The objective of this course is to familiarise the students with analytical solution techniques of nonlinear system of equations and thereby introduce chaos. Different types of turbulence will be taught including fluid, magneto-hydrodynamic and turbulence where physics of kinetic scales are involved. Stress will be given on numerical algorithms to solve such physics problems.
LO-1: The students will learn stability analysis and bifurcation diagram of different nonlinear maps in both one and two dimensions.
LO-2: The student will be able to demonstrate different types of nonlinear attractors using basic computational schemes.
LO-3: Theory of turbulence phenomena will be taught to understand the turbulence at different length and time scales occurring in nature.
LO-4: Numerical schemes for simulating turbulence phenomena at fluid as well as kinetic scales will be demonstrated.
Flows on the line
Bifurcations
Flows on the circle
Linear systems
Phase plane
Limit cycles
Bifurcation in two dimensional flows
Lorenz equations
One-dimensional maps
Fractals
Strange attractors
Symmetries and conservation laws
Probabilistic description of turbulence
Survey of probabilistic tools
Experimental laws of fully developed turbulence
Kolmogorov's theory of 1941
Turbulence phenomenology
Intermittency
Basic equations
Flux freezing
Applications of flux freezing
Motions of lines of forces in a vacuum
Validity of hydromagnetic equations
An exact solution
Goldreich - Sridhar theory
Cowling's theorem
Mean-field dynamo models
Biermann battery
Fluctuation dynamo
Particle-in-Cell Method
Revision of numerical integration
Working knowledge of numerical interpolation
Cloud-in-Cell algorithm
Electromagnetic PIC algorithms
Algorithms for Continuum Vlasov simulation
Piecewise Parabolic Method
Galerkin Methods
Vlasov Maxwell systems
Internal Test 1 (25 Marks)
Internal Test 2 (25 Marks)
Internal Test 3 (25 Marks)
Final Exam (50 Marks)
Final evaluation sheet will be prepared using the best TWO out of the three Internal Tests (25+25 = 50 Marks) + Final Exam (50 Marks).
Nonlinear Dynamics and Chaos by Steven H Strogatz
Turbulence: The Legacy of A N Kolmogorov by Uriel Frisch
Plasma Physics for Astrophysics by Russel Kulsrud
Plasma Physics via Computer Simulation by C K Birdsall
A Critical Comparison of Eulerian-Grid-Based Vlasov Solvers, by Arber and Vann; Journal of Computational Physics 180, 339–357 (2002)