Office: 202 S.W. Mudd
Mail: 200 S.W. Mudd, Mail Code: 4701
New York, NY 10027
B.S. (Mech. Eng.) Chiao-Tung University 1948. M.M.E. Cornell 1950
Ph.D. (Mathematics) Courant Institute, NYU 1959 (advisor: K.O. Friedrichs)
C. K. (John) Chu, the Fu Foundation Professor Emeritus of Applied Mathematics, is a pioneer in computational mathematics. His work in fluid dynamics, magnetohydrodynamics, and shock waves has garnered him international recognition while his zeal for his life’s work prompted him to work tirelessly to create a home for applied mathematics at Columbia. He is one of only seven mathematicians to receive an Honorary Doctor of Science Degree in Columbia’s 252-year history.
The son of parents with various degrees of Western education-his father received an MBA from NYU in 1929 and his mother spoke fluent English- he was born in Shanghai. He graduated in 1944 from St. John’s University High School, where the curriculum was half Chinese and half English. He was accepted at St. John’s University but decided to take the entrance examination for Chiaotung (now Jiaotong) University, the national science and engineering university.
“I didn’t think I had much of a chance, but I got in,” he said. After doing extremely well in his university studies, Chu came to the United States for graduate work. He received his master’s degree in 1950 from Cornell and accepted a job as an engineer at General Electric. “I was happy at GE,” he said, “and was all set in a special advanced program for engineers. And then, in 1953, a major incident occurred.”
That incident was a telephone call from Professor Chu’s advisor at Cornell, who had moved to Stevens Institute of Technology, offering him a position as an assistant professor there. The selling point was that Professor Chu would be near the Courant Institute of Mathematical Sciences. “On Friday, I interviewed for and accepted the position at Stevens and, on the following Monday, I interviewed with Richard Courant. He asked me what I needed for a salary and I said, 'I suppose if they are offering me an assistant professorship, my answer to you should be zero.’ I was accepted as a Ph.D. student and it changed my life.”
In 1959, he became the first Chinese student to receive a Ph.D. from Courant Institute. He taught at Pratt Institute and NYU Engineering before joining the Columbia Engineering faculty in 1963 as a visiting research scientist in the plasma physics laboratory. Within four years, he had been granted tenure and, a year later, was named a full professor. In 1999, he was named Fu Foundation Professor of Applied Mathematics. As a theoretician working with plasma physics, he was delighted when then dean Peter Likins asked him to form a new program in applied mathematics as a successor to the Mathematical Methods program already functioning well under Prof. Morton Friedman. Its first home was in Applied Physics and, in 1997, the name of the department was changed to Applied Physics and Applied Mathematics, fully recognizing the program.
Professor Chu recalls two exceptionally strong students, Robert Merton ’66 and Stephen H. Schneider ’66, ’71. Merton is a Nobel Laureate in economics and Schneider is professor of environmental biology and global change at Stanford and a MacArthur Fellow (the “Genius Award”).
Professor Chu, whose advice and guidance has helped hundreds of students for more than four decades, retired in 2003 but he still maintains contact with most of his 24 Ph.D. students and many of his former undergraduate students.
I am primarily interested in applied mathematics, particularly scientific computing. Applications include fluid dynamics, magnetohydrodynamics, shock waves, and plasma simulation. One of my research projects was plasma fusion simulation, in which the behavior of a magnetically confined plasma was simulated by the solution of a set of partial differential equations. Of particular interest are plasma equilibrium, stability, transport, and initial formation. Initial formation dynamics was for many years the specialty of the Columbia plasma physics group, and Columbia was noted for being a major supplier of Ph.D.'s in the field of plasma computation.
I have also directed much of my attention toward nonlinear waves and nonlinear phenomena in hydrodynamics, and their numerical computation. In collaboration with the NASA Goddard Institute for Space Studies, I was active in computational studies of global ocean circulation. In particular, that involved the calculation of behavior of intensified currents, such as the Gulf Stream, using special grids, and a self-consistent study of the thermohaline convection.
What excites me about computational physics is the understanding and discovery of physical phenomena and mathematical facts through the intelligent use of the computer. In a sense, computation parallels theory and experiment as a highly effective method to study physical phenomena.
APAM News Highlights
General Electric Company, 1950-53
Stevens Institute of Technology, assistant professor, 1953-57
Pratt Institute, associate professor, 1957-59
Courant Institute, NYU
- research assistant, 1956-58
- research associate, 1958-59
New York University, associate professor, 1959-63
- visiting research associate, 1963-65
- associate professor 1965-68
- professor, 1968-present
- chairman, plasma physics committee, 1966-67, 1970-71, 1974-77
- chairman, applied mathematics committee, 1978-present
- chairman, department of applied physics and nuclear engineering, 1982-83, 1985-88, 1995-1997
- resident professor, Wallach Hall, 1985-89
- Fu Foundation professor, 1992
John Simon Guggenheim Foundation Fellow, 1971-72
Senior NSF Fellow (awarded and declined), 1971-72
Fellow, American Physical Society, 1971-present
Fellow, Japan Society Promotion of Science, 1979
Sherman Fairchild Distinguished Scholar, Caltech, 1984
Who's Who in America, 1983-present
Columbia University Great Teacher Award, 1985
Shanghai Jiao Tong University, Advisory Professor, 1985-present
Academia Sinica, Inst. of Mechanics, Honorary Res. Prof., 1988-present
Wei Lun Foundation Lecturer, Chinese U. of Hong Kong, 1991
Hong Kong University, Honorary Professor of Mech. Eng., 1993
Honorary Doctor of Science Degree, Columbia University, 2006
Professional Society Memberships / Foundations / Activities
7th Symposium on Plasma Simulation, Co-chairman, 1975
AIAA Professional Development Course on Computation, Director
AIAA Journal, Assoc. Ed., 1966-68
American Physical Society, fellow
American University of Paris, Trustee, from 1999
Chinese U of Hong Kong, External examiner, 2000-2003
Chubb Foundation, trustee; chairman of board
Committee on Applied Mathematics, 1983-1986
Fluid Dynamics, 1970
Fluid Dynamics Division, Exec Committee, 1975
Fu Foundation, Executive Secy
Fusion Theory Annual Meetings, Exec Committee, 1978-80
John Simon Guggenheim Foundation, fellow
Physics of Fluids, Assoc. Ed., 1973-75
Plasma Physics Division, Exec Committee, 1982-83
P.Y. Chou Foundation, trustee
Rutgers University, Trustee, from 1999
San Siao Corporation, board of directors, from 1995
Shanghai Commercial Bank, Hong Kong, board of directors
Society of Industrial and Applied Mathematics
University Fusion Association, Exec Committee, 1983-1985
University Grants Committee of Hong Kong, Reviewer, from1999
Cornell University, Grad. School Aero. Eng., summer 1961
Univ. of Strathclyde, Glasgow, Math. Dept., summer 1966
Stanford University, Institute Plasma Studies, summer 1968
University of Oxford, Math. Inst., fall term 1971
University of Uppsala, Dept. Computer Sci., spring term 1972
University of Paris XI, Dept. of Math., summer 1974
University of Paris VI, Dept. of Mechanics, summer 1977
University of Nagoya, Inst. Plasma Physics, summer 1979
Shanghai Jiao-Tong University, summer 1979
University of Wisconsin, Math. Res. Center, summer 1981
Academia Sinica, Inst. of Mechanics, Peking, summer 1982.
National Tsing Hua University, Hsinchu, Taiwan, January 1985
Univ of Calif at Los Angeles, Mathematics, 1989-90
Hong Kong University of Science and Technology, spring 1993
Chinese University of Hong Kong, Ho SinHeng Visiting Professor, 1994 fall
Books and Chapters
1. Magneto-fluid-dynamics (with H. Grad). Chapter in Research Frontiers in Fluid Dynamics, (G. Temple and R. Seeger, eds.), Wiley-Interscience 1965.
2. Shock waves in plasma physics (with R.A. Gross). Chapter in Advances in Plasma Physics, Vol. 2, (A. Simon and W. B. Thompson, eds.), Wiley 1969.
3. Computational Fluid Dynamics (editor). AIAA Reprint Series, Vol. 4, 1968.
4. (Transl.-edit. with H. Cohen and B.D. Seckler) L.I. Sedov's Two-Dimensional Problems in Fluid Dynamics, Wiley-Interscience 1964.
5. Numerical Methods in Fluid Dynamics. Chapter in Advances in Applied Mechanics, Vol. 17 (C.S. Yih, ed.), Academic Press 1977.
6. Solitons Induced by Boundary Conditions. Chapter in Advances in Applied Mechanics, Vol. 27 (J. Hutchinson, T.Y. Wu, eds.), Academic Press, 1989.
Invited Papers and Review Papers
1. Some unconventional mathematical problems in magnetohydrodynamics. Symposium of Plasma Physics and Fluid Dynamics Div., American Phys. Soc., June 1962. New York U. Report AA-63-1.
2. Magneto-gas-dynamics. ASEE annual meeting, June 1964, paper 101-1.
3. Hydromagnetic shocks and ionizing shocks. 14th Japanese National Congress of Applied Mechanics, Sept. 1964. Proc. of the Congress, Science Council of Japan 1965, p. 1.
4. Some remarks on the stability of hydromagnetic shock waves. Amer. Math. Soc. Appl. Math. Symposium., April 1965. Proc. of Appl. Math. Symposium., Vol. 18, Amer. Math. Soc. 1967, p. 1.
5. Magnetohydrodynamic shock stability and numerical experiments. Symposium on "Discontinuous Waves," Strathclyde University, Scotland, June 1966.
6. Some problems in computational plasma physics. Amer. Phys. Soc. summer meeting, June 1970.
7. Numerical computation of nonlinear ion-acoustic phenomena. Second US-Japan Plasma Seminar, July 1971, Los Angeles.
8. Numerical solution of Navier-Stokes equations: survey and some observations. NASA Langley Symposium on Numerical Fluid Dynamics, October 1972.
9. Computational plasma dynamics. Poly. Inst. Brooklyn Symposium. on Computers and Fluids, January 1973.
10. Computational fluid dynamics: survey and observations. Soc. Ind. Appl. Math. national meeting, June 1973.
11. Time-dependent multidimensional MHD simulation of high-beta toroidal experiments. Amer. Phys. Soc. Div. of Plasma Phys. annual meeting, Albuquerque, NM, Nov. 1974.
12. Physical and numerical dissipation and dispersion in viscous compressible flow computation. Symposium on Unsteady Aerodynamics, Univ. of Arizona, March 1975. Proceedings of Symposium., p. 415-434.
13. Time-dependent computer simulation of an electromagnetic shock tube. 10th International Shock Tube Symposium., Kyoto, Japan, July 1975.
14. The implosion to equilibrium of toroidal plasmas. Am. Phys. Soc. Div. Plasma Phys. annual meeting, San Francisco, November 1976.
15. A survey of high-beta tokamak experiments. Oak Ridge Nat. Lab. Workshop on High-beta Tokamaks, Feb. 1977.
16. High-Beta Tokamaks by Pinch Techniques (with R.A. Gross). Proc. of Finite Beta Theory Workshop, Varenna, Sept. 1977. U.S. Dept. of Energy publication CONF-7709167, p. 187-92.
17. Computational Fluid Dynamics. Paper in S. Parter (ed.), Numerical Methods for Partial Differential Equations, Academic Press, 1979, p. 149-175.
18. Numerical simulation of reversed-field pinches. Am. Phys. Soc. spring meeting, Washington DC, May 1980.
19. Slow formation and equilibrium of compact tori. 3rd Symposium on Compact Toroids, Los Alamos, December 1980.
20. Some recent calculations relevant to the reversed-field pinch. US-Japan Conference on Reversed Field Pinch, Los Alamos, March 1981.
21. Numerical solution of tricomi-type equations. In R. Meyer, Multidimensional and Shock Flows, p. 147-159, Academic Press, 1982.
22. Simulation of plasma formation and attainment of equilibrium. In A.S. Kydes, ed., Energy Modelling and Simulation, p. 365-370, North-Holland, 1983.
23. High beta tokamak research (with R.A. Gross, G.A. Navratil et al). Nuc Fus 25, 1105 (1985).
24. High beta tokamak research at Columbia University (with G.A. Navratil). Paper presented to 2nd Natl. Conf. on Fusion Plasma Physics, China, Sept. 1985.
25. Chaos and macrostructures in fluid dynamics. Proc. 35th Anniversary Symposium, Institute of Mechanics, Academia Sinica, Peking (June 1986).
26. Accurate computation of ocean circulation flows. 3rd Nobeyama Workshop on Computat. Fluid Dynamics, Japan, 1989.
27. Magnetohydrodynamic shock waves revisited. 3rd Intern. Conf. on Hyperbolic Problems, Uppsala, Sweden, 1990.
28. Computational engineering I, II. Wei Lun Lectures, Chinese U. of Hong Kong, 1991.
29. Numerical mathematical aspects of ocean circulation modeling. Mini Symposium in Computational Science, HKU, August 1991.
30. MHD shocks and shocks in nonconvex systems. 4th Nobeyama Workshop on CFD, Japan, Sept. 1991.
31. Sensitivity study of reaction diffusion equations for film development (with K.C. Ng, and D. Ross). Mini Symposium on Inverse Problems, SIAM National Meeting, July 1992.
32. Computational fluid dynamics: a twenty-five year review. International Union of Pure and Applied Physics, 3rd Internationall Conference on Computational Physics. Fall 1995, Taiwan.
33. Mixed elliptic-hyperbolic equations - a new look. 6th Intern. Conf. on Hyperbolic Probs. June 96, Hong Kong.
34. Type-insensitive solutions of mixed elliptic-hyperbolic equations and their applications to nonlinear problems. Amer. Math. Soc. Annual Meeting, Jan 97, San Diego, Calif.
35. Mathematics, computing, and physics of fluids. Intern. Conf. on Physics Since Parity Symmetry Breaking, in Memory of CS Wu. Aug 97, Nanjing, China.
36. Computational results in the physics of fluids. Intern. Mtg for the 35th Anniv of the Institute of Mechanics, Academia Sinica, Beijing, Sept 97.
37. Practical ways of generating mathematical waves. First Intern. Congress of Chinese Mathematicians, Beijing, Dec 1998.
38. Generation of bubble K-P solitons. 1999 IMACS Intern. Conf., Athens Ga., April 1999.
Contributed Research Papers
1. 7 reports in General Electric Tech. Info. Series, 1952-56.
2. Type-insensitive finite difference methods (Ph.D. thesis). Presented Amer. Math. Soc., Apr. 1959; A.M.S. Notices 6, 556-38.
3. Type-insensitive approximation methods II: Galerkin's method. Presented Amer. Math. Soc., Jan. 1961; A.M.S. Notices 7, 576-189.
4. Magnetohydrodynamic nozzle flow with three transitions. Phys. Fluids 5, 550 (1962).
5. Steady magnetohydrodynamic flow past a non-conducting wedge (with Y.M. Lynn). AIAA Jour. 1, 1062 (1963).
6. Steady hyperbolic magnetohydrodynamic flow past infinite boundaries. Phys. Fluids 7, 707 (1964).
7. Charged aerosol energy converter (with A. Marks and E. Barreto). AIAA Jour. 2, 45 (1964).
8. Dynamics of ionizing wavers: shocks in transverse magnetic fields. Phys. Fluids 7, 1349 (1964).
9. Kinetic-theoretic description of the formation of a shock wave. Phys. Fluids 8, 12 (1965).
10. Kinetic-theoretic description of the shock wave formation II:monatomic molecules. Phys. Fluids 8, 1450 (1965).
11. Free-streaming plasma experiments (with B. Eastlund). Phys. Fluids 9, 161 (1966).
12. On the structure of Alfven shocks (with D.L. Turcotte). Zeit. Ang. Math. Mech. 17, 528 (1966).
13. Numerical experiments on hydromagnetic shocks and stability of the switch-on shock (with R.T. Taussig). Phys. Fluids 10, 249 (1967).
14. Alfven waves and induction drag on long cylindrical satellites (with R.A. Gross). AIAA Jour. 4 2209 (1966).
15. The high Mach number Rayleigh problem for the nonlinear Krook equation. Proc. 5th Symposium. on Rarified Gas Dynamics, Acad. Press 1967, p. 589.
16. Numerical solution of initial-value and initial-boundary-value problems for the Krook model Boltzmann equation. Y.W. Chen 60th Anniv. Vol., Academia Sinica, Taiwan 1970, p. 255.
17. A particle-in-cell method for model kinetic equations (with R.J. Mason). Proc. 7th Intern. Symposium. Rarefied Gas Dynamics, July 1970. Academic Press.
18. Formation of ion-acoustic collisionless shocks (with P.H. Sakanaka and T.C. Marshall). Phys. Fluids 14, 611 (1971).
19. Numerical study of the magnetic shock tube (with S.H. Schneider and B.P. Leonard). Phys. Fluids 14, 1103 (1971).
20. A modified particle-in-cell method for collisional plasmas (with W.P. Gula and R.J. Mason). Proc. 4th Conf. on Numerical Simulation of Plasmas. Naval Res. Lab. 1970, p. 390.
21. Plasma heating by strong shock waves (with Y.G. Chen, R.A. Gross, E. Halmoy, P. Moriette, S.H. Schneider). Proc. 4th IAEA Controlled Fusion Conf., Intern. Atomic Energy Agency 1971, Vol. 3, p. 241.
22. Computational study of shock waves. Dynamics of Ionized Gases, M.J. Lighthill, I. Imai, H. Sato (eds), U. Tokyo Press 1973, p. 101.
23. Numerical studies of the heat conduction equation with highly anisotropic tensor conductivity (with K.W. Morton and K.V. Roberts). Proc. 3rd Intern. Cong. Numerical Fluid Dynamics, Vol. II, Springer-Verlag 1973, p. 105.
24. Numerical studies of the heat conduction equation with highly anisotropic tensor conductivity II (with G. Johansson). Univ. of Uppsala, Dept. of Computer Science Report No. 40, 1972.
25. Effect of Krook model collisions on the two-stream instability (with W.P. Gula). Phys. Fluids 16, 1135 (1973).
26. Simulation of transient dc breakdown in a Penning mixture between two closely spaced parallel electrodes (with F.M. Lay). J. Appl. Phys., 44, 4008 (1973).
27. Simulation of cyclic operation of a gas panel device (with F.M. Lay and P.H. Haberland). IBM Jour. Res. Dev. 18, 244 (1974).
28. Boundary conditions in finite difference fluid dynamic codes (with A.Sereny). J. Comp. Phys. 15, 476 (1974).
29. Computational magnetohydrodynamics of time-dependent flows in toroidal systems (with H.C. Lui). Proc. 4th Intern. Conf. on Num. Meth. in Fluid Dynamics, Springer-Verlag, 1974, p. 263.
30. Two-dimensional magneto-hydrodynamic simulation of toroidal pinches (with H.C. Lui). Phys. Fluids 18, 1277 (1975).
31. Initial dynamics, equilibrium, and heating of belt pinches (with H.C. Lui, M.F. Reusch, R.T. Taussig). Proc. 3rd Conf. on High Beta Plasmas, Culham, 1974, Pergamon Press, 1976. p. 553.
32. Two-dimensional magnetohydrodynamic simulation of toroidal pinches II: Belt pinches (with H.C. Lui). Phys. Fluids 19, 1947 (1976).
33. Simulation of the Padua Eta-Beta Reversed Field Pinch (with W. Park). Nuclear Fusion 17, 1100 (1977).
34. Experimental and theoretical studies of belt pinches and high-beta tokamaks (with R.A. Gross, H.C. Lui, et al). Proc. 6th IAEA Conf. on Controlled Fusion, Intern. Atomic Energy Agency, Vienna (1977), Vol. I, p. 511.
35. Experimental and theoretical studies of high-beta tokamaks (with A. Aydemir, R.N. Byrne, et al). 7th IAEA Conf. on Controlled Fusion, Austria, 1978.
36. The Los Alamos Reversed-Field Pinch Program (with D.A. Baker, W. Park, et al). 7th IAEA Conf. on Controlled Fusion, Austria, 1978.
37. Formation of Hill's vortices and smoke rings (with H.C. Lui). Burns & Roe Tech. Report, 1980.
38. Suppression pool transients in BWR's under LOCA and SRV conditions (with J. Raymont). Proc. 5th Intern. Conf. on Struct. Mech. in Reactor Tech., Berlin, Aug. 1979.
39. Formation of LOCA jets and induced flows during vent-clearing (with T.T. Lee and H.C. Lui). Proc. 5th Intern. Conf. on Struct. Mech. in Reactor Tech., Berlin, Aug. 1979.
40. Transport and radiation studies of belt pinches an high-beta tokamaks (with R.N. Byrne). Phys. Fluids 22, 2418 (1979).
41. Two-dimensional simulation of the formation of the PPPL Spheromak (with A. Aydemir add H.C. Lui). Proc. U.S.-Japan Conf. on Compact Toruses, Princeton Plasma Phys. Lab., Dec. 1979.
42. Modeling ZT-40 with G2M diffusion code (with R.N. Byrne). Proc. 9th Euro. Conf. on Controlled Fusion, Oxford (1979), paper DP-2.
43. High-beta tokamak stability studies (with R.A. Gross, H.C. Lui, et al). Proc. 8th IAEA Conf. on Fusion, Bruxelles (1980), paper CN-38/L-4-2.
44. Simulation study for formation of spheromak and compact tori (with Grossmann, Furth, Jardin, Lui, et al). Proc. 8th IAEA Conf. on Fusion, Bruxelles (1980), paper CN-R-1.
45. Ballooning-mode-stable high-beta configurations for tokamaks (with K. Yamazaki). Japan J. Appl. phys.20, 665 (1981).
46. Two-dimensional simulation of the formation of the Princeton spheromak (with H.C. Lui and A. Aydemir). Phys. Fluids 24, 673 (1981).
47. High-beta tokamak physics studies (with R. A. Gross et al). Proc. 9th IAEA Conf. on Fusion, Baltimore (1982), paper CN-41/M-5-1.
48. Solitary waves generated by boundary motion (with L.W. Xiang, Y. Baransky). Comm. Pure Applied Math. 36, 495 (1983).
49. Minimum energy helical state for reverse-field pinches. Sherwood Theory Meeting, 1983.
50. Effects of q and high beta on tokamak stability (with DeLucia, Gross, Navratil, et al). Paper IAEA-CN-44/A-VI-2, Proceedings of l0th IAEA Conference (1984) on Plasma Physics and Fusion, IAEA Vienna, 1985, Vol. 1, p. 385.
51. Solitons induced by boundary conditions. SIAM summer meeting, Mini Symposium paper m/s 14. July, 1984.
52. Boundary generated solitary waves (with R.L. Chou). Proc. IMACS Congress, Aug. 1985, Oslo.
53. Nonlinear radial propagation of drift wave turbulence (with M. Prakash and A. Hasegawa). Phys Fluids 29, 2426 (1986).
54. Lagrangian turbulence in stokes flow (with J. Chaiken, M. Tabor, Q.M. Tan). Phys Fluids 30, 687 (1987).
55. Experimental and theoretical studies of circular cross-section high-beta tokamaks (with G. A. Navratil, A. Bhattacharjee, et al.). Proc. 11th Interntl. Conf. on Plasma Physics and Controlled Nuclear Fusion Vol. I, 299, IAEA, Vienna. 1987.
56. Relaxation phenomena in reversed-field pinches and tokamaks (with A. Bhattacharjee, Y.C. Kwok, et al.). Proc. 11th Interntl. Conf. on Plasma Physics and Controlled Nuclear Fusion Vol.II, 711, IAEA, Vienna. 1987.
57. Solitons and other macrostructures in fluid dynamics (with R.L. Chou). Proc. C.C. Lin Anniv. Symposium., World Scientific (1988), p. 230-236.
58. Some remarks on the accurate computation of ocean circulation. Proc. 5th Asian Conf. on Fluid Mech., Hong Kong 1989.
59. Boundary induced solitons according to the Boussinesq equation (with R. L. Chou). Phys. Fluids, A2, 1574 (1990).
60. Equilibrium response of ocean deep-water circulation to variations in Ekman pumping and deep-water sources (with F.L. Yin, I. Y. Fung). J. Phys. Oceanography, 22, 1129 (1992).
61. (organizer and speaker in session: Educating Applied Mathematicians) Applied Mathematics Program at Columbia University. S.I.A.M. 1993 Annual Meeting, July 1993.
62. An echocardiographic study of the fluid mechanics of obstruction in hypertrophic cardiomyopathy (with M. V. Sherrid et al). J. Amer. Coll. Cardiology 22, 816 (1993).
63. Domain decomposition for shallow water equations (with J. G. Chefter, D. E. Keyes). Contemporary Mathematics Vol 180 (Proc. 7th International Conf. on Domain Decomposition Methods in Sci. and Eng., Oct. 1993). American Math Soc, 1994. pp 485-490.
64. Unsteady evolution of vortex rings (with Q.Y. Ye), Phys. Fluids, 7, 795 (1995).