The effects of Ekman pumping on quasi-geostrophic Rayleigh-Benard convection

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 pumping. The fidelity of the model, obtained by an asymptotic reduction of the Navier-Stokes equations that implicitly enforces a pointwise geostrophic balance, is explored for the first time by comparisons of simulations against the findings of direct numerical simulations and laboratory experiments. Results from these methods have established Ekman pumping as the mechanism responsible for significantly enhancing the vertical heat transport. This asymptotic model demonstrates excellent agreement over a range of thermal forcing for Pr ~1 when compared with results from experiments and DNS at maximal values of their attainable rotation rates, as measured by the Ekman number (E ~ 10^{-7}); good qualitative agreement is achieved for Pr > 1. Similar to studies with stress-free boundaries, four spatially distinct flow morphologies exists. Despite the presence of frictional drag at the upper and/or lower boundaries, a strong non-local inverse cascade of barotropic (i.e., depth-independent) kinetic energy persists in the final regime of geostrophic turbulence and is dominant at large scales. For mixed no-slip/stress-free and no-slip/no-slip boundaries, Ekman friction is found to attenuate the efficiency of the upscale energy transport and, unlike the case of stress-free boundaries, rapidly saturates the barotropic kinetic energy. For no-slip/no-slip boundaries, Ekman friction is strong enough to prevent the development of a coherent dipole vortex condensate. Instead vortex pairs are found to be intermittent, varying in both time and strength.Comment: 20 pages, 10 figure

The sensitivity of rapidly rotating Rayleigh--B\'enard convection to Ekman pumping

The dependence of the heat transfer, as measured by the nondimensional Nusselt number $Nu$, on Ekman pumping for rapidly rotating Rayleigh-B\'enard 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 (CNH-QGM) and direct numerical simulations (DNS) of the incompressible fluid equations has mapped a wide range of the Rayleigh number $Ra$ - Ekman number $E$ parameter space within the geostrophic regime of rotating convection. Corroboration of the $Nu$-$Ra$ relation at $E = 10^{-7}$ from both methods along with higher $E$ covered by DNS and lower $E$ by the asymptotic model allows for this range of the heat transfer results. For stress-free boundaries, the relation $Nu-1 \propto (Ra E^{4/3} )^{\alpha}$ has the dissipation-free scaling of $\alpha = 3/2$ for all $E \leq 10^{-7}$. This is directly related to a geostrophic turbulent interior that throttles the heat transport supplied to the thermal boundary layers. For no-slip boundaries, the existence of ageostrophic viscous boundary layers and their associated Ekman pumping yields a more complex 2D surface in $Nu(E,Ra)$ parameter space. For $E<10^{-7}$ results suggest that the surface can be expressed as $Nu-1 \propto (1+ P(E)) (Ra E^{4/3} )^{3/2}$ indicating the dissipation-free scaling law is enhanced by Ekman pumping by the multiplicative prefactor $(1+ P(E))$ where $P(E) \approx 5.97 E^{1/8}$. It follows for $E<10^{-7}$ that the geostrophic turbulent interior remains the flux bottleneck in rapidly rotating Rayleigh-B\'enard convection. For $E\sim10^{-7}$, where DNS and asymptotic simulations agree quantitatively, it is found that the effects of Ekman pumping are sufficiently strong to influence the heat transport with diminished exponent $\alpha \approx 1.2$ and $Nu-1 \propto (Ra E^{4/3} )^{1.2}$.Comment: 9 pages, 14 figure

Numerically determined transport laws for fingering ("thermohaline") convection in astrophysics

We present the first three-dimensional simulations of fingering convection performed in a parameter regime close to the one relevant for astrophysics, and reveal the existence of simple asymptotic scaling laws for turbulent heat and compositional transport. These laws can straightforwardly be extrapolated to the true astrophysical regime. Our investigation also indicates that thermocompositional "staircases," a key consequence of fingering convection in the ocean, cannot form spontaneously in the fingering regime in stellar interiors. Our proposed empirically-determined transport laws thus provide simple prescriptions for mixing by fingering convection in a variety of astrophysical situations, and should, from here on, be used preferentially over older and less accurate parameterizations. They also establish that fingering convection does not provide sufficient extra mixing to explain observed chemical abundances in RGB stars.Comment: Submitted to ApJ Letters on October 29th. 15 pages, 4 figures. See Garaud 2010 for companion pape

Anelastic Versus Fully Compressible Turbulent Rayleigh-B\'enard Convection

Numerical simulations of turbulent Rayleigh-B\'enard convection in an ideal gas, using either the anelastic approximation or the fully compressible equations, are compared. Theoretically, the anelastic approximation is expected to hold in weakly superadiabatic systems with $\epsilon = \Delta T / T_r \ll 1$, where $\Delta T$ denotes the superadiabatic temperature drop over the convective layer and $T_r$ the bottom temperature. Using direct numerical simulations, a systematic comparison of anelastic and fully compressible convection is carried out. With decreasing superadiabaticity $\epsilon$, the fully compressible results are found to converge linearly to the anelastic solution with larger density contrasts generally improving the match. We conclude that in many solar and planetary applications, where the superadiabaticity is expected to be vanishingly small, results obtained with the anelastic approximation are in fact more accurate than fully compressible computations, which typically fail to reach small $\epsilon$ for numerical reasons. On the other hand, if the astrophysical system studied contains $\epsilon\sim O(1)$ regions, such as the solar photosphere, fully compressible simulations have the advantage of capturing the full physics. Interestingly, even in weakly superadiabatic regions, like the bulk of the solar convection zone, the errors introduced by using artificially large values for $\epsilon$ for efficiency reasons remain moderate. If quantitative errors of the order of $10\%$ are acceptable in such low $\epsilon$ regions, our work suggests that fully compressible simulations can indeed be computationally more efficient than their anelastic counterparts.Comment: 24 pages, 9 figure

Trends and Techniques in Visual Gaze Analysis

Visualizing gaze data is an effective way for the quick interpretation of eye tracking results. This paper presents a study investigation benefits and limitations of visual gaze analysis among eye tracking professionals and researchers. The results were used to create a tool for visual gaze analysis within a Master's project.Comment: pages 89-93, The 5th Conference on Communication by Gaze Interaction - COGAIN 2009: Gaze Interaction For Those Who Want It Most, ISBN: 978-87-643-0475-

Kinetic Energy Transport in Rayleigh--B\'enard Convection

The kinetic energy balance in Rayleigh--B\'{e}nard convection is investigated for the Prandtl number range $0.01\le Pr \le 150$ and for fixed Rayleigh number $Ra=5\cdot10^{6}$. The kinetic energy balance is divided into a dissipation, a production and a flux term. We discuss profiles of all terms and find that the different contributions to the energy balance can be spatially separated into regions where kinetic energy is produced and where kinetic energy is dissipated. Analysing the Prandtl number dependence of the kinetic energy balance, we show that the height-dependence of the mean viscous dissipation is closely related to the flux of kinetic energy. We show that the flux of kinetic energy can be divided into four additive contributions, each representing a different elementary physical process (advection, buoyancy, normal viscous stresses and viscous shear stresses). The behaviour of these individual flux contributions is found to be surprisingly rich and exhibits a pronounced Prandtl number dependence. Different flux contributions dominate the kinetic energy transport at different depth, such that a comprehensive discussion requires a decomposition of the domain into a considerable number of sub-layers. On a less detailed level, our results reveal that advective kinetic energy fluxes play a key role in balancing the near-wall dissipation at low Prandtl number, whereas normal viscous stresses are particularly important at high Prandtl number. Finally, our work reveals that classical velocity boundary layers are deeply connected to the kinetic energy transport, but fail to correctly represent regions of enhanced viscous dissipation

Dynamics of fingering convection II: The formation of thermohaline staircases

Regions of the ocean's thermocline unstable to salt fingering are often observed to host thermohaline staircases, stacks of deep well-mixed convective layers separated by thin stably-stratified interfaces. Decades after their discovery, however, their origin remains controversial. In this paper we use 3D direct numerical simulations to shed light on the problem. We study the evolution of an analogous double-diffusive system, starting from an initial statistically homogeneous fingering state and find that it spontaneously transforms into a layered state. By analysing our results in the light of the mean-field theory developed in Paper I, a clear picture of the sequence of events resulting in the staircase formation emerges. A collective instability of homogeneous fingering convection first excites a field of gravity waves, with a well-defined vertical wavelength. However, the waves saturate early through regular but localized breaking events, and are not directly responsible for the formation of the staircase. Meanwhile, slower-growing, horizontally invariant but vertically quasi-periodic gamma-modes are also excited and grow according to the gamma-instability mechanism. Our results suggest that the nonlinear interaction between these various mean-field modes of instability leads to the selection of one particular gamma-mode as the staircase progenitor. Upon reaching a critical amplitude, this progenitor overturns into a fully-formed staircase. We conclude by extending the results of our simulations to real oceanic parameter values, and find that the progenitor gamma-mode is expected to grow on a timescale of a few hours, and leads to the formation of a thermohaline staircase in about one day with an initial spacing of the order of one to two metres.Comment: 18 pages, 9 figures, associated mpeg file at http://earth.uni-muenster.de/~stellma/movie_small.mp4, submitted to JF