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Balanced ellipsoidal vortex equilibria in a background shear flow at finite Rossby number
- William J. McKiver
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- Journal:
- Journal of Fluid Mechanics / Volume 926 / 10 November 2021
- Published online by Cambridge University Press:
- 15 September 2021, A38
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We consider a uniform ellipsoid of potential vorticity (PV), where we exploit analytical solutions derived for a balanced model at the second order in the Rossby number, the next order to quasi-geostrophic (QG) theory, the so-called QG+1 model. We consider this vortex in the presence of an external background shear flow, acting as a proxy for the effect of external vortices. For the QG model the system depends on four parameters, the height-to-width aspect ratio of the vortex, $h/r$, as well as three parameters characterising the background flow, the strain rate, $\gamma$, the ratio of the background rotation rate to the strain, $\beta$, and the angle from which the flow is applied, $\theta$. However, the QG+1 model also depends on the PV, as well as the Prandtl ratio, $f/N$ ($f$ and $N$ are the Coriolis and buoyancy frequencies, respectively). For QG and QG+1 we determine equilibria for different values of the background flow parameters for increasing values of the imposed strain rate up to the critical strain rate, $\gamma _c$, beyond which equilibria do not exist. We also compute the linear stability of this vortex to second-order modes, determining the marginal strain $\gamma _m$ at which ellipsoidal instability erupts. The results show that for QG+1 the most resilient cyclonic ellipsoids are slightly prolate, while anticyclonic ellipsoids tend to be more oblate. The highest values of $\gamma _m$ occur as $\beta \to 1$. For large values of $f/N$, changes in the marginal strain rates occur, stabilising anticyclonic ellipsoids while destabilising cyclonic ellipsoids.
Balanced solutions for an ellipsoidal vortex in a rotating stratified flow
- William J. McKiver, David G. Dritschel
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- Journal:
- Journal of Fluid Mechanics / Volume 802 / 10 September 2016
- Published online by Cambridge University Press:
- 03 August 2016, pp. 333-358
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We consider the motion of a single ellipsoidal vortex with uniform potential vorticity in a rotating stratified fluid at finite Rossby number $\unicode[STIX]{x1D716}$. Building on previous solutions obtained under the quasi-geostrophic approximation (at first order in $\unicode[STIX]{x1D716}$), we obtain analytical solutions for the balanced part of the flow at $O(\unicode[STIX]{x1D716}^{2})$. These solutions capture important ageostrophic effects giving rise to an asymmetry in the evolution of cyclonic and anticyclonic vortices. Previous work has shown that, if the velocity field induced by an ellipsoidal vortex only depends linearly on spatial coordinates inside the vortex, i.e. $\boldsymbol{u}=\unicode[STIX]{x1D64E}\boldsymbol{x}$, then the dynamics reduces markedly to a simple matrix equation. The instantaneous vortex shape and orientation are encapsulated in a symmetric $3\times 3$ matrix $\unicode[STIX]{x1D63D}$, which is acted upon by the flow matrix $\unicode[STIX]{x1D64E}$ to provide the vortex evolution. Under the quasi-geostrophic approximation, the flow matrix is determined by inverting the potential vorticity to obtain the streamfunction via Poisson’s equation, which has a known analytical solution depending on elliptic integrals. Here we show that higher-order balanced solutions, up to second order in the Rossby number, can also be calculated analytically. However, in this case there is a vector potential that requires the solution of three Poisson equations for each of its components. The source terms for these equations are independent of spatial coordinates within the ellipsoid, depending only on the elliptic integrals solved at the leading, quasi-geostrophic order. Unlike the quasi-geostrophic case, these source terms do not in general vanish outside the ellipsoid and have an inordinately complicated dependence on spatial coordinates. In the special case of an ellipsoid whose axes are aligned with the coordinate axes, we are able to derive these source terms and obtain the full analytical solution to the three Poisson equations. However, if one considers the homogeneous case, whereby the outer source terms are neglected, one can obtain an approximate solution having a compact matrix form analogous to the leading-order quasi-geostrophic case. This approximate solution proves to be highly accurate for the general case of an arbitrarily oriented ellipsoid, as verified through comparisons of the solutions with solutions obtained from numerical simulations of an ellipsoid using an accurate nonlinear balance model, even at moderate Rossby numbers.
Effect of Prandtl’s ratio on balance in geophysical turbulence
- David G. Dritschel, William J. McKiver
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- Journal:
- Journal of Fluid Mechanics / Volume 777 / 25 August 2015
- Published online by Cambridge University Press:
- 21 July 2015, pp. 569-590
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The fluid dynamics of the atmosphere and oceans is to a large extent controlled by the slow evolution of a scalar field called ‘potential vorticity’ (PV), with relatively fast motions such as inertia-gravity waves playing only a minor role. This state of affairs is commonly referred to as ‘balance’. Potential vorticity is a special scalar field which is materially conserved in the absence of diabatic effects and dissipation, effects that are generally weak in the atmosphere and oceans. Moreover, in a balanced flow, PV induces the entire fluid motion and its thermodynamic structure (Hoskins et al., Q. J. R. Meteorol. Soc., vol. 111, 1985, pp. 877–946). While exact balance is generally not achievable, it is now well established that balance holds to a high degree of accuracy in rapidly rotating and strongly stratified flows. Such flows are characterised by both a small Rossby number, $\mathit{Ro}\equiv |{\it\zeta}|_{max}/f$, and a small Froude number, $\mathit{Fr}\equiv |{\bf\omega}_{h}|_{max}/N$, where ${\it\zeta}$ and ${\bf\omega}_{h}$ are the relative vertical and horizontal vorticity components, while $f$ and $N$ are the Coriolis and buoyancy frequencies. In fact, balance can even be a good approximation when $\mathit{Fr}\lesssim \mathit{Ro}\sim \mathit{O}(1)$. In this study, we examine how balance depends specifically on Prandtl’s ratio, $f/N$, in unforced freely evolving turbulence. We examine a wide variety of turbulent flows, at a mature and complex stage of their evolution, making use of the fully non-hydrostatic equations under the Boussinesq and incompressible approximations. We perform numerical simulations at exceptionally high resolution in order to carefully assess the degree to which balance holds, and to determine when it breaks down. For this purpose, it proves most useful to employ an invariant PV-based Rossby number ${\it\varepsilon}$, together with $f/N$. For a given ${\it\varepsilon}$, our key finding is that – for at least tens of characteristic vortex rotation periods – the flow is insensitive to $f/N$ for all values for which the flow remains statically stable (typically $f/N\lesssim 1$). Only the vertical velocity varies in proportion to $f/N$, in line with quasi-geostrophic (QG) scaling for which $\mathit{Fr}^{2}\ll \mathit{Ro}\ll 1$. We also find that as ${\it\varepsilon}$ increases towards unity, the maximum $f/N$ attainable decreases towards 0. No statically stable flows occur for ${\it\varepsilon}\gtrsim 1$. For all stable flows, balance is found to hold to a remarkably high degree: as measured by an energy norm, imbalance never exceeds more than a few per cent of the balance, even in flows where $\mathit{Ro}>1$. The vertical velocity $w$ remains a tiny fraction of the horizontal velocity $\boldsymbol{u}_{h}$, even when $w$ is dominantly balanced. Finally, typical vertical to horizontal scale ratios $H/L$ remain close to $f/N$, as found previously in QG turbulence for which $\mathit{Fr}\sim \mathit{Ro}\ll 1$.
Balance in non-hydrostatic rotating stratified turbulence
- WILLIAM J. McKIVER, DAVID G. DRITSCHEL
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- Journal:
- Journal of Fluid Mechanics / Volume 596 / 25 January 2008
- Published online by Cambridge University Press:
- 17 January 2008, pp. 201-219
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It is now well established that two distinct types of motion occur in geophysical turbulence: slow motions associated with potential vorticity advection and fast oscillations due to inertia–gravity waves (or acoustic waves). Many studies have theorized the existence of a flow for which the entire motion is controlled by the potential vorticity (or one ‘master variable’) – this is known as balance. In real geophysical flows, deviations from balance in the form of inertia–gravity waves or ‘imbalance’ have often been found to be small. Here we examine the extent to which balance holds in rotating stratified turbulence which is nearly balanced initially.
Using the non-hydrostatic fluid dynamical equations under the Boussinesq approximation, we analyse properties of rotating stratified turbulence spanning a range of Rossby numbers (Ro≡|ζ|max/f) and the frequency ratios (c≡N/f) where ζ is the relative vertical vorticity, f is the Coriolis frequency and N is the buoyancy frequency. Using a recently introduced diagnostic procedure, called ‘optimal potential vorticity balance’, we extract the balanced part of the flow in the simulations and assess how the degree of imbalance varies with the above parameters.
We also introduce a new and more efficient procedure, building upon a quasi-geostrophic scaling analysis of the complete non-hydrostatic equations. This ‘nonlinear quasi-geostrophic balance’ procedure expands the equations of motion to second order in Rossby number but retains the exact (unexpanded) definition of potential vorticity. This proves crucial for obtaining an accurate estimate of balanced motions. In the analysis of rotating stratified turbulence at Ro≲1 and N/f≫1, this procedure captures a significantly greater fraction of the underlying balance than standard (linear) quasi-geostrophic balance (which is based on the linearized equations about a state of rest). Nonlinear quasi-geostrophic balance also compares well with optimal potential vorticity balance, which captures the greatest fraction of the underlying balance overall.
More fundamentally, the results of these analyses indicate that balance dominates in carefully initialized simulations of freely decaying rotating stratified turbulence up to O(1) Rossby numbers when N/f≫1. The fluid motion exhibits important quasi-geostrophic features with, in particular, typical height-to-width scale ratios remaining comparable to f/N.
The stability of a quasi-geostrophic ellipsoidal vortex in a background shear flow
- WILLIAM J. McKIVER, DAVID G. DRITSCHEL
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- Journal:
- Journal of Fluid Mechanics / Volume 560 / 10 August 2006
- Published online by Cambridge University Press:
- 20 July 2006, pp. 1-17
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We consider the motion of a single quasi-geostrophic ellipsoid of uniform potential vorticity in equilibrium with a linear background shear flow. This motion depends on four parameters: the height-to-width aspect ratio of the vortex, $h/r$, and three parameters characterizing the background shear flow, namely the strain rate, $\gamma$, the ratio of the background rotation rate to the strain, $\beta$, and the angle from which the shear is applied, $\theta$. We generate the equilibria over a large range of these parameters and analyse their linear stability. For the second-order ($m\,{=}\,2$) modes which preserve the ellipsoidal form, we are able to derive equations for the eigenmodes and growth rates. For the higher-order modes we use a numerical method to determine the full linear stability to general disturbances ($m\,{>}\,2$).
Overall we find that the equilibria are stable over most of the parameter space considered, and where instability does occur the marginal instability is usually ellipsoidal. From these results, we determine the parameter values for which the vortex is most stable, and conjecture that these are the vortex characteristics which would be the most commonly observed in turbulent flows.
The quasi-geostrophic ellipsoidal vortex model
- DAVID G. DRITSCHEL, JEAN N. REINAUD, WILLIAM J. McKIVER
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- Journal:
- Journal of Fluid Mechanics / Volume 505 / 25 April 2004
- Published online by Cambridge University Press:
- 21 April 2004, pp. 201-223
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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.
The motion of a fluid ellipsoid in a general linear background flow
- WILLIAM J. McKIVER, DAVID G. DRITSCHEL
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- Journal:
- Journal of Fluid Mechanics / Volume 474 / 10 January 2003
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- 14 January 2003, pp. 147-173
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The study of the motion of a fluid ellipsoid has a long and fascinating history stretching back originally to Laplace in the late 18th century. Recently, this subject has been revived in the context of geophysical fluid dynamics, where it has been shown that an ellipsoid of uniform potential vorticity remains an ellipsoid in a background flow consisting of horizontal strain, vertical shear, and uniform rotation. The object of the present work is to present a simple, appealing, and practical way of investigating the motion of an ellipsoid not just in geophysical fluid dynamics but in general. The main result is that the motion of an ellipsoid may be reduced to the evolution of a symmetric, 3×3 matrix, under the action of an arbitrary 3×3 ‘flow’ matrix. The latter involves both the background flow, which must be linear in the Cartesian coordinates at the surface of the ellipsoid, and the self-induced flow, which was given by Laplace.
The resulting simple dynamical system lends itself ideally to both numerical and analytical study. We illustrate a few examples and then present a theory for the evolution of a vortex within a slowly varying background flow. We show that a vortex may evolve quasi-adiabatically, that is, it stays close to an equilibrium form associated with the instantaneous background flow. The departure from equilibrium, on the other hand, is proportional to the rate of change of the background flow.