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Dependence on kf L
We examine the dependence of the dynamo instability on the forcing wavenumber kf L. The amount of energy in the large scales increases as we increase kf L, since the domain over which the inverse cascade is present increases. This is seen from the energy spectra, see figure II.3. We adjust the value of the large scale friction in order to make sure we do not form a large scale condensate. Due to the construction of the governing equations (II.0.3) we see that the exact values of u2D and uz will depend on the value of kf L. We construct a new Reynolds number based on the results of the Ponomarenko dynamo (). The Ponomarenko dynamo consists of a swirling flow with the velocity field being u = U eθ + uz ez in cylindrical coordinates. The dynamo instability is due to the mean helicity induced by this vortical (screw-like) motion of the fluid. It is present only when both U, uz are non-zero. The Reynolds number is defined based on the velocity field defined as Up = U uz /(U 2 + u2z). We use this definition, Re = Up/(kf ν), Rm = Up/(kf η) where Up = u2D uz /(u22D + u2z).
Figure II.18 – The figure shows the normalized growth rate γ/(Upkf ) as a function of normalized kz /kf for a few diﬀerent kf L values for a) Helical flow, b) Nonhelical flow.
Figure II.18 shows the normalized growth rate γ/(Upkf ) as a function of the nor-malized kz /kf for diﬀerent values of kf . The chosen form of nondimensional quantities collapses all the curves together. The helical and nonhelical flow have a similar be-haviour. In general as kf is increased, both γ and the domain of kz unstable modes increase. We note that the most unstable mode scales like kz ∼ kf /3 as a function of kf . This is true for both the helical and the nonhelical flows. This implies that the most unstable mode scales with kf rather than the box length L.
Figure II.19 – Figures show maximum growth rate γmax/(Upkf ) for a) Helical, c) Non-helical flows and the normalized cut-oﬀ wavenumber kzc /kf for b) Helical, d) Nonhelical flow. Diﬀerent lines denote diﬀerent values of kf L.
on kf . In these set of parameters investigated the inverse cascade does not aﬀect the results, this is true as long as the largest shear is present at the forcing length scale. In the absence of large scale dissipation ν− the largest shear might be shifted to the large scale condensate whose dynamics are diﬀerent from what is observed here.
Conclusion – Part 1
Some conclusions one could take from the set of results presented here are, number which is a function of the Re. It becomes independent of Re at moderate values of Re examined here. This implies that the eﬀect of turbulence is minimal on the onset of the dynamo instability.
We have thus studied the kinematic dynamo in a quasi-twodimensional flow, which is an asymptotic limit of fast rotating flows. Such studies are also carried out in convective rotating flows [67, 68], showing the wide applicability of such an approach. The gain from developing such reduced models is that we can do numerics on smaller dimensional problems reducing the computational complexity. This could help us reach more realistic parameter regimes as in the case of the Earth, something which is not easy to attain in full 3D system of equations.
Saturation of the dynamo
Robert’s flow as an example
This section examines the saturation of the dynamo instability due to the Lorentz force. The Lorentz force modifies the base flow so that the eﬀect of the modified flow saturates the exponential growth of the magnetic field. It is interesting to note that the amplitude at which the magnetic field saturates depends on whether the underlying flow is laminar or turbulent. In this second part of the Chapter we look at the amplitude of the magnetic field saturation for diﬀerent flow regimes, laminar and turbulent.
We first examine a simple case for which the saturation amplitude can be calcu-lated analytically. This follows the weakly nonlinear analysis done elsewhere [69, 70]. We force the Navier Stokes equation with the forcing, f = f0 (cos (kf y) , sin (kf x) , cos (kf x) + sin (kf y)) (helical forcing). Considering the Navier Stokes equation in the Re ≪ 1, −ν∆u ≈ − 1 ∇ p + B2 + f + 1 B•∇B (II.8.1) ρ 2µ0 ρµ0.
The velocity field in the kinematic phase is given by u = f0/(kf2 ν) (cos (kf y) , sin (kf x) , cos (kf x) + sin (kf y)) when B ≪ 1.
For the full nonlinear problem we need to solve the Navier Stokes equation along with the induction equation. We take the form for the velocity field to be u = (U1 cos (kf y) , U2 sin (kf x) , U3 cos (kf x) + U4 sin (kf y)). The coeﬃcients U1, U2, U3, U4 are found by solving Navier Stokes and the induction equation. The induction equation at saturation is written as, 0 = ∇ × (u × B) + η∆B (II.8.2)
As done before (equation (II.3.5)) we use scale separation to simplify the calculation. We write B = B + b′ , the equation for b′ is, η∆b′ = − • ∇u (II.8.3) B
We take the large scale magnetic field to be of the form B = (B1, B2, 0)eiK•X . Using equation (II.8.3), we can calculate the small scale field to be b′ = 1/(ηkf ) U1B2 cos (kf y) , −U2B1 sin (kf x) , U3B1 cos (kf x) − U4B2 sin (kf y) . Using this we calculate the e.m.f term u × b′ . The governing equation for B gives us U1, U2, U3, U4, ∇ × u × b = −η∆ B (II.8.4)
We consider a flow driven by forcing f constant in time. Close to the threshold, the back reaction j × b is small. For an small Lorentz force we expand the velocity field as u = ub + ǫuc + O(ǫ2). Here u b denotes the base flow when there is no magnetic field and ǫ = (Rm − Rmc)/Rmc is the distance from the threshold. uc is the correction in the velocity field when we are above the threshold ǫ > 0. By doing a weakly nonlinear analysis we solve only for the order ǫ in the expansion of u. The equation for ub satisfies, ∂tub + ub • ∇ub = − 1 ∇p + ν∆ub + f (II.8.9) ρ.
SATURATION OF THE DYNAMO
The exponential growth phase ends when b becomes large enough to start modifying the underlying velocity. The governing equation for the Navier Stokes equation now reads, ∂tuc + uc • ∇ub + ub • ∇uc = − 1 ∇pc + ν∆uc + 1 j × b (II.8.10) ρ ρµ0ǫ.
Here pc is the correction in pressure due to the Lorentz force. Close to the threshold j × b ∼ O(ǫ) for a supercritical bifurcation. The Lorentz force can balance either the nonlinear term or the viscous term depending on the relative strength. The ratio of uc • ∇ub/(ν∆uc) is given by the Re, when Re ≪ 1 the underlying flow is laminar and the balance goes like.
Figure II.20 – Figures shows a sketch of a) Riga Dynamo, b) Karslruhe Dynamo, c) VKS dynamo.
Ponomarenko dynamo  while Karslruhe dynamo is based on the laminar theory of the Roberts flow  and the VKS dynamo based on the Von-Karman flow. In all the three cases the dynamo instability happens over a highly turbulent flow. However the dynamo instability thresholds of Riga and Karlsruhe dynamo are predicted very well by the laminar dynamo theory. This is because in these two cases the turbulent flows are constrained, leading to much smaller turbulent fluctuations. The turbulent fluctuations do not have a large aﬀect on the dynamo instability. In the VKS dynamo the turbulent fluctuations played an important part in the magnetic field that was generated. The mean flow alone does not predict the form of the magnetic field obtained, see [75, 76, 77].
The saturation energy of the magnetic field in all the three experiments is shown in figure II.21 taken from the . The magnetic energy is shown for the Riga experiment in ⋆, Karlsruhe in and the VKS experiment in •. All the three experiments show a critical Rmc ∼ 30 above which the dynamo instability occurs. For Rm > Rmc, there is a linear dependence of the amplitude of the saturated magnetic energy with the distance from the threshold. The value of the linear fitting parameter C diﬀers for diﬀerent scaling laws (equations (II.8.3),(II.8.6)), for the laminar scaling we expect C ∼ P m while for the turbulent scaling C ∼ 1. In the figure II.21 for both Riga and Karslruhe dynamo C = 1 while for the VKS dynamo C = 25. The reason for a high value of the fitting parameter C in the VKS dynamo was linked to the weak magnetic field intensity measured at the boundary. Thus in all three experiments we see the clear turbulent scaling while the laminar scaling would have predicted a factor 105 weaker magnetic field intensity.
Table of contents :
I.1 Rotating dynamos as a simple model
II Dynamo effect of quasi-twodimensional flows
II.1 Helical and Nonhelical flows
II.2 Dominant scales responsible for dynamo action
II.3 Helical dynamo
II.3.1 Dependence on Re
II.4 Nonhelical dynamo
II.4.1 Dependence on Re
II.5 Critical magnetic Reynolds number Rmc
II.6 Dependence on kf L
II.7 Conclusion – Part 1
II.8 Saturation of the dynamo
II.8.1 Robert’s flow as an example
II.8.2 Different scaling laws
II.9 On experimental dynamos
II.10 On numerical models of dynamo
II.11 Saturation of the 2.5D dynamo
II.12 Joule dissipation and dissipation length scale
II.13 Saturation in a thin layer
II.14 Conclusion – Part II
IIIKazantsev model for dynamo instability
III.1 Model development for 2.5D nonhelical flow
III.2 Model flow
III.3 Growth rate
III.4 Different limits
III.4.1 Limits Rm → ∞,Dr → 0
III.4.2 Rm → ∞,Dr → 0
III.4.3 Rm → ∞,Dr → ∞
III.5 Correlation function and energy spectra
III.6 Comparison with direct numerical simulations
III.6.1 White noise flows
III.6.2 Freely evolving flows
III.7 Conclusion-Part 1
III.8 Intermittent scaling of moments
III.9 α-dynamo for Kazantsev flow
III.10 Numerical results
III.10.1 Saturation/Nonlinear results
IV 3D Rotating flows and dynamo instability
IV.1 Parameter space
IV.1.1 Transition to the condensate
IV.2 Asymptotic limits
IV.3 Conclusion – Part 1
IV.4 Rotating dynamos
IV.4.1 Parameters of the study
IV.4.2 Critical magnetic Reynolds number
IV.4.4 Helical forcing case
IV.4.5 Structure of the unstable mode