On modified quaternionic analysis, irrotational velocity fields and new gradient dynamical systems in R3

2013 ◽  
Author(s):  
Dmitry Bryukhov
Keyword(s):  
Dynamical Systems ◽  
Velocity Fields ◽  
Quaternionic Analysis ◽  
Modified Quaternionic Analysis

Modified Quaternionic Analysis in R[sup 3] and Generalizations of Conformal Mappings of the Second Kind

2011 ◽  
Author(s):  
Dmitry Bryukhov ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Conformal Mappings ◽  
Quaternionic Analysis ◽  
Modified Quaternionic Analysis

scholarly journals Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics

2001 ◽  
Vol 8 (1/2) ◽  
pp. 69-94 ◽  
Author(s):  
C. Coulliette ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Ocean Circulation ◽  
Circulation Model ◽  
Velocity Fields ◽  
Ocean Circulation Model ◽  
Wind Stress Curl ◽  
Temporal Complexity ◽  
Lobe Dynamics ◽  
Spatio Temporal ◽  
Model Subject

Abstract. We study the flow obtained from a three-layer, eddy-resolving quasigeostrophic ocean circulation model subject to an applied wind stress curl. For this model we will consider transport between the northern and southern gyres separated by an eastward jet. We will focus on the use of techniques from dynamical systems theory, particularly lobe dynamics, in the forming of geometric structures that govern transport. By "govern", we mean they can be used to compute Lagrangian transport quantities, such as the flux across the jet. We will consider periodic, quasiperiodic, and chaotic velocity fields, and thus assess the effectiveness of dynamical systems techniques in flows with progressively more spatio-temporal complexity. The numerical methods necessary to implement the dynamical systems techniques and the significance of lobe dynamics as a signature of specific "events", such as rings pinching off from a meandering jet, are also discussed.

scholarly journals Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets

2002 ◽  
Vol 9 (3/4) ◽  
pp. 237-263 ◽  
Author(s):  
K. Ide ◽  
D. Small ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Vector Fields ◽  
Analytical Formula ◽  
Velocity Fields ◽  
Dynamical Systems Theory ◽  
Time Dependent ◽  
Time Interval ◽  
Data Set ◽  
Hyperbolic Trajectories

Abstract. In this paper we develop analytical and numerical methods for finding special hyperbolic trajectories that govern geometry of Lagrangian structures in time-dependent vector fields. The vector fields (or velocity fields) may have arbitrary time dependence and be realized only as data sets over finite time intervals, where space and time are discretized. While the notion of a hyperbolic trajectory is central to dynamical systems theory, much of the theoretical developments for Lagrangian transport proceed under the assumption that such a special hyperbolic trajectory exists. This brings in new mathematical issues that must be addressed in order for Lagrangian transport theory to be applicable in practice, i.e. how to determine whether or not such a trajectory exists and, if it does exist, how to identify it in a sequence of instantaneous velocity fields. We address these issues by developing the notion of a distinguished hyperbolic trajectory (DHT). We develop an existence criteria for certain classes of DHTs in general time-dependent velocity fields, based on the time evolution of Eulerian structures that are observed in individual instantaneous fields over the entire time interval of the data set. We demonstrate the concept of DHTs in inhomogeneous (or "forced") time-dependent linear systems and develop a theory and analytical formula for computing DHTs. Throughout this work the notion of linearization is very important. This is not surprising since hyperbolicity is a "linearized" notion. To extend the analytical formula to more general nonlinear time-dependent velocity fields, we develop a series of coordinate transforms including a type of linearization that is not typically used in dynamical systems theory. We refer to it as Eulerian linearization, which is related to the frame independence of DHTs, as opposed to the Lagrangian linearization, which is typical in dynamical systems theory, which is used in the computation of Lyapunov exponents. We present the numerical implementation of our method which can be applied to the velocity field given as a data set. The main innovation of our method is that it provides an approximation to the DHT for the entire time-interval of the data set. This offers a great advantage over the conventional methods that require certain regions to converge to the DHT in the appropriate direction of time and hence much of the data at the beginning and end of the time interval is lost.

On modified quaternionic analysis in $ {\Bbb R}^3 $

1998 ◽  
Vol 70 (3) ◽  
pp. 228-234 ◽  
Author(s):  
Sirkka-Liisa Eriksson-Bique ◽  
Heinz Leutwiler
Keyword(s):  
Quaternionic Analysis ◽  
Modified Quaternionic Analysis

scholarly journals Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics

2000 ◽  
Vol 7 (1/2) ◽  
pp. 59-85 ◽  
Author(s):  
C. Coulliette ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Ocean Circulation ◽  
Circulation Model ◽  
Velocity Fields ◽  
Ocean Circulation Model ◽  
Wind Stress Curl ◽  
Temporal Complexity ◽  
Lobe Dynamics ◽  
Spatio Temporal ◽  
Model Subject

Abstract. We study the flow obtained from a three-layer, eddy-resolving quasigeostrophic ocean circulation model subject to an applied wind stress curl. For this model we will consider transport between the northern and southern gyres separated by an eastward jet. We will focus on the use of techniques from dynamical systems theory, particularly lobe dynamics, in the forming of geometric structures that govern transport. By "govern", we mean they can be used to compute Lagrangian transport quantities, such as the flux across the jet. We will consider periodic, quasiperiodic, and chaotic velocity fields, and thus assess the effectiveness of dynamical systems techniques in flows with progressively more spatio-temporal complexity. The numerical methods necessary to implement the dynamical systems techniques and the significance of lobe dynamics as a signature of specific "events", such as rings pinching off from a meandering jet, are also discussed.

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