Beltrami flows

IX. Beltrami Flow as Archetypal Field Structure A Schauberger-Beltrami Connection ... 

Consequently, (4) represents a necessary and sufficient Beltrami condition. Since the Beltrami flow (1) describes parallel or antiparallel vorticity and velocity vectors, another useful formulation of the Beltrami condition is represented by the relation... 

In other words, in a solenoidal Beltrami field the vector lines are situated in the surfaces c = constant. This theorem was originally derived by Ballabh [4] for a Beltrami flow proper of an incompressible medium. For the sake of completeness, we mention that the combination of the three conditions (1), (2), and (3) only leads to a Laplacian field, that is better defined by a vector field that is both solenoidal (divergence-less) and lamellar (curlless). 

In the case of uniform c(grad c = 0), curl v = w (vorticity) is also a Beltrami field, possessing the same coefficient c. This type of vector field is called a Trkalian field, after Trkal, a Russian researcher who studied Beltrami flows... 

There are two known standard methods for decomposition of any smooth (differentiable) vector field. One is that attributed to Helmholtz, which splits any vector field into a lamellar (curl-free) component, and a solenoidal (divergenceless) component. The second, which divides a general vector field into lamellar and complex lamellar parts, is that popularized by Monge. However, the relatively recent discovery by Moses [7] shows that any smooth vector field— general or with restraints to be determined—may also be separable into circularly polarized vectors. Furthermore, this third method simplifies the otherwise difficult analysis of three-dimensional classical flow fields. The Beltrami flow field, which has a natural chiral structure, is particularly amenable to this type analysis. 

One of the underlying themes of this exposition is the suggestion that the Beltrami flow field could play an important but yet dimly suspected archetypal role in organizing matter and energy at a deeper level of nature. One indication of this might be the possible non-linear uncertainty principle cited earlier, which... 

An example of a Beltrami flow is the ABC flow, named after Arnold, Beltrami and Childress. It is defined by the velocity field... 

Geometric PDEs and DG theories of surfaces provide a natural and simple description for a solvent-solute interface. In 2005, Wei and his collaborators, including Michael Feig, pioneered the use of curvature-controlled PDEs for molecular surface construction and solvation analysis [120]. In 2006, based on DG, Wei and co-workers introduced the first variational solvent-solute interface the minimal molecular surface (MMS), for molecular surface representation [121-123]. With a constant surface tension, the minimization of surface free energy is equivalent to the minimization of surface area, which can be implemented via the mean curvature flow, or the Laplace-Beltrami flow, and gives rise to the MMS. The... 

However, it is well known that two-dimensional turbulence exhibits low dissipation at high Reynolds numbers due to spectral blocking (l.e., energy actually undergoes a reverse cascade from the small scales to the large scales which dramatically reduces the dissipation see Fjortoft [11]). In a Beltrami flow. 

However, since the energy cascade arises from the Lamb vector (a X u (which vanishes for a Beltrami flow), it follows that... 

Beltrami flows have low dissipation at high Reynolds numbers. Hence, Beltrami flows and two-dimensional flows each have low dissipation at high Reynolds numbers, but have helicity densities that are at the opposite extremes (the latter has a small normalized... 

As argued by Reed [4], the Beltrami vector field originated in hydrodynamics and is force-free. It is one of the three basic types of field solenoidal, complex lamellar, and Beltrami. These vector fields originated in hydrodynamics and describe the properties of the velocity field, flux or streamline, v, and the vorticity V x v. The Beltrami field is also a Magnus force free fluid flow and is expressed in hydrodynamics as... 

This vector field condition is sometimes referred to as Beltrami fluid flow, and was previously treated in a similar exposition by the author in 1995 [1], There it was indicated that Beltrami vector field flow is representative of a certain class of vector fields that are termed force-free. This type of field topology was first brought to prominence by Eugenio Beltrami in his 1889 paper Considerations on Hydrodynamics. [2], This type of morphology describes a regime of fluid... 

However, in a Beltrami field, the vorticity and velocity vectors are parallel or antiparallel, resulting in a zero Magnus force. The Beltrami condition (1) is therefore an equivalent way of characterizing a force-free flow situation, and vice versa. 

In order to model this type of flow field geometrically, Beltrami found that it was necessary to consider a three-dimensional circular axisymmetric flow in which the velocity and vorticity field lines described a helical pattern. This helicoidal flow field was unique in that the pitch of the circular helices decreased as the radius from the central axis increased. This produces a specialized shear effect between the field lines of successively larger cylindrical tubes constituting the respective helices. In the limit of such a field, the central axis of the flow also serves as a field line (see Fig. 3). 

Figure 8. Dissected diagram of the vector configuration of a pair of Beltrami vortex filaments formed in the current sheath of the plasma focus (v — flow velocity, B = local magnetic field, j = current density, (>) — vorticity, Bo — background magnetic field).

O. Bjorgum and T. Godal, On Beltrami Vector Fields and Flows, Part II, Universitet I Bergen... 

Additionally, the history of 5 O or 5D from the ice core can be used as a short cut to interpreting the borehole-temperature record (Paterson and Clarke 1978 Cuffey et al. 1992, 1994, 1995 Johnsen et al. 1995, Cuffey and Clow 1997 Johnsen et al. 1997 also see Beltrami and Taylor 1995). The records of ice from GISP2 and GRIP were used for central Greenland. A provisional relation between the of ice formed from accumulated snow and the surface temperature translates the record of the ice core into a provisional surface-temperature history, which is used to drive a time-dependent heat- and ice-flow model to predict modern temperature versus depth in the ice sheet. The provisional relation between of ice and temperature is then adjusted to optimize... 

The GPB and Laplace-Beltrami models discussed in the previous section were obtained from a variational principle applied to equilibrium systems. For chemical and biological systems far from equilibrium, it is necessary to incorporate additional equations (e.g., the Nernst-Planck equation) to describe the dynamics of charged particles. Various DG-based Nernst-Planck equations have derived from mass conservation laws in earlier work by Wei and co-workers [74, 75]. We outline the basic derivation here. For simplicity in derivation, we assume that the flow stream velocity vanishes ( v = 0) and we omit the chemical reactions in our present discussion. 

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