Multifractal structure of the vorticity gradient squared (click to enlarge)

In the forward enstrophy cascade
range, a linear drag causes the powerlaw exponent of the energy
spectrum to become steeper than the classical value of 3. Moreover,
the system becomes intermittent.
These are the results of the nonuniform stretching in the fluid
coupled with a linear drag. A theory based on finitetime Lyapunov
exponent is used to predict the energy spectrum exponent. The
intermittency is quantified by:
(i) the anomalous scaling of the vorticity structure functons,(ii) the scale dependence of the probability density of the vorticity increments, (iii) the multifractality of the viscous enstrophy dissipation. Intermittency in twodimensional turbulence with drag, Phys. Rev. E 71, 066313 (2005) [PDF] 
Energy
injection rate vs. drag coefficient (click to enlarge)

In the inverse energy cascade
range, a linear drag introduces a cutoff in the energy spectrum at
large scales rather than changing the powerlaw exponent significantly
from its classical value of 5/3. In this regime, we focus on the
dependence of the energy injection
rate on drag. The energy injection rate plays a crucial role in
Kraichnan's phenomenolgy, it is also of practical importance in many
engineering and meteorological applications. Our numerical simulations
reveal a new scaling regime in which the energy injection rate has a
powerlaw dependence on the drag coefficient, with a scaling exponent
of 1/3. Such scaling stems from
the nonlocal interaction between the smallscale forced mode and the
largescale eddies.
Forceddissipative twodimensional turbulence: a scaling regime controlled by drag, Phys. Rev. E 79, 045308(R) (2009) [PDF] 
Stability curves by various
methods (click to enlarge) 
We also study the stability of a twodimensional
flow forced by a sinusoidal body force (Kolmogorov flow) on a βplane.
We focus on the case where drag is the main dissipative mechanism.
Linear instability theory determines the part of the parameter space
where the flow is unstable to infinitesimal perturbations. On the other
hand, nonlinear stability analysis, establish the region in which the
flow is stable to arbitrary perturbations. Observing that there exists
a constraint on the time evolution of the difference between the energy
and enstrophy, we develop a new nonlinear stability method, the
energyenstrophy (EZ) method, which proves nonlinear stability in a
larger portion of the parameter space than the traditional energy
method.
Energyenstrophy stability of βplane Kolmogorov flow with drag, Phys. Fluids 20, 084102 (2008) [PDF] 
There have been a lot of interest in the onepoint velocity statistics of forced twodimensional turbulence. Specifically, is the velocity probability density function (PDF) Gaussian or nonGaussian?
It turns out the answer depends on the largescale dissipation that is required to remove the energy
injected by the forcing (at the small scales). For hypodrag or hypoviscosity, where damping only occurs
at the largescale modes, the velocity PDF is nonGaussian due to the strong vortices present in the
system. On the other hand, with the physically motivated linear (Ekman) drag or quadratic drag, the
velocity PDF is close to Gaussian and somewhat surprisingly, the background vorticity is an important
factor in shaping the velocity PDF even though vortices are visually dominant in the flow. Further information on the relation between the velocity PDF and the largescale drag is revealed by the vortex statistics obtained from a vortex census algorithm.
The movie on the right shows that the high velocity regions (which contirbute to the tail of the velocity PDF) comes from the background vorticity for linear drag but are associated with the vortices for hypodrag. Nonuniversal velocity probability densities in forced twodimensional turbulence: the effect of largescale dissipation, Phys. Fluids 22, 115102 (2010) [PDF] 
The breaking of internal gravity waves plays a role in deep ocean mixing. One route for internal waves
to dissipate is through parametric subharmonic instability (PSI), in
which waves at one temporal frequency impart energy to disturbances
with half that frequency and much smaller vertical spatial scale, thus
set the stage for turbulent mixing. Here we consider the rotationally
dominated case where the frequency of the pump wave is twice the local
inertial frequency, so that the recipient subharmonic is a
nearinertial oscillation. Our analytic estimate of the energy transfer rate compared favorably
with previous numerical studies and observational data.
The movie on the right shows the development of such nearinertial PSI, the parameters used corresponds to a M2 tidal beam at 28.8°N. Nearinertial parametric subharmonic instability, J. Fluid Mech. 607, 25 (2008) [PDF] 
Click image to see a plankton population being
stirred by a velocity field in a spatially varying environment
(MSMPEG4 version here) 
Plankton in the upper
ocean plays an essential role in the global carbon cycle by converting
carbon dioxide and other dissolved nutrients into particulate matter.
Thus, planktonic biomass is an important parameter in models of global
climate and climate change. In this study, we investigate the dynamics
of plankton population in a spatially heterogeneous environment, that
is, part of the environment is favorable to plankton growth while other
parts are considered to be hazardous. The plankton concentration in
such an environment can be modelled by the twodimensional
advectiondiffusion equation with a spatially varying logistic growth
term, the local growth rate can be positive or negative.
As a result of the interplay between the growth profile and the flow field, the plankton population can reach a statistical steady state or become extinct. In the limit of a rapidly decorrelating velocity field, we give theoretical prediction of the critical velocity above which the population extincts. In the case when the population survives, we derive upper and lower bounds on the biomass and productivity using variational arguments and direct inequalities. The movie on the right shows the effect of the velocity on the survival of the population, the region with positive growth rate is at the center part of the domain. Bounding biomass in the Fisher equation, Phys. Rev. E 75, 066304 (2007) [PDF] 
Click image to see the evolution of a
fast bimolecular reaction in a chaotic flow
(MSMPEG4 version here) 
When solutions of HCl
and NaOH are mixed, neutralization reaction occurs in which water and
salt are formed. Such acidbase reaction is an example of fast bimolecular
reactions in liquid phase. The evolution of a fast reaction is limited
by how quickly the reactants are brought into contact through diffusion.
When a reaction occurs in a chaotic flow, its progress may be promoted
due to enhancement in diffusion by the stretching and folding actions of
the flow.
The goal of this project is to predict the evolution of a fast bimolecular reaction in a chaotic flow when the properties of the flow are given. Interestingly, depending on the length scale of the velocity relative to the domain size, there are two different scenarios. If the velocity scale is comaprable to the domain size, the decay rate of the reactants is determined by the smallscale stretching statistics of the flow. On the other hand, when the velocity scale is small compared to the domain size, the progress of the reaction is given in terms of an effective diffusivity determined by the gross properties of the velocity. The movie on the right demonstrate the stretchingcontrolled case, where the color intensity represents the local concentration of the two reactants. Predicting the evolution of fast chemical reactions in chaotic flows, Phys. Rev. E 80, 026305 (2009) [PDF] 