Laboratory Experiments in Geophysical and Astrophysical Fluid Dynamics

When the main geomagnetic field undergoes a reversal in polarity it does so in no more than a few thousand years. Many hundreds of such reversals must have occurred over geological time (the age of...

In studies on geophysical fluid dynamics, it is common practice to take the Coriolis force only partially into account by neglecting the components proportional to the cosine of latitude, the so‐called traditional approximation (TA). This review deals with the consequences of abandoning the TA, based on evidence from numerical and theoretical studies and laboratory and field experiments. The phenomena most affected by the TA include mesoscale flows (Ekman spirals, deep convection, and equatorial jets) and internal waves. Abandoning the TA produces a tilt in convective plumes, produces a dependence on wind direction in Ekman spirals, and gives rise to a plethora of changes in internal wave behavior in weakly stratified layers, such as the existence of trapped short low‐frequency waves, and a poleward extension of their habitat. In the astrophysical context of stars and gas giant planets, the TA affects the rate of tidal dissipation and also the patterns of thermal convection.

Planets and stars are rotating objects and large scale fluid motions relative to the planets solid body rotation are constrained by rotation. Theoretical treatment of these fluid motions must, therefore, include rotation effects and it is for this reason that most of the advances in the understanding of rotating fluids have been made in the context of Geophysical Fluid Dynamics (GFD) and Astrophysical Fluid Dynamics (AFD). GFD can be considered a special case of AFD and many approximations made in GFD (see for instance Pedlosky [1]) are not valid in AFD [2]. For example, in GFD horizontal scales of motion L are much less than the radius of the planet, allowing the use of a constant or linearly varying Coriolis frequency f=2Q. The fluid layer depth H is much less than L so that only the component of Q perpendicular to the planets surface needs to be considered. The Boussinesq approximation has also only limited validity in AFD. All the lectures in this course use approximations current in GFD when problems of geophysical interest are considered.

This special issue brings together 19 review articles addressing a broad range of problems in geophysical and astrophysical fluid dynamics, reflecting both the current state of the art and ongoing research, with a special focus on the contribution of laboratory experiments. This foreword outlines the general context and key challenges of the field, and places the individual contributions in perspective.

Sue Bowler, Editor In these tough times – for everyone – it is worth thinking a bit about the real value of our sciences. Rather than accept decisions made on the basis of a snapshot of today’s successes, we should all be aware of our catalogue of achievements in the long term. And I don’t mean just the considerable astronomical and navigational heritage embodied by, for example, the Tompion clock now at home in the Royal Observatory Greenwich – replica of the original 17th-century world-leader. Much of this issue is dedicated to the more than half a century of science in a new field: geophysical and astrophysical fluid dynamics. This is a field largely sparked by the deceptively simple experiments devised and carried out some 60 years ago by Raymond Hide. This whole new field, arising from observing the flow of fluid between two cylinders, now illuminates studies of stellar and planetary interiors, climate modelling and even brings us better weather forecasts. None of these were goals of the original experiments. They arose from curiosity, ingenuity and hard work, coupled with a lovely appreciation of the significance of the results. These are the things that make good science – and they are what attracts people to science. Just look at the responses to telescopes at a music festival, and the chance to study astronomy at school, documented in this issue. Curiosity, fascination with physical phenomena and a drive to explore are what draw people into our physical sciences. And that in turn brings them excellent employment prospects and higher than usual salaries. UK science has been very successful for many decades – even centuries. If we can keep up investment in science, science will in turn reward UK industry as well as keep our science leading the world. Editorial NEws

Progress in geophysical fluid dynamics

Irregular buoyancy-driven flows occur in the atmospheres and fluid interiors of the Earth and other planets, and of the Sun and other stars, where they influence and often control the transfer of heat. Their presence is manifest in or implied by a wide variety of observed phenomena, including external magnetic fields generated by self-exciting magnetohydrodynamic (MHD ) dynamo action. Based on the laws of classical mechanics, thermodynamics and, in the case of electrically conducting fluids, electrodynamics, the governing mathematical equations are well known, but they are generally intractable owing to their essential nonlinearity. Computers play a key role in modern theoretical research in geophysical and astrophysical fluid dynamics, where ideas based on chaos theory are being applied in the analysis of models and the assessment of predictability. The aim of this paper is to provide a largely qualitative survey for non-specialists. The survey comprises two parts, namely a general introduction (Part I) followed by a discussion of two representative areas of research, both concerned with phenomena attributable to symmetry-breaking bifurcations caused by gyroscopic (Coriolis) forces (Part II), namely (a) large-scale waves and eddies in the atmospheres of the Earth, Jupiter and other planets (where, exceptionally, laboratory experiments have been influential), and (b) MHD dynamos. Various combinations of Faraday disc dynamos have been studied numerically as low-dimensional nonlinear electromechanical analogues of MHD dynamos, particularly in efforts to elucidate the complex time series of geomagnetic polarity reversals over geological time. The ability of the intensively studied Rikitake coupled disc dynamo system to behave chaotically appears to be a consequence of the neglect of mechanical friction, the inclusion of which renders the system structurally unstable.

We introduce n-dimensional equations of geophysical fluid dynamics (GFD) valid on rotating n-dimensional manifolds. Moreover, by using straight and twisted differential forms and an auxiliary velocity field, we introduce hierarchically-structured equations of (geophysical) fluid dynamics in which the equations are split into metric-free and metric-dependent parts. For these sets of equations we provide representations in local coordinate charts and we show that they conserve potential vorticity and that Kelvin’s circulation theorem holds. As such general n-dimensional formulations do not exist in vector calculus, we provide for both covariant and vector-invariant equations a representation on a rotating coordinate frame in an Euclidean space and compare these representations. This study reveals, among others, that the prognostic variables, described by straight and twisted differential forms, are independent of both metric and orientation. This makes them perfect descriptors of the fluid’s quantities of interest, as they assign, analogously to physical measurement devices, real valued numbers to finite distances, areas, or volumes. This is not the case for prognostic variables described by (vector) proxy fields, as they depend on metric and orientation. The new structuring reveals also important geometrical features of the equations of GFD. For instance, the dimensioned differential forms display the geometric nature of the fluid’s characteristics, while the equations’ structuring illustrates how the metric-free momentum and continuity equations geometrically interact and how they are connected by the metric-dependent Hodge star and Riemannian lift. Besides this geometric insight, this structuring has also practical benefits: since equations containing only topological structure are less complicated to implement and can be integrated exactly, our results may contribute to more efficient and accurate discretizations.

<p>We have studied theoretically the weakly nonlinear analysis in a rotating Rayleigh-Benard system of electrically conducting fluid in the presence of applied horizontal magnetic field with free-free boundary conditions [1]. This theoretical approach is carried near the onset of convection to study the flow behavior at the occurrence of cross rolls, which are perpendicular to the applied magnetic field. The nonlinear problem is solved by using the Fourier analysis of perturbations up to the O(ε<sup>8</sup>) to study the cross rolls visualization [2,3]. The dependence of the Nusselt number on the Rayleigh number, Ekman number, Elsasser number is extensively examined. The fluid flow is visualized in terms of kinetic energy, potential energy, streamlines, isotherms, and heatlines.</p><p> </p><p>References :</p><p>[1] P. H. Roberts and C. A. Jones , The Onset of Magnetoconvection at Large Prandtl Number in a Rotating Layer I. Finite Magnetic Diffusion, Geophysical and Astrophysical Fluid Dynamics, Vol. 92, pp. 289-325 (2000).</p><p>[2] H.L. Kuo, Solution of the non-linear equations of the cellular convection and heat transport,  Journal of Fluid Mechanics,  Vol.10, pp.611-630 (1961).</p><p>[3] Y. Rameshwar, M. A. Rawoof Sayeed, H. P. Rani, D. Laroze, Finite amplitude cellular convection under the influence of a vertical magnetic field, International Journal of Heat and Mass Transfer, Vol. 114, pp.  559-577 (2017).</p>

Planetary rotation organizes fluid motions into coherent, long-lived swirls,\nknown as large-scale vortices (LSVs), which play an important role in the\ndynamics and long-term evolution of geophysical and astrophysical fluids. Here,\nusing direct numerical simulations, we show that LSVs in rapidly rotating mixed\nconvective and stably stratified fluids, which approximates the two-layer,\nturbulent-stratified dynamics of many geophysical and astrophysical fluids,\nhave a generic shape and that their size can be predicted. We show that LSVs\nemerge in the convection zone from upscale energy transfers and can penetrate\ninto the stratified layer. At the convective-stratified interface, the LSV\ncores have a positive buoyancy anomaly. Due to the thermal wind constraint,\nthis buoyancy anomaly leads to winds in the stratified layer that decay over a\ncharacteristic vertical length scale. Thus LSVs take the shape of a\ndepth-invariant cylinder with a finite-size radius in the turbulent layer and\nof a penetrating half dome in the stratified layer. Importantly, we demonstrate\nthat when LSVs penetrate all the way through the stratified layer and reach a\nboundary that is no-slip, they saturate by boundary friction. We provide a\nprediction for the penetration depth and maximum radius of LSVs as a function\nof the LSV vorticity, the stratified layer depth, and the stratification. Our\nresults, which apply for cyclonic LSVs, suggest that LSVs in slowly rotating\nstars and Earth's liquid core are confined to the convective layer, while in\nEarth's atmosphere and oceans they can penetrate far into the stratified layer.\n

This chapter provides a brief overview of the importance of rotating flows in geophysical and astrophysical fluid dynamics, highlighting the role of rotation in tidal vortices, dust devils, tropical cyclones, and in the zonal winds on Jupiter and Saturn. The chapter also discusses how many of the key phenomena in rotating fluid dynamics, such as Ekman pumping, Taylor columns, and inertial waves, can be exposed using simple laboratory experiments. The emphasis throughout is on simple physical ideas, with mathematics taking a backseat. The discussion anticipates many of the phenomena which are exposed in subsequent chapters.

"Proceedings of the second SEDI symposium on reversals, secular variation and dynamo theory." Geophysical & Astrophysical Fluid Dynamics, 60(1-4), pp. 1–2

Friedmann's equation and the potential vorticity equation are generalised for turbulent motion. The generalised equations incorporate some new phenomena connected with turbulent transport of mass. It is proved that, if ▿×[S×Ω+S(▿·S)]≠0 where Ω is the absolute vorticity of the velocity and S is the turbulent density flux, then the Helmholtz-Kelvin theorem concerning the conservation of the velocity circulation around a closed path is violated and the potential vorticity is not a Lagrangian adiabatic invariant. The effects of this turbulent transport of mass on the creation or dissipation of vorticity discussed here is not equivalent to effects of baroclinicity or viscosity. Some possible implications of the new circulation theorem in geophysical and astrophysical fluid dynamics are discussed.

We study the nonlinear evolution of the magnetic buoyancy instability in rotating and non-rotating gas layers (with emphasis on the parameter range typical of spiral galaxies) using numerical solutions of non-ideal, isothermal MHD equations. The unstable magnetic field is either imposed through the boundary conditions or generated by an imposed α-effect. In the case of an imposed field, we solve for the deviations from the background state which satisfy periodic boundary conditions. We also include cosmic rays as a weightless fluid which exerts a dynamically significant pressure and somewhat amplifies magnetic buoyancy. This version of the instability is known as the Parker instability. Without rotation, systems with an imposed magnetic field evolve to a state with a very weak magnetic field, very different from the marginally stable eigenfunction, where the gas layer eventually becomes very thin as it is supported by the thermal and turbulent pressures alone. However, this does not happen when the magnetic field is maintained by the α-effect. Rotation fundamentally changes the development of the instability. A rotating system develops nonlinear oscillations, and the magnetic field direction changes even with an imposed magnetic field. We demonstrate that this is caused by the secondary α-effect at large altitudes as the gas flow produced by the instability becomes helical. The secondary α-effect has an anomalous sign with the α-coefficient being negative in the northern hemisphere, whereas the Coriolis force produces a positive α. The mean-field dynamo action outside the original gas layer can also lead to a switch in the magnetic field parity from quadrupolar (typical of the mean-field dynamo action in a thin layer) to dipolar. Altogether, the magnetic buoyancy instability and the mean-field dynamo action become separated as distinct physical effects in a nonlinear rotating system. We show that none of the assumptions used in analytic studies of the Parker instability is corroborated by numerical results.

This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing low-Mach dynamics, since they are soundproof. These models find wide use in many areas of fluid dynamics, including both geophysical and astrophysical fluid dynamics. Specifically, the Boussinesq, anelastic and pseudoincompressible equations are developed through a unified treatment valid for arbitrary Riemannian manifolds, thermodynamic potentials and geopotentials. By design, these formulations obey the 1st and 2nd laws of thermodynamics, ensuring their thermodynamic consistency. This general approach extends and unifies existing work, and helps clarify the thermodynamics of semi-compressible fluids. To further this goal, evolution equations are presented for a wide range of thermodynamic variables: entropy density s, specific entropy η, buoyancy b, temperature T, potential temperature θ and a generic entropic variable χ; along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation for the pressure perturbation (the Lagrange multiplier that enforces semi-compressibility) is developed for all three equation sets in the case of reversible dynamics, and for the Boussinesq/anelastic equations in the case of irreversible dynamics; and some discussion is given of the difficulty in formulating it for the pseudoincompressible equations with irreversible dynamics.