- Email: [email protected]
Plasma shield dynamics and target erosion in disruption simulation experiments
Fusion Engineering and Design 49 – 50 (2000) 389 – 395 www.elsevier.com/locate/fusengdes
Plasma shield dynamics and target erosion in disruption simulation experiments H. Wu¨rz a,*, S. Pestchanyi b, F. Kappler a a
Forschungszentrum Karlsruhe, IHM, P.O. Box 3640, D-76021 Karlsruhe, Germany b Troitsk Institute for Inno6ation and Fusion Research, 142092 Troitsk, Russia
Abstract Dynamics of carbon plasma shields and graphite target erosion experimentally studied at the recently upgraded plasma gun facility MK-200 UG at TRINITI Troitsk are compared with numerical results from FOREV-2 which allows a 2-D modeling of hot plasma target interaction. It is shown that turbulent processes are absent in the experimental plasma shields. Anomalous lateral losses of plasma mass are not occurring and plasma shield dynamics can be described by the multidimensional radiation magnetohydrodynamics (R-MHD) equations together with a consistent solution of the multidimensional magnetic field equations with classical magnetic field diffusion coefficient. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Plasma shield stability; Electron temperature; Electron density; Thomson scattering; Interferometry; Plasma; Separatrix
1. Introduction As is well known, a plasma shield from vaporized target material formed in front of the divertor plates during international thermonuclear experimental reactor (ITER) plasma disruptions and ELMs reduces the target heat load considerably because it converts the energy of the hot plasma into ionization hydrodynamic motion, and radiation of the plasma shield [1]. After formation of the plasma shield the energy finally arriving at the target surface is by direct heating by the hot particles not fully stopped in the plasma shield, by * Corresponding author. Tel.: + 49-72-47823619; fax: +4972-47824874. E-mail address: [email protected] (H. Wu¨rz).
radiation and by thermal energy transfer. A 2-D consistent modeling is required because of the rather complicated divertor geometry with possible tilting of targets in the poloidal plane and the asymmetrical power density profile of the impacting hot plasma across the SOL, because of the rather complex dynamics of the plasma shield [2], and because of re-radiation from the plasma shield to the side walls and other nearby components with possible damage to these components [3]. The physical properties and the long-term stability of plasma shields in the tokamak magnetic field have to be known for a reliable prediction of tokamak divertor erosion [2,4] but cannot be determined in existing tokamaks. Therefore experiments in disruption simulation facilities which allow to produce tokamak typical plasma
0920-3796/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 3 7 9 6 ( 0 0 ) 0 0 2 4 1 - 6
390
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
shields are necessary together with a 2-D numerical modeling of the plasma shield dynamics. This paper focuses on a discussion of the magnetohydrodynamics (MHD) of plasma shields and its consequences for target erosion. Most recently obtained experimental results at the upgraded plasma gun facility MK-200 UG [5] at TRINITI Troitsk and numerical results from the radiation magnetohydrodynamics (R-MHD) code FOREV2 [6] developed for a 2-D modeling of hot plasma target interaction including plasma shield dynamics and calculation of erosion are compared.
2. Brief theoretical background The system of equations used in FOREV-2 to solve the 2-D R-MHD problem of hot plasma target interaction with an external magnetic field is discussed in [6]. In the simulation experiments, the targets are either perpendicular to the guiding magnetic field given initially as Bb 0 =Bb (t =0)= (Bx0, 0, 0) or tilted with respect to Bx0.
2.1. 2 -D magnetohydrodynamics (MHD) The time evolution of Bb (t) inside of the plasma shield is described by magnetic field equations for the two components Bx and By and by appropriate boundary conditions at the target surface. The guiding magnetic field lines behave like elastic bands, which can be deformed by forces acting on them. The pressure of the plasma shield provides such a force. As a consequence the guiding magnetic field can be pushed away by the evolving plasma shield. This diamagnetic effect results in depletion of Bx in the center (at the position of the peak power density of the impacting hot plasma, the separatrix position). If the magnetic field is frozen in at the target the field lines are bent near the target and a y component (By ) of the magnetic field arises. The y component of the motion equation writes as
(ruy 1 ( B2 + 9a ruyu = (Bb 9a )By − P+ (t 2m0 m0 (y
(1)
with P, u , r the plasma shield density, velocity and pressure, Bb the magnetic field and 9a the gradient operator with 9a i
( ( +j (x (y
and i and j are the unit vectors in the x and y direction. The x and y dependence of By causes movement of cold plasma to the separatrix according to the first expression at the RHS of Eq. (1), whereas a plasma pressure profile P(y) peaked at the separatrix causes plasma movement away from the separatrix (second term at the RHS). In case of diamagnetism and frozen in condition at the target surface the magnetic force term (first term at RHS) dominates. If there is free movement of the magnetic field at the target surface there is no By component. In this case, the plasma shield is experiencing lateral forces due to the plasma pressure profile. This profile depends on the power density profile of the impacting hot plasma and on the momentum transfer from the hot plasma ions. The appropriate boundary condition for the evolution of the x component of the magnetic field at the target surface is important. In FOREV-2 the following boundary condition is used BBC (t+Dt)=BBC (t)+
Dt DBx (t) Dt +t
(2)
with DBx = Bx − BBC and Bx the magnetic field in the plasma shield close to the target, and t=L 2/ xm with L the target thickness (assumed as 1 cm) and xm the magnetic field diffusion coefficient of the target. The relaxation time t describes the time needed for adjustment of the magnetic field at the target surface in case of a diamagnetic effect in the plasma shield. Frozen in is described by values of the magnetic field diffusion coefficient xm below 105 cm2 s − 1, free movement is described by xm values typically above 106 cm2 s − 1. xm values in between describe a combination of both situations.
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
2.2. Turbulence in the plasma shield Turbulent processes in the plasma shield if existing result in anomalous transport of target plasma across magnetic field lines. In this case an increase in lateral losses of plasma mass from the plasma shield would occur resulting in plasma shield depletion reduced plasma shield stability and increased target erosion. Checking of occurrence of turbulent processes in the experimental plasma shields is done by comparing experimental results on evolution of plasma temperature and 2-D electron density distribution in plasma shields with numerical results from FOREV-2 based on a consistent 2-D MHD model using classical diffusion for the magnetic
391
field diffusion coefficient in the diffusive part of the 2-D magnetic field equations [6].
2.3. Target heat loads Angular dependent 2-D radiation transport with an improved and generalized forward reverse scheme is used to calculate radiative heat loads at the target. This time efficient scheme allows using multi frequency opacities. Calculations were performed with 69 group Planck opacities taking the most prominent lines for Li-, He- and H-like ions into account [6]. Thermal energy transfer from the plasma shield to the target is calculated via the electron heat conduction flux qel according to qel(x0, t)=
kDT(x, t) Dx cos b(x, t)
(3)
with k the electron heat conductivity coefficient, DT the temperature difference between the target surface and the plasma temperature in the first mesh of width Dx and b the inclination angle of the magnetic field lines in the poloidal plane at the target surface with b(x, t)= arc tan By (x, t)/Bx (x, t)+ a with a the target inclination angle. The heat conduction flux qel depends on the electron temperature increase at the first mesh and thus on the size of the used mesh. However, the limited accuracy of qel influences the plasma shield dynamics and the erosion only weakly because of a self regulating process which increases the radiative target heat load in case of low qel and vice versa.
3. Numerical results for perpendicular graphite targets
Fig. 1. Calculated plasma flow pattern and electron density distributions in a carbon plasma shield at 30 ms for different values of the magnetic field diffusion coefficient xm of the target.
The influence of different boundary conditions for the magnetic field at the target surface on plasma shield dynamics was investigated. The impacting hot plasma (ions of energy of 1 keV and Maxwellian distributed electrons of temperature of 0.3 keV) has a Gaussian power density profile of half width of 6 cm with peak power densities of up to 35 MW cm − 2 at the separatrix. The guiding magnetic field Bx0 is 2 T. Fig. 1 shows the plasma
392
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
Fig. 2. Calculated erosion profiles for a perpendicular graphite target for three different values of the magnetic field diffusion coefficient xm of the target.
Fig. 3. Calculated electron temperature profiles along the separatrix for the two different plasma flow regimes with xm = 2 ×104 and 106 cm2 s − 1.
flow pattern (arrows indicate the plasma flow Nc6 with Nc the carbon density and 6 the plasma velocity) and the plasma density distribution in the carbon plasma shield at 30 ms for free movement (xm =106 cm2 s − 1) and for the frozen in situation (xm =2 × 104 cm2 s − 1). Additionally shown are the pressure profiles of the plasma layer close to the target. Pressure profiles peaked at the separatrix drive an outward movement of the cold dense plasma layer close to the target. This cold plasma (temperature around 1 eV) is weakly confined by the magnetic field. Thus for
free movement there is a rather distinct flow of cold plasma away from the separatrix resulting in depletion of the plasma shield at the center. The flow direction remains unchanged for the whole duration of the hot plasma impact. For the frozen in situation there is a flow of cold plasma to the separatrix and along the separatrix upwards. At later times flow reversal occurs in the cold plasma layer close to the target. Fig. 2 shows erosion profiles for three different values of the magnetic field diffusion coefficient xm at the target surface. For free movement of the magnetic field (xm = 106 cm2 s − 1) because of plasma flow away from the separatrix the erosion profile is strongly peaked and the highest erosion value is achieved with a peak erosion value of 0.8 mm. For the frozen in situation (xm = 2× 104 cm2 s − 1) with its plasma flow to the separatrix the erosion in the center is about 0.3 mm and the erosion profile becomes more wide. For intermediate xm values (xm = 1.5× 105 cm2 s − 1) the plasma flow to the separatrix in comparison with the frozen in case is reduced but is still existing. Thus shielding at the separatrix is reduced and a peak erosion of 0.45 mm is obtained. The measured peak erosion value is 0.4 mm determined as average value after 15 shots. A xm value of 1.5× 105 cm2 s − 1 fits the experimental situation best. Fig. 3 shows plasma temperature profiles along the separatrix at two different times for the two different plasma flow regimes shown in Fig. 1. The plasma temperature profiles are significantly different for both cases. In case of plasma shield depletion at the separatrix (lateral losses of plasma mass) only a rather thin or no dense cold plasma layer is formed and the plasma temperature remains rather high along the separatrix. Differences in measured and calculated temperature profiles allow to identify abnormal lateral losses of plasma and thus indicate the existence of turbulent processes.
4. Plasma shield properties for perpendicular targets Local measurements of electron temperature and density were performed in a carbon plasma
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
shield by Thomson laser scattering (TS) along the separatrix [7]. A ruby laser beam (l= 0.6943 mm) was focused onto the plasma axis. Backward scat-
Fig. 4. Calculated plasma flow pattern and electron density distributions in a carbon plasma shield at 24 and 38 ms.
Fig. 5. Calculated profiles of electron density in a carbon plasma shield along the IF lines of sight at 30 ms and at 40 ms at different distances from the target as indicated in the figure.
393
tered light (at 160°) was detected by a set of photomultipliers. Optical interferometry (IF) was used for a measurement of line averaged electron density distributions along the target surface at different target distances [7]. The interferometer scheme consists of a Mach–Zehnder interferometer, a continuous gas laser (l= 0.51 mm), a highspeed camera as a recorder and associated optics. The scheme allows to study the evolution of the electron density in the plasma shield with 2-D spatial resolution of 1 mm and temporal resolution of 0.5 ms. FOREV-2 calculations were performed for a Gaussian power density profile and a peak power density of 35 MW cm − 2 at the separatrix. Fig. 4a and b show the calculated plasma flow pattern and the electron density distribution in the carbon plasma shield at 24 and 38 ms for classical diffusion. A value of xm of 1.5 × 105 cm2 s − 1 was used for the target. At early times and close to the target there is a flow of dense cold plasma towards the separatrix then along the separatrix up-stream and laterally outward thus forming lateral plasma jets outside of the separatrix. At 30 ms flow reversal occurs in the dense cold plasma close to the target and at 38 ms the plasma jets are rather well developed (Fig. 4b). At 24 ms the dense plasma layer at the separatrix has grown to a thickness of 2.1 cm whereas the lateral plasma jets extend over a length of about 6.5 cm. At 38 ms the dense plasma layer has grown to a thickness of 9 cm. Also indicated in Fig. 4a are lines of sight for the IF measurements and the Gaussian power density profile of the incoming hot plasma. Fig. 5 shows calculated electron density distributions in the carbon plasma shield (in y-direction) along the IF lines of sight at 30 and 40 ms for different distances from the target. The lateral plasma jets are clearly to be seen as density peaks outside of the central axis evolving at target distances larger than 2 cm. Fig. 6 shows a comparison of calculated and measured line averaged electron density distributions. The calculated values were obtained by averaging over the density distributions shown in Fig. 5. There is agreement between measurement and calculation at 10 ms as seen from Fig. 6. At 30 and 40 ms the calculated values at distances larger than 2 cm from the
394
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
Fig. 6. Comparison of measured and calculated line averaged electron density distributions in a carbon plasma shield at different times.
Fig. 7. Comparison of measured and calculated local electron temperature profiles along the separatrix.
target are also in good agreement with the measured values. Fig. 7 shows a comparison of calculated and measured local electron temperature profiles along the separatrix. The temperature increase at 30 ms occurs at a target distance of 1.5 cm and at 50 ms at 8 cm thus indicating the extension of the dense cold plasma along the separatrix. Fig. 8 shows a comparison of measured TS and calculated evolution of the local electron density and plasma temperature at the separatrix at 3 and 10 cm distance from the target. The sudden drop in
plasma temperature at 30 ms corresponds to a decrease of power density of the incoming hot plasma and to the arrival of the denser part of the plasma shield at the measurement position. At 3 cm distance from the target the density peak arrives about 6 ms earlier in the calculation than in the measurement and the calculated amplitude is about 60% larger. At 10 cm the TS values of electron density show a continuous increase with time whereas the calculation again shows the arrival of a dense plasma at 45 ms. Between 20 and 40 ms the calculated values are a factor of 2 smaller than the TS data but are in agreement with the IF data which indicate at distances above 7 cm density values are below 1017 cm − 3. The TS value on electron density at 40 ms and at a target distance of 2 cm with 2.5× 1017 cm − 3 is in agreement with the calculated value shown in Fig. 5. Unfortunately at 30 ms and at target distances below 3 cm TS data on electron densities are not available. Thus the IF data at 30 ms (Fig. 6) close to the target can not be checked. A comparison of the TS and IF data on electron density confirms the existence of lateral jets. At 30 ms and 3 cm distance the TS value is below 1017 cm − 3, the IF value is about 4 × 1017 cm − 3. Thus the density valley at the center as shown in the calculated density distributions along the target surface (see Fig. 5) is confirmed by the measurements. The calculated value at 3 cm distance at the separatrix is below 1017 cm − 3 and thus in good agreement with the TS value.
5. Conclusions The rather good agreement between the numerical results and the TS data allows to conclude that turbulent processes in the dense cold plasma layer which reduce the shielding efficiency are absent. Would they occur then there would exist a depletion of plasma in the center along the separatrix resulting in considerably higher plasma temperatures close to the target and larger peak erosion. A comparison of TS and IF data on electron density confirms the existence of the numerically predicted lateral plasma jets. The good agreement between numerical results and the IF
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
395
Fig. 8. Comparison of measured and calculated evolution of electron density and plasma temperature in a carbon plasma shield at the separatrix at 3 and 10 cm distance from the target.
data at larger distances from the target demonstrate that the expansion of the lateral plasma jets is correctly described by the 2-D modeling. Plasma shield dynamics given for the whole time duration of the pulsed heat load which is up to 50 ms can be adequately described by the consistent 2-D R-MHD model used in FOREV-2 with appropriate boundary conditions for the magnetic field at the target surface and with classical diffusion for the magnetic field diffusion coefficient. The plasma shields in the simulation experiments are tokamak typical. Therefore turbulent processes are also not occurring in the plasma shields produced in hot plasma divertor interactions during tokamak plasma disruptions. The long-term plasma shield dynamics and its stability under tokamak conditions and divertor erosion can thus be adequately described by FOREV-2.
Acknowledgements The authors thank Valeri Safronov and his team at TRINITI Troitsk for the outstanding results from simulation experiments performed in the frame of a TRINITI FZK cooperation supported additionally by the Russian German WTZ cooperation agreement under WTZ 524-96. Despite rather unfavorable conditions in those hot
plasma target experiments this team was able to apply Thomson scattering for measurement of local electron temperature and density in plasma shields.
References [1] H. Wu¨rz, et al., Plasma/surface interaction in ITER tokamak disruption simulation experiments, Fusion Technol. 32 (1997) 45 – 74. [2] S. Pestchanyi, et al., Results from 2-D radiation magnetohydrodynamic calculations of the interaction of ITER disruptive plasma with the slot divertor, Proceedings of the 24th European Conference on Controlled Fusion and Plasma Physics, Berchtesgaden, June 9 – 13, 1997, vol. 21A, Part III, 1997, pp. 277 – 281. [3] H. Wu¨rz, et al., Radiation in plasma target interaction events typical for ITER tokamak disruptions, Fusion Technol. 30 (1996) 739 – 744. [4] H. Wu¨rz, et al., A consistent 2-D analysis of erosion of the ITER slot divertor, Fusion Technol. 1 (1998) 271 – 274. [5] N.I. Arkhipov, et al., Study of structure and dynamics of shielding layer for inclined incidence of plasma stream at the MK-200 facilities, J. Nucl. Mater. 233 – 237 (1996) 767 – 770. [6] H. Wu¨rz, et al., Hot plasma interaction and quantification of erosion of the ITER slot divertor during disruptions and ELMs., Forschungszentrum Karlsruhe Report FZKA 6198, March 1999. [7] N.I. Arkhipov, et al., Study of plasma target interactions with plasma streams of power density of 40 MW cm − 2, Fusion Technol. 1 (1996) 507 – 510.
Plasma shield dynamics and target erosion in disruption simulation experiments H. Wu¨rz a,*, S. Pestchanyi b, F. Kappler a a
Forschungszentrum Karlsruhe, IHM, P.O. Box 3640, D-76021 Karlsruhe, Germany b Troitsk Institute for Inno6ation and Fusion Research, 142092 Troitsk, Russia
Abstract Dynamics of carbon plasma shields and graphite target erosion experimentally studied at the recently upgraded plasma gun facility MK-200 UG at TRINITI Troitsk are compared with numerical results from FOREV-2 which allows a 2-D modeling of hot plasma target interaction. It is shown that turbulent processes are absent in the experimental plasma shields. Anomalous lateral losses of plasma mass are not occurring and plasma shield dynamics can be described by the multidimensional radiation magnetohydrodynamics (R-MHD) equations together with a consistent solution of the multidimensional magnetic field equations with classical magnetic field diffusion coefficient. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Plasma shield stability; Electron temperature; Electron density; Thomson scattering; Interferometry; Plasma; Separatrix
1. Introduction As is well known, a plasma shield from vaporized target material formed in front of the divertor plates during international thermonuclear experimental reactor (ITER) plasma disruptions and ELMs reduces the target heat load considerably because it converts the energy of the hot plasma into ionization hydrodynamic motion, and radiation of the plasma shield [1]. After formation of the plasma shield the energy finally arriving at the target surface is by direct heating by the hot particles not fully stopped in the plasma shield, by * Corresponding author. Tel.: + 49-72-47823619; fax: +4972-47824874. E-mail address: [email protected] (H. Wu¨rz).
radiation and by thermal energy transfer. A 2-D consistent modeling is required because of the rather complicated divertor geometry with possible tilting of targets in the poloidal plane and the asymmetrical power density profile of the impacting hot plasma across the SOL, because of the rather complex dynamics of the plasma shield [2], and because of re-radiation from the plasma shield to the side walls and other nearby components with possible damage to these components [3]. The physical properties and the long-term stability of plasma shields in the tokamak magnetic field have to be known for a reliable prediction of tokamak divertor erosion [2,4] but cannot be determined in existing tokamaks. Therefore experiments in disruption simulation facilities which allow to produce tokamak typical plasma
0920-3796/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 3 7 9 6 ( 0 0 ) 0 0 2 4 1 - 6
390
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
shields are necessary together with a 2-D numerical modeling of the plasma shield dynamics. This paper focuses on a discussion of the magnetohydrodynamics (MHD) of plasma shields and its consequences for target erosion. Most recently obtained experimental results at the upgraded plasma gun facility MK-200 UG [5] at TRINITI Troitsk and numerical results from the radiation magnetohydrodynamics (R-MHD) code FOREV2 [6] developed for a 2-D modeling of hot plasma target interaction including plasma shield dynamics and calculation of erosion are compared.
2. Brief theoretical background The system of equations used in FOREV-2 to solve the 2-D R-MHD problem of hot plasma target interaction with an external magnetic field is discussed in [6]. In the simulation experiments, the targets are either perpendicular to the guiding magnetic field given initially as Bb 0 =Bb (t =0)= (Bx0, 0, 0) or tilted with respect to Bx0.
2.1. 2 -D magnetohydrodynamics (MHD) The time evolution of Bb (t) inside of the plasma shield is described by magnetic field equations for the two components Bx and By and by appropriate boundary conditions at the target surface. The guiding magnetic field lines behave like elastic bands, which can be deformed by forces acting on them. The pressure of the plasma shield provides such a force. As a consequence the guiding magnetic field can be pushed away by the evolving plasma shield. This diamagnetic effect results in depletion of Bx in the center (at the position of the peak power density of the impacting hot plasma, the separatrix position). If the magnetic field is frozen in at the target the field lines are bent near the target and a y component (By ) of the magnetic field arises. The y component of the motion equation writes as
(ruy 1 ( B2 + 9a ruyu = (Bb 9a )By − P+ (t 2m0 m0 (y
(1)
with P, u , r the plasma shield density, velocity and pressure, Bb the magnetic field and 9a the gradient operator with 9a i
( ( +j (x (y
and i and j are the unit vectors in the x and y direction. The x and y dependence of By causes movement of cold plasma to the separatrix according to the first expression at the RHS of Eq. (1), whereas a plasma pressure profile P(y) peaked at the separatrix causes plasma movement away from the separatrix (second term at the RHS). In case of diamagnetism and frozen in condition at the target surface the magnetic force term (first term at RHS) dominates. If there is free movement of the magnetic field at the target surface there is no By component. In this case, the plasma shield is experiencing lateral forces due to the plasma pressure profile. This profile depends on the power density profile of the impacting hot plasma and on the momentum transfer from the hot plasma ions. The appropriate boundary condition for the evolution of the x component of the magnetic field at the target surface is important. In FOREV-2 the following boundary condition is used BBC (t+Dt)=BBC (t)+
Dt DBx (t) Dt +t
(2)
with DBx = Bx − BBC and Bx the magnetic field in the plasma shield close to the target, and t=L 2/ xm with L the target thickness (assumed as 1 cm) and xm the magnetic field diffusion coefficient of the target. The relaxation time t describes the time needed for adjustment of the magnetic field at the target surface in case of a diamagnetic effect in the plasma shield. Frozen in is described by values of the magnetic field diffusion coefficient xm below 105 cm2 s − 1, free movement is described by xm values typically above 106 cm2 s − 1. xm values in between describe a combination of both situations.
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
2.2. Turbulence in the plasma shield Turbulent processes in the plasma shield if existing result in anomalous transport of target plasma across magnetic field lines. In this case an increase in lateral losses of plasma mass from the plasma shield would occur resulting in plasma shield depletion reduced plasma shield stability and increased target erosion. Checking of occurrence of turbulent processes in the experimental plasma shields is done by comparing experimental results on evolution of plasma temperature and 2-D electron density distribution in plasma shields with numerical results from FOREV-2 based on a consistent 2-D MHD model using classical diffusion for the magnetic
391
field diffusion coefficient in the diffusive part of the 2-D magnetic field equations [6].
2.3. Target heat loads Angular dependent 2-D radiation transport with an improved and generalized forward reverse scheme is used to calculate radiative heat loads at the target. This time efficient scheme allows using multi frequency opacities. Calculations were performed with 69 group Planck opacities taking the most prominent lines for Li-, He- and H-like ions into account [6]. Thermal energy transfer from the plasma shield to the target is calculated via the electron heat conduction flux qel according to qel(x0, t)=
kDT(x, t) Dx cos b(x, t)
(3)
with k the electron heat conductivity coefficient, DT the temperature difference between the target surface and the plasma temperature in the first mesh of width Dx and b the inclination angle of the magnetic field lines in the poloidal plane at the target surface with b(x, t)= arc tan By (x, t)/Bx (x, t)+ a with a the target inclination angle. The heat conduction flux qel depends on the electron temperature increase at the first mesh and thus on the size of the used mesh. However, the limited accuracy of qel influences the plasma shield dynamics and the erosion only weakly because of a self regulating process which increases the radiative target heat load in case of low qel and vice versa.
3. Numerical results for perpendicular graphite targets
Fig. 1. Calculated plasma flow pattern and electron density distributions in a carbon plasma shield at 30 ms for different values of the magnetic field diffusion coefficient xm of the target.
The influence of different boundary conditions for the magnetic field at the target surface on plasma shield dynamics was investigated. The impacting hot plasma (ions of energy of 1 keV and Maxwellian distributed electrons of temperature of 0.3 keV) has a Gaussian power density profile of half width of 6 cm with peak power densities of up to 35 MW cm − 2 at the separatrix. The guiding magnetic field Bx0 is 2 T. Fig. 1 shows the plasma
392
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
Fig. 2. Calculated erosion profiles for a perpendicular graphite target for three different values of the magnetic field diffusion coefficient xm of the target.
Fig. 3. Calculated electron temperature profiles along the separatrix for the two different plasma flow regimes with xm = 2 ×104 and 106 cm2 s − 1.
flow pattern (arrows indicate the plasma flow Nc6 with Nc the carbon density and 6 the plasma velocity) and the plasma density distribution in the carbon plasma shield at 30 ms for free movement (xm =106 cm2 s − 1) and for the frozen in situation (xm =2 × 104 cm2 s − 1). Additionally shown are the pressure profiles of the plasma layer close to the target. Pressure profiles peaked at the separatrix drive an outward movement of the cold dense plasma layer close to the target. This cold plasma (temperature around 1 eV) is weakly confined by the magnetic field. Thus for
free movement there is a rather distinct flow of cold plasma away from the separatrix resulting in depletion of the plasma shield at the center. The flow direction remains unchanged for the whole duration of the hot plasma impact. For the frozen in situation there is a flow of cold plasma to the separatrix and along the separatrix upwards. At later times flow reversal occurs in the cold plasma layer close to the target. Fig. 2 shows erosion profiles for three different values of the magnetic field diffusion coefficient xm at the target surface. For free movement of the magnetic field (xm = 106 cm2 s − 1) because of plasma flow away from the separatrix the erosion profile is strongly peaked and the highest erosion value is achieved with a peak erosion value of 0.8 mm. For the frozen in situation (xm = 2× 104 cm2 s − 1) with its plasma flow to the separatrix the erosion in the center is about 0.3 mm and the erosion profile becomes more wide. For intermediate xm values (xm = 1.5× 105 cm2 s − 1) the plasma flow to the separatrix in comparison with the frozen in case is reduced but is still existing. Thus shielding at the separatrix is reduced and a peak erosion of 0.45 mm is obtained. The measured peak erosion value is 0.4 mm determined as average value after 15 shots. A xm value of 1.5× 105 cm2 s − 1 fits the experimental situation best. Fig. 3 shows plasma temperature profiles along the separatrix at two different times for the two different plasma flow regimes shown in Fig. 1. The plasma temperature profiles are significantly different for both cases. In case of plasma shield depletion at the separatrix (lateral losses of plasma mass) only a rather thin or no dense cold plasma layer is formed and the plasma temperature remains rather high along the separatrix. Differences in measured and calculated temperature profiles allow to identify abnormal lateral losses of plasma and thus indicate the existence of turbulent processes.
4. Plasma shield properties for perpendicular targets Local measurements of electron temperature and density were performed in a carbon plasma
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
shield by Thomson laser scattering (TS) along the separatrix [7]. A ruby laser beam (l= 0.6943 mm) was focused onto the plasma axis. Backward scat-
Fig. 4. Calculated plasma flow pattern and electron density distributions in a carbon plasma shield at 24 and 38 ms.
Fig. 5. Calculated profiles of electron density in a carbon plasma shield along the IF lines of sight at 30 ms and at 40 ms at different distances from the target as indicated in the figure.
393
tered light (at 160°) was detected by a set of photomultipliers. Optical interferometry (IF) was used for a measurement of line averaged electron density distributions along the target surface at different target distances [7]. The interferometer scheme consists of a Mach–Zehnder interferometer, a continuous gas laser (l= 0.51 mm), a highspeed camera as a recorder and associated optics. The scheme allows to study the evolution of the electron density in the plasma shield with 2-D spatial resolution of 1 mm and temporal resolution of 0.5 ms. FOREV-2 calculations were performed for a Gaussian power density profile and a peak power density of 35 MW cm − 2 at the separatrix. Fig. 4a and b show the calculated plasma flow pattern and the electron density distribution in the carbon plasma shield at 24 and 38 ms for classical diffusion. A value of xm of 1.5 × 105 cm2 s − 1 was used for the target. At early times and close to the target there is a flow of dense cold plasma towards the separatrix then along the separatrix up-stream and laterally outward thus forming lateral plasma jets outside of the separatrix. At 30 ms flow reversal occurs in the dense cold plasma close to the target and at 38 ms the plasma jets are rather well developed (Fig. 4b). At 24 ms the dense plasma layer at the separatrix has grown to a thickness of 2.1 cm whereas the lateral plasma jets extend over a length of about 6.5 cm. At 38 ms the dense plasma layer has grown to a thickness of 9 cm. Also indicated in Fig. 4a are lines of sight for the IF measurements and the Gaussian power density profile of the incoming hot plasma. Fig. 5 shows calculated electron density distributions in the carbon plasma shield (in y-direction) along the IF lines of sight at 30 and 40 ms for different distances from the target. The lateral plasma jets are clearly to be seen as density peaks outside of the central axis evolving at target distances larger than 2 cm. Fig. 6 shows a comparison of calculated and measured line averaged electron density distributions. The calculated values were obtained by averaging over the density distributions shown in Fig. 5. There is agreement between measurement and calculation at 10 ms as seen from Fig. 6. At 30 and 40 ms the calculated values at distances larger than 2 cm from the
394
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
Fig. 6. Comparison of measured and calculated line averaged electron density distributions in a carbon plasma shield at different times.
Fig. 7. Comparison of measured and calculated local electron temperature profiles along the separatrix.
target are also in good agreement with the measured values. Fig. 7 shows a comparison of calculated and measured local electron temperature profiles along the separatrix. The temperature increase at 30 ms occurs at a target distance of 1.5 cm and at 50 ms at 8 cm thus indicating the extension of the dense cold plasma along the separatrix. Fig. 8 shows a comparison of measured TS and calculated evolution of the local electron density and plasma temperature at the separatrix at 3 and 10 cm distance from the target. The sudden drop in
plasma temperature at 30 ms corresponds to a decrease of power density of the incoming hot plasma and to the arrival of the denser part of the plasma shield at the measurement position. At 3 cm distance from the target the density peak arrives about 6 ms earlier in the calculation than in the measurement and the calculated amplitude is about 60% larger. At 10 cm the TS values of electron density show a continuous increase with time whereas the calculation again shows the arrival of a dense plasma at 45 ms. Between 20 and 40 ms the calculated values are a factor of 2 smaller than the TS data but are in agreement with the IF data which indicate at distances above 7 cm density values are below 1017 cm − 3. The TS value on electron density at 40 ms and at a target distance of 2 cm with 2.5× 1017 cm − 3 is in agreement with the calculated value shown in Fig. 5. Unfortunately at 30 ms and at target distances below 3 cm TS data on electron densities are not available. Thus the IF data at 30 ms (Fig. 6) close to the target can not be checked. A comparison of the TS and IF data on electron density confirms the existence of lateral jets. At 30 ms and 3 cm distance the TS value is below 1017 cm − 3, the IF value is about 4 × 1017 cm − 3. Thus the density valley at the center as shown in the calculated density distributions along the target surface (see Fig. 5) is confirmed by the measurements. The calculated value at 3 cm distance at the separatrix is below 1017 cm − 3 and thus in good agreement with the TS value.
5. Conclusions The rather good agreement between the numerical results and the TS data allows to conclude that turbulent processes in the dense cold plasma layer which reduce the shielding efficiency are absent. Would they occur then there would exist a depletion of plasma in the center along the separatrix resulting in considerably higher plasma temperatures close to the target and larger peak erosion. A comparison of TS and IF data on electron density confirms the existence of the numerically predicted lateral plasma jets. The good agreement between numerical results and the IF
H. Wu¨rz et al. / Fusion Engineering and Design 49–50 (2000) 389–395
395
Fig. 8. Comparison of measured and calculated evolution of electron density and plasma temperature in a carbon plasma shield at the separatrix at 3 and 10 cm distance from the target.
data at larger distances from the target demonstrate that the expansion of the lateral plasma jets is correctly described by the 2-D modeling. Plasma shield dynamics given for the whole time duration of the pulsed heat load which is up to 50 ms can be adequately described by the consistent 2-D R-MHD model used in FOREV-2 with appropriate boundary conditions for the magnetic field at the target surface and with classical diffusion for the magnetic field diffusion coefficient. The plasma shields in the simulation experiments are tokamak typical. Therefore turbulent processes are also not occurring in the plasma shields produced in hot plasma divertor interactions during tokamak plasma disruptions. The long-term plasma shield dynamics and its stability under tokamak conditions and divertor erosion can thus be adequately described by FOREV-2.
Acknowledgements The authors thank Valeri Safronov and his team at TRINITI Troitsk for the outstanding results from simulation experiments performed in the frame of a TRINITI FZK cooperation supported additionally by the Russian German WTZ cooperation agreement under WTZ 524-96. Despite rather unfavorable conditions in those hot
plasma target experiments this team was able to apply Thomson scattering for measurement of local electron temperature and density in plasma shields.
References [1] H. Wu¨rz, et al., Plasma/surface interaction in ITER tokamak disruption simulation experiments, Fusion Technol. 32 (1997) 45 – 74. [2] S. Pestchanyi, et al., Results from 2-D radiation magnetohydrodynamic calculations of the interaction of ITER disruptive plasma with the slot divertor, Proceedings of the 24th European Conference on Controlled Fusion and Plasma Physics, Berchtesgaden, June 9 – 13, 1997, vol. 21A, Part III, 1997, pp. 277 – 281. [3] H. Wu¨rz, et al., Radiation in plasma target interaction events typical for ITER tokamak disruptions, Fusion Technol. 30 (1996) 739 – 744. [4] H. Wu¨rz, et al., A consistent 2-D analysis of erosion of the ITER slot divertor, Fusion Technol. 1 (1998) 271 – 274. [5] N.I. Arkhipov, et al., Study of structure and dynamics of shielding layer for inclined incidence of plasma stream at the MK-200 facilities, J. Nucl. Mater. 233 – 237 (1996) 767 – 770. [6] H. Wu¨rz, et al., Hot plasma interaction and quantification of erosion of the ITER slot divertor during disruptions and ELMs., Forschungszentrum Karlsruhe Report FZKA 6198, March 1999. [7] N.I. Arkhipov, et al., Study of plasma target interactions with plasma streams of power density of 40 MW cm − 2, Fusion Technol. 1 (1996) 507 – 510.