Keith Julien | University of Colorado, Boulder (original) (raw)
Papers by Keith Julien
Physical review letters, Jan 19, 2014
Rapidly rotating Rayleigh-Bénard convection is studied by combining results from direct numerical... more Rapidly rotating Rayleigh-Bénard convection is studied by combining results from direct numerical simulations (DNS), laboratory experiments, and asymptotic modeling. The asymptotic theory is shown to provide a good description of the bulk dynamics at low, but finite Rossby number. However, large deviations from the asymptotically predicted heat transfer scaling are found, with laboratory experiments and DNS consistently yielding much larger Nusselt numbers than expected. These deviations are traced down to dynamically active Ekman boundary layers, which are shown to play an integral part in controlling heat transfer even for Ekman numbers as small as 10^{-7}. By adding an analytical parametrization of the Ekman transport to simulations using stress-free boundary conditions, we demonstrate that the heat transfer jumps from values broadly compatible with the asymptotic theory to states of strongly increased heat transfer, in good quantitative agreement with no-slip DNS and compatible ...
Physical Review E, 1996
We report a transition to hard turbulence in rapidly rotating Boussinesq convection at high Rayle... more We report a transition to hard turbulence in rapidly rotating Boussinesq convection at high Rayleigh and Taylor numbers. The probability density for vertical vorticity develops exponential tails, as in nonrotating hard-turbulent convection, whereas the temperature and velocity retain Gaussian distributions. The Nusseltnumber scaling with Rayleigh number for the rotating hard-turbulent state is identical to that for nonrotating laboratory experiments, viz., NuϳRa 2/7 . ͓S1063-651X͑96͒50306-5͔ PACS number͑s͒: 47.27.Te, 47.32.Cc, 47.27.Cn, 47.27.Eq Rayleigh-Bénard convection ͓1͔ is a common model problem for transitions to convective turbulence; the experiments of Libchaber and co-workers have delineated the transitions with increasing nondimensional Rayleigh number Ra ͓2-5͔. The hard-turbulent state at high Ra has drawn much attention ͓6͔; nevertheless, only recently ͑and partly through this work͒ has it been seen as an ubiquitous convective state, with manifestations spanning both large and small aspect ratio ͓2-5͔, two-dimensional ͑2D͒ flows ͓7͔, and even a sideheated geometry ͓8͔. Here we report an example of hard turbulence in a strongly rotating fluid of geophysical and astrophysical relevance with detailed dynamics dramatically different from the nonrotating case. This discovery sheds light on the workings of hard turbulence and aids in evaluating theories for convective heat transport.
Physical review letters, Jan 11, 2014
Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state k... more Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state known as geostrophic turbulence. This state is present for sufficiently large Rayleigh numbers representing the thermal forcing of the system, and is characterized by a leading order balance between the Coriolis force and pressure gradient. This turbulent state is itself unstable to the generation of depth-independent or barotropic vortex structures of ever larger scale through a process known as spectral condensation. This process involves an inverse cascade mechanism with a positive feedback loop whereby large-scale barotropic vortices organize small scale convective eddies. In turn, these eddies provide a dynamically evolving energy source for the large-scale barotropic component. Kinetic energy spectra for the barotropic dynamics are consistent with a k-3 downscale enstrophy cascade and an upscale cascade that steepens to k-3 as the box-scale condensate forms. At the same time the flo...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015
The linear theory for rotating compressible convection in a plane layer geometry is presented for... more The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e. incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilize the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.
Geophysical convection often occurs over areas occupied by ensembles of convecting plumes, and ma... more Geophysical convection often occurs over areas occupied by ensembles of convecting plumes, and may be influenced by rotation when the transit time of plumes is long compared to an inertial period. We examine the behavior of an ensemble of convective plumes, both with and without rotation to identify the influence of rotation on plume transports. Plumes are identified from numerical solutions of turbulent convection by conditionally sampled composites. The dynamical balances of these composite plumes are evaluated and compared with entraining plume models. Non-rotating and rotating plumes show many differences in their transports of mass, buoyancy and momentum. Rotation suppresses expansion of the turbulent plume by entrainment, but through plume-plume interactions enhances mixing between plume and ambient fluid. The resulting dilution of plume buoyancy diminishes the net buoyancy transport at high rotation. Plume acceleration is suppressed at high rotation, in part due to a pressure-gradient deceleration, reducing the net conversion of potential to kinetic energy.
We present the results of a combined experimental and numerical study in which MEMS micro-actuato... more We present the results of a combined experimental and numerical study in which MEMS micro-actuators were used to control the instabilities of a planar jet flow. The experimental growth rates of both the anti-symmetric and symmetric instabilities were measured, and a comparison made with numerical calculations based upon the assumption of non-parallel jet flow. In concluding, we shall discuss the implementation of a closed-loop control strategy for active supression of the instabilities.
Using DNS, we investigate the solution to a reduced system of nonlinear PDEs for rapidly rotating... more Using DNS, we investigate the solution to a reduced system of nonlinear PDEs for rapidly rotating convection: non-hydrostatic quasi-geostrophic equations (NHQGE). The NHQGE are derived asymptotically in the limit of rapid rotation from the Navier-Stokes equations under the Boussinesq approximation. Two distinct vertical scales are present: a small-scale occurring as a consequence of rotational alignment and large-scale due to convective forced motions. The resulting equations filter fast inertial waves and relax the need to resolve Ekman boundary layers, and are applicable to ocean deep turbulent convection, which, under thermal forcing, is characterized by thermal and vortical coherent structures that span the layer depth. Using a Chebyshev-Tau algorithm, we examine variation of heat transport as a function of scaled Rayleigh number and compare results from a single-mode theory. We also investigate the dynamics of the vortical structures and their effect on lateral mixing.
We discuss the derivation of a new reduced system of nonlinear PDEs for rapidly rotating Rayleigh... more We discuss the derivation of a new reduced system of nonlinear PDEs for rapidly rotating Rayleigh-Benard convection (RBC) in a cylinder and examine its numerical solution. The equations are derived asymptotically in the limit of rapid rotation from the Boussinesq equations. Numerical simulation of the full Boussinesq equations for such flow is restricted due to the existence of fast-propagating inertial waves, and due to Ekman boundary layers that become increasingly thin with increased rotation rate. In the rapidly rotating limit, such boundary layers are passive, and are filtered-out in the reduced equations. Numerical simulation of a similar set of reduced equations for an unbounded layer has allowed thorough investigation of RBC in the limit of rapid rotation and for large Rayleigh numbers. Here, we limit our discussion to pattern formation at slightly critical Rayleigh numbers but under rapid rotation, for which there remain unexplained phenomena.
The magnetorotational instability is investigated within the shearing box approximation in the la... more The magnetorotational instability is investigated within the shearing box approximation in the large Elsasser number regime. In this regime, which is of fundamental importance to astrophysical accretion disk theory, shear is the dominant source of energy, but the instability itself ...
The planetary geostrophic (PG) equations for large-scale oceanic flow are linked to the quasigeos... more The planetary geostrophic (PG) equations for large-scale oceanic flow are linked to the quasigeostrophic (QG) equations for mesoscale flow in a multiple-scales asymptotic expansion. The model describes the coupling of planetary-scale and mesoscale dynamics: eddy kinetic energy is generated by baroclinic instability of the planetary flow, and the resulting eddy buoyancy fluxes feed back on the planetary flow. Anisotropy of the planetary flow is seen to play a key role in allowing the two-way coupling. The resulting equations are amenable to theoretical and computational ...
Physical review letters, Jan 19, 2014
Rapidly rotating Rayleigh-Bénard convection is studied by combining results from direct numerical... more Rapidly rotating Rayleigh-Bénard convection is studied by combining results from direct numerical simulations (DNS), laboratory experiments, and asymptotic modeling. The asymptotic theory is shown to provide a good description of the bulk dynamics at low, but finite Rossby number. However, large deviations from the asymptotically predicted heat transfer scaling are found, with laboratory experiments and DNS consistently yielding much larger Nusselt numbers than expected. These deviations are traced down to dynamically active Ekman boundary layers, which are shown to play an integral part in controlling heat transfer even for Ekman numbers as small as 10^{-7}. By adding an analytical parametrization of the Ekman transport to simulations using stress-free boundary conditions, we demonstrate that the heat transfer jumps from values broadly compatible with the asymptotic theory to states of strongly increased heat transfer, in good quantitative agreement with no-slip DNS and compatible ...
Physical Review E, 1996
We report a transition to hard turbulence in rapidly rotating Boussinesq convection at high Rayle... more We report a transition to hard turbulence in rapidly rotating Boussinesq convection at high Rayleigh and Taylor numbers. The probability density for vertical vorticity develops exponential tails, as in nonrotating hard-turbulent convection, whereas the temperature and velocity retain Gaussian distributions. The Nusseltnumber scaling with Rayleigh number for the rotating hard-turbulent state is identical to that for nonrotating laboratory experiments, viz., NuϳRa 2/7 . ͓S1063-651X͑96͒50306-5͔ PACS number͑s͒: 47.27.Te, 47.32.Cc, 47.27.Cn, 47.27.Eq Rayleigh-Bénard convection ͓1͔ is a common model problem for transitions to convective turbulence; the experiments of Libchaber and co-workers have delineated the transitions with increasing nondimensional Rayleigh number Ra ͓2-5͔. The hard-turbulent state at high Ra has drawn much attention ͓6͔; nevertheless, only recently ͑and partly through this work͒ has it been seen as an ubiquitous convective state, with manifestations spanning both large and small aspect ratio ͓2-5͔, two-dimensional ͑2D͒ flows ͓7͔, and even a sideheated geometry ͓8͔. Here we report an example of hard turbulence in a strongly rotating fluid of geophysical and astrophysical relevance with detailed dynamics dramatically different from the nonrotating case. This discovery sheds light on the workings of hard turbulence and aids in evaluating theories for convective heat transport.
Physical review letters, Jan 11, 2014
Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state k... more Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state known as geostrophic turbulence. This state is present for sufficiently large Rayleigh numbers representing the thermal forcing of the system, and is characterized by a leading order balance between the Coriolis force and pressure gradient. This turbulent state is itself unstable to the generation of depth-independent or barotropic vortex structures of ever larger scale through a process known as spectral condensation. This process involves an inverse cascade mechanism with a positive feedback loop whereby large-scale barotropic vortices organize small scale convective eddies. In turn, these eddies provide a dynamically evolving energy source for the large-scale barotropic component. Kinetic energy spectra for the barotropic dynamics are consistent with a k-3 downscale enstrophy cascade and an upscale cascade that steepens to k-3 as the box-scale condensate forms. At the same time the flo...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015
The linear theory for rotating compressible convection in a plane layer geometry is presented for... more The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e. incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilize the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.
Geophysical convection often occurs over areas occupied by ensembles of convecting plumes, and ma... more Geophysical convection often occurs over areas occupied by ensembles of convecting plumes, and may be influenced by rotation when the transit time of plumes is long compared to an inertial period. We examine the behavior of an ensemble of convective plumes, both with and without rotation to identify the influence of rotation on plume transports. Plumes are identified from numerical solutions of turbulent convection by conditionally sampled composites. The dynamical balances of these composite plumes are evaluated and compared with entraining plume models. Non-rotating and rotating plumes show many differences in their transports of mass, buoyancy and momentum. Rotation suppresses expansion of the turbulent plume by entrainment, but through plume-plume interactions enhances mixing between plume and ambient fluid. The resulting dilution of plume buoyancy diminishes the net buoyancy transport at high rotation. Plume acceleration is suppressed at high rotation, in part due to a pressure-gradient deceleration, reducing the net conversion of potential to kinetic energy.
We present the results of a combined experimental and numerical study in which MEMS micro-actuato... more We present the results of a combined experimental and numerical study in which MEMS micro-actuators were used to control the instabilities of a planar jet flow. The experimental growth rates of both the anti-symmetric and symmetric instabilities were measured, and a comparison made with numerical calculations based upon the assumption of non-parallel jet flow. In concluding, we shall discuss the implementation of a closed-loop control strategy for active supression of the instabilities.
Using DNS, we investigate the solution to a reduced system of nonlinear PDEs for rapidly rotating... more Using DNS, we investigate the solution to a reduced system of nonlinear PDEs for rapidly rotating convection: non-hydrostatic quasi-geostrophic equations (NHQGE). The NHQGE are derived asymptotically in the limit of rapid rotation from the Navier-Stokes equations under the Boussinesq approximation. Two distinct vertical scales are present: a small-scale occurring as a consequence of rotational alignment and large-scale due to convective forced motions. The resulting equations filter fast inertial waves and relax the need to resolve Ekman boundary layers, and are applicable to ocean deep turbulent convection, which, under thermal forcing, is characterized by thermal and vortical coherent structures that span the layer depth. Using a Chebyshev-Tau algorithm, we examine variation of heat transport as a function of scaled Rayleigh number and compare results from a single-mode theory. We also investigate the dynamics of the vortical structures and their effect on lateral mixing.
We discuss the derivation of a new reduced system of nonlinear PDEs for rapidly rotating Rayleigh... more We discuss the derivation of a new reduced system of nonlinear PDEs for rapidly rotating Rayleigh-Benard convection (RBC) in a cylinder and examine its numerical solution. The equations are derived asymptotically in the limit of rapid rotation from the Boussinesq equations. Numerical simulation of the full Boussinesq equations for such flow is restricted due to the existence of fast-propagating inertial waves, and due to Ekman boundary layers that become increasingly thin with increased rotation rate. In the rapidly rotating limit, such boundary layers are passive, and are filtered-out in the reduced equations. Numerical simulation of a similar set of reduced equations for an unbounded layer has allowed thorough investigation of RBC in the limit of rapid rotation and for large Rayleigh numbers. Here, we limit our discussion to pattern formation at slightly critical Rayleigh numbers but under rapid rotation, for which there remain unexplained phenomena.
The magnetorotational instability is investigated within the shearing box approximation in the la... more The magnetorotational instability is investigated within the shearing box approximation in the large Elsasser number regime. In this regime, which is of fundamental importance to astrophysical accretion disk theory, shear is the dominant source of energy, but the instability itself ...
The planetary geostrophic (PG) equations for large-scale oceanic flow are linked to the quasigeos... more The planetary geostrophic (PG) equations for large-scale oceanic flow are linked to the quasigeostrophic (QG) equations for mesoscale flow in a multiple-scales asymptotic expansion. The model describes the coupling of planetary-scale and mesoscale dynamics: eddy kinetic energy is generated by baroclinic instability of the planetary flow, and the resulting eddy buoyancy fluxes feed back on the planetary flow. Anisotropy of the planetary flow is seen to play a key role in allowing the two-way coupling. The resulting equations are amenable to theoretical and computational ...