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Experiments on TCV tokamak have achieved high confinement mode (H-mode) operation with negative triangularity, and this mode shows quite different characteristics from those with the positive triangularity in experiment and simulation. Linear simulations for kinetic ballooning mode and peeling-ballooning(PB) mode without diamagnetic effect show that negative triangularity can enhance the instability of the ballooning mode and close access to the second stable region. However, the understanding of ELM for negative triangularity is not sufficient. Therefore, it is necessary to carry out further research on ELM with negative triangularity. In this work, based on a series of equilibria with different triangularities in Tokamak, the nonlinear characteristics of negative triangularity of PB mode is investigated. It is found that the negative triangularity can destabilize the PB mode by a larger unfavorable curvature region, which will reduce the instability threshold, and thus limiting the increase of pedestal height. In the nonlinear phase, the pressure perturbation intensity with negative triangularity will extend to the top area and the bottom area in the low field side and bring about an earlier ELM collapse. Meanwhile, modes with different toroidal mode numbers are more likely to be triggered off and then grow and replaces the initial unstable mode, showing more obvious turbulent transport characteristics, which can play a role in the ELM energy loss. [1] Wagner F, Becker G, Behringer K, Campbell D, Eberhagen A, Engelhardt W, Fussmann G, Gehre O, Gernhardt J, Gierke G v, Haas G, Huang M, Karger F, Keilhacker M, Klüber O, Kornherr M, Lackner K, Lisitano G, Lister G G, Mayer H M, Meisel D, Müller E R, Murmann H, Niedermeyer H, Poschenrieder W, Rapp H, Röhr H, Schneider F, Siller G, Speth E, Stäbler A, Steuer K H, Venus G, Vollmer O, Yü Z 1982 Phys. Rev. Lett. 49 1408
Google Scholar
[2] Zohm H 1996 Plasma Phys. Controlled Fusion 38 105
Google Scholar
[3] Snyder P B, Wilson H R, Ferron J R, Lao L L, Leonard A W, Osborne T H, Turnbull A D, Mossessian D, Murakami M, Xu X Q 2002 Phys. Plasmas 9 2037
Google Scholar
[4] Lao L L, Ferron J R, Miller R L, Osborne T H, Chan V S, Groebner R J, Jackson G L, La Haye R J, Strait E J, Taylor T S, Turnbull A D, Doyle E J, Lazarus E A, Murakami M, McKee G R, Rice B W, Zhang C, Chen L 1999 Nucl. Fusion 39 1785
Google Scholar
[5] Onjun T, Kritz A H, Bateman G, Parail V, Lonnroth J, Huysmans G 2004 Phys. Plasmas 11 3006
Google Scholar
[6] Laggner F M, Wolfrum E, Cavedon M, Dunne M G, Birkenmeier G, Fischer R, Willensdorfer M, Aumayr F, Team E M, Team A U 2018 Nucl. Fusion 58 046008
Google Scholar
[7] Sugihara M, Mukhovatov V, Polevoi A, Shimada M 2003 Plasma Phys. Controlled Fusion 45 L55
Google Scholar
[8] Wilson H R, Connor J W, Field A R, Fielding S J, Miller R L, Lao L L, Ferron J R, Turnbull A D 1999 Phys. Plasmas 6 1925
Google Scholar
[9] Saarelma S, Austin M E, Knolker M, Marinoni A, Paz-Soldan C, Schmitz L, Snyder P B 2021 Plasma Phys Contr F 63 105006
Google Scholar
[10] Austin M E, Marinoni A, Walker M L, Brookman M W, deGrassie J S, Hyatt A W, McKee G R, Petty C C, Rhodes T L, Smith S P, Sung C, Thome K E, Turnbull A D 2019 Phys. Rev. Lett. 122 115001
Google Scholar
[11] Pochelon A, Angelino P, Behn R, Brunner S, Coda S, Kirneva N, Medvedev S Y, Reimerdes H, Rossel J, Sauter O, Villard L, WÁGner D, Bottino A, Camenen Y, Canal G P, Chattopadhyay P K, Duval B P, Fasoli A, Goodman T P, Jolliet S, Karpushov A, Labit B, Marinoni A, Moret J M, Pitzschke A, Porte L, Rancic M, Udintsev V S, the T C V T 2012 Plasma Fusion Res. 7 2502148
Google Scholar
[12] Medvedev S Y, Kikuchi M, Villard L, Takizuka T, Diamond P, Zushi H, Nagasaki K, Duan X, Wu Y, Ivanov A A, Martynov A A, Poshekhonov Y Y, Fasoli A, Sauter O 2015 Nucl. Fusion 55 063013
Google Scholar
[13] Merle A, Sauter O, Medvedev S Y 2017 Plasma Phys. Controlled Fusion 59 104001
Google Scholar
[14] Crotinger J A, LoDestro L, Pearlstein L D, Tarditi A, Casper T A, Hooper E B 1997 Corsica: A comprehensive simulation of toroidal magnetic-fusion devices. Final report to the LDRD Program (Livermore, CA: Lawrence Livermore National Laboratory)
[15] Dudson B D, Umansky M V, Xu X Q, Snyder P B, Wilson H R 2009 Comput. Phys. Commun. 180 1467
Google Scholar
[16] Xu X Q, Dudson B, Snyder P B, Umansky M V, Wilson H 2010 Phys. Rev. Lett. 105 175005
Google Scholar
[17] Kaw P K, Valeo E J, Rutherford P H 1979 Phys. Rev. Lett. 43 1398
Google Scholar
[18] Greene J M, Chance M S 1981 Nucl. Fusion 21 453
Google Scholar
[19] Sauter O, Angioni C, Lin-Liu Y R 1999 Phys. Plasmas 6 2834
Google Scholar
[20] Sauter O, Angioni C, Lin-Liu Y R 2002 Phys. Plasmas 9 5140
Google Scholar
[21] Li G Q, Xu X Q, Snyder P B, Turnbull A D, Xia T Y, Ma C H, Xi P W 2014 Phys. Plasmas 21 102511
Google Scholar
[22] Freidberg J P 2014 IDEAL MHD (Cambridge: Cambridge University Press)
[23] Xu X Q, Ma J F, Li G Q 2014 Phys Plasmas 21 120704
Google Scholar
[24] Xia T Y, Xu X Q, Xi P W 2013 Nucl. Fusion 53 073009
Google Scholar
[25] Xu X Q, Xia T Y, Yan N, Liu Z X, Kong D F, Diallo A, Groebner R J, Hubbard A E, Hughes J W 2016 Phys. Plasmas 23 055901
Google Scholar
[26] Gui B, Xu X Q, Myra J R, D'Ippolito D A 2014 Phys. Plasmas 21 112302
Google Scholar
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图 1 不同三角形变位型的磁面 (a)
$ \delta =0; $ (b)$ \delta =-0.3 $ , 蓝色实线代表$ {\psi }_{{\rm{n}}}=0.4 $ 到1的磁面, 归一化磁通间隔为$ {\psi }_{{\rm{n}}}=0.1 $ , 其中$ {\psi }_{{\rm{n}}}=\left(\psi -{\psi }_{{\rm{a}}{\rm{x}}{\rm{i}}{\rm{s}}}\right)/\left({\psi }_{{\rm{s}}{\rm{e}}{\rm{p}}}-{\psi }_{{\rm{a}}{\rm{x}}{\rm{i}}{\rm{s}}}\right) $ , 绿色区域为坏曲率区域($\nabla {B}^{2}\cdot \nabla P > 0$ ), 黄色区域为好曲率区($\nabla {B}^{2}\cdot \nabla P < 0$ )[9,18], 黑色虚线为中平面位置Figure 1. Comparison of magnetic surfaces with varied δ: (a)
$ \delta =0; $ (b)$ \delta =-0.3 $ . Blue lines represent the magnetic surfaces from$ {\psi }_{{\rm{n}}}=0.4 $ to 1 with an interval of 0.1,$ {\psi }_{{\rm{n}}}=\left(\psi -{\psi }_{{\rm{a}}{\rm{x}}{\rm{i}}{\rm{s}}}\right)/\left({\psi }_{{\rm{s}}{\rm{e}}{\rm{p}}}-{\psi }_{{\rm{a}}{\rm{x}}{\rm{i}}{\rm{s}}}\right) $ is the normalized radial coordinate. The green areas show the unfavorable curvature regions where$\nabla {B}^{2}\cdot \nabla P > 0$ and yellow areas show the favorable curvature regions where$\nabla {B}^{2}\cdot \nabla P < 0$ . Black dashed line shows the position of the midplane.
图 2 压强剖面
$ {P}_{0} $ (黑色实线)和三角形变分别为$ \delta =-0.3 $ (红色实线),$ \delta =0.0 $ (蓝色点线)的平行电流剖面$ {J}_{\parallel 0} $ Figure 2. The pressure
$ {P}_{0} $ (black solid line) and parallel current$ {J}_{\parallel 0} $ profiles for cases$ \delta =0.0 $ (blue dotted line) and$ \delta =0.3 $ (red solid line).
图 3 不同三角形变(
$ \delta =-0.3—0 $ )位型的P-B模线性增长率模谱Figure 3. Linear growth rates versus toroidal mode number for
$\delta =-0.3$ –0.
图 4 不同三角形变(
$ \delta =-0.3—0 $ )位型外中平面上的局域磁剪切$ {s}_{{\rm{l}}} $ 在径向上的变化Figure 4. Profiles of local shear
$ {s}_{l} $ at the outer midplane for$\delta =-0.3$ –0.
图 5 不同三角形变位型下ELM能量损失的对数值随时间的演化
Figure 5. Time evolution of the logarithm of ELM size for different triangularity cases.
图 6 (a)
$ \delta =0 $ 和(b$\delta =-0.2$ 位型下,$ t=193{{\rm{\tau }}}_{{\rm{A}}} $ 时扰动压强的环向平均在极向截面的分布. 弱场侧顶部和底部区域的黑色虚线框显示了比较区域, 黑色点线表示中平面位置Figure 6. Distribution of the toroidal-averaged pressure perturbation at the poloidal cross section at
$ t=193{\tau }_{{\rm{A}}} $ for cases (a)$ \delta =0 $ and (b)$ \delta =-0.2 $ . Black dashed frames at the top and bottom areas in the low field side show the regions for comparison. Black dotted line shows the position of the midplane.
图 7 (a1)—(a3)
$\delta =0$ 和(b1)—(b3)$ \delta =-0.2 $ 在$t=100,~ 200, ~300{\tau }_{{\rm{A}}}$ 时在外中平面上的压强扰动Figure 7. Pressure perturbation at
$ t=100, 200, 300{\tau }_{{\rm{A}}} $ at the outer midplane for cases: (a1)–(a3)$\delta =0;$ (b1)−(b3)$\delta =-0.2$ .
图 8 (a)
$\delta =0$ 和 (b)$ \delta =-0.2 $ 位型下的环向模式演化Figure 8. Modes evolution for cases: (a)
$ \delta =0; $ (b)$ \delta =-0.2 $ . -
[1] Wagner F, Becker G, Behringer K, Campbell D, Eberhagen A, Engelhardt W, Fussmann G, Gehre O, Gernhardt J, Gierke G v, Haas G, Huang M, Karger F, Keilhacker M, Klüber O, Kornherr M, Lackner K, Lisitano G, Lister G G, Mayer H M, Meisel D, Müller E R, Murmann H, Niedermeyer H, Poschenrieder W, Rapp H, Röhr H, Schneider F, Siller G, Speth E, Stäbler A, Steuer K H, Venus G, Vollmer O, Yü Z 1982 Phys. Rev. Lett. 49 1408
Google Scholar
[2] Zohm H 1996 Plasma Phys. Controlled Fusion 38 105
Google Scholar
[3] Snyder P B, Wilson H R, Ferron J R, Lao L L, Leonard A W, Osborne T H, Turnbull A D, Mossessian D, Murakami M, Xu X Q 2002 Phys. Plasmas 9 2037
Google Scholar
[4] Lao L L, Ferron J R, Miller R L, Osborne T H, Chan V S, Groebner R J, Jackson G L, La Haye R J, Strait E J, Taylor T S, Turnbull A D, Doyle E J, Lazarus E A, Murakami M, McKee G R, Rice B W, Zhang C, Chen L 1999 Nucl. Fusion 39 1785
Google Scholar
[5] Onjun T, Kritz A H, Bateman G, Parail V, Lonnroth J, Huysmans G 2004 Phys. Plasmas 11 3006
Google Scholar
[6] Laggner F M, Wolfrum E, Cavedon M, Dunne M G, Birkenmeier G, Fischer R, Willensdorfer M, Aumayr F, Team E M, Team A U 2018 Nucl. Fusion 58 046008
Google Scholar
[7] Sugihara M, Mukhovatov V, Polevoi A, Shimada M 2003 Plasma Phys. Controlled Fusion 45 L55
Google Scholar
[8] Wilson H R, Connor J W, Field A R, Fielding S J, Miller R L, Lao L L, Ferron J R, Turnbull A D 1999 Phys. Plasmas 6 1925
Google Scholar
[9] Saarelma S, Austin M E, Knolker M, Marinoni A, Paz-Soldan C, Schmitz L, Snyder P B 2021 Plasma Phys Contr F 63 105006
Google Scholar
[10] Austin M E, Marinoni A, Walker M L, Brookman M W, deGrassie J S, Hyatt A W, McKee G R, Petty C C, Rhodes T L, Smith S P, Sung C, Thome K E, Turnbull A D 2019 Phys. Rev. Lett. 122 115001
Google Scholar
[11] Pochelon A, Angelino P, Behn R, Brunner S, Coda S, Kirneva N, Medvedev S Y, Reimerdes H, Rossel J, Sauter O, Villard L, WÁGner D, Bottino A, Camenen Y, Canal G P, Chattopadhyay P K, Duval B P, Fasoli A, Goodman T P, Jolliet S, Karpushov A, Labit B, Marinoni A, Moret J M, Pitzschke A, Porte L, Rancic M, Udintsev V S, the T C V T 2012 Plasma Fusion Res. 7 2502148
Google Scholar
[12] Medvedev S Y, Kikuchi M, Villard L, Takizuka T, Diamond P, Zushi H, Nagasaki K, Duan X, Wu Y, Ivanov A A, Martynov A A, Poshekhonov Y Y, Fasoli A, Sauter O 2015 Nucl. Fusion 55 063013
Google Scholar
[13] Merle A, Sauter O, Medvedev S Y 2017 Plasma Phys. Controlled Fusion 59 104001
Google Scholar
[14] Crotinger J A, LoDestro L, Pearlstein L D, Tarditi A, Casper T A, Hooper E B 1997 Corsica: A comprehensive simulation of toroidal magnetic-fusion devices. Final report to the LDRD Program (Livermore, CA: Lawrence Livermore National Laboratory)
[15] Dudson B D, Umansky M V, Xu X Q, Snyder P B, Wilson H R 2009 Comput. Phys. Commun. 180 1467
Google Scholar
[16] Xu X Q, Dudson B, Snyder P B, Umansky M V, Wilson H 2010 Phys. Rev. Lett. 105 175005
Google Scholar
[17] Kaw P K, Valeo E J, Rutherford P H 1979 Phys. Rev. Lett. 43 1398
Google Scholar
[18] Greene J M, Chance M S 1981 Nucl. Fusion 21 453
Google Scholar
[19] Sauter O, Angioni C, Lin-Liu Y R 1999 Phys. Plasmas 6 2834
Google Scholar
[20] Sauter O, Angioni C, Lin-Liu Y R 2002 Phys. Plasmas 9 5140
Google Scholar
[21] Li G Q, Xu X Q, Snyder P B, Turnbull A D, Xia T Y, Ma C H, Xi P W 2014 Phys. Plasmas 21 102511
Google Scholar
[22] Freidberg J P 2014 IDEAL MHD (Cambridge: Cambridge University Press)
[23] Xu X Q, Ma J F, Li G Q 2014 Phys Plasmas 21 120704
Google Scholar
[24] Xia T Y, Xu X Q, Xi P W 2013 Nucl. Fusion 53 073009
Google Scholar
[25] Xu X Q, Xia T Y, Yan N, Liu Z X, Kong D F, Diallo A, Groebner R J, Hubbard A E, Hughes J W 2016 Phys. Plasmas 23 055901
Google Scholar
[26] Gui B, Xu X Q, Myra J R, D'Ippolito D A 2014 Phys. Plasmas 21 112302
Google Scholar
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