Vortical control of forced two-dimensional turbulence

A new numerical technique for the simulation of forced two-dimensional turbulence (Dritschel and Fontane, 2010) is used to examine the validity of Kraichnan-Batchelor scaling laws at higher Reynolds number than previously accessible with classical pseudo-spectral methods,making use of large simulation ensembles to allow a detailed consideration of the inverse cascade in a quasi-steady state. Our results support the recent finding of Scott (2007), namely that when a direct enstrophy cascading range is well-represented numerically, a steeper energy spectrum proportional to k^(−2) is obtained in place of the classical k^(−5/3) prediction. It is further shown that this steep spectrum is associated with a faster growth of energy at large scales, scaling like t^(−1) rather than Kraichnan’s prediction of t^(−3/2). The deviation from Kraichnan’s theory is related to the emergence of a population of vortices that dominate the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum. An analytical model is proposed which closely matches the numerical spectra between the large scales and the forcing scale

Simply-connected vortex-patch shallow-water quasi-equilibria

This work is supported by a UK Natural Environment Research Council studentshipWe examine the form, properties, stability and evolution of simply-connected vortex-patch relative quasi-equilibria in the single-layer ƒ-plane shallow-water model of geophysical fluid dynamics. We examine the effects of the size, shape and strength of vortices in this system, represented by three distinct parameters completely describing the families of the quasi-equilibria. Namely, these are the ratio γ=L/LD between the horizontal size of the vortices and the Rossby deformation length; the aspect ratio λ between the minor to major axes of the vortex; and a potential vorticity (PV)-based Rossby number Ro=q′/ƒ, the ratio of the PV anomaly q′ within the vortex to the Coriolis frequency ƒ. By defining an appropriate steadiness parameter, we find that the quasi-equilibria remain steady for long times, enabling us to determine the boundary of stability λc=λc(γ, Ro), for 0.25≤γ≤6 and |Ro|≤1. By calling two states which share γ,|Ro| and λ ‘equivalent’, we find a clear asymmetry in the stability of cyclonic (Ro>0) and anticyclonic (Ro<0) equilibria, with cyclones being able to sustain greater deformations than anticyclones before experiencing an instability. We find that ageostrophic motions stabilise cyclones and destabilise anticyclones. Both types of vortices undergo the same main types of unstable evolution, albeit in different ranges of the parameter space, (a) vacillations for large-γ, large-Ro states, (b) filamentation for small-γ states and (c) vortex splitting, asymmetric for intermediate-γ and symmetric for large-γ states.Publisher PDFPeer reviewe

Impeded inverse energy transfer in the Charney--Hasegawa--Mima model of quasi-geostrophic flows

The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney--Hasegawa--Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber $\lambda$ (or free-surface elasticity). Herein, we derive a bound on the inverse energy transfer, specifically on the growth rate \d\ell/\dt of the characteristic length scale $\ell$ representing the energy centroid. It is found that \d\ell/\dt\le2\norm q_\infty/(\ell_s\lambda^2), where \norm q_\infty is the supremum of the potential vorticity and $\ell_s$ represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential energy dominated regime ($\ell\ge\ell_s\gg\lambda^{-1}$), the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.Comment: 8 pages, 2 figures, accepted for publication in JF

On the validity of Kraichnan scalings for forced two-dimensional turbulence

We examine the validity of the scaling laws derived by Kraichnan (1967) for forced two-dimensional turbulence. We use a new numerical technique (Dritschel & Fontane 2010) to reach higher Reynolds number than previously accessible with classical pseudo-spectral methods. No large scale friction or hypo-diffusion is used in order to avoid any distortion of the inverse cascade and to be in agreement with the theoretical framework used by Kraichnan. Both spectral and spatial forcing are considered and statistical convergence is obtained through large simulation ensembles. A steeper energy spectrum proportional to k^(-2) is observed for the inverse cascade in place of the classical k^(-5/3) prediction. This steepening is shown to be associated with a faster growth of energy at large scales, scaling like t^(-1) rather than Kraichnan's prediction of t^(-3/2). The deviation from Kraichnan's theory is related to the emergence of a vortex population dominating the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum

On the beta-drift of an initially circular vortex patch

The nonlinear inviscid evolution of a vortex patch in a single-layer quasi-geostrophic fluid and within a background planetary vorticity gradient is examined numerically at unprecedented spatial resolution. The evolution is governed by two dimensionless parameters: the initial size (radius) of the vortex compared to the Rossby deformation radius, and the initial strength of the vortex compared to the variation of the planetary vorticity across the vortex. It is found that the zonal speed of a vortex increases with its strength. However, the meridional speed reaches a maximum at intermediate vortex strengths. Both large and weak vortices are readily deformed, often into elliptical and tripolar shapes. This deformation is shown to be related to an instability of the instantaneous vorticity distribution in the absence of the planetary vorticity gradient β. The extremely high numerical resolution employed reveals a striking feature of the flow evolution, namely the generation of very sharp vorticity gradients surrounding the vortex and extending downstream of it in time. These gradients form as the vortex forces background planetary vorticity contours out of its way as it propagates. The contours close to the vortex swirl rapidly around the vortex and homogenize, but at some critical distance the swirl is not strong enough and, instead, a sharp vorticity gradient forms. The region inside this sharp gradient is called the ‘trapped zone’, though it shrinks slowly in time and leaks. This leaking occurs in a narrow wake called the ‘trailing front’, another zone of sharp vorticity gradients, extending behind the vortex

The quasi-geostrophic ellipsoidal vortex model

We present a simple approximate model for studying general aspects of vortex interactions in a rotating stably-stratified fluid. The model idealizes vortices by ellipsoidal volumes of uniform potential vorticity, a materially conserved quantity in an inviscid, adiabatic fluid. Each vortex thus possesses 9 degrees of freedom, 3 for the centroid and 6 for the shape and orientation. Here, we develop equations for the time evolution of these quantities for a general system of interacting vortices. An isolated ellipsoidal vortex is well known to remain ellipsoidal in a fluid with constant background rotation and uniform stratification, as considered here. However, the interaction between any two ellipsoids in general induces weak non-ellipsoidal perturbations. We develop a unique projection method, which follows directly from the Hamiltonian structure of the system, that effectively retains just the part of the interaction which preserves ellipsoidal shapes. This method does not use a moment expansion, e.g. local expansions of the flow in a Taylor series. It is in fact more general, and consequently more accurate. Comparisons of the new model with the full equations of motion prove remarkably close.Publisher PDFPeer reviewe

A conformal mapping approach to modelling two-dimensional stratified flow

Funding: This research received support from the UK Engineering and Physical Sciences Research Council (Impact Acceleration Account, both at the University of St Andrews and Newcastle University). This research received support from the UK Engineering and Physical Sciences Research Council (grants EP/R511778/1 and EP/R511584/1).Herein we describe a new approach to modelling inviscid two-dimensional stratified flows in a general domain. The approach makes use of a conformal map of the domain to a rectangle. In this transformed domain, the equations of motion are largely unaltered, and in particular Laplace's equation remains unchanged. This enables one to construct exact solutions to Laplace's equation and thereby enforce all boundary conditions. An example is provided for two-dimensional flow under the Boussinesq approximation, though the approach is much more general (albeit restricted to two-dimensions). This example is motivated by flow under a weir in a tidal estuary. Here, we discuss how to use a dynamically-evolving conformal map to model changes in the water height on either side of the weir, though the example presented keeps these heights fixed due to limitations in the computational speed to generate the conformal map. The numerical approach makes use of contour advection, wherein material buoyancy contours are advected conservatively by the local fluid velocity, while a dual contour-grid representation is used for the vorticity in order to account for vorticity generation from horizontal buoyancy gradients. This generation is accurately estimated by using the buoyancy contours directly, rather than a gridded version of the buoyancy field. The result is a highly-accurate, efficient numerical method with extremely low levels of numerical damping.Publisher PDFPeer reviewe

Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence

We study the small-scale behavior of generalized two-dimensional turbulence governed by a family of model equations, in which the active scalar $\theta=(-\Delta)^{\alpha/2}\psi$ is advected by the incompressible flow $\u=(-\psi_y,\psi_x)$. The dynamics of this family are characterized by the material conservation of $\theta$, whose variance is preferentially transferred to high wave numbers. As this transfer proceeds to ever-smaller scales, the gradient $\nabla\theta$ grows without bound. This growth is due to the stretching term $(\nabla\theta\cdot\nabla)\u$ whose ``effective degree of nonlinearity'' differs from one member of the family to another. This degree depends on the relation between the advecting flow $\u$ and the active scalar $\theta$ and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when $\nabla\u$ is a quantity of no smaller scales than $\theta$, so that it is insensitive to the direct transfer of the variance of $\theta$, which is nearly passively advected. This case corresponds to $\alpha\ge2$, for which the growth of $\nabla\theta$ is approximately exponential in time and non-accelerated. For $\alpha<2$, superlinear dynamics are realized as the direct transfer of entails a growth in \nabla\u, thereby enhancing the production of $\nabla\theta$. This superlinearity reaches the familiar quadratic nonlinearity of three-dimensional turbulence at $\alpha=1$ and surpasses that for $\alpha<1$. The usual vorticity equation ($\alpha=2$) is the border line, where \nabla\u and $\theta$ are of the same scale, separating the linear and nonlinear regimes of the small-scale dynamics. We discuss these regimes in detail, with an emphasis on the locality of the direct transfer.Comment: 6 journal pages, to appear in Physical Review

Ring configurations of point vortices in polar atmospheres

This paper examines the stability and nonlinear evolution of configurations of equalstrength point vortices equally spaced on a ring of constant radius, with or without a central vortex, in the three-dimensional quasi-geostrophic compressible atmosphere model. While the ring lies at constant height, the central vortex can be at a different height and also have a different strength to the vortices on the ring. All such configurations are relative equilibria, in the sense that they steadily rotate about the z axis. Here, the domains of stability for two or more vortices on a ring with an additional central vortex are determined. For a compressible atmosphere, the problem also depends on the density scale height H, the vertical scale over which the background density varies by a factor e. Decreasing H while holding other parameters fixed generally stabilises a configuration. Nonlinear simulations of the dynamics verify the linear analysis and reveal potentially chaotic dynamics for configurations having four or more vortices in total. The simulations also reveal the existence of staggered ring configurations, and oscillations between single and double ring configurations. The results are consistent with the observations of ring configurations of polar vortices seen at both of Jupiter’s poles [1].PostprintPeer reviewe

L’algorithme HyperCASL : une nouvelle approche pour la simulation des écoulements géophysiques

HyperCASL is a new fully Lagrangian algorithm combining Contour Dynamics and Vortex-In-Cell methods. After a description of the algorithm, an inter-code comparison is conducted using a two-dimensional inviscid unforced turbulence test-case. This enables us to point out the advantages of this new algorithm in terms of resolution, convergence, accuracy and computational efficiency compared to other existing methods