C. G. Phillips

Chilling to zero degrees disrupts pollen formation but not meiotic microtubule arrays in Triticum aestivum L.

Throughout the wheat-growing regions of Australia, chilling temperatures below 2 °C occur periodically on consecutive nights during the period of floral development in spring wheat (Triticum aestivum L.). In this study, wheat plants showed significant reductions in fertility when exposed to prolonged chilling temperatures in controlled environment experiments. Among the cultivars tested, the Australian cultivars Kite and Hartog had among the lowest levels of seed set due to chilling and their responses were investigated further. The developmental stage at exposure, the chilling temperature and length of exposure all influenced the level of sterility. The early period of booting, and specifically the +4 cm auricle distance class, was the most sensitive and corresponded to meiosis within the anthers. The response of microtubules to chilling during meiosis in Hartog was monitored, but there was little difference between chilled and control plants. Other abnormalities, such as plasmolysis and cytomixis increased in frequency, were associated with death of developing pollen cells, and could contribute to loss of fertility. The potential for an above-zero chilling sensitivity in Australian spring wheat varieties could have implications for exploring the tolerance of wheat flower development to chilling and freezing conditions in the field.

Spectral interactions of rapidly-rotating anisotropic turbulent viscous and thermal diffusion in the Earth’s core

Abstract Rapidly-rotating anisotropic turbulence in the Earth’s core is modelled by the viscous and thermal diffusion tensors, D ν ≔2ρν 0 I +ρν ΩΩ ΩΩ and D κ ≔κ 0 I +κ ΩΩ ΩΩ , where I is the unit tensor, Ω is the angular velocity and the coefficients ν ΩΩ and κ ΩΩ are spherically-symmetric. Toroidal–poloidal spectral interactions are derived for the anisotropic parts of the body forces associated with the mean anisotropic viscous stress tensor, D ν ·(∇ v ) S and with its symmetric part. The stress tensors are linear in the trace-free symmetric part of the velocity gradient. Techniques of vector and tensor spherical harmonic analysis are used to find the vector spherical harmonic components of the body forces. From these components the toroidal–poloidal field interactions, (ΩΩtt n ) , (ΩΩst n ) , (ΩΩts n ) , (ΩΩss n ) , (ΩtΩt n ) , (ΩsΩt n ) , (ΩtΩs n ) and (ΩsΩs n ) , of the toroidal ( t n ) and poloidal ( s n ) momentum equations are derived using computer algebra. The temperature spectral interactions are also derived for the mean heat flux given by the turbulent thermal diffusion tensor. These spectral interactions represent a computationally practical first step in incorporating anisotropic turbulence models into existing dynamically-consistent angular–spectral geodynamo codes.

Spherical anisotropic diffusion models for the Earth's core

Abstract Estimates of the molecular values of magnetic, viscous and thermal diffusion in the Earths core suggest turbulent small-scale velocity, magnetic and temperature fluctuations. Instability arguments (Braginsky, S.I., Meytlis, V.P., 1990. Local turbulence in the Earths core. Geophys. Astrophys. Fluid Dynam. 55 (1990) 71–87.) indicate that the viscous and thermal diffusions of the resulting turbulence are strongly anistropic, the directions of anisotropy being determined by the mean magnetic field, the rotation of the core, the mean temperature gradient and gravity. Physical principles and invariance arguments are used to constrain the forms of the turbulent viscous stress tensor to eight types and the turbulent heat flux vector to a single type. The models are interpreted in terms of angular momentum, kinetic energy and entropy exchange between the mean and fluctuating fields. For the development of the turbulence models in spherical geometries, general non-linear spectral expansions are derived for one of the turbulent viscous stress tensors, the turbulent heat flux vector and their divergences using vector and tensor spherical harmonic methods and poloidal–toroidal representations. The spectral expansions, which are complicated and would be difficult to derive using other methods, include all invariance terms and can be programmed directly from the explicit forms given, thus laying the foundation for future computational studies of core turbulence. They are particularly suitable for linearised problems, such as anisotropic convection and magnetoconvection. The spectral techniques employed are applicable to all eight turbulent viscous stress tensors.

Vector and scalar spherical harmonic spectral equations for rapidly rotating anisotropic alpha-effect dynamos

Spectral equations are derived for a mean field induction equation, with an α -effect, considered appropriate for rapid rotation, given by , where are Cartesian unit vectors, a 1(r, θ, φ), a 3(r, θ, φ) are scalar functions of position, (r, θ, φ) are spherical polar coordinates and R is the magnetic Reynolds number. The effect of rotation on convection for different boundaries and parameters is discussed. The effect of the flow structure on α for different geostrophic and near geostrophic models is analysed. The vector spherical harmonics where , , , the 2 × 3 matrix is a Wigner 3J coefficient and are scalar spherical harmonics, are used to derive the vector forms of the induction equation for this α -effect. The solenoidal condition is imposed by relating the formalism to the toroidal–poloidal harmonic formalism, and . The and components of the induction equation are thus derived in terms of , the components of F ; . These combined / , / vector spectral equations are then transformed into interaction type , , , and , , , equations for the isotropic and anisotropic components of α . As an application of the general spectral equations derived herein, the interaction equations can be specialised by restricting a 1 and a 3 to be proportional to r cos θ or cos θ, or restricting and α to be axisymmetric. These equations are then compared to those of previous works. The differences between the equations derived herein and those of past works provide corrections and account for, at least in part, the differences in numerical solutions of the past works.

From isotropic to rapidly rotating anisotropic alpha-squared dynamos

Mean field -dynamos in a sphere with an insulating exterior are considered for a steady -effect of the form , , derived by Moffatt (Dynamo action associated with random inertial waves in a rotating conducting fluid. J. Fluid Mech. 1970, 44, 705–719) for strong rotation. The unit vector is aligned with the angular velocity and as , . We consider the effect on axisymmetric magnetic fields of increasing rotation rate, i.e. , so that the -effect varies from isotropic to anisotropic, for two models. We find that both equatorially symmetric and antisymmetric critical solutions bifurcate from steady to oscillatory at some value of . The bifurcation is associated with the poleward migration of strong magnetic fields, leaving a region of weak field near the equator. This region increases with the rotation rate. No antidynamo action was found at the fast rotation limit . The -effect is re-scaled to for allowing all possible solutions to be studied by restricting to . In particular, we consider the mean field axisymmetric antidynamo limit at . Our results are in complete agreement with those in Phillips (Mean dynamos. Ph.D. Thesis, University of Sydney, 1993), where is considered.

Homogeneous planar and two-dimensional mean-field antidynamo theorems with zero mean flow

In an electrically conducting fluid, two types of turbulence with a preferred direction are distinguished: planar turbulence, in which every velocity in the turbulent ensemble of flows has no component in the given direction; and two-dimensional turbulence, in which every velocity in the turbulent ensemble is invariant under translation in the preferred direction. Under the additional assumptions of two-scale and homogeneous turbulence with zero mean flow, the associated magnetohydrodynamic alpha- and beta-effects are derived in the second-order correlation approximation (SOCA) when the electrically conducting fluid occupies all space. Limitations of the SOCA are well known, but alpha- and beta-effects of a turbulent flow are useful in interpreting the dynamo effects of the turbulence. Two antidynamo theorems, which establish necessary conditions for dynamo action, are shown to follow from the special structures of these alpha- and beta-effects. The theorems, which are analogues of the laminar planar velocity and two-dimensional antidynamo theorems, apply to all turbulent ensembles with the prescribed alpha- and beta-effects, not just the planar and two-dimensional ensembles. The mean magnetic field is general in the planar theorem but only two-dimensional in the two-dimensional theorem. The two theorems relax the previous restriction to turbulence which is both two-dimensional and planar. The laminar theorems imply decay of the total magnetic field for any velocity of the associated turbulent ensemble. However, the mean-field theorems are not fully consistent with the laminar theorems because further conditions beyond those arising from the turbulence must be imposed on the beta-effect to establish decay of the mean magnetic field. In particular, negative turbulent magnetic diffusivities must be restricted. It is interesting that there is no inconsistency in the alpha-effects. The failure of the SOCA with the two-scale approximation to simply preserve the laminar antidynamo theorems at the beta-effect level is a further demonstration of the restricted validity of the theory and shows that negative diffusivity effects derived by approximation methods must be treated cautiously.

Bounds on the infimum decay rate for axisymmetric incompressible dynamos

Abstract The variational lower bound v > 0.39π2 determined by Ivers (1984) for the infimum decay rate v of axisymmetric poloidal magnetic fields is corroborated numerically and an upper bound v < 0.66°2 established. This is achieved by correcting and extending results for certain flows considered by Chandrasekhar (1956).

Dynamo Problems in Spherical and Nearly Spherical Geometries

Hybrid vector spherical harmonic / poloidal-toroidal spherical spectral forms of the linearised magnetohydroynamic equations are described. The equations are highly structured with relatively few terms and form the basis of computer codes, which implement a wide range of dynamo problems in spherical and nearly spherical geometries.