chaos in space and time - virginia techsdross/frontiers/2013/midterm/... · chaos in space and time...
Post on 01-Oct-2020
2 Views
Preview:
TRANSCRIPT
Chaos in Space and Time • Chris Mehrvarzi, Alireza Sedighi, and Mu Xu • Faculty Sponsor: Prof. M. Paul • Midterm Presentation • ESM 6984 SS: Frontiers in Dynamical Systems
Why study chaos in space and time?
Many phenomena in nature are systems far-from equilibrium and exhibit chaotic dynamics in both space and time
NASA images
Falkowski Nature (2012) NASA image
Presentation Outline
Lattice map with “diffusive” coupling – Difference equations
Calculating Lyapunov vectors – System of ODEs
Transport in complex flow – Governing PDEs
Future work
Coupled Map Lattice 1D
)]())()((21[)( 11
)1( ni
ni
ni
ni
ni xfxfxfDxfx −++= −++
nix
f (xin ) = axi
n (1− xin )
λ = limn→∞
1n
ln $f (xin )
i=0
n−1
∑
])())()((21[)( )()(
11)(11
)()1( ni
ni
ni
ni
ni
ni
ni
ni
ni xxfxxfxxfDxxfx δδδδδ ʹ′−ʹ′+ʹ′+ʹ′= −−+++
nix 1−
nix 1+
Coupled Map Lattice 1D
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.3512
0.3514
0.3516
0.3518
0.352
0.3522
0.3524
0.3526
0.3528
0.353
0.3532
D
λ
Chaos⇒> 0λ
Lyapunov Vectors Covariant Lyapunov
Vectors Pros: • True direction in phase
space. • Reflect the direction of
perturbation • Test hyperbolicity Cons: • Difficult to calculate • Algorithm only recently
available(Ginelli (2007) and Pazo (2007))
Orthogonal Lyapunov Vectors
Pros: • Easy to calculate • Leading order Lyapunov
vector is in correct direction
• Can calculate fractal dimension
Cons • Lose all direction except
leading order
Lorenz System
𝑑𝑥/𝑑𝑡 =σ(𝑥−𝑦) 𝑑𝑦/𝑑𝑡 =𝑥(𝜌−𝑧)−𝑦 𝑑𝑧/𝑑𝑡 =𝑥𝑦−𝛽𝑧
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−16
−14
−12
−10
−8
−6
−4
−2
0
2
4
t
�
�1�2�3
𝜎=10 𝜌=28 𝛽= 8/3
Result
The direction of the second covariant Lyapunov vector and the direction of the tangent vector should be same.
Transport in Complex Flows
∂c∂t+ (u ⋅ ∇
)c = L∇
2cσ −1(∂
u∂t+ (u ⋅∇
)u) = −∇
p+∇2 u + RTz
(∂T∂t
+ (u ⋅∇)T ) =∇
2T
∇⋅u = 0
Boussinesq Equations Advection-Diffusion Equation
Ning et al. (2009) (a) (b)
L = Dκ
R = αgd3
νκΔT
Direct Numerical Simulations
100 101 102
100
101
102
t
M2
Le = 10−1
Le = 10−2
Le = 10−3
Slope = 1
[L]2∝D[T ]Diffusive Transport
Pr= 1 Pr= 1
Future Work
• Explore diagnostic tools for higher dimensional map lattice – Fractal dimension – Lyapunov spectrum – Covariant Lyapunov vectors
• Transport enhancement due to complex flow • Active transport
top related