the fate of alpha dynamos at large rmcameron/pres/limsi17cameron.pdf · 2017-02-01 · the fate of...
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The fate of alpha dynamos at large Rm
Alexandre Cameron & Alexandros Alexakis
LPS ENS
2nd February 2017
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Magnetic activity
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Observation
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Curie temperature
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Dynamo mechanism
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Where do large magnetic scales come from?
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Eugene Parker’s 1955 theory
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Eugene Parker’s 1957 theory
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M. Steenbeck, F. Krause, K.-H. Radler (1966)
Berechnung der mittleren Lorentz-Feldstarke v × B fur ein elektrischleitendes Medium in turbulenter, durch Coriolis-Krafte beeinflussterBewegung
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Moffatt’s Book (1978)
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Krause & Radler (1980)
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Alpha modeling is at the heart of todays Solar models!
A. Brun, M. Browning, M. Dikpati, H. Hotta, and A. Strugarek, Space Sci. Rev. 196, 101 (2015).M. Dikpati and P. A. Gilman, Astrophys. J. 649, 498 (2006).
A. R. Choudhuri, P. Chatterjee, and J. Jiang, Phys. Rev. Lett. 98, 131103 (2007).
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Formulation
∂tB = ∇× u× B + Rm−1∇2B,
u Velocity field of characteristic length-scale ` such that 〈u〉 = 0.B Magnetic field evolving in a domain of size L.
`� L
Averaging procedure for f(x) : 〈f 〉 =1
V`
∫V`
f (X + x)dx3
where V` a volume centered at X with `3 � V` � L3
Let B = 〈B〉+ b then
∂t〈B〉 = ∇× 〈u× b〉+ Rm−1∇2〈B〉,and
∂tb = ∇× (u× 〈B〉) +∇× (u× b− 〈u× b〉) + Rm−1∇2b
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Formulation
∂tB = ∇× u× B + Rm−1∇2B,
u Velocity field of characteristic length-scale ` such that 〈u〉 = 0.B Magnetic field evolving in a domain of size L.
`� L
Averaging procedure for f(x) : 〈f 〉 =1
V`
∫V`
f (X + x)dx3
where V` a volume centered at X with `3 � V` � L3
Let B = 〈B〉+ b then
∂t〈B〉 = ∇× 〈u× b〉+ Rm−1∇2〈B〉,and
∂tb = ∇× (u× 〈B〉) +∇× (u× b− 〈u× b〉) + Rm−1∇2b
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 13 / 36
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Formulation
∂tB = ∇× u× B + Rm−1∇2B,
u Velocity field of characteristic length-scale ` such that 〈u〉 = 0.B Magnetic field evolving in a domain of size L.
`� L
Averaging procedure for f(x) : 〈f 〉 =1
V`
∫V`
f (X + x)dx3
where V` a volume centered at X with `3 � V` � L3
Let B = 〈B〉+ b then
∂t〈B〉 = ∇× 〈u× b〉+ Rm−1∇2〈B〉,and
∂tb = ∇× (u× 〈B〉) +∇× (u× b− 〈u× b〉) + Rm−1∇2b
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 13 / 36
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Formulation
∂tB = ∇× u× B + Rm−1∇2B,
u Velocity field of characteristic length-scale ` such that 〈u〉 = 0.B Magnetic field evolving in a domain of size L.
`� L
Averaging procedure for f(x) : 〈f 〉 =1
V`
∫V`
f (X + x)dx3
where V` a volume centered at X with `3 � V` � L3
Let B = 〈B〉+ b then
∂t〈B〉 = ∇× 〈u× b〉+ Rm−1∇2〈B〉,and
∂tb = ∇× (u× 〈B〉) +∇× (u× b− 〈u× b〉) + Rm−1∇2b
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Formulation
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +∇× G + Rm−1∇2b
whereE = 〈u× b〉 and G = u× b− 〈u× b〉
Gradient expansion ...
E i = αij〈B〉j + βijk∇j〈B〉k + . . .
then for isotropic flows
∂t〈B〉 = α∇× 〈B〉+ (Rm−1 + β)∇2〈B〉+ . . . ,
Always unstable for α 6= 0 & L� ` with growth-rate γ
γ = αk − (Rm−1 + β)k2 + . . .
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Formulation
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +∇× G + Rm−1∇2b
whereE = 〈u× b〉 and G = u× b− 〈u× b〉
Gradient expansion ...
E i = αij〈B〉j + βijk∇j〈B〉k + . . .
then for isotropic flows
∂t〈B〉 = α∇× 〈B〉+ (Rm−1 + β)∇2〈B〉+ . . . ,
Always unstable for α 6= 0 & L� ` with growth-rate γ
γ = αk − (Rm−1 + β)k2 + . . .
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 14 / 36
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Formulation
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +∇× G + Rm−1∇2b
whereE = 〈u× b〉 and G = u× b− 〈u× b〉
Gradient expansion ...
E i = αij〈B〉j + βijk∇j〈B〉k + . . .
then for isotropic flows
∂t〈B〉 = α∇× 〈B〉+ (Rm−1 + β)∇2〈B〉+ . . . ,
Always unstable for α 6= 0 & L� ` with growth-rate γ
γ = αk − (Rm−1 + β)k2 + . . .
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Small Rm limit: Rm = u`/η � 1 and `/L = O(Rm2)
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,��∂tb = ∇× (u× 〈B〉) +����∇× G + Rm−1∇2b
b = Rm∇−2∇× (u× 〈B〉)E = α〈B〉 = 〈u× b〉
γ ' αk − Rm−1k2
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Small Rm limit: Rm = u`/η � 1 and `/L = O(Rm2)
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,��∂tb = ∇× (u× 〈B〉) +����∇× G + Rm−1∇2b
b = Rm∇−2∇× (u× 〈B〉)E = α〈B〉 = 〈u× b〉
γ ' αk − Rm−1k2
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Small Rm limit: Rm = u`/η � 1 and `/L = O(Rm2)
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,��∂tb = ∇× (u× 〈B〉) +����∇× G + Rm−1∇2b
b = Rm∇−2∇× (u× 〈B〉)E = α〈B〉 = 〈u× b〉
γ ' αk − Rm−1k2
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Multi-scale expansion (Childress 1969)
Expanding B = B0 + R B1 . . .
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Small τ approximation: correlation time τ � `/u
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +����∇× G +�����
Rm−1∇2b
b ' τ∇× (u× 〈B〉)E = α〈B〉 = 〈u× b〉
Assuming isotropy
α = −τ3〈u · ∇ × u〉
α is proportional to the helicity of the flow!
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Small τ approximation: correlation time τ � `/u
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +����∇× G +�����
Rm−1∇2b
b ' τ∇× (u× 〈B〉)E = α〈B〉 = 〈u× b〉
Assuming isotropy
α = −τ3〈u · ∇ × u〉
α is proportional to the helicity of the flow!
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 17 / 36
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Small τ approximation: correlation time τ � `/u
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +����∇× G +�����
Rm−1∇2b
b ' τ∇× (u× 〈B〉)E = α〈B〉 = 〈u× b〉
Assuming isotropy
α = −τ3〈u · ∇ × u〉
α is proportional to the helicity of the flow!
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 17 / 36
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Small τ approximation: correlation time τ � `/u
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +����∇× G +�����
Rm−1∇2b
b ' τ∇× (u× 〈B〉)E = α〈B〉 = 〈u× b〉
Assuming isotropy
α = −τ3〈u · ∇ × u〉
α is proportional to the helicity of the flow!
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The stances in the community
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Pro-Alpha groups
Measuring α by expansions and Numerical simulations
K.-H. Radler and M. Rheinhardt, Geophys. Astrophys. Fluid Dyn. 101, 117(2007).N. Kleeorin, I. Rogachevskii, D. Sokoloff, and D. Tomin, Phys. Rev. E 79, 046302(2009).M. Schrinner, K.-H. Radler, D. Schmitt, M. Rheinhardt, and U. R. Christensen,Geophys. Astrophys. Fluid Dyn. 101, 81 (2007).M. Rheinhardt and A. Brandenburg, Astron. Astrophys. 520, A28 (2010).S. Sur, A. Brandenburg, and K. Subramanian, Mon. Not. R. Astron. Soc. 385,L15 (2008).A. Brandenburg, K.-H. Radler, and M. Schrinner, Astron. Astrophys. 482, 739(2008).K. Subramanian and A. Brandenburg, Mon. Not. R. Astron. Soc. 445, 2930(2014).
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Contre-Alpha groups
S. Boldyrev, F. Cattaneo, and R. Rosner, Phys. Rev. Lett. 95, 255001 (2005).A. Courvoisier, D. W. Hughes, and S. M. Tobias, Phys. Rev. Lett. 96, 034503(2006).D. W. Hughes, Plasma Phys. Controlled Fusion 50, 124021 (2008).F. Cattaneo and D. W. Hughes, Mon. Not. R. Astron. Soc. 395, L48 (2009).F. Cattaneo and S. Tobias, Astrophys. J. 789, 70 (2014).D. Galloway and U. Frisch, Geophys. Astrophys. Fluid Dyn. 29, 13 (1984).
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Some quotes
“... We argue that the interpretation of these quantities (α, β) in terms ofthe evolution of the large-scale field may be fundamentally flawed.”F. Cattaneo and D. W. Hughes, Mon. Not. R. Astron. Soc. 395, L51 (2009).
“It is stressed that the connection of the mean electromotive force withthe mean magnetic field and its first spatial derivatives is in general neitherlocal nor instantaneous and that quite a few claims concerningpretended failures of the mean-field concept result from ignoringthis aspect.”K.-H. Radler Astron. Nachr. 335, 459 (2014)
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When can we neglect G?
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +����∇× G + Rm−1∇2b
whereE = 〈u× b〉 and G = u× b− 〈u× b〉
OK, when Rm� 1Maybe OK if τ � `/u if Rm is not to big.Not OK when small scale dynamo exists.
If G 6= 0 the linear relation E = α〈B〉+ . . . does not hold!
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When can we neglect G?
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +����∇× G + Rm−1∇2b
whereE = 〈u× b〉 and G = u× b− 〈u× b〉
OK, when Rm� 1Maybe OK if τ � `/u if Rm is not to big.Not OK when small scale dynamo exists.
If G 6= 0 the linear relation E = α〈B〉+ . . . does not hold!
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 22 / 36
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When can we neglect G?
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +����∇× G + Rm−1∇2b
whereE = 〈u× b〉 and G = u× b− 〈u× b〉
OK, when Rm� 1Maybe OK if τ � `/u if Rm is not to big.Not OK when small scale dynamo exists.
If G 6= 0 the linear relation E = α〈B〉+ . . . does not hold!
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 22 / 36
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When can we neglect G?
∂t〈B〉 = ∇× E + Rm−1∇2〈B〉,∂tb = ∇× (u× 〈B〉) +����∇× G + Rm−1∇2b
whereE = 〈u× b〉 and G = u× b− 〈u× b〉
OK, when Rm� 1Maybe OK if τ � `/u if Rm is not to big.Not OK when small scale dynamo exists.
If G 6= 0 the linear relation E = α〈B〉+ . . . does not hold!
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 22 / 36
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A simple example
q � k
bq = −ηq2 bq +αq bk ,
bk = αk bq +γSSD
bk
where γSSD
= ukk − ηk2 is the growth rate of the small scale dynamoobtained by setting α = 0.
γ =1
2
[γSSD− ηq2 ±
√γ2SSD
+ 4α2kq + 2γSSDηq2 + η2q4
]
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 23 / 36
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A simple example
γ =1
2
[γSSD− ηq2 ±
√γ2SSD
+ 4α2kq + 2γSSDηq2 + η2q4
]
γSSD
< 0 γSSD
> 0
γ ' α2kq/|γSSD| = O(q) γ ' γ
SSD= O(1)
bq/bk ' (|γSSD|/αk) = O(1) bq/bk ' (αq/|γ
SSD|) = O(q/k)
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 24 / 36
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The Floquet (Bloch) Formalism for periodic flows
Let u(x, t) be a spatially periodic flow of a given spatial period ` = 2π/k .
The Floquet theory states that the magnetic field can be decomposed asB(x, t) = e iq·xb(x, t) + c .c . where b(x, t) is a complex vector field thatsatisfies the same spatial periodicity as the velocity field u, and q is anarbitrary wave number.
∂t b = iq× (u× b) +∇× (u× b) + η(∇+ iq)2b
Fields with q = 0 and 〈b〉 = 0 correspond to purely small scale fields.For 0 < q/k < 1 the dynamo mode has in general a finite projection tothe large scales measured by 〈b〉.
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 25 / 36
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The Floquet (Bloch) Formalism for periodic flows
Let u(x, t) be a spatially periodic flow of a given spatial period ` = 2π/k .
The Floquet theory states that the magnetic field can be decomposed asB(x, t) = e iq·xb(x, t) + c .c . where b(x, t) is a complex vector field thatsatisfies the same spatial periodicity as the velocity field u, and q is anarbitrary wave number.
∂t b = iq× (u× b) +∇× (u× b) + η(∇+ iq)2b
Fields with q = 0 and 〈b〉 = 0 correspond to purely small scale fields.For 0 < q/k < 1 the dynamo mode has in general a finite projection tothe large scales measured by 〈b〉.
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 25 / 36
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The Floquet (Bloch) Formalism for periodic flows
Let u(x, t) be a spatially periodic flow of a given spatial period ` = 2π/k .
The Floquet theory states that the magnetic field can be decomposed asB(x, t) = e iq·xb(x, t) + c .c . where b(x, t) is a complex vector field thatsatisfies the same spatial periodicity as the velocity field u, and q is anarbitrary wave number.
∂t b = iq× (u× b) +∇× (u× b) + η(∇+ iq)2b
Fields with q = 0 and 〈b〉 = 0 correspond to purely small scale fields.For 0 < q/k < 1 the dynamo mode has in general a finite projection tothe large scales measured by 〈b〉.
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 25 / 36
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The Floquet (Bloch) Formalism for periodic flows
Let u(x, t) be a spatially periodic flow of a given spatial period ` = 2π/k .
The Floquet theory states that the magnetic field can be decomposed asB(x, t) = e iq·xb(x, t) + c .c . where b(x, t) is a complex vector field thatsatisfies the same spatial periodicity as the velocity field u, and q is anarbitrary wave number.
∂t b = iq× (u× b) +∇× (u× b) + η(∇+ iq)2b
Fields with q = 0 and 〈b〉 = 0 correspond to purely small scale fields.For 0 < q/k < 1 the dynamo mode has in general a finite projection tothe large scales measured by 〈b〉.
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 25 / 36
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A twofold gain
it provides with a clear way to disentangle dynamos that involve onlysmall scales (for which q/k ∈ Z3) from dynamos that involve largescales (0 < q/k � 1)
it allows to investigate numerically arbitrary large scale separationsq � k with no additional numerical cost.
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 26 / 36
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The examined Flows
u = U
sin(ky + φ2) + cos(kz + ψ3),sin(kz + φ3) + cos(kx + ψ1),sin(kx + φ1) + cos(ky + ψ2)
.(A) φi = ψi = 0 ie the helical ABC flow
(B) φi = ψi − π/2 = 0 a non-helical flowRm = U/kη
and γ is measured in units of Uk
(C) φi = ψi change randomly every time τRm = (U/kη)× (τUk) = U2τ/η
and γ is measured in units of U2k2τ
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 27 / 36
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Results: (A) ABC flow, q = 10−3
101 102
Rm
−0. 025
0. 000
0. 025
0. 050
0. 075
γ
R1 R2 R3
SSD
Floquet
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 28 / 36
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Results: (A) ABC flow
101 102
Rm
−0. 025
0. 000
0. 025
0. 050
0. 075
γ
R1 R2 R3
SSD
Floquet
γ E0/E = 〈b〉2/〈b2〉
10-6 10-5 10-4 10-3 10-2 10-1 100
q
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
γq
10-6 10-5 10-4 10-3 10-2 10-1 100
q
10-2410-2210-2010-1810-1610-1410-1210-1010-810-610-410-2100
E0
Etot
q2
q4
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 29 / 36
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Results: (B) Non-helical flow, q = 10−3
101 102
Rm
−0. 025
0. 000
0. 025
0. 050
0. 075
γ
R4
SSD
Floquet
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 30 / 36
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Results: (B) Non-helical flow
101 102
Rm
−0. 025
0. 000
0. 025
0. 050
0. 075
γ
R4
SSD
Floquet
γ E0/E = 〈b〉2/〈b2〉
10-6 10-5 10-4 10-3 10-2 10-1 100
q
10-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1
γq2
10-6 10-5 10-4 10-3 10-2 10-1 100
q
10-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1
E0
Etot
q2
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 31 / 36
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Results: (R) Random helical flow, q = 10−3
101 102
Rm
−0. 050
0. 000
0. 050
0. 100
0. 150
γ
R5
0. 50
0. 20
0. 10
0. 05
0. 02
Floquet
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 32 / 36
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Results: (C) Random helical flow
101 102
Rm
−0. 050
0. 000
0. 050
0. 100
0. 150
γ
R5
0. 50
0. 20
0. 10
0. 05
0. 02
Floquet 0 200 400 600
time
100
101
102
103
E
10−2 10−1 100 101
k
10−8
10−6
10−4
10−2
E
γ E0/E = 〈b〉2/〈b2〉
10-4 10-3 10-2 10-1 100
q
10-5
10-4
10-3
10-2
10-1
γ
q
10-4 10-3 10-2 10-1 100
q
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
E0
Etot
q2
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 33 / 36
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Large scale projection
q → εq � 1
γb = iεq× (u× b) +∇× (u× b) + η(∇+ iεq)2b
Let γ = γ0 + εγ1 + . . . and b = b0 + εb1 . . .
0th order
γ0 = γSSD
and 〈b0〉 = 0
1st order
γ0〈b1〉 = iq× 〈u× b0〉 and γ1 = i〈b†0 · (q× u× b0)〉
E0/E = 〈b〉2/〈b2〉 ∝ q2 if 〈u× b0〉 6= 0
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 34 / 36
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Large scale projection
q → εq � 1
γb = iεq× (u× b) +∇× (u× b) + η(∇+ iεq)2b
Let γ = γ0 + εγ1 + . . . and b = b0 + εb1 . . .
0th order
γ0 = γSSD
and 〈b0〉 = 0
1st order
γ0〈b1〉 = iq× 〈u× b0〉 and γ1 = i〈b†0 · (q× u× b0)〉
E0/E = 〈b〉2/〈b2〉 ∝ q2 if 〈u× b0〉 6= 0
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 34 / 36
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The Large scale behavior
Back to the gradient expansion ...
E i = E i0 + αij〈B〉j + βijk∇j〈B〉k + . . .
then for isotropic flows
∂t〈B〉 = E0 + α∇× 〈B〉+ (Rm−1 + β)∇2〈B〉+ . . . ,
whereE0 ∝ e(γSSD t)
and could be modeled as noise for a turbulent flow.
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 35 / 36
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The end
Thank you for your attention
A.Cameron & A. Alexakis Alpha Dynamos 2nd February 2017 36 / 36