Extending square conservation to arbitrarily structured C-grids with shallow water equations

The square conservation law is implemented in atmospheric dynamic cores on latitude–longitude grids, but it is rarely implemented on quasi-uniform grids, given the difficulty involved in constructing anti-symmetrical spatial discrete operators on these grids. Increasingly more models are being developed on quasi-uniform grids, such as arbitrarily structured C-grids. Thuburn–Ringler–Skamarock–Klemp (TRiSK) is a shallow water dynamic core on an arbitrarily structured C-grid. The spatial discrete operator of TRiSK is able to naturally maintain the conservation properties of total mass and total absolute vorticity and conserving total energy with time truncation error; the first two integral invariants are exactly conserved during integration, but the total energy dissipates when using the dissipative temporal integration schemes, i.e., Runge–Kutta (RK). The method of strictly conserving the total energy simultaneously, which means conserving energy in the round-off error over the entire temporal integration period, uses both an anti-symmetrical spatial discrete operator and a square conservative temporal integration scheme. In this study, we demonstrate that square conservation is equivalent to energy conservation in both a continuous shallow water system and a discrete shallow water system of TRiSK. After that, we attempt to extend the square conservation law to the TRiSK framework. To overcome the challenge of constructing an anti-symmetrical spatial discrete operator, we unify the unit of evolution variables of shallow water equations using the Institute of Atmospheric Physics (IAP) transformation, and the temporal derivatives of new evolution variables can be expressed by a combination of temporal derivatives of original evolution variables, which means the square conservative spatial discrete operator can be obtain by using original spatial discrete operators in TRiSK. Using the square conservative Runge–Kutta scheme, the total energy is completely conserved, and there is no influence on the properties of conserving total mass and total absolute vorticity. In the standard shallow water numerical test, the square conservative scheme not only helps maintain total conservation of the three integral invariants but also creates less simulation error norms.

A Basis of Casimirs in 3D Magnetohydrodynamics

We prove that any regular Casimir in 3D magnetohydrodynamics (MHD) is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\textrm{SDiff}(M)\ltimes \mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.

Extending Square Conservation to Arbitrarily Structured C-grids with Shallow Water Equations

The square conservation theory is widely used on latitude–longitude grids, but it is rarely implemented on quasi-uniform grids, given the difficulty involved in constructing anti-symmetrical spatial discrete operators on these grids. Increasingly more models are developed on quasi-uniform grids, such as arbitrarily structured C-grids. Thuburn–Ringler–Skamarock–Klemp (TRiSK) is a shallow water dynamic core on an arbitrarily structured C-grid. The spatial discrete operator of TRiSK is able to naturally maintain the conservation properties of total mass, total absolute vorticity and instantaneous total energy. The first 2 integral invariants are entirely conserved during integration, but the total energy dissipates when using the dissipative temporal integration schemes, i.e., Runge-Kutta. The method of strictly conserving the total energy simultaneously uses both an anti-symmetrical spatial discrete operator and square conservative temporal integration scheme. In this study, we demonstrate that square conservation is equivalent to energy conservation in both a continuous shallow water system and a discrete shallow water system of TRiSK, attempting to extend the square conservation theory to the TRiSK framework. To overcome the challenge of constructing an anti-symmetrical spatial discrete operator, we unify the unit of evolution variables of shallow water equations by Institute of Atmospheric Physics (IAP) transformation, expressing the temporal trend of the evolution variable by using the original operators of TRiSK. Using the square conservative Runge-Kutta scheme, the total energy is completely conserved, and there is no influence on the properties of conserving total mass and total absolute vorticity. In the standard shallow water numerical test, the square conservative scheme not only helps maintain total conservation of the three integral invariants but also creates less simulation error norms.

A new approach to design the ruled surface

This paper considers a kind of design of a ruled surface. The design interconnects some concepts from the fields of computer-aided geometric design (CAGD) and kinematics. Dual unit spherical Bézier-like curves on the dual unit sphere (DUS) are obtained by a novel method with respect to the control points. A dual unit spherical Bézier-like curve corresponds to a ruled surface by using Study’s transference principle and closed ruled surfaces are determined via control points and also, integral invariants of these surfaces are investigated. Finally, the results are illustrated by several examples and the motion interpolation was shown as an embodiment of this method.

The dual spatial quaternionic expression of ruled surfaces

In this paper, the ruled surface which corresponds to a curve on dual unit sphere is rederived with the help of dual spatial quaternions. We extend the term of dual expression of ruled surface using dual spatial quaternionic method. The correspondences in dual space of closed ruled surfaces are quaternionically expressed. As a consequence, the integral invariants of these surfaces and the relationships between these invariants are shown

Invariants of the Axisymmetric Flows of an Inviscid Gas and Fluid with Variable Density

Material conservation laws and integral invariants are constructed for the axisymmetric flows of an inviscid compressible gas and an ideal incompressible fluid with variable density$\rho(\mathbf{x},t)$. The functional independence of the new invariants from helicity is proven.