Blob Source

Plasma LCFS source is modelled by injection of filaments with Gaussian spatial profiles and properties drawn from predefined distribution functions for density, temperature, velocity, size and waiting time per metre. The blobs enter the domain at the LCFS with the Gaussian envelope,Env(t) = exp(t−t0)2

2τ2

, applied to density, temperature and radial velocity. The envelope,Env(t), has a temporal width defined by the time scale τ = ℓ/v, where ℓ is the blob size and v is the radial velocity of the blob. The amplitude of the temporal envelope is defined by Rt+∆t

t Env(t) at

each timestep to ensure that P

∆tEnv(t) = 1 over the temporal discretisation of

the simulation. These blobs are added to a linked list (and to the simulation) when the waiting time expires, and removed from the list when the time passes 9σ from the peak of the temporal envelope. Each timestep, the source termsSn, ST, SΩ are calculated based on the items in the blob list.

The distributions used in the simulation come from MAST probe measure- ments, using the same techniques as chapters 3 and 4; peak values are used to calculate the distribution functions shown in figure 6.1. Density and temperature are inferred by finding the distribution function ofIsat, normalising it so that it has

a mean value of unity, then multiplying by the mean values of density or tempera- ture as measured by the Nd:YAG [Walsh et al., 2003] system around the LCFS to get the blob temperature or density distribution. Waiting times are estimated by the distance between peaks in Isat, and, the polodial extent of the probe is used to

estimate the waiting time per metre. Preliminary simulations show that the floating potential measurements of particle-energy flux ΓE =−nT ∂_Byφ overestimate the real

flux by a factor of 2, therefore, a factor of 0.5 is introduced for the blob velocity distribution.

0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

1 2 3

Blob Peak Isat (A)

Probability 800 1000 1200 1400 1600 1800 2000 2200 2400 0 0.1 0.2 0.3 0.4 0.5

Blob Peak Radial Velocity (m/s)

Probability 0.5 1 1.5 2 2.5 3 x 106 0 0.2 0.4 0.6 0.8

Blob Waiting Time Density (ms)−1

Probability 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0 20 40 60 80 Blob Size (m) Probability

Figure 6.1: Blob distribution functions used in the hTOKER simulations. A factor of 0.5 is applied to the velocity source in light of preliminary simulations showing that the floating potential measurements overestimate the particle-energy flux ΓE =

−nT ∂yφ B .

Simulations of SOL turbulence including parallel currents associated with sheath boundary conditions are prone to an artificial steady state solution when implemented with steady-state particle density and temperature sources that favours the production of radially elongated streamers. The reason for this may be sought in a steady-state solution solution to linearised interchange equations of motion.

~

∇ ·J~=e~_{∇ ·}n ~vp+~vgi−~vge−~v_ke+~v_ki

= 0 (6.4)

gives the quasineutrality condition for the system, withvp the ion polarisation ve-

locity andvgi,ge the ion and electron drifts due to gravity. Following a linearisation n=n0+ ˜n, we have

n0∇ ·~ ~vp+n0∇ ·~ ~v_ki−~v_ke

+ (~vgi−~vge)∂yn,˜ (6.5)

sincen0 only has a gradient in the ˆx direction butn0≫n˜. The electron continuity equation, neglecting the motions due to drifts associated with the gravity is

∂n˜

∂t +~vE ·∇~n0+n0σ

eφ

Te

= 0, (6.6)

in which we can identify the electron drift frequency ωd = kyvd = kyκT_eBe₀, with κ=_−n1₀∂n0 ∂x, giving ˜ n n0 = eφ Te ωd−σ ω . (6.8)

Equation (6.8) may be substituted into the Fourier transform of (6.5) to find

k_⊥~vp+k_k ~v_ki−~vke +ky(~vgi−~vge) eφ Te ωd−σ ω , (6.9)

which in the steady-state limitω = 0 becomes the greatly simplified balance between sheath and diamagnetic rates

ωd=σ. (6.10)

This describes the balance between gravity and sheath currents leading to a well defined size of perturbation

ky =

eσB0

κTe

= ρiκl_k-1. (6.11)

The problem with a model that has such a well defined unstable mode number, and source terms supplying density and temperature at a constant rate is that a standing wave approximately this wavenumber will form in potential, and the density and temperature will be channelled along these streamlines in the x-direction further reinforcing the standing wave. In the real world, a LCFS will have turbulent fluxes crossing it and in particular, there will certainly be non steady-state sources of plasma potential emanating from processes inside the LCFS. The blob source terms used in hTOKER, which naturally vary in space and in time and provide a source of vorticity, are an attempt not only to realistically model the fluxes across the LCFS but to mitigate this unrealistic steady-state streamer class of behaviour.