3D Rotating flows and dynamo instability 

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Helical and Nonhelical flows

The velocity field is written in terms of the stream function ψ, vertical velocity uz, as u = ∇× ψˆez + uzˆez. The governing equations are:
∂tΔψ + u ·∇(Δψ) =νΔ2ψ − ν−Δψ + Δf , (II.1.1).
∂tuz + u ·∇uz =νΔuz + fz, (II.1.2).
where f , fz are forcing functions. We add a large scale friction ν− in the governing equation for ψ to model the Ekman friction. This large scale friction arises from the thin Ekman layers formed at boundaries of fast rotating flows and acts like a linear drag term [49, 50, 51]. The flow is generated by the body forcing terms f , fz through which we inject energy. Dissipation comes from both viscous terms and the linear friction term.
The presence or absence of mean helicity is important in determining whether a given flow can amplify a seed magnetic field at large scales. In this thesis we consider two different flows in the domain of 2.5D flows, one which has a mean helicity while the other without any mean helicity. We take the standard Roberts flow as the forcing which has a mean helicity, it has the form f = fz/kf = f0 cos (kfx) + sin (kf y) /kf . The forcing used to create a flow with zero mean helicity has the form f = f0  (kfx) + sin (kf y) /kf , fz = f0 sin (kfx) + cos (kf y) . Here f0 is the forcing amplitude and kf is the forcing wavenumber which injects energy at a single wavenumber. The main difference between the helical and the nonhelical forcing function is in fz which is π/2 shifted in both x and y directions. The helicity of the helical forcing hΔf fzi = kf with h·i denotes the average over space. The helicity of the nonhelical forcing is hΔf fzi = 0.

Dominant scales responsible for dynamo action

To study the dynamo instability one would like to know which scales are responsible for the amplification of the magnetic field. The discussion here is mostly speculative based on scaling laws. The dynamo instability is driven by the term B · ∇u which represents a transfer of energy from the kinetic field to the magnetic field through the shear of the velocity field ∇u. Thus a general outlook would be to look at which scale the shear ∇u is largest in the flow.
We need both the horizontal velocity field u2D and the vertical velocity field uz in order for the dynamo instability to exist. We denote the amplitude of the velocity field at a particular scale ℓ as u2D (ℓ) , uZ (ℓ) and the shear at a particular scale as S2D (ℓ) , SZ (ℓ). For 2D turbulence u2D behaves like u2D (ℓ) ∼ ℓ for scales between the forcing and the dissipation scales ℓf > ℓ > ℓ. Ideally we expect this scaling to arise at very large values of Re. Due to inverse cascade we expect that u2D (ℓ) ∼ l1/3 at scales larger than the forcing scale ℓ > ℓf . The vertical velocity uZ (ℓ) ∼ ℓ0 for scales smaller than the forcing scale ℓf > ℓ > ℓ. For the large scales ℓ > ℓf we have uZ (ℓ) ∼ ℓ−1. Thus we write the shear of both the two-dimensional flow and the vertical flow at different scales to be, S2D (ℓ) ∼ u2D (ℓ) ℓ ∼0.

Critical magnetic Reynolds number Rmc

The critical magnetic Reynolds number is defined as the minimum Rm necessary for a given Re to have a dynamo instability. Since we have an extra parameter kz we need to look at the available modes in the z-direction. The critical magnetic Reynolds number for an infinite layer which allows all possible kz modes, is defined as, Rmc (Re) = max n Rm s.t. γ ≤ 0 ∀kz o . (II.5.1).
For the helical flow since there always exists an unstable mode for any value of Re, we see that the Rmc for the infinite layer is zero i.e. for a given Rm there is a kz small enough that it is dynamo unstable. While for the nonhelical case it is nonzero and a function of the Re. We show the Rmc in figure II.15 as a function of Re for the nonhelical flow. We first look at the large Re number limit where we see that the critial Rmc saturates as a function of Re. This is similar to 3D flow where the Rmc was found to saturate at a large value of Re, see [59, 60, 61]. It is interesting to note here that the effect of increase in Re does not seem to affect the value of the threshold much. This is contrary to the 3D case where we see that Rmc ∼ Re for moderate values of the Re, implying that an increase in Re increases the value of Rmc. Finally, we note that the recent study of [62], where a 3D flow was considered with scale separation of kf L = 4 shows very little increase as one increases Re (note that the flow in the study [62] has mean helicity). The 2.5D case might indicate that rotation might help the dynamo instability in the turbulent regime. We will look at this in detail in chapter The two vertical lines in figure II.15 at values ReT1 and ReT2 denote transitions in the base state of the flow. ReT1 denotes a transition between one laminar state to another while ReT2 denotes a transition between a laminar state and a turbulent state.
The laminar nonhelical flow does not induce a dynamo instability through an α-effect but rather induces the dynamo instability through a β-effect. The β-effect comes at a higher order in the mean-field expansion (II.3.6), its value can be calculated only in a few cases analytically (see [63, 64, 65]). In order to find the value we need to expand the equations formally as done in [24]. However we find that at the lowest order one needs to invert the full induction operator. So analytically finding the value of β is difficult. The existence of the β-effect is seen from figure II.16, where γ the growth rate of the magnetic field is shown as a function of kz. A dashed line shows the scaling k2 z which is valid in the small values of kz. This scaling of growth rate is predicted by the β-effect. The β-effect amplifies the large scales of the magnetic field. The contour velocity field kf L = 4.
We have studied the dynamo threshold for an infinite layer, we now look at the implication of this study on a cubic box of size [2πL, 2πL, 2πL]. This geometry allows only for integral multiple of modes kz = 1. For a cubic geometry we only need to look at the kz = 1 mode and its integer multiples. The dynamo threshold for this domain is predicted by the most unstable mode among kz = 1 and its integer multiples. For the helical forcing, from figure II.10b, II.12b the most unstable mode is found to be kz = 1. It becomes unstable at Rm ≈ 2, slightly increasing as we increase the value of Re. Thus the critical magnetic Reynolds number for the cubic domain and the helical flow is through the kz = 1 mode and is around Rm ≈ 2. For the case of the nonhelical flow the most unstable mode is also close to kz = 1 mode and the critical magnetic Reynolds number (from the figure II.15) is found to be Rm ≈ 10. It stays constant even at large Re. The cubic geometry will be used later on in Chapter IV, when we study the effect of rapidly rotating flows on the dynamo instability.

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On experimental dynamos

We mention here briefly the results from various experimental dynamo studies and the scaling of the magnetic field close to the dynamo threshold. The experimental dynamos are the 1) Riga dynamo [72], 2) Karslruhe dynamo, [73] 3) VKS dynamo [74] are shown in figure II.20. The Riga dynamo is a based on the laminar theory of the Ponomarenko dynamo [66] while Karslruhe dynamo is based on the laminar theory of the Roberts flow [28] 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 [78]. 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 ∼ Pm 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.

10On numerical models of dynamo

Another important problem of the saturation of the dynamo instability is in the simulations of astrophysical systems. The geodynamo simulations consists of resolving the governing equations on a spherical domain. The simulations try to achieve an Earthlike parameter regime. Given the large values of the nondimensional parameters for the Earth, see Chapter I, the numerical simulations are not yet at the right zone of parameters. One of the main issues is with respect to the Re of the flow, the numerical simulations are restricted to moderate values of Re. Thus for simulating a flow of the inner core of the Earth, Pm ≈ 10−6 one needs to go to very large Re to study dynamo instability. Since computationally it is quite difficult to reach such large values of Re, numerical simulations are done at larger values of Pm, Pm ≥ 0.01 in fully periodic boxes [61] and Pm ≥ 0.05 in spherical domains [79, 80, 81].
The simulations at much larger values of Pm and moderate values of Re imply that the amplitude of the saturation of the magnetic field is affected by viscous dissipation. Indeed many studies [82, 83] have found that numerical geodynamo simulations are still in the regime of laminar flows. Thus numerical dynamo models are quite far from reaching the turbulent scaling required to model Earth like systems. Thus one of the question asked is when do we see the transition from the laminar to the turbulent scaling. In order to look at the dependence of the amplitude of the magnetic field as a function of the Prandt number Pm we use the 2.5D mode.

Saturation of the 2.5D dynamo

In the kinematic study presented in the first part of the Chapter we forced the velocity  field which excites the 0 mode along the z-direction. The 0 mode means that the flow is invariant along the z direction. Due to invariance along the z-direction the magnetic field is decomposed in kz modes along z-direction which are independent of each other. For a given domain [2πL, 2πL,H], represented in figure II.22, the unstable mode kz is a multiple of 2π/H. The back reaction of the most unstable is responsible for the saturation the dynamo instability. The back reaction through the Lorentz force j × b mode. Here the Lorentz force acts on the 0 mode and the 2kz mode for the velocity field. We write the velocity field as a sum of two vertical modes, the 0 and the 2kz mode as, u = v0 + v2. We denote the vorticity as, ! = ∇ × u = !0 + !2, with ∇×v0 = !0,∇×v2 = !2. By averaging along the vertical direction we can write the governing equations of the two modes separately as, ∂t!0 + N.L.(!0,!0)+N.L.(!2,!2) = νΔ!0 − ν−!0 + 1 ρL0(∇× (j × b)).

Table of contents :

Table of contents
I Introduction 
I.1 Rotating dynamos as a simple model
I.2 Turbulence
I.3 Rotation
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
III.11 Conclusion
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.4.3 Visualizations
IV.4.4 Helical forcing case
IV.4.5 Structure of the unstable mode
IV.5 Conclusion
Perspectives and conclusions
Appendix
A Derivation of Kazantsev model 
B Matched Asymptotics for the Kazantsev model 
B.0.1 Inner solution
B.0.2 Outer solution
B.0.3 Matching
C Numerical algorithm for multiplicative noise 
List of Figures
Bibliography

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