One-Dimensional Equation for a Two-Dimensional Bloch Electron in a Magnetic Field

1964 ◽  
Vol 136 (6A) ◽  
pp. A1647-A1649 ◽  
Author(s):  
J. Zak
Keyword(s):  
Magnetic Field ◽  
Two Dimensional ◽  
One Dimensional ◽  
Bloch Electron ◽  
Dimensional Equation

scholarly journals The influence of resistivity gradients on shock conditions for a Petschek reconnection geometry

2016 ◽  
Vol 34 (4) ◽  
pp. 421-425
Author(s):  
Christian Nabert ◽  
Karl-Heinz Glassmeier
Keyword(s):  
Magnetic Field ◽  
Necessary Conditions ◽  
The Other ◽  
Diffusion Region ◽  
Two Dimensional ◽  
Characteristic Velocity ◽  
Magnetohydrodynamic Equations ◽  
One Dimensional ◽  
The Magnetic Field ◽  
Alfven Velocity

Abstract. Shock waves can strongly influence magnetic reconnection as seen by the slow shocks attached to the diffusion region in Petschek reconnection. We derive necessary conditions for such shocks in a nonuniform resistive magnetohydrodynamic plasma and discuss them with respect to the slow shocks in Petschek reconnection. Expressions for the spatial variation of the velocity and the magnetic field are derived by rearranging terms of the resistive magnetohydrodynamic equations without solving them. These expressions contain removable singularities if the flow velocity of the plasma equals a certain characteristic velocity depending on the other flow quantities. Such a singularity can be related to the strong spatial variations across a shock. In contrast to the analysis of Rankine–Hugoniot relations, the investigation of these singularities allows us to take the finite resistivity into account. Starting from considering perpendicular shocks in a simplified one-dimensional geometry to introduce the approach, shock conditions for a more general two-dimensional situation are derived. Then the latter relations are limited to an incompressible plasma to consider the subcritical slow shocks of Petschek reconnection. A gradient of the resistivity significantly modifies the characteristic velocity of wave propagation. The corresponding relations show that a gradient of the resistivity can lower the characteristic Alfvén velocity to an effective Alfvén velocity. This can strongly impact the conditions for shocks in a Petschek reconnection geometry.

Wave packets, resonant interactions and soliton formation in inlet pipe flow

1993 ◽  
Vol 252 ◽  
pp. 1-30 ◽  
Author(s):  
Igor V. Savenkov
Keyword(s):  
Wave Packets ◽  
Three Dimensional ◽  
Short Wave ◽  
Nonlinear Interactions ◽  
Two Dimensional ◽  
Inlet Pipe ◽  
One Dimensional ◽  
Resonant Interactions ◽  
Dimensional Equation ◽  
Axisymmetric Disturbances

The development of disturbances (two-dimensional non-linear and three-dimensional linear) in the entrance region of a circular pipe is studied in the limit of Reynolds number R → ∞ in the framework of triple-deck theory. It is found that lower-branch axisymmetric disturbances can interact in the resonant manner. Numerical calculations show that a two-dimensional nonlinear wave packet grows much more rapidly than that in the boundary layer on a flat plate, producing a spike-like solution which seems to become singular at a finite time. Large-sized, short-scaled disturbances are also studied. In this case the development of axisymmetric disturbances is governed by single one-dimensional equation in the form of the Korteweg-de Vries and Benjamin-Ono equations in the long- and short-wave limits respectively. The nonlinear interactions of these disturbances lead to the formation of solitons which can run both upstream and downstream. Linear three-dimensional wave packets are also calculated.

scholarly journals Analytical solutions for the neutron transport using the spectral methods

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
Abdelouahab Kadem
Keyword(s):  
Analytical Solutions ◽  
Spectral Methods ◽  
Chebyshev Polynomials ◽  
Neutron Transport ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Multidimensional Problems ◽  
Dimensional Equation ◽  
Transport Problems

We present a method for solving the two-dimensional equation of transfer. The method can be extended easily to the general linear transport problem. The used technique allows us to reduce the two-dimensional equation to a system of one-dimensional equations. The idea of using the spectral method for searching for solutions to the multidimensional transport problems leads us to a solution for all values of the independant variables, the proposed method reduces the solution of the multidimensional problems into a set of one-dimensional ones that have well-established deterministic solutions. The procedure is based on the development of the angular flux in truncated series of Chebyshev polynomials which will permit us to transform the two-dimensional problem into a set of one-dimensional problems.

Split Mode Method for the Elliptic 2D Sine-Gordon Equation: Application to Josephson Junction in Overlap Geometry

1998 ◽  
Vol 09 (02) ◽  
pp. 301-323 ◽  
Author(s):  
Jean-Guy Caputo ◽  
Nikos Flytzanis ◽  
Yuri Gaididei ◽  
Irene Moulitsa ◽  
Emmanuel Vavalis
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Josephson Junction ◽  
Gordon Equation ◽  
Splitting Method ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Dimensional Solution ◽  
One Dimensional

We introduce a new type of splitting method for semilinear partial differential equations. The method is analyzed in detail for the case of the two-dimensional static sine-Gordon equation describing a large area Josephson junction with overlap current feed and external magnetic field. The solution is separated into an explicit term that satisfies the one-dimensional sine-Gordon equation in the y-direction with boundary conditions determined by the bias current and a residual which is expanded using modes in the y-direction, the coefficients of which satisfy ordinary differential equations in x with boundary conditions given by the magnetic field. We show by direct comparison with a two-dimensional solution that this method converges and that it is an efficient way of solving the problem. The convergence of the y expansion for the residual is compared for Fourier cosine modes and the normal modes associated to the static one-dimensional sine-Gordon equation and we find a faster convergence for the latter. Even for such large widths as w=10 two such modes are enough to give accurate results.

scholarly journals One-dimensional thermal equation modeled on a two-dimensional heat exchanger with variable solid-fluid interface

2021 ◽  
Vol 2090 (1) ◽  
pp. 012140
Author(s):  
Hideshi Ishida ◽  
Koichi Higuchi ◽  
Taiki Hirahata
Keyword(s):  
Heat Flux ◽  
Heat Exchanger ◽  
Model Equation ◽  
Computational Time ◽  
Two Dimensional ◽  
Fluid Interface ◽  
Thermal Equation ◽  
Fluid Equation ◽  
One Dimensional ◽  
Dimensional Equation

Abstract In this study, we are to present that a one-dimensional equation for vertically averaged temperature, modeled on a vertically thin, two-dimensional heat exchanger with variable top solid-fluid interface, recovers the two-dimensional thermal information, i.e. steady temperature and flux distribution on the top and temperature-fixed bottom faces. The relative error of these quantities is less than 5% with the maximum gradient of the height kept approximately below 0.5, while the computational time is reduced to 0.1–5%, when compared with direct two-dimensional computations, depending on the shape of the top face. The model equation, derived by the vertical averaging of the two-dimensional thermal conduction equation, is closed by an approximation that the heat exchanger is sufficiently thin in the sense that the second derivative of temperature with respect to the horizontal coordinate depends only on the coordinate. In this model equation, the fluid equation above the exchanger is decoupled by a conventional equation for the normal heat flux on the top surface. In principle, however, the coupling of the model and the fluid equation is possible through the temperature and heat flux on the top interface, recovered by the model equation. The type of mathematical modeling can be applicable to a wide variety of bodies with extremely small dimensions in some (coordinate-transformed) directions.

Parametric instability in the expanding solar wind

2021 ◽  
Author(s):  
Andrea Verdini ◽  
Roland Grappin ◽  
Francesco Malara ◽  
Leonardo Primavera ◽  
Luca Del Zanna
Keyword(s):  
Magnetic Field ◽  
Solar Wind ◽  
Parametric Instability ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Total Magnetic Field ◽  
One Dimensional ◽  
Preliminary Results ◽  
Dimensional Spectrum ◽  
N Wave

<p>Recent measurments of Parker Solar Probe show that alfvenic fluctuations in the solar wind often appear in the form of swithcback with constant total magnetic field. Our aim is to understand if and how such fluctuations can contribute to the heating or acceleration of the solar wind, via the Parametric Instability. The intability of one dimensional Alfvénic fluctuations has been extensively studied in both homogenoeus plasma and in the expanding solar wind, less so for the two-dimensional case which is closer to expected three-dimensional nature of switchbacks. In this work we study under which condition an Alfvén wave with a two dimensional spectrum (as introduced in Primavera et al ApJ 2019) can decay in the expanding solar wind and we will present preliminary results.</p>

Magnetometric resistivity (MMR) anomalies of two‐dimensional structures

Geophysics ◽  
1979 ◽  
Vol 44 (5) ◽  
pp. 947-958 ◽  
Author(s):  
E. Gomez Trevino ◽  
R. N. Edwards
Keyword(s):  
Magnetic Field ◽  
Integral Equation ◽  
Fourier Transform ◽  
Electric Field ◽  
Current Source ◽  
Two Dimensional ◽  
Galvanic Current ◽  
One Dimensional ◽  
The Magnetic Field ◽  
Surface Integrals

An inexpensive, rapid method has been developed for computing all three components of the magnetic field due to galvanic current flow from a point electrode in the vicinity of a conductive subsurface structure of infinite strike‐length and arbitrary cross‐section. For any three‐dimensional (3-D) structure, the magnetic field may be written as a sum of surface integrals over boundaries defining changes in conductivity by a direct modification of the Biot‐Savart law. The integrand of each surface integral includes the components of the electric field tangential to the boundary, which may be evaluated on the boundary using a standard integral equation technique. In the case of a two‐dimensional (2-D) structure, a reformulation of the theory by taking a one‐dimensional Fourier transform along the strike results in the reduction of both the surface integrals necessary to solve the integral equation for the electric field, and the integrals used in computing the magnetic field, to line integrals in wavenumber domain. We evaluate the integrals numerically and solve the integral equation for each of about ten wavenumbers; finally, we obtain the magnetic field in space domain through a concluding one‐dimensional inverse Fourier transform. Type curves and characteristic curves for the simple model of a buried horizontal cylinder beneath a thin layer of conductive overburden are constructed. In the absence of overburden, the half‐width of the anomaly is linearly related to the depth of the cylinder. In the presence of overburden, the form of the anomaly may be predicted in a simple manner from the corresponding anomaly in the absence of overburden, provided the distance from the current source is sufficiently large for most of the available current to have penetrated the overburden.

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