The Rayleigh–Taylor (RT) and Darrieus–Landau (DL) instabilities are studied in an inertial confinement fusion context within the framework of small critical-to-shell density ratio $DR$ and weak acceleration regime, i.e., large Froude number Fr. The quasi-isobaric analysis in Sanz *et al.* [Phys. Plasmas **13**, 102702 (2006)] is completed with the inclusion of non-isobaric and self-generated magnetic-field effects. The analysis is restricted to perturbation wavelengths $k\u22121$ larger than the conduction length scale at the ablation front, yet its validity ranges from wavelengths shorter and larger than the conduction layer width (distance between the ablation front and the critical surface). The use of a sharp boundary model leads to a single analytical expression of the dispersion relation encompassing both instabilities. The two new effects come into play by modifying the perturbed mass and momentum fluxes at the ablation front. The momentum flux (perturbed pressure at the spike) is the predominant stabilizing mechanism in the RT instability (overpressure) and the driving mechanism in the DL instability (underpressure). The non-isobaric effects notably modify the scaling laws in the DL limit, leading to an underpressure scaling as $\u223ck\u221211/15$ rather than $\u223ck\u22122/5$ obtained in the quasi-isobaric model. The magnetic fields are generated due to misalignment between pressure and density gradients (Biermann battery effect). They affect the hydrodynamics by bending the heat flux lines. Within the framework of this paper, they enhance ablation, resulting in a stabilizing effect that peaks for perturbation wavelengths comparable to the conduction layer width. The combination of parameters $DRFr2/3$ defines the region of predominance of each instability in the dispersion relation. It is proven that the DL region falls outside of the parameter range in inertial confinement fusion.

## I. INTRODUCTION

In direct-drive inertial confinement fusion (ICF),^{1,2} preserving the symmetry of the capsule is crucial for the efficiency of the implosion. The initial asymmetries are amplified by the hydrodynamic instabilities inherent to the process. Amongst others, the Rayleigh–Taylor (RT)^{3} and Darrieus–Landau (DL)^{4,5} instabilities play an important role since they compromise the integrity of the capsule.

The Rayleigh–Taylor instability has been thoroughly studied in the context of ICF.^{6–9} The laser energy, absorbed at the critical surface, is convected toward the cold dense shell by heat conduction. Mass is ablated off the shell and expands into the hot light corona, forming a structure commonly known as the ablation front. During the acceleration phase, this structure is RT unstable. In the linear regime, ablation stabilizes perturbations whose wavelengths are shorter than certain cutoff, $kco\u22121$, which depends on the Froude number Fr. This number is the governing dimensionless parameter in the ablative RT instability and stands for the ratio between ablative convection and the acceleration of the capsule. We restrict this analysis to the weak acceleration regime, which is mathematically characterized by $Fr\u226b1$. In this regime, the cutoff wavelength is large compared to the ablation-front scale length, $kcoLa\u226a1$, and the dispersion relation can be analytically derived exploiting the sharp boundary model (SBM).^{8,10–12} The main stabilizing mechanism is the “rocket effect,” by which a restoring overpressure is generated at the spike.

The analysis of unstable modes in the weak acceleration regime fails when the perturbation wavelength is comparable to the distance between the ablation front and the critical surface. This region of finite thickness is typically called the conduction layer. In the limit of zero acceleration, sufficiently long wavelengths undergo another type of instability known as Darrieus–Landau. This instability is generic for fronts where a dense fluid expands into a lighter one as it typically occurs in flames.^{13} An analytical study of the stability of ablation fronts that encompasses the transition between the RT and DL instabilities has been carried out in Ref. 14. In this reference, the dispersion relation was derived within the framework of a quasi-isobaric approximation. The analysis including compressible effects, which implies that the isothermal Mach number be unity at the critical surface, was not performed and is one of the results derived in the present paper.

During the development of these instabilities, magnetic (B) fields are generated due to the misalignment of gradients of density and pressure, known as baroclinic or the Biermann battery effect.^{15} In the particular case of the ablative RT instability, B-field generation has been investigated both in numerical studies^{16,17} and in experiments,^{18–20} in the latter reporting measurements of the order of several megagauss. Although there is a general agreement about the mechanism of generation of these B fields, their net effect on the hydrodynamics remains less clear. The presence of magnetic fields modifies the transport coefficients in a plasma. Thermal conduction perpendicular to the B field, $\kappa \u22a5$, is reduced, and a heat-flux component along isotherms is generated. The latter effect is known as Righi–Leduc and is important in the linear regime since it is of first order in B, Ref. 21. In essence, the Righi–Leduc effect deflects the heat flux lines, which in turn has a direct effect on the stabilization of the ablative RT instability. Depending on the sense of the self-generated B field, mass ablation (and therefore stabilization) can be either enhanced or diminished, as shown in Fig. 1. In Ref. 22, the RT instability was simulated with parameters relevant to fusion. It was reported that significant magnetization of the plasma could be attained even in the linear regime. The magnetic field is generated out of phase, with its peaks placed near the points where the ablation front is unperturbed. In a particular simulation setup, enhancement in the B-field generation and amplification was observed when the hydrodynamics was coupled with the induction equation. However, this unstable behavior was not acknowledged for every simulation setup studied, and a systematic analysis of the conditions of whether this feedback occurs remains undone.

In this paper, we derive a self-consistent linear stability analysis of the ablation front coupled to the adjacent coronal plasma in an ICF context, including both the effects of self-generated magnetic fields and compressibility. The main novelties include the analysis of the self-generated B-field effect on the RT instability for large Froude numbers and the derivation of the scaling laws of the DL instability with compressible effects. Under the assumptions of this analysis, the B field is found to play a stabilizing role, contrary to the simulation setup reported in Ref. 22. It is most important for short and intermediate wavelengths. The overpressure is enhanced by a maximum of 24% and the convective stabilization is doubled. It is barely noticeable for long wavelengths, which are DL unstable. In this wavelength range, the non-isobaric effects cause the driving underpressure to scale more strongly with the wave number *k*.

This paper is organized as follows: in Sec. II, the governing equations are presented and linearized. In Sec. III, the dispersion relation encompassing both the Rayleigh–Taylor and Darrieus–Landau instabilities with self-generated magnetic fields is derived. In Sec. IV, numerical results are discussed and the Darrieus–Landau instability is solved analytically in the small wave number limit. In Sec. V, the different limits in dispersion relation are explored and the application to a configuration of interest in ICF is discussed. Finally, in Sec. VI, conclusions are drawn.

## II. GOVERNING EQUATIONS

We consider a semi-infinite plasma slab expanding due to the deposition of laser energy in planar geometry (Fig. 2). Laser absorption is assumed to entirely take place at the critical density *n _{c}*. We choose a hydrodynamic description of a fully ionized, single-species plasma and we assume quasi-neutrality. We consider the atomic number

*Z*large enough that the ion pressure can be neglected while the energy flow is still dominated by electronic heat conduction and the ion energy equation is decoupled. With these hypotheses, the evolution of the electron density

*n*, ion velocity $v\u2192$, electron pressure

*p*, electron temperature

*T*, and magnetic-field intensity $B\u2192$ are given by the electron continuity, total momentum, and total energy conservation together with Faraday's law of induction, reading

where $m\xaf$ is the ion mass *m _{i}* divided by the plasma atomic number

*Z*, $E\u2192$ is the electric field, $g\u2192=ge\u2192x$ is the acceleration in the frame of reference of the ablation front and $j\u2192=\u2212enu\u2192$ stands for the current, with $u\u2192$ being the difference between electron and ion velocities. Electron inertia and ion heat flux have been neglected, and we assume the ideal gas equation of state,

*p*=

*nT*. For simplicity, laser intensity

*I*is deposited at the critical density, where $n[t,x\u2192c(t)]=nc$, and

*δ*denotes the Dirac delta. The electric field and the current are given by the electron momentum conservation, which stands for a generalized Ohm's law and the Ampére's law, respectively,

We use the Braginskii^{21} expressions and notations for the ion–electron friction force $R\u2192$ and electron heat flux $q\u2192e$. We assume small electron Hall parameter $\omega e\tau e=(eB/mec)\tau e$, with $\tau e\u221dT3/2/n$ being the electron collision time

In the linear stability analysis performed in this paper, only terms up to the first order in $|B\u2192|$ shall be retained. The Hall and Ettingshausen terms are then dropped since they are proportional to $|B\u2192|2$, and all the coefficients are taken in the unmagnetized limit. Besides, the thermoelectric effects cancel out when added up in Eq. (3), and the ion–electron friction becomes of second order in this same equation. We choose to rewrite $\gamma 0nT\tau e/me=K\xafT5/2$, with $K\xaf$ being the Spitzer conduction constant.

### A. Base flow

We consider a one-dimensional steady base flow. The only component of the velocity is the streamwise component *u*_{0} (not to be confused with the difference between electron and ion velocities $u\u2192$), and we assume $B\u21920=0\u2192$. The governing equations for the base flow become

with $p0=n0T0$.

As boundary conditions, we require that at $x\u2192\u2212\u221e$, we recover $n0=na,\u2009T0=Ta,\u2009u0=ua$, where the sub-index “*a*” denotes the state of the variables upstream of the ablation front (cold shell). Additionally, the laser energy is deposited at $x=xc$, where $n0(xc)=nc$. We assume, as typically is the case, a small density ratio $nc/na\u2261DR\u226a1$. Letting the sub-index “*c*” denote the value of the variables at the critical surface, we then have $uc/ua\u223c1/DR$, $Tc/Ta\u223c1/DR$. In the last estimation, we have assumed that $pc/pa\u223cO(1)$, which will be justified later.

We now normalize the variables with the values of reference in the cold shell. A length scale can be formed based on thermal conduction in the cold shell, $La=2K\xafTa7/2/5paua$. It also corresponds to the ablation front thickness. Two dimensionless numbers arise: the (isothermal) Mach number at the ablation front $Ma=m\xafnaua2/pa$ and the Froude number $Fr=ua2/gLa$. We will prove that the Mach number at the ablation front is $Ma2=DR/2$. The normalized continuity, momentum, and energy equations read

where $\varphi =2I/5paua$ is the normalized laser intensity. This system of equations is completed with the normalized boundary conditions $T0=u0=1$ at $x\u2192\u2212\u221e$, and we require $n0=DR$ at $x=xc$.

We integrate Eqs. (14) and (15) under the assumptions of a sharp boundary model. In this model, we exploit the disparity of thermal scale lengths at the critical surface and the ablation front. The characteristic conduction length at the critical surface is given by $Lc=2K\xafTc7/2/5pcuc$; therefore, we can estimate $La/Lc\u223cDR5/2\u226a1$. Consequently, we can decompose the base flow into two regions connected by the ablation front, which we choose to place at *x* = 0. To the left, *x* < 0, we find the cold shell, assumed to be homogeneous. To the right, *x* > 0, the hot conduction layer develops with $T0\u223cu0\u223cn0\u22121\u223cDR\u22121\u226b1$. The structure of the ablation front is therefore not retained and shrinks into a thin surface (i.e., SBM). This model holds as long as the wavelength of the unstable perturbations is longer than *L _{a}*, which is the case when the Froude number is large, thereby becoming the main assumption in the present derivation.

We now consider the region downstream of the ablation front. We rescale the variables, the spatial coordinate, and the laser intensity as $Ma2u0$, $Ma2T0,\u2009Ma\u22122n0,\u2009Ma5x$, and $Ma2\varphi $, respectively. To avoid excessive notation, the original names are kept for the rescaled variables. The ablation front is therefore seen as a sharp interface at *x* = 0 at which $u0=T0=Ma2\u22480$. We make the fairly restrictive assumption $Ma\u2009Fr\u226b1$, which makes it possible to integrate Eq. (14),

In Sec. V, we prove that if this assumption is not satisfied, neither the DL instability nor the self-generated B-fields affect the dispersion relation, and the unstable perturbations would undergo ablative RT instability in the isobaric limit, studied in Ref. 8. Equation (15) integrates into

with H standing for the Heaviside step function. In the integration of the equations, we have imposed $T05/2dT0/dx\u21920$ at the ablation front.

This stationary conduction layer would have to match a wider region where either non-stationary^{23,24} or geometrical^{25} effects become important and the plasma can expand isentropically to vacuum. In either case, the Chapman–Jouguet condition (isentropic sonic point) must be satisfied at the end of the conduction layer: $u02/T0|x\u2192\u221e=5/3$, which implies $u0|x\u2192\u221e=5/8$. This condition, together with the requirement that $dT0/dx\u21920$ at $x\u2192\u221e$, gives the value of the normalized laser intensity $\varphi =5/16$, in agreement with Ref. 25. This value must be seen as the level of absorbed laser intensity required to generate a given ablation pressure *p _{a}* and ablation rate $naua$. The base temperature has a maximum for $u0=1/2$; therefore, $dT0/dx$ must change sign in order to be able to reach $u0=5/8$. This implies that the laser intensity be deposited where $u0(xc)=1/2$. Consequently, we obtain that $DR=2Ma2$ and $pc/pa=1/2\u223cO(1)$, as assumed at the beginning of this section. We can finally write the equation governing the conduction layer

and we recall that $n0=1/u0,\u2009T0=u0\u2212u02,\u2009p0=1\u2212u0$. It should be noted that the velocity derivative is singular at the critical surface. Integrating Eq. (18) between the ablation front and the critical surface gives $xc\u22480.0117$. Profiles of the conduction layer are shown in Fig. 2. Temperature increases in the overdense plasma $(x<xc)$ and decreases in the underdense plasma $(x>xc)$, while the streamwise velocity increases monotonically until $u0(\u221e)=5/8$.

### B. Perturbed flow

We now linearize the equations to study the perturbed problem. We propose a modal decomposition in time and transversal coordinate as $exp\u2009(\gamma t+iky)$. The critical surface is also perturbed as $xc=xc0+\xi c\u2009exp\u2009(\gamma t+iky)$, with $xc0$ being its unperturbed position previously derived. When the diffuse structure of the ablation front is not considered but rather it is treated as a sharp interface, its position also needs to be perturbed as $xa=\xi a\u2009exp\u2009(\gamma t+iky)$. It is convenient to strain the *x* coordinate as

Consequently, the ablation front and critical surface are placed at *s* = 0 and $s=xc0$, respectively, at all orders, and we prevent the Dirac delta and derivatives from appearing in the perturbed equations. Notice that when referring to the base profiles, the variables *x* and *s* are interchangeable. Accordingly, we introduce the following ansatz for a generic variable $\varphi =\varphi 0(s)+\varphi 1(s)exp\u2009(\gamma t+iky)$, with $\varphi 1/\varphi 0\u226a1$. Particularly, we found convenient to expand the streamwise ion velocity as $u=u0(s)+{u1(s)+\gamma [\xi a+s(\xi c\u2212\xi a)/xc0]}exp\u2009(\gamma t+iky)$, and the electron pressure as $p=p0(s)+Ma2p1(s)exp\u2009(\gamma t+iky)$. We denote as *v*_{1}, *b*_{1} the transversal velocity and magnetic field, respectively, and we absorb the imaginary unit within them. We now recall the change of the derivatives as a consequence of introducing the straining. Denoting

and $\phi \u2032$ its derivative, we have

Inserting these ansätze into the governing Eqs. (1)–(4), linearizing and normalizing with the variables in the cold shell $(na,Ta,ua,La)$, we obtain

with the density perturbation being related to pressure and temperature through the linearized equation of state

The magnetic field *b*_{1} has been normalized with a reference $Bref$ derived from the Biermann battery term, $Bref=cm\xafua/eLa$. In the right-hand side of the induction Eq. (28), we have the magnetic field diffusion. It is inversely proportional to the magnetic Reynolds number $Rem=uaLa/Dm$, with *D _{m}* being the magnetic diffusivity given by $Dm=\gamma 0\alpha 0c2Ta2/10\pi e2pauaLa\u221dTa\u22123/2$.

It is important to highlight that, to the first order, the effect of the magnetic field on the hydrodynamics is restricted to the last term in brackets in the energy Eq. (27), which acts as a heat source. Two effects are accounted for here: the first one is the Righi–Leduc term and the second one comes from the difference between electron and ion enthalpy convection. They are proportional to $Ma2$; hence, the effect of the B field on the RT instability at the ablation front scale length is expected to be small as the ablative flow is highly subsonic there. The three numerical coefficients appearing in the Righi–Leduc term $(cR)$, current enthalpy convection $(cH)$ and Nernst term $(cN)$ take the value

The analytical expressions for the Mach and magnetic Reynolds numbers are

where *m _{p}* refers to the proton mass. Similarly, the ablation-front length scale length is

with Λ standing for the Coulomb logarithm. In the numerical applications in Sec. V, we take Λ = 5 and we consider the ratio $\gamma 0/Z$ to be equal to unity (*γ*_{0} is an increasing function of the atomic number). The Froude number then becomes

## III. LINEAR STABILITY ANALYSIS

We proceed to solve the governing Eqs. (24)–(28) imposing boundedness of the solution at $s\u2192\xb1\u221e$.

### A. Analysis of the cold shell

First, we analyze the cold shell. This region can be assumed to be isentropic since heat conduction is negligible. In fact, the thermal mode decays in a distance comparable to the ablation-front length scale; therefore, it is confined within the transition layer between the cold shell and the conduction layer and is not solved under the assumptions of this model.

The governing equations in this region assuming a small Mach number simplify to

The only bounded solution at $s\u2192\u2212\u221e$ corresponds to

with *A* being an arbitrary constant.

In order to connect with the hot conduction layer, mass and streamwise and transversal momentum fluxes must be continuous through the ablation front. Their corresponding perturbed quantities are $Cmass\u2261n1u0+n0u1$, $Cxmom\u2261n1u02+2n0u0u1+p1$ and $Cymom\u2261n0u0v1$, respectively. By means of the solution in the cold shell, we have at the ablation front

The perturbed mass and streamwise momentum fluxes can be combined in order to eliminate the constant *A*. In so doing, the dispersion relation is formed,

### B. Analysis of the conduction layer

The second step is to study the conduction layer. This would determine the value of the perturbed mass and momentum fluxes necessary for the dispersion relation Eq. (42).

We rescale the spatial coordinate and the base flow variables with their value of reference at the conduction layer, just as done in Sec. II A, as $Ma5s$ and $Ma2u0$, respectively. We perform a similar rescaling of the following variables: $Ma\u22125k,\u2009Ma\u22124n1,\u2009Ma\u22125b1$, $Ma3\xi a,\u2009Ma3\xi c,Ma\u22123\gamma $, $Ma\u221210Rem$, and $Ma\u2009Fr$. Again, we keep the same nomenclature for the rescaled variables so as to avoid overwriting. In so doing, the Mach number is absorbed in the governing Eqs. (24)–(29), and they no longer depend on it. The Righi–Leduc term in the energy equation is then of order unity.

Consistently with the assumptions made to derive the base flow, the redefined Froude number is considered to be large, and we drop the terms proportional to $Fr\u22121$. The redefined magnetic Reynolds number is much higher than in the cold shell (factor $Ma\u221210$). Typically, the conduction layer is highly conductive; therefore, we also drop the terms proportional to $Rem\u22121$.

We emphasize that the instability is driven by the ablation-front corrugation; hence, *ξ _{a}* is of leading order. Consequently, we divide all the perturbed variables by

*ξ*. The following eigenvalue related to the relative motion of the critical surface with respect to the ablation front is, therefore, introduced:

_{a}The rescaled governing equations are shown in the Appendix. They must be completed with the appropriate set of boundary conditions.

#### 1. Boundary conditions for the conduction layer

Mass and streamwise momentum fluxes must be continuous through the ablation front, $s=0+$. We consequently introduce a the eigenvalues related to perturbed mass, *f*, and streamwise momentum, *q*, fluxes as

The motivation behind these definitions comes from the fact that the dispersion relation (42), introducing again dimensional variables, becomes

The conservation of transversal momentum gives $v1|s=0+=\u2212\gamma +O(Ma2)$. The rest of the boundary conditions require temperature and heat flux to be zero at $s=0+$. Applying these boundary conditions allows one to obtain the shape of the profiles close to the ablation front depending upon the four eigenvalues ${f,q,h,r}$, which become the boundary conditions necessary to integrate the governing equations in the conduction layer (A1)–(A5)

and

The four eigenvalues ${f,q,h,r}$ are obtained while integrating the governing equations (A1)–(A5) ensuring four compatibility conditions required in the conduction layer. Three of them have a hydrodynamic nature, while the remaining one is imposed on the magnetic field.

#### 2. Hydrodynamic compatibility conditions

The first compatibility condition comes from the definition of the critical surface, where $n1(xc0)=0$. The second and third compatibility conditions come from canceling the two unbounded modes of hydrodynamic nature. One of them takes place at the critical surface, which is an isothermal sonic point. Here, canceling the unbounded mode implies requiring the perturbed isothermal Mach number to be unity,^{28} which is tantamount to impose $T1(xc0)=u1(xc0)$. This requirement allows one to rewrite the vanishing density condition as $p1(xc0)=2u1(xc0)$.

The second unbounded mode takes place at $s\u2192\u221e$, where $du0/ds\u21920$. It evolves as $exp\u2009(\lambda s)$. In the small *γ* limit (this limit is justified in Sec. III C), it corresponds to the single positive root of

where the sub-index “$\u221e$” denotes the value of the base profiles at infinity. It tends asymptotically to $\lambda =(3/2T\u221e5/2)1/3k2/3$ when $k\u226a1$ and $\lambda =k$ when $k\u226b1$.

Although the following does not stand for a compatibility condition but rather for a distinctive feature of the integration, we point out that, at the critical surface, the derivative of the base velocity jumps in value. Consequently, the derivative of the perturbed temperature is discontinuous and satisfies

with *u _{c}* standing for the value of

*u*

_{1}at the critical surface.

The fourth compatibility condition to be satisfied is related to the magnetic field and explained in Sec. III B 3.

#### 3. Magnetic compatibility conditions

The shape of the magnetic field close to the ablation front, Eq. (48), presents a singularity characterized by the eigenvalue *h*. It is a consequence of the B-field accumulation, convected by the Nernst term toward the ablation front. Yet, the value of the magnetic field at the ablation front must be null in order to match the profile in the cold side. This requires the existence of a boundary layer where diffusion becomes important.

We apply singular perturbation theory Ref. 29 to obtain the structure of this boundary layer. Rescaling the base velocity as $u0=\u03f5Bw$, with $\u03f5B\u2261Rem\u22121/5\u226a1$, introducing $\psi =b1/qk8/5$, and taking the leading order expansion in *ϵ _{B}*, Eq. (28) simplifies into

The term with “$cN\u22121$” accounts for the Nernst and bulk plasma convection. The former predominates, resulting in a net convection toward the ablation front. This equation must be solved with the boundary conditions $\psi (0)=0$ and $\psi (w\u2192\u221e)=(1+hRem1/5/w)/(cN\u22121)$. The solution is

The expansion of *ψ* for $w\u2192\u221e$ is

where $\Gamma (z)$ stands for the Euler gamma function.

This inner solution matches with the outer behavior as long as the exponential term converges. If the Nernst term is not taken into account, *c _{N}* = 0, this condition is not satisfied and we cancel the exponential term by setting $h=\u2212Rem\u22121/551/5/\Gamma (4/5)$, which tends to zero for infinite $Rem$. If the Nernst term is taken into account, then $cN\u22121>0$, and the exponential term converges. The magnetic field accumulates at the ablation front and presents a maximum at

*w*= 1.70, where $\psi |max=0.72hRem1/5$. The eigenvalue

*h*cannot be obtained from the analysis of this boundary layer alone, and the fourth condition is derived from the analysis of the critical surface.

At the critical surface, the Nernst convection velocity changes sign, altering the net B-field convection direction. To the left, the Nernst term overcomes the bulk plasma convection and the B field is convected toward the ablation front, while to the right, both velocities add up and the B field is convected outward. Integrating the induction equation to the left “–” and right “+” of this surface yields $b1+=b1\u2212\mu \u2212/\mu +$, where $\mu \xb1=u0\u2212cNn0\u22121T03/2dT0/ds|s=xc0\xb1$ is the net convection velocity. Inserting the numerical values gives $\mu +\u22480.55,\u2009\mu \u2212\u2248\u22120.60$ and hence $b1+\u2248\u22121.10b1\u2212$. This implies either null magnetic field at the critical surface or a singular current if $b1\u22600$. In the latter case, a diffuse boundary layer should take place to support the jump in B field. Singular perturbation theory shows that the width of such a layer would scale as $Rem\u22121$. Introducing $\eta =Rem(s\u2212sc0)$ and keeping the leading order terms in Eq. (28) gives $d2b1/d\eta 2=(\mu /8)db1/d\eta $, with $\mu =\mu \u2212<0$ for $\eta <0$ and $\mu =\mu +>0$ for $\eta >0$. The solution of this equation consists of a divergent exponential at both sides of the critical surface, $b1\u223c\u2009exp\u2009(\mu \xb1\eta /8)$, which cannot be matched with the outer solution unless it is forced to be zero.

In summary, when the Nernst term is taken into account, the eigenvalue *h* must ensure that $b1=0$ at the critical surface. Without Nernst, there is no singularity at the ablation front and we set *h* = 0.

### C. Resolution method

For simplicity, we rewrite the system of equations (A1)–(A5) in a matrix form. Letting $Y\u2192={u1,p1,v1,T1,dT1/ds,b1}T$ denote the state variables vector, we have

We exploit the smallness of the growth rate *γ*, following the same method as in Ref. 8. To justify this assumption, we proceed to the following estimations. In the RT instability, the term “*kg*” is predominant in Eq. (45) and we can state $\gamma \u223ckg$. From this equation, the cutoff can be estimated as $kcoLa\u223c(gLa/ua2)5/3$, which leads to $\gamma La/ua\u223c(kLa)4/5$. The latter estimation can be rewritten normalizing with the variables at the critical surface as $\gamma Lc/uc\u223cMa(kLc)4/5\u226a1$. Notice that a similar result, $\gamma Lc/uc\u223cMa$, would be obtained when the DL instability dominates and *q* < 0 becomes the predominant term. The conduction layer is, therefore, quasi-stationary (slow dynamics), and the terms proportional to *γ* in Eqs. (A1)–(A5) are small. This is a direct consequence of assuming the ablation front corrugation *ξ _{a}* to be of leading order. Another kind of instability, however, is contained within the structure of Eqs. (24)–(28). It corresponds to a faster dynamics, where the ablation remains unperturbed and $\gamma Lc/uc\u223cO(1)$. This limit is known as the magnetothermal instability

^{26,27}and is out of the scope of the present analysis.

We expand the state vector $Y\u2192$, eigenvalues ${f,q,h,r}$ and matrices *A*, *B*, *C* as

with $\u03d11$ depending on *k*, exclusively. The system in Eq. (54) yields a hierarchy of systems of linear differential equations. In order unity, we have

in order *γ*, we have

and so forth.

The eigenvalues ${fj,qj,hj,rj}$ are determined at each order “*j*” by numerically integrating the corresponding system, starting from the ablation front with the boundary conditions Eqs. (46)–(48), and imposing the four compatibility conditions previously described, that is, null perturbed density, isothermal Mach number, and magnetic field at the critical surface and cancelation of the hydrodynamic unbounded mode at infinity.

Bearing the smallness of *γ* into account, the dispersion relation in Eq. (45) yields the unstable root

We recall that, in dimensional variables, the eigenvalues ${fj,qj,hj,rj}$ depend on $kxc0=0.0117kLa(2na/nc)5/2$.

## IV. DISCUSSION OF RESULTS

The mechanisms of stabilization become clearer after inspection of Eq. (58). The quasi-steady overpressure *q*_{1} acts as a conservative force. It is positive for $kxc0>0.50$, meaning that the momentum flux increases at the crest (spikes) of the rippled ablation front and decreases in the valleys (bubbles), damping the growth of the unstable modes. It corresponds to the rocket effect (restoring force) mechanism, and, as discussed in Ref. 14, it is related to the quasi-steady rotational part of the velocity (vorticity *ω*) at the ablation front since $i\omega |s\u21920+=\u2212qk8/5$. The term proportional to *ku _{a}* is typically called “convective stabilization” and stands for a damping force. Both the mass flux and the nonsteady momentum flux are accounted for in this effect. The former corresponds to the “fire polishing effect,” by which the spikes evaporate more quickly than the bubbles, and is related to the potential part of the velocity at the ablation front, $v\u2192\u223cT5/2\u2207T$. It is interesting to notice that although this problem lacks a barotropic relation and, consequently, a potential and rotational velocity flow decomposition is pointless, it still makes sense locally at the ablation front, where the non-isobaric terms are small. Amongst the two stabilization mechanisms previously described, the rocket effect is the predominant under the assumptions considered in this paper. In fact, in the strict limit $Ma\u21920$, the term

*ku*can be neglected and the cutoff becomes $kcoLa=(gLa/ua2q1)5/3\u223cFr\u22125/3$.

_{a}All the eigenvalues tend to a finite value for large *k*. When the coupling with the induction equation is not considered, *f*_{1}, *q*_{1}, and *q*_{2} tend asymptotically to their corresponding values calculated in the SBM for ablative RT instability in Ref. 8, $q1=0.96,\u2009(1+f1+q2)/2\u22482$. Notice that the *q*_{1} defined there shall be divided by $(2/5)2/5$ for the comparison. It is interesting to observe that, contrary to the analysis of isobaric conduction layer in Ref. 14, both *f*_{1} and *q*_{1} present a maximum for $kxc0=2.63$ and $kxc0=2.24$, respectively. Another difference when taking into account non-isobaric terms lies in the ratio $\xi c/\xi a$. It becomes negative for $kxc0>1.80$ instead of tending monotonically to zero. The physical reason behind the critical surface oscillating out of phase with respect to the ablation front for perturbations confined in the latter lies in the vorticity mode. In the quasi-stationary conduction layer, vorticity is convected without decaying and reaches the critical surface. This vorticity trail excites the critical surface, which is set into motion in order to ensure the unit isothermal Mach number condition, that is $T1=u1$. We recall that this condition is not required if isobaricity is assumed. Instead, *T*_{1} must be null, leaving *u*_{1} free and the critical surface unexcited.^{14} The unsteady momentum flux *q*_{2} is positive for almost all of the wave numbers, helping to stabilize the motion. It presents a minimum around $kxc0=1$, where it becomes approximately zero.

For $kxc0<0.5$, the perturbed momentum flux *q*_{1} is negative, and the quasi-steady vorticity at the ablation front becomes the driving mechanism of the Darrieus–Landau instability. The critical surface and ablation front move synchronously. For smaller wave numbers, $kxc0<0.10$, the perturbed mass flux *f*_{1} is inverted and reinforces the DL instability. This effect is not observed in the isobaric model. The minimum values of *f*_{1} and *q*_{1} are –0.077 and –0.43 and take place at $kxc0=0.015$ and $kxc0=0.13$, respectively. Notice that *q*_{1} attains a lower minimum, $q1=\u22120.51$ at $kxc0=0.18$ in the isobaric model. Scaling laws for $k\u226a1$ are derived in Sec. IV A.

Under the assumptions of the sharp boundary model, the effect of the self-generated magnetic field is always stabilizing. The B-field effect is always present in the governing Eqs. (A1)–(A5) and does not depend upon any parameter. It increases both the momentum and mass fluxes, being more efficient for the former: the peak of *q*_{1} is increased by a 24% vs 11% for *f*_{1}. In Fig. 5, the profiles of the variables are shown for the perturbation at which *q*_{1} is maximum in the case coupled with Nernst. It can be seen that the magnetic profile at the right of the spike section is positive, in agreement with the stabilizing case in Fig. 1. The Nernst convection enhances the stabilizing effect of the B field. For perturbation wavelengths longer than the distance between ablation front and critical surface, $kxc0<1$, the B field is less effective, becoming totally negligible in the Darrieus–Landau instability region. The B-field effect is significantly stronger on the unsteady momentum flux *q*_{2}.

Asymptotic analysis indicates that, in the short wavelength limit, $k\u226b1$, both the non-isobaric and the B-field effects become negligible. In this limit, the perturbation is confined within a neighborhood around the ablation front of a characteristic length scale $k\u22121$. We can estimate $s\u223cO(k\u22121),\u2009u0\u223cT0\u223cO(k\u22122/5)$, and $T1\u223cu1\u223cO(k3/5)$. From induction Eq. (A5), equating B-field convection to the baroclinic term, we obtain $b1\u223ck8/5$. When plugged into the energy Eq. (A4), we obtain that both the Righi–Leduc term and the non-isobaric terms (all the terms in the left-hand side except for the divergence of velocity) turn out to be small of order $O(k\u22122/5)$. However, as shown in Fig. 3, the B-field effect is numerically more important than the non-isobaric effects, and *f*_{1} and *q*_{1} asymptote to their isobaric limit ($f1=1.02,\u2009q1=0.96$) significantly later (higher *k*) when the coupling with the induction equation is considered. This effect is even more remarkable for the higher-order eigenvalue *q*_{2} since its value remains tripled with respect to the isobaric limit ($q2=2.12$) for a significantly long wave number range. This has an important effect on the convective stabilization term in Eq. (58), which is enhanced from “$\u22122kua$” to “$\u22124kua$.”

### A. Analysis for $k\u226a1$: Darrieus–Landau instability

In this section, we consider the limit of large wavelength perturbations, $k\u226a1$, which are Darrieus–Landau unstable, and we derive analytically scaling laws for *f*_{1}, *q*_{1}, $\xi c/\xi a$, and *q*_{2}. The spatial domain *s* > 0 can be split into two distinguishable regions. The first one would correspond to the conduction layer properly said, where $s/xc0\u223cO(1)$, and $du0/ds\u223cO(1)$. The second one is a scale-free region adjacent to the conduction layer, where $s/xc0\u226b1,\u2009u0\u22485/8$ and $du0/ds\u226a1$.

We integrate the governing equations in the conduction layer. The transversal momentum equation is uncoupled from the rest of the system and yields $v1=ku0$. The conditions at the critical surface, $p1=2u1,\u2009T1=u1$, lead to consider *u*_{1}, *p*_{1}, and *T*_{1} of the same order, given by $q1k3/5$. At the same time, we can infer $q1k3/5\u223cf1k\u223cr1$, and we assume $q1k3/5\u226bk2$, which will be proven *a posteriori*. Continuity and streamwise momentum equations simplify to conservation of perturbed mass and momentum fluxes, which can then be integrated into

Applying the conditions at the critical surface allows one to obtain $f1k=2uc1$ and $q1k3/5=4uc1$, where $uc1$ is the first order [as given by the ansatz Eq. (55)] of the value of *u*_{1} at the critical surface. The magnetic field scales as $b1\u223cq1k8/5$, which is smaller than the other perturbed quantities. Consequently, the Righi–Leduc term can be neglected in the energy equation and the hydrodynamics becomes decoupled from the induction equation. The energy equation can be integrated once yielding

Imposing $T1=0$ at *s* = 0 and $T1=uc1$ at $s=xc0$ allows one to obtain $r1=8uc1$. The perturbed quantities in the subsonic region $(0<s<sc0)$ become then $u1/uc1=2u0$, $v1=ku0,\u2009p1/uc1=4(1\u2212u0),T1/uc1=4u0(1\u2212u0)$ and $n1=0$. Once the hydrodynamic profiles are derived, the induction equation can be integrated. The eigenvalue *h* yields $h=\u22123/8$, and the magnetic-field profile is

At the end of the conduction layer, we have $v1|s\u2192\u221e=u\u221ek=5k/8$ and, from Eq. (61), $dT0/ds|s\u2192\u221e=5uc1/8T\u221e5/2=4096uc1/4515$.

The relation between $uc1$ and *k* is derived from resolving the scale-free region. Let $\Phi \u2192={u1/u\u221e,v1/u\u221e,p1/p\u221e,T1/T\u221e}T$ denote the state variable vector, the solution in the scale-free region will consist of a linear combination of five modes. The first mode stands for vorticity being convected as a passive scalar by the bulk plasma, while the second mode is governed by transverse heat conduction

The third, fourth, and fifth modes are governed by longitudinal thermal conduction mixed with incompressible mode. Introducing $\beta =(3/2T\u221e5/2)1/3\u22483.84$, the third one reads

which is unbounded. The fourth and fifth modes are bounded complex conjugate modes

with

and

Since *u*_{1}, *p*_{1}, and *T*_{1} are of the same order of magnitude, the scaling laws in this region are given by the last two modes. In order to proceed with some order of magnitude estimations, we simply assume $\Phi \u2192=C4\Phi \u21924$. Matching this solution with the conduction layer yields $C4\u223ck2/3$ and $uc1\u223ck4/3$. The former condition has been derived by imposing $v1\u223ck$, while the latter comes from $dT1/ds\u223cuc1$. Consequently, in the scale-free region, we have $s\u223ck\u22122/3\u226b1$, *u*_{1}, *p*_{1}, and *T*_{1} peak at a much higher value with respect to their magnitude in the adjacent layer to the critical surface, $k2/3$ compared to $k4/3$. We can then solve the supersonic region assuming $u1=p1=T1=0,dT0/ds=5uc1/8T\u221e5/2$ and $v1=5k/8$ at *s* = 0, which gives precisely $\Phi \u2192=C4\Phi \u21924$, with

When compared to the numerical results, the scaling law is correct but there is a factor of 1.75 difference in the coefficient. The reason for this lies in the overlap between the conduction layer and the scale-free region. According to the previous results, in the former we have $v1\u226bT1$, while in the latter $T1\u226bv1$. Therefore, in the overlapping region we have $v1\u223cT1$, which takes place for $s\u223ck\u22121/3$. Since $du0/ds\u223c1/s2\u223ck2/3$ in this region, we cannot neglect the derivatives of the base profile when performing the coupling between both regions, and the matching must be done numerically. This yields the final result

and, consequently, we have $f1=\u22120.49(kxc0)1/3$, $q1=\u22125.8(kxc0)11/15$, and $\xi c/\xi a=1\u22122.0(kxc0)4/3$. They show good agreement with the numerical results (Fig. 6). Notice that these scaling laws differ from the ones derived in Ref. 14, where isobaricity was assumed, and $f1=kxc0$, $q1=\u2212(5kxc0/2)2/5,\u2009q2=1$ and $\xi c/\xi a=1\u2212(kxc0)2/2$. Of particular interest is the perturbed mass flux, which becomes negative (destabilizing) when the non-isobaric effects are considered. However, the driving mechanism of the instability, the momentum flux or vorticity *q*_{1}, is smaller in this case as it presents a stronger scaling on $kxc0$.

Finally, the next-order in the ansatz Eq. (55) can be obtained in a similar manner. The perturbed mass and momentum fluxes are conserved in the conduction layer, and, in the scale-free region, the variables evolve as given by the mode $\Phi \u21924$ too. However, the key point is to notice that it is the boundary condition at the ablation front, $v1=\u2212\gamma $, the one that imposes the scaling of the variables with *k* at this order. More precisely, the span-wise velocity is constant and equal to $\u2212k3/5$ in the conduction layer, which in turn leads to a constant $q2=0.66$.

## V. DISPERSION RELATION AND APPLICATION

The dispersion relation given in Eq. (58) can be rewritten in the form

We proceed to analyze this expression as a function of its two governing parameters: the Froude number Fr, defined in Sec. II A, and the critical-to-shell density ratio $nc/na\u2261DR$. We recall that the latter is equivalent to the Mach number at the ablation front: $DR=2Ma2$. It only appears through the eigenvalues *q*_{1}, *f*_{1}, and *q*_{2}, which are functions of $kxc0=0.066kLa/DR5/2$. In the derivation of Eq. (69), we have assumed both $Fr\u226b1$ and $DR\u226a1$. Under these assumptions, the rocket effect (perturbed pressure *q*_{1}) becomes the main stabilizing mechanism. The convective stabilization, in spite of being of lower order, is numerically significant for moderately large $Fr$. We recall that the cutoff as given by the rocket effect takes place for $kcoLa\u223c1/Fr5/3$. This expression holds as long as the perturbed pressure *q*_{1} is positive for this wave number. As seen in Fig. 3, the perturbed pressure changes sign for $kq1xc0\u22480.5$, or equivalently $kq1La\u22487.6DR5/2$, and becomes the driving DL instability mechanism. The ratio between these two wave numbers, $kq1/kco=(2.25DRFr2/3)5/2$, dictates the shape of the dispersion relation. Two different limits can therefore be identified depending on the value of $DRFr2/3$.

### A. RT limit

For $DRFr2/3\u226a1$, the perturbed pressure becomes negative for wave numbers smaller than the cutoff. The spectrum is totally dominated by the ablative RT instability. This limit is well described assuming isobaricity ($q1=0.96,\u2009f1=1.02$, and $q2=2.12$), and, to the leading order, the cutoff takes place at $kcoLa=1/(q1Fr5/3)$. This applies even in the part of the spectrum where $q1<0$. In this part, the vorticity generation is so weak that the RT term in Eq. (69) dominates over the “perturbed pressure” term, and the growth rate is simply $\gamma =kg$. In this limit, it is not necessary to take into account the critical surface and just the analysis of the ablation front suffices. As a result, the assumption made when deriving the base flow in Sec. II A, $MaFr\u226b1$, is well justified.

### B. DL limit

For $DRFr2/3\u226b1$, the perturbed pressure is negative for $kcoLa\u223c1/Fr5/3$; therefore, this becomes a poor estimation for the cutoff. It would rather take place close to $kq1La\u223cDR5/2$ at the very moment where *q*_{1} changes sign. In fact, we can estimate $q1\u223c1/(DRFr2/3)3/2\u226a1$ at the cutoff. Stabilization is still given by the rocket effect, but it takes place at a larger wave number, which is independent of Fr (see Fig. 7). This analysis is in agreement with the results shown in Fig. 7 in Refs. 14 and 30. For larger perturbations (smaller wave number), the spectrum is dominated by the DL instability, and the growth rate becomes $\gamma La/ua=|q1|1/2(kLa)4/5$. However, this regime does not apply for all the range of unstable modes. For very small wave numbers, the RT overcomes the DL. As derived in Sec. IV A, the underpressure is given by $q1=\u22120.78(kLa)11/15/DR11/6$ in this case. Estimating the two driving mechanisms in the square root in Eq. (69) yields a threshold wave number $kthLa=1.2DR11/8/Fr3/4$ below which the RT becomes the driving mechanism. Such a spectrum is sketched in Fig. 7(a). The largest perturbations undergo classic RT instability with $\gamma =kg$. For $k>kth$, the DL instability becomes predominant until we reach $k=kq1$, where the perturbed pressure becomes positive (vorticity changes sign) and the rocket effect stabilizes shorter perturbations.

A particular feature of this limit is that the dispersion relation can break into two unconnected branches for very large values of $DRFr2/3$, as shown in Fig. 7(b). This occurs when the RT unstable region, $k<kth$, is stabilized by the secondary mechanism in Eq. (69): convective stabilization. Convection can suppress the RT instability at a wave number $kcsLa\u223c1/Fr$. If $kcs<kth$, then perturbations smaller than $kcs\u22121$ are stable until the DL driving mechanism becomes strong enough to overcome the convective stabilization. This takes place at the wave number $kdlLa\u223cDR11/2$. Smaller perturbations undergo DL instability, and the rocket effect stabilizes those with $k>kq1$. However, this regime is not expected in ICF configurations since the condition $kcs<kth$ implies the restrictive $DR11/2Fr\u22731$, which corresponds to low acceleration levels.

Notice that this double branch shape can only occur if we take into account the non-isobaric effects, which induce the stronger scaling $q1=\u22125.8(kxc0)11/15$ compared to $q1=\u2212(5kxc0/2)2/5$ without them. This can be better seen in the limit of zero gravity (infinite Froude number) and long wavelength $kxc0\u21920$. Let $n\u221e$ denote the density far from the critical surface. The dispersion relation Eq. (69), introducing the laws derived for the eigenvalues in the long wavelength limit, would yield

in the isobaric case (Ref. 14) and

with non-isobaric effects as derived in this paper. The isobaric relation, Eq. (70), corresponds to the sharp boundary model of the DL instability derived in Refs. 4 and 5 in the $n\u221e/na\u226a1$ limit. The growth rate is linear with *k* and always positive. Introducing the non-isobaric effects, Eq. (71), notably modifies the dispersion relation. For $na>n\u221e$, convection stabilizes wave numbers smaller than the cutoff $kdlxc0=0.33(n\u221e/na)3$.

### C. ICF application

By means of Eqs. (31) and (34), the combination of parameters $DRFr2/3$ can be expressed as a function of physical variables that are more relevant in ICF as

It can be seen that, in the range of ICF applications, this parameter tends to be small (RT limit) or of the order of unity. Large values of it would correspond to specific low-acceleration experiments. A configuration of interest for ICF is shown in Fig. 8, where we have chosen $ua=1.2\u2009\mu m/ns,\u2009Ta=7.5\u2009eV$, $na=1024\u2009cm\u22123$ and $g=50\u2009\mu m/ns2$, which gives $Fr=50$ and $DRFr2/3=0.11$. The dispersion relation with self-generated B fields is compared to the case decoupled from induction and to the purely isobaric case. The DL mechanism comes into play by reducing the ablative stabilization around $kq1La=4.4\xd710\u22125$. It never overcomes the RT instability, but rather enhances the unstable behavior of these wavelengths. The effect of the B field comes into play for shorter wavelengths, when ablation becomes effective, and it has a significant stabilizing role, reducing the cutoff from $kLa=4.3\xd710\u22124$ to $kLa=3.6\xd710\u22124$.

### D. Summary of regimes

A schematic of the different regimes described in this section is plotted in Fig. 9. In this chart, the stability of a generic perturbation with wave number *kL _{a}* is given as a function of the Froude number and the critical-to-shell density ratio. It must be understood from an asymptotic analysis point of view, hence the transition from one region to another is blurry rather than a well-defined curve.

The two dotted areas show the combination of $Fr,\u2009DR$ for which the perturbation is unstable. One is RT dominated and the other one is DL dominated. The border between them, $DR11/8/Fr3/4=kLa$, corresponds to the perturbation wave number being equal to the threshold $kth$ as previously derived. The dashed curves do not delimit any stability region but rather they establish the transition between regimes.

For $DRFr2/3<1$, the spectrum is RT dominated; the cutoff is given by the rocket effect (dynamic pressure) and only depends upon the Froude number $(kLa\u223cFr\u22125/3)$. For $DRFr2/3>1$, the perturbation would undergo DL instability if $k>kth$ and RT otherwise. The cutoff appears whenever the DL driving mechanism (perturbed pressure) becomes positive, depending only on the density ratio $(kLa\u223cDR5/2)$.

The second dashed curve, $DR11/2Fr=1$, establishes the limit above which the secondary stabilizing mechanism (ablative convection) operates and breaks the unstable part of the spectrum into two unconnected regions: one RT unstable in the smallest *k* region and another DL unstable developing until the cutoff due to dynamic pressure.

## VI. CONCLUSION

In this paper, the hydrodynamic stability of a laser-accelerated target is investigated in the context of inertial confinement fusion. The stability analysis includes the coupling of the target to the ablated expanding plasma past the critical surface. The analysis is restricted to the linear regime and it is based upon two main assumptions: small critical-to-shell density ratio, $DR\u2261nc/na\u226a1$, and weak acceleration regime (large Froude number $Fr$). The two main novelties in this study are the inclusion of non-isobaric effects and the self-generated magnetic fields.

In the limit of perturbation wavelengths smaller than the distance between the ablation front and the critical surface (conduction layer width $xc0$), the expansion undergoes Rayleigh–Taylor (RT) instability driven by the acceleration. It is damped by mass ablation off the cold shell. The main stabilization mechanism corresponds to the overpressure generated at the spikes, rocket effect, which can stabilize the motion for wave numbers larger than a cutoff $kcoLa\u223cFr\u22125/3$. In the limit of large perturbation wavelengths, the Darrieus–Landau (DL) instability takes place. It is driven by the vorticity generated at the ablation front. It is closely related to the perturbed pressure at the spikes, which becomes negative (underpressure) and destabilizing. Asymptotic analysis allowed us to derive the scaling laws of the under-pressure *q*_{1} with the wave number. In this limit where the non-isobaric effects play an important role, making *q*_{1} scale as $q1=\u22125.8(kxc0)11/15$, compared to the isobaric case studied in Ref. 14, where $q1=\u2212(5kxc0/2)2/5$.

Magnetic fields are generated by the misalignment of density and pressure gradients, known as baroclinic or Biermann-battery effect. In the linear regime, they affect the hydrodynamic via the Righi–Leduc term. It acts as a heat source that deflects the heat flux lines. Under the SBM assumptions, the self-generated magnetic field has a stabilizing effect by enhancing ablation. It is most effective for perturbation wavelengths comparable to the conduction layer width, where the restoring overpressure can be increased up to 24%. The DL instability is barely affected by the self-generated B fields. In the opposite limit, small perturbation wavelength $kxc0\u226b1$, both the non-isobaric and B-field effects scale as $(kxc0)\u22122/5$. However, the B-field effect turns out to be numerically more important, and its stabilizing effect can even be observed for large $kxc0$. Particularly, the convective stabilization term is doubled to $\u22124kua$ for a wide wave number range. Finally, the Nernst term enhances the stabilizing effect of the self-generated magnetic fields.

The analysis of the dispersion relation reveals that the combination $DRFr2/3$ dictates the behavior of the spectrum. For $DRFr2/3\u226a1$, it is well described by the ablative RT instability in the isobaric regime, and the cutoff takes place for $kLa\u2248Fr\u22125/3$. In the opposite limit, $DRFr2/3\u226b1$, two regions can be defined. The long perturbations with $kLa<DR11/8/Fr3/4$ undergo RT instability, while the part of the spectrum with $kLa>DR11/8/Fr3/4$ is DL dominated. In this limit, the cutoff becomes independent of the Froude number: $kLa\u22487.6DR5/2$. The regime of application for ICF corresponds to $DRFr2/3\u22721$. When this parameter is close to unity, the DL effect operates by reducing the restoring overpressure and increasing the wave number at which ablation comes into play. It is precisely in this range, $DRFr2/3\u223c1$, where the effect of the self-generated B fields becomes more important. They enhance the stabilizing effect of ablation and can significantly reduce the cutoff.

## ACKNOWLEDGMENTS

This work was supported by the Department of Energy Office of Science, Fusion Energy Sciences program Grant Nos. DE-SC0016258 and DE-SC0014318. J. Sanz was also supported by the Spanish Ministerio de Economía y Competitividad, Project No. RTI2018-098801-B-I00. H. Aluie was also supported by U.S. DOE Grant Nos. DE-SC0020229, DE-SC0019329, U.S. NASA Grant No. 80NSSC18K0772, and U.S. NNSA Grant Nos. DE-NA0003856 and DE-NA0003914. This material is based upon work supported by the Department of Energy National NuclearSecurity Administration under Award No. DE-NA0003856, the University of Rochester, and the New York State Energy Research and Development Authority. We thank two referees for their valuable comments, which helped improve this paper. This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

## DATA AVAILABILITY

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

### APPENDIX: PERTURBED EQUATIONS IN THE CONDUCTION LAYER

The governing Eqs. (24)–(29) rescaled in the conduction region are

where, for convenience, the streamwise momentum and energy equations have been rewritten in a conservative form.