Poly-Dedekind sums associated with poly-Bernoulli functions

Apostol considered generalized Dedekind sums by replacing the first Bernoulli function appearing in Dedekind sums by any Bernoulli functions and derived a reciprocity relation for them. Recently, poly-Dedekind sums were introduced by replacing the first Bernoulli function appearing in Dedekind sums by any type 2 poly-Bernoulli functions of arbitrary indices and were shown to satisfy a reciprocity relation. In this paper, we consider other poly-Dedekind sums that are obtained by replacing the first Bernoulli function appearing in Dedekind sums by any poly-Bernoulli functions of arbitrary indices. We derive a reciprocity relation for these poly-Dedekind sums.

Eulerian and Lagrangian Comparison of Wind Jets in the Tokar Gap Region

The Lagrangian and Eulerian structure and dynamics of a strong wind event in the Tokar Gap region are described using a Weather Research and Forecasting (WRF) model hindcast for 2008. Winds in the Tokar Gap reach 25 m s−1 and remain coherent as a jet far out over the Red Sea, whereas equally strong wind jets occurring in neighboring gaps are attenuated abruptly by jump-like hydraulic transitions that occur just offshore of the Sudan coast. The transition is made possible by the supercritical nature of the jets, which are fed by air that spills down from passes at relatively high elevation. By contrast, the spilling flow in the ravine-like Tokar Gap does not become substantially supercritical and therefore does not undergo a jump, and also carries more total horizontal momentum. The Tokar Wind Jet carries some air parcels across the Red Sea and into Saudi Arabia, whereas air parcel trajectories in the neighboring jets ascend as they cross through the jumps, then veer sharply to the southeast and do not cross the Red Sea. The mountain parameter Nh/U is estimated to lie in the range of 1.0–4.0 for the general region, a result roughly consistent with a gap jet having a long extension, and supercritical flows spilling down from higher elevation passes. The strong event is marked by the formation of a feature with a vertical cellular structure in the upstream entrance region of the Tokar Gap, a feature absent from the more moderate events that occur throughout the summer. The cell contains descending air parcels that are fed into the Tokar Gap and one of the neighboring gaps. An analysis of the Bernoulli function along air parcel trajectories reveals an approximate balance between the loss of potential energy and gain of internal energy and pressure, with surprisingly little contribution from kinetic energy, along the path of the descending flow. The winds in all gaps attain the critical wind speed nominally required to loft dust into the atmosphere, though only the Tokar Gap has a broad, silty delta region capable of supplying particulate matter for dust storms.

Identities on poly-Dedekind sums

Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

Eulerian and Lagrangian Comparison of Primary and Secondary Wind Jets in the Tokar Gap Region

The Lagrangian and Eulerian structure and dynamics of a strong wind event in the Tokar Gap region are described using a WRF model hindcast for 2008. Winds in the Tokar Gap reach 25 m s-1 and remain coherent as a jet far out over the Red Sea, whereas equally strong wind jets occurring in neighboring gaps are attenuated abruptly by a jump-like hydraulic transition that occur just offshore of the Sudan coast. The transition is made possible by the supercritical nature of the jets, which are fed by air that spills down from passes at relatively high elevation. By contrast, the spilling flow in the ravine-like Tokar Gap does not become substantially supercritical and therefore does not undergo a jump, and also carries more total horizontal momentum. The Tokar Wind Jet carries some air parcels across the Red Sea and into Saudi Arabia, whereas air parcel trajectories in the neighboring jets ascend as they cross through the jumps, then veer sharply to the southeast and do not cross the Red Sea. The mountain parameter Nh/U is estimated to lie in the rage 1.0-4.0 for the general region, a result roughly consistent with a primary gap jet having a long extension, and supercritical jets spilling down from higher elevation passes. The strong event is marked by the formation of a cyclonic cell near the upstream entrance to the Tokar Gap, a feature absent from the more moderate events that occur throughout the summer. The cell contains descending air parcels that are fed into the primary and secondary jets. An analysis of the Bernoulli function along air parcel trajectories reveals an approximate balance between the loss of potential energy and gain of internal energy and pressure, with surprisingly little contribution from kinetic energy, along the path of the descending flow. All jets attain the critical wind speed nominally required to loft dust into the atmosphere, though only the Tokar Gap has a broad, delta region with plentiful deposits of silt.

Time dependence of advection-dominated accretion flow around a rotating compact object

ABSTRACT Time evolution of advection-dominated accretion flow (ADAF) around a rotating compact object is presented. The time-dependent equations of fluid including the Coriolis force along with the centrifugal and pressure gradient forces are derived. In this research, it is assumed that angular momentum transport is due to viscous turbulence and the α-prescription is used for the kinematic coefficient of viscosity. Moreover, the general relativistic effects are neglected. In order to solve the equations, we have used a self-similar solution. The solutions show that the behaviour of the physical quantities in a dynamical ADAF is different from that for a steady accretion flow. Our results indicate that the physical quantities are dependent of rotation parameter which is defined as the ratio of the intrinsic angular velocity of the central body to the angular velocity of disc. Also, the effect of rotation parameter on these quantities is different for co and counter-rotating flows. The solution shows that by increasing the rotation parameter a, inflow–outflow region approaches the central object for co-rotating flow and moves outwards for counter-rotating flow. We find that when flow is fully advection dominated (f → 1), the entire gas has positive Bernoulli function. Also, we suggest that the Bernoulli function becomes more positive when the effect of rotation on the structure of disc decreases.

Euler flows and singular geometric structures

Tichler proved (Tischler D. 1970 Topology 9 , 153–154. ( doi:10.1016/0040-9383(70)90037-6 )) that a manifold admitting a smooth non-vanishing and closed one-form fibres over a circle. More generally, a manifold admitting k -independent closed one-form fibres over a torus T k . In this article, we explain a version of this construction for manifolds with boundary using the techniques of b -calculus (Melrose R. 1993 The Atiyah Patodi Singer index theorem . Research Notes in Mathematics. Wellesley, MA: A. K. Peters; Guillemin V, Miranda E, Pires AR. 2014 Adv. Math. ( N. Y. ) 264 , 864–896. ( doi:10.1016/j.aim.2014.07.032 )). We explore new applications of this idea to fluid dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold's structure theorem and describe b -symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities (Miranda E, Oms C. 2018 Contact structures with singularities. https://arxiv.org/abs/1806.05638 ) and what we call b -Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance Etnyre J, Ghrist R. 2000 Trans. Am. Math. Soc. 352 , 5781–5794. ( doi:10.1090/S0002-9947-00-02651-9 )). These results provide a new technique to analyse the geometry of steady fluid flows on non-compact manifolds with cylindrical ends. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

A note on the stationary Euler equations of hydrodynamics

This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic Bernoulli function, then it can be rescaled to the Reeb vector field of a stable Hamiltonian structure. In particular, such a vector field has a periodic orbit unless the 3-manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis.

On the differences between two kinds of mean value formulas of number-theoretic error terms

In this paper, we shall consider differences between the "discrete" and the "continuous" mean value formulas for any number-theoretic error term. We first derive new type formulas of these differences, which are independent from the periodic Bernoulli function. As a direct consequence of this result, we shall give the alternative proofs of previous results in the case of the Dirichlet divisor and the circle problems, and improve the previous results of the differences in higher power cases. Furthermore, we apply our present result to the case of Rankin–Selberg's series.

Potential Vorticity Budgets in the North Atlantic Ocean

This paper focuses on potential vorticity (PV) budgets in the North Atlantic with an emphasis on the wind-driven subtropical gyre. Since PV is the key dynamical variable of the wind-driven circulation, these budgets are important to understand. PV is a conservative quantity on isopycnals and can only enter or exit through the boundaries, like the lateral topography or the surface. The latter fluxes are diagnosed and tested against the evolution of the PV content in an isopycnal layer. The former are computed using the Bernoulli function. The essential result is found for all the tested isopycnals, and the dominant feature of PV is recirculation, with very little added at the surface or the boundaries. Density coordinates are well suited to understanding PV circulation. A novel technique for computing the Bernoulli function is proposed. The Bernoulli function is governed by a simple elliptic equation and the solutions demonstrate the dominant contribution of PV advection.