Rossby-Haurwitz Wave Research Articles

Conjugate points along spherical harmonics

Utilizing structure constants, we present a version of the Misiolek criterion for identifying conjugate points. We propose an approach that enables us to locate these points along solutions of the quasi-geostrophic equations on the sphere S2. We demonstrate that for any spherical harmonics Ylm with 1≤|m|≤l, except for Y1±1 and Y2±1, conjugate points can be determined along the solution generated by the velocity field elm=∇⊥Ylm. Subsequently, we investigate the impact of the Coriolis force on the occurrence of conjugate points. Moreover, for any zonal flow generated by the velocity field ∇⊥Yl10, we demonstrate that proper rotation rate can lead to the appearance of conjugate points along the corresponding solution, where l1=2k+1.∈N Additionally, we prove the existence of conjugate points along (complex) Rossby-Haurwitz waves and explore the effect of the Coriolis force on their stability.

Spline Model: A Hydrostatic/Non-Hydrostatic Dynamic Core with Space-Time Second-Order Precision and Its Exact Tests

We present a new explicit quasi-Lagrangian integration scheme with the three-dimensional cubic spline function transform (transform = fitting + interpolation, referred to as the “spline format”) on a spherical quasi-uniform longitude–latitude grid. It is a consistent longitude–latitude grid, and to verify the feasibility, accuracy, convergence, and stability of the spline format interpolation scheme for the upstream point on the longitude–latitude grid, which may map a quasi-uniform longitude–latitude grid, a set of ideal, exact test schemes is adopted, which are recognized and proven to be effective internationally. The equilibrium flow test, cross-polar flow test, and Rossby–Haurwitz wave test are used to illustrate the spline scheme uniformity to the linear scheme and to overcome the over-dense grid in the polar region and the non-singularity of the poles. The cross-polar flow test demonstrates that the geostrophic wind crosses the polar area correctly, including the South Pole and North Pole. A non-hydrostatic, fully compressible dynamic core is used to complete the density flow test, demonstrating the existence of a time-varying reference atmosphere and that the spline format can simulate highly nonlinear fine-scale transient flows. It can be compared for the two results of the density flow test between the solution with the spline format and the benchmark reference solution with the linear format. Based on the findings, the non-hydrostatic dynamic core with the spline format is recommended for adoption. When simulated for the flow over an ideal mountain, through the “topographic gravity wave test”, the bicubic surface terrain and terrain-following height coordinates, time-split integration, and vector discrete decomposition can be derived successfully. These may serve as the foundations for a global, unified spline-format numerical model in the future.

Hydrodynamics of quantum vortices on a closed surface

We develop a neutral vortex fluid theory on closed surfaces with zero genus. The theory describes collective dynamics of many well-separated quantum vortices in a superfluid confined on a closed surface. Comparing to the case on a plane, the covariant vortex fluid equation on a curved surface contains an additional term proportional to Gaussian curvature multiplying the circulation quantum. This term describes the coupling between topological defects and curvature in the macroscopic level. For a sphere, the simplest nontrivial stationary vortex flow is obtained analytically and this flow is analogous to the celebrated zonal Rossby-Haurwitz wave in classical fluids on a nonrotating sphere. For this flow the difference between the coarse-grained vortex velocity field and the fluid velocity field generated by vortices is solely driven by curvature and vanishes in the corresponding vortex flow on a plane when the radius of the sphere goes to infinity. Published by the American Physical Society 2024

Rossby–Haurwitz Wave in a Rotating Bubble-Shaped Bose–Einstein Condensate

A Rossby–Haurwitz (RH) wave is an excitation mode of a fluid on a rotating spherical surface, which propagates westward in the rotating frame of reference. Motivated by the recent realization of the shell-shaped Bose–Einstein condensate in microgravity, we investigate the RH wave in a superfluid rotating on a spherical-surface geometry. We employ the point-vortex model and the three-dimensional Gross–Pitaevskii equation, and numerically demonstrate that RH waves can be observed in the system of a superfluid with quantized vortices.

Mixed-Precision for Linear Solvers in Global Geophysical Flows.

Semi‐implicit (SI) time‐stepping schemes for atmosphere and ocean models require elliptic solvers that work efficiently on modern supercomputers. This paper reports our study of the potential computational savings when using mixed precision arithmetic in the elliptic solvers. Precision levels as low as half (16 bits) are used and a detailed evaluation of the impact of reduced precision on the solver convergence and the solution quality is performed. This study is conducted in the context of a novel SI shallow‐water model on the sphere, purposely designed to mimic numerical intricacies of modern all‐scale weather and climate (W&C) models. The governing algorithm of the shallow‐water model is based on the non‐oscillatory MPDATA methods for geophysical flows, whereas the resulting elliptic problem employs a strongly preconditioned non‐symmetric Krylov‐subspace Generalized Conjugated‐Residual (GCR) solver, proven in advanced atmospheric applications. The classical longitude/latitude grid is deliberately chosen to retain the stiffness of global W&C models. The analysis of the precision reduction is done on a software level, using an emulator, whereas the performance is measured on actual reduced precision hardware. The reduced‐precision experiments are conducted for established dynamical‐core test‐cases, like the Rossby‐Haurwitz wavenumber 4 and a zonal orographic flow. The study shows that selected key components of the elliptic solver, most prominently the preconditioning and the application of the linear operator, can be performed at the level of half precision. For these components, the use of half precision is found to yield a speed‐up of a factor 4 compared to double precision for a wide range of problem sizes.

Precession resonance of Rossby wave triads and the generation of low-frequency atmospheric oscillations

The dynamics of the Earth's atmosphere is characterized by a wide spectrum of oscillations, ranging from hourly to interdecadal and beyond. The low-frequency component of the atmospheric variability cannot be understood solely in terms of linear atmospheric waves that have shorter timescales. A newly proposed mechanism, the precession resonance mechanism, is a regime of highly efficient energy transfer in the spectral space in turbulent systems. Here, we investigate the role of the precession resonance, and the alignment of dynamical phases, in the generation of low-frequency oscillations and the redistribution of energy/enstrophy in the spectral space using the barotropic vorticity equation. First, the mechanism and its ability to generate low-frequency oscillations are demonstrated in low-order models consisting of four and five nonlinearly interacting Rossby–Haurwitz waves. The precession resonance onset is also investigated in the full barotropic vorticity equation, and the results are in agreement with the reduced models. Efficiency peaks in the energy/enstrophy transfer also correspond to regimes of strong excitation of low-frequency oscillations. The results suggest that the organization of the dynamical phases plays a key role in the redistribution of energy in the spectral space, as well as the generation of low frequencies in the barotropic vorticity equation.

Stratospheric Planetary Flows from the Perspective of the Euler Equation on a Rotating Sphere

This article is devoted to stationary solutions of Euler’s equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the Earth. For the Euler equation, under appropriate conditions, rigidity results are established, ensuring that the solutions are either zonal or rotated zonal solutions. A natural analogue of Arnold’s stability criterion is proved. In both cases, the lowest mode Rossby–Haurwitz stationary solutions (more precisely, those whose stream functions belong to the sum of the first two eigenspaces of the Laplace-Beltrami operator) appear as limiting cases. We study the stability properties of these critical stationary solutions. Results on the local and global bifurcation of non-zonal stationary solutions from classical Rossby–Haurwitz waves are also obtained. Finally, we show that stationary solutions of the Euler equation on a rotating sphere are building blocks for travelling-wave solutions of the 3D system that describes the leading order dynamics of stratospheric planetary flows, capturing the characteristic decrease of density and increase of temperature with height in this region of the atmosphere.

Conjugate points in [formula omitted

Rossby-Haurwitz waves on the sphere S2 form a set of exact time-dependent solutions to the Euler equations of hydrodynamics and generate a family of non-stationary geodesics of the L2 metric in the volume preserving diffeomorphism group of S2. Restricting to a particular subset of Rossby-Haurwitz waves, this article shows that under certain conditions on the physical characteristics of the waves each corresponding geodesic contains conjugate points. In addition, a physical interpretation of conjugate points is given and links the result to the stability analysis of meteorological Rossby-Haurwitz waves.

Topography-induced locking of drifting Rossby–Haurwitz waves

Atmospheric blockings are persistent large-scale climate patterns that lock in a geographic position for days and even weeks. In principle, blockings might involve a large number of waves interacting non-linearly, and a conclusive description for their onset and duration is still elusive. In this paper, we introduce a simplified account for this phenomenon by means of a single-triad of Rossby–Haurwitz waves perturbed by one topographic mode. It is shown that the dynamical features of persistent flow patterns have zero measure in the phase space for an unperturbed triad, but such a measure becomes finite for the perturbed dynamics. By this account, we suggest that static inhomogeneities in the two-dimensional atmospheric layer are required for locking the flow patterns to the planetary rotation.

Tides in subsurface oceans with meridional varying thickness

Tidal heating can play an important role in the formation and evolution of subsurface oceans of outer-planet moons. Up until now tidal heating has only been studied in subsurface oceans of spatially uniform thickness. We develop a numerical model to consider oceans of spatially variable thickness. We use the Laplace Tidal Equations for the ocean and model the ice shell using membrane theory. The problem is solved using the commercial Finite Element software Comsol Multiphysics®. We use this new model to study the tidal response of Enceladus' ocean with a twofold objective: to understand how ocean thickness variations modify the tidal response of a subsurface ocean and to assess if tidal dissipation in an Enceladan ocean with varying ocean thickness can explain the high heat flux emanating from Enceladus' South Polar Terrain and the perdurance of a subsurface ocean. We consider the effect of meridional ocean thickness changes of spherical harmonic degree two and three as suggested by topography and gravimetry data. We observe that an ocean with degree two topography responds with the same eigenmodes as an ocean of constant thickness but resonances occur for thicker oceans. However, resonant ocean thicknesses are still thin compared to current estimates for Enceladus ocean thickness. Rossby-Haurwitz waves, excited by the obliquity tide for thick oceans of constant thickness, are not excited at the tidal frequency when oceans of variable thickness are considered. This result implies that the role of the obliquity tide in ocean tidal-dissipation might have been overestimated for Enceladus and other icy worlds. An antisymmetric, degree-three ocean thickness variation mixes the ocean modes excited in a constant thickness ocean by the eccentricity and obliquity tide.

Stability of Rossby–Haurwitz waves

AbstractExploiting a factorized form of the linear perturbation equation valid around Rossby–Haurwitz waves in the nonlinear barotropic vorticity equation system on the sphere, an elementary discrete numerical method for analysing the stability of these waves is proposed. In contrast with former approaches based on wave‐interaction theory and its cumbersome triad‐interaction coefficients, the discrete analysis proposed here is inspired by the spectral‐transform technique, where nonlinearities are treated in physical space. This approach, based only on standard linear algebra techniques, is simpler than former ones and allows a large number of degrees of freedom to be retained in perturbation space. The merits and possible limitations of the method are discussed; some validations and results are presented.

How I Learned to Love Normal-Mode Rossby–Haurwitz Waves

AbstractSeveral important developments in the 1960s showed the way to using spectral analysis to identify and describe atmospheric waves predicted by theory. Among these waves were normal-mode Rossby–Haurwitz waves (NMRHWs). What follows is, first, a brief outline of how the influence of these developments on analyses of data collected during the Line Islands Experiment led to work on NMRHWs. Next, theoretical expectations of free NMRHWs as described by Kasahara and Kasahara and Puri in the early 1980s are discussed. Finally, spectral analyses of observed vorticity fields are presented for easy comparison with those expectations. The similarity between these relatively simple model predictions and observations is unique in meteorology, where complexity is the general rule. Readily available routines coded in NCAR Command Language (NCL) were used to isolate NMRHWs. It should be noted that, while these routines provide approximations to the theoretical predictions, open-access software for exact solutions has become available.

Three-wave resonant interactions and zonal flows in two-dimensional Rossby-Haurwitz wave turbulence on a rotating sphere

This paper addresses three-wave interactions of Rossby-Haurwitz waves in two-dimensional turbulence on a rotating sphere. Zonal modes are often omitted from the resonant wave even when they satisfy the conditions for three-wave interactions, as they do not transfer any energy to other modes in a manner. However, the presence of zonal flows induces phase shifts in other modes, and it is not at all clear that their influence is negligible. Since it is expected that three-wave interactions govern the entire dynamics of turbulence if the rotation rate of the sphere is sufficiently high, by analogy with the theorem regarding three-wave interactions of Rossby waves on a beta plane with sufficiently large beta previously proven by Yamada and Yoneda [Physica D 245, 1 (2013)], an appropriate definition of the wave set was determined by comparing the time evolution of several wave sets on a rapidly rotating sphere. It was found that zonal waves of the form Y-l(m=0) exp(i omega t) with odd l, where Y(l)(m )are the spherical harmonics, should be considered for inclusion in the wave set to ensure that the dynamics of the wave set determine the overall dynamics of the turbulence on a rapidly rotating sphere. Consequently, it is suggested that the minimal wave set that must be considered in the discussion of the three-wave interaction of Rossby-Haurwitz waves is the set consisting of nonzonal waves and zonal waves of the form Y-l(0) exp(icot) with odd l.

Elementary properties of non-Linear Rossby-Haurwitz planetary waves revisited in terms of the underlying spherical symmetry

We revisit Rossby-Haurwitz planetary wave modes of a two-dimensional fluid along the surface of a rotating planet, as elements of irreducible representations of the so(3) Lie algebra. Key questions addressed are, firstly, why it is that the non-linear self-interaction of any Rossby-Haurwitz wave mode is zero, and secondly, why the phase velocity of these wave modes is insensitive to their orientation with respect to the axis of rotation of the planet, while at the same time the very rotation of the planet is a precondition for the existence of the waves. As we show, answers to both questions can be rooted in Lie group and representation theory. In our study the Rossby-Haurwitz modes emerge in a coordinate-free, as well as in a Ricci tensor rank-free manner. We find them with respect to a continuum of spherical coordinate systems, that are arbitrarily oriented with respect to the planet. Furthermore, we show that, in the same sense in which the Lie derivative of Ricci tensor fields is rankfree, the wave equation for Rossby-Haurwitz modes is rank-free. We find that, for each irreducible representation of so(3), there is a corresponding sufficient condition for existence of Rossby-Haurwitz modes as solutions that are separable with respect to space and time. This condition comes in the form of a system of equations of motion for the coordinate systems. Coordinate systems that move along with Rossby-Haurwitz modes emerge as special cases of these. In these coordinate systems the waves appear as stationary spatial fields, so that the motion of the coordinate system coincides with the wave phase propagation. The general solution of the existence condition is a continuum of moving spherical coordinate systems that precess about the axes of the Rossby-Haurwitz modes. Within a single irreducible representation of so(3), the waves are dispersionless.

Ocean tidal heating in icy satellites with solid shells

As a long-term energy source, tidal heating in subsurface oceans of icy satellites can influence their thermal, rotational, and orbital evolution, and the sustainability of oceans. We present a new theoretical treatment for tidal heating in thin subsurface oceans with overlying incompressible elastic shells of arbitrary thickness. The stabilizing effect of an overlying shell damps ocean tides, reducing tidal heating. This effect is more pronounced on Enceladus than on Europa because the effective rigidity on a small body like Enceladus is larger. For the range of likely shell and ocean thicknesses of Enceladus and Europa, the thin shell approximation of Beuthe (2016) is generally accurate to less than about 4%. Explaining Enceladus’ endogenic power radiated from the south polar terrain by ocean tidal heating requires ocean and shell thicknesses that are significantly smaller than the values inferred from gravity and topography constraints. The time-averaged surface distribution of ocean tidal heating is distinct from that due to dissipation in the solid shell, with higher dissipation near the equator and poles for eccentricity and obliquity forcing, respectively. This can lead to unique horizontal shell thickness variations if the shell is conductive. The surface displacement driven by eccentricity and obliquity forcing can have a phase lag relative to the forcing tidal potential due to the delayed ocean response. For Europa and Enceladus, eccentricity forcing generally produces greater tidal amplitudes due to the large eccentricity values relative to the obliquity values. Despite the small obliquity values, obliquity forcing generally produces larger phase lags due to the generation of Rossby–Haurwitz waves. If Europa’s shell and ocean are, respectively, 10 and 100 km thick, the tide amplitude and phase lag are 26.5 m and <1° for eccentricity forcing, and <2.5 m and <18° for obliquity forcing. Measurement of the obliquity phase lag (e.g. by Europa Clipper) would provide a probe of ocean thickness

On Liapunov and Exponential Stability of Rossby–Haurwitz Waves in Invariant Sets of Perturbations

In this work, the stability of the Rossby–Haurwitz (RH) waves from the subspace \(\mathbf {H}_{1}\oplus \mathbf {H}_{n}\) is considered (\(n\ge 2\)) where \(\mathbf {H}_{k}\) is the subspace of the homogeneous spherical polynomials of degree k. A conservation law for arbitrary perturbations of the RH wave is derived, and all perturbations are divided into three invariant sets \(\mathbf {M}_{-}^{n}\), \(\mathbf {M}_{0}^{n}\) and \(\mathbf {M} _{+}^{n}\) in which the mean spectral number \(\chi (\psi ^{\prime })\) of any perturbation \(\psi ^{\prime }\) is less than, equal to or greater than \( n(n+1) \), respectively. In turn, the set \(\mathbf {M}_{0}^{n}\) is divided into the invariant subsets \(\mathbf {H}_{n}\) and \(\mathbf {M}_{0}^{n}{\setminus } \mathbf {H}_{n}\). Quotient spaces and norms of the perturbations are introduced, a hyperbolic law for the perturbations belonging to the sets \(\mathbf {M}_{-}^{n}\) and \(\mathbf {M}_{+}^{n}\) is derived, and a geometric interpretation of variations in the kinetic energy of perturbations is given. It is proved that any non-zonal RH wave from \(\mathbf {H}_{1}\oplus \mathbf {H} _{n}\) (\(n\ge 2\)) is Liapunov unstable in the invariant set \(\mathbf {M} _{-}^{n}\). Also, it is shown that a stationary RH wave from \(\mathbf {H} _{1}\oplus \mathbf {H}_{n}\) may be exponentially unstable only in the invariant set \(\mathbf {M}_{0}^{n}{\setminus } \mathbf {H}_{n}\), while any perturbation of the invariant set \(\mathbf {H}_{n}\) conserves its form with time and hence is neutral. Since a Legendre polynomial flow \(aP_{n}(\mu )\) and zonal RH wave \(-\,\omega \mu +aP_{n}(\mu )\) are particular cases of the RH waves of \(\mathbf {H} _{1}\oplus \mathbf {H}_{n}\), the major part of the stability results obtained here is also true for them.

‘The Equator–Pole grid system’: an overset grid system for the sphere, with an optimal uniformity property

This article proposes a new grid system for the sphere, which consists of three orthogonal and almost uniform grids. The basic one is a latitude-longitude grid covering an annular band around the equator. For the rest of the sphere this grid is complemented by two grids covering the polar regions, and based on suitably modified stereographic coordinates. The rectangular structure of the grids makes highly efficient implementation on massively parallel computer systems possible. Numerical experiments with the new grid system are carried out for two advection examples, namely smooth deformational flow and rotation of the Cosine bell, and for the test problems 2, 3, and 6 from Williamson et al., concerning the non-linear shallow water equations. For problem 6, the Rossby–Haurwitz wave, we study conservation properties for mass and energy. The computational results compare favourably with results for other grids. Our focus is the definition of the grid system together with the connection between grids by overlapping and the use of centred interpolation formulas. The method of centred finite differences is used for the spatial discretisation of the differential equations, because it is well-known and easy to implement.

Lie group method for solving viscous barotropic vorticity equation in ocean climate models

In studying the problem of the nonlinear viscous barotropic non-divergent vorticity equation on f- and β- planes, the method of Lie group has been applied. The method reduces the number of independent variables by one, and consequently, for the case of three independent variables we applied the method successively twice and the nonlinear partial differential equation reduces to ordinary differential equation. Investigation of exact solutions of the viscous barotropic non-divergent vorticity equation on f- and β- planes, via the application of Lie group, provides large classes of new exact solutions which include both Rossby and Rossby–Haurwitz waves as special cases. Also, The Lie symmetries of the viscous barotropic non-divergent vorticity equation with two parameters F and β, are determined. The possible reductions of the viscous barotropic vorticity equation with two parameters F and β have been investigated by means of one- dimensional Lie subalgebras.

Shallow‐water equations on a spherical multiple‐cell grid

The shallow‐water equations (SWEs) are discretized on a spherical multiple‐cell (SMC) grid and validated with classical tests, including steady zonal flow over flat or hill floor, the Rossby–Haurwitz wave and the unstable zonal jet. The numerical schemes follow the conventional latitude–longitude (lat‐lon) grid ones at low latitudes but switch to a fixed‐reference direction system to define vector variables at high latitudes for reduction of polar curvature errors. Semi‐implicit schemes are used for both the Coriolis terms and the potential energy gradients. A C‐grid mass‐conserving advection–diffusion scheme is used for the thickness variable and mixed A‐ and D‐grid schemes are applied on the momentum equation. Tests demonstrate that the two reference directions work fine on the SMC grid and the SWEs model is stable as long as numerical noises are suppressed with enough smoothing. This implies that other reduced grids could be used for dynamical models if the vector polar problem is properly solved. The unstructured SMC grid also supports multi‐resolutions in mesh refinement style and a refined area is included to show grid refinement effects. For steady smooth flows, the refinement is almost unnoticeable while in unstable jet case, it may stir up instability ripples in the jet flow unless strong average is applied.

Mechanisms Linking Global 5-Day Waves to Tropical Convection

Abstract Reanalysis data and satellite-derived rainfall measurements are examined to determine possible mechanisms linking the “5 day” Rossby–Haurwitz wave to localized variations of tropical convection. The mechanisms in all regions rely on the modulation of zonal winds near the equator by the wave, but the nature of these mechanisms depends strongly on local topography and local climate. In the upper Amazon basin, the wave modulates the strength of prevailing easterlies and thus the upslope flow and associated convection on the eastern edge of the Andes. Similar modulation of upslope flow is involved off the Panamanian and Colombian Pacific coasts, but the deflection and confluence of low-level wind in the presence of the Andes and moisture transports across the Andes from the Amazon basin are also factors. Similar deflection and confluence of winds around and through the Maritime Continent lead to low-level divergence and convection anomalies over the eastern Indian Ocean. Anomalous moisture transports from the Congo basin to the eastern and northeastern Gulf of Guinea due to the wave affect atmospheric moisture over the Gulf of Guinea and thus convection in the region. Over oceanic convergence zones, modulations of the prevailing winds by the wave affect the overall wind magnitude, changing evaporation from the ocean surface and atmospheric moisture. Most of these mechanisms arise from the nonuniform nature of Earth’s surface and suggest that other external Rossby–Haurwitz waves may have similar interactions with convection.