Book contents
- Frontmatter
- Contents
- Preface
- List of Participants
- Magnetic Noise and the Galactic Dynamo
- On the Oscillation in Model Z
- Nonlinear Dynamos in a Spherical Shell
- The Onset of Dynamo Action in Alpha-lambda Dynamos
- Multifractality, Near-singularities and the Role of Stretching in Turbulence
- Note on Perfect Fast Dynamo Action in a Large-amplitude SFS Map
- A Thermally Driven Disc Dynamo
- Magnetic Instabilities in Rapidly Rotating Systems
- Modes of a Flux Ring Lying in the Equator of a Star
- A Nonaxisymmetric Dynamo in Toroidal Geometry
- Simulating the Interaction of Convection with Magnetic Fields in the Sun
- Experimental Aspects of a Laboratory Scale Liquid Sodium Dynamo Model
- Influence of the Period of an ABC Flow on its Dynamo Action
- Numerical Calculations of Dynamos for ABC and Related Flows
- Local Helicity, a Material Invariant for the Odd-dimensional Incompressible Euler Equations
- On the Quasimagnetostrophic Asymptotic Approximation Related to Solar Activity
- Simple Dynamical Fast Dynamos
- A Numerical Study of Dynamos in Spherical Shells with Conducting Boundaries
- Non-axisymmetric Shear Layers in a Rotating Spherical Shell
- Testing for Dynamo Action
- Alpha-quenching in Cylindrical Magnetoconvection
- On the Stretching of Line Elements in Fluids: an Approach from Differential Geometry
- Instabilities of Tidally and Precessionally Induced Flows
- Probability Distribution of Passive Scalars with Nonlinear Mean Gradient
- Magnetic Fluctuations in Fast Dynamos
- A Statistical Description of MHD Turbulence in Laboratory Plasmas
- Compressible Magnetoconvection in Three Dimensions
- The Excitation of Nonaxisymmetric Magnetic Fields in Galaxies
- Localized Magnetic Fields in a Perfectly Conducting Fluid
- Turbulent Dynamo and the Geomagnetic Secular Variation
- On-Off Intermittency: General Description and Feedback Model
- Dynamo Action in a Nearly Integrable Chaotic Flow
- The Dynamo Mechanism in the Deep Convection Zone of the Sun
- Shearing Instabilities in Magnetoconvection
- On the Role of Rotation of the Internal Core Relative to the Mantle
- Evolution of Magnetic Fields in a Swirling Jet
- Analytic Fast Dynamo Solution for a Two-dimensional Pulsed Flow
- On Magnetic Dynamos in Thin Accretion Disks Around Compact and Young Stars
- The Strong Field Branch of the Childress–Soward Dynamo
- Evidence for the Suppression of the Alpha-effect by Weak Magnetic Fields
- Turbulent Magnetic Transport Effects and their Relation to Magnetic Field Intermittency
- Proving the Existence of Negative Isotropic Eddy Viscosity
- Dynamo Action Induced by Lateral Variation of Electrical Conductivity
- Spherical Inertial Oscillation and Convection
- Hydrodynamic Stability of the ABC Flow
- Dynamos with Ambipolar Diffusion
- Subject Index
Proving the Existence of Negative Isotropic Eddy Viscosity
Published online by Cambridge University Press: 11 May 2010
- Frontmatter
- Contents
- Preface
- List of Participants
- Magnetic Noise and the Galactic Dynamo
- On the Oscillation in Model Z
- Nonlinear Dynamos in a Spherical Shell
- The Onset of Dynamo Action in Alpha-lambda Dynamos
- Multifractality, Near-singularities and the Role of Stretching in Turbulence
- Note on Perfect Fast Dynamo Action in a Large-amplitude SFS Map
- A Thermally Driven Disc Dynamo
- Magnetic Instabilities in Rapidly Rotating Systems
- Modes of a Flux Ring Lying in the Equator of a Star
- A Nonaxisymmetric Dynamo in Toroidal Geometry
- Simulating the Interaction of Convection with Magnetic Fields in the Sun
- Experimental Aspects of a Laboratory Scale Liquid Sodium Dynamo Model
- Influence of the Period of an ABC Flow on its Dynamo Action
- Numerical Calculations of Dynamos for ABC and Related Flows
- Local Helicity, a Material Invariant for the Odd-dimensional Incompressible Euler Equations
- On the Quasimagnetostrophic Asymptotic Approximation Related to Solar Activity
- Simple Dynamical Fast Dynamos
- A Numerical Study of Dynamos in Spherical Shells with Conducting Boundaries
- Non-axisymmetric Shear Layers in a Rotating Spherical Shell
- Testing for Dynamo Action
- Alpha-quenching in Cylindrical Magnetoconvection
- On the Stretching of Line Elements in Fluids: an Approach from Differential Geometry
- Instabilities of Tidally and Precessionally Induced Flows
- Probability Distribution of Passive Scalars with Nonlinear Mean Gradient
- Magnetic Fluctuations in Fast Dynamos
- A Statistical Description of MHD Turbulence in Laboratory Plasmas
- Compressible Magnetoconvection in Three Dimensions
- The Excitation of Nonaxisymmetric Magnetic Fields in Galaxies
- Localized Magnetic Fields in a Perfectly Conducting Fluid
- Turbulent Dynamo and the Geomagnetic Secular Variation
- On-Off Intermittency: General Description and Feedback Model
- Dynamo Action in a Nearly Integrable Chaotic Flow
- The Dynamo Mechanism in the Deep Convection Zone of the Sun
- Shearing Instabilities in Magnetoconvection
- On the Role of Rotation of the Internal Core Relative to the Mantle
- Evolution of Magnetic Fields in a Swirling Jet
- Analytic Fast Dynamo Solution for a Two-dimensional Pulsed Flow
- On Magnetic Dynamos in Thin Accretion Disks Around Compact and Young Stars
- The Strong Field Branch of the Childress–Soward Dynamo
- Evidence for the Suppression of the Alpha-effect by Weak Magnetic Fields
- Turbulent Magnetic Transport Effects and their Relation to Magnetic Field Intermittency
- Proving the Existence of Negative Isotropic Eddy Viscosity
- Dynamo Action Induced by Lateral Variation of Electrical Conductivity
- Spherical Inertial Oscillation and Convection
- Hydrodynamic Stability of the ABC Flow
- Dynamos with Ambipolar Diffusion
- Subject Index
Summary
We demonstrate the existence of a two-dimensional incompressible flow having a negative and isotropic eddy viscosity. Here, we understand by ‘eddy viscosity’ the sum of the molecular viscosity and of the small-scale flow contribution. The flow is deterministic, time-independent, space-periodic and has φ/3 rotational invariance. The eddy viscosity is calculated by multiscale techniques. The resulting equations for the transport coefficients are solved (i) by a Pade-resummed Reynolds number expansion and (ii) by direct numerical simulation. Results agree completely.
It is known that the action of a small-scale incompressible flow (having suitable symmetries) on a large-scale perturbation of small amplitude is ‘formally’ diffusive (Kraichnan 1976; Dubrulle & Frisch 1991). There are two essential assumptions. The first one is scale-separation: the ratio e between the typical length-scale of the basic flow and that of the perturbation is small. The second one is the absence of a large-scale AKA effect (Frisch et al 1987). If the basic flow is parity-invariant (i.e. has a center of symmetry), this condition is automatically satisfied. By ‘formally’ diffusive, we understand that, unlike the case of the eddy diffusivity for a passive scalar (Frisch 1989), the eddy viscosity tensor need not be positive definite. There are indeed examples of strongly anisotropic flows (e.g. the Kolmogorov flow), where some components of the tensor are negative, resulting in a large-scale instability (Meshalkin & Sinai 1961; Green 1974; Sivashinsky 1985; Sivashinsky & Yakhot 1985).
When the eddy viscosity tensor is isotropic, the equation for the perturbation reduces to an ordinary diffusion equation, with diffusion coefficient uE.
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- Solar and Planetary Dynamos , pp. 321 - 328Publisher: Cambridge University PressPrint publication year: 1994
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