Sea ice motion as a stochastic process

We use tools from statistical physics to develop a stochastic theory for the drift of a single Arctic sea-ice floe. Floe-floe interactions are modelled using a Coulomb friction term, with any change in the thickness or the size of the ice floe due to phase change and/or mechanical deformation being neglected. We obtain a Langevin equation for the fluctuating velocity and the corresponding Fokker-Planck equation for its probability density function (PDF). For values of ice compactness close to unity, the stationary PDFs for the individual components of the fluctuating velocity are found to be the Laplace distribution, in agreement with observations. A possible way of obtaining a more general model that accounts for thermal growth and mechanical deformation is also discussed.Comment: 5 pages, 2 figure

Penetrative Convection at High Rayleigh Numbers

We study penetrative convection of a fluid confined between two horizontal plates, the temperatures of which are such that a temperature of maximum density lies between them. The range of Rayleigh numbers studied is $Ra = \left[10^6, 10^8 \right]$ and the Prandtl numbers are $Pr = 1$ and $11.6$. An evolution equation for the growth of the convecting region is obtained through an integral energy balance. We identify a new non-dimensional parameter, $\Lambda$, which is the ratio of temperature difference between the stable and unstable regions of the flow; larger values of $\Lambda$ denote increased stability of the upper stable layer. We study the effects of $\Lambda$ on the flow field using well-resolved lattice Boltzmann simulations, and show that the characteristics of the flow depend sensitively upon it. For the range $\Lambda = \left[0.01, 4\right]$, we find that for a fixed $Ra$ the Nusselt number, $Nu$, increases with decreasing $\Lambda$. We also investigate the effects of $\Lambda$ on the vertical variation of convective heat flux and the Brunt-V\"{a}is\"{a}l\"{a} frequency. Our results clearly indicate that in the limit $\Lambda \rightarrow 0$ the problem reduces to that of the classical Rayleigh-B\'enard convection.Comment: 12 pages, 19 figure

Tailoring boundary geometry to optimize heat transport in turbulent convection

By tailoring the geometry of the upper boundary in turbulent Rayleigh-B\'enard convection we manipulate the boundary layer -- interior flow interaction, and examine the heat transport using the Lattice Boltzmann method. For fixed amplitude and varying boundary wavelength $\lambda$, we find that the exponent $\beta$ in the Nusselt-Rayleigh scaling relation, $Nu-1 \propto Ra^\beta$, is maximized at $\lambda \equiv \lambda_{\text{max}} \approx (2 \pi)^{-1}$, but decays to the planar value in both the large ($\lambda \gg \lambda_{\text{max}}$) and small ($\lambda \ll \lambda_{\text{max}}$) wavelength limits. The changes in the exponent originate in the nature of the coupling between the boundary layer and the interior flow. We present a simple scaling argument embodying this coupling, which describes the maximal convective heat flux.Comment: 6 pages, 6 figure

Turbulent Transport Processes at Rough Surfaces with Geophysical Applications

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Vortex shedding patterns, their competition, and chaos in flow past inline oscillating rectangular cylinders

The flow past inline oscillating rectangular cylinders is studied numerically at a Reynolds number representative of two-dimensional flow. A symmetric mode, known as S-II, consisting of a pair of oppositely-signed vortices on each side, observed recently in experiments, is obtained computationally. A new symmetric mode, named here as S-III, is also found. At low oscillation amplitudes, the vortex shedding pattern transitions from antisymmetric to symmetric smoothly via a regime of intermediate phase. At higher amplitudes, this intermediate regime is chaotic. The finding of chaos extends and complements the recent work of Perdikaris et al. [1]. Moreover it shows that the chaos results from a competition between antisymmetric and symmetric shedding modes. Rectangular cylinders rather than square are seen to facilitate these observations. A global, and very reliable, measure is used to establish the existence of chaos.Comment: Submitted to the Physics of Fluid

The role of grain-environment heterogeneity in normal grain growth: a stochastic approach

The size distribution of grains is a fundamental characteristic of polycrystalline solids. In the absence of deformation, the grain-size distribution is controlled by normal grain growth. The canonical model of normal grain growth, developed by Hillert, predicts a grain-size distribution that bears a systematic discrepancy with observed distributions. To address this, we propose a change to the Hillert model that accounts for the influence of heterogeneity in the local environment of grains. In our model, each grain evolves in response to its own local environment of neighbouring grains, rather than to the global population of grains. The local environment of each grain evolves according to an Ornstein-Uhlenbeck stochastic process. Our results are consistent with accepted grain-growth kinetics. Crucially, our model indicates that the size of relatively large grains evolves as a random walk due to the inherent variability in their local environments. This leads to a broader grain-size distribution than the Hillert model and indicates that heterogeneity has a critical influence on the evolution of microstructure.Comment: 24 pages, 8 figures, to be published in Acta Materiali

Thermal Convection over Fractal Surfaces

We use well resolved numerical simulations with the Lattice Boltzmann Method to study Rayleigh-B\'enard convection in cells with a fractal boundary in two dimensions for $Pr = 1$ and $Ra \in \left[10^7, 10^{10}\right]$. The fractal boundaries are functions characterized by power spectral densities $S(k)$ that decay with wavenumber, $k$, as $S(k) \sim k^{p}$ ($p < 0$). The degree of roughness is quantified by the exponent $p$ with $p < -3$ for smooth (differentiable) surfaces and $-3 \le p < -1$ for rough surfaces with Hausdorff dimension $D_f=\frac{1}{2}(p+5)$. By computing the exponent $\beta$ in power law fits $Nu \sim Ra^{\beta}$, where $Nu$ and $Ra$ are the Nusselt and the Rayleigh numbers for $Ra \in \left[10^8, 10^{10}\right]$, we observe that heat transport scaling increases with roughness over the top two decades of $Ra \in \left[10^8, 10^{10}\right]$. For $p$ $= -3.0$, $-2.0$ and $-1.5$ we find $\beta = 0.288 \pm 0.005, 0.329 \pm 0.006$ and $0.352 \pm 0.011$, respectively. We also observe that the Reynolds number, $Re$, scales as $Re \sim Ra^{\xi}$, where $\xi \approx 0.57$ over $Ra \in \left[10^7, 10^{10}\right]$, for all $p$ used in the study. For a given value of $p$, the averaged $Nu$ and $Re$ are insensitive to the specific realization of the roughness.Comment: 15 pages, 13 figure

A stochastic model for the turbulent ocean heat flux under Arctic sea ice

Heat flux from the upper ocean to the underside of sea ice provides a key contribution to the evolution of the Arctic sea ice cover. Here, we develop a model of the turbulent ice-ocean heat flux using coupled ordinary stochastic differential equations to model fluctuations in the vertical velocity and temperature in the Arctic mixed layer. All the parameters in the model are determined from observational data. A detailed comparison between the model results and measurements made during the Surface Heat Budget of the Arctic Ocean (SHEBA) project reveals that the model is able to capture the probability density functions (PDFs) of velocity, temperature and heat flux fluctuations. Furthermore, we show that the temperature in the upper layer of the Arctic ocean can be treated as a passive scalar during the whole year of SHEBA measurements. The stochastic model developed here provides a computationally inexpensive way to compute observationally consistent PDF of this heat flux, and has implications for its parametrization in regional and global climate models