Shock Waves and Ablation Dynamics

Abstract

When an intense laser is irradiated on a solid target, the laser energy is absorbed on the surface so that the material becomes plasma to expand into the vacuum region. Through the laser-plasma interaction, the laser energy heats the expanding region spreading by its sound velocity. As the result the expanding region has the temperature ~ 1 keV and the pressure reaches 100 Mbar (10TPa). Since the laser is absorbed near relatively high density (~cut-off density), the plasma can be assumed to be in LTE and hydrodynamic description is acceptable.

The surface pressure called ablation pressure drives strong shock waves in the solid material as if the solid is almost gas. The shock wave physics is briefly reviewed to use the Rankin-Hugoniot (RH) relation, although detail studied is needed for the equation of state of the compressed matter. By use of the ablation pressure, it is possible to accelerate a thin material to higher velocity like a rocket propulsion.

One dimensional hydrodynamics is reviewed for steady state and time dependent dynamics within the ideal fluid assumption. Deflagration and detonation waves are also explained as jump condition with energy deposition. The laser implosion dynamics is compered between stationary solutions, computational results, and the experimental data. The importance of validation of simulation codes is discussed.

3.1 Introduction

Intense lasers have been used to generate high pressure (~ 10–100 Mbar) and high material ablation velocity (~ 100–10,000 km/s) from solid target surface. With such parameters, even any solid materials are easily compressed by a shock wave generated on the surface by abrupt energy deposition and/or heating by laser energy absorption as seen in Volume 1. The Coulomb mean free path of hydrogen plasma is about 1 μm for 1 keV at the cut-off density of laser with its wavelength 0.35 μm, which is the third harmonics of glass laser and widely used to study energetic phenomena driven by intense lasers.

Therefore, it is acceptable to model the dynamical phenomena driven by laser heating with hydrodynamic model explained in Chap. 2. For detail or precise study of the plasma, atomic process explained in this book becomes important, while the theoretical study is in general carried out with help of computer simulations. Study here the basic principle of laser-driven plasmas, which can be studied without details of transport physics given later in this book.

In the present chapter, let us study a variety of analytical solutions of hydrodynamic phenomena generated in laser-produced plasmas. Here, we assume the laser produced plasma is governed by the classical one-fluid and one-temperature equations given in. In some case, we decouple the energy equation by assuming a constant temperature, adiabatic, given pressure, and so on. In some case, the heat conduction in the energy equation plays important role.

It is usually in these days that one can easily run an integrated hydrodynamic simulation code for laser plasma to compare the result to corresponding experimental result and others. In analyzing the experimental data directly or with help of such simulations, it is difficult to understand the plasma physically without the knowledge of plasma hydrodynamics. At first, one has to have enough knowledge about one-dimensional hydrodynamics of compressible fluid plasma. As good text books, the author recommend Refs [1, 2] for further study of the physics of compressible fluid dynamics.

Relating to the laser-plasma hydrodynamic phenomena, we can enumerate several analytical solutions found in a long history of hydrodynamics. In addition, we have to know some special solutions found in the history of laser plasma study. All of the solutions to be explained here are idealized one-dimensional solutions by assuming jump condition, stationary state, and self-similarity. The basic fluid equations are partial differential ones, but algebraic relations are obtained in the jump case such as a shock jump. In the stationary or self-similarity case, one can reduces the fluid equations to ordinary differential equations. Then, it is possible to solve analytically or numerically under one-dimensionally symmetric space. By combining these solutions, we can imagine the physics happening in the experiment or multi-dimensional simulation code. This is the reason why we have to study one-dimensional hydrodynamics of compressible fluid at first.

As we see here, a jump condition of fluid provides the relation between upstream and down-stream in a shock wave. If the energy is generated at the jump surface by a chemical reaction, the shock wave may change to a detonation wave, and the rarefactive jump is also physically possible foe entropy increase by chemical reaction . This is called deflagration wave (flame of candle, for example). The deflagration by laser heating maintains the ablated plasma expansion to the vacuum. The laser energy is absorbed at relatively lower density than the solid, and the electron and/or radiation energy transfer keep the solid surface ablation as schematically shown in Fig. 3.1.

Fig. 3.1

Schematics of time evolution of solid matter density and temperature. (a) before laser irradiation. (b) the laser energy is absorbed on the solid surface. The heat wave penetrates into the solid surface without hydrodynamic motion. (c) Rapid increase of the pressure in the surface area drives a shock wave going inside and expanding wave to the vacuum region

Then, the pressure of the ablation, ablation pressure, is extremely high to keep the shock wave propagating in the solid or keep accelerating the finite thickness target. This is called “ablative acceleration”. The shocked material is accelerated by rocket propulsion mechanism toward target center for a spherical fusion capsule. We can find a stationary solution of ablation structure to know its profiles as see later. This can be used to study the detail of the stability as will be shown in Vol. 3.

Finally, one-dimensional compressible hydrodynamics is also studied to know the Laval nozzle. The principle of jet engine is mentioned to relate the physics of ablation acceleration of nuclear fusion targets by intense laser irradiation.

3.2 Nonlinear Waves and Shock Waves

In Chap. 2, the linear analysis of the sound wave has been discussed. How the physics will change if the amplitude of the sound wave is high and nonlinear term cannot be neglected. Let’s consider a wave propagating to the right that satisfies the first term of (2.49) for simplicity. Such a wave propagating in only one direction is called a simple wave. Keeping the nonlinear convection term in (2.46) as in (2.24) and deriving a simple wave relation for velocity perturbation u₁, the following is obtained.

$$ \frac{\partial }{\partial t}{u}_1+\left({V}_s+{u}_1\right)\frac{\partial }{\partial x}{u}_1=0 $$

(3.1)

A nonlinear term by the convection

$$ {u}_1\frac{\partial {u}_1}{\partial x} $$

is added to the sound wave propagation equation. Consider to solve (3.1) under the initial condition at t = 0. Assume the initial wave velocity profile u₁ is as shown at the left edge in Fig. 3.2. The large part of u₁ propagates faster than the other part, and soon after the wave front becomes sharp to overlap at one point as shown with the wave form at the center in Fig. 3.2, Mathematically, the wave solution becomes a multivalued function as shown on the right. Such multivalued functions are not allowed physically. Actually, (2.46) inherently ignores the physics that becomes important when the velocity profile becomes sharp. It is the viscosity due to collision between micro particles in the fluid and Coulomb collisions in plasmas.

Fig. 3.2

Mathematical solution of nonlinear sinusoidal wave with viscosity

With the effect of viscosity, (2.46) becomes the Navier-Stokes equation (2.55). Therefore, it is physically reasonable to add the viscosity term to (3.1). Then, the equation will be given. In the frame moving with the sound velocity, x = V_st is easily derived as

$$ \frac{\partial }{\partial t}{u}_1+{u}_1\frac{\partial }{\partial x}{u}_1=\mu \frac{\partial^2}{\partial {x}^2}{u}_1 $$

(3.2)

This equation is called the Burgers equation. It is known that (3.2) has a shock wave solution in stationary sate. Note that the shock wave travels with additional velocity depending on the strength of the shock.

The shock front has a narrow region where the flow velocity changes due to the balance between fluid nonlinearity and viscosity. When the wave front becomes steep with nonlinearity, the viscosity term adjusts the gradient. This can be seen from the fact that RHS of (3.2) is in the form of diffusion. In microscopic view, when the flow velocity becomes steeps, it shows that the kinetic energy of the flow velocity is converted to the thermal energy by the collision between the molecules. Therefore, the entropy increases at the wave front from the upstream and down-stream. This is called shock wave heating.

Although the Burgers equation is a nonlinear partial differential equation, it can be transformed to a linear equation by Cole-Hopf transformation. For this purpose, introduce new variable ϕ

$$ {u}_1=-2\mu \frac{1}{\phi}\frac{\partial \phi }{\partial x} $$

(3.3)

Inserting (3.3) into (3.2), the following new equation to ϕ is obtained.

$$ \frac{\partial }{\partial x}\left(\frac{1}{\phi}\frac{\partial \phi }{\partial t}\right)=\mu \frac{\partial }{\partial x}\left(\frac{1}{\phi}\frac{\partial^2\phi }{\partial {x}^2}\right) $$

(3.4)

Integrating (3.4) with x, a linear equation to the new variable ϕ is obtained

$$ \frac{\partial \phi }{\partial t}=\mu \frac{\partial^2\phi }{\partial {x}^2}+f(t)\phi $$

(3.5)

Here, f(t) is an arbitrary function of time, and it is possible to set f(t) = 0. Then, the Eq. (3.5) becomes the following diffusion equation.

$$ \frac{\partial \phi }{\partial t}=\mu \frac{\partial^2\phi }{\partial {x}^2} $$

(3.6)

Since the RHS in (3.2) is small at the beginning (the initial condition in Fig. 3.2), the two terms on the LHS in (3.2) balance. As the slope of the wave front increases, the first term on LHS and RHS become balanced in (3.2). Since this is a diffusion equation, the amplitude of ϕ finally becomes small and it becomes flat and finishes Note that if the initial condistion is not sinusoidal but like u₁(0,x) = −tanh(x/L) with a large value of L, one can see the time evolution of u₁(t,x) to form a stationally shock wave, where the thickness of the shock front is deteremined as stationary solution of (3.2).

3.3 Shock Wave Jump Relation

In order to obtain more realistic relation to the shock waves it is required to solve the fluid equations in (20, 21, and 22) consistently. If there are no external force and no external energy source, it is possible to obtain the jump relation between the upstream and down-stream of the shock wave front. It is required to solve the fluid equation in the frame moving with the shock front. In this frame, the stationary state of a shock propagation can be assumed. To solve the jump relation or continuity relation in the shock frame is found to be mathematical relation between the given upstream condition and the solved downstream condition for a given strength of the shock wave. Such algebraic equations giving the jump relation is called Ranking-Hugoniot relation.

In many cases, it is possible to neglect the structure of the shock front being of order of mean-free-path. However, its structure becomes important in high-temperature plasma, since the Coulomb mean free path is proportional to the square of the temperature. In the plasma described with two-temperature model as shown in Chap. 2, it is known that the electron heat conduction and temperature relaxation make the shock structure about one hundred times wider than the length of Coulomb collision mean free path [1, 3]. When the length of such structure becomes the size of the fluid system, we have to consider the kinetic effect and solve Fokker-Planck equation to the plasma shock structure. It is also shown that the concentration of multi-species ion plasma is modified by the force of electric field produced by the abrupt change of pressure.

3.3.1 Rnaking-Hugoniot Relation

The Burgers Eq. (3.2) giving the solution of the shock wave is an equation specialized for a simple wave. Let’s find the relation of the shock wave exactly from the neutral fluid equations. Ignoring the wave front of the shock wave, that is, the region where the nonlinearity and the viscosity are balanced (narrow range of about the mean free path), the basic equations for the one-dimensional plane flow are derived from (2.23), (2.24), and (2.25) by neglecting external force and heating. The viscosity and conductivity are also neglected, since the physical quantities are constant in both sides of the shock front. These three equations can be rewritten as conservative equations as follows.

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

(3.7)

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

(3.8)

$$ \frac{\partial }{\partial t}\left(\varepsilon +\frac{1}{2}\rho {u}^2\right)+\frac{\partial }{\partial x}\left[\left(\varepsilon +P+\frac{1}{2}\rho {u}^2\right)u\right]=0 $$

(3.9)

Equation (3.8) is obtained by multiplying (3.7) by u and adding it to (2.24). Equation (3.9) can be derived by multiplying (3.7) by 1/2u², adding it to (3.8) multiplied by u, and adding it to (2.25).

Let’s consider a shock wave propagating to the left according to the custom. The Eqs. (3.7, 3.8, and 3.9) remain the same even in the system moving with the shock wave front, but the flow velocity becomes the velocity seen in the moving frame. Assume that the fluid is stationary and no time variation. In Fig. 3.3, the flow, and the change of variables across the shock front are schematically shown. The flowing fluid passes through the shock front and the deceleration of flow by the pressure accompanies the compression of flow. Since the viscosity and conduction are given by the spatial differentiation of the physical quantity, the spatially integrated relation of (3.7, 3.8, and 3.9) is sufficient for the jump relation of the shock wave.

Fig. 3.3

Physical quantities in the upstream and downstream of a shock front

Ignore the time derivative and require the conservation from the upstream (subscript 0 to the physical quantities) to the downstream (subscript 1 to the physical quantities). Spatial integration of (3.7, 3.8, and 3.9) gives the following relations.

$$ {\rho}_0{u}_0={\rho}_1{u}_1 $$

(3.10)

$$ {\rho}_0{u}_0^2+{P}_0={\rho}_1{u}_1^2+{P}_1 $$

(3.11)

$$ \left({\varepsilon}_0+{P}_0+\frac{1}{2}{\rho}_0{u}_0^2\right){u}_0=\left({\varepsilon}_1+{P}_1+\frac{1}{2}{\rho}_1{u}_1^2\right){u}_1 $$

(3.12)

These relations should be transformed into more easily understandable relations. It is natural to assume that the density ρ₀ and pressure P₀ of the front of the shock wave are given. Then, let the physical quantity that determines the strength of the shock wave be the pressure P₁ in the shocked region. As the equation of state, the ideal gas of (2.31) and (2.32) is assumed. Then, since (3.10, 3.11, and 3.12) has three unknowns and three relations, it can be solved. The relation of the pressure P₁ on the shocked region and the reciprocal (referred to as specific volume) of the density of the shocked region V₁ = 1/ρ₁ can be derived as follows.

$$ \frac{P_1}{P_0}=\frac{\left(\gamma +1\right){V}_0-\left(\gamma -1\right){V}_1}{\left(\gamma +1\right){V}_1-\left(\gamma -1\right){V}_0} $$

(3.13)

This relation is referred to as Rankin-Hugoniot (RH) relation. The curve given by (3.13) is also called Hugoniot curve or shock wave curve. This relation is shown with the line “RH” in Fig. 3.4. for the case with γ = 5/3. The line with “P” is the adiabatic curve (also referred as Poisson curve) and shows the relation (P ∝ V^−γ) for constant entropy. It is noted that the RH relation is along with the adiabatic curve for a weak shock and the entropy increases clearly near ρ₁~2ρ₀. This indicates that it is possible to compress fluid almost adiabatically by designing the compression with a series of shock wave production. This is called adiabatic compression by pressure tailoring.

Fig. 3.4

The Ranking-Hugoniot (RH) relation for a shock wave and Poisson relation (P) giving the adiabatic change

First, let’s explain the characteristics of the shock wave curve. At the limit of P₁ → ∞, there is the limited value for V₁, and its value can be obtained from eq. (3.13) as follows.

$$ {V}_1=\frac{\gamma -1}{\gamma +1}{V}_0,\kern1em {\rho}_1=\frac{\gamma +1}{\gamma -1}{\rho}_0 $$

(3.14)

In the case that the degree of freedom of the gas heated by the shock wave is N, from (2.33) the maximum compression rate is

$$ {\left.\frac{\rho_1}{\rho_0}\right|}_{max}=N+1 $$

(3.15)

For gases of monoatomic molecules moving three-dimensionally, the compression ratio is four times. The reason why the compression ratio increases as the degree of freedom increases is as follows. The shock wave is heated by converting kinetic energy to the internal energy.

How is the increase of the internal energy due to such energy conversion? For example, in diatomic molecule gas such as N₂ and O₂, the degree of freedom has three dimensions of translational motion and two dimensions of vibration and rotation of molecules, totaling 5 degrees of freedom. The five degree obtains energy, while only three degree of the translational motion contributes to the pressure. Therefore, even at the same speed, the temperature rise in the shocked region is only 3/5 times of the monoatomic molecule case. Since the pressure on the right side must support the ram pressure on the left side in (3.11), the density of the diatomic gas has to increase to maintain the pressure balance. This is also true when the ionization or phase transition happens at the shock front, the density jump depends on the details of equation of state.

The relation of the flow velocities is obtained from (3.10, 3.11, and 3.12) in the form.

$$ {u_0}^2=\frac{V_0}{2}\left[\left(\gamma -1\right){P}_0+\left(\gamma +1\right){P}_1\right] $$

(3.16)

$$ {u_1}^2=\frac{V_0}{2}\frac{{\left[\left(\gamma +1\right){P}_0+\left(\gamma -1\right){P}_1\right]}^2}{\left[\left(\gamma -1\right){P}_0+\left(\gamma +1\right){P}_1\right]} $$

(3.17)

As clear from Fig. 3.4, the shock wave curve is above the adiabatic curve for V₁ < V₀, indicating that the entropy increases through the shock wave surface. Although V₁ > V₀ is mathematically possible, it is not physically permitted in the normal ideal gas. This is because the entropy decreases across the discontinuous surface without any energy leakage from the shock front.

The dimensionless number characterizing the strength of the shock wave is the Mach number. The Mach number is the flow velocity divided by the sound velocity. In general, the Mach number M of a shock wave is defined as the value obtained by dividing the speed of the shock wave by the speed of sound ahead of the wave front.

$$ M=\frac{u_0}{V_S},\kern1em {V}_S=\sqrt{\gamma \frac{P_0}{\rho_0}} $$

(3.18)

The sound velocity given in (2.47) can be easily obtained in the ideal gas as above. Note that the Mach number M is always greater than unity.

When the temperature rises due to the shock wave is extremely high, for example, the molecular gas is dissociated and ionized to be in a plasma state as its atomic process will be explained in Chap. 5. This phenomenon is called shock wave ionization. Let’s look for the relation when the molecule of the main constituent is completely ionized by the shock wave like air, diatomic molecules. Let γ₀ be the specific heat ratio of the molecular gas, and γ₁ be the specific heat ratio of the plasma of the shock wave backside ionized completely. In the extremely strong shock waves, we can ignore both the pressure and the internal energy in the shock wave front, so we can see that all γ in (3.13) is good for γ₁. The compression ratio is determined only by the specific heat ratio of the rear side. However, note that (3.12) does not include the energy needed for the dissociation and ionization is not considered. It should be modeled.

The above discussion is clear in comparing Hugoniot curve of (3.13) with γ₁ = 5/3 to the experimental result for the liquid hydrogen shown in Fig. 3.5 [4] In Fig. 3.5, the orange line is RH curve for γ =5/3, while the marks with error bars are experimental data. The black lines are theoretical curves. It is suggested that the effective γ decreases around 30 GPa (0.3 Mbar), namely more freedom such as dissociation etc. increase the number of freedom N, [γ = (N + 2)/N].

Fig. 3.5

Shock Hugoniot data of liquid deuterium from an experiment and other experiments and theoretical curve. The orange curve is that from ideal RH relation. Reprint with permission from Ref. [4]. Copyright 1998 by American Physical Society

3.3.2 Structure of Shock Waves

The jump relation by Ranking-Hugoniot tells us nothing about the transition region of two states. In the case of neutral fluid, it is determined as the diffusive structure by the viscosity in general. It is well-know that the physical quantities continuously vary from the front region to the shocked region with a typical with of the molecular mean-free-path, which is very thin in the air and gases on grand. Of course, it is different for a very strong shock wave generated by the space shuttle in the re-entry to the earth atmosphere. In such a case, the kinetic effect apart from the local Maxwellian assumption should be solved to determine the structure; namely, Boltzmann equation has to be solved to obtain the detail of the shock structure.

The structure of the shock wave in the jump region, call hereafter it the shock front structure, in fully ionized plasmas was studied [3]. The stationary shock solution is solved with two fluid equations in (2.111, 2.112, 2.113, and 2.114) by coupling with two temperature energy Eqs. (2.107) and (2.108). In order to solve the steady state fluid equations, it is studied to find the property of the singularities in (velocity, temperature) space. The integration starts from the node point of the upstream region to the saddle point of the downstream region [3]. Such integration is demanded, because the plasma shock wave at high Mach-number has wide range of parameters. They are Debye length, Coulomb collision mean-free-path, dominant electron heat wave structure, and slow temperature relaxation between electrons and ions.

The details of the shock front structures are given in [3] for a variety of Mach number. The shock wave picture is measured in an experiment with use of optical imaging of the shock in gas, where the shadow of the probe light shows the region where the refractive index changes abruptly in space such as shock front. The shadow image of a bullet in the air is shown in Fig. 3.6. The bullet is flying from left to right at supersonic speed in the air. A strong shock wave is generated in front of the bullet, and it is also seen a turbulent flow behind the bullet. Since the radius of the bullet is small, the shock front shows the structure of a bow, and such shock wave is called a bow shock.

Fig. 3.6

An optical shadow image of a bullet flying in the air with supersonic velocity. The shock wave is generated in front of the bullet as well as both sides. Such shockwave is called a bow shock

Extremely high-pressure generated by intense lasers on a variety of solid materials allows us to study the shock waves in high-density matters. The progress of femto-second diagnostics by X-FEL (X-ray Free Electron Laser) made it possible to visualize the shock structure propagating in solid matters. Since the x-rays can propagate in matter whose density is higher than solid densities in general, it is possible to use the ultra-short pulse x-rays for the imaging diagnostics.

The temporal evolution of a shock wave in diamond is measured, yielding detailed information on shock dynamics, such as the shock velocity, the shock front width, and the local compression of the material [5]. It is reported that an intense laser with 150 ps pulse width and energy 130 mJ is focused on solid diamond at intensity of 10¹³ W/cm². The compression wave with density change of 10% (1.2 ns) and shock front width of 1 μm is observed as shown in Fig. 3.7. The XFEL pulse duration is 50 fs and the spatial resolution of the image is 0.5 μm. The progress of diagnostics has made it possible to study the shock wave, equation of state, and related physical properties in high-density plasmas.

Fig. 3.7

X-ray shadow images of a shock wave propagating in a diamond at 1.2 and 1.8 ns. The shock wave is generated by an intense laser. The two images below are the corresponding images after eliminating the back ground from the above two images. Reprinted by permission from Macmillan Publisher Ltd: Ref. [5], copyright 1993

It is shown for example at p. 474 in Ref. [1] that the thickness of the shock front obtained by solving equation with the viscosity is given in the form.

$$ \delta \sim {l}_0\frac{M}{M^2-1} $$

(3.19)

where l₀ is the mean free path in the upstream region and M is the Mach number of the shock wave. Note that the collision cross-section σ is constant in the mean-free path proportional 1/nσ.

To study the detail of the shock wave structure in high-density not only in single solid but also mixture material such as DT (deuterium-tritium), plastic CH (carbon-hydrogen), and so on, the fluid model is not appropriate, and the kinetic equation should be solved. The first trial to apply suck kinetic model to obtain the shock structure has been done by Mott-Smith as indicated at page. 476 in Ref. [1], where the structure is solved by assuming a linear combination of bi-Maxwellian of those in both constant regions. As more advance theoretical study, Fokker-Planck code is used to study not only the structure, but also the change of concentration due to local electric field in the shock front layer [6].

In the Fokker-Planck simulation in [6], the electrons are solved with fluid assumption, while the ions are solved with Fokker-Planck equations with Coulomb collisions between ions themselves and with electrons. In the collisional shock wave, the ion kinetic energy is converted to the thermal energy only via ion-ion collisional process and the ion viscosity determined the structure of ion density and temperature near the shock front. This thickness of the density variation of the ions at a shock front is roughly given as (3.19), although the mean-free-path of plasma strongly depends on the temperature.

As shown in Chap. 2.2, ion and electron mean-free paths are the same except Z and m_i dependence, while the diffusion coefficient is proportional to the product of the mean-free-path and the thermal velocity. Therefore, the electrons diffuse about $\sqrt{m_i/{m}_e}$ times wider than the ion-mean-free path. The electrons are heated via temperature relaxation after the abrupt heating of ions by viscosity; however, it is slow process. As the result, a strong plasma shock structure is given like as shown in Fig. 3.8 [6]. It consists of very sharp ion temperature and density jump with the thickness of the ion mean free path, and electron heat wave tongue in front of the ion jump and the temperature relaxation region behind the ion jump.

Fig. 3.8

Shock wave structure obtained by solving Fokker-Planck equations to multi-ion components with Coulomb collision effect. The electrons are assumed to be fluid. The red is the ion temperature profile and the blue is electron temperature one. Reprint with permission from Ref. [6]. Copyright 1998 by American Physical Society

In the Fokker-Planck calculation of Fig. 3.8, it is also found that strong electrostatic field is generated to cause nonuniform concentration of D and T ions in space [6]. The electric field is produced due to density and temperature gradients in the shock front. Since the electron inertial is much smaller than the ions, it is possible to neglect the electron inertial term in (2.114) to obtain the relation.

$$ eE+\frac{1}{n_e}\frac{\partial }{\partial x}\left({n}_e{T}_e\right)=0 $$

(3.20)

At the shock front the density and temperature of electrons increases abruptly and a strong electric field is generated from high to low pressure direction. Note that the almost charge neutrality is kept between electron and ion fluids and the change separation appears over the distance of Debye length characterized by the higher temperature. It is concluded that due to the difference of the Z/m_i in (2.112) for not hydrogen, but the mixture of deuterium and tritium, the component with larger Z/m_i (D⁺) shifts in the front region from the shocked region by the strong electric field, causing the change of concentration of multi-component ion plasma. This effect cannot be neglected in the fusion reaction of DT mixture fuel.

It is noted that we have assumed the simple equation of state even for dense plasmas so far. As will be discussed in Chap. 9, the equation of state (EOS) of matters near and over the solid density is not ideal (not simple), and EOS including many body interaction such as strongly coupling should be studied. In higher density and relatively lower temperature, the quantum effects also play important role in the thermodynamic properties. In some case, the phase transition occurs by a shock wave. In the phase transition, the latent or dissolved heat appear as energy release or absorption modifying the Ranking-Hugoniot relation.

3.4 Deflagration and Detonation Waves

3.4.1 Jump Relation with Energy Source

As described above, the discontinuous wave fronts where the density decreases from the upstream to the downstream cannot exist physically as long as the basic equations, (3.7), (3.8), and (3.9). This is because the entropy decreases from the upstream to the downstream of the jump. However, for example, if any chemical combustion or nuclear burning occur on the wave front, to increase the entropy at the jump layer, a solution of decreasing density is physically permitted. Such jump waves are called deflagration wave. In our surroundings, a good example is combustion waves.

When energy is generated in the wave front, the energy W generated per unit time and unit area appears in LHS of (3.12). Then, the relation (3.13) is modified as.

$$ \frac{P_1}{P_0}\kern1em =\kern1em \frac{\left(\gamma +1\right)-\left(\gamma -1\right){V}_1/{V}_0+2W/\left({J}_0{V}_0\right)}{\left(\gamma +1\right){V}_1/{V}_0-\left(\gamma -1\right)} $$

(3.21)

Here, J₀ = n₀u₀ is the fluid particle flux. For a given value W, the relation (3.21) is plotted in the (P, V) plane with thick solid line in Fig. 3.9. Note that there is no physical region between the points A and B because the value J₀² < 0 . For reference, the adiabatic curve with the entropyS=S₀ (constant) of the upstream region is also plotted as “P” (Poisson curve), the curve of S₀.

Fig. 3.9

The deflagration and detonation curve from the upstream (V₀, P₀) to the downstream

The physically meaningful solution is limited to the solution that transitions from the upstream (V₀, P₀) to the right point C, or the solution that to the left point D. The jump that transitions from the upstream to the right C is called deflagration wave. Since the propagation speed of the deflagration wave is slower than the sound speed in the upstream, a shock wave is generated in the upstream region by the pressure generation due to the energy release by combustion. On the other hand, the transition to compressible region D is called detonation wave.

3.4.2 Deflagration Waves

The basic structure of deflagration wave is shown in Fig. 3.10. The essential property of the deflagration is the same as those by chemical reaction, nuclear reaction, and laser heating in the critical region. The chemical and nuclear reaction rates are proportional to the higher power of the temperature; therefor, rapid heating happens when the temperature reaches the reaction temperature. Then, energy is generated in relatively narrow region as shown in Fig. 3.10. The same structure is also seen in the combustion waves in the jet and rocket engines. Energy is locally released in the region “b” in Fig. 3.10, and its energy is transported to the left conduction zone “a” by heat conduction. This is the basic structure of the deflagration waves and the heated gas by the reaction in the zone b tends to expand to relatively low-density region in the space or into the vacuum. The reaction of the exhaust of the expanding gas produces high pressure to generate shock waves toward left or sequential pulses of sound wave to accelerate the left region, for example, rocket propulsion.

Fig. 3.10

Schematic of the deflagration wave. Chemical reaction takes place as “H” and thermal conduction penetrates temperature as “T”. The fuel density drops as “C”

As can be seen in Fig. 3.9, the pressure hardly changes and the density drops due to the temperature distribution controlled by heat conduction. However, since heat conduction is not so high in a chemical reaction, the thermal conduction region “a” is not so wide and it looks as a sharp surface. The dimensionless quantity Z = a/b is called the Zel’dovich number. As shown below the thickness of the conduction zone “a” is relatively thin compraed to the size of the system.

It is well know that the deflagration wave driven by the internal reaction is unstable to the rippling of the burning front [2]. Therefore, it cannot remain as one-dimensional combustion wave for a long time and the burning front easily becomes unstable to increase the reaction front surface. The increase of the surface leads the increase of the enegy production rate to strength the shock wave in the front and finally the deflagration changes to the detonation to be described below. This explanation seems contradict against the stable frame of candle. The density of the wax from the solid to after-burn decreases along the flow. Thanks to the gravitation force, the candle can keep such a stable flame.

The deflagration wave produced by heating with laser has also the same structure. The heating region concentrates on the cutoff density of the laser, and the absorbed energy is transported to the left side by nonlinear electronic heat conduction, almost the same as Fig. 3.10. In Fig. 3.11, a snap shot of laser-deflagration is shown with 1-D hydro-simulation, when a thin flat plastic plate is irradiated with intense laser. This is obtained with the single-fluid, two-temperature fluid Eqs. (2.105, 2.106, 2.107, and 2.108).

Fig. 3.11

A deflagration structure, called as ablation structure in general, calculated by one-dimensional implosion code showing compressed shell and heat conduction region

The density is the highest in the wave front of thermal conduction (ablation surface). This is because the target is accelerating to the left by the ablation pressure, and the effective gravity (inertial force) is to the right direction. Due to the nonlinearity of the electron heat conduction, the ablating plasma rapidly expands. Since only the electrons are heated by laser, the electron temperature reaches 3 keV, but ions are heated up to about the half of the electron fluid by the temperature relaxation process. This wave front instability is very important for stable acceleration of the targets, and will be discussed in Volume 3.

The flame of candle in Fig. 3.12 is a slowly combusting wave in 3-D. It is famous Michael Faraday’s “The Chemical History of a Candle” [7]. He mentioned in 1850 as follows.

Fig. 3.12

A candle is a typical example of the deflagration wave

There is not a law under which any part of this universe is governed which does not come into play and is touched upon in these phenomena. There is no better, there is no more open door by which you can enter into the study of natural philosophy than by considering the physical phenomena of a candle.

Deflagration wave is important to know the mechanism of ablation pressure sustained by laser absorption in the expanding plasma. It is useful hear remind the science of a candle. Consider the mechanism that keeps a candle burning. Solid wax is heated by its own flame, melts and rises with decrease of the density by buoyancy (In Fig. 3.10, from solid wax of “C” to the zone “a”). When evaporated wax reaches the temperature of reaction (region “b”), it starts chemical reaction and fire appears. That reaction heat is what maintains the light as a flame.

The density of the wax expands from solid to vapor, and then expands further as it becomes even hotter as a result of the chemical reaction. We can regard the solid wax is the upstream state, and the flame heated by the chemical reaction corresponds to the downstream state. The zone “a” is the conduction zone, while it is assumed a jump from up- to down-stream. When we consider a flame that continues to burn in one dimension, it satisfies the discontinuity condition for the deflagration wave as shown below.

It is better to point out essential difference of deflagration waves between the chemical reaction and laser ablation surface. The combustion front of chemical reaction is unstable to two/three-dimensional deformation of the front, because the energy release rate increases as the area of the surface increases by the deformation. The energy release is due to the reaction of the fluid itself. On the other hand, in the laser heating deflagration, the energy input rate is not affected by the deformation of the ablation surface, and no instability is induced. In the case of ablation front, the instability is due to the inertial force. This is called Rayleigh-Taylor instability as will be study in Volume 3.

3.4.3 Detonation Waves

On the other hand, the wave that transitions to the left “D” in Fig. 3.9 is supersonic to the upstream fluid. Such a burning wave is an explosive combustion and it is called a detonation wave. Detonation waves are often seen in movies such as explosives. Since the temperature increased by the shock wave compression is higher than the ignition temperature of the explosive, the chemical reaction is taken place on the shock wave front and the energy is explosively released. This is an explosion phenomenon. The engine of the car uses the explosion pressure to run the car. The gasoline and air are injected to mix for chemical reaction into the piston, and the ignition is sputtered by a spark plug. Then, a detonation wave is generated from that point, and high pressure is generated. This pressure pushes down the cylinder and converts its work into kinetic energy of mechanical rotation of car wheels.

A schematic of the pressure and temperature profile of the detonation wave in a tube is shown in Fig. 3.13 [8]. The shock wave heats and compress the front to reach the ignition temperature of the fuel-air mixed gas (or unreacted explosive). Then, the chemical reaction heats the reaction zone to keep the shock propagation. The maximum temperature point in Fig. 3.13 corresponds to the point “D” in Fig. 3.9. Behind the reaction zone, low density expansion zone follows to continue the burned gas zone, which works strong pressure to the left wall.

Fig. 3.13

A schematic of the pressure and temperature profiles of a detonation wave in a tube [8]. (Courtesy of Dr. Minchinton)

It is well known that as mentioned above, the deflagration front of chemical reaction is hydrodynamically unstable. It is called Landau-Darrius instability and the deflagration front distorts to have fractal structure. Then, as the front surface (area in 3-D) increases, the energy generation rate due to the chemical reaction increases. Since the deflagration is subsonic and the strong sound wave or shock wave is formed in front of the deflagration wave. As the result, the increase of reaction energy enhances the strength of the shock. At the stage when the temperature becomes high enough to ignite the fuel, the deflagration wave transits to the detonation wave. This is called deflagration-detonation transition (DDT).

In Fig. 3.14, the temperature and pressure evolution around the time of DDT is shown from 2-D computer simulation of chemically-reacting gas [9]. In this case the deflagration wave is deformed by the boundary protruding at the center in the figure, it is seen that a reflected shock with higher pressure and temperature is generated at t = 10.48 ms. This reflected stronger shock wave locally ignites the fuel to change the burning to the detonation. The burning speed after this time increases dramatically and the temperature and pressure increased globally after DDT occurs.

Fig. 3.14

The time evolution of temperature and pressure showing the deflagration to ignite to change to a detonation by edge reflected shock heating [9]. From Journal of Combustion

3.4.4 Supernova Ia DDT

The physics of supernova type Ia explosion observed in space is thought to be a carbon nuclear burning type explosion, in which a white dwarf (WD) explodes by nuclear burning [10]. A white dwarf, the mother body of type Ia supernova explosion, is one of a binary system with a companion star, usually a main sequence star. The mass of the white dwarf is increases in time by peeling mass from the surface of the companion star with strong gravity. Before its mass reaches the critical mass of Chandrasekhar limit (approximately 1.4 times the solar mass), the gravity of WD is sustained by the Fermi pressure of the non-relativistic electrons.

However, the mass approaches the critical mass, the Fermi energy of the electrons near the center of WD becomes more than mc². The density dependence of the pressure shifts from the non-relativistic case of ρ^5/3 to the relativistic one of ρ^4/3, namely Fermi pressure becomes relatively soft. Then, the white dwarf cannot withstand further mass increase. As the result, the center of the white dwarf collapses to be higher temperature and higher density to ignite the nuclear fusion of carbon burning.

Since the mass of the white dwarf at the time of explosion is determined by the Chandrasekhar mass, it can be considered that the explosion energy is almost constant for Type Ia supernovae. Since the explosion itself becomes brighter than a galaxy typically consisting of 100 billions of starts, it becomes a standard light source from a far distant Universe. Observation data have impacted the argument of cosmology such as whether the universe is open or closed. At the present time, it is concluded from the observation of type Ia supernovae that the universe is further accelerating with increase of the expansion velocity. This fact concluded the universe is expanded by the dark energy, consisting of 70% of the energy in Universe.

A simple physical scenario of type Ia supernova explosion is as follows. The main components of white dwarfs before explosion are carbon and oxygen. If the temperature near the center exceeds several hundred keV, nuclear fusion reaction of carbon and oxygen occurs. However, the pressure rises caused by the nuclear reaction is still smaller than the degenerate pressure of the electrons, so the deflagration front does not explosively spread, propagating as a weak combustion wave. In Fig. 3.15, the one-dimensional simulation is shown for the propagation of the nuclear burning wave in the density 10¹⁰ g/cm³ [11]. It is surprising to know that the thickness of the wave is extremely small compared to the size of the white dwarf of thousands of kilometers.

Fig. 3.15

One-dimensional thermonuclear deflagration wave propagating near the center of a white dwarf characterized with extremely high-density and electron Fermi pressure. Reprinted with permission from Ref. [11]. Copyright by American Astronomical Society

If the wave front remains spherically symmetric, the stars will expand due to shock waves ahead of the deflagration wave, nuclear combustion will be terminated by this expansion. The nuclear deflagration front is hydrodynamically unstable same as chemical reaction, the deflagration front grows as fractal structure. As a result, the area of the deflagration front becomes many times larger than 4πR², where R is the average radius of the deflagration front. Then, the nuclear reaction energy released in unit time increases in proportion to the increase in area of the front, the shock wave is intensified.

When the temperature rise due to this shock wave exceeds the temperature at which the nuclear reaction takes place as shown in Fig. 3.13, the nuclear reaction transits to the detonation wave. Such a combustion scenario is called a delayed detonation model in astrophysics. Among the heavy elements in the universe, it is believed that the iron is mainly nuclear synthesized by type Ia supernovae. In the above scenario, unstable nuclear nickel 56 is produced by carbon and oxygen nuclear fusion and ⁵⁶Ni nuclear decays eventually to the iron in the universe.

3.5 Rarefaction Waves

A shock wave is generated by pushing its boundary by a piston. On the other hand, we have no stationary solution of the case when the piston moves to the opposite direction. If the piston disappears abruptly, the fluid expands into the vacuum with its sound velocity. As it is already mentioned, the rarefactive shock wave is not physically acceptable and it is required to find time-dependent solution of such expanding wave. Such solution is called rarefaction wave. It is well known that it is given by a self-similar solution. Here we consider two cases; one is adiabatic rarefaction wave and the other is rarefaction wave that expands isothermally due to overwhelming heat conduction. For the sake of simplicity, we assume that the fluid is initially located in the vacuum at t = 0.

Without viscosity and external force and assuming one-dimensional plane geometry, (2.23) and (2.24) can be written as follows.

$$ \frac{\partial \rho }{\partial t}+\rho \frac{\partial u}{\partial x}+u\frac{\partial \rho }{\partial x}=0 $$

(3.22)

$$ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{V_s^2}{\rho}\frac{\partial \rho }{\partial x}=0 $$

(3.23)

We try to find self-similar solution and reduce (3.22) and (3.23) to coupled ordinary differential equations according to the method explained in Chap. 2. From the dimensional analysis of (3.22) and (3.23), it is found that there is only one dimensionless variable ξ defined to be

$$ \xi =\frac{1}{c_0}\frac{x}{t} $$

(3.24)

Here c₀ is the sound velocity at t = 0 defined as

$$ {c}_0={\left.\sqrt{\frac{\partial P}{\partial \rho }}\right|}_{\rho ={\rho}_0} $$

(3.25)

Here ρ₀ is the initial density.

Let us assume that the solution of (3.22) and (3.23) can be written in the forms

$$ \rho ={\rho}_0R\left(\xi \right),\kern1em u={c}_0U\left(\xi \right),\kern1em {V}_s={c}_0V\left(\xi \right) $$

(3.26)

Then, using relations

$$ \frac{\partial \xi }{\partial t}=-\frac{\xi }{c_0t},\kern1em \frac{\partial \xi }{\partial x}=-\frac{1}{c_0t} $$

(3.27)

Equations (3.22) and (3.23) can be transformed to the following coupled ordinary differential equations.

$$ \left(U-\xi \right){R}^{\prime }+{RU}^{\prime }=0 $$

(3.28)

$$ \left(U-\xi \right){U}^{\prime }+{V}^2\frac{1}{R}{R}^{\prime }=0 $$

(3.29)

Here, the superscript (^′) represents the ordinary differentiation by ξ. For (3.28) and (3.29) to have a non-trivial solution, the determinant which can be made with the coefficients of differentiated variables needs to be zero. From this condition, the following relations are obtained.

$$ {\left(U-\xi \right)}^2={V}^2 $$

(3.30)

$$ U=\xi \pm V $$

(3.31)

For the case where the fluid is only in the region of 0 < x initially, the expansion front called as a rarefaction wave propagates in the fluid. This solution corresponds to the plus sign of (3.31).

3.5.1 Adiabatic Rarefaction Wave

The fluid expands adiabatically into the vacuum. Under the adiabatic condition the following relationship is satisfied.

$$ V={R}^{\frac{\left(\gamma -1\right)}{2}} $$

(3.32)

With use of (3.31), (3.29) can be modified like

$$ \frac{2}{\gamma -1}\frac{d}{d\xi}\left\{{R}^{\left(\gamma -1\right)/2}\right\}=\frac{d U}{d\xi} $$

(3.33)

Integrating this equation with the condition U = 0 at R = 1, the density profile is obtained.

$$ R={\left\{\frac{\gamma -1}{2}\left(\frac{2}{\gamma -1}-U\right)\right\}}^{\frac{2}{\left(\gamma -1\right)}} $$

(3.34)

The condition U = 0 at R = 1 is the point which propagating to the minus x-direction with the sound velocity c₀ and this corresponds to the front of the rarefaction wave.

In the moving frame with the rarefaction front, we take the plus solution of (3.31)

$$ U=\xi +V $$

(3.35)

Inserting (3.32) and (3.34 to 3.35), the flow velocity is obtained.

$$ U=\frac{2}{\gamma +1}\left(\xi +1\right) $$

(3.36)

Since U = 0 at the rear of the rarefaction wave, the normalized coordinate ξ should be ξ = −1, namely it is also clear that at this point the following relation is also satisfied.

$$ R\left(-1\right)=V\left(-1\right)=1 $$

(3.37)

In the present case of adiabatic expansion, the rarefaction wave expands with a finite velocity into vacuum and the expansion velocity of this front u_fand its ξ value ξ_f are given by setting R = 0 in (3.34) as

$$ {u}_f=\frac{2}{\gamma -1}{c}_0 $$

(3.38)

$$ {\xi}_f=\frac{\gamma +1}{\gamma -1} $$

(3.39)

This expansion front velocity can be explained physically as follows. The fluid flow is driven by enthalpy h. Namely, the enthalpy is converted to the kinetic energy of the flow:

$$ h\to \frac{1}{2}{u}^2 $$

(3.40)

Inserting the initial value of the enthalpy gives

$$ {\mathrm{u}}_{max}=\frac{2}{\gamma -1}{c}_0 $$

(3.41)

Such intuitive way gives the prediction of the solution (3.38).

Here we have solved mathematics for the rarefaction wave and the solution of (3.22) and (3.23) is given from (3.36) and (3.34) as

$$ u\left(t,x\right)=\frac{2}{\gamma +1}\left(\frac{x}{t}+{c}_0\right) $$

(3.42)

$$ \rho \left(t,x\right)={\rho}_0{\left\{1-\frac{\gamma -1}{\gamma +1}\left(\frac{x}{c_0t}+1\right)\right\}}^{\frac{2}{\left(\gamma -1\right)}} $$

(3.43)

It should be noted that the solutions are valid only for the region of the rarefaction wave defined by

$$ -1<\frac{x}{c_0t}<\frac{\gamma +1}{\gamma -1} $$

(3.44)

The density and velocity profiles at a given time is plotted in Fig. 3.16.

Fig. 3.16

A snap shot of density and velocity profiles of an adiabatic rarefaction wave

3.5.2 Isothermal Rarefaction Wave

As the degree of freedom increases, the specific heat ratio γ approaches unity, and as apparent from (3.38), the rarefaction wave spreads to infinity. However, when expanding while heating from the outside and keeping the initial temperature, the above analytical solution cannot be applied. Isothermal is possible, for example, by irradiating a solid surface with a high intensity laser, bringing it to a high-pressure state, and when a rarefaction wave is generated in the direction of laser irradiation in the vacuum. Because the electron heat conduction is dominant in the rarefaction region, the rarefaction wave expands while the temperature is almost uniform. It is useful to find an analytical solution for such a rarefaction wave. In this case as well, a self-similar solution exists.

The basic equations are same as (3.22, 3.23, 3.24, 3.25, 3.26, and 3.27) except (3.26). Then, (3.26) should be modified like

$$ \rho ={\rho}_0R\left(\xi \right),\kern1em u={c}_0U\left(\xi \right),\kern1em {V}_s={c}_0: const. $$

(3.45)

Although it is impossible to use (3.32) and (3.33), (3.30) gives the flow velocity.

$$ U=\xi +1 $$

(3.46)

Then, (3.28) and (3.29) turn out the same equation as

$$ {U}^{\prime }+\frac{1}{R}{R}^{\prime }=0 $$

(3.47)

This equation can be solved easily to give

$$ R={e}^{-U} $$

(3.48)

This gives the density profile as

$$ R={e}^{-U}\kern1em \to \kern1em \rho ={\rho}_0{e}^{-\left(\xi +1\right)} $$

(3.49)

This profiles are plotted in Fig. 3.16. The front of the rarefaction wave is U = 0 at ξ =  − 1.

In order to maintain such an isothermal rarefaction wave, it is necessary to constantly supply energy to the rarefaction wave from the outside. This is because the temperature does not change and the internal energy does not change, so it is necessary to constantly supply the increment of the kinetic energy obtained by the rarefaction wave as heat. Let’s calculate the rate of energy increase per unit time. Assuming that the energy increase rate is given as an energy flux Q from the outside,

$$ Q=\frac{d}{dt}{\int}_{-{c}_0}^{\infty}\left(\frac{1}{2}\rho {u}^2+\frac{1}{\gamma -1}P\right) dx $$

(3.50)

Inserting the solution above and change the integral with self-similar coordinate, (3.50) becomes

$$ Q={\rho}_0{c_0}^2{\int}_{-1}^{\infty}\left(\frac{1}{2}R{U}^2+\frac{1}{\gamma -1}R\right) d\xi $$

(3.51)

With the change of variable

$$ y=\xi +1 $$

(3.52)

(3.52) reduces to

$$ Q={\rho}_0{c_0}^3{\int}_0^{\infty}\left(\frac{1}{2}{y}^2{e}^{-y}+\frac{1}{\gamma -1}{e}^{-y}\right) dy $$

(3.53)

The integration of (3.53) is done easily to obtain

$$ Q=\frac{\gamma +1}{\gamma -1}{\rho}_0{c_0}^3 $$

(3.54)

Note that the coefficient of Q is 4 for γ = 5/3.

3.5.3 Shock Tube

Now, we knew that there is no jump solution with decreasing density along the flow, but the density decreases as the time-dependent rarefaction wave. The experiments of shock waves in gas have been carried out, for example, using the shock tube shown in Fig. 3.17. The cylindrical tube is divided into two parts by a thin film (diaphragm) and each pressure is set to a different pressure. When the thin film is burned out with a laser or the like, high-pressure gas expands to push the low-pressure part, and a shock wave is formed in the low-pressure part. However, in such experiments, only shock waves with a pressure jump of at most several atmospheres are possible to be generated. Therefore, a very strong shock wave was generated by high-explosives and so on. By using a high intensity laser to irradiate on gas or solid, it is now possible to generate high pressure more 10 times the bulk modulus B defined in (2.51). To study the high-pressure physics as to be discussed later, typically the diamond anvil cell for static high pressure or lasers are used to study the dynamical high-pressure property of solid material, such as internal state of the earth and planets.

Fig. 3.17

A schematics of shock tube. Pressure is initially different in both gases. The boundary of the gases is opened to shock wave travel to the right and the rarefaction wave goes to the left. Their trajectories ate plotted in above with a snap shot on the top

3.6 Ablation Pressure and Ablative Acceleration

3.6.1 Ablation Structure

Instead of the energy production by chemical reactions, the energy increase by intense laser can also keep the deflagration wave. In the case of Chapman-Jouguet deflagration wave, the exhaust velocity is the sound velocity at the rear of the deflagration wave. In the chemical reactions, the reacted molecule gas may have the temperature of the molecular bonding energy, namely T ~ 0.1 eV. Since the intense-laser can deposit much large amount of energy in the deflagration region, we can expect the temperature of more than 1 keV as seen below. Such high-temperature exhaust plasma can accelerate any matter to the velocity of sound speed, namely v~3 × 10⁷ cm/s as we already saw in Fig. 3.18.

Fig. 3.18

Schematics of a laser driven shock wave and ablation of a solid target. (a) The time evolution of shock and deflagration waves. (b) The snapshot of density, temperatures, and flow velocity at the time t₁. The broken curve in (a) represents a path of a fluid element

Consider a simple structure of the deflagration wave produced and maintained by constant heating of the ablating plasma by an intense laser. Assume that a stationary deflagration with a shock wave in the over-dense solid target and iso-thermal rarefaction wave into the vacuum region.

In Fig. 3.18a, the schematics of the trajectories of the shock front, deflagration zone, and the front of the rarefaction wave is plotted. The dashed line is a fluid particle trajectory from solid, shocked region, deflagration to the rarefaction wave. In the shocked region, fluids are accelerated to the left by the ablation pressure. The pressure generated by the deflagration is called ablation pressure.

In Fig. 3.18b, the rough snap shot at t = t₁ is drawn. The density (n) increases by the shock compression in the Zone-2, dramatically decreases in the deflation zone-3, and exponentially decreases in the Zone-4. The temperature increases by shock heating in Zone-2, abruptly increases in Zone-3 by electron heat conduction, and keeps almost constant in the expansion Zone-4. In two-temperature model [12], the ion fluid is heated by Coulomb collision in the subsonic Zone-3, while to be cool down in the expansion Zone-4 as roughly indicated in Fig. 3.18b.

The jump condition of the shock wave, ablation front, and the expansion into the vacuum region are shown as (V, P) diagram in Fig. 3.19. Fluid particles are compressed and accelerated forward by the shock wave from the point 1 to 2 and encounter the ablation surface. Then, they are heated in the direction of vacuum and accelerated as shown from point 2 to 3. After the passage of the Chapman-Jouguet (CJ) deflagration point 3, the particles are exhausted as the rarefaction wave into the vacuum. This is the same mechanism as rocket exhaust. The difference is that the rocket fuel is a chemical reaction and can only produce exhaust speeds on the order of 10 km/s, but with laser ablation it can be more than 1000 km/s.

Fig. 3.19

The P-V diagram for the two jumps of the shock wave and the following deflagration wave driven by laser heating. The density and temperature profiles are schematically shown in the inlet figure

Let’s calculate the laser heating energy needed to keep such almost stationary deflagration structure. The dominant energy is used to heat the exhausting plasma from the Chapman-Jouguet point and to heating the rarefaction wave expanding to the vacuum. The electron heat conduction penetrates in the deflagration structure and keeps the ablation of the dense material. Assuming the deflagration structure is stationary state, the inward heat flux is carried out as the ablation energy. We neglect the energy increase in the shocked region because it is a small fraction compared to the laser energy input. The energy flows out constantly into the rarefaction wave through the Chapman-Jouguet point.

3.6.2 Ablation Pressure

It is assumed that the rarefaction region has very good electron heat conduction and the temperature is constant in space. As derived in (3.54), energy flux required to maintain the isothermal rarefaction wave is evaluated to be,

$$ \frac{d{E}_{abl}}{dt}\approx {\left.4P{C}_s\right|}_{CJ} $$

(3.55)

Here, the right side is evaluated with the values at the CJ point of the ablation structure. Requiring that the laser must supply this energy, the relationship between laser absorption intensity and ablation pressure can be obtained.

By balancing the energy of the absorbed laser with the energy of (3.55), the scaling law of the following ablation pressure P_abl can be obtained. We use the fact that the pressure on the ablation surface is almost twice the pressure at the CJ point [12]. It is also assumed that the laser (wavelength is λ in μm unit) is absorbed at the critical density point. Then, the ablation pressure P_abl is calculated to be.

$$ {P}_{abl}\approx 12{\left(\frac{I_{14}}{\lambda_{\mu m}}\right)}^{2/3}\kern1em \left[ Mbar\right] $$

(3.56)

Here, it is assumed that the CJ point of the ablation is the cut-off density point of the laser, and I₁₄ is the absorbed laser intensity in the unit of 10¹⁴ W/cm². In early time of laser plasma experiment, the ablation pressure has been measured with foil targets as shown in Fig. 3.20 by four different laser wavelengths [13]. It is noted that shorter wavelength laser generates higher ablation pressure up to ~100 Mbar. The theoretical scaling law of (3.56) can well explain the experimental results.

Fig. 3.20

Ablation pressure as functions of laser intensity and its wavelength. The data are obtained in early time of laser plasma research. Reprinted with permission from Ref. [13]. Copyright by Institute of Physics

By solving the stationary solution, the mass flow velocity J₀ flowing into the ablation region can also be obtained. It is called mass ablation rate $\dot{m}$and given as

$$ \dot{m}\approx 1.5\times 1{0}^5{\left(\frac{I_{14}}{{\left({\lambda}_{\mu m}\right)}^4}\right)}^{1/3}\kern1em \left[g/\left(c{m}^2s\right)\right] $$

(3.57)

In Fig. 3.21, the experimental data are plotted for the cases of three different wavelength lasers [14]. It is seen that the theoretical result can explain the experimental data. Note that the mass ablation rate is important value to estimating the ablative stabilization to reduce the hydrodynamic instability of implosion. So, shorter wavelength is better from this point. The ablative stabilization will be discussed in Volume 3.

Fig. 3.21

Mass ablation frate at the ablation front as functions of laser intensity and laser wavelength. The data are obtained in early time of laser plasma research. Reprint from Ref. [14]. Copyright 2012, with permission from Elsevier

3.6.3 Rocket Model

By using the relation of ablation pressure obtained above, we can derive an equation for acceleration of an object (thin foil) of finite mass by laser ablation pressure. This is called a “rocket model” and corresponds to a simple evaluation of rocket design. Now, when the thin foil is moving at the velocity V(t), the following relation is obtained with the mass of the accelerating part ahead of the ablation front as M (t).

$$ M\frac{dV}{dt}={P}_{abl} $$

(3.58)

Since mass decreases with the rate given in (), the following is given

$$ M(t)={M}_0-\dot{m}t $$

(3.59)

where M₀ is the initial mass. Assuming that $\dot{m}$ and P_abl are constant, the Eq. (3.58) can be integrated and the following relation is obtained.

$$ V={V}_0\mathit{\ln}\left(\frac{1}{\varepsilon}\right) $$

$$ \varepsilon =1-\frac{t}{\tau_0}\ \left(=\frac{M(t)}{M_0}\right) $$

(3.60)

where

$$ {V}_0={P}_{abl}/\dot{m\approx 2{C}_s},\kern1em {\tau}_0={M}_0/\dot{m} $$

(3.61)

(3.60) show that the velocity is only a function of the remaining mass. V₀ is a value about two times the sound velocity at the CJ point, C_s. That is, the maximum speed is determined by how high the temperature can be achieved in the deflagration. Acceleration distance can be calculated,

$$ d={d}_0\left(1-\varepsilon +\varepsilon ln\varepsilon \right) $$

(3.62)

where

$$ {d}_0={V}_0{\tau}_0 $$

(3.63)

As a result of acceleration, the fraction of the input energy to the kinetic energy is called hydrodynamic efficiency η_H.

$$ {\eta}_H\equiv \frac{\frac{1}{2}M{V}^2}{E_{ab}}={\eta}_0\frac{\varepsilon {\left( ln\varepsilon \right)}^2}{1-\varepsilon } $$

(3.64)

With absorbed laser intensity I_ab, the non-dimensional coefficient η₀ is given to be

$$ {\eta}_0=\frac{{\left({P}_{ab l}\right)}^2}{\dot{2m}{I}_{ab}} $$

(3.65)

Figure 3.22 shows the dependence of η_H/η₀ upon ε found in (3.65). The value of η₀ is 1/2 for the isothermal from CJ point to vacuum. Therefore, it can be η_h = 0.3 at maximum for ε ≈ 30 %. Accordingly, if the thin foil is accelerated to ablate about 80%, the mass of the foil can be accelerated to about 1.6 times the sound velocity of the CJ point with an energy efficiency of 30 to 40%.

Fig. 3.22

Normalized rocket kinetic energy efficiency as a function of the mass ablation fraction

For example, even if the sound velocity at the C-J point is low, the rocket which fly to the space must escape from the earth’s gravity and fly far. This is called the Earth escape velocity and its value is about 11.2 km/s (40,300 km/h). In the reaction of chemical fuel, the maximum sound velocity does not exceed this value. Therefore, by designing the final mass of the rocket is small, the escape speed can be achieved. This corresponds to the fact that the spacecraft part flying into space is a very small tip part compared to the main body at the time of launch.

3.7 Ablation Structure in Acceleration Phase

In the early stage of laser fusion experiments, so called glass-micro-balloon (GMB) has been used as a spherical capsule to confine deuterium-tritium mixture fusion fuel gas [15]. Even for a very thin shell glass, it is seen that the glass is ablated by the laser heating explained in Chap. 2 (Vol. 1). The ablation pressure drives the shock waves and acceleration of the glass plasma to further accelerating the fuel by the shock waves. An example of one-dimensional simulation of the implosion dynamics of such a GMB is shown in Fig. 3.23.

Fig. 3.23

The radius-time diagram of the Lagrange fluid trajectories obtained with one-dimensional implosion code. A thin shell glass micro-balloon filled with DT fuel is imploded by laser irradiation to generate fusion neutrons. The snap shots at six different time indicated the r-t diagram are plotted on the right for electron temperature and density

In the radius-time diagram, the lines are fluid Lagrange grids and totally, 160 grids are used. The line crossing the grids is the trajectory of the cut-off density. The details of the parameters are given in Ref. [15]. A gaussian shape pulse is irradiated with about 10 kJ energy of green lasers. The top on the right is the time evolution of electron temperature and the bottom is the plasma density at 6 timings shown with arrows on the right in the r-t diagram. The ablation pressure generates a shock wave to accelerate the glass plasma and heat and compress the DT fuel plasma. It is seen that almost stationary ablation structure is propagating to the center of the target.

3.7.1 Stationary Accelerating Ablation Front

To precisely study the hydrodynamic stability of such ablation structure, stationary accelerating ablation structure has been used in spherical geometry as the back-ground implosion dynamics [16]. One-fluid one-temperature hydrodynamic equations in spherical geometry have been used including the nonlinear thermal conduction. When the equations are normalized by the physical quantities at the CJ point of the deflagration wave, the normalized equations governing the stationary solution are as follows [16].

$$ \frac{\partial }{\partial \overset{\sim }{r}}\left({\tilde{r}}^2{\overset{\sim }{\rho}}_0{\tilde{u}}_0\right)=0 $$

(3.66)

$$ {\overset{\sim }{\rho}}_0{\tilde{u}}_0\frac{\partial }{\partial \overset{\sim }{r}}{\tilde{u}}_0=-\frac{\partial }{\partial \overset{\sim }{r}}{\overset{\sim }{P}}_0+G{\overset{\sim }{\rho}}_0 $$

(3.67)

$$ \frac{3}{2}{\overset{\sim }{\rho}}_0{\tilde{u}}_0\frac{\partial }{\partial \overset{\sim }{r}}{\overset{\sim }{T}}_0=-\frac{{\overset{\sim }{P}}_0}{{\tilde{r}}^2}\frac{\partial }{\partial \overset{\sim }{r}}\left({\tilde{r}}^2{\tilde{u}}_0\right)+\frac{K_0}{{\tilde{r}}^2}\frac{\partial }{\partial \overset{\sim }{r}}\left({\tilde{r}}^2{{\overset{\sim }{T}}_0}^{5/2}\frac{\partial }{\partial \overset{\sim }{r}}{\overset{\sim }{T}}_0\right), $$

(3.68)

where G is the normalized gravity given by the inertial force. Here, ${\overset{\sim }{\rho}}_0,{\tilde{u}}_0,{\overset{\sim }{T}}_0$are normalized density, flow velocity, and temperature, respectively, and they are functions only of normalized spatial coordinate $\overset{\sim }{r}$. Normalized physical quantities are defined as follows using the physical quantities at the CJ sonic point.

$$ {\overset{\sim }{\rho}}_0=\frac{\rho_0}{\rho_s},\kern1em {\tilde{u}}_0=\frac{u_0}{C_s},\kern1em {\overset{\sim }{T}}_0=\frac{T_0}{T_s} $$

$$ \overset{\sim }{r}=r/{r}_s,\kern1em \overset{\sim }{t}=t/\left({r}_s/{C}_s\right) $$

(3.69)

Here, ${\overset{\sim }{\rho}}_0,{\tilde{u}}_0,{\overset{\sim }{T}}_0$ are the unity at the sonic point of ablation. C_s is the sound velocity at the sonic point where ${\tilde{u}}_0={u}_0/{C}_s=1$ is satisfied. Here, we also showed the normalized time to be used in the stability analysis in Vol. 3.

In normalizing as above, the following two dimensionless coefficients appeared.

$$ G=\frac{g{r}_s}{C_s^2},\kern1em {K}_0=\frac{K\left(T={T}_s\right)}{\rho_s{C}_s{r}_s{A}^{-1}} $$

(3.70)

Here, the numerator of the definition of Κ₀ is the thermal conduction coefficient at the sonic point, and A is the atomic number of the matter.

Equations (3.66, 3.67, and 3.68) are integrated numerically from the CJ sonic point toward the upstream side. For a given value of G, we obtain the following density ratio R_ρ by varying the value of Κ₀.

$$ {R}_{\rho }=\frac{\rho_a}{\rho_s}=\frac{\left( density\ at\ ablation\ front\right)}{\left( density\ at\ sonic\ point\right)} $$

(3.71)

Regards this value as the eigenvalue of the integration, by changing the value of K₀.

The resultant density profiles are shown in Fig. 3.24 for the fixed density ration R_ρ=50 and five different G [16]. It seems that the ablation front is discontinuous; however, the profiles are all continuous. This is because the heat conduction coefficient is proportional to the 5/2-th power of the temperature, so it has a steep structure as shown in Fig. 2.8.

Fig. 3.24

Stationary solution of compressed shell and expanding ablation profiles driven by Spitzer’s nonlinear heat conduction from the right boundary of the sonic point. Each profile is obtained for a fixed density ratio of 50 and different strength of the inertial force. Reprint with permission from Ref. [16]. Copyright 1998 by American Institute of Physics

The density profile in the left of the ablation front decreases rapidly as G increases. The thin shell GMB, the density profile in Fig. 3.23 is like the case with large G. The spatial structure of the compressed region is like the Earth’s atmosphere due to the gravity. In addition, when passing through that surface, the plasma is rapidly heated by the electron heat conduction, resulting in a dramatically varying structure. Namely, the density sharply decreases and temperature increases drastically. This is a typical profile of laser heating ablation. Note that the ablation profile is sensitive to the energy transport physics. As mentioned in Chap. 6 the profile changes for the case that the nonlocal electron transport and/or radiation one is dominant in the deflagration region.

3.8 Implosion Dynamics and Ablation Profiles in Experiments

3.8.1 Implosion Dynamics

The dynamics of implosion has been measured experimentally by the framing camera of the self-emitting x-ray as shown in Fig. 3.25. In the implosion experiment, a CH polymer polystyrene micro-balloon of radius 226 μm and the thickness 8 μm is irradiated with a squared laser pulse of 1.6 ns width with a picket fence of 0.2 ns and the main pulse of 1,6 ns 0.4 ns after the picket pulse. The laser wavelength is 0.53 μm. The time evolution of x-rat emitting mainly from the imploding high-density shell plasma and the heated fuel gas at the final compression time are measured as shown at the right in Fig. 3.25. The x-ray emission is also calculated for the corresponding implosion with one-dimensional hydrodynamic simulation code ILESTA-1D [15]. It was confirmed that the implosion dynamics agrees well, while the final strong x-ray emission by heated fuel plasma in the stagnation phase is not clearly observed in the experiment. This is due to the hydrodynamic instability in the final stagnation phase.

Fig. 3.25

Implosion dynamics of a spherical shell target with DD fuel. Both images show x-ray self-emission obtained with one-dimensional simulation and from the experiment in the same condition. The global image is same, while the final compression is very week in the experiment, suggesting hydrodynamic instability in the stagnation phase

The implosion diagnostic image shown in Fig. 3.25 is rather old and the diagnostics of small scale and short time has progressed rapidly. More precision technique, for example, has demonstrated the measurement of self-emission x-ray shadowgraph, which provides a method to measure the ablation-front trajectory and low-mode nonuniformity of a target imploded by directly illuminating a fusion capsule with laser beams [17]. The technique uses time-resolved images of soft x-rays (>1 keV) emitted from the coronal plasma of the target imaged onto an x-ray framing camera to determine the position of the ablation front. This method has been used to accurately measure the ablation-front radius, image-to-image timing, and absolute timing. Angular averaging of the images provides an average radius measurement and an error in velocity of 3%. This technique has already been used as the diagnostics of implosion experiments at the Omega Laser Facility and the National Ignition Facility.

The experimental data can be used widely in the laser-produced plasmas to verify and validate (V&V) the physics-integrated simulation codes. The simulation code should be improved by checking its prediction via comparison with corresponding experiments.

In Fig. 3.26, the series of the x-ray self-emission images from an implosion experiment is shown at the top. Each image is time integrated over ~40 ps, and interstrip timing is ~250 ps. The clear green circles are the surface of ablation front. The implosion is done with 19.6 kJ laser of 60 beams OMEGA facility. The target has an 867.8 μm outer diameter with a 26.8 μm thick CH ablator covered by 0.1 μm of aluminum and filled with deuterium at 10.5 atm. The laser pulse shape is plotted in Fig. 3.26a with the solid line. It has one 100-ps picket pulse to set the initial condition of target implosion, and the step-like main pulse of 2 ns duration.

Fig. 3.26

The top: Sequential photographs of self-emitting x-ray from ablation fronts imploding by laser irradiation. (a): Time evolution of laser intensity and the trajectory of ablation front. The red marks are experimental data and the dashed line is from simulation. (b): The velocity of the ablation front from data with red mark and the blue line from simulation. Reprinted with permission from Ref. [17]. Copyright by Cambridge University Press

The trajectory of the ablation front is compared to the corresponding one-dimensional integrated code LILAC. The dashed line is the trajectory from LILAC simulation and the red marks are taken from the experiment data shown on the top. The imploding ablation front velocity is also compared in Fig. 3.26b. The maximum velocity is ~2 × 10⁷ cm/s, well reproduced in LILAC code. The discrepancy in the early phase is due to the sensitivity of modeling target material with fluid approximation. It is clear in both comparisons that the simulation code well reproduces the trajectory, meaning that the energetics of the implosion dynamics is predictable with LILAC code.

3.8.2 Back Light Imaging

By use of external x-ray source more precise distribution of the density can be measured. This method is called X-ray back-lighting. To measure the density profile, plane target is used. The target is C + H polymer polystyrene. The thickness is 40 μm and the width is 200 μm. The main drive-laser is irradiated on the target surface. At the same time, another laser irradiates the titanium plate of 20 μm to produce a small titanium plasma near the target. Filtering x-rays so that only the x-ray of narrow energy band around 4.85 keV can transmit through the C + H plasma from the side, the shadow image of the accelerating foil plasma is measured. The spatial distribution of the transmission intensity as a function of time can be obtained with a high-speed X-ray camera and the data are processed to the time evolution of the density distribution. The principle is simple. Electrons in K shell of carbon (C) in the C + H plasma absorb X rays coming from the titanium plasma. With the known energy of X-ray (=4.85 keV), the absorption coefficient (=19.72 cm²/g), and the initial width of the foil 200 μm, it is easy to obtain the density from the ratio of transmission. With such data the spatial distribution of density has been obtained experimentally as shown in Fig. 3.27 [18].

Fig. 3.27

R-T diagram from experiment and ILESTA code. The black marks are taken from time evolution of x-ray backlighting measurement. The mass center is plotted with black line. The flow lines of Lagrangian meshes from 1-D ILESTA code are plotted with water color lines. Reprint with permission from Ref. [18]. Copyright 1998 by American Institute of Physics

The experimental time evolution of the foil acceleration dynamics and that obtained with the ILESTA-1D code are compared. The trajectories of the foil ablation front and rear surface are plotted for experimental data (marks with error bars) and ILESTA-1D (dashed and dash dot lines) for the density of 0.5 g/cm³. The trajectory of the center of mass is also plotted to compare. The ablation surface starts to move downward before 1 ns, and the shock wave is produced at the same time. The shock wave reaches the rear surface around 2.2 ns, and a rarefaction wave having the initial velocity of the shocked region expands downward. In detail, the rarefaction wave propagates upward in Fig. 3.27 to the ablation surface. Then, gradually the density profile becomes a steady acceleration density one as shown in Fig. 3.24. As can be seen from this comparison, the experiment can be well reproduced with sufficient precision, and this code is widely used in the Japanese laser plasma community as the standard code for design and proposal of experiments and further analysis after a variety of experiments.

In Fig. 3.28 the density distribution of the experiment and two simulations are plotted for the time of 3 ns in Fig. 3.27. In the simulations, the results are shown for two cases with the diffusion model (Spitzer-Harm: SH) or kinetic model (Fokker-Planck equation: FP) as discussed later soon. In the present experiment, the laser intensity is relatively low at 7× 10¹³ W/cm², and the wavelength is short as 0.35 μm of the third harmonics. It is seen that the diffusion approximation sufficiently reproduces experimental data in this case.

Fig. 3.28

The snap shot of the density profile around t = 3 ns in Fig. 3.27 is shown from experimental data with error bars, comparing to the corresponding simulation profile with solid line. Reprint with permission from Ref. [18]. Copyright 1998 by American Institute of Physics

3.8.3 Hydrodynamics of the Final Compression

In Fig. 3.29 a stretched view of the compression phase of the radius-time diagram of Fig. 3.23 is shown. After the shock front collides at the central singular point, the shock wave is reflected to decelerate the following fuel gas to stagnate the DT fuel plasma. Then, it collides at the boundary of high-density glass plasma and reflected toward the center again. Since the reflected shock becomes higher pressure via energy conversion from fluid kinetic energy to the thermal energy, the temperature and pressure of the fuel gas increases more to decelerate the glass plasma to finally push back outward as seen in Fig. 3.29. Of course, the assumption of one-dimensional spherical symmetry is too idealistic. It is natural to think that the lower-density fuel plasma cannot decelerate the heavier glass plasma and the glass plasma may penetrate directly to the central region repelling the DT fuel plasma. As the result, the fuel plasma cannot obtain enough pressure work by the glass plasma not to be heated up to enhance fusion reaction. Such physics is called hydrodynamic instability and mixing. Material mixing by hydrodynamic instability in the stagnation phase is critical issue to explain the experimental results as shown in for example [15]. The instability and resultant turbulent mixing will be studied and discussed later. However, it is informative to know more about one-dimensional hydrodynamics in the final compression phase as shown in Fig. 3.29.

Fig. 3.29

A stretched r-t diagram near the final compression of Fig. 3.23. When the shock wave collides the center, a reflected shock is generated to propagate outward to collide the contact surface with the falling glass shell plasma. This shock is again reflected by the higher density of the glass plasma by gaining energy from the kinetic energy of the glass plasma. This shock compress and heat further the DT plasma to enhance the DT fusion yield

3.9 Ablation and Nozzle

3.9.1 Laser Ablation by Heat Conduction

It is useful to know a general property of the stationary ablation structure as shown in Fig. 3.18 driven by nonlinear electron heat wave, whose (P, V) diagram is plotted in Fig. 3.19. Let us consider for the case of plane geometry. From (2.23) and (2.27), the following two conservation relations are obtained.

$$ \rho u={J}_0: const. $$

(3.72)

$$ \rho {u}^2+P=2{P}_0: const., $$

(3.73)

where the constants are given by the values at the sonic point defined with the subscript “0” as

$$ {u}_0^2=\frac{P_0}{\rho_0} $$

(3.74)

Chapman-Jouguet deflagration wave is given by knowing how to obtain the structure connecting from subsonic region to the sonic point.

Let us discuss the structure in normalized flow velocity U = u/u₀ and normalized temperature T = (P/P₀)/(ρ/ρ₀). From (3.72) and (3.73), the following simple relation is derived.

$$ T=-{\left(U-1\right)}^2+1 $$

(3.75)

Note that this relation gives a monotonic increase of the flow velocity in the subsonic region and requires the temperature is maximum at the sonic point (U = 1).

$$ dT=-2\left(U-1\right) dU $$

(3.76)

In the case of laser ablation structure, the electron heat conduction determines the structure. So, the ablation structure depends on the physics of energy transport. In the case of diffusion-type heat conduction, the temperature change is managed by heat flux Q

$$ Q(x)=-K(x)\frac{\partial T}{\partial x} $$

(3.77)

Therefore, the absorbed laser energy must supply this heat flux at the sonic point x = x₀ with the laser heating rate S in the form.

$$ \frac{\partial }{\partial x}Q= S\delta \left(x-{x}_0\right) $$

(3.78)

This assumption allows the solution continues to increase the flow velocity and expands to the vacuum region as the isothermal rarefaction wave.

When we assume the stationary state even for the supersonic region, the temperature must go down in the supersonic region as given in solution (3.75). This is not realistic model for the ablation structure expanding into the vacuum. As a simple model, it is appropriate to assume that the stationary deflagration wave is continuously connects the iso-thermal rarefaction wave.

Such one-dimensional stationary model can be applicable by changing the size of the one-dimensional geometry like a nozzle and spherical geometry as seen below.

3.9.2 Laval Nozzle

It is well-known that even with one-dimensional flow, but if the flow is in a nozzle with the cross-sectional area changing in space, it is possible to obtain a solution continuously transiting the sonic point from subsonic to supersonic. In this case, we have another valuable the cross-sectional area A(x). Then, (3.72) is replaced by the following relation.

$$ \rho uA= const. $$

(3.79)

In the present case, we have to use Bernoulli relation of steady state flow (2.97) to compressible flow given as

$$ \frac{1}{2}{u}^2+\int \frac{dP}{\rho }= const. $$

(3.80)

Introducing the sound speed “a” and local Mach number M defined as

$$ {a}^2\equiv \frac{d P}{d\rho},\kern2.5em M=\frac{u}{a} $$

(3.81)

Then, (3.80) is rewritten as

$$ \frac{d\rho}{\rho }=-{M}^2\frac{d u}{u} $$

(3.82)

This indicates that the density decreases along with the increase of the flow velocity. It is possible to change (3.79) as

$$ \frac{d\rho}{\rho }+\frac{d u}{u}+\frac{d A}{A}=0 $$

(3.83)

Then, the following relation is obtained.

$$ \left(1-{M}^2\right)\frac{du}{u}=-\frac{dA}{A} $$

(3.84)

This indicates that in the subsonic region (M < 1), the flow velocity increases with the decrease of the pipe size A, and the flow comes to the sonic point by keeping increase if A is minimum value at the sonic point M = 1. Then, in the supersonic region, the increase of flow velocity is maintained by increasing the size of the pipe. Such pipe is called Laval nozzle as shown in Fig. 3.30.

Fig. 3.30

Structure of a Laval nozzle and flow velocity changing from subsonic to supersonic

It is useful to estimate how high supersonic flow velocity of gas is generated as the maximum. Let us assume that the high-pressure with very low velocity flow is generated out of the subsonic edge in Fig. 3.30. Assuming that the gas is adiabatic and P ∝ ρ^γ is satisfied. Then, (3.80) at the left boundary should conserve to reduce the relation.

$$ {u}_{max}=\sqrt{\frac{2}{\gamma -1}}{V}_s\left({x}_0\right) $$

(3.85)

where V_s(x₀) is the sound velocity at the left boundary. Roughly speaking, the sound velocity increases with the temperature, increase of the gas temperature injecting to Laval nozzle gives higher exhausting velocity. In the jet engine, the fuel combustion in front of the nozzle increases the temperature of the gas to increase the pressure and the temperature to covert the heating energy to the large momentum flow. The total momentum flux is proportional to ρuAu. Increase of ρuA and exhausts with u_max gives us the maximum propulsion force.

Gas jet generated with such Laval nozzle is widely used for laser-plasma experiments. In the laser wake-field acceleration of charged particles, ultra-intense-short pulse laser propagates in low density gas and the gas jet is used to provides such almost constant density gas. The gas is spontaneously ionized when the laser interacts with the gas atom and plasma waves are generated as a wake of laser passage. In Fig. 3.31 an example of the gas density profile after the exhaust from the gas jet nozzle is shown. It is expected to give an almost constant density plasma in a vacuum chamber [19].

3.9.3 Solar Wind (Parker Solution)

Although the temperature of the sun surface is about 6000 degrees, there is a corona region whose temperature is one million degrees in the outer layer, and plasma always flows out into outer space from the surface. This is called solar wind. The same thing happens with other stars, which is called stellar wind. Generally, hydrogen, helium and other heavy elements are ejected from the sun surface. It is huge amount of one million tons per second, and the solar wind is falling against the earth as well. The speed of the solar wind is as high as 300 to 900 km/s. The temperature is about 10 to one million degrees.

Fig. 3.31

A typical experimental data of gas density ejected from the exit of gas jet nozzle for use to laser wake-field acceleration experiment. Reprint from Ref. [19]. Copyright 2012, with permission from Elsevier

Regarding solar wind, Parker elucidated the above Laval nozzle idea by applying it to solar wind. He has chosen the effect of spreading the cross-sectional area A to the spherical shape, and that M = 1 is gives with the radius at which the deceleration effect by the Sun gravity successfully transitions from subsonic to supersonic.

Let’s follow the Parker’s solution. First, the equation of continuity is

$$ 4\pi {r}^2\rho u= const.=\left|\frac{dM}{dt}\right| $$

(3.86)

where M is the mass of the sun. We assume that the temperature of the solar wind is constant (T: constant). Then, the pressure is proportional to the density, and the sound speed a is also constant as

$$ {a}^2=\frac{P}{\rho }= const. $$

(3.87)

Then, the equation of motion is

$$ u\frac{d u}{d r}=-\frac{a^2}{\rho}\frac{d\rho}{d r}-\frac{GM}{r^2} $$

(3.88)

Differentiation of (3.86) gives

$$ 2\frac{d r}{r}+\frac{d\rho}{\rho }+\frac{d u}{u}=0 $$

(3.89)

Substituting this into (3.88) and erasing dρ yields the following equation. 。

$$ \left({M}^2-1\right)\frac{1}{u}\frac{du}{dr}=\frac{2}{r}\left(1-\frac{r_c}{r}\right) $$

(3.90)

$$ {r}_c=\frac{GM}{2{a}^2} $$

(3.91)

Here, r_c is the point at which the solar wind becomes sonic speed. When a typical value is entered, it is about 2.5 times the sun radius.

Finally, it is useful to note about mathematics in obtaining the solution of Laval nozzle and the solar wind. It is easily found that for example, for a given temperature, numerical integration to the radius r cannot be extended to the supersonic region. As shown in Fig. 3.32, most of the integral solution of (3.90) cannot pass the sonic point, because the sonic point is mathematically singular point [20]. In general, it is recommended to find the pass from the sonic point to give physically reasonable solution at the solar surface and the infinity.

Fig. 3.32

The velocity-radius diagram to obtain the solution of the solar wind generated on the sun surface as subsonic flow to pass the sonic point to accelerate to supersonic. The integration path passing the saddle point provides uniquely the physically acceptable solution. Reprint from Ref. [20] with kind permission from Springer Science + Business Media

3.9.4 Singularity and Saddle Points

Now, in the case of Laval nozzle, an analytical solution has been shown already, but here we show that there exists a solution in which a differential equation having a singularity at the sonic point in (3.90) continues from subsonic to supersonic. For that purpose, we investigate the properties of differential curves at singular point (r = r_c, u = a).

There are two types of singularities: saddle and node points. With a saddle point, the integral curve is uniquely determined in two-directional space in (3.90), but not uniquely determined if it is a node. It is understood that this sonic point is a saddle point, and there are two integral curves connecting from subsonic to supersonic speed. Let us see the mathematics below.

Now, we study mathematical properties of the singular point in (3.90)

$$ r={r}_c+\delta r $$

(3.92)

$$ u=a+\delta u $$

(3.93)

Then, Taylor expansion (3.90) is carried out with (3.91) and (3.92) to make (3.90) linearized equation around the singular pint.

$$ \frac{du}{dr}=2{\left(\frac{a}{r_c}\right)}^2\frac{\delta r}{\delta u} $$

(3.94)

The small deviations should be on the integral lines, namely

$$ \frac{\delta u}{\delta r}=\frac{du}{dr} $$

(3.95)

So, after all, the integral from the singular point must have the slope of the following two directions.

$$ \frac{du}{dr}=\pm \sqrt{2}\frac{a}{r_c} $$

(3.96)

The actual integral curve is as shown the thick solid line A-B in Fig. 3.32. The line C-D is non-physical integration path.

In Fig. 3.32, the solar wind solution increases from subsonic to supersonic with radius, which corresponds to the solution of + sign in (3.96). Another integral curve is Bondi’s solution (1952) representing spherically symmetric accretion flow. The fundamental Eqs. (3.86) and (3.87) are the same, although the sign of the flow velocity is negative, namely matters fall on the surface of stars. The stationary solution of the so-called accretion flow, in which matters are accumulated from the surroundings, can be obtained at the same time. Actually, accretion is not spherical symmetry in the universe, in the case of a binary star system a white dwarf peels off the surface of a companion star and become heavier. In such a case, it is of a form of disk and it has an angular momentum at the same time. It rotates around the white dwarf and loses its angular momentum with viscosity and accretes matters on the surface of the white dwarf. Therefore, note that the Bondi solution is a very rough approximate solution.