# Workshop Programme

## for period 8 - 12 December 2008

### IUTAM - Rotating Stratified Turbulence and Turbulence in the Atmosphere and Oceans

8 - 12 December 2008

Timetable

Monday 8 December | ||||

08:30-09:25 | Registration | |||

09:15-09:20 | Ben Mestel - Welcome | |||

09:20-09:30 | Hunt, J (UCL) |
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Introduction | Sem 1 | |||

09:30-10:00 | Rhines, P (Washington) |
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Eddies and circulation: lessons from oceans, atmospheres and the GFD lab | Sem 1 | |||

Planetary fluids inherit the angular momentum of their planets. Concentrated and diluted, and fragmented into potential vorticity (PV), angular momentum controls the circulation under a limited budget of energy. In particular, wherever energy decays, planetary PV takes control of the dynamics. This so-called β-control is manifested in the ‘stiffness’ and scale-dependent ‘PV elasticity’ of rotating fluids which leads to many kinds of wave motion, and to limitation of turbulent mixing. The fluid finds ways, for example PV staircases and their attendant jets, of dealing with limited energy and PV control. Momentum rearrangement that follows stirring of the PV field by eddies is a key process. We see numerous zonal jets on the rapidly rotating, gas giants (strong β-control), and a weaker β-control over Earth’s subpolar atmospheric jet streams, where the kinetic energy density averages 10^6 J m^-2. The Earth’s oceans, operating at lower kinetic energy levels (10^4 J m^-2). have selected jets of much finer scale, and a dominant energy-containing eddy mode manifested as dimples on the sea surface, simply marching westward. These are strongly nonlinear baroclinic Rossby waves which do not obey the simple rules of geostrophic turbulence, namely, expansion of scale laterally and vertically toward a barotropic state, and coalescence into sparsely distributed hard-core vortices. A second mode of oceanic eddy that is widespread is the (equivalent-) barotropic mode of geostrophic flow, ‘tall’ eddies which are highly coordinated with bottom topography. Here we describe field observations and simulations from the GFD laboratory. These demonstrate Rossby wave propagation, induction of zonal circulation and inhibition of mixing which leads to the ‘ozone hole’ in the terrestrial southern stratosphere; also transition between Rossby waves and solitary eddies which transport fluid (as in the world’s oceans), topographic production of eddies and waves, with steering by PV waveguides (formed by topography and circulation). Using a new laboratory technique known as ‘optical altimetry’ we now can see the interaction of unbalanced flows (downslope winds, gravity- and inertial waves) with the energy-containing geostrophic eddies, as seen in the upward radiation of gravity waves as storms encounter the Greenland’s icy topography. |
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10:00-10:30 | Thompson, AF (Cambridge) |
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Jet formation and transport in baroclinic turbulence with simple topography | Sem 1 | |||

Jets are a well-known feature of the Southern Ocean's Antarctic Circumpolar Current. Evidence from satellite altimetry and numerical models suggests that zonal jets are also a robust feature of the mid-latitude ocean basins. The characteristics of mid-latitude and Southern Ocean jets differ significantly, with the latter having narrow, ribbon-like appearances and a greater tendency to merge, migrate and meander. Topography, lack of meridional boundaries, and variations in the strength and vertical structure of mean flows all contribute to the dissimilarity of mid-latitude and Southern Ocean jets. This study considers the influence of simple topography--sinusoidal ridges and bumps--on the formation and transport properties of coherent structures (jets and eddies) in forced-dissipative, quasi-geostrophic turbulence. The experimental framework is a series of two-layer, baroclinically-unstable simulations in a doubly-periodic domain. Transport and mixing properties are diagnosed using the Nakamura effective diffusivity. In simulations with zonal ridges, the upper layer, in particular, feels a competition between the imposed topographical scale (ridge separation) and the Rhines scale. This can lead to unsteady jet structure and vertical misalignment of transport barriers. Three-dimensional topography with a sufficiently large amplitude may induce periodic bursts of high eddy kinetic energy related to baroclinic instability acting on topographically steered non-zonal mean flows. These episodes allow large-scale reorganization of the jet structure. It is likely that these features play a key role in the dynamic nature of Southern Ocean jets. |
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10:30-11:00 | Esler, G (UCL) |
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The turbulent equilibration of an unstable baroclinic jet | Sem 1 | |||

The evolution of an unstable baroclinic jet, subject to a small perturbation, is examined numerically in a quasi-geostrophic two-layer channel model. After a period of initial wave growth, wave breaking leads to two-dimensional turbulence within each layer, and to the eventual equilibration of the flow. The equilibrated flow must satisfy certain constraints; its total momentum is conserved, its total energy is bounded and the flow must be realizable via some area preserving (diffusive) rearrangement of the potential vorticity field of the initial flow. A theory is introduced that predicts the equilibrated flow in terms of the initial flow parameters. The idea is that the final state minimizes available potential energy, subject to constraints on the total momentum and total energy, and the further constraint that the potential vorticity changes through a process of complete homogenization within well-delineated regions in each layer. Within a large region of parameter space, the theory is found to accurately predict the cross-channel structure and strength of the equilibrated jet, the regions where potential vorticity mixing takes place, and total eddy mass (temperature) fluxes. |
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11:00-11:30 | Coffee | |||

11:30-12:00 | Carnevale, GF (UC, San Diego) |
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Inertial instability in rotating and stratified flow | Sem 1 | |||

The unfolding of inertial instability in vortices in a uniformly rotating and stratified fluid is studied through numerical simulations. The vortex dynamics during the instability is examined in detail. We demonstrate that the instability is stabilized via redistribution of angular momentum in a way that produces a new equilibrated vortex with a stable velocity profile. Based on extrapolations from the results of a series of simulations in which the Reynolds number and strength of stratification are varied, we arrive at a construction based on angular momentum mixing that predicts the infinite-Reynolds-number form of the equilibrated vortex toward which inertial instability acting alone would drive an unstable vortex. The essential constraint is conservation of total absolute angular momentum. The construction can be used to predict the total energy loss during the equilibration process. |
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12:00-12:30 | Billant, P (École Polytechnique) |
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Zigzag instability of vortex arrays in stratified and rotating fluids | Sem 1 | |||

We investigate the three-dimensional stability of columnar vertical vortex arrays (Karman street, double symmetric row of vortices) in a stratified and rotating fluid by means of an asymptotic theory for long-vertical wavelength and well-separated vortices. It is found that both the Karman street and the double symmetric row are unstable to the zigzag instability when the fluid is stratified independently of the background rotation. The zigzag instability bends the vortices with almost no internal deformation and ultimately slices the flow into horizontal layers. The results are in excellent agreement with direct numerical stability analyses. They may explain the formation of layers commonly observed in stratified flows. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-14:30 | Meunier, P (CNRS) |
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Instabilities of a columnar vortex in a stratified fluid | Sem 1 | |||

The stability of a columnar vortex in a stratified fluid is addressed theoretically and experimentally. In a first part, we show that the addition of a small tilt angle between the vortex axis and the density gradient direction changes drastically the structure and stability of the vortex. For Froude numbers larger than one, a strong axial flow is created around the critical radius where the angular velocity equals the Brunt-Vaisala frequency. For large Reynolds numbers, this axial flow is sufficiently localized in the critical layer to become unstable with respect to the shear instability. Theoretical predictions for the structure of the critical layer and the instability characteristics are compared with experimental results. In a second part, we analyse the stability of a vortex aligned along with the density gradient direction. We show that the linear modes of the vortex are weakly unstable for small Froude numbers. These modes are radiative at large radii, which means that the vortex generates internal waves spontaneously. At moderate Froude numbers, these radiative modes cease to be unstable. However, a resonance between the radiative modes and the Kelvin modes of the vortex becomes possible leading to instability in narrow wavenumber bands. |
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14:30-15:00 | Davidson, P (Cambridge) |
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Structure formation in rotating turbulence | Sem 1 | |||

We consider decaying rotating turbulence at a Rosby number of around unity. We show that the large-scale columnar structures so evident in many experiments and some simulations are created by linear wave propagation and not the result of triad interactions. We also discuss the reasons for the predominance of cyclones, and the mechanisms by which anisotopy feeds down to the small scales. |
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15:00-16:00 | Tea | |||

16:00-16:30 | Tran, CV (St Andrews) |
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Local transfer and spectra of a diffusive field advected by large-scale incompressible flows | Sem 1 | |||

This study revisits the problem of advective transfer and spectra of a diffusive scalar field in large-scale incompressible flows in the presence of a (large-scale) source. By ``large scale'' it is meant that the spectral support of the flows is confined to the wave-number region $k<k_d$, where $k_d$ is relatively small compared with the diffusion wave number $k_\kappa$. Such flows mediate couplings between neighbouring wave numbers within $k_d$ of each other only. It is found that the spectral rate of transport (flux) of scalar variance across a high wave number $k>k_d$ is bounded from above by $Uk_dk\Theta(k,t)$, where $U$ denotes the maximum fluid velocity and $\Theta(k,t)$ is the spectrum of the scalar variance, defined as its average over the shell $(k-k_d,k+k_d)$. For a given flux, say $\vartheta>0$, across $k>k_d$, this bound requires $$\Theta(k,t)\ge \frac{\vartheta}{Uk_d}k^{-1}.$$ This is consistent with recent numerical studies and with Batchelor's theory that predicts a $k^{-1}$ spectrum (with a slightly different proportionality constant) for the viscous-convective range, which could be identified with $(k_d,k_\kappa)$. Thus, Batchelor's formula for the variance spectrum is recovered by the present method in the form of a critical lower bound. The present result applies to a broad range of large-scale advection problems in space dimensions $\ge2$, including some filter models of turbulence, for which the turbulent velocity field is advected by a smoothed version of itself. For this case, $\Theta(k,t)$ and $\vartheta$ are the kinetic energy spectrum and flux, respectively. |
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16:30-17:00 | Shrira, V (Keele) |
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Singular focusing of sub-inertial internal waves on the `non-traditional’ beta-plane | Sem 1 | |||

On the `non-traditional' beta-plane internal waves propagating poleward partly penetrate through the inertial latitude and turn into sub-inertial waves which propagate further poleward trapped in the narrowing waveguides around the local minima of the buoyancy frequency. The present work addresses basic open questions about wave evolution and, in particular, its possible role in deep ocean mixing. |
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17:00-17:45 | Gong Show | |||

17:45-18:45 | Wine Reception | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Tuesday 9 December | ||||

09:30-10:00 | McWilliams, JC (UC, Los Angeles) |
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Turbulent fluid dynamics at the margins of rotational and stratified control | Sem 1 | |||

Geophysical fluid dynamicists have developed a mature perspective on the dynamical influence of Earth's rotation, while most other areas of fluid dynamics can safely disregard rotation. Similarly, geophysical problems usually arise under the influence of stable density stratification at least as importantly as velocity shear. In this talk the dominant turbulence and wave behaviors in the rotating and non-rotating, stratified and non-stratified fluid-dynamical realms are described, and particular attention is given to their borderlands, where rotational and stratified influences are significant but not dominant. Contrary to the inverse energy cascade of geostrophic turbulence toward larger scales, a forward energy cascade develops within the borderlands from the breakdown of diagnostic force balances, frontogenesis, and frontal instabilities, and then it continues further through the small-scale, non-rotating, unstratified (a.k.a. universal) realm until it dissipates at the microscale. In particular, this submesoscale cascade behavior is of interest as a global route to kinetic and available-potential energy dissipations in the oceanic general circulation, as well as an energy source for microscale material mixing across stably-stratified density surfaces. |
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10:00-10:30 | Bartello, P (McGill) |
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Quasigeostrophic and stratified boussinesq turbulence | Sem 1 | |||

Turbulence generated by random isotropic i.c.'s in the strongly stratified and rapidly rotating environment that characterises large-scale atmosphere/ocean flows, leads to nonlinear geostrophic adjustment via a downscale cascade of wave energy and a first-order decoupling of geostrophic modes from linear wave modes. The march toward ever higher resolution in "realistic" models has renewed the community's interest in smaller-scale stratified turbulence with weak rotation, where this large-scale decoupling is reduced. Proceeding to even smaller scales, i.e. L~H~U/N, where L and H are the horizontal and vertical length scales and N is the Brunt-Vaissalla frequency, 3-d overturning may occur (Thorpe 1977). Our recent non-rotating simulations of stratified flow show that, if the Reynolds number is sufficiently high, then the characteristic vertical scale decreases until the usual Froude number (U/NH) reaches order unity from below, where it saturates. Increasing the stratification decreases the Froude number based on the horizontal scale. Rescaling the equations in this way yields hydrostatic flow where the vertical advection term is not small. On the other hand, if the vertical scale of the flow is set by viscosity, weak rotation or, for some period of time, by the initial conditions, the classical quasi-horizontal layered flow of Riley, Metcalfe & Weissman (1981) is recovered. We also simulated how this picture changes with the addition of varying rotation rates in order to observe the transition from stratified to quasigeostrophic (QG) turbulence. Throughout this work we examined the statistics of linear wave and vortical (PV) modes separately. It was shown that, although they form a complete basis, they can only be expected to have dynamical relevence when U/NH<<1. While this is the case in QG flow, and for strongly-stratified decay of isotropic i.c.'s, its application must be restricted to the buoyancy range in stratified turbulence without rotation. In the QG limit, we removed the omega-equation contribution to the vertical velocity from the total field, to study the degree of balance obtained at various Rossby numbers. Finally, with parameterisation of "wave drag" in balanced models in mind, we explicitly calculated the energy drain on the linear vortical modes exerted by the linear wave modes as a tentative preliminary step. Predictability, as a function of Rossby and Froude numbers is an ongoing focus. |
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10:30-11:00 | Scott, R (St Andrews) |
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Effects of large-scale energy dissipation in geostrophic turbulence | Sem 1 | |||

We compare the distinct effects of frictional damping and radiative, or thermal, damping on the equilibration of two-dimensional geostrophic turbulence. The spatial distribution of energy in both physical and spectral space is examined with particular attention to the distribution of coherent vortices, which are found to be ubiquitous with either form of large-scale energy dissipation. Consideration of the stochastically forced vorticity equation suggests that in the case of frictional damping, maximum vorticity values depend on the damping coefficient $r$ through \qext$\sim r^{-1/2}$, while in the case of thermal damping \qext is approximately independent of damping coefficient. These are well-supported by numerical experiments. The difference between frictional and thermal damping becomes striking in simulations of forced shallow water turbulence on the sphere. While shallow-water models have been successful in reproducing the formation of robust, and fully turbulent, latitudinal jets similar to those observed on the giant planets, they have to date consistently failed to reproduce prograde (superrotating) equatorial winds. Here it is demonstrated that shallow water models not only can give rise to superrotating winds, but do so very robustly, provided that the physical process of large-scale energy dissipation by radiative relaxation (thermal damping) is taken into account. With appropriate choice of thermal damping rate, equatorial superrotation can be achieved at apparently any deformation radius. |
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11:00-11:30 | Coffee and Posters | |||

11:30-12:00 | Falkovich, G (Weizmann Institute of Science) |
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Two-dimensional turbulence inside a strong vortex | Sem 1 | |||

I will present experimental results on turbulence generated in thin fluid layers in the presence of a large-scale coherent flow, or a spectral condensate. It will be shown that the condensate modifies the third-order velocity moment in a much wider interval of scales than the second one. The modification may include the change of sign of the third moment in the inverse cascade. This observation may help resolve a controversy on the energy flux in meso-scale atmospheric turbulence (10-500 km): to recover a correct energy flux from the third velocity moment one needs first to subtract the coherent flow. We find that the condensate also increases the velocity flatness. I will briefly present a theoretical consideration for condensate-turbulence interaction. |
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12:00-12:30 | Sugimoto, N (Keio) |
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Spontaneous gravity wave radiation from nearly balanced rotational flows in a shallow water system | Sem 1 | |||

Spontaneous gravity wave radiation from nearly balanced rotational flows is investigated in shallow water system on a rotating sphere. Gravity waves are very important for their role on global material circulation in the middle atmosphere. It is suggested that gravity waves are radiated from strong rotational flows, such as polar night jet, sub-tropical jet, and typhoon in recent observational studies. This radiation process is called as a spontaneous gravity wave radiation, since gravity waves are spontaneously radiated from initial balanced rotational flows. Although there are several numerical studies, this process has not been fully understood. In the present study, we use the most simplified system of shallow water that includes both gravity waves and rotational flows. In addition, the spectral-like three point combined compact difference (sp-CCD) scheme is used, which has high accuracy as well as the spherical harmonics model. This model allows us to estimate gravity wave amplitude with high accuracy. We investigate dependencies of positions of the jet on spontaneous gravity wave radiation. To discuss on the conditions of gravity wave radiation and propagation, we use the analogy with the theory of the aero-acoustic sound wave radiation (Lighthill theory). We also show results of two layer shallow water system in which small internal gravity waves are spontaneously radiated from baloclinic instability. Finally, we will mention recent theoretical study of spontaneous gravity wave radiation from simple vortex pairs. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-14:30 | Busse, FH (Bayreuth) |
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Unusual properties of turbulent convection and dynamos in spherical shells | Sem 1 | |||

The dynamics of convecting fluids in rotating spherical shells is governed at Prandtl numbers of the order unity by the interaction between differential rotation and roll-like convection eddies. While the differential rotation is driven by the Reynolds stresses of the eddies, its shearing action inhibits convection and causes phenomena such as localized convection and turbulent relaxation oscillations. The response of the system is enriched in the case of dynamo action. Lorentz forces may brake either entirely or partially the geostrophic differential rotation and give rise to two rather different dynamo states. Bistability of turbulent dynamos exists for magnetic Prandtl numbers of the order unity. While the ratios between mean magnetic and kinetic energies differ by a factor of 5 or more for the two dynamo states, the mean convective heat transports are nearly the same. They are much larger than in the absence of a magnetic field. |
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14:30-15:00 | Tobias, S (Leeds) |
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Angular momentum transport and dynamos in magnetised rotating stratified systems | Sem 1 | |||

In this presentation I shall discuss how the presence of a weak magnetic field may suppress the generation of large-scale zonal flows in stably stratified domains. I shall also discuss the characteristics of magnetic field generated via dynamo action in shallow-water magnetohydrodynamics |
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15:00-16:00 | Tea and Posters | |||

16:00-16:30 | Yoden, S (Kyoto) |
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Jet formation in decaying two-dimensional turbulence on a rotating sphere | Sem 1 | |||

Jet formation in decaying two-dimensional turbulence on a rotating sphere is reviewed from the view point of wave mean-flow interaction for both shallow-water case and non-divergent case as the limit of Fr (Froude number) going to zero. A series of computations are performed to confirm the behavior of zonal mean zonal flow generation on the parameter space of the rotation rate Omega and Fr. When the flow is non-divergent and Omega is large, intense retrograde circumpolar jets tend to emerge in addition to a banded structure of mean zonal flows with alternating flow directions. As Fr increases, circumpolar jets disappear and a retrograde jet emerges in the equatorial region. The appearance of the intense retrograde jets can be understood by the angular momentum transport associated with the generation, propagation, and absorption of Rossby waves. When the flow is non-divergent, long Rossby waves tend to be absorbed near the poles. In contrast, when Fr is large, Rossby waves can hardly propagate poleward and tend to be absorbed near the equator. The direction of the equatorial jet, however, is not always retrograde. Our ensemble experiments showed the emergence of a prograde jet, though less likely. This result is contrasted with the previous studies that reported retrograde equatorial jets in all the cases for shallow-water turbulence. Furthermore, a mean zonal flow induced by wave-wave interactions was examined using a weakly nonlinear model to investigate the acceleration mechanisms of the equatorial jet. The second-order acceleration is induced by the Rossby and mixed Rossby-gravity waves and its mechanisms can be categorized into two types. |
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16:30-17:00 | Cho, JYK (QMUL) |
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Magnetohydrodynamic shallow-water turbulence on the sphere | Sem 1 | |||

Motivated by astrophysical-geophysical applications, we have performed a series of high Reynolds number simulations of magnetohydrodynamic shallow-water turbulence (MHDSWT) on a rotating sphere. MHDSWT is the simplest turbulence model that allows the effects of differential rotation, stratification and magnetic field to be studied over long simulation times. A systematic exploration of the full physical and numerical parameter-space shows novel as well as consistent behavior, compared with those of pure hydrodynamic (HD) and 2-D MHD counterparts. In the case without rotation and weak magnetic field strength, the turbulent evolution is sensitive to initial conditions, with the strongest dependence on the peak of the initial energy spectrum. With increasing magnetic field strength, the flow field is more susceptible to loss of balance, and the field blows up in finite time. In addition, the pronounced zonal structures observed in differentially-rotating HD systems do not form. |
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17:00-18:30 | Parellel discussion sessions | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Wednesday 10 December | ||||

09:30-10:00 | McIntyre, ME (Cambridge) |
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Beyond Lighthill: three ways to stay close to balance | Sem 1 | |||

This talk proposes to discuss the recently discovered cases of hybrid propagating structures consisting of vortex dipoles and co-moving gravity waves undergoing wave capture. It is shown how these cases fall outside the scope of the Lighthill theory of spontaneous imbalance and, concomitantly, outside the scope of shallow-water dynamics. Besides the fact that going from shallow water to continuous stratification allows disparate vertical scales -- small for inertia-gravity waves and large for vortical motion -- the key points are (a) that in contrast with cases covered by the Lighthill theory the wave source in these cases feels an order-unity radiation reaction, hence cannot be prescribed in advance, when Rossby numbers R \sim 1; (b) that cases of this sort will supply exceptions to the general rule that spontaneous imbalance is exponentially small in R; and (c) that unsteady vortical motion in continuous stratification can stay close to balance thanks to three quite separate mechanisms. These are (1) the near-suppression, by Lighthill interference, of large-scale imbalance (inertia-gravity waves of large horizontal scale) where "large" means relative to Rossby deformation lengths L_D characterizing the vortical motion, (2) the low-group-velocity flaccidity hence near-steadiness of L_D-wide jets that meander and form loops, Gulf-Stream-like, on streamwise scales >> L_D, and (3) the dissipation of small-scale imbalance by wave capture leading to wave breaking, generically likely in an environment of random shear and straining. Shallow-water models include the first two mechanisms but exclude the third. |
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10:00-10:30 | Vanneste, J (Edinburgh) |
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Inertia-gravity-wave generation: a geometric-optics approach | Sem 1 | |||

The generation of inertia-gravity waves in complex flows is examined using a geometric-optics approach. This approach considers the dynamics of a small-scale wavepacket in prescribed time-dependent, balanced flow. The wavepacket is assumed to be in the so called wave-capture regime, where the wave intrinsic frequency is negligible compared with the Doppler shift. The dynamics is reduced to a number of ordinary differential equations describing the evolution of the wavepacket position, of the wavevector, and of three scalar fields describing the wavepacket amplitude and polarisation. The approach clearly identifies two classes of wave-generation processes: unbalanced instabilities, associated with linear interactions between inertia-gravity waves, and spontaneous generation, associated with a conversion between vortical and inertia-gravity modes. Applications to idealised and realistic flows are discussed. |
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10:30-11:00 | Bustamante, M (Warwick) |
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Dynamics of nonlinear resonance clusters in atmosphere and oceans | Sem 1 | |||

We regard as a starting point barotropic vorticity equation, for two types of boundary conditions: first, on a rotating sphere, second, in a rectangular basin. Our main goal is to study the dynamical systems describing small clusters of nonlinear resonances, the smallest one being a triad. For both types of boundary conditions, the geometrical and topological structures of the clusters are presented. It is shown that most frequently met clusters are integrable. It is also shown that dynamical phases, usually regarded as equal to zero or constants, play a substantial role in the dynamics of resonance clusters. Indeed, their effects are: (i) To reduce the period of energy exchange $\tau$ within a cluster by 20% and more. (ii) To diminish, at time scales of the order of $\tau$, the variability of wave energies by 25% and more. (iii) To generate a new time scale $T >> \tau$ in which we observe a considerable energy exchange within a cluster, as well as an increase in the variability of wave energies for each and every mode. |
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11:00-11:30 | Coffee and Posters | |||

11:30-12:00 | Macaskill, C (Sydney) |
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Vortex simulations of 2D turbulence in confined domains | Sem 1 | |||

This talk considers the evolution of 2D turbulence in a confined domain with slip boundary conditions imposed. Several domain shapes are considered, both regular (e.g. a circle, a square) and irregular (e.g. random coastlines). The CASL (Contour-Advective Semi-Lagrangian) technique is employed, taking as the initial condition a random assembly of vortex patches. It is known that the initial angular momentum is important in determining whether the very long time state is dipolar or monopolar when no-slip boundary conditions are considered. Although this is not the case for slip boundary conditions, the initial total circulation plays a similar role. We explore the dependence of the final state on the initial circulation for a range of geometries. Some examples are shown where trapping of vortices in 'bays', caused by domain-scale interactions, can influence the long time evolution. Recent work on 2D turbulence in a periodic box has shown how the rate of dipole/monopole interactions can be related to the time rate of decay of enstrophy at intermediate to long times (once the initial very strong interactions are over). The presence of the domain boundary provides a mechanism for enhancing the rate of such interactions, as monopoles of opposite sign tend to hug the boundaries, travel in opposite directions and then meet to form dipoles which are then launched into the flow. Accordingly, we consider the effect of domain shape on the frequency of dipole formation and dipole/monopole interactions and the corresponding rate of decay of enstrophy. Finally, we discuss extensions of this work to the 3D quasi-geostrophic case. |
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12:00-12:30 | Dritschel, DG (St Andrews) |
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The hyper-CASL algorithm: an accurate, efficient new approach to modelling complex atmospheric and oceanic flows | Sem 1 | |||

We describe a powerful new approach to the simulation of layerwise-2D geophysical flows. Our approach combines the CASL algorithm, a hybrid contour-dynamics method ideally suited for the Lagrangian advection of potential vorticity (PV) contours across an Eulerian fixed grid, with a standard vortex-in-cell method. The latter is here used to compensate for errors arising from contour regularisation or `surgery' appearing at the level of the grid. In this way, (essential) dissipation occurs only at sub-grid scales, and there is no direct impact of surgery on the computed velocity field, which is obtained on the grid by standard spectral or (compact) finite-difference methods. This is particularly important for long-term simulations, where error accumulation needs to be minimised. This new approach, coined the `hyper-CASL algorithm', also allows for general non-conservative effects, by design. For instance, one may take account of radiative (thermal) effects which attract a given flow to some thermal equilibrium distribution, or one can account for `Ekman pumping' which converts wind stress at the ocean's surface into a vorticity source or sink there. In fact, very general forms of forcing or damping can be accounted for (e.g. stochastic, moisture, small-scale convection, etc.). Hyper-CASL is not limited to PV, but can be applied to other tracers such as chemical species in the atmosphere, including moisture. In particular, modelling water vapour by contours and subgrid-scale convection by Lagrangian particles may be a timely opportunity to improve the forecasting of precipitation. Hyper-CASL is illustrated here in long-term simulations of quasi-geostrophic and shallow-water turbulence. These simulations reveal unprecedented detail, in particular vortices spanning a very wide range of scales and significantly affecting flow statistics. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-18:30 | Excursion | |||

19:00-23:00 | Conference Dinner at St John's College (pre dinner drinks from 19:00) |

Thursday 11 December | ||||

09:30-10:00 | Vallis, GK (Princeton) |
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Is geostrophic turbulence relevant for tropospheric dynamics? | Sem 1 | |||

Geostrophic turbulence is a model for understanding the large-scale structure and flow of energy in stably-stratified, rapidly rotating flows that, typically, are baroclinically unstable or driven externally by small-scale convection. Inverse energy cascades and the production of zonality by the beta effect robust predictions for such flows. Is such a model relevant for the Earth's troposphere? On the one hand it appears that such a model must surely be relevant because of its generality, but on the other hand the inverse cascade is noticeable by its absence in the Earth's atmosphere, and zonal jets do not appear as well formed as those on some other planets. We will discuss these issues, and compare some turbulent simulations from a quasi-geostrophic model with those from a primitive-equation model in parameter settings including, but not restricted to, the Earth's troposphere. Joint work with P. Zurita-Gotor. |
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10:00-10:30 | Kimura, Y (Nagoya) |
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Spectral study of stably stratified turbulence | Sem 1 | |||

The energy spectrum of stably stratified turbulence is studied numerically by solving the 3D Navier-Stokes equations under the Boussinesq approximation pseudo-spectrally. The resolution is $1024^3$. Using toroidal-polodial decomposition (Craya-Herring decomposition), the velocity field is divided into the vortex mode and the wave mode. For randomly forced flows, we show that there is a sharp wave number transition in the energy spectra of the vortex and wave modes. With the initial kinetic energy zero, the vortex-mode first develop a $k_{¥perp}^{-3}$ spectrum for the whole wavenumber range studied. Here, $k_{¥perp}¥equiv ¥sqrt{k_x^2+k_y^2}$. Then at large $k_{¥perp}$, a $k_{¥perp}^{-5/3}$ part appears with a sharp transition in the wave number. Meanwhile the wave-mode spectra show a $k_{¥perp}^{-2}$ first, and then $k_{¥perp}^{-5/3}$ part appears at high $k_{¥perp}$. Spectra for different values of the Brunt-Vaisala frequency are investigated. We present evidence that the $k_{¥perp}^{-3}$ part at the large scale in the vortex-mode spectra may be characterized as 2d turbulence. The small scale part is a Kolmogorov spectrum $¥sim ¥epsilon^{2/3}k_{¥perp}^{-5/3}$, where $¥epsilon$ is the horizontal energy dissipation of the vortex-mode. Finally, we discuss decaying stratified turbulence. We note that our present high resolution simulations support earlier simulations (Kimura & Herring 1996) in which the decay of total energy was found to be $E(t)¥sim t^{-1}$. An explanation for such a decay is proposed in terms of the shape of the toroidal and polodial energy spectra. |
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10:30-11:00 | Nazarenko, S (Warwick) |
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Triple cascade and zonal flows in beta-plane turbulence | Sem 1 | |||

At low amplitudes, the beta-plane model has three conservation laws: the energy, the potential enstrophy and the zonostrophy (an additional invariant found by Balk, Nazarenko and Zakharov in 1991). Application of a standard Fjortoft argument to this system leads to a conclusion that these 3 invariants must cascade in an anisotropic fashion, each cascade occupying its own sector in the 2D k-space. It appears that the energy cascade sector corresponds to generation of large-scale zonal flows in the system. I will discuss some recent numerics aimed at checking conservation of zonostrophy and the triple cascade property. |
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11:00-11:30 | Coffee and Posters | |||

11:30-12:00 | Miyazaki, T (Electro-Communications, Japan) |
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Equilibrium state of quasi-geostrophic point vortices | Sem 1 | |||

The statistics of quasi-geostrophic point vortices is investigated theoretically and numerically, in order to understand fundamental aspects of quasi-geostrophic turbulence. Numerical simulations of N-vortex system (N = 2000 − 8000) in an infinite fluid domain are performed using a fast special-purpose computer (MDGRAPE-3) for molecular dynamics simulations.[1] The vortices of the same strength are initially located randomly and uniformly in a cubic box, and we choose the state with the highest multiplicity for fixed angular momentum. The axi-symmetric equilibrium state is obtained after about 20 turn over time. The probability density distribution of the center region resembles that of the purely two-dimensional point vortices. The three-dimensional effect appears near the upper and lower lids in the tighter concentration of vortices around the axis of symmetry (End-effect). The most probable vortex distributions are determined based on the maximum entropy theory.[2] We search for the state of maximum Shanon-entrophy under the constraints of vertical vorticity distribution, angular momentum and energy (mean-field approximation). The theoretical predictions agree quite well with the numerical results.[3] We investigate the influence of energy on the equilibrium state, in some detail. Larger vortex clouds have lower energy, larger entropy and lager angular momentum. Each vertical layer has same contribution to the entropy and the angular momentum, whereas the center region has stronger influence to the energy than the lids. Therefore, the distribution in the center region expands radially for lower energy and shrinks for higher energy. In order to keep the angular momentum unchanged, the distribution near the lids should shrink for lower energy and should expand for higher energy. References [1] Yatsuyanagi Y., Kiwamoto Y., Tomita H., Sano M.M., Yoshida T., Ebisuzaki. T.: Dynamics of Two-Sign Point Vortices in Positive and Negative Temperature States. Phys. Rev. Lett. 94:054502, 2005 [2] Kida S.: Statistics of the System of Line Vortices. J. Phys. Soc. Jpn. 39(5):1395–1404, 1975 [3] Hoshi S., Miyazaki T.: Statistics of Quasi-geostrophic Point Vortices. FDR (Subbmitting), 2008. |
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12:00-12:30 | Carton, X (Bretagne Occidentale) |
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Influence of external shear and strain on baroclinic vortex alignment | Sem 1 | |||

The influence of an external strain (or shear) field on the evolution of two identical vortices at different depths is investigated in a two-layer quasi-geostrophic model. The external deformation can include a time-varying component. Using point vortex modeling, symmetric equilibrium positions for the vortex doublet are identified and their stability is computed. A time- varying external field can induce resonances on both harmonic and subharmonic frequencies. A slow time amplitude equation is then computed for the vortex trajectories around their equilibrium positions ; its solutions are compared successfully to numerical simulations of a baroclinic point vortex model. The evolution from regular to chaotic trajectories when the external field amplitude grows, is analyzed. Finite-area vortices exhibit the classical regimes of alignment or of co-rotation when submitted to external strain and rotation, but they can also remain stationary at the location of the neutral points for the equivalent point vortices, or oscillate around these positions. Another regime is evidenced, in which originally very distant vortices are advected towards the center of the plane by the large-scale flow, and finally merge. Thus, in a dense field of coherent structures, neighboring vortices can favor the vertical alignment of two central vortices, despite the erosion that they induce on these vortices. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-14:30 | Herring, J (N.C.A.R.) |
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Spectra and distribution functions of stably stratified turbulence | Sem 1 | |||

We consider homogeneous stably stratified turbulence both decaying and randomly forced cases. Our tools include direct numerical simulations (DNS) and elements of statistical theory as expressed by two-point closures. Our DNS--at $1024^3$--permits a large scales Taylor micro scale $R_{\lambda}\sim 300$ The size distribution of such large scales is closely related to conservation principles, such as angular momentum, energy, and scalar variance; and we relate these principles to our DNS results. Stratified turbulence decays more slowly than isotropic turbulence with the same initial conditions. We offer a simple explanation in terms of the diminution of energy transfer to small scales resulting from phase-mixing of gravity waves (Kaneda 1998). Enstrophy structures in stratified flows (scattered pancakes) are distinctly different from those found from isotropic turbulence (vortex tubes), and we show examples of the transition from isotropic turbulence enstrophy structures to those of strongly stratified turbulence. We discuss briefly changes in the probability distribution functions for velocity and vorticity for stratified turbulence concluding that stratification induces a return towards Gaussianity for these quantities. For the forced case, we examine the modification of the inertial range induced by strong stratification ($k_{\perp}^{-5/3}\rightarrow \sim k_{\perp}^{-2}$) for the wave component, and ($k_{\perp}^{-5/3}\rightarrow\sim k_{\perp}^{-3}$) for the vortical component. Here, $k_{\perp}$ is the horizontal wave number. These DNS findings are discussed from the perspective of two-point closure. |
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14:30-15:00 | Riley, JJ (Washington) |
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Stratified turbulence: a possible interpretation of some geophysical turbulence measurements | Sem 1 | |||

Stratified turbulence is a flow dominated by stable density stratification, and characterized by low internal Froude numbers and high Reynolds numbers; in the present study it is also assumed that the Rossby number is large. Flows in such parameter regimes occur often in the oceans and in the atmosphere. Results of direct numerical simulations of stratified turbulence, some scaling arguments, and implications regarding laboratory experiments will be presented. The results suggest that stratified turbulence is a possible interpretation of some oceanic and atmospheric measurements. |
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15:00-16:00 | Tea | |||

16:00-16:30 | Zeitlin, V (LMD-ENS) |
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Once again about the parallel between stratification and rotation in hydrodynamics and between both and external magnetic field in MHD | Sem 1 | |||

On the example of simplified 2D models I will discuss the mathematical background of the well-known parallel between the effects of stratification, rotation and external magnetic field. Special emphasis will be made on the fact that Hamiltonian structure of the corresponding hydrodynamical systems is the same, and the consequences of this fact in what concerns both the coherent structures (stationary nonlinear waves/vortices) and the equilibrium spectra of wave turbulence of anisotropic waves proper to each effect (internal gravity waves, gyroscopic waves, and Alfven waves, respectively) |
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16:30-17:00 | Sommeria, J (LEGI-CNRS) |
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Application of statistical mechanics to the modeling of potential vorticity or density mixing | Sem 1 | |||

Statistical mechanics of vortex patches can provide models for potential vorticity mixing with the constraints of fluid incompressibility and energy conservation. This leads to the self-organisation of turbulence into large coherent structures. We can apply a similar approach to the vertical mixing of fluid elements with different densities. This leads to a model of vertical mixing which involves the competition of an eddy diffusivity with sedimentation of fluid elements. In addition to these two processes, the straining of fluid elements by eddies must be included. The behaviour of the resulting model will be discussed for different cases of vertical mixing in a stratified fluid. |
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17:30-18:30 | Parallel discussion sessions | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Friday 12 December | ||||

09:30-10:00 | Winters, KB (UC, San Diego) |
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Bounds on the energetics and entropy production of a buoyancy forced ocean | Sem 1 | |||

We show that the volume averaged rate of entropy production in horizontal convection, i.e., $\chi \equiv \kappa < |{\nabla} b |^2 >$ where $b$ is the buoyancy, is bounded from above by $4.57 H^{-1} \kappa^{2/3} \nu^{-1/3} b_{\rm max}^{7/3}$. Here $H$ is the depth of the container, $\kappa$ is the molecular diffusion, $\nu$ the kinematic viscosity and $b_{\rm max}$ the maximum buoyancy difference prescribed on the surface. The rate of entropy production is directly related, via the volume integrated diapycnal buoyancy flux, to the energetics of irreversible mixing and the rate of energy transfer between available and background potential energy in the Boussinesq limit. The bound implies that the rate of generation of available potential energy by horizontal convection is no larger than $\kappa^{1/3}$ in the limit $\kappa \to 0$ at fixed $Pr=\nu/\kappa$. The bound on the energetics of mixing reinforces and strengthens the statement of Paparella and Young (2002) that horizontal convection is nonturbulent in limit of vanishing fluid viscosity and diffusivity. In the context of a model ocean, insulated at all boundaries except at the upper surface where the buoyancy is prescribed, the bounds on the energy transfer rates in the mechanical energy budget imply that buoyancy forcing alone is insufficient by at least three orders of magnitude to maintain observed or inferred average oceanic dissipation rates and that additional energy sources such as winds, tides or (perhaps) bioturbation are necessary to sustain nontrivial levels of turbulent dissipation and mixing in the world's oceans |
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10:00-10:30 | Hua, BL (IFREMER) |
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Simulations of stratified turbulence surrounding Meddy structures | Sem 1 | |||

Recently available geosismic data in the North-East Atlantic ocean have revealed ubiquituous pancake-like layers of density anomalies of 30-60 meters thickness, surrounding Meddy structures within the Mediterranean Water outflow. The depths of pancake-like density layering unambiguously coincide with potential energy spectra in $k_h^{-5/3}$,-- where $k_h$ is the horizontal wavenumber--, for horizontal spatial scales less than 1km, reminiscent of non-rotating stratified turbulence results. Using ultra-high 3D resolution simulations of a Meddy structure on the Earth Simulator, with a horizontal grid size down to 100m and a vertical grid size of 3m, we have been able to reproduce the pancake-like layering surrounding the eddy. Furthermore, potential and kinetic energy with $k_h^{-5/3}$ inertial ranges are simulated. A rationale is proposed for the formation mechanism of the layering and we discuss the influence of rotation on stratifed turbulence properties. |
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10:30-11:00 | Lelong, MP (NorthWest Research Associates, USA) |
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Internal tide generation from surface tide/geostrophic Eddy interactions | Sem 1 | |||

Most studies of internal tide generation have focused on the role of the barotropic tide interacting with topography. Our study focuses on a mechanism which has not been considered up to now, namely the interaction of a barotropic tide with a baroclinic eddy field. The interaction of a barotropic tidal current and a mesoscale eddy field is investigated to determine the ambient conditions under which energy is transferred most efficiently from barotropic to baroclinic tides. We examine the transfer of energy as a function of latitude and eddy field structure. The existence of critical latitudes is also discussed in this context. |
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11:00-11:30 | Coffee | |||

11:30-12:00 | Staquet, C (Joseph Fourier Grenoble) |
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Propagation of the internal tide from a continental shelf: laboratory and numerical experiments | Sem 1 | |||

Internal gravity waves in the deep ocean have been given much importance for the last ten years, since oceanographers realized the important role of fluid mixing in raising the cold abyssal water masses. Mixing in the deep ocean, below 2 or 3 kms, is indeed thought to be mostly due to nonlinear internal gravity waves. In the abyss, internal waves result mainly from the interaction of the surface tide with the variable bottom topography and are referred to as the internal tide. The thermal equilibrium of the deep ocean thus depends upon local and strongly intermittent processes -the breaking internal tide- which occur on temporal scales of the order of one minute and on spatial scales of the order of ten meters. This gigantic difference in spatial and temporal scales requires the parameterization of the mixing processes in large scale circulation models. A direct study of the breaking internal tide is therefore needed. The present study focuses upon the academic situation of a uniformly stratified ocean, when the surface tide interacts with an idealized two-dimensional continental shelf. Numerical experiments have been conducted in close correspondance with the laboratory experiments, the geometrical and physical parameters being the same. The laboratory experiments have been performed on the rotating Coriolis platform in Grenoble. The nonlinear non-hydrostatic finite-volume numerical code developed at MIT was used to closely model the experiments. In this simplified situation, the internal wave field organizes as a rectilinear wave beam with tidal frequency. We shall discuss the generation mechanism and structure of the wave beam, relying on theoretical modelling of internal wave emission by an oscillating body source. We shall also investigate the nonlinear dynamics of the tide, when harmonics and parametric sub-harmonics are generated. The localization and quantification of mixing induced by the internal tide will eventually be considered. |
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12:00-12:30 | Straub, D (McGill) |
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2d-3d energy exchanges in a homogeneous thin-aspect-ratio ocean | Sem 1 | |||

Because of its thin aspect ratio, large scale ocean circulation is generally assumed to obey hydrostatic dynamics. In the idealized case of a homogeneous (unstratified) fluid, this implies that the horizontal pressure gradient force is independent of depth. Homogeneous models of ocean circulation typically assume the flow to be two dimensional. However, it can be shown that such a two dimensional flow is unstable to 3d perturbations. Following saturation of the 3d perturbations, a 2d-to-3d energy transfer persists. After a brief description of the instability mechanism, we i) test its robustness to nonhydrostatic effects and ii) address its potential role in the damping of midlatitude ocean gyres. Initially 2d turbulence in a Boussinesq fluid is shown numerically to be unstable to 3d perturbations. Although non-hydrostatic pressure effects act to slow the growth, a robust growth in nonetheless observed even at unit aspect ratio. The perturbations are found to saturate at a level proportional to the domain aspect ratio. Thus, in the thin aspect ratio limit, nearly 2d dynamics are recovered --- provided the 3d modes are not externally forced. We then consider the classic wind-driven ocean gyre problem for a homogeneous thin aspect ratio fluid on a beta plane. A large-scale stochastic forcing is applied to the 3d modes. This produces a background sea of near-inertial oscillations which interact with the depth averaged flow. It is found that 2d-to-3d energy transfers coupled with a forward cascade of 3d energy provides a robust dissipation mechanism for the gyres. We speculate that similar dynamics may also apply in a stratified ocean. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-14:30 | Caulfield, CP (Cambridge) |
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Richardson number-dependent bounds on buoyancy flux | Sem 1 | |||

Parameterizing the mixing of a stratified fluid subject to shear is a fundamental challenge for models of environmental and industrial flows. In particular, it is of great value to parameterize the efficiency of turbulent mixing, in the sense of the proportion of the kinetic energy converted into potential energy (through irreversible mixing of fluid of different density) compared to the total amount converted to both potential energy and internal energy (through viscous dissipation). Various competing models have been presented to relate the mixing efficiency to bulk properties of the flow, especially through different Richardson numbers, which quantify the relative importance of buoyancy and shear within the flow. One promising approach is to construct rigorous bounds on the long-time average of the buoyancy flux (i.e. the mixing rate) within simple model stratified shear flows, imposing physically reasonable constraints on the model flow fields. In this talk, we apply this technique to stably stratified Couette flow, imposing as a constraint the mixing efficiency, and show that a bound on the long-time average of the buoyancy flux can only be found for a certain range of bulk Richardson numbers. For sufficiently strong stratifications, it appears that no bound exists, suggesting that it is not possible to find a statistically steady state. We discuss the implications of this result for various classical stratified shear turbulence models. |
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14:30-15:00 | Chomaz, JM (LadHyX) |
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Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid | Sem 1 | |||

This work investigates the three-dimensional stability of a horizontal flow sheared horizontally, the hyperbolic tangent velocity profile, in a stably stratified fluid. In an homogeneous fluid, the Squire theorem states that the most unstable perturbation is two-dimensional. When the flow is stably stratified, this theorem does not apply and we have performed a numerical study to investigate the three-dimensional stability characteristics of the flow. When the Froude number, Fh, is varied from ∞ to 0.05, the most unstable mode remains two-dimensional. However, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude number for Fh 1. This means that the stronger the stratification, the smaller the vertical scales that can be destabilized. This loss of selectivity of the two-dimensional mode in horizontal shear flows stratified vertically may explain the layering observed numerically and experimentally. Extension to transient and nonlinear behaviour are presented. |
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15:00-16:00 | Tea | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |