Dynamo eﬀect of quasi-twodimensional flows
Velocity and magnetic fields that have many symmetries cannot be sustained by dy-namo action, this is shown by the anti-dynamo theorems of Cowling , Zeldovich  and others [19, 20, 21], see also [22, 23, 24] for detailed discussions. Some examples are, axisymmetric magnetic fields cannot be sustained by dynamo action , purely two-dimensional flow and parallel shear flows (like u = U (y)eˆx) do not give rise to the dynamo instability . Toroidal flows in a spherical geometry do not give rise to dynamo instability , purely radial flows cannot sustain a magnetic field . We look at the proof for the case of a purely two dimensional flow u(x, y) = (u, v, 0). The magnetic field can be written as B = beikz z + c with b = (bx, by , bz ), we define b2D = (bx, by , 0). The equation for bz is written as, ∂tbz + (u • ∇) bz = η∆bz , (II.0.1) which is similar to the heat equation with no source term and the z-component of the magnetic field dies out bz → 0. The resulting field has only two components b2D = ∇ × (aez ) which is written in terms of the vector potential a. The governing equation of a can be written as, ∂ta + (u • ∇) a = η∆a, (II.0.2) which implies a decays to zero. Thus a purely two dimensional flow does not give rise to a dynamo instability.
A simple flow that can result in a dynamo instability in a Cartesian domain is the 2 5D-flow. The flow can be written as u = (u(x, y), v(x, y), w(x, y)), here all the three components of the velocity field are non-zero but they only depend on two directions x, y. So the 2 5D flow is independent of the z-direction. The vertical velocity uz adds the necessary amplification term in order for the magnetic field to amplify, diﬀering from the case of purely 2-dimensional flows. Since the flow is invariant along the z direction we can decompose the magnetic field into Fourier modes along the z direction as B(x, y, z) = b(x, y)eikz z + c . The induction equation for each mode can be written as, ∂tb + u • ∇b + ikz uz b = b • ∇u + η ∆ − kz2 b (II.0.3)
The solenoidality condition gives ∇ + ikz eˆz • b = 0.
The main motivation for the study of this type of flow in this thesis is to un-derstand the eﬀect of fast rotation on the dynamo instability. We know from the Taylor-Proudmann theorem that very fast rotating flow leads to a constraint ∂z u → 0 as Ω → ∞, see [25, 26, 27]. Thus one can consider the 2 5D flow as a limit of infinite rotation. Such kind of flows have been studied previously by many people. The classic example where such a flow was used to study the dynamo instability dates back to Roberts . He proposed four diﬀerent sets of laminar flows with which he could find the dynamo instability for suﬃciently large magnetic Reynolds number Rm. One of those laminar flows was explained theoretically using the scale separation α-dynamo model in the work of . The physical explanation actually dates back to Eugene Parker . Soward  studied the dynamo instability for the α-dynamo in the limit of large Rm for the laminar flows. These flows showed that there cannot be any fast dynamo action (where the growth rate of the magnetic field asymptotes to zero as one increased Rm, ). A time varying version of the quasi-twodimensional flow with chaotic structure was used to show the existence of a finite growth rate in the limit of large Rm, see [33, 34]. The freely evolving 2 5D flows obtained from solving Navier-Stokes was first studied by , and later on by [36, 37]. We take this flow as a starting step to understand later on in the thesis the dynamo instability aris-ing from fully three-dimensional rotating flows. We first look at the solutions of the Navier-Stokes equation without the magnetic field.
The x, y component of the velocity field ux, uy behave like 2D turbulence given by the Kraichnan-Leith-Batchelor theory [38, 39, 40]. In 2D turbulence there are two conserved positive definite quantities when dissipation is zero, they are the energy U2D = 1/(LxLy ) u 2x + u2y dxdy and the enstrophy W2D = 1/(LxLy ) ωz2 dxdy. ωz is the vorticity along the z direction ωz = ∂xuy − ∂y ux. Two-dimensional turbulence exhibits the dual-cascade picture where the energy U2D cascades to large scales while enstrophy W2D cascades to small scales. For historical context and detailed discussions see the reviews [41, 42, 43, 44, 45]. The vertical velocity uz acts like a passive scalar being advected by ux, uy . The square of the vertical velocity is a positive conserved quantity when the dissipation in zero, in the finite dissipation case it cascades to small scales.
A model spectra of energy is shown in figure II.1 where the forcing injects energy and enstrophy at an intermediate scale. We define the 2D velocity field spectra E2D (k) as E2D (k′ ) = |uˆx(k)|2 + |uˆy (k)|2 δ|K|,k′ dk, where uˆx, uˆy are the Fourier transforms of ux, uy respectively. δ|K|,k′ is the Kronecker delta, taking the value 1 when |k| = k′ and zero otherwise. Figure II.1a shows the E2D (k) dual cascade picture with power law behaviours in the two inertial range of scales. The power laws are calculated using Kolmogorov arguments, if we denote kf as the injection scale then E 2D (k) ∼ ǫ2Ω/3k−3 for k > kf due to the forward cascade of enstrophy with ǫΩ being the enstrophy injection rate. E2D (k) ∼ ǫ22D/3k−5/3 for k < kf due to the inverse cascade of energy with ǫ2D being the energy injection rate of the 2D components of the velocity field. Figure II.1b shows the model spectra of EZ (k). The vertical velocity spectra at scales larger than the forcing scale k > kf follows the prediction EZ (k) ∼ ǫZ k −1/ǫ1Ω/3 where ǫZ is the injection rate of u2z. For scales larger than the forcing scale, k < kf , EZ (k) ∼ k+1 due to equipartition of energy. The modes are at equipartition since there is no cascade to large scales, similar to the behaviour of scales larger than the forcing scale in 3D turbulence see [47, 48]. At equipartition all the modes have equal energy and doing the spherical shell averaging we get the prediction of EZ (k) ∼ ǫZ k+1 /ǫ1Ω/3.
Helical and Nonhelical flows
The velocity field is written in terms of the stream function ψ, vertical velocity uz , as u = ∇ × ψeˆz + uz eˆz . The governing equations are, ∂t∆ψ + u • ∇ (∆ψ) =ν∆2ψ − ν− ∆ψ + ∆fψ , (II.1.1)
∂tuz + u • ∇uz =ν∆uz + fz , (II.1.2)
where fψ , 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 fψ , 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 con-sider 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 fψ = fz /kf = f0 cos (kf x) + sin (kf y) /kf . The forcing used to create a flow with zero mean helicity has the form fψ = f0 cos (kf x) + sin (kf y) /kf , fz = f0 sin (kf x) + 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 func-tion is in fz which is π/2 shifted in both x and y directions. The helicity of the helical forcing ∆fψ fz = kf with denotes the average over space. The helicity of the nonhelical forcing is ∆fψ fz = 0.
The nondimensional numbers for the hydrodynamic equations (II.1.1), (II.1.2) are, the Reynolds number Re = u2 1/2 /(kf ν), the Reynolds number related to the linear friction ν− denoted as Re− = u2 1/2 kf /ν− , the forcing wavenumber kf L. Here u2 1/2 is the r.m.s of the velocity field averaged over space and time. It is a measured quantity resulting from a particular choice of the forcing amplitudes and dissipation parameters. Hence in order to understand the behaviour we have three independent parameters to change, kf L, Re− , Re.
We conduct numerical simulations in a box [2πL, 2πL] for diﬀerent values of Re. First we increase the value of Re keeping Re− fixed. Here the forcing amplitude is fixed at f0 = 1 and ν is decreased in order to increase the value of Re. We show the spectra of E2D (k), EZ (k) both for the helical and the nonhelical flow in figure II.2 for diﬀerent values of Re keeping the large scale dissipation ν− fixed. For small Re the length scales k > kf are more steeper than the predicted exponents of E2D (k) ∼ k−3, EZ (k) ∼ k−1. As we increase Re we start to see that the spectra tends more towards the Kraichnan-Leith-Batchelor theory. This was studied previously by  where they showed that in the limit of large Re the energy spectra at small scales tend to the KLB prediction.
We notice that the spectra seems to behave similarly for both the helical and the nonhelical forcing.
Figure II.3 shows the energy spectra E2D (k), EZ (k) as a function of the normalized wavenumber k/kf for both the helical and the nonhelical forcing. Here the value of forcing wavenumber kf L is changed to see the behaviour of scales larger than the forcing scale. Re is kept almost constant, by increasing the scale separation between the box size and injection scale, Re− is increased. For the 2D components, scales larger than the forcing scale k > kf show that E2D ∼ k−5/3. This corresponds to spectrum formed from the inverse cascade of energy. There is no large scale condensate due to the presence of the friction term. The vertical velocity spectrum follows the equipartition spectra EZ ∼ k+1. Similar to the small scales the large scale behaviour of both E2D , EZ seem to be independent of the presence or absence of any mean helicity injection by the forcing.
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 > ℓν . 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 > ℓν . For the large scales ℓ > ℓf we have uZ (ℓ) ∼ ℓ−1 scaling represents the low Rm limit where it can be shown that α ∼ uRm. For large values of Rm the expansion in 1/Rm breaks down and the theory is no longer valid. Numerically we see that the value of α saturates and becomes independent of Rm.
The α-coeﬃcient is finite at large Rm, the question one could ask is whether it plays a role in the dynamo instability at large Rm. We show in figure II.8 the magnetic field spectra EB (k) for diﬀerent values of Rm mentioned in the legend for the parameter kz = 0 25, Re ≈ 530. The magnetic spectra are rescaled so that the total magnetic energy is 1. For low values of Rm the spectra of the magnetic field show the signature of an α-eﬀect due to the presence of a peak at the largest scale. As we increase Rm we see that the magnetic field is increasingly stronger in the smaller scales. The α-eﬀect becomes less important at large Rm even though the α coeﬃcient is non zero as seen in figure II.7b. See the recent work  for a detailed picture.
Dependence on Re
To quantify the dependence on the Re we focus on two diﬀerent quantities, firstly γmax defined as the maximum growth rate over all kz for a given Re, Rm. Next we define kzc as the maximum wavenumber at which one can sustain the dynamo instability for a given Re, Rm. We illustrate this in figure II.9.
Figure II.10 shows both the maximum growth rate γmax and the cut-oﬀ wavenu-meber kzc as a function of Rm, Re. Figure II.10a shows that γmax increases initially as Rm3 for low Rm and then saturates at large values of Rm becoming independent of Rm. γmax does not vary much with Re. Figure II.10b shows that kzc increases initially as Rm2 for small values of Rm before transitioning to another regime at large Rm where it scales like Rm1/2. The prediction at low Rm for both γmax ∼ Rm3 and kzc ∼ Rm2 can be obtained from the α-theory (see equation (II.3.9)). We mention also that kz at which γmax is found, increased initially with Rm as kz ∼ Rm2 before satu-rating at the value kz ≈ 1. The curve kzc at large Rm shows the scaling kzc ∼ Rm1/2
This is diﬀerent from the α-scaling of Rm2 implying that for large Rm the cut-oﬀ length scale kzc is not controlled by α-eﬀect. There seems to be a weak dependence of kzc on the Re in the large Rm limit. As one increases Re we see that kzc decreases, this is expected since increase in Reynolds number correspond to increase in the noise in the system making it diﬃcult for the dynamo action to take place. Hence small wavelength modes (large kz ) become less and less unstable as Re is increased. By doing an empirical fit we find that kzc ∼ Rm1/2Re−3/8. However one should look at much larger values of Re in order to conclude a powerlaw dependence on Re.
We now consider the nonhelical forcing. Figure II.11 shows γ as a function of kz for diﬀerent values of Rm mentioned in the legend. Contrary to the case of the helical flow, there is a minimum Rm to have dynamo unstable modes. This minimum Rm is the critical magnetic Reynolds number Rmc for an infinite domain, as changing kz changes the vertical extent of the system. Rmc is in general a function of Re. Close to the threshold of the dynamo instability, the first kz mode that becomes unstable is found at a value kz ≈ 1. As we increase Rm more modes become unstable similar to the case of the helical flow. Unlike the helical flow, γ does not reach a saturation even for the largest value of Rm we have explored. For comparable Re, Rm values, the nonhelical flow always gives a smaller value of γ.
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 Rm s.t. γ ≤ 0 ∀kz (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 , 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  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 IV where we consider a 3D flow subject to global rotation.
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 . 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 kz2 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 of the magnetic energy in the two diﬀerent laminar regimes are shown in figure II.17. The large scale structures of the magnetic field vary over a larger length scale than the 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.
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
III Kazantsev 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
Perspectives and conclusions
A Derivation of Kazantsev model
B Matched Asymptotics for the Kazantsev model
B.0.1 Inner solution
B.0.2 Outer solution
C Numerical algorithm for multiplicative noise
List of Figures