A geometric look at momentum flux and stress in fluid mechanics

We develop a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form valued 2-forms, and their divergence as a covariant exterior derivative. We review the necessary tools of differential geometry and obtain the corresponding coordinate-free form of the equations of motion for a variety of inviscid fluid models -- compressible and incompressible Euler equations, Lagrangian-averaged Euler-$\alpha$ equations, magnetohydrodynamics and shallow-water models -- using a variational derivation which automatically yields a symmetric momentum flux. We also consider dissipative effects and discuss the geometric form of the Navier--Stokes equations for viscous fluids and of the Oldroyd-B model for visco-elastic fluids

A geometric look at MHD and the Braginsky dynamo

This is the author accepted manuscript. The final version is available from Taylor & Francis via the DOI in this recordThis paper considers magnetohydrodynamics (MHD) and some of its applications from the perspective of differential geometry, considering the dynamics of an ideal fluid flow and magnetic field on a general three-dimensional manifold, equipped with a metric and an induced volume form. The benefit of this level of abstraction is that it clarifies basic aspects of fluid dynamics such as how certain quantities are transported, how they transform under the action of mappings (for example the flow map between Lagrangian labels and Eulerian positions), how conservation laws arise, and the origin of certain approximations that preserve the mathematical structure of classical mechanics. First, the governing equations for ideal MHD are derived in a general setting by means of an action principle, and making use of Lie derivatives. The way in which these equations transform under a pull back, by the map taking the position of a fluid parcel to a background location, is detailed. This is then used to parameterise Alfv´en waves using concepts of pseudomomentum and pseudofield, in parallel with the development of Generalised Lagrangian Mean theory in hydrodynamics. Finally non-ideal MHD is considered with a sketch of the development of the Braginsky αω-dynamo in a general setting. Expressions for the α-tensor are obtained, including a novel geometric formulation in terms of connection coefficients, and related to formulae found elsewhere in the literature.Leverhulme TrustScience and Technology Facilities Council (STFC

Biometry of late Quaternary coccoliths from the Southern Cadiz region

The Cadiz region lies between the Iberian borderland and Morocco, west of the Strait of Gibraltar and the Western Mediterranean. Core GeoB9064-1 (35°24,91’N 6°50,72’W) is located in the southwest at a depth of 702 m, close to the Al Arraich mud volcano field 30 km off the Moroccan margin„ and has a length of 544 cm. Like most coastal regions, the southern Cadiz region is characterised by a coccolith assemblage dominated by the placoliths Emiliania huxleyi and Gephyrocapsa muellerae.Late Quaternary fluctuations are pronounced in this core, as shown by geochemical (TOC and CaC03) and XRF analysis (K, Mg, Fe, etc.), but also in abundances of coccoliths and more particularly Emiliania huxleyi. These can be related to upwelling and/or bottom currents.Colmenero-Hidalgo (2002) has split up Emiliania huxleyi in a larger coldwater and smaller warmwater variety based on a 4 µm cut-off value. Colmenero-Hid algo (2004) identified a deglacial decrease in the larger coldwater variety.In this study, 100 Emiliania huxleyi and 100 Gephyrocapsa muellerae lengths were measured in 30 samples. Comparison of the biometry of Emiliania huxleyi and Gephyrocapsa muellerae shows that these both species have similar fluctuations and both become smaller during the Holocene, revealing the splitting of Emiliania huxleyi in two morphotypes, to be more complex. A new method to tackle this problem is proposed