributed proton flux $\Psi$
can be obtained from the response function of RCF optical density (OD) to
proton dose, and which has an almost linear relationship in the low-dose
region. The inhomogeneity of the initial proton flux $\Psi_{0}$ is ignored,
see also Fig. blueS1 in Supplementary.

### 3D radiation-magnetohydrodynamic (RMHD) simulation

The three-dimensional (3D) RMHD simulations in this work are carried out by
using the code “FLASH” Fryxell2000 , which has been developed to include many
high energy density physics modeling capabilities including laser energy
deposition, multi-temperature ($T_{e}\neq T_{i}\neq T_{rad}$), anisotropic
electron thermal conductivity, and multi-group radiation transport etc. We
have also further extended the code with the self-consistent magnetic field
modeling capabilities including the Biermann battery, Nernst advection and so
on, which is the reason why we called it as ”RMHD”.

The initial conditions of 3D RMHD simulations are taken following those in the
experiments. The size of the simulation box is $4\times 6\times 1.5$ $\rm
mm^{3}$ in all $(x,y,z)$ directions and the solid CH/Cu foil targets with
thicknesses $\rm 30\ \mu m$ in z direction and areas of $1.0\times 2.0$ $\rm
mm^{2}$ in respectively x and y directions are placed at the position $z=0.2\
\rm{mm}$. The ns laser pulses with energy of 800J (a multiplier is used to
account for the scattering caused by parameter instabilities and match the
plasma morphology in the experiments), flat-top temporal profile of duration
0.9 ns plus 0.1ns rising and falling times, and focal spots consistent with
experiments are incident on the foil targets at an oblique angle of about 45∘.
The equation of state (EOS) and opacity of solid foil materials (CH and Cu)
come from the code BADGER Heltemes2012 and IONMIX Macfarlane1989 ,
respectively. The initial densities of the solid foil targets are set to
$\rho_{\rm CH}=1.02$ g/cm3 and $\rho_{\rm Cu}=8.92$ g/cm3, and the rest is
filled with background He gas with a density of $2\times 10^{-6}$ g/cm3. All
the initial temperatures are set to be uniform as room temperatures $T_{0}\sim
290$ K.

In the simulations, an adaptive grid is used. And the geometric-optical
approximation of inverse bremsstrahlung absorption is used to represent laser
heating. The Riemann solver of the MHD equations uses HLLC, and the slope
limiter uses minmod. The Courant (CFL) number is self-adaptive, the typical
value is 0.4. Fully implicit electron thermal conductivity is also adopted,
and the flux limiter is selected as a typical 0.06. The boundary conditions
for fluid, electron thermal conductivity and radiation transport are all set
to open. The simulation results of plasma properties are shown in
Supplementary Fig. blueS2.

### Self-consistent RMHD-PIC simulation

The fully-kinetic PIC simulations in this work are performed in two-
dimensional (2D) xy plane with the code “EPOCH” Arber2015 . It has been
identified that the 3D geometric effect has little effect on the reconnection
rate in such a strongly-driven regime Rosenberg2015b . In order to connect the
PIC simulation with the RMHD simulation in a self-consistent manner, a self-
similar transformation is proposed and applied. That is, before the kinetic
effects become important, RMHD is used to simulate the macroscopic states of
the plasmas, and then PIC is used to simulate the subsequent kinetic
processes, while ensuring a self-consistent connection between them. Just like
the self-similarity principle of the magnetohydrodynamic equations
Remington1999 ; Ryutov2000 ; Ryutov2001 , we choose a set of free parameters
to spatially scale down the plasma parameters obtained from the above RMHD
simulation to the relative small scale that is practical for kinetic PIC
simulation, where the plasma properties including $\beta$, $\beta_{\rm ram}$,
the Mach number and the Alfvénic Mach number, etc. are all kept conserved
between RMHD and PIC. The Coulomb collisions are also well included by making
$L/\lambda_{ei}$ conserved.

The principle of this method comes from the self-similarity of the ideal MHD
equations Remington1999 ; Ryutov2000 ; Ryutov2001 , that is, under a set of
specific parameter transformation,

$\displaystyle\textbf{r}_{1}$ $\displaystyle=a\
\textbf{r}_{0},\quad\rho_{1}=b\ \rho_{0},\quad p_{1}=c\ p_{0},$ (7)
$\displaystyle\textbf{v}_{1}=$ $\displaystyle\sqrt{c/b}\
\textbf{v}_{0},\quad\textbf{B}_{1}=\sqrt{c}\ \textbf{B}_{0},\quad
t_{1}=a\sqrt{b/c}\ t_{0},$

the form of the ideal MHD equations remains unchanged. $a$, $b$ and $c$ are
the free transformation parameters. r, $\rho$, $p$, v, B and $t$ respectively
represent length, density, pressure, velocity, magnetic field and time. The
subscripts 0 and 1 respectively represent two different systems connected by
this parameter transformation. This means that under similar initial and
boundary conditions, the two systems have the same evolution characteristics.
This scale transformation is often used in laboratory astrophysics to compare
experiment results with astrophysical events.

The condition for the establishment of this self-similarity is that the
plasmas is polytropic and ideal, which means that viscosity, electric
resistivity and heat conductivity are not dominant, i.e., the Reynolds number
$R_{e}\equiv Lv/\nu$, the magnetic Reynolds number $R_{m}\equiv Lv/\eta$ and
the Peclet number $P_{e}\equiv Lv/\chi$ are far greater than 1.0. These
dimensionless numbers, before the Cu and CH plasma bubbles touch each other,
are listed in Table 1(a). Both $R_{e}$ and $R_{m}$ are far greater than 1.0.
Although $P_{e}$ is less than 1.0, considering that the plasma corona is
approximately isothermal (see Supplementary Fig. blueS2) and the suppression
of the electron heat flow by the magnetic fields, the coronal $P_{e}$ will be
greater than 1.0. And the coronal plasma is approximately an ideal gas, so it
is also polytropic. Therefore, under limited approximations, the laser-driven
expanding plasmas can be regarded as an ideal polytropic gas, where the self-
similar transformation can be applied.

In the context of PIC simulations, since the plasma density has the relation
$\rho\sim m_{i}n_{i}$ and pressure $p\sim nT$, the transformation parameters
should be chosen as $b=f_{m}f_{n}$ and $c=f_{n}f_{T}$, where $f_{m}$, $f_{n}$
and $f_{T}$ are the transformation parameters of ion mass, plasma density and
plasma temperature respectively. As for the transformation parameter $a$ of
length, in order to correctly account for the two-fluid effect, it should be
ensured that $L/d_{i}$ (on the order of 100 in typical laser-ablated plasmas)
remains unchanged under the similar transformations, and this constraint gives
$a=\sqrt{f_{m}/f_{n}}$, which also maintains compatibility with the Ampere’s
law $\textbf{J}=\nabla\times\textbf{B}$. In this way, the self-similar
transformation of ideal RMHD equations is self-consistently converted into the
language of PIC simulation, that is, the transformation parameters $a$, $b$
and $c$ of RMHD variables are converted into $f_{m}$, $f_{n}$ and $f_{T}$ of
PIC particle variables,

$a=\sqrt{f_{m}/f_{n}},\quad b=f_{m}f_{n},\quad c=f_{n}f_{T}.$ (8)

Other variables are also transformed according to the corresponding
parameters, the fluid velocity
$\textbf{v}_{1}/\textbf{v}_{0}=\sqrt{f_{T}/f_{m}}=c_{s,1}/c_{s,0}$, i.e., the
ratio of the ion sound velocity of the two systems. And the time
$t_{1}/t_{0}=f_{m}/\sqrt{f_{n}f_{T}}=f_{m}/\sqrt{c}=\omega_{ci,1}^{-1}/\omega_{ci,0}^{-1}$,
the ratio of the ion cyclotron time of the two systems.

We apply this novel transformation methodology to the kinetic simulation of
hybrid magnetic reconnection, and transform the results of the RMHD simulation
at $t=0.8$ ns to the PIC simulation, see Supplementary Fig. blueS2 and Fig.
blueS3. Balance the computing power and the universality, we choose the mass
of Cu ions in the PIC simulation to be $m_{\rm Cu}=1000m_{e}$, and the mass of
CH ions $m_{\rm CH}$ and He ions $m_{\rm He}$ are calculated proportionally,
which gives $f_{m}\sim 0.0085$. The parameters of number density $f_{n}=10$
and temperature $f_{T}=5$. These parameters lead to $t_{1}/t_{0}\sim 0.0012$,
i.e., 1.2 ps in PIC simulations represents 1.0 ns in RMHD simulations and
experiments. In the PIC simulations, the ion charge state $\bar{Z}$ is always
consistent with the RMHD simulations. The whole simulation box is divided into
$1600\times 2400$ grids, and 100 macro-particles per species are employed in
each grid. Open boundary conditions are also applied in each direction. Other
set of transformation parameters hardly affect the final results.

Although the introduction of collision effects will destroy the self-
consistency of similar transformations (due to the bad-scaled transport
coefficient $\nu$, $\eta$ and $\chi$), as long as the dimensionless parameters
$R_{e}$, $R_{m}$ and $P_{e}$ are far greater than 1.0, then this deviation is
negligible. In the hybrid collisional-collisionless magnetic reconnection, the
collision effect of Cu plasma may be important near the current sheet, where
$\lambda_{ei}\sim 2\delta$. The commonly used parameter to evaluate the
strength of the collision effect is $L/\lambda_{ei}$. In the PIC simulation,
we keep this parameter consistent with the RMHD simulation and experiments.
Because of $\lambda_{ei}\propto T_{e}^{2}/n_{e}$, the transformation parameter
of the collision frequency is
$\nu_{ei,1}/\nu_{ei,0}=f_{T}^{2}/\sqrt{f_{m}f_{n}}$. By this way, the binary
Coulomb collision Nanbu1998 can be appropriately included.

### Synthetic proton radiography simulation

In order to directly compare with the experimental proton radiography images,
we perform synthetic, numerical proton radiography simulation based on the
magnetic fields obtained from the self-consistent combined MHD-PIC simulation.
The main steps are as follows: protons with a point source are magnified by
point projection, then enter the area filled with magnetic fields and deflect
by the Lorentz force. After escaping these magnetic fields, they perform a
uniform linear motion and are finally deposited in the simulated RCF stacks.
These protons are regarded as PIC macro-particles, and their energy is
consistent with the experiments. The geometry and magnification rate of the
system are also consistent with the experiments. The deflection motion of
these protons in the magnetic fields adopts the Boris algorithm, and the
magnetic field fluxes are equivalent to the path-integrated magnetic field
$\psi$ that extracted from the experimental images. These protons deposit
energy in the simulated RCF stacks and gradually slow down until they stop.
The stopping power of different materials come from the NIST Standard
Reference Database NIST , and the scattering effect is ignored. The accuracy
of this code has been well benchmarked.

## Data availability

All data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Code availability

FLASH is an open radiation MHD simulation code for plasma physics and
astrophysics, developed by the DOE NNSA-ASC OASCR Flash Center at the
University of Chicago. The code is available for download from
http://flash.uchicago.edu/site/flashcode/.

EPOCH is the Extendable PIC Open Collaboration project to develop a UK
community advance relativistic EM PIC code. The open source code is available
for download from https://cfsa-pmw.warwick.ac.uk/EPOCH/epoch.

PROBLEM solver is an open source Python implementation of the proton
radiography reconstruction algorithm of Bott (2017), the source code is
available for download from https://github.com/flash-center/PROBLEM.

## References

## References

  * (1) Zweibel, E. G. & Yamada, M. Magnetic Reconnection in Astrophysical and Laboratory Plasmas. _Annu. Rev. Astron. Astrophys._ 47, 291–332 (2009).
  * (2) Yamada, M., Kulsrud, R. & Ji, H. Magnetic reconnection. _Rev. Mod. Phys._ 82, 603–664 (2010).
  * (3) Schrijver, C. J. _et al._ Large-scale coronal heating by the small-scale magnetic field of the Sun. _Nature_ 394, 152–154 (1998).
  * (4) Piran, T. The physics of gamma-ray bursts. _Rev. Mod. Phys._ 76, 1143–1210 (2004).
  * (5) Biskamp. D. Collisional and collisionless magnetic reconnection. _Phys. Plasmas_ 4, 1964–1968 (1997).
  * (6) Shibata, K. Chromospheric Anemone Jets as Evidence of Ubiquitous Reconnection. _Science_. 318, 1591–1595 (2007).
  * (7) Sweet, P. A. The Production of High Energy Particles in Solar Flares. _Nuovo Cim_ 8, 188–196 (1958).
  * (8) Parker, E. N. The solar flare phenomenon and the theory of reconnection and annihilation of magnetic fields. _Astrophys. J. Suppl. Ser._ 8 177–211 (1963).
  * (9) Cassak, P. A. & Fuselier, S. A. Reconnection at Earth’s Dayside Magnetopause. in _Magnetic Reconnection: Concepts and Applications_ (eds. Gonzalez, W. & Parker, E.) 213–276 (Springer International Publishing, 2016).
  * (10) Petrukovich, A., Artemyev, A. & Nakamura, R. Magnetotail Reconnection. in _Magnetic Reconnection: Concepts and Applications_ (eds. Gonzalez, W. & Parker, E.) 277–313 (Springer International Publishing, 2016).
  * (11) Hesse, M., Birn, J. & Kuznetsova, M. Collisionless magnetic reconnection: Electron processes and transport modeling. _J. Geophys. Res. Sp. Phys._ 106, 3721–3735 (2001).
  * (12) Hesse, M., Neukirch, T., Schindler, K., Kuznetsova, M. & Zenitani, S. The Diffusion Region in Collisionless Magnetic Reconnection. _Space Sci. Rev._ 160, 3–23 (2011).
  * (13) Li, L. _et al._ Magnetic reconnection between a solar filament and nearby coronal loops._Nat. Phys._ 12, 847–851 (2016).
  * (14) Yang, B. & Chen, H. Filament Eruption and Its Reformation Caused by Emerging Magnetic Flux. _Astrophys. J._ 874, 96 (2019).
  * (15) Nilson, P. M. _et al._ Magnetic reconnection and plasma dynamics in two-beam laser-solid interactions. _Phys. Rev. Lett._ 97, 255001 (2006).
  * (16) Li, C. K. _et al._ Observation of megagauss-field topology changes due to magnetic reconnection in laser-produced plasmas. _Phys. Rev. Lett._ 99, 055001 (2 007).
  * (17) Zhong, J. _et al._ Modelling loop-top X-ray source and reconnection outflows in solar flares with intense lasers. _Nat. Phys._ 6, 984–987 (2010).
  * (18) Fiksel, G. _et al._ Magnetic reconnection between colliding magnetized laser-produced plasma plumes. _Phys. Rev. Lett._ 113, 105003 (2014).
  * (19) Rosenberg, M. J. _et al._ Slowing of Magnetic Reconnection Concurrent with Weakening Plasma Inflows and Increasing Collisionality in Strongly Driven Laser-Plasma Experiments. _Phys. Rev. Lett._ 114, 205004 (2015).
  * (20) Rosenberg, M. J. _et al._ A laboratory study of asymmetric magnetic reconnection in strongly driven plasmas. _Nat. Commun._ 6, 6190 (2015).
  * (21) Shibata, K. & Tanuma, S. Plasmoid-induced-reconnection and fractal reconnection. _Earth Sp. Sci._ 53, 473–482 (2001).
  * (22) Shibata, K. & Takasao, S. Fractal Reconnection in Solar and Stellar Environments. in _Magnetic Reconnection: Concepts and Applications_ (eds. Gonzalez, W. & Parker, E.) 373–407 (Springer International Publishing, 2016).
  * (23) Uzdensky, D. A. & Loureiro, N. F. Magnetic Reconnection Onset via Disruption of a Forming Current Sheet by the Tearing Instability. _Phys. Rev. Lett._ 116, 105003 (2016).
  * (24) Samtaney, R. _et al._ Formation of plasmoid chains in magnetic reconnection. _Phys. Rev. Lett._ 103, 105004 (2009).
  * (25) Uzdensky, D. A., Loureiro, N. F. & Schekochihin, A. A. Fast magnetic reconnection in the plasmoid-dominated regime. _Phys. Rev. Lett._ 105, 235002 (2010).
  * (26) Dong, Q.-L. _et al._ Plasmoid Ejection and Secondary Current Sheet Generation from Magnetic Reconnection in Laser-Plasma Interaction. _Phys. Rev. Lett._ 108, 215001 (2012).
  * (27) Hare, J. D. _et al._ Anomalous Heating and Plasmoid Formation in a Driven Magnetic Reconnection Experiment. _Phys. Rev. Lett._ 118, 085001 (2017).
  * (28) Petrasso, R. D. _et al._ Lorentz mapping of magnetic fields in hot dense plasmas. _Phys. Rev. Lett._ 103, 085001 (2009).
  * (29) Gao, L. _et al._ Precision mapping of laser-driven magnetic fields and their evolution in high-energy-density plasmas. _Phys. Rev. Lett._ 114, 215003 (2015).
  * (30) Zhao, Z. H. _et al._ Three-dimensional synchronous proton radiography for dynamic magnetic fields in laser-produced high-energy-density plasmas. submitted.
  * (31) Bott, A. F. A. _et al._ Proton imaging of stochastic magnetic fields. _J. Plasma Phys._ 83, 905830614 (2017).
  * (32) Graziani, C., Tzeferacos, P., Lamb, D. Q. & Li, C. Inferring morphology and strength of magnetic fields from proton radiographs. _Rev. Sci. Instrum_. 88, 123507 (2017).
  * (33) Tubman, E. R. _et al_. Observations of pressure anisotropy effects within semi-collisional magnetized plasma bubbles. _Nat. Commun._ 12, 334 (2021).
  * (34) Fryxell, B. _et al._ FLASH : AN ADAPTIVE MESH HYDRODYNAMICS CODE FOR MODELING ASTROPHYSICAL THERMONUCLEAR FLASHES. _The Astrophysical Journal Supplement Series_ 131, 273-334 (2000).
  * (35) Heltemes, T. A. & Moses, G. A. BADGER v1.0: A Fortran equation of state library. _Comput. Phys. Commun._ 183, 2629–2646 (2012).
  * (36) Macfarlane, J. J. IONMIX-A CODE FOR COMPUTING THE EQUATION OF STATE AND RADIATIVE PROPERTIES OF LTE AND NON-LTE PLASMAS. _Comput. Phys. Commun._ 56, 259–278 (1989).
  * (37) Fox, W., Bhattacharjee, A. & Germaschewski, K. Fast Magnetic Reconnection in Laser-Produced Plasma Bubbles. _Phys. Rev. Lett._ 106, 215003 (2011).
  * (38) Totorica, S. R., Abel, T. & Fiuza, F. Nonthermal Electron Energization from Magnetic Reconnection in Laser-Driven Plasmas. _Phys. Rev. Lett._ 116, 095003 (2016).
  * (39) Xu, Z. _et al._ Characterization of magnetic reconnection in the high-energy-density regime. _Phys. Rev. E_ 93, 033206 (2016).
  * (40) Remington, B. A., Arnett, D., Drake, R. P. & Takabe, H. Modeling Astrophysical Phenomena in the Laboratory with Intense Lasers. _Science_ 284, 1488–1493 (1999).
  * (41) Ryutov, D. D., Drake, R. P. & Remington, B. A. Criteria for Scaled Laboratory Simulations of Astrophysical MHD Phenomena. _Astrophys. J. Suppl. Ser._ 127, 465–468 (2000).
  * (42) Ryutov, D. D., Remington, B. A., Robey, H. F., Drake, R. P. & Introduction, I. Magnetohydrodynamic scaling: From astrophysics to the laboratory*. _Phys. Plasmas_ 8, 1804 (2001).
  * (43) Nanbu, K. & Yonemura, S. Weighted Particles in Coulomb Collision Simulations Based on the Theory of a Cumulative Scattering Angle. _J. Comput. Phys._ 145, 639-654 (1998).
  * (44) Arber, T. D. _et al._ Contemporary particle-in-cell approach to laser-plasma modelling. _Plasma. Phys. Controlled. Fusion._ 57, 113001 (2015).
  * (45) Fitzpatrick, R. Collisionless magnetic reconnection with arbitrary guide field. _Phys. Plasmas_ 11, 4713 (2004).
  * (46) Loureiro, N. F. & Boldyrev, S. Collisionless Reconnection in Magnetohydrodynamic and Kinetic Turbulence. _Astrophys. J._ 850, 182 (2017).
  * (47) Hosseinpour, M. A fast tearing mode instability driven by agyrotropic electron pressure. _Adv. Sp. Res._ 54, 955–960 (2014).
  * (48) NIST Standard Reference Database, https://www.nist.gov/pml/stopping-power-range-tables-electrons-protons-and-helium-ions, (2017).

## Acknowledgements

This work is supported by Science Challenge Project, No. TZ2018005; National
Natural Science Foundation of China, Grant Nos. 12135001, 11825502 and
11921006; the Strategic Priority Research Program of Chinese Academy of
Sciences, Grant No. XDA25050900; BQ acknowledges support from National Natural
Science Funds for Distinguished Young Scholar, Grant No. 11825502. The
simulations are carried out on the Tianhe-2 supercomputer at the National
Supercomputer Center in Guangzhou.

## Author contributions

B.Q. and S.P.Z. proposed and were in charge of the research campaign as
principle investigators. Z.H.Z., Y.X., Z.L., W.P.Y., W.Q.Y. and B.Q. carried
the simulations, the data analysis. H.H.A., Z.H.Z., J.X., C.W., J.J.Y, Z.Y.X.,
Z.H.F., A.L.L., W.M.Z. and B.Q. carried out the experiments. B.Q., and Z.H.Z.
wrote the paper. W.B.P. S.P.Z. and X.T.H. contribute to theoretical
interpretations of the results.

## Competing interests

The authors declare no competing interests.

## Additional information

Correspondence and requests for materials should be addressed to B.Q.