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Question: How does the specific-heat ratio affect the average motion of the bubble?

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Context 1: Paper Info

Title: Specific-heat ratio effects on the interaction between shock wave and heavy-cylindrical bubble: based on discrete Boltzmann method
Publish Date: May 29, 2023
Author List: Yanbiao Gan (from School of Liberal Arts and Sciences, Hebei Key Laboratory of Trans-Media Aerial Underwater Vehicle, North China Institute of Aerospace Engineering), Yudong Zhang (from School of Mechanics and Safety Engineering, Zhengzhou University)

Figure

Figure 1: Research orientation and tasks of DBM 
Figure 2: Sketch of D2V16 model The numbers in the figure represent the index i in Eq  (3) 
Figure 3: The computational configuration of the shock-bubble interaction 
In the figure, results from odd rows are experimental, and the even rows indicate DBM simulation results The typical wave patterns and bubble's main characteristic structures are marked out in the figures Numbers in the pictures represent the time in µs Schlieren images of DBM results are calculated from the density gradient formula, i e , |∇ρ|/|∇ρ| max , with |∇ρ| = (∂ ρ/∂ x) 2 + (∂ ρ/∂ y)2  At t = 0µs, the incident shock wave impacts the upstream interface, and subsequently generates a transmitted shock (TS) propagating downstream in the bubble and a reflected shock wave moving upward in ambient gas The incident shock wave travels downstream contin-
The definitions and the corresponding physical meanings of the common TNE quantities in DBM, where the operator ∑ ix,iy indicates integrating over all the fluid units and multiply the unit area dxdy From a certain perspective, the TNE strength is increasing; While from a different perspective, the TNE strength, on the other hand, may be decreasing It is one of the concrete manifestations of the complexity of non-equilibrium flow behavior 
Figure 4: Snapshots of schlieren images of the interaction between a shock wave and a heavy-cylindrical bubble The odd rows represent experimental results from Ref  [31] with permission, and the even rows are DBM simulation results The typical wave patterns and the bubble's main characteristic structure are marked out in the figures Numbers in the picture represent the time in µs 
Figure 5: The temporal variations of length and width of the bubble The symbols represent DBM results and the lines are experimental The definition of the length and the width of the bubble can be seen in the illustration Experimental results are obtained from Fig  12, in Ref  [31] with permission 
Figure 6: Density contours and particle tracer images at three different moments (i e , t = 0 07,t = 0 11, and t = 0 16) with various specific-heat ratios The odd rows represent density contours, and the even rows are tracer particle images 
Figure 9: Vorticity contours at t = 0 134, with various specific-heat ratios The arrows in the vorticity image point out the apparent difference between case γ = 1 4 and case γ = 1 09 
Figure 11: Density contours (first row) and mixing degree M (second row) at several typical moments 
Figure 13: (a) Temporal evolution of D * 3,1 and D * 4,2  (b) Temporal evolution of D * 2 and D * 3  Lines with different colors represent the cases with various specificheat ratios 
The development of schemes for checking TNE state, extracting TNE information and describing corresponding TNE effects in DBM 

abstract

Specific-heat ratio effects on the interaction between a planar shock wave and a two-dimensional heavy-cylindrical bubble are studied by the discrete Boltzmann method  Snapshots of schlieren images and evolutions of characteristic scales, being consistent with experiments, are obtained  The specific-heat ratio effects on some relevant dynamic behaviors such as the bubble shape, deformation process, average motion, vortex motion, mixing degree of the fluid system are carefully studied, as well as the related Thermodynamic Non-Equilibriums (TNE) behaviors including the TNE strength, entropy production rate of the system 
Specifically, it is found that the influence of specific-heat ratio on the entropy production contributed by non-organized energy flux (NOEF) is more significant than that caused by non-organized momentum flux (NOMF)  Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary 
The effects of specific-heat ratio on various TNE quantities show interesting differences  These differences consistently show the complexity of TNE flows which is still far from clear understanding 

Introduction

The applications of shock-accelerated inhomogeneous flows (SAIFs) are of significant value in biomedicine, energy utilization, and astrophysics fields, including but not limited to scenarios such as the impact of shock waves on kidney stones, the interaction between shock waves with foams, the impacting of detonation wave with burning flames in supersonic combustion systems, the formation of supernova remnants, etc  
Shock-bubble interaction (SBI) is one of the most fundamental problems in the research of SAIFs  Its applications and academic research are interdisciplinary  Generally, there are two kinds of problems encountered in SBI research: (i) The geometry of shock waves, the shape of material interfaces, and the structure of container are complex in the actual scene 
They will result in various wave patterns and significantly affect the flow morphology and bubble's evolution  (ii) There usually exist multi-physics coupling problem in the engineering application of SBI  Such as the supersonic combustion machines  When the shock waves passing through the reactants, it may lead to phase transition and chemical reactions, making the flow morphology more complex and inducing small structure (or fast-changing pattern)  
In an underwater explosion experiment, the interaction between shock waves and bubbles may refer to the cavitation and annihilation effects  The other scene is the inertial confinement fusion (ICF), in which the laser ablation, electron heat conduction, self-generated electromagnetic field, radiation, and many other factors may complicate the investigation of hydrodynamic instabilities  
Commonly, research on SBI mainly includes three methods: theoretical derivation, experiment, and numerical simulation  As a fundamental research method, theoretical research can provide a clear understanding of physical processes  In 1960, Rudinger et al  developed a theory that permits computing the response of bubbles to accelerations  
In order to describe the formation and evolution processes of vortex structure quantitatively, many scholars have developed circulation models   However, theoretical works provide limited information  Meanwhile, in the late stage of SBI evolution, the bubble deformation and flow morphology dominated by the developed Richtmyer-Meshkov instability (RMI) and Kelvin-Helmholtz instability (KHI) are difficult to be predicted accu-rately 
As the research method closest to engineering application, the experimental results are often regarded as standard results to verify the rationality and accuracy of theoretical and numerical works  To study the SBI process accurately, the scholars have made a series of improvements to experimental equipment or technique, including the generation techniques of different types of shock waves, interface formation methods, schlieren facilities, and image recognition techniques  
Among these, two of important and valuable works are performed by Ding et al   Based on the soap film technique, they formed kinds of initial interfaces with different curvatures through the wire-restriction method and captured the wave patterns and interface evolution with high-speed schlieren photography  
Other works, such as evolutions of a spherical gas interface under reshock conditions , developments of a membrane-less SF 6 gas cylinder under reshock conditions , and interactions of a cylindrical converging shock wave with an initially perturbed gaseous interface , are also performed by many other scholars 
However, we know that the experimental studies mainly depend on the experimental platform  When studying some complex and demanding condition problems, it takes much work to build the experimental platform  In this situation, numerical simulation research becomes an option  Generally, there are three kinds of physical modeling methods (or models) for SBI numerical research, i e , the macroscopic, mesoscopic, and microscopic modeling methods 
Most of the existing numerical researches on SBI are related to the macroscopic modeling methods (such as the Euler and Navier-Stokes (NS) models) based on the continuous hypothesis (or equilibrium and nearequilibrium hypothesis)   For example, presented the computational results on the evolution of the shock-accelerated heavy bubbles through the multi-fluid Eulerian equation  
There also exist a few SBI works based on the mesoscopic modeling method, such as the Direct Simulation Monte Carlo method   The microscopic modeling methods such as the Molecular dynamics (MD) simulation, is capable of capturing much more flow behaviors but restricted to smaller spatiotemporal scales because of its huge computing costs 
In the numerical research on SBI, three points need to be concerned  (i) Investigation of kinetic modeling that describes the non-continuity/non-equilibrium flows  Most of the current researches are based on macroscopic models  However, there exist abundant small structure (and fast-changing patterns) behaviors and effects such as the shock wave, boundary layer, material defects, etc 
For cases with small structures, the mean free path of molecules cannot be ignored compared to the characteristic length, i e , the non-continuity (discreteness) of the system is pronounced, which challenge the rationality and physical function of the macroscopic models based on the continuity hypothesis 
For cases with fast-changing patterns, the system dose not have enough time to relax to the thermodynamic equilibrium state, i e , the system may significantly deviate from the thermodynamic equilibrium state  Therefore, the rational-ity and physical function of the macroscopic models based on the hypothesis of thermodynamic equilibrium (or near thermodynamic equilibrium) will be challenged 
(ii) Improvement of method that describes the evolution characteristics of bubbles and flows morphology  Most of the studies describe bubble characteristics and flows morphology from a macroscopic view  The mesoscopic characteristics such as the kinetic effects which help understand the kinetic process, are rarely to be studied 
(iii) Further studies of effects of specific-heat ratio on SBI process  The specific-heat ratio is an essential index for studying the compressibility of the gas  Research from Igra et al  has shown that the differences in the specific-heat ratio of bubbles would cause various wave patterns and pressure distribution inside the bubbles during the interaction process  
Besides, many works on hydrodynamic instability have also demonstrated the importance of investigating the specific-heat ratio effect   Among these, Chen et al  investigated the specific-heat ratio effects on temperature gradient and the TNE characteristics of compressible Rayleigh-Taylor (RT) system  
For the above three points, in this work we apply the recently proposed discrete Boltzmann method (DBM)   The Lattice Boltzmann Method (LBM) research has two complementary branches   One aims to work as a kind of new scheme for numerical solving various partial differential equation(s)  The other aims to work as a kind of new method for constructing kinetic model to bridge the macro and micro descriptions 
The two branches have different goals and consequently have different rules  The current DBM is developed from the second branch of LBM and focusing more on the Thermodynamic Non-Equilibrium (TNE) behaviors that the macro modeling generally ignore  It breaks through the continuity and near-equilibrium assumptions of traditional fluid modeling, discards the lattice gas image of standard LBM, and adds various methods based on phase space for checking, exhibiting, describing and analyzing the non-equilibrium state and resulting effects 
More information extraction technologies and analysis methods for complex field are introduced with time  The numerical simulation includes three parts, as shown in Fig    (1) Physical modelling, (2) Algorithm design, (3) Numerical experiments and analysis of complex physical fields  The research of equation algorithm corresponds to the part (2) of the above three parts 
The DBM aims at parts (1) and (3) of the three mentioned above  It belongs to a physical model construction method rather than a numerical solution for the equations  The tasks of DBM are to: (i) Ensure the rationality of the physical model (theoretical model) and balance the simplicity for the problem to be studied; (ii) Try to extract more valuable physical information from massive data and complex physical fields 
Based on the coarse-grained modeling method of nonequilibrium statistical physics, the DBM aims to solve the following dilemma: (i) The traditional hydrodynamic modelings are based on the continuous hypothesis (or near-equilibrium hypothesis)  They only concern the evolution of three conserved kinetic moments of the distribution function, i e  the density, momentum and energy, so their physical functions are insufficient 
(ii) The situation that the MD can be used is restricted to too small spatial-temporal scales  The physical requirement for the modeling is that except for the Hydrodynamic Non-Equilibriums (HNE), the most related TNE are also needed to be captured  Theoretically, the Boltzmann equation is suitable for all-regime flows, including the continuum regime, slip regime, transition regime, and free molecule flow regime 
Based on the Chapman-Enskog (CE) multiscale analysis , through retaining various orders of Kn number (or considering different orders of TNE effects), the Boltzmann equation can be reduced to the various orders of hydrodynamic equations  They can be used to describe the hydrodynamic behaviors, i e , the conservations of mass, momentum and energy , in corresponding flow regimes 
Because what the traditional hydrodynamic equations describe are only the conservation laws of mass, momentum and energy  Consequently, it should be pointed out that, the information lost in the traditional hydrodynamic equations increases sharply with increasing the Kn number  With increasing the Kn number, to ensure the describing capability not to decrease significantly, the more appropriate hydrodynamic equations should be the Extended Hydrodynamic Equations (EHEs) which include not only the evolution equations of conserved kinetic moments but also the most relevant nonconserved kinetic moments of distribution function 
For convenience of description we refer the modeling method that derives EHEs from the fundamental kinetic equation to Kinetic Macroscopic Modeling (KMM) method  It is clear that, the complex process of CE expansion is necessary and the simulation is still based on the macroscopic equations in KMM  As a comparison, the DBM is a kind of Kinetic Direct Modeling (KDM) method 
In DBM modeling, the CE analysis is only used to quickly determine which kinetic moments should keep values unchanged, the final EHEs are not needed, and the simulation is not based on the complicated EHEs  As the TNE degree of the flow to be described rises gradually, the complexity of the derivation process and difficulty of numerical simulation in the KMM method increase sharply 
However, in the DBM method, to describe flows in a one-order more deeper depth of TNE, only two more related kinetic moments need to be added  Since without needing to derive and solve the EHEs, as the TNE degree deepens, the complexity of the DBM approach increases much slower than that of KMM method 
The core step in DBM modeling is to provide a feasible scheme for detecting, describing, presenting, and analyzing TNE effects and behaviors beyond traditional macroscopic modeling  Based on the non-equilibrium statistical physics, we can use the non-conservative moments of ( f − f eq ) to describe how and how much the system deviates from the thermodynamic equilibrium state and to check corresponding effects due to deviating from the thermodynamic equilibrium 
The non-conservative moments of ( f − f eq ) open a phase space, and this space and its subspaces provide an intuitive geometric correspondence for describing complex TNE system properties  The development of schemes for checking TNE state, extracting TNE information and describing corresponding TNE effects in DBM are seen in Table  
Actually, this set of TNE describing methods has been applied in many kinds of complex fluid systems such as hydrodynamic instability system , combustion and detonation systems , multiphase flow system , plasma system , etc  Besides the scheme for detecting, describing, presenting, and analyzing TNE effects and behaviors, the DBM incorporates other methods for analyzing the complex physical field 
One of them is the tracer particle method  The introduction of the tracer particle method makes the gradually blurred interface appear clearly   The rest of the paper is structured as follows  Section 2 shows Year Scheme for investigating TNE effects and behaviors Before 2012 Two classes of LBMs did not show a significant difference in physical function 
2012 Use the non-conservative moments of ( f − f eq ) to check and describe TNE   This is the starting point of current DBM approach  2015 Open TNE phase space based on non-conservative moments of ( f − f eq ) and define a TNE strength using the distance from a state point to the origin  This is the starting point of the phase space description method  
2018 Extend the distance concepts in phase space to describe the TNE difference/similarity of TNE states and kinetic processes   2021 Further extend the phase space description methodology to any set of system characteristics   the modeling method  Then, the numerical simulations and results are presented in Section 3, which includes two subsections 
Section 4 concludes the paper  Other complementary information is given in the Appendix 

Model construction

Based on the Bhatnagar-Gross-Krook (BGK) singlerelaxation model, a two-fluid DBM with a flexible specific-heat ratio is presented in this part  From the origin Boltzmann equation to a DBM, four fundamental steps are needed: (i) Simplification and modification of the Boltzmann equation according to the research requirement 
(ii) Discretization of the particle velocity space under the condition that the reserved kinetic moments keep their values unchanged  (iii) Checking the TNE state and extracting TNE information  (iv) The selection/design of the boundary conditions 

Simplification and modification of the Boltzmann equation

As we know, the collision term in the original Boltzmann contains high dimensional distribution functions  Therefore, the direct solution to it needs too much computing consumption  The most common method to simplify the collision operator is to introduce a local equilibrium distribution function ( f eq ) and write the complex collision operator in a linearized form, i e , the original BGK collision operator − 1 τ ( f − f eq ), where τ is the relaxation time  
The original BGK operator describes the situation where the system is always in the quasi-equilibrium state  Namely, it characterizes only the situation where the Kn number of the system is small enough and f ≈ f eq   The currently used BGK operator for non-equilibrium flows in the field is a modified version incorporating the meanfield theory description  
Based on the above considerations, the simplified Boltzmann equation describing the SBI process is where the two-dimensional equilibrium distribution function is ) where ρ, T , v, u, I, R, and η are the mass density, temperature, particle velocity vector, flow velocity vector, the number of the extra degrees of freedom including molecular rotation and vibration inside the molecules, gas constant, and a free parameter that describes the energy of the extra degrees of freedom, respectively 
The specific-heat ratio is flexible by adjusting parameter I, i e , γ = (D + I + 2)/(D + I), where D = 2 represents the two-dimensional space 

Discretization of the particle velocity space and determination of f σ ,eq i

The continuous Boltzmann equation should be discretized for simulating  Specifically, the continuous velocity space can be replaced by a limited number of particle velocities  So that the values of continuous kinetic moments can be obtained from the summation form of kinetic moments  In this process, it requires the reserved kinetic moments, which are used to characterize the system behaviors, keep their values unchanged after discretizing the velocity space 
Namely, f the reserved kinetic moments  According to the CE analysis, f can be expressed by f eq   Therefore, in the process of discretization, the reserved kinetic moments of f eq should keep their values unchanged, i e , where i represents the kind of discrete velocities and α (α = x or y) is the direction in cartesian coordinate 
To simulate the interaction between two different fluids, a two-fluid DBM should be constructed  Based on the singlerelaxation model, the discrete two-fluid Boltzmann equation can be written as : where σ represents the types of material particle and f σ ,eq i = f σ ,eq i (ρ σ , u, T)  In two-fluid DBM, the macroscopic quantities of the mixture and each component are
where ρ σ and u σ are the mass density and flow velocity of the component σ , respectively  ρ and u represent the mass density and flow velocity of the mixture, respectively  There exist two kinds of temperature (internal energy) definitions in two-fluid DBM because the definition of temperature (internal energy) depends on the flow velocity to be chosen as a reference 
The first definition is by choosing the velocity of the mixture to be a reference, i e , So that the expressions of temperature of component σ and mixture are where We can also choose the flow velocity of component as a reference, i e , , where u σ is the flow velocity of component σ   The corresponding definitions of temperature for component σ and the mixture are
where ∆E * I is It is clear to see that these two definitions of temperature for mixture are the same, but for temperature of component σ are different  We choose the first definition in this manuscript  To solve the Eq  ( ), it is necessary to determine the values of f σ ,eq i   Its values depend on the reserved kinetic moments which characterize the main system behaviors 
In DBM modeling, the CE multiscale analysis is used to determine quickly the reserved kinetic moments  Specifically, when constructing a DBM which only the first order term of Kn number is retained (i e , only the first order TNE effects are retained), seven kinetic moments should be reserved, i e , the M 0 , M 1 , M 2,0 , M 2 , M 3,1 , M 3 , M 4,2  
Two more kinetic moments ( M 4 and M 5,3 ) are needed when the second order TNE is considered   However, it should be noted that the function of CE analysis in DBM modeling is only to determine the kinetic moments that need to be preserved  Whether or not to derive the hydrodynamic equations does not affect the DBM simulation 
The kinetic moments used in our physical modeling are shown in the Appendix B  Their expressions can be obtained by integrating v and η with continuous-form f eq   For better understanding, the Appendix C gives the two-fluid hydrodynamic equations recovered from the Boltzmann equation  The kinetic moments in Appendix B can be written in matrix form, i e , C • f σ ,eq = fσ,eq , (
where C is the matrix of discrete velocity and feq represents the kinetic moments  A proper discrete velocity model is needed to confirm the values of f σ ,eq i   The f σ ,eq can be obtained by solving the inverse matrix, i e , f σ ,eq = C −1 • fσ,eq , where C −1 is the inverse matrix of C  It is very convenient to obtain the inverse matrix of C through some mathematical softwares such as Mathematica, etc 
The D2V16 model is chosen in this paper, its sketches can be seen in Fig    The specific values of D2V16 are given in the following equations: where "cyc" indicates cyclic permutation and c is an adjustable parameter of the discrete velocity model  The sketch of η in D2V16 is η i = η 0 for i = 1 − 4, and η i = 0 for i = 5 − 16 

Checking the TNE state and extracting TNE information

Many physical quantities can characterize the degree of TNE in a fluid system, such as relaxation time, Kn number, viscosity, heat conduction, the gradients of macroscopic quantity, etc  They are all helpful to characterize the TNE strength and describe the TNE behaviors of a fluid system from their perspectives 
But it is not enough only relying on these quantities  Besides the above physical quantities describing the TNE behaviors, in DBM modeling, we can also use the non-conservative moments of ( f − f eq ) to characterize the TNE state and extract TNE information from the fluid system  Fundamentally, four TNE quantities can be defined in a firstorder DBM, i e , ∆ σ * 2 , ∆ σ * 3,1 , ∆ σ * 3 , and ∆ σ * 4,2  
Their definitions can be seen in Table , where v * i = v i − u represents the central velocity and u is the macro flow velocity of the mixture  Physically, ∆ σ * 2 = ∆ σ * 2,αβ e α e β and ∆ σ * 3,1 = ∆ σ * 3,1 e α represent the viscous stress tensor (or non-organized momentum flux, NOMF) and heat flux tensor (or non-organized energy flux, NOEF), respectively 
The e α (e β ) is the unit vector in the α (β ) direction  The later two higher-order TNE quantities contain more condensed information  Specifically, and it indicates the flux information of ∆ σ * 2   To describe the TNE strength of the whole fluid system, some TNE quantities contained more condensed information are also defined, i e ,
Other TNE quantities can be defined based on specific requirements  All the independent components of TNE characteristic quantities open a highdimensional phase space, and this space and its subspaces provide an intuitive image for characterizing the TNE state and understanding TNE behaviors   It should be emphasized that: (i) The TNE strength/intensity/degree is the most basic parameter of non-equilibrium flow description; And any definition of non-equilibrium strength/intensity/degree depends on the research perspective 
(ii) The physical meaning of D * m,n is the TNE strength of this perspective  (iii) From a certain perspective, the TNE strength is increasing; While from a different perspective, the TNE strength, on the other hand, may be decreasing  It is normal, one of the concrete manifestations of the complexity of non-equilibrium flow behavior 
Strictly speaking, those TNE intensity and effect descriptions that do not account for the research perspective are not correct  Do not explain the research perspective, the corresponding is not dependent on the research perspective 

Numerical simulations and results

In this section, we first validate the DBM code by comparing the DBM results with experimental results  Then, the effects of specific-heat ratio on the dynamic process and TNE behaviors on SBI are investigated 

Comparison with experimental results

In the following part, we use a first-order two-fluid DBM to simulate the interaction between a planar shock wave with a 2-D heavy-cylindrical bubbles, and compare the DBM results with the experimental results from Ref    The computational configuration can be seen in Fig    In a flow field which is filled with Air, there is a static bubble composed of 26% Air and 74% SF 6  
A shock with Ma = 1 2 would pass through the bubble from left to right  The initial conditions of ambient gas are ρ 0 = 1 29kg/m 3 , T 0 = 293K, p 0 = 101 3kPa  Ignoring the pressure difference between interior gas and ambient gas, the initial parameters of the bubble are ρ bubble = 4 859kg/m 3 , p bubble = 101 3kPa,
and T 0 = 293K  For simulating, these actual physical quantities should be transferred to dimensionless parameters  This process can refer to the Appendix A  The dimensionless conditions of macroscopic quantities of the fluid field in initial time are (ρ, T, u x , u y ) bubble = (4 0347, 1 0, 0 0, 0 0), (ρ, T, u x , u y ) 1 = (1 3416,
1 128, 0 3616, 0 0), (ρ, T, u x , u y ) 0 = (1 0, 1 0, 0 0, 0 0), where the subscript "0" ("1") represents downstream (upstream) region  In two-fluid DBM code, the distribution function f Air is used to describe the ambient gas, i e , Air  The f bubble characters the bubble which is a mixture that composed of Air and SF 6  
The grid number is N x × N y = 800 × 400, where the N x and N y are grid number in x and y direction, respectively  This grid size has passed the mesh convergence test  The below results also show that it is sufficient to meet the requirements of the following research problem  Other parameters used for the simulation are: c = 1 0, η Air = η bubble = 10 0,
I Air = 3, I bubble = 15, ∆x = ∆y = 1 2 × 10 −4 and ∆t = 1 × 10 −6   The viscosity effect is feeble compared to the shock compression effect, so it does not significantly affect the deformation of the bubble  Therefore, in this part, the relaxation time τ is set sufficiently small  The inflow (outflow) boundary condition is used in the left (right) boundary, and the periodic boundary is adopted in the y direction 
The first-order forward difference scheme is used to calculate the temporal derivative, and the second-order nonoscillatory nonfree dissipative scheme is adopted to solve the spatial derivative in Eq  ( )   Two quantitative comparisons between experimental results and DBM simulations are shown in the following part, including snapshots of schlieren images and evolutions of characteristic scales for the bubble 
The first is shown in Fig  non-organized momentum flux (NOMF) uously to form a diffracted shock (DS)  As TS propagates, it will split into three branches due to the considerable pressure perturbations caused by the gradual decay of the DS strength   Afterward, as shown in the subfigure at about t = 128µs, two high pressure regions (ROH) generate because of the interaction of these branches 
Subsequently, at about t = 148µs, the two ROHs meet, causing the shock focusing  On the one hand, at about t = 168µs, the shock focusing causes the generation of downstream-propagating second transmitted shock (STS) and upward-moving rarefaction wave  On the other hand, it will produce high pressure region inside the bubble, which later leads to a jet structure, as shown at about t = 288µs 
At about t = 428µs, due to the deposited vorticity, there will produce a pair of counter-rotating vortexes at the pole region of the bubble  The further development of the vortex pair and the effect of viscosity decrease the amplitude of the jet  Finally, the jet structure disappears  The second quantitative comparison is the interface structure described by the length and width of the bubble, as shown in Fig   
The experimental data are extracted from Fig  , in Ref    Quantitative agreements between DBM simulation and experimental results are seen  For the profile of bubble width, there are mainly two stages  At an early time (t < 150µs), it decreases to a minimum value because of the shock compression effect 
After the shock wave passes through the bubble (t > 150µs), the developed vortex pair caused by the deposited vorticity gradually dominates the growth of bubble width  Different from width evolution, the temporal variation of length experiences three stages  In the early stages (t < 150µs), it decreases quickly due to the shock compression effect 
Then, the jet structure emerges, which results in a growth in length (150µs < t < 250µs)  Because the upstream interface moves faster than the downstream interface, the bubble length would decrease at 250µs < t < 500µs  In the third stage (t > 500µs), the vortex pair forms and then leads to a continuous development of bubble length 
Both the length and width experience oscillations in the later stages due to complex wave patterns  The quantitative agreements between DBM simulation and experimental results indicate the following two facts: (i) the order of TNE considered in the current DBM is sufficient, (ii) the choosing of discrete velocities and spatial-temporal steps and simulation parameters like the relaxation times is suitable for characterizing the deformation of bubble, wave patterns, main characteristics of flow morphology 

Effects of specific-heat ratio on SBI

The major of current works on SBI research have not focused on specific-heat ratio effects  In this part, the simulation parameters are fine-adjusted based on the parameters in Section 3 1 to highlight the influence of specific-heat ratio  Through adjusting the extra degree of freedom I, five cases with various specific-heat ratios of the bubble are simulated, i e , γ = 1 4,
1 28, 1 18, 1 12, and 1 09  Two kinds of analysis methods, including tracer particle method and two-fluid model, are used to characterize qualitatively the macroscopic behaviors such as the shape, deformation process, mixing degree, etc  The related TNE behaviors are also studied 

Effects of specific-heat ratio on jet shape, deformation process, and average motion

We first observe the specific-heat ratio effect on the bubble shape from the view of density contour and images of particle tracer visually  As shown in Fig  , pictures with three typical moments are plotted, i e , t = 0 07,t = 0 11, and t = 0 16  The odd rows represent density contours and the even rows are tracer particle images 
It can be seen that the specific-heat ratio significantly affects the length and shape of the jet structure  The smaller the specific-heat ratio is, the stouter the jet structure can be seen  The reason is that the specific-heat ratio significantly changes the propagation speed of shock waves and wave patterns inside the bubble 
The specific-heat ratio also influences the vortex structure in early stage but contributes little effects to it in later stage  In the later stage, for cases with different specific-heat ratios, the differences in vortex pairs are almost invisible  Then, the effects of specific-heat ratio on deformation process are analyzed 
Shown in Fig  are the evolutions of characteristic scales which used to describe the bubble size, i e , width and length  It can be seen that the smaller the specific-heat ratio of bubble, the smaller the bubble width and length  For the fluid with smaller specific-heat ratio, it is easier to be compressed 
Therefore, the characteristic scales of bubbles with smaller specific-heat ratio tend to be compressed smaller  It can also be seen that the case with the largest specific-heat ratio reaches the minimum characteristic scales firstly  The reason is that the shock wave propagates faster in case with larger specific-heat ratio 
Through the method of tracer, information on the average motion of the bubble is easy to obtain  Shown in Fig  are the average position and average velocity of the bubble, with different specific-heat ratios  It is found that, in the shock compression stage (t < 0 03), the effect of specific-heat ratio contributes little to the average motion 
However, when the shock wave passes through the bubble (t > 0 03), a larger specific-heat ratio speeds up the average motion of the bubbles  The reason is that the bubbles with smaller specific-heat ratio need more energy to compress their size, so their translational energy is smaller 

Effects of specific-heat ratio on vortex motion

Vorticity is one of the most important physical quantities in describing the vortex motion  In the 2-D case, the vorticity can be calculated by the following equation: The positive (negative) value of ω represents the positive (negative) direction along the z axis  Vorticity contours at t = 0 134, with various specific-heat ratios, are shown in Fig   
The discernable difference between cases with various specific-heat ratios can be observed  The arrows in the vorticity images point out the obvious differences around the interface between case γ = 1 4 and case γ = 1 09  That is to say, there exists influences of specific-heat ratio on the rotational motion of the bubble 
The strength of vorticity is described by circulation Γ, where Γ = ∑ ω∆x∆y  Γ + = ∑ ω| ω>0 ∆x∆y is the positive circulation and Γ − = ∑ ω| ω<0 ∆x∆y represents the negative circulation  Figure shows the temporal evolution of circulations on SBI process  It can be seen that the values of Γ are equal to zero all the time because the values of Γ + and Γ − are the same 
But they point in the opposite direction  In the shock compression stage (t < 0 03), the specific-heat ratio effect contributes little to the circulation of the bubble  When the shock wave sweeps through the bubble (t > 0 03), the specific-heat ratio affects the value of circulation obviously  The cases with a smaller specific-heat ratio experiences a larger range of amplitude of change, which is caused by its good compressibility 

Effects of specific-heat ratio on mixing degree

The mixing process is a fundamental research content on SBI  In two-fluid DBM, the mixing degree at each fluid unit can be defined as M = 4 • M A M B , where M σ represents the mass fraction of component σ   The higher the value of M, the higher the mixing amplitude  Images of density (first cow) and mixing degree M (second row) at several typical moments are shown in Fig   
As can be seen, the mass mixing occurs in the region where two media contact  In addition, the mixing degree M g described the whole fluid field can be defined where the symbol "−" indicates integrating the M σ over the whole fluid field and then dividing the grid size N x • N y   Shown in Fig  is the temporal evolution of the global mixing degree M g  
As can be seen, temporal profiles of the global mixing degree show two stages: t < 0 03 and t > 0 03  When t < 0 03, there is almost no difference between cases with various specific-heat ratios  But for t > 0 03, the stronger the specific-heat ratio effect, the larger the mixing degree  Actually, there are mainly two indicators that measure the global mixing degree: the amplitude of mixing and the area of the mixing zone between two fluids 
At the stage t > 0 03, the shock compression dominates the mix by enhancing the mixing amplitude and increasing the area of the mixing zone simultaneously  In this stage, the specific-heat ratio effect contributes little to the mix  However, when the shock passes through the bubble, the deformation of interface and the evolution of vortex core both significantly raise the area of the mixing zone 
As can be seen in Fig  , the smaller specific-heat ratio of bubble, the stronger global mixing degree of fluid field  Intuitively, for the fluid with smaller specific-heat ratio, it is easier to deform and compress, which is beneficial for the fluid mixing  It can also be explained by the diffusion formula, i e ,  
The specific-heat ratio affects both the temperature T and the gradient of density simultaneously  Therefore, these two aspects comprehensively influence the material diffusion between the two fluids  Due to the complex reflected shock wave, the global mixing degree shows a tendency for oscillating growth 

Effects of specific-heat ratio on TNE behaviors

The investigation of TNE behaviors is of great importance for understanding the kinetics process on SBI  These TNE quantities describe the fluid system deviating from the thermodynamic state from their own perspectives  The effects of specific-heat ratio on global TNE strength, i e , D * 2 , D * 3 , D * 3,1 , and D * 4,2 , are shown in Fig   
It can be seen that the effects of specific-heat ratios on various TNE quantities are different  Theoretically, the influence of specific-heat ratio on the non-equilibrium effect is reflected in two aspects: transport coefficient and macroscopic quantity gradient  For example, on the one hand, the specific-heat ratio reduces heat conductivity, while on the other hand, it enhances the temperature gradient 
Therefore, the effect of specific heat ratio on NOEF is the comprehensive result of the competition between the two aspects  As shown in Fig  , the smaller the specific-heat ratio, the stronger strength of D * 3,1   It indicates that the specific-heat ratio increase the strength of D * 3,1 by raising the heat conductivity  
For the strength of D * 3 , as shown in Fig  (b), it is seen that it decreases as the specific-heat ratio becomes small  The reason is that a smaller specific-heat ratio decreases the temperature gradient  Effects of specific-heat ratio on D * 4,2 show two-stage  In the shock compression stage (t < 0 03), the smaller specificheat ratio, the larger the strength of D * 4,2  
But the situation is reversed at the stage t > 0 03  For strength of D * 2 , the specificheat effects are more significant in later stage 

Effects of specific-heat ratio on entropy production rate

and entropy production The concepts of entropy are commonly used in complex flows   In DBM, there are two kinds of en- tropy production rates, i e , ṠNOEF and ṠNOMF   They are key factors in compression science field  The former is induced by temperature gradient and the NOEF (∆ * 3,1 )  The latter is affected by velocity gradient and the NOMF (∆ * 2 ) 
The entropy production rates are defined by the following formulas : Integrating the ṠNOEF and ṠNOMF over time t, the entropy generations over this period of time are obtained, i e , S NOEF = t 0 ṠNOEF dt and S NOMF = t 0 ṠNOMF dt  Plotted in Fig  ) and 14(b) are the temporal evolution of ṠNOMF and ṠNOEF , respectively 
The evolution of entropy generation rate is related to two aspects: (i) the propagation of the shock wave, and (ii) the deformation of the bubble  The former generates a macroscopic quantity gradient, and the latter makes the contact interface wider, longer, and deformed  Depending on the location of the shock wavefront, there exist two critical moments in this SBI process: (i) at around t = 0 03, the shock wave just sweeps through the bubble, and (ii) at t = 0 06, the shock wave exits the flow field 
Therefore, the temporal evolution of the entropy production rate shows three stages, i e , t < 0 03, 0 03 < t < 0 06, and t > 0 06  At the stage t < 0 03, the shock compression stage, the shock effects compress the bubble  It generates the large macroscopic quantity gradients, resulting in a quick increase of ṠNOMF  
At around t = 0 03, the shock wave passed through the bubble  So the values of ṠNOMF decreases  The values of ṠNOMF would continue to decrease due to the gradually wider contact interface caused by the diffusion effect  At around t = 0 06, the shock wave comes out of the flow field so that the values of ṠNOMF drops rapidly 
In the third stage, i e , t > 0 06, because of the diffusive effect, the general trend of ṠNOMF is downward  However, it shows an oscillatory trend due to the influence of various reflected shock waves  The specific heat ratio indirectly changes the value of ṠNOMF by changing the velocity gradient  The smaller the specific-heat ratio, the larger ṠNOMF  
Different understanding can be seen in Fig  , where the temporal evolution of ṠNOEF is plotted  In the first stage (t < 0 03), cases with different specific-heat ratios show various trends  At the stage where the bubble deformation is not very large, i e , 0 03 < t < 0 06, values of ṠNOEF fluctuate near the average value 
In the third stage (t > 0 06), evolutions of ṠNOEF in cases with larger specific-heat ratios show an apparent growing tendency  Differently, the values of ṠNOEF in cases with smaller specific-heat ratios remain almost unchanged  The influence of specific heat ratio on the ṠNOEF , similar with the effect on NOEF, is also affected by the heat conductivity and the temperature gradient 
It can be seen that, except for the case of γ = 1 09, the larger the specific-heat ratio, the higher entropy production rate ṠNOEF   The temporal evolutions of ṠNOEF of case γ = 1 09 and case γ = 1 12 are very similar  Consequently, the specific-heat ratio increases the ṠNOEF by raising the temperature gradient 
Further understanding can be seen in Fig  , where the entropy productions over this period are plotted  For convenience, the sum and difference between S NOMF and S NOEF are also plotted in the figure  The variation range of S NOEF is larger than that of S NOMF   It indicates that the influence of specific-heat ratio on S NOEF is more significant than that on S NOMF  
Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary  Specifically, it can be seen that the entropy production contributed by NOMF increases with re- duced specific-heat ratio  But the entropy production caused by NOEF first reduces with decreasing specific-heat ratio and then approaches to a saturation value 
The S NOEF in case γ = 1 09 is almost the same with it in case γ = 1 12  When the specificheat ratio γ is smaller than a threshold value γ c (γ c ≈ 1 315), the entropy production induced by NOEF is more significant than that caused by NOMF  However, in the case of γ > γ c , the situation reverses  The temporal evolution of the total entropy production (S NOMF +S NOEF ) is similar to the S NOEF profile 
The difference between S NOMF and S NOEF increases with decreasing specific-heat ratio 

Conclusions

Specific-heat ratio effects on the interaction between a planar shock wave and a 2-D heavy-cylindrical bubble are studied by a two-fluid DBM which has a flexible specific-heat ratio and includes several schemes for analyzing the complex physical fields  Besides the HNE that NS easily describes, the DBM pays more attention to the related TNE that NS is not convenient to describe 
First, both the snapshots of schlieren images and evolutions of characteristic scales from DBM simulation are compared with those from experiment  The quantitative agreements between them indicate the following two facts: (i) the order of TNE considered in the current DBM is sufficient, (ii) the choosing of discrete velocities, spatial-temporal steps, and simulation parameters like the relaxation times are suitable for the following physical researches 
Then, five cases with various specific-heat ratios are simulated  Several analysis methods for complex physical fields, including the description scheme of TNE behaviors, tracer particle method, and two-fluid model, are used to characterize the effects of specific-heat ratio on the bubble shape, deformation process, average motion, vortex motion, mixing degree of the fluid system, TNE strength, and entropy production 
Specifically, for bubble shape, bubbles with different specific-heat ratios display various jet structures  The smaller the specific-heat ratio is, the stouter the jet structure can be seen  For the case with smaller specific-heat ratio, the fluid is easier to be compressed  So, the characteristic scales of bubbles with smaller specific-heat ratio tend to be compressed smaller 
For the bubble, the smaller the specific-heat ratio, the slower average motion  In the shock compression stage, the specific-heat ratio contributes little effects to the vortex motion  Differently, after the shock passes through the bubble, it significantly influences the vorticity around the interface and the corresponding amplitude of circulation due to the development of KHI 
The larger the difference in specific-heat ratio between the bubble and ambient gas, the higher the degree of material mixing  Effects of specific-heat ratio on various TNE quantities are different  These differences consistently show the complexity of TNE flows which is still far from a clear understanding 
In addition, it is found that the temporal evolution of the entropy production rates ṠNOMF and ṠNOEF both show three stages because of the influence of the shock wave location  The smaller the specific-heat ratio, the larger the velocity gradient, which indirectly enhances the strength of ṠNOMF   The specific-heat ratio increases the ṠNOEF by raising the temperature gradient 
The influence of specific-heat ratio on S NOEF is more significant than that on S NOMF   Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary  Specifically, the entropy production contributed by NOMF increases with reduced specific-heat ratio  But the entropy production caused by NOEF first reduces with decreasing specific-heat ratio and then approaches to a saturation value 
When the specific-heat ratio γ is smaller than a threshold value γ c (γ c ≈ 1 315), the entropy production induced by NOEF is more significant than that caused by NOMF  However, in the case of γ > γ c , the situation reverses  The fundamental research in this paper helps to understand the interaction mechanism between shock waves and bubbles in ICF, supersonic combustors, underwater explosions, etc 
The effects of viscosity and heat conduction on the interaction between shock waves and bubbles will be studied in the following work  where the subscript "m, n" means that the m-order tensor is contracted to n-order tensor  According to the CE multiscale analysis, the Boltzmann-BGK equation can be reduced to the hydrodynamic equations 
In the following part, the derivation process from Boltzmann-BGK equation to a two-fluid hydrodynamic equation are shown  More details can see the reference presented by Zhang et al    The discrete Boltzmann equation for component σ is (C 1) In Eq  C 1 there are two equilibrium distribution functions, i e , f σ ,seq = f σ ,seq (ρ σ , u σ , T σ ) and f σ ,eq = f σ ,eq (ρ σ , u, T ) 
For convenience, S σ i is defined as We perform the CE expansion around the f σ ,seq   That is, the distribution function f σ i can be expanded as where ε is a coefficient referring to Knudsen number  The partial derivatives of time and space can also be expanded to Substituting the above four equations into Eq 
(C 1), we can obtain (C 6) When retaining to ε terms, the following equation is obtained When retaining to ε 2 terms, we can obtain where M 2,αβ ( f σ ,(1) ) = ∑ i v iα v iβ f σ ,(1) i and M 3,1,α ( f σ ,(1) ) =   Substituting Eq  (C 14) into the above three equations, and replacing the time derivatives with the space derivatives, we obtain It should be noted that the ability to recover the corresponding level of macroscopic fluid mechanics equations is only part of the physical function of DBM 
The corresponding to the physical functions of DBM is the EHEs, which, in addition to the conserved moments evolution equations corresponding to the three conservation laws of mass, momentum and energy, also includes some of the most closely related nonconserved moments evolution equations  We refer the EHEs derivation based on kinetic equation to as KMM 
The necessity of the expanded part, the evolution equations of the relevant non-conserved moments, increases rapidly as increasing the degree of non-continuity/non-equilibrium  As the degree of non-continuity/non-equilibrium increases, the complexity will rapidly make KMM simulation studies, deriving and solving EHE, impossible 
Context 2: Margaret Way (b  Brisbane d  Cleveland, Queensland, Australia ) was an Australian writer of romance novels and women's fiction  A prolific author, Way wrote more than 120 novels since 1970, many through Mills & Boon, a romance imprint of British publisher Harlequin UK Ltd , owned by Harlequin Enterprises 

Biography
Before her marriage, she was a well-known pianist, teacher, vocal coach and accompanist  She began writing when her son, Laurence Way, was born, a friend took a pile of Mills & Boon books to her, she read all and decided that she also could write these types of novels  She began to write and promote her country with her stories set in Australia  She sold her first novels in 1970  Margaret Way lives with her family in her native Brisbane  Beginning in 2013, Margaret began to self-publish, releasing her first "e-book" mid-July



ground_truth: The specific-heat ratio affects the average motion of the bubble. The bubbles with smaller specific-heat ratios have slower average motion.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Where was Margaret Way born and where did she die?

Read the following text and answer briefly.

Context 2: 

Margaret died on the 10th of August 2022 in Cleveland, Queensland 

Bibliography

Single Novels
King Country (1970)
Blaze of Silk (1970)
The Time of the Jacaranda (1970)
Bauhinia Junction (1971)
Man from Bahl Bahla (1971)
Summer Magic (1971)
Return to Belle Amber (1971)
Ring of Jade (1972)
Copper Moon (1972)
Rainbow Bird (1972)
Man Like Daintree (1972)
Noonfire (1972)
Storm Over Mandargi (1973)
Wind River (1973)
Love Theme (1974)
McCabe's Kingdom (1974)
Sweet Sundown (1974)
Reeds of Honey (1975)
Storm Flower (1975)
Lesson in Loving (1975)
Flight into Yesterday (1976)
Red Cliffs of Malpara (1976)
Man on Half-moon (1976)
Swan's Reach (1976)
Mutiny in Paradise (1977)
One Way Ticket (1977)
Portrait of Jaime (1977)
Black Ingo (1977)
Awakening Flame (1978)
Wild Swan (1978)
Ring of Fire (1978)
Wake the Sleeping Tiger (1978)
Valley of the Moon (1979)
White Magnolia (1979)
Winds of Heaven (1979)
Blue Lotus (1979)
Butterfly and the Baron (1979)
Golden Puma (1980)
Temple of Fire (1980)
Lord of the High Valley (1980)
Flamingo Park (1980)
North of Capricorn (1981)
Season for Change (1981)
Shadow Dance (1981)
McIvor Affair (1981)
Home to Morning Star (1981)
Broken Rhapsody (1982)
The Silver Veil (1982)
Spellbound (1982)
Hunter's Moon (1982)
Girl at Cobalt Creek (1983)
No Alternative (1983)
House of Memories (1983)
Almost a Stranger (1984)
A place called Rambulara (1984)
Fallen Idol (1984)
Hunt the Sun (1985)
Eagle's Ridge (1985)
The Tiger's Cage (1986)
Innocent in Eden (1986)
Diamond Valley (1986)
Morning Glory (1988)
Devil Moon (1988)
Mowana Magic (1988)
Hungry Heart (1988)
Rise of an Eagle (1988)
One Fateful Summer (1993)
The Carradine Brand (1994)
Holding on to Alex (1997)
The Australian Heiress (1997)
Claiming His Child (1999)
The Cattleman's Bride (2000)
The Cattle Baron (2001)
The Husbands of the Outback (2001)
Secrets of the Outback (2002)
With This Ring (2003)
Innocent Mistress (2004)
Cattle Rancher, Convenient Wife (2007)
Outback Marriages (2007)
Promoted: Nanny to Wife (2007)
Cattle Rancher, Secret Son (2007)
Genni's Dilemma (2008)
Bride At Briar Ridge (2009)
Outback Heiress, Surprise Proposal (2009)
Cattle Baron, Nanny Needed (2009)

Legends of the Outback Series
Mail Order Marriage (1999)
The Bridesmaid's Wedding (2000)
The English Bride (2000)
A Wife at Kimbara (2000)

Koomera Crossing Series
Sarah's Baby (2003)
Runaway Wife (2003)
Outback Bridegroom (2003)
Outback Surrender (2003)
Home to Eden (2004)

McIvor Sisters Series
The Outback Engagement (2005)
Marriage at Murraree (2005)

Men Of The Outback Series
The Cattleman (2006)
The Cattle Baron's Bride (2006)
Her Outback Protector (2006)
The Horseman (2006)

Outback Marriages Series
Outback Man Seeks Wife (2007)
Cattle Rancher, Convenient Wife (2007)

Barons of the Outback Series Multi-Author
Wedding At Wangaree Valley (2008)
Bride At Briar's Ridge (2008)

Family Ties Multi-Author
Once Burned (1995)

Hitched! Multi-Author
A Faulkner Possession (1996)

Simply the Best Multi-Author
Georgia and the Tycoon (1997)

The Big Event Multi-Author
Beresford's Bride (1998)

Guardian Angels Multi-Author
Gabriel's Mission (1998)

Australians Series Multi-Author
7  Her Outback Man (1998)
17  Master of Maramba (2001)
19  Outback Fire (2001)
22  Mistaken Mistress (2002)
24  Outback Angel (2002)
33  The Australian Tycoon's Proposal (2004)
35  His Heiress Wife (2004)

Marrying the Boss Series Multi-Author
Boardroom Proposal (1999)

Contract Brides Series Multi-Author
Strategy for Marriage (2002)

Everlasting Love Series Multi-Author
Hidden Legacy (2008)

Diamond Brides Series Multi-Author
The Australian's Society Bride (2008)

Collections
Summer Magic / Ring of Jade / Noonfire (1981)
Wife at Kimbara / Bridesmaid's Wedding (2005)

Omnibus in Collaboration
Pretty Witch / Without Any Amazement / Storm Over Mandargi (1977) (with Lucy Gillen and Margaret Malcolm)
Dear Caliban / Heart of the Eagle / Swans' Reach (1978) (with Jane Donnelly and Elizabeth Graham)
The Bonds of Matrimony / Dragon Island / Reeds of Honey (1979) (with Elizabeth Hunter and Henrietta Reid)
The Man Outside / Castles in Spain / McCabe's Kingdom (1979) (with Jane Donnelly and Rebecca Stratton)
Winds From The Sea / Island of Darkness / Wind River (1979) (with Margaret Pargeter and Rebecca Stratton)
Moorland Magic / Tree of Idleness / Sweet Sundown (1980) (with Elizabeth Ashton and Elizabeth Hunter)
The Shifting Sands / Portrait of Jaime / Touched by Fire (1982) (with Jane Donnelly and Kay Thorpe)
Head of Chancery / Wild Heart / One-Way Ticket (1986) (with Betty Beaty and Doris Smith)
Heart of the Scorpion / The Winds of Heaven / Sweet Compulsion (1987) (with Janice Gray and Victoria Woolf)
One Brief Sweet Hour / Once More With Feeling / Blue Lotus (1990) (with Jane Arbor and Natalie Sparks)
Marry Me Cowboy (1995) (with Janet Dailey, Susan Fox and Anne McAllister)
Husbands on Horseback (1996) (with Diana Palmer)
Wedlocked (1999) (with Day Leclaire and Anne McAllister)
Mistletoe Magic (1999) (with Betty Neels and Rebecca Winters)
The Australians (2000) (with Helen Bianchin and Miranda Lee)
Weddings Down Under (2001) (with Helen Bianchin and Jessica Hart)
Outback Husbands (2002) (with Marion Lennox)
The Mother's Day Collection (2002) (with Helen Dickson and Kate Hoffmann)
Australian Nights (2003) (with Miranda Lee)
Outback Weddings (2003) (with Barbara Hannay)
Australian Playboys (2003) (with Helen Bianchin and Marion Lennox)
Australian Tycoons (2004) (with Emma Darcy and Marion Lennox)
A Mother's Day Gift (2004) (with Anne Ashley and Lucy Monroe)
White Wedding (2004) (with Judy Christenberry and Jessica Steele)
A Christmas Engagement (2004) (with Sara Craven and Jessica Matthews)
A Very Special Mother's Day (2005) (with Anne Herries)
All I Want for Christmas    (2005) (with Betty Neels and Jessica Steele)
The Mills and Boon Collection (2006) (with Caroline Anderson and Penny Jordan)
Outback Desire (2006) (with Emma Darcy and Carol Marinelli)
To Mum, with Love (2006) (with Rebecca Winters)
Australian Heroes (2007) (with Marion Lennox and Fiona McArthur)
Tall, Dark and Sexy (2008) (with Caroline Anderson and Helen Bianchin)
The Boss's Proposal (2008) (with Jessica Steele and Patricia Thayer)
Island Heat / Outback Man Seeks Wife / Prince's Forbidden Virgin / One Night Before Marriage / Their Lost-and-found Family / Single Dad's Marriage Wish (2008) (with Robyn Donald, Marion Lennox, Carol Marinelli, Sarah Mayberry and Anne Oliver)
Australian Billionaires (2009) (with Jennie Adams and Amy Andrews)
Cattle Baron : Nanny Needed / Bachelor Dad on Her Doorstep (2009) (with Michelle Douglas)

External links
Margaret Way at Harlequin Enterprises Ltd

Australian romantic fiction writers
Australian women novelists
Living people
Year of birth missing (living people)
Women romantic fiction writers
Context 3: \section{Introduction}

Ultracold neutral plasmas studied in the laboratory offer access to a regime of plasma physics that scales to describe thermodynamic aspects of important high-energy-density systems, including strongly coupled astrophysical plasmas \cite{VanHorn,Burrows}, as well as terrestrial sources of neutrons \cite{Hinton,Ichimaru_fusion,Atzeni,Boozer} and x-ray radiation \cite{Rousse,Esarey}   Yet, under certain conditions, low-temperature laboratory plasmas evolve with dynamics that are governed by the quantum mechanical properties of their constituent particles, and in some cases by coherence with an external electromagnetic field    

The relevance of ultracold plasmas to such a broad scope of problems in classical and quantum many-body physics has given rise to a great deal of experimental and theoretical research on these systems since their discovery in the late 90s   A series of reviews affords a good overview of progress in the last twenty years \cite{Gallagher,Killian_Science,PhysRept,Lyon}   Here, we focus on the subset of ultracold neutral plasmas that form via kinetic rate processes from state-selected Rydberg gases, and emphasize in particular the distinctive dynamics found in the evolution of molecular ultracold plasmas   

While molecular beam investigations of threshold photoionization spectroscopy had uncovered relevant effects a few years earlier \cite{Scherzer,Alt}, the field of ultracold plasma physics began in earnest with the 1999 experiment of Rolston and coworkers on metastable xenon atoms cooled in a magneto optical trap (MOT) \cite{Killian}   

This work and many subsequent efforts tuned the photoionization energy as a means to form a plasma of very low electron temperature built on a strongly coupled cloud of ultracold ions   Experiment and theory soon established that fast processes associated with disorder-induced heating and longer-time electron-ion collisional rate processes act to elevate the ion temperatures to around one degree Kelvin, and constrain the effective initial electron temperature to a range above 30 K \cite{Kuzmin,Hanson,Laha}   

This apparent limit on the thermal energy of the electrons can be more universally expressed for an expanding plasma by saying that the electron correlation parameter, $\Gamma_e$, does not exceed 0 25, where, 
\begin{equation}
\Gamma_e = \frac{e^2}{4\pi \epsilon_0 a_{ws}}\frac{1}{k_B T_e}
\label{eqn:gamma_e}
\end{equation}
defines the ratio of the average unscreened electron-electron potential energy to the electron kinetic energy   $a_{ws}$ is the Wigner-Seitz radius, related to the electron density by, $\rho_e = 1/(\frac{4}{3} \pi a_{ws}^3)$   These plasmas of weakly coupled electrons and strongly coupled ions have provided an important testing ground for ion transport theory and the study of electron-ion collision physics \cite{Strickler} 

Soon after the initial reports of ultracold plasmas formed by direct photoionization, a parallel effort began with emphasis on the plasma that forms spontaneously by Penning ionization and electron-impact avalanche in a dense ultracold Rydberg gas \cite{Mourachko}   This process affords less apparent control of the initial electron temperature   But, pulsed field-ionization measurements soon established that the photoionized plasma and that formed by the avalanche of a Rydberg gas both evolve to quasi-equilibria of electrons, ions and high-Rydberg neutrals \cite{Rolston_expand,Gallagher}   

Early efforts to understand plasmas formed by Rydberg gas avalanche paid particular attention to the process of initiation   Evolution to plasma in effusive atomic beams was long known for high-Rydberg gases of caesium and well explained by coupled rate equations \cite{Vitrant}   But, low densities and ultracold velocity distributions were thought to exclude Rydberg-Rydberg collisional mechanisms in a MOT   

In work on ultracold Rydberg gases of Rb and Cs, Gallagher, Pillet and coworkers describe the initial growth of electron signal by a model that includes ionization by blackbody radiation and collisions with a background of uncooled Rydberg atoms \cite{Mourachko,Gallagher,Li,Comparat,Tanner}  This picture was subsequently refined to include many-body excitation and autoionization, as well as attractive dipole-dipole interactions \cite{Viteau,Pillet}, later confirmed by experiments at Rice \cite{Mcquillen}   

The Orsay group also studied the effect of adding Rydberg atoms to an established ultracold plasma   They found that electron collisions in this environment completely ionize added atoms, even when selected to have deep binding energies \cite{Vanhaecke}   They concluded from estimates of electron trapping efficiency that the addition of Rydberg atoms does not significantly alter the electron temperature of the plasma   

Tuning pair distributions by varying the wavelength of the excitation laser, Weidem\"uller and coworkers confirmed the mechanical effects of van der Waals interactions on the rates of Penning ionization in ultracold $^{87}$Rb Rydberg gases \cite{Amthor_mech}   They recognized blackbody radiation as a possible means of final-state redistribution, and extended this mechanical picture to include long-range repulsive interactions \cite{Amthor_model}   This group later studied the effects of spatial correlations in the spontaneous avalanche of Rydberg gases in a regime of strong blockade, suggesting a persistence of initial spatial correlations \cite{RobertdeSaintVincent}   

Robicheaux and coworkers have recently investigated the question of prompt many-body ionization from the point of view of Monte Carlo classical trajectory calculations \cite{Goforth}   For atoms on a regular or random grid driven classically by an electromagnetic field, they find that many-body excitation enhances prompt ionization by about twenty percent for densities greater than $5 6 \times 10^{-3}/(n_0^2 a_0)^3$, where $n_0$ is the principal quantum number of the Rydberg gas and $a_0$ is the Bohr radius   They observed that density fluctuations (sampled from the distribution of nearest neighbour distances) have a greater effect, and point to the possible additional influence of secondary electron-Rydberg collisions and the Penning production of fast atoms not considered by the model, but already observed by Raithel and coworkers \cite{Knuffman}   

The Raithel group also found direct evidence for electron collisional $\ell$-mixing in a Rb MOT \cite{Dutta}, and used selective field ionization to monitor evolution to plasma on a microsecond timescale in ultracold $^{85}$Rb $65d$ Rydberg gases with densities as low as $10^8$ cm$^{-3}$ \cite{WalzFlannigan}   Research by our group at UBC has observed very much the same dynamics in the relaxation of Xe Rydberg gases of similar density prepared in a molecular beam \cite{Hung2014}   In both cases, the time evolution to avalanche is well-described by coupled rate equations (see below), assuming an initializing density of Penning electrons determined by Robicheaux's criterion \cite{Robicheaux05}, applied to an Erlang distribution of Rydberg-Rydberg nearest neighbours   

Theoretical investigations of ultracold plasma physics have focused for the most part on the long- and short-time dynamics of plasmas formed by direct photoionization \cite{PhysRept,Lyon}   In addition to studies mentioned above, key insights on the evolution dynamics of Rydberg gases have been provided by studies of Pohl and coworkers exploring the effects of ion correlations and recombination-reionization on the hydrodynamics of plasma expansion \cite{Pohl:2003,PPR}   Further research has drawn upon molecular dynamics (MD) simulations to reformulate rate coefficients for the transitions driven by electron impact between highly excited Rydberg states \cite{PVS}, and describe an effect of strong coupling as it suppresses three-body recombination \cite{Bannasch:2011}   MD simulations confirm the accuracy of coupled rate equation descriptions for systems with $\Gamma$ as large as 0 3   Newer calculations suggest a strong connection between the order created by dipole blockade in Rydberg gases and the most favourable correlated distribution of ions in a corresponding strongly coupled ultracold plasma \cite{Bannasch:2013}   

Tate and coworkers have studied ultracold plasma avalanche and expansion theoretically as well as experimentally   Modelling observed expansion rates, they recently found that $^{85}$Rb atoms in a MOT form plasmas with effective initial electron temperatures determined by initial Rydberg density and the selected initial binding energy, to the extent that these parameters determine the fraction of the excited atoms that ionize by electron impact in the avalanche to plasma \cite{Forest}   This group also returned to the question of added Rydberg atoms, and managed to identify a crossover in $n_0$, depending on the initial electron temperature, that determines whether added Rydberg atoms of a particular initial binding energy act to heat or cool the electron temperature \cite{Crockett}    

Our group has focused on the plasma that evolves from a Rydberg gas under the low-temperature conditions of a skimmed, seeded supersonic molecular beam   In work on nitric oxide starting in 2008 \cite{Morrison2008,Plasma_expan,Morrison_shock,PCCP}, we established an initial kinetics of electron impact avalanche ionization that conforms with coupled rate equation models \cite{Saquet2011,Saquet2012,Scaling,haenelCP} and agrees at early times with the properties of ultracold plasmas that evolve from ultracold atoms in a MOT   We have also observed unique properties of the NO ultracold plasma owing to the fact that its Rydberg states dissociate \cite{Haenel2017}, and identified relaxation pathways that may give rise to quantum effects \cite{SousMBL,SousNJP}   The remainder of this review focuses on the nitric oxide ultracold plasma and the unique characteristics conferred by its evolution from a Rydberg gas in a laser-crossed molecular beam   


\section{Avalanche to strong coupling in a molecular Rydberg gas}

\subsection{The molecular beam ultracold plasma compared with a MOT}

When formed with sufficient density, a Rydberg gas of principal quantum number $n_0>30$ undergoes a spontaneous avalanche to form an ultracold plasma \cite{Li,Morrison2008,RobertdeSaintVincent}   Collisional rate processes combine with ambipolar hydrodynamics to govern the properties of the evolving plasma   For a molecular Rydberg gas, neutral fragmentation, occurs in concert with electron-impact ionization, three-body recombination and electron-Rydberg inelastic scattering   Neutral dissociation combined with radial expansion in a shaped distribution of charged particles, can give rise to striking effects of self-assembly and spatial correlation \cite{Schulz-Weiling2016,Haenel2017}    

The formation of a molecular ultracold plasma requires the conditions of local temperature and density afforded by a high mach-number skimmed supersonic molecular beam   Such a beam propagates at high velocity in the laboratory, with exceedingly well-defined hydrodynamic properties, including a propagation-distance-dependent density and sub-Kelvin temperature in the moving frame \cite{MSW_tutorial}   The low-temperature gas in a supersonic molecular beam differs in three important ways from the atomic gas laser-cooled in a magneto-optical trap (MOT) 

The milli-Kelvin temperature of the gas of ground-state NO molecules entrained in a beam substantially exceeds the sub-100 micro-Kelvin temperature of laser-cooled atoms in a MOT   However, the evolution to plasma tends to erase this distinction, and the two further characteristics that distinguish a beam offer important advantages for ultracold plasma physics:  Charged-particle densities in a molecular beam can exceed those attainable in a MOT by orders of magnitude   A great many different chemical substances can be seeded in a free-jet expansion, and the possibility this affords to form other molecular ultracold plasmas, introduces interesting and potentially important new degrees of freedom governing the dynamics of their evolution 


\subsection{Supersonic molecular beam temperature and particle density}

Seeded in a skimmed supersonic molecular beam, nitric oxide forms different phase-space distributions in the longitudinal (propagation) and transverse coordinate dimensions   As it propagates in $z$, the NO molecules reach a terminal laboratory velocity, $u_{\parallel}$, of about 1400 ${\rm ms^{-1}}$, which varies with the precise seeding ratio   

The distribution of $v_{\parallel}$, narrows to define a local temperature, $T_{\parallel}$, of approximately 0 5 K   The beam forms a Gaussian spatial distribution in the transverse coordinates, $x$ and $y$   In this plane, the local velocity, $v_{\perp}(r)$ is defined for any radial distance almost entirely by the divergence velocity of the beam, $u_{\perp}(r)$   Phase-space sorting cools the temperature in the transverse coordinates, $T_{\perp}$ to a value as low as $\sim 5$ mK \cite{MSW_tutorial}   

The stagnation pressure and seeding ratio determine the local density distribution as a function of $z$   For example, expanding from a stagnation pressure of 500 kPa with a 1:10 seeding ratio, a molecular beam propagates 2 5 cm to a skimmer and then 7 5 cm to a point of laser interaction, where it contains NO at a peak density of $1 6 \times 10^{14}$ cm$^{-3}$   

Here, crossing the molecular beam with a laser beam tuned to the transition sequence, ${\rm X} ~^2 \Pi_{1/2} ~N'' = 1 \xrightarrow{\omega_1} {\rm A} ~^2\Sigma^+ ~N'=0  \xrightarrow{\omega_2} n_0 f(2)$ forms a Gaussian ellipsoidal volume of Rydberg gas in a single selected principal quantum number, $n_0$, orbital angular momentum, $\ell = 3$, NO$^+$ core rotational quantum number, $N^+ = 2$ and total angular momentum neglecting spin, $N=1$   

A typical $\omega_1$ pulse energy of 2 $\mu$J and a Gaussian width of 0 2 mm serves to drive the first step of this sequence in a regime of linear absorption   Overlapping this volume by an $\omega_2$ pulse with sufficient fluence to saturate the second step forms a Rydberg gas ellipsoid with a nominal peak density of $5 \times 10^{12}$ cm$^{-3}$  \cite{Morrison2008,MSW_tutorial}   Fluctuations in the pulse energy and longitudinal mode of $\omega_1$ cause the real density to vary   For certain experiments, we find it convenient to saturate the $\omega_1$ transition, and vary the density of Rydberg gas by delaying $\omega_2$   An $\omega_1$-$\omega_2$ delay, $\Delta t$, reduces the Rydberg gas density by a precise factor, $e^{-\Delta t/\tau}$, where $\tau$ is the 200 ns radiative lifetime of NO ${\rm A} ~^2\Sigma^+ ~N'=0$ \cite{Carter,Hancock} 


\subsection{Penning ionization}

The density distribution of a Rydberg gas defines a local mean nearest neighbour distance, or Wigner-Seitz radius of $ a_{ws} =  \left(3/4 \pi \rho \right)^{1/3} $, where $\rho$ refers to the local Rydberg gas density   For example, a Rydberg gas with a density of $ \rho_0=0 5 \times 10^{12}$ cm$^{-3} $ forms an Erlang distribution \cite{Torquato 1990} of nearest neighbour separations with a mean value of $ 2 a_{ws}=1 6$  $\mu$m   

A semi-classical model \cite{Robicheaux05} suggests that 90 percent of Rydberg molecule pairs separated by a critical distance, $ r_c = 1 8 \cdot 2 n_0^2 a_0 $ or less undergo Penning ionization within 800 Rydberg periods   We can integrate the Erlang distribution from $ r=0 $ to the critical distance $r = r_c$ for a Rydberg gas of given $n_0$, to define the local density of Penning electrons ($ \rho_e$ at $t=0$) produced by this prompt interaction, for any given initial local density, $\rho_0$ by the expression:
\begin{equation}
\rho_e(\rho_0,n_0) = \frac{0 9}{2} \cdot 4 \pi \rho_0 ^2\int_0^{r_{c}} r^2 \mathrm{e}^{-\frac{4\pi}{3}\rho_0 r^3}\mathrm{d}r \quad 
\label{eqn:Erlang}
\end{equation}

Evaluating this definite integral yields an equation in closed form that predicts the Penning electron density for any particular initial Rydberg density and principal quantum number 
\begin{equation}
\rho_e(\rho_0,n_0) =\frac{0 9 \rho_0}{2}(1-\mathrm{e}^{-\frac{4\pi}{3}\rho_0 r_c^3}) \quad 
\label{Eq:PenDens}
\end{equation}
\begin{figure}[h!]
\centering
\includegraphics[scale=0 33]{Penning_Latice pdf}
\caption{Distributions of ion-ion nearest neighbours following Penning ionization and electron-impact avalanche simulated for a predissociating molecular Rydberg gas of initial principal quantum number, $n_0$, from 30 to 80, and density of 10$^{12}$ cm$^{-3}$   Dashed lines mark corresponding values of $a_{ws}$  Calculated by counting ion distances after relaxation to plasma in 10$^6$-particle stochastic simulations  Integrated areas proportional to populations surviving neutral dissociation }
\label{fig:PL}
\end{figure}

Prompt Penning ionization acts on the portion of the initial nearest-neighbour distribution in the Rydberg gas that lies within $r_c$   When a molecule ionizes, its collision partner relaxes to a lower principal quantum number, $n'<n_0/\sqrt{2}$   This close-coupled interaction disrupts the separability of Rydberg orbital configurations in the Penning partner   This causes mixing with core penetrating states that are strongly dissociative   Penning partners are thus very likely to dissociate, leaving a spatially isolated distribution of ions   We refer to the spatial correlation that results as a Penning lattice \cite{Sadeghi:2014}   The extent of this effect varies depending on the local density and the selected initial principal quantum number   Figure \ref{fig:PL} shows the degree to which Rydberg gases with initial principal quantum numbers from 30 to 80 form a Penning lattice for an initial density of $1 \times 10^{12} ~{\rm cm}^{-3}$   

\subsection{Spontaneous electron-impact avalanche}

The electrons produced by prompt Penning ionization start an electron impact avalanche   The kinetics of this process are well described by a set of coupled rate equations that account for state-to-state electron-Rydberg inelastic scattering, electron-impact ionization and three-body ion-electron recombination \cite{PPR,Saquet2011,Saquet2012,Scaling} using detailed rate coefficients,  $k_{ij}$, $k_{i,ion}$ and $k_{i,tbr}$ validated by MD simulations \cite{PVS}   
\begin{eqnarray}
-\frac{d\rho_i}{dt}&=&\sum_{j}{k_{ij}\rho_e\rho_i}-\sum_{j}{k_{ji}\rho_e\rho_j} \nonumber\\
&& +k_{i,ion}\rho_e\rho_i-k_{i,tbr}\rho^3_e 
  \label{level_i}
\end{eqnarray}
\noindent and,
\begin{equation}
\frac{d\rho_e}{dt}=\sum_{i}{k_{i,ion}\rho_e^2}-\sum_{i}{k_{i,tbr}\rho^3_e}
  \label{electron}
\end{equation}

The relaxation of Rydberg molecules balances with collisional ionization to determine an evolving temperature of avalanche electrons to conserve total energy per unit volume  
\begin{equation}
E_{tot}=\frac{3}{2}k_BT_e(t)\rho_e(t)-R\sum_i{\frac{\rho_i(t)}{n_i^2}},
  \label{energy}
\end{equation}
Here, for simplicity, we neglect the longer-time effects of Rydberg predissociation and electron-ion dissociative recombination \cite{Saquet2012} 

Such calculations show that the conversion from Rydberg gas to plasma occurs on a timescale determined largely by the local Penning electron density, or Penning fraction, $P_f = \rho_e/\rho_0$, which depends on the local density of Rydberg molecules and their initial principal quantum number   

Avalanche times predicted by coupled rate equation calculations range widely   For example, in a model developed for experiments on xenon, simulations predict that a Rydberg gas with $n_0 = 42$ at a density of $8 8 \times 10^8 ~{\rm cm}^{-3}$ ($P_f = 6 \times 10^{-5}$) avalanches with a half time of  40 $\mu$s \cite{Hung2014}   At an opposite extreme, rate equations estimate that a Rydberg gas of NO with $n_0=60$ at a density of $1 \times 10^{12} ~{\rm cm}^{-3}$ ($P_f = 0



ground_truth: Margaret Way was born in Brisbane and died in Cleveland, Queensland, Australia.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What kind of ultracold neutral plasmas does this study focus on?

Read the following text and answer briefly.

Context 3: 3$) rises to plasma in about 2 ns \cite{Saquet2012}   

\begin{figure}[h!]
\centering
\includegraphics[width=  49 \textwidth]{SFI_n=49 pdf}
   \caption{Contour plots showing SFI signal as a function the applied field for an $nf(2)$ Rydberg gas with an initial principal quantum number, $n_0=49$   Each frame represents 4,000 SFI traces, sorted by initial Rydberg gas density   Ramp field beginning at 0 and 150 ns (top, left to right), and  300 and 450 ns (bottom) after the $\omega_2$ laser pulse   The two bars of signal most evident at early ramp field delay times represent the field ionization of the $49f(2)$ Rydberg state respectively to NO$^+$ X $^1\Sigma^+$ cation rotational states, $N^+=0$ and 2   The signal waveform extracted near zero applied field represents the growing population of plasma electrons   
   }
\label{fig:SFI}
\end{figure}

Selective field ionization (SFI) probes the spectrum of binding energies in a Rydberg gas   Applied as a function of time after photoexcitation, SFI maps the evolution from a state of selected initial principal quantum number, $n_0$, to plasma \cite{Haenel2017}   Figure \ref{fig:SFI} shows SFI spectra taken at a sequence of delays after the formation of $49f(2)$ Rydberg gases of varying density     

Here, we can see that a $49f(2)$ Rydberg gas with an estimated initial density $\rho_0 = 3 \times 10^{11} ~{\rm cm}^{-3}$ relaxes to plasma on a timescale of about 500 ns   Observations such as these agree well with the predictions of coupled rate-equation calculations   We can understand this variation in relaxation dynamics with $\rho_0$ and $n_0$ quite simply in terms of the corresponding density of prompt Penning electrons these conditions afford to initiate the avalanche to plasma   

Figure \ref{fig:scaled_rise} illustrates this, showing how rise times predicted by coupled rate-equation simulations for a large range of initial densities and principal quantum number match when plotted as a function of time scaled by the ultimate plasma frequency and fraction of prompt Penning electrons   The dashed line gives an approximate account of the scaled rate of avalanche under all conditions of Rydberg gas density and initial principal quantum number in terms of the simple sigmoidal function:

\begin{equation}
\frac{\rho_e}{\rho_0} = \frac{a}{b+e^{-c\tau}},
  \label{scaledEq1}
\end{equation}
where,
\begin{equation}
\tau = t \omega_e P_f^{3/4},
  \label{scaledEq2}
\end{equation}
in which $\omega_e$ is the plasma frequency after avalanche, $P_f$ is the fraction of prompt Penning electrons, and $a = 0 00062$,  $b =   0 00082$ and $c =     0 075$ are empirical coefficients   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{sim_analytical_density pdf}
   \caption{Rise in fractional electron density as a function of time scaled by the plasma frequency, $\omega_e$ and fraction, $\rho_e(t=0)/\rho_0 = P_f$, of prompt Penning electrons   Simulation results shown for $n_0 = 30$, 50 and 70 with initial densities, $\rho_0 = 10^9,~10^{10},~10^{11},~{\rm and}~10^{12}~{\rm cm}^{-3}$   
   }
\label{fig:scaled_rise}
\end{figure}


\subsection{Evolution to plasma in a Rydberg gas Gaussian ellipsoid}

As outlined above, the local density and principal quantum number together determine the rate at which a Rydberg gas avalanches to plasma   Our experiment crosses a 2 mm wide cylindrically Gaussian molecular beam with a 1 mm diameter TEM$_{00}$ $\omega_1$ laser beam to produce a Gaussian ellipsoidal distribution of molecules excited to the A $^2\Sigma^+$ $v=0, ~N'=0$ intermediate state   A larger diameter $\omega_2$ pulse then drives a second step that forms a Rydberg gas in a single $n_0f(2)$ state with the spatial distribution of the intermediate state   

We model this shaped Rydberg gas as a system of 100 concentric ellipsoidal shells of varying density \cite{haenelCP}   Coupled rate equations within each shell describe the avalanche to plasma   This rate process proceeds from shell to shell with successively longer induction periods, determined by the local density as detailed above   The rising conversion of Rydberg molecules to ions plus neutral dissociation products conserves the particle number in each shell   We assume that local space charge confines electrons to shells, conserving quasi-neutrality   Electrons exchange kinetic energy at the boundaries of each shell, which determines a single plasma electron temperature   

\begin{figure}[h!]
\centering
\includegraphics[width=  5 \textwidth]{shell_model_100}
   \caption{(top frame) Cross-sectional contour diagram in the $x,y$ plane for $z=0$ describing the distribution of ion plus electron density over 100 shells of Gaussian ellipsoid with initial dimensions, $\sigma_x= 0 75$ mm and $\sigma_y= \sigma_z = 0 42$ mm and an initial $n_0 = 50$ Rydberg gas density, $\rho_0 = 2 \times 10^{11}$ cm$^{3}$ after an evolution time of 100 ns   (bottom frame) Curves describing the (dashed) ascending ion and (solid) descending Rydberg gas densities of each shell as functions of evolution time, for $t=20$, 40, 60, 80 and 100 ns   
   }
\label{fig:shell}
\end{figure}

The upper frame of Figure \ref{fig:shell} shows contours of NO$^+$ ion density after 100 ns obtained from a shell-model coupled rate-equation simulation of the avalanche of a Gaussian ellipsoidal Rydberg gas of nitric oxide with a selected initial state, $50f(2)$ and a density of $2 \times 10^{11}$ cm$^{-3}$   Here, we simulate a relaxation that includes channels of predissociation at every Rydberg level and redistributes the energy released to electrons, which determines a uniform rising electron temperature for all shells   

For comparison, the lower frame plots curves describing the ion density of each shell as a function of time from 20 to 100 ns, as determined by applying Eq \ref{scaledEq1} for the local conditions of initial Rydberg gas density   This numerical approximation contains no provision for predissociation   Coupled rate-equation simulations for uniform volumes show that predissociation depresses yield to some degree, but has less effect on the avalanche kinetics \cite{Saquet2012}   Therefore, we can expect sets of numerically estimated shell densities, scaled to agree with the simulated ion density at the elapsed time of 100 ns to provide a reasonable account of the earlier NO$^+$ density profiles as a function of time    


\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{Shell_pop}
   \caption{Global population fractions of particles as they evolve in the avalanche of a shell-model ellipsoidal Rydberg gas with the initial principal quantum number and density distribution of Figure \ref{fig:shell}
   }
\label{fig:shell_yields}
\end{figure}

For each time step, the difference, $\rho_0 - \rho_e$ defines the neutral population of each shell   We assign a fraction of this population to surviving Rydberg molecules, such that the total population of NO$^*$ as a function of time agrees with the prediction of the shell-model simulation, as shown in Figure \ref{fig:shell_yields}   We consider the balance of this neutral population to reflect NO$^*$ molecules that have dissociated to form N($^4$S) $+$ O($^3$P)   Figure \ref{fig:shell} plots these surviving Rydberg densities as a function of radial distance for each evolution time   At the initial density of this simulation, note at each time step that a higher density of Rydberg molecules encloses the tail of the ion density distribution in $x$    

\subsection{Plasma expansion and NO$^+$ - NO$^*$ charge exchange as an avenue of quench}

We regard the ions as initially stationary   The release of electrons creates a radial electric potential gradient, which gives rise to a force, $-e\nabla \phi_{k,j}(t)$, that accelerates the ions in shell $j$ in direction $k$ according to \cite{Sadeghi 2012}:

\begin{align}
\frac{-e}{m'}\nabla \phi_{k,j}(t) = & \frac{\partial u_{k,j}(t)}{\partial t} \notag  \\
= & \frac{k_BT_e(t)}{m'\rho_j(t)} \frac{\rho_{j+1}(t) - \rho_j(t)}{r_{k,j+1}(t) - r_{k,j}(t)},
  \label{dr_dt}
\end{align}

\noindent where $\rho_j(t)$ represents the density of ions in shell $j$   

The instantaneous velocity, $u_{k,j}(t)$ determines the change in the radial coordinates of each shell, $r_{k,j}(t)$, 
\begin{equation}
\frac{\partial r_{k,j}(t)}{\partial t}=u_{k,j}(t) = \gamma_{k,j}(t) r_{k,j}(t),
  \label{dr_dt}
\end{equation}
\noindent which in turn determines shell volume and thus its density, $ \rho_j(t)$    
The electron temperature supplies the thermal energy that drives this ambipolar expansion   Ions accelerate and $T_e$ falls according to: 

\begin{equation}
\frac{3k_B}{2}\frac{\partial T_e(t)}{\partial t}= -\frac{m'}{\sum_{j}{N_j}}\sum_{k,j}{N_j u_{k,j}(t)\frac{\partial u_{k,j}(t)}{\partial t}},
  \label{dr_dt}
\end{equation}
\noindent where we define an effective ion mass, $m'$, that recognizes the redistribution of the electron expansion force over all the NO$^+$ charge centres by resonant ion-Rydberg charge exchange, which occurs with a very large cross section \cite{PPR}   
\begin{equation}
m' =\left (1+ \frac{\rho^*_{j}(t)}{ \rho_j(t)}\right) m ,
  \label{dr_dt}
\end{equation}
\noindent in which $\rho^*_{j}(t)$ represents the instantaneous Rydberg density in shell $j$ 

The initial avalanche in the high-density core of the ellipsoid leaves few Rydberg molecules, so this term has little initial effect   Rydberg molecules predominate in the lower-density wings   There, momentum sharing by charge exchange assumes a greater importance   

We see this most directly in the $\omega_2$ absorption spectrum of transitions to states in the $n_0 f(2)$ Rydberg series, detected as the long-lived signal that survives a flight time of 400 $\mu$s to reach the imaging detector   The balance between the rising density of ions and the falling density of Rydberg molecules depends on the initial density of electrons produced by prompt Penning ionization   As clear from Eq \ref{Eq:PenDens}, this Penning fraction depends sensitively on the principal quantum number, and for all principal quantum numbers, on the initial Rydberg gas density   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{w2_spectra}
   \caption{Double-resonant spectra of nitric oxide Rydberg states in the $nf$ series converging to NO$^+$ $v=0$, $N^+=2$ (designated, $nf(2)$), derived from the late-peak signal obtained after a flight time of 400 $\mu$s by scanning $\omega_2$ for $\omega_1$ tuned to NO A $^2\Sigma^+$ $v=0$, $N'=0$ for initial $nf(2)$ densities from top to bottom of 0 07, 0 10, 0 13, 0 19, 0 27, 0 30, 0 32 and $3 \times 10^{12}$ cm$^{3}$   
  }
\label{fig:w2_spectra}
\end{figure}

Figure \ref{fig:w2_spectra} shows a series of $\omega_2$ late-signal excitation spectra for a set of initial densities   Here, we see a clear consequence of the higher-order dependence of Penning fraction - and thus the NO$^+$ ion - NO$^*$ Rydberg molecule balance - on $n_0$, the $\omega_2$-selected Rydberg gas initial principal quantum number   This Penning-regulated NO$^+$ ion - NO$^*$ Rydberg molecule balance appears necessary as a critical factor in achieving the long ultracold plasma lifetime required to produce this signal   We are progressing in theoretical work that explains the stability apparently conferred by this balance   


\subsection{Bifurcation and arrested relaxation}

Ambipolar expansion quenches electron kinetic energy as the initially formed plasma expands   Core ions follow electrons into the wings of the Rydberg gas   There, recurring charge exchange between NO$^+$ ions and NO$^*$ Rydberg molecules redistributes the ambipolar force of the expanding electron gas, equalizing ion and Rydberg velocities   This momentum matching effectively channels electron energy through ion motion into the overall $\pm x$ motion of gas volumes in the laboratory   The internal kinetic energy of the plasma, which at this point is defined almost entirely by the ion-Rydberg relative motion, falls   Spatial correlation develops, and over a period of 500 ns, the system forms the plasma/high-Rydberg quasi-equilibrium dramatically evidenced by the SFI results in Figure \ref{fig:SFI}   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{Bifurcation pdf}
   \caption{$x,y$ detector images of ultracold plasma volumes produced by 2:1 aspect ratio ellipsoidal Rydberg gases with selected initial state, $40f(2)$ after a flight time of 402 $\mu$s over a distance of 575 mm   Lower frame displays the distribution in $x$ of the charge integrated in $y$ and $z$   Both images represent the unadjusted raw signal acquired in each case after 250 shots   
   }
\label{fig:bifurcation}
\end{figure}

In the wings, momentum redistribution owing to cycles of ion-Rydberg charge transfer retards radial expansion \cite{Pohl2003,PPR}   By redirecting electron energy from ambipolar acceleration to $\pm x$ plasma motion, NO$^+$ to NO$^*$ charge exchange dissipates electron thermal energy   This redistribution of energy released in the avalanche of the Rydberg gas to plasma, causes the ellipsoidal Rydberg gas to bifurcate \cite{Schulz-Weiling2016,Haenel2017}, forming very long-lived, separating charged-particle distributions   We capture the electron signal from these recoiling volumes on an imaging detector as pictured in Figure \ref{fig:bifurcation}   Here, momentum matching preserves density and enables ions and Rydberg molecules to relax to positions that minimize potential energy, building spatial correlation   

The semi-classical description of avalanche and relaxation outlined above forms an important point of reference from which to interpret our experimental observations   The laser crossed molecular beam illumination geometry creates a Rydberg gas with a distinctively shaped high-density spatial distribution   This initial condition has an evident effect on the evolution dynamics   We have developed semi-classical models that explicitly consider the coupled rate and hydrodynamic processes governing the evolution from Rydberg gas to plasma using a realistic, ellipsoidal representation of the ion/electron and Rydberg densities \cite{haenelCP}   No combination of initial conditions can produce a simulation that conforms classically with the state of arrested relaxation we observe experimentally  


\subsection{A molecular ultracold plasma state of arrested relaxation}

Thus, we find that spontaneous avalanche to plasma splits the core of an ellipsoidal Rydberg gas of nitric oxide  As ambipolar expansion quenches the electron temperature of this core plasma, long-range, resonant charge transfer from ballistic ions to frozen Rydberg molecules in the wings of the ellipsoid quenches the ion-Rydberg molecule relative velocity distribution  This sequence of steps gives rise to a remarkable mechanics of self-assembly, in which the kinetic energy of initially formed hot electrons and ions drives an observed separation of plasma volumes  These dynamics redistribute ion momentum, efficiently channeling electron energy into a reservoir of mass-transport  This starts a process that evidently anneals separating volumes to a state of cold, correlated ions, electrons and Rydberg molecules  

We have devised a three-dimensional spin model to describe this arrested state of the ultracold plasma in terms of two, three and four-level dipole-dipole energy transfer interactions (spin flip-flops), together with Ising interactions that arise from the concerted pairwise coupling of resonant pairs of dipoles \cite{SousMBL,SousNJP}   

The Hamiltonian includes the effects of onsite disorder owing to the broad spectrum of states populated in the ensemble and the unique electrostatic environment of every dipole   Extending ideas developed for simpler systems \cite{Burin1,Sondhi}, one can make a case for slow dynamics, including an arrest in the relaxation of NO Rydberg molecules to predissociating states of lower principal quantum number 

Systems of higher dimension ought to thermalize by energy transfer that spreads from rare but inevitable ergodic volumes (Griffiths regions) \cite{Sarang2, Roeck_griffith, RareRegions_rev, Thermal_inclusions}   However, a feature in the self-assembly of the molecular ultracold plasma may preclude destabilization by rare thermal domains:  Whenever the quenched plasma develops a delocalizing Griffiths region, the local predissociation of relaxing NO molecules promptly proceeds to deplete that region to a void of no consequence 

In summary, the classical dynamics of avalanche and bifurcation appear to create a quenched condition of low temperature and high disorder in which dipole-dipole interactions drive self-assembly to a localized state purified by the predissociation of thermal regions   We suggest that this state of the quenched ultracold plasma offers an experimental platform for studying quantum many-body physics of disordered systems  \\

\section{Acknowledgments}
This work was supported by the US Air Force Office of Scientific Research (Grant No  FA9550-17-1-0343), together with the Natural Sciences and Engineering research Council of Canada (NSERC), the Canada Foundation for Innovation (CFI) and the British Columbia Knowledge Development Fund (BCKDF)  





ground_truth: A subset that form via kinetic rate processes from state-selected Rydberg gases.