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In 1990, Masato Nagata published the first ever computation of fully nonlinear, finite-amplitude invariant solutions in planar Couette flow. In his paper, he mixed dynamical systems theory, fluid dynamics, and computational science, and it has garnered nearly three hundred citations over the years and has had an enormous impact on the analysis of turbulence as a dynamical system. In this interview, Prof. Nagata shares some of the long, arduous, and occasionally joyful journey to this landmark contribution.

We met at the Fields Institute shortly before the ICTAM meeting in Montreal in August 2016. Nagata was passing through Toronto on his way from Tianjin, where he is currently at Tianjin University and is Professor Emeritus at Kyoto University. He has held positions in the United States, England, and Japan.

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*DSWeb: *You got your Ph.D. at UCLA in 1983. How did you get there?

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Masato: *I wanted to study geophysical fluid dynamics in the US and had applied to several places. In the end, I chose UCLA to work with Professor Friedrich Busse, getting a Research Assistantship.

*DSWeb: *What was the topic of your research?

*Masato: *We were trying to find three-dimensional invariant solutions in shear flows like plane Couette flow, which lack a linear instability. That was considered one of the great open questions in fluid mechanics for decades before the 1980s. We had little knowledge of the system, and we failed. So we started to study stratified flow, with the shear induced by a temperature gradient. The shear profile was cubic instead of linear. This flow can have linear instabilities, but its velocity profile is anti-symmetric, it has the same symmetry as plane Couette flow, and I could use the same computer code. In the limit of zero Prandtl number we can avoid the effect of temperature disturbances. The results were published in 1983 in the Journal of Fluid Mechanics [1]--that was my first paper in JFM.

*DSWeb: *How did you compute your solutions with the computers of the 1980s and long before the introduction of Krylov subspace methods?

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Masato: *I wrote my own code, including matrix inversion and an eigenvalue solver. It ran on a singe CPU, day and night. The largest matrix I could handle was about 1000 by 1000. But I exploited a lot of symmetry reduction, so the actual number of degrees of freedom was 8 or 16 times greater. Another problem was that there were many turning points along solution branches. Every time I encountered a turning point, I had to go in by hand and change the continuation parameter. Later, I invented better methods, like quadratic extrapolation.

*DSWeb: *Did that first paper include solutions in the zero Prandtl number limit? [In this limit the flow is still driven by differential heating, but heat advection is negligible as compared to conduction.]

*Masato: *The paper concerned only the zero Prandtl number limit. I found a bifurcation to a spanwise-independent flow, and from there to a three-dimensional "cat's eye" flow. But then, I moved to Newcastle. There, I joined the group of Professor P.H. Roberts, now at UCLA. He allowed me to study any flow, as long as it was related to rotation or magnetic fields. I chose to work on Taylor-Couette flow in the narrow-gap limit. In that limit, the base flow becomes linear, like in planar Couette flow. I studied Taylor vortices and two types of three-dimensional flows that bifurcate from it: wavy vortex flow and twist vortices. The results were published in JFM [2].

For small rotation rate, I found a supercritical bifurcation to Taylor vortex flow followed by a subcritical bifurcation to three-dimensional flow. The latter can be continued to zero rotation rate. But this was at very small truncation levels, so I only presented it in JFM [3] as a conjecture that the solution could exist in pure Couette flow.

*DSWeb: *And then you gradually increased the resolution?

*Masato: *Yes. Even though the resolution was low, the computations were slow, and there were many turning points on the continuation curves.

I submitted the result to JFM, but the referees asked me to further increase the resolution. Actually, they did not believe it at first. I kept increasing the resolution and finally the toughest referee commented that *I am inclined to believe in the solution*. It took five years to get this accepted by the journal [4].

*DSWeb: *Why do you think this was? Was it too different from the fluid mechanics they knew?

*Masato: *Actually, I also submitted it to Science, but it was rejected. The comment was that there were only a few people interested in this. Last year, I participated in a meeting for Professor Javier Jiménez's 70th birthday. I was talking to Bruno Eckhardt and I told him this story. He said I was too early....

*DSWeb: *But you got your revenge, you now have about 300 citations to this one paper [4].

*Masato: *These days there are many people interested in this work. Recently, I found out that Fabian Waleffe looked at my paper around 1990, when he was a postdoc with Professor Jiménez. Fabian's Ph.D. topic was the elliptic instability, it had nothing to do with three-dimensional shear flows. I was very glad to hear that

Javier and Fabian had already noticed the importance of the paper when it was not appreciated very much by other people! Fabian went on to formulate his ideas about the self-sustaining process.

*DSWeb: *This was high-risk research..

*Masato: *Yes, high risk. As I mentioned, I failed to find a solution for plane Couette flow as a Ph.D. student. Professor Busse had warned me that I might not find a solution and my Ph.D. might fail. But I still wanted to do it. Luckily, I could write my thesis even without that central result.

*DSWeb: *Do you know where he got the idea to look for solutions in Couette flow?

*Masato: *I think he had a lot of analysis in astrophysical sciences and he also had a good connection with D.D. Joseph. I think he also worked on Taylor-Couette flow analysis--not numerics. From that I think he had the idea that there may be three-dimensional invariant solutions in shear flows.

*DSWeb: *What are your current interests?

*Masato: *I would like to reproduce the large-scale motions that were observed recently in a rotating plane Couette flow experiment at KTH [5].

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At this point, Prof. Nagata shows me the experimental results. One can easily see that the computation of invariant solutions that reproduce this behaviour will be quite challenging due to the wide separation of spatial scales--even with present-day algorithms and computers. And so the journey continues....

[1] M. Nagata and F. H. Busse, Three-dimensional tertiary motions in a plane shear layer,

[2] M. Nagata, Bifurcations in Couette flow between almost corotating cylinders,

[3] M. Nagata, On wavy instabilities of the Taylor-vortex flow between corotating cylinders, *J. Fluid Mech.* 188 (1988), 585–598.

[4] M. Nagata, Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity, *J. Fluid Mech.* 217 (1990), 519–527.

*Before
the 1980s it was well known that rotating and differentially heated flows often transition from laminar to more complicated behaviour through supercritical bifurcations. Both laboratory experiments and linear stability analysis had been conducted for such phenomena as Rayleigh-Bénard convection and Taylor-Couette flow. For some elementary flows driven purely by shear, in contrast, no linear instabilities were found. For planar Couette flow (the flow of a viscous fluid between two moving parallel walls), there even exists a proof of the asymptotic stability of the laminar profile for any Reynolds number. Yet, as Nagata and Busse in [1] put it: "It is widely believed that similar sequences of bifurcations exist in problems of plane-parallel shear flow. But those bifurcations are usually not observable because they occur mostly subcritically." Prof. Nagata managed to find invariant solutions in plane Couette flow that coexist with a stable laminar flow by using a homotopy approach. He first added differential heating and then rotation to planar Couette flow, forcing a supercritical bifurcation. Then he tracked the resulting, finite-amplitude solution, and others born in secondary and tertiary instabilities, away from the bifurcation point. Finally, he tried to track them in the homotopy parameter (for instance the rate of rotation) to recover solutions for the original, shear-driven flow. Completing these steps took great patience as solution curves fold, terminate, or become impossible to resolve numerically and there is no a priori guarantee that any given solution exists in the original flow. Moreover, Prof. Nagata obtained his central result using single CPU computers with megabytes of memory.*

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