Self-Similar Solutions of Compressible Fluids

Abstract

Strong shock waves are used to compress and heat any matters in the laboratory. The ablation pressure by intense laser is used to compress even solid matters. In plane geometry, it is easier to design multi-shocks to compress the matters, while it is more beneficial to use the spherical compression. No simple solutions are available to know the trajectories of shocks in one-dimensional spherical symmetry. Here we see several analytical solutions with the self-similar method. The method is to find new governing solution of ordinary differential equation from partial differential fluid equations. The self-similar method is known before the birth of computer.

The blast wave is the most famous one. Here, we review the basic method to derive several self-similar solutions allowing the spherical implosion, useful to laser driven implosion. The isobaric solution provides uniform pressure and spark-main fuel structure, and isochoric solution gives us uniform density profile at the maximum compression. It is shown that even including thermal conduction, it is possible to find a solution of ablation structure. This is an extended solution more appropriate compared to the steady state solutions shown in the previous chapter.

The blast waves are widely used from laser experiments to supernova remnants (SNRs). SNRs are blast waves driven by the matters exploding by supernova explosion. A self-similar solution with forward and reverse shock waves is found to explain many observation data of SNRs. A numerical simulation shows that the solution of ejecta-driven shock changes from Chevalier’s self-similar solution to the other Sedov-Taylor one. The self-similarity is one of the key physics controlling nonlinear hydrodynamic equations.

4.1 Introduction

4.1.1 Strong Shock Reflection

Consider strong shock limit where P₁ ≫ P₀ and P₀ can be neglected in the Ranking-Hugoniot relation in (3.13). In the case of irradiation of intense laser on solid matters, this limit is possible to be realized. However, as we see in Fig. 2.3, the equation of state (EOS) is not ideal. This means an effective specific heat γ is a function of density and pressure even in LTE. For example, aluminum is almost incompressible up to the 100 kBar (1 Bar = 10⁵ Pa) and it becomes compressible over a few hundred k-Bar. The solid matters are hard because of Coulomb repulsive force in lower pressure than the Bulk modulus shown in Chap. 2. Over this pressure, most of solids have the dominant pressure of electron Fermi pressure. The Fermi pressure P_F is proportional to the density as P_F ∝ ρ^5/3.

Neglecting P₀, (3.16) gives the following scaling of achievable pressure by stagnation of a shock wave by a rigid boundary as shown in Fig. 4.1.

$$ {P}_1\approx {\rho}_0{u}_0^2 $$

(4.1)

The following simple formula is obtained

$$ P\ \left[ Mbar\right]=\frac{\rho_0}{\left[g/c{m}^3\right]}{\left\{\frac{u_0}{\left[{10}^6 cm/s\right]}\right\}}^2 $$

(4.2)

If we can compress and accelerate a matter to the density of 100 g/cm³ and the velocity of 3 × 10⁷cm/s, an extremely high pressure around 100 Gbar can be achieved by the stagnation. It is hard to realize such initial condition with ρ₀ and u₀ in plane geometry, but the following idea is used.

Fig. 4.1

The kinetic energy of fluid flow is converted to the thermal energy by collision with a rig wall. The density increases in the reflecting shock wave, but it is limited by the RH relation in plane geometry

4.1.2 Tailored Compression

Even in plane geometry, however, it is possible to compress matters to extremely high-density by use of multi shock waves, namely sequential shock compression. In Fig. 3.4, we have seen that the Ranking-Hugoniot curve is almost the same curve as the adiabatic curve for lower compression case. So, if we can shape the pressure to make shock waves so that each shock wave is that of compression of about two-times the density of the rear of the previous shock wave, it is possible to compress plasma under almost adiabatic condition.

In Fig. 4.2, this tailored pressure compression is shown in x-t diagram (a) and a snap shot of the density at a time t_s is plotted on the right (b). In this hydrodynamics, all shock waves are designed to arrive at x = 0 at the time of maximum compression (t_m). The initial shock trajectory is plotted with blue and the sequential shocks are plotted with yellow lines in (a). Since a shock is always stronger than the front shock, the generated shock velocity is faster than the front one. If we can increase the number of shocks, it is equivalent to the adiabatic compression and the maximum density is proportional to the available maximum pressure. By use of a finite strength shock for the first one (blue), the adiabatic compression is possible with the entropy determined by the first shock wave.

Fig. 4.2

Schematics of tailored pressure for almost adiabatic compression in plane geometry. After the first shock wave shown in blue, many subsequent shock waves are generated. The density profile becomes like (b) at t = t_s. The maximum density is expected at t = t_m

Of course, the design of the tailored compression is not so simple mainly because of non-ideal equation of state. As shown in Fig. 3.5, the phase transition from solid to plasma has complicated process with an effectively different freedom of physical condition. It has been carried out to study the equation of state of shocked solid materials theoretically, computationally, and experimentally. This issue will be discussed in this text later.

4.1.3 Hollow Shell Implosion

When an intense laser irradiates a solid material, it is possible to generate ~100 Mbar of so-called ablation pressure on the surface of a spherical target for nuclear fusion experiment shown, for example, in Fig. 4.3a. By use of spherical convergence geometry, it is possible to enhance the density and pressure. It is, however, hard to evaluate the effect of the spherical geometry, because as shown in Fig. 4.3b, the compressed and accelerated plasma of DT ice by shock waves is decelerated by the decrease of the surface shown with blue arrows (∝r²), depending on the thickness of the shell and its pressure. In addition, self-pressure works to expands the shell as green force in Fig. 4.3b to reduce the density. It is true that we can achieve higher compression and extreme pressure if the thickness of the plasma shell is thin enough and the pressure is as low as possible, namely keeping low entropy state. As shown in Fig. 2.3, the lowest pressure at higher density is given by Fermi pressure and it is impossible to keep the pressure lower than the Fermi pressure. The Fermi pressure decelerate the shell velocity and force to expand the shell as shown in Fig. 4.3b.

Fig. 4.3

(a) A cut-view of a typical target structure of spherical laser fusion implosion. The ablator is irradiated by intense laser to generate the ablation pressure to compress the fusion fuel DT ice at the center of the target. The gas is residual gas with low pressure. (b) is the time the DT plasma is imploded near the center. It feels the force as shown with blue and green arrows

Therefore, it is not clear how the accelerated matter by shock waves converges toward the central singularity point. Theoretical guideline of the spherical effect on the shock dynamics is given as self-similar solution of the ideal fluid equations. This is explained in this Chapter.

To obtain the image of one-dimensional fluid dynamics of laser implosion, one example obtained with the physics-integrated implosion code ILESTA [1] is shown for the case of a hollow plastic shell implosion. In Fig. 4.4, time evolution of fluid elements (Lagrangian trajectories) is plotted in radius and time diagram. A thin plastic shell (CH) of diameter 500 μm and thickness of 10 μm is irradiated by green laser (λ_L = 0.53μm) with energy of 8 kJ. The shapes of input (P_in) and the absorbed (P_abs) pulses are plotted in the inlet figure.

Fig. 4.4

RT-diagram of dynamics of fluid elements in one-dimensional spherical implosion simulation. Plastic hollow shell is imploded by the laser power shown in the inlet

In Fig. 4.4, the initial shock wave arrives at the rear-side of the shell at the point (1). The shock breaks through the shell and the shocked plasma expands as the rarefaction wave. The trajectory of the laser cut-off density is plotted with the line (2). As seen in the inlet, the peak of absorbed laser power is around t = 2.5 ns, and the shell plasma keeps to shrink toward the center of the target. The mean velocity of the plasma shell is kept almost constant and it is estimated to be u₀ ≈ 3 × 10⁷cm/s.

In Fig. 4.5, the trajectories of all fluid elements are plotted in the density and pressure diagram, where the cold curve of plastic shown with the blue dashed line from (0) to (1) is taken from Fig. 2.3. The initial condition of the simulation is all below the point (0). Since the laser intensity increases continuously, the plastic of solid CH is compressed almost adiabatically. The outer plastic is then ablated into vacuum, while it decreases the density more than two order of magnitude from the point (2) to (3). Note that the pressure is kept almost constant as the characteristic of the deflagration wave. This pressure is the ablation pressure to be discussed soon. It is seen that the ablation pressure is about 20 ~ 30 Mbar in Fig. 4.5. After the exhaust of plasma in the deflagration wave, the rarefaction wave expands to the vacuum while keeping a constant temperature, where the red dashed line is the constant temperature line (P ∝ ρ) in Fig. 4.5.

Fig. 4.5

Time evolution of all fluid elements in Fig. 4.4 are plotted in density-pressure diagram. The initial condition is near the point (0). The blue dashed line is the cold curve. The red dashed cure is a constant temperature line

In the present simulation, the radiation preheat by d doped silicon is included. The radiation pre-heat increases the entropy of the plastic shell plasma. As the shrink of the shell radius, the density increases in the phase of the point (4) thanks to the effect of compression by the spherical convergence. The density increases from (2) to (5), then a strong shock wave generated at the center converted the shell kinetic energy to the compression energy as shown from the point (5) to (6). It is found that from (2) to (5) the compression is almost adiabatic and its adiabat α defined as

$$ P=\alpha {P}_c $$

(4.3)

is about α = 8 in (2)–(5) and α = 10 at the maximum density.

In the final convergence of the kinetic energy to the thermal energy, the maximum pressure is obtained in the simulation. By use of the velocity u₀ and density ρ₀ in the simulation, the point (5) in Fig. 4.5, rough evaluation (4.2) gives the pressure as

$$ P=27\ \left[ Gbar\right]\ \mathrm{for}\ {\rho}_0=30\ \left[g/ cc\right] $$

(4.4)

The simulation resulted the maximum pressure of about 40 Gbar, higher than the value above due to the spherical effect. As seen at the point (6) the density profile of the maximum pressure is clear to have a structure with the density from 30 to 100 g/cc. This is because the center of the core is the part initially expanded by the shock at the time (1) in Fig. 4.4 and its entropy increases higher by the second shock at 2.4 ns. This means the final core has the central spark of high-temperature and the surrounding of high-density region automatically. The isobaric profile is commonly seen in the self-similar solution to be discussed later.

It is well-known that to achieve the extremely high-pressure for nuclear fusion ignition, so-called tailored pulse is required [2]. Also, the implosion dynamics should be designed so that the DT ice plasma region in Fig. 4.3 is protected from the entropy increase by shock wave and radiation pre-heat. The material of the ablator should be selected by taking account that it does not emit x-ray radiation or it should be shielded by some idea. As the case of Fig. 4.4, the radiation pre-heat prohibited the low-adiabat compression of the adiabat, α < 2~3.

The hydrodynamic stability of implosion is very critical especially for thin-shell and low-adiabatic implosion. This means that to achieve expected high-density compression, it is required to develop three-dimensional hydrodynamics code with important physics integrated in the code. For example, HYDRA and ASTER are used as the standard three-dimensional integrated codes [3, 4], respectively At least two-dimensional simulation code is necessary to analyze the implosion experiments from which we can obtain limit data indirectly. The hydrodynamic instability of implosion is hot topics and to be discussed in details in Volume 3.

4.1.4 Analytic Solution of Spherical Implosion

Any kind of computer simulation code should be verified through comparing to the code with the corresponding experiment. The comparison has been carried out for a simple part of the plasma dynamics as explained in the previous chapter. Since the implosion dynamics with help of the singularity of spherical geometry is important to the application of high-energy-density physics in laser plasma, it is useful to know some analytic solutions of spherical dynamics. Possible idealistic solutions have been found by altering the time and space dependent partial differential equations to coupled ordinary differential equations. This method is to find self-similar solution by finding the similarity variable.

In the previous chapter, we studied analytic solutions with the steady state assumption in the appropriate moving frame. In the present chapter, on the other hand, we will review the self-similar solutions describing spherical implosion and explosion dynamics. It is also shown that a self-similar solution is also found to the time dependent ablation structure under the idealized boundary condition.

4.2 Basic Equations for Self-Similar Solutions

Try to study the physics of spherical compression by a strong shock wave propagating from the outer sphere in a uniform density fluid. It is surprising to know that theoretical works were published in early time by Guderley, Landau, others as described in the books Chap. XII in [5] and Chap. 6.7 in [6], where the method of self-similar-solution has been used to reduce nonlinear partial differential equations to ordinary differential equations as an eigen value problem. They obtained approximate analytical solution, and the mathematical method is described in [5, 6]. The hydrodynamic stability of the Guderley solution is studied in [7], where the property of the Guderley solution is numerically solved to obtain the solution. The Guderley solution gives the fluid dynamics in the converging phase of shock wave and the fluid dynamics after the shock reflection at the center, the singular point. It is reasonable to expect the geometrical effect, namely the shocked matter moves toward the center to be compressed and heated adiabatically by the geometrical effect and the central singularity effect.

Of course, the spherically symmetric hydrodynamics is naive assumption and there is no proof that the spherical symmetry can be reasonably achieved even with highly precision technology. This is a big issue to be discussed relating to the laser fusion in Volume 3. The present understanding is that it is not possible because of the hydrodynamics instability and turbulent mixing generated by the thermal noise on the target and pressure nonuniformity by laser ablation as will be discussed later. But, analytical solution such as self-similar solution is very useful to know the fluid dynamics in such extreme condition. The solution can be used to verify the accuracy of the hydrodynamic simulation code under idealized condition.

4.2.1 Self-Similar Solutions

Compression of matters with shock waves can be evaluated by using the RH relations and rarefaction wave in the plane geometry. It is not trivial to predict the propagation of shock wave in the spherical geometry, because time dependent geometry effects on the shock front continuously change the density, flow velocity, and pressure in time, in space as well. In solving an idealized nonlinear coupled equations, so-called the method of self-similar solution has been applied in many cases. The type of the fluid dynamics consists of two-types [8].

• 1. Implosion and collapsing cavities

• 2. Converging and diverging shocks

Following Refs. [6, 9], let us summarize the solutions useful for relating to the laser implosion.

The self-similar solution is based on the dimensional analysis of the basic equations as we have already shown a simple case of the nonlinear heat conduction in Chap. 2. In what follows, we consider only the case of adiabatic dynamics except shock jump surface. Including the shock front in the self-similar solutions, the strong jump limit is assumed to connect two different self-similar solutions. Note that the adiabatic assumption means that the entropy S = P/ρ^γ is constant in time for each fluid particle, while the spatial variation of S(r) is allowed as initial condition.

Let us follow the notation in [9]. The basic equations are (2.20) – (2.22) with Q = 0, namely adiabatic condition dS/dt = 0 is assumed. In the spherical geometry, the equations are given in Euler description as

$$ \frac{\partial \rho }{\partial t}+\frac{\partial }{\partial r}\left(\rho u\right)+\frac{2\rho u}{r}=0 $$

(4.5)

$$ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial r}+\frac{1}{\gamma \rho}\frac{\partial }{\partial r}\left(\rho {c}^2\right)=0 $$

(4.6)

$$ \frac{\partial c}{\partial t}+u\frac{\partial c}{\partial r}+\frac{\gamma -1}{2}c\left(\frac{\partial u}{\partial r}+\frac{2u}{r}\right)=0 $$

(4.7)

where c is the sound velocity.

$$ {c}^2=\gamma \frac{P}{\rho } $$

(4.8)

We take the similarity variable ξ with an unknown constant α in the form.

$$ \xi =\frac{r}{{\left|t\right|}^{\alpha }} $$

(4.9)

Note that ξ is non-dimensional variable with use of typical time t₀ (t → t/t₀) and r₀ (r → r/r₀) to be defined in applying the solutions to some real problem [6, 9].

Assume the following solution from the dimensional analysis.

$$ u\left(r,t\right)=\alpha \frac{r}{t}U\left(\xi \right) $$

(4.10)

$$ c\left(t,r\right)=\alpha \frac{r}{t}C\left(\xi \right) $$

(4.11)

$$ \rho \left(r,t\right)={\rho}_0{r}^{\kappa }N\left(\xi \right) $$

(4.12)

Note that α and κ are constants and α is the eigen value as see below. In the analysis, the constant α is determined like an eigen value to obtain any expected solution of U(ξ) and C(ξ), while the parameter κ is given so that the density profile is reasonable as expected. As we see soon, for a given test value of α, we can obtain the function U(ξ) and C(ξ) satisfying an appropriate boundary condition in the case of collapsing cavities. In the shock case, the jump relation connects the solution of U(ξ) and C(ξ) from subsonic to supersonic points.

Inserting (4.10, 4.11, and 4.12) to (4.5, 4.6, and 4.7) and using the simple relations

$$ \frac{\partial f\left(\xi \right)}{\partial t}=-\alpha \frac{\xi }{t}\frac{d f\left(\xi \right)}{d\xi},\frac{\partial f\left(\xi \right)}{\partial r}=\frac{\xi }{t}\frac{d f\left(\xi \right)}{d\xi} $$

(4.13)

Then, (4.6) and (4.7) are reduced to the following two ordinary differential equations and one conservation relation [6],

$$ \frac{dU}{d\left( ln\xi \right)}=\frac{G\left(U,C\right)}{D\left(U,C\right)} $$

(4.14)

$$ \frac{dC}{d\left( ln\xi \right)}=\frac{F\left(U,C\right)}{D\left(U,C\right)} $$

(4.15)

where

$$ D\left(V,C\right)={C}^2-{\left(1-U\right)}^2 $$

(4.16)

$$ F\left(U,C\right)=C\left\{\left(1-U\right)\left(\frac{1}{\alpha }-U\right)+U\left[\lambda +\left(\gamma -1\right)\left(U-1\right)\right]-{C}^2+\frac{\varepsilon }{2\gamma}\frac{C^2}{U-1}\right\} $$

(4.17)

$$ G\left(U,C\right)=U\left(1-U\right)\left(\frac{1}{\alpha }-U\right)-{C}^2\left[3U+\left(\kappa -2\lambda \right)/\gamma \right] $$

(4.18)

$$ \lambda =\frac{1}{\alpha }-1,\varepsilon =\kappa \left(\gamma -1\right)+2\lambda $$

(4.19)

And the conservation law provides the density profile as follows.

$$ N\left(\xi \right)={K}_3{\left(\alpha {\xi}^{1/\alpha }C\right)}^A{\left(1-U\right)}^B, $$

(4.20)

$$ A=\frac{\mu \left(\kappa +3\right)}{\beta },B=\frac{\left(\kappa +\mu \lambda \right)}{\beta },\mu =\frac{2}{\left(\gamma -1\right)},\beta =3-\mu \lambda $$

(4.21)

where K₃ is constant.

These equations are used by Guderley to solve the shock wave converging to the center and reflected by the center in the spherical geometry [9]. It is amazing to know that he solved this problem in 1942 [9]. Almost the same time, Taylor and Sedov has independently solved the problem of the blast wave, which is the shock dynamics of point source explosion in air [10,11,12]. In this chapter, let us briefly review the two cases relating to the laser-driven implosions and blast waves. The first one is isobaric and isochoric implosion, and the second one is Taylor-Sedov explosion.

4.3 Self-Similar Implosion (Isobaric)

The mathematical method proposed by Guderley is applied to obtain an implosion dynamics to form almost constant pressure with higher temperature at the center behind the reflected shock wave at the maximum compression time [13]. The basic equations are (4.5, 4.6, and 4.7). In the implosion phase, we find the solution with the density like Fig. 4.2b, while spatial profile of the entropy increasing from the rear to the front of the imploding shell, so that the high-temperature with relatively low-density central core is expected. The free parameter of κ in (4.12) is set κ = 3.

The general properties of coupled differential Eqs. (4.14) and (4.15) is studied. At first

$$ D=0,C=-U+1 and\ U-1 $$

(4.22)

D = 0 gives a singularity condition. Integrating (4.14) and (4.15) on these lines, the derivative of U and C diverges except for G = 0 and F = 0. This means if we can find the point in the (U, C) plane where G = F = 0, the integration can be proceeded normally. This point is also a singular point (S).

Solving the algebraic couple equations

$$ G\left(U,C;\alpha \right)=0 $$

(4.23)

$$ F\left(U,C;\alpha \right)=0 $$

(4.24)

on the line (4.22), it is possible to obtain the singular point (S) in a form.

$$ \alpha =\alpha \left({U}_s,{C}_s\right) $$

(4.25)

In the isobaric implosion [13], it is found that α = 0.7 gives the expected solution with reasonable profiles. In finding the exponent as an eigen value problem, the following integral is solved in (U, C) space with a trial value of α from (4.25).

$$ \frac{dU}{dC}=\frac{G\left(U,C;\alpha \right)}{F\left(U,C;\alpha \right)} $$

(4.26)

It is concluded in [13] that the integration path of implosion phase (t < 0) and explosion phase (t > 0) should take the path as shown in Fig. 4.6. The point B and D are the singular points.

Fig. 4.6

The integral trajectories to allow the self-similar solution of isobaric implosion and explosion dynamics in (U, C) space. The implosion (t < 0) is in U > 0 region and explosion (t > 0) is in U < 0 region. Reprinted with permission from Ref. [6]. Copyright 1998 by Oxford University Press

In the implosion phase (t < 0), the integration path from F (r → 0) to B to O (r → ∞) gives the solution. In the explosion phase (t > 0), the high-temperature central core is E(r → 0) to the rear of the reflection shock S₁ to the state in front of the shock wave S₂ connecting to the central high-temperature core E (r → 0). Note that the velocity is defined as in (4.10) and U > 0 in t < 0 means the velocity is negative, namely in implosion phase. For t > 0, the strong shock jump from the RH relation is assumed, and the fluid is strongly decelerated by the shock from U ≈  − 3.5 to U ≈  − 0.5.

It is important to know the properties of the singular points in (U, C) plane. It is easily found that the singular point B in Fig. 4.6 is the saddle point and the integration path can cross the singular line (4.22) only the path shown in a stretched view Fig. 4.7 near the singular point B. It shows a general property of integration paths started from different points in (U, C) plane. The point B is indicated as the point S, the saddle point in Fig. 4.7. We found that this is the same as the saddle point in Fig. 3.32.

Fig. 4.7

A schematic of a saddle point in the integration paths in two-dimensional space. The integration path can smoothly continue over the saddle point as the orange line

On the other hand, the singular point of O, where U = C = 0 is the other type of singular point. It is a node point. Near the node point, all integral paths converge to the node point or all paths can start all direction from the node point. In the present isobaric implosion, the solution should be an integral path from the point O to smoothly transit the singular point B on the saddle path shown with orange in Fig. 4.7. Then, all normalized functions, U(ξ), C(ξ), and N(ξ) are obtained by numerically integrating (4.14) and (4.15).

The time evolution of the density profile is given as shown in Fig. 4.8 [13]. Note that the solution is a hollow shell with the front ξ_F = 0.96 and the shock front ξ_S = 0.198. Their trajectories are plotted with the dashed lines, r(t) = |t|^αξ, where ξ = ξ_F and ξ_S. To make the central hot spark, the density and entropy profiles in the implosion phase is assumed at t = −1 as shown in Fig, 4.8. At the time of void closure (t = 0), the solution continues to the explosion phase (t > 0). A strong shock wave traveling to the outward is generated in the still-imploding fluid. The spatial profiles of the density, temperature, and pressure after the implosion are shown in the inset figure at t = 1.6. So-called isobaric central ignition profile is formed.

Fig. 4.8

The density profiles given by the self-similar solution for the isobaric implosion. Reprint with permission from Ref. [13]. Copyright 1998 by American Physical Society

4.4 Guderley Self-Similar Solution

The Guderley solution is also an implosion dynamics. The difference from the above is that in the implosion phase, a strong shock wave is assumed to travel in a uniform gas (κ = 0). Then, it is found that the eigen-value is α = 0.688 [9].

In this case, the self-similar solution starts from the shock front (ξ = 1). The imploding shock front is the point “A” in Fig. 4.6 obtained by the RH relation for strong shock limit. The integral path has to come to the singular point (saddle point) on the line U + C = 1 same as the previous case. The integration path goes to the point “O”. In the explosion phase, it takes the path in negative U and jump from “S₁” in the negative U to “S₂” in the positive U, then it goes to the large C region in Fig. 4.6.

In Fig. 4.9, the time evolution of the density in (r, t) space is shown. The density behind the converging shock wave increases with the radius because of spherical geometry effect. The density becomes flat at t = 0 with negative velocity, but U = 0 in 0 < ξ < 1. Then, the reflected shock is produced and propagates to have the snap shot at t = 1 in Fig. 4.9. The maximum density is 32ρ₀.

Fig. 4.9

The density profiles given by the self-similar solution of converging shock wave in gas with constant density in spherical geometry, Guderley solution. Reprinted with permission from Ref. [6]. Copyright 1998 by Oxford University Press

It is noted that hydrodynamic stability is studied to this Guderley solution numerically. It is reported that the 3-D perturbation at the shock front oscillates and relatively stable to the hydrodynamics instability [7].

4.5 Isochoric Implosion

In Ref. [14], a self-similar implosion to provide with a uniform density profile behind the reflected shock is tried to find. This is the modification of the above isobaric implosion, and such uniform density is called “isochoric”, isochoric implosion. The initial condition of the entropy is almost null at the front with finite implosion velocity. This indicates that the front point (ξ = 1) should be (U, C) = (1, 0). Then, the integration path has to pass the saddle point on U + C = 1 in Fig. 4.6 to converge to the point “O”.

In the explosion phase (t > 0), the shock wave travel outward while keeping high-density behind the shock. We find a solution that the produced shock propagates outward with falling fluid in front and compressed fluid behind. In Ref. [14], the solution is found by starting the mathematical definition used in Ref. [8], where the similarity variable x = t/r^λ. This corresponds to α = 1/λ in the form of ξ = r/t^α.

From the asymptotic solution given in [6], κ is found to be given as a function of α = 1/λ in the form.

$$ \kappa =2\left(1/\alpha +1\right)/\left(\gamma -1\right) $$

(4.27)

This relation is obtained by requiring the density is flat in the asymptotic relation at t = 0. The eigen-value α = 0.789 is obtained. Then, the density profile must be steep as κ = 6.801 to allow very low adiabat at the front.

The time evolution of the density from the resultant self-similar solution is plotted in Fig. 4.10 for the implosion phase (left) and explosion phase (right). About 400 times the initial density is achieved with almost flat profile.

Fig. 4.10

Time evolution of isochoric implosion (left) and explosion (right) based on a self-similar solution. Reprinted with permission from Ref. [14]. Copyright by IAEA

In this paper, the authors tried to find the target design so that the high-density compression is approximately realized by laser irradiation on a spherical target. They try to find the laser pulse shape for a standard laser fusion target shown in Fig. 4.11. To form the initial condition with U, C, and N profiles in Fig. 4.10, one Mbar pressure is loaded at the DT ice surface neat t = 10 ns. The optimized radius-time evolution of all fluid elements obtained with HYDRA code is plotted in Fig. 4.12 with the time evolution of corresponding pressure at the out surface of the DT ice. To keep the pressure reproduces the self-similar solution, 700 Mbar pressure is generated at the end of the optimized tailored laser pulse.

Fig. 4.11

The optimized target structure to realize an isochoric implosion with HYDRA 1-D simulation code. Reprinted with permission from Ref. [14]. Copyright by IAEA

Fig. 4.12

Flow diagram and the pressure history of the optimized implosion to realize the isochoric density profile after the maximum compression. Reprinted with permission from Ref. [14]. Copyright by IAEA

The initial shock wave travels in the ablator and collides the DT ice surface (t = 10 ns). Then, the rarefaction wave starts to expand to the inside of the shell around t = 19 ns to decrease the density like in Fig. 4.12. The rarefaction wave front comes to the DT ice surface to decrease the pressure near t = 20 ns. With continuous increase of laser power, the pressure increases to keep the self-similar solution. It is impossible to control the evolution after t > 0 in the self-similar solution. Actually, HYDRA simulation gave the density evolution after the shock formation as shown in Fig. 4.13. It is clear that almost the same profile as the self-similar solution in Fig. 4.10 is realized. The optimized laser pulse shape is given in [14], which is a highly tailored pulse with intensity from ~3 × 10¹² to ~4 × 10¹⁵ W/cm². It is concluded that this isochoric implosion can be designed with the total laser energy of ~500 kJ.

Fig. 4.13

HYDRA 1-D simulation result of the density evolution modeling the self-similar solution of Fig. 4.10 (right). Reprinted with permission from Ref. [14]. Copyright by IAEA

4.6 Self-Similar Solution – Homogeneous Dynamics

In early time of laser fusion research, adiabatic self-similar solutions have been studied for the implosion and the final stagnation phases. R. Kidder has published a series of papers to show self-similar solutions of hydrodynamics of spherical implosion by tailored adiabatic compression [15]. Such solution is called homogeneous adiabatic flow. Hydrodynamic stability is also studied to the self-similar solution of implosion. The solution is applied to study the physics of ignition and nuclear burn as a pioneering work in laser fusion [16]. The homogeneous adiabatic flow is also applied to model the stagnation dynamics near the maximum compression of implosion [17] to study the hydrodynamic stability of the final compression phase.

Both of self-similar solutions are important to study the dynamics of implosion and analytically study the stability of the solutions. The self-similar solution and its stability analysis can be useful not only to design the implosion dynamics but also to use the verification and validation (V&V). The self-similar solutions to be discussed here is obtained by the method of variable separation, not using the similarity variable in (4.9). The mathematics to obtain the self-similar solutions in this case is easier than the previous case, since the solution is adiabatic and not necessary to take into account the jump by shock wave.

Here, two different self-similar solutions are derived for adiabatic assumption, Namely, it is possible to apply any cases with different spatial distribution of entropy for each fluid element in the imploding or stagnating plasmas. The solution of the stagnation dynamics is explained at first to derive the equations, and the implosion dynamics is explained by use of the same equation with different separation constant.

4.6.1 Stagnation Dynamics

The self-similar solutions for the spherical geometry studied so far always have singularity at t = 0 and we allow the fact that the solutions diverge to infinity. Of course, such divergence is allowed only mathematically and we have to consider neglected physics such as thermal conduction, viscosity, etc. We assume, however, that even such non-adiabatic physics plays important role near t = 0, the solution will be approximately continuous over the time t = 0.

In laser plasmas, another type of self-similar solution is used to describe the final implosion phase, so-called stagnation phase [1]. This solution models the hydrodynamics seen in Fig. 3.29. This self-similar method is introduced to model supernova explosion hydrodynamics [18]. In this case, the similarity variable in the form of (4.9) is not assumed, but ordinary differential equation is found by the mathematical method of separation of variables. Instead of Euler type basic equations in (2.23, 2.24, and 2.25), Lagrangian type Eqs. (2.20, 2.21, and 2.22) is used to obtain the trajectory of each fluid element R(t, r₀), where r₀ is Lagrange coordinate of each fluid element. So, the density ρ(t, r), flow velocity u(t, r), and pressure P(t, r) are assumed in a functional form of A(r₀)B(t). This means the Lagrangian coordinate r₀ corresponds to the similarity variable ξ.

In spherical one-dimensional system, Lagrangian type Eqs. (2.20, 2.21, and 2.22) without the force and heat source can be given in the form with the Lagrangian coordinate R(t, r₀),

$$ \frac{d\ }{dt}\rho +\frac{\ \rho }{R^2}\frac{\partial }{\partial R}\left({R}^2u\right)=0 $$

(4.28)

$$ \rho \frac{d}{dt}u=-\frac{\partial }{\partial R}P $$

(4.29)

$$ \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left(P\ {\rho}^{-\gamma}\right)=0 $$

(4.30)

The radius R is the radius of fluid element located at r = r₀ at t = 0 and defined as

$$ \frac{d}{dt}R=u $$

(4.31)

The time evolution of the radius of a fluid element is defined with the scale function F(t), corresponding to the above B(t). We try to model the stagnation dynamics shown in Fig. 3.29. Smoothing the shock traveling effect in the stagnation dynamics, it is possible to obtain the functional form of F(t) so that the kinetic energy is converted to the thermal pressure to re-bounce the converging dynamics to our-ward.

Setting t = 0 at the maximum compression and the radius of each fluid element to be ξ, the time dependence of each Lagrange mesh is defined as

$$ R\left(t,\xi \right)=\xi F(t) $$

(4.32)

Then, F(t = 0) = 1 is required.

It is easily to derive the following relation from conservation law of mass and entropy.

$$ \rho \left(t,\xi \right)=\Phi \left(\xi \right)F{(t)}^{-3} $$

(4.33)

$$ P\left(t,\xi \right)=\Pi \left(\xi \right)F{(t)}^{-3\gamma } $$

(4.34)

where Φ(ξ) and Π(ξ) are the profiles of density and pressure at t = 0. For the case of deceleration to the center, the equation to the time evolution is derived from (4.29) in the form.

$$ {F}^{3\gamma -2}\frac{d^2}{dt^2}F=\frac{1}{\tau^2} $$

(4.35)

where τ is a characteristic time constant for the stagnation dynamics. It is order of nano-sec in laser implosion and sec in supernova explosion.

We can assume any reasonable density profile Φ(ξ) at the maximum compression, while the pressure should satisfy the force balance relation,

$$ \frac{d}{d\xi}\varPi \left(\xi \right)=-\frac{\xi }{\tau^2}\varPhi \left(\xi \right) $$

(4.36)

It is easy to solve (4.35) and for the case with γ = 5/3, we obtain.

$$ F\left(\mathrm{t}\right)={\left\{1+{\left(\frac{t}{\tau}\right)}^2\right\}}^{1/2} $$

(4.37)

One can easily confirm that this F(t) is a good approximation of the dynamics of the contact surface in Fig. 3.29. It provides approximate model for convergence (t < 0) and expansion (t > 0) as shown in [t/τ, F(t)] diagram (t-r diagram) in Fig. 4.14.

Fig. 4.14

Normalized t-r diagram of fluid elements in the deceleration phase, where t = 0 is the maximum compression

4.6.2 Kidder’s Implosion Dynamics

In 1974, just after the proposal of tailored implosion for laser fusion by Nuckolls et al., Kidder proposed to use a self-similar solution for theoretical design of laser implosion to achieve extremely high-density [16]. It is the theory of homogeneous isentropic compression and its application to laser fusion. Homogeneous compression indicates that the target material with any layered structure converges uniformly with a constant entropy S given as the initial condition. The solution allows any entropy distribution in space.

The equations are same from (4.28) to (4.34), while the variable separation constant in (4.35) has negative sign and (4.37) becomes as follow.

$$ F\left(\mathrm{t}\right)={\left\{1-{\left(\frac{t}{\tau}\right)}^2\right\}}^{1/2} $$

(4.38)

(4.38) is rewritten in the relation of circle in (t/τ, F)

$$ {F}^2+{\left(t/\tau \right)}^2=1 $$

(4.39)

It is clear that the fluid is at rest at the beginning, t = 0, and dF/dt → ∞ at t/τ = 1.

At the maximum compression, t = τ, implosion velocity becomes infinity and all fluid elements converge at the central singular point. Of course, this is mathematical solution and density of (4.33) and pressure (4.34) becomes infinity.

In Fig. 4.15, virtual trajectories of fluid elements located in normalized space, ξ/R₀, is plotted, where R₀ is the initial radius of target and τ is regarded as the implosion time. This solution can be applicable to any density profile at t = 0, for a spherical solid ball, gas target, shell target, and so on. The application of the solution to laser fusion experiment, the critical issues are

• 1. How to generate the initial condition satisfying (4.36).

• 2. How high-density can be achieved even if the finite maximum pressure is demanded.

• 3. How to generate the ablation pressure as shown in Fig. 4.12.

Kidder has applied the solution of a hollow shell DT target to study the possible scenario, such as required laser pulse shape and its total energy, for demonstration of ignition, burn, and fusion energy production [19].

Fig. 4.15

Normalized t-r diagram of fluid elements in uniform implosion dynamics

4.7 Self-Similar Solution of Ablation Dynamics

It is already shown that the deflagration and isothermal rarefaction waves are generated, when an intense laser is absorbed by solid target. Then, it is assumed that the deflagration wave is in stationary state. It is better if we can find an self-similar solution of the ablation structure valid to describe the time evolution of ablation plasma. The author found the self-similar solution of the ablation dynamics to the basic equations of one-fluid two-temperature fluid model [20], while the paper was published in Japanese. Therefore, I would like to show the self-similar solution for the ablating plasma. The nonlinear electron heat conduction and electron-ion temperature relaxation are included within the ideal plasma assumption.

We assume that the plasma is fully ionized with ion charge Z and its mass m_i. We solve (2.105, 2.106, 2.107, and 2.108) to the ion number density n, velocity u, and ion and electron temperatures, T_i and T_e. The specific heat ratio γ = 5/3 is assumed. Then, ρ = m_in, ε_i = 3/2(T_i/m_i), ε_e = 3/2(ZT_e/m_i). We try to find a self-similar solution in the plane geometry in the x-coordinate. Then, it is possible to reduce them to the following four coupled equations after neglecting source terms and ion thermal conduction much smaller than electron one.

$$ \frac{\partial n}{\partial t}+\frac{\partial (nu)}{\partial x}=0 $$

(4.40)

$$ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{1}{m_in}\frac{\partial }{\partial x}\left[n\left(Z{T}_e+{T}_i\right)\right]=0 $$

(4.41)

$$ \frac{\partial {T}_i}{\partial t}+u\frac{\partial {T}_i}{\partial x}+\frac{2}{3}{T}_i\frac{\partial u}{\partial x}-{\nu}_0\frac{n\left({T}_e-{T}_i\right)}{{T_e}^{\frac{3}{2}}}=0 $$

(4.42)

$$ \frac{\partial {T}_e}{\partial t}+u\frac{\partial {T}_e}{\partial x}+\frac{2}{3}{T}_e\frac{\partial u}{\partial x}+{\nu}_0\frac{n\left({T}_e-{T}_i\right)}{Z{T_e}^{\frac{3}{2}}}-\frac{2}{3n}{K}_0\frac{\partial }{\partial x}\left({T_e}^{\frac{5}{2}}\frac{\partial }{\partial x}{T}_e\right)=0 $$

(4.43)

where ν₀ and K₀ are physical constants defined in (2.109) and (2.110) with Coulomb log. In the following calculations we solve the case with Z = 1 and lnΛ = 10 for simplicity.

It is important to note that if we try to find the solution with a given density at the ablation front, it is hard to obtain the self-similar solution because the density is fixed by this condition. Instead, we use the property of the ablation front that the density, velocity, and temperature profiles are very steep and the ablation structure may not be affected even if we adopt an ideal condition. The mass ablation rate, nu, and ablation pressure, nT_e and nT_i, are finite, while n is infinity and u, T_e, and T_i are null. This assumption is acceptable as seen, for example, in Figs. 3.23 and 2.24. Therefore, we find a self-similar solution satisfying the conditions.

$$ \left\{\begin{array}{c}n=\infty, \kern1.5em u,{T}_e,{T}_i=0\\ {}\kern-1em nu,n\left({T}_e+{T}_i\right): finite\ \end{array}\right.\ at\ x=0 $$

(4.44)

4.7.1 Dimensional Analysis

The dimensional analysis of (4.40)–(4.43) requires the following dependence to the coordinate x and time t.

$$ u\sim \frac{x}{t},{T}_e,{T}_i\sim {\left(\frac{x}{t}\right)}^2 $$

(4.45)

In the present analysis, the temperature relaxation and electron thermal conduction by Coulomb binary collision are determined by the collision time discussed in Sect. 2.1. They both are function of the collision frequency ν in the form.

$$ \nu \propto \frac{n}{T_e^{3/2}} $$

(4.46)

Since (4.40) and (4.41) are homogeneous to the density n, (4.46) gives the dimension of the density n as the condition that a self-similar solution is possible. Inserting ν~1/t to (4.46), it is found that the density should be proportional in the form.

$$ n\sim \frac{x^3}{t^4} $$

(4.47)

We introduce the similarity variable ξ same as (4.9).

$$ \xi =\frac{x}{t^{\alpha }} $$

(4.48)

Then, we can assume the following functional form.

$$ n={K}_0{m}_i^{5/2}{t}^{3\alpha -4}g\left(\xi \right) $$

(4.49)

$$ u={t}^{\alpha -1}v\left(\xi \right) $$

(4.50)

$$ {T}_i={m}_i{t}^{2\left(\alpha -1\right)}{\tau}_i\left(\xi \right) $$

(4.51)

$$ {T}_e={m}_i{t}^{2\left(\alpha -1\right)}{\tau}_e\left(\xi \right) $$

(4.52)

Inserting (4.49, 4.50, 4.51, and 4.52) into (4.40, 4.41, 4.42, and 4.43), the following coupled ordinary differential equations are obtained.

$$ \left(v-\alpha \xi \right){g}^{\prime }+\left[{v}^{\prime }+\left(3\alpha -4\right)\right]g=0 $$

(4.53)

$$ \left(v-\alpha \xi \right){v}^{\prime }+\frac{1}{g}{\left[g\left({\tau}_e+{\tau}_i\right)\right]}^{\prime }+\left(\alpha -1\right)v=0 $$

(4.54)

$$ \left(v-\alpha \xi \right){\tau}_i^{\prime }+\left[\frac{2}{3}{v}^{\prime }+2\left(\alpha -1\right)\right]{\tau}_i-{\mu}_0\frac{g\left({\tau}_e-{\tau}_i\right)}{\tau_e^{3/2}}=0 $$

(4.55)

$$ \left(v-\alpha \xi \right){\left({\tau}_e+{\tau}_i\right)}^{\prime }+\left[\frac{2}{3}{v}^{\prime }+2\left(\alpha -1\right)\right]\left({\tau}_e+{\tau}_i\right)-\frac{2}{3g}{\left({\tau}_e^{\frac{5}{2}}{\tau_e}^{\prime}\right)}^{\prime }=0 $$

(4.56)

where the dash (^′) means the derivative by ξ. In (4.55), μ₀ = m_iK₀ν₀ (=6.417).

4.7.2 Integration

We have to solve the coupled Eqs. (4.53, 4.54, 4.55, and 4.56) by starting from the boundary shown in (4.44). To numerically integrate them, approximated analytic solutions are necessary near the ablation front (x = 0). By use of the approximate boundary condition (4.44), (4.53, 4.54, 4.55, and 4.56) reduce to the following equations for ξ ≈ 0.

$$ vg={J}_0: const. $$

(4.57)

$$ g\left({\tau}_e+{\tau}_i\right)={P}_0: const. $$

(4.58)

$$ {\tau}_e={\tau}_i $$

(4.59)

$$ v{\left({\tau}_e+{\tau}_i\right)}^{\prime }+\frac{2}{3}{v}^{\prime}\left({\tau}_e+{\tau}_i\right)-\frac{2}{3g}{\left({\tau}_e^{\frac{5}{2}}{\tau_e}^{\prime}\right)}^{\prime }=0 $$

(4.60)

It is possible to set J₀ = 1 by re-scaling the definition of ξ, but P₀ should be determined so that the solution is acceptable. As we see below, the constant P₀ is determined as the eigen-value problem so that the solution satisfies the boundary condition at large ξ point.

Now, it is possible to solve (4.60) for ξ ≪ 1, and the following relation is obtained.

$$ {\tau}_e={\tau}_i={\left(\frac{25}{2}\xi \right)}^{2/5} $$

(4.61)

$$ g=1/v=\frac{P_0}{2}{\left(\frac{25}{2}\xi \right)}^{-2/5} $$

(4.62)

Consider an asymptotic solution for ξ → ∞. Since we are interested in a solution with finite total energy of the ablation plasmas, namely the heat flux from ξ → ∞ should be finite.

$$ {\tau}_e^{\frac{5}{2}}{\tau}_e^{\prime }={S}_0\ \left(\xi \to \infty \right) $$

(4.63)

where S₀ is a constant to be determined after integration. The boundary condition (4.63) is satisfied with the asymptotic solution of the electron temperature.

$$ {\tau}_e\propto {\xi}^{\frac{2}{7}},{\tau}_i=0\ \left(\xi \to \infty \right) $$

(4.64)

Consider how the integration path is determined by refereeing the same type of plane of Fig. 4.6. In the present case, we consider the velocity and temperature diagram (V, T) as defined below.

$$ g\left(\xi \right)=\frac{\Gamma \left(\xi \right)}{\xi },v\left(\xi \right)=\xi V\left(\xi \right),{\tau}_e\left(\xi \right)+{\tau}_i\left(\xi \right)={\xi}^2T\left(\xi \right) $$

(4.65)

Transform (4.53) and (4.54) to the coupled equations for the new variables.

$$ \left(V-\alpha \right)\frac{dln\Gamma}{ dln\xi}+\frac{dV}{ dln\xi}=4\left(1-\alpha \right) $$

(4.66)

$$ T\frac{d ln\Gamma}{d ln\xi}+\left(V-\alpha \right)\frac{d V}{d ln\xi}=-\frac{d}{d\xi}\left(\xi T\right)-\left(V-1\right)V $$

(4.67)

Consider the property of (4.66) and (4.67) in the asymptotic limit (ξ → ∞). From the asymptotic relation of (4.64), T and the pressure term in (4.67) vanishes.

$$ T\propto \propto {\xi}^{-12/7},\frac{d}{d\xi}\left(\xi T\right)\propto {\xi}^{-12/7}\to 0\ \left(\xi \to \infty \right) $$

(4.68)

Eliminating the velocity derivative term in (4.66) and (4.67), we obtain the following equation to the density.

$$ \frac{dln\Gamma}{ dln\xi}=\frac{\left(V-1\right)V+4\left(1-\alpha \right)\left(V-\alpha \right)}{{\left(V-\alpha \right)}^2-T} $$

(4.69)

To satisfy the rarefaction wave condition that the density decreases monotonically with increase of ξ.

$$ \frac{dln\Gamma}{ dln\xi}<0\ \left(\xi \to \infty \right) $$

(4.70)

For the case with α > 9/8, the numerator is positive for any value of V and the condition (4.70) requires

$$ T>{\left(V-\alpha \right)}^2 $$

(4.71)

In this case, the physically reasonable solution is possible only when the integration path from subsonic near ablation front continuously goes to supersonic region of asymptotic flow with the path shown with the dashed lines in Fig. 4.16 This requirement gives unique solution for a given α (>9/8). It is also clear from () that the density decreases exponentially for ξ → ∞. It is also possible to obtain the solution with exponential decay for 1 < α < 9/8.

Fig. 4.16

Similarity functions velocity and temperature. From both boundary conditions, the integration path is found to take the path shown with dotted line to converge the singular point (α, 0)

On the other hand, self-similar solution for α < 1, from (4.49, 4.50, 4.51, and 4.52) it is clear that at t = 0,

$$ {T}_e,{T}_i,u\to \infty $$

(4.72)

The solution has singularity at t = 0. It is important to know that the solutions of implosion dynamics of the self-similar solutions in this Chapter have been obtained with α < 1.

In [20], further new variables are introduced to solve (4.53, 4.54, 4.55, and 4.56) numerically with functions of slowly varying in the lnξ-ccoordinate, but we skip discussing this method here.

4.7.3 Classical Absorption Case (α = 5/4)

Although the heat flux is deposited near the critical surface, for example, short wavelength laser is irradiated with not so high-intensity, where the classical absorption is dominant process to hear the electron. Here we assume that the heat flux coming from ξ → ∞ is approximately equal to the laser energy flux. The heat flux is given as

$$ Q={K}_0{T}_e^{5/2}\frac{\partial }{\partial x}{T}_e\propto {t}^{6\alpha -7}{S}_0 $$

(4.73)

where S₀ is defined in (4.63) and obtained after solving the self-similar solution. It is clear that we can obtain the solution for the constant energy deposition for the case with α = 7/6. In Ref. [20], the classical absorption for DT plasma is assumed to evaluate the absorbing power for a constant laser intensity. It is shown that α = 5/4 is approximated [20].

In Fig. 4.17, nondimensional ablation structure is shown after solving the eigen value problem to the normalized pressure P₀. It is found that P₀ = 2.7227. This is regarded better solution than the stationary solution shown in the previous chapter, while the self-similar solution gives approximately the same scaling laws for the ablation pressure and so on as a function of absorbed laser intensity within our interest for laser plasmas.

Fig. 4.17

A self-similar solution of laser ablation plasma for the case of α =5/4 corresponding to the classical absorption of a constant intensity laser irradiation. The self-similar solution can be found by assuming that the density is infinite and the others are null in the moving frame, while the product pressure and mass ablation rate are kept finite. The profiles are normalized density (g), flow velocity (v), electron and ion temperatures (τ_e,τ_i) and electron heat flux (S)

4.8 Blast Wave (Taylor-Sedov Solution)

After the implosion and energy concentration at the compressed small central area, it is also important to know how this energy spreads hydrodynamically in space spherically. It is surprising that with some idealization of the problem, it is possible to find another self-similar solution as mentioned above. This is well known as the one-dimensional mathematical solution in spherically symmetry when a point energy source is released spontaneously at the center (r = 0). This is called a blast wave and a spherical wave with a strong shock front propagating outward.

It is well summarized on a brief history of study of the blast wave in Ref. [21], so let us follow the description. A blast wave follows the rapid and localized release of a large amount of energy in a medium. The physics community got seasonably interested in the dynamics of such shocks in air in the early 1940s. Taylor [10], von Neumann [12] and Sedov [11] independently understood that, because of the global conservation of mass and energy, the extension R of the blast had to grow with time like a power law (time)^α with α = 2/5 [or 2/(N + 2) in dimension N. From a few publicly available snapshots of the blast at different times, Taylor could estimate within 10% the strength of the Trinity detonation in 1945, at the time classified information [22].

Remarkably, the hydrodynamic description of the flow inside the blast, now known as the Taylor–von Neumann– Sedov solution (or Taylor-Sedov solution), is self-similar in time, depending only on the rescaled radial distance r = R(t). This similarity is of the first kind [22], i.e., driven by global invariants, and all exponents can be derived by dimensional analysis. This solution found widespread relevance beyond its initial realm, notably in plasma physics to describe laser-induced shocks and in astrophysics for the evolution of supernova remnants.

When the energy of the explosion is E₀ and the density of the surrounding gas of uniform density is ρ₀, the basic equations are one-dimensional equations of (4.5, 4.6, and 4.7). In this case, a strong shock wave propagates in the gas. Since the total energy of blast wave should conserve, the dimensional analysis requires the relation:

$$ {E}_0\sim \rho {u}^2{r}^3 $$

(4.74)

Inserting (4.10, 4.11, and 4.12) into (4.74), the similarity valuable is obtained as

$$ \alpha =\frac{2}{5},\xi ={\left(\frac{\rho_0}{E_0}\right)}^{1/5}\frac{r}{t^{2/5}} $$

(4.75)

Here uniform density K = 0 is assumed.

The radius of the strong shock wave front is defined as

$$ R(t)={\xi}_0{\left(\frac{E_0}{\rho_0}\right)}^{1/5}{t}^{2/5} $$

(4.76)

where ξ₀ gives the shock front and ξ₀ is the eigenvalue of this mathematical problem.

The equations to be solved are the same as (4.14) and (4.15). In the present case, the eigen value of the consistent solution is determined so that the following energy conservation is satisfied.

$$ {E}_0={\int}_0^R4\pi {r}^2\rho \left(\varepsilon +\frac{u^2}{2}\right) dr $$

(4.77)

The integration path in (U, C) plane is shown in Fig. 4.18a [9]. In this case, the shock RH relation allows to jump over the singular curve and the point A in Fig. 4.18a is found to satisfy (3.13). The shocked region finally goes to the node singular point (U, C) = (0, ∞) as shown in the Fig. 4.18a.

Fig. 4.18

(a) Taylor-Sedov solution for a strong point explosion in the U, C plane. Parameters:γ = 5/3, α = 2/5, Κ = 0. (b) Density ρ, pressure P and velocity U of the Taylor-Sedov solution as a function of radius r. The label A denotes the values at the shock front. Reprinted with permission from Ref. [6]. Copyright 1998 by Oxford University Press

The normalized blast wave profile is obtained in Fig. 4.18b. Three quantities abruptly increase at the shock front and the velocity and density disappear at the center. Note that the temperature becomes infinity at the center, and the product pressure of the density and temperature is kept finite.

The calculation has been done with computer, however, in the former Soviet Union where there was no computer, surprisingly Sedov analytically solved the problem and found the eigenvalue. The approximate analytical solution of the eigen-value is given in the form in [23].

$$ {\xi}_0={\left[\frac{75\left(\gamma -1\right){\left(\gamma +1\right)}^2}{16\pi \left(3\gamma -1\right)}\right]}^{1/5} $$

(4.78)

The value agrees well with the numerical calculation values by Taylor. In Table 4.1, both results are compared for three different specific heat γ. Both gives reasonable γ-dependence. For the case with large γ, the internal freedom of the gas is at most N = 3 (x, y, z translational motions) and γ = (N + 2)/N gives γ = 5/3. Additional freedoms like molecular rotation and vibration yields N = 5 and γ = 7/5 = 1.4. Increase of freedom means more need for thermal energy for all freedoms and the fraction of the energy going to the shock kinetic energy decreases. The decrease of the eigen-value in Table indicates the shock speed gets slow.

It should be noted that both results give the eigen-value is almost equal to unity and this result encourages us in comparing some experimental result to the theoretical self-similarity. Only with the dimensional analysis, it is reasonable to assume ξ₀ = 1 for an approximate solution. Without solving the complicated equations for the spatial profile, it is possible to compare with the experimental data.

Let’s calculate the energy partition ratio of the self-similar solution. The fractional ratio of the thermal energy and kinetic flow energy to the explosive energies does not change in time. Although thermal energy escapes as energy such as radiation, kinetic energy is preserved and spreads to space. That proportion is obtained with the following integration

$$ F=\frac{\int_0^R4\pi {r}^2\rho \varepsilon dr}{\int_0^R4\pi {r}^2\rho \frac{u^2}{2} dr} $$

(4.79)

The result is obtained as.

$$ F\kern1em =6.1\ \left(\gamma =1.2\right),\kern1em =3.5\ \left(\gamma =1.4\right),\kern1em =2.5\ \left(\gamma =5/3\right) $$

(4.80)

The ratio of conversion to the internal energy increases as γ approaches unity. The physical reason is clear as already mentions regarding to the Table 4.1. It was γ = (N + 2)/N, where N is the internal degree of freedom of gas. The greater the degree of internal freedom, the lower the proportion of compatible energy going to the flow kinetic energy.

Table 4.1 The eigen value of self-similar solutions depends on the value of the specific heat γ

Let’s itemize the features of this solution.

• 1. Flow velocity is null and the gradient of pressure is also null at the center (r = 0) to satisfy the boundary condition. As the result, the density is also zero at the center.

• 2. Because the pressure is finite at the center, it turns out that the temperature diverges to infinity.

• 3. When the blast wave arrives, the delta function-like force −∂P/∂r works outward, while the opposite-directional force toward the center works immediately. After a while the force disappears.

• 4. The solution shows that the temperature becomes infinity at the center cannot happen physically. This is because there is a self-similar solution only by neglecting heat conduction. In fact, in the early time of explosion radiative heat conduction waves such as X-rays (this is historically called fireball) spread and blast wave is generated after a while.

• 5. Self-similar solution is not applicable at any time, but because the blast wave ionizes surrounding gas, it will cool down while losing energy by radiation such as X-rays from partial ionized plasma. As the density of the blast wave front is higher, it is easier to cool, so the pressure at the front decreases and the kinetic energy is accumulated near the front and shocked matter expands like a shell.

4.9 Laser Blast Wave and Dissipation

4.9.1 Laser Experiments

By focusing and irradiating high-intensity short-pulse laser to a solid target surface placed in gas, the ablating plasma from the solid creates a clean blast wave in the gas as shown in Fig. 4.19a, where the blast wave image is taken by the darkfield shadow imaging [24]. Nd glass laser of 200 J and 5 ns pulse is irradiated on a foil to heat up about 1 keV. The gas is nitrogen gas of 5-torr, and the laser is irradiated from the right. The laser-irradiated target is an aluminum plate. It is measured that the ablating plasma expands with the velocity of 700 km/s. The shock wave is collisional shocks and the white ring in Fig. 4.19a is due to the refraction of the diagnostic laser beam at the shock front with abrupt electron density jump.

Fig. 4.19

Blast waves produced by laser irradiation on solid target in gas. (a) the case with nitrogen gas. This is a case of energy conserving blast wave. (b) the same experiment but the gas is higher Z xenon. Due to radiation energy loss, the blast wave front becomes unstable to hydrodynamic instability. Reprint with permission from Ref. [23]. Copyright 1998 by American Physical Society

In Fig. 4.20, the measured radius of the blast wave is plotted as a function of time. It is seen that for t = 6–18 ns the front moves at a constant velocity corresponding to the velocity of the ablating plasma, a blast wave not yet having formed. Around 25 ns it is clear that the blast wave velocity change to the time dependence given by the blast wave in (4.76) in proportion to t^-2/5. As we see later, the early time evolution may be a blast wave induced by the ejecta-driven, where the ejecta means the ablating plasma.

Fig. 4.20

Time evolution of the blast wave front from early ejecta motion to the phase of Taylor-Sedov blast wave for a long duration. Reprint with permission from Ref. [23]. Copyright 1998 by American Physical Society

In the case of the nitrogen gas, it is evaluated that the specific heat γ = 1.3 ± 0.1. On the other hand, when the same blast wave experiment is carried out in xenon gas, the image in Fig. 4.19b is measured. This is due to an instability of radiative blast wave predicted relating to astrophysical objects. It is concluded in Ref. [25] that because of the radiation cooling effect, the effective specific heat becomes lower than the previous nitrogen case, γ = 1.06 ± 0.02. This is the experimental evidence that the blast wave is unstable for γ < 1.2. This is called Vishiniac instability.

For a variety of energy sources, the self-similar solution of the blast wave can be plotted on a single space of pressure and the scale range parameter S as shown below. From (4.76) and (4.74) with dimensional relation ρu²~P, it is easy to derive the following functional relation after eliminating the time.

$$ P=\frac{A}{S^3},S=\frac{R}{{\left({E}_0\right)}^{1/3}} $$

(4.81)

where A is a numerical constant of for a given γ and S is the scaled range parameter. In Fig. 4.21, the data from NRL experiment and NIF laser with 10 kJ irradiations are plotted with orange and green solid circles in (P, S) diagram [26]. For comparison to another type of blast wave, those produced by high-explosive and nuclear explosion are also shown with red dashed and solid blue lines. For comparison of extremely different energy explosion, the radius is in unit of meter and energy is in the unit of kiloton of TNT (KT = 4x10¹² J). It is found that for small S the relation (P ∝ S⁻³) is satisfied, while in large S region the relation tends to another relation (P ∝ S⁻¹). Note that A ~ 10⁷ in these units with the value of NRL laser experiment (S, P) = (10, 10,000).

Fig. 4.21

Scaling law for the pressure at the blast wave front as a function of the scaled rage parameter. Three different scale experiments from laser, high explosive, and nuclear explosive are found to be in the same scaling law. Reprint with permission from Ref. [26]. Copyright 1998 by American Institute of Physics

Although the explosion energy in Fig. 4.21 varies from 100 [J] to one [KT], while the change of S is about 4x10⁻⁴ in (energy)⁻³ and the radius of the same pressure for the laser blast wave of 1 cm as seen in Fig. 4.19 is 27 m in KT blast wave.

4.9.2 Dissipative Blast Waves

The radiation loss from shocked plasma decreases the temperature of the plasma and the pressure also decreases. Note that the self-similarity relation of Taylor-Sedov blast wave is first derived from the dimensional analysis of the system with the energy conservation relation. Namely,

$$ {E}_0\propto {\rho}_0{D}^2{r}^d\propto \frac{r^{d+2}}{t^2} $$

(4.82)

where d is the dimension of the system. The power law dependence on the system dimension is given as

$$ R(t)\propto {t}^{\alpha }, $$

(4.83)

$$ \alpha =2/\left(d+2\right) $$

(4.84)

The power α = 2/5 (3-D, spherical), =1/2 (2-D, cylindrical), = 2/3 (1-D, planar).

The blasts caused by an intense explosion seen above like Taylor-Sedov blast wave are the prototypical example of self-similarity driven by conservation laws. In dissipative media, however, the energy conservation is violated, yet it is known that a distinctive self-similar solution appears. It hinges on the decoupling of random and coherent motion permitted by a broad class of dissipative mechanisms. This enforces a peculiar, layered structure in the shock as shown in [21]. It has been derived by the full hydrodynamic solution, validated by a microscopic approach based on molecular dynamics simulations.

When the thermal energy of the blast wave escapes or dissipates from the system by any process, for example, radiation loss, the blast wave changes from the energy conservative solution to the momentum conservation one. The dimensional analysis in this case is given as

$$ {M}_0\propto {\rho}_0D{r}^d\propto \frac{r^{d+1}}{t} $$

(4.85)

where M₀ is the total momentum of the expanding matter and α in (4.84) is given as

$$ \alpha =1/\left(d+1\right) $$

(4.86)

In [21], the particle dynamics is solved by molecular dynamics simulation for a model system of granular gas consisting of identical spherical grains with the same radius and mass, where inelastic binary collisions conserve momentum but dissipate kinetic energy. The physical property of the results does not depend on specific dissipation mechanisms. In Fig. 4.22, the density, velocity, and temperature profiles are plotted as a function of the normalized radius. The cross marks with “Cons.” are the results of the energy conservation case of the Taylor-Sedov solution, while the open circles with “Dissip.” are the results obtained with the molecular dynamic simulation. The temperature decreases due to the dissipation inducing the increase of the density to keep the pressure jump at the blast wave.

Fig. 4.22

The simulation result of a dissipative blast wave is shown with solid circles in a self-similar evolution. The corresponding Taylor-Sedov solution is also plotted with cross marks. Strong dissipation in the shocked region leads the solution from energy conservative one (Taylor-Sedov) to momentum conservative self-similar solution. Reprint with permission from Ref. [21]. Copyright 1998 by American Physical Society

In Fig. 4.22, the orange zone is the shock front, and the next zone is cooling region. After the zone, cold fluid zone shown with the two dashed lines follows. Since the velocity is kept high after the shock front, the central fluid easily flows the cold fluid to generate the cavity region after the thin shell structure of the dissipative blast wave. In Fig. 4.22, the solid lines are theoretical curves obtained from the fluid equations by modeling a dissipation term proportional to the product of the density and temperature [21].

More general model of cooling blast wave has been solved by assuming homogeneous self-similar cooling in the equation of energy. It is mathematically shown that for assuming the radius of the shock front of the blast wave, R(t) ∝ t^α, there is self-similar solution physically reasonable for a given α in the rage α = 1/4 ~ 2/5 as given above for spherically symmetric case [27]. The power law of the blast radius α is also called a deceleration parameter.

4.9.3 Radiation Effect on Blast Waves

Let us consider the case where the radiation cooling by Bremsstrahlung and radiative recombination becomes important in the blast wave evolution. It is known that the both cooling rates are roughly proportional to the density times the square root of the electron temperature. So, the above dissipation model can be applicable to predict that the same type of the momentum conserving blast wave becomes dominant after the Taylor-Sedov blast wave and the power law of the time evolution of the blast wave radius changes from α = 2/5 to 1/4 for spherical geometry (d = 3).

The study of blast waves produced by intense lasers in gases has also been done in the laboratory with better diagnostic instruments. A systematic scan of laser produced blast waves was performed and the structure of blast waves was examined over a wide range of drive laser energy. Lasers with energies ranging from 10–1000 J irradiating a pin target in either xenon or nitrogen gas, creating a spherical blast wave [28].

A strongly radiating blast wave in xenon gas is observed while blast waves in nitrogen more closely approximate a pure Taylor–Sedov wave as already seen in Fig. 4.19. Radiation emitted from the hot expanding shell ionizs the gas ahead of the shock wave, leading to a radiative ionization precursor. The precursor is the same as the preheating tongue in case of electron preheating in shock structure as show in Fig. 3.8. Furthermore, energy loss by radiation in an optically thin system will increase the rate of deceleration of the blast wave.

These blast waves exhibit significant energy loss through radiation while propagating in xenon as evidenced by interferometric imaging revealing radiative precursors and deceleration parameter α well below those of an energy-conserving wave. Thinning of the blast wave shell from radiative cooling is observed through comparison of shocks launched in gases of differing atomic number. Shell thinning is also measured in cylindrical geometry (N = 2), when the gas density is altered, indicating the influence of conditions within the pre-shock medium [29]. These results are compared with radiative-hydrodynamic simulations.

One-dimensional simulations of blast wave evolution in xenon were carried out using a radiation hydrodynamics code in which radiation transport was calculated using a multi-group implicit Monte Carlo (IMC) technique, and the individual plasma components are treated in LTE. Figure 4.23 shows the simulated radial profiles of the electron density compared with the experimental data at different times in the blast wave evolution [29]. This discrepancy is likely a result of non-LTE effects associated with the ion fluid since the ions typically take several nanoseconds to thermalize. The radiative precursor is larger for the higher density in agreement with the experimental profile but extends further ahead of the shock than measured.

Fig. 4.23

Time evolution of the electron density of a cylindrical blast wave generated by laser irradiation in gas. The dotted lines are corresponding simulation result with radiation transport. Reprint from Ref. [29] with permission from Institute of Physics

This indicates that in the precursor region the code does not accurately model the physics because of a significant departure from LTE caused by the detailed atomic physics of the radiating shock. The lower deceleration parameter of α = 0.44 compared with α = 0.47 with high density. Such effect of non-LTE atomic process has been studied with use of the collisional radiative equilibrium (CRE) code the physics, where CRE will be discussed later. The previous experimental data [29] is analyzed with CRE non-LTE code and it concluded that non-LTE code can explain the experimental data [30].

4.10 Blast Waves in Supernova Remnants

4.10.1 Supernova Remnants (SNRs)

The blast wave generated by an explosion of supernova is very important for study of astrophysics mainly related to the heavy element production in Universe. There are a variety of types of the supernova explosions. Most of they are classified as Type Ia and Type II the explosion mechanisms of which are very different, but the explosion energies are almost the same as 10⁴⁴ [J]. It is surprising to know that this energy can be imagined by comparing to the rest mass energy of the Sun, Mc² = 2x10⁴⁷ [J], where the mass of the Sun is M = 2x10³³ [g] and c is the speed of light.

In Type-Ia, the explosion energy is produced by nuclear fusion reactions in a white dwarf as explained in Fig. 3.15. On the other hand, Type II supernova explosion is triggered by gravitational collapse of massive start and the energy of about 10⁴⁶ [J] is generated as the energy of neutrinos. Since the mean free path of the neutrinos is much larger than the column density of the star, only 1% of the neutrino energy is converted to the matter explosion energy.

After such supernova explosions, the matter of exploded star and the surrounding gas continue to emit a wide range of electromagnetic waves over then of thousand years. Such remnant of the supernova explosion is called supernova remnant (SNR). In Fig. 4.24, X-ray image of the supernova remnant SN1006 is shown, where the image was taken by the x-ray satellite Chandra [31]. Its explosion was recorded many places in 1006. The detail of this remnant is given, for example, in [31]. Its distance from the Earth is about 7200 light-years (~ 7 × 10¹⁶ km). SN1006 was the brightest supernova (SN) witnessed in the human history. As of 1000 years later, it stands out as an ideal laboratory to study Type Ia supernova and the shocks in supernova remnants.

Three-dimensional supernova explosions have been studied for a variety of so-called progenitors (stars before explosion). According to [32], the density structure of progenitor of Type II is given approximately in the power law form,

$$ \rho (r)={\rho}_0{\left(\frac{r}{r_0}\right)}^{-3} $$

(4.87)

where ρ₀ = 10¹⁰ [g/cm³] and r₀ = 10⁷ [cm]. The surface of the star has about r = 10¹³ [cm] and ρ(r) = 10⁻⁸ [g/cm³]. The shock wave generated inside the propagates to propagate outward and in the idealistic limit like the power law density profile, we can also find a self-similar solution of the shock propagation as described at p. 812 in [5].

In [32], the averaged shock wave velocity is shown to be about 10⁴ km/s, 3% of the speed of light. So, roughly speaking, the shock wave arrives at the surface of the progenitor after 3 h, then the star starts to shine. Once the shock front arrives at the surface of the star, the shocked material expands in a low density inter-stellar medium (ISM). This is the same as the free expansion of the ablation plasma by laser irradiation as seen in Fig. 4.20, where free expansion is measured until about 20 ns and then the self-similar blast wave was generated. Since the density ratio of the expanding material called “ejecta” to the density in ISM is extremely high, the ejecting material pushes the ISM with the same velocity. This is because the velocity of the ejecta is faster than the sound velocity of compressed ISM in front of the contact surface. Such pushing is called “snowplow”.

In Fig. 4.25, one-dimensional simulation of the gravitational collapsing supernova 1987A is shown [33]. Note that the surface of the progenitor has a sharp density drop at the surface because of strong stellar wind. In the figure, the stage numbers correspond to the time since explosion as (0) t = 0 s; (1) 8.96 s; (2) 1.67 × 10² s; (3) 1.06 × 10³ s; (4) 3.33 × 10³ s; (5) 7.46 × 10³ s. It is seen that the shock wave arrives at the center around 2 h. The density profile at the time of the shock arrival at the surface is almost flat profile and due to the expansion, this density decreases as the volume of the matter increases with the expansion velocity of 10,000 km/s. Then, the snowplow continues for a long time.

It is said that such a snowplow phase continues until the total mass of snow-plowed gas becomes almost the same mass of the effective ejecta M.

$$ \frac{4}{3}\pi {R}^3{\rho}_{ISM}=M $$

(4.88)

where ρ_ISM is the mass density of ISM. Assuming the ejecta is expanding with the velocity 10⁴ km/s, ρ_ISM as one hydrogen per cm³, and demanding the mass M is equal to the solar mass, then the timing to satisfy (4.88) is easily evaluated as about 200 years. Note that this simple value is too early as see below. It takes long time to detach the ejecting material and the autonomous propagation of the blast wave as the self-similar solution. This is because the density of the ejecting material is very high compare to the density of ISM.

The snow-plowed gas forms Sedov-Taylor self-similar blast wave after the deceleration of the contact surface begins. At the same time, the deceleration of the front region of the ejecting material pushes the following ejecta material to produce a reverse shock wave propagating toward the central region, where the density increase. This is unstable scheme to hydrodynamic instability. The Rayleigh-Taylor instability of the contact surface has been studied to compare the observation images.

It is noted that the velocity of the collisionless shock of SN1006 in Fig. 4.24 is U_s = 3000 km/s evaluated through the observation over 11 years. Typical values for the SNRs are given in [34] and (t, R_sh, U_sh) = (100, 2, 8000), (1000, 5, 2000), (10,000, 12.5, 500), where t is the time after explosion in the unit of years, R_sh is the shock front radius in units of pc (=3.1 × 10¹⁶ m) and U_sh is the velocity of the shock front in units of km/s, respectively.

Fig. 4.24

The x-ray emission image of the supernova 1006 observed by Chandra x-ray satellite (NASA). Reprint from Ref. [31] with kind permission from Springer Science + Business Media

Fig. 4.25

Time evolution of the density profiles as a function of mass from the center. The shock wave is generated at t = 0. The stage numbers correspond to the time since explosion as (0) t = 0 s; (1) 8.96 s; (2) 1.67 × 10² s; (3) 1.06 × 10³ s; (4) 3.33 × 10³ s; (5) 7.46 × 10³ s. Around the time of 2 h, the shock wave arrives at the surface of the star to start shining. Reprinted with permission from Ref. [33]. Copyright by American Astronomical Society

Fig. 4.26

Self-similar solutions of density, pressure, and velocity of the ejector-dominant blast wave of a typical Type Ia supernova remnant. The radius r = R_c is the contact surface. The radii of the reverse shock front and the blast wave are 0.935 and 1.181. Reprinted with permission from ref. [39]. Copyright by American Astronomical Society

Fig. 4.27

The simulation result of time evolution of two shock fronts are plotted with the black solid lines. The shock fronts of the present ejector-driven self-similar solution are plotted with gray solid lines. The dotted two lines are analytic solution [40]. Credit: Fraschetti, Federico, et al., A&A, 515, A104, 2010, reproduced with permission © ESO

Fig. 4.28

The simulation result of time evolution of density profiles. The thin red lines show the present and Taylor-Sedov blast wave solutions [40]. Credit: Fraschetti, Federico, et al., A&A, 515, A104, 2010, reproduced with permission © ESO

Fig. 4.29

X-ray image of Tycho SNR by Chandra x-ray satellite [NASA]

Fig. 4.30

Simulation result of star formation starting from low-density molecular cloud in self-gravitational system. The self-similar solution by Larson-Penston agreed well in the early stage. Reprinted with permission from Ref. [43]. Copyright by American Astronomical Society

In Fig. 4.24, the spatial x-ray image of SN1006 near the sharp shock front has been analyzed in detail relating to the physics of cosmic-ray generation and confinement near the shock surface by magnetic field [35]. The typical temperature and density at the rear of the shock front is T_i ∼ 15 keV (T_e ∼ 0.5–1 keV) and the number density n_i ∼ 1 cm⁻³ and the corresponding proton Coulomb mean free path is estimated to be 4 × 10¹⁷ m (= 41 light years). The diameter of SN1006 is about 3 × 10¹⁷ m (= 32 light years). The shock front width which is here defined to be that of the sharp intensity front in Fig. 4.24 is about 1.2 × 10¹⁵ m (=0.12 light years) [35]. As a result, it is concluded that the thickness of the shock front of SN1006 SNR is less than 1% of the radius and the Coulomb mean free path is 400 times longer than the shock front thickness, namely, the shock should be the collisionless shock.

Most of the shock waves are collisionless in Universe. To demonstrate the formation of collisionless shock with intense lasers, a model experiment was proposed [36]. With use of NIF and OMEGA lasers, the collisionless shock generation has been demonstrated experimentally with use of counter streaming ablation plasma [37, 38]. The magnetic field is generated by Weibel instability and the collisionless shock wave is generated after the nonlinear amplification of magnetic field.

In studying such collisionless shock waves, it is possible to use most of the hydrodynamics relations such as Ranking-Hugoniot relation except the shock wave structure. Instead of the particle collisions, Larmor motion of charge particles in turbulent magnetic field plays the role of dissipation. For example, a proton Larmor radius with a velocity of 10³ km/s in the 3μG magnetic field is 3 × 10⁷ m, extremely shorter than the radius of SNR and even shorter than the front thickness of the x-ray image, where the magnetic field in ISM is about 1 ~ 10 μG.

It should be noted that the big and essential difference of the SNR shocks compared with the laser-driven shocks seen above is not the difference in the energy, the difference of 42 orders of magnitude, but the difference in physics. The shock wave of the SNRs is a collisionless plasma shock, but the laser shock wave is in general the collisional hydrodynamic shock wave. In the laser shock, the mean free path is almost the same as the thickness of the shock front. In addition, the collisionless shock accompanies the electric and magnetic fields which play an essential role in accelerating the charged particle, namely, the origin of cosmic rays [38].

Finally, let us evaluate the scale range parameter S in (4.81) for SN1006 blast wave. Assume the explosion energy is 10⁴⁴ [J] and its radius 3x10¹⁷ [m], we obtain the value of S ~ 10⁷ [m/KT^1/3]. The pressure of the blast wave is evaluated roughly as P~4 × ρ_ISM(U_Sh)². Then we obtain P~10⁻¹²[atm] and the pressure evaluated with A = 10⁷ is P~10⁻¹⁴[atm]. They are different of two orders of magnitudes and main cause of the difference is over estimate of the radius due to rapid expansion in the early snow-plow phase with expansion velocity of 10,000 km/s. The point is very far right in Fig. 4.21. This is because of extremely low pressure of ISM gas. It is not so bad evaluation. Note that the self-similar solution assumed adiabatic condition, while in later time around 100,000 years, it is said that the radiation cooling effect as shown below becomes important and the blast wave front continue to expand as a thin shell structure.

4.10.2 Self-Similar Solution of SNRs

We see that the blast waves of SNRs are ejecta-driven blast wave and not the point-energy driven one like laser-driven blast wave seen in Fig. 4.19a. It is, however, surprising that even in such a case, Chevalier has found a group of self-similar solutions including the whole structure of expanding ejecta, reverse-shock, and blast wave propagating outward [39]. The mathematical method is almost the same as Sect. 4.2.

The ejecting matter is assumed to have the density profile in the radial direction like r^-n initially, where n is a given constant. Assuming a uniform expansion, the time and spatial evolution of the ejecta density profile is given in the form;

$$ \rho \propto {t}^{-3}{\left(r/t\right)}^{-n} $$

(4.89)

where t⁻³ is due to uniform expansion and r/t is due to the assumption of constant velocity to each Lagrange fluid. The density profile of the surrounding gas (ISM) is assumed in the form.

$$ \rho \propto {r}^{-s} $$

(4.90)

where s is a given constant. It is mentioned in [39] that Type-I supernovae are given for (n, s) = (7,0), while Type II is better for s = 2 modeling the expanding envelope by the wind.

Since the time dependence of the radius of the contact surface, R_c(t) should be the same for both in (4.89) and (4.90). Inserting R_c(t) in the both and requiring both density has the same time dependence, it is easy to obtain the time dependence of R_c(t) in the form.

$$ {R}_c(t)=A{t}^{\left(n-3\right)/\left(n-s\right)} $$

(4.91)

where A is a constant. Note that in the case of (n, s) = (5, 0), R_c(t) ∝ t^2/5 and it is the case of Sedov-Taylor solution and it is studied the case n > 5 for s = 0. In this case, the expansion velocity is faster than the case of n = 5 due to the ejector snow plow effect.

In Fig. 4.26, dimensionless profiles of the density (ρ), pressure (p), and velocity (u) profiles of the self-similar solution are plotted in the normalized coordinate, r/R_c(t). The radii of the reverse shock front and the blast wave are 0.935 and 1.181. The velocity is almost uniform, while the density changes rapidly around the contact surface.

Consider an applicability of the ejecta-driven self-similar solution. It is clear that the mass of the ejecta (4.89) diverges near r = 0, while the total mass of the supernova ejecta is limited, for example, about the solar mass in Type I supernova explosion. It is clear that when the effective ejecta mass becomes near the solar mass, the present self-similar solution is not applicable. The ejecta velocity will reduce due to the energy transfer to the blast wave. It is clear that the total kinetic energy of the blast wave region E_BW can be shown as

$$ {E}_{BW}\propto {\left(\frac{d{R}_c}{dt}\right)}^2{R}_c^3\propto {t}^{6/7} $$

It is reasonable to consider that after the present solution breaks down, the blast wave continues expanding by following Sedov-Taylor solution as far as the adiabatic flow is assumed.

The trajectories of a blast wave (forward shock front; R_BW) and the reverse shock front (R_RS) were shown for further study with 3-D hydrodynamic simulation and one-dimensional result of radius-time evolution is plotted in Fig. 4.27 [40], where (n, s) = (7,0) and γ = 5/3. The characteristic time (t_ch) and radius (r_ch) is calculated so that the ejected mass becomes the same as the mass of the shocked gas (ISM) [41].

In Fig. 4.27, the simulation result of time evolution of (R_BW, R_RS) are plotted with the black solid lines. The present ejector-driven self-similar solution is plotted with gray solid lines. The dotted two lines are analytic solution. It can be said that the ejector-driven phase is over before t = t_ch, while the dynamics in the early phase (t < t_ch) well agrees with the simulation. Since the mass of the ejector is limited, the R_RS travels toward the center after t_ch. After t_ch, R_BW becomes slower to fit to Sedov-Taylor solution.

In Fig. 4.28, the time evolution of the density profiles obtained by the simulation is plotted as a function of the radius normalized by R_BW(t), where t_ch =1950 years. In the figures, the ejector-dominant solution and Sedov-Taylor solution are also plotted with thin red lines. It is clearly seen that the simulation profiles smoothly transit around t = t_ch.

It is clear that the contact surface is unstable to Rayleigh-Taylor instability in the ejector-driven phase seen in Figs. 4.26 and 4.28. Since R_c(t) ∝ t^4/7 and the contact surface is kept decelerating by the pressure gradient force. Three-dimensional simulation has been done to study the physics of observation data, for example in [40]. In Fig. 4.29, X-ray image of Tycho SNR observed by Chandra x-ray satellite is shown. Very different of SN1006, Tycho SNR is sure to be affected by Rayleigh-Taylor instability. It is concluded that the growth of Rayleigh-Taylor instability is rather independent of the seeding by the non-uniformity in the explosion phase [42]. It is out of the topics in the present volume and to be discussed in later chapters.

Finally, it is useful to point out that the self-similar solution is studied in many situations in astrophysics. Even with the gravitational force, self-similar solution of star formation was found to explain the time evolution of gravitational collapse of molecular cloud. A famous solution is called the Larson-Penston (LP) type similarity solution [43]. Such star formation has been studied also with computational method to compare with the self-similar solutions. The LP assumed that in early phase of contraction the proto-star is optically thin and the radiation cooling makes the system uniform temperature in time evolution.

Numerical simulation with reasonable opacity of the cloud and radiation transport has been carried out [43]. The time evolution of the density, temperature and velocity are compared to the LP solution to find the good agreement in the early phase before the density approaches about 10²⁰ cm³ over more than ten-order of magnitude evolution. In Fig. 4.30, the simulation result is shown for density, temperature, velocity and hydrogen molecule concentration distribution for time from 0 to 7 timings [43]. It is confirmed that the density and velocity profiles are well explained with LP solution up to the time of 5. After the time 5, the dissociation of hydrogen molecule makes the system relatively optically thick system to apart from the LP self-similar solution.