The survival probability for a periodic non-autonomous Ornstein–Uhlenbeck process is calculated analytically using two different methods. The first uses an asymptotic approach. We treat the associated Kolmogorov Backward Equation with an absorbing boundary by dividing the domain into an interior region, centered around the origin, and a “boundary layer” near the absorbing boundary. In each region we determine the leading-order analytical solutions, and construct a uniformly valid solution over the entire domain using asymptotic matching. In the second method we examine the integral relationship between the probability density function and the mean first passage time probability density function. These allow us to determine approximate analytical forms for the exit rate. The validity of the solutions derived from both methods is assessed numerically, and we find the asymptotic method to be superior.

We use direct numerical simulations to study convection in rotating Rayleigh-Bénard convection in horizontally confined geometries of a given aspect ratio, with the walls held at fixed temperatures. We show that this arrangement is unconditionally unstable to flow that takes the form of wall-adjacent convection rolls. For wall temperatures close to the temperatures of the upper or lower boundaries, we show that the base state undergoes a Hopf bifurcation to a state comprised of spatiotemporal oscillations - 'wall modes' - precessing in a retrograde direction. We study the saturated nonlinear state of these modes, and show that the velocity boundary conditions at the upper and lower boundaries are crucial to the formation and propagation of the wall modes: asymmetric velocity boundary conditions at the upper and lower boundaries can lead to prograde wall modes, while stress-free boundary conditions at both walls can lead to wall modes that have no preferred direction of propagation.

We study the hydrodynamic interaction between a microswimmer and a deformable interface when the swimmer can stochastically reorient itself. We consider a force- and torque-free swimmer, modeled as a slender body, that can execute random orientation tumbles or active Brownian rotations in the plane of the deformable interface. When the swimmer is in the more viscous fluid, our analysis shows that both tumbles and Brownian rotations acting on timescales comparable to that of interface deformations can lead to a pusher-type swimmer rotating away from the interface, while enhancing its attraction towards the interface. In turn, the intrinsic orientational stochasticity of the microswimmer favors a stronger migration of pushers towards the interface at short times, but migration away from the interface in the long-time limit. However, irrespective of the viscosity ratio of the two fluid medium, the tendency of a pusher to align parallel to the interface is suppressed; the results for puller-type swimmers are the opposite. Our study has potential consequences for the residence time of swimming microorganisms near deformable boundaries.

Motivated by wind blowing over water, we use asymptotic methods to study the evolution of short wavelength interfacial waves driven by the combined action of these flows. We solve the Rayleigh equation for the stability of the shear flow, and construct a uniformly valid approximation for the perturbed streamfunction, or eigenfunction. We then expand the real part of the eigenvalue, the phase speed, in a power series of the inverse wavenumber and show that the imaginary part is exponentially small. We give expressions for the growth rates of the Miles (J. Fluid Mech., vol. 3, 1957, pp. 185–204) and rippling (e.g. Young & Wolfe, J. Fluid Mech., vol. 739, 2014, pp. 276–307) instabilities that are valid for an arbitrary shear flow. The accuracy of the results is demonstrated by a comparison with the exact solution of the eigenvalue problem in the case when both the wind and the current have an exponential profile.

We study the orientation dynamics of two-dimensional concavo-convex solid bodies that are denser than the fluid through which they fall under gravity. We show that the orientation dynamics of the body, quantified in terms of the angle 𝜙 relative to the horizontal, undergoes a transcritical bifurcation at a Reynolds number and a subcritical pitchfork bifurcation at a Reynolds number . For , the concave-downwards orientation of 𝜙=0 is unstable and bodies overturn into the 𝜙=𝜋 orientation. For , the falling body has two stable equilibria at 𝜙=0and𝜙=𝜋 for steady descent. For , the concave-downwards orientation of 𝜙=0 is again unstable and bodies that start concave-downwards exhibit overstable oscillations about the unstable fixed point, eventually tumbling into the stable 𝜙=𝜋 orientation. The at which the subcritical pitchfork bifurcation occurs is distinct from the 𝑅⁢𝑒 for the onset of vortex shedding, which causes the 𝜙=𝜋 equilibrium to also become unstable, with bodies fluttering about 𝜙=𝜋. The complex orientation dynamics of irregularly shaped bodies evidenced here are relevant in a wide range of settings, from the tumbling of hydrometeors to the settling of mollusk shells.

The Thorndike et al. (1975, ) theory of the ice thickness distribution, g(h), treats the dynamic and thermodynamic aggregate properties of the ice pack in a novel and physically self-consistent manner. Therefore, it has provided the conceptual basis of the treatment of sea-ice thickness categories in climate models. The approach, however, is not mathematically closed due to the treatment of mechanical deformation using the redistribution function ?, the authors noting The present theory suffers from a burdensome and arbitrary redistribution function ?. Toppaladoddi and Wettlaufer (2015, ) showed how ? can be written in terms of g(h), thereby solving the mathematical closure problem and writing the theory in terms of a Fokker-Planck equation, which they solved analytically to quantitatively reproduce the observed winter g(h). Here, we extend this approach to include open water by formulating a new boundary condition for their Fokker-Planck equation, which is then coupled to the observationally consistent sea-ice growth model of Semtner (1976, ) to study the seasonal evolution of g(h). We find that as the ice thins, g(h) transitions from a single- to a double-peaked distribution, which is in agreement with observations. To understand the cause of this transition, we construct a simpler description of the system using the equivalent Langevin equation formulation and solve the resulting stochastic ordinary differential equation numerically. Finally, we solve the Fokker-Planck equation for g(h) under different climatological conditions to study the evolution of the open-water fraction.Plain Language Summary A quantitative understanding of the evolution of the thickness distribution of sea ice is necessary to accurately predict changes in the Arctic ice cover. In the original formulation of the governing equation for the thickness distribution by Thorndike et al., the treatment of the redistribution term-which represents the mechanical deformation of ice by rafting and ridging-is referred to as arbitrar y and burdensome. Using an analogy with Brownian motion, we have recast the redistribution term, closed the original theory and incorporated the process of open water formation to produce seasonal predictions of the thickness distribution. Using our theory we show that a second peak in the thickness distribution emerges in summer, which is consistent with observations. Furthermore, we explore how the greenhouse gas and oceanic heat flux forcings impact the open-water fraction and mean thickness, and the relative sensitivities to these forcings.

The Landauer principle sets a fundamental thermodynamic constraint on the minimum amount of heat that must be dissipated to erase one logical bit of information through a quasistatically slow protocol. For finite time information erasure, the thermodynamic costs depend on the specific physical realization of the logical memory and how the information is erased. Here we treat the problem within the paradigm of a Brownian particle in a symmetric double-well potential. The two minima represent the two values of a logical bit, 0 and 1, and the particle's position is the current state of the memory. The erasure protocol is realized by applying an external time-dependent tilting force. We derive analytical tools to evaluate the work required to erase a classical bit of information in finite time via an arbitrary continuous erasure protocol, which is a relevant setting for practical applications. Importantly, our method is not restricted to the average work, but instead gives access to the full work distribution arising from many independent realizations of the erasure process. Using the common example of an erasure protocol that changes linearly with time acting on a double-parabolic potential, we explicitly calculate all relevant quantities and verify them numerically.

We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the beam equation describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain lengths of the confined sheet; we explain this degeneracy using an asymptotic analysis valid for long sheets, and show that it corresponds to the switching of the sheet between symmetric and antisymmetric buckling modes. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets concentrate the bending energy towards the soft regions of the sheet. We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation.

When wind blows over water, ripples are generated on the water surface. These ripples can be regarded as perturbations of the wind field, which is modelled as a parallel inviscid flow. For a given wavenumber k, the perturbed streamfunction of the wind field and the complex phase speed are the eigenfunction and the eigenvalue of the so-called Rayleigh equation in a semi-infinite domain. Because of the small air–water density ratio, ρa/ρw≡ϵ≪1, the wind and the ripples are weakly coupled, and the eigenvalue problem can be solved perturbatively. At the leading order, the eigenvalue is equal to the phase speed c0 of surface waves. At order ϵ, the eigenvalue has a finite imaginary part, which implies growth. Miles (J. Fluid Mech., vol. 3, 1957, pp. 185–204) showed that the growth rate is proportional to the square modulus of the leading-order eigenfunction evaluated at the so-called critical level z=zc, where the wind speed is equal to c0 and the waves extract energy from the wind. Here, we construct uniform asymptotic approximations of the leading-order eigenfunction for long waves, which we use to calculate the growth rate as a function of k. In the strong wind limit, we find that the fastest growing wave is such that the aerodynamic pressure is in phase with the wave slope. The results are confirmed numerically.

Biota are found in glaciers, ice sheets and permafrost. Ice bound micro-organisms evolve in a complex mobile environment facilitated or hindered by a range of bulk and surface interactions. When a particle is embedded in a host solid near its bulk melting temperature, a melted film forms at the surface of the particle in a process known as interfacial premelting. Under a temperature gradient, the particle is driven by a thermomolecular pressure gradient toward regions of higher temperatures in a process called thermal regelation. When the host solid is ice and the particles are biota, thriving in their environment requires the development of strategies, such as producing exopolymeric substances (EPS) and antifreeze glycoproteins (AFP) that enhance the interfacial water. Therefore, thermal regelation is enhanced and modified by a process we term bio-enhanced premelting. Additionally, the motion of bioparticles is influenced by chemical gradients influenced by nutrients within the icy host body. We show how the overall trajectory of bioparticles is controlled by a competition between thermal regelation and directed biolocomotion. By re-casting this class of regelation phenomena in the stochastic framework of active Ornstein-Uhlenbeck dynamics, and using multiple scales analysis, we find that for an attractive (repulsive) nutrient source, that thermal regelation is enhanced (suppressed) by biolocomotion. This phenomena is important in astrobiology, the biosignatures of extremophiles and in terrestrial paleoclimatology.