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Transient random convection

On the generation of convective cells of like-sign of vorticity in strong radial temperature gradients Florin Spineanu and Madalina Vlad Association EURATOM-MEdC Romania National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania

F. Spineanu – TTG - Padova –

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Transient random convection

A model for the reversal of toroidal rotation • the increase of the gradient of the pressure triggers formation of cells of convection (similar to Rayleigh-Benard (RB), but with a single sign of vorticity); • the envelope of the periferic (same direction!) velocities of the convection cells is equivalent to poloidal velocity. The gradients that sustain the convection rolls also sustain this poloidal velocity against damping due to the magnetic pumping. • the fast increase of poloidal flow induces a high time derivative of the radial electric field; • the neoclassical polarization creates a series of parallel accelerations (kiks on each bounce) leading to an increase of the toroidal precession or its reversal; the source of energy is the work done by the radial electric field which is itself fastly growing. • the diffusion transfers on resistive scale the toroidal momentum.

F. Spineanu – TTG - Padova –

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Transient random convection

If we want improved confinement we better look for poloidal rotation Sheared poloidal rotation is highly efficient in reducing or suppressing the instabilities by: • shift of the eigenmode relative to the resonant surface • coupling of radial harmonics, leading to flow of the perturbation energy toward damped modes. Alternatively, the reduction of the turbulence radial correlation length by sheared flow. However vpol is damped by parallel viscosity (magnetic pumping) in ∼ τii . In the model described here the poloidal velocity is generated and sustained by the gradients of temperature and pressure. This drive can be higher than the magnetic damping. ((F.Spineanu, M.Vlad, H-mode TTG Meeting Oxford 2011, http://arxiv.org/pdf/1202.4426.pdf )

F. Spineanu – TTG - Padova –

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Transient random convection

An imperfect and transitory, Rayleigh-Benard-type, bifurcation

• generation of coherent structures consisting of a chain of rolls of convection in the meridional section of the plasma (as in the bifurcation conduction/convection in a Rayleigh-Benard classical instability • the role of the buoyancy is taken by the baroclinic term • They are supposed to have the same direction of rotation (equivalently, the same sign of the toroidal vorticity) due to the presence of the nonlinearity of the KdV type which breaks the reflection invariance of the vectorial nonlinearity, thus allowing single-sign chain of vortices

Single sign vorticity in non-neutral plasma (Durkin)

F. Spineanu – TTG - Padova –

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Transient random convection

It is clear that the generation of the chain of convection rolls must be a transitory event. A marginal stability regime can be established. Formulation of the problem We consider a layer of sheared rotation connex to a region dominated by drift-type turbulence. In the drift-turbulent region there is stochastic generation of strong vortical structures, on a range of space scales. Since the background has a gradient of vorticity a vortex will move in a direction which depends on the sign of its vorticity relative to the one of the background: (Shecter Dubin) • the prograde vortices are moving toward the maximum of vorticity and • the retrograde vortices are moving toward the minimum of vorticity. The prograde vortices will be absorbed by the sheared layer and they will contribute with their vorticity content to the momentum of the background flow. This is a source of momentum which sustains the sheared

F. Spineanu – TTG - Padova –

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Transient random convection

flow, in particular sustains the sheared poloidal rotation against the damping due to the magnetic pumping.

The large scale envelope of the perturbations of a sheared flow The later is described by the Rayleigh-Kuo equation, also known as the barotropic equation   ∂φ d2 U ∂φ ∂ ∂ 2 2  β φ × n ) · ∇ ] ∇ + +U ∇⊥ φ − + ε [(−∇ ⊥ ⊥ ⊥φ = 0 2 ∂y ∂t ∂y dx ∂y Here the coefficients are   Ti 1 L2  non-dimensional β ≡ Ωi 1 + Te Ln U0 ε≡

1 Te 1 1 non-dimensional U0 |e| B L

The analysis on multiple scales starts by introducing the variables on

F. Spineanu – TTG - Padova –

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Transient random convection

scales separated by the parameter ε: T1 = εt, T2 = ε2 t Y1 = εy, Y2 = ε2 y This gives for the derivatives ∂ ∂ ∂ ∂ → +ε + ε2 ∂t ∂t ∂T1 ∂T2

(1)

∂ ∂ ∂ 2 ∂ → +ε +ε ∂y ∂y ∂Y1 ∂Y2

(2)

The solution is adopted to be of the form φ = φ(1) + εφ(2) + ε2 φ(3) + · · ·

(3)

F. Spineanu – TTG - Padova –

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Transient random convection

After the expansion one obtains the equation ∂A ∂ 2 A ∂ 2 A 2 + + |A| A=0 + i 2 2 ∂t ∂x ∂y for the envelope of the perturbation field. The dynamics flow-vortices in the k space We take the geometry x ≡ y

≡

transversal on the layer of flow, radial in tokamak longitudinal, streamwise, poloidal in tokamak

Physical fields of the k-space representation of the dynamics A scalar function A (x, y, t) which is the complex amplitude of slow spatial variation of the envelope of the eddies of drift turbulence. From A (x, y, t) the components of the velocity are derived as  /B. The multiple space-time scale analysis leads vE = −∇A (x, y) × n F. Spineanu – TTG - Padova –

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Transient random convection

to the NSEq. The general structure of such this equation is i

∂A (x, y, t) = ΔA (x, y, t) − γ |A| A∗ ∂t

The interesting term is the nonlinearity (the last term) which may be seen as derived from the a self-interaction potential V [A (x, y, t)] of the Landau-Ginsburg type, i.e. ϕ4 -type. The later potential expresses the fact that the system is led to stay, at equilibrium, in some fixed, constant magnitude A and any departure is penalized by a higher magnitude of the action functional. Looking for an analytical model for A directly in the k-space we will use the structure which is suggested above. We expect to have a diffusion in k−space and the same self-limitting

F. Spineanu – TTG - Padova –

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Transient random convection

nonlinear term ∂A (kx , ky , t) i = ∂t



2

2

∂ ∂ + 2 ∂kx2 ∂ky



A (kx , ky , t) − γ |A| A∗ + S

and we have added a source that represents the interaction of the basic flow A (x, y, t) with the vortices that are randomly generated in the drift turbulence and are joining the flow. The field of the incoming vortices and of trapped convection The addition of drops of vorticity is represented by the k-space scalar function φ φ (kx , ky ) Several space scales are present in the structure of the function φ (x, y) in real space. The reason is that we must consider that the vortices that are continuously generated in the turbulent region have two characteristics: F. Spineanu – TTG - Padova –

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Transient random convection

• the space extension of individual vortices coming into the layer has a wide range of values. The range extends from vortices of the dimension ρs up to convective events representing streamers with cuasi-closed (roll-type) geometry and spatial extension given by the inverse of kθ = m/r for m of only few units. • the spatial distribution of the place where these vortical structures are reaching the layer of sheared flow is random over practically the circumference of the initial poloidal rotation. The assumptions lead to periodicity of φ in k-space. (1)

• One periodicity can be at the level of a smallest vortex, ky . This means that a sequence of maxima of φ (x, y) appear with spatial periodicity 2π (1)

ky and this corresponds to the addition of smallest vortices into the F. Spineanu – TTG - Padova –

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Transient random convection

flow layer. It is also implied that the spatial distribution of the places of contact, where the vortex is absorbed by the layer, is uniform. (2)

• Further, on a longer space scale there is another periodicity, ky . This means that the sequence on the lower scale (with much denser spatial granulations of vorticity, 2π (1) ) is modulated on a ky

longer space scale with a periodicity 2π (2)

ky

This is because larger vortices (but not yet convection rolls) are absorbed into the flow layer. • Etc. • Finally we can have the largest (accessible) spatial scale where a convective event occurs as a result of a stochastic event, or a F. Spineanu – TTG - Padova –

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Transient random convection

streamer sustained by the baroclinic term and having the close-up property. 2π (n)

ky

This periodicity in the y variable is subject to a combination with the periodicity in the x direction. The dependence is supposed to be linear: this implies for analytical description a hyperbolic operator  2  2 ∂ ∂ − 2 φ (kx , ky ) = source ∂kx2 ∂ky ∂2 2 ∂kx

∂2 ∂ky2

where the D ’Alamebertian operator kx ky ≡ − is the signature of the simultaneous and correlated periodicities on the poloidal and radial directions of the field φ representing the incoming drift vortices - up to convection rolls, generated in the drift

F. Spineanu – TTG - Padova –

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Transient random convection

turbulence region, moving on the gradient of vorticity-profile, and finally absorbed into the rotation layer. It is expected that |A|2 becomes a periodic function in the spectrum, retaining a periodic accumulation of energy on spectral sub-domains that are separated on a constant scale in k-space. This implies that 

2

2

∂ ∂ + 2 ∂kx2 ∂ky



2

2

|A| + β |A| ≈ 0

where β is a measure of the periodicity, i.e. of the intervals between 2 space scales in real space where the energy of the perturbed field |A| is significantly accumulated due to incoming (“falling”) drops of vorticity coming from the turbulence region. We can ignore the less significant periodicity in x (radial) direction

F. Spineanu – TTG - Padova –

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Transient random convection

and retain



∂2 ∂2 − 2 2 ∂kx ∂ky



∂2 2 φ (kx , ky ) ≈ −β 2 |A| ∂ky

Returning to the equation for A (kx , ky , t) we can specify the interaction between the two fields A and φ in the simplest way S = 2φ (kx , ky , t) A (kx , ky , t)

F. Spineanu – TTG - Padova –

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Transient random convection

It results: The Davey - Stewartson system in k - space  2  2 ∂ ∂A (kx , ky , t) ∂ ∗ = i + , k , t) − γ |A| A + 2φA A (k x y ∂t ∂kx2 ∂ky2   2 2 ∂ ∂2 ∂ 2 φ (k − , k , t) ≈ −γ |A| x y ∂kx2 ∂ky2 ∂ky2 Several analytical solutions are available (0412005 Fokas, Theta Function Solutions Chow). The solutions for the DS-I equation are obtained using the Hirota method, by Chow.   θ4 (αx, τ ) θ1 (βy, τ1 ) − θ1 (αx, τ ) θ4 (βy, τ1 ) A = λ2 exp (−iΩt) θ4 (αx, τ ) θ4 (βy, τ1 ) + θ1 (αx, τ ) θ1 (βy, τ1 ) and an expression for φ in terms of the Riemann θ functions, with τ a purely imaginary variable. It is possible to express the solution in

F. Spineanu – TTG - Padova –

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Transient random convection

terms of Jacobi elliptic functions.   S1 − S exp (−iΩt) A = λ2 1 + S1 S S1

=

S

=

√

ksn (rx, k)  k1 sn (ry, k1 )

with the relationships defining the constants r (1 + k) = s (1 + k1 ) λ22

= =

−r 2 k + s2 k1

 2 2 r 1+k

k1

2

(1 + k1 )

−



k (1 + k)

2

   E 2 2 2 Ω = 4r 1 − − r 1 + k + 2s2 k1 K F. Spineanu – TTG - Padova –

18 There are analytical solutions that have a very interesting structure: On y = longitudinal, streamwise, it results with very long wavelengths Transient random convection

A (x, y, t)

→

A (kx , ky , t)

=

located at ky ≈ 0 on ky and at kx in an interval around some finite kx(0)

CONCLUSIONS The drift turbulence can generate randomly vortices and convection cells. These evolve differently (according to the relative sign of the vorticity) in a background of vorticity gradient. The incoming vortices will deposit their vorticity sustaining the large scale angular momentum of the poloidal rotation (a kind of discrete Reynolds stress). The process is transient on very short time scale and the polarization radial electric field has high time derivative, sustaining the toroidal acceleration of ion bananas. F. Spineanu – TTG - Padova –