We investigate numerically the evolution of a baroclinic vortex in a two-level surface quasi-geostrophic model. The vortex is composed of two circular patches of uniform buoyancy, one located at each level. We vary the vortex radii, the magnitude of buoyancy, and the vertical distance between the two levels. We also study different radial profiles of buoyancy for each vortex. This article considers two main situations: firstly, initially columnar vortices with like-signed buoyancies. These vortices are contra-rotating, are linearly unstable and may break. Secondly, we consider initially tilted vortices with opposite-signed buoyancies, which may align vertically. Numerical experiments show that (1) identical contra-rotating vortices break into hetons when initially perturbed by low azimuthal modes; (2) unstable, vertically asymmetric, contra-rotating vortices can stabilise nonlinearly more often than vertically symmetric ones, and can form quasi-steady baroclinic tripoles; (3) co-rotating vortices can align when the two levels are close to each other vertically, and when the vortices are initially horizontally distant from each other by less than three radii; (4) for initially more distant vortices, two such vortices rotate around the plane center; and(5) in all cases, the vortex boundaries are disturbed by Rossby waves. These results compare favorably to earlier results with internal quasi-geostrophic vortices. Further modelling efforts may extend the present study to fully three-dimensional ocean dynamics.

Interactions between the rotating stratified oceanic and atmospheric flows and topography play a fundamental role in Earth's climate. Here we use laboratory experiments in a differentially-heated rotating annulus to explore stratified flow-topography interactions in a dynamical regime of strong background geostrophic turbulence. A localised small-scale topographic ridge is differentially-rotated at a range of angular velocities around the base of the annulus to impose a relative velocity between the stratified fluid and the small ridge. Considering the idealised setup of the laboratory configuration, the experiments exhibit rich dynamics that include, but are not limited to, lee waves, internal bores, baroclinic instabilities, boundary currents, large-scale gyres, blocking, and geostrophic eddies. Despite the complicated nature of the circulations, several bulk properties and features of the system are able to be characterised globally by relatively simple parameters. We find that the most useful parameter for describing the flows is the internal Froude number ( F r n ), which is the ratio between the imposed ridge velocity and the internal wave phase speed in the stratified fluid. Standing features that are predominantly barotropic and stationary relative to the ridge are only able to occur when the imposed ridge velocity is less than the internal wave phase speed ( | F r n | < 1 ). This implies that the flow-topography interactions leading to stationary dynamics are primarily internal stratified processes, which steer the flow and shape the background geostrophic turbulence such that it can support standing barotropic features.

In this study, we develop a Smoothed Particle Hydrodynamics (SPH) 2D-model for simulating fully submerged granular flows and their arising water waves. The granular particles are characterised by a non-Newtonian flow pattern, following a Casson constitutive law, generalised by applying the infinitesimal strain theory to avoid numerical singularities inherited from the original law. The implementation of this rheological model on the weakly compressible viscous Navier-Stokes equations enables the simultaneous modelling of the motion of granular flows and their resulting water waves, establishing a monolithic representation of fluid-structure coupling. The novelty of this model lies in the numerical continuity of the generalised rheological model based mainly on the yield stress criterion, which is computed purely from the mechanical properties of granular materials, including internal friction, cohesion, and viscosity coefficients. The proposed SPH model is validated through two benchmarks available in the literature, representing a submarine landslide along an inclined plane and an immersed granular column collapse. The outcomes of our study illustrate the effectiveness of the proposed model in accurately predicting the motions of submerged granular masses and their resulting water waves, which is crucial for accurately predicting the behaviour of underwater landslides and other natural hazards.

The full Coriolis force consists of terms proportional to the sine and cosine of latitude. The latter, referred to as the non-traditional Coriolis terms, couple the zonal and vertical momentum equations, and are often neglected. When considering the incompressible Euler equations at the equator in the presence of the full Coriolis force, it can be shown that all dynamical fields can be diagnosed from the velocity potential, which itself is the solution to an elliptic equation that depends on latitude, momentum drag, and radiative damping. In this work, we present solutions to these equations for several different types of momentum drag and radiative damping. An important insight of our work is the combined effect that both rotation and radiation have on providing a mechanism for ascent and descent of air away from a localised source of heating.

A classic stability problem relevant to many applications in geophysical and astrophysical fluid mechanics is that of Kolmogorov flow, a unidirectional purely sinusoidal velocity field written here as u = ( 0 , sin ⁡ x ) in the infinite ( x , y ) -plane. Near onset, instabilities take the form of large-scale transverse flows, in other words flows in the x-direction with a small wavenumber k in the y-direction. This is similar to the phenomenon known as zonostrophic instability, found in many examples of randomly forced fluid flows modelling geophysical and planetary systems. The present paper studies the effect of incorporating a magnetic field B 0 , in particular a y-directed “vertical” field or an x-directed “horizontal” field. The linear stability problem is truncated to determining the eigenvalues of finite matrices numerically, allowing exploration of the instability growth rate p as a function of the wavenumber k in the y-direction and a Bloch wavenumber ℓ in the x-direction, with − 1 / 2 < ℓ ≤ 1 / 2 . In parallel, asymptotic approximations are developed, valid in the limits k → 0 , ℓ → 0 , using matrix eigenvalue perturbation theory. Results are presented showing the robust suppression of the hydrodynamic Kolmogorov flow instability as the imposed magnetic field B 0 is increased from zero. However with increasing B 0 , further branches of instability become evident. For vertical field there is a strong-field branch of destabilised Alfvén waves present when the magnetic Prandtl number P m < 1 , as found recently by A.E. Fraser, I.G. Cresswell and P. Garaud (J. Fluid Mech. 949, A43, 2022), and a further branch for 1 $ ]]> P m > 1 in the presence of an additional imposed x-directed fluid flow U 0 . For horizontal magnetic field, a branch of field-driven, tearing mode instabilities emerges as B 0 increases. The above instabilities are present for Bloch wavenumber ℓ = 0 ; however allowing ℓ to be non-zero gives rise to a further branch of instabilities in the case of horizontal field. In some circumstances, even when the system is hydrodynamically stable arbitrarily weak magnetic fields can give growing modes, via the instability taking place on large scales in x and y. Detailed comparisons are given between theory for small k and ℓ, and numerical results.

Rotating convective turbulence is ubiquitously found across geophysical settings, such as surface and subsurface oceans, planetary atmospheres, molten metal planetary cores, magma chambers, magma oceans, and basal magma oceans. Depending on the thermal and material properties of the system, buoyant convection can be driven thermally or compositionally, where a Prandtl number ( Pr = ν / κ i ) defines the characteristic diffusion properties of the system, with κ i = κ T representing thermal diffusion and κ i = κ C representing chemical diffusion. These numbers vary widely for geophysical systems; for example, the liquid iron undergoing thermal-compositional convection in Earth's core is defined by P r T ≈ 0.1 and P r C ≈ 100 , while a thermally-driven liquid silicate magma ocean is defined by P r T ≈ 100 . Currently, most numerical and laboratory data for rotating convective turbulent flows exists at Pr = O ( 1 ) ; high Pr rotating convection relevant to compositionally-driven core flow and other systems is less commonly studied. Here, we address this deficit by carrying out a broad suite of rotating convection experiments made over a range of Pr values, employing water and three different silicone oils as our working fluids ( Pr = 6, 41, 206, and 993). Using measurements of flow velocities (Reynolds, Re) and heat transfer efficiency (Nusselt, Nu), a baroclinic torque balance is found to describe the turbulence regardless of Prandtl number so long as Re is sufficiently large ( Re ≳ 10 ). Estimated turbulent scales are found to remain close to onset scales in all experiments, a result that may extrapolate to planetary settings. Lastly, we use our data to build Pr-dependent predictive nondimensional and dimensional scaling relations for rotating convective velocities that can be applied across a broad range of geophysical fluid dynamical settings.

The temporal spectrum of the solar activity is more than just the main cycle. It contains different timescales, which can be considered as continuous components of the activity spectrum. The possibility of finding a realistic spectrum of the solar magnetic activity variation is analysed for several versions of a simple model for solar activity based on the original idea of E. Parker. In particular, we study the original set of partial differential equations with two versions of suppression of the dynamo action and the fourth-order dynamical system obtained by truncating the Parker equations. We show that the effects included in the models, i.e. the nonlinear dynamo suppression and the dynamical chaos, as well as random fluctuations of the dynamo drivers, are quite sufficient to obtain the main solar cycle and the continuous components of the spectrum similar to the observed ones. However, the capabilities of the approach under consideration substantially vary from one model to another. Each model reproduces a continuous component of the spectrum in a specific parameter range. This study has confirmed the view that the examination of various solar dynamo models with the aim to find a reasonable combination of main activity cycle and continuous spectrum of solar activity can be used as an additional test of their validity.

An analytical solution is presented to study a plane solitary wave propagating past a concentric segmented arc-shaped breakwater using the matched eigenfunction and separation of variable methods. The undetermined potential coefficients are obtained based on matching conditions. The numerical results of this study are found to agree well with previous calculation results. The major factors (including the number of arc-shaped breakwaters, opening angle, incident angle, and gap width) that affecting the hydrodynamic loads and diffracted wave surface are discussed. The results indicate that the shielding effect of a segmented two-arc breakwater with a small gap width is better than that of an unsegmented arc-shaped breakwater. However, the arrangement of the segmented arc impacts the sheltering effect. Numerical results provide a valuable reference for the hydrodynamic analyses and structural design of segmented arc-shaped breakwaters.

Here we consider a model of an isolated vortex to understand the vertical dynamics induced by mesoscale eddies in the ocean. We use the analytical solutions to a balanced model for an ellipsoid of uniform potential vorticity to examine how the vertical motions induced depend on the vortex shape and its orientation, i.e. whether the vortex is vertically upright or tilted with respect to the vertical axis. The motion induced by the vortex can be divided into two kinds: (1) the interior flow which acts on the vortex itself and (2) the exterior flow which acts on its surroundings. For an upright ellipsoid, there are no self-induced vertical motions and the vortex rotates steadily about the vertical axis. However, for a tilted ellipsoid we find solutions exist where the vortex rotates about the vertical axis, while the vertical motions cause the tilt angle of the vortex to oscillate. This effect is stronger as the tilt angle is increased. Considering the exterior flow, there exists an exterior vertical velocity for the upright and tilted ellipsoids. However, the dynamics induced by the exterior vertical velocity is very different for the upright and tilted cases. We find that for an upright ellipsoidal vortex, the vertical motions are largest for vortices with high horizontal eccentricity and a vertical height-to-width aspect ratio near unity, vanishing as the horizontal cross-section of the vortex becomes circular. Instead for the tilted case, the vertical motions are largest when the horizontal cross section is circular, and for strongly prolate vortices, with the largest vertical motions occurring when the tilt angle is .

We consider a three-vortex interaction which leads to the vertical alignment of two like-signed quasi-geostrophic vortices in a continuously stratified, rotating fluid. The interaction is close to the classical collapse interaction of three co-planar vortices except that the vortices centres move on close but different horizontal planes. The vertical alignment of vortices helps create larger structures and contributes in physical space to the inverse energy cascade observed in spectral space in geostrophic turbulence.