Stratified Flow Past an Obstacle

The laboratory experiments of Browand and Winant (Geophysical Fluid Dynamics, 1972), in which a cylinder is towed horizontally in a uniformly stratified fluid, will be used as a case study in combining laboratory, numerical, and theoretical approaches to idealized flows with geophysical applications.

This talk will focus first on the steady, low topographic Froude number regime characterized by blocking, upstream propagation of long gravity waves and stratified hydraulic control. Laboratory and numerical results will be used to motivate a new theoretical framework that describes the observed accelerated, asymmetric flow over the obstacle crest in terms of reduced gravity single layer hydraulics. The active layer is subcritical upstream, controlled at the crest, and supercritical downstream.

These ideas are then extended to oscillating flows and a new scaling analysis is introduced that clarifies the distinction between linear flows characterized by internal wave radiation and nonlinear flows that exhibit a mix of wave-­‐like and hydraulic-­‐like features. These nonlinear flows oscillate between states in which supercritical jets separate from the lee of the obstacle and are broken apart by a series of shear instabilities. The theoretical and scaling arguments are compared to both laboratory and numerical experiments of oscillating stratified flow past a cylinder.

On the Development of a Scalable Fully-Implicit Stabilized Unstructured FE Capability for Resistive MHD with Integrated Adjoint Error-Estimate

The resistive magnetohydrodynamics (MHD) model describes the dynamics of charged fluids in the presence of electromagnetic fields. MHD models are used to describe important phenomena in the natural physical world and in technological applications. This model is non-self adjoint, strongly coupled, highly nonlinear and characterized by multiple physical phenomena that span a very large range of length- and time-scales. These interacting, nonlinear multiple time-scale physical mechanisms can balance to produce steady-state behavior, nearly balance to evolve a solution on a dynamical time-scale that is long relative to the component time-scales, or can be dominated by just a few fast modes. These characteristics make the scalable, robust, accurate, and efficient computational solution of these systems extremely challenging. For multiple-time-scale systems, fully-implicit methods can be an attractive choice that can often provide unconditionally-stable time integration techniques. The stability of these methods, however, comes at a very significant price, as these techniques generate large and highly nonlinear sparse systems of equations that must be solved at each time step.

This talk describes recent progress on the development of a scalable fully-implicit stabilized unstructured finite element (FE) capability for 3D resistive MHD with integrated adjoint error- estimation capability. The brief discussion considers the development of the stabilized FE formu- lation and the underlying fully-coupled preconditioned Newton-Krylov (NK) nonlinear iterative solver. To enable robust, scalable and efficient solution of the large-scale sparse linear systems generated by the Newton linearization, fully-coupled multilevel preconditioners are employed. The stabilized FE formulation and robust fully-coupled NK iterative solvers enable the solution of a wide range of flow conditions that include incompressible, low Mach number approximations, Boussinesq, anelastic, and low Mach number compressible flow. In addition the fully-implicit NK formulation allows the development of adjoint-based error-estimation methods. We present some recent representative results employing the adjoint methods to simple Navier-Stokes and resistive MHD verification problems as well as a RANS turbulence model.

We then briefly consider two sets of recent simulation results with relevance to geophysical and astrophysical flows. The first is the break-up of thin Sweet-Parker current sheets into smaller plasmoids that has been the subject of attention as a possible mechanism for fast reconnection in resistive MHD. Various studies, both theoretical and numerical, have shown that the fast formation of small structures is not only possible, but in fact unavoidable for large enough Lundquist numbers. In this study, we have used state-of-the-art computational capabilities to perform simulations of the Fadeev island coalescence problem in the high Lundquist number regime to investigate if thin current sheets dynamically formed in this strongly non-uniformly driven problem are prone to break- up by fast plasmoid instabilities. Our numerical simulations confirm that plasmoid break-up of dynamically formed current sheets occur for S > 106 (with L. Chacon and D. Knoll LANL). Second we present some very recent results for high Rayleigh number thermal convection in cylindrical geometries of various aspect ratios that have relevance to aspects of the SpinLab experiments.

#This work was supported by the DOE office of Science Advanced Scientific Computing Research – Applied Math Research program at Sandia National Laboratory.

Inverse Cascade in Anisotropic Flows

We examine the inverse cascade of kinetic energy to large scales in rotating stratified turbulence as occurs in the oceans and in the atmosphere, while varying the relative frequency of gravity to inertial waves, N/f . Using direct numerical simulations with grid resolutions up to 1024^3 points, we find that the transfer of energy from three-dimensional to two-dimensional modes is most efficient in the range 1/2 ≤ N/f ≤ 2, in which resonances disappear. In this range, the cascade is faster than in the purely rotating case, and thus the interplay between rotation and stratification helps creating large scale structures. The ensuing inverse cascade follows a −5/3 spectral law with an approximately constant flux.The purely stratified case will also be examined in this context being limit of infinite N/f.

Baroclinic Critical Layers and Zombie Vortices in Couette-Taylor Flow

We report a new mechanism for creating vortices in a class of flows that are linearly stable and believed, by most researchers, to be also finite-amplitude stable. The vortices should form in stably-stratified Couette flows (both plane and circular), and in protoplanetary disks around forming protostars. Our study was motivated by the fact that protoplanetary disks must have flow instabilities that are capable of transporting angular momentum radially outward so that the protostars can accrete gas and grow into stars. The mechanism that we discovered allows small-amplitude perturbations (i.e., with small volumes and Rossby numbers) to form vortices that are large in volume and amplitude (with a Rossby number of order unity). The mechanism works by exciting neutrally stable baroclinic critical layers, which differ from the usual barotropic critical layers in uni-directional flows (responsible for the much-discussed but rarely-observed Kelvin’s cats-eye vortices). The singularities in the former layers are in their vertical velocities, while the latter are in their stream-wise velocities. The energy of the vortices becomes large, and it is supplied by the kinetic energy of the background shear flow. The vortices we found have an unusual property: a vortex that grows from a single, local perturbation triggers a new generation of vortices to grow at nearby locations. After the second generation of vortices grows large, it triggers a third generation. The triggering of subsequent generations continues ad infinitum so that a front dividing the vortex- dominated flow from the unperturbed flow advances until the entire domain fills with large vortices. The vortices do not advect across the region, the front of the vortex- populated fluid does. The region in protoplanetary disks where we have found this new mechanism is thought to be stable; thus, in the astrophysical literature this region is called the dead zone. Because the vortices we report here arise in the dead zone, grow large, and spawn new generations of vortices that march across the domain, we refer to them as zombie vortices. We consider the mechanism of the zombie vortices’ growth and advance in a proposed lab experiment: circular Couette flow with a vertically stably- stratified Boussinesq fluid (i.e., salt water) with a density that is linear with height. Because this flow is nearly homogenous, the first vortex formed by the initial instability self-replicates in an approximately spatially self-similar manner and fills the domain with a lattice of 3D vortices, which persists, despite the fact that the flow is turbulent.

A Pressurized Cryogenic Nitrogen Cell to Study Rotating Turbulent Convection

Due to its fluid properties, cryogenic nitrogen can be used to study flows at high Reynolds and Rayleigh numbers in a compact apparatus compared to standard test fluids. We present the design of a new convection experiment with rotation that uses cryogenic nitrogen, and discuss a few scientific problems that can be addressed with it.

The apparatus consists of a 0.5m diameter 1m tall pressure vessel that can hold a pressure up to 35bar, enclosed in a vacuum jacket to improve insulation, and mounted on a rotating table. Several ports on the vessel provide space for different diagnostics and/or optical access to visualize the flow. By using cryogenic nitrogen from its freezing point up to the critical point, we can greatly vary its fluid properties and tune them for different experiment.

Liquid nitrogen below 77K is particularly useful to study highly convective flows where the effect of rotation is important, i.e. at high Rayleigh numbers and low Rossby numbers. In particular with a 20cm diameter liquid nitrogen cell of aspect ratio 1/2 we plan to study the heat transfer scalings in the geostrophic regime at Rayleigh numbers of order 1011 and to test different theories [1, 2, 3].

Nitrogen close to its critical point would yield the highest Rayleigh numbers flows, allowing us to inves- tigate highly turbulent convective cells with large aspect ratios. In such region of the parameter space very limited data is available and several outstanding questions are still open. In particular, we are interested in the large scale structures of the flow and in studying the existence of polygonal convective cells observed in simulations [4]. With a 37cm diameter cell with top and bottom sapphire plates, and aspect ratio 8, we plan to visualize the flow at Rayleigh number as high as 1011 − 1012.

By using both liquid and gaseous nitrogen along the saturated vapor curve, these experiments can be done also in the case of two phase convection.


  1. [1]  E. M. King, S. Stellmach, J. Noir, U. Hansen, and J. M. Aurnou, “Boundary layer control of rotating convection systems.,” Nature, vol. 457, pp. 301–4, Jan. 2009.
  2. [2]  E. M. King, S. Stellmach, and J. M. Aurnou, “Heat transfer by rapidly rotating Rayleigh–B ́enard con- vection,” Journal of Fluid Mechanics, vol. 691, pp. 568–582, Jan. 2012.
  3. [3]  K. Julien, E. Knobloch, A. M. Rubio, and G. M. Vasil, “Heat Transport in Low-Rossby-Number Rayleigh- B ́enard Convection,” Physical Review Letters, vol. 109, p. 254503, Dec. 2012.
  4. [4]  J. Bailon-Cuba, M. S. Emran, and J. Schumacher, “Aspect ratio dependence of heat transfer and large- scale flow in turbulent convection,” Journal of Fluid Mechanics, vol. 655, pp. 152–173, May 2010.

Plasma and Liquid Sodium Laboratory Dynamo Experiments at UW Madison

First measurements of plasma temperature, density, and flow have been made on the Madison Plasma Dynamo Experiment (MPDX) that allow the particle and energy confinement as well as the plasma conductivity (η) and viscosity (ν) to be estimated. The MPDX is designed to create large flowing plasmas with high magnetic Reynolds number Rm = vL/η >> 1000, and an adjustable fluid Reynolds number 10 < Re = vL/ν < 1000, in the regime where the kinetic energy of the flow exceeds the magnetic energy (MA = v/vA >> 1). Simulations provide scenarios for generating large scale “slow” dynamos and small scale “fast” dynamos to be studied. Confinement is provided by alternating rings of 4 kG permanent magnets lining the vessel walls. Stirred is induced using anodes and thermally emissive Lanthanum hexaboride (LaB6) cathodes inserted in the confining magnetic multicusp edge of the plasma in a method first developed by the Plasma Couette Experiment (PCX) at UW Madison. An overview of plasma flows in PCX and MPDX will be presented as well as several experimental setups designed to achieve dynamo in MPDX. Resent results studying the vector turbulent EMF (the β effect) in the Madison Dynamo Experiment (MDE), a liquid sodium experiment at UW Madison will also be presented.

Low-Dimensional Modeling of Turbulent Convection Roll Dynamics

Turbulence is of tremendous importance in a wide range of astrophysical and geophysical flows. Unfortunately, the equations of motion are notoriously difficult to solve. I will introduce an approach to low-dimensional modeling of turbulent flows that focuses on the the large, coherent flow structures which often occur, such as convection rolls in the atmosphere or ocean currents. These structures and their dynamics can be described with relatively few variables using a model consisting of stochastic ordinary differential equations. As a model system to test this approach, we use Rayleigh-Benard convection experiments, in which a container is filled with water and heated from below. Buoyancy drives a flow which organizes into a roll-shaped circulation. This convection roll exhibits a wide range of dynamics including erratic meandering, spontaneous flow reversals, and several oscillation modes, all of which are reminiscent of phenomena observed in astro/geophysical flows. A simple model of stochastic motion in a potential quantitatively reproduces all of these observed flow dynamics. The potential term is a direct function of boundary geometry (i.e. topography), and is found to accurately predict the different flow dynamics observed in experiments with different boundary geometries. This approach may lead to more general and relatively easy to solve models for turbulent flows with potential applications to climate, weather, and even the turbulent dynamo that generates Earth’s magnetic field.

Horizontal shear in the rotating, stratified ocean : Linear theory and nonlinear evolution

Submesoscale instabilities and mixing are poorly understood. We focus our work on barotropic shear with Rossby and Froude numbers of O (1). Instabilities and nonlinear cascades are possible in this regime even though stable stratification is significant. We have demonstrated previously (Arobone and Sarkar, JFM 2012) that the linear stability of the shear layer shows new aspects for strong stratification and moderate rotation rates. In this regime stratification acts to stabilize the inertial instability but greatly increase the range of vertical wavenumbers unstable to barotropic instability when Ro ∼ −1.

Nonlinear simulations are used to explore the shear layer with Ro(t). Coherent structure evolution varies greatly between cases with different moderate anticyclonic values of Ro0, but cyclonic rotation and strong anticyclonic rotation modify the flow in a more straightforward manner. Possible instability mechanisms, e.g. elliptic, zigzag, inertial, and barotropic instabilities are related to the simulation results. Enstrophy budgets from the simulations show a marked transition corresponding to sign reversal of centerline absolute vorticity, consistent with the linear modification of barotropic instability when Ro ∼ −1. New results analyzing saturation of inertial instability in the presence of strong stratification will be presented, noting strong differences from unstratified flows.

Why did the 2010 Eyjafjallajökull volcanic eruption cloud last so long?

The global economic consequences of the relatively small Eyjafjallajok̈ ull eruption in the spring of 2010 caught the world off guard. That the eruption cloud lasted for several months rather than weeks, efficiently disrupting air travel and the holiday plans of thousands of Northern Europeans, drew arguably more attention and a certainly garnered a highly emotional response. The unexpected longevity of this eruption cloud was touted to be the consequence of unusual ”perfect-storm-like” weather patterns that also conspired to produce the very dry conditions leading to the massive Russian fires later that summer. It was called ”an anomaly”. However, this anomaly nearly repeated itself the following year in the form of the 2011 Grimsvoẗ n eruption cloud. Indeed, in the geological record, possibly 45% of explosive eruptions produced long-lasting clouds similar to the 2010 Eyjafjallajok̈ ull event, which is clearly not so unusual.

A major reason that the behavior of the 2010 Eyjafjallajok̈ ull eruption cloud was surprising is that ”standard” models for how ash sedimentation works (i.e., heavy particles fall out of the cloud faster than light particles) are incomplete with significant consequences not just for assessing hazards to air traffic, but also for understanding, for example, the effect of volcanism on climate. Observations of the 2010 Eyjafjallajok̈ ull and 2011 Grimsvoẗ n umbrella clouds, as well as the structure of atmospheric aerosol clouds from the 1991 Mt Pinatubo event, suggest that an additional key process in addition to particle settling is the production of internal layering. I will use analog experiments on turbulent particle-laden umbrella clouds understood with simple models to show that this layering occurs where natural convection driven by particle sedimentation and the differential diffusion of primarily heat and fine particles give rise to a large scale instability leading to this layering. This “particle diffusive convection” strongly influences cloud longevity where volcanic umbrella clouds are enriched in fine ash. More generally, however, volcanic cloud residence times will depend on ash fluxes related to both individual particle settling and diffusive convection. I will discuss a new sedimentation model that includes both sedimentation processes captures real-time measurements of the rate of change of particle concentration in the 1982 El Chichon, 1991 Mt Pinatubo and 1992 Mt Spurr ash-clouds. Finally, although we have made progress to understanding how volcanic ash clouds ultimately work, there remain some fundamental problems that I will discuss, depending on time.