scholarly journals A computational study of residual KPP front speeds in time-periodic cellular flows in the small diffusion limit

2015 ◽  
Vol 311-312 ◽  
pp. 37-44 ◽  
Author(s):  
Penghe Zu ◽  
Long Chen ◽  
Jack Xin
Keyword(s):  
Computational Study ◽  
Diffusion Limit ◽  
Small Diffusion ◽  
Cellular Flows ◽  
Kpp Front Speeds ◽  
Time Periodic

Metastability for a generalized Burgers equation with applications to propagating flame fronts

1999 ◽  
Vol 10 (1) ◽  
pp. 27-53 ◽  
Author(s):  
X. SUN ◽  
M. J. WARD
Keyword(s):  
Flame Front ◽  
Principal Eigenvalue ◽  
Vertical Channel ◽  
Front Propagation ◽  
Slow Motion ◽  
Diffusion Limit ◽  
Asymptotic Results ◽  
Small Diffusion ◽  
Generalized Burgers Equation ◽  
Exponentially Small

In the small diffusion limit ε→0, metastable dynamics is studied for the generalized Burgers problemformula hereHere u=u(x, t) and f(u) is smooth, convex, and satisfies f(0)=f′(0)=0. The choice f(u)=u2/2 has been shown previously to arise in connection with the physical problem of upward flame-front propagation in a vertical channel in a particular parameter regime. In this context, the shape y=y(x, t) of the flame-front interface satisfies u=−yx. For this problem, it is shown that the principal eigenvalue associated with the linearization around an equilibrium solution corresponding to a parabolic-shaped flame-front interface is exponentially small. This exponentially small eigenvalue then leads to a metastable behaviour for the time- dependent problem. This behaviour is studied quantitatively by deriving an asymptotic ordinary differential equation characterizing the slow motion of the tip location of a parabolic-shaped interface. Similar metastability results are obtained for more general f(u). These asymptotic results are shown to compare very favourably with full numerical computations.

scholarly journals Advection of a passive scalar by a vortex couple in the small-diffusion limit

1994 ◽  
Vol 270 ◽  
pp. 219-250 ◽  
Author(s):  
Joseph F. Lingevitch ◽  
Andrew J. Bernoff
Keyword(s):  
Boundary Layer ◽  
Diffusion Equation ◽  
Particle Method ◽  
Passive Scalar ◽  
Outer Edge ◽  
Diffusion Limit ◽  
Asymptotic Results ◽  
Small Diffusion ◽  
Diffusive Boundary ◽  
Large Peclet Number

We study the advection of a passive scalar by a vortex couple in the small-diffusion (i.e. large Péclet number, Pe) limit. The presence of weak diffusion enhances mixing within the couple and allows the gradual escape of the scalar from the couple into the surrounding flow. An averaging technique is applied to obtain an averaged diffusion equation for the concentration inside the dipole which agrees with earlier results of Rhines & Young for large times. At the outer edge of the dipole, a diffusive boundary layer of width O(Pe−½) forms; asymptotic matching to the interior of the dipole yields effective boundary conditions for the averaged diffusion equation. The analysis predicts that first the scalar is homogenized along the streamlines on a timescale O(Pe−$\frac{1}{3}$). The scalar then diffuses across the streamlines on the diffusive timescale, O(Pe). Scalar that diffuses into the boundary layer is swept to the rear stagnation point, and a finite proportion is expelled into the exterior flow. Expulsion occurs on the diffusive timescale at a rate governed by the lowest eigenvalue of the averaged diffusion equation for large times. A split-step particle method is developed and used to verify the asymptotic results. Finally, some speculations are made on the viscous decay of the dipole in which the vorticity plays a role analogous to the passive scalar.

Surface and deep ocean connectivity inferred from robust extraction of coherent sets in ocean flow using models and sparse, scattered, and incomplete float data with transfer operator and dynamic Laplacian methods.

2021 ◽  
Author(s):  
Gary Froyland ◽  
Ryan Abernathey ◽  
Michael Denes ◽  
Shane Keating
Keyword(s):  
Coherent Structures ◽  
Finite Time ◽  
Transfer Operator ◽  
Deep Ocean ◽  
Difficult Problem ◽  
Lagrangian Coherent Structures ◽  
Diffusion Limit ◽  
Small Diffusion ◽  
Transport And Mixing ◽  
Atmospheric Vortices

<p>Transport and mixing properties of the ocean's circulation is crucial to dynamical analyses, and often have to be carried out with limited observed information. Finite-time coherent sets are regions of the ocean that minimally mix (in the presence of small diffusion) with the rest of the ocean domain over the finite period of time considered. In the purely advective setting (in the zero diffusion limit) this is equivalent to identifying regions whose boundary interfaces remain small throughout their finite-time evolution. Finite-time coherent sets thus provide a skeleton of distinct regions around which more turbulent flow occurs. Well known manifestations of finite-time coherent sets in geophysical systems include rotational objects like ocean eddies, ocean gyres, and atmospheric vortices. In real-world settings, often observational data is scattered and sparse, which makes the difficult problem of coherent set identification and tracking challenging. I will describe mesh-based numerical methods [3] to efficiently approximate the recently defined dynamic Laplace operator [1,2], and rapidly and reliably extract finite-time coherent sets from models or scattered, possibly sparse, and possibly incomplete observed data. From these results we can infer new chemical and physical ocean connectivities at global and intra-basin scales (at the surface and at depth), track series of eddies, and determine new oceanic barriers.</p><p>[1] G. Froyland. Dynamic isoperimetry and the geometry of Lagrangian coherent structures. <em>Nonlinearity</em>, 28:3587-3622, 2015</p><p>[2] G. Froyland and E. Kwok. A dynamic Laplacian for identifying Lagrangian coherent structures on weighted Riemannian manifolds. <em>Journal of Nonlinear Science</em>, 30:1889–1971, 2020.</p><p>[3] Gary Froyland and Oliver Junge. Robust FEM-based extraction of finite-time coherent sets using scattered, sparse, and incomplete trajectories. <em>SIAM J. Applied Dynamical Systems</em>, 17:1891–1924, 2018.</p>

Sign in / Sign up

Export Citation Format

Share Document