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text: Projection

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text: annotated

text: bibliog-

text: raphy

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text: books

text: and

text: reviews,

text: Optimization

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text: Censor

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text: Mansour,

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text: tures

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text: Optimization:

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text: NY,

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text: P.L.

text: Combettes,

text: Solving

text: monotone

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text: compositions

text: of

text: non-

text: expansive

text: averaged

text: operators,

text: Optimization

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text: Douglas

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text: H.H.

text: Rachford,

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label: paragraph


text: (cid:90) (cid:90) √ ≡ − ≡ √M − S d4x g ρ ( x ) uµ uν , ρ ( x ) dτ δ ( x x (τ) ) (2.1) M g and S is the sum of the Einstein-Hilbert action and a correction due to unparticles U (cid:90) (cid:34) (cid:18) (cid:19) (cid:35) 1 √ A κ2∗ −D2 1−dU −1 S = d4x g 1 + − dU R. (2.2) U 2κ2 ( 2d 1 ) sin ( π d ) κ2 Λ2 U U U In the above expression D2 stands for the D’Lambertian, ≡ 16π5/2 Γ ( d + 1/2 ) A −U (2.3) dU ( 2π )2dU Γ ( dU 1 ) Γ ( 2dU ) (cid:18) (cid:19) and ≡ 1 Λ dUV κ∗ U . (2.4) Λ M U U The strength of the coupling constant is determined by the mass scale M which U replaces the Planck mass. By assuming a static source the authors found the analytical (cid:34) (cid:35) (cid:18) (cid:19) solution for the spherically symmetric line elements. They are given by − − − − 2M ± R 2dU 2 g 1 = g = 1 1 s,v (2.5) rr 00 r r (cid:34) (cid:35) where 1− − M2 κ2 − 2dU 2 Γ ( d 1/2 ) Γ ( d + 1/2 ) R = Γ Pl. −s,v Λ2 2dU ; Γ = U U , s,v U π2dU 1 U U Γ ( 2dU ) k are the coupling of the s, v unparticles with gravity and M is the plank mass. The s,v Pl plus signal is taken for the scalar (tensor) unparticle case (R ) and the minus one for the s − vector case (R ). These solutions has horizon curves deﬁned by g 1 = 0 v rr r 1 H M = ± − . (2.6) 2 1 ( R /r )2dU 2 s,v H The authors in Refs [29, 30] distinguish between two regimes. The gravity dominated (cid:28) (GD) regime is deﬁned by R r and the ungravity dominated (UGD) regime by (cid:28) s,v H r R . In the GD regime, for both cases, the mass parameter reduces to standard H s,v M = r /2. However in the UGD regime the mass is given by H − ≈ ±1 r2dU 1 M H − . (2.7) 2 R2dU 2 s,v – 3 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: √M

text: S

text: g

text: ρ

text: x

text: uµ

text: uν

text: ρ

text: x

text: dτ

text: δ

text: x

text: x

text: (τ)

text: M

text: g

text: and

text: S

text: is

text: the

text: sum

text: of

text: the

text: Einstein-Hilbert

text: action

text: and

text: a

text: correction

text: due

text: to

text: unparticles

text: U

text: A

text: κ2∗

text: −1

text: S

text: g

text: dU

text: R.

text: U

text: sin

text: π

text: d

text: κ2

text: Λ2

text: U

text: U

text: U

text: In

text: the

text: above

text: expression

text: stands

text: for

text: the

text: D’Lambertian,

text: Γ

text: d

text: A

text: −U

text: dU

text: 2π

text: Γ

text: dU

text: Γ

text: and

text: Λ

text: dUV

text: κ∗

text: U

text: Λ

text: M

text: U

text: U

text: The

text: strength

text: of

text: the

text: coupling

text: constant

text: is

text: determined

text: by

text: the

text: mass

text: scale

text: M

text: which

text: U

text: replaces

text: the

text: Planck

text: mass.

text: By

text: assuming

text: a

text: static

text: source

text: the

text: authors

text: found

text: the

text: analytical

text: solution

text: for

text: the

text: spherically

text: symmetric

text: line

text: elements.

text: They

text: are

text: given

text: by

text: R

text: g

text: g

text: s,v

text: rr

text: r

text: r

text: where

text: 1−

text: κ2

text: Γ

text: d

text: Γ

text: d

text: R

text: Γ

text: Pl.

text: −s,v

text: Λ2

text: Γ

text: U

text: U

text: s,v

text: U

text: U

text: U

text: Γ

text: k

text: are

text: the

text: coupling

text: of

text: the

text: s,

text: v

text: unparticles

text: with

text: gravity

text: and

text: M

text: is

text: the

text: plank

text: mass.

text: The

text: s,v

text: Pl

text: plus

text: signal

text: is

text: taken

text: for

text: the

text: scalar

text: (tensor)

text: unparticle

text: case

text: (R

text: and

text: the

text: minus

text: one

text: for

text: the

text: s

text: vector

text: case

text: (R

text: These

text: solutions

text: has

text: horizon

text: curves

text: deﬁned

text: by

text: g

text: v

text: rr

text: r

text: H

text: M

text: R

text: /r

text: s,v

text: H

text: The

text: authors

text: in

text: Refs

text: distinguish

text: between

text: two

text: regimes.

text: The

text: gravity

text: dominated

text: (GD)

text: regime

text: is

text: deﬁned

text: by

text: R

text: r

text: and

text: the

text: ungravity

text: dominated

text: (UGD)

text: regime

text: by

text: s,v

text: H

text: r

text: R

text: In

text: the

text: GD

text: regime,

text: for

text: both

text: cases,

text: the

text: mass

text: parameter

text: reduces

text: to

text: standard

text: H

text: s,v

text: M

text: r

text: However

text: in

text: the

text: UGD

text: regime

text: the

text: mass

text: is

text: given

text: by

text: H

text: ±1

text: M

text: H

text: s,v

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Therefore, for the vector case, the mass parameter becomes negative in the UGD regime. In fact for the entire region r < R this parameter is negative. This fact lead H v the authors of Ref. [30] to ignore this region. However it is a known fact that a negative mass parameter indicates black hole solution with nontrivial topology [34]. This suggests that beyond the fractalization of the horizon the vector unparticle can in fact contribute to changes in the black hole topology. Since the UGDR regime must be contained in a full quantum gravity theory, it is expected that quantum ﬂuctuations in geometry should enhance quantum ﬂuctuations in topology [35]. In fact in the incoming sections we will see that this is a very rich region, with phase transitions and stable black holes. The Hawking temperature can also be computed. It is given by [36] κ T = , (2.8) H 2π (cid:34) (cid:18) (cid:48) (cid:19) (cid:35) where κ is the surface gravity which in this case is given κ = g (r )/2 and we get 00 H (cid:20) (cid:21) (cid:16) (cid:17) − 1 ± − R 2dU 2 s,v TdU = 4π r 1 ± Rs,v 2dU−2 1 ( 2du 1 ) rH . (2.9) H rH In the GD regime the temperature reduces to the usual expression for the Hawking temperature. However in the UGD regime it is given by − ≈ 2d 1 U T . (2.10) dU 4πr H By comparing this with the Hawking temperature for a Schwarzschild black hole in D spacetime dimensions [37] − ≈ D 3 T (2.11) D 4πr H and than they argue that unparticle gravity leads to a fractalization of the event horizon with ≡ − d = 2d ; d D 2. (2.12) H U H In order to enforce the fractalization of the event horizon in the UGD regime the authors compute the entropy of the system. At this point we must be careful. It is very easy to see that the standard expression for entropy S = πr2 is not correct for this H corresponds to dS = 2πr which is very diﬀerent of dM/T as given above. Therefore the correct procedure is to obtain the entropy by integrating dS = dM/T . In the UGD regime we use Eqs. (2.10) and (2.7) to obtain – 4 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Therefore,

text: for

text: the

text: vector

text: case,

text: the

text: mass

text: parameter

text: becomes

text: negative

text: in

text: the

text: UGD

text: regime.

text: In

text: fact

text: for

text: the

text: entire

text: region

text: r

text: R

text: this

text: parameter

text: is

text: negative.

text: This

text: fact

text: lead

text: H

text: v

text: the

text: authors

text: of

text: Ref.

text: to

text: ignore

text: this

text: region.

text: However

text: it

text: is

text: a

text: known

text: fact

text: that

text: a

text: negative

text: mass

text: parameter

text: indicates

text: black

text: hole

text: solution

text: with

text: nontrivial

text: topology

text: This

text: suggests

text: that

text: beyond

text: the

text: fractalization

text: of

text: the

text: horizon

text: the

text: vector

text: unparticle

text: can

text: in

text: fact

text: contribute

text: to

text: changes

text: in

text: the

text: black

text: hole

text: topology.

text: Since

text: the

text: UGDR

text: regime

text: must

text: be

text: contained

text: in

text: a

text: full

text: quantum

text: gravity

text: theory,

text: it

text: is

text: expected

text: that

text: quantum

text: ﬂuctuations

text: in

text: geometry

text: should

text: enhance

text: quantum

text: ﬂuctuations

text: in

text: topology

text: In

text: fact

text: in

text: the

text: incoming

text: sections

text: we

text: will

text: see

text: that

text: this

text: is

text: a

text: very

text: rich

text: region,

text: with

text: phase

text: transitions

text: and

text: stable

text: black

text: holes.

text: The

text: Hawking

text: temperature

text: can

text: also

text: be

text: computed.

text: It

text: is

text: given

text: by

text: κ

text: T

text: H

text: 2π

text: where

text: κ

text: is

text: the

text: surface

text: gravity

text: which

text: in

text: this

text: case

text: is

text: given

text: κ

text: g

text: (r

text: and

text: we

text: get

text: H

text: R

text: s,v

text: TdU

text: 4π

text: r

text: Rs,v

text: rH

text: H

text: rH

text: In

text: the

text: GD

text: regime

text: the

text: temperature

text: reduces

text: to

text: the

text: usual

text: expression

text: for

text: the

text: Hawking

text: temperature.

text: However

text: in

text: the

text: UGD

text: regime

text: it

text: is

text: given

text: by

text: U

text: T

text: dU

text: H

text: By

text: comparing

text: this

text: with

text: the

text: Hawking

text: temperature

text: for

text: a

text: Schwarzschild

text: black

text: hole

text: in

text: D

text: spacetime

text: dimensions

text: D

text: T

text: D

text: H

text: and

text: than

text: they

text: argue

text: that

text: unparticle

text: gravity

text: leads

text: to

text: a

text: fractalization

text: of

text: the

text: event

text: horizon

text: with

text: d

text: d

text: D

text: H

text: U

text: H

text: In

text: order

text: to

text: enforce

text: the

text: fractalization

text: of

text: the

text: event

text: horizon

text: in

text: the

text: UGD

text: regime

text: the

text: authors

text: compute

text: the

text: entropy

text: of

text: the

text: system.

text: At

text: this

text: point

text: we

text: must

text: be

text: careful.

text: It

text: is

text: very

text: easy

text: to

text: see

text: that

text: the

text: standard

text: expression

text: for

text: entropy

text: S

text: is

text: not

text: correct

text: for

text: this

text: H

text: corresponds

text: to

text: dS

text: which

text: is

text: very

text: diﬀerent

text: of

text: dM/T

text: as

text: given

text: above.

text: Therefore

text: the

text: correct

text: procedure

text: is

text: to

text: obtain

text: the

text: entropy

text: by

text: integrating

text: dS

text: dM/T

text: In

text: the

text: UGD

text: regime

text: we

text: use

text: Eqs.

text: and

text: to

text: obtain

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: References [1] N. Aronszajn, Theory of Reproducing Kernels Transactions of the American Mathematical Society, 68, No. 3 (1950), pp.337-404 [2] L. Schwartz, Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math., 13 (1964), p. 115-256. [3] S. Saitoh et Y. Sawano, Theory of Reproducing Kernels and Applications, Springer, Developments in Mathematics, 2016 [4] J-L. Basdevant, Les principes variationnels en physique, Vuibert, 2014 [5] D. Renard, Groupes et représentations, Ecole polytechnique, 2009 [6] S. Weinberg, The Quantum Theory of Fields, Vol 1 Foundations, Cambridge University Press, 1995 15

text: References

text: N.

text: Aronszajn,

text: Theory

text: of

text: Reproducing

text: Kernels

text: Transactions

text: of

text: the

text: American

text: Mathematical

text: Society,

text: No.

text: L.

text: Schwartz,

text: Sous-espaces

text: hilbertiens

text: d’espaces

text: vectoriels

text: topologiques

text: et

text: noyaux

text: associés

text: (noyaux

text: reproduisants),

text: J.

text: Analyse

text: Math.,

text: p.

text: S.

text: Saitoh

text: et

text: Y.

text: Sawano,

text: Theory

text: of

text: Reproducing

text: Kernels

text: and

text: Applications,

text: Springer,

text: Developments

text: in

text: Mathematics,

text: J-L.

text: Basdevant,

text: Les

text: principes

text: variationnels

text: en

text: physique,

text: Vuibert,

text: D.

text: Renard,

text: Groupes

text: et

text: représentations,

text: Ecole

text: polytechnique,

text: S.

text: Weinberg,

text: The

text: Quantum

text: Theory

text: of

text: Fields,

text: Vol

text: Foundations,

text: Cambridge

text: University

text: Press,


text: 𝑥 𝑛 𝑥 0 𝑥 𝑖+1 𝑥 𝑖 𝑥 1 Figure 2: Adding measures limits the possibilities, as the wave function collapse occurs at every point . x i Lemma 2.2. is a positive deﬁnite kernel under a boundary condition where is K l Feyn P a positive constant: 2 2 max | ( )| ≤ π and ( ) ≡ ≥ 3 · π K x , x +1 K x , x A i=1...n Feyn i i lP Feyn i i lP P P itiviPtyrofoorf:anKyFseeyqnuiesnhceer(mait)ia≤n. ∈LeCtNS. := Ni=1 Nj=1 aiajK(xi, xj), we must show its pos- i i N The proof in the general case amounts to apply the multiplicity of to recover K Feyn the elementary pieces ( ). K x , x +1 Feyn i i Let (x ) =2··· −1 belonging to a trajectory joining x1 to x . Thanks to equation (2), i i N N we may express the sum adequately based on the ( ). Let be the S K x , x +1 K Feyn i i tr.sup. matrix of the ﬁrst diagonal of the normalized Gram matrix and the largest module of α the ( )  K x , x +1 Feyn i i     0 K (x1, x2)/A 0 . . . 0  Feyn  0 0 ( ) 0 K := KFeyn x2, x3 /A . . . α := max |K (x , x +1)|. tr.sup. . . . . . . . . . . . . . . . . . . . . . . i=1...n Feyn i i 0 0 0 0 . . . 9 ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: 𝑥

font: default

text: 𝑛

text: 𝑥

text: 𝑥

text: 𝑥

text: 𝑖

text: 𝑥

text: Figure

text: Adding

label: caption

text: measures

text: limits

text: the

text: possibilities,

text: as

text: the

text: wave

text: function

text: collapse

text: occurs

text: at

text: every

text: point

text: x

text: i

text: Lemma

text: is

text: a

text: positive

text: deﬁnite

text: kernel

text: under

text: a

text: boundary

text: condition

text: where

text: is

text: K

text: l

text: Feyn

text: P

text: a

text: positive

text: constant:

text: max

text: π

text: and

text: π

text: K

text: x

text: x

text: K

text: x

text: x

text: A

text: Feyn

text: i

text: i

text: lP

text: Feyn

text: i

text: i

text: lP

text: P

text: P

text: ∈LeCtNS.

text: aiajK(xi,

text: xj),

text: we

text: must

text: show

text: its

text: pos-

text: i

text: i

text: N

text: The

text: proof

text: in

text: the

text: general

text: case

text: amounts

text: to

text: apply

text: the

text: multiplicity

text: of

text: to

text: recover

text: K

text: Feyn

text: the

text: elementary

text: pieces

text: K

text: x

text: x

text: Feyn

text: i

text: i

text: Let

text: (x

text: =2···

text: −1

text: belonging

text: to

text: a

text: trajectory

text: joining

text: to

text: x

text: Thanks

text: to

text: equation

text: i

text: i

text: N

text: N

text: we

text: may

text: express

text: the

text: sum

text: adequately

text: based

text: on

text: the

text: Let

text: be

text: the

text: S

text: K

text: x

text: x

text: K

text: Feyn

text: i

text: i

text: tr.sup.

text: matrix

text: of

text: the

text: ﬁrst

text: diagonal

text: of

text: the

text: normalized

text: Gram

text: matrix

text: and

text: the

text: largest

text: module

text: of

text: α

text: the

text: K

text: x

text: x

text: Feyn

text: i

text: i

text: 0

text: K

text: 0

text: Feyn

text: K

text: KFeyn

text: /A

text: α

text: max

text: |K

text: (x

text: x

text: tr.sup.

text: Feyn

text: i

text: i

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ! We rewrite thXe dXouble sum : X (cid:16) (cid:17) S N N · † · N † · S := a a K(x , x ) = A a Id + (K )l + (K )l a i j i j tr.sup. tr.sup. =1 =1 =1 i j l We poXsit(cid:16) (cid:17) R := N (K )l + (K† )l implying that A·a† (Id + R) a = S ≥ 0 ⇔ kak2 ≥ |a†Ra| tr.sup. tr.sup. =1 l We thus have to limit †R : X a a − |a†Ra| ≤ 2kak2 · N (α/A)k = 2α/A · kak2 · 1 1(−α/A)N+1 α/A =1 l Consequently: − 2 · k k2 · 1 (α/A)N+1 ≤ · k k2 ⇒ ≥ 0 α a 1 − A a S α/A We restrict ourselves to convergent sums for which 1 and we take the limit in : α/A < N 2 2 ≤ · (1 − ) ⇔ ≤ A =: = π ⇒ ≥ 0 α A α/A α 3 kp S l P 2 It thus suﬃces that: max | ( )| ≤ π = A. i=1...n KFeyn xi, xi+1 lP 3 (cid:4) Remark: This boundary condition is furthermore necessary and suﬃcient for an inﬁnite number of measurements. If the sequence ( ) is taken constant equal to 1 and the such a x that ( ) = , then: i i ! K xX, x +1 α X X X Feyn i i S = N · A + (α/A)| i−j| = N · A + 2A N N−!i(!α/A)l = N · A + 2α N 1 − 1(α−/A)N+1−i 6= =1 =1 =1 α/A i j i l i − = · + 2α − 1 (α/A)N+1 N A 1 − N α 1 − !! α/A α/A − ≥ 0 ⇔ ≥ 2α 1 − α/A · 1 (α/A)N+1 −−−−−→ 2α S A 1 − α/A N 1 − α/A N→+∞ 1 − α/A Remarks: • is the Hilbert space of complex-valued functions which is the completion of the V Feyn · · one generated by the wave functions ( ) := ( ) of a particle which has ψ K x0, r0,t0 Feyn been measured in (which may be the point of its destructive measurement). x0 • is a positive deﬁnite kernel thanks to a concatenation property of the trajec- K Feyn tories. 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: rewrite

text: thXe

text: dXouble

text: sum

text: X

text: S

text: N

text: N

text: N

text: S

text: a

text: a

text: K(x

text: x

text: A

text: a

text: Id

text: (K

text: )l

text: (K

text: )l

text: a

text: i

text: j

text: i

text: j

text: tr.sup.

text: tr.sup.

text: i

text: j

text: l

text: We

text: R

text: N

text: (K

text: )l

text: (K†

text: )l

text: implying

text: that

text: A·a†

text: (Id

text: R)

text: a

text: S

text: |a†Ra|

text: tr.sup.

text: tr.sup.

text: l

text: We

text: thus

text: have

text: to

text: limit

text: †R

text: X

text: a

text: a

text: |a†Ra|

text: N

text: (α/A)k

text: α/A

text: l

text: Consequently:

text: k

text: k

text: α

text: a

text: A

text: a

text: S

text: α/A

text: We

text: restrict

text: ourselves

text: to

text: convergent

text: sums

text: for

text: which

text: and

text: we

text: take

text: the

text: limit

text: in

text: α/A

text: N

text: A

text: π

text: α

text: A

text: α/A

text: α

text: kp

text: S

text: l

text: P

text: It

text: thus

text: suﬃces

text: that:

text: max

text: π

text: A.

text: KFeyn

text: xi,

text: lP

text: Remark:

text: This

text: boundary

text: condition

text: is

text: furthermore

text: necessary

text: and

text: suﬃcient

text: for

text: an

text: inﬁnite

text: number

text: of

text: measurements.

text: If

text: the

text: sequence

text: is

text: taken

text: constant

text: equal

text: to

text: and

text: the

text: such

text: a

text: x

text: that

text: then:

text: i

text: i

text: K

text: xX,

text: x

text: α

text: X

text: X

text: X

text: Feyn

text: i

text: i

text: S

text: N

text: A

text: (α/A)|

text: i−j|

text: N

text: A

text: N

text: N−!i(!α/A)l

text: N

text: A

text: 2α

text: N

text: α/A

text: i

text: j

text: i

text: l

text: i

text: 2α

text: N

text: A

text: N

text: α

text: α/A

text: α/A

text: 2α

text: α/A

text: 2α

text: S

text: A

text: α/A

text: N

text: α/A

text: N→+∞

text: α/A

text: Remarks:

text: is

text: the

text: Hilbert

text: space

text: of

text: complex-valued

text: functions

text: which

text: is

text: the

text: completion

text: of

text: the

text: V

text: Feyn

text: one

text: generated

text: by

text: the

text: wave

text: functions

text: of

text: a

text: particle

text: which

text: has

text: ψ

text: K

text: Feyn

text: been

text: measured

text: in

text: (which

text: may

text: be

text: the

text: point

text: of

text: its

text: destructive

text: measurement).

text: is

text: a

text: positive

text: deﬁnite

text: kernel

text: thanks

text: to

text: a

text: concatenation

text: property

text: of

text: the

text: trajec-

text: K

text: Feyn

text: tories.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Contents 1 Introduction 1 2 All We Need to Know about the G(2) Group 2 3 The Thick Center Vortex and Casimir Scaling 4 4 The Conﬁnement Behavior of the G(2) Group Using The Thick Center Vortex Model 10 5 The internal SU(3) Thick Center Vortex within the G(2) Gauge Theory 17 6 Conclusion 19 1 Introduction Quantum Chromo dynamics (QCD) at low energies is dominated by non-perturbative phenomena of the quark conﬁnement and spontaneous chiral symmetry breaking(SCSB). Quarks and gluons as the building blocks of a gauge theory for the strong interaction are not present in the QCD spectrum. This leads to the conﬁnement phenomena [1]. Now a days from lattice gauge theory simulations we know that the conﬁnement phenomenon is present in a non abelian gauge theory [2–5]. So we must search for a model or mechanism to satisfy the conﬁnement problem in a non abelian gauge theory and it is not essential to have a super symmetric theory for example to have conﬁnement [6]. A good model or theory of conﬁnement should convey all of the properties of the conﬁnement which is present in the lattice gauge theory calculation. One of the way to study the problems of conﬁnement in a theory is to obtain the potential part of the interaction energy between the static sources. The kinetic part of energy can be eliminated if we study the quark when they are massive [7, 8]. Studying static potential can describe the properties of the conﬁnement in a gauge theory. According to the Regge trajectories from experimental data [9–12] and also the results from the lattice gauge theory [1, 4, 5] , the phenomenological models for the quark conﬁnement are introduced [1, 8, 13, 13] . In these models, the QCD vacuum is ﬁlled with some topological conﬁgurations conﬁning colored objects. The most popular candidates among these topological ﬁelds are monopoles and vortices. Other candidates include instantons, merons, calorons, dyons. However Greensites shows that only the vortex model could convey a diﬀerence in-area-law [15] and other models such as monopoles gas, caloron ensemble, or dual abelian Higgs actions cannot convey the law. Each of these defects has some advantages relative to the others and the conﬁnement problem can be studied with in these models. For example calaron can explain the relation of the conﬁnement to the temperature or the thick center vortex – 1 – ##LTLine## ##LTLine##

text: Contents

text: Introduction

text: All

text: We

text: Need

text: to

text: Know

text: about

text: the

text: Group

text: The

text: Thick

text: Center

text: Vortex

text: and

text: Casimir

text: Scaling

text: The

text: Conﬁnement

text: Behavior

text: of

text: the

text: Group

text: Using

text: The

text: Thick

text: Center

text: Vortex

text: Model

text: The

text: internal

text: Thick

text: Center

text: Vortex

text: within

text: the

text: Gauge

text: Theory

text: Conclusion

label: section

text: Introduction

text: Quantum

text: Chromo

text: dynamics

text: (QCD)

text: at

text: low

text: energies

text: is

text: dominated

text: by

text: non-perturbative

text: phenomena

text: of

text: the

text: quark

text: conﬁnement

text: and

text: spontaneous

text: chiral

text: symmetry

text: breaking(SCSB).

text: Quarks

text: and

text: gluons

text: as

text: the

text: building

text: blocks

text: of

text: a

text: gauge

text: theory

text: for

text: the

text: strong

text: interaction

text: are

text: not

text: present

text: in

text: the

text: QCD

text: spectrum.

text: This

text: leads

text: to

text: the

text: conﬁnement

text: phenomena

text: Now

text: a

text: days

text: from

text: lattice

text: gauge

text: theory

text: simulations

text: we

text: know

text: that

text: the

text: conﬁnement

text: phenomenon

text: is

text: present

text: in

text: a

text: non

text: abelian

text: gauge

text: theory

text: So

text: we

text: must

text: search

text: for

text: a

text: model

text: or

text: mechanism

text: to

text: satisfy

text: the

text: conﬁnement

text: problem

text: in

text: a

text: non

text: abelian

text: gauge

text: theory

text: and

text: it

text: is

text: not

text: essential

text: to

text: have

text: a

text: super

text: symmetric

text: theory

text: for

text: example

text: to

text: have

text: conﬁnement

text: A

text: good

text: model

text: or

text: theory

text: of

text: conﬁnement

text: should

text: convey

text: all

text: of

text: the

text: properties

text: of

text: the

text: conﬁnement

text: which

text: is

text: present

text: in

text: the

text: lattice

text: gauge

text: theory

text: calculation.

text: One

text: of

text: the

text: way

text: to

text: study

text: the

text: problems

text: of

text: conﬁnement

text: in

text: a

text: theory

text: is

text: to

text: obtain

text: the

text: potential

text: part

text: of

text: the

text: interaction

text: energy

text: between

text: the

text: static

text: sources.

text: The

text: kinetic

text: part

text: of

text: energy

text: can

text: be

text: eliminated

text: if

text: we

text: study

text: the

text: quark

text: when

text: they

text: are

text: massive

text: Studying

text: static

text: potential

text: can

text: describe

text: the

text: properties

text: of

text: the

text: conﬁnement

text: in

text: a

text: gauge

text: theory.

text: According

text: to

text: the

text: Regge

text: trajectories

text: from

text: experimental

text: data

text: and

text: also

text: the

text: results

text: from

text: the

text: lattice

text: gauge

text: theory

text: the

text: phenomenological

text: models

text: for

text: the

text: quark

text: conﬁnement

text: are

text: introduced

text: In

text: these

text: models,

text: the

text: QCD

text: vacuum

text: is

text: ﬁlled

text: with

text: some

text: topological

text: conﬁgurations

text: conﬁning

text: colored

text: objects.

text: The

text: most

text: popular

text: candidates

text: among

text: these

text: topological

text: ﬁelds

text: are

text: monopoles

text: and

text: vortices.

text: Other

text: candidates

text: include

text: instantons,

text: merons,

text: calorons,

text: dyons.

text: However

text: Greensites

text: shows

text: that

text: only

text: the

text: vortex

text: model

text: could

text: convey

text: a

text: diﬀerence

text: in-area-law

text: and

text: other

text: models

text: such

text: as

text: monopoles

text: gas,

text: caloron

text: ensemble,

text: or

text: dual

text: abelian

text: Higgs

text: actions

text: cannot

text: convey

text: the

text: law.

text: Each

text: of

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text: (cid:29) Figure 4. A schematic view of the vortex thickness relative to a planer Wilson loop with T R. x is the position of the center of the vortex. R is the vortex thickness and relates to a, b. y(x) is C the nearest distance to the time like side of the Wilson loop. − b α(cid:126)n(x) = Nn[1 tanh(ay(x) + ], (4.1) c i R Nn is the normalization number for the vortex type n and i is due to the Cartan i generators. a and b are the free constants of the proﬁle, and y(x) is (cid:26) − | − | x R x > x y(x) = − | − | ≤ , (4.2) x R R x x (cid:29) A schematic view of the vortex thickness relative to a planer Wilson loop with T R is shown in ﬁgure 4. y(x) is the nearest distance of x from time like side of the Wilson loop. R is the vortex thickness and relates to a, b. The normalization constants Nn are C i obtained from the maximum ﬂux condition where the loop contains the vortex completely exp(iα(cid:126)n.H(cid:126) ) = Z I, (4.3) n With ∈ 2πin zn = e N ZN, (4.4) – 11 – ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

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text: 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Re(gr(α)) Figure 6. The imaginary versus real value of g (α(x)) for the G(2) gauge theory. The elements r of G(2) are real and no imaginary part is present.The real value of g (α(x)) have values from 1 − r to 0.28. The minimum can be explain with the SU(3) center elements. So the SU(3) vortices − explain the value 0.28. 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Re gr(α) ⊕ Figure 7. Instead of G(2) group the imaginary versus real value for a symmetry with SU(3) ⊕ U(1) SU(3) structure is considered here. This leads to the presence of the imaginary parts. The − imaginary value of this ﬁgure are zero for 1 and 0.14 which can be explain the maximums of − Reg (α(x)) at 1 and 0.14. r The free parameters of the model are considered as a = 0.05, b = 4, f = 0.1 for the present numerical calculation. Figure 5 showes the fundemental potentail. The asymptotic screening behavior is obtained which it is also observed using lattice gauge theory [43]. An – 14 – )α(rg mI ))α(rg(mI ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Re(gr(α))

text: Figure

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text: imaginary

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text: for

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text: gauge

text: theory.

text: The

text: elements

text: r

text: of

text: are

text: real

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text: present.The

text: real

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text: have

text: values

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text: r

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text: minimum

text: can

text: be

text: explain

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text: the

text: center

text: elements.

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text: vortices

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text: value

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text: gr(α)

text: Figure

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text: can

text: be

text: explain

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text: maximums

text: of

text: Reg

text: (α(x))

text: at

text: and

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text: free

text: parameters

text: of

text: the

text: model

text: are

text: considered

text: as

text: a

text: b

text: f

text: for

text: the

text: present

text: numerical

text: calculation.

text: Figure

text: showes

text: the

text: fundemental

text: potentail.

text: The

text: asymptotic

text: screening

text: behavior

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text: obtained

text: which

text: it

text: is

text: also

text: observed

text: using

text: lattice

text: gauge

text: theory

text: An

text: )α(rg

text: mI

text: ))α(rg(mI

text: ##LTFigure##

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text: 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -0.5 0 0.5 1 Re(gr(α)) Figure 8. The imaginary versus real value of Reg (α(x)) for the SU(3) group. The real values − − r change from 0.5 to 1. 0.5 is the nontrivial center element value for this group. 12 fundemental adjoint 10 27 8 6 4 2 0 0 10 20 30 40 50 60 70 80 90 100 R Figure 9. The potential of static sources of quark anti quark with a G(2) gauge theory in the fundamental and adjoint and the 27 representations. The thick center vortex model is applied to the internal SU(3) vortices. The behavior of conﬁnement in the Casimir region and the asymptotic region can be explained using these internal vortices. ⊕ ⊕ intermediate linear potential is observerd. Also the potential of a SU(3) U(1) SU(3) − ⊕ symmetry is ploted. The second G(2) slope is the slope of 3 ality region of the SU(3) – 15 – )R(V ))α(rg(mI ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Re(gr(α))

text: Figure

text: The

text: imaginary

text: versus

text: real

text: value

text: of

text: Reg

text: (α(x))

text: for

text: the

text: group.

text: The

text: real

text: values

text: r

text: change

text: from

text: to

text: is

text: the

text: nontrivial

text: center

text: element

text: value

text: for

text: this

text: group.

text: fundemental

text: adjoint

text: R

text: Figure

text: The

text: potential

text: of

text: static

text: sources

text: of

text: quark

text: anti

text: quark

text: with

text: a

text: gauge

text: theory

text: in

text: the

text: fundamental

text: and

text: adjoint

text: and

text: the

text: representations.

text: The

text: thick

text: center

text: vortex

text: model

text: is

text: applied

text: to

text: the

text: internal

text: vortices.

text: The

text: behavior

text: of

text: conﬁnement

text: in

text: the

text: Casimir

text: region

text: and

text: the

text: asymptotic

text: region

text: can

text: be

text: explained

text: using

text: these

text: internal

text: vortices.

text: intermediate

text: linear

text: potential

text: is

text: observerd.

text: Also

text: the

text: potential

text: of

text: a

text: symmetry

text: is

text: ploted.

text: The

text: second

text: slope

text: is

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text: slope

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text: )R(V

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text: 2.4 k /k 2.2 14 7 k /k 27 7 2 1.8 1.6 1.4 1.2 1 0.8 0 5 10 15 20 25 30 35 40 R Figure 10. The potential ratio for k /k and k /k . Upper than 43 percent Casimir scaling is 14 7 27 7 observed. ⊕ U(1) SU(3) symmetry. We plot real versus imaginary part of the g (α) for G(2) in ﬁgure r 6 and in the ﬁgure 7 we plot this ratio considering two SU(3) subgroups of G(2) considering positivie ﬂux for both of the subgroup instead of considering the two subgroups with true ﬂuxes which destroy the imaginary part of each other as in the ﬁgure 6. This ﬁgure help us to understand the behavior of Reg (α) and its maximum and minimum. The same r ﬁgure is ploted for the SU(3) group in ﬁgure 8. In this ﬁgure the maximum is 1 and the − minimum is 0.5 which is due to the maximum value of the ﬂux for this group. This value 2πin obtained setting n = 0 in e 3 . The minimum is also obtained by n = 1. In the G(2) gauge theory also the properties of g (α) can be explained by its SU(3) subgroups. The r   absolute minimum is obtained [46, 47] from the SU(3) minimum as    (cid:48) ×  g (α) d 0 0 1 1 R min r Reg (α) = T r(eiαH) = T r 0 1 0 r min min 7 7 (cid:48) × 0 0 g (α) d R min r 1 − × − × − Reg (α) == ( 0.5 3 + 1 0.5 3) = 0.28. (4.9) r min 7 (cid:48) Which g (α) is the minimum value of the ﬂux in the SU(3) subgroup. Despite the R min maximum and the minimum of the Reg (α) can be obtained using the above equation 4.9 r the local minimum and maximum of Reg (α) can be exist. In the ﬁgure 11 the Reg (α) is r r – 16 – ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: k

text: /k

text: k

text: /k

text: R

text: Figure

text: The

text: potential

text: ratio

text: for

text: k

text: /k

text: and

text: k

text: /k

text: Upper

text: than

text: percent

text: Casimir

text: scaling

text: is

text: observed.

text: symmetry.

text: We

text: plot

text: real

text: versus

text: imaginary

text: part

text: of

text: the

text: g

text: (α)

text: for

text: in

text: ﬁgure

text: r

text: and

text: in

text: the

text: ﬁgure

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text: plot

text: this

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text: considering

text: two

text: subgroups

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text: positivie

text: ﬂux

text: for

text: both

text: of

text: the

text: subgroup

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text: the

text: two

text: subgroups

text: with

text: true

text: ﬂuxes

text: which

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text: the

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text: part

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text: each

text: other

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text: in

text: the

text: ﬁgure

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text: ﬁgure

text: help

text: us

text: to

text: understand

text: the

text: behavior

text: of

text: Reg

text: (α)

text: and

text: its

text: maximum

text: and

text: minimum.

text: The

text: same

text: r

text: ﬁgure

text: is

text: ploted

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text: group

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text: ﬁgure

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text: this

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text: the

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text: value

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text: setting

text: n

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text: e

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text: minimum

text: is

text: also

text: obtained

text: by

text: n

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text: the

text: gauge

text: theory

text: also

text: the

text: properties

text: of

text: g

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text: can

text: be

text: explained

text: by

text: its

text: subgroups.

text: The

text: r

text: absolute

text: minimum

text: is

text: obtained

text: from

text: the

text: minimum

text: as

text: g

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text: d

text: R

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text: r

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text: (α)

text: T

text: r(eiαH)

text: T

text: r

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text: min

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text: d

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text: the

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text: r

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text: local

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text: In

text: the

text: ﬁgure

text: the

text: Reg

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text: model can explain the Casimir scaling. Among these models I used the thick center vortex model to study the conﬁnement problem[16]. The center vortex model was initially introduced by ’t Hooft[13]. It is able to explain the conﬁnement of quark pairs at the asymptotic region, but it is not able to explain the conﬁnement at intermediate distances especially for the higher representations. The model then is modiﬁed to the thick center vortex model by Greensite, Faber, etc[16]. Within this model one can obtain the Casimir scaling and N-ality behavior of the gauge theory. This is done before for the SU(2), SU(3) and SU(4) [18–20]. Using another modiﬁcation of the model to the domain vacuum structures it is possible to describe the properties of gauge theory without non trivial center such as the G(2) gauge theory[21, 22]. In this article I try to show that G(2) has SU(3) subgroups decomposition which can explain the conﬁnment properties at intermediate and asymptotic regions. In the next section the G(2) group properties are introduced especially the topological properties for the presence of the topological solitonic structures. In section III the thick center vortex model is introduced and then the Casimir ratios of the G(2) are obtained analytically. Obtaining the G(2) Casimir ratios exactly with the use of its decomposition to the SU(3) subgroup leads to the conjecture that any properties of the conﬁnement should be governed by its SU(3) subgroups. To study this I obtain the behavior of the conﬁnement in the G(2) gauge theory in the section IV by applying the thick center vortex model to the SU(3) subgroups of the G(2) group. In the section V the properties of such ”internal vortices” are explained. 2 All We Need to Know about the G(2) Group The exceptional Lie group G(2) is the auto morphism group of the octonion algebra [35–39]. The group G(2) is a simply connected, compact group. The G(2) is its own covering group and its center is trivial. As it is clear the rank of the G(2) is 2 and it has 14 generators. In the Cartan sub algebra which is the representation which we have the most simultaneous diagonal generators for the generators of a group, the G(2) has two diagonal generators and the remaining 12 generators are not diagonal. Since there are 14 generators × the adjoint representation of G(2) can be introduced by 14 14 matrices. So it is a real group of 14 dimensional. Also its fundamental representation is 7 dimensional. Since it is the subgroup of SO(7) with rank 3 and 21 generators, its elements can be obtained by SO(7) elements obeyed the 7 constrained reduced to 14 generators of the fundamental × representation. If we consider Us the 7 7 real orthogonal matrices with determinant 1, then † UU = 1, (2.1) is the constrained matrices that are elements of SO(7). Within the constrained called cubic constrained which is – 2 –

text: model

text: can

text: explain

text: the

text: Casimir

text: scaling.

text: Among

text: these

text: models

text: I

text: used

text: the

text: thick

text: center

text: vortex

text: model

text: to

text: study

text: the

text: conﬁnement

text: The

text: center

text: vortex

text: model

text: was

text: initially

text: introduced

text: by

text: ’t

text: It

text: is

text: able

text: to

text: explain

text: the

text: conﬁnement

text: of

text: quark

text: pairs

text: at

text: the

text: asymptotic

text: region,

text: but

text: it

text: is

text: not

text: able

text: to

text: explain

text: the

text: conﬁnement

text: at

text: intermediate

text: distances

text: especially

text: for

text: the

text: higher

text: representations.

text: The

text: model

text: then

text: is

text: modiﬁed

text: to

text: the

text: thick

text: center

text: vortex

text: model

text: by

text: Greensite,

text: Faber,

text: Within

text: this

text: model

text: one

text: can

text: obtain

text: the

text: Casimir

text: scaling

text: and

text: N-ality

text: behavior

text: of

text: the

text: gauge

text: theory.

text: This

text: is

text: done

text: before

text: for

text: the

text: and

text: Using

text: another

text: modiﬁcation

text: of

text: the

text: model

text: to

text: the

text: domain

text: vacuum

text: structures

text: it

text: is

text: possible

text: to

text: describe

text: the

text: properties

text: of

text: gauge

text: theory

text: without

text: non

text: trivial

text: center

text: such

text: as

text: the

text: gauge

text: In

text: this

text: article

text: I

text: try

text: to

text: show

text: that

text: has

text: subgroups

text: decomposition

text: which

text: can

text: explain

text: the

text: conﬁnment

text: properties

text: at

text: intermediate

text: and

text: asymptotic

text: regions.

text: In

text: the

text: next

text: section

text: the

text: group

text: properties

text: are

text: introduced

text: especially

text: the

text: topological

text: properties

text: for

text: the

text: presence

text: of

text: the

text: topological

text: solitonic

text: structures.

text: In

text: section

text: III

text: the

text: thick

text: center

text: vortex

text: model

text: is

text: introduced

text: and

text: then

text: the

text: Casimir

text: ratios

text: of

text: the

text: are

text: obtained

text: analytically.

text: Obtaining

text: the

text: Casimir

text: ratios

text: exactly

text: with

text: the

text: use

text: of

text: its

text: decomposition

text: to

text: the

text: subgroup

text: leads

text: to

text: the

text: conjecture

text: that

text: any

text: properties

text: of

text: the

text: conﬁnement

text: should

text: be

text: governed

text: by

text: its

text: subgroups.

text: To

text: study

text: this

text: I

text: obtain

text: the

text: behavior

text: of

text: the

text: conﬁnement

text: in

text: the

text: gauge

text: theory

text: in

text: the

text: section

text: IV

text: by

text: applying

text: the

text: thick

text: center

text: vortex

text: model

text: to

text: the

text: subgroups

text: of

text: the

text: group.

text: In

text: the

text: section

text: V

text: the

text: properties

text: of

text: such

text: ”internal

text: vortices”

text: are

text: explained.

text: All

text: We

text: Need

text: to

text: Know

text: about

text: the

text: Group

text: The

text: exceptional

text: Lie

text: group

text: is

text: the

text: auto

text: morphism

text: group

text: of

text: the

text: octonion

text: algebra

text: The

text: group

text: is

text: a

text: simply

text: connected,

text: compact

text: group.

text: The

text: is

text: its

text: own

text: covering

text: group

text: and

text: its

text: center

text: is

text: trivial.

text: As

text: it

text: is

text: clear

text: the

text: rank

text: of

text: the

text: is

text: and

text: it

text: has

text: generators.

text: In

text: the

text: Cartan

text: sub

text: algebra

text: which

text: is

text: the

text: representation

text: which

text: we

text: have

text: the

text: most

text: simultaneous

text: diagonal

text: generators

text: for

text: the

text: generators

text: of

text: a

text: group,

text: the

text: has

text: two

text: diagonal

text: generators

text: and

text: the

text: remaining

text: generators

text: are

text: not

text: diagonal.

text: Since

text: there

text: are

text: generators

text: the

text: adjoint

text: representation

text: of

text: can

text: be

text: introduced

text: by

text: matrices.

text: So

text: it

text: is

text: a

text: real

text: group

text: of

text: dimensional.

text: Also

text: its

text: fundamental

text: representation

text: is

text: dimensional.

text: Since

text: it

text: is

text: the

text: subgroup

text: of

text: with

text: rank

text: and

text: generators,

text: its

text: elements

text: can

text: be

text: obtained

text: by

text: elements

text: obeyed

text: the

text: constrained

text: reduced

text: to

text: generators

text: of

text: the

text: fundamental

text: representation.

text: If

text: we

text: consider

text: Us

text: the

text: real

text: orthogonal

text: matrices

text: with

text: determinant

text: then

text: UU

text: is

text: the

text: constrained

text: matrices

text: that

text: are

text: elements

text: of

text: Within

text: the

text: constrained

text: called

text: cubic

text: constrained

text: which

text: is


text: exceptional group among these groups.Two properties of trivial center and its own corver- ing group are interesting for us to study the properties of the conﬁnement in the gauge theories with such symmetry. 3 The Thick Center Vortex and Casimir Scaling The idea of vortex model for the conﬁnement is due to ’t Hooft and Mandelstam [13, 14, 23–25]. They used the Nielson Oleson vortex solution to obtain the conﬁnement properties [26]. In the dual superconductivity the string between the sources of abelian electric charges is due to the abelian magnetic charge condensation and the string obey the vortex type equations(The string between quark-antiquark in a meson for example). In the center vortex picture presence of vortices in the vacuum is due to the center elements and their ﬂuctuation in the number of center vortices linked to the loop leads to an area law Wilson loop and a linear potential or string like behavior. In the dual superconductor the vortices have electric ﬂux, but in the center vortex picture the vorices have magnetic ﬂux. Kronfeld and etc suggested test ’t Hooft theory in the lattice gauge theory[27]. The thick center vortex model was introduced by Del Debbio, Faber, J. Greensite and Olejn´ık to obtain the intermediate behavior of the quark potential using the lattice gauge theory (LGT) results [28]. The center vortex is a topological ﬁeld conﬁguration which is line like in D=3 dimen- sional and surface like in D=4 dimension and have some ﬁnite thickness. A discontinuity in the background gauge transformation related to the gauge group center leads to a center vortex. The center vortex creation linked to a Wilson loop, in the fundamental repre- sentation of SU(N) changed the Wilson loop holonomy by an element of the gauge group center. −→ 2πni W (C) e N W0(C), (3.1) The conﬁnement is obtained from random ﬂuctuations in the Linking number. A vortex piercing a Wilson loop contribute with a center element Z somewhere between the group elements of the gauge group −→ W (C) = T r[UUU...U] T r[UU....(Z)U]. (3.2) Center elements commute with all members of the group, so the location of Z in eq. 3.2 can be changed by changing the place of discontinuity which leads to a vortex formation. We can obtain an equation for the string tension σ assuming that vortices are thin and pierce Wilson loops in single plaquette with independent probability f, for example for the SU(2) group we get (cid:89) (cid:104) (cid:105) { − − }(cid:104) (cid:105) − (cid:104) (cid:105) W (C) = (1 f) + f( 1) W (C) = exp[ σ(C)A] W (C) . (3.3) 0 0 – 5 – ##LTLine##

text: exceptional

text: group

text: among

text: these

text: groups.Two

text: properties

text: of

text: trivial

text: center

text: and

text: its

text: own

text: corver-

text: ing

text: group

text: are

text: interesting

text: for

text: us

text: to

text: study

text: the

text: properties

text: of

text: the

text: conﬁnement

text: in

text: the

text: gauge

text: theories

text: with

text: such

text: symmetry.

text: The

text: Thick

text: Center

text: Vortex

text: and

text: Casimir

text: Scaling

text: The

text: idea

text: of

text: vortex

text: model

text: for

text: the

text: conﬁnement

text: is

text: due

text: to

text: ’t

text: Hooft

text: and

text: Mandelstam

text: They

text: used

text: the

text: Nielson

text: Oleson

text: vortex

text: solution

text: to

text: obtain

text: the

text: conﬁnement

text: properties

text: In

text: the

text: dual

text: superconductivity

text: the

text: string

text: between

text: the

text: sources

text: of

text: abelian

text: electric

text: charges

text: is

text: due

text: to

text: the

text: abelian

text: magnetic

text: charge

text: condensation

text: and

text: the

text: string

text: obey

text: the

text: vortex

text: type

text: equations(The

text: string

text: between

text: quark-antiquark

text: in

text: a

text: meson

text: for

text: example).

text: In

text: the

text: center

text: vortex

text: picture

text: presence

text: of

text: vortices

text: in

text: the

text: vacuum

text: is

text: due

text: to

text: the

text: center

text: elements

text: and

text: their

text: ﬂuctuation

text: in

text: the

text: number

text: of

text: center

text: vortices

text: linked

text: to

text: the

text: loop

text: leads

text: to

text: an

text: area

text: law

text: Wilson

text: loop

text: and

text: a

text: linear

text: potential

text: or

text: string

text: like

text: behavior.

text: In

text: the

text: dual

text: superconductor

text: the

text: vortices

text: have

text: electric

text: ﬂux,

text: but

text: in

text: the

text: center

text: vortex

text: picture

text: the

text: vorices

text: have

text: magnetic

text: ﬂux.

text: Kronfeld

text: and

text: etc

text: suggested

text: test

text: ’t

text: Hooft

text: theory

text: in

text: the

text: lattice

text: gauge

text: The

text: thick

text: center

text: vortex

text: model

text: was

text: introduced

text: by

text: Del

text: Debbio,

text: Faber,

text: J.

text: Greensite

text: and

text: Olejn´ık

text: to

text: obtain

text: the

text: intermediate

text: behavior

text: of

text: the

text: quark

text: potential

text: using

text: the

text: lattice

text: gauge

text: theory

text: (LGT)

text: results

text: The

text: center

text: vortex

text: is

text: a

text: topological

text: ﬁeld

text: conﬁguration

text: which

text: is

text: line

text: like

text: in

text: dimen-

text: sional

text: and

text: surface

text: like

text: in

text: dimension

text: and

text: have

text: some

text: ﬁnite

text: thickness.

text: A

text: discontinuity

text: in

text: the

text: background

text: gauge

text: transformation

text: related

text: to

text: the

text: gauge

text: group

text: center

text: leads

text: to

text: a

text: center

text: vortex.

text: The

text: center

text: vortex

text: creation

text: linked

text: to

text: a

text: Wilson

text: loop,

text: in

text: the

text: fundamental

text: repre-

text: sentation

text: of

text: SU(N)

text: changed

text: the

text: Wilson

text: loop

text: holonomy

text: by

text: an

text: element

text: of

text: the

text: gauge

text: group

text: center.

text: W

text: (C)

text: e

text: N

text: The

text: conﬁnement

text: is

text: obtained

text: from

text: random

text: ﬂuctuations

text: in

text: the

text: Linking

text: number.

text: A

text: vortex

text: piercing

text: a

text: Wilson

text: loop

text: contribute

text: with

text: a

text: center

text: element

text: Z

text: somewhere

text: between

text: the

text: group

text: elements

text: of

text: the

text: gauge

text: group

text: W

text: (C)

text: T

text: r[UUU...U]

text: T

text: r[UU....(Z)U].

text: Center

text: elements

text: commute

text: with

text: all

text: members

text: of

text: the

text: group,

text: so

text: the

text: location

text: of

text: Z

text: in

text: eq.

text: can

text: be

text: changed

text: by

text: changing

text: the

text: place

text: of

text: discontinuity

text: which

text: leads

text: to

text: a

text: vortex

text: formation.

text: We

text: can

text: obtain

text: an

text: equation

text: for

text: the

text: string

text: tension

text: σ

text: assuming

text: that

text: vortices

text: are

text: thin

text: and

text: pierce

text: Wilson

text: loops

text: in

text: single

text: plaquette

text: with

text: independent

text: probability

text: f,

text: for

text: example

text: for

text: the

text: group

text: we

text: get

text: W

text: (C)

text: f)

text: f(

text: W

text: (C)

text: exp[

text: σ(C)A]

text: W

text: (C)

text: ##LTLine##


text: Figure 1. The weight diagram of G(2) group and its decomposition to SU(3) group in the weight space. { } { } ⊕ { } ⊕ { } 7 = 3 ¯3 1 (3.12) Then we have ⊕ ⊕ H{ } = H{ } H{ } H{ } (3.13) 7 3 ¯3 1 Multiplying two Cartan generators ⊗ { ⊕ ⊕ } ⊗ { ⊕ ⊕ } H{ } H{ } = H{ } H{ } H{ } H{ } H{ } H{ } (3.14) 7 i 7 j 3 i ¯3 i 1 i 3 j ¯3 j 1 j And we need trace of the above expression. We hold only the acceptable matrix product { ⊗ } { ⊕ ⊕ } T r H{ } H{ } = T r H H H H H H , (3.15) 7 i 7 j 3i 3j ¯3i ¯3j 1i 1j Using the eq. 3.9 we have − { ⊗ } d (N2 1)d d T r H{ } H{ } = 2C{2 } −r + 0 = 2 r −r + 0 = 1, (3.16) 7 i 7 j 3 N2 1 2N2 N2 1 Where d is the dimension of the representaion of the SU(3) subgroup and N is the r dimension of the subgroup. The eq. 3.16 is exactly the Casimir eigen value for the G(2) in fundamental representation. So we obtain the Casimir eigen value of G(2) using its decomposition to SU(3) subgroup. Here we do the same for the adjoint representation of G(2) group. The G(2) root diagram and its decomposition to its SU(3) subgroup is shown in ﬁgure 2. Again we have { } { } ⊕ { } ⊕ { } 14 = 8 3 ¯3 , (3.17) and then – 8 – ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: The

text: weight

text: diagram

text: of

text: group

text: and

text: its

text: decomposition

text: to

text: group

text: in

text: the

text: weight

text: space.

text: ¯3

text: Then

text: we

text: have

text: H{

text: H{

text: H{

text: H{

text: ¯3

text: Multiplying

text: two

text: Cartan

text: generators

text: H{

text: H{

text: H{

text: H{

text: H{

text: H{

text: H{

text: H{

text: i

text: j

text: i

text: ¯3

text: i

text: i

text: j

text: ¯3

text: j

text: j

text: And

text: we

text: need

text: trace

text: of

text: the

text: above

text: expression.

text: We

text: hold

text: only

text: the

text: acceptable

text: matrix

text: product

text: T

text: r

text: H{

text: H{

text: T

text: r

text: H

text: H

text: H

text: H

text: H

text: H

text: i

text: j

text: Using

text: the

text: eq.

text: we

text: have

text: d

text: d

text: T

text: r

text: H{

text: H{

text: −r

text: r

text: −r

text: i

text: j

text: Where

text: d

text: is

text: the

text: dimension

text: of

text: the

text: representaion

text: of

text: the

text: subgroup

text: and

text: N

text: is

text: the

text: r

text: dimension

text: of

text: the

text: subgroup.

text: The

text: eq.

text: is

text: exactly

text: the

text: Casimir

text: eigen

text: value

text: for

text: the

text: in

text: fundamental

text: representation.

text: So

text: we

text: obtain

text: the

text: Casimir

text: eigen

text: value

text: of

text: using

text: its

text: decomposition

text: to

text: subgroup.

text: Here

text: we

text: do

text: the

text: same

text: for

text: the

text: adjoint

text: representation

text: of

text: group.

text: The

text: root

text: diagram

text: and

text: its

text: decomposition

text: to

text: its

text: subgroup

text: is

text: shown

text: in

text: ﬁgure

text: Again

text: we

text: have

text: ¯3

text: and

text: then

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Under consideration for publication in J. Plasma Phys. 1 Generation of large-scale vorticity in rotating stratiﬁed turbulence with inhomogeneous helicity: mean-ﬁeld theory † N. K L E E O R I N AND I. R O G A C H E V S K I I Department of Mechanical Engineering, Ben-Gurion University of the Negev, P. O. Box 653, 84105 Beer-Sheva, Israel Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden (Received 30 May 2020; revised ; accepted ) We discuss a mean-ﬁeld theory of generation of large-scale vorticity in a rotating density stratiﬁed developed turbulence with inhomogeneous kinetic helicity. We show that the large-scale nonuniform ﬂow is produced due to ether a combined action of a density strat- iﬁed rotating turbulence and uniform kinetic helicity or a combined eﬀect of a rotating incompressible turbulence and inhomogeneous kinetic helicity. These eﬀects result in the formation of a large-scale shear, and in turn its interaction with the small-scale turbu- lence causes an excitation of the large-scale instability (known as a vorticity dynamo) due to a combined eﬀect of the large-scale shear and Reynolds stress-induced generation of the mean vorticity. The latter is due to the eﬀect of large-scale shear on the Reynolds stress. A fast rotation suppresses this large-scale instability. 1. Introduction A large-scale nonuniform ﬂow or diﬀerential rotation in a helical small-scale turbulence can result in generation of a large-scale magnetic ﬁeld by αΩ or α2Ω mean-ﬁeld dynamo (see, e.g., Moﬀatt 1978; Parker 1979; Krause & R¨adler 1980; Zeldovich et al. 1983; Ruz- maikin et al. 1988; Ru¨diger et al. 2013). The kinetic α eﬀect is related to a kinetic helicity produced, e.g., by a combined action of uniform rotation and density stratiﬁed or inho- mogeneous turbulence. Formation of the nonuniform ﬂows is caused, e.g., by a rotating anisotropic density stratiﬁed turbulence or turbulent convection. The latter eﬀect is also related to a problem of generation of large-scale vorticity by a turbulent ﬂow, and has various applications in geophysical and astrophysical ﬂows (see, e.g., Lugt 1983; Pedlosky 1987; Chorin 1994). It has been suggested by Moiseev et al. (1983), that the generation of the large-scale vorticity in a helical turbulence occurs due to the kinetic alpha eﬀect. This idea is based on an analogy between the induction equation for magnetic ﬁeld and the vorticity equa- tion (Batchelor 1950). The latter implies that a large-scale instability is associated with ∇× the term (αW ) in the equation for the mean vorticity, W , similarly to the mean-ﬁeld ∇× equation for the magnetic ﬁeld, B, where the key generation term is (αB), see Moi- seev et al. (1983). A mean-ﬁeld equation for the vorticity has been derived by Khomenko et al. (1991) using the functional technique for a compressible helical turbulence. It has been shown there that the mean vorticity grows exponentially in time due to the kinetic alpha eﬀect. † Email address for correspondence: gary@bgu.ac.il ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Under

text: consideration

text: for

text: publication

text: in

text: J.

text: Plasma

text: Phys.

text: Generation

label: title

text: of

text: large-scale

text: vorticity

text: in

text: rotating

text: stratiﬁed

text: turbulence

text: with

text: inhomogeneous

text: helicity:

text: mean-ﬁeld

text: theory

text: N.

text: K

text: L

text: E

text: E

text: O

text: R

text: I

text: N

text: AND

text: I.

text: R

text: O

text: G

text: A

text: C

text: H

text: E

text: V

text: S

text: K

text: I

text: I

text: Department

label: author

text: of

text: Mechanical

text: Engineering,

text: Ben-Gurion

text: University

text: of

text: the

text: Negev,

text: P.

text: O.

text: Box

text: Beer-Sheva,

text: Israel

text: Nordita,

text: KTH

text: Royal

text: Institute

text: of

text: Technology

text: and

text: Stockholm

text: University,

text: Roslagstullsbacken

text: Stockholm,

text: Sweden

text: (Received

label: date

text: May

text: revised

text: accepted

text: We

label: abstract

text: discuss

text: a

text: mean-ﬁeld

text: theory

text: of

text: generation

text: of

text: large-scale

text: vorticity

text: in

text: a

text: rotating

text: density

text: stratiﬁed

text: developed

text: turbulence

text: with

text: inhomogeneous

text: kinetic

text: helicity.

text: We

text: show

text: that

text: the

text: large-scale

text: nonuniform

text: ﬂow

text: is

text: produced

text: due

text: to

text: ether

text: a

text: combined

text: action

text: of

text: a

text: density

text: strat-

text: iﬁed

text: rotating

text: turbulence

text: and

text: uniform

text: kinetic

text: helicity

text: or

text: a

text: combined

text: eﬀect

text: of

text: a

text: rotating

text: incompressible

text: turbulence

text: and

text: inhomogeneous

text: kinetic

text: helicity.

text: These

text: eﬀects

text: result

text: in

text: the

text: formation

text: of

text: a

text: large-scale

text: shear,

text: and

text: in

text: turn

text: its

text: interaction

text: with

text: the

text: small-scale

text: turbu-

text: lence

text: causes

text: an

text: excitation

text: of

text: the

text: large-scale

text: instability

text: (known

text: as

text: a

text: vorticity

text: dynamo)

text: due

text: to

text: a

text: combined

text: eﬀect

text: of

text: the

text: large-scale

text: shear

text: and

text: Reynolds

text: stress-induced

text: generation

text: of

text: the

text: mean

text: vorticity.

text: The

text: latter

text: is

text: due

text: to

text: the

text: eﬀect

text: of

text: large-scale

text: shear

text: on

text: the

text: Reynolds

text: stress.

text: A

text: fast

text: rotation

text: suppresses

text: this

text: large-scale

text: instability.

text: Introduction

text: A

text: large-scale

text: nonuniform

text: ﬂow

text: or

text: diﬀerential

text: rotation

text: in

text: a

text: helical

text: small-scale

text: turbulence

text: can

text: result

text: in

text: generation

text: of

text: a

text: large-scale

text: magnetic

text: ﬁeld

text: by

text: αΩ

text: or

text: α2Ω

text: mean-ﬁeld

text: dynamo

text: (see,

text: e.g.,

text: Moﬀatt

text: Parker

text: Krause

text: R¨adler

text: Zeldovich

text: et

text: al.

text: Ruz-

text: maikin

text: et

text: al.

text: Ru¨diger

text: et

text: al.

text: The

text: kinetic

text: α

text: eﬀect

text: is

text: related

text: to

text: a

text: kinetic

text: helicity

text: produced,

text: e.g.,

text: by

text: a

text: combined

text: action

text: of

text: uniform

text: rotation

text: and

text: density

text: stratiﬁed

text: or

text: inho-

text: mogeneous

text: turbulence.

text: Formation

text: of

text: the

text: nonuniform

text: ﬂows

text: is

text: caused,

text: e.g.,

text: by

text: a

text: rotating

text: anisotropic

text: density

text: stratiﬁed

text: turbulence

text: or

text: turbulent

text: convection.

text: The

text: latter

text: eﬀect

text: is

text: also

text: related

text: to

text: a

text: problem

text: of

text: generation

text: of

text: large-scale

text: vorticity

text: by

text: a

text: turbulent

text: ﬂow,

text: and

text: has

text: various

text: applications

text: in

text: geophysical

text: and

text: astrophysical

text: ﬂows

text: (see,

text: e.g.,

text: Lugt

text: Pedlosky

text: Chorin

text: It

text: has

text: been

text: suggested

text: by

text: Moiseev

text: et

text: al.

text: that

text: the

text: generation

text: of

text: the

text: large-scale

text: vorticity

text: in

text: a

text: helical

text: turbulence

text: occurs

text: due

text: to

text: the

text: kinetic

text: alpha

text: eﬀect.

text: This

text: idea

text: is

text: based

text: on

text: an

text: analogy

text: between

text: the

text: induction

text: equation

text: for

text: magnetic

text: ﬁeld

text: and

text: the

text: vorticity

text: equa-

text: tion

text: (Batchelor

text: The

text: latter

text: implies

text: that

text: a

text: large-scale

text: instability

text: is

text: associated

text: with

text: the

text: term

text: (αW

text: in

text: the

text: equation

text: for

text: the

text: mean

text: vorticity,

text: W

text: similarly

text: to

text: the

text: mean-ﬁeld

text: equation

text: for

text: the

text: magnetic

text: ﬁeld,

text: B,

text: where

text: the

text: key

text: generation

text: term

text: is

text: (αB),

text: see

text: Moi-

text: seev

text: et

text: al.

text: A

text: mean-ﬁeld

text: equation

text: for

text: the

text: vorticity

text: has

text: been

text: derived

text: by

text: Khomenko

text: et

text: al.

text: using

text: the

text: functional

text: technique

text: for

text: a

text: compressible

text: helical

text: turbulence.

text: It

text: has

text: been

text: shown

text: there

text: that

text: the

text: mean

text: vorticity

text: grows

text: exponentially

text: in

text: time

text: due

text: to

text: the

text: kinetic

text: alpha

text: eﬀect.

text: Email

text: address

text: for

text: correspondence:

text: gary@bgu.ac.il

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Generation of large-scale vorticity in helical turbulence 3 In the present study we develop a mean-ﬁeld theory of the generation of large-scale vorticity in a rotating density stratiﬁed turbulence with inhomogeneous helicity and large Reynolds numbers. To derive the mean-ﬁeld equation for the vorticity, the spectral τ approach was applied here for a large Reynolds number turbulence. We have shown that a non-uniform large-scale ﬂow is produced in a rotating fully developed turbulence due to either inhomogeneous kinetic helicity or a combined eﬀect of a density stratiﬁed ﬂow and uniform kinetic helicity. An interaction of the turbulence with the formed large- scale shear causes an excitation of the large-scale instability resulting in the generating of the mean vorticity (vorticity dynamo). On the other hand, a fast rotation suppresses this large-scale instability. The present study of the dynamics of large-scale vorticity in a rotating density stratiﬁed helical turbulence demonstrates that the mean-ﬁeld equation ∇ × for the large-scale vorticity does not contain the (αW ) term as was previously suggested by Moiseev et al. (1983). 2. Eﬀect of rotation on the Reynolds stress To study an eﬀect of rotation on the Reynolds stress in a rotating, density stratiﬁed and inhomogeneous turbulence, we apply a mean-ﬁeld approach and use the Reynolds averaging. In the framework of this approach, the velocity and pressure are separated into the mean and ﬂuctuating parts. 2.1. Equation for velocity ﬂuctuations To determine the Reynolds stress, we use equation for ﬂuctuations of velocity u, which is obtained by subtracting equation for the mean ﬁeld from the corresponding equation for the instantaneous ﬁeld: ∇ ∂u − · ∇ − · ∇ − p × = (U )u (u )U + 2u Ω + UN. (2.1) ∂t ρ 0 Here p are ﬂuctuations of ﬂuid pressure, U is the mean ﬂuid velocity, and Eq. (2.1) is written in the reference frame rotating with the angular velocity Ω. The ﬂuid velocity for a low-Mach-number ﬂuid ﬂow satisﬁes the continuity equation written in the anelastic approximation: div (ρ U) = 0 and div (ρ u) = 0. The mean ﬂuid density and pressure 0 0 with the subscript ”0” correspond to the hydrostatic basic reference state, given by the ∇ equation: P = ρ g. The nonlinear term UN which includes the molecular viscous 0 0 force, ρ F (u), is given by 0 ν (cid:104) · ∇ (cid:105) − · ∇ UN = (u )u (u )u + F (u). ν The derivation of the equation for the Reynolds √stress includes the following steps: (i) use new variables for ﬂuctuations of velocity v = ρ u; (ii) derivation of the equation (cid:104) 0 (cid:105) for the second moment of the velocity ﬂuctuations v v in the k space; (iii) application i j (cid:104) (cid:105) of the spectral τ approach (see Sect. ) and solution of the derived equation for v v i j in the k space; (iv) returning to the physical space to obtain formula for the Reynolds stress as the function of the rotation rate Ω. Here the angular brackets denote ensemble averaging. 2.2. Equation for the Reynolds stress Applying a multi-scale approach (Roberts & Soward 1975) and using Eq. (A 2) de- rived in Appendix , we obtain equation for the correlation function: f (k, K, t) = (cid:104) (cid:105) − ij v (k , t)v (k , t) , where k = k + K/2, k = k + K/2, and K correspond to the i 1 j 2 1 2 large scales, while k to the small ones. The equation for f (k, K, t) is given by ij ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Generation

text: of

text: large-scale

text: vorticity

text: in

text: helical

text: turbulence

text: In

text: the

text: present

text: study

text: we

text: develop

text: a

text: mean-ﬁeld

text: theory

text: of

text: the

text: generation

text: of

text: large-scale

text: vorticity

text: in

text: a

text: rotating

text: density

text: stratiﬁed

text: turbulence

text: with

text: inhomogeneous

text: helicity

text: and

text: large

text: Reynolds

text: numbers.

text: To

text: derive

text: the

text: mean-ﬁeld

text: equation

text: for

text: the

text: vorticity,

text: the

text: spectral

text: τ

text: approach

text: was

text: applied

text: here

text: for

text: a

text: large

text: Reynolds

text: number

text: turbulence.

text: We

text: have

text: shown

text: that

text: a

text: non-uniform

text: large-scale

text: ﬂow

text: is

text: produced

text: in

text: a

text: rotating

text: fully

text: developed

text: turbulence

text: due

text: to

text: either

text: inhomogeneous

text: kinetic

text: helicity

text: or

text: a

text: combined

text: eﬀect

text: of

text: a

text: density

text: stratiﬁed

text: ﬂow

text: and

text: uniform

text: kinetic

text: helicity.

text: An

text: interaction

text: of

text: the

text: turbulence

text: with

text: the

text: formed

text: large-

text: scale

text: shear

text: causes

text: an

text: excitation

text: of

text: the

text: large-scale

text: instability

text: resulting

text: in

text: the

text: generating

text: of

text: the

text: mean

text: vorticity

text: (vorticity

text: dynamo).

text: On

text: the

text: other

text: hand,

text: a

text: fast

text: rotation

text: suppresses

text: this

text: large-scale

text: instability.

text: The

text: present

text: study

text: of

text: the

text: dynamics

text: of

text: large-scale

text: vorticity

text: in

text: a

text: rotating

text: density

text: stratiﬁed

text: helical

text: turbulence

text: demonstrates

text: that

text: the

text: mean-ﬁeld

text: equation

text: for

text: the

text: large-scale

text: vorticity

text: does

text: not

text: contain

text: the

text: (αW

text: term

text: as

text: was

text: previously

text: suggested

text: by

text: Moiseev

text: et

text: al.

text: Eﬀect

text: of

text: rotation

text: on

text: the

text: Reynolds

text: stress

text: To

text: study

text: an

text: eﬀect

text: of

text: rotation

text: on

text: the

text: Reynolds

text: stress

text: in

text: a

text: rotating,

text: density

text: stratiﬁed

text: and

text: inhomogeneous

text: turbulence,

text: we

text: apply

text: a

text: mean-ﬁeld

text: approach

text: and

text: use

text: the

text: Reynolds

text: averaging.

text: In

text: the

text: framework

text: of

text: this

text: approach,

text: the

text: velocity

text: and

text: pressure

text: are

text: separated

text: into

text: the

text: mean

text: and

text: ﬂuctuating

text: parts.

text: Equation

text: for

text: velocity

text: ﬂuctuations

text: To

text: determine

text: the

text: Reynolds

text: stress,

text: we

text: use

text: equation

text: for

text: ﬂuctuations

text: of

text: velocity

text: u,

text: which

text: is

text: obtained

text: by

text: subtracting

text: equation

text: for

text: the

text: mean

text: ﬁeld

text: from

text: the

text: corresponding

text: equation

text: for

text: the

text: instantaneous

text: ﬁeld:

text: ∂u

text: p

text: (U

text: )u

text: (u

text: )U

text: Ω

text: UN.

text: ∂t

text: ρ

text: Here

text: p

text: are

text: ﬂuctuations

text: of

text: ﬂuid

text: pressure,

text: U

text: is

text: the

text: mean

text: ﬂuid

text: velocity,

text: and

text: Eq.

text: is

text: written

text: in

text: the

text: reference

text: frame

text: rotating

text: with

text: the

text: angular

text: velocity

text: Ω.

text: The

text: ﬂuid

text: velocity

text: for

text: a

text: low-Mach-number

text: ﬂuid

text: ﬂow

text: satisﬁes

text: the

text: continuity

text: equation

text: written

text: in

text: the

text: anelastic

text: approximation:

text: div

text: (ρ

text: U)

text: and

text: div

text: (ρ

text: u)

text: The

text: mean

text: ﬂuid

text: density

text: and

text: pressure

text: with

text: the

text: subscript

text: ”0”

text: correspond

text: to

text: the

text: hydrostatic

text: basic

text: reference

text: state,

text: given

text: by

text: the

text: equation:

text: P

text: ρ

text: g.

text: The

text: nonlinear

text: term

text: UN

text: which

text: includes

text: the

text: molecular

text: viscous

text: force,

text: ρ

text: F

text: (u),

text: is

text: given

text: by

text: ν

text: UN

text: (u

text: )u

text: (u

text: )u

text: F

text: (u).

text: ν

text: The

text: derivation

text: of

text: the

text: equation

text: for

text: the

text: Reynolds

text: √stress

text: includes

text: the

text: following

text: steps:

text: (i)

text: use

text: new

text: variables

text: for

text: ﬂuctuations

text: of

text: velocity

text: v

text: ρ

text: u;

text: (ii)

text: derivation

text: of

text: the

text: equation

text: for

text: the

text: second

text: moment

text: of

text: the

text: velocity

text: ﬂuctuations

text: v

text: v

text: in

text: the

text: k

text: space;

text: (iii)

text: application

text: i

text: j

text: of

text: the

text: spectral

text: τ

text: approach

text: (see

text: Sect.

text: and

text: solution

text: of

text: the

text: derived

text: equation

text: for

text: v

text: v

text: i

text: j

text: in

text: the

text: k

text: space;

text: (iv)

text: returning

text: to

text: the

text: physical

text: space

text: to

text: obtain

text: formula

text: for

text: the

text: Reynolds

text: stress

text: as

text: the

text: function

text: of

text: the

text: rotation

text: rate

text: Ω.

text: Here

text: the

text: angular

text: brackets

text: denote

text: ensemble

text: averaging.

text: Equation

text: for

text: the

text: Reynolds

text: stress

text: Applying

text: a

text: multi-scale

text: approach

text: (Roberts

text: Soward

text: and

text: using

text: Eq.

text: (A

text: de-

text: rived

text: in

text: Appendix

text: we

text: obtain

text: equation

text: for

text: the

text: correlation

text: function:

text: f

text: (k,

text: K,

text: t)

text: ij

text: v

text: (k

text: t)v

text: (k

text: t)

text: where

text: k

text: k

text: k

text: k

text: and

text: K

text: correspond

text: to

text: the

text: i

text: j

text: large

text: scales,

text: while

text: k

text: to

text: the

text: small

text: ones.

text: The

text: equation

text: for

text: f

text: (k,

text: K,

text: t)

text: is

text: given

text: by

text: ij

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##