Keith Julien

Keith Julien
University of Colorado Boulder | CUB · Department of Applied Mathematics (College of Arts and Sciences)

Professor

About

154
Publications
7,955
Reads
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3,057
Citations
Additional affiliations
July 2008 - present
University of Colorado Boulder
Position
  • Professor
Description
  • Dr. Julien's primary area of research is focussed in the mathematical geo- and astro-physical sciences. Specifically, the modeling of dynamical processes and instabilities occurring in geophysical and astrophysical flows. Examples include protoplanetary disks, stably and unstably stratified flows, shear flows and boundary effects in turbulent convection. Particular emphasis is placed on the identification of reduced PDE models that accurately describe flow dynamics in extreme regimes.
Education
October 1986 - November 2019
University of Cambridge
Field of study
  • Applied Mathematics and Theoretical Physcis
October 1983 - June 1986
King's College University of London
Field of study
  • Applied Mathematics and Theoretical Physcis

Publications

Publications (154)
Preprint
Full-text available
Bifurcation analysis of confined salt-finger convection using single-mode equations obtained from a severely truncated Fourier expansion in the horizontal is performed. Strongly nonlinear staircase-like solutions having, respectively, one (S1), two (S2) and three (S3) regions of well-mixed salinity in the vertical direction are computed using numer...
Preprint
Full-text available
Instabilities of fluid flows often generate turbulence. Using extensive direct numerical simulations, we study two-dimensional turbulence driven by a wavenumber-localised instability superposed on stochastic forcing, in contrast to previous studies of state-independent forcing. As the instability growth rate increases, the system undergoes two tran...
Preprint
Standard spectral codes for full sphere dynamics utilize a combination of spherical harmonics and a suitableradial basis to represent fluid variables. These basis functions have a rotational invariance not present ingeophysical flows. Gyroscopic alignment - alignment of dynamics along the axis of rotation - is ahallmark of geophysical fluids in the...
Preprint
Full-text available
The competition between turbulent convection and global rotation in planetary and stellar interiors governs the transport of heat and tracers, as well as magnetic-field generation. These objects operate in dynamical regimes ranging from weakly rotating convection to the `geostrophic turbulence' regime of rapidly rotating convection. However, the la...
Article
Standard spectral codes for full sphere dynamics utilize a combination of spherical harmonics and a suitable radial basis to represent fluid variables. These basis functions have a rotational invariance not present in geophysical flows. Gyroscopic alignment - alignment of dynamics along the axis of rotation - is a hallmark of geophysical fluids in...
Article
A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal sc...
Preprint
A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal sc...
Article
The competition between turbulent convection and global rotation in planetary and stellar interiors governs the transport of heat and tracers, as well as magnetic field generation. These objects operate in dynamical regimes ranging from weakly rotating convection to the “geostrophic turbulence” regime of rapidly rotating convection. However, the la...
Article
Full-text available
The observational absence of giant convection cells near the Sun’s outer surface is a long-standing conundrum for solar modelers. We herein propose an explanation. Rotation strongly influences the internal dynamics, leading to suppressed convective velocities, enhanced thermal-transport efficiency, and (most significantly) relatively smaller domina...
Article
Full-text available
Salt-finger convection provides a key mixing process in geophysical and astrophysical fluid flows. Because of its small characteristic spatial scale and slow diffusive time scale, this process must be parameterized in geophysical and astrophysical models, where relations linking background gradients to fluxes are required. To obtain such relations,...
Preprint
This paper considers the dominant dynamical, thermal and rotational balances within the solar convection zone. The reasoning is such that: Coriolis forces balance pressure gradients. Background vortex stretching, baroclinic torques and nonlinear advection balance jointly. Turbulent fluxes convey what part of the solar luminosity that radiative diff...
Article
Full-text available
In this paper, we investigate and develop scaling laws as a function of external nondimensional control parameters for heat and momentum transport for nonrotating, slowly rotating, and rapidly rotating turbulent convection systems, with the end goal of forging connections and bridging the various gaps between these regimes. Two perspectives are con...
Preprint
Buoyancy-driven convection is likely the dominant driver of turbulent motions in the universe, and thus, is widely studied by physicists, engineers, geophysicists and astrophysicists. Maybe unsurprisingly, these different communities discuss the gross convective behaviors in different ways, often without significant cross-talk existing between them...
Preprint
Full-text available
Rotating Rayleigh-B\'enard convection is investigated numerically with the use of an asymptotic model that captures the rapidly rotating, small Ekman number limit, $Ek \rightarrow 0$. The Prandtl number ($Pr$) and the asymptotically scaled Rayleigh number ($\widetilde{Ra} = Ra Ek^{4/3}$, where $Ra$ is the typical Rayleigh number) are varied systema...
Article
Numerical simulations of quasi-static magnetoconvection with a vertical magnetic field are carried out up to a Chandrasekhar number of $Q=10^{8}$ over a broad range of Rayleigh numbers $Ra$ . Three magnetoconvection regimes are identified: two of the regimes are magnetically constrained in the sense that a leading-order balance exists between the L...
Article
Full-text available
The heat transfer scaling theories for Rayleigh‐Bénard convection (RBC) are reviewed and discussed for configurations with and without rotation and magnetic fields. Scaling laws are a useful tool in studying and characterizing geophysical flows as they provide a basis for extrapolation to extreme parameter regimes that remain unobtainable by curren...
Article
A method is presented for constructing energy-conserving Galerkin approximations in the vertical coordinate of the full quasigeostrophic model with active surface buoyancy. The derivation generalizes the approach of Rocha et al. (2016) [1] to allow for general bases. Details are then presented for a specific set of bases: Legendre polynomials for p...
Preprint
Full-text available
Numerical simulations of quasi-static magnetoconvection with a vertical magnetic field are carried out up to a Chandrasekhar number of $Q=10^8$ over a broad range of Rayleigh numbers $Ra$. Three magnetoconvection regimes are identified: two of the regimes are magnetically-constrained in the sense that a leading-order balance exists between the Lore...
Article
We present results from an asymptotic magnetohydrodynamic model that is suited for studying the rapidly rotating, low-viscosity regime typical of the electrically conducting fluid interiors of planets and stars. We show that the presence of sufficiently strong magnetic fields prevents the formation of large-scale vortices and saturates the inverse...
Article
Full-text available
Large-scale coherent structures such as jets in Rayleigh–Bénard convection and related systems are receiving increasing attention. This paper studies, both numerically and theoretically, the process of jet formation in two-dimensional salt-finger convection. The approach utilizes an asymptotically derived system of equations referred to as the modi...
Article
Geophysical fluid dynamics is inherently multiscale, with timescales from seconds to millenia and with space scales from millimeters to tens of thousands of kilometers. The most general models of the dynamics are single scale in the sense that there is one set of governing equations, for example, the Navier-Stokes equations, that applies at all sca...
Article
Full-text available
The piston shock problem is a prototypical example of strongly nonlinear fluid flow that enables the experimental exploration of fluid dynamics in extreme regimes. Here we investigate this problem for a nominally dissipationless, superfluid Bose-Einstein condensate and observe rich dynamics including the formation of a plateau region, a non-expandi...
Preprint
A method is presented for constructing energy-conserving Galerkin approximations to the full quasigeostrophic model with active surface buoyancy. The derivation generalizes the approach of Rocha et al. (2016) to allow for general bases. Details are then presented for a specific set of bases: Legendre polynomials for potential vorticity and a recomb...
Article
Full-text available
Many geophysical and astrophysical phenomena are driven by turbulent fluid dynamics, containing behaviors separated by tens of orders of magnitude in scale. While direct simulations have made large strides toward understanding geophysical systems, such models still inhabit modest ranges of the governing parameters that are difficult to extrapolate...
Article
The quasi-geostrophic dynamo model (QGDM) is a multiscale, fully nonlinear Cartesian dynamo model that is valid in the asymptotic limit of low Rossby number. In the additional limit of small magnetic Prandtl number investigated here, the QGDM is a self-consistent, asymptotically exact form of an $\unicode[STIX]{x1D6FC}^{2}$ large-scale dynamo. This...
Preprint
Full-text available
We present results from an asymptotic magnetohydrodynamic model that is suited for studying the rapidly rotating, low viscosity regime typical of electrically conducting planetary and stellar fluid systems. We show that sufficiently strong magnetic fields saturate large-scale turbulent flows at a finite length-scale that is independent of the geome...
Article
Full-text available
Motivated by the recent discovery of subsurface oceans on planetary moons and the interest they have generated, we explore convective flows in shallow spherical shells of dimensionless gap width ɛ2≪1 in the rapid rotation limit E≪1, where E is the Ekman number. We employ direct numerical simulation (DNS) of the Boussinesq equations to compute the l...
Article
The magnetorotational instability is widely believed to be responsible for outward angular momentum transport in astrophysical accretion discs. The efficiency of this transport depends on the amplitude of this instability in the saturated state. We employ an asymptotic expansion based on an explicit, astrophysically motivated time-scale separation...
Article
The piston shock problem is a prototypical example of strongly nonlinear fluid flow that enables the experimental exploration of fluid dynamics in extreme regimes. We investigate this problem for the case of a nominally inviscid, superfluid Bose-Einstein condensate and observe rich dynamics including the formation of a plateau region, a sharp shock...
Article
Full-text available
The effect of domain anisotropy on the inverse cascade occurring within the geostrophic turbulence regime of rapidly rotating Rayleigh-B\'enard convection (RRBC) is investigated. In periodic domains with square cross-section in the horizontal a domain-filling dipole state is present. For rectangular periodic domains a Kolmogorov-like flow consistin...
Article
The dependence of the heat transfer, as measured by the nondimensional Nusselt number Nu, on Ekman pumping for rapidly rotating Rayleigh-Bénard convection in an infinite plane layer is examined for fluids with Prandtl number Pr=1. A joint effort utilizing simulations from the composite non-hydrostatic quasi-geostrophic model and direct numerical si...
Article
Full-text available
We present a numerical spectral method to solve systems of differential equations on an infinite interval $y\in (-\infty, \infty)$ in presence of linear differential operators of the form $Q(y) \left(\partial/\partial_y\right)^b$ (where $Q(y)$ is a rational fraction and $b$ a positive integer). Even when these operators are not parity-preserving, w...
Article
Many geophysical and astrophysical phenomena are driven by massively-turbulent, multiscale fluid dynamics. These fluid systems are often both too remote and too complex to fully grasp without employing forward models. While attempts to directly simulate geophysical systems have made important strides, such models still inhabit modest ranges of the...
Article
Compositional convection is thought to be an important energy source for magnetic field generation within planetary interiors. The Prandtl number, Pr, characterizing compositional convection is significantly larger than unity, suggesting that the inertial force may not be important on the small scales of convection as long as the buoyancy force is...
Article
Full-text available
A simple model of nonlinear salt-finger convection in two dimensions is derived and studied. The model is valid in the limit of small solute to heat diffusivity ratio and large density ratio, which is relevant to both oceanographic and astrophysical applications. Two limits distinguished by the magnitude of the Schmidt number are found. For order o...
Article
Full-text available
Compositional convection is thought to be an important energy source for magnetic field generation within planetary interiors. The Prandtl number, $Pr$, characterizing compositional convection is significantly larger than unity, suggesting that the inertial force may not be important on the small scales of convection. We develop asymptotic dynamo m...
Article
Numerical simulations of 3D, rapidly rotating Rayleigh-Benard convection are performed using an asymptotic quasi-geostrophic model that incorporates the effects of no-slip boundaries through (i) parameterized Ekman pumping boundary conditions, and (ii) a thermal wind boundary layer that regularizes the enhanced thermal fluctuations induced by pumpi...
Article
Most large-scale planetary magnetic fields are thought to be driven by low Rossby number convection of a low magnetic Prandtl number fluid. Here kinematic dynamo action is investigated with an asymptotic, rapidly rotating dynamo model for the plane layer geometry that is intrinsically low magnetic Prandtl number. The thermal Prandtl number and Rayl...
Article
An efficient, spectral numerical method is presented for solving problems in a spherical shell geometry that employs spherical harmonics in the angular dimensions and Chebyshev polynomials in the radial direction. We exploit the three-term recurrence relation for Chebyshev polynomials that renders all matrices sparse in spectral space. This approac...
Article
Full-text available
We consider a close relative of plane Couette flow called Waleffe flow in which the fluid is confined between two free-slip walls and the flow driven by a sinusoidal force. We use a reduced model of such flows constructed elsewhere to compute stationary exact coherent structures in this flow in periodic domains with a large spanwise period. The com...
Article
Full-text available
It is a well established result of linear theory that the influence of differing mechanical boundary conditions, i.e., stress-free or no-slip, on the primary instability in rotating convection becomes asymptotically small in the limit of rapid rotation. This is accounted for by the diminishing impact of the viscous stresses exerted within Ekman bou...
Conference Paper
Full-text available
Doubly diffusive processes play a fundamental role in many physical phenomena of substantial geophysical and astrophysical importance. Despite much research basic questions, e.g. how primary and secondary instabilities saturate and how mean fields are generated, remain unresolved. In the salt-finger regime, we use a systematic asymptotic procedure...
Article
Full-text available
The onset of dynamo action is investigated within the context of a newly developed low Rossby, low magnetic Prandtl number, convection-driven dynamo model. The model represents an asymptotically exact form of an $\alpha^2$ mean field dynamo model in which the small-scale convection is represented explicitly by the finite amplitude, single mode conv...
Article
Here, the effects of surface waves on submesoscale instabilities are studied through analytical and linear analyses, as well as nonlinear large eddy simulations of the wave-averaged Boussinesq equations. The wave averaging yields a surface intensified current (Stokes drift) which advects momentum, adds to the total Coriolis force, and induces a Sto...
Article
Full-text available
The influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is investigated for the case of stress-free mechanical boundary conditions. It is shown that whereas the leading order system satisfies fixed temperature boundary conditions implicitly, a double boundary laye...
Article
An accurate description of turbulent core convection is necessary in order to build robust models of planetary core processes. Towards this end, we focus here on the physics of rapidly rotating convection. In particular, we present a closely coupled suite of advanced asymptotically-reduced theoretical models, efficient Cartesian direct numerical si...
Article
Full-text available
Numerical simulations of an asymptotically reduced version of the Craik–Leibovich (CL) equations are described. By filtering surface waves, the CL equations facilitate simulations of surface-wave driven or surface-wave modified phenomena in the upper ocean—most notably, Langmuir circulation (LC)—with time scales long relative to the period of the d...
Article
Full-text available
A convection-driven multiscale dynamo model is developed in the limit of low Rossby number for the plane layer geometry in which the gravity and rotation vectors are aligned. The small-scale fluctuating dynamics are described by a magnetically modified quasi-geostrophic equation set, and the large-scale mean dynamics are governed by a diagnostic th...
Article
Full-text available
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 scali...
Article
Full-text available
A reduced description of shear flows motivated by the Reynolds number scaling of lower-branch exact coherent states in plane Couette flow (Wang J, Gibson J and Waleffe F 2007 Phys. Rev. Lett. 98 204501) is constructed. Exact time-independent nonlinear solutions of the reduced equations corresponding to both lower and upper branch states are found f...
Article
Full-text available
Rapidly rotating Rayleigh-B\'enard 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 tr...
Article
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 scali...
Article
The interactions between boundary layer turbulence, including Langmuir turbulence, and submesoscale processes in the oceanic mixed layer are described using large-eddy simulations of the spindown of a temperature front in the presence of submesoscale eddies, winds, and waves. The simulations solve the surface-wave-averaged Boussinesq equations with...
Article
In rapidly rotating convection four flow regimes with distinct characteristics have been identified via simulations of asymptotically reduced equations as a function of a reduced Rayleigh number RaE4/3 and Prandtl number sigma (K. Julien, A. Rubio, I. Grooms, and E. Knobloch, "Statistical and physical balances in low Rossby number Rayleigh-Benard c...
Article
Full-text available
Exact coherent states of a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force (a flow first introduced by F. Waleffe, Phys. Fluids 9, 883 (1997)) are computed using equations obtained from a large Reynolds-number asymptotic reduction of the Navier-Stokes equations. The redu...
Article
A linear stability analysis for compressible convection in a plane layer geometry both with and without the influence of rotation is presented. For the rotating cases we employ the tilted -plane geometry that allows for varying angles between the rotation and gravity vectors. The stability criteria for compressible and anelastic ideal gases is comp...
Article
Full-text available
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 turbulen...
Article
Full-text available
A reduced description of shear flows consistent with the Reynolds number scaling of lower-branch exact coherent states in plane Couette flow [J. Wang et al., Phys. Rev. Lett. 98, 204501 (2007)] is constructed. Exact time-independent nonlinear solutions of the reduced equations corresponding to both lower and upper branch states are found for Waleff...
Article
Full-text available
A weak-wave turbulence theory for rapidly rotating flows is developed in this paper. The governing equations are an asymptotically reduced set of equations that are derived from the incompressible Navier-Stokes equations. These equations are applicable for rapidly rotating flow regimes and are best suited to describe anisotropic dynamics of rotatin...
Conference Paper
The Craik–Leibovich (CL) equations are a surface-wave filtered version of the instantaneous Navier–Stokes equations in which the rectified effects of the surface waves are captured through a so-called ``vortex force'' term: the cross-product of the Stokes, or Lagrangian, mass drift associated with the filtered surface waves and the filtered vortici...
Article
The rotating cylindrical annulus geometry was first developed by Busse (J. Fluid Mech., vol. 44, 1970, pp. 441–460) as a simplified analogue for studying convection in rapidly rotating spherical geometries. Although it has provided a more tractable two-dimensional model than the sphere, it is formally limited to asymptotically small slopes and thus...
Conference Paper
A primary challenge in physical oceanography is the systematic representation of non-hydrostatic boundary-layer (BL) turbulence in numerical models and stability analyses of hydrostatic flows. Here, we use asymptotic analysis to derive a multiscale PDE model that captures the coupling between wind-driven Langmuir turbulence and O(10)-km submesoscal...
Article
A multiscale algorithm is proposed for simulations of the spatially-extended dynamics of Langmuir circulation, a wind- and surface-wave-driven convective flow that dominates vertical transport and mixing on the scale of the 0 (100)-m deep ocean surface boundary layer. The algorithm is motivated by multiple scale asymptotic analysis of the master pa...
Article
Full-text available
We demonstrate, via simulations of asymptotically reduced equations describing rotationally constrained Rayleigh-Bénard convection, that the efficiency of turbulent motion in the fluid bulk limits overall heat transport and determines the scaling of the nondimensional Nusselt number Nu with the Rayleigh number Ra, the Ekman number E, and the Prandt...
Conference Paper
he ocean surface boundary layer (BL), the upper 50-100 meters of the water column that experiences the direct impact of atmospheric forcing, plays a pivotal role in climate dynamics, pollutant dispersal, and marine ecosystem viability. A vexing challenge in numerical ocean modelling, particularly for climate applications, is parameterizing shear- a...
Article
Here we investigate the effects of fluid properties on the morphology and dynamics of convection in the Earth's outer core. The results of two quasi-geostrophic convection simulations are carried out at comparable convective velocities for fluids in which the ratio between the kinematic viscosity and thermal diffusivity (the Prandtl number, Pr) is...