110. To Find the Distance between Two Points, α 1 β 1 γ 1 , α 2 β 2 γ 2 in Trilinear Coordinates

1902 ◽  
Vol 2 (34) ◽  
pp. 189
Keyword(s):  
Trilinear Coordinates

1694. Trilinear Coordinates and Statics

1943 ◽  
Vol 27 (277) ◽  
pp. 214
Author(s):  
N. M. Gibbins
Keyword(s):  
Trilinear Coordinates

Three-vortex quasi-geostrophic dynamics in a two-layer fluid. Part 1. Analysis of relative and absolute motions

2013 ◽  
Vol 717 ◽  
pp. 232-254 ◽  
Author(s):  
M. A. Sokolovskiy ◽  
K. V. Koshel ◽  
J. Verron
Keyword(s):  
Vortex Motion ◽  
Point Vortex ◽  
Bottom Layer ◽  
Phase Portraits ◽  
Fundamental Result ◽  
Barotropic Fluid ◽  
Oscillation Frequencies ◽  
Trilinear Coordinates ◽  
Analytical Expressions

AbstractThe results presented here examine the quasi-geostrophic dynamics of a point vortex structure with one upper-layer vortex and two identical bottom-layer vortices in a two-layer fluid. The problem of three vortices in a barotropic fluid is known to be integrable. This fundamental result is also valid in a stratified fluid, in particular a two-layer one. In this case, unlike the barotropic situation, vortices belonging to the same layer or to different layers interact according to different formulae. Previously, this occurrence has been poorly investigated. In the present work, the existence conditions for stable stationary (translational and rotational) collinear two-layer configurations of three vortices are obtained. Small disturbances of stationary configurations lead to periodic oscillations of the vortices about their undisturbed shapes. These oscillations occur along elliptical orbits up to the second order of the Hamiltonian expansion. Analytical expressions for the parameters of the corresponding ellipses and for oscillation frequencies are obtained. In the case of finite disturbances, vortex motion becomes more complicated. In this case we have made a classification of all possible movements, by analysing phase portraits in trilinear coordinates and by computing numerically the characteristic trajectories of the absolute and relative vortex motions.

Chart of nuclei on trilinear coordinates.

1947 ◽  
Vol 24 (9) ◽  
pp. 436 ◽  
Author(s):  
D. E. Hull ◽  
W. H. Sullivan
Keyword(s):  
Trilinear Coordinates

Triangles and trilinear coordinates

1998 ◽  
Vol 29 (4) ◽  
pp. 501-509
Author(s):  
H. Brian Griffiths ◽  
Adrian Oldknow
Keyword(s):  
Trilinear Coordinates
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