To achieve the high-confinement mode in most magnetic fusion machines, such as the ITER tokamak, it is necessary to improve the fueling efficiency and fueling penetration depth. Three main methods have been used to fuel tokamak machines: gas puffing (GP),^[1] pellet injection (PI),^[2] and supersonic molecular beam injection (SMBI).^[3] In gas puffing, its penetration depth is very shallow and its fueling efficiency is very low. Most of the puffed molecules remain outside of the last-closed flux surface during GP, and the molecules penetrate into the plasma injected mainly through thermal diffusion (i.e, local cooled). Although pellet injection (PI) has a deep penetration depth and high fueling efficiency, it is expensive. Therefore, it is desirable to find a new fueling method with high efficiency but low cost. Supersonic molecular beam injection (SMBI) is an improvement on GP with a radial directed supersonic group velocity generated by a special designed Laval nozzle. In SMBI, the injected molecules interact with the fusion plasma directly, which differs from PI with the stripping (ablation) and shielding process. Different from GP, whose fueling process is dominated by diffusion, convection plays a key role in SMBI owing to the high directed group velocity of the SMB. Experiments have proven that SMBI has higher fueling efficiency and deeper injection than GP. Although PI has a high penetration depth, it costs more than SMBI.

SMBI is not only a fueling method but also an efficient way to control plasma behavior. It can control the plasma density,^[4] trigger the L–H transition and improve confinement,^[5] and simultaneously control and mitigate the edge-localized mode (ELM).^[6] Even more, it can be used to experimentally investigate non-local heat transport,^[7] impurity transport,^[8] and particle transport.^[9] Considering all of these advantages of SMBI, we believe that the fueling problems about the penetration depth of SMBI must be studied.

To improve the fueling performance of SMBI, some experimental investigations^[3,4,10] have been performed to improve its penetration depth and fueling efficiency, such as by converting from low (magnetic) field side injection to high (magnetic) field side injection, changing the SMB speed, temperature, and particle flux intensity, as well as the background electron density and temperature. The effects of plasma density and temperature on penetration depth can be researched experimentally by using methods such as fast TV cameras and charge-exchange recombination spectroscopy.^[11,12] However, such experimental methods are limited in their abilities to directly measure the penetration front of neutral particles, particle transport, and various collision reactions. Some simulations have been performed in order to improve the penetration depth, such as by comparing GP and SMBI^[13] and by studying the effect of SMBI flux.^[14]

Large-scale real-time numerical simulations can provide detailed temporal–spatial information on neutral front propagation and collision reactions, because such simulations are not constrained by the complicated technical problems present in experiments. There exist some codes for performing this kind of simulation, such as B2,^[15] EPIC,^[16] UEDGE,^[17,18] BOUT/BOUT++,^[19–21] and TOKES.^[22] The BOUT++ code is used to study edge plasma physics, with its physical transport models based on two fluid MHD equations. The BOUT++ code is a software platform for simulating two-dimensional (2D) and three-dimensional (3D) fluid plasma in an arbitrary curvilinear coordinate system. With the BOUT++ framework, many practical subroutine modules have been developed, such as the trans-neut^[23] module, which issued for studying macroscopic transport dynamics in 3D simulations, and the TPSMBI code,^[24] used for studying the radial transport dynamics of SMBI. Performing a 3D spatial–temporal simulation of the real tokamak geometry with X-point would require massive CPU time and number of tasks that need to be submitted, then this paper simply assumes toroidal symmetry for some basic studies, such as radial propagation of neutral particles, because the toroidal asymmetry makes little difference to the radial propagation of neutral particles. Therefore, 2D simulation via trans-neut is applied to study the fueling problems of SMBI. The simulation considers four kinds of main collision reactions: those among hydrogen molecules, atoms, electrons, and ions. It also accounts for particle recycling, the sheath effect, and the local constant flux boundary condition.

In the present paper, we will use the trans-neut module of the BOUT++ code to study how the background plasma affects SMBI penetration in the HL-2A tokamak. In order to understand how the plasma density and the temperature profiles affect the SMBI penetration depth and fueling efficiency, we will perform simulations through the developed trans-neut module. Section 2 introduces the physical model. Section 3 gives the boundary conditions of the numerical calculations. Section 4 illustrates the simulated results. Finally, Section 5 summarizes the principal results.

It is to study the fueling physics of SMBI by using the MHD equations. Tokamak plasma contains multiple species, including hydrogen molecules, atoms, ions, and electrons. Thus, the SMBI fueling problem in HL-2A tokamak plasmas relates to sophisticated physical processes. Considering the difficulties in the calculation and simulation, four main kinds of collision reactions between various species are considered during SMBI: the molecular dissociation reaction (molecular-dissociated reaction), the atom-ionizing reaction, the charge-exchange reaction between atoms and ions, and the recombination reaction between electrons and ions. Our physical model considers some primary physical effects, including perpendicular plasma density diffusion, atom diffusion, heat diffusion and conduction, parallel plasma density convection, molecule radial density convection, energy interchange between ions and electrons, and parallel ion viscosity. This model uses the seven-field model deduced from the Braginskii equations, and it solves for the molecule and atom density, ion and electron temperatures, parallel plasma velocity, and molecule radial velocity. The transport equations of the model are given as follows:^[25]

Plasma transport is described in Eqs. (1)–(4), which use the quasi-neutral condition ( . Atom transport is given in Eq. (5), and molecule transport is included in Eqs. (6) and (7), where the molecule pressure is, is the molecule temperature (usually room temperature), and k is Boltzmann’s constant. Our simulation uses field-aligned coordinates (x, y, z), which can be transformed into the common orthogonal toroidal coordinate (ψ, θ, ξ) via coordinate conversion.^[19–21]

In this simple model, all source and sink terms of particle and heat are due to atomic and molecular reactions, such as molecules dissociation, atoms ionization, ion–atom charge exchange, and electron–ion recombination. The interactions between atom, molecule and plasma are included self-consistently. This model captures some basic transport physics during SMBI on both perpendicular and parallel directions, including: i) with a constant injection flux of molecule density at the outermost boundary flux surface, neutral molecules and atoms propagate inwards continuously due to molecule convection and atom diffusion effect; ii) both locally peaked plasma density and locally decreased plasma temperature profiles are formed due to both particle fueling and heat sinking effects of SMBI via neutrals dissociation and ionization, even though there is strong parallel convection and conduction transport; iii) the locally peaked plasma density will dramatically increase the losing rates of molecule and atom which prevents the further penetration of SMBI; iv) propagating front of molecules stagnates due to the total molecule dissociation rate balancing with the molecule injection rate at the SMBI source; v) plasma density blobs (i.e., source) and ion temperature holes (i.e., sink) propagate on the parallel direction due to parallel convection effects.

Our simulation uses the real magnetic geometry of the HL-2A tokamak with X-point (Fig. 1). This geometry can be divided into three solving regions: the plasma core and edge region, the scrape-off-layer (SOL) region, and the private flux and divertor plate region. In the simulation, different boundary conditions are set in each solving region.

Fig. 1.

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	Fig. 1. (color online) SMBI configuration of HL-2A (a) and the different solving regions on the poloidal cross-section of HL-2A (b).

Neumann radial boundary conditions are set for all evolving quantities, such as for , , , and , except for , , and at the innermost boundary flux surface of the core side. For , , and , the fixed-gradient or flux-driven boundary conditions are given as follows:

We also investigate how the plasma temperature affects the fueling of SMBI. For this case, the plasma density gradient is set constant as 500, while the electron and ion temperature gradient is set to 12550, 25100, 50200, 75300, or 100400. Also, the plasma first evolves into five steady states for each set of , . Then the same SMBI is turned on in order to study the SMB penetration depth at various initial temperature profiles.

On the innermost magnetic surface of the private flux region, the following Dirichlet boundary conditions are applied: and . At the outermost flux surface in the SOL and private flux regions, Neumann boundary conditions are also used for all evolving quantities except , , , and . The Dirichlet boundary conditions are used for and , the same as in the private flux region. While the particle recycling boundary conditions are applied to and , as shown below:

On the divertors, the sheath boundary conditions (12)–(14) and the particle-recycling boundary conditions (15) are used, as shown below:

The applied local constant flux boundary condition is used to simulate SMBI. It is given that the poloidal flux of molecules renders an exponential distribution. To avoid numerical instability due to a sharp gradient in the poloidal direction, the model uses an exponential profile of particle flux. Therefore, the local constant flux boundary condition is modified as follows:

Using the trans-neut module of BOUT++, we have investigated how the initial plasma density and temperature profiles affect the SMBI penetration depth on which the fueling efficiency strongly depends. The radial plasma densities and temperatures are initialized to be a set of linear profiles with different core plasma densities and temperatures. The principal results and the explanations of the present simulations are summarized as follows: (i)

The penetration depth of SMB could be weakened with denser background core plasma. The injected SMB will be blocked in the SOL region when the core plasma density is large enough. This is due to two main reasons. One reason is that the dense background plasma makes the injected SMB molecules collide more frequently with the background plasma, and then the speed of the injected SMB front is slowed down more quickly. Therefore, the injection depth will be weakened by the dense plasma. The other reason is that the molecule dissociation rate is proportional to the plasma density as displayed in Figs. 4 and 5. Larger core plasma density means more molecules will be dissociated in the edge region before penetrating into the deeper region. The SMB front propagates inwards until the overall molecule dissociation rate (i.e., sink) is balanced by the SMBI rate (i.e., source). As the SMBI rate is constant, the molecule dissociation rate determines the penetration depth of the SMB. A larger core plasma density increases the molecule dissociation rate at the edge, decreasing the penetration depth during SMBI.

In a word, both the high density and the high temperature of the background plasma make the neutral particles in the injected SMB interacted more frequently with the background species and blocked the SMB front. It is because the dissociation rate of molecules strongly depends not only on the background plasma density but also on the background plasma temperature. Both the denser density and the higher temperature of the background plasma will make the neutral particles in the beam front dissociated more easily and quickly. When more neutral injected molecules are dissociated and then ionized before propagating inwards, the plasma density at the edge region will increase and the injection depth of the SMB will decrease further in a positive feedback.

It is more difficult to penetrate inside the separatrix in high parameter plasma discharges, especially in H-mode discharges characterized by the high density and temperature in the edge region. Thus, it is necessary and urgent to find a new method to improve the SMB penetration depth and its fueling efficiency especially in the high-confinement regime. Unfortunately, the penetration depth of SMBI decreases in high-parameter plasma conditions in the present simulation, which reduces its fueling efficiency to be comparable to GP. It is not acceptable but it encourages us to improve SMBI technology urgently. To achieve the high-confinement mode in most magnetic fusion machines, such as the ITER tokamak, the fueling efficiency and fueling penetration depth must be improved, but it remains a great technical challenge to increase the penetration depth and fueling efficiency of SMBI in H-mode. Except for the effects of the background plasma conditions on the injection depth, the inherent parameters of the injected SMB (such as the injection density, injection velocity, injection width, and so on) could also affect the injection depth of the SMB. The effects of the SMB injection densities and injection velocities on the penetration depth have been done by Zhou et al.^[14] The preliminary simulations of the effects of the injection density and injection width have been tested, which indicates that the deeper injection depth could be reached with smaller density and larger width of SMB. To find a way to increase the fueling efficiency and suggest some future experiments in the H-mode plasma conditions, more systematical and detailed simulations may be required by varying the SMBI parameters in H-mode plasma with pedestals.