We demonstrate a methodology for diagnosing the multiscale dynamics and energy transfer in complex HED flows with realistic driving and boundary conditions. The approach separates incompressible, compressible, and baropycnal contributions to energy scale-transfer and quantifies the direction of these transfers in (generalized) wavenumber space. We use this to compare the kinetic energy (KE) transfer across scales in simulations of 2D axisymmetric vs fully 3D laser-driven plasma jets. Using the FLASH code, we model a turbulent jet ablated from an aluminum cone target in the configuration outlined by Liao *et al.* [Phys. Plasmas, **26** 032306 (2019)]. We show that, in addition to its well known bias for underestimating hydrodynamic instability growth, 2D modeling suffers from significant spurious energization of the bulk flow by a turbulent upscale cascade. In 2D, this arises as vorticity and strain from instabilities near the jet's leading edge transfer KE upscale, sustaining a coherent circulation that helps propel the axisymmetric jet farther ( $ \u2248 25 %$ by $ 3.5$ ns) and helps keep it collimated. In 3D, the coherent circulation and upscale KE transfer are absent. The methodology presented here may also help with inter-model comparison and validation, including future modeling efforts to alleviate some of the 2D hydrodynamic artifacts highlighted in this study.

## I. INTRODUCTION

High speed jets arise in many astrophysical and high energy density (HED) applications.^{1} In astrophysics, supersonic hydrodynamic jets can arise as an initially collimated outflow or an otherwise quasi-spherical supersonic blast wave propagating into an inhomogeneous medium. Jets appear on sources of all scales of astrophysics, including supernovae evolution.^{2} In inertial confinement fusion (ICF), jets result from the interaction between a shock and density perturbations such as those due to the fill-tube^{3} or due to voids.^{4,5} These jets can trigger the mixing of ablator and fuel, which ultimately degrade the ignition yield.^{6,7} Jets can often be characterized as complex multi-scale flows involving instabilities and turbulence, and in many cases, instabilities trigger turbulent mixing.^{8,9}

Turbulence is common in complex HED flows in astrophysics and ICF. In the evolution of core-collapse supernovae, for example, turbulence is generated due to instabilities between the inner core and the outgoing shock.^{10–12} Energy transfer between scales is believed to play a key role in such simulations and may account for differences between 2D and 3D.^{13–15} In particular, it is believed that upscale (inverse) transfer of kinetic energy in 2D may help to increase the efficacy of turbulent transport of heat to the shock to prevent it from stalling, making explosions easier to achieve than in 3D.

Instabilities are also common in complex HED flows in astrophysics and ICF. For example, the Rayleigh-Taylor (RT) instability in ICF has been a significant hindrance to ignition.^{16} In ICF, 2D simulations are the main “work horse” for experimental design^{17–19} as routine 3D simulations are prohibitively expensive. A capsule-only 3D simulation requires tens to hundreds of millions of CPU hours on supercomputers even without incorporating the hohlraum and laser physics.^{20,21} The tradeoffs between 2D and 3D flow physics are still not widely appreciated in ICF simulations. The primary objective of this study is to highlight the differences in energy scale-transfer how they affect the travel of laser-driven jets between 2D and 3D.

Length scales in complex flows are generally characterized by a wavenumber energy spectrum. Yet, the energy transfer pathways and interactions between different scales are often subtle and cannot be diagnosed using the spectrum alone. In this work, we investigate two processes that transfer energy across scales: deformation work, Π, and baropycnal work, Λ.

Deformation work is due to the multi-scale nature of the velocity field and is the sole mechanism for energy transfer across scales (or cascade) in constant-density (incompressible) flows described by the Navier–Stokes equation.^{22,23} In 3D incompressible homogeneous turbulence, the main mechanism behind Π is believed to be vortex stretching,^{24–28} which transfers energy downscale. Kraichnan^{29} predicted and subsequent studies^{30,31} have demonstrated that an incompressible homogeneous turbulent flow, when restricted to two spatial dimensions, will tend to transfer energy in the opposite direction, i.e., upscale rather than downscale, due to vorticity conservation in 2D. However, HED flows involve significant density variations and do not conserve vorticity, even when restricted to 2D, due to baroclinicity. Therefore, extending such phenomenologies^{29,31} from incompressible homogeneous flows to inhomogeneous flows with variable density remains dubious.^{32} A main contribution in this work is to demonstrate, via a direct measurement of Π, that an upscale cascade can indeed arise in 2D hydrodynamic models of HED systems, despite the lack of vorticity conservation.

The second energy transfer pathway we analyze here is baropycnal work Λ, which results from pressure and density variations at different scales.^{33,34} The direction of energy transfer by Λ depends on the source of pressure gradient and density variations.^{32,35–38} Typical cases include (i) a baroclinic flow wherein Λ transfers energy downscale into small-scale velocity fluctuations due to opposing directions of pressure and density gradients,^{32} and (ii) a shock wherein pressure and density gradients are aligned and Λ transfers energy upscale into the large-scale pressure field.^{34,36,38}

Previous works have utilized coarse-graining methods that are common in large eddy simulation modeling^{39} to decompose length scales and analyze deformation work^{40,41} and baropycnal work.^{33,38,42} Lees and Aluie^{42} conducted simulations of compressible turbulence in 3D with varying levels of compressibility. Their results demonstrated that in 3D homogeneous compressible turbulence, Π transfers energy downscale to the small-scale turbulence, while Λ transfers energy upscale into large-scale potential energy residing in the pressure field. Zhao *et al.*^{32} analyzed the energy scale-transfer in baroclinically unstable flows by simulating the RT instability and the ensuing late-stage turbulence in 2D and 3D. They found that in both 2D and 3D, Λ transfers energy downscale by releasing potential energy stored at the largest scales and depositing it over a wide range of smaller scales in the fully turbulent regime. Energy transfer by Π, on the other hand, exhibits opposing behaviors in 2D vs 3D.^{32} In 2D, upscale transfer by Π leads to higher bulk kinetic energy at large scales, whereas in 3D, the energy transfer is downscale.

Here, we extend and demonstrate the methodology to high energy density (HED) flow regimes by taking advantage of laser-driven jets. We reference the simulation configuration by Liao *et al.,*^{43} which proposes a design to set up a turbulent dynamo operating on the OMEGA-EP laser.^{44,45} This design, sketched in Fig. 1, shines multiple laser beams on the inner wall of a cone target to create a supersonic turbulent jet. In Sec. II, we describe the model configurations, address the sensitivity to initial conditions in 2D and 3D, and report on the jets' evolution and properties. In Sec. III, we introduce the coarse-graining method for scale decomposition and define the energy scale-transfer terms, Π and Λ. In Sec. IV, we analyze KE at large scales and show that the corresponding difference between 2D and 3D can be quantified by Π and Λ. In Sec. V, we characterize the turbulent region at the jet's leading edge using subscale stress and discuss KE scale-transfer in that region. In Sec. VI, we decompose Π and Λ into contributions from the divergence-free and irrotational components of the flow to examine the mechanism causing the 2D jet's leading edge to remain narrow. We summarize the results in the concluding Sec. VII.

## II. NUMERICAL SIMULATIONS

### A. Simulation configurations

FLASH is a multiphysics radiation-hydrodynamics code used frequently for HED applications. It provides a comprehensive environment for full-physics laser driven simulations by solving governing equations in three-temperature unsplit hydrodynamics on an adaptive mesh refinement (AMR) grid.^{46,47} The laser driven simulation module includes features of laser ray tracing,^{48} electron thermal conduction with Spitzer conductivities,^{49} and radiation diffusion with tabulated opacity.^{50} Liao *et al.*^{43} used the FLASH code to develop a test platform for laser-driven turbulent dynamos that matches realizable OMEGA-EP experimental configurations in 3D.^{43} We conduct our 3D simulations using a similar configuration that generates turbulence by the interaction of laser ablated jets. To simulate the analogous cone target in 2D, we employ cylindrical coordinates, which significantly modifies the configuration of Liao *et al.* in that the laser deposition becomes azimuthally symmetric. To approximate this modification in 3D, within Cartesian coordinates, we simulate a quasi-azimuthally symmetric configuration in which 256 laser beams deposit energy symmetrically about the center vertical axis of the target. We investigate sensitivity to the laser drive and show the validity of comparing 2D and 3D simulations below. In both 2D and 3D geometries, UV laser beams (351 nm wavelength) deliver a total power of 0.5 TW over a 3.5 ns duration using square pulses. The simulations use multigroup diffusion for radiation transport using six groups.

As shown in Fig. 1, the simulation domain is 0.25 cm wide and 0.75 cm high, and the target occupies 0.3 cm of height from the top boundary of the domain. The main target is 2.7 g/cc aluminum notched into a hollow cone with an interior space of 0.2 cm diameter at the cone base and 0.274 cm of height filled with a pseudo-vacuum of $ 10 \u2212 6 \u2009$ g/cc helium. The vacuum region extends from the cone's opening to the bottom of the domain over 0.45 cm. The equations of state (EOS) and opacity of both the aluminum cone and helium pseudo-vacuum are obtained from the atomic database IONMIX.^{51}

Due to the nature of cylindrical coordinates, the 2D computational domain takes an axisymmetric boundary at *x* = 0 in Fig. 1 as the center axis, for which both normal and toroidal vector components change sign. The boundaries everywhere else are outflow, which specifies a zero gradient to allow the flow to leave the domain freely. The 3D computational domain takes outflow boundaries in all directions. In this work, all visualizations of the 2D domain are mirrored about the axisymmetric boundary (*x* = 0) to compare to the slice of the 3D domain.

Applying adaptive mesh refinement,^{52} we set the initial static grid size of a mesh block to be 16 in all directions, with a maximum refinement level of 5 in both 2D and 3D. Each refinement level doubles the grid size at the finest resolution. Since the computational domain's aspect ratio is 1:6 in 2D and 1:3 in 3D, we use three mesh blocks in 3D aligned in the vertical direction and six mesh blocks in 2D. The effective mesh resolution can be calculated as $ ( grid \u2009 size \u2009 o f \u2009 single \u2009 block ) \xd7 2 ( max \u2009 refinement \u2009 level \u2212 1 ) \xd7 ( number \u2009 o f \u2009 blocks )$, which gives the resolution 256 × 1536 in 2D and $ 256 \xd7 256 \xd7 768$ in 3D. The mesh refinement criterion used is the modified Löhner's estimator,^{50} for which we set the density and electron temperature to be the refinement variables and the threshold value to trigger refinement is 0.8. This grid mesh configuration resolves turbulence structures that appear around the leading edge of the jet at late times. In the Appendix, we report on our analysis of numerical convergence (see Fig. 20).

### B. Sensitivity to laser drive

Figure 1 shows initial configurations for runs L5R090 and L6R720, where the label “L” denotes the y-location of laser deposition in millimeters, and the label “R” is the radius of laser deposition in micrometers. Due to the inherent azimuthal symmetry in the 2D simulations, which requires only one laser beam to illuminate the entire target. In 3D, the beams are separated with equal angle azimuthally on the cone. Matching the 2D and 3D configurations of the initial flow is crucial for making comparisons. Here, we list a few laser-drive parameters that play an important role in the initial flow's development.

First, the number of laser rays traced in the simulation significantly affects the ablated flow's smoothness at early times. As shown in Figs. 2(a) and 2(c), the same number of rays reveals distinct early time velocity profile differences. The 2D initial flow using $ 4096 \u2009$ rays in the single beam manifests large-scale non-uniformity in the reflected-energy region ( $ y > 0.5325$ cm, highlighted by arrows in Fig. 2), whereas this issue is absent in 3D due to using a higher number of laser beams, each with $ 4096 \u2009$ rays. With a sufficient number of rays shown in Fig. 2(b), the early time velocity profile off the ablated surface agrees in 2D and 3D.

Second, the radius of laser deposition has an important role in matching the early time evolution in 2D and 3D. In run L5R090, the reflected laser energy appears in the domain at $ y > 0.5325$ cm in Fig. 2, highlighted by white arrows. However, the fraction of reflected energy differs between 2D and 3D, which leads to differences in the mass ablated at different locations along the cone surface. This can be quantified by comparing the kinetic energy at $ y > 0.5325$ cm, *KE ^{up}* to kinetic energy over the entire domain,

*KE*. Their ratio, $ K E u p / K E tot$, is listed in Table I and shows a mismatch between 2D and 3D for run L5R090. Using a larger radius for the laser deposition area minimizes ablation from reflected radiation and, therefore, minimizes differences between 2D and 3D in the early time flow. This is achieved in run L6R720 in Fig. 3, where the fraction $ K E u p / K E tot$ is the same in 2D and 3D in Table I. Since the y-momentum,

^{tot}*G*, is central to the jet's traveling distance, we also show a similar ratio $ G y u p / G y tot$ in Table II, which agrees between 2D and 3D in run L6R720. Despite increasing the laser radius in run L6R720, it is still necessary to use a sufficient number of rays in 2D to match the early time solutions from the 2D and 3D simulations, as discussed in the previous paragraph.

_{y}Run . | KE(y > 0.5325 cm)
. ^{up} | KE(y < 0.5325 cm)
. ^{down} | KE
. ^{tot} | $ K E tot + I E tot$ . | $ K E u p / K E tot$ . |
---|---|---|---|---|---|

2D L5R090 | $ 2.27 \xd7 10 7$ | $ 2.96 \xd7 10 8$ | $ 3.19 \xd7 10 8$ | $ 1.26 \xd7 10 9$ | 7% |

3D L5R090 | $ 6.12 \xd7 10 6$ | $ 2.42 \xd7 10 8$ | $ 2.48 \xd7 10 8$ | $ 1.26 \xd7 10 9$ | 2% |

2D L6R720 | $ 7.58 \xd7 10 7$ | $ 1.70 \xd7 10 8$ | $ 2.46 \xd7 10 8$ | $ 1.29 \xd7 10 9$ | 31% |

3D L6R720 | $ 5.82 \xd7 10 7$ | $ 1.27 \xd7 10 8$ | $ 1.85 \xd7 10 8$ | $ 1.29 \xd7 10 9$ | 31% |

Run . | KE(y > 0.5325 cm)
. ^{up} | KE(y < 0.5325 cm)
. ^{down} | KE
. ^{tot} | $ K E tot + I E tot$ . | $ K E u p / K E tot$ . |
---|---|---|---|---|---|

2D L5R090 | $ 2.27 \xd7 10 7$ | $ 2.96 \xd7 10 8$ | $ 3.19 \xd7 10 8$ | $ 1.26 \xd7 10 9$ | 7% |

3D L5R090 | $ 6.12 \xd7 10 6$ | $ 2.42 \xd7 10 8$ | $ 2.48 \xd7 10 8$ | $ 1.26 \xd7 10 9$ | 2% |

2D L6R720 | $ 7.58 \xd7 10 7$ | $ 1.70 \xd7 10 8$ | $ 2.46 \xd7 10 8$ | $ 1.29 \xd7 10 9$ | 31% |

3D L6R720 | $ 5.82 \xd7 10 7$ | $ 1.27 \xd7 10 8$ | $ 1.85 \xd7 10 8$ | $ 1.29 \xd7 10 9$ | 31% |

Run . | $ G y u p$(y > 0.5325 cm) . | $ G y down$(y < 0.5325 cm) . | $ G y tot$ . | $ G y u p / G y tot$ . |
---|---|---|---|---|

2D L5R090 | 4.90 | 16.69 | 21.58 | 23% |

3D L5R090 | 2.10 | 14.84 | 16.94 | 12% |

2D L6R720 | 8.89 | 16.86 | 25.75 | 35% |

3D L6R720 | 6.29 | 12.71 | 19.01 | 33% |

Run . | $ G y u p$(y > 0.5325 cm) . | $ G y down$(y < 0.5325 cm) . | $ G y tot$ . | $ G y u p / G y tot$ . |
---|---|---|---|---|

2D L5R090 | 4.90 | 16.69 | 21.58 | 23% |

3D L5R090 | 2.10 | 14.84 | 16.94 | 12% |

2D L6R720 | 8.89 | 16.86 | 25.75 | 35% |

3D L6R720 | 6.29 | 12.71 | 19.01 | 33% |

*u*is the y-component of velocity,

_{y}*ρ*is mass density, and

*V*is the volume of the physical domain. As shown in Fig. 4, run L5.5R360 in 3D converges using 128 laser beams, while L6R720, which uses a larger beam radius, requires only 64 beams. A larger beam radius makes it easier to approximate azimuthal symmetry with fewer beams as expected. Considering the three factors discussed above, we use run L6R720 in the rest of the paper.

*t*= 3.5. Figure 5 plots kinetic energy KE, internal energy IE, and the total energy E

^{tot}being the sum of the KE and IE. Internal energy is also almost identical between 2D and 3D, differing by 2.1% at $ t = 3.5$ ns. To quantify the similarity between 3D quasi-azimuthal symmetry and 2D azimuthal symmetry, we take the standard deviation of the averaged kinetic energy along the azimuthal direction,

*θ*is the azimuthal angle, $ \u27e8 \cdots \u27e9 \theta $ is the domain average at an azimuthal coordinate,

*N*is the number of

*θ*bins, and the overbar denotes the average over all

*θ*. We find a percentage of 2.9% by comparing $ \u27e8 K E \u27e9 \theta std$ to the domain averaged kinetic energy, $ \u27e8 K E \u27e9$, at $ t = 3.5$ ns in 3D. This percentage quantifies the $ \u27e8 K E \u27e9$ variance in the azimuthal direction. Naturally, this variance is identically 0% for the 2D simulation.

### C. Jet traveling distance

All visualizations in the remainder of this paper are shown from run L6R720 at time $ t = 3.5$ ns, which is approximately when the jet reaches the bottom of the domain and are shown at the cross section in the *xy*-plane at *z* = 0. In Fig. 6, the jet exhibits turbulent structures in both 2D and 3D, especially at the leading edge, which we characterize using the subscale stress discussed in Sec. V. It is evident from the velocity profiles in Fig. 6 that the jet travels farther in 2D than 3D, despite having almost identical domain-integrated kinetic energy (Fig. 5). This indicates that KE resides at different scales between the two configurations, which is most evident at the leading edge of the jets. A shock that is reflected from the jet's leading edge, traveling toward the cone, can be seen as a discontinuity in the velocity at $ y \u2248 0.15 \u2009$ to $ \u2009 0.2$ cm in Fig. 6 (see also Fig. 2 in Ref. 53).

### D. Compressibility metrics

For completeness, we report on the compressibility levels of the flow using two quantities. First, in Fig. 9, we show the pointwise Mach number, $ M = | u | / c$, where $ | u |$ is magnitude of the local velocity in the lab frame and *c* is the local sound speed. The Mach number in 2D is generally greater than in 3D, especially in the pre-shocked region at $ y \u2248 0.15$– $ 0.3$ cm, which is consistent with the 2D jet being generally faster than in 3D. To quantify the differences in Fig. 9, we show the bulk Mach number, $ M = \u27e8 | u | \u27e9 / \u27e8 c \u27e9$, in Appendix Fig. 21(a), where $ \u27e8 \cdots \u27e9$ is the domain average. A difference of $ 12.1 %$ between 2D and 3D occurs at the end of the simulation ( $ t = 3.5$ ns).

Second, in Fig. 10, we calculate the time series of $ | \u2207 \xb7 u | rms$ and $ | \u2207 \xd7 u | rms$, which are more sensitive to the flow at smaller scales due to shocks and turbulence. Both $ | \u2207 \xb7 u | rms$ and $ | \u2207 \xd7 u | rms$ are greater in 2D than in 3D. However, Fig. 10(b) shows that $ | \u2207 \xd7 u | rms$ in 2D starts decreasing after $ 1$ ns, which we will show is due to an upscale vortical energy transfer discussed in Sec. VI A. In contrast, $ | \u2207 \xd7 u | rms$ in 3D increases over the entire duration of the simulation as seen from Fig. 10(b). From the ratio $ | \u2207 \xb7 u | rms / | \u2207 \xd7 u | rms$ [see the Appendix Fig. 21(b)], which is a measure of compressibility levels at small scales, we can infer that the 2D jet also exhibits higher small-scale compressibility levels than its 3D counterpart at $ t = 3.5$ ns.

## III. ANALYZING MULTISCALE DYNAMICS AND ENERGY TRANSFER

### A. Coarse graining

The coarse-graining approach allows for analyzing the dynamics at different scales in a complex flow, which has proven to be a natural and versatile framework to understand scale interactions.^{40,54–57} The approach is standard in partial differential equations and distribution theory.^{58,59} It became common in large eddy simulation (LES) modeling^{39} of turbulence, thanks to the foundational works of Leonard^{60} and Germano.^{61} Reference 62 provides an overview of coarse-graining and its connection to other methods in physics.

*n*-dimensions as

^{63}) $ G ( s )$ is normalized, $ \u222b d n s \u2009 G ( s ) = 1$. It is an even function such that $ \u222b s \u2009 G ( s ) \u2009 d n s = 0$, which ensures that local averaging is symmetric. $ G ( s )$ also has its main support (or variance) over a region of order unity in diameter, $ \u222b d n \u2009 s \u2009 | s | 2 \u2009 G ( s ) = O ( 1 )$. The dilated version of the kernel, $ G \u2113 ( r ) = \u2113 \u2212 n G ( r / \u2113 )$, is a function of dimensional position vector $r$ and inherits all those properties, except that its main support is over a region of diameter $\u2113$.

The scale decomposition in Eq. (4) is essentially a partitioning of scales for a field into large scales ( $ \u2273 \u2113$), captured by $ a \xaf \u2113$, and small scales ( $ \u2272 \u2113$), captured by the residual $ a \u2032 \u2113 = a \u2212 a \xaf \u2113$. In the following analyses, we use the Boxcar kernel in Eq. (6) for coarse-graining. With non-periodic boundary conditions, such as with our simulation domains here, filtering near the boundary requires a choice for the fields beyond the boundary. It was shown in previous works that a natural choice is to extend the domain beyond the physical boundaries with values compatible with the boundary conditions.^{64–66} However, over the times we analyze the jets in this work, flow across the domain boundaries is negligible. With consideration to the Helmholtz decomposition we use in Sec. VI and the ease of its implementation, we mirror (or reflect) the variables across the boundary when coarse-graining. In the Appendix Figs. 22–24, we show that mirroring has negligible effect on the results compared to extending the variables beyond the domain boundaries using the true boundary conditions.

### B. Mechanisms of energy transfer

^{34}Several different decompositions have been proposed in the literature, such as $ 1 2 \rho \xaf \u2113 | u \xaf \u2113 | 2$ (e.g., Refs. 67–70) and $ 1 2 | ( \rho u ) \xaf \u2113 | 2$ (e.g., Refs. 37 and 71–73). However, Zhao and Aluie

^{65}showed that these scale decompositions violate the “inviscid criterion,” which requires that viscous effect be negligible at sufficiently large scales. They also demonstrated that the Favre decomposition,

^{34,74}$ 1 2 \rho \xaf \u2113 | u \u0303 \u2113 | 2$, satisfies the inviscid criterion, where the Favre filtered velocity is density-weighted according to

^{34,75}

^{61}is

^{34,38}is

$ \Pi \u2113 ( x )$ and $ \Lambda \u2113 ( x )$ contain all information needed to quantify the exchange of energy between the two sets of scales, $ > \u2113$ and $ < \u2113$. Since we have complete knowledge of the dynamics at all scales resolved in a simulation, $ \Pi \u2113 ( x )$ and $ \Lambda \u2113 ( x )$ can be calculated exactly at every point $x$ in the domain and at any instant in time *t*. It is often not possible from simulations or observations to resolve all scales present in the real system. Therefore, computing $ \Pi \u2113 ( x )$ and $ \Lambda \u2113 ( x )$ is only measuring the dynamical coupling between scales present in the data.

In many instances, standard tools that were developed and used in the turbulence literature to the study of cross-scale transfer are only strictly valid to analyze homogeneous isotropic incompressible flows. Consequently, calculations of the energy transfer rates in HED hydrodynamic applications that use these tools may give ambiguous results for inhomogeneous flows, such as the jet in Fig. 6. The problem arises because there are several possible definitions for the scale-transfer terms, $ \Pi \u2113 ( x )$ and $ \Lambda \u2113 ( x )$, as we now elaborate.

Definitions (9) and (10) for the scale-transfer of energy in budget (8) are not unique with other definitions possible (see examples in Refs. 41 and 64) The difference between any two of these definitions is a divergence term, $ \u2207 \xb7 ( \cdots )$, which amounts to a reinterpretation of which terms in budget (8) represent transfer of energy across scales and which terms redistribute (or transport) energy in space, $ J \u2113 ( x )$. There is an infinite number of ways to reorganize terms in budget (8) and, thus, an infinite number of possible definitions for the transfer of kinetic energy between scales. This freedom in defining $ \Pi \u2113 ( x )$ can be thought of as a *gauge freedom.*^{64}

In a homogeneous flow, spatial averages of all these definitions are equal because their difference is a divergence that is zero, $ \u27e8 \u2207 \xb7 ( \cdots ) \u27e9 = 0$. On the other hand, if one considers inhomogeneous flows, such as the jet in Fig. 6, or if one wishes to analyze the cascade locally in space without spatial averaging, then such definitions can differ qualitatively as well as quantitatively (see Ref. 64 for examples). Definitions (9) and (10) are proper measures of the energy scale-transfer because they satisfy two important physical criteria: (i) Galilean invariance and (ii) vanish in the absence of subscale motion.^{76} Using such criteria to choose the scale-transfer definitions may be thought of as *gauge fixing*. Regarding the second criterion, both scale-transfer terms vanish identically at every location $x$ when $\u2113$ is the grid scale or smaller.^{76} The latter is a physically important constraint since scales smaller than $\u2113$ should not influence the larger scale flow if those scales do not exist.

^{77}corresponding to length scale $\u2113$,

## IV. ENERGY TRANSFER IN THE ENTIRE DOMAIN

We have already shown in Fig. 5 that KE in 2D and 3D is almost identical at *t* = 3.5 ns, differing by only 0.47%. However, the distribution of this KE among length scales differs significantly between 2D and 3D.

### A. Cumulative energy

Coarse KE in Eq. (8), $ \rho \u2113 | u \u0303 \u2113 | 2 / 2$, quantifies the cumulative KE at all scales larger than $\u2113$. Taking $ k \u2113 = L / \u2113 = 1$, Fig. 11 shows that KE in 2D at scales larger than $ L = 0.25$ cm exceeds that in 3D. This difference of $ 4.80 \xd7 10 9 \u2009$ erg/cm^{3} in KE content at large scales is highlighted in Table III, which is greater by more than an order of magnitude compared to the $ 2.96 \xd7 10 8 \u2009$ erg/cm^{3} (0.47% difference) in total KE. This is consistent with the bulk jet flow in 2D being faster than in 3D (Fig. 6). Indeed, Fig. 25 in the Appendix shows that most of this difference between 2D and 3D is from the y-component of KE, $ \rho \u2113 | ( u y ) \u0303 \u2113 | 2 / 2$, due to the flow in the axial direction. Figure 11 also shows that for larger $ k \u2113 = 2 , 4$ values, the difference between 2D and 3D diminishes. This is to be expected since increasing $ k \u2113$ is equivalent to including smaller scales in coarse KE metric, which converges to the total KE, $ \rho \u2113 | u \u0303 \u2113 | 2 / 2 \u2192 \rho | u | 2 / 2$ in the limit $ k \u2113 \u2192 \u221e$ or, equivalently, $ \u2113 \u2192 0$. We remind the reader that total KE is almost identical in 2D and 3D (Table III and Fig. 5).

Cumulative energy ( $ erg / cm 3 )$ . | $ k \u2113 = 1$ . | KE
. ^{tot} |
---|---|---|

$ K E 2 D$ | $ 4.09 \xd7 10 10$ | $ 6.89 \xd7 10 10$ |

$ K E 3 D$ | $ 3.61 \xd7 10 10$ | $ 6.86 \xd7 10 10$ |

$ K E 2 D \u2212 K E 3 D$ | $ 4.80 \xd7 10 9$ | $ 2.96 \xd7 10 8$ |

Cumulative energy ( $ erg / cm 3 )$ . | $ k \u2113 = 1$ . | KE
. ^{tot} |
---|---|---|

$ K E 2 D$ | $ 4.09 \xd7 10 10$ | $ 6.89 \xd7 10 10$ |

$ K E 3 D$ | $ 3.61 \xd7 10 10$ | $ 6.86 \xd7 10 10$ |

$ K E 2 D \u2212 K E 3 D$ | $ 4.80 \xd7 10 9$ | $ 2.96 \xd7 10 8$ |

### B. Kinetic energy transfer at selected scale

To explain the aforementioned large scale ( $ \u2113 > L = 0.25$ cm) KE difference between 2D and 3D, we analyze the KE scale-transfer, $ \Pi \u2113$ and $ \Lambda \u2113$ in Eqs. (9) and (10). Figure 12 shows the time series of domain-averaged $ \Pi \u2113$ and $ \Lambda \u2113$ evaluated at $ k \u2113 = L / \u2113 = 1$. In both 2D and 3D, we find $ \Pi < 0$ when domain-averaged, indicating a transfer of KE upscale from scales smaller than *L* to larger scales due to the ablative expansion and spread as the jet propagates. On the other hand, $ \Lambda > 0$ in both 2D and 3D when domain-averaged, indicating downscale transfer. As discussed in Sec. I and in Refs. 32 and 34, energy by baropycnal work Λ arises from the release (or storage, if $ \Lambda < 0$) of potential energy due to pressure gradients acting against density variations. Much of the pressure gradient arises near the ablation front as expected.

Table IV shows that the large scale KE difference between 2D and 3D can be attributed to differences in their KE scale-transfer. $ \Pi \u2113$ and $ \Lambda \u2113$ represent energy transfer *rates* (in erg/s/cm^{3}). Integrating $ \Pi \u2113$ and $ \Lambda \u2113$ over the duration of the simulation from *t* = 0 to 3.5 ns, we see in Table IV that there is $ 4.60 \xd7 10 9 \u2009$ erg/cm^{3} excess energy received by the large scales ( $ \u2113 > L = 0.25$ cm) in 2D compared to 3D. This value matches the large scale KE difference of $ 4.80 \xd7 10 9 \u2009$ erg/cm^{3} between 2D and 3D in Table III remarkably well,^{78} to within 4%. Since Π and Λ are the only terms that transfer KE across scales using our coarse-graining decomposition [Eq. (8)], we can infer that differences between 2D and 3D in the jet's bulk speed and the distance traveled (Fig. 6) are due to differences in KE scale-transfer. In the rest of this paper, we explore a physical origin for these differences in KE scale-transfer between 2D and 3D.

Energy transferred ( $ erg / cm 3 )$ . | $ \u222b \u27e8 \Pi \u27e9 d t$ . | $ \u222b \u27e8 \Lambda \u27e9 d t$ . | $ \u222b \u27e8 \Pi \u27e9 d t$ + $ \u222b \u27e8 \Lambda \u27e9 d t$ . |
---|---|---|---|

2D | $ \u2212 1.64 \xd7 10 10$ | $ 1.02 \xd7 10 10$ | $ \u2212 6.23 \xd7 10 9$ |

3D | $ \u2212 1.22 \xd7 10 10$ | $ 1.06 \xd7 10 10$ | $ \u2212 1.63 \xd7 10 9$ |

Difference: 2D–3D | $ \u2212 4.11 \xd7 10 9$ | $ \u2212 3.87 \xd7 10 8$ | $ \u2212 4.60 \xd7 10 9$ |

Energy transferred ( $ erg / cm 3 )$ . | $ \u222b \u27e8 \Pi \u27e9 d t$ . | $ \u222b \u27e8 \Lambda \u27e9 d t$ . | $ \u222b \u27e8 \Pi \u27e9 d t$ + $ \u222b \u27e8 \Lambda \u27e9 d t$ . |
---|---|---|---|

2D | $ \u2212 1.64 \xd7 10 10$ | $ 1.02 \xd7 10 10$ | $ \u2212 6.23 \xd7 10 9$ |

3D | $ \u2212 1.22 \xd7 10 10$ | $ 1.06 \xd7 10 10$ | $ \u2212 1.63 \xd7 10 9$ |

Difference: 2D–3D | $ \u2212 4.11 \xd7 10 9$ | $ \u2212 3.87 \xd7 10 8$ | $ \u2212 4.60 \xd7 10 9$ |

## V. SUBSCALE STRESS AND THE TURBULENT REGION

From Fig. 6, there are notable differences between the 2D and 3D jets near the leading edge, where the flow is suggestive of turbulence. To objectively quantify the turbulence intensity, it is traditional to apply a Reynolds decomposition,^{23} $ u i = \u27e8 u i \u27e9 + u i \u2032$, to separate the velocity field into a mean part $ \u27e8 u i \u27e9$ and a fluctuating component $ u i \u2032$, where $ \u27e8 \cdots \u27e9$ is an ensemble average. The Reynolds stress tensor, $ \u27e8 u i \u2032 u j \u2032 \u27e9$, is typically used to quantify the turbulent fluctuations.^{22,23,79}

^{80,81}It is possible to use the analog of the Reynolds stress $ \u27e8 u i \u2032 u j \u2032 \u27e9$ within the coarse-graining approach, which is the subscale stress (per unit mass) as shown by Germano,

^{61}

^{,}

^{61}defined in Eq. (11). The subscale stress quantifies the momentum flux contribution from subscales,

^{39,41}i.e., scales smaller than $\u2113$. Taking the length scale $ \u2113 = L / 2$ (i.e., $ k \u2113 = 2$), the subscale stress magnitude at 3.5 ns is shown in Fig. 13. Since the dominant velocity component is in the y-direction, the magnitude of $ \tau \xaf \u2113 ( u i , u j )$ in both 2D and 3D is predominantly from by the tensor component $ \tau \xaf \u2113 ( u y , u y )$. Comparing 2D and 3D in Fig. 13, we see that the high-intensity subscale stress occurs at the front of the jets but with a marked difference at the leading edge along the jet axis. The 2D jet has pronounced stress at the leading edge between approximately $ x = \u2212 0.05 \u2009$ and $ 0.05$ cm, which is absent in the 3D jet. The structures giving rise to this stress will be discussed in Sec. VI B. The stress morphology in Fig. 13 justifies a focus on the lower half of the domain (y < 0.375 cm, highlighted boxes in Fig. 13) in an attempt to glean insight into differences in the KE scale-transfer between 2D and 3D.

## VI. HELMHOLTZ DECOMPOSITION

Applying the Helmholtz decomposition to the velocity, $ u = u d + u s$, partitions it into a dilatational component, $ u d$, and a solenoidal component, $ u s$. Their curl-free ( $ \u2207 \xd7 u d = 0$) and divergence-free ( $ \u2207 \xb7 u s = 0$) properties^{82} allow us to explore the dominant flow components, such as shocks and vorticity, which may contribute to the jet traveling farther in 2D than in 3D. To perform the Helmholtz decomposition, the domain is mirrored to create periodic boundary conditions. While our choice for dealing with the boundary conditions by mirroring is not unique, we shall see that our results are physically meaningful. Dilatational and solenoidal specific kinetic energy (i.e., per unit mass), $ K d = u i d u i d$ and $ K s = u i s u i s$, respectively, are shown in Figs. 14 and 15.

For both the 2D and 3D jets, we can see from *K ^{d}* in Fig. 14 that there is a marked discontinuity (underscored by a horizontal black line in Fig. 14) at the same

*y*-location where the velocity is discontinuous in Fig. 6. To leading order, the structure of

*K*in Fig. 14 is similar between 2D and 3D aside from a difference in their

^{d}*y*-location.

Relative to *K ^{d}*, the solenoidal flow

*K*in Fig. 15 shows obvious differences between 2D and 3D. The highest intensity solenoidal flow in 2D is more collimated along the jet axis compared to that in 3D. There is also significant post-shock ( $ y \u2248 0$ to $ 0.13$ cm) solenoidal flow activity near the leading edge of the 2D jet, which suggests the presence of vorticity that is absent near the leading edge of the 3D jet. This suggests that differences in the KE scale-transfer are significantly influenced by the solenoidal component in the turbulent region near the leading edge, which we shall now analyze.

^{s}### A. Deformation and Baropycnal Work

Using the Helmholtz decomposition, we can investigate the contribution of the solenoidal and dilatational flow to Π and Λ, which we denote by $ \Pi sol , \u2009 \Pi dil , \u2009 \Lambda sol$, and $ \Lambda dil$. These are obtained by replacing $u$ with $ u s$ or $ u d$ in Eqs. (9) and (10), respectively. Note that $ \Pi sol + \Pi dil \u2260 \Pi $ since the decomposition also yields transfer terms involving both $ u s$ and $ u d$, which we do not analyze here and focus instead on the pure solenoidal behavior.

The opposing signs of $ \Pi sol$ between 2D and 3D in Fig. 16 show the qualitative contrast of the dynamics. In 3D, deformation work from the vortical flow transfers KE downscale, $ \u27e8 \Pi \u2113 sol \u27e9 > 0$, whereas it transfers KE upscale in 2D, which sustains large-scale coherent vortical structures near the leading edge of the 2D jet as we shall see in Figs. 17–19.

The upscale transfer of KE in 2D by $ \Pi sol$ in Fig. 16 is similar to the upscale cascade seen in 2D variable density Rayleigh-Taylor turbulence.^{32} It is also similar to 2D constant-density turbulence^{30,31,83} due to the absence of vortex stretching and strain self-amplification.^{28} Even though the magnitudes of $ \Pi sol$ and $ \Lambda sol$ in Fig. 16 are smaller than total Π and Λ over the entire domain in Fig. 12, the vorticity dynamics at the leading edge can have a disproportionate effect on the jet's traveling distance and bulk flow evolution as we shall discuss below.

### B. Circulation mechanism

To understand the mechanistic cause of differences in $ \Pi sol$ between 2D and 3D, we probe its tensorial components. Figure 17 visualizes $ \Pi \u2113 sol$ at $ k \u2113 = 2$. We can see obvious differences between 2D and 3D in the sign of $ \Pi \u2113 sol$ at the leading edge of the jet, which seem to correlate with differences seen in Fig. 15. As defined in Eq. (9), $ \Pi sol$ is a contraction between the velocity gradient tensor, $ \u2212 \u2202 j ( u s \u0303 ) i$, and the subscale stress tensor, $ \rho \xaf \tau \u0303 ( u i s , u j s )$. To understand the sign of $ \Pi sol$, we focus on the tensor components $ i , j = 2$ (*y*-components), which make the dominant contribution to the spatially averaged $ \Pi sol$.

The subscale stress component $ \rho \xaf \tau \u0303 ( u y s , u y s )$ is mathematically guaranteed to be positive semi-definite at every point in the domain.^{34,77,84} This is shown in Fig. 18. Therefore, the sign of $ \Pi \u2113 sol$ from the *y*-components is the same as that of $ \u2212 \u2202 y ( u s \u0303 ) y$. Figure 18 visualizes $ \u2202 y ( u s \u0303 ) y$ with red/blue indicating positive/negative values, implying upscale/downscale KE transfer ( $ \Pi \u2113 sol$ from $ i , j = 2$ components is negative/positive). Overlapping the colormap in Fig. 18 is a gray-scale map of $ \tau \u0303 ( u y s , u y s )$, which, being positive semi-definite, highlights regions of strong sub-scale stress and, therefore, strong scale-transfer.

Both panels of Fig. 18 at $ y > 0.15$ cm are roughly similar to leading order in regions of strong stress. In those highlighted regions at the jet flanks at $ y > 0.15$ cm, we see that $ \u2202 y ( u s \u0303 ) y$ is mostly negative, implying downscale transfer in both 2D and 3D. The main differences are seen at the flanks of the 2D jet's leading edge, $ | x | = 0.03$ to $ 0.05$ cm and $ y < 0.1$ cm, where $ \u2202 y ( u s \u0303 ) y$ is mostly positive, implying an upscale transfer in 2D that is absent in 3D.

At the flanks of the leading edge ( $ | x | = 0.03$ to $ 0.05$ cm) in 2D, we see in Fig. 18 that $ \u2202 y ( u s \u0303 ) y$ switches sign along the y-direction between $ y \u2248 0.05 \u2009$ and $ 0.1$ cm. This is suggestive of vortical motion. Figure 19 visualizes the flow streamlines superposed over $ \u2202 y ( u s \u0303 ) y$. From the streamlines in Fig. 19, we can see clearly two coherent vortical structures at the leading edge of the 2D jet, which are absent from the 3D jet. The streamlines in Fig. 19 are consistent with observations we made above that the flow at $ y > 0.15$ cm is roughly similar between 2D and 3D (see also Fig. 29 in Appendix). The main difference is at $ y < 0.15$ cm, where the coherent roll-up structure in the 2D jet at $ y \u2248 0.05$ cm is co-located with the upscale energy transfer seen at that same location in Fig. 17. This is indicative of the coherent roll-up being energized by the smaller scale turbulence created post-shock. Such an upscale scale-transfer is absent in 3D, which cannot sustain the coherent roll-up. Vorticity within this roll-up creates an effective velocity drift^{84} that helps propel the 2D jet farther and keep it collimated relative to its 3D counterpart.

## VII. CONCLUDING REMARKS

In this paper, we demonstrated a methodology for diagnosing the multiscale dynamics and energy transfer in complex HED flows with realistic driving and boundary conditions. While it is well known that 2D modeling underestimates the proneness of the flow to instabilities, what we have shown here is that 2D modeling also suffers from significant spurious energization of the bulk flow by instabilities. The energization of the roll-up structures via an upscale transfer from smaller scale turbulence in Figs. 17–19 brings to the fore some of the hydrodynamics artifacts associated with 2D modeling, reinforcing recent findings.^{32}

In HED applications such as ICF, 2D simulations remain to be the main “work horse” for experimental design^{17–19} as routine 3D simulations are prohibitively expensive.^{3} Our hope from this work is to highlight the tradeoff between 2D and 3D flow physics, which may not be as widely appreciated as tradeoffs from approximating other system components such as the laser drive or the hohlraum. While computational realities may prevent the community from routinely conducting 3D hydrodynamic modeling in the foreseeable future, we hope that this work (see also Ref. 32) highlights the need to alleviate some of the hydrodynamic artifacts associated with 2D models.

By demonstrating the applicability of coarse-graining for comparing the multiscale dynamics and energy transfer between 2D and 3D, we have shown that this methodology can help with inter-model comparison and validation. We believe that future attempts at alleviating some of the 2D hydrodynamic artifacts would have to use this approach, at least in some fashion, for testing and model development. After all, the approach has plenty in common with large eddy simulation modeling, which is a well-established field in fluid dynamics.^{39}

## ACKNOWLEDGMENTS

This work was supported by CMAP, an NSF Physics Frontiers Center, under Grant No. PHY-2020249. Partial support from Grant Nos. DE-SC0020229, DE-SC0019329, and CBET-2143702 is also acknowledged. H.A. was also supported by US DOE Grant Nos. DE-SC0014318, DE-SC0019329, US NSF Grant Nos. OCE-2123496, PHY-2206380, US NASA Grant No. 80NSSC18K0772, and US NNSA Grant Nos. DE-NA0003856, DE-NA0003914, DE-NA0004134. J.S. was supported by Nos. DE-SC0019329, DE-NA0003914, and DE-NA0004134. Computing time was provided by NERSC under Contract No. DE-AC02–05CH11231.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Hao Yin:** Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (lead); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). **Jessica K. Shang:** Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (lead); Writing – review & editing (equal). **Eric G. Blackman:** Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). **Gilbert Collins:** Funding acquisition (lead); Project administration (lead); Resources (equal); Writing – review & editing (equal). **Hussein Aluie:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (lead); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

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

### APPENDIX: SUPPLEMENTARY FIGURES

Figure 20 shows that the average traveling speed converges at refinement level 5, which is chosen for the runtime parameter stated in simulation configurations in Sec. II A.

Figure 21 provides two compressibility metrics supplemental to Sec. II D, where the 2D jet presents higher compressibility levels at the end of the simulation ( $ t = 3.5$ ns). Figure 21(a) is the bulk Mach number that compares the visualization of the Mach number in Fig. 9. A difference of $ 12.1 %$ between 2D and 3D occurs at the end of the simulation ( $ t = 3.5$ ns). Figure 21(b) is the root mean square (rms) ratio of divergence of velocity [Fig. 10(a)] to curl of velocity [Fig. 10(b)]. The ratios for 2D and 3D are both below one implying the significance of the divergence-free part of the flow even with the strong shock in the domain.

Figures 22–24 compare the coarse-grained fields with two boundary conditions: zero-gradient boundary and mirrored boundary. Figure 22 visualizes the coarse y-velocity, and Fig. 23 visualizes the y-gradient of coarse y-velocity. Figure 24 compares a lineout for Figs. 22 and 23 and demonstrates the differences between the two boundary conditions are negligible. The mirrored boundary is selected to be the boundary condition for the simulations described in Sec. III A.

Figure 25 shows the evolution of dominant component (y-component) of the kinetic energy at various scales. There is more kinetic energy in 2D at large scales compare to 3D, which is consistent with the evolution of total kinetic energy in Fig. 11.

Figure 26 provides that, at *t* = 3.5 ns, $ \Pi \u2113$ transfers energy upscale and $ \Lambda \u2113$ mainly transfers energy downscale as discussed in Sec. IV B.

Figure 27 shows the deformation work and baropycnal work due to the dilatational flow at *t* = 3.5 ns. Energy transfer directions are the same in 2D and 3D for both $ \u27e8 \Pi \u2113 dil \u27e9$ and $ \u27e8 \Lambda \u2113 dil \u27e9$, which follows the dynamics in homogeneous compressible turbulence.^{42}

Figure 28 shows the components of deformation work, $ \u27e8 \Pi \u2113 \u27e9$, calculated with cross terms at *t* = 3.5 ns. All components transfer energy downscale in both 2D and 3D, highlighting the contrast of dynamics of $ \u27e8 \Pi sol \u27e9$ discussed in Sec. VI A.

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*A First Course in Turbulence*

*Turbulent Flows*

*Flash User's Guide*

*A Guide to Distribution Theory and Fourier Transforms*

*Ten Lectures on Wavelets*

^{3}) and Table IV ( $\n4.60\n\xd7\n\n\n10\n9\n\u2009$ erg/cm

^{3}) is, at least in part, due to temporal discretization errors when calculating the integrals $\n\n\u222b\n\Gamma \n.$

*On Helmholtz's Theorem in Finite Regions*