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ELMs and filaments

Filamentation of a strongly sheared rotation layer Florin Spineanu Association EURATOM-MEdC Romania

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

Vortex nucleation in strongly sheared velocity layers Basic facts (in general supported by observations, still to be verified by experiments) • in H mode a layer of plasma at the edge rotates poloidally with strong radial shear: this means a vorticity sheet • there is a current sheet superposed on the vorticity sheet • the current-vorticity sheet is unstable and breaks up into filaments The mechanism of filamentation has the same nature as the instability of anomalous polytropic (Chaplygin) gases. The parallel dynamics is essential (is not collisional or Landau saturation). Previous works: Ott, Trubnikov, Bulanov and Sasorov.

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

Vortex nucleation in protoplanetary accreation disks

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

Vortex nucleation from unstable strongly sheared velocity layers The case of the planetary atmosphere

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

Non-neutral plasma: Breaking of a ring of vorticity into filaments

Durkin, Schecter, Dubin, etc.

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

Stability of sheared layers

Two kinds of perurbations:

Rossby waves

cat’s eye, Both these instabilities preserve the geometry of the flow.

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

Breaking of a sheared velocity layer into filaments of both signs

Flierl and Zabusky

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

There is a sort of current sheet at the edge superposed on the sheared velocity layer

There is a current sheet superposed on the sheared layer

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

Breaking into localised structures of the density distribution in a layer of current The width of the layer is initially L0 and it evolves to a profile L which is variable along the direction y of the layer (poloidal). x Unperturbed state: A = Az (x) = −LB0 ln cosh L . Using the notation nL (t, y) ρ (t, y) = n0 L0 we have the usual density conservation ∂ ∂ρ + (ρv) = 0 ∂t ∂y The current density is jz = en (viz − vez )

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

where viz − vez =

cB0 const = 2πenL (t, y) nL

The equation of motion is ∂v ∂v +v ∂t ∂y

= =

1 (−jz Bx ) nmi c ∂A e (viz − vez ) mi c ∂y

When the system is invariant along the z direction then the generalized momenta of the electrons and of ions are conserved e mi viz + A = c e me vez − A = c

const const

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

Then the equation of motion is ∂v e ∂A ∂v 2 1 ∂ρ +v = (viz − vez ) = c0 3 ∂t ∂y mi c ∂y ρ ∂y

(1)

The solution of the equations of the “Chaplygin” fluid The two equations

∂ ∂ρ + (ρv) = 0 ∂t ∂y

and

∂v ∂v 1 ∂ρ +v = c20 3 ∂t ∂y ρ ∂y is obtained using a hodograph transformation. The formulas are nL n0 L0 v c0

=

ρ (t, y) =

=

−

sinh (|τ |) cosh (τ ) − cos χ

sin χ sinh (|τ |)

F. Spineanu – TTG Culham 2009 –

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ELMs and filaments

where τ

=

χ

=

t