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This text focuses on conservation laws in magnetohydrodynamics,
gasdynamics and hydrodynamics. A grasp of new conservation laws is
essential in fusion and space plasmas, as well as in geophysical
fluid dynamics; they can be used to test numerical codes, or to
reveal new aspects of the underlying physics, e.g., by identifying
the time history of the fluid elements as an important key to
understanding fluid vorticity or in investigating the stability of
steady flows. The ten Galilean Lie point symmetries of the
fundamental action discussed in this book give rise to the
conservation of energy, momentum, angular momentum and center of
mass conservation laws via Noether's first theorem. The advected
invariants are related to fluid relabeling symmetries - so-called
diffeomorphisms associated with the Lagrangian map - and are
obtained by applying the Euler-Poincare approach to Noether's
second theorem. The book discusses several variants of helicity
including kinetic helicity, cross helicity, magnetic helicity,
Ertels' theorem and potential vorticity, the Hollman invariant, and
the Godbillon Vey invariant. The book develops the non-canonical
Hamiltonian approach to MHD using the non-canonical Poisson
bracket, while also refining the multisymplectic approach to ideal
MHD and obtaining novel nonlocal conservation laws. It also briefly
discusses Anco and Bluman's direct method for deriving conservation
laws. A range of examples is used to illustrate topological
invariants in MHD and fluid dynamics, including the Hopf invariant,
the Calugareanu invariant, the Taylor magnetic helicity
reconnection hypothesis for magnetic fields in highly conducting
plasmas, and the magnetic helicity of Alfven simple waves, MHD
topological solitons, and the Parker Archimedean spiral magnetic
field. The Lagrangian map is used to obtain a class of solutions
for incompressible MHD. The Aharonov-Bohm interpretation of
magnetic helicity and cross helicity is discussed. In closing,
examples of magnetosonic N-waves are used to illustrate the role of
the wave number and group velocity concepts for MHD waves. This
self-contained and pedagogical guide to the fundamentals will
benefit postgraduate-level newcomers and seasoned researchers
alike.
With This Ring is a Bible study for couples, classes, and
counselors who are exploring the Biblical teachings about marriage.
Many of the passages studied are familiar, but the perspective and
emphasis is uniquely designed to cause serious reflection on the
changes needed for our marriages to grow stronger and more pleasing
to the Lord.
Many Christians secretly battle powerful, deadly spiritual forces
-- often without realizing the source or the solution to the misery
they have endured all too long. Are you one of those? Free Indeed
is a devotional workbook for those believers who are struggling
with destructive, "besetting" sins. Each daily reading calls
attention to a specific principle that leads a disciple of the Lord
into the freedom that only the Son can give. The daily format
includes a brief Scripture passage, reflections, and a sample
prayer to express the heart of the liberating truth focus for the
day. The author himself has experienced how the truth of God's Word
brought salvation and deliverance during the time when he was
addicted to drugs, fascinated with the occult, and obsessed with
suicidal thoughts. You too can be transformed by the truth of
Scripture and receive the joyous freedom of the Lord.
This is the story of Keith Stanworth, a man who was shaped by his
mother and father, the land he grew up on, the people around him,
and the faith he was taught. More than that, it is the story of a
man who took full responsibility for what he became-for good or
bad. When he was a young man, his teachers, coaches, and leaders
influenced him greatly. He wanted the chance to use his love of
sports to help others shape their lives and so decided to become a
coach, focusing his abilities and dedication on shaping young
people. He was raised in the outdoors and couldn't have imagined a
career without time on a mountain-a fact that led him to his home
with the Forest Service. In both careers, he had some of the most
wonderful-and frightening-experiences of his life. This memoir
shares the story of the life of Keith Stanworth. Sometimes people
see the same things differently, but this is the way he saw it.
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