12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

MIXING TRANSITION IN THE MAGNETIC RAYLEIGH-TAYLORINSTABILITY

Antoine Briard,
CEA, DAM, DIF, 91297 Arpajon, France
antoine.briard@cea.fr

Benoı̂t-Joseph Gréa
CEA, DAM, DIF, 91297 Arpajon, France
LMCE, Université Paris-Saclay, France,
benoit-joseph.grea@cea.fr

Florian Nguyen
CEA, DAM, DIF, 91297 Arpajon, France
florian-nguyen@orange.fr

ABSTRACTparallel B0 , with elongated fingers separated by a distanceλc = 2πB20 /(A g), which is somehow representative of the
Crab nebula structure (Stone & Gardiner (2007a,b); Carlyle& Hillier (2017)).
The knowledge of this typical distance λc between structures and their growth rate α may be used to evaluate the intensity of the ambient magnetic field, under some strong hypothesis (Ryutova (2010); Hillier (2018)).
Note that in this configuration with a mean magnetic
parallel to the interface, the interchange mode, such thatk · B 0 = 0, is not damped at all. This is the reason why
magnetic shear has also been investigated (Carlyle & Hillier,2017) where the direction of B 0 , still in the plane of theinterface, is different in the lower and upper fluids.

The effect of a mean magnetic field on the developmentof the Rayleigh-Taylor instability is investigated using highresolution direct numerical simulations. The magnetic field isapplied and maintained perpendicular to the interface between
two miscible fluids. It initially inhibits small-scale shear instabilities so that mixing is reduced, and hence structures arestrongly stretched in the vertical direction and rapidly grow.
At some point, when the typical vertical velocity exceeds the
Alfvèn velocity imposed by the mean magnetic field, the flow
experiences a rapid transition to turbulence, shear instabilities
develop, structures break and mixing occurs.

INTRODUCTION

In this work, we rather focus on the less investigated framwork where the mean magnetic field is perpendicular to theinterface: this configuration remains statistically axisymetric.
In this case, the growth rate of a single mode of wavelength
2π/k is damped, but the instability still occurs (Chandrasekhar(1961)): unlike the tangential magnetic field, there is no critical wavelength for which the instability is totally suppressed.
For a fully perturbed interface, Jun et al. (1995) observedwith 2D simulations that, depending on the magnitude of B0 ,the MRTI can be either enhanced or reduced with respect tothe non-magnetic case. However, it is not clear how the 2D
configuration affects the results: indeed, it is known that in thepurely hydrodynamic case, 2D and 3D simulations of the RTI
yield quite different results (Dimonte, 2004).
In order to shed some light on the MRTI, we propose toquantify how the vertical mean magnetic field affects mixing
at small scales and the dynamics of large scales, thanks tohigh resolution DNS. In particular, we are interested in thetransition between the early damped phase, where magnetic
effects inhibit shear instabilities between structures and hence
turbulent mixing, and a later regime where turbulence is strong
enough to break the smooth elongated fingers.

Perturbations at the interface between two fluids such thatthe direction of the acceleration is opposite to the mean density
gradient may become unstable with the baroclinic productionof vorticity: this is the Rayleigh-Taylor instability (RTI). In thecase of miscible fluids, a turbulent mixing zone emerges andgrows with time. If these two fluids have magnetic properties,
the magnetohydrodynamics (MHD) equations are required, in
addition of the Navier-Stokes equations, to describe the fulldynamics. The magnetic Rayleigh-Taylor instability (MRTI)
can play an important role in several astrophysical systems,
such as for example the formation of the Crab nebula, theemergence of magnetic fluxes from the Sun interior, the accretion onto compact objects, the expansion of young supernova
remnants, ... (see Carlyle & Hillier (2017); Hillier (2018) andreferences therein).
For inviscid and incompressible fluids with a mean magnetic field applied parallel to the interface, a perturbation of
wavevector k parallel to the interface is unstable only if B0(the mean magnetic field intensity scaled asp
the Alfvèn velocity) is smaller than the critical value Bc = A g/k, where
A is the Atwood number, k = |kk | the wavenumber, and g themagnitude of the gravitationnal acceleration (Chandrasekhar(1961)). Early simulations have revealed that ascending anddescending structures are greatly smoothed when B0 < Bc ,changing the usual picture of growing mushrooms into elongated fingers (Jun et al. (1995)). This observation was confirmed later in 3D direct numerical simulations (DNS), where
an initial perturbed interface develops, in the presence of a

EQUATIONS AND NUMERICAL SETUP
The motion of the two incompressible fluids, initially at
rest in an unstable configuration and separated by a flat interface, is given by the following set of equations within the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
non-magnetic case. Shortly after t = 0, a variety of bubblelike structures appear. Merging and shearing between ascending and descending structures lead to turbulent mixing and agreat diversity of scales, which can be deduced from the second image of figure 1 with intermediate values of C (light blue
and yellow). Later on, the turbulent mixing zone L(t) eventually grows like L ∼ t 2 (Youngs, 1994).
If a vertical magnetic field is added to the previous configuration, the development of structures is dramatically altered, as revealed in figure 2. Indeed, instead of growing
bubbles, the interface perturbations are stretched, without any
mixing, which gives birth to these remarkable smooth fingers.
The mean magnetic field inhibits small-scale shear instabilities
between structures, which hence eventually grow faster thanin the non magnetic case. When turbulence becomes strong
enough to overcome the damping imposed by the magnetic
field, fingers develop bubbles at their edges. Later on, the usual
picture of a turbulent Rayleigh-Taylor mixing zone is recovered, with bubbles of different sizes, and complex interacting
patches of scalar of different intensities, as can be seen on thesides of the DNS box in the third image of figure 2.
To better quantify the complex dynamics of the MRTI, weinvestigate in figure 3 the time evolution of the mixing zone
size, defined as

Boussinesq approximation:∇ p − 2A gCnn3∂t + u · ∇ − ν∇2 u = −∇∇ × B) × B,+ (∇B · ∇)uu,∂t + u · ∇ − η∇2 B = (B∂t + u · ∇ − κ∇2 C = 0,
∇ · u = 0,

∇ · B = 0,

(1)
(2)
(3)
(4)

where u is the velocity field, B the total magnetic field scaled
as a velocity, with B = b + B0 n 3 and n 3 the upward unit vector,
C the concentration field, p is the reduced pressure, and ν, ηand κ are respectively the kinematic viscosity, magnetic diffusivity and molecular diffusivity. For simplicity, we chooseν = κ = η.
The pseudo-spectral code STRATOSPEC, previously
used to investigate turbulent mixing within the Faraday instability (Briard et al. (2020)), is employed here to solve equations (1)-(4). The domain is a triply periodic cubic box ofsize 2π, and time advancement is realized using a third-orderstrong stability preserving Runge-Kutta scheme. To evaluate
the effects of the magnitude of B0 , DNS with N = 10243 grid
points are performed. A summary of the simulations is provided in Table 1.

Z

L(t) = 6
Table 1. Parameters of the DNS: Number of points N, Atwood number times gravitational acceleration A g, Mean vertical magnetic field intensity B0 , Diffusion coefficients ν.

Label

N

Ag

B0

ν =κ =η

R1B0

10243

0.5

0

2 × 10−4

R1B01

10243

0.5

0.10

2 × 10−4

R1B02

10243

0.5

0.20

2 × 10−4

R1B05

10243

0.5

0.50

2 × 10−41z − S (x, y)
1 + tanh
,
2
σ

(6)

where h·i refers to the horizontal average. At the instability onset, the non magnetic case (black) obviously increases the most
rapidly, since there is no damping by the mean magnetic field.
This is in agreement with the linear theory (Chandrasekhar,1961) and with early 2D simulations by Jun et al. (1995). For
B0 6= 0, the more intense B0 , the slower the growth of L(t).
But later one, when turbulence becomes sufficiently intense to overcome the magnetic tension, structures break andone eventually recovers the usual picture of the RTI, with anonlinear saturation that yields L(t) = 2αA gt 2 , with α thegrowth rate of the ”bubbles”, which is approximately α = 0.02
in R1B0.
It is possible to be more specific about the statement thatstructures break when ”turbulence becomes strong enough”.
In fact, such a criterion was already invoked in Jun et al.
(1995). Considering the (vertical) interface between ascending
and descending smooth structures in the case of a sufficiently
intense vertical B0 , this amounts to a kind of magnetic KelvinHelmholtz configuration, without any gravity. The inviscid linear stability analysis of the magnetic Kelvin-Helmholtz flow
was performed in Chandrasekhar (1961), and in the present
situation this eventually yields that the structures are destabilized if

At t = 0, only the scalar field is perturbed at the interface
according toC(xx,t = 0) =hCi (1 − hCi) dz,(5)with σ = 0.02, a small parameter that ensures at least 20 points
along z for the lowest resolution. S (x, y) is a deformation surface that is determined by a variance and a peak wavenumber:the same deformation is used for all simulations at a given resolution, with the peak wavenumber 40 ≤ k p ≤ 50, and an infrared slope like k4 .
With such an initialization, scalar perturbations induce
velocity fluctuations through the buoyancy term 2A gC in(1), which in turn produce magnetic fluctuations through thestretching term B0 ∂z u in (2).

V3 > B0 ,(7)

namely if the typical turbulent vertical velocity V3 exceeds
the Alfvènvelocity B0 . To verify this criterion, we chooseqV3 = u23 , where u23 is the vertical kinetic energy. The comparison between the DNS results and the simple criterion (7)is presented in figure 4.
The ratio V3 /B0 is presented for three different values of
B0 . For the lowest value B0 = 0.10, V3 overcomes B0 quite
early, and a change of regime is observed around t ' 3, which
was already seen in figure 3. This change of dynamics illus-

TRANSITION TO TURBULENT MIXINGFirst, in order to illustrate the impact of a vertical meanmagnetic field on a developing mixing zone, the 3D concentration field C of DNS R1B0 is presented in figure 1 in the2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1. Time evolution of the concentration field C (from left to right) of DNS R1B0 10243 , without any magnetic field. The color
scale indicates the value of C, that varies between 0 (dark blue) and 1 (red); pure fluids are transparent.

Figure 2. Time evolution of the concentration field C (from left to right) of DNS R1B02 10243 , with the vertical mean magnetic field
intensity B0 = 0.2. The color map is the same as figure 1.

7

3.5
2

6

3
1

5

2.5

4

2
0
0

3

2

4

1.5

2

1

1

0.5

0

0
0

2

4

6

8

10

0

Figure 3. Effects of the mean magnetic field intensity B0 in
the R1 simulations. Mixing zone size L (6).

2

4

6

8

10

Figure 4. Effects of the mean magnetic field intensity B0
in the R1 simulations. Normalized vertical velocity with the
Alfven velocity V3 /B0 , see criterion (7).

trates the transition between smooth growing fingers, and turbulent mixing of sheared bubbles: this will be better characterized in terms of anisotropy and mixing later on.
For B0 = 0.20, the transition happens later since the magnetic damping is stronger. Finally, for the strongest Alfvènvelocity B0 = 0.5, turbulence is never intense enough to overcome the mean magnetic field and generate small-scale shear
instabilities between structures. Hence, the criterion (7), originating from inviscid linear stability analysis, provides a lower
bound for the transition to turbulence in the MRTI.
Now that we have shown that transition to turbulencecorresponds to a slowing down of the turbulent mixing zone

growth, which is well characterized by the ratio of the verticalturbulent velocity and Alfvèn velocity, we turn our attention to
mixing, and define the so-called mixing parameterΘ = 1−6LZ

hc2 idz,(8)

where hc2 i refers to the horizontal average of the concentration variance. This mixing parameter quantifies how well thetwo fluids are mixed together, with Θ = 1 for perfectly mixed3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

1

1.2

0.8

1

0.6

0.8

0.4

0.6

0.2

0.4

0

0.2
0

2

4

6

8

10

0

Figure 5. Effects of the mean magnetic field intensity B0 in
the R1 simulations. Mixing parameter Θ defined in (8).

Ecc (kk ) sin3 θ dθ dk,k ) sin θ dθ dk
0 Ecc (k6810When there is an effective transition to turbulence withsmall-scale shear instabilities, sin2 γ is significantly reduced,
which is correlated to the increase of Θ in figure 5, due to thereturn to isotropy of the smallest scales of the flow. Except
for B0 = 0.5, where the Alfvèn velocity is so strong that thesmooth elongated fingers continue to grow without mixing.
An interesting feature here, different from the observations made regarding Θ, is that even in the asymptotic state,
the final value of sin2 γ increases with B0 , which shows that
there is still a strong imprint, in the fully turbulent regime, ofthe vertical anisotropy imposed by the mean magnetic field.

CONCLUSIONSThe Rayleigh-Taylor instability submitted to a mean vertical magnetic field has been investigated with the use of highresolution DNS. The dramatic effect of the Lorentz force on
the overall dynamics is to initially inhibit small-scale shear
instabilities between ascending and descending structures, sothat mixing is strongly damped. Instead of turbulent bubbles,smooth elongated fingers emerge, that grow rapidly since there
is no small scale mixing.
When turbulence becomes strong enough, meaning thatthe typical vertical velocity of the fluid exceeds the Alfvènvelocity, shear eventually induces mixing, fingers break andtraditional bubbles are recovered. The mixing and global
anisotropy parameters were investigated to better quantify thistransition to turbulence, and the outcome is interesting. It
seems that there is an optimal value of B0 such that mixing
remains as effective as the non-magnetic case. This is possibly
because with a long transient where smooth elongated fingers
grow, a larger surface for mixing becomes available when theflow transitions to turbulence.
Regarding anisotropy, the conclusion is somehow different, since even in the fully turbulent regime, the verticalanisotropy of the concentration field is greater for larger B0 ,which is a permanent imprint of the mean vertical magnetic
field.
The perspectives of this study are to investigate moredeeply two-point statistics in order to provide scale-by-scale
information about anisotropy, but also spectral scalings. We
wish also to perform simulations at better resolution to increase the level of turbulence, and investigate if the observations made regarding the ”booster effect” of the vertical meanmagnetic field persist.

R∞Rπ
0

4

Figure 6. Effects of the mean magnetic field intensity B0 in
the R1 simulations. Global anisotropy parameter sin2 γ definedin (9).

fluids. In classical RTI, this parameter can be related to thegrowth rate α of the mixing zone under the Boussinesq approximation (Gréa (2013)).
We investigate in figure 5 this mixing parameter for different values of the mean vertical magnetic field B0 . For the nonmagnetic case, Θ tends to 0.8 after a transient regime, value
which is typical for the RTI (see fig9 in Gréa (2013) and references therein). For a weak magnetic field B0 = 0.10, thetransient regime is longer, but the final state remains close to
the hydrodynamic case. Nevertheless, the final value of Θ isslightly smaller, showing that the initial damping of the meanmagnetic field is reflected in the long-time dynamics.
More interestingly, this behavior is not monotonous withincreasing B0 . Indeed, when the mean magnetic field intensity
is further increased to B0 = 0.20, the transient regime is once
again longer, but the final value of Θ is the same as in the nonmagnetic case. Therefore, according to these global quantities,
it seems that there is an optimal B0 for a given configurationthat boosts turbulence: this observation was already made in
2D simulations (Jun et al., 1995).
Finally, for B0 = 0.50, as already seen in figure 4, thereis never a transition to turbulence, so that magnetic tension always inhibits small-scale shearing instabilities, and no mixing
occurs. Remark that the strong initial decrease of Θ reflects thelength of the transient regime where structures grow without
significant small-scale shear. For different initial conditions,with bigger perturbations for instance, the initial value of Θ
could be smaller.
Finally, we analyze the anisotropy at the level of the scalar
field. This is done thanks to the following parametersin2 γ = R0∞ R0π

2(9)where Ecc (kk ) = ĉ(kk )ĉ(kk ) is the spectral variance density, ĉ isthe Fourier transformof the concentration fluctuation c = C −RR
hCi, so that k2 Ecc sin θ dθ dk = c2 , with k · n 3 = k cos θ . In
particular, for an isotropic field, sin2 γ = 2/3.
This global anisotropy parameter is presented in figure
6. Remarks comparable to those regarding Θ can be made,namely that the stronger B0 , the longer the transient regime.
In this transient regime, sin2 γ first increases, which reflects thestretching of the initial interface perturbations under the effects
of gravity. The mean magnetic field amplifies this anisotropy
by stretching even more the structures. Note that values like
2/3 ≤ sin2 γ ≤ 1 correspond to vertically elongated structures.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
REFERENCES

instability. Physics of Fluids 25, 015118.
Hillier, A. 2018 The magnetic Rayleigh-Taylor instability insolar prominences. Rev. Mod. Plasma Phys. 2, 1–47.
Jun, B.-I., Norman, M.L. & Stone, J.M. 1995 The non-linear
growth of the magnetic Rayleigh-Taylor instability. The Astrophysical Journal 453, 332–349.
Ryutova, M. et al. 2010 Observation of plasma instabilities inquiescent prominences. Solar Phys. 267, 75–94.
Stone, J.M. & Gardiner, T. 2007a The magnetic RayleighTaylor instability in three dimensions. The AstrophysicalJournal 671, 1726–1735.
Stone, J.M. & Gardiner, T. 2007b Nonlinear evolution of themagnetohydrodynamic Rayleigh-Taylor instability. Physics
of Fluids 19, 094104.
Youngs, D.L. 1994 Numerical simulation of mixing by
Rrayleigh-Taylor and Richtmyer-Meshkov instabilities.
Laser Particle Beams 12, 725–750.

Briard, A., Gostiaux, L. & Gréa, B.-J. 2020 The turbulent Faraday instability in miscible fluids. Journal of Fluid Mechanics 883, A57.
Carlyle, J. & Hillier, A. 2017 The non-linear growth of themagnetic Rayleigh-Taylor instability. Astronomy & Astrophysics 605, A101.
Chandrasekhar, S. 1961 X: The stability of superimposed fluids: the Rayleigh-Taylor instability. In Hydrodynamic andHydromagnetic Stability, pp. 428–477. Oxford University
Press.
Dimonte, G. et al. 2004 A comparative study of the turbulent Rayleigh-Taylor instability using high-resolution threedimensional numerical simulations: The Alpha-Group collaboration. Physics of Fluids 16, 1668.
Gréa, B.-J. 2013 The rapid acceleration model and the growthrate of a turbulent mixing zone induced by Rayleigh-Taylor

5