Mechanical and Aerospace Engineering, UC San Diego
Mechanical and Aerospace Engineering, UC San Diego
Engineering Mechanics, Zhejiang University
Research Interests
Theoretical and numerical studies of differential equations and dynamical systems relevant to mixing and transport processes in environmental & geophysical flows and their impacts.
Associate Professor
Assistant Professor
Postdoctoral Associate
Postdoctoral Researcher
Arizona State University
Arizona State University
Massachusetts Institute of Technology
UC San Diego
Other Affiliations
Affiliated Member
Mathematical, Computational and Modeling Sciences Center
Mechanical & Aerospace Engineering, ASU
Grants and Fellowships

PI: "Collaborative Research: Theories and experiments on scalar mixing in chaotic flows", 2012-2015
CoPI: "MCTP: Mathematics Mentoring Partnership Between ASU and Maricopa County Community College District", 2012-2017
CoPI: "CMG: Multiscale Modeling of Urban Atmospheres in a Changing Climate", 2009-2014 (webpage)
CoPI: "SCREMS: Visualization of Complex Spatio-Temporal Multiscale Fluid Dynamic Phenomena", 2009-2010
PI: A2C2, ASU, "High resolution simulation of terminal winds at the Hong Kong International Airport", 2011-2012 (CPU hours)
2012: Staff member, Woods Hole Oceanographic Institution Summer Geophysical Fluid Dynamics Program
2009-2010: ProjectNExT Fellow



Fluid Stirring, Coherent Structures

Fluid transport is the result of the interplay between stirring and diffusion processes. In nature additional chemical and physical processes impose further complexities to transport problems. In the past decade the use of dynamical systems methods has become popular to identify coherent structures as a result of stirring. Our group is interested in new applications that use information based on the identified coherent structures, and new models using measures from these coherent structures. Some on-going projects include identifying turbulence structures in atmospheric flows for aviation safety; pattern formation and modeling for pollutant transport; mixing dynamics for charged flows in earth's ionosphere; and modeling diffusion-reaction in ecological systems based on identification of the coherent structures. Overall the question that needs to be addressed is: given some resolved flow (from model or observations) what can we tell about the transport dynamics at different scales and what will that bring to our understandings of physical/chemical/biological/ecological processes in the environment that may have major impact on human activities?

Irreversible Mixing
We live in an environment where turbulence is ubiquitous. With diffusion, turbulence redistributes scalars which irreversibly mix. To quantify irreversible mixing one is typically interested in the rate of mechanical energy dissipation and the decay rate of scalar variance. Our group is interested in using novel mathematical tools to estimate these quantities. One direction we have taken is the use of integral constraints and variational calculus to generate optimal bounds on energy dissipation and scalar variance. We use this approach to study irreversible mixing in environmental and geophysical flows. Recent projects include quantification of energy dissipation rate subject to novel physical constraints and optimal bound calculation on scalar variance. We are interested in generalizing these results to turbulent flows subject to different forcing mechanisms and seek the use of these quantifications in real parameterizations.
Gravity Waves
Internal gravity waves (IGW) is the stratified analog to surface waves. Energy is converted from the global flow to supply wave motion. It's significance in the ocean is such that after the generation at the site IGW can propagate away and break. This could lead to elevated mixing in the ocean interior, which helps homogenizing heat, salt and nutrient from different bodies of water. Experiments, simulations and analytical models have been developed to address the generation of IGW in the lab or in the ocean. Our group is interested in estimating the total rate of energy conversion and its modal composition of wave energy. Some past projects involve modeling the energy conversion rate given complex topography, and finding the energy pathways when IGW reflect at the boundaries. Our interest is to further develop theoretical and numerical tools for better models on IGW conversion in large-scale simulations.


Calculus For Engineers II Syllabus  
Methods of integration, applications of calculus, elements of analytic geometry, improper integrals, Taylor series.
Calculus For Engineers III Syllabus  
Vector-valued functions of several variables, partial derivatives, multiple integration.
MAT275(FA11) Modern Differential Equations Syllabus  
Introduces differential equations, theoretical and practical solution techniques. Applications. Problem solving using MATLAB. Credit is allowed for only MAT 275 or 274 toward a mathematics degree.
Applied Linear Algebra Syllabus  
Solving linear systems, matrices, determinants, vector spaces, bases, linear transformations, eigenvectors, norms, inner products, decompositions, applications. Problem solving using MATLAB.
Intro Chaos/Nonlinear Dynamics Syllabus  
Properties of nonlinear dynamical systems; dependence on initial conditions; strange attractors; period doubling; bifurcations; symbolic dynamics; Smale-Birkhoff theorem; and applications.
Applied Partial Differential Equations Syllabus  
Second-order partial differential equations, emphasizing Laplace, wave, and diffusion equations. Solutions by the methods of characteristics, separation of variables, and integral transforms.
Differential Equations II Syllabus  
Methods of characteristics. Classification of second order partial differential equations. Fourier Series, convergence theorems, Fourier transforms. Separation of variables. Greens identities. Fundamental solutions and Greens functions, sturm-Liouville problems. Harmonic Functions, maximum Principles.
Applied Dynamical Systems Methods Syllabus  
Applies modern dynamical systems methods to fluid mechanics: bifurcations, normal forms, nonlinear dynamics, pattern formation, mixing, and Lagrangian chaos.


In the news
'Model simulates flow patterns of urban PM', Environmental Health Perspectives, NIH
'Urban wind flows deposit pollutants in repetitive patterns', Wired
'Wind Concentrates Pollutants with Unexpected Order in an Urban Environment', AIP Newswise
'The skeleton of water', The Economist
'Finding Order in the Apparent Chaos of Currents', New York Times
Research project highlights
Ionospheric dynamics

Dynamics of ion density and electric field subject to inertial gravity wave forcing during a wave breaking event.

Reference: The response of charged density due to breaking inertial gravity wave in the lower regions of ionosphere (2014), Tang, W. & Mahalov, A., Phys. Plasmas, accepted

Variability of scalar reaction in coherent structures (right click to play flash movie)

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Varability of scalar reaction dependent on coherent structures. Reaction speed is enhanced in hyperbolic regions and suppressed in elliptic regions. The variability of the reaction rates can be modeled with finite-time Lyapunov exponents characterizing early-time separation of trajectories.

Reference: Dependence of advection-diffusion-reaction on flow coherent structures (2013), Tang, W. & Luna, C., Phys. Fluids, 25, 106602

Stochastic coherent structures in realistic urban flows

(Physics of fluids cover) Stochastic Lagrangian coherent structures in an urban flow based on realistic diffusivity. Signatures of structures still visible even when full scale of randomness near the atmospheric boundary layer is considered.

Reference: The geometry of inertial particle mixing in urban flows, from deterministic and random displacement models (2012), Tang, W., Knutson, B., Mahalov, A & Dimitrova, R., Phys. Fluids, 24, 063302

Turbulent structures near Hong Kong International Airport (right click to play flash movie)

Evolution of radial velocity detected by LIDAR along with backward-time FDFTLE extracted from the retrieved 2D velocity field. Also shown are Hovmoller diagrams at various ranges indicated by the dotted lines in the movies. First movie: recirculation bubble detachment and regeneration next to a mountain peak. Second movie: a persistent ridge of updraft originated inside a mountain gap.

Reference: Lagrangian Coherent Structure analysis of terminal winds detected by LIDAR. Part II: structure evolution and flight data analyses (2011). Tang, W. Haller, G. & Chan, P.W., J. Appl. Meteorol. Clim., 50, 2167-2183

Lagrangian signatures of a jet stream and balloon measurements

Comparison between the repelling LCS generated from model data and atmospheric characteristics estimated by weather balloon measurements. Color contour is the Forward-time DLE. Black line originating from the big island is the balloon trajectory, green curve is refractive index structure constant Cn2 and blue curve is the dissipation rate.

Reference: Lagrangian Coherent Structures Near a Subtropical Jet Stream (2010), Tang, W., Mathur, M., Haller, G., Hahn, D.C. & Ruggiero, F.H., J. Atmos. Sci., 67, 7, 2307-2319.

Finite-size Pollutant Particle Transport in Urban Street Canyon (right click to play flash movie)

Attracting Lagrangian Coherent Structures (LCS) inside an urban street canyon for finite-size inertial particles. The LCS are computed from the slow manifoid velocity derived from simulation data. The forward-time motion shows that an inertial particles is attracted to the local maxima of the Direct Lyapunov Exponent field. The backward-time motion shows the results for inversion of the finite-size particle using different schemes.

Reference: Locating an atmospheric contamination source using slow manifolds (2009), Tang, W., Haller, G., Baik, J.-J. & Ryu, Y.-H., Phys. Fluids, 21, 043302.



Baik, J.-J. Seoul National University  
Calhoun, R. MAE, ASU
Caulfield, C.P. DAMTP, University of Cambridge
Chan, P.W. Hong Kong Observatory
del-Castillo-Negrete, D., Oak Ridge National Lab
Dimitrova, R., CEE, Notre Dame
Fernando, H.J.S., CEE, Notre Dame
Hahn, D.C. Hanscom AFB
Haller, G. ETC Zurich
Kerswell, R.R. Math, University of Bristol
Llewellyn Smith, S.G. MAE, UC San Diego
Ouellette, N, ME, Yale
Mahalov, A. Math, ASU
Peacock, T. ME, MIT
Ruggiero, F.H. Hanscom AFB
Stocker, R. CEE, MIT
Taylor, J.R. DAMTP, University of Cambridge  
Thiffeault, J.-L., Mathematics, Wisconsin  
Young, W.R. SIO  


Graduate Students   Undergrads  
  Brent Knutson, PhD Candidate, Applied Mathematics   Christopher Luna, Undergraduate, Math & Physics
  Phillip Walker, PhD Candidate, Applied Mathematics   Aditya Dhumuntarao, Undergraduate, Math & Physics
  Kimberly Jones, MSc, Applied Mathematics    
  Tim Lai, Undergraduate, Mathematics Angelica Deibel, Undergraduate, Mathematics
  Christian Wake, Undergraduate, Mathematics Hershey Kelley, Undergraduate, Mathematics
  James Upton, Undergraduate, Mathematics Craig Gassaway, Undergraduate, Math & Physics
    Juan Durazo, Undergraduate, Mathematics
  Inez Ibarra, Biosciences High School Lena Tamrat, Biosciences High School


Refereed Journal Publications
18. Lagrangian Coherent Structure Analysis of Terminal Winds: Three-Dimensionality, Intramodel Variations, and Flight Analyses, (2015), Knutson, B., Tang, W. & Chan, P.W., Advances in Meteorology, 2015, 816727, [pdf] DOI:10.1155/2015/816727
17. Application of short-range LIDAR in early alerting for low-level windshear and turbulence at Hong Kong international airport (2014), Hon, K.K., Chan, P.W., Chiu, Y.Y. & Tang, W., Advances in Meteorology, 2014, 162748, [pdf] DOI: 10.1155/2014/162748
16. The response of charged density due to breaking inertial gravity wave in the lower regions of ionosphere (2014), Tang, W. & Mahalov, A., Phys. Plasmas, 21, 042901[pdf] DOI: 10.1063/1.4870760
15. Dependence of advection-diffusion-reaction on flow coherent structures (2013), Tang, W. & Luna, C., Phys. Fluids, 25, 106602, [pdf] DOI: 10.1063/1.4823991
14. Stochastic Lagrangian dynamics for charged flows in the E-F regions of ionosphere (2013), Tang, W. & Mahalov, A., Phys. Plasmas, 20, 032305, [pdf] DOI:10.1063/1.4794735
13. Finite-time statistics of scalar diffusion in Lagrangian coherent structures (2012), Tang, W. & Walker, P., Phys. Rev. E, 86, 045201(R).[pdf] DOI: 10.1103/PhysRevE.86.045201PRE Kaleidoscope
12. The geometry of inertial particle mixing in urban flows, from deterministic and random displacement models (2012), Tang, W., Knutson, B., Mahalov, A & Dimitrova, R., Phys. Fluids, 24, 063302. [pdf] DOI:10.1063/1.4729453 Phys. Fluids cover
11. Lagrangian Coherent Structure analysis of terminal winds detected by LIDAR. Part II: structure evolution and flight data analyses (2011), Tang, W., Haller, G. & Chan, P.W., J. Appl. Meteor. Clim., 50, 2167-2183[pdf] DOI:10.1175/2011JAMC2689.1
10. Lagrangian Coherent Structure analysis of terminal winds detected by LIDAR. Part I: turbulence structures (2011), Tang, W., Haller, G. & Chan, P.W., J. Appl. Meteor. Clim., 50, 325-338[pdf] DOI:10.1175/2010JAMC2508.1
9. Lagrangian dynamics in stochastic inertia-gravity waves (2010), Tang, W., Taylor, J.E. & Mahalov, A., Phys. Fluids, 22,126601. [pdf] doi:10.1063/1.3518137
8. Lagrangian Coherent Structures near a subtropical jet stream (2010), Tang, W., Mathur, M., Haller, G., Hahn, D.C., Ruggiero, F.H., J. Atmos. Sci.,67, 7, 2307-2319[pdf] DOI: 10.1175/2010JAS3176.1
7. Lagrangian Coherent Structures and internal wave attractors (2010), Tang, W., Peacock, T., Chaos, 20, 017508. [pdf] DOI:10.1063/1.3273054
6. Accurate extraction of LCS over finite domains, with applications to flight data analyses over Hong Kong International Airport (2010), Tang, W., Chan, P.W. & Haller, G., Chaos, 20, 017502. [pdf] DOI: 10.1063/1.3276061
5. A prediction for the optimal stratification for turbulent mixing (2009), Tang, W., Caulfield, C.P. & Kerswell, R.R., J. Fluid Mech., 634, 487-497. [pdf] DOI:10.1017/S0022112009990711
4. Locating an atmospheric contamination source using slow manifolds (2009), Tang, W., Haller, G., Baik, J.-J. & Ryu, Y.-H., Phys. Fluids, 21, 043302. [pdf] URL: DOI: 10.1063/1.3115065
3. Bounds on dissipation in stress-driven flow in a rotating frame (2005) Tang, W., Caulfield, C.P. & Young, W.R., J. Fluid Mech. 540, 373-391. [pdf] DOI:10.1017/S0022112005005926
2. Bounds on dissipation in stress-driven flow (2004) Tang, W., Caulfield, C.P. & Young, W.R., J. Fluid Mech. 510, 333-352. [pdf] DOI: 10.1017/S0022112004009589
1. Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in a plane stratified Couette flow (2004) Caulfield, C.P., Tang, W. & Plasting, S.C., J. Fluid Mech. 498, 315-332. [pdf] DOI: 10.1017/S0022112003006797


Concept based learning workshop for high school students

This workshop is aimed at engaging high-achieving high school students with continued motivation for further mathematical studies. Mathematical models are introduced along with development of physical concepts so the students will learn both simultaneously. Various types of ODE models for natural systems are formed and the concept of dynamics is developed through discussions of the meaning of the models. Students in this workshop get hands-on experience in applying mathematical and numerical tools to analyze the dynamical behaviors of the models. Topics such as the Taylor series expansion, vector calculus, linear algebra, differential equations and phase portraits, suitable to the problems presented, are connected so they will not be perceived as disjointed math subjects in future studies. MATLAB is used for numerically modeling the dynamical systems.

High school students in this workshop also get the opportunity to meet with successful undergraduate students in applied mathematics on experiences and tips being a math and science major. Interactions with college students and faculty help the participating students learn from role models and be informed with career opportunities in STEM fields. Past participants attend UC Berkeley, UCLA, U Chicago, Harvey Mudd College and the Barrett Honors College at ASU.

A presentation on applied dynamics can be located here.

Workshop notes can be located here.

Workshop Participants, 2011 Workshop Participants, 2012
2013 Workshop Participants, 2013 Workshop Participants, 2014


Applied Math internship for high school students

High school students can also participate in research activities at our lab. Throughout the course of the internship the students will be exposed to different mathematical concepts linked to solve applied problems in different disciplines. Concept based learning of mathematical topics is further solidified through hands on research questions.

Currently students from BioSciences High School, Phoenix have benefitted from participating in this internship.


Last Update: 3/14/2015