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 \({\sf n}=n\), particle mass \(m_p\) and temperature \(T_{\sf n}=10\) eV, for a timescale of \(\tau _{\sf n}=L_{mfp}/v_{\sf n}\) where the mean free path \(L_{mfp}\approx L_0\) and the neutral speed typically is \(v_{\sf n}\). For then \(v_{\sf n}=\sqrt {|e|T_{\sf n}/m_p} =\sqrt {|e|/m_e} \cdot \sqrt {m_e/m_p} \sqrt {T_{\sf n}}\), ie. \(v_{\sf n}\approx 4 \times 10^5 \times (1/40) \times \sqrt {10}\approx 3 \times 10^4\). Thus \(\tau _{\sf n}\approx 10^{-6}\) 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 \(c_{se}\approx 40 v_{\sf n} \approx 10^6\) 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 \(\tau _i\approx \tau _{\sf n}\approx 10^{-6}\) 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 \(D=3\), \(N_3 \approx 1\,000\), 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.