Order-of-magnitude estimates for tokamak edge modelling

Suppose plasma number density $n\approx {10}^{18}$  m${}^{-3}$. Order of magnitude dimensions are $L_0\approx 0.1$ m for SOL thickness, reactor minor and major radii say $a=3$ m and $R_0=10$ m, so the volume of SOL $\approx 4{\pi }^{2}a{R}_0{L}_0\approx 100$ m${}^3$. It follows that the total number of electrons $\approx {10}^{20}$.

The shortest timescale is inverse $|e|B/m_e$, the electron cyclotron frequency, where the ratio of charge to mass for the electron is $|e|/m_e=1.76\times {10}^{11}$ C kg${}^{-1}$, and $B\approx 10$ T, so ${\tau }_{ce}\approx {10}^{-12}$ s. Hence the number of particle-steps to evolve $1$ s of physical time is ${10}^{20+12+1}$ (assuming of order $10\pi$ timesteps per orbit), which assuming $1000$ flop per update, needs a total of ${10}^{36}$ flop. Thus to complete a computation in $1$ s on Exascale machine, only one $1$ particle-step in ${10}^{18}$ is allowed. This implies for example only $100$ superparticles in the volume can be used supposing each has a weight ${10}^{18}$, which is unlikely to be adequate because electrostatic and other effects will produce a noise level that swamps any physical effects. However if $3$ or $4$ months (approx. ${10}^7$ s) are allowed, then a calculation with ${10}^9$ particles may be performed if memory-access bandwidth constraints can be satisfied, when the noise levels may be manageable.

The situation may be shown to be a million times easier for neutrals considered separately, assuming neutral density $\mathsf{n}=n$, particle mass $m_p$ and temperature $T_{\mathsf{n}}=10$ eV, for a timescale of ${\tau }_{\mathsf{n}}=L_{mfp}/v_{\mathsf{n}}$ where the mean free path $L_{mfp}\approx L_0$ and the neutral speed typically is $v_{\mathsf{n}}$. For then $v_{\mathsf{n}}=\sqrt{|e|T_{\mathsf{n}}/m_p}=\sqrt{|e|/m_e}\cdot \sqrt{m_e/m_p}\sqrt{T_{\mathsf{n}}}$, ie. $v_{\mathsf{n}}\approx 4\times {10}^5\times (1/40)\times \sqrt{10}\approx 3\times {10}^4$. Thus ${\tau }_{\mathsf{n}}\approx {10}^{-6}$ s, ie. a million times longer than the electron cyclotron timescale.

Suppose a fluid model is employed instead, ie. the electron distribution is represented by its first $3$ moments. If the electron temperature $T_e\approx 10$ eV, then the thermal speed $c_{se}\approx 40v_{\mathsf{n}}\approx {10}^6$ m s${}^{-1}$. Supposing the SOL to be sampled at a uniform $1$ mm interval, then the number of sample-points $\approx {10}^{11}$, and the timestep for an explicit scheme $\approx ({10}^{-3}/{10}^6)\approx {10}^{-9}$ s, so the number of sample-point updates is ${10}^{11+9}$. Assuming $1000$ flop each update, this means one second of physical time needs ${10}^{23}$ flop or ${10}^5$ s$\approx 1$ d of Exascale machine time, if memory-access bandwidth constraints can be satisfied. If an implicit scheme is used to simulate plasma ions as a fluid on a drift type timescale$\approx L_0/c_{si}$, then possibly the timestep ${\tau }_i\approx {\tau }_{\mathsf{n}}\approx {10}^{-6}$ s, ie. a thousand times larger, and calculations lasting only a few minutes might suffice.

Another way of looking at this is to suppose that numerical problem is $D$-dimensional, $1000$ flop are needed for each sample update and $N_D$ samples per spatial dimension and $N_D^2$ time samples are allowed. Then to update such a model in $1$ s requires $N_D^{D+2}\approx {10}^{15}$. Thus if $D=3$, $N_3\approx 1000$, and $N_5\approx 100$. It seems that accuracy controlled, unstructured, implicit fluid models should be possible, although for the more complex models $1000$ flop per update may be an underestimate by orders of magnitude.