2. Sausage and MRTI growth rates

Consider a long, cylindrical liner with constant density $\rho$ and dynamic viscosity $\eta$ . Let there be a geometrical perturbation at the outer liner–vacuum interface of the form $\xi = \xi (z)$ . The outer radius of the liner as a function of the axial distance is $R = R_{\mathrm{out}} + \xi (z)$ , where $R_{\mathrm{out}}$ is the mean outer radius. Similarly, let $R_{\mathrm{in}}$ be the unperturbed inner radius of the liner. Consider the liner to be thick enough such that feedthrough effects are neglected and that the thermal vacuum pressures for $r\lt R_{\mathrm{in}}$ and $r\gt R_{\mathrm{out}}$ are equal. The instability growth rate for this set-up was previously derived by Dai et al. (Reference Dai, Sun, Wang, Zeng and Zou2023). We refer the reader to Dai et al. (Reference Dai, Sun, Wang, Zeng and Zou2023) for a detailed mathematical derivation of the growth rate. The growth rate for a liner with perturbation wavenumber $k$ and outer radius $R_{\mathrm{out}}$ is given by simplifying (23) in Dai et al. (Reference Dai, Sun, Wang, Zeng and Zou2023) as follows:

(2.1) \begin{align} \gamma ^2 + 2\nu k^2\left (2 - \frac {{{I}_{1}}(kR_{\mathrm{out}})}{kR_{\mathrm{out}} {{I}_{0}}(kR_{\mathrm{out}})}\right )\gamma - \left (\frac {2P_M}{\rho R_{\mathrm{out}}} + g\right )\frac {{{I}_{1}}(kR_{\mathrm{out}})}{{{I}_{0}}(kR_{\mathrm{out}})}k = 0, \end{align}

where $\gamma$ is the growth rate, $\nu = \eta / \rho$ is the kinematic viscosity, $P_M = {\unicode{x03BC}} _0 I^2 / 8 \pi ^2 R_{\mathrm{out}}^2$ is the magnetic pressure, ${\unicode{x03BC}} _0$ is the permeability of free space, $I$ is axial drive current magnitude, $g$ is the external acceleration of the liner, $h$ is the liner thickness and ${I}_{n}$ is the $n$ th-order modified Bessel function of the first kind.

For a liner with $kR_{\mathrm{out}} \gg 1$ (i.e. the short wavelength limit), ${{I}_{1}}(kR_{\mathrm{out}})/$ ${{I}_{0}}(kR_{\mathrm{out}}) \rightarrow 1$ , which simplifies the dispersion relation as

(2.2) \begin{align} \gamma ^2 + 4\nu k^2 \gamma - \left (\frac {2P_M}{\rho R_{\mathrm{out}}} + g\right )k = 0. \end{align}

Here, $2P_M/(\rho R_{\mathrm{out}})$ is the component of the growth rate due to the classical sausage instability and $g$ is the component due to MRTI. Explicitly solving for the growth rate via the quadratic formula, the growth rate becomes

(2.3) \begin{align} \gamma = 2\nu k^2\left (\sqrt {\frac {1}{4\nu ^2k^3}\left (\frac {2P_M}{\rho R_{\mathrm{out}}} + g\right ) + 1}-1\right )\!. \end{align}

In their original asymptotic derivation for the small wavelength regime, Dai et al. assumed the first term under the square root

(2.4) \begin{align} \frac {1}{4\nu ^2k^3}\left (\frac {2P_M}{\rho R_{\mathrm{out}}} + g\right ) \ll 1 , \end{align}

which is an unstated part of their derivation. Applying a Taylor series to the square root term ( $\sqrt {x + 1} \approx x/2 + 1$ for $x \ll 1$ ), one derives (36) of Dai et al. (Reference Dai, Sun, Wang, Zeng and Zou2023) as follows:

(2.5) \begin{align} \gamma \approx \left (\frac {P_M}{2}\frac {1}{R_{\mathrm{out}}} + \frac {\rho g}{4}\right )\frac {1}{\eta k}. \end{align}

This asymptotic form is useful since it highlights an inverse relation to $k$ and $\eta$ , suggesting that decreasing perturbation wavelength will always decrease the growth rate for small wavelengths. However, we will show that for this asymptotic form is not valid for all small wavelength regimes. Rather, for small $\eta$ , increases in $k$ will lead to an increase in growth, even in the $kR_{\mathrm{out}} \gg 1$ regime. This is a particularly important result in investigating the effect of surface roughness on MagLIF liners.

Consider (2.3) once again. For $P_M/\nu ^2 \gg 1$ or $g/\nu ^2\gg 1$ , the assumption (2.4) is not necessarily true, as is the case for large $P_M$ (strong compression) or small but non-zero $\nu$ . This suggests the left-hand side of (2.4) is a non-dimensional parameter that requires additional consideration.

We define

(2.6) \begin{align} g_{\mathrm{eff}} \equiv \frac {2P_M}{\rho R_{\mathrm{out}}} + g, \end{align}

i.e. an effective gravitational acceleration consisting of the acceleration on the surface perturbation associated with the pure sausage instability $2P_M / \rho R_{\mathrm{out}}$ and acceleration associated with the acceleration of the whole liner $g$ (i.e. the pure MRTI mode). Substituting this definition into (2.3) results in

(2.7) \begin{align} \gamma = 2\nu k^2\left (\sqrt {\frac {g_{\mathrm{eff}}k}{4\nu ^2k^4} + 1} - 1\right )\!. \end{align}

An alternative rearrangement is

(2.8) \begin{align} \gamma = \sqrt {g_{\mathrm{eff}}k + 4\nu ^2 k^4} - 2\nu k^2. \end{align}

Consider the inviscid limit, $\nu \rightarrow 0$ , $\gamma \rightarrow \sqrt {g_{\mathrm{eff}}k}$ . For stages of the implosion when the pressure in the liner is constant (i.e. for $R_{\mathrm{in}} \lt r \lt R_{\mathrm{out}}$ ), $g\approx 0$ and $g_{\mathrm{eff}}\approx 2P_M/(\rho R_{\mathrm{out}})$ , which simplifies the growth rate into the classical sausage instability (Thorne & Blandford Reference Thorne and Blandford2017). This occurs in situations such as during stagnation. For thin liners with liner thickness $h \ll R_{\mathrm{out}}$ , the pressure profile in the liner for $R_{\mathrm{in}} \lt r \lt R_{\mathrm{out}}$ is approximately linear. Thus, $g\approx P_M/(\rho h) \gg 2P_M/(\rho R_{\mathrm{out}})$ , consequently resulting in $g_{\mathrm{eff}}\approx g$ . In this case, (2.8) simplifies into the purely classical MRTI (Weis et al. Reference Weis, Zhang, Lau, Schmit, Peterson, Hess and Gilgenbach2015). Henceforth, we refer to arbitrary $\gamma$ as $\gamma _{\mathrm{visc}}$ and $\gamma _{\mathrm{inv}} = \sqrt {g_{\mathrm{eff}}k}$ to emphasise the significance of viscosity.

Both the forms of (2.7) and (2.8) elucidate physical intuition that was not apparent before. Note that $\gamma _{\mathrm{visc}}$ is only a function of two parameters: $g_{\mathrm{eff}}k = \gamma _{\mathrm{inv}}^2$ and $\nu k^2$ , i.e. only a function of gravitational and viscous forces, respectively. In fact, these two forces have directly competing effects, which is most easily seen in (2.8). Gravitational forces act as a growth mechanism for MRTI (as is the case for classical RTI), while viscous forces act as a damping mechanism, with no coupling between the two physics. In the highly viscous limit as $\nu \rightarrow \infty$ , $\gamma _{\mathrm{visc}}\rightarrow 0$ and viscous effects will completely eliminate instability growth. Furthermore, the growth component $\gamma _{\mathrm{inv}}$ is proportional to $\sqrt {k}$ and will dominate at lower $k$ , while the damping component is proportional to $k^2$ and will dominate at larger $k$ . This suggests that the ratio of these two effects is an important parameter to understand which physics will dominate instability behaviour.

We define the non-dimensional Galilei number as

(2.9) \begin{align} {\textit{Ga}} \equiv \frac {g_{\mathrm{eff}}}{4\nu ^2k^3} = \frac {g_{\mathrm{eff}}\lambda ^3}{32\pi ^3 \nu ^2}, \end{align}

which quantifies the effect of gravitational versus viscous forces. Here, $\lambda = 2\pi /k$ is the perturbation wavelength. This ratio explicitly appears in (2.7) as the first term under the square root. Rewriting (2.5) with the definition of $Ga$ , Dai et al.’s original growth rate expression becomes

(2.10) \begin{align} \gamma _{\mathrm{visc}} \approx \left (\frac {g_{\mathrm{eff}}^2}{2\nu }\right )^{1/3}\left (\frac {Ga}{8}\right )^{1/3}. \end{align}

Similarly, (2.7) can be rewritten in two different forms

(2.11) \begin{align} \gamma _{\mathrm{visc}} = \left (\frac {g_{\mathrm{eff}}^2}{2\nu }\right )^{1/3}\left (\frac {\sqrt {Ga + 1} - 1}{{\textit{Ga}}^{2/3}}\right ) \end{align}

or, equivalently,

(2.12) \begin{align} \tilde {\gamma } = \frac {\gamma _{\mathrm{visc}}}{\gamma _{\mathrm{inv}}} = \sqrt {\frac {1}{Ga} + 1} - \sqrt {\frac {1}{Ga}}. \end{align}

Note that, under this normalisation and definition of $g_{\mathrm{eff}}$ , (2.2) is equivalently

(2.13) \begin{align} \tilde {\gamma }^2 + \frac {2}{\sqrt {Ga}}\tilde {\gamma } - 1 = 0, \end{align}

in which the corresponding normalised growth rate is more easily derived.

Consider (2.11) and (2.12). Since $Ga$ is a function of $g_{\mathrm{eff}}$ and $\nu$ , it is not immediately apparent how changing either of these parameters affects the nominal growth rate $\gamma _{\mathrm{visc}}$ in (2.11). However, in (2.12), the dependence on $g_{\mathrm{eff}}^2/2\nu$ is eliminated and the growth rate ratio becomes solely a function of $Ga$ . Therefore, when determining if a sausage instability/MRTI flow configuration requires viscous modelling, one can explicitly quantify the decrease in growth due to viscosity based only on $Ga$ . This is analogous to the well-known classification of laminar or turbulent flows based on the Reynolds number $Re$ , the ratio of inertial to viscous forces. In fact, one can define $Re \equiv u/(2\nu k)$ and the dimensionless Froude number as $Fr \equiv u\sqrt {k}/\sqrt {g_{\mathrm{eff}}}$ and to define $Ga$ as

(2.14) \begin{align} \textit{Ga} \equiv \left (\frac {\textit{Re}}{\textit{Fr}}\right )^2, \end{align}

where $Fr$ is the ratio of inertial and gravitational forces and $u$ is some characteristic speed.

Note that, under the approximation that $Ga \ll 1$ , $\sqrt {Ga + 1} \approx Ga/2 + 1$ , and thus (2.10) is approximately equal to (2.11). This implies that (2.5) is calculated under the assumption that $Ga \ll 1$ . Physically, this assumes an accelerating implosion such that viscous forces dominate, akin to laminar flow in low Reynolds number regimes. This is not necessarily true even under the small wavelength limit. In fact, we find in our viscous MagLIF liner simulations that $Ga \gg 1$ , akin to turbulent flows in high Reynolds number regimes, invalidating both (2.5) and (2.10).

Dai et al.’s asymptotic form of (2.10) offers immediate physical insights that our derived (2.11) does not. First, it is immediately clear that as $Ga\rightarrow 0$ , $\gamma _{\mathrm{visc}}\rightarrow 0$ . Since $Ga\propto k^{-3}\nu ^{-2}$ , one finds that $\gamma _{\mathrm{visc}}\propto (\nu k)^{-1}$ , as was the original conclusion by Dai et al. (Reference Dai, Sun, Wang, Zeng and Zou2023), which is not apparent from (2.11). However, that conclusion is only valid for $Ga \ll 1$ . For $Ga \gg 1$ , (2.11) simplifies to

(2.15) \begin{align} \gamma _{\mathrm{visc}} \approx \left (\frac {g_{\mathrm{eff}}^2}{2\nu }\right )^{1/3}\textit{Ga}^{-1/6} = \gamma _{\mathrm{inv}}. \end{align}

Therefore, for large $Ga$ , the growth rate approaches the inviscid regime ( $\gamma _{\mathrm{visc}} \approx \gamma _{\mathrm{inv}}$ ).

The modes of maximum growth for viscous instability are found from (2.7). The wavenumber that maximises the growth rate is

(2.16) \begin{align} k_{\mathrm{max}} = \max _{\arg k} \gamma _{\mathrm{visc}} = \left (\frac {g_{\mathrm{eff}}}{32\nu ^2}\right )^{1/3}, \end{align}

with maximum growth rate

(2.17) \begin{align} \gamma _{\mathrm{max}} = \left (\frac {g_{\mathrm{eff}}^2}{16\nu }\right )^{1/3}. \end{align}

This corresponds to

(2.18) \begin{align} Ga_{\mathrm{max}} = \max _{\arg Ga} \gamma _{\mathrm{visc}} = 8. \end{align}

Under these conditions, $\tilde {\gamma } = 1/\sqrt {2}$ .

Equation (2.18) defines two regimes of the viscous flow. For $Ga\lt Ga_{\mathrm{max}} = 8$ , an increase in wavenumber $k$ will lead to a decrease in the growth since viscous forces dominate, as predicted by Dai et al. (Reference Dai, Sun, Wang, Zeng and Zou2023). However, for $Ga \gt Ga_{\mathrm{max}}$ , the inverse is true, where an increase in $k$ decreases the growth rate since gravitational forces dominate. Equivalently, $k$ that approaches $k_{\mathrm{max}}$ will increase the growth rate. For systems in which instability growth should be minimised, perturbations of wavelength corresponding to wavenumber near $k_{\mathrm{max}}$ should be avoided. This classification for viscosity-dominated flows ( $Ga \ll 8$ ) and gravity-dominated flows ( $Ga \gg 8$ ) is similar to the classical classification of laminar (small $Re$ ) and turbulent (large $Re$ ) flows.

Figure 2.

(a) Viscous growth rates $\gamma _{\mathrm{visc}}$ via (2.11). (2.10) and (2.15) are overlaid at corresponding values of $g_{\mathrm{eff}}^2 / 2\nu$ . Note that the viscous growth rate reaches a max value at $Ga = 8$ . (b) Ratio of growth rates $\gamma _{\mathrm{visc}}/\gamma _{\mathrm{inv}}$ via (2.12). As $Ga$ decreases, the flow becomes more viscous, damping the MRTI growth. (c) Growth rates for nominal $k$ values at $g=10$ (arbitrary units). Equations (2.10) and (2.15) are overlaid at corresponding values of $\nu$ .

Figure 2 plots the growth rates in different forms to demonstrate various characteristics and trends. Figure 2(a) calculates the nominal growth rate $\gamma _{\mathrm{visc}}$ versus $Ga$ . Figure 2(a) visually depicts the three regimes of flow discussed: viscosity-dominated flows for $Ga\ll 8$ , gravity-dominated flows for $Ga\gg 8$ and transitional flows for $Ga\sim 8$ . Dai et al.’s asymptotic expression (2.10) is depicted in dotted black lines which are overlaid on top of the exact (2.11) growth rates. They accurately approximate growth rates for the small $Ga$ regime/viscosity-dominated regime, as discussed previously. Similarly, overlaid cyan circle-marked lines are calculated via the inviscid growth rate (2.15) and accurately approximate the growth rates for the large $Ga$ regime/gravity-dominated flows. For the transitional $Ga\sim 8$ regime, neither (2.10) nor (2.15) accurately predict the exact growth rate.

Figure 2(b) plots $\tilde {\gamma } = \gamma _{\mathrm{visc}}/\gamma _{\mathrm{inv}}$ from (2.12). As viscosity increases, $Ga$ will decrease, causing a decrease in $\tilde {\gamma }$ , demonstrating that viscosity has a damping effect on instability growth compared with the inviscid case, as is seen typically in non-magnetised RTI (Chandrasekhar Reference Chandrasekhar1961). Figure 2(c) plots the growth rates against nominal wavenumber $k$ for various values of $\nu$ at $g=10$ (chosen arbitrarily). The highly viscous growth rate (2.10) only agrees for $k\gg k_{\mathrm{max}}$ while the inviscid growth (2.15) only agrees for $k\ll k_{\mathrm{max}}$ .

Alternatively, we introduce a time scale that corresponds to a physical process more directly relevant for MagLIF implosions. Let $t_{\mathrm{comp}} = \sqrt {R_{\mathrm{out}}/g_{\mathrm{eff}}}$ , which is a characteristic time scale for the compression of the liner. Similarly, we define an alternative Galilei number as

(2.19) \begin{align} {\textit{Ga}}^* \equiv \frac {g_{\mathrm{eff}}R_{\mathrm{out}}^3}{4\nu ^2}, \end{align}

where the characteristic length is now associated with the liner outer radius, which is assumed to be much larger than the perturbation wavelength. Under these parameters, (2.11) can be equivalently rewritten as

(2.20) \begin{align} \gamma _{\mathrm{visc}} = 2\nu k^2 \left (\sqrt {\frac {{\textit{Ga}}^*}{(kR_{\mathrm{out}})^3} + 1} - 1\right )\!, \end{align}

or, non-dimensionally,

(2.21) \begin{align} \tilde {\gamma }^* = \gamma _{\mathrm{visc}}t_{\mathrm{comp}} = (kR_{\mathrm{out}})^2 \left (\sqrt {\frac {1}{{\textit{Ga}}^*} + \frac {1}{(kR_{\mathrm{out}})^3}} - \sqrt {\frac {1}{{\textit{Ga}}^*}}\right )\!. \end{align}

The form presented in (2.21) may be useful for practical design applications. For instance, let $t_{\mathrm{MRTI}} = 1/\gamma _{\mathrm{visc}}$ be the characteristic time scale for MRTI growth. If $\tilde {\gamma }^* = t_{\mathrm{comp}}/t_{\mathrm{MRTI}}\ll 1$ , then MRTI growth will be negligible compared with the compression time scale since compression occurs before MRTI growth can occur. Hence, (2.21) is a more meaningful quantity for design compared with (2.12). However, all the expressions presented are identically equivalent and may be used for a variety of purposes.

3. Comparison with FLASH simulations

Figure 3.

Initial set-up of the imploding liner simulations for $n=4$ and $n=8$ modes. An axial current of $5$ MA is applied on the outer surface of the aluminium liners.

To validate the theory presented, we conduct simplified cylindrical liner implosion simulations (depicted in figure 3) loosely based on the experimental set-up described by Sinars et al. (Reference Sinars2010, Reference Sinars2011). The set-up itself is not directly a MagLIF simulation as there is no preheating nor axial magnetic field, but is still a relevant process for MagLIF. The simulation is composed of a 1 mm thick aluminium liner surrounded by vacuum with outer radius $R_{\mathrm{out}} = 3.2$ mm. The computational radial domain spans $r\in [0,4.8]$ mm and the axial domain spans $z\in [0,2.4]$ mm, with periodic boundaries in the axial direction. The liner outer radius is initially geometrically perturbed by an $n$ mode perturbation, where $n = 2.4\text{ [mm]}/\lambda$ . In this work, we evaluate two sets of test cases: the $n=4$ mode and the $n=8$ mode. The implosion is conducted with a constant, uniform drive current of $I=5$ MA applied at the outer surface of the liner. This corresponds to $Ga\eta ^2 \approx 1.8\times 10^{6}$ and $Ga\eta ^2 \approx 2.3\times 10^{5}$ for each mode, respectively. This value was chosen to represent a lower ramp current the liner will undergo during the onset of the sausage instability/MRTI. These simulations incorporate thermal conduction and magnetic diffusion physics, but are not significant during the simulation time.

For each set of test cases, we simulate the implosion with constant viscosity in the liner. The viscosities tested are $\eta \in [0,100,1000]$ g cm⁻¹ s⁻¹. All simulations are conducted on the radiative hydrodynamics simulation code FLASH (Fryxell et al. Reference Fryxell, Olson, Ricker, Timmes, Zingale, Lamb, MacNeice, Rosner, Truran and Tufo2000), which has recently been validated for pulser-drive ICF target design (Ellison et al. Reference Ellison2025a ). An adaptive mesh refinement (AMR) algorithm is used to automatically refine the computational mesh in regions in which the magnitude of the second derivative-based estimator exceeds a certain density threshold. We define the AMR refinement levels such that $\Delta r = \Delta z$ with a maximum cell size of $50\,{\unicode{x03BC}}$ m and a minimum cell size $6.25\,{\unicode{x03BC}}$ m. These mesh refinement parameters are the same as those presented in MRTI MagLIF benchmark simulations presented by Ellison et al. (Reference Ellison2025a ), which demonstrated good agreement between FLASH simulations and experiments. All simulations are seeded with a perturbation amplitude of $50\,{\unicode{x03BC}}$ m; these perturbed regions are refined via the AMR algorithm prior to the beginning of the simulation. The simulation is run for a total of approximately $125$ ns. The Equation of State (EOS) of the liner is tabulated from common databases such as SESAME and PROPACEOS, following the methodology of validation benchmarks presented by Ellison et al. (Reference Ellison2025a ). We do not consider an ideal gas EOS with large polytropic index $\gamma$ to mimic an incompressible fluid. Although this would be more aligned with the derivation of the analytical growth rates, the speed of sound of the simulated flow would be artificially large and difficult to resolve computationally. Furthermore, we show the results of these simulations are still consistent with the derived theory despite using a more realistic material EOS.

Figure 4.

Snapshot of the liner density at $t\approx 125$ ns for the $n=4$ mode perturbation simulations (top row) and the $n=8$ mode perturbation simulation (bottom row). Increasing viscosity (presented in CGS units) clearly decreases the nonlinearity of the perturbation and its amplitude.

Figure 5.

(Top row) Fast Fourier transform time histories for the initial $n=4$ mode perturbation simulations. (Middle row) Fast Fourier transform time histories for the $n=8$ mode perturbation simulations. (Bottom row) Comparison of the simulated amplitudes versus theoretical amplitude $\xi _{\mathrm{theo}}$ via (2.20) and Dai amplitude $\xi _{\mathrm{Dai}}$ via (2.5). The linear growth of all simulated cases matches almost identically with (2.20), while only $\eta =1000$ matches with (2.5). No $\xi _{\mathrm{Dai}}$ is plotted for $\eta =0$ since (2.5) predicts an infinite growth rate.

Figure 4 depicts a snapshot of each test case discussed at the end of the simulation $t\approx 125$ ns. All qualitative behaviour seen aligns with what has been discussed. As viscosity increases, $Ga$ (or ${\textit{Ga}}^*$ ) decreases, thereby decreasing the growth of the main $n$ mode. Interestingly, viscosity seems to surpress nonlinearities as well. This is seen most prominently for the $n=8$ mode, where the nonlinear MRTI spikes of the inviscid case are curved. On the other hand, the nonlinear MRTI spikes of the $\eta =100$ case are completely straight and parallel to the radial axis. When increasing viscosity further for the $\eta =1000$ case, the perturbation remains mostly linear and maintains a nearly sinusoidal shape.

The first two rows of figure 5 plot the fast Fourier transform (FFT) amplitudes of the liner interface at each time step for every case. Here, we define the interface as the density contour where $\rho = 2$ g cc⁻¹. The bottom row compares the amplitude of the largest mode with what is predicted by theory (2.20). The FFTs exhibit interesting behaviour. In the inviscid test cases, there is significant growth of the harmonics of the principal mode ( $n=4$ and $n=8$ , respectively) as well as other various `noisy’ non-integer modes. This may result from numerical seeds, such as a discrepancy between the Cartesian mesh versus the curved perturbation interface, and may lead to higher nonlinear, multimodal growth than expected. In-depth studies isolating the effects of numerical MRTI growth versus physical growth is an ongoing area of research (Tranchant et al. Reference Tranchant, Hansen, Michta, Garcia-Rubio, Rahman, Ney, Ruskov and Tzeferacos2025); however, the amplitude growth is still well predicted by the growth of the principal mode. When the viscosity increases to $\eta =100$ , only the principal mode and its corresponding harmonics remain. At the largest viscosity $\eta =1000$ , only the $n$ and $2n$ modes remain, showing clear suppression of higher modes due to increased viscosity and lower $Ga$ . This nonlinear behaviour is not predicted with the theory presented; although $Ga$ values that can induce this behaviour are not realistic in MagLIF implosions, other disciplines may find it useful to investigate this phenomenon more deeply.

In the bottom row of figure 5, there is clear agreement between the theoretical amplitude $\xi _{\mathrm{theo}}$ expected from (2.20) and the simulated perturbation. The theoretical growth rate is calculated using a density of $\rho = 3.6$ g cc⁻¹, which corresponds to the post-shocked aluminium density. During the simulation, the external acceleration $g$ is much smaller than the acceleration associated with the sausage instability due to the relatively small driving current, as evidenced by the minimal movement of the inner liner throughout the simulation. Therefore, the theoretical growth rates are calculated for when $g\approx 0$ and $g_{\mathrm{eff}} \approx 2P_M/(\rho R_{\mathrm{out}})$ . In the $n=8$ cases, nonlinear saturation seems to appear once $\xi = 0.3\lambda$ , with the theoretical amplitude significantly overpredicting the simulated results. Before this threshold, the theoretical and simulated amplitudes show good agreement. Note that, in the $\eta =1000$ case, in which we qualitatively do not see much nonlinearity in the liner outer interface from figure 4, the theoretical and simulated amplitudes match well (albeit slightly under predicting) throughout the entirety of the simulations. Nonetheless, the linear growth region matches well, validating the discussions from § 2. On the other hand, $\xi _{\mathrm{Dai}}$ from (2.5) is only accurate for the highest viscosity test case $\eta =1000$ . This corresponds to values of $Ga \approx 2$ and $Ga \approx 0.2$ for the $n=4$ and $n=8$ cases, respectively, again agreeing with our earlier discussion. At smaller $\eta$ (corresponding to larger $Ga$ ), the growth is vastly overestimated.

4. Evaluating viscosity impact on implosions relevant for MagLIF

Let us define a ‘significant’ change as when $\tilde {\gamma } = \gamma _{\mathrm{visc}}/\gamma _{\mathrm{inv}} \lt C$ , where $C$ is some constant (say, $C=0.95$ for a 5 % decrease in growth rate). Then the condition for viscosity to become significant from (2.12) is

(4.1) \begin{align} \sqrt {1 + {\textit{Ga}}} - 1 \lt C\sqrt {\textit{Ga}}. \end{align}

For a given perturbation wavelength $kR_{\mathrm{out}}$ , this is equivalent to

(4.2) \begin{align} {\textit{Ga}} \lt \alpha , \end{align}

where

(4.3) \begin{align} \alpha = \frac {4C^2}{1-C^2}. \end{align}

Looking at the expression for $Ga$ , this becomes apparent when

(4.4) \begin{align} \nu \gt \sqrt {\frac {g_{\mathrm{eff}}}{4\alpha k^3}}. \end{align}

4.1. Case I: sausage instability-dominated growth

Consider phases in which the sausage instability will dominate, with $g_{\mathrm{eff}} \approx 2P_M/(\rho R_{\mathrm{out}})$ , such as during stagnation. Alternatively, during the acceleration phase of cylindrical implosions for thick liners with aspect ratio $AR = R_{\mathrm{out}}/h \sim \mathcal{O}(1)$ , an order of magnitude approximation for the acceleration $g$ is

(4.5) \begin{align} g \approx \frac {P_M}{\rho h } \sim \mathcal{O}\left (\frac {P_M}{\rho R_{\mathrm{out}}}\right )\!. \end{align}

Therefore, the MRTI component will still be comparable to the sausage instability component associated with $2P_M/(\rho R_{\mathrm{out}})$ . The following discussion is based on an order of magnitude analysis; we refer to this regime as the sausage instability-dominated regime which implicitly includes cases when the MRTI component is comparable to the sausage instability component.

Via the definition for magnetic pressure and $g_{\mathrm{eff}} \approx 2P_{M}/(\rho R_{\mathrm{out}})$ , we can relate dynamic viscosity and drive current from (4.4) with

(4.6) \begin{align} \eta \gt \sqrt {\frac {{\unicode{x03BC}} _0}{16\pi ^2}}\sqrt {\frac {\rho I^2 }{\alpha (kR_{\mathrm{out}})^3}} \approx 10^{-4}\sqrt {\frac {\rho I^2 }{\alpha (kR_{\mathrm{out}})^3}}. \end{align}

The controlling parameters in determining the significance of viscosity for MagLIF implosions is the density of the material, the current, and the wavelength of perturbation. Note that this is the formula for viscosity in SI units. The equivalent formula in CGS units is

(4.7) \begin{align} \eta \gt 10^{-2}\sqrt {\frac {\rho I^2 }{\alpha (kR_{\mathrm{out}})^3}}. \end{align}

Figure 6.

Critical viscosity threshold $\eta _C$ such that there will be a 5 % decrease in MRTI growth $kR_{\mathrm{out}}\in (0,50)$ (top) and $kR_{\mathrm{out}}\in (10,10^4)$ (bottom) versus drive current $I$ . For currents relevant for MagLIF of the order of $1{-}100$ MA, the critical viscosity threshold is of order $\mathcal{O}(10^2)-\mathcal{O}(10^5)$ for $kR_{\mathrm{out}}\in (0,50)$ . For very large $kR_{\mathrm{out}}$ , $\eta _C$ decreases by several orders of magnitude.

As a concrete illustrative example, consider an implosion liner of aluminium. Consider a perturbation such that $kR_{\mathrm{out}} \sim \mathcal{O}(10)$ . The density of aluminium at initial implosion will be $\rho \sim \mathcal{O}(1)$ g cc⁻¹ $\sim \mathcal{O}(10^3)$ kg m $^{-3}$ . We now define the critical viscosity $\eta _C$ as the minimum viscosity such that there will be a 5 % change in the MRTI growth rate compared with the inviscid case. Thus, $C=0.95$ and $\alpha \sim \mathcal{O}(10)$ . The resulting critical viscosity is

(4.8) \begin{align} \eta _{C,\mathrm{SI}} \gt 10^{-5}I,\quad \eta _{C,\mathrm{CGS}} \gt 10^{-4}I. \end{align}

For MagLIF implosions, the ramp current $I\sim \mathcal{O}(1)$ MA, resulting in $\eta _{C, \mathrm{CGS}} \sim \mathcal{O}(10^2)$ during the initial onset of MRTI. At peak currents ranging from tens to hundreds of MA, this critical viscosity increases by several orders of magnitude.

Figure 6 plots the critical viscosities of the aforementioned example over a range of driving currents and wavenumbers. At $I=1$ MA – as is the typical minimum current for modern MagLIF implosions – and for perturbation wavelengths corresponding to $kR_{\mathrm{out}} \lt 50$ , $\eta _C\sim \mathcal{O}(10^2)$ g cm⁻¹s⁻¹, matching the order of magnitude analysis discussed. As designs improve, the driving current will increase to larger values, increasing the critical viscosity required to affect MRTI growth. For $kR_{\mathrm{out}} \gt \mathcal{O}(10^2)$ , the critical viscosity decreases, demonstrating viscosity’s ability to dampen shorter wavelength modes more easily. For instance, for $I=10$ MA with $kR_{\mathrm{out}}=10^4$ , $\eta _C \sim \mathcal{O}(10^{-2})$ . If the viscosity of the imploding liner is orders of magnitude greater or lower than $\eta _{C}$ for a given drive current and perturbation wavelength range, designers will be easily able to determine whether viscosity will be an important design consideration or not.

To estimate values of $\eta$ , we consider the viscosity model described in Bergeson et al. (Reference Bergeson, Baalrud, Ellison, Grant, Graziani, Killian, Murillo, Roberts and Stanton2019), which models the viscosity of an unmagnetised plasma in a high-energy density regime. We list the relevant equations in Appendix A. Again, as an illustrative example, consider an imploding aluminium plasma liner with average charge state $\bar {Z}_i = 2$ . This is fair approximation for the low temperatures of the liner at the onset of the implosion.

Figure 7.

Aluminium viscosity for hypothetical mass densities and ion temperatures. For the example discussed considering $kR_{\mathrm{out}} \lt 50$ , $\eta \gt \eta _C = 10^2$ only when $T_i \gt 10^7$ K, suggesting viscosity will not influence MRTI growth in the initial stages of implosion.

Figure 7 plots the viscosity over a range of mass densities and ion temperatures. In all cases, $\eta$ only reaches a magnitude of $\mathcal{O}(10^2)$ at temperatures greater than $10^7$ K $\approx 1$ keV; this is largely independent of the density of the aluminium liner (i.e. how compressed it is). Consider liners with outer radii of the order of millimetres. These results suggest that for $kR_{\mathrm{out}} \lt 50$ (corresponding to perturbations with wavelength of the order of $0.1$ mm, as may be the case with surface roughness), the effects of viscosity are largely insignificant and can be neglected for simpler analysis in the initial formation of the sausage instability/MRTI. Therefore, viscosity is not a viable mechanism or design parameter for mitigating perturbation growth in MagLIF liners under this perturbation wavelength range.

On the other hand, consider finer resolution perturbations, such as perturbations seeded by the electrothermal instability (ETI). The ETI may be able to seed perturbations with characteristic wavelength of the order of $1{-}10$ ${\unicode{x03BC}}$ m (Peterson et al. Reference Peterson, Yu, Sinars, Cuneo, Slutz, Koning, Marinak, Nakhleh and Herrmann2013), which would correspond to the approximate range $kR_{\mathrm{out}} \in (10^2,10^4)$ . From figure 6, for currents $I\in (1,100)$ MA, the critical viscosity ranges reaches as low as $10^{-2}$ g cm⁻¹s⁻¹ at the shortest perturbation wavelengths. These are attainable viscosity values at low temperatures, as seen in figure 7 for $\rho = 3$ g cc⁻¹, where plasmas with ion temperature of the order of $10^4$ K $\sim 1$ eV have $\eta \sim 10^{-2}$ g cm⁻¹s⁻¹. Therefore, for the finest seeds of ETI, viscosity is suspected to be able to dampen instability growth.

This current analysis does not include the effects of magnetisation on viscosity which introduces anisotropic viscous diffusion for magnetised plasmas (Braginskii Reference Braginskii1958), which we leave for future work. However, we suspect that the unmagnetised viscosity will predict a higher viscosity coefficient compared with the magnetised case, as thus the conclusions drawn from this work should be applicable.

4.2. Case II: MRTI-dominated growth

Consider thin liners with aspect ratio $AR = R_{\mathrm{out}}/h \gg 1$ , $g_{\mathrm{eff}} \approx g \approx P_M/(\rho h)$ since $g \gg 2P_M/(\rho R_{\mathrm{out}})$ . Following a similar methodology to § 4.1, we can relate the critical dynamic viscosity and drive current from (4.4) as

(4.9) \begin{align} \eta \gt \sqrt {\frac {{\unicode{x03BC}} _0}{32\pi ^2}}\sqrt {\frac {\rho I^2}{\alpha (kR_{\mathrm{out}})^2 (kh)}}, \end{align}

or, in terms of the aspect ratio $AR$ ,

(4.10) \begin{align} \eta \gt \sqrt {\frac {{\unicode{x03BC}} _0}{32\pi ^2}}\sqrt {\frac {(AR)\rho I^2}{\alpha (kR_{\mathrm{out}})^3}}. \end{align}

We define the critical viscosity governed by the sausage instability acceleration $g_{\mathrm{eff}}\sim \mathcal{O}(P_M/\rho R_{\mathrm{out}})$ in § 4.1 as $\eta _{C,\mathrm{SI}}$ . Similarly, we define the critical viscosity governed by the thin liner MRTI acceleration $g_{\mathrm{eff}} \sim \mathcal{O} (P_M / \rho h)$ as $\eta _{C,\mathrm{MRTI}}$ . The ratio of the two critical viscosities is

(4.11) \begin{align} \frac {\eta _{C,\mathrm{MRTI}}}{\eta _{C, \mathrm{SI}}} = \sqrt {\frac {AR}{2}}. \end{align}

Interestingly, the critical viscosity is enhanced for $AR \gg 1$ , meaning a larger viscosity is required to dampen thin, accelerating liners. This results from thin liners experiencing larger effective acceleration under the same magnetic pressure compared with thick liners due to the steeper pressure gradient in the liner, requiring a larger viscosity to counteract these forces. Section 4.1 concluded that viscosity will not be significant in damping instability growth since realistic values of unmagnetised plasma viscosity are significantly smaller than $\eta _C$ except in the shortest perturbation wavelengths. However, as $AR$ increases towards larger values (thinner liners), $\eta _C$ increases proportional to $\sqrt {AR}$ . Therefore, viscosity in very thin liners with $AR \gg 1$ will be unable to dampen the finest modes as well. However, at high $AR$ , feedthrough effects may become important, violating one of the initial assumptions of this work. Future work must incorporate these effects before making conclusive statements regarding the effect of viscosity for thin liners.