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E-grāmata: Large-Scale Perturbations of Magnetohydrodynamic Regimes: Linear and Weakly Nonlinear Stability Theory

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  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Physics 829
  • Izdošanas datums: 28-Jun-2011
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642181702
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Large-Scale Perturbations of Magnetohydrodynamic Regimes: Linear and Weakly Nonlinear Stability TheoryPalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Physics 829
  • Izdošanas datums: 28-Jun-2011
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642181702
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New developments for hydrodynamical dynamo theory have been spurred by recent evidence of self-sustained dynamo activity in laboratory experiments with liquid metals.

The emphasis in the present volume is on the introduction of powerful mathematical techniques required to tackle modern multiscale analysis of continous systems and there application to a number of realistic model geometries of increasing complexity.

This introductory and self-contained research monograph summarizes the theoretical state-of-the-art to which the author has made pioneering contributions.
1 Introduction
1(14)
1.1 Notation
13(2)
2 Large-Scale Passive Scalar Transport
15(10)
2.1 The Linear Stability Problem
15(3)
2.1.1 The Governing Equations
16(1)
2.1.2 Asymptotic Expansion of a Large-Scale Stability Mode
17(1)
2.2 The Kernel of the Passive Scalar Transport Operator and Solvability of Auxiliary Problems
18(2)
2.2.1 Stability of Passive Scalar Admixture Transport
19(1)
2.2.2 The Hierarchy of Equations for a Large-Scale Mode
19(1)
2.2.3 Solvability of the Order ε0 Equation
20(1)
2.3 Perturbation of the Neutral Mode
20(3)
2.3.1 Solution of the Order ε0 and ε1 Equations
21(1)
2.3.2 Eddy Diffusion
21(1)
2.3.3 Enhancement of Molecular Diffusion
22(1)
2.4 Conclusions
23(2)
3 Steady-Flow Dynamos
25(36)
3.1 Kinematic Magnetic Dynamos: The Statement of the Problem
26(2)
3.1.1 The Eigenvalue Problem for the Magnetic Induction Operator
26(1)
3.1.2 Asymptotic Expansion of Large-Scale Magnetic Modes
27(1)
3.1.3 The Hierarchy of Equations for the Large-Scale Modes
27(1)
3.2 The Kernel of the Magnetic Induction Operator and Solvability of Auxiliary Problems
28(1)
3.3 Magnetic α-Effect
29(6)
3.3.1 Solution of the Order ε0 Equations
30(3)
3.3.2 The Solvability Condition for the Order ε1 Equations
33(2)
3.4 Magnetic Eddy Diffusion in Steady Parity-Invariant Flows
35(3)
3.4.1 Solution of the Order ε1 Equations in the Absence of α-Effect
35(1)
3.4.2 The Solvability Condition for the Order ε2 Equations: The Operator of Magnetic Eddy Diffusion
36(2)
3.5 Complete Asymptotic Expansion of Large-Scale Magnetic Modes
38(3)
3.6 How Rare is Negative Magnetic Eddy Diffusivity?
41(5)
3.7 Strongly Negative Magnetic Eddy Diffusivity
46(5)
3.8 Perturbation of an Oscillatory Short-Scale Magnetic Mode
51(5)
3.8.1 Solvability Condition for Auxiliary Problems
52(1)
3.8.2 Order ε1 Equation: The α-Effect
53(1)
3.8.3 Order ε1 Equations in the Absence of the α-Effect
54(1)
3.8.4 Order ε2 equations in the Absence of α-Effect: Eddy Diffusion Revisited
55(1)
3.9 Magnetic Field Generation with Moderate Scale Separation
56(3)
3.10 Conclusions
59(2)
4 Time-Periodic Large-Scale Magnetic Dynamos
61(32)
4.1 The Floquet Problem for Large-Scale Magnetic Modes
62(5)
4.1.1 Statement of the Problem
62(1)
4.1.2 Asymptotic Expansion of Large-Scale Magnetic Modes
63(1)
4.1.3 Solvability of Auxiliary Problems
64(3)
4.1.4 Solution of the Floquet Problem for a Generic Time-Periodic Parity-Invariant Flow
67(1)
4.2 Magnetic Eddy Diffusion in Time-Periodic Parity-Invariant Flows of the Simple Structure
67(7)
4.2.1 The Hierarchy of Equations for Flows of the Simple Structure
67(2)
4.2.2 Solution of Order ε0 Equations
69(2)
4.2.3 Solution of Order ε1 Equations
71(1)
4.2.4 The Solvability Condition for Order ε2 Equations: The Operator of Magnetic Eddy Diffusion
72(1)
4.2.5 Solution of Order ε2 Equations
73(1)
4.3 Complete Asymptotic Expansion of Large-Scale Modes for Flows of the Simple Structure
74(3)
4.4 Computation of the Tensor of Magnetic Eddy Diffusion
77(4)
4.5 Is Periodic Time Dependence of Flows Favourable for Negative Magnetic Eddy Diffusivity?
81(4)
4.6 The Limit of High Frequencies for Flows of the Simple Structure
85(5)
4.7 Conclusions
90(3)
5 Convective Plan Form Dynamos in a Layer
93(22)
5.1 Statement of the Problem
94(5)
5.1.1 Boundary Conditions
94(1)
5.1.2 Neutral Magnetic Modes
95(1)
5.1.3 The Symmetry About a Vertical Axis
95(1)
5.1.4 Asymptotic Expansion of Large-Scale Magnetic Modes and the Hierarchy of Equations for Large-Scale Magnetic Modes
96(1)
5.1.5 Solvability of Auxiliary Problems
97(2)
5.2 Magnetic Eddy Diffusion in Flows, Symmetric About a Vertical Axis
99(4)
5.2.1 Solution of Order ε0 Equations
99(1)
5.2.2 The Solvability Condition for Order ε1 Equations: Insignificance of the α-Effect
100(1)
5.2.3 Solution of Order ε1 Equations
101(1)
5.2.4 The Solvability Condition for Order ε2 Equations: The Operator of Magnetic Eddy Diffusion
102(1)
5.3 Complete Asymptotic Expansion of Large-Scale Magnetic Modes
103(2)
5.4 Enhancement of Molecular Diffusion in Plane-Parallel Flows in a Layer
105(1)
5.5 Convective Plan Forms in a Layer of a Non-rotating Fluid
106(3)
5.5.1 Analytic Expressions
106(1)
5.5.2 Additional Symmetries
107(1)
5.5.3 Magnetic Eddy Diffusion Operator for Convective Plan Forms in a Layer
108(1)
5.6 Numerical Investigation of Magnetic Eddy Diffusion
109(4)
5.7 Conclusions
113(2)
6 Linear Stability of Steady MHD States to Large-Scale Perturbations
115(20)
6.1 Statement of the Linear Stability Problem
116(2)
6.1.1 The Governing Equations
116(1)
6.1.2 Asymptotic Expansion of Large-Scale MHD Stability Modes
117(1)
6.1.3 The Hierarchy of Equations for the Large-Scale Modes
117(1)
6.2 Solvability of Auxiliary Problems
118(2)
6.3 The Combined MHD α-Effect
120(3)
6.3.1 Solution of Order ε0 Equations
121(1)
6.3.2 The Solvability Condition for Order ε1 Equations: The Operator of the Combined MHD α-Effect
122(1)
6.4 Complete Expansion of Large-Scale MHD Modes in the Presence of the α-Effect
123(4)
6.4.1 Solution of Order ε1 Equations
123(1)
6.4.2 Solution of Order εn, n > 1 Equations
124(3)
6.5 MHD Eddy Diffusion in Steady Parity-Invariant MHD States
127(2)
6.5.1 Solution of Order ε1 Equations in the Absence of α-Effect
127(1)
6.5.2 The Solvability Condition for Order ε2 Equations: The Operator of the Combined MHD Eddy Diffusion
128(1)
6.6 Complete Expansion of Large-Scale MHD Stability Modes in the Absence of the α-Effect
129(3)
6.7 How Rare is Negative MHD Eddy Diffusivity?
132(2)
6.8 Conclusions
134(1)
7 Weakly Nonlinear Stability of MHD Regimes to Large-Scale Perturbations
135(26)
7.1 Statement of the Problem
136(4)
7.1.1 The Governing Equations
137(2)
7.1.2 Asymptotic Expansion of Weakly Nonlinear Large-Scale Perturbations
139(1)
7.1.3 The Hierarchy of Equations for the Perturbations
139(1)
7.2 Solvability of Auxiliary Problems
140(3)
7.3 Large-Scale Instability in the Presence of the MHD α-Effect
143(4)
7.3.1 Solution of Order ε0 Equations
143(2)
7.3.2 The Solvability Condition for Order ε1 Equations: Linear Equations for Weakly Nonlinear Perturbations
145(2)
7.4 Equations for Weakly Nonlinear Perturbations in the Absence of the MHD α-Effect
147(10)
7.4.1 Solution of Order ε1 Equations
147(4)
7.4.2 Bounds for Linear Problems with the Operator of Linearisation
151(4)
7.4.3 Mean-Field Equations for Large-Scale Perturbations
155(2)
7.5 Computation of Coefficients of the Combined MHD Eddy Operators
157(2)
7.6 Conclusions
159(2)
8 Weakly Nonlinear Stability of Forced Thermal Hydromagnetic Convection
161(58)
8.1 Statement of the Problem
163(5)
8.1.1 The Governing Equations for the Thermal Hydromagnetic Convection
163(1)
8.1.2 Boundary Conditions
163(1)
8.1.3 Equations for Perturbations of a CHM Regime
164(2)
8.1.4 Asymptotic Expansion of Weakly Nonlinear Large-Scale Perturbations
166(1)
8.1.5 The Hierarchy of Equations for Perturbations
166(2)
8.2 Mathematical Tools
168(8)
8.2.1 Reconstruction of the Flow from the Vorticity
168(2)
8.2.2 Solvability of Auxiliary Problems
170(5)
8.2.3 A Bound for a Smooth Vector Field, Solenoidal in Fast Variables
175(1)
8.3 The Combined Magnetic and Kinematic α-Effects
176(8)
8.3.1 Solution of Order ε0 Equations
177(3)
8.3.2 Solvability of Order ε1 Equations
180(4)
8.4 The Symmetries of the CHM Regime Guaranteeing Insignificance of the MHD α-Effect
184(3)
8.5 Mean-Field Equations for Large-Scale Perturbations in the Case of Insignificant α-Effect
187(12)
8.5.1 Solution of Order ε1 Equations
187(7)
8.5.2 A Solvability Condition for Order ε2 Equations: The Equation for the Mean Magnetic Perturbation
194(2)
8.5.3 A Solvability Condition for Order ε3 Equations: The Equation for the Mean Flow Perturbation
196(3)
8.6 Large-Scale Perturbations of Convective Hydromagnetic Regimes Near a Point of Bifurcation
199(15)
8.6.1 Asymptotic Expansion
202(1)
8.6.2 The Order ε0 Equation and Solvability Conditions for Auxiliary Problems
203(1)
8.6.3 The Combined α-Effect
204(3)
8.6.4 Insignificant α-Effect: Solution of Order ε1 Equations
207(1)
8.6.5 Mean-Field Equations Derived from Order ε2 Equations
208(2)
8.6.6 Amplitude Equations Derived from Order ε2 Equations
210(4)
8.7 Computation of Coefficients of the MHD Eddy Operators
214(2)
8.8 Conclusions
216(3)
9 Weakly Nonlinear Stability of Free Thermal Hydromagnetic Convection
219(24)
9.1 Statement of the Problem
220(1)
9.2 Neutral Short-Scale Linear Stability Modes
221(2)
9.3 Solvability of Auxiliary Problems
223(1)
9.4 The Combined α-Effect
224(3)
9.5 Amplitude Equations in the Absence of Significant α-Effect
227(6)
9.5.1 Solvability Conditions for Order ε1 Equations
227(3)
9.5.2 Solution of Order ε1 Equations
230(3)
9.6 Solvability Conditions for Order ε2 and ε3 Equations
233(2)
9.7 An Alternative Amplitude Equation for Steady Symmetric CHM Regimes
235(4)
9.8 Non-zero Initial Mean Flow Perturbation <<Vo>>h
239(1)
9.9 Conclusions
240(3)
10 Magnetic Field Generation by a Two-Scale Flow in an Axisymmetric Volume
243(34)
10.1 Mathematical Tools
244(9)
10.1.1 Functional Sobolev Spaces Wq(ω)
244(1)
10.1.2 The Scale of Functional Sobolev Spaces Hq(ω) for a Fluid Inside a Dielectric
245(5)
10.1.3 Bounds in the Norms of Sobolev Spaces
250(3)
10.2 The Kinematic Dynamo Problem
253(5)
10.2.1 Statement of the Problem
253(1)
10.2.2 Formal Asymptotic Expansion of Large-Scale Magnetic Modes
254(1)
10.2.3 The Hierarchy of Equations for Large-Scale Magnetic Modes
254(1)
10.2.4 Solvability Condition for an Equation ∂2{h}/∂Φ2 = g
255(1)
10.2.5 Derivation of the Limit Operator
256(2)
10.3 Uniform Boundedness and Convergence of Resolvents
258(5)
10.3.1 Uniform Boundedness of Resolvents
258(2)
10.3.2 Convergence of the Resolvents
260(3)
10.4 Complete Asymptotic Expansion of Magnetic Modes with Internal Scaling
263(6)
10.4.1 Solution of the Hierarchy of Equations in all Orders
263(2)
10.4.2 Convergence of Asymptotic Series
265(4)
10.5 Physical Properties of the Axisymmetric Dynamo
269(5)
10.5.1 The Dynamo Mechanism and the Magnetic Reynolds Numbers
270(1)
10.5.2 Comparison with Braginsky's Dynamo
271(1)
10.5.3 Helicity of the Flow Potential and the Coefficient of Magnetic α-Effect
272(2)
10.6 Conclusions
274(3)
11 Magnetic Field Generation by a Two-Scale Flow in a Sphere
277(20)
11.1 Bounds in the Norms of Sobolev Spaces
277(4)
11.2 The Spherical Kinematic Dynamo
281(5)
11.2.1 Statement of the Problem
281(1)
11.2.2 Derivation of the Limit Operator
281(5)
11.3 Uniform Boundedness and Convergence of Resolvents
286(2)
11.3.1 Uniform Boundedness of Resolvents
286(1)
11.3.2 Convergence of the Resolvents
287(1)
11.4 Complete Asymptotic Expansion of Magnetic Modes
288(3)
11.4.1 Solution of the Hierarchy of Equations in all Orders
289(2)
11.4.2 Convergence of Asymptotic Series
291(1)
11.5 Physical Properties of the Spherical Dynamo
291(3)
11.5.1 The Axisymmetric Dynamo
292(1)
11.5.2 Helicity of the Flow Potential and its Relation to the Magnetic α-Effect
293(1)
11.6 Conclusions
294(3)
12 Concluding Remarks
297(6)
Appendix A Mean-Field and Amplitude Equations for Bifurcations of Three-Dimensional MHD Regimes 303(6)
References 309(16)
Index 325
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