ALBERT

Description: A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to simulate shear-induced instabilities in an internal solitary wave (ISW). A three-layer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck-Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steady-state, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin-Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (Ric) and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61-95), namely Ric = 0.1±0.01, is not a sufficient condition for instability. It is also shown that the critical value of L x/λ required for instability, where Lx is the length of the region in a wave in which Ri 〈 1/4 is the half-width of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that √σiTw (where √σi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to Ri 〈 1/4) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1-29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61-95), respectively, namely Lx/λ ≥0.86 and √σiTw 〉 5, are shown to be valid only for a limited parameter range. © Cambridge University Press 2011.

Description: We examine the basic properties and stability of isolated vortices having uniform potential vorticity (PV) in a non-hydrostatic rotating stratified fluid, under the Boussinesq approximation. For simplicity, we consider a uniform background rotation and a linear basic-state stratification for which both the Coriolis and buoyancy frequencies, f and N, are constant. Moreover, we take f/N〈〈 1, as typically observed in the Earth's atmosphere and oceans. In the small Rossby number 'quasi-geostrophic' (QG) limit, when the flow is weak compared to the background rotation, there exist exact solutions for steadily rotating ellipsoidal volumes of uniform PV in an unbounded flow (Zhmur & Shchepetkin, Izv. Akad. Nauk SSSR Atmos. Ocean. Phys., vol. 27, 1991, pp. 492-503; Meacham, Dyn. Atmos. Oceans, vol. 16, 1992, pp. 189-223). Furthermore, a wide range of these solutions are stable as long as the horizontal and vertical aspect ratios λ and μ do not depart greatly from unity (Dritschel et al.,J. Fluid Mech., vol. 536, 2005, pp. 401-421). In the present study, we examine the behaviour of ellipsoidal vortices at Rossby numbers up to near unity in magnitude. We find that there is a monotonic increase in stability as one varies the Rossby number from nearly -1 (anticyclone) to nearly +1 (cyclone). That is, QG vortices are more stable than anticyclones at finite negative Rossby number, and generally less stable than cyclones at finite positive Rossby number. Ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability. The converse is true for an anticyclone. For all Rossby numbers, stability is reinforced by increasing λ towards unity or decreasing μ An unstable vortex often restabilises by developing a near-circular cross-section, typically resulting in a roughly ellipsoidal vortex, but occasionally a binary system is formed. Throughout the nonlinear evolution of a vortex, the emission of inertia-gravity waves (IGWs) is negligible across the entire parameter space investigated. Thus, vortices at small to moderate Rossby numbers, and any associated instabilities, are (ageostrophically) balanced. A manifestation of this balance is that, at finite Rossby number, an anticyclone rotates faster than a cyclone. © 2014 Cambridge University Press.

Description: We examine the form, properties, stability and evolution of doubly-connected (two-vortex) relative equilibria in the single-layer f-plane quasi-geostrophic shallow-water model of geophysical fluid dynamics. Three parameters completely describe families of equilibria in this system: the ratio γ = L/LD between the horizontal size of the vortices and the Rossby deformation length; the area ratio α of the smaller to the larger vortex; and the minimum distance σ between the two vortices. We vary 0〈 γ ≤ 10 and 0.1 ≤ α ≤ 1.0, determining the boundary of stability σ = σc γ, α). We also examine the nonlinear development of the instabilities and the transitions to other near-equilibrium configurations. Two modes of instability occur when σ σc : a small-γ asymmetric (wave 3) mode, which is absent for α 〉 0.6; and a large- γ mode. In general, major structural changes take place during the nonlinear evolution of the vortices, which near σc may be classified as follows: (i) vacillations about equilibrium for γ 〉 2. 5; (ii) partial straining out, associated with the small- γ mode, where either one or both of the vortices get smaller for γ ≤ 2. 5 and α ≤ 0. 6; (iii) partial merger, occurring at the transition region between the two modes of instability, where one of the vortices gets bigger, and (iv) complete merger, associated with the large- γ mode. We also find that although conservative inviscid transitions to equilibria with the same energy, angular momentum and circulation are possible, they are not the preferred evolutionary path. © 2013 Cambridge University Press.

Description: In this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number Pm = ν/η . When Pm ≪ 1, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilize is a novel hybrid contour-spectral method, the 'combined Lagrangian advection method', formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing Pm for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given. © 2012 Cambridge University Press.

Description: The structure of zonal jets arising in forced-dissipative, two-dimensional turbulent flow on the β-plane is investigated using high-resolution, long-time numerical integrations, with particular emphasis on the late-time distribution of potential vorticity. The structure of the jets is found to depend in a simple way on a single nondimensional parameter, which may be conveniently expressed as the ratio LRh=Lε, where LRh = √U/β and Lε D (ε/ β3) 1/5 are two natural length scales arising in the problem; here U may be taken as the r.m.s. velocity, β is the background gradient of potential vorticity in the north-south direction, and ε is the rate of energy input by the forcing. It is shown that jet strength increases with LRh=Lε, with the limiting case of the potential vorticity staircase, comprising a monotonic, piecewise-constant profile in the north-south direction, being approached for LRh/Lε ∼ O(10). At lower values, eddies created by the forcing become sufficiently intense to continually disrupt the steepening of potential vorticity gradients in the jet cores, preventing strong jets from developing. Although detailed features such as the regularity of jet spacing and intensity are found to depend on the spectral distribution of the forcing, the approach of the staircase limit with increasing LRh=Lε is robust across a variety of different forcing types considered. . © 2012 Cambridge University Press.

Description: A new numerical scheme for obtaining the steady-state form of an internal solitary wave of large amplitude is presented. A stratified inviscid two-dimensional fluid under the Boussinesq approximation flowing between horizontal rigid boundaries is considered. The stratification is stable, and buoyancy is continuously differentiable throughout the domain of the flow. Solutions are obtained by tracing the buoyancy frequency along streamlines from the undisturbed far field. From this the vorticity field can be constructed and the streamfunction may then be obtained by inversion of Laplace's operator. The scheme is presented as an iterative solver, where the inversion of Laplace's operator is performed spectrally. The solutions agree well with previous results for stratification in which the buoyancy frequency is a discontinuous function. The new numerical scheme allows significantly larger amplitude waves to be computed than have been presented before and it is shown that waves with Richardson numbers as low as 0.062 can be computed straightforwardly. The method is also extended to deal in a novel way with closed streamlines when they occur in the domain. The new solutions are tested in independent fully nonlinear time-dependent simulations and are verified to be steady. Waves with regions of recirculation are also discussed. © 2011 Cambridge University Press.

Description: Polar vortex vacillations are investigated using long-term simulations of potential-vorticity (PV)-based shallow-water (SW) models for the stratosphere. In the models examined, mechanical forcing is applied through a time-independent topography mimicking tropospheric excitation of the stratosphere. Thermal forcing is applied through a linear relaxation of the mass field to a time-independent equilibrium state mimicking the radiative relaxation taking place in the stratosphere. The SW equations in the PV, velocity divergence, and acceleration divergence representation are solved for a range of resolutions using the “diabatic contour-advective semi-Lagrangian” (DCASL) algorithm and a standard pure semi-Lagrangian (SL) algorithm. Using very different numerical algorithms enables the determination of the degree of numerical sensitivity and the properties of the vacillations with much greater accuracy than in previous related studies. The focus here is on the Lagrangian or material evolution of the polar vortex. The authors examine quasi-Lagrangian diagnostics based on equivalent latitude, the mass enclosed by PV contours, and the terms involved in its time evolution. The PV field forms the basis for calculating quasi-Lagrangian diagnostics. Variations in the mass enclosed by a PV contour are associated with nonconservative processes such as diabatic heating, friction, and irreversible small-scale mixing. Generally, the mass of the polar vortex increases under the action of diabatic mass fluxes, whereas it decreases under the action of dissipative mass fluxes. The results herein differ from previous results reported at T42 resolution by Rong and Waugh in which a spectral transform algorithm is used to solve the SW equations in a vorticity–divergence–mass representation, and in which dissipation is provided by explicitly damping vorticity using hyperdiffusion. Except for the first large-amplitude oscillation, there is little sign of a clear, systematic phase shift between the dissipative and diabatic mass fluxes across the edge of the polar vortex, as proposed by Rong and Waugh as the main mechanism responsible for the vacillations. Concomitant with the absence of a phase shift, the vacillations tend to decay and occur intermittently. Rather than a phase shift, inherent fluctuations in both the diabatic and mass fluxes across the edge of the polar vortex appear to be responsible for the vacillations.