Two-dimensional turbulence with drag

This is motivated by the fact that in most laboratory experiments and geophysical applications of two-dimensional turbulence, friction is inevitably an important effect. The cornerstone of two-dimensional hydrodynamics is the existence of a dual cascade in which enstrophy forward cascade to smaller scales and energy inverse cascade to larger scales.

vorticity gradient squared
Multifractal structure of the vorticity gradient squared (click to enlarge)
In the forward enstrophy cascade range, a linear drag causes the power-law 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 non-uniform stretching in the fluid coupled with a linear drag. A theory based on finite-time 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 two-dimensional turbulence with drag, Phys. Rev. E 71, 066313 (2005) [PDF]

energy injection rate vs. drag
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 power-law 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 power-law dependence on the drag coefficient, with a scaling exponent of 1/3. Such scaling stems from the nonlocal interaction between the small-scale forced mode and the large-scale eddies.

Forced-dissipative two-dimensional 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 two-dimensional flow forced by a sinusoidal body force (Kolmogorov flow) on a β-plane. 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 energy-enstrophy (EZ) method, which proves nonlinear stability in a larger portion of the parameter space than the traditional energy method.

Energy-enstrophy stability of β-plane Kolmogorov flow with drag, Phys. Fluids 20, 084102 (2008) [PDF]

high velocity bubbles
Click image to see the evolution of the high velocity regions (MS-MPEG4 version here)
There have been a lot of interest in the one-point velocity statistics of forced two-dimensional turbulence. Specifically, is the velocity probability density function (PDF) Gaussian or non-Gaussian? It turns out the answer depends on the large-scale dissipation that is required to remove the energy injected by the forcing (at the small scales). For hypo-drag or hypo-viscosity, where damping only occurs at the large-scale modes, the velocity PDF is non-Gaussian 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 large-scale 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 hypo-drag.

Non-universal velocity probability densities in forced two-dimensional turbulence: the effect of large-scale dissipation, Phys. Fluids 22, 115102 (2010) [PDF]

Internal gravity waves

Click image to see the development of small-scale structures in PSI (MS-MPEG4 version here)
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 near-inertial 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 near-inertial PSI, the parameters used corresponds to a M2 tidal beam at 28.8°N

Near-inertial parametric subharmonic instability
, J. Fluid Mech. 607, 25 (2008) [PDF]

Plankton population dynamics

Click image to see a plankton population being stirred by a velocity field in a spatially varying environment (MS-MPEG4 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 two-dimensional advection-diffusion 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]

Fast chemical reactions in chaotic flows

fast chemical reaction in chaotic flow
Click image to see the evolution of a fast bimolecular reaction in a chaotic flow (MS-MPEG4 version here)
When solutions of HCl and NaOH are mixed, neutralization reaction occurs in which water and salt are formed. Such acid-base 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 small-scale 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 stretching-controlled 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]

More to come . . .