Rotating Turbulent Dynamos Kannabiran Seshasayanan

Rotating turbulent dynamos Kannabiran Seshasayanan

To cite this version:

Kannabiran Seshasayanan. Rotating turbulent dynamos. Physics [physics]. Université Pierre et Marie Curie - Paris VI, 2017. English. NNT : 2017PA066158. tel-01646437

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Remerciements

Avant d’arriver en France, je ne savais pas beaucoup la culture de ce beau pays. Quand même j’ai décidé de rester et apprendre la langue, la culture et faire une grosse partie de ma vie à Paris. J’ai la chance de sentir une partie de la culture. Je remercie tous ceux qui ont contribué à cela. Je remercie mon directeur de thèse, Alexandros Alexakis pour son soutien pendant toute la durée de ma thèse, sa confiance sur moi de m’accorder la liberté d’explorer les différents thèmes de recherche et surtout sa patience de me laisser prendre le temps nécessaire. Je le remercie pour avoir une vision long temps sur ma carrière et de me faire apprendre les fonctionnement de la communauté scientifique. Pour ses expressions courtes et fortes ! Je n’oublierai certainement pas son expression “What !!”. Je remercie Steve Tobias et Andreas Tilgner d’avoir accepté d’être rapporteurs pour ma thèse. Je remercie aussi Laurette Tuckerman et Dario Vincenzi pour accepter d’être les examinateurs à la soutenance. Maintenant, il faut que je parle de l’équipe Physique nonlinéaire. J’avais la grande plaisir de fait passé tout ce dernière trois ans avec cet équipe. J’avais la chance de travailler/discuter avec tout les membres. D’abord je remercie François Pétrélis de m’envoyer vers Alex quand j’avais demandé des conseils pour le stage de Master. Et aussi pour son cours de l’effet dynamo où j’avais appris beaucoup de chose et m’avait convaincu de faire une thèse sur ce sujet. J’avais travaillé avec lui sur l’intermittence ON-OFF. Je remercie Christophe Gissinger pour me laisser joué avec les gouttes de Couder pendant long temps. J’ai appris beaucoup de chose sur l’instabilité de Faraday et l’analyse des images avec la camera rapide. Je le remercie aussi pour les discussions sur l’instabilité dynamos, le manip de VKS, et ces conseils sur les films français !!. Je remercie le chef de l’équipe Stephan Fauve pour ses conseils, ses idées brillant et ses analyses critiques de ma recherche. Surtout pour ses blagues divers. Je remercie beaucoup Benoît Semin pour sa patience de corriger tous les fautes d’orthographe dans mes mails. Ces bonbons gratuit qu’il a ramené de l’Alsace/Lorraine. Ses réponses sur toutes les questions sur touts les sujets (vraiment tous les sujets !!). Après, je remercie les thésards. Guillaume Michel pour bien organiser les séminaires de groupe, ses réponses rapide pour toutes mes questions sur l’administration, l’École doctorale etc. Je remercie Michael Pereira pour son bon humeur, son énergie et pour les fêtes qu’on avait fait dans les conférences (surtout à BIFD !!). J’ai eu la chance de travailler avec Basile Gallet sur le sujet de saturation de l’effet dynamo. Je le remercie pour sa patience et sa vision claire sur le sujet. Je remercie les étudiants d’Alex, Vassili, Mira, Santiago pour les discussions. Je remercie encore Vassili pour la collaboration sur le travail de réduction du seuil de

i ii dynamo. Je remercie également les postdocs de l’équipe, Gustavo, Paola, Sandeep, Vishy et Eric. Avec l’équipe, on avait beaucoup discuter et partager notre vie scientiﬁque avec les membres du Laboratoire de Radio Astronomie (LRA). Je remercie les personnels de ce labo pour les discussions. Ludovic Petitdemange pour les discussions sur la politique, la science et d’autre choses. Florence Marcotte pour les informations intéressantes sur tout le monde. Raphael Raynaud pour son bon humeur. Alexandre Cameron pour les discussions et de me donner ses programmes de regriding. Je remercie les secrétaires du LPS, Annie Ribaudeau et Nora Sadaoui pour leur aide pendant ces trois ans. Je remercie mes amis à la Résidence Robert GARRIC (Maison Internationale) où j’avais habité pendant ces trois ans. J’ai eu la chance de rencontrer beaucoup de gens de cultures diﬀérentes. Je remercie l’administration de cette maison pour la facilité de séjour. Je remercie la Maison de l’Inde où j’ai célébré la plupart des festivals indiens. Tout les week-ends j’ai passé beaucoup de temps avec mes amis, Rajiv, Anirudh, Mafalda, Bade, Carolyna, Sviat. Ils ont eu la malchance d’écouter et faire semblant de rire de mes mauvaises blagues !! Je remercie aussi mes autre amis Shreyas, Chakri, Ganga, Anchal, Lawrence, Venkatesh pour les bon moments qu’on avait passé ensemble. கைட எ ெபேறாக ந ெதெகாேற. ச- யாேவா ெமாைக ேபாட ஒ நல ெபாேபாகாக இத. Contents

Table of contents iii

I Introduction 1 I.1 Rotating dynamos as a simple model ...... 2 I.2 Turbulence ...... 4 I.3 Rotation ...... 5

II Dynamo effect of quasi-twodimensional flows 9 II.1 Helical and Nonhelical flows ...... 11 II.2 Dominant scales responsible for dynamo action ...... 15 II.3 Helical dynamo ...... 16 II.3.1 Dependence on Re ...... 18 II.4 Nonhelical dynamo ...... 20 II.4.1 Dependence on Re ...... 21 II.5 Critical magnetic Reynolds number Rmc ...... 21 II.6 Dependence on kf L ...... 25 II.7 Conclusion - Part 1 ...... 26 II.8 Saturation of the dynamo ...... 27 II.8.1 Robert’s flow as an example ...... 27 II.8.2 Different scaling laws ...... 28 II.9 On experimental dynamos ...... 30 II.10 On numerical models of dynamo ...... 31 II.11 Saturation of the 2.5D dynamo ...... 32 II.12 Joule dissipation and dissipation length scale ...... 37 II.13 Saturation in a thin layer ...... 40 II.14 Conclusion - Part II ...... 43

IIIKazantsev model for dynamo instability 45 III.1 Model development for 2.5D nonhelical flow ...... 47 III.2 Model flow ...... 51 III.3 Growth rate ...... 52 III.4 Different limits ...... 54 III.4.1 Limits Rm ,D 0 ...... 54 → ∞ r → III.4.2 Rm ,D 0 ...... 54 → ∞ r → III.4.3 Rm ,D ...... 56 → ∞ r → ∞ III.5 Correlation function and energy spectra ...... 57 III.6 Comparison with direct numerical simulations ...... 61

iii iv CONTENTS

III.6.1 White noise flows ...... 61 III.6.2 Freely evolving flows ...... 63 III.7 Conclusion-Part 1 ...... 64 III.8 Intermittent scaling of moments ...... 65 III.9 α-dynamo for Kazantsev flow ...... 67 III.10 Numerical results ...... 72 III.10.1 Saturation/Nonlinear results ...... 73 III.11 Conclusion ...... 78

IV 3D Rotating ﬂows and dynamo instability 81 IV.1 Parameter space ...... 85 IV.1.1 Transition to the condensate ...... 89 IV.2 Asymptotic limits ...... 93 IV.3 Conclusion - Part 1 ...... 98 IV.4 Rotating dynamos ...... 100 IV.4.1 Parameters of the study ...... 101 IV.4.2 Critical magnetic Reynolds number ...... 102 IV.4.3 Visualizations ...... 105 IV.4.4 Helical forcing case ...... 107 IV.4.5 Structure of the unstable mode ...... 108 IV.5 Conclusion ...... 110

Perspectives and conclusions 113

Appendix

A Derivation of Kazantsev model 117

B Matched Asymptotics for the Kazantsev model 119 B.0.1 Inner solution ...... 119 B.0.2 Outer solution ...... 120 B.0.3 Matching ...... 120

C Numerical algorithm for multiplicative noise 123

List of Figures 125

Bibliography 129 Chapter I Introduction

Dynamo instability refers to the generation and sustenance of the magnetic field by a conducting fluid. It was one of the mechanism proposed by Larmor [1] for the existence of the Earth’s magnetic field. The growth or decay of an infinitesimal magnetic field in a conducting fluid is a competition between two different processes. The first process involves the stretching and folding of magnetic field lines by the velocity field. These field lines reinforce the existing magnetic field to give way for an instability. This process is in direct competition with joule dissipation (resistance against the movement of charged particles). When the stretching and folding is sufficiently stronger than the dissipation, dynamo instability occurs and infinitesimal perturbations of the magnetic field grow exponentially. The magnetic field of the Earth and Sun are shown in figures I.1a, I.1b. Even though dynamo instability is given as the explanation for these observed magnetic fields, there are a lot of open questions. For example, in the case of the Earth dynamo, different groups work on looking at what is the dominant forcing mechanism for the underlying flow. The different physical processes that could drive the internal flow could be due to thermal convection, precession [2, 3] etc. A difficulty in studying the dynamo instability is the wide range of non-dimensional parameters seen in different astrophysical objects. Table I.1 shows the different range of parameters for different astrophysical objects. Except Mars which is no longer an active dynamo, the other astrophysical objects have magnetic fields due to dynamo instability. It is clear that numerically and experimentally we cannot attain these values of the parameters (at least in the near future). Studies thus look for scaling laws and try to extrapolate the observations in labs to the parameters of astrophysical objects. Obtaining scaling laws for the different regimes of the dynamo instability and even the underlying flow is thus one of the main objectives. Astrophysical systems including Earth and Sun have numerous physical processes occurring near the core. The outer core is possibly the main location where the flow responsible for the dynamo instability is present. Trying to model such a system taking into account the complicated geometry is difficult and tedious. In what is to follow we choose a simplified model of a rotating turbulent flow to understand dynamo instability. The objective is to understand the importance of rotation and turbulence on the dynamo instability.

1 2 Chapter I. Introduction

(a) (b)

Figure I.1 – The figure a) shows the radial component of the magnetic field of Earth taken from [4] in mT (data taken from Oersted initial field model [5]). The map represents the field at the surface of the Earth’s core (3000 km below the crust). Figure b) shows a magnetogram image of the sun, which measures the magnetic field in the solar photosphere. Black and white indicating opposite polarities of the magnetic field of Sun. Figure taken from https://sohowww.nascom.nasa.gov/data/realtime/hmi_ mag/512/.

Table I.1 – Nondimensional numbers for certain Astrophysical objects. Values taken from [6, 7]. Mars has no longer an active core and its magnetic ﬁeld is residual.

Object Re (UL/ν) Rm (UL/η) P m (ν/η) Ro (U/(2ΩL)) B (T) | | Sun 1013 1011 10 2 10 2 10 4 ∼ ∼ ∼ − ∼ − ∼ − Earth 108 100 10 6 10 7 10 4 ∼ ∼ ∼ − ∼ − ∼ − Galaxy 105 1019 1914 100 10 10 ∼ ∼ ∼ ∼ ∼ − Mars 10 9 10 14 − − − − ∼ − − − Jupiter 1010 104 10 6 10 5 5 10 4 ∼ ∼ ∼ − ∼ − ∼ × −

I.1 Rotating dynamos as a simple model

In this thesis, we consider the simplified model of a fluid in a cubic container subject to rotation and an external forcing. Due to the limited computational powers, we consider only periodic boundary conditions for both the velocity and the magnetic field. This simplification helps us push for large range of parameters, meaning turbulent and fast rotating limits. The idea then is to look at the dynamo instability generated in the parameter space. A sketch of the domain is shown in figure I.2.

The governing equations of an incompressible MHD ﬂow subject to global rotation I.1. Rotating dynamos as a simple model 3

Figure I.2 – Figure shows a sketch of the domain. f denotes the forcing here represented by a 4 4 vertically invariant vortices. Ω denotes the global rotation. × in the rotating frame of reference are,

1 1 ∂ u + u ∇u = ∇p 2 (Ω u) + j B + ν∆u + f, (I.1.1) t · − ρ − × ρ × ∇ u =0, (I.1.2) · ∂ B =∇ (u B) + η∆B, (I.1.3) t × × ∇ B =0. (I.1.4) ·

Here u is the velocity field, B the magnetic field. p is the total pressure modified by the centrifugal force. Ω the rotation vector taken along the z-direction Ω = Ωez. f is the forcing. j is the current related to the magnetic field as, j = 1 ∇ B. ν is the µ0 × kinematic viscosity. The magnetic diffusion coefficient is η = 1/(µ0σ), with µ0 being the permeability of vacuum and σ the electrical conductivity of the fluid. The dynamo instability generated by such flows is the main subject of the thesis. The non-dimensional parameters in the system are, 1) Rosby number : Ratio of inertia to Coriolis force, 2) Reynolds number : Ratio of advection to diffusion/dissipation of the flow, 3) Magnetic Reynolds number: Ratio of the stretching by the flow to joule dissipation, 4) the forcing length scale or the domain size kf L. The Prandtl number is given by the ratio of two diffusivity P m = ν/η = Rm/Re. We have four different control parameters in the system. Very large Re of the system means that the flow is highly turbulent and small Ro number means that rotation plays a very important role in the hydrodynamic flow. In particular we look for the large Re and low Ro limit of the system. 4 Chapter I. Introduction

I.2 Turbulence

The problem with understanding dynamo instability is that the underlying Re is very high, Re 1. This falls into the regime of turbulence which is a highly nonlinear ≫ process. Only a few exact universal properties of turbulence is known, most of the results are attributed to Andrei Kolmogorov [8, 9, 10, 11, 12]. Figure I.3 shows the classical picture attributed to Richardson showing the transfer of energy. The ﬁgure on the left I.3a shows the energy spectrum E(k) deﬁned as,

E(k) = u 2(k)dk, (I.2.1) k =k | | Z| | ZZ withu ˆ(k) being the three-dimensional Fourier transform of the velocity field u (r). The one dimensional energy spectra E(k) is a sum of the energy of all modes that lie on a sphere of radius k in the wavenumber space. The figure I.3a shows that energy from large scales is transferred to smaller and smaller scales until it reaches the dissipation scales. This cascade is represented in terms of vortical structures in figure I.3b. Energy

Injection

Transfer

Large Dissipation scales Inertial Range scales Dissipation

(a) (b)

Figure I.3 – The figure on the left shows the Richardson cascade [13] with the transfer of energy . The figure on the right shows the same picture in terms of vortices taken from [14]. is injected at the large scales L, they correspond to a typical length scale of the domain. It is then transferred to smaller scales ℓ < L. The rate of transfer of energy ǫ is constant and in these scales dissipation is negligible. When the cascade reaches the dissipation 3/4 scale ℓν defined as ℓν = LRe− the injected energy is dissipated as heat. The larger the Re the smaller the dissipation scale ℓν and the intermediate range L < ℓ < ℓν becomes larger. The intermediate scales or the inertial range corresponds to the range of scales where viscosity or the domain do not play a role. These intermediate scales I.3. Rotation 5 only see the energy it receives and is far away from both the injection length scale L and the dissipation scale ℓν. In this inertial range, the two-point correlation function can be written in terms of the transfer rate ǫ as,

[u (x + r) u (x)]2 Cǫ2/3r2/3, (I.2.2) − ∼ D E where C 2 is the Kolmogorov constant [15]. We can write this in terms of the energy ≈ spectrum as, 2/3 5/3 E(k) = C2ǫ k− , (I.2.3) with C 0.76C, see [16]. This is the phenomenological 5/3rd law of turbulence and 2 ≈ is not an exact result. An exact result though exists for the third-order structure function for a homogeneous isotropic turbulence, it can be stated as, 4 (e (u (x + r) u (x)))3 = ǫ r. (I.2.4) r · − −5 Dh iE This is known as the Kolmogorov’s -4/5th law. This result is valid in the large Re → ∞ limit. The E(K) k 5/3 can be derived from the above expression if the turbulent ∼ − inertial range is self-similar. The picture of a self-similar law in the inertial range is only an approximation. The reason being that close to the dissipation scales, the turbulent velocity field is both spatially and temporally intermittent. The small scale structures depend on the large integral length scales and are not universal. Thus each flow with a particular geometry and a particular forcing mechanism has those imprints on the small scales. Intermittency has also been measured in the inertial range, suggesting that the scales in the inertial range are also intermittent. Intermittency makes statistics of the moments of the velocity differences to differ from the self-similar law and is an active area of research. Due to these properties no theoretical model for turbulence exists. Thus most studies on turbulence use experiments and numerical simulations. Nevertheless many simple models of turbulence have been proposed, using models for closure, eddy-viscosity, EDQNM, LES models etc. Such models are being used actively in research and also in industrial processes and are based on some simplifications. They do not capture the dynamics of small scales leaving turbulence models far from reality.

I.3 Rotation

The effect of global rotation on the flow is through two different terms as seen in equation (I.1.1). One is to modify the pressure by the centrifugal force. The other is due to the Coriolis force. The Coriolis force is responsible for suppressing certain types of interactions among the different scales of the underlying flow. The exact nature of its effects are still widely contested. At fast rotation rates, one of the main effects of Coriolis force is seen by the Taylor-Proudmann theorem. Fluid columns behave like solid objects and they move together when a part of the column is displaced, also sketched in figure I.4. The theorem leads to the constraint,

∂zu = 0, (I.3.1) 6 Chapter I. Introduction

Taylor Column

Rotating Tank

Figure I.4 – Figure shows a sketch of the a rotating tank. The Taylor column moves along with the spherical object. in the fast rotating limit. This condition is obtained by taking the curl of equation (I.1.1). Thus the underlying flow has a tendancy to bidimensionalize, and the flow becomes independent along the coordinate of the rotating axis. For a given rotation though, the flow could show both three-dimensional and two-dimensional behaviour. This makes it non-trivial to understand the effect of rotation on the flow. Also there are other predictions which show that the effect of rotation could be explained from weakly nonlinear interactions (also known as weak wave turbulence theory). Thus rotating turbulent flows remain an active area of research. Global rotation does not directly affect the induction equation, see equation (I.1.2). Its effect is through the modification of the velocity field in the induction equation. Hence it is of importance to understand the underlying hydrodynamic flow before looking at the dynamo instability. I.3. Rotation 7

Thesis setup

The rest of the thesis is based on diﬀerent limits of the rotating dynamo study. It involves the interplay between global rotation and turbulence and their eﬀects on the dynamo instability. The thesis is divided into three Chapters. Each Chapter is then divided into two sections. Each section focussing on a particular problem, has its own introduction and conclusion. The thesis is written as follow,

In Chapter II, we study the dynamo instability in the limit of very fast rotation • rates. We simplify the system to a quasi-twodimensional flow. Here the dynamo instability is driven by a flow which is uniform along the z-axis. Two main questions we try to look at in this Chapter are, 1. The different regimes of dynamo instability in a quasi-twodimensional flow and 2. The different scaling laws of the saturation amplitude of the magnetic field close to the threshold of the dynamo instability.

In Chapter III, we look at the theoretical models of a quasi-twodimensional • flow. Theoretically one could model turbulence as a fluctuating field, with which one can make considerable analytical progress. We restrict to the case of a Kazantsev type model. Here the velocity field is white in time and Gaussian. Two main questions we try to look at in this Chapter are, 1) The Kazantsev predictions for the quasi-twodimensional nonhelical flow and 2) Intermittency effects on the growth rate of the magnetic field due to a fluctuating velocity field. The theoretical results obtained in this Chapter are then compared to the results obtained through numerical simulations and to the results of Chapter II.

In Chapter IV, we look at the rotating dynamo instability in a 3-dimensional • domain. Here we study the full system of equations. The two main questions we try to look at in this Chapter are, 1. The different regimes of a rotating flow in the parameter space of Rosby-Reynolds numbers and 2. The dynamo instability arising from these rotating flows. In particular we look at the threshold of the dynamo instability, the effect of global rotation on the dynamo instability threshold. Finally we compare the fast rotating three-dimensional flow to the reduced order models of Chapter II, III. 8 Chapter I. Introduction Chapter II Dynamo effect of quasi-twodimensional flows

Velocity and magnetic fields that have many symmetries cannot be sustained by dynamo action, this is shown by the anti-dynamo theorems of Cowling [17], Zeldovich [18] 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 [17], purely two-dimensional flow and parallel shear flows (like u = U(y)eˆx) do not give rise to the dynamo instability [18]. Toroidal flows in a spherical geometry do not give rise to dynamo instability [19], purely radial flows cannot sustain a magnetic field [21]. We look at the proof for the case of a purely two dimensional flow u(x, y) = (u, v, 0). ikzz The magnetic field can be written as B = be + c.c with b = (bx, by, bz), we define b2D = (bx, by, 0). The equation for bz is written as, ∂ b + (u ∇) b = η∆b , (II.0.1) t z · z z which is similar to the heat equation with no source term and the z-component of the magnetic field dies out b 0. The resulting field has only two components z → b = ∇ (ae ) which is written in terms of the vector potential a. The governing 2D × z equation of a can be written as,

∂ a + (u ∇) a = η∆a, (II.0.2) t · 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)eikzz +c.c. The induction equation for each mode can be written as, ∂ b + u ∇b + ik u b = b ∇u + η ∆ k2 b. (II.0.3) t · z z · − z 9 10 Chapter II. Dynamo effect of quasi-twodimensional flows

The solenoidality condition gives ∇ + ik eˆ b = 0. z z · The main motivation for the study of this type of flow in this thesis is to understand 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 ∂ u 0 z → 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 [28]. 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 [29]. The physical explanation actually dates back to Eugene Parker [30]. Soward [31] 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,[32]). 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 [35], and later on by [36, 37]. We take this flow as a starting step to understand later on in the thesis the dynamo instability arising 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 2 2 2 U2D = 1/(LxLy) ux + uy dxdy and the enstrophy W2D = 1/(LxLy) ωz dxdy. ωz is the vorticity along the z direction ω = ∂ u ∂ u . Two-dimensional turbulence RR z x y − y x RR 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 [46]. 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) 2 2 ′ as E2D (k′) = uˆx(k) + uˆy(k) δ k ,k dk, whereu ˆx, uˆy are the Fourier transforms | | | | | | of ux, uy respectively. δ k ,k′ is the Kronecker delta, taking the value 1 when k = k′ RR | | | | 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 k as the injection scale then E (k) ǫ2/3k 3 for f 2D ∼ Ω − k > kf due to the forward cascade of enstrophy with ǫΩ being the enstrophy injection rate. E (k) ǫ2/3k 5/3 for k < k due to the inverse cascade of energy with ǫ 2D ∼ 2D − f 2D 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 > k follows the prediction E (k) ǫ k 1/ǫ1/3 where ǫ is the f Z ∼ Z − Ω Z injection rate of u2. For scales larger than the forcing scale, k < k , E (k) k+1 due z f Z ∼ II.1. Helical and Nonhelical flows 11

E (k) E(k) 2D z −5/3 k F +1 F k k−1 Energy −3 k Dissipation Passive scalar Enstrophy Dissipation

k k (a) (b)

Figure II.1 – The ﬁgure shows the model energy spectra of a) E2D (k) and b) EZ (k), as a function of the wavenumber k. The dashed lines denote scaling laws. F denoted energy injection. 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 E (k) ǫ k+1/ǫ1/3. Z ∼ Z Ω II.1 Helical and Nonhelical ﬂows

The velocity field is written in terms of the stream function ψ, vertical velocity uz, as u = ∇ ψeˆ + u eˆ . The governing equations are, × z z z 2 ∂ ∆ψ + u ∇ (∆ψ) =ν∆ ψ ν−∆ψ + ∆f , (II.1.1) t · − ψ ∂ u + u ∇u =ν∆u + f , (II.1.2) t z · z z z 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 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 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 12 Chapter II. Dynamo effect of quasi-twodimensional flows 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 ∆f f = k with denotes the average over space. The helicity of the h ψ zi f h·i nonhelical forcing is ∆f f = 0. h ψ zi The nondimensional numbers for the hydrodynamic equations (II.1.1), (II.1.2) are, 2 1/2 the Reynolds number Re = u /(kf ν), the Reynolds number related to the linear 2 1/2 friction ν− denoted as Re− = u kf /ν−, the forcing wavenumber kf L. Here 2 1/2 u 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.

0 10 0 Re=46.26 10 k -1 Re=46.26 Re=106.19 Re=106.19 k -3 Re=241.16 Re=241.16 Re=530.33 Re=530.33 -5 10 Re=1149.25 Re=1149.25 ) ) k k (

( -5 D z

2 10 E E 10-10

10-15 10-10 100 101 102 103 100 101 102 103 k k

(a) (b)

0 10 100 k -3 k -1

10-5 ) ) k k (

( -5 D z

2 Re=32.18 10 Re=32.18 E E Re=70.63 Re=70.63 -10 10 Re=151.08 Re=151.08 Re=320.14 Re=320.14 Re=669.89 Re=669.89 Re=1388.3 Re=1388.3 10-15 10-10 100 101 102 103 100 101 102 103 k k

Figure II.2 – The figure shows the energy spectra E2D (k),EZ (k) for the helical flow on top and the nonhelical flow on bottom. Darker shades denote increasing values of Re. Black dotted lines denote scaling laws.

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 > k are more steeper than the predicted exponents of E (k) k 3,E (k) k 1. f 2D ∼ − Z ∼ − As we increase Re we start to see that the spectra tends more towards the Kraichnan- Leith-Batchelor theory. This was studied previously by [52] where they showed that in the limit of large Re the energy spectra at small scales tend to the KLB prediction. II.1. Helical and Nonhelical flows 13

0 10 100 k -5/3 k +1 10-5 ) ) k k (

( -5 D z

2 10 E E 10-10 Re=106.19, Re− =53.09 Re=106.19, Re− =53.09 Re=91.75, Re− =183.5 Re=91.75, Re− =183.5 Re=97.19, Re− =388.77 Re=97.19, Re− =388.77 10-15 10-10 10-2 10-1 100 101 102 10-2 10-1 100 101 102 k/kf k/kf (a) (b) 100 100

k -5/3 k +1 10-5 ) ) k

k -5 (

( 10 D z 2 E E 10-10 Re=70.63, Re− =35.31 Re=70.63, Re− =35.31 Re=81.27, Re− =162.54 Re=81.27, Re− =162.54 Re=90.91, Re− =363.66 Re=90.91, Re− =363.66 10-15 10-10 10-2 10-1 100 101 102 10-2 10-1 100 101 102 k/kf k/kf (c) (d)

Figure II.3 – The figure shows the energy spectra E2D(k),EZ (k) for the helical flow on top and the nonhelical flow on bottom as a function of the normalized wavenumber k/kf . Darker shades denote increasing values of Re−. Black dotted lines denote scaling laws. 14 Chapter II. Dynamo effect of quasi-twodimensional flows

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 > k show that E k 5/3. This corresponds to spectrum f 2D ∼ − 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 E k+1. Similar to the small scales the large scale behaviour of both E ,E Z ∼ 2D Z seem to be independent of the presence or absence of any mean helicity injection by the forcing.

6 27 6 4.5 24 5 5 21 3.0

4 18 4 1.5 15 0.0 3 3 12

1.5

2 9 2 3.0 6 1 1 4.5 3 6.0 0 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6

(a) (b)

6 36 6 3.2 32 2.4 5 5 28 1.6 4 24 4 0.8 20 0.0 3 3

16 0.8

2 12 2 1.6

8 2.4 1 1

4 3.2

0 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6

Figure II.4 – Figure shows the contours of 2D energy E2D (x, y) and the vertical velocity uz(x, y) for the helical flow on top (figures a) and b)) and the nonhelical flow on bottom (figures c) and d)).

Figure II.4 shows the 2D energy E2D (x, y) and the vertical velocity uz for both the helical and the nonhelical ﬂow for the case of kf L = 16. The energy has large scale vortices present signifying more energy at large scales while the vertical velocity is dominated at the forcing scale kL k L = 16. ∼ f II.2. Dominant scales responsible for dynamo action 15

II.2 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 S (ℓ) ,S (ℓ). For 2D turbulence u behaves like u (ℓ) ℓ for scales between the 2D Z 2D 2D ∼ 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 u (ℓ) l1/3 at scales 2D ∼ larger than the forcing scale ℓ > ℓ . The vertical velocity u (ℓ) ℓ0 for scales smaller f Z ∼ than the forcing scale ℓ > ℓ > ℓ . For the large scales ℓ > ℓ we have u (ℓ) ℓ 1. f ν f Z ∼ − Thus we write the shear of both the two-dimensional flow and the vertical flow at different scales to be, 0 u2D (ℓ) l : ℓf > ℓ > ℓν, S2D (ℓ) 2/3 (II.2.1) ∼ ℓ ∼ ( l− : ℓ > ℓf , 1 uZ (ℓ) l− : ℓf > ℓ > ℓν, SZ (ℓ) 2 (II.2.2) ∼ ℓ ∼ ( l− : ℓ > ℓf .

Figure II.5 – The ﬁgure illustrates the behaviour of S2D (ℓ) and SZ (ℓ) as a function of ℓ based on the equations (II.2.1), (II.2.2).

Figure II.5 illustrates the behaviour of both S2D (ℓ) ,SZ (ℓ). It can be concluded that the dominant scales for maximum shear for u2D is at ℓ . ℓf . It is to be noted 3 that for the resolutions considered here the spectra is steeper than k− (at scales

ℓf > ℓ > ℓν), so the shear S2D is largest at the forcing length scale. While for the vertical velocity uz we have the shear SZ (ℓ) being the largest at the dissipation scales. Thus a priori we do not have one particular scale which could be responsible for the dynamo instability. Note that we need both S2D ,SZ to have the dynamo instability. 16 Chapter II. Dynamo effect of quasi-twodimensional flows

II.3 Helical dynamo

We now concentrate on the case of the helical forcing, we calculate the growth rate γ of an initial seed magnetic ﬁeld deﬁned as,

1 B 2 (t) γ = lim log | | . (II.3.1) t t B 2 (0) →∞ h| | i For turbulent flows we calculate γ by doing a linear fit in a log-linear plot of the time series of the magnetic energy. The growth rate γ depends on the nondimensional parameters Re, Rm, kf L, Re−. For the first part we look at the dependence on Re, Rm, kz L. The growth rate γ is shown in figure II.6 as a function of kz for different values of Rm. The dynamo instability exists for all Rm, though the domain of the

0.5 0.23 23.13 0.4 0.58 46.26 1.16 92.52 0.3 2.31 185.04 5.78 370.09 0.2 γ 0.1

0.0

−0.1

−0.2 0 1 2 3 4 5 6 kz

Figure II.6 – The figure shows the growth rate γ for the helical flow as a function of kz for different values of Rm as mentioned in the legend. The Reynolds numbers for these set of simulations are Re 46, Re 23. ≈ − ≈ instability shrinks as one decrease Rm, due to increase in dissipation. The dynamo instability mechanism is given by the α-effect, it being a mean-field effect amplifies the magnetic field at large scales. We can analytically derive the growth rate γ in the limit of large scale separation and small Rm limit, see [53, 54, 55, 56, 24]. Scale separation here is between the length scale of the velocity field and the length scale of the magnetic field k k . The magnetic field is written as a sum of a large scale z ≪ f component and a small scale component B = B0 + b. We can write the governing equations of the large scale and the small scale field separately as,

∂ B = ∇ u b + η∆B , (II.3.2) t 0 × h × i 0 ∂ b = (B ∇) u + ∇ (u b) ∇ u b + η∆b, (II.3.3) t 0 · × × − × h × i where denotes the spatial averaging over the length scales of the velocity ﬁeld. In h·i the limit of large scale separation or “short sudden” approximation we can approximate II.3. Helical dynamo 17 the above equations as,

∂ B = ∇ u b + η∆B , (II.3.4) t 0 × h × i 0 η∆b (B ∇) u. (II.3.5) − ≈ 0 · The electromotive force u b is the term responsible for the ampliﬁcation of the h × i large scale magnetic ﬁeld B0. It can be expanded a linear function of B0 as,

u b = αB + β∇B + . (II.3.6) h × i 0 0 ··· The first term on the right corresponds to the α-effect and the value of α can be | | shown to be related to the mean helicity of the flow. Using this theory we can show that the growth rate γ can be written as,

γ = αk ηk2. (II.3.7) z − z in the scale separation k /k 1 and small Rm 1 limit. f z ≫ ≪ 101 100

100 10−1

−1 −2 10 α γ 10 Rm Rm =0.2 −3 10−2 10 Rm =0.5 Re =46.26 Rm =20 Re =106.19 10−4 10−3 10−3 10−2 10−1 100 101 10−3 10−2 10−1 100 101 102 103 kz Rm (a) (b)

Figure II.7 – The figure a) shows the growth rate γ as a function of Rm along with the prediction αkz shown as dashed lines for different values of Rm mentioned in the legend. The figure b) shows α as a function of Rm for two different values of Re mentioned in the legend. The dashed line in figure b) denotes the scaling law Rm.

In order to test this prediction we calculate the α-coefficient numerically. The value of the α coefficient can be obtained by considering the e.m.f u b generated from h × i applying a constant magnetic field B0, see [57]. The numerical result is obtained by solving the following equations,

α B = u b , (II.3.8) · 0 h × i ∂ b = ∇ (u b) + (B ∇) u + η∆b. (II.3.9) t × × 0 · We show in figure II.7a the growth rate γ (calculated previously) along with the prediction αkz valid for small kz. The value of α found from solving equation (II.3.9). In figure II.7b, we show the value of α as a function of Rm for two different values of Re as mentioned in the legend along with the scaling Rm shown by the dashed line. The 18 Chapter II. Dynamo effect of quasi-twodimensional flows

101 Rm=1.66 0 10 Rm=33.15 10−1 Rm=265.16 ) Rm=1062.18 k

( 10−2 B −3 E 10

10−4

10−5 100 101 102 103 k

Figure II.8 – The figure shows the magnetic energy spectra EB (k) for different values of Rm as mentioned in the legend. The parameters for these set of simulations are k = 0.25, Re 530. Darker shades denote higher value of Rm. z ≈ 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 k = 0.25, Re 530. The magnetic spectra are rescaled so that the total magnetic z ≈ 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 [58] for a detailed picture.

II.3.1 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 c kz 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- c meber kz 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 c of Rm. γmax does not vary much with Re. Figure II.10b shows that kz 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 γ Rm3 and max ∼ kc Rm2 can be obtained from the α-theory (see equation (II.3.9)). We mention also z ∼ that k at which γ is found, increased initially with Rm as k Rm2 before satu- z max z ∼ rating at the value k 1. The curve kc at large Rm shows the scaling kc Rm1/2 z ≈ z z ∼ II.3. Helical dynamo 19

γ (Rm, Re) c max (Rm, Re) kz

Figure II.9 – Figure shows an illustrative curve of γ as a function of kz for a given c Re, Rm. It shows both γmax and kz

100 1 Rm2 101 10−1

100 c −2 z

max 10 k

γ Re=46.26 Re=46.26 3 Re=106.19 10−1 Re=106.19 10−3 Rm Re=241.16 Rm2 Re=241.16 Re=530.33 10−2 Re=530.33 10−4 10−2 10−1 100 101 102 103 104 10−2 10−1 100 101 102 103 104 Rm Rm (a) (b)

Figure II.10 – The figures show a) the maximum growth rate γmax and b) the cut-off c wavenumber kz as a function of Rm for different values of Re mentioned in the legend for the helical forcing case. Dashed black lines denote scaling laws. 20 Chapter II. Dynamo effect of quasi-twodimensional flows which can be explained by a balance between the dissipation and the stretching term,

c b u kz 1/2 η 2 b = Rm . (II.3.10) ℓz ∼ ℓf ⇒ kf ∼

This is different from the α-scaling of Rm2 implying that for large Rm the cut-off c length scale kz is not controlled by α-effect. There seems to be a weak dependence c c of kz on the Re in the large Rm limit. As one increases Re we see that kz 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 kc Rm1/2Re 3/8. However one should look at z ∼ − much larger values of Re in order to conclude a powerlaw dependence on Re.

II.4 Nonhelical dynamo

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

0.5 4.03 0.4 8.07 12.1 0.3 16.13 32.26 0.2 64.53

γ 129.05 0.1 258.1 0.0

−0.1

−0.2 0 1 2 3 4 5 6 7 kz

Figure II.11 – Figure shows the growth rate γ as a function of kz for diﬀerent values of Rm mentioned in the legend for the nonhelical forcing.

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 k 1. As we increase Rm more modes become unstable similar z ≈ 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. II.5. Critical magnetic Reynolds number Rmc 21

II.4.1 Dependence on Re

c Figure II.12 shows γmax and kz as a function of Rm for diﬀerent values of Re. γmax

100 102 Re=32.18 1 Re=70.63 Rm2 Re=151.08 c 10−1 z 101 k max γ Re=32.26 Re=70.4 Re=152.12 10−2 100 100 101 102 103 104 100 101 102 103 104 Rm Rm (a) (b)

Figure II.12 – The figure shows a) the maximum growth rate γmax and b) the cut-off c wavenumber kz as a function of Rm for different values of Re as mentioned in the legend. increases as we increase Rm and does not saturate even for the largest values of Rm that we have explored. This non-saturation of γmax means that we have not yet reached c large enough values of Rm. γmax is independent of Re for the values studied here. kz shows the scaling Rm1/2 as a function of Rm, which is similar to the helical case (see c figure II.10b). It has a weak dependence on Re, as we increase Re the value of kz decreases. By doing an empirical fitting we find that kc Rm1/2Re 1/4, though one z ∼ − needs to look at much larger values of Re in order to conclude a powerlaw dependence on Re. Figure II.13a shows the magnetic field spectra for different values of Rm with kz = 1, we see that the shape of the spectra does not change much at large scales while it gets stretched towards smaller scales as we increase Rm. For all values of Rm the magnetic field spectra is concentrated more on the small scales. The form of the unstable mode is similar with the helical case (see figure II.8) except at low Rm where the dynamo instability is not present for the nonhelical case. Figure II.13b shows the magnetic energy spectra for different values of the Re. There does not seem to be much of a dependence on Re and the spectra seem to collapse on top of each other. We will later on study the shape of the spectra in detail in the next Chapter. Figure II.14 shows the contours of the magnetic energy field for different values of Rm. The magnetic field gets concentrated in thin filaments whose length scale decreases as we increase Rm.

II.5 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 22 Chapter II. Dynamo effect of quasi-twodimensional flows

101 100 100 10−1 −1 10 10−2 ) ) k k ( ( 10−2 −3 B

B 10 9.44

Rm E

E = −3 10 Rm =18.89 10−4 Rm =515.59, Re =32.18 Rm =151.08 Rm =565.83, Re =70.63 10−4 10−5 Rm =605.2 Rm =605.2, Re =151.08 10−5 10−6 100 101 102 100 101 102 103 k k (a) (b)

Figure II.13 – Figure shows the magnetic energy spectra EB (k) as a function of k for a) different values of Rm with Re 530, b) different values of Re with Rm being ≈ almost constant. The parameter kz = 1, darker shades correspond to larger values of Rm in a), Re in b). In figure b) all curves collapse on each other.

(a) (b) (c)

Figure II.14 – Figure shows the contour of the magnetic energy for different values of Rm a) Rm 32, b) Rm 1030, c) Rm 2060. The parameter Re 32. ≈ ≈ ≈ ≈ II.5. Critical magnetic Reynolds number Rmc 23 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,

Rm (Re) = max Rm s.t. γ 0 k . (II.5.1) c ≤ ∀ z n o 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

ReT1 ReT2 60 Laminar state - 1 50 Laminar state - 2 Turbulent state 40 c 30 Rm 20

0 10−1 100 101 102 103 104 Re

Figure II.15 – Figure shows the critical magnetic Reynolds number Rmc as a function of Re. The black vertical dotted lines denote transitions in the underlying base ﬂow.

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 Rm Re for moderate values of c ∼ 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 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 [24]. However we find that at the lowest order one 24 Chapter II. Dynamo effect of quasi-twodimensional flows 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 2 kz 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

100 Rm =21.8 10−1 10−2 −3 10 2 kz 10−4 γ 10−5 10−6 10−7 10−8 10−3 10−2 10−1 100 101 kz

Figure II.16 – Figure shows the growth rate γ as a function of kz. The black dashed 2 line denotes the scaling law kz . 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

(a) (b)

Figure II.17 – Figure shows the contour of the magnetic energy for two different laminar flows a) Re < Re 5.4 < Re , b) Re 0.53 < Re . T1 ≈ T2 ≈ T1 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 II.6. Dependence on kf L 25 the helical forcing, from figure II.10b, II.12b the most unstable mode is found to be k = 1. It becomes unstable at Rm 2, slightly increasing as we increase the value of z ≈ Re. Thus the critical magnetic Reynolds number for the cubic domain and the helical flow is through the k = 1 mode and is around Rm 2. For the case of the nonhelical z ≈ 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.

II.6 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 ([66]). The Ponomarenko dynamo consists of a swirling flow with the velocity field being u = Ueθ + uzez 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 2 2 is defined based on the velocity field defined as Up = Uuz/(U + uz). We use this 2 2 definition, Re = Up/(kf ν), Rm = Up/(kf η) where Up = u2D uz/(u2D + uz).

0.14 0.10 kf L =4, Rm = Re =70.4 0.12 kf L =8, Rm = Re =81.27 kf L =16, Rm = Re =90.91 0.10 ) )

f 0.05 f k k 0.08 p p U U

( ( 0.06 0.00

γ/ γ/ 0.04 kf L =4, Rm = Re =106.19

kf L =8, Rm = Re =91.75 0.02 −0.05 kf L =16, Rm = Re =97.19 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 kz/kf kz/kf (a) (b)

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 normalized 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 behaviour. 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 k k /3 as a function of z ∼ f kf . This is true for both the helical and the nonhelical flows. This implies that the 26 Chapter II. Dynamo effect of quasi-twodimensional flows most unstable mode scales with kf rather than the box length L. c Figure II.19 shows the maximum growth rate γmax and the cut-off wavenumber kz as a function of Rm for a few different kf . There seems to be almost no dependence

100 101

−1 0 10 10 1 ) 2

f Rm k f p

U −2 −1 /k ( 10 10 c z / k

max −3 k L =4.0, Re = Rm =106.19 −2 k L =4.0, Re = Rm =106.19 γ 10 f 10 f kf L =8.0, Re = Rm =91.75 kf L =8.0, Re = Rm =91.75

kf L =16.0, Re = Rm =97.19 kf L =16.0, Re = Rm =97.19 10−4 10−3 10−1 100 101 102 103 104 10−1 100 101 102 103 104 Rm Rm (a) (b) 101

10−1 ) f k p f 1 2 U 0 Rm /k ( 10 c z / k

−2 max 10 k L =4.0, Re = Rm =70.4 k L =4.0, Re = Rm =70.4

γ f f

kf L =8.0, Re = Rm =81.27 kf L =8.0, Re = Rm =81.27

kf L =16.0, Re = Rm =90.91 kf L =16.0, Re = Rm =90.91 10−1 101 102 103 104 101 102 103 104 Rm Rm (c) (d)

Figure II.19 – Figures show maximum growth rate γmax/(Upkf ) for a) Helical, c) Non- c helical flows and the normalized cut-off wavenumber kz/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.

II.7 Conclusion - Part 1

Some conclusions one could take from the set of results presented here are,

1. The helical case can be predicted by the α-dynamo for small Rm. For large Rm the behaviour is more similar to the nonhelical case.

2. The nonhelical ﬂow for an inﬁnite layer displays a critical magnetic Reynolds II.8. Saturation of the dynamo 27

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.

3. In the presence of large scale friction, which limits the formation of large scale condensate, we ﬁnd that the inverse cascade does not aﬀect the dynamo instability. The scales that drive the dynamo instability are the forcing scales.

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.

II.8 Saturation of the dynamo

II.8.1 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 calculated 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, ≪ 1 B2 1 ν∆u ∇ p + + f + B ∇B. (II.8.1) − ≈ −ρ 2µ0 ! ρµ0 ·

2 The velocity field in the kinematic phase is given by u = f0/(kf ν) (cos (kf y) , sin (kf x) , cos (k x) + sin (k y)) when B 1. f f ≪ 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) × × 28 Chapter II. Dynamo effect of quasi-twodimensional flows

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′ = B ∇u. (II.8.3) − · iK x We take the large scale magnetic field to be of the form B = (B1, B2, 0)e · . Using equation (II.8.3), we can calculate the small scale field to be b′ = 1/(ηkf ) U1B2 cos (kf y) , U B sin (k x) ,U B cos (k x) U B sin (k y) . Using this we calculate the e.m.f − 2 1 f 3 1 f − 4 2 f term u b . The governing equation for B gives us U ,U ,U ,U , h × ′i 1 2 3 4 ∇ u b = η∆B. (II.8.4) × h × i − 2 2 2 2 We find that U1 = k/(νk + B2/(ρηµ0),U2 = k/(νk + B1/(ρηµ0), U3 = U2/k, U4 = U1/k. Since the magnetic energy is mostly concentrated at the largest mode we use 2 B2 B . We take K to be the length scale over which B varies, the amplitude of the ≈ magnetic field is found to be,

2 B 2 = νkf (Rm Rmc) , (II.8.5) µ0ρη − 2 where Rm = f0/(ρνk )/( Kkf η). We non-dimensionalize and end up with, p B2 2 2 = P m (Rm Rmc) , (II.8.6) µ0ρη kf − with the factor P m. The scaling law follows the classical pitch fork bifurcation and close to the threshold the square of the amplitude scales linearly with the distance from the threshold.

II.8.2 Different scaling laws The above example considered the simple case of Roberts flow. Following the calculation made in [71], one can derive the scaling laws of the magnetic field for both laminar and turbulent flows. We write the full MHD equations as,

1 b2 1 ∂tu + u ∇u = ∇ p + + ν∆u + f + b ∇b, (II.8.7) · − ρ 2µ0 ! ρµ0 · ∂ b =∇ (u b) + η∆b. (II.8.8) t × × 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 × 2 u = ub + ǫuc + O(ǫ ). Here ub denotes the base flow when there is no magnetic field and ǫ = (Rm Rm )/Rm is the distance from the threshold. u is the correction in − c c c 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, 1 ∂ u + u ∇u = ∇p + ν∆u + f. (II.8.9) t b b · b − ρ b II.8. Saturation of the dynamo 29

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, 1 1 ∂tuc + uc ∇ub + ub ∇uc = ∇pc + ν∆uc + 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 u ∇u /(ν∆u ) is given by the Re, when Re 1 the underlying ﬂow is laminar and c · b c ≪ the balance goes like, 1 ν∆uc jb. (II.8.11) ∼ ρµ0ǫ One should then substitute this expression into the induction equation. One can then get the scaling law, ρν B2 (Rm Rm ) . (II.8.12) ∝ σL2 − c We can non-dimensionalize the above equation to get,

B2L2 2 P m (Rm Rmc) . (II.8.13) ρµ0η ∝ − The above scaling is the laminar scaling for the dynamo instability. For the turbulent scaling Re 1, the viscous correction is negligible compared to ≫ the correction from the nonlinear term. The balance for the Lorentz force comes from, 1 uc ub jb. (II.8.14) ∇ ∼ ρµ0

Here ub is the turbulent base ﬂow and the nonlinear term does not depend on the viscosity. By assuming that the dynamo instability is a supercritical bifurcation, we can write the normal form for the amplitude of the magnetic ﬁeld as,

2 ρ B 2 2 (Rm Rmc) . (II.8.15) ∝ µ0σ L − By non-dimensionalizing we get,

B2L2 2 (Rm Rmc) . (II.8.16) ρµ0η ∝ − This is the turbulent scaling expected at large Re where viscosity does not play any role in the saturation of the magnetic field. The laminar (II.8.13) and the turbulent scaling (II.8.16) differ from each other by a factor P m. This has consequences for the amplitude of the saturation of the magnetic field. 30 Chapter II. Dynamo effect of quasi-twodimensional flows

II.9 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

(a) (b) (c)

Figure II.20 – Figures shows a sketch of a) Riga Dynamo, b) Karslruhe Dynamo, c) VKS dynamo.

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 Rm 30 above which the dynamo instability occurs. For Rm > Rm , c ∼ c 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. II.10. On numerical models of dynamo 31

15 ρ / 2 10 R 2 σ 0 µ > 2

0 26 30 34 38 42 46 Rm

Figure II.21 – Figure shows the amplitude of the saturation of the magnetic ﬁeld in the Riga experiment (in ⋆), Karlsruhe experiment (in ) and the VKS experiment (in ). Figure taken from [78]. •

In conclusion the kinematic dynamo theory works well to predict the dynamo threshold, while the nonlinear saturation has to come from considering a highly turbulent base ﬂow.

II.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 Earth- like 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, P m 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 P m, P m 0.01 in fully periodic ≥ boxes [61] and P m 0.05 in spherical domains [79, 80, 81]. ≥ The simulations at much larger values of P m 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 P m we use the 2.5D model. 32 Chapter II. Dynamo effect of quasi-twodimensional flows

II.11 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 ×

Figure II.22 – Figure shows a description of the domain [2πL, 2πL, H]. The left part shows only the kinematic problem as studied in the first part of the Chapter where the velocity field is blue and magnetic field is red. The right side shows the back reaction through the Lorentz force. for a magnetic field of kz mode gives rise to two harmonics the 2kz mode and the 0 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 = v + v . We denote the vorticity as, ω = ∇ u = ω + ω , with 0 2 × 0 2 ∇ v = ω , ∇ v = ω . By averaging along the vertical direction we can write the × 0 0 × 2 2 governing equations of the two modes separately as,

1 ∂ ω + N.L.(ω , ω )+N.L.(ω , ω ) = ν∆ω ν−ω + (∇ (j b)), (II.11.1) t 0 0 0 2 2 0 − 0 ρL0 × × 1 1 ∂ v + N.L.(v , v ) = ∇p + ν∆v ν−v + (j b) 2Ω v . (II.11.2) t 2 0 2 − ρ 2 − 2 ρL2 × − × 2

The operator N.L.(u, u) is the nonlinear u ∇u in the Navier Stokes equation. · L0 is the 0 mode projection and is the 2k mode projection of the Lorentz force. L2 z We write the equation for v0 in terms of ω0 (equation (II.11.1)) since it shows that ω (or v ) is not aﬀected by global rotation. The Coriolis force Ω v in equation 0 0 × 2 (II.11.2) suppresses the 2kz mode. In the fast rotation limit, the amplitude of v2 is obtained by balancing the Coriolis force with the Lorentz force. v2 scales inversely with Ω as v B2/(ρµ LΩ). In the limit of large rotation rates v will have negligible 2 ∼ 0 2 contribution to the saturation of the magnetic ﬁeld. Thus we only consider the 0 mode of the Lorentz force to saturate the dynamo instability. Neglecting the 2kz mode, the governing equations of the saturated dynamo problem II.11. Saturation of the 2.5D dynamo 33 are,

∂ ∆ψ + (∇ ψeˆ ) ∇ ∆ψ = ν ∆2ψ ν ∆ψ + ∆f t × z · − − ψ − 1 eˆ (∇ (J B)) , (II.11.3) ρ h z · × × iz 1 ∂tuz + (∇ ψeˆz) ∇uz = ν ∆uz + fz + (B ∇) Bz , (II.11.4) × · µ0ρ h · iz ∂ b + ( ψeˆ ) b + u ik b = b ( ψeˆ + u eˆ ) t ∇ × z · ∇ z z · ∇ ∇ × z z z + ν ∆ k2 b. (II.11.5) P m − z Here denotes the z averaging to get the z independent component of the Lorentz h·iz force. We consider the helical forcing with fψ = fz/kf = f0 cos (kf x) + sin (kf y) /kf with the prefactor f0 controlling the amplitude of the forcing. Since we are interested in the dependence of the scaling laws on P m we fix the ratio of ν/η = P m. We fix the dimensions of the domains, H/L = 5. This gives the value kz L = 0.2 for the most unstable mode. The forcing function gives rise to the α dynamo for the Rm values examined here. The study is restricted to values of Rm close to the dynamo threshold Rm & Rmc. We define the kinetic and magnetic Reynolds number as, Re = Vc/(kf ν), Rm = Vc/(kf η) with Vc being the velocity field in the kinematic stage. Finally, we note that by restricting the study to values close to Rm < √2Rmc, we do not excite any other integer multiple of kzL mode, see figure II.10b. The numerical runs are performed in the following way. We first compute the velocity field until it reaches its steady state and then we turn on the magnetic field with a small amplitude. For each set of simulations we first fix P m and we change the amplitude of the forcing f0. The forcing f0 changes Re, Rm. The large scale friction is kept constant at ν− = 0.2 and the forcing wavenumber at kf L = 4. The onset of the dynamo instability Rmc is found by varying f0. Figure II.23 shows the value of Rmc obtained as a function of P m for all the values of P m examined in the study. The value of Rmc remains almost a constant for over seven decades of P m. This weak variation is attributed to scale separation [62] and the 2.5D nature of the flow as seen in the first part of this Chapter. Figure II.24 shows a typical time series of the evolution in time of the kinetic and magnetic energy. The magnetic energy is denoted as B 2 where denotes | | · spatial average. The parameters for the runs are Re 1.03e + 4, P m 4.2e 5. ≈ ≈ − The magnetic field initially increases exponentially until its amplitude becomes large enough to modify the velocity field. The insets show the time series of 2D and the z- component of the kinetic energy denoted as, u 2, u2 respectively. When the magnetic | 2D | z field becomes strong, the velocity field starts to get modified and saturates at a different value, seen more clearly in the uz signal. We denote the time averaged magnetic energy as B 2 . The normalized magnetic | | energy B 2 /(ρµη2k2) as a function of Rm is shownD in figureE II.25. The dotted line | | f is a linearD fitE through the data points and the x-intercept of this fit gives us Rmc. The slope of the linear fit is the quantity of interest and is denoted as SL. SL would depend on the underlying flow and the magnetic Prandt number P m, as seen from the laminar (II.8.13) and turbulent scaling (II.8.16). We write B 2 /(ρµ η2k2) S (Rm Rm ) | | 0 f ∼ L − c where Rmc stays almost a constant for all the values of P m explored. According to 34 Chapter II. Dynamo effect of quasi-twodimensional flows

0.5

0.4

0.3 c

Rm 0.2

0.1

0.0 10−5 10−4 10−3 10−2 10−1 100 101 102 103 P m

Figure II.23 – Figure shows Rmc as a function of P m.

100

10−1 |B|2

10−2

10−3 2 | 4.0 25 B

| −4 3.5 10 3.0 20 2.5 15 −5 2.0 10 1.5 10 1.0 2 5 2 −6 0.5 |u2D| uz 10 0.0 0 0 400 800 1200 0 400 800 1200 10−7 0 200 400 600 800 1000 1200 1400 t

Figure II.24 – Figure shows the growth and saturation of the magnetic energy B 2 | | when Rm is above the threshold Rmc. The insets show the kinetic energy in 2D, u 2 and the energy in the vertical velocity, u2. The parameters corresponding to | 2D | z the run are Re 10304, P m 4.2e 5. ≈ ≈ − II.11. Saturation of the 2.5D dynamo 35

0.0030 P m =1.4E-03 0.0025

0.0020 2 f i k 2 2 | η 0 B 0.0015 | h ρµ 0.0010

0.0005

0.0000 0.34 0.35 0.36 0.37 0.38 0.39 Rm

Figure II.25 – The figure shows the normalized amplitude of the magnetic field B 2 /(ρµ η2k2) as a function of Rm. The dotted line indicates the linear fit through | | 0 f the data points.

−3 5 ×10

4 2 f

i 3 k 2 2 | η 0 B |

h 2 ρµ

0 0.00 0.01 0.02 (Rm − Rmc)

2 2 2 Figure II.26 – Figure shows the normalized magnetic field B /(ρµ0η kf ) as a func- | | 2 3 tion of (Rm Rmc) for different values of P m. The friction is ν−/(ηkf ) = 5.1 10− − 3 3 2 and the symbols are: , P m = 1.4 10− ; ▽, P m = 7.0 10− ; ⋆, P m = 4.0 10− ; , 2 • ⋄ P m = 8.9 10− . The different dashed lines correspond to linear fits through the respective data points. 36 Chapter II. Dynamo effect of quasi-twodimensional flows equations (II.8.13) and (II.8.16), SL follows,

P m1 : Re 1, P m 1, S ≪ ≫ (II.11.6) L ∼ P m0 : Re 1, P m 1. ( ≫ ≪

From the calculations from the section II.8.1 we can ﬁnd the behaviour of SL for the laminar ﬂow to be,

2 B kf 1 | |2 2 = 2P m (Rm Rmc) . (II.11.7) ρµ η k s kz 1 − 0 f 1 + − 1+ ν νk2 s f

− kz ν 1 where Rmc = k 1 + 2 1 + . In the laminar regime ν 1 we f 2νkf − 1+ ν ≫ q v νk2 u f u find that Rm 0.32 which matchest with the values computed from the numerical c ≈ simulations shown in figure II.23. The above expression shows the laminar scaling of the slope S P m1 with P m. L ∝ In figure II.26, we show the normalized magnetic field B 2 /(ρµ η2k2) as a func- | | 0 f tion of (Rm Rm ) for different values of P m. The dotted lines denote linear fits − c through respective data points. We fit only through points which are 10% away from the threshold. As we increase the value of P m we see that the slope of each linear fit increases. We now plot the quantity SL as a function of P m in figure II.27. The black solid line denotes the laminar theory which is taken from equation (II.11.7) and it scales like P m1. The black dashed line shows the scaling P m0 which is the turbulent scaling in the limit of large Re and small P m. We see that in the limit of very small P m the slope of the quantity SL becomes constant. The saturation mechanism by the nonlinear term (turbulent scaling) becomes dominant starting from P m 10 3 and ∼ − downwards. Lower than this value the quantity SL remains constant. Close to the value P m 0.03 the flow undergoes a transition from a laminar to a turbulent flow ≈ which explains the slight discontinuity in the data points. The inset corresponds to the variation of the quantity SL as a function of the non-dimensionalized large scale ν− dissipation defined as 2 . In the inset we show runs performed for different values of ηkf ν keeping the value of P m = 1.7e 4 fixed. This value of P m corresponds to dark − − circle on the main figure. As seen from the inset, the large scale dissipation coefficient has very little effect on SL, which confirms the independence of SL on the dissipation parameters. In order to observe the turbulent scaling of P m0, simulations need to be done at 3 very low values of P m < 10− . This could be one of the reasons why it is not easy to observe the turbulent scaling in 3D numerical simulations in periodic boxes [61] or spherical domains [83]. An interesting behaviour was found for the data point corresponding to the value P m 0.03. The flow behaves like a turbulent flow when the magnetic field is almost ≈ negligible, but it becomes laminar once magnetic field becomes strong enough. This is shown in figure II.28, the plot shows the magnetic energy B 2 as a function of | |

II.12. Joule dissipation and dissipation length scale 37

103 10−1 102 P m1: laminar scaling 1 10−2 10 P m =1.7E-04 10−3 10−2 L ν−

S 0 10 2 ηkf

10−1

−2 P m0: turbulent 10 scaling

10−3 10−5 10−4 10−3 10−2 10−1 100 101 102 103 P m

Figure II.27 – Figure shows the quantity SL as a function of P m. The red circles denote turbulent dynamos while blue triangles denote laminar dynamos. The black solid line denotes the laminar theory with the scaling P m1 while black dashed line denotes a scaling law of P m0. The inset shows the slope as a function of the non-dimensionalized 2 parameter ν−/(ηkf ) which measures the large scale friction coefficient. time t. The inset shows both u 2 and u2 as a function of time. Initially the | 2D | z magnetic field is weak and the underlying flow is turbulent. Since Rm > Rmc we see an exponential growth in the magnetic field, once the Lorentz force becomes strong enough to modify the underlying flow we see that the turbulent flow is laminarized. Such kind of behaviour is seen in dynamo simulations at large P m where the Lorentz force is strong enough to kill the turbulent fluctuations, see [84].

II.12Joule dissipation and dissipation length scale

We examine Joule dissipation as a function of P m. The energy injected is balanced by the total dissipation in the system. From the system of equations in (II.11.3), (II.11.4), (II.11.5) we see that,

2 2 2 η J + ν ω + ν− ω = f u , (II.12.1) | | | | | z | · D E D E D E D E where on the left we have the dissipative terms and on the right is the injected power. The above relation states that the time averaged dissipation rate should equal the injection rate of energy into the system. Figure II.29 shows the normalized viscous dissipation rate deﬁned as (ν u 2 + | | ν ∇ψ 2 )/ f u and the joule dissipation deﬁned as η J 2 / f u as a function − | | · | | · of the distanceD fromE the threshold (Rm Rm )/Rm . The sum ofD the twoE normalized − c c dissipation rate is 1. As we move away from the threshold Rmc, we have larger 38 Chapter II. Dynamo effect of quasi-twodimensional flows

100 2 10−1 |B|

10−2 4.0 2

| −3 10 3.5 2 2 |u2D| u

B z | 3.0 −4 2.5 10 2.0 1.5 10−5 1.0 0.5 0.0 10−6 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 t

Figure II.28 – Figure shows the time series of the magnetic energy B 2 as a function | | of time t. The inset shows the energy in the 2D, u 2 and energy in the vertical | |2D component of the velocity ﬁeld u 2. | z|

101

100

10−1

10−2 (Rm − Rmc)/(Rmc)

10−3 Viscous Joule 10−4 10−4 10−3 10−2 10−1 100 (Rm − Rmc)/(Rmc)

Figure II.29 – Figure shows normalized dissipation rate of the magnetic (Joule) and the kinetic energy (Viscous) as a function of (Rm Rm )/Rm . The black line gives − c c the linear scaling from the threshold. II.12. Joule dissipation and dissipation length scale 39 amplitude of B2 leading to larger Joule dissipation, which increases linearly with the distance from the threshold. This is expected since B2 (Rm Rm ) and if the ∼ − c dissipation length scale remains constant close to the threshold we expect J 2 (Rm ∼ − Rmc).

1.8 1.6 1.4 1.2 1.0 /L d

ℓ 0.8 0.6 0.4 Pm = 6.8e-4 Pm = 0.159 0.2 Pm = 0.089 Pm = 0.955 0.0 0.00 0.02 0.04 0.06 0.08 0.10 (Rm − Rmc)/Rmc

Figure II.30 – Figure shows the dissipation length scale ℓd/L as a function of the distance from threshold (Rm Rm )/Rm for a few diﬀerent values of P m as mentioned − c c in the legend.

We further examine how the magnetic dissipation length scale changes as a function of both Rm, P m. The dissipation length scale for the magnetic field is defined as 2 2 1/2 ℓd/L = ( B / J ) , is shown in figure II.30 as a function of the distance from the threshold for a few values of P m. The dissipation length scale ℓd/L is larger than 1 since the magnetic field is largest at the k = 0 mode (k being the 2D wavenumber). This is the mode which only varies along the z-direction and is dissipated only by its variation in z whose length scale 1/kz is larger than L. The figure II.30 shows that the dissipation length scale is almost a constant close to the threshold. This implies that the form of the unstable mode does not change close to the threshold and is expected to be similar to the kinematic dynamo spectra. The reasoning for this goes as follows, the modification of the magnetic field spectra from the kinematic stage to the saturated stage should be given by the effect of the velocity correction in the induction equation. The velocity correction here is the difference between the velocity in the kinematic and saturated regime. The velocity correction itself comes from the Lorentz force which scales like B2 (Rm Rm ). Thus the correction in ∼ − c the velocity field U scales like U B2. The correction in the magnetic field is then c c ∼ proportional to Uc B. Thus in the saturated state the correction in the magnetic field scales like B3 which is small close to the threshold (compared to B). This means that the spectra of the magnetic field in the kinematic computations give a good estimate of the dissipation in the saturated state. Figure II.31 shows the ratio of the dissipation length scale of the magnetic field ld/L as a function of P m. The dissipation length scale seems to stay almost a constant for several decades of P m. Thus the effect of turbulence is quite minimal in changing 40 Chapter II. Dynamo effect of quasi-twodimensional flows the dissipative length scale of the magnetic field. Also the dissipation, scales with the length scale of the box (and kzL) showing that all the dissipation is concentrated at the largest scales of the magnetic field.

1.6 1.4 1.2 1.0

/L 0.8 d ℓ 0.6 0.4 0.2 0.0 10−5 10−4 10−3 10−2 10−1 100 101 102 103 P m

Figure II.31 – Figure shows the dissipation length scale of the magnetic ﬁeld ld/L as a function of P m.

Given that the dynamo instability mechanism is due to the α effect, the magnetic field is concentrated at the largest scale. The small scale magnetic field b is induced by the stretching of the large scale magnetic field by the velocity field. From the induction equation one can get, η∆b B ∇u. Component wise we expect, ∼ · b /u Bk 1 for the 2D field, while b /u Bk 1. We define the spectra of the to- 2D 2D ∼ − z z ∼ − tal magnetic field and its individual components as, E (k) = b 2dk dk , B kx+ky=k | 2D | x y B 2 B 2 E (k) = b dkxdky, E (k) = bzR dkxdky.R Similarly we 2D kx+ky=k | 2D | Z kx+ky=k | | can define theR totalR velocity field spectra and itsR individualR components as, E(k) = u 2dk dk , E (k) = u 2dk dk , E (k) = u 2dk dk . kx+ky=k | | x y 2D kx+ky=k | 2D | x y Z kx+ky=k | z| x y B 2 B 2 DueR to inductionR we expect E /ER kR− and E /E k− . FigureR II.32aR shows 2D 2D ∼ Z Z ∼ in a) the ratio of the total magnetic energy spectra to the total kinetic energy spec- B B tra EB (k)/E(k), b) for the individual components E2D (k)/E2D (k),EZ (k)/E2D (k). In both the plots we see that the spectrum follows the scaling of the dotted line which 2 is k− . We find that the saturated magnetic field has a very similar structure to the kinematic α-dynamo field close to the threshold.

II.13 Saturation in a thin layer

We have so far considered the saturation of the dynamo instability in boxes with large aspect ratio, H/L = 5. We now focus on a small aspect ratio domain (thin layer) with H/L = 1/9, which is sketched in figure II.33. The smallest mode which becomes unstable is kz L = 4.5. The α effect explains the mean field dynamo effect in the limit of large scale separation, in thin layers we no longer have scale separation as k k . This z ∼ f II.13. Saturation in a thin layer 41

10−2 10−2 B E (k) 10−3 10−3 2D E2D(k) 10−4 −2 10−4 −2 B k k E (k) Z ) ) k E (k)

k −5 −5 ( Z

( 10 10 B E E 10−6 10−6

10−7 10−7

10−8 10−8 100 101 102 100 101 102 k k (a) (b)

Figure II.32 – Figure a) shows the ratio of the spectra of the total magnetic energy EB (k) with the total kinetic energy E(k) . Figure b) shows the ratios of the planar and the vertical magnetic energy spectra with the planar and the vertical kinetic energy B B E (k) E (k) spectra, 2D , Z as a function of k. The black dashed line denotes the scaling E2D (k) EZ (k) k 2. The parameters for the run are P m 4.25 10 5, Rm 0.42. − ≈ × − ≈ is also seen from ﬁgures II.10b, where we had shown that the α-dynamo prediction no longer works when scale separation is absent. We continue the study using the helical forcing. We saturate the dynamo instability by the 0-mode projection of the Lorentz force.

Figure II.33 – Figure shows the thin layer domain. The aspect ratio of this geometry is H/L = 1/9.

Figure II.34 shows the normalized B 2 /(ρµ η2k2) as a function of Rm for a | | 0 f Re 50. At this Re the flow is turbulent. The critical magnetic Reynolds number is ∼ found to be Rm 220 and the parameters of the run are, k L = 4, P m = 4.75. The c ≈ f large value of Rmc shows the difficulty in obtaining a dynamo instability in such thin layers as compared to the dynamo instability driven by the α effect. As seen from the previous section the magnetic field saturates like the square root of the distance from the threshold. The slope of the linear fit is S 13. L ≈ The time series of the magnetic energy is shown in figure II.35 for four different values of Rm as mentioned in the legend. The time series is very intermittent since the magnetic energy stays low for most of the time with intermittent bursts when it reaches order 10 2 magnitude. This behaviour of the magnetic field happens very ∼ − 42 Chapter II. Dynamo effect of quasi-twodimensional flows

900 800 700 600 2 f i

k 500 2 2 | η 0 B | h

ρµ 400 300 200 100 0 180 200 220 240 260 280 300 Rm

Figure II.34 – Figure shows the normalized magnetic energy B 2 /(ρµ η2k2) as a | | 0 f function of Rm for the thin layer problem. The dashed line shows the linear ﬁt through the data points. Here P m = 4.75 and the slope of the linear ﬁt is S 13. L ≈

10−1 10−3 10−5 10−7 10−9 10−11 −13 10 2 | 10−15 B

| −17

10 10−19 10−21 10−23 Rm =220.8 −25 Rm =221.4 10 Rm =222.0 −27 10 Rm =222.6 10−29 0 10000 20000 30000 40000 50000 60000 t

Figure II.35 – Figure shows the magnetic energy B 2 as a function of time t for | | different values of Rm as mentioned in the legend. Darker shades of blue denote larger values of Rm. II.14. Conclusion - Part II 43 close to the instability threshold. In the figure the Rm varies by less than 1% and as we increase the value of Rm we see that the magnetic field fluctuates less. This kind of intermittency is known as the On-Off intermittency [85, 86, 87]. The magnetic energy fluctuates between an “OFF” phase and an “ON” phase. The background turbulent velocity field acts like a noise in the system. This can be reproduced by simple stochastic models [88, 89]. One such model which explains the two phases clearly is,

X˙ = µX X nX + ζX, (II.13.1) − | | where ζ is a white noise with ζ(t)ζ(t ) = 2Dδ(t t ). When we neglect the noise, h ′ i − ′ we see that there are three solutions, we have X = 0 stable solution for µ < 0 and X = µ1/n stable solution for µ > 0. The change of stability is due to the pitch-fork ± bifurcation at µ = 0. In the presence of noise, close to the onset of the bifurcation the equation exhibits On-off intermittency behaviour. The intermittency occurs when we are close to the threshold for small positive values of µ < D. Let us consider the evolution of the logarithm of X, y = log(X), which follows the equationy ˙ = µ+ζ eny. − For X 1, y is large and negative, which implies that the nonlinear term (here eny) ≪ is very small. So when X 1, y follows a brownian motion with a small positive ≪ drift which marks the off phase,y ˙ µ + ζ. For µ small it takes long time for the ≈ system to escape from the Off phase. Due to the positive drift the variable X reaches the state X 1, which is the ON phase where the system. The variable y reaches ∼ a value close to zero and the nonlinearity eny restricts the system from crossing or overshooting the value y 0. Large positive fluctuations do not modify the system ∼ due to the nonlinearity while large negative fluctuations take the system back to the state X 1 or y 0. We can write the probability distribution function for the ≪ ≪ equation (II.13.1), as

1 µ n − Dn µ n D 1 X /(nD) P (X) = µ µ X − e− , (II.13.2) nD D Γ( Dn ) for µ > 0. This shows that for µ & 0 we have P (X) X 1. Close to the threshold the ∼ − system spends a long time close to the solution X = 0, while the statistical average can be computed as X = XP (X)dX µ. Even though the system stays for long time h i ∝ close to X = 0, it visits the nonlinear stage X O(1) enough times to have X µ. R ∼ h i ∝ Figure II.36 shows the probability distribution function (pdf) of the total magnetic energy B2 for diﬀerent values of Rm. We see that the pdf of the magnetic energy as 1 we approach the threshold scales like P (B2) B2 − . This is true for Rm Rm ∼ ≥ c close to the threshold of the dynamo instability. As we move away from the dynamo instability we see that the pdf no longer peaks at the value B2 0, the peak shifting ∼ to larger values as we increase Rm.

II.14 Conclusion - Part II

In the second part of this Chapter we have looked at the saturation of the dynamo instability though a quasi-twodimensional model. A 2.5D ﬂow leads to a magnetic 44 Chapter II. Dynamo effect of quasi-twodimensional flows

Figure II.36 – Figure shows the probability distribution function of B2 for diﬀerent values of Rm as mentioned in the legend. The curves are shifted along the y-axis for 1 comparison. The black dotted line indicates the scaling B2 − . Darker shades of red indicate larger values of Rm.

field growth at the most unstable wavenumber kz allowed by the domain length in the z direction. This magnetic field at kz mode has a back reaction on the velocity field through the Lorentz force. The Lorentz force can be decomposed into 2kz and 0 mode along the z-direction. The fast rotating limit only allows the z independent component of the Lorentz force to be of importance for the saturation. In this limit, we find that the 0-mode projection of the Lorentz force saturates the dynamo instability. Using such a system we studied how the saturation amplitude of the magnetic field changes as a function of P m. The two different regimes of saturation, one controlled by viscosity and the other independent of the viscosity are identified. The implications from the study is that to attain a scaling independent of the viscosity it is necessary to do simulations at P m 10 3. Such turbulent flow is still out of reach for fully ≤ − three dimensional simulations in periodic boxes or spherical domains. It also explains the mismatch between the current numerical simulations and laboratory experiments. The study also shows that it might be useful thus to consider reduced order models in order to attain more Earth-like parameters, see [68]. The reduced order models need lesser computational power to reach the extreme values and could help extrapolate existing results to a wider range of parameters. Finally we show the existence of ON-OFF intermittency in dynamo saturation in thin layers in the background of a turbulent flow. This intermittent behaviour is only found for thin layer geometries where Rm 1 and it seems to be not present for the ≫ α-dynamo (large aspect ratio) where Rm < 1. Even though the simulations involving the α-dynamo were done at a much larger Re. One possible reason could be that the α-effect involves some form of averaging over fast scales. We return to this subject in the second part of the next Chapter. Chapter III Kazantsev model for dynamo instability

One could model the turbulent flow as a velocity field with zero mean and with fluctuations modelled as random noise. Such a model was first proposed by Kazantsev [90] for the study of dynamo instability, almost the same time Kraichnan [91] proposed it to study the advection of a passive scalar by a turbulent flow. Analytically it is one of the few models in turbulence that can be exactly solved. Such stochastic models are also used to study statistics of moments in turbulent flows [92, 93]. In the Kazantsev model, the velocity fluctuations are taken to be Gaussian distributed and white in time with a particular dependence in spatial coordinates. In the following study we take such a model for the 2.5D flows that were considered in the previous Chapter II and try to obtain analytical results to compare with the existing numerical results. Before proceeding to the model it is important to note the drawbacks in taking a flow that is white in time and Gaussian to model turbulent flows. The two-point velocity correlation function in a turbulent velocity fields does not have a Gaussian distribution. The existence of a cascade of a quantity shows that the P.D.F of the correlation function is skewed, which is seen from the Kolmogorov 4/5-th law in three dimensional turbulence, see [8, 14]. The property of being δ-correlated implies that the turn over time scale of the velocity field is infinitesimal. This is not valid at any scale of the velocity field in the turbulent flow. This model though could capture the large Rm limit of a dynamo instability, that is, when the velocity field varies over a smaller time scale compared to the magnetic diffusion time scale, ℓ/u ℓ2/η (in the large Rm ≪ limit). Thus the assumption that the velocity field is Gaussian and the δ-correlated model does not model turbulent flows. Even with these limitations we will see that the model does capture certain aspects of the instability. The Kazantsev model considers a velocity field which is white in time and gaussian distributed. For variables that are Gaussian distributed, the mean and the second order moment determines the statistics of all higher order moments. We consider a velocity field that is homogeneous and isotropic in 3D. Under these assumptions the second order moment of the velocity field is written as,

i j ij u (x, t) u x + r, t′ = g (r) δ t t′ . (III.0.1) − D E 45 46 Chapter III. Kazantsev model for dynamo instability

The homogeneity property is used when gij is considered independent of x, gij contains the information of the spatial structure of the velocity field. The Holder exponent ζ of the flow is defined by expanding the second order correlation function gij,

gij (r) = g rζ g + . (III.0.2) 0 − 2 ··· Here g0, g2 are constants which depend on the flow structure. When the underlying flow is smooth, also true in the viscous part of a turbulent spectra, the Holder exponent is ζ = 2. A Holder exponent of ζ < 2 is said to model a rough flow. Kazantsev [90] studied the dynamo instability for flows with Holder exponent 0 < ζ 2 and found ≤ that the dynamo instability exists only for ζ > 1. Then [94] considered the model of the dynamo instability driven by the turbulent inertial length scales. The exponent in the turbulent inertial range was taken to be ζ = 4/3 obtained by computing the turbulent diffusivity D v r r4/3 (also the Richardson law). Other authors have contributed ∼ r ∼ in understanding the structure and the form of the unstable mode at different limits of Prandtl and Reynolds numbers [95, 96, 97, 98, 99, 100]. Studies were also done on the geometric properties of passive scalar advection [101], and the magnetic field lines [102]. Nonlinear models have also been used to look at the saturated regime of the instability arising from a Kazantsev model [103]. More recently, there have been numerical simulations that have been done to compare with the model [104, 105, 106]. The two cases studied which are of importance to our work is the standard 3D isotropic homogeneous model and the 2D isotropic homogeneous model. We refer to the study [99] for a more general analysis of the model for different dimensions and varying degree of compressibility. The 3D and the 2D model differ fundamentally since the former can give rise to a dynamo instability while the latter does not. In the case of 3D, for large P m the underlying unstable mode forms a spectra of k3/2 at scales larger than the viscous scales. While for the 2D model, it was shown that it cannot sustain a dynamo instability. This is in agreement with the Zheldovsky anti-dynamo theorem [18]. The initial condition problem of how a seed magnetic field decays was studied by [107]. For an initial magnetic field localized in the wavenumber space in 2D the magnetic field gets stretched and grows due to increasing number of modes becoming unstable. This happens until the magnetic energy hits the dissipation length scale after which the dissipation becomes effective and the magnetic energy decreases. The magnetic field forms a spectra k2 until the dissipation scales. We sketch these predictions in figure III.1 which shows the two types of the magnetic energy spectra formed by 2D and 3D flows. The energy spectra is given by E(k) while the magnetic energy spectra by EB (k), the viscous dissipation scale is denoted by kν and the magnetic dissipation scale by kη. The seed magnetic field is denoted by dotted lines which evolve into the 2D or 3D magnetic field spectra. The 2D magnetic field has transient growth until it reaches the dissipative scale kη before decaying while the 3D magnetic field grows with time. In the first part of the Chapter we look develop the 2.5D nonhelical Kazantsev model and understand the structure of the magnetic spectra. We will compare the results of the 2.5D model with the 2D and the 3D model and understand the effect of anisotropy for the dynamo instability. The findings of this theory will then be compared to new and existing numerical results that were obtained in the previous Chapter. III.1. Model development for 2.5D nonhelical flow 47

2D 3D

(a) (b)

Figure III.1 – The figure shows a sketch of the Kazantsev prediction for the magnetic field decay/amplification for a) 2D and b) 3D fields. The energy spectra is denoted by

E(k) while the magnetic energy spectra is denoted by EB (k). The dotted black line denotes the initial magnetic ﬁeld.

III.1 Model development for 2.5D nonhelical ﬂow

For the 2.5D model, we make the following assumptions that the velocity field is homogeneous and isotropic in 2D, x y. The flow is independent of the z-direction, − hence it is anisotropic (uniform along the z-direction as compared to dependence in x, y). We need to derive the general form of the second order correlation function gij for this flow. We recall the form of such functions for an isotropic, homogeneous flows in 3D and 2D dimensions. In 3D isotorpic and homogeneous flows, the general expression for the second order correlation function gij can be written as,

rirj rk gij (r) = g (r) δij + (g g ) δij + H (r) ǫijk . (III.1.1) LL LL − NN − r2 r ij ijk Where δ is the Kronecker delta tensor and ǫ is the Levi-Civita tensor. Here gLL (r) gives the longitudinal auto correlation of the velocity field component along the r direction. The quantity gNN (r) denotes the transverse auto correlation function of one velocity field along r and the other field perpendicular to r. A general 2D flow retains the above form except that it does not posses any helicity, H(r) = 0. For an ∇ ij incompressible flow, we have the constraint u = 0 which gives the criterion g,i = ij · i g,j = 0. Here the subscript ,i denotes differentiation with respect to r . Substituting the incompressibility constraint into (III.1.1) we find the relation gNN = gLL′ +(d 1)gLL /r. ′ − Here the prime denotes differentiation with r and d is the dimension of the flow taking the value 2 or 3 depending on whether we look at a 2D or a 3D flow. These relations can also be found in standard books on turbulence [108]. The current study involved three components of velocity field which depends on two directions which is different from the case of (III.1.1), thus we need to find a new expression for the correlation function. One possible way (see [109, 110]) to derive the form for gij is to write the velocity field in terms of scalar functions (another way is to consider dyadic vectors, see [111, 112, 113]). For 3D flows we can write the velocity 48 Chapter III. Kazantsev model for dynamo instability

ﬁeld in terms of poloidal and toroidal scalar functions as,

u (x, t) = ∇ (eˆ ∇P (x, t)) + eˆ ∇T (x, t) , (III.1.2) − × × v × v where eˆ is an arbitrary direction and Pv,Tv are the polenoidal and toroidal scalar functions. For 2D we can write it in terms of the stream function ψ,

u (x, t) = ∇ (ψ (x, t) eˆ ) . (III.1.3) × z The procedure in order to derive equation (III.1.1) goes as follows, firstly, we write the quantity ui (x, t) uj (x + r, t) where implies time averaging or ensemble T h·iT averaging. Then using the arguments of isotropy and homogeneity for both the 3D and the 2D case, leads to the correlation function being written in terms of products of scalar functions. This would lead to the final expression of the form (III.1.1). We now find the generalized second order correlation function gij for a 2.5D flow following such an approach. We start by writing the velocity field in terms of three scalar functions, ψ, φ, uz as u (x, t) = ∇ (ψ (x, t) eˆ ) + ∇φ + u (x, t) eˆ . (III.1.4) × z z z Calculating the function uiuj and using the properties of homogeneity and isotropy (in x, y) we get, i j ij ij ij r r i3 j3 g (r) =gLL (r) δ gLL (r) gNN (r) δ + (gZ (r) gNN (r)) δ δ − − − r2 ! − rj ri rp rp + g (r) δi3 δj3 + g (r) ǫ3jpδi3 ǫ3ipδj3 , (III.1.5) c r − r p r − r where r = (x, y, 0), since the field is independent of z-direction. The definition of the scalar functions gLL , gNN , gc, gz are,

g (r) = (eˆ u)(u′ eˆ ) , g (r) = (eˆ u)(u′ eˆ ) , LL r · · r T Z z · · z T g (r) = (eˆ u)(u′ eˆ ) , g (r) = (eˆ u)(u′ (eˆ eˆ )) , (III.1.6) c z · · r T p z · · z × r T g (r) = ((eˆ eˆ ) u)(u′ (eˆ eˆ )) ,. NN z × r · · z × r T here u is the velocity field at a point x + r at time t and u′ is the velocity field at a point x at time t. Physically, the quantity gLL measures the longitudinal auto correlation function of the two-dimensional velocity field and the quantity gNN measures the transverse auto correlation of the two dimensional velocity field. The functions gc and gp are the cross correlation between the two-dimensional velocity field and the vertical velocity field. The function gZ gives the autocorrelation of the vertical velocity field. In particular, the function gp is related to the helicity of the velocity field. Since we consider a velocity field that is nonhelical, we take gp (r) = 0. The incompressibility condition for the velocity field is written as ∂xux + ∂yuy = 0. This implies that the ij ij correlation function has the constraints g,i = g,j = 0. Putting this constraint into equation (III.1.5) we get,

gNN (r) = gLL (r) + gLL′ (r) r, (III.1.7) gc (r) = 0. (III.1.8) III.1. Model development for 2.5D nonhelical flow 49

This helps us to rewrite the expression for gij as,

i j ij ij ij r r i3 j3 g (r) = g (r) δ + g′ (r) r δ + g (r) g (r) g′ (r) r δ δ . LL LL − r2 z − LL − LL (III.1.9)

Thus one needs only two functions gLL (r) and gz (r) to completely determine the statistics of the velocity field. Now we move on to construct the second order correlation function of the magnetic field. As studied in the previous chapter, the magnetic field in this configuration takes the form B = beikzz + c.c. Since b is complex, we define the second order correlation function Hij as,

bi (x + r, t) † bj (x, t) = Hij (r, t) , (III.1.10) where the symbol † denotes complex conjugate. We can write the magnetic ﬁeld in terms of complex scalar functions Φ, Ψ, bz as,

b = ∇Φ + (Ψeˆ ) + b eˆ , (III.1.11) ∇ × z z z where ∇ = (∂x, ∂y, 0). Following similar arguments as before and using the properties of homogeneity and isotropy, one can derive the form of Hij. Considering only the mirror symmetric components we get,

rirj Hij (r, t) =H (r) δij H (r) H (r) δij + H (r) H (r) δi3δj3 LL − LL − NN − r2 Z − NN rj ri + iH (r) δi3 + δj3 . (III.1.12) c r r The reason for considering only the mirror symmetric part goes as follows, given that the velocity field is mirror symmetric we can show that the mirror symmetric part of the magnetic field is sufficient to study the instability. This can be seen from the induction equation, in the absence of mirror asymmetric terms the term u b leads to × decoupled equations for the mirror symmetric and asymmetric parts of the correlation function. This point is elaborated in detail in Appendix A. In studies which involve the kinetic helicity [114, 115, 116], the mirror asymmetric terms in the magnetic field (magnetic helicity) is also taken into account. Going back to the expression for Hij (r) in equation (III.1.12), the functions HLL ,HNN ,Hc,HZ are scalar functions that depend only on the magnitude r. Their definitions are,

H (r, t) = (eˆr b†)(b′ eˆr) , Hc (r, t) = (eˆz b†)(b′ eˆr , LL · · T · · T D E D E H (r, t) = ((eˆz eˆr) b†)(b′ (eˆz eˆr)) , H (r, t) = (eˆz b†)(b′ eˆz) . NN × · · × T Z · · T D E D (III.1.13)E where b denotes the magnetic ﬁeld at a point x + r and time t, b′ denotes the magnetic

field at a point x and time t. The function HLL is the longitudinal auto correlation function of the two dimensional magnetic field and HNN is the transverse auto correlation function of the two-dimensional magnetic field. The function Hc is the cross 50 Chapter III. Kazantsev model for dynamo instability correlation function between the two-dimensional magnetic field with the vertical magnetic field bz. HZ is the auto-correlation function of vertical magnetic field bz. We can simplify the correlation function (III.1.13) using the solenoidality property of the magnetic field. ∇ B = 0 implying, · ∂ b (x, y, t) + ∂ b (x, y, t) = ik b (x, y, t) . (III.1.14) x x y y − z z This translates into the following constrains for Hij, Hij ik H3j = 0, Hij ik Hi3 = 0. (III.1.15) ,i − z ,j − z Substituting the above condition into the equation (III.1.13) we have the following two conditions,

Hc (r) k H (r) = H′ (r) + , (III.1.16) z Z c r

HLL (r) HNN (r) k H (r) = H′ (r) + − . (III.1.17) − z c LL r

Thus one needs only two functions HLL and HZ to describe the second order magnetic ij field correlation function H . We look at the case of kz = 0, where the magnetic field behaves like a 2.5D velocity field, as B becomes independent of z. When kz = 0 we ij ij see that H,i = H,j = 0, which implies HNN = HLL + HLL′ r. Thus simplifying the expression for the second order correlation function of the magnetic field Hij into an expression similar to that of the correlation function of the velocity field gij. We come ij back to the case of kz = 0 after we derive the governing equation for H , since it has strong implications for the existence of the dynamo instability. Given the form of the correlation function Hij (r, t), we now need to find its governing equation starting from the induction equation, ∂ B = ∇ (u B) + η∆B. (III.1.18) t × × For a given mode B = beikzz + c.c, this simplifies to, ∇ ∇2 2 ∂tb = ( + ikzeˆz) (u b) + η kz b, (III.1.19) ⊥ × × ⊥ − where ∇ denotes (∂x, ∂y). Here the velocity field is prescribed and from it one can obtain the⊥ second order correlation function gij. The governing equation for Hij can be derived in terms of gij using the Gaussian property of the velocity field. This effectively helps us write the higher order moments in terms of the second order moments that we have defined. Following [99] we use the Furutsu-Novikov theorem ([117, 118]) to derive the governing equation for Hij as a function of gij. The exact derivation is explained in Appendix A. The governing equations for HLL ,HZ are,

HLL′ 2 ∂ H 2η + g (0) g H′′ + 3 + k 2η + g (0) g H = g′′ H t LL − LL − LL LL r z Z − Z LL − LL LL h i HLL 2 g′ 2H′ + 3 3k H g′ + 2η + g (0) g k H , − LL LL r − z c LL r LL − LL z c (III.1.20)

1 1 2 ∂ H 2η + g (0) g H′′ + H′ H + k 2η + g (0) g H = k g′ H . t c− LL − LL c r c − r2 c z Z − Z c − z Z LL h i (III.1.21) III.2. Model flow 51

The quantity gLL (0) is the total energy of the velocity ﬁeld in 2D and the quantity gZ (0) is the total energy of the z component of the velocity ﬁeld. In a domain which is homogeneous, these terms depend on the frame of reference from which they are measured and do not modify the dynamo instability. Thus the above equations are independent of gLL (0) , gZ (0). There are three special cases for the governing equations of Hij, we discuss each case in detail here,

1. The case of kz = 0 : The governing equation (III.1.20) becomes identical to a magnetic field driven by a 2D flow ([99]), thus decaying in the long time limit. The vertical magnetic field equation (III.1.21) also decays. The case kz = 0 does not give rise to a dynamo instability. Thus we need k = 0 for dynamo instability z 6 to exist.

2. The case of no shear in uz : When the z-component of the velocity ﬁeld is zero or constant, it implies g (0) g = 0. Substituting this into the governing Z − Z equations (III.1.20),(III.1.21) we ﬁnd that Hc is no longer driven by HLL . This

again implies that Hc decays in the long time limit which means HLL will also decay.

3. The case of no shear in u - This implies that g (0) g = 0. This case does 2D 2D − 2D not lead to a dynamo instability. The component bz can be amplified by the stretching of bx, by by uz. But from the induction equation, the magnetic field components bx, by are advected by uz and dissipated by the ohmic dissipation with no amplification from the stretching term. Thus both bx, by decay in the long time limit which makes bz to decay in the long time limit. All the three cases above are related to the anti-dynamo theorem [18]. The velocity field should have shear in both u and u and k = 0 for the existence of dynamo 2D z z 6 instability.

Next we consider a model flow in order to fix gLL , gZ . When the velocity correlation functions are fixed, we look for unstable solutions of equations (III.1.20), (III.1.21).

III.2 Model ﬂow

We consider a ﬂow which is homogeneous and isotropic in 2D given in terms of ψ, uz as,

k0 ψ (r, t) =ζ1 (t) sin sin (φ1 (t)) x + cos (φ1 (t)) y + φ2 (t) , (III.2.1) 2 ! h i k0 uz (r, t) =ζ2 (t) cos sin (φ1 (t)) x + cos (φ1 (t)) y + φ2 (t) . (III.2.2) 2 ! h i where φ1 (t) , φ2 (t) are random variables which are uniformly distributed over [0, 2π] and render the flow homogeneous and isotropic. ζ1 (t) and ζ2 (t) are random variables that are Gaussian distributed and δ correlated in time ζ (t) ζ (t ) = Θ δ (t t ), h 1 1 ′ i 1 − ′ ζ (t) ζ (t ) = Θ δ (t t ) and they are not correlated ζ (t) ζ (t ) = 0. The h 2 2 ′ i 2 − ′ h 1 2 ′ i 52 Chapter III. Kazantsev model for dynamo instability wavenumber k0 defines a typical length scale for the velocity field. In spatial coordinates, the velocity field is a simple single wavenumber flow that is both isotropic and homogeneous. Given the velocity field, we can calculate the correlation function from its definition gij (r) δ (t t ) = ui (x + r, t) uj (x, t ) . The correlation function − ′ ′ T of the velocity field is calculated to be,

r i j r ij k0Θ1 ij J0′ k0 2 ij r r J0′ k0 2 k0 r g (r) = δ + δ J0′′ k0 4 ( − r − r2 r − 2 2 ) Θ r + 2 J k δi3δj3, (III.2.3) 2 0 0 2 where J0 is the Bessel function of the ﬁrst kind and J0′ stands for its derivative. The functions g2D , gZ are then,

k0Θ1 r Θ2 r g (r) = J ′ k , g (r) = J k . (III.2.4) 2D − 4r 0 0 2 Z 2 0 0 2 We can expand the above functions at small r to be, g (r) = g (0) D r2 + E r4 O r6 , g (r) = g (0) D r2 + E r4 O r6 . 2D 2D − 1 1 − Z Z − 2 2 − (III.2.5)

2 4 2 where g2D (0) = k0Θ1/16, gZ (0) = Θ2/2,D1 = k0Θ1/512,D2 = k0Θ2/32. At small scales, the velocity field is smooth and behaves like g r2, g r2. 2D ∼ Z ∼ We note that D1 has dimensions of inverse time and defines the dynamical time scale τd 1/D1 that we will use to non-dimensionalize our system. Accordingly, the ≡ 2 magnetic Reynolds number is defined as the ratio of the diffusion time scale 1/ηk0 to the dynamical time scale, Rm D /(k2η) = k2/k2, where k is the dissipation ≡ 1 0 d 0 d length scale of the magnetic field k k D /η = k √Rm. A third dimensionless d ≡ 0 1 0 parameter can be defined by the ratio ofp the vertical velocity field gradients to the planar velocity field gradients, which we will quantify as Dr = D2/D1. The quantity 2 2 Dr depends on the ratio of the amplitudes of k0Θ1 and Θ2 as Dr = 16Θ2/ Θ1k0 . Thus the nondimensionalized control parameters are, the Floquet mode kz/k 0, the magnetic Reynolds number Rm and Dr. We define the growth rate γ of the second order correlation function as,

1 H (t) 1 H (t) γ = lim log | LL | = lim log | c | . (III.2.6) t t H (0) t t H (0) →∞ | LL | →∞ | c | This γ is twice the growth rate of the magnetic ﬁeld. We look to study γ as a function of the control parameters of the system.

III.3 Growth rate

γt γt We substitute HLL (r, t) = e hLL (r) and Hc (r, t) = e hc (r) in the equations (III.1.20),

(III.1.21) to get an eigen value problem with the eigenfunctions being hLL , hZ and the eigenvalue γ. The boundary conditions are h (0) = 0, h (0) = 0, h ( ) = LL′ c LL ∞ III.3. Growth rate 53

0, h ( ) = 0, which are obtained from the boundedness of the magnetic energy and c ∞ symmetry properties. The largest eigenvalue of the system γ controls the long time evolution of the magnetic ﬁeld correlation functions. The resulting set of equations is solved numerically using a chebyshev spectral method. The functions gLL (r) , gZ (r), h (r), h (r) are deﬁned over a domain r [0, ), we discretize the domain r LL Z ∈ ∞ ∈ [0, rmax]. The equations (III.1.20),(III.1.21) in this truncated basis can now be reduced to a linear matrix eigenvalue problem. We compute the largest positive eigenvalue of the discretized matrix using standard linear algebra software. We have checked the convergence of the resulting eigenvalue in terms of the number of basis functions used and the domain size rmax.

3 Rm=0.5 2.5 Rm=0.75 Rm=2 2 Rm=4 Rm=20 d

τ 1.5 Rm=200

γ Rm=20000 1 Rm=∞

0.5

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 k /k z d

Figure III.2 – The ﬁgure shows the normalized growth rate γτd as a function of kz/kd for diﬀerent values of Rm as shown in the legend. Larger values of Rm is shown by darker shades of blue. The dark black line denotes the curve obtained in the limit Rm . → ∞

Figure III.2 shows the normalized growth rate γτd as a function of kz/kd for diﬀerent values of the magnetic Reynolds number Rm mentioned in the legend. The critical Rm for the presence of dynamo instability (at least one kz unstable mode) is found to be Rm 0.45. Close to Rm , the unstable mode occur at a value k 0.18k 0.12k . c ≈ c z ≈ d ≈ 0 Instability occurs over a range of modes kmin < kz < kmax with kmin, kmax being functions of Rm. We remind that k k √Rm, thus the largest wavenumber k d ∝ 0 max for which there is a dynamo instability increases like k k √Rm. The smallest max ∝ 0 wavenumber at which dynamo instability occurs kmin decreases as we increase Rm, approaching k 0 as Rm is increased. The growth rate of each mode k increases min → z as we increase Rm, reaching an asymptotic value at large Rm. For large Rm the growth rate curves tend to the black dark line which gives γ in the limit of Rm . → ∞ The maximum value of γ over all values of kz, Rm is found to be γτd = 3, is obtained for Rm and k 0. We discuss the asymptote curve (Rm ) in the next → ∞ z → → ∞ section. 54 Chapter III. Kazantsev model for dynamo instability

III.4 Diﬀerent limits

We now look at the diﬀerent possible limits of the parameters of the problem. We consider three diﬀerent cases, of (Rm , k 0), (Rm ,D 0) and (Rm ,D ). → ∞ z → → ∞ r → → ∞ r → ∞ III.4.1 Limits Rm ,D 0 → ∞ r → We start from the governing equation (III.1.20), (III.1.21) and take the limit of η 0 → which is the limit of Rm . The limiting procedure consists of the following ˜ 2→ ∞ rescaling,r ˜ = r kd, t = tηkd = t/D1. We can now expand the correlation functions in the following way,

g (˜r) =g (0) ηr˜2 + O(η2r˜4), (III.4.1) 2D 2D − g (˜r) =g (0) D ηr˜2 + O(η2r˜4). (III.4.2) Z Z − r In the limit η 0, we need to solve only for the ﬁrst order in η, we end up getting, →

2 hLL′ 2 2 γτ h 2 +r ˜ h′′ + 3 + k˜ 2 + D r˜ h = 2h d LL − LL r˜ z r LL LL h i hLL 2 2 2˜r 2h′ + 3 + 6˜rk˜ h + 2 +r ˜ k˜ h , (III.4.3) LL r˜ z c r˜ z c 2 1 1 2 2 γτ h 2 +r ˜ h′′ + h′ h + k˜ 2 + D r˜ h = 2k˜ D rh˜ . (III.4.4) d c − c r˜ c − r˜2 c z r c z r LL h i In this limit the eigen value γ does not depend on k0 or on the exact form of the flow (form of gij). It only depends on the local structure of the velocity field governed by the parameter Dr. This is because the length scales at which the dynamo instability takes places is much smaller than the velocity scale, k /k 1. d 0 ≫ The eigen values for the black solid curve in figure III.2 are obtained by solving equations (III.4.3), (III.4.4). It is to be noted that the limit of η 0 does not → correspond to the equations when η = 0. Once the limit Rm is taken, we take → ∞ the limit k 0 and find that the growth rate γ 3. In order to get this solution z → → analytically we do a matched asymptotic expansion which is explained in Appendix B. If we take this limit in the opposite sense k 0 and Rm , we find that the z → → ∞ dynamo instability does not exist. The limit k 0 taken first does not lead to a z → dynamo since solving for the lowest order of kz in equations (III.1.20), (III.1.21), leads to γ < 0 for finite Rm. Thus we have a set of noncommuting limits,

3 = lim lim γ = lim lim γ = 0. (III.4.5) kz 0 Rm 6 Rm kz 0 → →∞ →∞ → We mention here that the anti-dynamo theorem is still respected since it corresponds to the second limiting procedure above.

III.4.2 Rm ,D 0 → ∞ r → Figure III.3 shows the dependence of the normalized growth rate γτd as a function of k /k for different values of D . Figure III.3a shows the case of finite Rm 1.95e+05, z d r ≈ and figure III.3b shows the case of infinite Rm . In this section we consider the → ∞ III.4. Different limits 55

3 3

2.5 0.00016 2.5 0.0016 0.016 2 2 0.16 0.00016 2D 1.6 0.0016

d 16 d

τ 1.5 τ 1.5 0.016

γ 160 γ 0.16 16000 1.6 1 1600000 1 16 2D 1600 160000 0.5 0.5

0 0 10−6 10−4 10−2 100 10−4 10−3 10−2 10−1 100 k /k k /k z d z d (a) (b)

Figure III.3 – The figures show the normalized growth rate γτd as a function of kz/kd for different values of D for a) the case of finite Rm 1.95e+05, b) the case of infinite r ≈ Rm . The black arrow with the annotation 2D shows the limit of D 0. The → ∞ r → values of Dr are mentioned in the legend and darker shades of blue correspond to smaller values of Dr. limits Rm and D 0. The limit D 0 is the limit of the vertical velocity → ∞ r → r → uz going to zero. Thus the flow becomes two-dimensional and from the anti-dynamo theorem we expect that the dynamo instability to disappear. This is seen from the figure III.3a, for a finite Rm, the limit of D 0 marked by the arrow 2D shows that r → γ 0. While in the limit of Rm , the growth rates tend to an asymptotic limit → → ∞ as seen in figure III.3b. Thus we have another set of noncommuting limits, D 0 r → and Rm , for a finite k < 1 we can say, → ∞ z 0 < lim lim γ = lim lim γ = 0. (III.4.6) Dr 0 Rm 6 Rm Dr 0 → →∞ →∞ → The limit on the left leads to an asymptotic curve in the limit D 0, which is seen r → from the figure III.3b, where the curves corresponding to small Dr approach a limiting behaviour. We have already looked at the limit Rm by expanding equations → ∞ (III.1.20), (III.1.21) into equations (III.4.3), (III.4.4) by looking at the lowest order in η. Similarly we can capture the limit of D 0 taken after the limit Rm r → → ∞ starting from the equations (III.4.3), (III.4.4). We apply the following set of rescaling, √D r˜ r˜, h √D h˜ . We get the limit D 0 by taking the lowest order in D , r → c → r c r → r leading to the following set of equations,

2 hLL′ 2 2 γτ h r˜ h′′ + 3 + k˜ 2 + r˜ h =2h (III.4.7) d LL − LL r˜ z LL LL h i hLL 2r˜ 2h′ + 3 +8r˜k˜ h˜ , LL r˜ z c 2 1 1 2 2 γτ h˜ r˜ h˜′′ + h˜′ h˜ + k˜ 2 + r˜ h˜ =2k˜ rh˜ . (III.4.8) d c − c r˜ c − r˜2 c z c z LL h i The above set of equations corresponds to taking the limit Rm first and → ∞ then the limit D 0. The obtained growth rate γ is shown in figure III.4a. This r → also corresponds to the asymptotic curve in the figure III.3b marked by the 2D arrow. The above equations are thus valid in the parameter range of, 1 D Rm 1. The ≫ r ≫ − 56 Chapter III. Kazantsev model for dynamo instability

3 3

2.5 2.5

2 2 d d τ τ

1.5

1.5 γ γ

1 1 Rm 0.5 0.5 → ∞ Rm → ∞ Rm=1.95e+05

0 0 −4 −2 0 10−4 10−2 100 102 10 10 10 ˜ kz √Dr kz/kd (a) (b)

Figure III.4 – The ﬁgures show the normalized growth rate γτd. In a) for the case of Rm and then D 0 as a function of k /k . In b) the case of (Rm and → ∞ r → z d → ∞ then D ) and (ﬁnite Rm and D ), as a function of √D k /k . r → ∞ r → ∞ r z d

1 expansion fails when Dr becomes of the same order as Rm− , for Dr values smaller 1 than Rm− dissipation eﬀects become stronger and the dynamo instability disappears in agreement with the Zeldovich anti-dynamo theorem.

III.4.3 Rm ,D → ∞ r → ∞ We now look at the limit Rm and then D . This corresponds to the → ∞ r → ∞ limit of zero shear in the 2D flow ∇u 0, relative to the gradients in the vertical | 2D | → velocity uz. Figures III.3a, III.3b show that as Dr becomes larger, the unstable modes shifts to smaller and smaller values. This is because as we increase Dr we increase the amplitude of vertical velocity uz, thus we need magnetic field correlated over larger length scale along z-direction. From the figures III.3a, III.3a one can notice that for large values of Dr the curve seems to be self similar with a horizontal shift. This gives an idea about a possible rescaling of the variable kz/kd in terms of Dr to collapse the curves on top of each other. In both the finite and the infinite Rm cases, there seems to be a domain of dynamo unstable modes when we take the limit D . r → ∞ One possible reason why the instability exists in the limit of D is because r → ∞ the amplitude of the horizontal velocity u2D is kept constant when we take the limit D . Thus the normalization of the growth rate γ is made using τ which is based r → ∞ d on the shear of u2D . If we take note of all the three components of the velocity field, 2 2 we need to normalize the growth rate as γ/ D1 + D2. Thus as Dr becomes larger we see that γτ / 1 + D2 0. q d r → To study the dependence on the large D case we look at the limit Rm and p r → ∞ then the limit D . Similar to the previous section we can expand the equations r → ∞ (III.1.20), (III.1.21) for the finite Rm and equations (III.4.3), (III.4.4) for infinite Rm to look at the lowest order in 1/Dr. We do not show the resulting equations, but we show the solutions of the resulting equations in figure III.4b. It shows γτd as a function of √D k /k for both finite Rm and Rm as mentioned in the legend. r z d → ∞ These curves reproduce the limit D in the figures III.3a, III.3a. The two limit r → ∞ Rm and D are found to be commutative. → ∞ r → ∞ III.5. Correlation function and energy spectra 57

III.5 Correlation function and energy spectra

We now look at the form of the correlation functions hLL (r) and hc (r) and the spectra of the magnetic ﬁeld. We have discussed in the introduction of the chapter that the spectra seen for the 2D Kazantsev model is a k2 spectra formed between velocity scale 3/2 k0 and the dissipation scale kd. The 3D Kazantsev model predicts a k spectra between k0 and the dissipation scale kd. In the system 2.5D we have three diﬀerent scales of importance k0, kz, kd. We know from the previous section that only a particular or- dering of these quantities will lead to a dynamo instability. The range of values for the dynamo instability are, c k √D k

In general the functions hLL (r) and hc (r) can be expanded for both large r and small r. At small r we have,

(γ + 2ηk2 8D )a 4ηk b h (r) =a z − 1 0 − z 1 r2 + O r4 , (III.5.1) LL 0 − 16η (γ + 2ηk2)b 2k D a h (r) =b r z 1 − z 2 0 r3 + O r5 , (III.5.2) c 1 − 16η where a0, b1 are constants related to the eigenvalue γ. In the large r limit we get the behaviour,

√γ/2η+k2 r h (r) e− z , (III.5.3) LL ∼ √γ/2η+k2 r h (r) e− z . (III.5.4) c ∼ We can do a similar expansion in the limit of large Rm (equations (III.4.3), (III.4.4)), where we have the rescalingr ˜ = rkd and k˜z = kz/kd. The behaviour of the functions hLL (˜r) and hc (˜r), in the smallr ˜ limit are found to be,

(γ + 2k˜2 8)˜a 4k˜ ˜b h (˜r) =a ˜ z − 0 − z 1 r˜2 + O(˜r4), (III.5.5) LL 0 − 16 (γ + 2k˜2)˜b 2k˜ D a˜ h (˜r) = ˜b r˜ z 1 − z r 0 r˜3 + O(˜r5), (III.5.6) c 1 − 16 wherea ˜0, ˜b1 are some constants. In the larger ˜ we see,

√Dr k˜zr˜ h (˜r) e− , (III.5.7) LL ∼ √Dr k˜zr˜ h (˜r) e− . (III.5.8) c ∼

We concentrate on the limit of large scale separation between k0, kz, kd and look at the limit Rm , where considerable analytical progress can be made. We find three → ∞ distinct range of scales that display different behaviour for the functions hLL (˜r) , hc (˜r). Values ofr ˜ 1 correspond to scales below dissipative scales, where the value 1 denotes ≪ the dissipation scale due to the rescalingr ˜ = r/rd used. For larger ˜ which is the regime r˜ 1/k˜ , corresponds to correlations in 2D much larger than the correlations along the ≫ z z direction. The behaviour at these two limits are already known from small and large 58 Chapter III. Kazantsev model for dynamo instability r˜ expansions (see (III.5.7), (III.5.8)). Between the dissipation scale and 1/kz, there is an intermediate range of scales 1 r˜ 1/k . The scaling in this intermediate ≪ ≪ z range of scales can be obtained by using matched asymptotics, the details of which are given in the Appendix B. By doing the matched asymptotics we also get the maximum growth rate γτ = 3 in the limit of k˜ 0 independent of the value of D . This value d z → r is in accordance with results shown in figures III.2, III.3a, III.3b. In general we can write the behaviour of the functions hLL (r) and hc (r) for large Rm in the following form,

2 4 1 1 1 1 c1r + O(r ) if r c4r if r − ≪ kd ≪ kd 1 1 1 0 1 1 hLL = c2r− if r , hc = c5r if r ,  kd ≪ ≪ kz  kd ≪ ≪ kz  c3r 1  c2r 1 e− if r e− if r ≫ kz ≫ kz   (III.5.9)   where c1,c2,c3,c4,c5,c6 can be calculated from the equations (III.5.1), (III.5.2), (III.5.3), (III.5.4) and from Appendix B. We look at the eigenfunction found from the numerical eigen value calculation. Shown in ﬁgure III.5 are the correlation function for the magnetic energy hLL , hc calculated numerically for the set of equations (III.4.3), (III.4.4) for the parameters k˜z = 0.005,Dr = 1. In the ﬁgure we note that the dissipation

0 10 r0 ˜ h r LL (˜) h r 10−2 c(˜) r˜−1

10−4 ) r

(˜ r0 c ˜

h −6 , 10 ) r (˜

LL 1 h −8 r˜ 10

10−10

−12 10 10−6 10−4 10−2 100 102 104 r˜

Figure III.5 – The figure shows the correlation functions hLL (˜r), hc (˜r) as a function of r˜ for the parameters k˜z = 0.005,Dr = 1. The dotted black lines denote the predicted exponents in their respected range of scales. The dissipation length scale is given by r˜ = 1. scale is given byr ˜ = 1. The black dotted lines give the expected theoretical scaling summarized in equation (III.5.9). Given the behaviour of the magnetic field correlation functions as a function of r we now look at the magnetic field spectra. The two-point correlation function is related to the energy spectra by the Wiener-Khintchine theorem, see [119]. For a function III.5. Correlation function and energy spectra 59

M(r), we deﬁne its isotropic Fourier transform as,

∞ M (k) = k rM(r)J0 (kr) dr. (III.5.10) Z0 Due to the anisotropy alongc the z-direction, we define two different magnetic field spectra. One gives the magnetic energy spectra of the components in the 2D plane B denoted as E2D (k), while the other gives the magnetic energy spectra for the vertical B component of the magnetic field denoted as EZ (k). The relation of the magnetic energy spectra with hLL , hc is given by,

B ∞ E2D (k) = k r 2hLL (r) + rhLL′ (r) + rkzhc(r) J0 (kr) dr, (III.5.11) 0 Z B ∞ 1 hc(r) EZ (k) = k r hc′ (r) + J0 (kr) dr. (III.5.12) 0 kz r Z B B It is interesting to get the different power laws for E2D (k) ,EZ (k) for the three different regimes k k , k k k , k k . From the equations (III.5.11), (III.5.12) it is ≪ z z ≪ ≪ d ≫ d not trivial to get the different scaling laws since we need to integrate over the range of r. For small r which corresponds to large k we do the following representation of √Drkzr˜ 2n √Drkzr˜ 2n+1 hLL , hc as, hLL (˜r) = e− n∞=0 hnr˜ , hc (˜r) = e− n∞=0 gnr˜ . We find the relation for small k as, P P

B k 3 E2D (k) =c1 + O(k ), (III.5.13) kz k3 EB (k) =c + O(k3), (III.5.14) Z 2 3 kz where c1 and c2 are some constants that are independent of k. For large k values, we expect the magnetic field spectra to decay exponentially above the dissipation scale kd. This is related to the behaviour of the functions hLL , hc at small r where the flow is smooth. We use the steepest descent method to integrate and find the dominant behaviour at large k for the integrals (III.5.11), (III.5.12) to be,

B k/kd 3/2 E2D (k) =e− c˜1 + O(k− ) , (III.5.15) B k/kd 3/2 EZ (k) =e− c˜2 + O(k− ) , (III.5.16) wherec ˜1,c ˜2 are some constants independent of k. In the intermediate regime between kd and kz, we get the expected spectral exponent from the match asymptotics (in Appendix B). We write out the behaviour of the spectra in the respective length scales as,

1 3 k if k kz k if k kz B 0 ≪ B 0 ≪ E (k) = k if kz k kd ,E (k) = k if kz k kd , 2D  ≪ ≪ Z  ≪ ≪ e k/kd if k k e k/kd if k k − ≫ d − ≫ d   (III.5.17)   60 Chapter III. Kazantsev model for dynamo instability

1 1 10 10 k0 k0 0 0 10 10

(k) − (k) 1 −1 B Z 10 B 2D 10 1 E E k k =0.2 k =0.2 z z k3 k =0.1 k =0.1 z − z −2 2 10 k =0.05 10 k =0.05 z z k =0.025 k =0.025 z z

k =0.005 − k =0.005 −3 z 3 z 10 10 −3 −2 −1 0 −3 −2 −1 0 10 10 10 10 10 10 10 10 k/k k/k d d (a) (b)

B B Figure III.6 – Figures show in a) E2D , b) EZ as a function of k/kd for diﬀerent values of kz as mentioned in the legend. Lighter shades of blue correspond to smaller values of kz with the parameter Dr = 1.

B B These predicted behaviour for E2D ,E2D are well reproduced from the numerical simulations shown in figure III.6. It shows the two spectra for different values of kz with the expected scaling shown in black thick lines. In the limit of large scale separation, k k , we recover the theoretical predictions for the intermediate range of scales z ≪ d k k˜ 1. z ≪ ≪ We finally sketch the theoretical predictions in the limit of large scale separation between kz and kd in figure III.7. Figure III.7a shows the correlation functions B B hLL (r) , hc (r) and figure III.7b shows the spectra E2D (k) ,E2D (k). The scaling laws

(a) (b)

Figure III.7 – Figures show in a) the correlation functions hLL (r) , hc (r) as a function B B of r, b) the spectra of the magnetic field E2D ,EZ as a function of k. Both the figures correspond to the case of large scale separation k k . z ≪ d are shown by solid black lines in the respective regimes and generalized to take into account Dr. The general form of the unstable mode is thus understood from the figure III.7. We now try to reproduce these results with direct numerical simulations. III.6. Comparison with direct numerical simulations 61

III.6 Comparison with direct numerical simulations

III.6.1 White noise ﬂows

We solve the governing equation (equation (III.1.18)) in a 2D periodic domain using Fourier spectral methods. To do numerical simulations for a flow that is homogeneous and isotropic would require an infinitely long domain which is numerically not possible. A periodic domain is homogeneous but not isotropic, since there are two preferred directions x, y. We note that an axi-symmetric flow is isotropic, but is not homogeneous. Hence we restrict our study to numerically simulate the delta correlated velocity field that is only homogeneous but not isotropic. We expect the results to only match qualitatively. We consider a velocity field of the form,

ψ (x, y, t) = ζ3 (t) sin (φ3 (t)) cos (kf x + φ4 (t)) + cos (φ3 (t)) sin (kf y + φ4 (t)) /kf , h (III.6.1)i

uz (x, y, t) = ζ4 (t) sin (φ3 (t)) sin (kf x + φ4 (t)) + cos (φ3 (t)) cos (kf y + φ4 (t)) , h (III.6.2)i where ζ3 (t) , ζ4 (t) are two Gaussian distributed random variables satisfying the relations, ζ (t) ζ (t ) = δ (t t ) , ζ (t) ζ (t ) = δ (t t ), ζ (t) ζ (t ) = 0. φ (t) , φ (t) h 3 3 ′ i − ′ h 4 4 ′ i − ′ h 3 4 ′ i 3 4 are uniformly distributed random variables in the interval [0, 2π]. The simulations are done in a domain [2πL, 2πL] with kf L being the forcing wavenumber. The above system is homogeneous and invariant under π/2 rotations. We numerically integrate the induction equation for the above flow. Due to the presence of a multiplicative noise term, we use the Stratonovich formulation. We solve for the modified governing equation, the modification done in order to take into account the multiplicative noise term. This is explained in detail in Appendix C, also see [120, 121]. The growth rate γτd is shown as a function of kz/kd for different values of Rm in figure III.8. The length scale of the velocity field is taken as kf L = 1. The exact values of γ do not match, but it matches qualitatively with the results of figure III.2. The discrepancy is possibly due to the non-isotropy of the underlying flow. The time series of the growth of the magnetic energy shows a lot of intermittent behaviour when Rm Rmc a subject which will be studied in detail in the second part of this chapter. ∼ B In figure III.9, we show the growing unstable mode (magnetic spectra) in a) E2D , b) EB for two different values of k . The k 21 for the parameters corresponding Z z d ∼ to the spectra. The black solid lines show the expected theoretical spectra, the case k = k = 1 k , the spectra show the scaling EB k0,EB k0. While in the z 0 ≪ d 2D ∼ Z ∼ case k = 9 k k , the spectra show the scaling EB k1, EB k3. We seem to z ∼ d ≫ 0 2D ∼ Z ∼ have a good reproduction of the theoretical estimates for both the growth rates and the form of the unstable spectra. We finally show the magnetic field structure of the unstable growing mode in figure III.10. It shows the contours of the magnetic energy in 2D in a) b 2 = b b +b b and | 2D | x† x y† y for the vertical component in b) b 2, for the case of Rm 210, k /k = 0.35, k = 1. | z| ≈ z d 0 We see that the magnetic field is concentrated in small scales in the form of thin filamental structures. Their thickness decreases as we increase Rm. 62 Chapter III. Kazantsev model for dynamo instability

3.0 Rm=6.67 2.5 Rm=13.33 Rm=213.33 Rm=426.67 2.0

d 1.5 γτ

1.0

0.5

0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 kz/kd

Figure III.8 – Figure shows the normalized growth rate γτd as a function of kz/kd for diﬀerent values of Rm, for the case of white noise. Darker shades of blue correspond to larger values of Rm.

k = 1 0 k = 1 10−1 k0 z 10−1 k z kz = 9 kz = 9 10−2 10−2 ) k1 ) k k ( ( D B B Z 2 10−3 10−3 E E k3 10−4 10−4

10−5 10−5 100 101 102 100 101 102 k k (a) (b)

B B Figure III.9 – Figures show the magnetic ﬁeld spectra, a) E2D , b) EZ as a function of k for two diﬀerent values of kz as mentioned in the legend. The parameters are k = 1, k 21. The black solid lines show the expected power laws. 0 d ≈ III.6. Comparison with direct numerical simulations 63

(a) (b)

Figure III.10 – Figures show the contours of, a) the two dimensional magnetic energy density b 2, b) the vertical magnetic energy density b 2. The parameters of the | 2D | | z| simulation are Rm 210, k /k = 0.35, k = 1. ≈ z d 0

III.6.2 Freely evolving ﬂows

Now we consider the results of the forced Navier Stokes equation presented in the previous Chapter II. The flow resulting from a constant nonhelical forcing with a fixed spatial structure, would be a non-homogeneous and non-isotropic flow. The dynamo instability considered here is driven by a freely evolving chaotic/turbulent flow obtained from resolving the 2.5D Navier Stokes equations. We consider the forcing length case of kf L = 4, but since an inverse cascade forms we cannot associate one particular length scale for the velocity field. We reproduce the figure II.11 in Chapter II by rescaling

3.0 Rm=8.07 2.5 Rm=12.1 Rm=16.13 2.0 Rm=32.26 Rm=64.53

d Rm=129.05 1.5

γτ Rm=258.1 Rm=516.2 1.0 Rm=1032.41

0.5

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 kz/kd

Figure III.11 – Figure shows the normalized growth rate γτd as a function of kz/kd for different values of Rm for the case of a turbulent flow. Darker shades of blue correspond to larger values of Rm. 64 Chapter III. Kazantsev model for dynamo instability with kd defined here as kd = kf √Rm, where the magnetic Reynolds number is defined as Rm = u/(kf η), with u being the root-mean-square velocity. The kinetic Reynolds 2 number is defined as Re = u/(kf ν), and the diffusion time scale is τd = 1/(ηkf ). The growth rate curves in figure III.11 seem to match qualitatively ( a similar dependence with kz, Rm) with the ones found from theoretical calculations shown in figure III.2. This is despite the fact that the assumptions in the theoretical calculations are much stronger. We mention that the exact values of growth rates and the critical magnetic Reynolds number do not match between the two cases. B B In figure III.12, we show the magnetic field spectra, a) E2D (k), b) EZ (k) as a function of k for a few different values of kz. The solid black lines denote the theoretical scaling laws in the respective domains. The spectra shown here correspond to a simulation run of Rm 1020, Re 32 taken after t 1000 nonlinear time scales. ≈ ≈ ≈ Due to the development of an inverse cascade the value of the velocity length scale is not clear. We take it to be the forcing length scale k 4 and we find k 80. For 0 ∼ d ∼ these parameters, we find the scaling k0 when k k k and the scalings k1, k3 z ∼ 0 ≪ d when k k k . z ∼ d ≫ 0

10−1 10−1 k0 k0 10−2 10−2

) −3 ) 10 k k ( 1 ( k 3

D k B B Z 2 −4 E

E 10 10−3 kz = 1 kz = 1 kz = 3.5 10−5 kz = 3.5 kz = 15 kz = 15 10−4 10−6 100 101 102 100 101 102 k k (a) (b)

B B Figure III.12 – Figures show the magnetic field spectra, a) E2D , b) EZ as a function of k for two different values of kz as mentioned in the legend for the case of the forced Navier Stokes flow (turbulent flow). Lighter shades correspond to smaller values of kz. The black solid lines denote scaling laws from the theoretical predictions.

Thus the theoretical results seem to agree qualitatively in nonhelical flows where the velocity field has a finite correlation time. The shape of the unstable spectra will be of use later on in the Chapter IV where we will study the dynamo instability driven by a fast rotating flow. We will see that these theoretical results hold even for fast rotating 3D nonhomogeneous and nonisotropic flows with zero mean helicity.

III.7 Conclusion-Part 1

In the first part of this chapter we have studied the Kazantsev model for nonhelical flows. Such class of flows help us to model turbulent fluctuations and helps one to make considerable analytical progress. We have studied the form of the unstable spectra in a 2.5D flow and compared it with its counterparts in 2D and 3D flows. This setup helped us to study the dynamo instability close to the anti-dynamo theorem of the III.8. Intermittent scaling of moments 65

Zeldovich. We derived a set of simple one dimensional coupled differential equations to study the dependence of the growth rate and the most unstable mode as a function of the parameters of the system. We could also find the growth rates at extreme values of the parameters, which is quite difficult to attain numerically. To compare with the theoretical results, we performed numerical simulations. Nu- merical simulation were done in restricted geometries for two kinds of flows, one which has delta-correlated statistics and the other which has finite correlation time (freely evolving turbulent 2.5D flow). In both cases the growth rates only matched qualitatively due to the non-isotropy of the flow (in the second case also non-homogeneous). The form of the most unstable modes seem to be well captured by the theoretical analysis. The present theoretical study was limited only for smooth flows. An interesting ex- tension would be to study the dynamo instability driven by rough flows that resembles the turbulent flow under fast rotation. This is the case for low P m number flows. One could also look at the helical version of this study by considering the magnetic helicity in the equations and try to get the results of the previous chapter for the helical forcing case. Finally a theoretical Kazantsev model for a fully anisotropic 3D flow could be carried out by considering ψ, uz to have some dependence in the z-direction. Such problems could model the fully 3D rotating flow, but the resulting equations might no longer be a function of a one-dimensional variable (in general of two variables r, z), needing more work to be understood.

III.8 Intermittent scaling of moments

The white-noise model considered in the first part of this Chapter used the second- order moment to calculate the growth rate. It is to note that close to the threshold, the second order moment does not predict the correctly the onset of the instability. This is because, close to the dynamo instability we expect that the magnetic energy fluctuates due to the fluctuations in the velocity field. In such a scenario the different moments of the magnetic field is expected to scale differently. A simple example to show the effect of fluctuations on the growth of the magnetic field, is to consider xt+1 = vα xt.

Let vα be a ﬂuctuating parameter taking the value,

2,P (vα = 2) = 0.3, vα = (III.8.1) (0.5,P (vα = 0.5) = 0.7.

n One can then calculate the growth rate of different moments as γ = 1 log xt , n t x0 with denoting the statistical average. We find that the first moment D decays E ex- h·i ponentially in time with a decay rate γ = 0.0513. The second moment grows 1 − exponentially in time with a growth rate γ2 = 0.318. The higher moments n > 2, also grow exponentially in time. So a fluctuating growth leads to moments which grow at different rate. This poses a problem for calculating a threshold of an instability, with different moments predicting different thresholds. We show that the exact threshold is actually given by the full nonlinear problem. 66 Chapter III. Kazantsev model for dynamo instability

We will follow the case of a simple one-dimensional multiplicative noise model which gives a simple picture for the full nonlinear problem. We consider the model,

x˙ = µx x3 + ζ(t)x, (III.8.2) − which is a pitchfork bifurcation model with a multiplicative noise. The PDF of the above equation is found to be,

1 µ 2 − 2D µ 2 D 1 X /(2D) P (X) = µ µ X − e− , (III.8.3) 2D D Γ( 2D ) for µ > 0 and P (X) = δ(X) for µ < 0. The n-moment given by, Xn = Γ( µ + h i 2D n )/Γ( µ ) (2D)n/2 from which we can ﬁnd the ﬁrst few moments to be, X µ, X2 = 2 2D h i ∝ µ for µ 0. For µ < 0 all the moments are zero. Clearly the bifurcation occurs at ≥ µ = 0. If we consider the linearized equation which is usually studied to look for the onset of instability we get,

X˙ = µX + ζX, (III.8.4) whose PDF has the following form in the variable Y = log(X),

2 1 (µ+Y −Y0) P (Y ) = e− Dt , (III.8.5) √πDt where Y0 is the initial position t = 0. We can compute the growth rate of the moment Xn to be, h i 2 Xn = Xn e(µn+Dn )t, (III.8.6) h it h i0 The growth rate of each moment µn + Dn2 is a nonlinear function of n, positive values of growth rate indicate the onset of instability. Each moment bifurcates at separate (n) values of µ, we denote µc as the value of µ at which the n-th moment has a positive (n) growth rate. We find that µc = (nD), thus each moment predicts a different − (1) threshold for the onset of the instability. The first moment predicts µc = D, the (2) (0) − second moment predicts µc = 2D and the zeroth moment predicts µc = 0. While − in the presence of nonlinearity (III.8.2), we see that the threshold is predicted only by the 0-th moment. The 0-th moment (or n = 0) gives the linear growth rate of the log 1 X t of the variable, growth rate = log | | . t X 0 | | Figure III.13 shows the bifurcationD diagramE of X as a function of µ with the vertical dotted lines showing the predictions for the instability threshold for the different moments. Thus the second moment predicts a lower value for the instability threshold. Hence while solving for the linear part of the nonlinear problem one should look at the 0-th moment in order to find the exact threshold. This is true for a simple one- dimensional model, the question which we ask here is whether for Kazantsev model where the flow is modelled as white noise, shows similar behaviour. The kinematic dynamo study is a linear version of the nonlinear MHD equations. The growth rates of III.9. α-dynamo for Kazantsev flow 67

Figure III.13 – Figure shows the bifurcation diagram of a pitchfork bifurcation in the presence of a multiplicative noise. The vertical lines denote the predictions of the onset from the linearized equations. the second-order moments Hij that we calculated, correspond to the kinematic study. They might not be predicting the exact threshold when we solve for the full problem where nonlinearity saturates the exponential growth phase. In order to show this both analytically and numerically we consider the helically forced 2.5D fluctuating velocity field. We first derive the scaling of different moments analytically and calculate their threshold. Later we compare it with the full nonlinear problem solved numerically.

III.9 α-dynamo for Kazantsev ﬂow

Let us consider a Gaussian distributed white noise flow v = ζ(t)u, where u is a function of space only. The noise ζ satisfies the relationship ζ(t)ζ(t ) = 2Dδ(t t ). In the limit h ′ i − ′ of scale separation, where the velocity field is present at small length scales compared to the domain size, and in the presence of kinetic helicity, the magnetic field grows at the largest scale of the domain. In this limit we can use the α-effect to explain dynamo instability. The magnetic field can be written as B = B + b. We consider h i the induction equation in a general 3D flow written as,

∂ B = ∇ ( v b ) + η∆ B , (III.9.1) t h i × h × i h i ∂ b η∆b = ∇ (v B) ( v b ) . (III.9.2) t − × × − ∇ × h × i As mentioned in section II.3 in Chapter II we can simplify the equation (III.9.2) in the limit of small Rm (Rm 1) to, ≪ ∂ b η∆b = ∇ (v B ) = ζ(t) B ∇u. (III.9.3) t − × × h i h i · In order to simplify the above expression we consider that the ﬂow is 2π periodic in all 3/2 ik r directions. We deﬁne the Fourier transform as, bˆ (k) = (2π)− b (r) e · dr, uˆ (k) = 3/2 u r ik r r k (2π)− ( ) e · d . We can write the above equation (III.9.3RRR ) for each mode as, RRR

∂ bˆ (k) + ηk2bˆ (k) = ζ(t)i B kuˆ (k) . (III.9.4) t h i · 68 Chapter III. Kazantsev model for dynamo instability

The velocity Fourier mode uˆ (k) is responsible in sustaining the magnetic Fourier mode bˆ (k). For equation (III.9.4) we can write the solution for bˆ (k) as,

2 bˆ (k) = Yk (t) ηk bˆr (k) , (III.9.5) where bˆr is the solution of the equation (III.9.4) when ζ (t) = 1. We ﬁnd bˆr = i B kuˆ (k) /(ηk2). Y (t) is the solution of the equation, h i · k dY (t) k + ηk2Y (t) = ζ (t) . (III.9.6) dt k

2 Yk (t) is a standard Ornstein-Uhlenbeck process with a damping rate ηk . Using scale separation we have written the small scale magnetic field as a product of an Ornstein Uhlenbeck process and a spatially varying function. The statistics of Yk is given by its probability distribution function. The large scale magnetic field is driven by the interaction between the small scale velocity field and the small scale magnetic field. We find,

3 v b =ζ(t)(2π)− Yk(t)û( k) i B kû(k), (III.9.7) h × i k − × h i · X where each wavevector k contributes to the e.m.f in the equation (III.9.1) for the large scale magnetic field. If the velocity field is of a single wavenumber, the above expression can be simplified to,

v b =ζ(t)Y (t) ηk2α B , (III.9.8) h × i k · h i where α is the alpha-tensor obtained when the noise ζ(t) = 1. It can be written as,

3 kq α = (2π)− i (uˆ ( k) uˆ (k)) . (III.9.9) pq ηk2 − × p α depends only on the spatial structure of the flow. We now look to solve for the large scale magnetic field (equation (III.9.1)). Substituting the relation (III.9.8) into the equation for the large scale magnetic field we get,

∂ B = ζ (t) Y (t) ηk2α B ηK2 B , (III.9.10) t h i k h i − h i where the large scale magnetic field B varies over a wavelength K. We now need to h i diagonalize the α tensor to find the most unstable mode. We decompose the α tensor into symmetric and anti-symmetric components, the anti-symmetric component leads to an uniform advection term. More precisely ζ(t)Y (t)∇ (α B ) = ζ(t)Y (t)U k × a h i k α · ∇ B where α corresponds to the anti-symmetric part of the α tensor, U is a vector h i a α associated with the anti-symmetric component of the α tensor. This advection term ′ ′ ′ ζ(t )Y (t )iUα Kdt can be taken into account by multiplying the equation by e k · and does not affect the rest of the discussion for the dynamo instability.R Thus we only consider the symmetric part of α and diagonalize to obtain, α B = s s h i (α B , α B , α B ), where α , α , α are the eigenvalues. We find the largest 1 h 1i 2 h 2i 3 h 3i 1 2 3 two of the three eigenvalues, say α1, α2 and the most unstable mode is then along the III.9. α-dynamo for Kazantsev flow 69 z-direction, B = B¯ eiKz. For positive α , α we calculate the most unstable mode to h i 1 2 be, Bp = √α1B¯1 + √α2B¯2 and it satisfies,

∂ B = ζ (t) Y (t) ηk2αB ηK2B , (III.9.11) t p k p − p where α = √α1α2. The large scale ﬁeld Bp at a time t can be thus written as,

ηk2αI(t) ηK2t Bp (t) = Bp (0) e − , (III.9.12)

t with I(t) = 0 ζ (t) Yk (t) dt. The above equation gives the evolution of the large scale ηk2αI(t) magnetic fieldR Bp (t) as a function of time t starting from Bp (0). The term e is ηK2t responsible for the amplification and the term e− for decay due to dissipation. In order to obtain the different moments of the large scale magnetic field, we need the statistics of the quantity I (t). It is interesting to note that the statistics of the quantity I(t) has been studied in detail by [122]. It was studied in the context of a particle in a viscous fluid subject to a random force ζ(t) with damping due to the viscous forces (here the damping rate is ηk2). The quanitity I(t) is then the injected power into the particle by the random force ζ(t), see also the footnote. 1 The probability distribution function of the quantity I(t) at long time can be written as,

t tg(i) P (I = ti) →∞ e− , (III.9.14) ∼ where means that the above exponent is at highest order in t. i is the mean injection ∼ power in the long time t, i = I(t)/t and g(i) is called the rate function. The above expression implies that I(t) at long times follows a law of large deviation. It is known that the long time statistics of the function I(t) has a little dependence on the initial condition of Yk(t), [122]. If the initial condition is zero, Yk (0) = 0 then the rate function is found to be,

ηk2D i 2 g (i) = 1 , (III.9.15) 4i D − for positive i and is inﬁnite for negative i. If the initial condition of Yk (0) is taken to be the equilibrium distribution, then the rate function g(i) is equal to,

ηk2D i 2 i 1 1 , < , 4i D D 3 g (i) =  − (III.9.16)  i i 1 ηk2 1 2 , > . D D 3 −  1For a particle with velocity Y (t) satisfying Y˙ = −ηk2Y + ζ(t). The kinetic energy of the particle then follows the equation,

Y˙2 = −ηk2Y 2 + ζ(t)Y, (III.9.13) 2 where the term −ηk2Y 2 is the dissipation rate due to the viscous forces and ζ(t)Y (t) is the energy injection rate of the random force ζ(t). 70 Chapter III. Kazantsev model for dynamo instability

For the calculations forward we take the case of an initial condition Y (0) = 0. The large deviation function is given by (III.9.15). The statistics of the function I(t) is given by the large deviation function P (I) from equation (III.9.14). We denote the statistical average of diﬀerent moments of the n large scale magnetic ﬁeld Bp as Bp . Here the statistical average is taken over s h·is all realizations of the variable I(Dt). TheE moments are given by,

n nηk2(αK βK2)it ηnK2t B e − − P (i)di h p is ∝ ∼ Z t(g(i) nηk2(αK βK2)i) ηnK2t e− − − − di. (III.9.17) Z In the large t limit, the integral in the above equation (III.9.17) can be calculated by the Laplace method (saddle point method/steepest descent method). We ﬁnd the saddle point i (n) from the condition g (i ) nαηk2K = 0, to be i = D/√1 4DnαK. We c ′ c − c − get the condition that for ic to be bounded, 4DnKα < 1. The growth rate of the n-th moment is given by,

λ = g(i ) + ni αnηk2K nηK2 n − c c − ηk2 = nηK2 + 1 √1 4DnKα . (III.9.18) − 2 − − Thus the growth rate of different moments λn scale nonlinearily with n. Figure III.14 1 shows the growth rate λn as a function of n for the parameters, k = 4, η = 4.0,D = 2 , for different values of Rm as mentioned in the legend. Rm is defined as, Rm =

0.025 Rm =2.50E-01 6 Rm =8.33E-02 0.020 Rm =2.50E-04 Rm =7.14E-02 Rm =1.25E-04 Rm =6.25E-02 0.015 4 Rm =8.33E-05 Rm =5.56E-02 0.010 Rm =4.55E-02 n n 2 λ 0.005 λ 0.000 0 −0.005 −2 −0.010 0 2 4 6 8 10 0 2 4 6 8 10 n n

(a) (b)

Figure III.14 – Figures show the growth rate of diﬀerent moments λn as a function of the moment n for the parameters k = 4, η = 4.0 for diﬀerent values of Rm for a) K/k = 2.5e 4, b) K/k = 1.25e 1. Darker shades of blue correspond to larger values − − of Rm.

α/(ηk). Two different values of scale separation are shown, one in figure III.14a, anotherp in figure III.14b. For small values of scale separation, we see that the growth III.9. α-dynamo for Kazantsev flow 71 rate scales linearly with n and as we reduce scale separation, λn becomes nonlinear with n. This can be seen from expanding the equation (III.9.18),

DU 2 K K λ = nηK2 + ηk2 n (1 )+ n − η k − k D2U 4 K2 K n2 (1 )2 + , (III.9.19) η2 k2 − k ··· where the higher order term n2 becomes larger as we increase the scale separation. As mentioned in the set-up of this problem (section III.8), we are interested in the growth rate of the log of the magnetic ﬁeld deﬁned as,

λn log(Bp) λ0 = lim = lim h i. (III.9.20) n 0 n t t → →∞

We can calculate the quantity λ0 from equation (III.9.18) and find that the logarithm of the magnetic field grows like (ηk2αKD ηK2)t. This implies that the growth rate − of the magnetic field as predicted by the logarithm (the zeroth moment) is given by, (ηk2αKD ηK2)t e − . We incorporate this into the previous figure by plotting the ratio λn/n λn in figure III.15. The value of n (n = 0) corresponds to the growth rate of the magnetic field as predicted by the logarithm (zero moment).

0.005 1.0 Rm =2.50E-01 Rm =8.33E-02 0.004 Rm =2.50E-04 0.8 Rm =7.14E-02 Rm =1.25E-04 0.6 Rm =6.25E-02 0.003 Rm =8.33E-05 Rm =5.56E-02 0.4 Rm =4.55E-02

/n 0.002 /n 0.2 n n

λ 0.001 λ 0.0 0.000 −0.2

−0.001 −0.4 0 2 4 6 8 10 0 2 4 6 8 10 n n

(a) (b)

Figure III.15 – Figures show the growth rate of diﬀerent moments λn/n as a function of the moment n for the parameters k = 4, η = 4.0, for diﬀerent values of Rm, in a) K/k = 2.5e 4, b) K/k = 1.25e 1. Darker shades of blue correspond to larger values − − of Rm.

We look at the instability threshold predicted by each moment n, the instability is predicted when the growth rate becomes positive λn > 0. From the growth rate λn in equation (III.9.18), we ﬁnd the critical value of α for the moment λn > 0 to be,

K K2 αc (n) = 1 n . (III.9.21) k2D − k2 ! 72 Chapter III. Kazantsev model for dynamo instability

The true onset is given by the growth of the logarithm and is given by, αc(0) = K/(k2D). We will show that this is the onset which predicts the instability in the presence of a nonlinear saturation term. For large scale separation, see figure III.15a, λn n stays constant with n, thus as we increase Rm we see that all the moments predict the same threshold. While for the case of figure III.15b, corresponding to the case of small scale separation, each moment predicts a different threshold.

III.10 Numerical results

We now look to numerical simulations of the dynamo instability driven by multiplicative noise. We solve for the induction equation, see Appendix C. Since we need to do statistical average over many realizations of the noise, we do numerical computations on a 2.5D flow. The 2.5D flow unlike the 3D flow, allows one to have much longer integration times helping us to do better statistical averaging. Since the calculation presented before is based on the scale-separation argument, we consider the velocity to be Robert’s flow (also the helical version of the forcing used in Chapter II). The velocity field is taken as v = ζ(t)U (cos (ky) , sin (kx) , cos (kx) + sin (ky)). We solve the induction equation numerically in a periodic box of size [2πL, 2πL] using Fourier pseudo-spectral method. k is the length scale of the velocity field. To capture the large scale separation limit K/k 1, we start with K/k = ≪ 0.01. We show in figure III.16 the time series of the total magnetic energy B 2 = B B | | † †dxdy. From the figure we see that the time series of the magnetic energy is RR 10−7 Rm=1.32e-03 Rm=1.28e-03 Rm=1.25e-03 2 | B |

10−8

0 100 200 300 400 500 600 700 800 t

Figure III.16 – Figure shows the total magnetic energy B 2 as a function of time | | t for diﬀerent values of Rm as mentioned in the legend, for the case of K/k = 0.01. Darker shades of blue correspond to larger values of Rm.

fluctuating in time. We can calculate the different moments of the large scale magnetic field B n and calculate its growth rate λ . We do the statistical averaging of the h| | i n growth rate by cutting the time series into small intervals and averaging the growth rate over the individual pieces. Figure III.17 shows the quantity λn/n as a function of n for the case of K/k = 0.01, for different values of Rm. The numerical values are III.10. Numerical results 73

0.010 theoretical,Rm=1.67e-03 Rm=1.67e-03 theoretical,Rm=1.32e-03 Rm=1.32e-03 0.008 theoretical,Rm=1.28e-03 Rm=1.28e-03 theoretical,Rm=1.25e-03 Rm=1.25e-03

0.006 n n λ 0.004

0.002

0.000

0 2 4 6 8 n

Figure III.17 – Figure shows the quantity λn/n as a function of n for different values of Rm as mentioned in the legend for the case of K/k = 0.01. Numerical values are shown by data points while theoretical values are shown by lines. shown as data points while the theoretical values (from equation (III.9.18)) are shown by lines. We mention that the theoretical values here also include the β-effect. The point n = 0 corresponds to the growth of the logarithm of the magnetic field. The theoretical and the numerical results agree very well for the different values of Rm examined. As we can see in the limit of large scale separation K/k = 0.01 1, the ≪ growth rates λn do not show nonlinearity with respect to n, in other words we find that λ n. We need to increase the scale separation in order to see λ scaling nonlinearly n ∝ n with n. This is also seen from the theory, see figures III.15, III.15. We show in figures III.18a, III.18b, III.18c the quantity λn/n for three different values of K/k = 0.04, 0.1, 0.25. The theoretical predictions are only shown for the figure III.18a because the theory is no longer valid when the scale separation is small. As we reduce scale separation the theoretical predictions move farther away from the numerical results. Even though the theoretical results agree at large scale separation, they will not be used in what is to follow. Moving from plots III.18a, III.18b, III.18c we reduce the scale separation and we start to see that λn/n is no longer a constant with n. Thus we start to see nonlinear scaling when K/k O(1). ∼ When λn scales nonlinearily with n, the threshold of the dynamo instability is not the same for different n. With the above numerical results, we can calculate the threshold from the linearized (kinematic dynamo problem) system. In the next section we will add the nonlinearity to see which moments predicts the threshold. To check the dependence on the form of the nonlinearity we will consider two different forms of nonlinearity.

III.10.1 Saturation/Nonlinear results

When the nonlinearity due to the Lorentz force 1 (∇ B) B is taken into account µ0 × × the ﬂow does not remain 2D. The Lorentz force excites the 3D modes of the velocity 74 Chapter III. Kazantsev model for dynamo instability

0.12 1.0 theoretical,Rm=1.67e-02 Rm=1.67e-02 Rm=2.50e-01 Rm=6.25e-02 0.8 0.10 theoretical,Rm=1.39e-02 Rm=1.39e-02 Rm=8.33e-02 Rm=5.56e-02 theoretical,Rm=1.32e-02 Rm=1.32e-02 0.6 Rm=7.14e-02 0.08 theoretical,Rm=1.28e-02 Rm=1.28e-02 0.4

n 0.06 n n n λ λ 0.2 0.04 0.0 0.02 −0.2 0.00 −0.4 0 2 4 6 8 0 2 4 6 8 n n

(a) (b) 0.8

0.6

0.4 n

n 0.2 λ

0.0 Rm=2.50e-01 Rm=1.92e-01 −0.2 Rm=2.22e-01 Rm=1.79e-01 Rm=2.00e-01 −0.4 0 2 4 6 8 n

(c)

Figure III.18 – Figure shows the quantity λn/n as a function of n for, a) K/k = 0.04, b) K/k = 0.1, c) K/k = 0.25 for diﬀerent values of Rm as mentioned in the legend. In a) the theoretical predictions are shown in solid lines while numerical simulations are shown by points. In b) and c) we only show numerical data as both data points and lines. III.10. Numerical results 75

field. The Lorentz force by the magnetic field B, which has a variation kz = K along the z direction, is composed of two different modes kz = 0 and kz = 2K. The component 2K induces the 3D modes in the velocity field. A similar set-up is explained in section II.11. In order to do long time statistics we try to remain in the 2D domain. We consider thus two types of nonlinearity, one which is phenomenological and the other which is on similar lines with the approach taken in section II.11, where we only consider the back reaction through the kz = 0 mode. We start with the phenomenological nonlinearity, the governing equation consists of only the induction equation written as, ∂B = v B B 2B + η 2B. (III.10.1) ∂t ∇ × × − | | ∇ The nonlinearity is cubic in B and respects solenoidality condition, ∇ B = 0. A simple · argument for considering a cubic nonlinear term can be understood by looking at the Lorentz force. We know that the Lorentz force scales like B2, thus any correction ∼ 2 in the velocity field, denoted as vc, due to the back reaction should scale like B , that is v B2. This velocity correction is responsible for the saturation of the dynamo c ∼ instability. Including this velocity correction in the induction equation leads to a term v B, which scales like B3. This argument is only phenomenological, since we did c × not take into account the spatial structure of the magnetic field, so the above equation (III.10.1) is a model for the nonlinear dynamo instability. From the previous section we choose the scale separation to be K/k = 0.25, where we see nonlinear scaling of moments. In figure III.19, we show the time series of the total magnetic energy B 2 for different values of Rm. Close to the threshold | |

10−3

10−5

10−7

10−9

−11

 10 2 | −13 B 10 | 10−15

10−17 Rm=2.08e-01 10−19 Rm=2.04e-01 Rm=1.92e-01 10−21 0 100 200 300 400 500 600 700 800 t

Figure III.19 – Figure shows the total magnetic energy B 2 as a function of time | | t for diﬀerent values of Rm as mentioned in the legend for the case of K/k = 0.25. Darker shades of blue correspond to larger values of Rm.

Rm 0.2, we see that the signal is very intermittent, needing long time series for ≈ averaging. We now show in ﬁgure III.20 the time averaged magnetic energy denoted as B 2 as a function of Rm. The predictions from the linear results are shown by | | D E 76 Chapter III. Kazantsev model for dynamo instability

−5 2.0×10

1.5 2

| 1.0 B | 

Linear ﬁt 0.5 Rmc for n = 0

Rmc for n = 1

Rmc for n = 2 K/k = 0.25 0.0 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 Rm

Figure III.20 – Figure shows the total magnetic energy B 2 as a function of Rm | | for the case of K/k = 0.25. The predictions from the linear simulations are shown by thick vertical lines for n = 0, 1, 2. The error bars around them are shown by dashed vertical lines. The linear fit through the data points is shown by the thick black line. vertical solid lines for n = 0, 1, 2. These predictions are calculated by extrapolating the numerical results (shown in figure III.18c). The error in calculating the thresholds from the linear calculations are shown by vertical dashed lines. All the errorbars shown here are calculated using a bootstrap method with 95%-confidence interval (For bootstrap method see [123]). The linear fit through the data points is given by the thick black line and its x-intercept gives the actual threshold for the dynamo instability. The error bar associated with the x-intercept of the linear fit is given by black vertical dashed lines and they lie in the domain of the Rmc predicted by the n = 0 moment of the linear calculation. Note that the prediction of the n = 2 moment grossly underestimates the threshold. We now consider the second type of nonlinearity, the MHD equations with only the kz = 0 projection of the Lorentz force. Following section II.11 we write the governing equations as, 1 1 ∂ v + v v = p + ν ∆v + f + (J B) , (III.10.2) t · ∇ −ρ∇ ρ h × iz ∂ B = (v B) + η ∆B. (III.10.3) t ∇ × × Where J = 1 B is the current and indicates averaging along the z-direction µ0 ∇ × h·iz which in effect only takes the kz = 0 mode of the Lorentz force. The problem thus remains 2D and we can do long time averaging. The forcing used here is f = ζ (t) f0 (cos (ky), sin (kx) , cos (kx) + sin (ky)). ζ(t) is a white noise. The velocity field in the limit of Re 1 is similar to an Ornstein-Uhlenbeck process and ≪ t νk2(t τ) it can be approximated to v (t) = v (0) + 0 e− − f (τ) dτ. The correlator of the 2 ′ velocity field can be written as, v (t) v (t ) = Df e νk t t , where D is a tensor h R′ i νk2 − | − | f which describes the spatial dependencies. The velocity field is thus correlated over a time scale (νk2) 1 and as ν we get back the delta-correlated white-noise. Such − → ∞ III.10. Numerical results 77 type of models have been numerically studied, see [104, 106], where they studied the kinematic dynamo problem (equations (III.10.2), (III.10.3) without the Lorentz force).

We deﬁne the kinetic Reynolds number as Re = f0/k/(νk) and the magnetic Reynolds number as Rm = f0/k/(ηk). Figure III.21 showsp the time series of the time averaged magnetic energy B 2 for diﬀerent values of Rm, obtained for this second p| | type of nonlinearity. We keepD theE scale separation at K/k = 0.25, expecting to see a

101 10−1 10−3 10−5 10−7

 −9 2 10 | −11 B

| 10 10−13 10−15 −17 10 Rm=4.26e+00 −19 Rm=4.08e+00 10 Rm=3.92e+00 10−21 0 500 1000 1500 2000 2500 t Figure III.21 – Figure shows the total magnetic energy B 2 as a function of time | | t for different values of Rm as mentioned in the legend, for the force Navier Stokes equation with K/k = 0.25. Darker shades of blue correspond to larger values of Rm. similar nonlinear scaling consistent with the previous phenomenological nonlinearity. The Reynolds number is kept constant at Re = 0.05. Comparing the above time series with the time series from the previous type of nonlinearity (shown in figure III.19), we see that close to the threshold the saturation amplitude is different. This is expected since the form of the nonlinearity changes the saturation amplitude of an instability. Similar to the phenomenological nonlinear simulations (figure III.19), we see that the time series become quite intermittent close to the threshold Rm 4. ≈ Figure III.22 shows the time averaged magnetic energy B 2 as a function of | | Rm. We again compare the threshold from the nonlinear saturatedD E simulations with the threshold from the linear calculations. The linear theory was done separately by only considering the equations (III.10.2), (III.10.3) without the Lorentz force coupling. The growth rates of the different moments were calculated and the dynamo threshold Rmc was found for each moment through extrapolation. Since the study is quite similar to the ones presented in the previous sections, we do not need to discuss the results of the linear calculation. Going back to figure III.22, the prediction of the threshold for the different moments are shown by vertical lines. The threshold predicted by the saturated dynamo problem is given by the x-intercept of the linear fit (black thick line in figure) through the data points. The error bar associated with the x-intercept lies close to the predictions from the linear theory of the n = 0 moment. Once again the n = 2 moment grossly underestimates the threshold. 78 Chapter III. Kazantsev model for dynamo instability

2.0

1.5 2

| 1.0 B | 

Linear ﬁt 0.5 Rmc for n = 0

Rmc for n = 1

Rmc for n = 2 K/k = 0.25 0.0 3.7 3.8 3.9 4.0 4.1 4.2 4.3 Rm

Figure III.22 – Figure shows the total magnetic energy B 2 as a function of Rm for | | the forced Navier Stokes equation, with K/k = 0.25, Re = 0. 05. The predictions from the linear simulations are shown by thick vertical lines for n = 0, 1, 2. The error bars around them are shown by dashed vertical lines. The linear ﬁt through the data points is shown by the thick black line.

Thus in the two different types of nonlinearity that we have studied, the n = 0 moment predicts the correct threshold. The error in the prediction of the threshold by the n = 2 moment is of the order of 12% form the actual threshold in the first type of ≈ nonlinearity (see figure III.20). It is 6% from the actual threshold in the second type ≈ of nonlinearity (see figure III.22). Thus with regards to models of turbulent dynamo like the Kazantsev model, the threshold should be understood by looking at the n = 0 moment. One could thus extend this question as to what happens when we consider a more general kind of flow with different noise terms v = ζ1 (t) u1 + ζ2 (t) u2. Also one could look at the prediction of the Kazantsev nonhelical dynamo (see first part of this Chapter) and compare it with direct numerical simulations of the saturated dynamo.

III.11 Conclusion

In this Chapter we have studied two different aspects of the dynamo instability driven by fluctuating velocity fields to model turbulent flows. The specific model considered here is the Kazantsev flow and we have tried to look at both the dynamo driven by scale separation and the one which is driven by the local shear. In the second part of this Chapter we had looked at the growth rate predicted by different moments of the magnetic field, in a kinematic dynamo problem driven by noise. In the limit of large scale separation, we derive a theoretical expression for the growth rate of the different moments of the magnetic field. The theoretical expression shows that the n- th moment of the magnetic field has a growth rate which is nonlinear in n. Thus each moment predicts its own threshold for the dynamo instability. This is of importance in the context of the Kazantsev model where the dynamo instability is usually predicted using the second moment of the magnetic field. III.11. Conclusion 79

We then use numerical simulations which matches with the theoretical calculations in the large scale separation. In the small scale separation, the growth rate becomes nonlinear with respect to n. We numerically calculate the threshold predicted by each moment. Using two different forms of nonlinearity, we confirm that the n = 0 or the growth rate predicted by the logarithm of the magnetic field, predicts correctly the threshold. The second moment grossly underestimates the threshold, depending on the system of equations considered the second moment has an error 5 10% ≈ − from the actual threshold. This study used simplified models of turbulence (quasi 2- dimensional, white noise) to model the study of dynamo instability. Ideally it would be interesting to look at a fully three-dimensional turbulent flow and see the effects of velocity fluctuations on the growth rate of different moments. Though quantifying the effects of a turbulent velocity field would require a long time series. 80 Chapter III. Kazantsev model for dynamo instability Chapter IV 3D Rotating flows and dynamo instability

In the previous chapters we have concentrated on quasi-twodimensional flows and the dynamo instability driven by such flows. They model the limit of fast rotating flows. In this Chapter we try to examine 3D rotating flows and rotating dynamos. We approach the fast rotating limit to test the validity of the assumption of the quasi- twodimensional approach. In the first part of this Chapter, we will concentrate on the effect of Rotation on the hydrodynamic flows driven by the Helically/Nonhelically forced Roberts flow. Rotation is one of the main mechanisms by which geophysical/atmospheric flows which are highly turbulent, cascade energy to large scales. Rotating flows become increasingly anisotropic as the rotation rate is increased, through the effect of the Coriolis force. In a homogeneous flow without walls, the Coriolis force does not inject any energy into the system, it affects the way energy is distributed among different scales. The simple observation of the effect of Coriolis force is through the Taylor- Proudmann theorem [25, 26, 27], which essentially states that in a fast rotating fluid if we displace a packet of fluid the whole column of liquid moves along with it. In other words the flow becomes independent along the coordinate of the rotation axis. We consider the rotation axis to be along the z-direction, Ω = Ωeˆz where Ω is the global rotation frequency. At large Ω or small Ro the underlying flow is quite different from the standard 3D flow, due to the presence of an inverse cascade of energy. Also rotation allows linear waves to propagate in the system. We know that for the linearised inviscid equation, we can find plane wave solutions, known as inertial i(k x σt) waves, of the form e · − . These inertial waves have been observed in experiments and numerical simulations [124, 125, 126, 127, 128]. The dispersion relation for the plane wave solution is,

ez k k σ = 2Ω · = 2Ω k = 2Ω cos θ, (IV.0.1) k k | | where θ is the angle between ez and k. The component parallel to the rotation direction 2 is denoted as k = k ez, the component perpendicular is denoted as k = (kx + 2 1/2 k · ⊥ ky) . Thus the frequency of propagation is related to the angle with respect to the

81 82 Chapter IV. 3D Rotating flows and dynamo instability z-direction. For the planar two-dimensional flow θ = 90◦, the inertial waves have zero frequency σ = 0 and are steady in time. For waves that are not two-dimensional, the frequency increases linearly with Ω. Fast rotating turbulence can be described by weak wave turbulence theory, where the interaction between the highly dispersive inertial waves becomes more important than the classical turbulent nonlinear interactions, [129, 130]. It is interesting to note that weak-wave-turbulence theory predicts that the energy is not transfered to the kz = 0 mode, while there is a pile up of energy in the modes close kz = 0, [131]. Even though weak wave turbulence theory in rotating flows has not been confirmed, a tendency to bidimensionalize as the fluctuations along the direction of rotation is suppressed, observed in both experiments and numerical simulations. There have been many studies using both experimental and numerical methods to study rotating turbulence. Experimentally it is quite difficult to realize fast rotating flows due to mechanical constraints, also the effect of solid walls makes it difficult to create a homogeneous rotating flow. Numerically the problem lies in the fact that both fast rotating low Ro and high Re seem to be out of reach with the existing computational power. Hence a full parametric study of rotating turbulence at all possible regimes is still far from being realized. Experimental and numerical studies on rotating flows have concentrated on decaying turbulence under global rotation and forced turbulence under rotation. Decaying rotating turbulence was studied experimentally by many different groups, [132, 133, 134, 135, 124, 136]. In such systems, when confinement effects are strong, it is observed that turbulence decayed faster as one increased the rotation rate due to the dissipation at the boundaries from Ekman layers, [49, 50]. When confinement effects are weak it is observed that rotating turbulence decays slower than classical turbulence. This is because the flow cascades energy to large scales which dissipate much slowly as compared to the dissipation through a forward cascade of energy. These were confirmed by numerical studies by [137]. Forced turbulence was experimentally studied by [138, 139, 140, 141, 142, 143] and studied numerically by [144, 145, 146, 147, 148, 149]. When the system is forced at large enough rotation rates, a steady inverse cascade towards large scales becomes present. There have been many different studies trying to look at the power law scaling of the energy spectra for scales smaller than the forcing length scale. Different scaling laws have been observed and they have been attributed either to weak wave turbulence theory or two-dimensionalization at low Ro (or helicity flux by [150, 151]). In particular we note that the scale at which the inertial time scale is equal to that of the eddy turn over time scale denoted as ℓΩ is known as the Zeman scale, [152]. Thus this gives rise to two inertial ranges in the scales smaller than the forcing scale, one in which the flow is affected by rotation ℓ ℓ ℓ and scales that fluctuate fast f ≫ ≫ Ω enough to not see the effect of rotation ℓ ℓ ℓ . To study these scales would Ω ≫ ≫ ν require very large experimental set-ups and large scale numerical simulations, as one requires a large scale separation between ℓf , ℓΩ, ℓν. For scales larger than the forcing scale, at low enough Ro we know that the large scales start to grow due to the inverse cascade. Most of the energy injected at the forcing scale goes to large scales which needs to be dissipated to attain a statistically steady state. One possible mechanism of dissipation is through the large scale friction due to the formation of Ekman layer in the presence of solid boundaries [49, 153, 83

154, 155, 50]. The Ekman layers are thin layers located near the solid boundary whose thickness scales like 1/Ω1/2. If the Ekman friction is strong enough, the inverse cascade does not lead to the formation of a condensate and most of the energy is dissipated at the largest scale by a linear drag term [156, 157, 135, 143]. For laminar boundary layers, the Ekman friction coefficient is proportional to Ω1/2. This large scale friction is reminiscent of quasi 2D turbulence in thin layer flows where the vertical confinement creates a large scale dissipation modelled as a bulk linear drag term. Such a suppression of the large scale condensate in rotating turbulence is observed in rotating experiments in large aspect ratio systems, see [158], where the small vertical confinement also plays a role in the large scale friction. They are also observed in rotating convection [159] when the rotation rates are moderate, (condensate reappears at low enough Ro see [160]). In the absence of Ekman friction the inverse cascade leads to the formation of a large scale condensate. Such studies model systems where the solid boundaries are far from the bulk of the flow or when confinement effects are weak. The formation of condensates have been reported in many forced rotating turbulence studies, see [161, 143]. The inverse cascade can then saturate to a statistically steady state by two different mechanisms, which have been reported on different types of system. We denote them as Mechanism I and Mechanism II. Mechanism I, one possible way of saturating the condensate is through dissipation at the largest mode by viscosity. Here all the energy is dissipated by viscosity at the largest coherent vortex. The amplitude of this large scale vortex/condensate scales like, U Re1/3 and it grows as we increase ∝ f viscosity, see [162, 163]. Mechanism II, a second way to saturate the condensate is by the formation of strong counter rotating vortices which locally cancels global rotation and cascades energy to small scales. In this case the amplitude of the condensate scales like U ΩL making the local Rosby number Ro O(1). Such kind of saturation have ∼ ∼ been observed in other studies [149, 164]. In both the scenarios the large scale condensate leads to the reduction of energy injection due to the sweeping effet. Here the mean flow due to the condensate advects the forcing scale vortices making them decorrelate from the forcing. This decorrelation of the velocity and the forcing leads to a suppression of the energy input. Sweeping effect of large scales leading to decorrelation of small scales have been observed in many systems, see [165, 166, 167, 162, 163, 168]. Here we report the observation of different types of saturation obtained in the case of a constant forcing which is independent along the direction of rotation. We denote f0 the amplitude of the external forcing and U the r.m.s value of the velocity field. kf denotes the forcing wavenumber and Ref , Rof denote the Reynolds and Rosby number based on the forcing amplitude. The figure IV.1 shows a sketch of 2 the normalized energy U kf /f0 as a function of the Rosby Rof for different regimes c obtained in the system under study. For large Rof > Rof we see that the system does not cascade any net inverse cascade to large scales, the energy follows standard 3D 2 c Kolmogorov turbulence with U kf /f0 O(1), [14]. For Rof < Rof we start to see ∼ c the large scales grow due to the inverse cascade, the vertical line at Rof denotes this transition. The transition is smooth at low Re and approaches a critical behaviour at large enough Re. The critical behaviour implying that the normalized energy scales c like a powerlaw of the distance from the threshold Rof . As we reduce Rof for large 84 Chapter IV. 3D Rotating flows and dynamo instability

Re >Re Re >Re 4 3 > 2 1 Mechanism II

Re 4

Re 3 Mechanism I Re 2

condensate no condensate

2 Figure IV.1 – Figure shows an illustration of the normalized energy U kf /f0 as a function of Rof . The dashed vertical line shows the transition from a flow with no inverse cascade to a flow with an inverse cacade. The circle shaded region corresponds 2 to the transition regime. The dashed curved line shows the scaling 1/Rof . The different colored lines denote different values of Re.

2 fixed Re, we see the appearance of 1/Rof scaling which corresponds to the saturation when the anticyclonic vortex is large enough to locally cancel the effect of global rotation (Mechanism II). The anti-cyclonic vortex will cascade energy to small scales, its amplitude is given U 2 Ω2L2. For a fixed Re at very small Ro , we see that U 2 ∼ f becomes independent of Rof , as the saturation now comes from the viscous dissipation of the largest mode (Mechanism I). This is because the co-rotating vortex amplitude increases to a value where the condensate amplitude becomes large enough that viscous dissipation saturates the inverse cascade. The amplitude of the condensate is given as U 2 (f 2L2/(k ν))2/3. This scaling law comes from taking into account sweeping, ∼ 0 f ǫ f 2/(Uk ) and dissipation at the largest scale ǫ νU 2/L2. Here ǫ denotes energy ∼ 0 f ∼ injection/dissipation rate and f the amplitude of the forcing. The regimes of the two different types of saturation mechanisms is controlled by the Ref , Rof and forcing length scale kf L of the system. In this study we are interesting in looking at the phase space picture of rotating flows under the control parameters. We have two control parameters in the system Ref , Rof and we look at the steady saturated turbulence regime as we change these two control parameters. Both the helical and the nonhelical cases are studied separately. Some of the main questions we try to answer in this study are,

1. How the transition from forward to inverse cascade occur and how it behaves as a function of Ro and Re ?

2. Whether the presence or absence of helicity is of importance in steady rotating turbulent ﬂow ? IV.1. Parameter space 85

3. What are the asymptotic regimes of weak and fast rotating ﬂows at large values of Re ?

4. What are the diﬀerent saturation mechanisms of the condensate ?, which regime of parameters do they occur ?, and what is the scaling of the amplitude of the condensate in each regime ?

Figure IV.1 already gives a sketch of the results, we proceed to describe the system and look into each question in detail.

IV.1 Parameter space

The governing equation for a ﬂuid element in the rotating reference frame is written as,

1 ∂ u + u ∇u = ∇p + ν∆u 2Ω u + f. (IV.1.1) t · −ρ − ×

For the forcing term f we consider two types of forcing, one with mean helicity (helical forcing) and the one with zero helicity (nonhelical forcing). The helical forcing is fh = f0 (cos (kf y) , sin (kf x) , cos (kf x) + sin (kf y)), and the nonhelical forcing is fnh = f0 (cos (kf y) , sin (kf x) , sin (kf x) + cos (kf y)). The parameter f0 controls the amplitude of the forcing, kf controls the wavenumber at which energy is injected into the flow. The Coriolis term on the right 2Ω u modifies the underlying flow due × to the global rotation. The control parameters in this system of equations are, the

Reynolds number based on the forcing amplitude as Ref = f0/kf /(kf ν), the Rosby q number based on the forcing amplitude Rof = f0/kf kf /(2Ω), the forcing wavenumber kf L. Thus for the problem of rotating homogeneousq turbulence, we have three control parameters which can be varied. We show in figure IV.2 the set of numerical data points that have been done for a) the Helical Roberts flow, b) the Nonhelical Roberts flow on the log-log Ro Re plane. f − f The figure shows symbols that correspond to simulations that lead to hydrodynamic steady states. Darker symbols corresponding to larger values of Rof . Larger symbols correspond to larger values of Ref , the largest symbols correspond to simulation runs of size 5123 points. Different symbols correspond to different behaviour of the flow, corresponds to flows that are steady laminar, N corresponds to chaotic behaviour of the velocity field close to the laminar-turbulent threshold, corresponds to turbulent flow, • corresponds to flows that have condensate arising from the inverse cascade of energy. In figure IV.2, we have shifted the points corresponding to Ω = 0, Ro = to the f ∞ values Rof = 100, in order for them to appear along with other points that correspond to the finite rotation limit. The symbols denote hyperviscous runs which model the limit Ref and are obtained when we replace the laplacian in the equation (IV.1.1) → ∞ 4 with hyperdissipation ∆ . The hyperviscosity runs represent a large Ref value, but in order for them to appear in the same figure, they are shifted to a value Ref = 1000. The symbols ⋆ denote the simulations done using the reduced quasi-twodimensional of the governing equation at Ro 0. The governing equations for this reduced model f → 86 Chapter IV. 3D Rotating flows and dynamo instability

102 102

101 101

f 100 f 100 Ro Ro

10−1 10−1

10−2 10−2 100 101 102 100 101 102 Ref Ref (a) (b)

Figure IV.2 – The figures show Rof as a function of Ref for all the examined numerical runs for a) the case of helical flow and b) the case of the nonhelical flow. Larger symbols denote larger values of Ref and darker symbols correspond to larger values of Rof . Different symbols correspond to different behaviour of the flow. are,

1 ∂ u + u ∇u = ∇p + ν∆u + f, (IV.1.2) t 2D 2D · 2D −ρ 2D 2D

where the subscript 2D denotes independent of z. Note u2D = (ux, uy, uz)(x, y) has all three compoenents of the velocity field. Here the effect of infinite rotation is that the velocity field in this reduced model is independent of z. The quasi-2D simulations 3 (Rof = 0 limit) is placed at the position Rof = 10− to appear in the same figure. This quasi-2D simulations are similar to studies in Chapters II,III. The two different forcing mechanisms seem to have a similar effect on the range of values of Ref , Rof that have been examined. Meaning similar behaviour is obtained for the same parameters, whether in the presence or absence of mean helicity. For small values of Ref the flow is laminar. The base flow for both the helical and the nonhelical forcing can be written as,

1 u = 2 fh, (IV.1.3) νkf 1 1 u f = 2 nh + O( 3 ). (IV.1.4) νkf ν

The laminar ﬂow for the helical forcing is exactly proportional to the forcing, since the nonlinear term is exactly zero. This is because the helical forcing used here has the following properties, ∇ f = k f and ∇ f = 0. These imply that the nonlinear × h f h · h term u ∇u = u (∇ u) + ∇(u2/2) is zero and the laminar ﬂow is proportional · − × × to the forcing. This is not true for the nonhelical forcing since ∇ f = f , which × nh 6 nh implies that the nonlinear term would contribute to higher order correction in the 1/ν expansion. IV.1. Parameter space 87

As we increase the value of Ref , the flow undergoes a transition to turbulence though a linear instability. The base laminar flow is two-dimensional, and we can decompose the unstable modes into Floquet modes along the z-direction eikzz. We find that the most unstable mode is actually kz = 0, i.e. the two-dimensional base flow is unstable to two-dimensional perturbations. The instability threshold is found to be at Rec 1.278 for both the flows and is independent of Ω. For the values of Re close f ≈ f to the onset of the instability, denoted by N points in figure IV.2, the flow is neither laminar or fully turbulent. At Re > Rec we have turbulence denoted by the symbols . For small values of Ro < 1 and large enough values of Re , the flow becomes • f f quasi-twodimensional with the cascade of energy going to large scales. Condensates are formed where the energy is accumulated at the large scales and for such behaviour we use the symbols . We look closely at the different behaviour of the flows as we fix either Ref or Rof and vary the other. Figure IV.3 shows the time series of the total energy u 2 = | | 1/(2πL)3 u u dxdydz. Figure IV.3a shows the different simulation runs for the · nonhelicalRRR flow for a fixed Ref = 100 and different values of Rof as mentioned in the legend. For weak rotation rates Ro 1, we see that the total energy u 2 behaves f ≥ | | like standard 3D turbulent flow where u 2 f /k . As we reduce Ro < 1 we | | ∼ 0 f f start to see the formation of condensates, the flow has an initial growth of u 2 due | | to the inverse cascade, and then it saturates. For a description on the growth phase of the large scales due to the inverse cascade see [169]. For large enough Ref , the amplitude of the energy increases as we increase the global rotation. We had identify the condensate regimes apart from the turbulent flows in figure IV.2 by looking at the difference in the saturation amplitude of u 2 . | | In figure IV.3b, we show different simulation runs of the helical forcing case for a fixed Rof = 0.2 and different values of Ref as mentioned in the legend. As mentioned previously there is a transition from the laminar flow at a value Re 1.278. The value f ≈ Ref = 2.5 corresponds to the chaotic behaviour N points, the velocity field has phases of steady and fluctuating behaviour in time. Since the Rof is small, for sufficiently large Ref we have the development of an inverse cascade of energy and the energy gets accumulated at large scales. The kinetic energy u 2 increases as we increase Re . | | f We now move on to draw out the relations between other quantities namely, Rev, Rov, Red, Rod for the points in figures IV.2. We show in figures IV.4 Rov as a function of Rev for both the helical and the nonhelical forcing. The laminar and turbulent results are shifted by the value of U which depends only on Ref with a weak dependence on the rotation rate Rof . While for the condensates there is a jump from the turbulent state as U increases from the inverse cascade. Each point in the figures IV.4 can be mapped directly to the figures IV.2, the shift in the points is proportional to the value U, which changes as we change Ref , Rof . The hyperviscous runs denoted by are placed at Rev = 5000. The quasi-twodimensional results denoted by ⋆ are placed at Rov = 0.008. The reduced models follow the behaviour of the DNS results. For 3D nonrotating turbulent flows we can expect that the velocity field scales 1/2 like U f 1/2k− and does not vary much with the Re, as is seen from the weakly ∼ f rotating flows (large Rov) in figure IV.4. As we increase rotation, we start to see that the velocity field increases because of the inverse energy cascade and there is a jump in the value of Re at about Ω 2. This is seen from the deviation in the alignment of v ∼ 88 Chapter IV. 3D Rotating flows and dynamo instability

102 102 Rof =∞ Rof =0.333

Rof =1 Rof =0.2 1 Rof =0.5 10

E 1 E 0 2 10 2 10 | | u u | | D D

10−1 Ref =1.25 Ref =50 Ref =2.5 Ref =100

Ref =12.5 100 10−2 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 t t (a) (b)

Figure IV.3 – The figures show the time series of the total spatial averaged energy u 2 as a function of time. The figure a) shows runs at different values of Ro with | | f Re f = 100 for the case of helical flow. The figure b) shows different values of Ref with fixed Rof = 0.2 for the case of the nonhelical flow.

102 102

101 101 v v 100 100 Ro Ro

10−1 10−1

10−2 10−2 100 101 102 103 104 100 101 102 103 104 Rev Rev (a) (b)

Figure IV.4 – The figures show Rov as a function of Rev for the examined numerical runs for a) the case of helical flow and b) the case of the nonhelical flow. Larger symbols denote larger values of Ref and darker symbols correspond to larger values of Rof . Different symbols correspond to different behaviour of the flow. IV.1. Parameter space 89 data points as compared to figure IV.2. For small values of Reu and as we reduce Rou, we see that there is a small deviation both horizontally and vertically due to the higher value of U. This difference becomes larger as we increase Reu, giving a large difference between the turbulent state and the condensate state. For the values of parameters studied here the Ro Re parameter space seems to have a one-to-one mapping with v − v Ro Re parameter space. f − f We next show in figures IV.5 Rod as a function of Red for the helical and nonhelical forcing. The phase space of the non-dimensional numbers based on the dissipation/injection energy rate. The laminar and the turbulent data points follow the

102 102

101 101

d 100 d 100 Ro Ro

10−1 10−1

10−2 10−2

100 101 102 100 101 102 Red Red (a) (b)

Figure IV.5 – The figures show Rod as a function of Red for the examined numerical runs for a) the case of helical flow and b) the case of the nonhelical flow. Larger symbols denote larger values of Red and darker symbols correspond to larger values of Rod. Different symbols correspond to different behaviour of the flow. expected behaviour with little dependence on Rod. The turbulent flow dissipation energy due to the forward energy cascade to small scales where the dissipation is given 3 by uf kf where uf is the amplitude of the velocity field at the injection length scale kf . For small rotation rates this the dissipation rate in the turbulent flow is independent of Ω. As we increase the rotation rate the data points start to move to smaller dissipation rates. This is due to the formation of the condensate which sweeps the forcing length scales to decrease the energy injection. Also the fluctuations along the direction of rotation (z) gets suppressed leading to a partial two-dimensionalization of the turbulent flow. In the figure IV.5, the condensate states have a much lower dissipation as compared to the turbulent states for the same amplitude of the forcing. Again for the parameters studied here each point can be mapped to the figures IV.2, IV.4. The hyperviscous points denoted by are placed at Red = 500 and the quasi-twodimensional simulations denoted by ⋆ are placed at Rod = 0.005 for comparison.

IV.1.1 Transition to the condensate In the previous section we have studied the transition from the laminar flow to the turbulent flow through a linear instability. We now look at the transition from a 90 Chapter IV. 3D Rotating flows and dynamo instability turbulent flow to a condensate regime as we increase the rotation rate. We look at both rotating flows under study, the helical and the nonhelical forced Roberts flow to see the effects of helicity on the transition. It is important to note that the both helical and nonhelical Roberts flow are independent of z direction and in the limit of very small Ro will admit to a quasi 2D flow in periodic boundary conditions. Thus there seems to be a transition from a flow that is cascading energy to small scales to a flow that starts to form a large scale condensate. The flow here transitions from one state of a turbulent flow to another as we increase the control parameter Rof . Most of the rotating flows that have been studied looked at a limited range of control parameters and no clear picture on the type of transition was obtained. Such phenomena exists in other systems that have been studied such as flows in thin layers, flows in the presence of rotation and/or stratification, 2D MHD systems, a three dimensional flow subject to a strong magnetic field, using numerical, experimental and observations [170, 171, 172, 173, 174, 175, 176, 177, 161, 178, 179, 180, 181, 182, 183, 184, 185, 186]. A few of these studies showed that the transition from a forward cascade to an inverse cascade occurs through a phase transition like phenomena [182, 183, 185, 186]. In these studies, a critical value of the control parameter was found where the rate of energy flux cascading to large scales is zero. Close to this critical value of the parameter, the inverse energy flux scales like a power law of the distance of the control parameter to the critical point. This was found to be quite robust in the limit of very large domain sizes (here kf L). The critical point in the systems examined was found to be a function of the dissipation at the small scales.

30 30 10 10 25 8 25 8 6 6 20 4 20 4 2 2 D D 2 2 2 15 0 2 15 0

U 0.4 0.6 0.8 U 0.4 0.6 0.8

Ref =100.0 Ref =100.0 10 10 Ref =50.0 Ref =50.0

Ref =25.0 Ref =25.0 5 5 Ref → ∞ Ref → ∞

Rof → 0 Rof → 0 0 0 10−3 10−2 10−1 100 101 102 103 10−3 10−2 10−1 100 101 102 103 Rof Rof (a) (b)

2 Figure IV.6 – The figures show the energy at the large scales U2D as a function of Rof for different values of Ref as mentioned in the legend. The figures correspond to, a) the case of helical flow and b) the case of the nonhelical flow. The insets show the same parameters in a linear-linear plot zoomed near the value Rof = 0.6.

In this study we look at such a transition for a fully 3D flow subject to global rotation for both the helical and nonhelical forcing. In order to calculate the amount of energy accumulated at the large scales due to the inverse cascade, we look at the 2 energy at the largest 2D mode denoted as U2D . We show in figures IV.6 the quantity 2 U2D as a function of Rof for different values of Re as mentioned in the legend for a) IV.1. Parameter space 91 the helical flow, b) the nonhelical flow. We see that close to Ro 0.5 there is a f ∼ transition from a turbulent state to a condensate state. This is seen to be independent of the presence of helicity in the flow. The inset in the figures show the transition more clearly in a linear-linear plot zoomed close to the transition. For the largest Ref , the transition is found close to the Rosby number Ro 0.6. f ≃ For low values of Ref , the transition from the turbulent flow to the condensate is smooth. For a finite Ref and a small Rof , the large scale condensate dissipates all the energy at the largest mode, this corresponds to Mechanism I. Here the velocity field is expected to scale like U 2k /f Re2/3 (k L)4/3, becoming independent of Ro . f 0 ∝ f f f The quasi-2D simulations denoted by the points ⋆ follow this scaling and dissipates the energy through Mechanism I. As we increase the value of Ref , this transition 2 becomes sharper and the value of U2D moves to larger values. This happens until it reaches the scaling U ΩL which corresponds to the Mechanism II mentioned in the 2D ∼ Introduction. The limit of large Re is modelled by the hyper viscosity runs and f → ∞ they are shown by black hexagons in the figures IV.6. The low Rof hyperviscous runs follows Mechanism II, the dissipation of energy occurring due to the forward cascade formed by the contrarotating vortex. We mention here that in previous studies of forced rotating turbulence a particular type of forcing, known as the Taylor-Green forcing, leads to a hysteresis behaviour at large enough Ref , see [149, 164]. The transition observed here is more closer to the observed results of [182, 183, 185, 186], where the control parameter scales like a power law of the distance from the threshold. In both the previous studies of rotating turbulence [149, 164], the forcing wavenumber was kf L = 2 and the Taylor-Green forcing depends on z-direction with its projection on the 2D plane (the kz = 0 mode) is zero. This could explain why in the current study we do not observe any hysterical behaviour (or why they do not observe a critical transition). Though figure IV.6 gives the picture of the transition in terms of Ref , Rof , one should ideally also look at the limit of large kf L. So that the box size is large enough to reach the thermodynamic limit in the inverse cascade regime as was done in previous studies of [182, 183, 185, 186]. The dependence of the transition to the condensate regime on kf L is yet to be studied and we leave it for future studies. We mention that in the studies of [182, 183, 185, 186], there was a large scale dissipation used to saturate the inverse cascade. Here, even in the absence of the large scale dissipation we get a critical behaviour for the parameter range explored here. For large values of Ref , we start to see large oscillations in the condensate amplitudes, also seen from the large error bars in figures IV.6a,IV.6b. In figures IV.7, we show the time series of the quantity U 2 and the square of the amplitude of the 2 condensate U2D as a function of time for the nonhelical forcing case. The figure IV.7a shows them for a value Ref = 100, Rof = 0.556 which is close to the threshold at which the transition to the condensate occurs (see figure IV.6b). The insets in the figures correspond to contours of vertical vorticity ω = e (∇ u). They are placed close to z z · × vertical dashed lines which correspond to the time instance at which the vorticity field was computed. The time series shows that the flow oscillates between a condensate regime and a normal 3D turbulent regime. This is seen in the insets. The inset at time t 1491 shows one corotating vortex surrounded by the turbulent flow. The ∼ inset at time t 1929 does not have any large scale vortex and behaves closer to a 3D ∼ 92 Chapter IV. 3D Rotating flows and dynamo instability

9 8 U 2 U 2 7 2D 6 5 4 3 2 1 0 0 500 1000 1500 2000 2500 t

(a) 25 U 2 U 2 20 2D

0 0 500 1000 1500 2000 t

(b)

Figure IV.7 – The figures show the time series of the total energy U 2 and the square 2 of the amplitude of the large condensates U2D corresponding to, a) Ref = 100, Rof = 0.556 close to the threshold and b) Ref = 100, Rof = 0.357 after the transition (far 2 from the threshold). The darker shade of blue corresponds to U2D . The insets show the vertical vorticity ωz contours at times corresponding to the dashed vertical black lines. IV.2. Asymptotic limits 93 turbulent flow. Thus the flow oscillates between two bistable states and the amount of time spent in the condensate state increases as we approach the threshold. This is different from the hysterical nature of the Taylor-Green forcing [149, 164]. More studies need to be done in order to understand the time spent in the two bistable states as we approach the transition. 2 2 The figure IV.7b shows the time series of the two quantities U ,U2D for a value Ref = 100, Rof = 0.357 which is slightly far from the transition. As in figure IV.7a, we see large fluctuations in the velocity field which correlate with the fluctuations in 2 the two dimensional condensate U2D . At these values of the parameters the flow is 2 in a condensate state given by the large value of U2D . The bursts that we observe correspond to larger amplitudes of condensates. From the three insets shown at times t 796, 1351, 1727 we cannot conclude much about their spatial structure. ∼

IV.2 Asymptotic limits

We have in the previous sections, looked at the transitions between regimes of laminar (Re < Rec) to turbulence (Re > Rec) and slow rotating ﬂows (Ro > 1) to fast rotating ﬂows (Ro < 1) (also the transition to the condensate regime). In this section we look

102 102 −2 −2 Rof Rof

101 101 2 2 U U

100 100

10−3 10−2 10−1 100 101 102 10−3 10−2 10−1 100 101 102 Rof Rof (a) (b)

2 Figure IV.8 – The figures show the energy U as a function of Rof for the case of a) helical forcing and b) nonhelical forcing. Larger symbols denote larger values of Ref and darker symbols correspond to larger values of Rof . Different symbols correspond to different behaviour of the flow. at the behaviour of global quantities in the limits of large Re and low Ro 0 → ∞ → flows. 2 We first look at the energy U shown in figure IV.8, as a function of Rof for both types forcing. We have shown the Ro = 0 case by ⋆ points and the case of hyperviscous runs by points. For small values of Ref , when viscosity is dominant, we get the 2 2 2 2 laminar scaling U f0 /(νkf ) . The behaviour at large Ref values depends on Rof . ∼ 2 For Rof > 1 and large Ref , the value U becomes independent of Ref . This is the standard 3D turbulent flow where the total energy U 2 can be given by U 2 f /k . ∝ 0 f 94 Chapter IV. 3D Rotating flows and dynamo instability

This is well seen from the asymptote to the hyperviscous simulation results, which are 2 shown by the symbols . For small values of Rof < 1, we see that the energy U increases due to the inverse cascade to large scales. For values of Rof close to the 2 2 2 2 threshold Rof 0.5, the quantity U brieﬂy follows a scaling U Ω L . This scaling ∼ 2 ∝ is shown by the dashed line which denotes the scaling Rof− . The independence of Ref is also seen in the asymptote to the hyperviscous simulations which are denoted by the symbols.

Rof = 0.02 Rof = 0.02 102 102 Rof → 0 1 Rof → 0 1 Ref Ref

101 101 2 2 U U

100 100

10−1 10−1 100 101 102 103 100 101 102 103 Ref Ref (a) (b)

2 Figure IV.9 – The ﬁgures show the total energy U as a function of Ref for the case 1 of a) helical forcing and b) nonhelical forcing. The black line denotes the scaling Ref .

2 For a given Ref and as Rof is decreased, the energy U saturates at a value that 2 depends on Ref . Figures IV.9 show the energy U as a function of Ref for two smallest 1 values of Rof . The black solid line shows the scaling Ref . The data points do not follow the scaling of U 2k /f Re2/3 (k L)4/3 predicted for mechanism I. Here we f 0 ∝ f f find U 2k /f Re . Such a scaling could be achieved when the injection energy stays f 0 ∝ f constant, implying the balance between dissipation at the largest scale with the energy 2 2 3/2 1/2 injection νU /L f0 /kf . We will see that the sweeping effect saturates at large 2 ∼ Ref or large U , possibly due to insufficient scale separation between the forcing scale and the box size. More studies need to be done with large kf L to verify whether we 2 2/3 4/3 get the scaling U kf /f0 Ref (kf L) . ∝2 2 Figure IV.10 shows U2D Rof as a function of Rof for the case of a) helical and b) nonhelical forcing. The figure uses the same symbols of figure IV.6. As seen from the figure the scaling predicted by mechanism II of U 2 Ω2 is possibly seen only for a few ∼ values of Rof . In order to confirm this scaling over a larger range of Rof , one would have to do numerics at large Ref . We look at the injection/dissipation rate as a function of Rof in figure IV.11 for both the helical and the nonhelical forcing. At low Ref , the laminar scaling follows ǫ f 3/(ν5k5) where as we increase the viscosity ν, the dissipation increases. Since the ∝ 0 f laminar flow is two-dimensional, it is not affected by rotation and ǫ is independent of Rof . This happens until the transition to the turbulent state occurs. In the large Ref and weak rotation case we have the scaling ǫ f 3/2/k1/2 becoming independent of ν. ∼ f IV.2. Asymptotic limits 95

3.0 3.0 Ref =100.0 Ref =100.0 2.5 Ref =50.0 2.5 Ref =50.0 Ref =25.0 Ref =25.0 2.0 Ref → ∞ 2.0 Ref → ∞

2 f Rof → 0 2 f Rof → 0

Ro 1.5 Ro 1.5 D D 2 2 2 2

U 1.0 U 1.0

0.5 0.5

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Rof Rof (a) (b)

2 2 Figure IV.10 – The ﬁgures show the rescaled energy of the condensate U2D Rof as a function of Rof for the case of a) helical forcing and b) nonhelical forcing. The black 0 dashed line denotes the scaling Rof .

100 100 ǫ ǫ

10−1 10−1

10−3 10−2 10−1 100 101 102 10−3 10−2 10−1 100 101 102 Rof Rof (a) (b)

Figure IV.11 – The figures show the energy injection/dissipation rate ǫ as a function of Rof for the case of a) helical forcing and b) nonhelical forcing. Larger symbols denote larger values of Ref and darker symbols correspond to larger values of Rof . Different symbols correspond to different behaviour of the flow. 96 Chapter IV. 3D Rotating flows and dynamo instability

We note that the turbulent dissipation rate ǫ for the helical and the nonhelical flows are slightly different. The helical flow is more efficient in drawing more energy from the forcing as compared to the nonhelical flow, for the same parameters of the system 2 (Ref ). Figure IV.12a shows the energy U as a function of time for the helical and the nonhelical forcing. Similarly, figure IV.12b shows the dissipation rate ǫ as a function of time. These figures clearly show that the total energy and the dissipation rate are higher in the case of helical flow for the same set of parameters, in the non-rotating problem Ω = 0.

3.5 1.2

1.0 3.0 0.8

2.5 0.6 2 ǫ U 2.0 0.4 0.2 1.5 helical helical 0.0 nonhelical nonhelical 1.0 −0.2 0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2 t ×103 t ×103 (a) (b)

Figure IV.12 – The ﬁgures show the time series of the total energy U 2 and the dissipation/injection rate of energy ǫ as a function of time t for the case of a) helical forcing and b) nonhelical forcing. The darker line corresponds to the nonhelical forcing case.

Going back to figure IV.11, as Ω increases we reach the quasi-twodimensional scaling where the injected energy becomes smaller as we increase Ref due to sweeping effect, as discussed previously for figure IV.8. The sweeping effect saturates at the largest Re with no substantial reduction below ǫ 10 1, this is possibly due to lim- f ≃ − ited scale separation between the forcing scale and the box size. In order to verify this, a more detailed study with varying scale separation is required. 3 A nondimensional dissipation rate can be constructed from ǫ and U as ǫ/(kf U ). 3 Figure IV.13 shows ǫ/(kf U ) as a function of Rev for both helical and nonhelical forcing. Low Rev values correspond to the laminar flow where viscosity dominates, we get the scaling for the laminar solution where the normalized dissipation scales like 1 Rev− . As we reduce the viscosity (increasing Rev) we see two different branches that separate, one branch consists of the turbulent data points and the other consists of • condensates . The turbulent regime corresponds to Rof , Rov > 1 where the effect 3 of rotation is weak. ǫ/(U kf ) becomes independent of the Rev as expected from the 0 Kolmogorov theory. The black solid lines help us identify the scaling Rev in the large Re regime. The turbulent data points asymptote the hyperviscous points v • representing the limit Re . f → ∞ The second branch consisting of condensate points, depends on both Rof and Ref values. We see two behaviours given by the two different saturation mechanisms. For IV.2. Asymptotic limits 97

100 100 −1 −1 Rev Re0 Rev v Re0 10−1 10−1 v ) ) 3 3

U −2 U −2

f 10 f 10 k k ( ( ǫ/ ǫ/ 10−3 10−3

−4 −4 10 −1 10 −1 Rev Rev 100 101 102 103 104 105 100 101 102 103 104 105 Rev Rev (a) (b)

3 Figure IV.13 – The figures show the normalized dissipation ǫ/(kf U ) as a function of Rov for the case of a) helical and b) nonhelical forcing. Larger symbols denote larger values of Ref and darker symbols correspond to larger values of Rof . Different symbols correspond to different behaviour of the flow. The black thick lines help to identify different scaling laws. Red dashed lines connect the same Rof points. Three different red dashed lines connect the data points of Ω = 1, Rof = 1.0, Ω = 2, Rof = 0.5 and Ω = 3, Rof = 0.333.

a given Ref , increasing the effect of rotation (or lower values of Rof ), we reach the 1 viscous scaling of Rev− . Here the condensate saturates from the viscous dissipation of the large scale condensate. This corresponds to mechanism I, where the normalized dissipation rate follows ǫ/(k U 3) Re 1. This scaling is also followed by the points f ∼ v− ⋆ corresponding to the quasi-twodimensional simulations. For intermediate values of 0 Rof we see the data points occupying the region between the two branches Rev and 1 Rev− . For a given value of Rof we find that as we increase the inertial range (increasing Rev), the dissipation at the viscous scales becomes more important than the dissipation from the condensate. This is due to formation of the contra-rotating vortex cascading 0 energy to small scales, corresponds to mechanism II. This leads to a scaling of Rev, like the turbulent scaling for a fixed Rof and large enough Rev. Three different lines corresponding to fixed Rof values of 1, 0.5, 0.333 are denoted by dashed red lines. They 1 initially follow a scaling close to Rev− for moderate values of Rev, but at large values of Rev they become independent of Rev. Thus the two limiting scenarios leads to different types of saturation mechanisms, one which has Re with Ro finite f → ∞ f corresponding to the hyperviscous simulations leads to saturation by Mechanism II. The other has Ro with Re finite corresponding to the quasi-twodimensional f → ∞ f simulations leads to saturation by Mechanism I. 3 We show in figure IV.14 the normalized dissipation rate ǫ/(kf U ) as a function of Rov for the case of a) helical forcing, b) nonhelical forcing. For weak rotation rates (Ro 1), as we increase Re we see that the normalized dissipation reduces until v ≫ f it reaches the large turbulent limit, where it becomes independent of the Ref . For large rotation rates Rov < 1, the normalized dissipation rate decreases as we increase the Ref (due to larger U and smaller ǫ). For large enough Ref and a finite Rof , 98 Chapter IV. 3D Rotating flows and dynamo instability

100 100

10−1 10−1

) −2 ) −2

3 10 3 10 U U f f k k ( 10−3 ( 10−3 ǫ/ ǫ/

10−4 10−4

10−5 10−5 10−2 10−1 100 101 102 10−2 10−1 100 101 102 Rov Rov (a) (b)

3 Figure IV.14 – The figures show the normalized dissipation ǫ/(kf U ) as a function of Rov for the case of a) helical forcing and b) nonhelical forcing. Larger symbols denote larger values of Ref and darker symbols correspond to larger values of Rof . Different symbols correspond to different behaviour of the flow. The vertical dotted lines correspond to the value Rov = 3 around which the condensates lie.

the condensates fall around the vertical dotted line at the value Rov = 3. This is the saturation mechanism II due to the contra rotating vortex cancelling the effect of global rotation at a value Ro 1. As we increase the Re , the condensate points try v ∼ v to approach this vertical dashed line. Finally, we show the spectra of the velocity field E(k) as a function of k in figure IV.15. The figure IV.15a shows the spectra for the nonrotating case with Rof = , Ref = 200 for both helical and the nonhelical forcing. The dotted line with the ∞ 5/3 scaling k− is the 3D Kolmogorov spectra for comparison. Figure IV.15b shows the spectra for the fast rotating case with Rof = 0.02, Ref = 200 for both the types of 2 forcing. The dotted line represents the scaling k− for comparison. The fast rotating case shows the large scale condensate due to the inverse cascade of energy, the energy being concentrated at the k = 1 mode.

IV.3 Conclusion - Part 1

In the first part of this Chapter we have studied the effect of global rotation on fluid turbulence. We have examined the different flow regimes that arise in the large phase space of Rosby number and Reynolds number. The limiting cases of small Ro limit has been studied using the 2.5D model. While the large Re has been studied using hyperviscous models. These models help us to extend the phase space behaviour as simulations of extreme values of nondimensional numbers cannot be simulated with the existing computational power. In this study we have looked at rotating flows under constant in time forcing with both helical and nonhelical configurations that are z independent. Irrespective of the existence of mean helicity of the forcing or not we end up with a similar phase space behaviour of the resulting flow for the control parameters Rof , Ref . This is possibly IV.3. Conclusion - Part 1 99

102 0 10 101 100 10−1 −1 ) ) 10 k k ( 10−2 ( −2 E E 10 k−5/3 10−3 −3 −2 10 helical helical k 10−4 nonhelical nonhelical 10−4 10−5 100 101 102 100 101 102 k k (a) (b)

Figure IV.15 – The figures show the spectra of energy E(k) as a function of k for the case of a) the nonrotating case with Ro = , Re = 200 and b) fast rotating case f ∞ f with Rof = 0.02, Ref = 200. The lighter shade corresponds to the helical case. The dotted lines represent power laws for comparison. due to the z-independence of the forcing which lets the presence or absence of helicity decouple from global rotation. We identify four regimes laminar, chaotic, turbulent and condensate states in the different regions of the phase space. The region of their existence is quite different from the case of the Taylor Green forcing [149] and is also attributed to the z-independence of the forcing. We find the transition to turbulence given by a linear instability of the underlying quasi-twodimensional laminar flow to perturbations that are also 2D.

At large enough Ref we find turbulent flow with a forward cascade of energy, which in the small Rof starts to cascade energy to large scales. The transition to the condensate regime takes place at a value of Ro 0.5. As we increase the Re , f ∼ f the transition becomes more sharper with a possible indication that in the limit of very large Ref it leads to a second order critical phase transition like behaviour. At large Ref close to the transition, we see oscillations where some duration of time the energy cascades to large scales, while in other instances of time it cascades energy to small scales. This is again in contrast to previous results of Taylor-Green forcing in [149, 164] where a subcritical behaviour was observed. The differences with that study are attributed to the z-dependence of the forcing and the low forcing wavenumber kf L = 2 used in their study of the TG flow. In order to understand the limiting case of Ref , much larger simulations need to be carried out in order to confirm with the hyperviscous predictions. Also the nature of the transition should also be looked at when we increase the box size or the forcing length scale kf L, as have been done in previous studies, to look at the thermodynamic limit.

At large values of Ref and small values of Rof we see two different types of saturation as mention in figure IV.1. The first mechanism (mechanism I) corresponds to the energy being dissipated at the largest scale of the system due to the formation of a large vortex. This large vortex dissipates energy by viscosity and saturates the inverse cascade. The second mechanism (mechanism II) corresponds to the formation 100 Chapter IV. 3D Rotating flows and dynamo instability of the anti-cyclonic vortex which locally cancels the effect of rotation and starts to cascade energy to small scales. We have thus identified these two regimes and their domain of existence in the phase space of Re Ro . Fixed Ro and large Re leads f − f f f to the saturation mechanism II, while fixed Ref and large Rof leads to the saturation mechanism I. In conclusion we note the importance of detailed phase space studies which helps to map out the system over the whole domain of parameters. Using simplified/reduced models (hyperviscous/quasi-twodimensional) we can get an estimate of the asymptotic limits of extreme values of the control parameters of the system (here rotating flows).

IV.4 Rotating dynamos

We now look at the dynamo instability driven by a rotating turbulent flow. We have so far looked at the different types of flows that could arise at different limits of Re, Ro. Each set of Re, Ro will lead to a critical magnetic Reynolds number above which the dynamo instability occurs. In the remaining part of this Chapter we will look at the dynamo instability threshold as one changes the Rosby number Ro and Reynolds number Re. In the limit of very small Ro, we would expect our results to match with the studies in Chapter II where the limit of Ro was modelled by a → ∞ quasi-twodimensional flow. Due to the constraint of computational resources we restrict our study to a few different cases of Ω namely, Ω = 0, 1, 3, 50, for both the helical and the nonhelically ∞ forced flows. The choice of these values comes from the fact that the transition to the condensate occurs for a Ω 2, thus we examine the transition Ω = 1 3. The case ∼ → of Ω = corresponds to the 2.5D case studied previously in Chapter II. The value ∞ Ω = 50 is expected to be similar to the case of the 2.5D flows and finally Ω = 0 is chosen in order to compare with a standard nonrotating flow. As we cross the threshold Ω 2, there is a decrease in the turbulent fluctuations which could possibly reduce the ∼ dynamo instability threshold. In theory we do not know apriori whether the dynamo instability driven by a rotating flow would have a smaller Rm as its threshold. For a given Re we want to understand which of the global rotation rates gives a lower dynamo threshold. The problem of lowering the dynamo threshold is of importance in the dynamo community since experimentally dynamo instability is difficult to observe. In most experiments which have been demonstrated, the dynamo instability have been either flows that are constrained or by using ferromagnetic materials. The difficulty arises due to the low value of the magnetic Prandtl number for liquid metals (P m 10 5), which requires a highly turbulent flow for achieving a nominal value ∼ − of Rm O(1), necessary for having the dynamo instability. A highly turbulent flow ∼ dissipates as the cubic power of the Reynolds number, ǫ Re3 where ǫ denotes the ∝ energy injection rate. This makes it very expensive to do experiments at very large Re. The critical magnetic Reynolds number for a given geometry and a given forcing, depends on the Reynolds number of the flow, Rmc = f(Re). This was studied by [59, 60, 61] where they found that for initial increase of Re above the laminar threshold, the Rmc was found to increase linearly with Re. For large enough Re, the increase in IV.4. Rotating dynamos 101 critical magnetic Reynolds number Rmc saturates and becomes independent of Re at large enough Re. This was also found to be the case for the 2.5D flows in Section II.5. turb We denote the saturated Rmc at large Re by Rmc = limRe Rm. This number →∞ depends on the parameters of how the turbulent flow is forced and on the domain turb (geometry and boundary conditions). So the quantity Rmc is not universal for all kinds of turbulent flow. This is seen quite clearly in the recent study of [62] where the forcing length scale/domain size was varied to find an optimal length scale which reduces the dynamo threshold. In this second part of the Chapter we look at both the rotating dynamo driven by helical and nonhelical forcing. We show that rotation reduces the dynamo instability threshold and the form of the unstable mode changes as we move from the weakly rotating flow to a strongly rotating flow.

IV.4.1 Parameters of the study The governing equations are the Navier Stokes equation with the induction equation written as, 1 ∂ u + u ∇u = ∇p 2Ω u + ν∆u + f, (IV.4.1) t · − ρ − × ∂ B =∇ (u B) + η∆B. (IV.4.2) t × × f is the forcing that sustains the flow. We first look at the nonhelical forcing case, f = f0 (cos(kf y), sin(kf x), cos(kf y) + sin(kf x)) and later on look at the helical forcing case. The simulation is done in a periodic domain of size [2πL, 2πL, 2πL]. As mentioned previously, we are interested in reducing the energy injection of the underlying flow, thus we use the Reynolds numbers based on the energy injection rate ǫ = ν ∇u 2 defined (1/3) (1/3) | (1/|3) as Red = (ǫ/kf ) /(kf ν), Rmd = (ǫ/kf ) /(kf η) and Rod = (ǫ/kf ) kf /(2Ω). 3 4 3 turb The critical injected power can now be defined as ǫc = ρ(2πL) kf η Rmc where turb Rmc = limRed Rmd. In order to reach asymptotically large values of Re , →∞ → ∞ we also use hyperviscosity where the Laplacian of the viscous term in equation (IV.4.1) is replaced by ∆4 (used already in the first part of this Chapter). The other limit we want to reach is the very small Ro 0 limit, where we use the 2.5D approxima- → tion. Such aymptotic models helped us reach extreme parameter regimes to study the hydrodynamic equation (IV.4.1), as seen in the first part of the Chapter. We show in table IV.1 the largest values of the examined parameter for the set of runs used for the normal and the hyperviscous simulations. Figure IV.16 shows 2 3 the normalized energy U /(f0/kf ) and the normalized dissipation rate ǫ/(U kf ) as a function of Red for different values of Ω. The points denoted by ⋆ at Red = 200 denote the hyperviscous results which represent the large Re 1 limit. They are connected d ≫ to the corresponding normal viscous simulations by dotted lines. For low Red we see the laminar scaling and rotation does not affect the quantities since the laminar flow is independent of z. For Re 3 there is a transition to a 3D turbulent state d ∼ and rotation starts to affect the flow quantities. For weak rotation rates the scaling follows standard Kolmogorov theory, where we expect the dissipation and the total energy to become independent of viscosity. We see that the normalized energy and the injection/dissipation rate become constant at large enough Re. The hyperviscous runs 102 Chapter IV. 3D Rotating flows and dynamo instability

Table IV.1 – Numerical parameters of the simulations. For all runs f0 = 1, L = 1 and kf = 4. N notes the grid size. The reported values are for the largest values of Re (regular viscosity), Rmturb is based on the hyperviscous runs. The Ω = corresponds c ∞ to the 2.5D simulations.

Ω Rod Red RoU ReU N 0 210 580 512 ∞ ∞ 1 1 200 3 600 512 3 0.19 110 2.4 1440 512 50 0.011 55 0.18 920 256 0 60 0 950 2048 ∞ give the asymptotic saturation for the weak rotation rates Ω = 0, 1 and they match with the large Red simulations (with normal viscosity). For higher values of Ω 3 we start to see inverse cascade in the system leading ≥ to a much larger value of U 2. The injection of energy starts to reduce as the flow becomes more bi-dimensional and becomes anisotropic with lesser fluctuations along the z direction. The saturation of the condensate growth occurs due to the formation of counter rotating vortex which locally cancels the effect of the global rotation. This leads to the scaling of the velocity of the condensate like U ΩL as explained in the ∼ first part of this Chapter. For the case of the 2.5D flow, corresponding to Ω = , the ∞ dissipation/injection scales linearly with friction ν or inversely with Re. Thus in the limit of large rotation rates we expect a laminar behaviour for the dissipation. This makes it quite difficult to converge the hyperviscous simulations as the condensate amplitude increases to very large values and since the dissipation of the condensate is from viscosity, the hyperviscous simulations become physically irrelevant at very large Ω. So we restrict the hyperviscous simulations for Ω = 0, 1, 3. Mean helicity plays an important role for the large scale dynamo action (the amplification of large scale magnetic fields). The normalized helicity is defined as ρ = u ω /( u2 1/2 ω2 1/2). In figure IV.17, we show the normalized helicity as H h · i a function of time for the nonhelical forcing as we change Ω for the case of the hyperviscous runs. For small values of Ω the helicity fluctuations are small, oscillating over the eddy time-scale L/U 0.2. While for the case of Ω = 3 we start to see large ≃ fluctuations that have much longer time scale than the eddy turn over time L/U 0.2. ≃ These fluctuations are due to the formation of the large scale condensate [187], with the condensate fluctuating over a much larger time scale. There have been certain studies of helicity fluctuations that could possibly generate dynamo instability but they correspond to low Re laminar flows with scale separation. Since the condensate is found at the largest scale of the domain there is no scale separation between the velocity field and the magnetic field. Thus we cannot conclude from this whether the dynamo instability is enhanced or not.

IV.4.2 Critical magnetic Reynolds number

c We show in ﬁgure IV.18 the critical magnetic Reynoolds number Rmd as a function of Red for diﬀerent values of Ω. The hyperviscous runs are shown by ⋆ points at IV.4. Rotating dynamos 103

Ω =0 100 Ω =1 2 10 Ω =3 Ω =50

10−1 ) Ω = ∞ f ) 3 /k U

0 1 f 10 f k −2 ( ( Red ≫ 1

10 / 2

ǫ/ Ω =0 U Ω =1 10−3 Ω =3 100 Red ≫ 1 Ω =50 Ω = ∞ 10−4 100 101 102 103 100 101 102 103 Red Red (a) (b)

2 Figure IV.16 – Figure a) shows the normalized energy U /(f0/kf ) and ﬁgure b) shows 3 normalized dissipation rate ǫ/(U kf ) as a function of the Reynolds number Red for diﬀerent values of the rotation rate as mentioned in the legend. The points denoted by ⋆ symbols at Red = 2000 denote hyperviscosity runs.

0.4 Ω =0 0.3 Ω =1 0.2 Ω =3 0.1 H

ρ 0.0 −0.1 −0.2 −0.3 −0.4 0 500 1000 1500 2000 2500 3000 3500 4000 4500 t

Figure IV.17 – Figure shows the relative helicity ρH as a function of time t for diﬀerent values of Ω mentioned in legend. Darker shades of blue correspond to larger values of Ω. 104 Chapter IV. 3D Rotating flows and dynamo instability

Re = 2000, they represent the large Re 1 limit. The case of Ω = 0 shows an d d ≫ 40 Ω =0 35 Ω =1 Ω =3 30 Ω =50 25 Ω = ∞ c d 20 Rm 15

10 Red ≫ 1 5 0 100 101 102 103 Red

c Figure IV.18 – Figure shows the critical magnetic Reynolds number Rmd as a function of Red for diﬀerent values of Ω shown in the legend. The points denoted by ⋆ at Re = 2000 denote hyperviscosity runs.

c initial increase with respect to Red and for very large Red we expect that Rmd becomes independent of Red, its limit is captured by the hyperviscous runs. For weak rotation rate Ω = 1 we see that rotation increases the threshold showing an initial inhibitory effect of the dynamo instability, this behaviour is also seen for the asymptotically large Red value captured by the hyperviscous runs. c For the Ω = 3 runs, we see a large drop in the value of Rmd, also there is no increase due to turbulence fluctuations and the threshold remains constant for all values of Red in the turbulent regime. The hyperviscous run at Ω = 3 also shows the same threshold. For higher values of Ω, Ω = 50, we see that the behaviour is very similar to the case ∞ of Ω = 3. Thus the gain from the reduction in turbulent fluctuations due to rotation c is very well captured by the case of Ω = 3. The ratio between the Rmd for the case of Ω = 0 and the case Ω = 3 for the hyperviscous runs is approximately 12. 3 ∼ The injected power ǫ scales like Red implying a reduction in the power required for a dynamo instability by a factor of 2 103 between Ω = 0 and 3 and a factor of 8 103 · · between Ω = 1 and 3 (see Fig. IV.18). c This gain in the Rmd value goes along with the drop in the turbulent fluctuations, seen in figure IV.16b. We show in figure IV.19a the form of the compensated kinetic energy spectra for the two cases Ω = 0, 3 taken from the hyperviscous runs. The kinetic 2 5/3 2 1/3 enstrophy spectra shows the Kolmogorov scaling E(k)k k− k k for the no 2 ∼ ∼ rotation case. The enstrophy spectrum Ekk giving a measure of the stretching and is largest near the viscous scales. In the case of Ω = 3, the energy is concentrated at the largest modes of the system due to the inverse cascade of energy. The enstrophy spectra measuring the stretching rate, is also peaked at the large scale (more clearly at the forcing scale). Thus the stretching due to the small scales, which are incoherent, is much larger in the case of Ω = 0. While for the case of Ω = 3, we see that the stretching is predominantly in the large scales which are coherent in time. Now we IV.4. Rotating dynamos 105

102 100

101 10−1 ) ) k k ( 0 −2

10 +1/3 ( 10

E k M 2 k

10−1 10−3 Ω =0 Ω =0 Ω =3 Ω =3 10−2 10−4 100 101 102 100 101 102 k k (a) (b)

Figure IV.19 – Figure shows in a) the compensated kinetic energy spectra k2E(k) in b) the magnetic energy spectra for the two different cases of Ω = 0, 3 as mentioned in the legend for the hyperviscous runs. The dashed black line shows the scaling k1/3. look at the magnetic energy spectra M(k) shown in figure IV.19b, which shows the Ω = 0 case having a lot of energy at the smaller scales, thus having larger dissipation. The prediction of the magnetic energy spectra does not seem to follow the Kazantsev prediction of k+3/2 (expected till the dissipative scales). This is possibly due to the fact that here we are in the limit of P m 1 while the theory is supposed to work ≪ for the large P m limit. This was remarked in the study of [59, 188], where they find a qualitatively different behaviour for the case of P m 1 dynamo and the P m 1 ≫ ≪ dynamo. For the case of Ω = 3, we see that the magnetic energy M(k) is concentrated more at the large scales and thus the dissipation is much smaller as compared to the case of Ω = 0. The prediction from the Kazantesev theory for the problem of 2.5D is that M(k) k0 (see Chapter III) seem to be present in some form at the large scales ∼ of M(k). We do mention that the theory is made for the case of P m 1 and the case ≫ of P m 1 is not yet know. ≪

IV.4.3 Visualizations

Now we look at the structure of the kinetic and the magnetic energy in three dimensional space. We already have an idea about the energy distribution among different scales from the information in the kinetic/magnetic energy spectra (figure IV.19). In figure IV.20 we show the color contours of the vertical vorticity ωz and the vertical current jz with blue and red colors denoting positive and negative values respectively. The figure corresponds to the case of Ω = 0 with the parameters Re 106, Rm 19. d ≈ d ≈ The velocity field and the magnetic field as seen from the figure, resides mostly in the small scales with no preferred direction. The velocity field behaves like a standard 3D homogeneous turbulence. The magnetic field is driven by the small scales of the velocity field and do not have any preferred direction. In figure IV.21, we show the color contours of the vertical vorticity and vertical current for the case of Ω = 3 for the parameters Re 60, Rm 2.25. We see d ≈ d ≈ that the vertical vorticity (figure IV.21a) has a large scale corotating vortex aligned with the global rotation and a counter rotating vortex which spins opposite to the

IV.4. Rotating dynamos 107 global rotation. The counter rotating vortex is less organised and is responsible for the energy cascade towards smaller scales due to its three-dimensionality. The figure IV.21b shows the vertical current jz which shows the magnetic energy concentrated at the largest scales, mostly in the kz = 1 mode along the z-direction. We see that the eigenmode is aligned along with the co-rotating vortex. This shows that rotation leads to a magnetic unstable mode which is coherent and aligned with the global rotation. We also note here that the k = 1 mode is the most unstable mode in the Ω = z ∞ simulations (which is the 2.5D configuration), implying that the Ω = 3 simulations already show some quasi-twodimensional behaviour. We finally note that similar unstable modes have also been found in dynamos driven by rotating convection flows, see [189, 190]. Thus the unstable modes seem to be more general than the particular choice of forcing used in this study.

IV.4.4 Helical forcing case We now look to the dynamo instability driven by a helical forcing as a function of global rotation. The helical forcing is given by the Roberts ﬂow, f = f0 (cos(kf y), sin(kf x), sin(kf y) + cos(kf x)). We solve the numerical equations (IV.1.1), (IV.0.1) 2 with this forcing. We show in ﬁgure IV.22 the normalized energy U /(f0/kf ) and

Ω =0 100 Ω =1 2 10 Ω =3 Ω =50

10−1 ) f ) 3 /k U

0 1 f 10 f k −2 ( (

10 / 2 ǫ/ U Ω =0 10−3 Ω =1 100 Ω =3 Ω =50 10−4 100 101 102 103 100 101 102 103 Red Red (a) (b)

2 Figure IV.22 – The ﬁgure a) shows the normalized total velocity squared U /(f0/kf ) 3 and b) shows normalized dissipation rate ǫ/(U kf ) as a function of the Reynolds number Red for diﬀerent values of the rotation rate as mentioned in the legend for the helical forcing case.

3 the normalized dissipation ǫ/(U kf ) as a function of Red for different values of Ω as mentioned in the legend for the helical forcing case. The behaviour is quite similar to the nonhelical results already shown in figure IV.16. c Figure IV.23 shows the critical magnetic Reynolds number Rmd as a function of Red for different values of Ω. These results are qualitatively much different from the results from the nonhelical case shown in figure IV.18. Firstly we see that the helical dynamo has a much lower threshold than the nonhelical case, in particular Ω = 0 does not see any effect of turbulence, this corresponds to the recent study of [62]. d Comparing the values of Rmc , the dynamo instability driven by the α mechanism seems to be most feasible in laboratory experiments. Secondly the gain from the effect of rotation seems to be quite moderate, the ratio between the largest values is at most 108 Chapter IV. 3D Rotating flows and dynamo instability

2.0

Ω =0 Ω =1 1.5 Ω =3 Ω =50 c d 1.0 Rm

0.5

0.0 100 101 102 103 Red

c Figure IV.23 – Figure shows the critical magnetic Reynolds number Rmd as a function of Red for diﬀerent values of Ω shown in the legend for the helical forcing case.

1.5. Rotation thus gives a reduction in the energy consumption at the threshold by ∼ at-most a factor of 3. Experimentally though this factor would account for the cost ∼ of subjecting the ﬂow to global body rotation to create the rotation rate Ω = 3, the extra dissipation coming from the Ekman friction associated with this rotation rate. c Nevertheless we see a reduction in the threshold Rmd hinting that the form of the unstable mode changes in fast rotating ﬂows.

IV.4.5 Structure of the unstable mode

We now look at the form of the unstable mode for the cases of Ω = 0 and Ω = 3. We show in ﬁgure IV.24 the kinetic and the magnetic energy spectra for the two values of Ω. The kinetic energy spectra behaviour is similar to the nonhelical study presented

102 100

10−1 101

+1/3 10−2 ) k ) k k ( 0 10 ( 10−3 E M 2

k 10−4 10−1 Ω =0 10−5 Ω =0 Ω =3 Ω =3 10−2 10−6 100 101 100 101 k k (a) (b)

Figure IV.24 – Figure shows in a) the compensated kinetic energy spectra k2E(k) in b) the magnetic energy spectra for the two different cases of Ω = 0, 3 as mentioned in the legend for the hyperviscous runs for the helical forcing. The dashed black line shows the scaling k1/3. IV.4. Rotating dynamos 109 in figure IV.19. For the case of Ω = 0 we see that the stretching is concentrated at the small scales while for the case Ω = 3 we see that the stretching is concentrated at large scales. The magnetic energy spectra for Ω = 0 is largest at the k = 1 mode as expected from α mechanism. While for the case Ω = 3, we see that the magnetic c energy is distributed equally among a few large scale modes. The reduction in Rmd could be explained by the apparent reduction in the energy of the small scale modes for the Ω = 3, thus having lesser dissipation as compared to the Ω = 0 case. The form of the unstable mode thus changes when we change global rotation. For Ω = 3, the unstable mode is more flat at the large scales as was seen from the nonhelical case, see figure IV.19b. We now look at the visualizations and see whether the spatial structure of the magnetic field is much different between the cases of Ω = 0 and Ω = 3. The vertical component of the vorticity and the current are shown in figure IV.25 for the case of

(a) (b)

Figure IV.25 – Figure shows in a) the vertical vorticity ωz in b) the vertical current jz for the case of Ω = 0 for the helical forcing case. The blue and red colors correspond to positive and negative values with darker shades denoting larger values.

Ω = 0 and Re 110, Rm 0.7. As we can see the vorticity seems to quite similar d ∼ d ∼ to the case of the nonhelical flow (in figure IV.20a), but the structure of the magnetic field is radically different from the case of the nonhelical flow (in figure IV.20b). This is due to the α effect where the magnetic field is dominated by the largest mode in the system. The turbulent flow is homogeneous with no preferred direction and the generated magnetic field also has no preferred direction. Since the structure of the c magnetic field is largely dominated by the large scales, the Rmd is thus quite small as compared to the nonhelical case. Next we show the vertical component of the vorticity ωz and the current jz in figure IV.26, for the case of Ω = 3 and Re 70, Rm 0.5. The vorticity ω consists of d ∼ d ∼ z the co-rotating and the contra-rotating vortices as was seen in the nonhelical case (in figure IV.21a). The vertical component of the current jz consists of helical flux tubes 110 Chapter IV. 3D Rotating flows and dynamo instability

(a) (b)

Figure IV.26 – Figure shows in a) the vertical vorticity ωz in b) the vertical current jz for the case of Ω = 3 for the helical forcing case. The blue and red colors correspond to positive and negative values with darker shades denoting larger values. of alternating signs, not very different from the nonhelical case (in figure IV.21b), but it does not exactly match its structure. The magnetic field again is dominated by the mode kz = 1, concentrated near the shear layers surrounding the columnar vortex of the velocity field. In this case the unstable mode has a clear preferential direction along z.

IV.5 Conclusion

In this Chapter we have studied rotating flows driven by helical and nonhelical forcing, and the dynamo instability driven by such flows. The first part of this Chapter we have looked at the steady state regimes of forced homogeneous rotating turbulence as a function of the Rosby number Ro and the Reynolds number Re. In the second part of this Chapter we have looked at rotating dynamos where the dynamo instability is driven by the flow studied in the first part of this Chapter. In the absence of rotation, helical forcing drives the dynamo by the α effect while the nonhelical flow drives it through the small scale dynamo effect giving a larger critical Rm. The effect of rotation in both the cases increased the amplitude of the velocity field due to the formation of the large scale condensate leading to a large value of the critical magnetic Reynolds number based on the velocity field Rmu. When normalized with the injection rate, in both the forcing cases, we find a reduction in the critical Rmd for the dynamo instability. c In the case of the nonhelical flow, the apparent drop in the Rmd from a nonrotating to a rotating flow was by a factor of 10 (see figure IV.18). Also the form of the unstable mode is different for the nonrotating and the rotating flow (figures IV.20b,IV.21b). The IV.5. Conclusion 111 rotating nonhelical flow created a large scale magnetic field but not by an α-dynamo mechanism. This could be seen from looking at the spectra of the magnetic field which did not show a preferential amplification of the largest k = 1 mode, also the velocity and the magnetic fields are concentrated at large scales. While the α-dynamo instability predicts a preferential amplification of the largest k = 1 mode in the presence of scale c separation. The reduction is the Rmd is attributed to a) suppression of turbulent fluctuations and b) the formation of large scales which have long correlation times making them more effective in performing a constructive refolding of the magnetic field lines. In the case of the helical rotating flow there is an apparent drop in the threshold Rmc as one increases the rotation rate (see figure IV.23). The drop is 1.5 is much d ≈ less compared to the nonhelical case. The drop in the critical magnetic Reynolds c number Rmd is a much smaller factor, since the dynamo driven by the helical flow in scale separation kf L = 4 is already very efficient [62]). This drop is accompanied by a change in the form of the unstable mode between the nonrotating and the rotating case (see figures IV.25b,IV.26b). In the nonrotating case the magnetic field did not have a preferred direction, while in the rotating case the unstable mode becomes more aligned along the direction of rotation. The change in the form of the unstable mode is also seen from the magnetic field spectra (figure IV.24b). Independent of the presence or absence of helicity, we have a lower threshold for the dynamo as we change Ω. This lowering of the threshold is linked to the transition to the inverse cascade regime occuring around Ω 2, Ro 0.5 found in the first part ≈ ≈ d of this Chapter (see figure IV.6). We thus obtain the following ranking of lowest Rmc among the flows that have been studied in this chapter,

Rotating helical flow • Non rotating helical flow • Rotating nonhelical flow • Non rotating helical flow. • We should mention that this whole Chapter has taken idealized conditions of periodic boundary conditions on the three directions. In general, more realistic boundary conditions of rotating flows are subject to no slip boundary conditions on all surfaces. For the study presented here, the flow is required to reach quasi-twodimensional behaviour where the injected power is much lower than the standard three-dimensional turbulence. Rotation rates where such kind of flows have been obtained in laboratory water-tank experiments, see [161, 143]. As mentioned previously, the recent work of [168] has shown that the measured dissipated power in rotating turbulence decreases by a factor of 10 at the highest rotation examined due to a two-dimensionalization of the flow, as compared to standard nonrotating turbulence. For this lowest rotation rates achieved, there were no big enhancement of the viscous dissipation due to Ekman layers observed. So suppressing turbulent fluctuations and decreasing energy dissipation by adding global rotation is indeed feasible experimentally. The additional energy cost for maintaining the rotation is probably minimal compared to the large 112 Chapter IV. 3D Rotating flows and dynamo instability gain of the order of 103 due to the suppression of turbulent fluctuations in the case of nonhelical flows. Since the effect of solid boundaries has not been investigated, we might expect some reordering in the above list if one were to take the solid walls into account. An other issue that needs to be considered is that the design of the domain and the forcing should guarantee that all three velocity components are present, so that the flow becomes 2.5D and not 2D. This difficulty however can also be overcome by the proper design of the forcing mechanism that amplifies all velocity components. To answer these questions, further work needs to be pursued with simulations in more realistic domains. Perspectives and conclusions

In this thesis we have studied rotating dynamos as a theoretical tool to understand dynamo instabilities in astrophysical objects. The use of periodic boundary conditions helped us to do numerical simulations over large range of parameters. The study was aimed at covering the parameter space and observing the dominant behaviour in the different regimes of the parameter space. For this we used simulations and theoretical tools to do a systematic study. We began by studying quasi-twodimensional flows which are obtained in the limit of fast rotation. The hydrodynamic flow was first studied for two types of forcing, one has non-zero mean helicity (helical forcing) and the other which has zero mean helicity (nonhelical forcing). The dynamo instability resulting from such type of flows were studied in detail, in particular, the importance of helicity at low and large Rm was investigated. We found that the helical/nonhelical flow gave a dynamo threshold which stayed almost constant as we increased Re. Later we looked at the saturation of this instability in the quasi-twodimensional geometry. Saturation of an unstable magnetic field involved the back reaction through the Lorentz force. The Lorentz force for the magnetic field had one z-invariant component and another which varied along the z direction. Only the z invariant projection of the Lorentz force was used and was found to saturate the growth of the magnetic field. The amplitude of the saturated magnetic field was then studied as a function of Rm, P m. At high P m, the saturation amplitude followed the laminar scaling. At low P m, the saturation amplitude followed the turbulent scaling. The turbulent scaling was found to appear at 3 a value of P m . 10− , explaining why they are still not observed in three-dimensional simulations. The P m, Re of the underlying flow clearly affects the scaling of the magnetic field. The second part of the thesis concerned the modelling of the dynamo instability using a Kazantsev model. The Kazantsev model considers a velocity field to be a white Gaussian noise. The flow was considered to be isotropic and homogeneous in the x y directions. Under these assumptions the induction equation was reduced − to a one-dimensional coupled system of equations. The dynamo instability was then studied for a wide range of control parameters. The form of the most unstable mode was derived analytically and the spectra of the unstable mode was found. The theoretical calculations were then compared with numerical simulations. The theoretical predictions on the form of the unstable mode were found to match with the numerical results. Next, the intermittency effect on the growth rate of the magnetic field due to a fluctuating velocity was investigated. A theoretical model using scale separation was derived using large deviation theory. The theory showed that the different mo-

113 114 Chapter IV. 3D Rotating flows and dynamo instability ments of the magnetic field grow at different rates. In effect, each moment predicts a different threshold for the dynamo instability. Numerical simulations were then used to calculate the exact threshold of the kinematic dynamo problem. Then two different forms of nonlinear saturation were considered. In both the cases we found out that the exact threshold is given by the 0-moment (or the growth rate predicted by the log of the magnetic field) of the linear kinematic problem. In the last part of this thesis we considered the fully rotating 3D flow. The hydrodynamic part was first studied as a function of Re, Ro for both the helical and nonhelical forcing. The study concentrated on two different questions. One was the nature of the transition from a forward cascade to an inverse cascade. The other question was how the saturation of the inverse cascade took place. We showed that for the particular choices of forcing used, the transition from a forward to inverse cascade is a critical transition. The transition becoming sharper as we increased the Re and the condensate amplitude approached a power law scaling as a distance from the critical point. Then we found that there are two different saturation mechanisms for the inverse cascade for the types of flow investigated here. For Ro below the critical point and for large Re, the saturation of the inverse cascade occurred through the formation of a contra-rotating vortex. This vortex locally cancels the effect of rotation and cascades energy to small scales. For a fixed Re and very low Ro, a second type of saturation mechanism through the formation of a coherent large scale vortex was found. The saturation comes from the viscous dissipation at the largest scale of the box (the scale of the coherent vortex). In the second part of the Chapter, we looked at the dynamo instability resulting from both the helical and the nonhelical rotating flow. We particularly concentrated on the transition regime of the inverse cascade Ro 0.5, Ω 2. The nonhelical rotating flow showed a large drop in the critical f ∼ ∼ magnetic Reynolds number across Ω = 1 3. This drop in Rm corresponds to a drop → of about 8 103 factor in the injected power needed for the dynamo instability. Thus × rotation played a positive role for this type of forcing. For the helical forcing we found the drop in Rm to be moderate, of about 1.5. The form of the unstable mode in both the helical and nonhelical forcing changed as we cross the transition. Thus rotation plays an important effect in the dynamo instability (either the Rm or the form of the unstable mode).

Outlook

The rotating turbulent flow was used to study the effect of rotation and turbulence on the dynamo instability. For the rotating dynamo more simplified models like the quasi-twodimensional and the hypervisous models helped reach extreme values of the parameters. Such models along with systematic numerical simulations helped us map out the parameter space and extract the different behaviour. Future work would be to study the saturation mechanisms of the magnetic field for the dynamo instability in a rotating turbulent flow. In the Earth dynamo it is conjectured that the amplitude of the magnetic field is expected to scale with the rotation rate [191], what is known in the literature as the Strong field dynamo. These have not yet been shown to exist at the low Prandtl number limit or in simplistic IV.5. Conclusion 115 systems as the one examined here. 116 Chapter IV. 3D Rotating flows and dynamo instability Appendix A Derivation of Kazantsev model

In order to derive the equations (III.1.20),(III.1.21), we follow a procedure similar to the one mentioned in [99]. We start with the index form of the induction equation (III.1.18) written as,

∂ bi = bm∂mui um∂mbi i k u bi + η ∂k∂k k2 bi. (A.0.1) t − − z z − z where ∂i denotes the derivative with respect to the coordinate xi. Next we write the equation for the magnetic correlation function Hij (r) = bi (x + r) † bj (x) , which reads as, D E

∂ Hij 2η ∆ k2 Hij = ∂ Cikj (r, t) Ckij (r, t) Cjki ( r, t) † + Ckji ( r, t) † t − − z k − − − − 3ij 3ji † i3j j3i † + ikz C (r, t) C ( r, t) C (r, t) + C ( r, t) , − − − − (A.0.2) where the quantity Ckij is the triple product average deﬁned as Ckij (r, t) = uk (x + r, t)

bi (x + r, t) † bj (x, t) . This triple product average can be simpliﬁed usingD the Furutsu- Novikov theorem ([117E, 118]), which can be written as,

Ckij (r, t) = uk (x + r, t) bi (x + r, t) † bj (x, t) δ bi (x + r, t) † bj (x, t) k m = dx′dt′ u (x + r, t) u x′, t′ . (A.0.3) δum (x , t ) Z * ′ ′ + D E The above expression can be simpliﬁed by using the delta-correlation property of the † δ (bi(x+r,t)) bj (x,t) velocity correlator. The term δum(x′,t′) can be simpliﬁed by taking the * + functional derivative of the governing equation of the two point magnetic correlation function bi † bj. Integrating it with respect to time and taking the statistical average 117 118 Appendix A. Derivation of Kazantsev model we end up with the following,

1 Ckij (r, t) = gkl (r, t) gkl (0, t) Hij (r, t) gkj (r, t) Hil (r, t) gki (0, t) Hlj (r, t) 2 − ,l − ,l − ,l +ik Hij (r, t) gk3 (0, t) gk3 (r, t) . (A.0.4) z − We mention here that the Furutsu-Novikov theorem follows the Stratanovich interpre- tation of the noise (and not Ito). Substituting the last expression for the triple point averages into the equation (A.0.4) and after some long but trivial calculation we can find the equation for Hij (r). Now given the equation for Hij that can be obtained from both (A.0.2) and (A.0.4), ij we look at constructing the equations for scalar functions of H (in terms of HLL ,HNN etc). The procedure to express the tensor Hij in terms of scalar functions is mentioned in [110]. It can then be shown that the correlation tensor Hij has the general form written out in equation (III.1.12). We mention here that only the mirror symmetric part of the correlation function Hij is important in the discussion. This is because the helical part of the magnetic field is not coupled to the governing equations of the nonhelical part. One simple way to see this is to take the equation (A.0.2), now we use the form of Ckij from (A.0.4). The scalar function corresponding to helicity (both velocity and magnetic fields) is pseudo-scalar while the nonhelical components are proper scalar. If we look at an equation governing the proper scalar function in Hij, it can be made up of two kinds of terms. One form of the term is a product of two proper scalar functions, more precisely a product of one proper scalar function in gij and one in Hij. The other way is to construct it using the product of two pseudo scalar functions, one pseudo scalar function in gij and the other from Hij. Since there are no pseudo scalar functions in gij the pseudo scalar functions in Hij do not enter the governing equations of the proper scalar functions in Hij. Hence we consider the magnetic ij correlation function H made of only the proper scalar terms, HLL ,HNN ,HZ ,Hc. Due to the solenoidal condition, we stick with two of these quantities HLL ,Hc and their governing equation derived using equations (A.0.2), (A.0.4) is mentioned in equations (III.1.20),(III.1.21). Appendix B Matched Asymptotics for the Kazant- sev model

B B We are interested in finding the behaviour of the functions E2D ,EZ in the intermediate region k k k . In this process we would would like to find the value of γ in z ≪ ≪ d the limit of k k . From the numerics we can see that the value of γ is 3 in the z ≪ d limit of small kz and independent of the value of Dr, see figures III.2, III.3. In the limit Rm , the governing equations are given by equations (III.1.20),(III.1.21). → ∞ Since the equations are rescaled with k , the small parameter now is k˜ 1. The d z ≪ idea here is to find the inner solution of the equation by expanding in terms of powers of k˜z to find the solutions of the equations (III.4.3), (III.4.4). Then we compute the outer solution by rescaling the variabler ˜ tor ˆ = √Drkzr˜. This rescaling would then provide us with a new set of equations for the outer solution. The behaviour of the inner solution is valid in the regionr ˜ 1 while the outer solution is valid in the region ≪ r˜ 1/k˜ . The matching will take place in the intermediate range of scales, to get the ≫ z exponents and the eigenvalue γ.

B.0.1 Inner solution

We do asymptotics for k˜ 1 with h = H + k˜2H + and h = D k˜ G + k˜2 z ≪ LL 0 z 1 ··· c r z 0 z G + ), the equation for zeroth order in k˜ satisfy, 1 · z

2 H0′ H0 H0′ r˜ H′′ + 7 (γ 8) + 2H′′ + 6 =0, (B.0.1) 0 r˜ − − r˜2 0 r˜ h i h i 2 G0′ G0 G0′ G0 r˜ G′′ + (γ + 1) + 2G′′ + 2 2 =2˜rH. (B.0.2) 0 r˜ − r˜2 0 r˜ − r˜2 h i h i Now we write the homogeneous solution to the equations using hypergeometric functions F deﬁned as F (a, b, c, d) = Γ(c)/(Γ(b)Γ(c b)) 1 tb(1 t)c b 1/(1 tz)adt, 2 1 2 1 − 0 − − − − R 3 √1 + γ 3 √1 + γ r˜2 H (˜r) = C F , + , 2; , (B.0.3) 0 1 2 1 2 − 2 2 2 − 2 h1 √1 + γ 1 √1 + γ r˜2 i G (˜r) = C r˜ F , + , 2; , (B.0.4) 0H 2 2 1 2 − 2 2 2 − 2 h i 119 120 Appendix B. Matched Asymptotics for the Kazantsev model where G0 = G0H +G0I , with G0H denoting the homogeneous solution and G0I denoting the inhomogeneous solution. G0I can be found and expressed in terms of integrals using the Wronskian. The asymptotics for larger ˜ is found out to be,

1 2√γ+1 G (˜r) = C r˜− . (B.0.5) 0I 1 (γ + 1)

B.0.2 Outer solution For the larger ˜ limit, we could rescaler ˆ k˜ r˜, but in order to get rid of the dependence → z on D at the lowest order we do the following rescaling,r ˆ √D k˜ r˜. This ends up r → r z with the following set of equations,

2 2 hLL 2 2 γh 2D k˜ +r ˆ h′′ + 3 + 2k˜ +r ˆ h = 8h LL − r z LL rˆ z LL LL h i rˆ 4 ˜2 +4ˆrhLL′ + 8 hc + Drkz hc, (B.0.6) √Dr rˆ p 2 2 hc′ hc 2 2 γh 2D k˜ +r ˆ h′′ + + 2k˜ +r ˆ h = 2ˆr D h . (B.0.7) c − r z c rˆ − rˆ2 z c r LL h i p Since k˜ 1 we again expand the quantities H (ˆr) ,H (ˆr) in powers of k˜ , z ≪ LL c z h = Hˆ (ˆr, γ) + k˜2Hˆ r,ˆ γ, k˜ , D + k˜4Hˆ r,ˆ γ, D + , (B.0.8) LL 0 z 1 z r z 2 r ··· h p p i h = D Gˆ (ˆr, γ) + k˜2Gˆ r,ˆ γ, D + k˜4Gˆ r,ˆ γ, D + . (B.0.9) c r 0 z 1 r z 2 r ··· p h p p i With this expansion above, the equation at the leading order becomes independent of D with the assumptions being D k˜2 1, k˜2 1. The leading order equations are, r r z ≪ z ≪ ˆ 2 H0′ 2 (γ 8) Hˆ rˆ Hˆ ′′ + 3 +r ˆ Hˆ 4ˆrHˆ ′ = 8ˆrGˆ , (B.0.10) − 0 − 0 rˆ 0 − 0 0 h ˆi ˆ 2 G0′ G0 2 γGˆ rˆ Gˆ′′ + +r ˆ Gˆ = 2ˆrHˆ . (B.0.11) 0 − 0 rˆ − rˆ2 0 0 h i The smallr ˆ behaviour of the functions Hˆ0, Gˆ0 can be obtained by expanding in powers ofr ˆ. By direct substitution it can be shown that a simple power law expansion fails for any value of γ and the expansion for smallr ˆ contains logarithmic corrections.

B.0.3 Matching We have to rescale the inner and outer variable to match the solutions at an intermediate range. The large r form for the inner solution reads like,

√1+γ 1 1 1 √1+γ 1 1 1 H (r) = r− f (γ) + f (γ) + O + r m (γ) + m (γ) + O , 0 1 r3 2 r5 r7 1 r3 2 r5 r7 h i h (B.0.12)i

1 √1+γ 1 1 1 1+√1+γ 1 1 1 G (r) = r − f˜ (γ) + f˜ (γ) + O + r m˜ (γ) +m ˜ (γ) + O , 0 1 r 2 r3 r5 1 r 2 r3 r5 h i h (B.0.13)i 121 for γ = 3. For γ = 3 the coeﬃcients f , m and f˜, m˜ diverge. In this case the expansion 6 i i i i involves logarithmic corrections to the power laws. A successful matching with the outer solution (that also includes logarithmic corrections) becomes only possible for γ = 3. The power law behaviours for the correlation functions are then a direct consequence of this eigenvalue and the properties of the hypergeometric functions. Thus in the intermediate region r r 1/k the solution has the exponents h d ≪ ≪ z LL ∼ r 1, h √D k r0. Using Wiener-Khintchine we can ﬁnd the corresponding behaviour − c ∼ r z in the spectral space to be, EB k0 + 1 √D k /k2 and EB (k) 1 √D k k0. 2D ∼ 2 r z Z ∼ 2 r z 122 Appendix B. Matched Asymptotics for the Kazantsev model Appendix C Numerical algorithm for multiplicative noise

We consider an equation with a multiplicative noise term,

dX = a(X) + b(X)ζ(t). (C.0.1) dt

ζ(t) is a Gaussian white in time noise satisfying the condition ζ(t)ζ(t ) = 2Dδ(t t ). h ′ i − ′ Integrating both sides of equation (C.0.1) in time we get,

tn+1 tn+1 X(tn+1) = X(tn) + a(X(t′))dt′ + b(X(t′))dW (t′). (C.0.2) Ztn Ztn Here W (t) is the Wiener process [192]. The second integral is ill defined and could lead to different interpretations (mathematicians use the Ito protocol while physicists use the Stratanovich protocol). Here we follow the Stratanovich formulation. We take t t in equation (C.0.2) to be small, we can then expand the coefficients n+1 − n a(X, t), b(X, t) in Taylor series. This gives,

1 2 a(X(t′)) = a(X(t )) + a′(X(t )) X(t′) X(t) + a′′(X(t )) X(t′) X(t) + , n n − 2 n − ··· (C.0.3) 1 2 b(X(t′)) = b(X(t )) + b′(X(t )) X(t′) X(t) + b′′(X(t )) X(t′) X(t) + . n n − 2 n − ··· (C.0.4)

We denote tn+1 tn = ∆n and the Wiener noise increments as W (tn+1) W (tn) = − 1/2 − ∆Wn. The Wiener noise W (t) varies over a time scale t , thus care should be taken when truncating the series. We expand (C.0.2),

tn+1 X(t ) =X(t ) + a(X(t ))∆n + b(X(t ))∆W + a′(X(t )) X(t′) X(t ) dt′+ n+1 n n n n n − n Ztn tn+1 b′(X(t )) X(t′) X(t ) dW (t′) + . (C.0.5) n − n ··· Ztn 123 124 Appendix C. Numerical algorithm for multiplicative noise

We can substitute X(t ) X(t ) from the above equation into the integrals. We get, ′ − n tn+1 X(t ) =X(t ) + a(X(t ))∆n + b(X(t ))∆W + a′(X(t ))a(X(t )) (t′ t )dt′+ n+1 n n n n n n − n Ztn tn+1 tn+1 tn+1 a′(X(tn))b(X(tn)) dt′ ζ(t′′)dt′′ + b′(X(tn))a(X(tn)) ζ(t′) t′ tn dt′ tn tn tn − Z Z ′ Z tn+1 t b′(X(t ))b(X(t )) ζ(t′)dt′ ζ(t′′)dt′′ + . (C.0.6) n n ··· Ztn Ztn We truncate it to the lowest order in ∆n. We look at the order in ∆n of each of the above integrals. We ﬁnd,

tn+1 1 2 2 t′ tn dt′ = ∆n = O(∆n ), (C.0.7) tn − 2 Z ′ tn+1 t 3/2 dt′ ζ(t′′)dt′′ = O(∆n ), (C.0.8) Ztn Ztn tn+1 3/2 ζ(t′) t′ tn dt′ = O(∆n ), (C.0.9) tn − Z ′ tn+1 t tn+1 2 ζ(t′)dt′ ζ(t′′)dt′′ = ∆W (t′)dW (t′) = ∆Wn /2 = O(∆n). (C.0.10) Ztn Ztn Ztn Keeping only the integrals of the order ∆n and truncating higher order terms we get,

1 2 3/2 X(t ) =X(t ) + a(X(t ))∆n + b(X(t ))∆W + b′(X(t ))b(X(t ))∆W + O(∆n ). n+1 n n n n 2 n n n (C.0.11)

We can extend this to the induction equation to keep only the lowest order in dt. List of Figures

I.1 Earth and the Sun’s magnetic ﬁeld...... 2 I.2 Sketch of the physical domain in which the rotating turbulent dynamo is studied...... 3 I.3 Sketch of the Richardson cascade showing transfer of energy to small scales from large scales...... 4 I.4 Sketch of a rotating tank showing the Taylor column moving along with a spherical object...... 6

II.1 Sketch of the energy spectra E2D (k),EZ (k) for a two dimensional ﬂow. 11

II.2 Energy spectra E2D (k),EZ (k) for helical and nonhelical ﬂow for different Re...... 12

II.3 Energy spectra E2D (k),EZ (k) for helical and nonhelical ﬂow for different k/kf , Re−...... 13

II.4 Contours of 2D energy E2D (x, y) and the vertical velocity ﬁeld uz(x, y) for helical and nonhelical ﬂow...... 14

II.5 Figure illustrates the shear S2D (ℓ) ,SZ (ℓ) as a function of ℓ...... 15 II.6 Growth rate γ as a function of kz for different values of Rm for the helical flow...... 16 II.7 Figure a) shows Growth rate γ as a function of Rm with the prediction αkz for different Rm values. Figure b) shows α as a function of Rm. . . 17

II.8 Magnetic energy spectra EB (k) for different values of Rm for kz = 0.25, Re 530...... 18 ≈ c II.9 Figure illustrates γmax, kz for a γ kz curve...... 19 c − II.10 Figure shows γmax, kz for different values of Rm for the helical forcing. 19 II.11 Figure shows γ as a function of kz for different values of Rm for the nonhelical forcing...... 20 c II.12 Figure shows γmax, kz as a function of Rm for different values of Re for the nonhelical forcing...... 21

II.13 Figures show the magnetic field EB (k) as a function of k. Figure a) shows for different values of Rm with Re 530, figure b) shows for ≈ different values of Re with Rm almost constant for the nonhelical forcing. 22 II.14 Figures show the contour of the magnetic field for a) Rm 32, b) ≈ Rm 1030, c) Rm 2060...... 22 ≈ ≈ II.15 Figure shows the critical magnetic Reynolds number Rmc as a function of Re...... 23 II.16 Figure shows the growth rate γ as a function of kz for the β-dynamo flow...... 24

125 126 LIST OF FIGURES

II.17 Contour of the magnetic field for two different laminar flows...... 24 II.18 Normalized growth rate γ/(Upkf ) as a function of normalized kz/kf for a few different kf L values for Helical and Nonhelical forcing...... 25 c II.19 Figures show γmax/(Upkf ), kz/kf as a function of Rm for different kf L for Helical and Nonhelical forcing...... 26 II.20 Figures show a sketch of a) Riga Dynamo, b) Karslruhe Dynamo, c) VKS dynamo...... 30 II.21 Figure shows the saturation of the magnetic field for the Riga Dynamo, the Karslruhe Dynamo and the VKS dynamo...... 31 II.22 Figure shows a sketch of the domain with the kinematic velocity and magnetic field along with the Lorentz force...... 32 II.23 Figure shows Rmc as a function of P m...... 34 II.24 Time series of B 2, u 2, u2 is shown for the parameters Re | | | 2D | z ≈ 10304, P m 4.2e 5...... 34 ≈ − II.25 Figure shows the normalized amplitude of the magnetic field B 2 /(ρµ η2k2) | | 0 f as a function of Rm...... 35 II.26 Figure shows the normalized amplitude of the magnetic field B 2 /(ρµ η2k2) | | 0 f as a function of (Rm Rm ) for different values of P m...... 35 − c II.27 Figure shows the normalized amplitude of the magnetic field B 2 /(ρµ η2k2) | | 0 f as a function of (Rm Rm ) for different values of P m...... 37 − c II.28 Figure shows the time series of the magnetic energy B 2, 2D Kinetic | | energy u 2 e and kinetic energy in z-component u 2...... 38 | |2D | z| II.29 Figure shows the normalized dissipation rate of the magnetic (Joule) and the kinetic energy (Viscous) as a function of (Rm Rm )/Rm . . . 38 − c c II.30 Figure shows the dissipation length scale ℓd/L as a function of the distance from threshold (Rm Rm )/Rm for a few different values of − c c P m...... 39 II.31 Figure shows the dissipation length scale of the magnetic field ld/L as a function of P m...... 40 EB (k) II.32 Figure shows the ratio of the spectra in a) total E(k) , b) component B B E (k) E (k) wise 2D , Z as a function of k...... 41 E2D (k) EZ (k) II.33 Figure shows the thin layer domain...... 41 II.34 Figure shows the normalized magnetic energy B 2 /(ρµ η2k2) as a | | 0 f function of Rm for the thin layer problem. Here P m = 4.75...... 42 II.35 Figure shows the time series of the magnetic energy B 2 for a fer | | different Rm...... 42 II.36 Figure shows the probability distribution function of B2 for different 1 values of Rm along with the scaline B2 − ...... 44

III.1 Kazantsev theory for 2D and 3D dynamo instability. Decay/Growth shown for an infinitesimal magnetic field...... 47 III.2 Normalized growth rate γτd as a function of kz/kd for different values of Rm. Eigenvalue solver for the Kazantsev problem...... 53 III.3 Normalized growth rate γτd as a function of kz/kd for different values of D shown for a) finite Rm 1.95e+05 b) infinite Rm ...... 55 r ≈ → ∞ LIST OF FIGURES 127

III.4 Normalized growth rate γτd shown as a function of kz/kd for two different limiting procedure. First procedure Rm and then D → ∞ r → 0. Second procedure (Rm and then D ), (finite Rm and → ∞ r → ∞ Dr )...... 56 → ∞ ˜ III.5 Correlation functions hLL (˜r) , hc (˜r) for the parameters kz = 0.005,Dr = 1 for the Kazantsev model found from eigenvalue problem...... 58 B B III.6 Spectra E2D ,EZ for different kz of the Kazantsev model found from eigenvalue problem...... 60 B B III.7 Sketch of correlation functions hLL (r) , hc (r) and spectra E2D ,EZ for the theoretical Kazantsev model...... 60 III.8 Normalized growth rate γτd as a function of kz/kd. Numerical simulations of Kazantsev flow...... 62 B B III.9 Magnetic spectra E2D (k),EZ (k) is shown as a function of k. Numer- ical simulations of Kazantsev flow...... 62 III.10 Contours b 2, b 2 for the parameters Rm 210, k /k = 0.35, k = | 2D | | z| ≈ z d 0 1. Numerical simulations of Kazantsev flow...... 63 III.11 Normalized growth rate γτd as a function of kz/kd for forced Navier Stokes flow (turbulent)...... 63 B B III.12 E2D ,EZ as a function of k for forced Navier Stokes flow (turbulent). . 64 III.13 Pitch fork in the presence of multiplicative noise along with predictions of the onset from the linearized equations...... 67 III.14 λn as a function of n for different values of Rm, K/k from theoretical model...... 70 III.15 λn/n as a function of n for different values of Rm, K/k from theoretical model...... 71 III.16 Time series of total magnetic energy B 2 for different values of Rm | | with K/k = 0.01 using numerical simulations...... 72 III.17 λn/n as a function of n for different values of Rm with K/k = 0.01 using numerical simulations...... 73 III.18 λn/n as a function of n for different values of K/k, Rm using numerical simulations...... 74 III.19 Time series of B 2 in a phenomenological nonlinear induction equa- | | tion...... 75 III.20 Saturation of B 2 in a phenomenological nonlinear induction equation. 76 | | III.21 Time series B 2 in forced Navier Stokes equation...... 77 | | III.22 Saturation of B 2 in forced Navier Stokes equation...... 78 | | 2 IV.1 Illustration of the normalized energy U kf /f0 as a function of Rof indicating different scalings for different saturation mechanisms...... 84 IV.2 Figure shows the examined parameters in the phase space of Ro Re , f − f for both helical and nonhelical forcing...... 86 IV.3 Figure shows the time series of the energy u 2 for a) different values | | of Rof with Ref = 100, b) different values of Re f with fixed Rof = 0.2. 88 IV.4 Figure shows the examined parameters in the phase space of Ro Re , v− v for both helical and nonhelical forcing...... 88 128 LIST OF FIGURES

IV.5 Figure shows the examined parameters in the phase space of Ro Re , d− d for both helical and nonhelical forcing...... 89 2 IV.6 Transition from forward to inverse cascade, shown as U2D as a function of Rof for different values of Ref . Both helical and nonhelical forcing show similar behaviour...... 90 2 2 IV.7 Time series of U ,U2D for a) close to the transition Ref = 100, Rof = 0.556 and b) far from transition Ref = 100, Rof = 0.357 ...... 92 2 IV.8 Figure shows U as a function of Rof for helical and nonhelical forcing. 93 2 IV.9 Saturation mechanism I shown through U as a function of Ref with 1 Ref scaling, for helical and nonhelical forcing...... 94 2 2 IV.10 Possible saturation mechanism II shown through U2D Rof as a function 0 of Rof with Ref scaling, for helical and nonhelical forcing...... 95 IV.11 Figure shows ǫ as a function of Rof for helical and nonhelical forcing. 95 IV.12 Difference in helical and nonhelical forcing. Figure shows the time series of U 2, ǫ for helical and nonhelical forcing...... 96 3 IV.13 Figure shows the normalized dissipation rate ǫ/(kf U ) as a function of Rev for helical and nonhelical forcing. Two different scalings 1 0 of Rev− , Rev follow the different saturation mechanisms...... 97 3 IV.14 Figure shows the normalized dissipation rate ǫ/(kf U ) as a function of Rov for helical and nonhelical forcing...... 98 IV.15 Figure shows the energy spectra for both helical and nonhelical forcing for a) the nonrotating case with Ro = , Re = 200 and b) fast f ∞ f rotating case with Rof = 0.02, Ref = 200...... 99 IV.16 Figure shows the normalized energy and normalized dissipation rate 2 3 U /(f0/kf ), ǫ/(U kf ) as a function of Red for different Ω (Nonhelical flow)...... 103

IV.17 Figure shows the time series of the relative helicity ρH for different Ω (Nonhelical flow)...... 103 c IV.18 Figure shows the critical magnetic Reynolds number Rmd as a function of Red (Nonhelical flow)...... 104 IV.19 Figure shows the compensated kinetic energy spectra k2E(k) and the magnetic energy spectra M(k) for both Ω = 0, 3 (Nonhelical flow). . . . 105 IV.20 Figure shows in a) the vertical vorticity ωz in b) the vertical current jz for the case of Ω = 0 (Nonhelical flow)...... 106 IV.21 Figure shows in a) the vertical vorticity ωz in b) the vertical current jz for the case of Ω = 3 (Nonhelical flow)...... 106 IV.22 Figure shows the normalized energy and normalized dissipation rate 2 3 U /(f0/kf ), ǫ/(U kf ) as a function of Re for different Ω (Helical flow). . 107 c IV.23 Figure shows the critical magnetic Reynolds number Rmd as a function of Red (Helical flow)...... 108 IV.24 Figure shows the compensated kinetic energy spectra k2E(k) and the magnetic energy spectra M(k) for both Ω = 0, 3 (Helical flow)...... 108 IV.25 Figure shows in a) the vertical vorticity ωz in b) the vertical current jz for the case of Ω = 0 (Helical flow)...... 109 IV.26 Figure shows in a) the vertical vorticity ωz in b) the vertical current jz for the case of Ω = 3 (Helical flow)...... 110 Bibliography

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Sujet : Dynamos Turbulent en Rotation

Résumé : Dans cette thèse, nous étudions l’effet de la turbulence en rotation sur l’instabilité dynamo. Nous étudions les différentes limites de la turbulence en rotation numérique- ment et théoriquement. D’abord, nous avons considéré l’effet dynamo engendré par les écoulements quasi-bidimensionnel (un écoulement avec trois composantes de vitesse qui dépendent de deux directions), qui modélise la limite de rotation très rapide. Nous avons étudié l’amplitude de saturation du champ magnétique en fonction du nombre de Prandt magnétique pour ce type d’écoulement. Un modèle théorique est développé et comparé avec les résultats numériques. Nous avons aussi regardé l’effet d’une vitesse bruitée sur le taux de croissance des différents moments du champ magnétique. Nous avons étudié l’écoulement 3D en rotation globale pour différents régimes du paramètre de contrôle. Pour l’écoulement hydrodynamique, nous avons étudié la transition vers une cascade inverse et les différents types de saturation de la cascade inverse. Nous avons regardé l’instabilité dynamo de ces écoulements. Nous avons montré que la rotation modifie le mode le plus instable et dans certains cas peut réduire le seuil de l’instabilité dynamo.

Mots clés : mecanique des ﬂuides, turbulence, physique nonlineaire, magnétohydrody- namique, l’instabilité de dynamo, physique statistique

Subject : Rotating Turbulent Dynamos

Abstract : In this thesis, we study the effect of rotating turbulent flows on the dynamo instability. We study the different limits of rotating turbulence using numerical simulations and theoretical tools. We first look at the dynamo instability driven by quasi- twodimensional flows (flows with three components varying along two directions), which models the limit of very fast rotation. We look at the saturation amplitude of the magnetic field as a function of the magnetic Prandtl number for such flows. A theoretical model for the dynamo instability is later developed and compared with the numerical results. We also study the effect of a fluctuating velocity field on the growth rate of different moments of the magnetic field. The three dimensional rotating flow is then studied for different range of parameters. For the hydrodynamic problem, we study the transition to an inverse cascade and the different saturation mechanism of the inverse cascade. Later the dynamo instability driven by such flows is investigated. We show that the effect of rotation modifies the most unstable mode and in some cases can reduce the dynamo threshold.

Keywords : ﬂuid mechanics, turbulence, nonlinear physics, magnetohydrodynamics, dynamo instability, statistical physics