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ring vorticity
Ring - type vorticity distribution arising as stationary coherent flows in a field theoretical description of the plasma in strong magnetic field Florin Spineanu, Madalina Vlad, Virgil Baran National Institute of Laser, Plasma and Radiation Physics, Romania
[email protected]
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ring vorticity
1.
There is turbulence and there are structures. In 2D the relaxation ends up with coherent flows. • separation of vorticity, vortex merging
2.
• ring-type vorticity in zonal flows • ring-type vorticity in H-mode in tokamak • ring-type vorticity in tropical cyclones
3.
Structures mean priviledged states. Variational approach is needed. Statistics of point-like vortices is one way (however limited).
4.
Field theoretical formulation is possible and is highly suggestive. Applied to localized vortices. Applied to ring-type vorticity.
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The limit of statistics: coherent structures
Simulation of CHM
Potential vorticity CHM
eq.(Khukharin)
(Horton)
Vortex encounters and mergings. How much universality? Widom Rawlingson phase transition. We can use just a notation: vorticity concentration. Invoked in Rayleigh Benard and in Kelvin Helmholtz as in geophysics etc.
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Coherent structures in fluids and plasmas (numerical 1)
D. Montgomery, W.H. Matthaeus, D. Martinez, S. Oughton, Phys. Fluids A4 (1992) 3.
Numerical simulations of the Euler equation.
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MHD relaxation (numerical)
R. Kinney, J.C. McWilliams, T. Tajima Phys. Plasmas 2 (1995) 3623.
Numerical simulations of the MHD equations.
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First stop for partial conclusion The structures are not accidental and are not an exotic exception to the turbulence The structures are a destination for any evolution of a system that is not strongly driven. The structures are priviledged states, distinct compared all other possible states, by the fact that the system chooses them from a huge number of alternatives. (See the stationary Euler fluid states). Conservation laws are not sufficient. We need variational instruments. Until here: monopolar and dipolar vortices.
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Annular robust structures: ring of vorticity
Cuasi-stable rings in ocean flow (Gulfstream)
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Annular robust structures: the zonal flows
Zonal flows condensation from turbulence. Hasegawa McLennan Kodama 1978.
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Annular robust structures: the radial electric field in tokamak in H-mode
Zone of ExB velocity in
Poloidal flow of the plasma in
H-mode
H-mode (Burrell 1997)
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Annular robust structures: the eye-wall replacement
An annular zone of high vorticity advances towards the eye-wall
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Annular robust structures: the eye-wall replacement (2)
An annular zone of high vorticity advances towards the eye-wall (Willoughby Black: the tropical cyclone Gilbert)
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Second stop for a partial conclusion The ring - type vortices in plane arise in many physical situations They are structures, as are the monopolar and dipolar vortices Several cases where the rings are dynamically connected with monopolar vortices
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Two dipoles
Dipolar
Dipolar, too.
(see wrl’s).
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Two dipoles
Dipolar
Dipolar, too.
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Two faces of the dipole structure: pair of vortices and respectively axisymmetric. The latter is a ring of vorticity. There are sudden transitions between them: saw-teeth in tokamak and possibly the H-mode.
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0.1
Theory of point-like vortices and the statistical approach
According to Joyce Montgomery (Journal of Plasma Physics, 1973) The physical quantities describing the two-dimensional fluid dynamics are ψ
≡
streamfunction
v
≡
velocity
ω ez
=
vorticity (perp. on the plane)
with the equations v
=
−∇ψ × ez
ω
=
Δψ
The formal solution of the last equation, connecting the vorticity and F. Spineanu – Les Houches 2013 –
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the streamfunction, can be obtained using the Green function for the Laplace operator Δx,y G (x, y; x , y ) = δ(x − x )δ (y − y ) where (x , y ) is a reference point in the plane. Then the Green function has the explicit expression G (x, y; x , y )
≡ =
G (r; r ) 1 ln (|r − r |) 2π
after a normalization of the distances in plane by the length of the side L of the square domain. Using the Green function for the laplacian in the plane we invert the
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equation relating ω and ψ : ψ = dx dy G (r; r ) ω (r ) 1 ln (|r − r |) ω (r ) = dx dy 2π Consider now the discretization of the vorticity field ω (x, y) in a discrete set of 2N point like vortices each carrying the elementary quantity ω0 of vorticity which can be positive or negative ωi = ±ω0 N vortices with the vorticity + ω0 and N vortices with the vorticity − ω0 We will need to use the circulation γi for the elementary vortex i, F. Spineanu – Les Houches 2013 –
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instead of ωi (the vorticity) due to the units implicitely contained in δ (x − xi ) δ (y − yi ) ∼ 1/l2 . dxdyω (x, y) γ = S
= S
dxdy [(∇ × v) · ez ] v · dl
= Γ
with units m2 /s. The current position of a point-like vortex is (x, y) at the moment t. The total vorticity is ω (x, y) =
2N
γi δ (x − xi ) δ (y − yi )
i=1
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from which we derive the streamfunction solution by inverting the Laplacian Δψ (x, y)
=
ψ
=
ψ
=
2N
i=1 −1
Δ
= or
γi δ (x − xi ) δ (y − yi ) ω
dx dy
1 ln (|r − r |) ω (r ) 2π
2N 1 dx dy ln (|r − r |) γi δ (x − xi ) δ (y − yi ) 2π i=1 2N
1 ψ (r) = ln (|r − ri |) γi 2π i=1
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The velocity of the k-th point-vortex is vk
= =
− ∇ψ|r=rk × ez 2N
1 rk − ri − γi ez 2 × 2π |rk − ri | i=1
or dxk dt dyk dt
=
vx(k) = −
2N i=1
=
vy(k)
γi
1 yk − yi 2π |rk − ri |2
2N
1 xk − xi = γi 2 2π |r − r | k i i=1
The entropy is used in the theory of statistical mechanics of point-like vortices . The quantity W represents the volume occupied in the space of all F. Spineanu – Les Houches 2013 –
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states of a system consisting of N particles distiguishable by the distribution function f corresponding to the occupation numbers {ni }. It is equal to the number of ways to distribute N distinguishable molecules among K cells such that there are ni of them in the cell i. N! Ω {ni } = n1 !n2 !...nK ! N! N! W= + − ni ! ni ! S = ln W = −
i
i
n+ i
i
ln n+ i
−
i
− n− ln n i i
Joyce Montgomery 1973 as Ni+ Ni− = const F. Spineanu – Les Houches 2013 –
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The two equations ln Ni+
+α +β
ln Ni−
−
+
φij
j
+α −β
φij
j
Nj+ Nj+
−
Nj−
−
Nj−
=
0
=
0
are obtained variationally under the constraints i
i
E
>
0 , const
Ni+
=
N = const
Ni−
=
N = const
It results Δψ + sinh (ψ) = 0
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Wonderful. But hardly useful further (plasma, MHD, rings). And what actually are these elementary vortices ? We understood something fundamental: the real content of the theory is: matter, field, interaction. (Smile. We just met a Field Theory).
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1.
Abelian, Chern-Simons, scalar self-interaction of fourth order. The Liouville equation. The current density in tokamak plasma (cf. J.B. Taylor).
2.
Non-Abelian, Chern-Simons, scalar self-interaction of order four. The sinh-Poisson equation. The incompressible Navier-Stokes fluid in the absence of dissipation and viscosity, i.e. the Euler fluid (cf. Montgomery et al.).
3.
Non-Abelian, Chern-Simons, scalar self-interaction of sixth order. An equation that seems able to describe correctly the stationary states of a 2D plasma in strong magnetic field and of the tropical cyclone.
4.
Abelian, Chern-Simons, scalar self-interaction of sixth order. An equation giving states or ring-type vorticity distribution, stabilized by a topological constraint. Possible physical applications: the sheared velocity layers of H-mode and of Internal Transport Barriers in tokamak.
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Lagrangian for the Euler fluid: Non-Abelian sl (2, C), Chern-Simons, 4th order L
=
2 −εμνρ T r ∂μ Aν Aρ + Aμ Aν Aρ + (1) 3 1 1 2 † † † iT r Ψ D0 Ψ − T r (Di Ψ) Di Ψ + T r Ψ , Ψ 2 4
where Dμ Ψ = ∂μ Ψ + [Aμ , Ψ] The equations of motion are 1 2 1 † iD0 Ψ = − D Ψ − Ψ, Ψ , Ψ 2 2
(2)
i Fμν = − εμνρ J ρ 2
(3)
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The Hamiltonian density is 1 2 1 H = T r (Di Ψ)† (Di Ψ) − T r Ψ† , Ψ 2 4
(4)
Using the notation D± ≡ D1 ± iD2 † † T r (Di Ψ) (Di Ψ) = T r (D− Ψ) (D− Ψ) + 1 † † Tr Ψ Ψ, Ψ , Ψ 2 Then the energy density is 1 † H = T r (D− Ψ) (D− Ψ) ≥ 0 2
(5)
and the Bogomol’nyi inequality is saturated at self-duality D− Ψ = 0
† ∂+ A− − ∂− A+ + [A+ , A− ] = Ψ, Ψ
(6) (7)
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The static solutions of the self-duality equations : the algebraic ansatz: Ai =
r
Aai Ha , Ψ =
a=1
r
ψ a Ea + ψ M E−M
a=1
r M 2 † a 2 |ψ | Ha + ψ H−M Ψ ,Ψ =
(8)
a=1
1 2 −M 2 for ρ1 ≡ ψ , ρ2 ≡ ψ
ρ2 = const ρ−1 1
(9)
Δ ln ρ1 + 2(ρ1 − ρ−1 1 ) = 0
(10)
Δψ + γ sinh (βψ) = 0.
(11)
We then have
The water we drink is self-dual
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Lagrangian for 2D plasma in strong magnetic field: Non-Abelian sl (2, C), Chern-Simons, 6th order • gauge field, with “potential” Aμ , (μ = 0, 1, 2 for (t, x, y)) described by the Chern-Simons Lagrangean; • matter (“Higgs” or “scalar”) field φ described by the covariant kinematic Lagrangean (i.e. covariant derivatives, implementing the minimal coupling of the gauge and matter fields)
† • matter-field self-interaction given by a potential V φ, φ with 6th power of φ; • the matter and gauge fields belong to the adjoint representation of the algebra sl (2, C)
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L
=
2 −κεμνρ tr ∂μ Aν Aρ + Aμ Aν Aρ 3 † μ −tr (D φ) (Dμ φ) † −V φ, φ
(12)
Sixth order potential † 1 † 2 † 2 tr φ, φ φ φ . , φ − v φ, φ , φ − v V φ, φ† = 2 4κ (13) The Euler Lagrange equations are Dμ Dμ φ =
∂V ∂φ†
−κενμρ Fμρ = iJ ν
(14) (15)
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The energy can be written as a sum of squares. The self-duality eqs. D− φ
=
F+−
=
0
1 2 † † ± 2 v φ − φ, φ , φ , φ κ
(16)
The algebraic ansatz : in the Chevalley basis [E+ , E− ]
=
H
[H, E± ]
=
±2E±
tr (E+ E− )
2 tr H
=
1
=
2
(17)
The fields φ = φ1 E+ + φ2 E− A+ = aH, A− = −a∗ H
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Equations for the components of the density of vorticity (here for + ) 1 1 2 − Δ ln ρ1 = − 2 (ρ1 − ρ2 ) 2 (ρ1 + ρ2 ) − v 2 κ 1 1 − Δ ln ρ2 = 2 (ρ1 − ρ2 ) 2 (ρ1 + ρ2 ) − v 2 2 κ Δ ln (ρ1 ρ2 ) = 0
(18) (19)
introduce a single variable v 2 /4 ρ1 = ρ≡ 2 v /4 ρ2
(20)
and obtain 1 1 − Δ ln ρ = − 2 4
2
v 1 1 1 ρ− ρ+ −1 κ ρ 2 ρ 2
(21)
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This simplest form of the equation governing the stationary states of the CHM eq. Δψ +
1 sinh ψ (cosh ψ − 1) = 0 2
The ’mass of the photon’ is v2 1 m= = κ ρs κ
≡
cs
v2
≡
Ωci
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Coherent structures in plasma L-H
Streamfunction
vorticity shows a
The velocity changes
small ring
sign along radius.
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Coherent structures in plasma L-H (2)
Streamfunction
vorticity shows a
The velocity changes
small ring
sign along radius.
Now we have to do for the ring structure what we have done for the structures of Euler fluid and of plasma/atmosphere: can-we find a field theory that describes rings ? [Yes]
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Abelian dominance The last Lagrangian In certain cases the model collapses to an Abelian structure, where (φ, Aμ ) are complex scalar functions 1 ∗ 2 L = (D μ φ) (Dμ φ) + κεμνρ Aμ Fνρ − V |φ| 4 where ∂φ + ieAμ φ Dμ φ = μ ∂x and
2
V |φ|
2 e2 2 2 = 2 |φ| |φ| − v 2 κ
with metric g μν = (1, −1, −1)
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The equations of motion ∂V ∂φ∗
Dμ Dμ φ
=
−
1 μνρ ε Fνρ 2
=
Jρ
where J μ = ie [φ∗ (Dμ φ) − (Dμ φ)∗ φ]
From the second equation of motion B = − κe ρ one finds A0 =
κ B 1 ∂ [phase of (φ)] − 2 2 2e |φ| e ∂t
In a field theory one can obtain the energy-momentum tensor by writing the action with the explicit presence of the metric g μν
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followed by variation of the action to this metric. Tμν
=
∗
∗
(Dμ φ) (Dν φ) + (Dμ φ) (Dν φ)
∗ 2 −gμν (Dλ φ) (Dλ φ) − V |φ|
The energy is the time-time (00) component of this tensor
∗ ∗ 2 2 E = d r (D0 φ) (D0 φ) + (Dk φ) (Dk φ) + V |φ| 2 2 κ B ∂ |φ| ∗ 2 + 2 2 + (Dk φ) (Dk φ) + V |φ| = d2 r ∂t 4e |φ| 2
The second term imposes that B and |φ| vanish in the same points. 2 Then the magnetic flux lies in a ring around the zeros of |φ| .
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The SELF-DUALITY The energy is transformed similar to the Bogomolnyi form
2 2 E = d r |(Dx ± iDy ) φ|
2 2 2 κ −1 e ∂ |φ| 2 ∗ 2 + φ B ± φ |φ| − v + 2e κ ∂t 1 ±ev 2 Φ + dl · J 2 r=∞
Restrict to the states 1. static (∂/∂t ≡ 0); 2. the current goes to zero at infinity such that the last integral is zero.
Then the energy consists of a sum of squared terms plus an additional term that has a topological nature, proportional with the total magnetic flux through the area. F. Spineanu – Les Houches 2013 –
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Taking to zero the squared terms we get (Dx ± iDy ) φ
=
0 2
eB
=
m2 |φ| ∓ 2 v2
2
|φ| 1− 2 v
The mass parameter is 2 v m ≡ 2e2 κ These are the equations of self-duality and the energy in this case is bounded from below by the flux
E ≥ ev 2 |Φ|
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The equation for the ring-type vortex The first of the two SD equations can be written eAk = ±εkj ∂j ln |φ| + ∂ k [phase of φ] Replacing the potential in the second SD equation we get 2 2 |φ| |φ| Δ ln |φ|2 − m2 2 −1 =0 2 v v equation that is valid in points where |φ| = 0. For these points there is an additional term, a Dirac δ coming from taking the rotational operator applied on the term containing the phase of φ.
Δψ = exp (ψ) [exp (ψ) − 1] + 4π
N
δ (x − xj )
j=1
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The return of the topological constraint At infinity (|φ| v) the covariant derivative term goes to 0 Dk φ → 0 at r → ∞ ∂k φ + ieAk φ → 0 dl · ∇ ln (φ) = i d (phase of φ) = 2πin
(22)
r=∞
The flux is
2π n e The magnetic flux is discrete, integer multiple of a physical quantity. The topological constraint is ensured by a mapping from the circle at infinity into the circle representing the space of the internal phase of the field φ in the asymptotic region, S 1 → S 1 classified according to the first homotopy group,
1 π1 S = Z Φ=
d2 r (∇ × A) =
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The (plane) ring-type vortical structures The field theoretical model naturally provides such solutions. In the language of FT the physical model must have two vacua. 2
Then there are domains (|φ| = v1 , v2 , ...) and domain walls. An infinite domain wall can be rolled up into a circular step-like structure, separating a region of one vacuum from the region of the other vacuum. The transition looks like a kink. At the transition the vorticity has a maximum. This is a ring-type planar vortex. An example: |φ|2
=
0 (Coulombian vacuum) and
|φ|2
=
v 2 (Higgs vacuum)
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Such structure corresponds to a change of direction of the velocity of the flow, smoothly centered on a circle.
Two flows with opposite directions: in the center and at the edge of the tokamak (Candy)
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Conclusions • The planar rings of vorticity are structures. • There is a field theoretical description that shows that the rings are extrema of an action functional • this means that the rings belong to the family of priviledged states and the flow will try to organize itself in such coherent pattern. • the interpretation of the rings of vorticity: the point-like elementary vortices have both short range and long range interaction. The ring in Field Theoretical model separates the domains of two distinct vacua. F. Spineanu – Les Houches 2013 –
