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The energy leaking through a slot in the hohlraum filled with low-Z foams is a typical dynamic problem of the ablated high-Z plasmas. In this paper, we develop a simplified one-dimensional model to study the expansion-reverse process of the ablated Au plasmas, which corresponds to the closing-reopening process of a slot. Our work shows that its physical mechanism is the ablation pressure competing with radiation pressure difference and the material pressure of low-Z foams. The analytical formulas for the reverse time and reverse distance of the Au plasma are deduced, respectively, indicating that the cubic value for each of both peak temperatures is proportional to the density of the low-Z foams. The main conclusions of analytic theory are verified by numerical simulation through using the modified radiation-hydrodynamic program MULTI. It is shown that the power exponents of scaling law in high-Z plasma ablation keep unchanged in a wide range of density of low-Z foams. The range of validity of the model is discussed.
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Keywords:
- high-Z plasmas /
- ablation pressure /
- confined ablation /
- low-Z foams
[1] Davidson R C 2004 National Task Force on High Energy Density Physics (Washington, DC: Office of Science and Technology Policy) pp1, 2
[2] Meng G W, Wang J G, Wang X R, Li J H, Zhang W Y 2016 Matter Rad. Extremes 1 249
Google Scholar
[3] Lindl J D, Amendt P, Berger R L, Glendinning S G, Glenzer S H, Hann S W, Kauffman R L, Landen O L, Suter L J 2004 Phys. Plasmas 11 339
Google Scholar
[4] Remington B A, Drake R P, Takabe H, Arnett D 2000 Phys. Plasmas 7 1641
Google Scholar
[5] Ensman L, Burrows A 1992 Astrophys. J. 393 742
Google Scholar
[6] Blondin J M, Wright E B, Borkowski K J, Reynolds S P 1998 Astrophys. J. 500 342
Google Scholar
[7] Vink J 2012 Astron. Astrophys. Rev. 20 49
Google Scholar
[8] Laming J M, Grun J 2002 Phys. Rev. Lett. 89 125002
Google Scholar
[9] Pound M W, Kane J O, Ryutov D D, Remington B A, Mizuta A 2007 Astrophys. Space Sci. 307 187
Google Scholar
[10] Mizuta A, Kane J O, Pound M W, Remington B A, Ryutov D D, Takabe H 2006 Astrophys. J. 647 1151
Google Scholar
[11] Armitage P J, Livio M 1998 Astrophys. J. 493 898
Google Scholar
[12] Maccarone T J 2014 Space Sci. Rev. 183 101
Google Scholar
[13] Marshak R E 1958 Phys. Fluids 1 24
Google Scholar
[14] Zeldovich Y B, Raizer Y P 1967 Physics of Shock Waves and High Temperature Hydrodynamics Phenomena, Part II (New York: Academic) pp238－240
[15] Meng G W, Li J H, Yang J M, Zhu T, Zou S Y, Wang M, Zhang W Y 2013 Phys. Plasmas 20 092704
Google Scholar
[16] Pakula R, Sigel R 1985 Phys. Fluids 28 232
Google Scholar
[17] Shussman T, Heizler S I 2015 Phys. Plasmas 22 082109
Google Scholar
[18] Kaiser N, Meyer-ter-Vehn J, Sigel R 1989 Phys. Fluids B 1 1747
[19] Hammer J H, Rosen M D 2003 Phys. Plasmas 10 1829
Google Scholar
[20] Hurricane O A, Hammer J H 2006 Phys. Plasmas 13 113303
Google Scholar
[21] Back C A, Bauer J D, Landen O L, Turner R E, Lasinski B F, Hammer J H, Rosen M D, Suter L J, Hsing W H 2000 Phys. Rev. Lett. 84 274
Google Scholar
[22] Back C A, Bauer J D, Hammer J H, Lasinski B F, Turner R E, Rambo P W, Landen O L, Suter L J, Rosen M D, Hsing W H 2000 Phys. Plasmas 7 2126
Google Scholar
[23] Hoarty D, Willi O, Barringer L, Vickers C, Watt R, Nazarov W 1999 Phys. Plasmas 6 2171
Google Scholar
[24] Guymer T M, Moore A S, Morton J, Kline J L, Allan S, Bazin N, Benstead J, Bentley C, Comley A J, Cowan J, Flippo K, Garbett W, Hamilton C, Lanier N E, Mussack K, Obrey K, Reed L, Schmidt D W, Stevenson R M, Taccetti J M, Workman J 2015 Phys. Plasmas 22 043303
Google Scholar
[25] 李三伟, 杨东, 李欣, 等 2018 中国科学: 物理学 力学 天文学 48 065202
Li S W, Yang D, Li X, et al. 2018 Sci. Sin.: Phys. Mech. Astron. 48 065202
[26] 蓝可, 贺贤土, 赖东显, 李双贵 2006 55 3789
Google Scholar
Lan K, He X T, Lai D X, Li S G 2006 Acta Phys. Sin. 55 3789
Google Scholar
[27] Jones O S, Schein J, Rosen M D, Suter L J, Wallace R J, Dewald E L, Glenzer S H, Campbell K M, Gunther J, Hammel B A, Landen O L, Sorce C M, Olson R E, Rochau G A, Wikens H L, Kaae J L, Kilkenny J D, Nikroo A, Regan S P 2007 Phys. Plasmas 14 056311
Google Scholar
[28] Orzechowski T J, Rosen M D, Kornblum H N, Porter J L, Suter L J, Thiessen A R, Wallace R J 1996 Phys. Rev. Lett. 77 3545
Google Scholar
[29] Yang J M, Meng G W, Zhu T, Zhang J Y, Li J H, He X A, Yi R Q, Xu Y, Hu Z M, Ding Y N, Liu S Y, Ding Y K 2010 Phys. Plasmas 17 062702
Google Scholar
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Google Scholar
[31] Moore A S, Cooper A B R, Schneider M B, MacLaren S, Graham P, Lu K, Seugling R, Satcher J, Klingmann J, Comely A J, Marrs R, May M, Widmann K, Glendinning G, Castor J, Sain J, Back C A, Hund J, Baker K, Hsing W W, Foster J, Young B, Young P 2014 Phys. Plasmas 21 063303
Google Scholar
[32] Meng G W, Zou S Y, Wang M 2019 Phys. Plasmas 26 022708
Google Scholar
[33] Hall G N, Jones O S, Strozzi D J, Moody J D, Turnbull D, Ralph J, Michel P A, Hohenberger M, Moore A S, Landen O L, Divol L, Bradley D K, Hinkel D E, Mackinnon A J, Town R P J, Meezan N B, Hopkins L B, Izumi N 2017 Phys. Plasmas 24 052706
Google Scholar
[34] Schneider M B, MacLaren S A, Widmann K, Meezan N B, Hammer J H, Yoxall B E, Bell P M, Benedetti L R, Bradley D K, Callahan D A, Dewald E L, Doppner T, Eder D C, Edwards M J, Guymer T M, Hinkel D E, Hohenberger M, Hsing W W, Kervin M L, Kikenny J D, Landen O L, Lindl J D, May M J, Michel P, Milovich J L, Moody J D, Moore A S, Ralph J E, Regan S P, Thomas C A, Wan A S 2015 Phys. Plasmas 22 122705
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Google Scholar
[36] 曾先才, 姜荣洪, 常铁强 1991 强激光与粒子束 3 477
Zeng X C, Jiang R H, Chang T Q 1991 High Power and Particle Beams 3 477
[37] Ramis R, Schmalz R, Meyer-ter-Vehn J 1988 Comput. Phys. Commun. 49 475
Google Scholar
[38] Ramis R, Meyer-ter-Vehn J 2016 Comput. Phys. Commun. 203 226
Google Scholar
[39] Ramis R 2017 J. Comput. Phys. 330 173
Google Scholar
[40] Pasley J, Nilson P, Willingale L, Haines M G, Notley M, Tolley M, Neely D, Nazarov W, Willi O 2006 Phys. Plasmas 13 032702
Google Scholar
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图 1 (a)物理模型的简化; (b)一维模型的示意图; (c)波系示意图
Figure 1. (a) Simplification of physical model; (b) one-dimensional model; (c) the wave system.
图 2 Au等离子体的左界面在不同
${d_1}$ 和${d_2}$ 条件下的(a)位移和(b)速度随时间的变化; 折返时间和折返距离分别随(c)${d_1}$ (取${d_2} = 1\;{\rm{cm}}$ ), (d)${d_2}$ (取${d_{\rm{1}}} = 50 \;{\rm{cm}}$ )的变化Figure 2. (a) Displacement and (b) velocity of the left interface of Au plasmas versus time under the condition of different
${d_1}$ and${d_2}$ . The reverse time and distance of Au plasmas versus (c)${d_1}$ with${d_2} = 1\;{\rm{cm}}$ and (d)${d_2}$ with${d_{\rm{1}}} = 50 \;{\rm{cm}}$ .
图 3 在
$T_{\rm{r}}^{} = 16 \;{\rm{MK}}$ ,${\rho _{\rm{1}}} = 0.15\;{\rm{g}} \cdot {\rm{c}}{{\rm{m}}^{ - 3}}$ 条件下,${\rm{0}}.02 \;{\text{μs}}$ 时网格的温度、速度、密度和压强随网格编号n的变化Figure 3. Temperature, velocity, density and pressure versus cell number n at 0.02 μs under the condition of
$T_{\rm{r}}^{} = 16 \;{\rm{MK}}$ and${\rho _{\rm{1}}} = 0.15\;{\rm{g}} \cdot {\rm{c}}{{\rm{m}}^{ - 3}}$ .
图 4 理论预测(取
$\xi = \eta = 1$ )的(a)折返时间和(b)折返距离随辐射源温度${T_{\rm{r}}}$ 的变化Figure 4. Theoretical prediction (with
$\xi = \eta = 1$ ) of (a) reverse time and (b) reverse distance versus${T_{\rm{r}}}$ .
图 5 折返时间和折返距离分别在不同的密度
$ {\rho _{\rm{1}}}$ (a) 0.05, (b) 0.5, (c) 1 g·cm–3下与辐射源温度Tr的变化关系; (d) 参数ξ和η随$ {\rho _{\rm{1}}}$ 的变化Figure 5. Reverse time and distance versus Tr under different density
$ {\rho _{\rm{1}}}$ of (a) 0.05, (b) 0.5, and (c) 1 g·cm–3. (d) ξ and η versus$ {\rho _{\rm{1}}}$ .
图 6 折返时间和折返距离的(a)峰值温度
${T_{\rm{m}}}$ 和(b)峰值温度的三次方$T_{\rm{m}}^{\rm{3}}$ 随密度${\rho _{\rm{1}}}$ 的变化Figure 6. (a) The peak temperature
${T_{\rm{m}}}$ and (b)$T_{\rm{m}}^{\rm{3}}$ of reverse time and distance versus${\rho _{\rm{1}}}$ .
图 A1 (a)辐射温度和Au等离子体密度的空间分布; (b)
$ {t_{\rm{s}}} $ 随辐射源温度${T_{\rm{r}}}$ 的变化; (c)烧蚀压和(d)烧蚀质量随时间的变化Figure A1. (a) Temperature and density versus distance; (b)
${t_{\rm{s}}}$ versus${T_{\rm{r}}}$ ; (c) ablation pressure versus time; (d) ablated mass versus time.表 A1 b和l的拟合值随
${T_{\rm{r}}}$ 的变化Table A1. b and l versus
${T_{\rm{r}}}$ ${T_{\rm{r}}}$/MK b l $l - b$ 6 –0.47817 0.53663 1.01480 8 –0.47479 0.52883 1.00362 10 –0.46818 0.52192 0.99009 12 –0.47633 0.51626 0.99259 14 –0.47161 0.51122 0.98283 16 –0.47083 0.50764 0.97846 
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[1] Davidson R C 2004 National Task Force on High Energy Density Physics (Washington, DC: Office of Science and Technology Policy) pp1, 2
[2] Meng G W, Wang J G, Wang X R, Li J H, Zhang W Y 2016 Matter Rad. Extremes 1 249
Google Scholar
[3] Lindl J D, Amendt P, Berger R L, Glendinning S G, Glenzer S H, Hann S W, Kauffman R L, Landen O L, Suter L J 2004 Phys. Plasmas 11 339
Google Scholar
[4] Remington B A, Drake R P, Takabe H, Arnett D 2000 Phys. Plasmas 7 1641
Google Scholar
[5] Ensman L, Burrows A 1992 Astrophys. J. 393 742
Google Scholar
[6] Blondin J M, Wright E B, Borkowski K J, Reynolds S P 1998 Astrophys. J. 500 342
Google Scholar
[7] Vink J 2012 Astron. Astrophys. Rev. 20 49
Google Scholar
[8] Laming J M, Grun J 2002 Phys. Rev. Lett. 89 125002
Google Scholar
[9] Pound M W, Kane J O, Ryutov D D, Remington B A, Mizuta A 2007 Astrophys. Space Sci. 307 187
Google Scholar
[10] Mizuta A, Kane J O, Pound M W, Remington B A, Ryutov D D, Takabe H 2006 Astrophys. J. 647 1151
Google Scholar
[11] Armitage P J, Livio M 1998 Astrophys. J. 493 898
Google Scholar
[12] Maccarone T J 2014 Space Sci. Rev. 183 101
Google Scholar
[13] Marshak R E 1958 Phys. Fluids 1 24
Google Scholar
[14] Zeldovich Y B, Raizer Y P 1967 Physics of Shock Waves and High Temperature Hydrodynamics Phenomena, Part II (New York: Academic) pp238－240
[15] Meng G W, Li J H, Yang J M, Zhu T, Zou S Y, Wang M, Zhang W Y 2013 Phys. Plasmas 20 092704
Google Scholar
[16] Pakula R, Sigel R 1985 Phys. Fluids 28 232
Google Scholar
[17] Shussman T, Heizler S I 2015 Phys. Plasmas 22 082109
Google Scholar
[18] Kaiser N, Meyer-ter-Vehn J, Sigel R 1989 Phys. Fluids B 1 1747
[19] Hammer J H, Rosen M D 2003 Phys. Plasmas 10 1829
Google Scholar
[20] Hurricane O A, Hammer J H 2006 Phys. Plasmas 13 113303
Google Scholar
[21] Back C A, Bauer J D, Landen O L, Turner R E, Lasinski B F, Hammer J H, Rosen M D, Suter L J, Hsing W H 2000 Phys. Rev. Lett. 84 274
Google Scholar
[22] Back C A, Bauer J D, Hammer J H, Lasinski B F, Turner R E, Rambo P W, Landen O L, Suter L J, Rosen M D, Hsing W H 2000 Phys. Plasmas 7 2126
Google Scholar
[23] Hoarty D, Willi O, Barringer L, Vickers C, Watt R, Nazarov W 1999 Phys. Plasmas 6 2171
Google Scholar
[24] Guymer T M, Moore A S, Morton J, Kline J L, Allan S, Bazin N, Benstead J, Bentley C, Comley A J, Cowan J, Flippo K, Garbett W, Hamilton C, Lanier N E, Mussack K, Obrey K, Reed L, Schmidt D W, Stevenson R M, Taccetti J M, Workman J 2015 Phys. Plasmas 22 043303
Google Scholar
[25] 李三伟, 杨东, 李欣, 等 2018 中国科学: 物理学 力学 天文学 48 065202
Li S W, Yang D, Li X, et al. 2018 Sci. Sin.: Phys. Mech. Astron. 48 065202
[26] 蓝可, 贺贤土, 赖东显, 李双贵 2006 55 3789
Google Scholar
Lan K, He X T, Lai D X, Li S G 2006 Acta Phys. Sin. 55 3789
Google Scholar
[27] Jones O S, Schein J, Rosen M D, Suter L J, Wallace R J, Dewald E L, Glenzer S H, Campbell K M, Gunther J, Hammel B A, Landen O L, Sorce C M, Olson R E, Rochau G A, Wikens H L, Kaae J L, Kilkenny J D, Nikroo A, Regan S P 2007 Phys. Plasmas 14 056311
Google Scholar
[28] Orzechowski T J, Rosen M D, Kornblum H N, Porter J L, Suter L J, Thiessen A R, Wallace R J 1996 Phys. Rev. Lett. 77 3545
Google Scholar
[29] Yang J M, Meng G W, Zhu T, Zhang J Y, Li J H, He X A, Yi R Q, Xu Y, Hu Z M, Ding Y N, Liu S Y, Ding Y K 2010 Phys. Plasmas 17 062702
Google Scholar
[30] Cooper A B R, Schneider M B, MacLaren S A, Moore A S, Young P E, Hsing W W, Seugling R, Foord M E, Sain J D, May M J, Marrs R E, Maddox B R, Lu K, Dodson K, Smalyuk V, Graham P, Foster J M, Back C A, Hund J F 2013 Phys. Plasmas 20 033301
Google Scholar
[31] Moore A S, Cooper A B R, Schneider M B, MacLaren S, Graham P, Lu K, Seugling R, Satcher J, Klingmann J, Comely A J, Marrs R, May M, Widmann K, Glendinning G, Castor J, Sain J, Back C A, Hund J, Baker K, Hsing W W, Foster J, Young B, Young P 2014 Phys. Plasmas 21 063303
Google Scholar
[32] Meng G W, Zou S Y, Wang M 2019 Phys. Plasmas 26 022708
Google Scholar
[33] Hall G N, Jones O S, Strozzi D J, Moody J D, Turnbull D, Ralph J, Michel P A, Hohenberger M, Moore A S, Landen O L, Divol L, Bradley D K, Hinkel D E, Mackinnon A J, Town R P J, Meezan N B, Hopkins L B, Izumi N 2017 Phys. Plasmas 24 052706
Google Scholar
[34] Schneider M B, MacLaren S A, Widmann K, Meezan N B, Hammer J H, Yoxall B E, Bell P M, Benedetti L R, Bradley D K, Callahan D A, Dewald E L, Doppner T, Eder D C, Edwards M J, Guymer T M, Hinkel D E, Hohenberger M, Hsing W W, Kervin M L, Kikenny J D, Landen O L, Lindl J D, May M J, Michel P, Milovich J L, Moody J D, Moore A S, Ralph J E, Regan S P, Thomas C A, Wan A S 2015 Phys. Plasmas 22 122705
Google Scholar
[35] Lan K, Liu J, Li Z C, Xie X F, Huo W Y, Chen Y H, Ren G L, Zheng C Y, Yang D, Li S W, Yang Z W, Guo L, Li S, Zhang M Y, Han X Y, Zhai C L, Hou L F, Li Y K, Deng K L, Yuan Z, Zhan X Y, Wang F, Yuan G H, Zhang H J, Jiang B B, Huang L Z, Zhang W, Du K, Zhao R C, Li P, Wang W, Su J Q, Deng X W, Hu D X, Zhou W, Jia H T, Ding Y K, Zheng W G, He X T 2016 Matter Radiat. Extremes 1 8
Google Scholar
[36] 曾先才, 姜荣洪, 常铁强 1991 强激光与粒子束 3 477
Zeng X C, Jiang R H, Chang T Q 1991 High Power and Particle Beams 3 477
[37] Ramis R, Schmalz R, Meyer-ter-Vehn J 1988 Comput. Phys. Commun. 49 475
Google Scholar
[38] Ramis R, Meyer-ter-Vehn J 2016 Comput. Phys. Commun. 203 226
Google Scholar
[39] Ramis R 2017 J. Comput. Phys. 330 173
Google Scholar
[40] Pasley J, Nilson P, Willingale L, Haines M G, Notley M, Tolley M, Neely D, Nazarov W, Willi O 2006 Phys. Plasmas 13 032702
Google Scholar
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