ASYMPTOTIC ESTIMATION FOR MINIMAL PLASMA RADIUS IN A SPHERICALLY SYMMETRIC MAGNETIZED TARGET FUSION REACTOR MODEL

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1 ASYMPTOTIC ESTIMATION FOR MINIMAL PLASMA RADIUS IN A SPHERICALLY SYMMETRIC MAGNETIZED TARGET FUSION REACTOR MODEL MICHAEL LINDSTROM Astract. Nonlinear conservation laws with moving oundaries often arise in applications such as gas and plasma dynamics and more recently in the context of nuclear fusion reactors. Gaining insight into the nature of solutions to such models and their dependence upon design parameters is often done numerically. Instead of relying upon numerics, in this paper we use formal asymptotics to study a model of a magnetized target fusion reactor for producing energy. We will asymptotically solve the compressile euler equations with two free moving oundaries, with an algeraic coupling of fluid pressure at one of the free oundaries. The model consists of an intense pressure eing imparted on the oundary of a large sphere holding molten lead-lithium over a very short time scale, in the middle of which is a smaller concentric sphere containing plasma. A localized disturance quickly propagates towards the plasma, wherey part of the energy is reflected and part compresses the plasma. The plasma pressure is modelled y a magnetic pressure, dependent upon its radius. Both the outer oundary of the lead-lithium sphere and the plasma oundary are free. Using matched asymptotics and various techniques for solving linear hyperolic systems including Riemann Invariants and using the velocity potential formulation of the linear acoustic equations, we estimate the maximum compression of the plasma. This maximum compression is otained in two ways: with an energy argument originally attriuted to Lord Rayleigh in studying ule dynamics and a new asymptotic argument for the maximum compression of a ule sujected to the Rayleigh-Plesset equation. This estimation gives insights into the factors relevant in designing an energy-producing reactor. Keywords: Compressile Euler equations, hyperolic conservation laws, free oundaries, formal asymptotics, magnetized target fusion 1. Introduction. Developing technology suitale for harvesting fusion energy and using it for power is an exciting and active area of research. Thanks to advances in technology, various projects worldwide have gotten closer to achieving this aim [4]. Recently, a paper was written illustrating a numerical framework that could e used in simulations of magnetized target fusion; in the paper, a simplified model of one particular reactor design was investigated [1]. In this paper, we investigate the model further from an analytic perspective. Mathematical models of fluids often deal with nonlinear conservation laws such as the Euler or Navier Stokes equations. These equations are often difficult to solve or approximate analytically. In this paper, we will e dealing with the hyperolic conservation laws for mass and momentum conservation given y the compressile Euler equations (see [5] for relevant ackground to these equations). There is an additional ostacle we deal with, which are the free oundaries that arise naturally in our application. Free oundary, or moving oundary prolems, entail a physical oundary (possily the domain of a PDE or an interface) evolving with time due to the conditions at the oundary itself. Instead of the oundary position eing prescried in the prolem or eing a known function of time, it must e solved for within the prolem as its evolution depends upon the nature of the solution at any given time, where the solution evolves as a PDE in oth space and time. A common example is that of a Stefan prolem descriing the position of the oundary of an ice cue as it melts: the velocity of the oundary depends on the jump in heat flux at the interface which changes with time [14]. Another example, which we will deal with, relates to ule dynamics and the motion of the oundary of a gas ule in a fluid (see [] for a thorough introduction to ule dynamics). This work is novel in that asymptotic analysis allows us to overcome oth the ostacle of the nonlinear conservation laws and the difficulty of the free oundaries to yield Mathematics Department, University of British Columia, mlrtlm@math.uc.ca 1

2 insightful results for the application. In our context, the fluid will e lead-lithium etween the free oundaries of an inner spherical plasma all and an outer spherical shell impacted y pistons. The paper is organized into the following sections: this introduction, where we overview the physical motivation and context for the prolem; section, where we outline the equations and nondimensionalize them; section 3, where we use formal asymptotic analysis to study the system; section 4, where we validate the asymptotics y comparing with numerical results; and finally section 5 where we conclude and consider future work A Design for a Magnetized Target Fusion Reactor. Nuclear fusion is a complicated process wherey lighter elements fuse together to produce heavier elements. Fusion technology researchers often consider the use of deuterium or tritium fuels, producing helium and free neutrons upon fusion [1]. The released energy can e collected and used. Many factors impact fusion, ut three very ig factors in determining if a deuterium or tritium fuel can fuse to yield more energy output than input are pressure (or particle density), temperature, and confinement time [11]. In general, the greater the density and temperature that can e achieved and for the longer the time, the more effective the device. The design we consider is ased on a Canadian fusion research company, General Fusion [8] [9]. The apparatus eing engineered consists of a large sphere of molten lead with a vertical axis that is empty through which a toroidal plasma will e fired. Pistons will impart a large pressure to the outer sphere over a fast time scale. This will generate a disturance in the fluid which will propagate and focus radially inward, reaching the centre. The plasma will e fired into the central axis at a precise time so as to e at the centre when the pressure wave reaches it. Our ojective in this paper is to predict the minimal radius of the plasma, as this is a key parameter in how efficient such a design could potentially e. Other factors such as confinement time are not analyzed. 1.. Physical Assumptions. Plasma pressures have oth a gas pressure and magnetic pressure component. The gas pressure is due to the particle density and the magnetic pressure arises due to magnetic fields. In many plasmas, the magnetic pressure is often much larger initially. Although the plasma in the device is toroidal [8], we take the plasma to e spherical where its major radius is the radius of the sphere in the model and we assume spherical symmetry for all components. The plasma pressure is modelled y the magnetic pressure (scaling inversely with the fourth power of its radius) [17], where we neglect the gas pressure completely. The pressure imparted y the pistons is descried y a aseline pressure plus a very large pressure modulated y a Guassian with a very short time-scale. The pressure of lead-lithium is taken to e linear in density. Our model will neglect mixing effects and radiation losses. These simplifications allow us to write more tractale equations and gain qualitative insight into how the most asic engineering parameters influence the compression of the plasma.. Equations and Nondimensionalization. Here we present the equations relevant to this study and nondimensionalize the system so that formal asymptotics can e applied. The values and explanations of the constants used are provided in tales 1 and..1. Euler Equations. Within the lead-lithium, we work with the Euler equations for mass and momentum conservation. Denoting the mass density y ϱ, the local fluid velocity y V and the pressure y P in a space-time coordinate system (X, T ), we have: ϱ T + X (ϱv) = 0 and (ϱv) T + X P + X (ϱv V) = 0

3 Variale Description Value C s Sound speed of lead at 1 atm 090 m/s ϱ 0 Density of lead at 1 atm kg/m 3 P atm Atmospheric pressure Pa P plasma,0 Initial plasma pressure 5 MPa P max Maximum piston pressure GPa Γ Coefficient for magnetic pressure growth 8000 N m T 0 Guassian decay time scale 45 µs R inner,0 Initial plasma radius 0. m R outer,0 Initial lead sphere radius 1.5 m Tale 1: Characteristic constants for this application. which with spherical symmetry, so that the vector quantities only have components in the radial direction and V = V, reduce to (.1) ϱ T + (ϱv ) R + R (ϱv ) = 0 and (ϱv ) T + P R + (ϱv ) R + R (ϱv ) = 0. The space-time coordinates are now (R, T ). We will model the equation of state of lead-lithium y linearizing its pressure P aout its density at atmospheric pressure ϱ 0. From experimental data[15], the sound speed C s of lead at atmospheric pressure is 090 m/s. The square of the sound speed is the derivative of P with respect to density ϱ so this allows us to express P (ρ) as an approximate linear function. We specify P (ϱ) = Cs (ϱ ϱ 0 ) + P atm (.). Note that we are approximating the equation of state for lead-lithium y the linearized equation of state for solid lead-lithium... Boundary Conditions. Due to the impact of the pistons and the interaction of the wave with the plasma oundary, this is a moving oundary prolem. We denote R L to e the radius of the inner wall of the lead-lithium sphere (the plasma radius), and R R to e the radius of the outer wall of the lead-lithium sphere. The local fluid velocity at the inner and outer oundaries matches the velocities of these walls respectively: (.3) V R (T ) dr L dt = V (R L(T ), T ) and V R (T ) dr R dt = V (R R(T ), T ). At any given time, the lead-lithium exists etween R L and R R. At the inner wall, the pressure is given y the plasma magnetic pressure. This gives (.4) P L (T ) = P (R L (T ), T ) = Γ R L (T ) 4 = P plasma,0 (R L (T )/R L ()) 4. This is an algeraic coupling of the fluid pressure and plasma pressure at this wall. At the outer wall, the pressure is given y the piston pressure which is modelled y a Gaussian pressure in addition to a aseline minimum applied pressure (.5) P R (T ) = (P max P plasma,0 )e T /T 0 + Pplasma,0. 3

4 Variale Notation/Equation Characteristic Scale Density ϱ ϱ = ϱ 0 11, 340 kg m 3 Pressure P P = Pmax GPa Distance R R = Router,0 1.5 m Time T T = R ϱ/ P 3.57 ms Velocity V V = P / ϱ 40. m s 1 Tale : Characteristic dimensions for the system. The first three entries are chosen y the theoretical operating conditions. The last two entires can e derived from the first three..3. Initial Conditions. The maximum piston pressure occurs at T = 0, ut we allow for negative times relative to this. The system egins at T = where (.6) V (R, ) = 0, P (R, ) = Γ R L () 4, ϱ(r, ) = P (R, ) P plasma,0 C s + ϱ 0 with (.7) R L () = R inner,0 and R R () = R outer,0..4. Nondimensionalization. To nondimensionalize, we will look at (.1) with the change of variales ϱ = ϱρ, V = V v, P = P p, R = Rr, T = T t where the ars denote characteristic dimensional quantities and ρ, v, p, r, and t are dimensionless. We wind up with: (.8) ϱ T ρ ϱ V t + R (ρv) ϱ V r + R r (ρv) = 0 and ϱ V T (ρv) t + P R p ϱ V ϱ V r + R (ρv ) r + R r (ρv ) = 0. By matching dimensional terms in (.8) 1, we have ϱ/ T = ϱ V / R so that V = R/ T (.9). By matching dimensional terms in (.8), we have ( ϱ V )/ T = P / R giving ϱ = ( P T )/( R V ) (.10) and ( ϱ V )/ T = ( ϱ V )/ R giving the same as in (.9). Equations (.9) and (.10) give good guidance as to reasonale scalings for the system, and we also have some freedom. We will choose R = 1.5 m, the initial outer radius; ϱ = kg/m 3, the density of lead at atmospheric pressure; P = GPa, the maximum pressure imparted on the outer wall. Sustituting (.9) into.10, we have ϱ = ( P T )/( R ) so that T = R ϱ/ P = 3.57 ms. Using this now known value of T in (.9) gives V = 40. m/s. A tale summarizing this is given in tale..5. Selecting an Asymptotic Parameter. Through the nondimensionalization, various dimensionless parameters appear. Upon considering the physics of the prolem, an intense pressure imparted over a very short time-scale (45µs 3.57 ms), we will define ϵ = T 0 / T = such that the Gaussian component of the pressure ecomes e (t T /T 0 ) = e t /ϵ. With respect to this value of ϵ, the different dimensionless parameters have characteristic orders. For example, in dimensionless form, p = c (ρ 1)+d where c = Cs ϱ/ P 4.7 and d = P atm / P The density of leadlithium corresponding to the maximum pressure is 1.04, which is well descried y ρ = 1 + O(ϵ). If ρ = 1 + O(ϵ) corresponds to p = O(1) then c = O(ϵ 1 ). Thus, it is reasonale to write p(ρ) = ϵ (ρ 1) + aϵ5/ where = and a =.84. In fact, since we are aiming for a leadingorder ehaviour of the system, we will neglect aϵ 5/ entirely since it is so small. Also due to its 4

5 Fig. 1: The geometry we consider is spherically symmetric, with r L and r R free oundaries. negligile size, O(ϵ 3/ ), we neglect the dimensionless P plasma,0 term in the pressure condition at the right wall. Discarding these terms can e justified a posteriori y noting that in all the analysis to come, only pressures that are O(ϵ) alance in the relevant equations. Proceeding in a similar way with the remaining equations and oundary conditions, and making use of the fact that knowing either p or ρ is equivalent, the final set of equations and conditions we consider are presented elow, valid for r L < r < r R, < t <. A diagram of our geometry is given in figure 1. (.11) ρ t + (ρv) r + r (ρv) = 0, (ρv) t + ϵ ρ r + (ρv ) r + r (ρv ) = 0 (.1) p = (ρ 1) ϵ (.13) v L (t) = dr L dt = v(r L (t), t), v R (t) = dr R dt = v(r R (t), t) (.14) p L (t) = p(r L (t), t) = γϵ7/ r L (t) 4, p R(t) = p(r R (t), t) = e t /ϵ (.15) v(r, ) = 0, p(r, ) = γϵ7/ r L () 4 ρ(r, ) = 1 (.16) r L () = χϵ 1/, r R () = 1 We summarize the constants in the tale 3. Our main ojective is to estimate the smallest value of r L which we denote r. 3. Formal Asymptotic Analysis. Before presenting the asymptotic analysis, we will outline the asic scalings, physical mechanisms, and mathematical strategies. Much of the analysis is done on the equations of linear acoustics as the asymptotics allow for linear equations. The analysis has 5

6 Variale Meaning Formula Value T ϵ pulse time scale 0 T C scaled sound speed s ϵ V γ scaled magnetic pressure coefficient χ scaled initial inner radius P plasma,0 R 4 inner,0 Tale 3: Dimensionless parameters of the system. P R 4 ϵ 7/ 3.5 R inner,0 R 1.19 ϵ three distinct prolems. First (region I), over a very fast time scale O(ϵ), a narrow pressure/velocity pulse is formed of width O( ϵ) which moves radially inward without focusing. Riemann invariants are used to solve the system. Secondly (region II), the pulse moves radially inward over a time of O( ϵ) covering an O(1) distance, growing in amplitude y a factor of 1/ ϵ due to focusing where we use the velocity potential and a transformation that turns the linear acoustic equations for a sphere into one-dimensional wave equations. Finally, in region III, the pulses interact with the plasma over a time scale O(ϵ) and spatial scale O( ϵ) with much of the energy reflecting ut a small portion of useful kinetic energy remaining which we find indirectly y various integrals of the oundary conditions. By an energy argument and also y matched asymptotics involving the Rayleigh-Plesset equation, we compute the plasma radius at which the velocity of the plasma oundary switches from negative (i.e. compressing) to zero at leading order, which we define as the minimum plasma radius. The actual space and time scales over which this velocity turnaround point occur are O(ϵ) and O(ϵ 5/4 ) respectively Region I Pulse Formation Setup. To analyze the formation of the pulse, we rescale time so that t = ϵτ, and make the ansatzes that ρ 1+ϵρ 1 +ϵ l ρ and v ϵ m 0 v 0 +ϵ m 1 v 1. Seeing as the peak impulse pressure of p = O(1) corresponds to ρ = 1 + O(ϵ) these expansions are justified. In order to capture the pulse formation, we also define y = r 1 ϵ. This leads to ρv ϵ m 0 v n 0 + ϵ m 1 v 1 + ϵ m0+1 ρ 1 v 0 and ρv ϵ m v0, r = 1 + ϵ 1/ y, /r ϵ 1/ y, r = ϵ n y and t = ϵ 1 τ. Using this in (.11), we have to various leading orders: ϵ 1 (ϵρ 1 + ϵ l ρ ) τ + ϵ n (ϵ m0 v 0 + ϵ m1 v 1 + ϵ m0+1 ρ 1 v 0 ) y + (ϵ m0 v 0 + ϵ m1 v 1 + ϵ m0+1 ρ 1 v 0 ) = 0 ϵ 1 (ϵ m0 v 0 + ϵ m1 v 1 + ϵ m0+1 ρ 1 v 0 ) τ + ϵ n 1 (ϵρ 1 + ϵ l ρ ) y + ϵ 1/ (ϵ m0 v 0) y + (ϵ m0 v 0) = 0. Choosing a rich alance to leading order, we choose m 0 n = 0 and m 0 1 = n so that m 0 = n = 1/. This gives ρ 1,τ + v 0,y + ϵ l 1 ρ τ + ϵ m1 1/ v 1,y + ϵ 1/ v 0 = 0 ϵ 1/ (v 0,τ + ρ 1,y ) + ϵ m1 1 v 1,τ + ϵ l 3/ ρ,y + ϵ 1/ (v 0) y = 0. We choose another rich alance of terms with l 1 = m 1 1/ = 1/ so that l = 3/, and m = 1. We thus arrive at ρ 1 + ϵρ 1 + ϵ 3/ ρ, v ϵ 1/ v 0 + ϵv 1 6

7 with t = ϵτ and y = r 1. ϵ 1/ To deal with oundary conditions, we first need to find an asymptotic expression for r R 1 + ϵ k r R0 (τ). To leading order v R = dr R dt = O(ϵ 1/ ). Thus, r R (t) = ϵk r R0 (t) = O(ϵ1/ ). Using d/dt = ϵ 1 d/dτ, we have ϵ k 1 r R0 (τ) = O(ϵ1/ ) so that k = 3/. It is worth mentioning that when we write r R0 (τ) we really mean r R0 (t = ϵτ) and not r R0 (t = τ)! Throughout this paper we use the shorthand that a dependent variale evaluated at a rescaled variale should e interpreted y the proper scalings in the original (r, t) variales. The outer wall changes y a distance O(ϵ 3/ ). If we denote y R as the position of the outer oundary in the y coordinates, then y R = r R 1 = ϵr ϵ 1/ R0 (τ). The oundary condition that must e upheld is p R (τ) = ϵ (ρ 1) = e τ so that ρ 1 (r R (τ), τ) + ϵ 1/ ρ (r R (τ), τ) = 1 e τ. We can now perform a Taylor expansion (seeing as the prolem is linear and the forcing terms are continuous and differentiale). Keeping just a few terms, we otain (ρ 1 (0, τ) + ρ 1y (0, τ)(ϵr R0 ) + ϵ 1/ ρ (0, τ)) = e τ which implies (3.1) ρ 1 (0, τ) = 1 e τ and ρ (0, τ) = 0. As far as initial conditions go, (3.) ρ 1 (y, ) = ρ (y, ) = v 0 (y, ) = v 1 (y, ) = Riemann Invariants. Both the leading order and first correction terms can e written in the form ( ) ( ) ( ) ( ) ρτ 0 1 ρy f(y, τ) (3.3) + = 0 g(y, τ) v τ and such prolems can e solved effectively with a technique known as Riemann Invariants [3]. Oserve the matrix matrix has eigenvalues ± with corresponding left eigenvectors l ± = (±, 1). Upon left multiplying (3.3) y l ±, we otain (±ρ + v) τ ± (±ρ + v) y = ±f + g. Thus, the system has characteristics. Denoting c ± = ±ρ + v, we have that along dy/dτ = ±, dc ± /dτ = ±f + g h ±. Given that ρ = v = 0 at τ =, (so c ± = 0 at τ = ) we can compute c + (y, τ) = 0 + τ h(ỹ( τ), τ)d τ where ỹ( τ) = y + ( τ τ) descries the straight pathway along the rightgoing characteristic starting at τ = and reaching (y, τ). Thus, (3.4) c + (y, τ) = τ v y h + (y + ( τ τ), τ)d τ. To find c (y, τ), we need to know its value at the point along the τ axis whose leftgoing characteristics eventually reach (y, τ). As c + is known along the τ axis, and ρ is also specified there, we can find c. The leftgoing characteristic path leading to (y, τ) can e descried y (ỹ( τ), τ) = ((τ τ), τ) for τ + y/ τ τ. From c + (0, τ + y/) = ρ(0, τ + y/) + v(0, τ + y/) we get c (0, τ + y/) = ρ(0, τ + y/) + v(0, τ + y/) = c + (0, τ + y/) ρ(0, τ + y/). We thus have (3.5) c (y, τ) = c + (0, τ + y/) ρ(0, τ + y/) + 7 τ τ+y/ h ((τ τ), τ)d τ

8 Given c ±, we can recover (3.6) ρ = 1 (c+ c ) and v = 1 (c+ + c ) Leading Order. At leading order, using (3.) and (3.1) 1 we have ( ρ1,τ v 0,τ ) + ( ) ( ) ρ1,y = 0, ρ(0, τ) = 1 e τ, ρ(y, ) = v(y, ) = 0. Thus, c + (y, τ) = 0 y (3.4) since the forcing function h = 0. And c (y, τ) = 1 e (τ+y/) = e (τ+y/) y (3.5). We then use (3.6) to get v 0,y (3.7) ρ 1 (y, τ) = 1 e (τ+y/) and (3.8) v 0 (y, τ) = 1 e (τ+y/) Correction. At the next order, we otain ( ρτ v 1τ ) + ( ) ( ) ρy = v 1y ( ) v0 0 suject to ρ (0, τ) = 0, ρ (y, ) = v 1 (y, ) = 0 as per (3.1) and (3.). In this case, h ± = v 0. By (3.4), c + (y, τ) = τ ( )v 0 (y + ( τ τ), τ)d τ = We can evaluate this integral with a sustitution u = τ + y/ τ to get Similarly, y (3.5), c + = τ+y/ e u du = τ y/ τ e ( τ+y/ τ) d τ. π e u du = erfc( τ y/). π τ π c (y, τ) = erfc( τ y/) 0+ ()v 0 ((τ τ), τ)d τ = τ+y/ erfc( τ y/)+ y e (τ+y/). We once again use (3.6) to furnish ρ (y, τ) = y e (τ+y/) and v 1 (y, τ) = 8 π erfc( τ y/) + y e (τ+y/).

9 Region I Results. To leading order, the density and velocity are descried y leftgoing plane waves. Due to the scalings, we have een looking on such a small scale that the system hasn t quite noticed the spherical nature of the prolem and everything appears flat (for y = O(1)). The divergent terms with y prefactors, however, actually result from the spherical geometry as we shall see in the following section. We have (3.9) ρ 1 + ϵ 1 e (τ+y/) ϵ 3/ y e (τ+y/) and 1/ 1 π (3.10) v ϵ e (τ+y/) + ϵ( erfc( τ y/) + y e (τ+y/) ). A plot of these solutions at τ = 1 is given in figure. In the r-coordinates, the characteristic widths of these pulses are O( ϵ). The plot includes numerical validation of the results which are discussed in more detail in section Region II Spherical Focusing Setup. Having otained the asic shape of the pulses eing formed, we can now analyze how these pulses ehave as they move towards the centre of the sphere. From region I, we have ρ 1 + ϵρ 1 + ϵ 3/ ρ and v ϵ 1/ v 0 + ϵv 1 and we shall make a similar ansatz here. We will consider r = O(1) and choose a time scale t = ϵ k T. With these scalings, (.11) ecomes: ϵ k (ϵρ 1 + ϵ 3/ ρ ) T + (ϵ 1/ v 0 + ϵv 1 ) r + r (ϵ1/ v 0 + ϵv 1 ) = 0 ϵ k (ϵ 1/ v 0 + ϵv 1 ) T + ϵ 1 (ϵρ 1 + ϵ 3/ ρ ) r + (ϵv 0) r + r (ϵv 0) = 0. Clearly k = 1/ yields a suitale alance. The equations are: ρ 1,T + v 0,r + r v 0 = 0 and v 0,T + ρ 1,r = 0 where the identical equations are upheld for ρ and v 1. The initial and oundary conditions are more sutle here. They cannot e written down clearly ecause of the different time scales (typically, to match at T = 0, we would require τ ut in this case, the solutions found in region I tend to 0). However, y solving the hyperolic systems at different orders, we can find the functional form of the solution, which allows us to find the solutions y matched asymptotics Spherical Linear Acoustic Limit. For the two leading orders, the systems of PDEs that need to e solved are of the form (3.11) ρ T + v r + r v = f and v T + ρ r = g. Such equations also appear in linear acoustic prolems, often with f = g = 0 (and in our case we will not need to explicitly solve such an equation with f, g 0). 9

10 0 =0.557,χ=1.19,γ=3.5,ε=0.016,t=ε/ =0.557,χ=1.19,γ=3.5,ε=0.016,t=ε/ Velocity v num ε 0.5 v 0 ε 0.5 v 0 + ε v 1 Density ρ num 1+ε ρ ε ρ 1 + ε 1.5 ρ r r Velocity =χ=γ=1,ε=0.001 v num ε 0.5 v 0 ε 0.5 v 0 + ε v r Density ρ num 1+ε ρ 1 1+ε ρ 1 + ε 1.5 ρ =χ=γ=1,ε= r Fig. : Plots depicting the profiles of the velocity and density at a very small value of t to validate the asymptotics of region 1. In the case of the second row, the two term asymptotic expansions agree so well with the numerics the plots cannot e distinguished. To solve this, we will make use of a velocity potential ϕ(r, T ) = r v(s, T )ds. With this sustitution, we have (3.1) ρ T + ϕ rr + r ϕ r = f and ϕ rt + ρ r = g. Integrating (3.1) from r =, assuming ρ = 0 at r =, we have ϕ T (r, T ) + ρ(r, T ) = r g(s, T )ds G(r, T ) so that ϕ T T = ρ T +G T. Using (3.1) 1 here now gives ϕ T T = ( ϕ rr r ϕ r + f) + G T so (3.13) ϕ T T = (ϕ rr + r ϕ r) + G T f. Note that y taking a time derivative of (3.11) 1 and a space derivative of (3.11) we have (3.14) ρ T T = v rt r v T + f T and v rt = g r ρ rr. 10

11 Equations (3.14) and (3.14) 1 can then e used in (3.11) 1 to otain ρ T T = (g r ρ rr ) r (g ρ r ) + f T so that (3.15) ρ T T = (ρ rr + r ρ r) + f T r g g r. A symmetry argument now helps to solve (3.13) and (3.15). If ϕ = Φ/r and ρ = K/r then ϕ rr + r ϕ r = Φ rr r and similarly for K. Thus we can transform these equations into one-dimensional wave equations [16]: (3.16) Φ T T = Φ rr + S 1 (r, T ), K T T = K rr + S (r, T ) with S 1 (r, T ) = r(g T f), and S (r, T ) = r(f T r g g r) Leading Order and Correction in Outer Region. The system of equations for ρ 1 and v 0 and ρ and v 1 are the same, and in oth cases the source terms are fortunately zero. With no source terms, and with only leftgoing waves (since there are no sources going right), the solutions of (3.16) are K = P (r + T ) and Φ = Q(r + T ) respectively. Then, eing mindful of the fact that the pulses generated have a width of size O( ϵ) and are leftgoing originating near r = 1, we can write the solutions to the systems of equations as ρ 1 = v 0 = ϵ( Q 0 ( r 1+T ϵ ) ϵ ) ), and v 1 = ϵ( Q 1 ( r 1+T ϵ ) ϵr Q 0( r 1+T r ϵr Q 1( r 1+T ϵ ) r ). r 1+T P1( ϵ ) r, ρ = r 1+T P( ϵ ) r, We will now take these solutions and express them in the inner (y, τ)-coordinates. We will apply the Van-Dyke matching of inner and outer solutions [6], and since we have a two-term expansion for the inner solution, we also take two terms in the outer solution terms. We have t = ϵτ = ϵ 1/ T so T = ϵ 1/ τ and r = 1 + ϵ 1/ y so that r 1+T ϵ = y + τ. ρ 1 = P 1 (y + τ) 1 + ϵ 1/ y = P 1 (y + τ) ϵ 1/ y P 1 (y + τ), ρ = P (y + τ) 1 + ϵ 1/ y = P (y + τ) ϵ 1/ y P (y + τ), v 0 = Q 0 (y + τ) 1 + ϵ 1/ y ϵ1/ Q0 (y + τ) (1 + ϵ 1/ ) = Q 0 (y + τ) + ϵ 1/ ( Q 0 (y + τ) Q0 (y + τ)), v 1 = Q 1 (y + τ) 1 + ϵ 1/ y ϵ1/ Q1 (y + τ) (1 + ϵ 1/ ) = Q 1 (y + τ) + ϵ 1/ ( Q 1 (y + τ) Q1 (y + τ)) Now we write (1 + ϵρ 1 + ϵ 3/ ρ ) outer = (1 + ϵρ 1 + ϵ 3/ ρ ) inner 1 + ϵ( 1 ) + ϵ 3/ ( y e (τ+y/) e (τ+y/) ) = 1 + ϵ( P 1 (y + τ)) + ϵ 3/ ( y P 1 (y + τ) + P (y + τ)) so that P 1 (x) = 1 e (x/) and P (x) = 0. Doing the same for the velocity, we write: 11

12 (ϵ 1/ v 0 + ϵv 1 ) outer = (ϵ 1/ v 0 + ϵv 1 ) inner 1/ 1 π ϵ e (τ+y/) + ϵ( erfc( τ y/) + y e (τ+y/) ) = ϵ 1/ Q0 (y + τ) + ϵ( Q 0 (y + τ) Q0 (y + τ) + Q 1 (y + τ)) giving us Q 0 (x) = π erfc(x/) and Q 1 (x) = 0 so that Q1 is constant. Since P i (x/ ϵ) = P i (x) and Q i (x/ ϵ) = Q i (x), we have that during this initial phase of focusing, (3.17) ρ 1 + ϵ( 1 r 1+T r e ( ) ϵ ) + o(ϵ 3/ ) and (3.18) v ϵ 1/ ( 1 r e ( r 1+T ϵ ) π 1 + T ) + ϵ( erfc( r r )) + o(ϵ). ϵ Leading Order and Correction in Inner Region. As the pulses move inwards, due to the 1/r and 1/r terms, the amplitudes grow and the asymptotic expansions we arrived at aove lose their validity. To descrie the region where the pulses have grown and arrive at the inner wall, we rescale space and time. We let r = ϵ 1/ σ and t t s = ϵˆt, where t s is a shifting parameter such that the peak of the leftgoing pressure wave will have reached the plasma wall at r = χϵ 1/ at ˆt = 0. We anticipate different scalings for the asymptotic expansions. From r = O(1) to r = O( ϵ), the first perturation term in the density perturation given in (3.17) should have grown from O(ϵ) to O( ϵ). Similarly, from (3.18), the velocity amplitude should have grown from O( ϵ) to O(1) and the velocity term that was O(ϵ) should also ecome O(1) since it grows like 1/r. Therefore, in this next inner region, we write ρ 1 + ϵ 1/ ρ 1 + ϵρ and v v 0 + ϵ 1/ v 1. Using r = ϵ 1/ σ and t = ϵ 1 ˆt, (.11) ecomes ϵ 1 (ϵ 1/ ρ 1 + ϵρ )ˆt + ϵ 1/ (v 0 + ϵ 1/ (v 1 + ρ 1 v 0 )) σ + ϵ 1/ σ (v 0 + ϵ 1/ (v 1 + ρ 1 v 0 )) = 0 ϵ 1 (v 0 + ϵ 1/ (v 1 + ρ 1 v 0 ))ˆt + ϵ 3/ (ϵ 1/ ρ 1 + ϵρ ) σ + ϵ 1/ (v 0) σ + ϵ 1/ σ v 0 = 0 which upon looking at alancing terms yield two sets of PDEs: (3.19) ρ 1,ˆt + v 0,σ + σ v 0 = 0, v 0,ˆt + ρ 1,σ = 0 and (3.0) ρ,ˆt + v 1,σ + σ v 1 = (ρ 1 v 0 ) σ σ ρ 1v 0, v 1,ˆt + ρ,σ = (ρ 1 v 0 )ˆt (v 0) σ σ v 0. By virtue of the fact these PDEs can e solved and matched to (as done in the following paragraphs), we can e confident in the scalings chosen. We will now consider oundary conditions. For a 1

13 timescale that is O(ϵ), given a velocity v = O(1), the inner wall can only move a distance O(ϵ) and thus the inner wall position remains O( ϵ) (recall that r L () = χ ϵ). We can find the leading order nonzero contriution to the inner wall motion in the (σ, ˆt)-coordinates. We define σ L = r L / ϵ. Then σ L (ˆt) = χ + ˆt ϵv(σl (ŝ), ŝ)dŝ. The factor of ϵ comes from the fact that dr L dt = v(r L (t), t) = dϵ1/ σ L dϵˆt = ϵ 1/ dσ L dˆt. Thus, ˆt ˆt σ L (ˆt) = χ + ϵv(χ + O( ϵ), ŝ)dŝ = χ + ϵ 1/ v(χ, ŝ)dŝ + O(ϵ). As the pressure at the inner wall is given y p = γϵ 7/ /r 4 L, for r L = O( ϵ), p = O(ϵ 3/ ) = ϵ (ϵ1/ ρ 1 (σ L, ˆt)+ϵρ (σ L, ˆt)+...) so that y a Taylor expansion, ϵ 1/ ρ 1 (χ, ˆt)+ϵ(ρ 1σ (χ, ˆt) ˆt v 0(χ, ŝ)dŝ+ ρ (χ, ˆt)) + o(ϵ) = 0. This gives us oundary conditions ˆt (3.1) ρ 1 (χ, ˆt) = 0 and ρ (χ, ˆt) = ρ 1σ (χ, ˆt) v 0 (χ, ŝ)dŝ. If we again consider (3.19) in potential form with ϕ 0 (σ, ˆt) = σ v 0(ˆσ, ˆt)dˆσ then y integrating the equation from σ = to σ = χ, ϕ 0ˆt (χ, ˆt) + ρ 1 (χ, ˆt) = 0 where we used ρ 1 = ϕ 0 = 0 at σ =. Since ρ 1 (χ, ˆt) = 0, we must have (3.) ϕ 0ˆt (χ, ˆt) = 0. Similarly y integrating (3.0) we have ϕ 1ˆt (χ, ˆt) + ρ (χ, ˆt) = χ ( (ρ 1v 0 )ˆt (v 0) σ σ v 0)dσ which with (3.1) gives ˆt χ (3.3) ϕ (χ, 1ˆt ˆt) = ρ 1σ (χ, ˆt) v 0 (χ, ŝ)dŝ + ( (ρ 1 v 0 )ˆt (v 0) σ σ v 0)dσ. We can find the form of the solution to (3.19) using the solutions of (3.16) with S 1 = S = 0. We note that there are oth incoming signals (from σ = ) and outgoing signals (from the interaction of the pulses with the plasma) so we have waves propagating in oth directions. Also, we are on a spatial scale of O( ϵ) so that it is not necessary to rescale the argument of the solutions like we did in the previous section. As ˆt, we will neglect the rightgoing wave and consider only the incoming wave. We write ρ 1 = P + 1 (σ+ˆt) σ, v 0 = Q+ 0 (σ+ˆt) σ Q+ 0 (σ+ˆt) σ. Expressing the outer solutions in (r, T )-coordinates in the inner coordinates and going to leading order, and using the fact that any function of r 1+T is a function of σ χ ϵ + ˆt, we have (1 + ϵ 1/ ρ 1 ) inner = (1 + ϵ 1/ ρ 1 ) outer P ϵ 1/ 1 ( σ χ + ˆt) σ = 1 + ϵ( 1 e ( σϵ1/ giving us a leading order solution in the inner region for the density (3.4) ρ 1 = 1 σ χ σ e ( +ˆt) (ˆt ). 13 σ χ +ˆt) )

14 Also, Q + 0 ( σ χ + ˆt) Q + 0 ( σ χ + ˆt) σ so that v 0 = 1 σ χ σ e ( +ˆt) + π σ erfc( ( σ χ + ˆt)). These solutions are only valid for ˆt. (v 0 ) inner = (ϵ 1/ v 0 + ϵv 1 ) outer σ = ϵ 1/ 1 ( +ˆt) π ) + ϵ( σ χ e ( σϵ1/ σ ϵ erfc(σ χ To get solutions for other times, we first find the potential as ϕ 0 = σ v 0(σ, ˆt)dσ so that ϕ 0 = π σ χ σ erfc( ˆt) (ˆt ). From equation (3.1) 1, in order for ρ 1 to e 0 along σ = χ and to have the incoming solution given in (3.4), the solution is (3.5) ρ 1 = 1 σ χ σ (e ( +ˆt) χ σ ( e +ˆt) ) For the potential, we note that ϕ 0 (χ, ) = π χ erfc( ) = 0 so that y (3.), ϕ 0(χ, ˆt) = 0 is the oundary condition for all ˆt. Using this, and the wave solution to ϕ 0, we must have that π ϕ 0 = σ (erfc( χ σ ˆt) erfc( σ χ ˆt)) so (3.6) v 0 = + ˆt)) π σ (erf(σ χ ˆt) erf( χ σ ˆt)) 1 σ χ σ (e ( +ˆt) χ σ ( + e +ˆt) ) In writing the aove equation, we used the fact that erfc( x y) erfc(x y) = (1 erf( x y)) (1 erf(x y)) = erf(x y) erf( x y). We note that the leading order density perturation and velocity profiles grow in amplitude like 1/r, or close to such scaling, at their peak values as the profiles move radially inwards. This is consistent with known results for fluid dynamics with spherical symmetry [10]. Given the equation of state (.1), this also means the peak pressure grows like 1/r. To e more precise, the asymptotic profile for ρ 1 and p do grow like 1/r, ut ecause v 0 in the inner region also includes an error function term that reduces the magnitude of the Gaussian peak, the scaling may e slightly smaller. A typical profile of the velocity and pressure, computed numerically [1] is depicted in figure 3. We also oserve that v 0 and ρ 1 tend to 0 as ˆt and the total distance the plasma wall moves due to these leading order terms is O(1) O(ϵ) = O(ϵ). More preciesly the leading order displacement solely due to the v 0 term is (3.7) δ 0 = ϵ v 0 (χ, ˆt)dˆt = ϵ e ˆt dˆt = πϵ χ χ. Any nonnegligile compression will e a result of the correction terms. Equations (3.0), (3.1), and (3.3) give insights into what the effects of these correction terms are, which we discuss in the following portion of this paper Resultant Velocity Field. Given that most of the localized disturances are reflected, the driving force for compression, if a sustantial compression is to occur, must come from a residual negative radial velocity of the inner wall after the waves come in. We egin y considering the 14

15 Fig. 3: The pulses are moving to the left. The pressure peak grows like 1/r as the pulses move inwards. The velocity has a peak with negative value and one with positive value. The negative value grows roughly like 1/r. systems given y (3.0) in a time-independent regime (a long time after the v 0 and ρ 1 pulses have interacted with the inner wall). We then find that v 1σ + σ v 1 = 0 and ρ σ = 0 so that v 1 = A/σ and ρ = B. Physically, ρ (σ = ) = 0 so that B = 0, ut the constant A cannot e determined. We can, however determine ϕ 1 = σ A ˆσ dˆσ = A/σ (ˆt ). This suggests that the correction term leaves a velocity field ehind that may do further work to compress the plasma. Stricly speaking, we show that the value of ϕ 1 approaches a constant at σ = χ as ˆt ut section 4.3 validates this residual velocity field numerically. Oserve that χϕ 1 (χ, ) = A and ϕ 1 (χ, ) = ϕ 1 (χ, ) + ϕ 1ˆt (χ, ˆt)dˆt. At ˆt =, v 1 = 0, and ϕ 1 (χ, ) = 0. Then y using (3.3), we have ϕ(χ, ) = ˆt [ ρ 1σ (χ, ˆt) ˆt χ v 0 (χ, ŝ)dŝ + = ρ 1σ (χ, ˆt) v 0 (χ, ŝ)dŝdˆt }{{} χ T 1 (v 0) σ dσdˆt } {{ } T 3 χ ( (ρ 1 v 0 )ˆt (v 0) σ σ v 0)dσ]dˆt χ σ v 0dσdˆt. }{{} T 4 15 (ρ 1 v 0 )ˆt dσdˆt } {{ } T

16 We can simplify these terms and there is cancellation. Indeed, T 1 = T = 4 e ˆt χ = 4 π χ = 4 π χ ( 1 = T 3 = = T 4 = χ χ e ˆt 0dσ = 0 χ χ ˆt ˆt χ e ŝ dŝdˆt = ˆte ˆt erf(ˆt)dˆt erf(ˆt) + (ρ 1 v 0 )ˆt dσdˆt = (v 0) σ dσdˆt = χ 4 e ˆt dˆt χ = π χ σ v 0dσdˆt = (y parts) 4 π ˆt ˆte (1 erf( ˆt))dˆt χ 1 e ˆt dˆt) = π π χ v 0 (χ, ˆt) dˆt χ (ρ 1 v 0 )ˆt dˆtdσ σ v 0dσdˆt Adding all terms together gives us ϕ(χ, ˆt) = χ σ v 0dσdˆt > 0. Thus, A < 0 and there is a remaining velocity field in the negative radial direction that can compress the plasma. Otaining a simple expression for the exact value of this integral is not possile, ut y taking note of the characteristic shape of the integrand, we can approximate it with high precision. We remark that (3.8) π v0 = [ σ (erf(σ χ ˆt) erf( χ σ ˆt)) 1 σ (e ( 1 σ (e ( 1 σ (e ( π 1 σ σ χ +ˆt) χ σ ( + e +ˆt) ) σ χ +ˆt) χ σ ( + e +ˆt) ) (δ(ˆt σ χ ) + δ(ˆt χ σ )) σ χ +ˆt) χ σ ( + e +ˆt) )] We justify the first approximation y noting the terms with erf are modulated y a 1/σ which decays to 0 faster as σ than terms with 1/σ and that for σ χ, the two error functions are eing sutracted and have similar arguments. The second approximation comes from the fact that a Gaussian is dominated y its ehaviour near its maximum and there is little overlap etween the two Gaussians. The final approximation comes from approximating each Gaussian y its area modulating a delta function centred at its maximum. 16

17 (, χ) Iapprox Inumerical (0.557, 1.19) (1.11, 1.19) (0.88, 1.19).5.8 (0.557,.38) (0.557, 0.595) Tale 4: Verifying the numerical validity of the approximation given in (3.8) y douling/halving the governing parameters and computing I with the approximation of v 0 given in (3.8) and the numerically integrated value. From the approximation in (3.8), ϕ 1 (χ, ) = I = = χ χ χ 1 π σ σ (δ(ˆt σ χ π σ 3 (δ(ˆt σ χ π σ 3 dσ = π χ. ) + δ(ˆt χ σ ))dσdˆt ) + δ(ˆt χ σ ))dˆtdσ Therefore A π χ. For the remainder of this paper, we will take this approximations as the value of I. Tale 4 computes the integral χ σ v 0dσdˆt using (3.8) and numerically, and the results show good agreement. With this A, the long-time velocity profile is v ϵv 1 = ϵ π χσ (3.9) Region II Results. After the pulses have interacted with the inner wall, there remains a residual velocity field which remains as a time-independent solution of the asymptotic equations. We find that ρ 1 + o(ϵ) and v π χσ ϵ 1/ + o(ϵ 1/ ) Region III An Energy Argument. At this point we now compute the remaining kinetic energy (all the particles with a negative radial velocity), assuming there are no reflections at the right oundary and that all the useful energy is in the vicinity of the inner wall. The kinetic energy density is E = 1 ρv, and to leading order this is E = 1 (1)(ϵ1/ v 1 ) = ϵ v 1. The total kinetic energy is 4π r R r L Er dr, which in the inner variales at leading order is E = ϵ5/ χ v 1(σ) σ dσ = 4π ϵ 5/ 4 χ χ dσ σ. The total possile compression energy is (3.30) E = 4π ϵ 5/ 4 χ 3. The work done in compressing the plasma from r L = χ ϵ to r L = r the minimum radius P dv is 4π r χ ϵ P (r L)rL dr L = 4π r χ γ dr ϵ rl L giving us the total work done (3.31) W = 4πγϵ 7/ ( 1 r ϵ 1/ χ ). 17

18 Equating E = W in (3.30) and (3.31) implies 4π ϵ 5/ 4 χ = 4πγϵ 7/ ( 1 3 r with (3.3) r 4 χ 3 γ π ϵ. ϵ 1/ χ ) so that r = O(ϵ) This sort of energy argument was originally used y Lord Rayleigh in considering the compression of a ule in an incompressile fluid [13]. We present a formal asymptotic argument for this result in the next two susections Outer Region for Motion of Plasma Boundary. To proceed with formal asymptotics, we need to consider higher orders of the equations of mass and momentum conservation. By taking the equations of (.11) to two higher orders than presented in (3.0) with ρ 1 + ϵ 1/ ρ 1 + ϵρ + ϵ 3/ ρ 3 + ϵ ρ 4 and v v 0 + ϵ 1/ v 1 + ϵv + ϵ 3/ v 3 we otain the following alances: (3.33) ρ 3,ˆt + v,σ + σ v = (ρ v 0 + ρ 1 v 1 ) σ σ (ρ v 0 + ρ 1 v 1 ) (3.34) v,ˆt + ρ 3,σ = (ρ 1 v 0 + v 0 v 1 ) σ σ (ρ 1v 0 + v 0 v 1 ) (ρ v 0 + ρ 1 v 1 )ˆt (3.35) ρ 4,ˆt + v 3,σ + σ v 3 = (ρ 3 v 0 + ρ v 1 + ρ 1 v ) σ σ (ρ 3v 0 + ρ v 1 + ρ 1 v ) v 3,ˆt + ρ 4,σ = (ρ v 0 + ρ 1 v 0 v 1 + v 0 v + v 1) σ σ (ρ v 0 + ρ 1 v 0 v 1 + v 0 v + v 1) (3.36) (ρ 3 v 0 + ρ v 1 + ρ 1 v )ˆt From our previous considerations of the ehaviour of the solutions as ˆt, we oserve that (3.33) and (3.34) can also reach a steady-solution with ρ 3 = 0 and v = constant/σ since ρ 1, ρ, v 0 all tend to 0 which eliminates all forcing terms in the equations. Looking at (3.35) and (3.36) similarly, we actually have residual forcing terms that do not vanish as ˆt, namely the terms with v 1. After a long time, we infer that ρ 1 + ϵ ρ 4 and v ϵ 1/ v By scaling so that t = ϖ, an O(1) time-scale we otain a new PDE system to evolve, which serves as an outer region for a prolem descriing the motion of the plasma radius. Our alance is: (3.37) v 1,σ + σ v 1 = 0, v 1,ϖ + ρ 4,σ + v 1σ + σ v 1 = 0. Equation (3.37) 1 gives an effective incompressiility to the lead-lithium and we can proceed to solve (3.37) as done to derive the Rayleigh-Plesset equation for ule dynamics []. From (3.37) 1 we have v 1 = f(ϖ)/σ so that upon sustiting this into (3.37) we have f σ + ρ 4σ f σ = 0 which can e 5 integrated from σ = to σ = σ L (ϖ) the position of the plasma inner radius in the σ coordinates to get f σ L + ρ 4 (σ L ) + f σl 4 = 0 where we used ρ 4 (, ϖ) = 0. We have ρ 4 (σ L, ϖ) = 0 as well 18

19 ecause if r = O( ϵ), p = O(ϵ 3/ ) so that the density at the left oundary is 1 + O(ϵ 5/ ). We then get (3.38) df dϖ = f σl 3. Additionally, the velocity at the inner wall dσ L dϖ = v 1(σ L, ϱ) (due to the scaling in this regime there is no extra factor of ϵ to deal with) so (3.39) dσ L dϖ = f σl. Dividing (3.38) and (3.39) we find that df dσ L = f σ L which has solution f = C σ L for a constant C. Thus we can treat v 1 as a function of the inner wall position where v 1 (σ L ) = C/σ 3/ L after using the solution in conjunction with (3.39). Given π (3.40) v 1 (χ) = χ 3 from equation (3.9), we have C = π χ 3/ (3.41) v 1 (σ L ) = π and with respect to these outer coordinates χ 3/ σ 3/ L At this point, the system has not felt the effects of the plasma pressure and the wall is rapidly accelerating inwards Inner Region for Motion of Plasma Boundary. Numerous alances are possile for the system of (.11) and it is possile to arrive at the scaling we choose here considering all possile alances and solutions and choose the one that can match to the outer region, ut we will consider a physical argument here to otain the alance. We seek a alance of terms where the pressure of the plasma resists the compression (i.e. the pressure gradient alances the momentum flux). We consider r = O(ϵ c ). If r = O(ϵ c ) then p = O(ϵ 7/ 4c ) and ρ = 1 + O(ϵ 9/ 4c ). Based ϵ on the growth predicted y (3.41) this would give a scaling of v = O( 1/ ) = O(ϵ 5/4 3c/ ). (ϵ c /ϵ 1/ ) 3/ The form of the scaling as written comes from the velocity growing like 1/r 3/ L ut eing O( ϵ) when r = O( ϵ). To alance the pressure/density ρ term in the momentum equation with the v terms, we require 7/ 4c = 5/ 3c so that c = 1. Now we let r = ϵ 1 z. Then v ϵ 1/4 v 1, ρ 1 + ϵ 1/ ρ 1, and to alance the scales of velocity times time equals distance, we pick t = ϵ 5/4 ˆϖ where ˆϖ measures a time with respect to an aritrary reference point. The resulting alance is given y (3.4) v 1,z + z v 1 = 0 and v 1, ˆϖ + ρ 1,σ + (v 1) z + z v 1 = 0. Using the functional form of v 1 implied y (3.4) 1, v 1 = g( ˆϖ)/zL, in (3.4) and integrating as efore, this time up to z L, the position of the wall in the inner coordinates gives us g z L + γ + g = zl 4 zl 4 0 where we used ρ 1 (z L, ˆϖ) = γ/zl 4. It follows then that (3.43) dg d ˆϖ = γ + g / z 3 L 19.

20 (3.44) dz L d ˆϖ = g zl so that after dividing the equations we get dg z L g which we separate to get g γ+g / dg = dzl z L with solution γ + g / = Dz L for a constant D. We can solve for g, otaining g = Dz L γ (the negative sign giving the correct direction of the wall) which finally gives us dz L = γ+g / (3.45) v 1 = Dz L γ zl. Oserve that when z L = zl = γ/d the velocity is 0 to leading order and this will e our leading order estimate to the compression. To find D, we need to peform matched asymptotics etween the inner and outer regions with respect to the functional dependence of the wall velocity upon the wall position Matching to Determine Minimum Radius. We now put the outer variales of (3.41) in inner variales to match the ehaviours as z L and σ L 0. ϵ 1/4 (v 1 ) inner = ϵ 1/ (v 1 ) outer ϵ 1/4 D z 3/ L = ϵ 1/ π χ 3/ (ϵ 1/ z L ) 3/ yielding D z 3/ L zl = γ/d = 4 χ 3 γ = π χ 3/ z 3/ L so that D = π 4 χ 3. In the z variale, we have a minimum radius of π and we thus find r = 4 χ 3 γ π ϵ consistent with (3.3) found with the energy argument. While the energy argument was assumed true for incompressile fluids, it seems in the asymptotic limit compressiility does not influence the compression to leading order. Note that y having quarter powers of ϵ, it may ecomes difficult to differentiate different asymptotic terms without taking ϵ to e extremely small. When v = O(ϵ 1/4 ), this is very close to v = O(ϵ 1/ ), the sound speed magnitude and the wave-like ehaviour of the equations may e close to reaking down. Indeed, when ϵ is too large, the local velocity magnitude can even exceed the sound speed during the compression phase Dimensional Expression for Plasma Compression. We can finally go ack to the dimensional parameters in our prolem. Using tales and 3, we find the minimum radius is (3.46) R L C4 s P plasma,0 R 7 inner,0 ϱ3 0 πp 4 maxr 4 outer,0 T 0 0

21 with (3.47) (3.48) (3.49) (3.50) T 0 R outer,0 P max ϱ 0 1 C s T 1/4 0 P 1/4 max R 1/ outer,0 ϱ1/4 0 P plasma,0 Rinner,0 4 ϱ7/4 0 Pmax 11/4 R 1/ outer,0 T 7/4 0 R inner,0 ϱ 1/4 0 R 1/ outer,0 T 1/ 0 = O(1) = O(1) = O(1). Equation (3.3) gives the approximate expression for the minimum radius, which requires that ϵ is small (3.47) and that, γ, and χ all e O(1) as given in equations (3.48) through (3.50). 4. Comparison with Numerics. In previous work [1], a finite volume numerical framework was developed to solve the nonlinear conservation laws with the moving oundaries. Using this numerical framework, we run simulations on the asymptotic prolem formulated in this paper. Our validation is done in multiple stages: we examine the profiles of the pulses during their formation oth numerically and asymptotically y comparing the profiles of the resultant variales; then, we study the growths of the amplitudes of the pulses as they move radially inward as predicted y the asymptotics and numerics; we next proceed to study the existence of the residual velocity field with the numerics and the growth of the inner wall velocity with inner wall position; we oserve that the velocity at the inner wall is qualitatively consistent with the numerical results; and finally, we compute the minimal radius in the asymptotic and numerical workings for various values of, γ, χ, and ϵ. We remark that the asymptotic prolem we have considered is primarily concerned with a single direction of information propagation: the pulse travels towards the plasma wall, it is reflected, and then everything else governing the plasma wall takes place locally near the wall and there is no influence from leftgoing waves. We need to exercise caution with this numerically y removing reflections that occur at the outer wall. This allows for etter agreement etween the asymptotics and numerics ut we make two remarks. We first note that if these reflections are not removed then a reflection occurs at the outer wall and disturances return to hit the plasma again which drastically reduces the compression. Such effects are noticeale for ϵ The second point to make is that even in attempting to remove the reflections at the outer wall, there is likely still a small numerical error that remains Pulse Formation. We consider two parameter regimes here. One is the for the parameter set of interest, and the other is for a much smaller ϵ with other dimensionless parameters set to unity. We fix an end time t = ϵ/ and plot the profiles of v and ρ as computed numerically, and with two different levels of asymptotic accuracy. From the plots, we are ale to verify that the successive terms of the asymptotic expansion yield higher accuracies, and that the asymptotics and numerics are in good agreement. See figure. 4.. Focusing. Here, we consider the asymptotically predicted growth rates for the peak amplitudes of the velocity and pressure. From the asymptotic predictions, oth of these amplitudes 1

22 9 =χ=γ=1,ε= =χ=γ=1,ε=0.001 Maximum Pressure Value Numerical Peak Asymptotic Peak Minimum Velocity Value Numerical Peak Asymptotic Peak Maximum Position Minimum Position Fig. 4: Numerical and asymptotic descriptions of the growth in peak values for the minimum value of velocity and peak value of pressure. Based on a est fit, the numerically oserved scalings are 1/r 0.99 and 1/r 0.90 for the left and right plot respectively. should grow inversely with the position of these peaks. This validation is rather delicate: these growths should e upheld in the limit ϵ 0; however, there are difficulties in getting the numerics to give highly accurate results on regions where r 1. What we choose to do is pick a very small value of ϵ, 0.001, and consider the growth of the amplitudes on the region r > 0.1 O( ϵ). We plot the predictions as given y a perfect growth of 1/r ased on the initial peak pressure and minimum velocity, and the numerical results for the peak amplitude vs position for this ϵ value. The plots are given in figure 4 and we oserve strong consistency Residual Velocity Field. Our prediction of a steady velocity field resulting from the disturances interacting with the plasma was ased primarily on intuition, as we did not formally find the solution for v 1 and ρ for general ˆt; we simply showed that the potential approaches a constant value along the oundary. Numerically, however, we can validate the scaling. After a long time with respect to the ˆt time-scale so that v 0 has had its full effect, we plot the profile of the velocity versus the radial position. As the inner wall moves inward, in its wake there is a velocity profile that scales like 1/rL. See figure Outer Region descriing Inner Wall Velocity. This validation is rather delicate as there are a numer of sources of error. Firstly, the inner wall position at which the oundary condition of (3.40) is etter approximated y σ = χ ϵ π χ as per (3.7) ut to leading order we have taken it as σ = χ. In the case considered in figure 6, this already amounts to an error of 17% in estimating the oundary condition which will lead to an error in the constants otained. We additionally know that v ϵv 1 + ϵv so that there is an O(ϵ) error term in the radial position that is accrued over a time scale of O(1). Throwing these effects together makes it very hard to get a clean fit etween numerical and asymptotic results. In finding the slope of the log v L vs log r L plot, a least squares fit shows the numerical scaling is v L 1/rL 1.49 which is completely consistent with the asymptotic scaling Qualitative Agreement of Inner Wall Velocity. The asymptotics predict a series of phenomena at the inner wall: firstly, the wall is stationary until the pressure pulse reaches it wherey it takes on a Gaussian shape and decreasing in speed; secondly, a smaller asymptotic term

23 Velocity Profile at t=0.0784, =γ=χ=1, ε = Numerical Asymptotic Velocity v L Position r Fig. 5: A plot of the numerical and asymptotic velocity profile after much of the pulses have reflected off of the inner wall. The two are nearly indistinguishale. Wall Velocity Growth with Position at =γ=χ=1, ε = 0.005, from t= to Inner Wall Velocity v L Numerical Asymptotic Inner Wall Position r L Fig. 6: A plot of the numerical and asymptotic velocity profile for how the inner wall velocity implicitly depends on the inner wall position. descriing the wall velocity remains for some time (as the velocity acquires the residual profile); thirdly, the wall rapidly speeds up; and finally, the wall is stopped aruptly on a small spatial scale when the plasma pressure finally takes over. We oserve all of these phenomena in figure Predictions for Plasma Compression. In this section, we verify that the fundamental asymptotic predictions are consistent with the numerics. Tale 5 presents the results. We remark that verifying the asymptotic limit is not trivial. As ϵ 0, the numerics, for modest discretizations, lose accuracy due to the very small radial positions under consideration. As a result, we cannot not take ϵ too small. We also have to ensure that quantities such as 4 χ 3 γ/π, etc. remained roughly O(1) for these values of ϵ. Picking ϵ too large also leads to prolems: for one, the higher order 3