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Scale-dependent Rayleigh-Taylor dynamics with variable acceleration by group theory approach

This work focuses on Rayleigh-Taylor instability (RTI)driven by acceleration with power-law time-dependence. We review the existing theoretical approaches, apply the group theory approach to solve this long-standing problem, and yield the unified framework for the scale-dependent dynamics of Rayleigh-Taylor (RT) bubbles and RT spikes. For the early-time linear dynamics we provide the dependence of RTI evolution on the acceleration parameters and the initial conditions. For the late-time nonlinear dynamics, we find a continuous family of asymptotic solutions, directly link the interface velocity to the interface morphology and the interfacial shear, derive solutions for the regular bubbles and for the singular spikes, and study stability of these solutions. The properties of special nonlinear solutions in the RT family are scrupulously described, including the critical, the Taylor, the Layzer-drag, and the Atwood solutions. It is shown that the fastest Atwood bubble is regular and stable, and the fastest Atwood spike is singular and unstable. The essentially multi-scale and interfacial character of RT dynamics is demonstrated. The former can be understood by viewing RT coherent structure of bubbles and spikes as a standing wave with the growing amplitude. The latter implies that RT flow has effectively no motion of the fluids away from the interface and has intense motion of the fluids near the interface, with shear-driven vortical structures appearing at the interface. Our theory agrees with available observations, and elaborates extensive benchmarks for future research and for better understanding of RT driven phenomena in plasmas.

RTI and RT mixing are extremely challenging to study in theory, experiments, and simulations [1,2]. RT experiments use advanced technologies to meet tight requirements for the flow implementation, diagnostics and control [13][14][15][16][17][18]. RT simulations employ highly accurate numerical methods and massive computations to track unstable interfaces, capture small-scale processes and enable large span of scales [19][20][21]. In theory we have to reliably grasp non-equilibrium RT dynamics, identify universal properties of the asymptotic solutions, and capture the symmetries of the unstable flow [11,12]. Significant success was achieved recently in the understanding of RTI and RT mixing with constant acceleration [1,2]. In particular, group theory approach [11,12] found the multi-scale character of nonlinear RTI and the order in RT mixing, and explained a broad set of experiments in plasmas and fluids which observed that RT mixing may depart from scenario of canonical turbulence and may keep order at high Reynolds numbers up to 3.2 million [13][14][15][16][17][18].
In realistic circumstances RT relevant phenomena are often driven by variable acceleration [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18]22]: In supernova blast the acceleration is induced by strong variable shocks and RT mixing with variable acceleration enables the conditions for the synthesis of heavy and intermediate mass elements. In inertial confinement fusion, the shocks' induced accelerations and decelerations of the ablation front influence the formation of hot spot. In nano-fabrication, the transformation of materials under high strain rate is governed by RTI with variable acceleration.
In this work we focus on RTI subject to variable acceleration, and we consider accelerations with power-law time-dependence [1,22,23]. On the side of fundamentals, power-law functions are important to study since they may result in new invariant and scaling properties of the dynamics [10,22]. As regards to applications, power-law functions can be used to adjust the acceleration's time-dependence in realistic environments and to ensure practicality of theoretical results [5][6][7][8][9]22]. To solve the long-standing problem of the scale-dependent early-time and late-time RT dynamics with variable acceleration, we apply group theory approach [11,12,22]. This approach worked remarkably well for constant and impulsive accelerations [11][12][13][14]. In the case of variable acceleration, it revealed the new mechanism for energy accumulation and transport at small scales in supernovae [22]. Furthermore, group theory approach found the dependence of the dynamics on the exponent of the acceleration's power-law [22]. Particularly, the scale-dependent interfacial dynamics is of Rayleigh-Taylor type and is driven by the acceleration for the acceleration exponents greater than (−2). It is of Richtmyer-Meshkov type and is driven by the initial growth rate for the acceleration exponents smaller than (−2) [22,23].
Here, we consider RTI for a three-dimensional spatially extended periodic flow with hexagonal symmetry in the plane normal to the acceleration, and we solve the boundary value problem involving boundary conditions at the interface and at the outside boundaries of the domain [10][11][12]22]. For the earlytime linear dynamics we provide the dependence of RTI growth-rate on the acceleration's parameters. For the late-time non-linear dynamics, we find a continuous family of asymptotic solutions, directly link the interface dynamics to the interface morphology and the interfacial shear, and derive the solutions for the regular bubbles and for the singular spikes. The family contains the special solutions for the bubbles and for the spikes, including the critical, the convergence limit, the Taylor, the Layer-drag, the Atwood and the flat solutions. The fastest Atwood bubble is regular and stable, whereas the fastest Atwood spike is singular and unstable. We also reveal the interfacial and multi-scale character of the scale-dependent RT dynamics, and analyze the mechanism of transition from the scale-dependent dynamics to the self-similar mixing for RT bubbles and for RT spikes. Our theory agrees qualitatively with available observations, and elaborates extensive quantitative benchmarks for future research [1,2,[13][14][15][16][17][18][19][20][21]. The properties of the scale-dependent RT dynamics found by our theory -the fields of the velocity and pressure, the interplay of interface morphology and shear with the flow acceleration, the interface growth and growth-rateserve for better understanding of RT driven plasma phenomena in nature and technology [1,2,[5][6][7][8][9][10]17,18].
By using the Heaviside step-function    H , we represent the flow fields in the entire domain as         H  E  ,  P  ,  ,  H  E  ,  P  ,  ,  E  ,  P  ,  , l h v v v [11,12,22]. At the interface, the balance of fluxes of mass and normal and tangential components of momentum and energy obey the boundary conditions: , the normal component of velocity and the pressure are continuous at the interface, and the tangential velocity component and the enthalpy can be discontinuous at the interface [10][11][12][22][23][24].
The outside boundaries have no influence on the dynamics, and the flow is free from mass sources. This leads to: [27][28][29][30]. Particularly, in the linear regime, the RTI growth-rate was derived in a continuous approximation as well as through the solution of the boundary value problem Eqs. (1)(2)(3)(4) with account of the effects of surface tension, viscosity, mass diffusion, mass ablation, magnetic fields, compressibility, thermal conductivity, stratification, finite size of the domain, and other factors [3,4,11,12,[27][28][29][30]. For RT flows with various symmetries and with variable acceleration group theory approach was employed [11][12][13][14]18,22,23]. Weakly-nonlinear stages of RTI were studied theoretically and empirically with the focus on the effect of initial spectra on RTI growth-rate [30][31][32].
Nonlinear dynamics of RTI is a long-standing problem, investigated by Davies & Taylor [4], Layzer [33], Birkhoff [34], Garabedian [35]. Group theory approach analyzed a broad class of Rayleigh-Taylor problems, including dynamics of two-and three-dimensional RT flows with various symmetries [11,12,37]. In case of constant acceleration, it reconciled with one another the approaches [4,[33][34][35][36][37], and identified the properties of nonlinear RT dynamics, including, e.g., the tendency to keep isotropy in the plane normal to the acceleration, the discontinuity of the dimensional crossover, and the multi-scale character of the dynamics of nonlinear RT bubbles [11,12].
The interactions of multiple scales further trigger transition of the scale-dependent RT dynamics to the self-similar mixing [13][14][15][16][17][18]. The transition may occur via the growth of horizontal scales of RT coherent structure, as found by the empirical models, including the interpolation, the buoyancy-drag, and the bubble merge models [39][40][41]47]. It may also occur due to the dominance of the vertical scale, as first found by group theory approach through the momentum model [11,42]. By further assuming that the self-similar RT mixing is similar to canonical turbulence due to its large Reynolds number, the interpolation and turbulence models were developed; they substituted the growth-rate of RT mixing in turbulent scaling laws and computed turbulence effects [25,[43][44][45][46][47]. By being harmonious with the theory of canonical turbulence [10,48] and with models [39][40][41][43][44][45][46][47], and by analyzing symmetries and invariants of RT dynamics, group theory approach identified, through the momentum model, the properties of RT mixing not considered in other studies [11,13,25,42]. Particularly, it found that in case of constant acceleration the self-similar RT mixing may exhibit order, since it has stronger correlations, steeper spectra, weaker fluctuations, and stronger sensitivity to deterministic conditions when compared to canonical turbulence [11,12]. These results explained the high Reynolds number experiments in plasmas and fluids [13][14][15]17,18].

Sub-section 2.3 -Group theory approach for Rayleigh-Taylor dynamics
In this work, we apply group theory approach to yield the unified framework for the scaledependent RT dynamics with variable acceleration in the early-time linear and the late-time nonlinear regimes [11][12][13][14]22,23]. We solve the governing equations Eqs. (1)(2)(3)(4) by employing theory of space groups, by using canonical forms of the Fourier series and spatial expansions, and by accounting for non-locality and singularities of the interface dynamics [12,37,38,49]. For the early-time regime, we provide the dependence of RT evolution on the acceleration parameters and the initial conditions. For the late-time regime, we find nonlinear solutions for regular RT bubbles and for singular RT spikes, and provide physics interpretations of mathematical attributes of the boundary value problem Eqs. (1)(2)(3)(4). By analyzing properties of regular RT bubbles and singular RT spikes, we further outline the mechanism of the flow transition to the self-similar mixing. We address to the future the establishment of a direct link of the scale-dependent RT dynamics to the self-similar RT mixing within group theory approach.

Sub-section 2.3.1 -Space groups
For spatially periodic flows, space groups can be applied, since RT dynamics is invariant with respect to a group G, whose generators are translations in the plane, rotations and reflections [11,12,49].
These groups are also known as Fedorov and/or Schoenflies groups, and they are commonly used in physics [10,49]. While there are 7 one-dimensional and 17 two-dimensional space groups, only some of these groups should be considered. Particularly, groups relevant to structurally stable RT dynamics must have the anisotropy in the acceleration direction and the inversion in the normal plane, such as groups of hexagon mm p6 or square mm p4 for a three-dimensional (3D) flow, and group 11 pm in a twodimensional (2D) flow [10][11][12]49]. We use international classification of the space groups, where p stands for periodicity in one (two) direction(s), and, for each of spatial directions, 1 stands for unit element, m stands for mirror plane of reflection, and   4 6 stands for   4 6 -fold axis of rotation [10,49].
By using the techniques of group theory, we apply irreducible representations of a relevant group to expand the flow fields as Fourier series. We further make spatial expansions in a vicinity of a regular point at the interface (i.e., the tip of the bubble or the tip of the spike). The governing equations are then reduced to a dynamical system in terms of surface variables and moments. The system's solutions are sought, and their stability is studied [11,12,37,38]. More details on group theory approach can be found in works on RTI with constant acceleration [11,12,14,22,37,38].

Sub-section 2.3.2 -Fourier series and spatial expansions
We consider incompressible dynamics with negligible stratification and density variations. For incompressible ideal fluids, the spatial period (wavelength)  of the initial perturbation defines the length-scale of the early-time linear and the late-time nonlinear dynamics. We focus on the large-scale coherent dynamics with length scale  , presuming that shear-driven interfacial vortical structures are small with length scales   [11,12].
For the large-scale coherent structure, the fluid motion is potential in the bulk, and the velocity of We consider three-dimensional coherent structure with hexagonal symmetry in the plane normal to the acceleration, Figure 1 [10][11][12]22,23,49]. In order to make a Fourier series expansions of the flow fields, we recall that for group mm p6 with hexagonal lattice the spatial periods      [11,37,49].
For convenience, we perform the derivations in the non-inertial frame of reference moving with the velocity   t v in the z -direction relative to the laboratory frame of reference [11,12,23,37]. Then, the Fourier series expansions of the potentials   l h  have the form:  is the principal curvature at the bubble (spike) tip. Cross terms appear in high orders [11,12,22,23,37].

Sub-section 2.3.3 -Dynamical system
Upon substituting these expansions in the governing equations and further expanding the equations in the vicinity of the regular point of the interface -the tip of the bubble (spike) -we derive from the governing equations a dynamical system in terms of moments and surface variables [11,12,22,23,37]. For group mm p6 , , and the dynamical system is:   in Eqs.(7) [22]. According to the boundary conditions in Eqs. (3) and Eqs. (7), the shear value can be arbitrary.
In dynamical system Eqs.  [22,23]. We find solutions for RT bubbles and RT spikes in the early time linear regime,      0 t t and in the late time nonlinear regime, where p I is the modified Bessel function of the p th order, Table 1 [22,23]. For constant acceleration, 0  a , this solution reproduces the classical result. Figure 2 illustrates the solutions in Eqs.(10) for the early-time dynamics of RT bubbles and RT spikes at various values of the acceleration exponent a . In the linear regime, the growth and the growth-rates of the bubbles and the spikes are nearly symmetric.
Note that Richtmyer-Meshkov dynamics with 2   a is driven by the initial growth-rate 0 v and is independent of the acceleration; its early-time solution is the contribution of these initial-growth-rate driven terms is negligible when compared to those induced by the acceleration in Eqs. (10).

Sub-section 3.1.2 -Effect of initial conditions
Solutions for linearized system Eqs.(9) describe, as time evolves, the dynamics of bubbles with To study how the bubbles and spikes are being formed, we consider the very early time dynamics, 0 t t  [22,23]. The solution for Eqs.(9) linearized for This suggests the following changes of the morphology and the velocity of the interface near the tip.
, the interface becomes more curved and its velocity increases.

Sub-section 3.2.1 -Asymptotic behavior of nonlinear solutions
For the late-time dynamics, we seek asymptotic solutions in the form of power-law functions where the exponents   , are defined from the dominant balance conditions in system Eqs. (7). By substituting these expressions into the system Eqs. (7), we find that for 2   a , the dominant balance conditions require the power-law exponents to be   This leads [22] to asymptotic nonlinear solutions with the dominant balance conditions require the power-law exponents to be and lead to asymptotic nonlinear solution with , the contribution of these terms is negligible, and Rayleigh-Taylor nonlinear dynamics is independent of the initial growth-rate.

Sub-section 3.2.2 -Family of nonlinear asymptotic solutions
To find for . Each of the moments is an infinite sum of the weighted Fourier amplitudes [11,12,22,[36][37][38]. To solve the closure problem, group theory is further applied. Particularly, the nonlinear asymptotic solutions are not unique and form a continuous family. Their multiplicity is due to the singular and non-local character of RT dynamics [35][36][37][38]. The number of the family parameters is set by the flow symmetry. For group mm p6 the dynamics is , the interface morphology is captured by the principal curvature  , and the family has one parameter [11,12,37,38]. The multiplicity of the nonlinear asymptotic solutions is also due to the presence of shear  at the interface Eqs.
depend on the curvature  as (see Table 2 for the explicit expressions): In the following sub-sections we study in details the properties of bubbles and spikes in the scaledependent nonlinear dynamics of RTI with variable acceleration.

Sub-section 3.3.1 -Nonlinear solutions
For nonlinear bubbles, we consider solutions with     0 The solutions for RT bubbles with   , where the sub-script stands for critical. The range of the velocity function is

Sub-section 3.3.2 -Special solutions
The family of asymptotic solutions for nonlinear RT bubbles with 0   k has some special solutions. We call these solutions the critical bubble, the convergence limit bubble, the Taylor bubble, the This solution is special, since the bubbles in the family of solutions cannot be more curved than the critical bubble. For the critical bubble the velocity is zero and the shear function achieves its maximum value. The large shear can alone maintain the pressure at the interface, Figure 5.
, since the amplitudes of the heavy fluid are , and the amplitudes of the light fluid are . For any Atwood number, the convergence limit bubble is less curved and has than larger velocity and smaller shear when compared to the critical bubble, Figure 5.

C. The Taylor bubble
The family has a solution with  k We call this solution the Taylor bubble since its curvature is the same as in the solution which can be found in Davies & Taylor 1950, upon modifying the  is the first zero of the Bessel function 1 J [4,11,12,37]. For the Taylor bubble, the curvature, the velocity, and the shear function are Note that for the Taylor bubble the second Fourier amplitude of the heavy fluid is zero at . This property does not hold for 1  N . The ratios are For any Atwood number, the Taylor bubble is less curved and has larger velocity and smaller shear when compared to the convergent limit bubble and the critical bubble, Figure 5.

D. The Layzer-drag bubble
The bubble family has a solution with velocity . We call the solution the Layzer-drag bubble since the drag model applies this velocity re-scaling with the Atwood number to the single-mode Layzer's first-order approximation at 1  A [33,39]. Experiments and simulations tend to compare well with this re-scaling [33,39,47]. For the Layzer-drag bubble, the curvature, the velocity, and the shear function are The dependence of the Layzer-drag bubble curvature on the Atwood number is cumbersome, For fluid with very similar densities   0 A , the solution for the Layzer-drag bubble is: The Layzer-drag bubble is more curved for fluids with similar densities

E. The Atwood bubble
We call the fastest solution in the bubble family as the 'Atwood bubble' to emphasize its complex dependence on the Atwood number, For fluid with very similar densities   0 A the curvature, the velocity and the shear function of the Atwood bubble are: The curvature, the velocity and the shear function of this bubble equal to zero. The ratio is Note that while the velocities of the flat and the critical bubbles are both zero, the shear of the flat bubble is zero, whereas the shear of the critical bubble has the maximum value.

Sub-section 3.3.3 -Stability of solutions
To analyze asymptotic stability of the solutions in the family, we slightly perturb  [37,38], and we address it to the future.

Sub-section 3.4.1 -Nonlinear solutions
For nonlinear spikes, we consider solutions with     0    Figure 3.
The universality of nonlinear dynamics of RT spikes and the link between the interface dynamics, interface morphology and the interfacial shear are clearly seen from these plots. Figure 6d

Sub-section 3.4.2 -Special solutions
. We call this solution the Layzer-drag spike since the drag model applies such re-scaling with the Atwood number for the velocity of the nonlinear spike to the Layzer first order approximation of the velocity of the nonlinear bubble at 1  A [33,39]. Experiments and simulations tend to compare well with this re-scaling [39,47].
The Layzer-drag spike is more curved for fluids with very different densities . We call this solution the 'Atwood spike' due to its complex dependence on the Atwood number, For the Atwood spike, the curvature is regular, The Atwood spike is less curved than the Layzer-drag spike for   The curvature, the velocity and the shear function of this spike equal to zero. The ratio is Note that while the value  Table 1 [22,23]. The nonlinear dynamics is more complex. From the properties of the solutions' family Eqs. (15), including the special solutions for RT bubbles Eqs. (16)(17)(18)(19)(20)(21) and for RT spikes Eqs. (23)(24)(25)(26)(27)(28)(29), we find that for incompressible ideal fluids the asymptotic nonlinear RT dynamics is influenced by the interplay of the acceleration, the interface morphology and the interfacial shear, Table 2

B. RT spikes
Similarly to RT bubbles, RT spikes move slower for larger magnitudes of their curvatures, Eqs. (15,(23)(24)(25)(26)(27)(28)(29), Figure 6. For fluids with a finite density ratio, the shear of the strongly curved critical spike can alone maintain the pressure at the interface reducing the spike velocity to zero value, Figure 6.
In contrast to RT bubbles, RT spikes are singular, Figure 3, Figure 6. The velocity and the shear values of the convergence limit and the Taylor spikes become singular approaching negative infinity at some Atwood numbers, Figure 7. Those of the Layzer-drag spike approach negative infinity for the Atwood number   1 A , with the shear being more singular than the velocity, Figure 7.
The fastest moving Atwood spike is singular for any Atwood number, Eqs. (27,28), Figure 7. For the Atwood spike, the velocity and shear both approach infinite values in case of fluids with a finite density ratio 1 0   A , and the shear is even more singular than the velocity in case of fluids with highly contrasting densities   1 A . The singular character and the instability of the Atwood spike solution suggest that RT spike, upon achieving a finite curvature value, can become unstable and move faster than it is prescribed by the nonlinear asymptotic dynamics Eqs. (14).

Sub-section 3.5.2 -Single-scale and multi-scale dynamics
By analyzing properties of the early-time linear dynamics Eqs.(10), we find that for ideal incompressible fluids and for the interface perturbations with very small initial amplitude RT, dynamics is singe-scale and is defined by the horizontal length scale, which is the spatial period (the wavelength) of the interface perturbation, Figure 2.
The nonlinear dynamics is more complex. From the properties of the solutions' family in Eqs. (15), we find that for ideal incompressible fluids the asymptotic nonlinear RT dynamics is essentially multi-scale and is defined by the two macroscopic scales -the horizontal length scale (the wavelength) and the vertical length scale (the amplitude). This can be understood by viewing RT coherent structure as a standing wave, whose nonlinear dynamics is influenced by its wavelength and its growing amplitude.

A. RT bubbles
In the family of the asymptotic solutions describing nonlinear RT bubbles, we define the fastest stable solution as the physically significant solution, The Atwood bubble has the invariant property, , Eqs. (20). The left-hand side of the expression is a function of time, and of the vertical and the horizontal length scales and their derivatives, whereas the right-hand side is unity. This invariance implies that the two macroscopic scales -the wavelength  and amplitude 0 z -contribute to the multi-scale dynamics of nonlinear RT bubbles [11,12,22,37,38].

B. RT spikes
The fastest Atwood spike has no invariant property similar to that of the fastest Atwood bubble.
The singular character of the dynamics of the (unstable) fastest Atwood spike with suggests that the nonlinear RT spike can move faster than the nonlinear asymptotic dynamics prescribes, Eqs. (27,28). As such, it is influenced even more by the vertical scale (the amplitude) when compared to the nonlinear RT bubbles. The dynamics of the nonlinear RT spike is thus multi-scale, with the two macroscopic length scale contributing -the vertical length scale (amplitude) and the horizontal length scale (wavelength). To our knowledge, this is the first report of the multi-scale character of nonlinear dynamics of RT spikes based on the properties of nonlinear asymptotic solutions for the boundary value problems Eqs. (1-4).
Group theory approach studies properties of self-similar RT dynamics within the frame of momentum model [11,14,22,42,55]. This model has the same symmetries and scaling transformations as the governing equations and yields asymptotic solutions for the scale-dependent and the self-similar dynamics, up to constant.
The important outcome of the momentum model is that RT dynamics can transit from the scaledependent nonlinear regime to the scale-invariant self-similar mixing regime when the vertical scale (the amplitude) is the dominant scale, in excellent agreement with high Reynolds number experiments [11,14,42]. This mechanism compliments the traditional merger mechanism, in which the transition to the self-similar mixing is believed to occur via the growth of horizontal scales [39,40,41]. Moreover, for RT coherent structures with hexagonal symmetry the growth of horizontal scales may not be feasible, since it may lead to anisotropy of the highly isotropic hexagonal pattern [56]. Ref. [56] provides the detailed classification of the formation of patterns in RT flows and possible discrete transitions with the increase of the wavelength(s) of RT coherent structures under the influence of modulations. These results excellently agree with existing experiments [13][14][15][16][17][18].
We address to the future the identification of a direct link between the analytical results in the present paper and momentum model. Here we analyze how the multi-scale character of the nonlinear dynamics is associated with and may lead to the acceleration of RT bubbles and RT spikes.

A. RT bubbles
Recall that in laboratory frame of reference the interface is described near the tip of the bubble as   Eqs. (14,15). The invariant property of the Atwood bubble Eqs. (20) suggests that the amplitude 0 z and the spatial period  contribute independently to the multi-scale dynamics of the nonlinear bubble. Hence the amplitude 0 Gt v [11,14,22,42,55].
For fluids with very different densities, , the Atwood spike is highly singular, suggesting that its dynamics can only be self-similar, in agreement with experiments [4,[13][14][15][16][17][18][19]. To our knowledge, this is the first analysis of the mechanism of transition to self-similar mixing on the basis of properties of rigorous solutions for nonlinear dynamics of RT bubbles and RT spikes in Eqs. (1-4).

Sub-section 3.5.4 -Interfacial dynamics
By accurately accounting for the interplay of harmonics and by systematically connecting the interface velocity to the interfacial shear in a broad range of the acceleration parameters, our analysis finds that RT dynamics is essentially interfacial: In both the linear and nonlinear regimes, for both RT bubbles and RT spikes, the flow has intense motion of the fluids in a vicinity of the interface and has effectively no motion away from the interface. The velocity field is potential in the fluid bulk. Due to the presence of shear at the interface, shear-driven vortical structures develop at the interface. Figure 8 illustrate the velocity fields in the vicinity of the tips of RT bubble and RT spike.
This velocity pattern is in excellent agreement with experiments and simulations [13][14][15][16][17][18][19][20][21]. For realistic fluids, in order to accurately grasp the formation and the dynamics of vortical structures, one should also account for the influence of viscosity, compressibility, surface tension, mass flux across the interface, magnetic field, and other factors. We address these tasks to the future.

Sub-section 3.5.5 -Effect of interfacial shear
Our theory finds that the interfacial shear plays the important role in RT dynamics. The presence of interfacial shear is fully consistent with the conversation laws, including the continuity of tangential parameters. According to our results, the shear function is regular in the region of bubbles and is singular in the region of spikes. Hence, one expects a more intense production of vortical structures for RT spikes when compared to RT bubbles, in excellent agreement with existing observations [13][14][15][16][17][18][19][20][21]. To our knowledge, this is the first theoretical explanation of the bubble/spike asymmetry of vortical structures in RT flow.

Sub-section 3.5.6 -Formal properties of nonlinear solutions
By directly linking the interface velocity, the flow fields, the interfacial shear function, and the interface morphology in RT nonlinear dynamics, our theory finds the physics interpretation and resolves the long-standing problem of multiplicity of solutions for nonlinear RTI [33][34][35][36][37]. According to our results, for RT dynamics with variable acceleration, there is a continuous family of asymptotic nonlinear solutions, with each solution having its own curvature, velocity, the Fourier amplitudes, the flow fields, and the interfacial shear [22]. The multiplicity of nonlinear asymptotic solutions for the boundary value problem is due to the presence of shear at the interface, and is the essential property of RT dynamics associated with the governing equations and the boundary conditions at the interface Eqs.(1-4) [11,22,35]. In the continuous family of nonlinear solutions, asymptotic solutions with negative curvature and positive velocity correspond to regular RT bubbles (which move up and concave down), whereas asymptotic solutions with positive curvature and negative velocity correspond to singular RT spikes (which move down and concave up), Figure 1, Figure 8. To our knowledge, our work is the first to report that asymptotic solutions describing dynamics of the nonlinear RT spikes and the nonlinear RT bubbles with variable acceleration form a continuous family.

Sub-section 3.5.7 -Effect of the acceleration parameters
According to our results, for a broad range of the acceleration parameters, where RTI is driven by an explosion or an implosion with acceleration exponents by blast waves [22,[50][51][52][53][54]. Our results thus provide new extensive theory benchmarks for the future.

Sub-section 3.5.8 -Comparisons with existing observations
Significant efforts are undertaken in RT community to better understand the scale-dependent RT dynamics and the self-similar RT mixing. See for details edited research books [1,2] and experimental and numerical papers [11-25,45-47,55,-60]. As discussed earlier in Sections 3.1.1-3.5.7, our theory is in excellent qualitative agreement with the existing observations [11-25,45-47,55,-60], and provides new extensive quantitative benchmarks for experiments and simulations. To date, to the best of the authors' knowledge, accurate experiments and simulations are free from systematic investigations of RT dynamics driven by variable acceleration for three-dimensional flow with hexagonal symmetry [13][14][15][16][17][18][19][20][21].
In this work we consider RT dynamics with variable acceleration for three-dimensional flow with hexagonal symmetry, since it is important to study from the physics perspectives and since it has a number of advantages when compared to other space groups. For instance, hexagonal group is the most isotropic, when compared to other space groups; for structures with hexagonal symmetry, transitions to super-structures with larger periods lead to a loss of isotropy and may thus be infeasible [10][11][12] [11,12,22].
Our theory finds the effect of the interplay of the acceleration, the interface morphology and the interfacial shear on the scale-dependent dynamics of RTI with variable acceleration. Our results suggest that this interplay can be more complex than it is traditionally seen by the empirical Layzer-type buoyancy-drag models [25,39,41,47]. A direct link of the scale-dependent RT dynamics and to the selfsimilar RT is thus the important task. Group theory approach can be applied to solve this task, to be done in the future [11,14,22].
Our theory focuses on RT dynamics in ideal incompressible fluids. In realistic circumstances, other factors should be accounted for, including, e.g., viscosity, compressibility, surface tension, magnetic field. Qualitative and quantitative influences of these factors should be systematically studied.
For instance, viscosity may influence the instability growth-rate in the linear regime as well as the realization of asymptotic solutions in the nonlinear regime. Particularly, while the Layzer-drag bubble has a smaller velocity when compared to the Atwood bubble, it also has a larger shear and may thus 'survive' under the viscous effects, Figure 5. Likewise, viscosity may 'stabilize' for a finite time and realize the nonlinear spike with asymptotic velocity a t G v  , before the spike transits to the self-similar mixing, Figure 7.

Sub-section 3.5.9 -Theory benchmarks
Our analysis elaborates qualitative and quantitative benchmarks of RT dynamics not discussed before. These are the fields of velocity and pressure, the interface morphology and the curvatures of the bubble and spike, the interfacial shear and its link to the velocity and the curvature of the bubble and the spike, the spectral properties of the velocity and pressure, as well as the interface growth and growth-rate.
By diagnosing the dependence of these quantities on the density ratio, the flow symmetry, and the acceleration's exponent and strength, by identifying their universal properties, and by accurately measuring departures of data in real fluids from theoretical solutions in ideal fluids, one can further advance knowledge of RT dynamics in realistic environments, achieve better understanding of RT relevant processes, from supernovae to fusion, and improve methods of numerical modeling and experimental diagnostics of interfacial dynamics in fluids, plasmas and materials under conditions of low and in high energy densities.

Section 4 -Discussion and conclusion
In this work, we focused on the scale-dependent dynamics of Rayleigh-Taylor instability driven by variable acceleration with power-law time-dependence for the acceleration exponents greater than (-2), and studied the spatially periodic three-dimensional flow with hexagonal symmetry in the plane normal to the acceleration direction. We applied group theory approach to solve the boundary value problem, by expanding the flow fields with the use of irreducible representations of the group, by deriving the dynamical system from the governing equations, and by finding solutions for the system to describe dynamics of RT coherent structure of bubbles and spikes in the linear and nonlinear regimes, Eqs.(1-29), Table 1 and Table 2 The former can be understood by viewing RT coherent structure as a standing wave with the growing amplitude. The latter implies that RT flow has effectively no motion of the fluids away from the interface and intense motion of the fluids near the interface, with shear-driven vortical structures appearing at the interface, Figure 8. Our theory provides the unified framework for the scale-dependent RT dynamics Eqs. . Our solutions describe in the linear and nonlinear regime the RT bubbles, which move up and concave down, and the RT spikes, which move down and concave up. We find: In the linear regime RT bubbles and RT spikes grow nearly symmetrically, Figure 2. In the nonlinear regime, the dynamics of RT bubbles is regular and the dynamics of RT spikes is singular, Figure 3. We also reveal that RT bubbles and RT spikes may further accelerate due to the dominance of the vertical length scale and thus transit from the scale-dependent to the self-similar regime. Our results are congruent with previous theoretical studies [11,12, and with available observations [13][14][15][16][17][18][19][20][21], and elaborate extensive theory benchmarks for future experiments and simulations.
The group theory approach finds the multiplicity of nonlinear solutions for three-dimensional coherent dynamics of RT bubbles and RT spikes with variable acceleration, and associates this multiplicity with the presence of the interfacial shear. Our approach can be further applied to find solutions for three-dimensional (3D) flows with other symmetries and for two-dimensional (2D) flows [12,37,49,56]. We address the study of these problems to the future. As a brief summary -3D flows tend to conserve isotropy in the plane, 3D highly symmetric dynamics is universal, and the dimensional 3D-2D crossover is discontinuous [12,37,49,56]. Within the frame of group theory approach, we can also directly link the scale-dependent dynamics and the self-similar mixing, to be done in the future [11,22].
Our work studies the dynamics of ideal incompressible fluids. Our approach can be further applied to systematically analyze the influence of surface tension, viscosity, compressibility and other factors on RT dynamics with variable acceleration. We address these studies to the future. Our work focuses on the acceleration-driven RT dynamics with the acceleration exponents 2   a . For 2   a the dynamics is driven by the initial growth-rate and is Richtmyer-Meshkov (RM) type, whereas at 2   a , the transition occurs from RT type to RM type dynamics with varying acceleration strength [22,23]. We provide in the future the detailed analysis of 3D and 2D flows with various symmetries for acceleration exponents 2   a [22].
Possible applications of our theory in high energy density plasmas include supernovae remnants, inertial fusion and nano-fabrication, among others [5][6][7][8][9][57][58][59][60]. For instance, the explosion of corecollapse supernova is driven by a blast wave, which is a variable strong shock [5,22]. The blast-waveinduced acceleration is a power-law function of time, with the acceleration exponents . According to our results, the scale-dependent interfacial dynamics driven by such accelerations is of RT type, and it is sub-exponential in the linear regime, and is decelerating in the nonlinear regime, Figure   2, Figure 3. In the self-similar mixing, the dynamics can be Richtmyer-Meshkov type and be subdiffusive [22]. The latter implies that the energy transport at small scales may occur via localization and trapping thus influencing the synthesis of heavy mass elements in the supernova blast [22]. Our results can thus be applied to explain the abundance of chemical elements and the richness of structures observed in supernovae remnants [5].
Our group theory approach finds that Rayleigh-Taylor dynamics can in principle be controlled, in excellent agreement with experiments in fluids and plasmas at Reynolds number up to 6 10 2 3  . [14][15][16][17][18]58] For inertial fusion, this suggests new opportunities for control of plasma flows by means of variable acceleration and deterministic conditions [6,14,[57][58][59][60]. Particularly, in the inertial confinement fusion, it may be worth to scratch the target in order to pre-impose proper deterministic conditions and to gain better control of fluid instabilities and the interfacial mixing [6,[57][58][59][60]. This may help the existing methods, that are focused on a fine polishing of ICF targets in order to fully eliminate RTI [6,57].
The results of group theory approach can also be applied in nano-fabrication, by describing the dynamics and morphology of RT unstable interface in the scale-dependent regime, and the scaling and properties as well as the sensitivity to deterministic conditions in the self-similar regime [9,[57][58][59][60].
To conclude, we studied the problem of RTI with variable acceleration by applying group theory approach. We linked the interface dynamics to the interfacial shear, revealed the interfacial and multiscale character of RT dynamics, and elaborated new theory benchmarks for future research.

Section 5 -Acknowledgements
SIA thanks for support the University of Western Australia, AUS (project grant 10101047); and the National Science Foundation, USA (award 1404449).

Section 6 -Data availability
The methods, the results and the data presented in this work are freely available to the readers in the paper and on the request from the authors.

Section 7 -Author's contributions
The authors contributed to this work as follows: SIA -conceptualization, formal analysis, methodology, investigation, project administration, resources, supervision, writing; KCW -investigation, preparation of figures.