Surface instability of static liquid metal in magnetized fusion plasma

Understanding surface instability in magnetized fusion plasma supports the appropriate implementation and handling of liquid metal as plasma facing components (PFCs) in future fusion reactors. A Lagrange equation describing a viscous liquid surface deformation in a magnetized plasma is derived using Rayleigh’s method. Its solution justifies the general instability criterion and helps in characterizing the key interactions driving such instability under fusion conditions. Surface tension and gravity, especially with the poloidal angles of the lower part of a plasma chamber, mainly stabilize the liquid surface at small and large disturbance wavelengths, respectively. The sheath electric field and the external tangential magnetic field cause the liquid surface to disintegrate at an intermediate wavelength. Practically, a magnetic confinement fusion (MCF) device requires a strong magnetic field for confinement. The study suggests that such a strong field dominates the rest and governs instability. In addition, this implies that the configuration of a static planar free liquid surface is difficult to adopt as a candidate for handling the liquid metal as PFCs in next step MCF devices.


Introduction
To accomplish long-run fusion energy production, the utilization of a liquid as plasma facing components (PFCs) is strongly suggested [1], because such a surface is not permanently damaged. The surface material removal can be reversible by possible simultaneous self-healing and self-replenishment owing to liquid mobility. Furthermore, liquid, covering on solid with suitable wetting, can protect the inner structure * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of fusion devices from heat deposition. It is also possible to transport the deposited heat and plasma particles away concurrently. Consequently, the lifetime of PFCs is longer, and normal operations can be proceeded with longer period, and less disruption and instability. However, several conditions must be carefully considered. It has been recommended that the liquid is made of metal with a low melting temperature and must be compatible with fuels, be abundant as a resource and have acceptable radioactive and chemical toxicity. Past and current experiments on liquid metal PFCs have been actively carried out in several machines, e.g. HT-7 [2][3][4][5][6], EAST [3,4,7], TJ-II [8,9], T-11M [10][11][12] and T-10 [11,12] for lithium (Li), ISTTOK [13,14] for gallium (Ga), and FTU [15,16] for tin (Sn). The feeding configuration of liquid metal PFCs can be either static (no-flowing), locked in a porous structure with slow capillary motion, or dynamic (flowing or free-fall).
The installation can be at the bottom (HT-7, T-11M, T-10, FTU and TJ-II), at the low-field-side mid-plane (EAST and T-11M), and at the top (T-10 and ISTTOK) of the device chamber. The most promising feeding designs of liquid lithium (Li) PFCs, recommended for future fusion devices, are owing to either capillary action or controlled flowing. The former design [11,12,[17][18][19] adopts capillary force to firmly hold liquid Li in a porous structure. This results in the reduction of instability and associated droplet splashing [11]. However, lithium deuteride reduces Li self-replenishment flow rate, against capillary [17]. In addition, this configuration provides slow heat transfer, so that it is strongly damaged by repetitive heat loads [19]. The latter design adopts controlled flowing, e.g. the LiMIT system [4,5,20,21], which maintains the Li flow by electromagnetic force along guiding channels, and the FLiLi system [7,22,23], which maintains the thin laminar Li flow by electromagnetic force and gravity along guiding a plate. It appears that the controlled flowing design provides excellence heat and particle removal [7]. However, wettability against temperature evolution is of concern. Inappropriate wetting leads to instability with droplet splashing [5]. In addition, the mechanism related to thermoelectric magnetohydrodynamics using mainly in this design crucially depends on temperature gradient between liquid surface and coolant [20]. This implies that heat transfer between liquid Li and coolant needs attention.
The study of plasma-liquid interactions plays a crucial role in understanding the interactions between a liquid and magnetized fusion plasma. The issue is concentrated in this work, starting by assuming that a liquid surface is installed somewhere in a fusion chamber. The liquid surface is quickly charged by a plasma; then, the plasma sheath (a region of a self-induced electric field) is formed. It is discovered in the study that the repulsion by the sheath electric field drives instability against the attraction by surface tension. In addition, in the magnetized fusion plasma, the interaction between the charged liquid and the magnetic field plays an important role in de-stabilizing surface. In addition, the installation site, depending on the poloidal angle of the liquid surface, becomes a factor for a surface undergoing instability against gravity. The viscosity of the liquid controls the growth rate and the triggering timescale of instability via dissipative damping.
It has been known for some time that the liquid metal PFCs using the static free surface configuration is easily unstable in a fusion plasma, resulting in droplet release and major disruption as observed in HT-7 [5], compared to the configurations using capillary action and controlled flowing, in turn. Nevertheless, the current study was intentionally carried out the study of the stability of a static free liquid surface in a magnetized plasma, which later discusses the feasibility of its usage in EU DEMO and CFETR. Section 2 describes the way to determine the set of several perturbed quantities and the Lagrangian (section 2.1), the Lagrange equation (section 2.2), and the dispersion relation and the associated unstable solution (section 2.3). Section 3 provides and discusses the results, for which the poloidal angle, electron temperature, external tangential magnetic field strength, and surface tension are varied, including the damping effect by viscosity. The conclusion is provided in last section.

Methodology
The stability analysis of liquid metal PFCs in plasmas has been conducted by several authors [20,[24][25][26][27][28][29]. In the current study, the focus is on the static planar liquid surface exposed to a magnetized plasma under the influences of a sheath electric field (E Φ ) and an external magnetic field (B 0 ). The current study adopts the Lagrange equation to derive the dispersion relation of the dynamics of an infinitesimally perturbed surface, which was proposed by Lord Rayleigh [30][31][32][33]. The diagram in figure 1 clarifies the structure of the problem.
First, the infinitesimal deviation of the planar liquid surface, originally at z = 0, is described by where η(t) ∝ exp(ω ′ t) is the time-dependence small deviation of the surface, and k x and k y are the components of the disturbance wave vector in the x-and y-directions.
The motion of the liquid is represented by the velocity potential. The liquid is assumed to be an incompressible and irrotational fluid. The velocity potential (ϕ v ) of the perturbed liquid is described by [34,35] ϕ v (x, y, z, t) = A(t) cosh(k(z + ℓ)) cos(k x x) cos(k y y), (2) where A(t) ∝η(t) = ∂η(t) ∂t is the time-dependence amplitude of ϕ v , ℓ is the thickness of the liquid, and k = √ k 2 x + k 2 y is the resultant wave number. To determine A(t), the expression, i.e. δz = ∂δz ∂t = − ( ∂ϕv ∂z ) z=0 [34], is used, and as a result, Because, the liquid can be a viscous fluid, the theory of a potential flow may not be accurate. Fortunately, an infinitesimal deviation is applied just to find the onset conditions, not the time evolution, of a surface instability. Furthermore, the perturbed velocity potential is moved very slightly from its equilibrium. In this case, the theory of a potential flow is still a reasonable approximation [34,36,37] that is implemented throughout the study. Even though the plasma sheath is thin (the size of which is estimated as 10λ D where λ D is a Debye length [38]), the magnitude of the perturbation applied on the liquid surface is relatively smaller. Therefore, the electrical potential (Φ) is approximately varied linearly near the liquid surface. At z = L ≈ 10λ D (estimated as the position of the sheath edge [38]), the plasma is electrically floated by ϕ f and the liquid surface is set up to have zero potential. The electrical potential near the surface up to the sheath edge is approximated by Φ = ϕf L z. The perturbed electrical potential due to the deviated surface (ϕ) is described by [35] ϕ(x, y, z, t) = B(t) sinh(k(z − L)) cos(k x x) cos(k y y), where B(t) ∝ η(t) is the time-dependence amplitude of ϕ.

Lagrangian (L)
The Lagrangian (L) representing the deformation of the charged liquid surface in the magnetized plasma consists of the kinetic energies (KEs), the potential energies of deformation contributed from a gravity (PE g ), surface tension (PE s ) and the sheath electric field (PE sh ), and that induced by the interaction between the surface and the external magnetic field (U ind ). All energies are averaged over the wavelengths of the x-and y-directions, i.e. λ x = 2π kx and λ y = 2π ky , respectively. The KE is derived using ϕ v in equation (2) and its derivative, as follows [30,34], where ρ is the liquid mass density. The potential energy of deformation due to the gravitational field is derived as follows [34], where δz is of equation (1), and g is the magnitude of the gravitational acceleration. The surface energy involving deformation depends on the curvature of the disturbance, described as [34], where σ is the surface tension at the liquid surface.
The electrical potential energy associated with the sheath electric field drives the surface evolution. The sheath electric field is initially perpendicular to the charged surface under the magnetic field [38,39]. The electrical potential energy is determined by [35], where ϵ 0 is the vacuum permittivity. The magnetic field (B) perturbs the charge distribution on the surface, and the surface charges re-locate on the surface [40]. This affects the stability. To derive the additional induced energy (U ind ) by the existence of the magnetic field, the liquid is assumed to be a perfect conducting fluid, which is a good approximation of a liquid metal. In this case, the ideal Ohm's law [41], i.e. E + v × B = 0, where E is the associated electric field and v is the liquid velocity, is adopted. Because the considered liquid is static, there is no bulk liquid motion in this case. Only the perturbed z-component velocity (δz) at the liquid surface is available. The additional electric field (E 1 ) is induced by the existence of the external magnetic field (B 0 ). Using the ideal Ohm's law, where x,0 + B 2 y,0 and B ⊥,0 = B z,0 are the tangential and the perpendicular magnetic field components corresponding to the liquid surface, respectively. E 1 results in the surface charge re-location in the parallel direction on the liquid surface [40] and this leads to the alteration of the surface charge configuration. From this, an average change in the additional induced energy with respect to time ( ∂U ind ∂t ) is written as, where µ 0 is the vacuum permeability. Substituting equations (1) and (14) into equation (15) leads to , this results in and the additional induced energy is An alternative representation regarding equation (18) suggests that a part of the total energy of the perturbed surface is interchanged with the tangential magnetic flux. This amount of energy is converted to work being the origin of a magnetic tension, and possibly the initiation of a shear Alfvén wave. This arises because the magnetic flux is deformed in such a way as to respond to the surface deformation through the frozen-in flux mechanism [42]. This promotes instability. Furthermore, the coupling between the liquid surface and the magnetic field clearly depends on the curvature of the disturbance, similar to the surface tension. Equations (7), (8), (10), (13) and (18) are combined to form the Lagrangian (L) as follows, where

Dispersion relation
If ω ′ is imaginary, the solution, described by η(t), is oscillatory with a damping term, i.e. exp ( , which indicates a stable equilibrium. Therefore, to justify an unstable equilibrium, ω ′ has to be real, as follows For an unstable growth, G < 0. The general solution is in the form of .
If a perturbation disturbs a liquid initially in a static equilibrium, c 1 = c 2 = 1 2 . This leads to the compact form of the solution, i.e.
The instability exhibits as hyperbolically exponential growth of the surface. The damping term exp reduces the global growth rate of the instability. Therefore, the growth slows down but does not disappear. Still, the onset of the surface instability is initiated under the condition of G < 0. In other words, the initiation criteria of the charged liquid surface instability in a magnetic field in the cases of µ = 0 and µ ̸ = 0 are the same; however, their growth rates are not.

Result and discussion
It is well known that surface tension generally opposes surface instability [30,31,34,36]. Only a dense fluid, e.g. a liquid, beneath a dilute fluid, e.g. gas, is stable under gravity. This situation tends to be preserved even if the dilute fluid is an unmagnetized plasma at a low temperature and the liquid is aligned horizontally at the bottom of the plasma chamber. For a magnetized fusion plasma, the surface instability is also governed by the sheath electric field and the external tangential magnetic field. Either the installation of liquid PFCs or the formation of a molten liquid layer on solid PFCs at various poloidal angles leads to different wavelengths that are sensitive to the liquid being unstable.
The effect of gravity to the existence of liquid surface instability is considered via the poloidal positions of the plasma chamber by transforming g → −g sin θ in the term G in equation (25). This is named effective gravitational acceleration, where 0 • ⩽ θ < 360 • is the range of the poloidal angles with respect to the counter-clockwise direction and θ = 0 • is at the outer mid-plane of the poloidal cross-section. It is assumed that the liquid surface faces toward the center of the plasma chamber. The instability criterion of the charged liquid surface in the magnetized plasma, i.e. G < 0, is written in terms of the resultant wave number (k), as follows 2ϵ 0 ϕ f 2 coth(kL) It suggests that surface tension, surface electrical potential (generally controlled by electron and ion temperatures), tangential magnetic field strength, and poloidal angles govern the surface instability via the first, the second (i.e. the first and second terms in the bracket), and the third terms, in turn. For generalization, equation (28) is normalized by ρgL and a dimensionless parameter, x = kL. With this, the instability occurs at which where µ0ρgL . This makes each term in equation (29) dimensionless, and weighed with respect to gravity. Also, A ′ , B ′ , and C ′ are always positive. Figure 2 provides the trendlines and the associated labels of f (x) where A ′ = B ′ = C ′ = 1 (moderated profile) for further analysis. Such a profile refers to the situation that the same magnitude of each interaction is exerted on the perturbed surface. The sheath electric field and the external tangential magnetic field tend to de-stabilize the surface with weakly exponential and linear strengths. In contrast, the surface tension opposes instability and stabilizes the perturbed surface by parabolic strength. The stable and unstable are represented by f(x) > sin θ and f(x) < sin θ, in turn. In the poloidal cross-section of the plasma chamber, 0 • ⩽ θ < 360 • satisfies −1 ⩽ f(x) ⩽ 1. The stabilization at small x, i.e. large λ, is associated with the low curvature of the perturbed surface. In this case, the surface tension and the magnetic effect do not dominate; therefore, gravity is the main stabilization mechanism, especially at θ → 270 • . If the plasma temperature is weak, the perturbed liquid can be stabilized by gravity (as well as by surface tension) against the external tangential magnetic field. For the stabilization at large x, a liquid surface with small λ has large curvature, so that surface tension opposes the surface instability against electrostatic and magnetic effects. Therefore, the surface instability emerges at a relatively intermediate range of λ, where effective gravity and surface tension are weaker than the combination of electrostatic and magnetic effects. This is described by f(x = x stable ) = sin θ. Figure 2 also illustrates that the liquid surfaces installed at the bottom (θ = 270 • ) and the top (θ = 90 • ) of the plasma chamber are the most stable, i.e. the smallest x stable , and unstable, i.e. the largest x stable , respectively. The small λ is required for the stabilization by surface tension in the upper part of the chamber because stabilization by gravity is too benign. Furthermore, there is x = x crit at the minimum point of the trendline, and this satisfies ( This is involved with x crit , with the resultant disturbance by all interactions being in the optimized condition. Subsequently, this leads to the existence of the most effective wavelength (λ crit ) among the sensitive λ, associated with x stable , for the surface to be unstable. Notably, x crit is independent of θ. In this condition, x crit = 0.726 and λ crit = 27.5πλ D . Figures 3(a) and (b) depict x stable and x crit , respectively, under variations of A ′ , B ′ , and C ′ . It appears that x stable and x crit decrease with increasing magnitudes of surface tension (see the thick lines). This implies that the surface becomes stable within a wider range of λ due to the greater surface tension. In contrast, the strength increments of the sheath electric field and external tangential magnetic field increase x stable and x crit , i.e. shorten λ stable and λ crit . It appears that both interactions are asymptotic with the same tendency and independence of θ (see the overlapped lines). If either the external tangential magnetic field, the sheath electric field or both are strong, the surface is unstable at small x or large λ, where the stabilization due to surface tension and gravity are subdued. As shown in the areas in the upper and the lower parts of the trendlines in figure 3(a), representing the surface being stable (x > x stable ) and unstable (x < x stable ), it becomes clear that the stability under smaller x stable , i.e. larger λ stable , corresponds to stronger surface tension, weaker sheath electric field and weaker external tangential magnetic field, and vice versa. Referring to the interaction between a liquid with a relatively low temperature weakly magnetized plasma, the surface should be mostly stable. However, the stabilization can be achieved only in the lower part of the plasma chamber, approximately −1 ⩽ sin θ < −0.1. Beyond this limit, the effective gravity at the midplane and the upper part of the chamber cannot keep a static liquid stable. This strongly suggests that the bottom of the plasma chamber is the best place to install static planar liquid PFCs. Therefore, from this point, only the situation with θ = 270 • is considered.
In general, any perturbation on a liquid surface can be thought of as the superposition of all possible values of λ. The sensitive λ are in the range of the associated x stable . Among the sensitive λ, there may be λ crit corresponding to x crit (equation (30)). Figure 4 represents x crit > 0 by the colored bar and x crit < 0 by the dark blue color. As mentioned before, x crit > 0 relates to the most effective λ, i.e. the natural wavelength of the system, for which all interactions strongly promote surface instability. However, x crit < 0, i.e. no existing   where either the sheath electric field, external tangential magnetic field or both are large. From this, x stable should be considered as the stability criterion, and x crit should be used to characterize the development of surface instability. As can be clearly seen in figures 5(a) and (b), the range of sensitive λ values is generally 0 < x < x stable . If the external tangential magnetic field is relatively strong, there can be two positive conditions for real x stable , i.e. x stable,1 > 0 and x stable,2 > 0, where x stable,1 < x stable,2 . In this case, the range of sensitive λ values is in x stable,1 < x < x stable,2 near x stable = x crit . For example, if A ′ = 1.0, B ′ = 0.001 and C ′ = 10, 0.1 < x < 9.9 for instability. With regards to effective gravity, x stable is increased, when sin θ is increased. This expands the range of sensitive λ values; however, λ crit remains unchanged. So far, the model is generalized by A ′ , B ′ and C ′ . To quantitatively describe surface instability, A ′ , B ′ and C ′ are substituted by plasma and material parameters. A planar floating ) [38] where T e is the electron temperature, β is the ion-to-electron temperature ratio, m e is the electron mass and m i is the plasma ion mass, are adopted to describe the sheath electric field strength on a liquid metal in a quasi-neutral plasma. The plasma is of deuterium (D; m i = 3.344 × 10 −27 kg [45]) in HT-7 and of deuterium-tritium (DT;m i = 4.175 × 10 −27 kg [45]) in EU DEMO and CFETR with n = 2 × 10 18 m −3 and β = 1.0. The liquid metal is liquid Li, whose surface tension and mass density are σ Li = 0.399 N m −1 [46] and ρ Li = 0.52 × 10 3 kg · m −3 [45], respectively. Notably, a static liquid is assumed in the model and it can be used in the case of no initial flow and low T e , i.e. natural convection in the liquid is negligible. Therefore, an approximation using the model is reasonable for edge fusion plasma near PFCs. Figure 6 . This implies that a static planar liquid Li, with either a very flat or a very wavy surface, is required for stability in such plasma profiles. Notably, if either the sheath electric field, external tangential magnetic field or both are relatively strong, x stable,2 ≈ 2x crit and λ crit ≈ 2λ stable,2 .
For instability, 1 where µ = µ Li = 5.541 × 10 −2 Pa · s [51], σ = σ Li , ρ = ρ Li and θ = 270 • . λ stable and λ crit are implemented for determining k to characterize instability in terms of ℓ. The sensitive thickness (ℓ) of the liquid Li in the plasma profiles of HT-7, EU DEMO and CFETR are approximated. Notably, the edge tokamak plasma is large in B ||,0 but low in T e , i.e. B ′ ≪ A ′ ≪ C ′ . This implies that the surface instability of liquid metal PFCs is promoted mainly by the magnetic field. With the parameters of HT-7, the calculation suggests that it is difficult to keep the static liquid Li stable for all possible values T e at B ||,0 = 1.8 T, corresponding to ℓ stable → ∞. This is supported by the camera observation in HT-7 that droplets were always observed nearby the static liquid lithium limiter (LLL), thickness (ℓ) of which is 0.003 m [5]. Most instabilities tend to occur at all positive ℓ, mainly under ineffective λ, because ℓ crit → ∞ too. For EU DEMO and CFETR, figures 7 and 8 suggest that static liquid Li is never stable in EU DEMO and CFETR.
With T e = 5 eV, the surface is stable only up to 0.01 T of B ||,0 . In addition, the parameter space between ℓ stable , B ||,0 , and T e shown in figure 8 suggests that implementing liquid Li as PFCs in magnetic confined fusion devices cannot rely where ℓ > 0, which is stable against magnetized fusion plasma. The approximate boundary in equation (32) is valid only for a liquid Li surface installed at θ = 270 • exposed in DT plasma with n = 2 × 10 18 m −3 in the ranges 0.01 ⪅ T e ⪅ 100 eV and 0.001 ⪅ B ||,0 ⪅ 10 T. The initial development of the surface growth is discussed by consideration of η(t), described by equation (27) normalized to be a unit amplitude. The plots of η(t) with n = 10 19 m −3 of hydrogen (H) plasma and several selected T e and B ||,0 are depicted in figure 9. For convenience, k = 1 and ℓ = 0.001 m selectively. The assumption of µ = 0 gives rise to too rapid growth of the perturbed surface. In practice, viscosity  , and modifies the growth rate, i.e. √ F 2 4D 2 − G D , of the disturbed surface. This leads to a gentler effective growth rate (γ eff ). As illustrated in the subplot of figure 9, this results in two distinct developments during the initial formation of instability. One is the quiescence phase, which represents a slow response with a slight reduction in the surface amplitude by the damping effect. This lasts for the delay timescale (t d ). Another is the creeping-up phase, where the surface starts to grow not so rapidly because of viscosity. The numerically fits associated with this phase using exp(γ eff t) are depicted in figures 10(a) and (b). A triggering timescale (τ ) is roughly estimated by the combination of t d and 1/γ eff , i.e. τ ≈ t d + 1 γ eff . Regarding the variation of t d under the effect of a magnetized plasma, the main plots of figures 9 and 10(a), (b) suggest that the variation of B ||,0 strongly modifies t d . An increase in B ||,0 by one order leads to a decrease in t d from one-half to a few orders, especially for B ||,0 > 1 T. For very strong B ||,0 , t d ≪ 1/γ eff . It is surprising that an increase in T e hardly competes with B ||,0 in terms of increasing γ eff and decreasing t d . For example, t d is not greatly modified if T e < 100 eV. An increase in the liquid thickness (ℓ) shortens t d but causes γ eff to increase. Simply, a thick liquid layer becomes unstable more quickly than a thin liquid layer. In this study so far, there should be a coupling between a perturbed charged liquid surface to a magnetic tension so that disruption initiation should be involved. It is informed by figure 12(b) of [5] that implementing a thick liquid Li layer in HT-7 results in current quench and a major disruption in the timescale to be smaller than from using a thin liquid Li layer.
Previously, HT-7 with D plasmas has been implemented to investigate a static free surface LLL [2][3][4][5]. Its configuration corresponds to that assumed in the current study. A static LLL with thickness of ℓ = 0.003 m was installed at the bottom (θ = 270 • ) of HT-7 [3]. Figure 11 in [5] shows that the abnormal peaks of MHD activities, which trigger just before major disruption, have been observed in HT-7 equipped with the static LLL. This supports the outcome using the model that a liquid surface growth in an edge fusion plasma should be dominantly in connection with the interaction between the perturbed surface and the magnetic field. Subsequently, the growth is associated MHD instability leading to disruption. Regarding the triggering of the surface instability, the calculated triggering timescale is τ ≈ t d + 1 γ eff = 1.74 × 10 −7 s. Unfortunately, the HT-7 camera observed the droplet splashing in milliseconds in the time resolution, so the triggering of the surface instability could not be seen.
Assuming that a static planar liquid Li is used as the PFCs in EU DEMO (B ||,0 = 5.86 T, T e = 5 eV and x crit = 4.03 × 10 3 ) and CFETR (B ||,0 = 6.5 T, T e = 5 eV and x crit = 4.95 × 10 3 ), the liquid Li of 10 −4 ⩽ ℓ ⩽ 10 −2 m should become very rapidly unstable at the approximated triggering timescale of τ ≈ 1.62 × 10 −8 s, due to the very strong magnetic field producing a very large x crit . This implies that the static free surface configuration of liquid metal PFCs is hardly attractive in future magnetic confinement fusion devices.

Conclusion
The theoretically conducted investigation determines the triggering of surface evolution leading to instability where a static planar liquid metal is exposed to a magnetized fusion plasma. In general, the combined effect of the sheath electric field and the magnetic field plays a crucial role in driving instability at intermediate λ values. The instability can evolve either with or without a natural wavelength. Due to the magnetic field in a fusion plasma being relatively strong, the liquid surface becomes unstable and grows mainly by the magnetic effect, indicating possible relationships between the liquid surface growth and the magnetic perturbation related to MHD activities and disruption, as observed in HT-7, for example. This also implies that in the next step fusion devices, e.g. EU DEMO and CFETR, liquid metal PFCs should not rely on solely a static planar surface configuration. The boundary line of the external tangential magnetic field (B ||,0 ) and the electron temperature (T e ), i.e. T e ≈ (1.96 × 10 29 )B 15.8 ||,0 , justifies the stability of a liquid Li surface in a magnetized fusion plasma. Liquid viscosity provides damping that delays the growth (the quiescence phase) and globally reduces the initial growth rate (the creeping-up phase). The timescale of the instability initiation is in the order of 10 −8 s. It takes longer for a thinner liquid than a thicker one before surface instability is initiated. As stronger magnetic field provides a larger initial effective growth rate, so the surface evolves more quickly. In addition, either a very flat or a very wavy surface is required for stabilization under a strong magnetic field. The former is due to gravity, especially if the surface is installed in the lower part of the plasma chamber with the condition of −1 ⩽ sin θ < −0.1, whereas the latter is due to surface tension.