School of Mathematics and Statistics
University of St Andrews
St Andrews KY16 9SS, Scotland
Chuong V. Tran
Office: 306
Tel: +44 1334 463741
Fax: +44 1334 46 3748
Email: cvt1@standrews.ac.uk
Web site:
http://wwwvortex.mcs.stand.ac.uk/~chuong/
Education
B.Sc. in Mathematical
Physics, 1995 (Alberta)
M.Sc. in Physics, 1996 (Toronto)
Ph.D. in Physics, 2001 (Toronto)
Work Experience
01/2001  06/2001:
Postdoctoral Fellow, Department of Physics, University of Toronto
07/2001  08/2004: Postdoctoral Fellow (Pacific Institute for the Mathematical Sciences Postdoctoral Fellow
from 04/2002 to 03/2004), Department of Mathematical and Statistical
Sciences, University of Alberta
09/2004  08/2005: Temporary Lecturer, Mathematics Institute, University of
Warwick
09/2005  08/2010: RCUK Academic Fellow, School of Mathematics and
Statistics, University of St Andrews
09/2010  Present: Lecturer, School of Mathematics and Statistics,
University of St Andrews
Professional Affiliation
London Mathematical Society
Teaching
Previous years:
MT1001  Introductory Mathematics
MT2001  Mathematics
MT3503  Complex Analysis
MT4005  Linear and Nonlinear Waves
MT4508  Dynamical Systems
MT5802  Advanced Analytic techniques
20172018:
MT1001  Introductory Mathematics
MT5802  Advanced Analytic techniques
(For prospective students, the module evaluation for last year can be seen
here)
List of courses offered
by the department and relevant information
Research
Research
interests
GFD, MHD, turbulence, stability, mixing and transport, NavierStokes
theory
Publications

CVT and X. Yu: A geometrical approach
to the problem of NavierStokes regularity,
submitted.

CVT and X. Yu: Regularity of
NavierStokes flows with bounds for the velocity gradient along streamline
and an effective pressure,
submitted.

CVT and X. Yu: Note on ProdiSerrinLadyzhenskya regularity criteria for the NavierStokes equations,
J. Math. Phys.
58,
011501 (2017).
(online)
The classical L^r(0,T;L^s) regularity criteria for both velocity and
pressure have received numerous improvements. Roughly speaking, there
are 2 types of improvements: weakening the criteria by (i) replacing
the Lebesgue L^s norm by a weaker (but almost equivalent) norm and (ii)
adding a logarithmic factor. This study features a new and substantial
type of improvement: adding factors of negative powers of the (scaling
invariant) L^3 and H^{1/2} norms of the velocity.

CVT and X. Yu: Regularity of
NavierStokes flows with bounds for the pressure,
Appl. Math. Lett.
67,
2127 (2017).
(online)

CVT and X. Yu: Pressure moderation and
effective pressure in NavierStokes flows,
Nonlinearity
29,
29903005 (2016).
(online)
In the evolution equation for the Lq norm of the velocity u, viscous
effects are found to be strong enough to neutralize those of the physical
pressure p all over space except in a neighbourhood Omega of high fluid
velocity. In the limit of large q or singularity picture, the measure of
Omega vanishes. On the other hand, the pressure in the driving term in
this equation is determined up to a fictitious pressure P(x,u,t) of
relatively broad form. In other words, we can replace p by p+P in the
evolution equation for the Lq norm of u. This allows us to vary P within
its admissible class for optimal estimates of the said term. Regularity
criteria can be derived in terms of bounds for the effective pressure
p+P, among other things, within the vanishingly small set Omega.

CVT and X. Yu: Logarithmic improvement
of regularity criteria for the NavierStokes equations in terms of pressure,
Appl. Math. Lett.
58,
6268 (2016).
(online)

CVT and X. Yu:
Depletion of nonlinearity in the pressure force driving NavierStokes
flows,
Nonlinearity
28,
12951306 (2015).
(online)
A longstanding issue in
mathematical fluid dynamics is concerned with whether solutions of
the 3D NS equations evolving from smooth (but otherwise arbitrary)
initial velocity fields remain globally smooth (regular) in time.
Decades of active research since Leray's seminal studies in the
1930s have resulted in a rich literature. Yet, the prospect of a
definitive answer to the problem remains remote, leading to its
designation as one of the millennium prize problems by the
Clay Mathematics Institute.
This study features some standard and nonstandard analysis of the
pressure force that drives NS flows. This force has apparent nonlinear
depletion, which may give rise to mild growth of local momentum. It is
hoped that such depletion can be fully exploited in the near future,
enabling one to show that viscous effects are adequate in regularising
the dynamics.

L.A.K. Blackbourn and CVT:
Inertialrange dynamics and scaling laws of twodimensional
magnetohydrodynamic turbulence in the weakfield regime,
Phys. Rev. E
90,
023012 (2014).
(online)
A graphic from this paper was used
in the Physical Review E
kaleidoscope for August 2014.

CVT, X. Yu and L.A.K. Blackbourn:
Twodimensional magnetohydrodynamic turbulence in the limits of infinite
and vanishing magnetic Prandtl number,
J. Fluid Mech.
725,
195215 (2013).
(online)

CVT, X. Yu and Z. Zhai: On global
regularity of 2D generalized magnetohydrodynamic equations,
J. Differential Equations
254,
41944216 (2013).
(online)
Since vortex stretching is absent
in 2D, the vorticity in 2D MHD (driven by the Loretz force) remains smooth
so long as the magnetic field is smooth. Hence global regularity is possible
in the presence of sufficiently strong magnetic diffusion (to regularize the
magnetic field) and in the absence of momentum diffusion. On the other hand,
since the induction equation is linear in the magnetic field (true in 3D too),
global regularity is possible in the presence of sufficiently strong momentum
diffusion (to control the vortex amplification by the Lorentz force) and in
the absence of magnetic diffusion. This paper derives regularity criteria
involving either (or both) diffusion channel. The results have been improved
and cited by a number of authors, and the paper has been marked as a
"highly cited paper" by Web of Science (14 citations in 2014, then 8 or 9 in each of the following years).

CVT, X. Yu and Z. Zhai: Note on solution
regularity of the generalized magnetohydrodynamic equations with partial
dissipation,
Nonlinear Anal.
85,
4351 (2013).
(online)

CVT and X. Yu: Bounds for the number
of degrees of freedom of magnetohydrodynamic turbulence in two and three
dimensions, Phys. Rev. E
85,
066323 (2012).
(online)

L.A.K. Blackbourn and CVT: On
energetics and inertialrange scaling laws of twodimensional
magnetohydrodynamic turbulence,
J. Fluid Mech.
703,
238254 (2012).
(online)

CVT and L.A.K. Blackbourn: A dynamical
systems approach to fluid turbulence,
Fluid Dyn. Res.
44,
031417 (2012).
(online)

L.A.K. Blackbourn and CVT:
Effects of friction on twodimensional NavierStokes turbulence,
Phys. Rev. E
84,
046322 (2011).
(online)

CVT, L.A.K. Blackbourn and R.K. Scott:
Number of degrees of freedom and energy spectrum of surface quasigeostrophic
turbulence, J. Fluid Mech.
684, 427440
(2011). (online)

CVT and D.G. Dritschel:
Energy dissipation and resolution of steep gradients in
onedimensional Burgers flows, Phys.
Fluids 22, 037102 (2010). (online)

CVT, D.G. Dritschel and R.K. Scott:
Effective degrees of nonlinearity in a family of generalized models of
twodimensional turbulence, Phys.
Rev. E
81, 016301 (2010). (online)
The advection term in the equation
governing the motion of a fluid system is apparently quadratic in the
velocity. However, the dynamics of the fluid small scales may not be
nonlinear, let alone quadratic! This paper introduces the notion of
effective degrees of nonlinearity in a family of generalized models of
twodimensional turbulence. In this family, the dynamical behaviour of
the small scales is wideranging, from highly superlinear to linear and
probably to sublinear. This has farreaching implications on turbulence
transfer.

CVT and L. Blackbourn: Number of degrees of freedom of twodimensional turbulence, Phys. Rev. E
79, 056308 (2009). (online)
This paper derives upper bounds
for the number of degrees of freedom of twodimensional NavierStokes
turbulence. The bounds scale linearly and almost linearly with the
Reynolds number. These results are at odds with existing bounds that
scale superlinearly with the Reynolds number. It is demonstrated that
the "extra" dependence on the Reynolds number of these superlinear bounds
is a removable artifact, brought about by the use of flowdependent
forcing.

CVT: The number of degrees of freedom of
threedimensional NavierStokes turbulence, Phys. Fluids
21, 125103 (2009). (online)

D.G. Dritschel, R.K. Scott, C. Macaskill,
G.A. Gottwald and CVT: Vortex selfsimilarity and the evolution of unforced inviscid twodimensional turbulence, Advance in
Turbulence XII  Proceedings of the 12th Euromech European Turbulence Conference
132, 461464
(2009).

D.G. Dritschel, R.K. Scott, C. Macaskill,
G.A. Gottwald and CVT: Late time evolution of unforced inviscid twodimensional turbulence, J. Fluid Mech.
640, 215233
(2009). (online)

CVT: Constraints on scalar
diffusion anomaly in threedimensional flows having bounded velocity
gradients, Phys. Fluids
20, 077103 (2008).
(online)
The subtle notion of diffusion
anomaly of a passive scalar in turbulence flows is unnecessarily
confusing. This paper discusses the simple fact that for flows with bounded
velocity gradients, the gradient of the scalar may grow at most exponentially
in time. Hence the notion of diffusion anomaly does not apply to this case.

CVT: Local transfer and
spectra of a diffusive field advected by largescale incompressible
flows, Phys. Rev. E
78, 036310 (2008).
(online)
This study features a rigorous
examination of the dynamics of a passive scalar advected by largescale
incompressible flows. For a steady nonzero flux of the scalar variance
through the inertial range, the spectrum of the scalar variance in this
range must not be steeper than k^{1}. The critical k^{1} scaling is
the Batchelor result.

D.G. Dritschel, R.K. Scott, C. Macaskill,
G.A. Gottwald and CVT: Unifying theory for vortex dynamics in twodimensional
turbulence, Phys. Rev. Lett.
101, 094501 (2008).
(online)
This editors' suggestion article
formulates a theory of inviscid vortex dynamics in two dimensions at late
times.

CVT: Constraints on inertial
range scaling laws in forced twodimensional NavierStokes turbulence,
Phys. Fluids
19, 108109 (2007).
(online)

CVT: An upper bound for passive
scalar diffusion in shear flows,
Phys. Fluids
19, 068104 (2007).
(online)
Shear flows weakly amplify the gradient
of a passive scalar because the component of this gradient along streamlines
is conserved. This paper makes use of this conservation law to derive an
upper bound for the scalar variance diffusion rate that scales as 1/3 power
of the diffusivity.

D.G. Dritschel, CVT and R.K. Scott:
Revisiting Batchelor's theory of twodimensional turbulence,
J. Fluid Mech.
591,
379391 (2007).
(online)
This paper reports numerical results
supporting our earlier mathematical finding that enstrophy dissipation in
twodimensional turbulence vanishes in the inviscid limit. The paper was
on the list of most downloaded papers in the month of its publication.

CVT and D.G. Dritschel:
Largescale dynamics in twodimensional Euler and surface quasigeostrophic
flows, Phys. Fluids
18, 121703
(2006).
(online)

CVT and D.G. Dritschel: Vanishing enstrophy dissipation in twodimensional
NavierStokes turbulence in the inviscid limit,
J. Fluid Mech.
559,
107116 (2006).
(online)
Batchelor's classical theory of
twodimensional NavierStokes turbulence is premised on the hypothesis
that the enstrophy dissipation rate remains nonzero in the inviscid limit.
From there the theory predicts the k^{1} scaling for the enstrophy
spectrum in the inertial range, among other things. This paper proves
that for powerlaw enstrophy spectra, including the k^{1} scaling of
Batchelor, the enstrophy dissipation rate approaches zero (uniformly
in time) in the inviscid limit. For the k^{1} scaling, this approach is
shown to be logarithmic in the Reynolds number, so slow that Batchelor's
hypothesis may turn out to be a good approximation for some practical
purposes. Nonetheless, his prediction and hypothesis are incompatible and
the theory is wrong at the very fundamental level. As is well known, smooth
solutions of the twodimensional Euler equations remain smooth globally
in time. Consequently, the palinstrophy remains finite in finite times.
In fact, growth of palinstrophy may not be more rapid than double
exponential in time. Now in the inviscid limit, NavierStokes solutions
tend to the corresponding Euler solutions. This necessarily means that
the enstrophy dissipation rate tends to zero in that limit (for finite
times). That said, one might still attempt to salvage Batchelor's theory
by the possibility that the limiting rate could be nonzero in infinite
time. This paper rules out this possibility, thus establishing vanishing
enstrophy dissipation in the inviscid limit for all times.

CVT and D.G. Dritschel:
Impeded inverse energy transfer in the CharneyHasegawaMima model of
quasigeostrophic flows,
J. Fluid Mech.
551,
435443 (2006).
(online)

CVT: Diminishing
inverse transfer and noncascading dynamics in surface quasigeostrophic
turbulence, Physica D
213, 7684
(2006).
(online)

CVT:
Enstrophy dissipation in freely evolving twodimensional turbulence,
Phys. Fluids
17, 081704
(2005).
(online)
The enstrophy dissipation rate is
of fundamental importance in the classical theory of twodimensional
NavierStokes turbulence, which postulates that in the inviscid limit,
this rate remains nonzero and becomes independent of the Reynolds number.
Despite its significance, decades of research had failed to render
a quantitative knowledge of the enstrophy dissipation rate. This paper
derives its first and only upper bound. The bound is valid for all
Reynolds numbers and expressible in terms of the initial vorticity
field only.

CVT and J.C. Bowman: Largescale
energy spectra in surface quasigeostrophic turbulence,
J. Fluid Mech.
526,
349359 (2005).
(online)

CVT, T.G. Shepherd and
H.R. Cho: Extensivity of twodimensional turbulence,
Physica D
192,
187195 (2004).
(online)

CVT: Nonlinear transfer and
spectral distribution of energy in alpha turbulence,
Physica D
191,
137155 (2004).
(online)

CVT and J.C. Bowman: Robustness
of the inverse cascade in twodimensional turbulence,
Phys. Rev. E
69,
036303 (2004).
(online)

CVT and J.C. Bowman: Energy budgets
in CharneyHasegawaMima and surface quasigeostrophic turbulence,
Phys. Rev. E
68,
036304 (2003).
(online)

CVT and J.C. Bowman: On the dual
cascade in twodimensional turbulence,
Physica D
176,
242255 (2003).
(online)

CVT, T.G. Shepherd and
H.R. Cho: Stability of stationary solutions of the forced NavierStokes
equations on the twotorus,
Discrete Contin. Dyn. Syst. B
2,
483494 (2002).
(online)

CVT and T.G. Shepherd: Constraints
on the spectral distribution of energy and enstrophy dissipation in forced
twodimensional turbulence,
Physica D
165,
199212 (2002).
(online)
 CVT: Extensive chaos and complexity of
twodimensional turbulence, PhD thesis, University of Toronto (2001).
Recent Invited
Talks
The pressure force in NavierStokes
flows, Warwick EPSRC Symposium on PDEs in Fluid Mechanics
(Warwick, September 2016)
Regularity of NavierStokes flows
with bounds for the pressure, Recent Advances in Hydrodynamics
(Banff, June 2016)
A physicist's approach to NavierStokes
regularity, Third Scottish PDE Colloquium
(Strathclyde, June 2015)
Growth of velocity norms in
NavierStokes flows, Clay Mathematics Institute Workshop on
Geometry and Fluids
(Oxford, April 2014)
Global regularity of MHD turbulence
with partial hyperdissipation, IUTAM Symposium on Understanding
Common Aspects of Extreme Events in Fluids
(Dublin, July 2012)
A dynamical systems approach to
fluid turbulence, Fourth International Symposium on Bifurcations
and Instabilities in Fluid Dynamics
(Barcelona, July 2011)
A quantitative description
of turbulence, Canadian Mathematical Society Meeting
(Edmonton, June 2011)
The number of degrees of freedom of
threedimensional NavierStokes turbulence, Fields Institute
Workshop on the Dynamics in Environmental and Geophysical flows
(Waterloo, July 2009)
Refereeing for
Physica D, Journal of Fluid Mechanics,
Physics of Fluids, Physical Review Letters, Physical Review E, Journal of
the Atmospheric Sciences, Discrete and Continuous Dynamical Systems,
Journal of Mathematical Physics, Geophysical and Astrophysical Fluid
Dynamics, Applied Mathematics and Computation, SIAM Journal on Applied
Mathematics, Europhysics Letters, Journal of Physics A, Applied Mathematical
Modelling, Applied Mathematics Letters, Journal of Mathematical Analysis
and Applications, Nonlinear Analysis, Nonlinearity, Zeitschrift fuer
Naturforschung A, Physical Review Fluids
Thank you for visiting my home page.
How about a tour of my bigger home at the link below?
Vortex Dynamics Research Group
Grand problems in fluid dynamics
are concerned with the inviscid limit.
There arise the subtle notion of dissipation anomaly
and the tricky issue of solution regularity.
Visit this page again for ... some turbulence limerics.