Temperature dependence of parametric instabilities in the context of the shock-ignition approach to inertial confinement fusion Download: 843次
1. Introduction
The principal constituents of shock ignition (SI), converging shocks and their returns in spherical or cylindrical geometry, are based on ideas suggested some time ago[1–3], before it was applied for concrete applications in inertial confinement fusion (ICF)[4–8]. SI is an alternative scheme to ignite pre-compressed fuel in an efficient and robust way, relaxing conditions on the degree of compression and illumination symmetry with respect to standard direct or indirect drive[9–11]. Differently from standard approaches it relies on a short, high-intensity pulse to drive an additional shock which collides with the rebound of the compression shock and allows the creation of a hotspot and thereby ignition of the fuel. Basically SI is a redistribution of the available driver energy by shaping the temporal pulse structure (see Figure
Fig. 1. The temporal evolution of the intensity in the case of conventional drive (blue curve) and SI drive (red curve). In the standard approach to ICF the driver is responsible for fuel assembly and a high velocity, , for igniting the fuel due to the creation of a hotspot. In the SI scenario the main drive is responsible for fuel assembly but at a lower velocity, , preventing ignition. The short high-intensity shock-inducing pulse launched at a later time will reach the fuel at stagnation and ignite it. (Note: the curves in this cartoon drawing are not to scale.)
One of the important results of kinetic simulations of LPI in the framework of SI is the fact that the laser energy is absorbed not at the critical density via inverse Bremsstrahlung but by collective effects in the low-density plasma corona[19–24]. This affects the gain considerably as far as ignition is concerned[11]. More detailed investigations are required concerning the coupling of kinetic simulation results from LPI and radiation hydrodynamic simulations for the compression and ignition phase. This paper considers in some detail the role of the coronal electron plasma temperature as far as the SI scenario is concerned. It is complementary to previous publications[19, 20], which have mostly concentrated on the fundamental scenario of parameteric instabilities in the framework of SI. Originally, cavitation at the quarter critical density was identified as a fundamental aspect of LPI for the kind of parameter regime used in the simulations. The present work shows that this phenomenon is strongly temperature dependent. Similarly, the relative production of hot electrons by SRS and TPD is affected by the electron temperature. The coronal plasma temperature is therefore an important issue as present-day preliminary experiments for SI-LPI operate in a regime of lower electron temperature. The coronal plasma electron temperature is expected to be of the order of , whereas the installations used for LPI in the context of SI achieve much less at present (LULI: ; PALS: ; LIL: ; OMEGA: ). Care has to be taken when extrapolating physics behaviour from low temperature to the higher operating temperature expected for SI.
Most of the experimental activity initially concentrated on the hydrodynamic aspects of SI. However, in recent years, several experiments have been performed of relevance or at least related to LPI aspects of SI[10]. Experiments have been performed that concern beam propagation[25–27], SBS[25, 28–35], SRS[36–39] and TPD (Refs. [40, 41] and references therein). It seems that in general energy losses due to backscattering are dominated by Brillouin rather than Raman. The creation of hot electrons by TPD and SRS at the quarter critical density has been observed. The main problem is that one has to fulfil simultaneously the conditions imposed by plasma scale length, electron temperature and laser intensity, as given by possible future experiments on ignition-scale facilities such as NIF and LMJ. Therefore, it is difficult to be conclusive as far as the present experimental effort is concerned. The present experiments should be compared carefully with multi-dimensional kinetic simulations in order to benchmark the codes and to make predictive simulations for the SI parameters.
Fig. 2. Localization of the various parametric instabilities in the plasma profile. The figure represents a realistic profile. The one used in the simulations is smaller (see Section 2 ).
Table 1. Summary of the simulations. Here, refers to the laser intensity, is the electron plasma temperature. All simulations are at , i.e., a laser wavelength of . The fully relativistic PIC code emi2D[42] was used for all simulations; for all simulations. The reduced intensity case i8 will not be discussed in the text as the results show the same scenario as the corresponding high-intensity case h8.
|
The remainder of this paper is organized as follows. Section
2. Simulation setup
The simulations used in this paper are summarized in Table
The simulation box is in the parallel direction (laser propagation direction) and in the transverse direction. For a wavelength of this corresponds to by . The simulated plasma fills the box completely in the transverse direction but has a length of only in the parallel direction, being surrounded by a vacuum region on both side. The plasma profile is exponential in the propagation direction of the laser with a scale length of and extends from up to with 60 particles per cell at the highest density. It should be noted that in reality the gradient scale length is of the order of a few hundred microns. However, previous simulations in 1D have shown that does not affect the underlying scenario and physics strongly but only affects the time scales. The plasma temperature is a much more stringent parameter for LPI-SI simulations then the gradient scale length. The boundary conditions are periodic in the transverse direction and open in the parallel direction. The incident laser light is p-polarized in normal incidence. The pulse length is infinite with a ramp-up time of a few laser cycles. The spatial discretization is (in the laser propagation direction), (in the transverse direction) and , respecting the Courant–Friedrichs–Lewy condition for explicit integration of the PIC equations. Relativistic normalization is used throughout, i.e., the time and spatial coordinate are normalized by the laser frequency and vacuum -vector , respectively. Collisions are not accounted for. The simulation times are too short and the temperatures too high for collisions to have much of an effect on the parametric instabilities analysed in the following simulations. The typical electron–ion collision frequency is given as , with in and in keV. The use of, as worst-case scenario, the highest plasma density, , and the lowest temperature, , employed in the simulations results in a collision time of . This is twice the simuation time. Moreover, for the considered intensities the growth rates of the instabilities are also much larger than the typical collision frequencies.
Figure
Fig. 3. The profiles of the plasma and the incident laser beam. The parameters are given in Section 2 .
3. Analysis of simulation results
3.1. Characterization of the parametric instabilities involved
The two main instabilities of interest here are SRS and TPD instability. The resonant SRS process consists of the decomposition of a laser photon () into a backscattered frequency-downshifted photon () and a forward travelling electron plasma wave () (EPW). It fulfils the following conditions for the frequency and wavevector:
The frequency of the EPW follows from the dispersion relation as
The TPD is the decomposition of the laser photon () into two plasmons (). The plasmon frequencies are determined by the dispersion relation for the electron plasma waves, Equation (
Figure
Fig. 4. Geometry of the -vectors involved in the TPD instability. The decay of a photon into two plasmons can be realized in two possible ways while preserving energy and momentum. This particular geometry applies in 2D and helps with the interpretation of the phase space diagrams. In reality, 3D, the number of possible -vectors is infinite lying on an asymmetric cone around the laser -vector.
The threshold for the TPD instability in an inhomogeneous plasma profile[43, 44] is temperature dependent and is given as
Here, , is in units of kilo electron volts and is the laser intensity in units of ; and are given in units of microns. The TPD instability is excited in the vicinity of the quarter critical density and develops as an absolute instability as the slow plasma waves do not escape the resonance. In contrast, the threshold for SRS excitation near is not a function of and is given by[17, 45]
Depending on the electron temperature, the thresholds for the two instabilities can be quite similar and develop in competition. A high-frequency hybrid instability (HFHI) can develop[46, 47], leading to a co-existence of electrostatic and electromagnetic modes. The growth rate for Raman is given as[48], with . Here, and are the damping rates due to Landau damping and collisions, respectively. In the case when Raman is above threshold, i.e., , the growth rate reduces to . Even for the lowest temperature used in the simulations, , the collisional damping rate is two orders of magnitude smaller than the Landau damping rate for EPWs, which, for backward SRS, is given as
Here, is the electron plasma wavevector and one has the following expression for in practical units:
Evaluation of these expressions at a density of and for results in a Landau damping rate of , i.e., much larger than the collision rate quoted in Section
At very low electron plasma temperature, a few hundred eV, the Langmuir decay instability (LDI) can play an important role as it saturates TPD and SRS activity. Although of limited importance in the case of SI which operates in multi-keV conditions, it could be relevant for present-day LPI experiments for SI, as these take place at a much lower temperature. Therefore it is presented in some more detail.
The LDI was predicted in the 1960s[49, 50] and in LPI experiments it was verified more recently by direct observation[36, 51–53], although its experimental existence had already been conjectured before[54], and it has been observed indirectly due to the ion-acoustic wave (IAW) damping on SRS[55–59].
The LDI induces a decay of the pump plasma wave into an IAW characterized by (travelling in the direction of the original EPW) and an anti-Stokes daughter EPW (travelling in the opposite direction) having a frequency close to the original frequency (downshifted by ) and a wavevector given by , where the correction has the form
Here, is the Debye length, which in practical units is given as , with in eV and in . A necessary condition for LDI to take place is therefore that[49, 60]
For all the cases considered in Table
The maximum growth rate is given by
Here, is the electrostatic field associated with the EPWs, represent the two plasmons, and
For the intensity considered in this study SBS takes place in the weak-coupling regime even for the lowest temperature used, i.e., Equation (
In the following the simulation results are analysed with respect to the behaviour of the reflectivity of the incident laser beam, the induced activity of the parametric instabilities, and the phase space and Poynting vector. These issues are of course strongly interdependent.
3.2. Overall scenario
Two fundamental issues have to be addressed as far as the simulations are concerned.
The relative importance of TPD and SRS. One would expect that the colder the plasma the stronger the TPD. Which mechanism is saturating the TPD and SRS activity at the quarter critical density
With respect to the first point it is indeed found that the higher the temperature the more pronounced SRS becomes, although it is a negligible energy loss mechanism as far as reflectivity is concerned. With respect to the second point one can observe a clear transition from LDI-induced saturation of Raman at to saturation due to cavitation and density fluctuations for temperatures of and above. For the low-temperature case c8 there is LDI and TPD activity but no SRS.
SBS is present for any temperature but decreases in importance the higher the temperature. The general conclusion of this set of simulations is that the plasma temperature plays a crucial role as far as the LPI scenario is concerned. Already a factor of two in the electron plasma temperature can significantly affect the relative importance of the parametric instabilities and their effect on the plasma dynamics and laser absorption. Realistic simulations and future experiments for SI require LPI to take place at the right temperature.
The overall energy balance is affected by the amount of the energy of the incident laser beam reflected due to SRS and SBS, Section
3.3. Reflectivity
Figure
Fig. 5. Reflectivities (, i.e., reflected intensity over incident intensity at the centre of the speckle in the transverse direction) for the cases (a) c8, (b) h8, (c) h7 and (d) h9. The curves are ‘filled’ as the laser temporal period is resolved. The blue curve corresponds to SBS-like frequencies, summing the range –. The red curve corresponds to SRS-like frequencies, summing the range –. No frequencies are present in the interval –. Note: the time on the axis refers to the moment the reflected light crosses the boundary of the computational box; as the quarter critical density is located at , the light was actually refelected earlier.
The total reflectivities, i.e., the SRS and SBS contributions combined, reduce strongly as the temperature is increased. For the cases c8 and h8 the average reflectivity is of the order of 10%–15%, whereas for the high-temperature cases, h7 and h9, the overall reflectivity is reduced to 1%–2% only.
SBS develops everywhere in the profile, up to . At SBS is strongly inhibited due to the fact that the laser beam is randomized due to the cavitation process and LDI, and the associated strong density fluctuations (see, e.g., Figure
By contrast, SRS develops predominantly at . The simulated time scales are short for the SBS evolution. It is therefore unclear what the saturation mechanisms for SBS are on longer time scales. The bursty behaviour of the reflectivities as visible in Figure
Fig. 7. Frequency spectra for the cases (a) c8, (b) h9 and (c) a zoom of (b). Note: (a) and (b) are on log scale whereas (c) is on linear scale.
Figure
Fig. 8. Two-dimensional Fourier spectra of the electromagnetic field evaluated in the vicinity of for the cases c8 (a, c) and h9 (b, d) taken at times (a, b) and (c, d).
Figure
3.4. Electron-related mode activity: SRS, TPD and LDI
As already mentioned, there is almost no SRS for the cases c8 and h8. Strong SRS is present for higher temperatures above . In general, it can be said that there is no SRS activity in the low-density part of the plasma corona. SRS is concentrated near the quarter critical density. The high-temperature case h9 has very strong absolute SRS activity but basically no TPD.
As can be seen from Table
Table 2. Temperature-dependent occurrence of LPI phenomena. The number of stars gives a rough ‘visual’ interpretation of the strength of the process occurring, with strongest and weakest. The numbers in the columns and are calculated from the corresponding Equations (5 ) and (6 ). The thresholds have to be compared with the laser intensity, which in units of is 1.2 for all cases. CAV cavitation.
|
As the SBS reflectivity data (see Figure
At a later stage SRS still does not develop due to the strong, irregular small-scale density modulations induced by TPD. For the cold case c8, TPD is saturated by LDI which develops on the EPWs generated originally by TPD.
In the high-temperature cases (h7 and h9) the threshold for SRS is lower than for TPD and strong absolute SRS develops which induces cavitation and leads to saturation.
Figure
Fig. 9. Poynting vector for the case c8 at . The ‘hole’ behind the density layer around is clearly visible.
Fig. 10. Fourier transform of the ion density corresponding to Figure 11 . (a) Case c8 at , (b) case h8 at , (c) case h7 at and (d) case h9 at . It should be noted that the axes for the various cases differ as the -vectors become shorter as the temperature increases.
The EPWs generated by TPD and the secondary waves generated by LDI evolve into turbulence, as observed in other recent simulations for SI[18, 70–72] (see the discussion of real-space figures in Section
3.5. Plasma cavitation
The role of cavitation was clearly identified in previous work related to SI[19–21]. It is a fundamental mechanism of LPI which acts as a converter, transferring energy from the laser into kinetic energy of the plasma, and was originally identified in SBS activity[73]. Cavities present strong local plasma perturbations which can act as a dynamic random phase plate and induce coherence loss to the incident laser beam, i.e., they act as a means of plasma smoothing[68]. They are also essential in saturating parametric instabilities at the quarter critical density. Finally, they are an important mechanism for laser energy absorption. Figure
Fig. 11. Ion density near the quarter critical density (located at . (a) Case c8 at , (b) case h8 at , (c) case h7 at and (d) case h9 at . It should be noted that the colour scale used is not the same for each of the four sub-figures in order to enhance the visibility of the structures.
The high-temperature case h9, in particular, Figure
As discussed above (see Section
The creation of the cavities is therefore an intricate interdependence of laser intensity, plasma profile, plasma temperature and growth rate of absolute Raman at . From previous and present simulations it follows that the optimum is around for cavity creation. Figure
Fig. 12. The transverse electron phase space as a function of the laser propagation direction for (a) case c8 at , (b) case h8 at , (c) case h7 at and (d) case h9 at . The time slice for h9 is taken at an early time as the electrons start to recirculate quickly.
3.6. Laser absorption into hot electrons
SRS, TPD and cavitation are all sources of hot electron production. However, the hot electrons differ as far as temperature and propagation direction are concerned[19, 20].
As discussed above, TPD is the dominant process in the cold case c8. As shown in Figure
As was pointed out previously[10], the temperature characterizing the hot tail of the electron distribution function is less than and is not dependent on the laser intensity but is determined by the resonant interaction of the electrons with the plasma waves generated by TPD and SRS. The laser intensity affects the number of hot electrons produced. The relevant parameters affecting the hot electron production are the phase velocity and the initial bulk electron temperature. Present and previous simulation work[19, 20] indicates that the hot electron temperature increases with the initial bulk temperature of the plasma. However, a more detailed analysis would be required to quantify this relationship.
4. Conclusions
The relative importance of the various phenomena (cavitation, LDI, SRS, SBS and TPD) is summarized in Table
An important conclusion of the simulation work presented is the necessity to have better absorption models for the laser beam in radiation hydrodynamic simulation for shock ignition. It is clear that a large fraction, or even most of it, dependent on the LPI parameters, is not absorbed via inverse Bremsstrahlung at the critical density, but rather by collective effects in the low-density corona. This has to be accounted for in hydrodynamic simulations of the implosion phase in a realistic way. Reliable integrated simulations for SI are necessary.
SI experiments and simulations therefore currently have some important caveats.
Another important issue is how these instabilities are affected in the case of multiple overlapping beams as the driver[75].
Much more detailed experiments and simulations are needed to determine the presence and relative importance of the various participating instabilities.
A very interesting issue is to determine experimentally where exactly the laser energy is absorbed: in the low-density plasma corona, at the critical surface or at both locations (in which case the ratio would be important). Possible hot electrons have to be attributed clearly to either SRS or TPD. The distribution functions and directionality of the hot electrons will help in this respect. The simulations clearly show the importance of cavitation at the quarter critical density. The cavities are in general of the order of a few wavelengths. It should be possible to perform interferometry at to image the presence of cavities.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
Article Outline
S. Weber, C. Riconda. Temperature dependence of parametric instabilities in the context of the shock-ignition approach to inertial confinement fusion[J]. High Power Laser Science and Engineering, 2015, 3(1): 010000e6.