Planetary dynamo

Click the image to see how radial magnetic fields at different spatial scales evolve

Scaling of the geomagnetic secular variation time scales The time variation of the geomagnetic field over the range of years to centuries, known as the secular variation, is believed to be originated from the magnetohydrodynamics inside the Earth's outer core. Since the magnetic field at different spatial scales usually vary on different time scales, it is common to investigate the secular variation spectrum, i.e. the secular variation as a function of the spherical harmonics degree 𝑙. An 𝑙-dependent time scale τsv(𝑙) has been constructed from the ratio of the magnetic power spectrum to the secular variation spectrum measured at the Earth's surface. τsv(𝑙) is only defined for regions above the core–mantle boundary (CMB), but it is often assumed to be representative of time scales related to the dynamo inside the outer core. Here we introduce a magnetic time-scale spectrum τ(𝑙,𝑟) that is valid for all radius 𝑟 above the inner core and reduces to τsv(𝑙) above the CMB. We study τ(𝑙,𝑟) in a numerical geodynamo model, focusing on the large scales (small 𝑙). We find that both τ(𝑙,𝑟) and τsv(𝑙) scale like 𝑙-1 at the CMB. However, the scaling of τ(𝑙,𝑟) undergoes a sharp transition just below the CMB and become shallower than 𝑙-1 inside the outer core. Therefore τsv(𝑙) becomes unreliable in estimating time scales inside the outer core. A frozen-flux argument that ignores magnetic diffusion and the radial derivative in the induction equation is often used to explain the scaling of τsv(𝑙). Here we find that away from the boundaries, magnetic diffusion is indeed small but ∂𝐵/∂𝑡 is controlled by the radial derivative in the induction term. Near the CMB, magnetic diffusion affects ∂𝐵/∂𝑡 significantly. Hence the frozen-flux argument is not applicable to τ(𝑙,𝑟) anywhere inside the outer core. We have used either a no-slip or a stress-free condition at the CMB and find that the above results are independent of the velocity boundary conditions employed. Scaling of the geomagnetic secular variation timescales, Geophys J. Int., 239, 1 (2024)

Click the image to see how the magnetic energy spectrum and other properties in a Jupiter dynamo model varies with depth

Magnetic energy spectrum and the dynamo radius of Jupiter Jupiter's magnetic field is generated by the fluid motion of the electrically conducting metallic hydrogen inside the dynamo region of the giant planet. It is believed that the transition from molecular hydrogen near the planetary surface to metallic hydrogen in the interior dynamo region is sharp but nonetheless a smooth one. An interesting question is then: at what depth—the dynamo radius—does generation of magnetic field become significant? We examine the magnetic energy spectrum (roughly speaking, magnetic energy per spherical harmonic degree) as a function of depth in a numerical dynamo model of Jupiter. We discover that the shape of the magnetic energy spectrum becomes invariant inside the dynamo region and so it provides a way to define the dynamo radius. We also find that the magnetic Reynolds number Rm∼O(100)≫1 at this dynamo radius characterized using the magnetic energy spectrum. Can we infer the dynamo radius from observations at the surface? In the case of the Earth, Lowes (1974) assumed the magnetic energy spectrum (for the large scales) is flat at the core–mantle boundary—the so-called white source hypothesis. Then, using the magnetic energy spectrum observed at the Earth's surface, he derived a procedure that gives an accurate estimate for the Earth's dynamo radius. In our dynamo model of Jupiter, we find that the magnetic energy spectrum in the dynamo region to be shallow, but not exactly flat. In this case, we conclude that Lowes' procedure will give a lower bound to the dynamo radius. Characterising Jupiter's dynamo radius using its magnetic energy spectrum, Earth Planet. Sci. Lett. 530, 115879 (2020)

Dynamics of atmospheric water vapor

Click the image to see the Lagrangian and Eulerian formulations of advection–condensation in a channel flow

Condensation of moisture: Lagrangian versus Eulerian formulation In a stochastic Lagrangian formulation of water vapor transport, moist air is represented by an ensemble of parcels tagged with a humidity variable and advected by a velocity whose small-scale turbulent component is modeled by a random process. Each parcel condenses individually as it moves through the saturation humidity field. The 'observed' humidity at a particular location is the result of averaging over the parcels nearby. On the other hand, in an Eulerian formulation, moisture is represented by a coarse-grained field on a numerical grid. The field value at a grid point is interpreted as an average over many fluid parcels. If there is no subgrid modeling, condensation simply occurs when the field value exceeds the local saturation limit. Thus in this case, the averaging operation precedes condensation. This results in loss of small-scale moisture variability and causes a coarse-resolution Eulerian model to produce large regions of high humidity. In this porject, we demonstrate this theoretically and numerically using idealized advection–condensation models. While a stochastic Lagrangian model can account for local moisture fluctuation, it comes with a high computational cost. Practically, all climate and weather models adopt the Eulerian approach, with small-scale flow represented by an eddy diffusion. Hence, we ask the question: is it possible for an Eulerian model to mimic its stochastic Lagrangian counterpart by incorporating subgrid-scale condensation? This can be achieve by assuming the humidity inside a given grid box has a probability distribution. Here, we show that if we specify this assumed distribution by requiring its moments to match those of the Lagrangian model (up to a certain order), the Eulerian model indeed can produce results similar to the Lagrangian model. Based on this result, we discuss how stochastic transport models could be used to parameterize condensation in atmospheric models. A stochastic Lagrangian basis for a probabilistic parameterization of moisture condensation in Eulerian models , J. Atmos. Sci., 75, 3925 (2018)

Click the image to see interesting features developed in the advection–condnesation of moist parcels in a cellular flow

Advection–condensation in a large-scale circulation Atmospheric water vapor controls the absorption of outgoing long-wave radiation and thus affects the Earth's climate. According to the advection-condensation paradigm, the key features of atmospheric humidity distribution are determined by the interplay between the processes of transport and condensation, with cloud-scale microphysics negligible. Employing this idea, previous studies have successfully reconstructed the humidity field using wind fields from observation. In this project, we developed an analytically tractable advection-condensation model that mimics the Hadley cell. We consider an overturning circulation in a square domain with a moisture source at the bottom boundary. Moist air parcels are advected through a saturation humidity field and precipitation occurs when their humidity exceeds the local saturation value. Small-scale turbulence is modeled by white noise. This stochastic model produces many interesting features reminiscent of the atmosphere: a boundary layer near the bottom that resembles the planetary boundary layer a "tropical" region of high precipitation near the rising arm of the circulation the humidity inside these boundary layers has a bimodal distribution a relative humidity minimum developed near the center of the cell Using matched asymptotics, we find that as the strength of the large-scale circulation increases: precipitation becomes more intense and more localized around the "Tropics" the total moisture content in the system decreases The effect of coherent stirring on the advection–condensation of water vapour, Proc. R. Soc. A 473, 20170196 (2017)

Chaotic advection and reactive mixing

Click the image to see the evolution of a fast bimolecular reaction in a chaotic flow	Fast chemical reactions in chaotic flows 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 reaction rate of a fast bimolecular reaction in a chaotic flow when the properties of the flow are given. Depending on the length scale of the velocity relative to the domain size, there are two different scenarios. If the velocity scale is comparable 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)
Click the image to see a plankton population being stirred by a velocity field in a spatially varying environment	Plankton population dynamics 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 modeled 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)

Rotating, stratified (geophysical) flows

Click the image to see the evolution of vortices with different horizontal and vertical aspect ratios λ0 and μ0	Ageostrophic ellipsoidal vortex An ellipsoidal volume of uniform potential vorticity in a rotating stratified fluid is often used an idealized model for vortices in the Earth's atmosphere and oceans. Most previous works are limited to the zero Rossby number quasi-geostrophic (QG) regime where inertia-gravity waves are filtered out. The goal of this research is to expand our mostly QG-based knowledge by considering an ellipsoidal vortex in a non-hydrostatic Boussinesq system at finite Rossby number. Our key findings are: the stability of the vortex increases with the Rossby number, thus, a cyclone is generally more stable than an anticyclone, ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability; an oblate vortex with near-circular cross section tends to be more stable, as a result, an unstable vortex often evolves and restabilizes itself by developing a near-circular cross section; ageostrophic effects cause an anticyclone to rotate faster than its cyclonic counterpart; throughout the nonlinear evolution of the vortex, the amount of spontaneous inertia-gravity wave emission is negligible, thus, the vortex and any associated instabilities are ageostrophically balanced. Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation, J. Fluid Mech. 762, 196 (2015)
Click the image to see the development of small-scale structures in PSI	Inertia-gravity wave 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 rotation-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 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)

Two-dimensional forced-dissipative turbulence

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. However, in most laboratory experiments and geophysical flows, the presence of a large-scale dissipative mechanism significantly modifies the classical phenomenological picture.

Multifractal structure of the vorticity gradient squared (click to enlarge)	Intermittency 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: the anomalous scaling of the vorticity structure functons the scale-dependence of the probability density of the vorticity increments the multifractality of the viscous enstrophy dissipation Intermittency in two-dimensional turbulence with drag, Phys. Rev. E 71, 066313 (2005)
Energy injection rate vs. drag coefficient (click to enlarge)	Energy injection rate 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 theory predicts 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)
Stability curves by various methods (click to enlarge)	Energy-enstrophy stability We 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 the 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)
Click the image to see the evolution of the high velocity regions	Velocity probability distribution function There have been a lot of interest in the one-point velocity statistics in forced two-dimensional turbulence. 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 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 is essentially controlled by the background turbulent vorticity instead of the visually dominant vortices. Hence, in contrast to conventional wisdom, the Gaussianity of the PDF is not a result of the application of the central limit theorem to the vortices. The movie on the right shows that the high velocity regions (which contribute 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)