Geostrophic Modes Research Articles

The Elbert range of magnetostrophic convection. I. Linear theory

In magnetostrophic rotating magnetoconvection, a fluid layer heated from below and cooled from above is equidominantly influenced by the Lorentz and the Coriolis forces. Strong rotation and magnetism each act separately to suppress thermal convective instability. However, when they act in concert and are near in strength, convective onset occurs at less extreme Rayleigh numbers (, thermal forcing) in the form of a stationary, large-scale, inertia-less, inviscid magnetostrophic mode. Estimates suggest that planetary interiors are in magnetostrophic balance, fostering the idea that magnetostrophic flow optimizes dynamo generation. However, it is unclear if such a mono-modal theory is realistic in turbulent geophysical settings. Donna Elbert first discovered that there is a range of Ekman (, rotation) and Chandrasekhar (, magnetism) numbers, in which stationary large-scale magnetostrophic and small-scale geostrophic modes coexist. We extend her work by differentiating five regimes of linear stationary rotating magnetoconvection and by deriving asymptotic solutions for the critical wavenumbers and Rayleigh numbers. Coexistence is permitted if and . The most geophysically relevant regime, the Elbert range, is bounded by the Elsasser numbers . Laboratory and Earth’s core predictions both exhibit stationary, oscillatory, and wall-attached multi-modality within the Elbert range.

Gravity Wave Generation in Balanced Sheared Flow Revisited

Abstract The generation of internal gravity waves from an initially geostrophically balanced flow is diagnosed in nonhydrostatic numerical simulations of shear instabilities for varied dynamical regimes. A nonlinear decomposition method up to third order in the Rossby number (Ro) is used as the diagnostic tool for a consistent separation of the balanced and unbalanced motions in the presence of their nonlinear coupling. Wave emission is investigated in an Eady-like and a jet-like flow. For the jet-like case, geostrophic and ageostrophic unstable modes are used to initialize the flow in different simulations. Gravity wave emission is in general very weak over a range of values for Ro. At sufficiently high Ro, however, when the condition for symmetric instability is satisfied with negative values of local potential vorticity, significant wave emission is detected even at the lowest order. This is related to the occurrence of fast ageostrophic instability modes, generating a wide spectrum of waves. Thus, gravity waves are excited from the instability of the balanced mode to lowest order only if the condition of symmetric instability is satisfied and ageostrophic unstable modes obtain finite growth rates. Significance Statement We aim to understand the generation of internal gravity waves in the atmosphere and ocean from a flow field that is initially balanced, i.e., free from any internal gravity waves. To examine this process, we use simulations from idealized numerical models and nonlinear flow decomposition method to identify waves. Our results show that a prominent mechanism by which waves can be generated is related to symmetric or ageostrophic instabilities of the balanced flow possibly occurring during frontogenesis. This process can be a significant mechanism to dissipate the energy of the geostrophic flow in the ocean.

Flow structures, nonlinear inertial waves and energy transfer in rotating spheres

We investigate flow structures, nonlinear inertial waves and energy transfer in a rotating fluid sphere, using a Galerkin spectral method based on helical-wave decomposition (HWD). Numerical simulations of flows in a sphere are performed with different system rotation rates, where a large-scale forcing is employed. For the case without system rotation, the intense vortex structures are tube-like. When a weak rotation is introduced, small-scale structures are reduced and vortex tubes tend to align with the rotation axis. As the rotation rate increases, a large-scale anticyclonic vortex structure is formed near the rotation axis. The structure is shown to be led by certain geostrophic modes. When the rotation rate further increases, a cyclone and an anticyclone emerge from the top and bottom of the boundary, respectively, where two quasi-geostrophic equatorially symmetric inertial waves dominate the flow. Based on HWD, effects of spherical confinement on rotating turbulence are systematically studied. It is found that the forward cascade becomes weaker as the rotation increases. When the rotation rate becomes larger than some critical value, dual energy cascades emerge, with an inverse cascade at large scales and a forward cascade at small scales. Finally, the flow behavior near the boundary is studied, where the average boundary layer thickness gets smaller when system rotation increases. The flow behavior in the boundary layer is closely related to the interior flow structures, which create significant mass flux between the boundary layer and the interior fluid through Ekman pumping.

Near-resonant instability of geostrophic modes: beyond Greenspan's theorem

Abstract

Dispersion analysis of compatible Galerkin schemes on quadrilaterals for shallow water models

Compatible Galerkin methods (the Galerkin analogue of an Arakawa C-Grid) are growing in popularity for simulating geophysical fluid flows, due to their desirable characteristics, including but not limited to: energy conservation, higher-order accuracy, steady geostrophic modes and the absence of spurious stationary modes, such as pressure modes. However, these characteristics still do not guarantee good wave dispersion properties. In this work, we study the dispersion properties of two compatible Galerkin families for the 2D linear rotating shallow water equations on quadrilaterals: the Qn−Λk family from finite element exterior calculus and the newly developed MGDn family. These families are the extensions to quadrilaterals of the PnC−Pn−1DG and GDn−DGDn−1 pairs, respectively, studied for the 1D linear shallow water equations in [13]. A major finding from that paper was that the PnC−Pn−1DG pair has spectral gaps for n≥2 and the GDn−DGDn−1 does not. These spectral gaps are non-dimensional wavenumbers where the dispersion relationship is double-valued, and lead to anomalous dispersion and noise in numerical simulations. On quadrilaterals, previous work [24,26] on the Qn−Λk family for inertia waves with n=2 and for gravity waves for arbitrary n has indicated the presence of spectral gaps, in the form of line discontinuities. The investigation of these gaps for the Qn−Λk family is extended in this paper to inertia-gravity waves for arbitrary n, including plots of the dispersion relationship for n=2 when using the lumping developed in [26,35] that eliminates the spectral gaps. Additionally, the MGDn family is studied (including the use of reduced quadrature), which is found to be free of spectral gaps. For both families asymptotic convergence rates are established, effective resolutions determined and plots of the dispersion relationships for a range of n and Rossby radii are shown. Finally, a pair of numerical simulations are run to investigate the consequences of the spectral gaps and highlight the main differences between the two families.

Dispersion analysis of compatible Galerkin schemes for the 1D shallow water model

In this work, we study the dispersion properties of two compatible Galerkin schemes for the 1D linearized shallow water equations: the PnC−Pn−1DG and the GDn−DGDn−1 element pairs. Pn is the order n Lagrange space, Pn−1DG is the order n−1 discontinuous Lagrange space, GDn is the order n Galerkin difference space, and DGDn−1 is the order n−1 discontinuous Galerkin difference space. Compatible Galerkin methods have many desirable properties, including energy conservation, steady geostrophic modes and the absence of spurious stationary modes, such as pressure modes. However, this does not guarantee good wave dispersion properties. Previous work on the P2C−P1DG pair has indeed indicated the presence of spectral gaps, and it is extended in this paper to the study of the PnC−Pn−1DG pair for arbitrary n. Additionally, an alternative element pair is introduced, the GDn−DGDn−1 pair, that is free of spectral gaps while benefiting from the desirable properties of compatible elements. Asymptotic convergence rates are established for both element pairs, including the use of inexact quadrature (which diagonalizes the velocity mass matrix) for the PnC−Pn−1DG pair and reduced quadrature for the GDn−DGDn−1 pair. Plots of the dispersion relationship and group velocities for a wide range of n and Rossby radii are shown. A brief investigation into the utility of mass lumping to remove the spectral gaps for the P3C−P2DG pair is performed. Finally, a pair of numerical simulations are run to investigate the consequences of the spectral gaps and highlight the main differences between the two elements.

Comparison of 2D triangular C-grid shallow water models

An ideal two-dimensional (2D) shallow water model should be able to simulate correctly various types of waves including pure gravity and inertia-gravity waves. In this paper, two different triangular C-grid methods are considered, and their dispersion of pure gravity waves, frequencies of inertia-gravity waves and geostrophic balance solutions are investigated. The proposed C-grid methods employ different spatial discretization schemes for coupling shallow water equations together with the various reconstruction techniques for tangential velocity estimation. The proposed reconstruction technique for the second method, which is analogous to a hexagonal C-grid scheme, is shown to be energy conservative and satisfies the geostrophic balance exactly while it supports the unphysical geostrophic modes for hexagonal C-grid. Because of the importance of the application of 2D shallow water models on fully unstructured grids, particular attention is also given to various types of isosceles triangles that may appear in such grids. For the gravity waves, the results of the phase speed ratio of the computed phase speeds over the analytical one are shown and compared. The non-dimensional frequencies of various modes for inertia-gravity waves are also investigated and compared in terms of being monotonic and isotropic respect to the continuous solution. The analyses demonstrate some advantages of the first method in phase speed behaviour for gravity waves and monotonicity of inertia-gravity dispersion. The results of the dispersion analysis are verified through a number of numerical tests. The first method, which is shown to have a better performance, examined through more numerical tests in presence of various source terms and results confirm its capability.

Tilted incompressible Coriolis modes in spheroids

The incompressible flow of a uniform fluid, which fills a rigid spheroid rotating about an arbitrary axis fixed in an inertial frame, is dominated at small Rossby and Ekman numbers by the rotation through the Coriolis force. The effects of rotation on the flow can be found by treating the Coriolis force modified by a pressure gradient as a skew-symmetric bounded linear operator $\boldsymbol{{\mathcal{C}}}$ acting on smooth inviscid incompressible flows in the spheroid. It is shown that the space of incompressible polynomial flows of degree $N$ or less in the spheroid is invariant under $\boldsymbol{{\mathcal{C}}}$ for any $N$. The skew symmetry of $\boldsymbol{{\mathcal{C}}}$ implies the Coriolis operator $\boldsymbol{{\mathcal{C}}}$ is non-defective for such flows with an orthogonal set of eigenmodes (inertial and geostrophic modes) which form a basis for the finite-dimensional space of spheroidal polynomial flows. The eigenmodes are tilted if the rotation axis is not aligned with the symmetry axis of the spheroid. The non-defective property of $\boldsymbol{{\mathcal{C}}}$ enables enumeration of the modes and proof of their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spheroidal polynomial flows under $\boldsymbol{{\mathcal{C}}}$ and completeness of the Coriolis modes, is that the solution of the polynomial Poisson–Neumann problem, i.e. Poisson’s equation with Neumann boundary condition and polynomial data, in a spheroid is a polynomial. The Coriolis modes of degree one and all geostrophic modes are explicitly constructed. Only the modes of degree one have non-zero angular momentum in the boundary frame.

Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid.

Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. When the fluid is incompressible, inviscid, and contained in a rigid container, these modes satisfy Poincaré's equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are, therefore, solutions of an ill-posed boundary-value problem. In this paper, we investigate the mathematical side of this problem. We first show that the Poincaré problem can be formulated in the Hilbert space of square-integrable functions, with no hypothesis on the continuity or the differentiability of velocity fields. We observe that with this formulation, the Poincaré operator is bounded and self-adjoint, and as such, its spectrum is the union of the point spectrum (the set of eigenvalues) and the continuous spectrum only. When the fluid volume is an ellipsoid, we show that the inertial modes form a complete base of polynomial velocity fields for the square-integrable velocity fields defined over the ellipsoid and meeting the boundary conditions. If the ellipsoid is axisymmetric, then the base can be identified with the set of Poincaré modes, first obtained by Bryan [Philos. Trans. R. Soc. London 180, 187 (1889)PTRMAD1364-503X10.1098/rsta.1889.0006], and completed with the geostrophic modes.

Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a tri-axial ellipsoid

Inertial waves often occur in geophysics and astrophysics since fluids dominated by rotation are common. A simple model to study inertial waves consists of a uniform incompressible fluid filling a rigid tri-axial ellipsoid, which rotates about an arbitrary axis fixed in an inertial frame. The waves are due to the Coriolis force, which can be treated mathematically as a skew-symmetric bounded linear operator acting on smooth inviscid flows in the ellipsoid. It is shown that the space of incompressible polynomial flows in the ellipsoid of degree N or less is invariant under . The symmetry of thus implies the Coriolis operator is non-defective with an orthogonal set of eigenmodes – Coriolis modes – in the finite-dimensional space of inviscid polynomial flows in the ellipsoid. The modes with non-zero eigenvalues are the inertial modes; the zero-eigenvalue modes are geostrophic. This shows the Coriolis modes are polynomials, enables their enumeration and leads to proof of their completeness by using the Weierstrass polynomial approximation theorem. The modes are tilted if the rotation axis is not aligned with a principal axis of the ellipsoid. A basic tool is that the solution of the polynomial Poisson-Neumann problem, i.e. Poisson’s equation with Neumann boundary condition and polynomial data, in an ellipsoid is a polynomial. The tilted Coriolis modes of degree one are explicitly constructed and shown to be the only modes with non-zero angular momentum in the boundary frame. All tilted geostrophic modes are also explicitly constructed.

The interaction between synoptic‐scale balanced flow and a finite‐amplitude mesoscale wave field throughout all atmospheric layers: weak and moderately strong stratification

The interaction between locally monochromatic finite‐amplitude mesoscale waves, their nonlinearly induced higher harmonics, and a synoptic‐scale flow is reconsidered, both in the tropospheric regime of weak stratification and in the stratospheric regime of moderately strong stratification. A review of the basic assumptions of quasi‐geostrophic theory on an f‐plane yields all synoptic scales in terms of a minimal number of natural variables, i.e. two out of the speed of sound, gravitational acceleration and Coriolis parameter. The wave scaling is defined so that all spatial and temporal scales are shorter by one order in the Rossby number, and by assuming their buoyancy field to be close to static instability. WKB theory is applied, with the Rossby number as scale separation parameter, combined with a systematic Rossby‐number expansion of all fields. Classic results for synoptic‐scale‐flow balances and inertia‐gravity‐wave (IGW) dynamics are recovered. These are supplemented by explicit expressions for the interaction between mesoscale geostrophic modes (GMs), a possibly somewhat overlooked agent of horizontal coupling in the atmosphere, and the synoptic‐scale flow. It is shown that IGW higher harmonics are slaved to the basic IGW, and that their amplitude is one order of magnitude smaller than the basic‐wave amplitude. GM higher harmonics are not that weak and they are in intense nonlinear interaction between themselves and the basic GM. Compressible dynamics plays a significant role in the stratospheric stratification regime, where anelastic theory would yield insufficient results. Supplementing classic derivations, it is moreover shown that, in the absence of mesoscale waves, quasi‐geostrophic theory holds also in the stratospheric stratification regime.

Accuracy analysis of mimetic finite volume operators on geodesic grids and a consistent alternative

Many newly developed climate, weather and ocean global models are based on quasi-uniform spherical polygonal grids, aiming for high resolution and better scalability. Thuburn et al. (2009) and Ringler et al. (2010) developed a C staggered finite volume/difference method for arbitrary polygonal spherical grids suitable for these next generation dynamical cores. This method has many desirable mimetic properties and became popular, being adopted in some recent models, in spite of being known to possess low order of accuracy. In this work, we show that, for the nonlinear shallow water equations on non-uniform grids, the method has potentially 3 main sources of inconsistencies (local truncation errors not converging to zero as the grid is refined): (i) the divergence term of the continuity equation, (ii) the perpendicular velocity and (iii) the kinetic energy terms of the vector invariant form of the momentum equations. Although some of these inconsistencies have not impacted the convergence on some standard shallow water test cases up until now, they may constitute a potential problem for high resolution 3D models. Based on our analysis, we propose modifications for the method that will make it first order accurate in the maximum norm. It preserves many of the mimetic properties, albeit having non-steady geostrophic modes on the f-sphere. Experimental results show that the resulting model is a more accurate alternative to the existing formulations and should provide means of having a consistent, computationally cheap and scalable atmospheric or ocean model on C staggered Voronoi grids.

Resonant fast–slow interactions and breakdown of quasi-geostrophy in rotating shallow water

In this paper we investigate the possibility of fast waves affecting the evolution of slow balanced dynamics in the regime $Ro\sim Fr\ll 1$ of a rotating shallow water system, where $Ro$ and $Fr$ are the Rossby and Froude numbers respectively. The problem is set up as an initial value problem with unbalanced initial data. The method of multiple time scale asymptotic analysis is used to derive an evolution equation for the slow dynamics that holds for $t\lesssim 1/(fRo^{2})$, $f$ being the inertial frequency. This slow evolution equation is affected by the fast waves and thus does not form a closed system. Furthermore, it is shown that energy and enstrophy exchange can take place between the slow and fast dynamics. As a consequence, the quasi-geostrophic ideology of describing the slow dynamics of the balanced flow without any information on the fast modes breaks down. Further analysis is carried out in a doubly periodic domain for a few geostrophic and wave modes. A simple set of slowly evolving amplitude equations is then derived using resonant wave interaction theory to demonstrate that significant wave-balanced flow interactions can take place in the long-time limit. In this reduced system consisting of two geostrophic modes and two wave modes, the presence of waves considerably affects the interactions between the geostrophic modes, the waves acting as a catalyst in promoting energetic interactions among geostrophic modes.

Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere

AbstractWe consider incompressible flows in the rapid-rotation limit of small Rossby number and vanishing Ekman number, in a bounded volume with a rigid impenetrable rotating boundary. Physically the flows are inviscid, almost rigid rotations. We interpret the Coriolis force, modified by a pressure gradient, as a linear operator acting on smooth inviscid incompressible flows in the volume. The eigenfunctions of the Coriolis operator$\boldsymbol{{\mathcal{C}}}$so defined are the inertial modes (including any Rossby modes) and geostrophic modes of the rotating volume. We show$\boldsymbol{{\mathcal{C}}}$is a bounded operator and that$-\text{i}\boldsymbol{{\mathcal{C}}}$is symmetric, so that the Coriolis modes of different frequencies are orthogonal. We prove that the space of incompressible polynomial flows of degree$N$or less in a sphere is invariant under$\boldsymbol{{\mathcal{C}}}$. The symmetry of$-\text{i}\boldsymbol{{\mathcal{C}}}$thus implies the Coriolis operator is non-defective on the finite-dimensional space of spherical polynomial flows. This enables us to enumerate the Coriolis modes, and to establish their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spherical polynomial flows under$\boldsymbol{{\mathcal{C}}}$and completeness, is that the solution of the polynomial Poisson–Neumann problem, i.e. Poisson’s equation with a Neumann boundary condition and polynomial data, in a sphere is a polynomial. We also enumerate the Coriolis modes in a sphere, with careful consideration of the geostrophic modes, directly from the known analytic solutions.

A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed-sphere grids

Abstract. A new algorithm is presented for the solution of the shallow water equations on quasi-uniform spherical grids. It combines a mimetic finite volume spatial discretization with a Crank–Nicolson time discretization of fast waves and an accurate and conservative forward-in-time advection scheme for mass and potential vorticity (PV). The algorithm is implemented and tested on two families of grids: hexagonal–icosahedral Voronoi grids, and modified equiangular cubed-sphere grids. Results of a variety of tests are presented, including convergence of the discrete scalar Laplacian and Coriolis operators, advection, solid body rotation, flow over an isolated mountain, and a barotropically unstable jet. The results confirm a number of desirable properties for which the scheme was designed: exact mass conservation, very good available energy and potential enstrophy conservation, consistent mass, PV and tracer transport, and good preservation of balance including vanishing ∇ × ∇, steady geostrophic modes, and accurate PV advection. The scheme is stable for large wave Courant numbers and advective Courant numbers up to about 1. In the most idealized tests the overall accuracy of the scheme appears to be limited by the accuracy of the Coriolis and other mimetic spatial operators, particularly on the cubed-sphere grid. On the hexagonal grid there is no evidence for damaging effects of computational Rossby modes, despite attempts to force them explicitly.

The route to dissipation in strongly stratified and rotating flows

AbstractWe investigate the route to dissipation in strongly stratified and rotating systems through high-resolution numerical simulations of the Boussinesq equations (BQs) and the primitive equations (PEs) in a triply periodic domain forced at large scales. By applying geostrophic scaling to the BQs and using the same horizontal length scale in defining the Rossby and the Froude numbers, $\mathit{Ro}$ and $\mathit{Fr}$, we show that the PEs can be obtained from the BQs by taking the limit ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} \rightarrow 0$. When ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} $ is small the difference between the results from the BQ and the PE simulations is shown to be small. For large rotation rates, quasi-geostrophic dynamics are recovered with a forward enstrophy cascade and an inverse energy cascade. As the rotation rate is reduced, a fraction of the energy starts to cascade towards smaller scales, leading to a shallowing of the horizontal spectra from ${ k}_{h}^{- 3} $ to ${ k}_{h}^{- 5/ 3} $ at the small-scale end. The vertical spectra show a similar transition as the horizontal spectra and we find that Charney isotropy is approximately valid also at larger wavenumbers than the transition wavenumber. The high resolutions employed allow us to capture both ranges within the same simulation. At the transition scale, kinetic energy in the rotational and in the horizontally divergent modes attain comparable values. The divergent energy is several orders of magnitude larger than the quasi-geostrophic divergent energy given by the $\Omega $-equation. The amount of energy cascading downscale is mainly controlled by the rotation rate, with a weaker dependence on the stratification. A larger degree of stratification favours a downscale energy cascade. For intermediate degrees of rotation and stratification, a constant energy flux and a constant enstrophy flux coexist within the same range of scales. In this range, the enstrophy flux is a result of triad interactions involving three geostrophic modes, while the energy flux is a result of triad interactions involving at least one ageostrophic mode, with a dominant contribution from interactions involving two ageostrophic and one geostrophic mode. Dividing the ageostrophic motions into two classes depending on the sign of the linear wave frequency, we show that the energy transfer is for the largest part supported by interactions within the same class, ruling out the wave–wave–vortex resonant triad interaction as a mean of the downscale energy transfer. The role of inertia-gravity waves is studied through analyses of time-frequency spectra of single Fourier modes. At large scales, distinct peaks at frequencies predicted for linear waves are observed, whereas at small scales no clear wave activity is observed. Triad interactions show a behaviour which is consistent with turbulent dynamics, with a large exchange of energy in triads with one small and two large comparable wavenumbers. The exchange of energy is mainly between the modes with two comparable wavenumbers.

Analysis of a mixed finite‐element pair proposed for an atmospheric dynamical core

AbstractWe present a numerical dispersion analysis for the P2 − P1DG finite‐element pair applied to the linear shallow‐water equations in one dimension. The aim is to provide insight into the numerical dispersion properties of the RT1 and BDFM1 finite‐element pairs in two dimensions, which have recently been proposed for horizontal discretisations of atmospheric dynamical cores with quasi‐uniform grids. This is achieved via analysis of a one‐dimensional RT1 element. Whilst these finite‐element pairs have been shown to have many desirable properties that extend properties of the C grid to non‐orthogonal quadrilateral and triangular grids, including stationary geostrophic modes on the f plane, and a 2:1 ratio of velocity to pressure degrees of freedom (a necessary condition for the absence of spurious mode branches), it is also important to have appropriately physical numerical wave propagation. In the absence of Coriolis force, we compute the group velocity for P2 − P1DG. We find that, as well as dropping to zero at the grid‐scale, which also occurs for the C‐grid finite‐difference method, the group velocity also drops to zero in a narrow band around kh = π which corresponds to eigenmodes with a wavelength close to two element widths. This is a potential problem because it increases the amount of wavenumber space that needs to be filtered. In this one‐dimensional case, we find that this particular issue can be removed by a small modification of the equations, namely partially lumping the mass matrix, in such a way that the other favourable properties of the scheme are not affected. We discuss various symmetric and asymmetric modifications of the mass matrix, and show that both such modifications preserve energy conservation (having modified the definition of discrete kinetic energy). Finally we illustrate our findings with numerical experiments, and discuss the potential to extend this modification to two‐dimensional schemes.

Mixed finite elements for numerical weather prediction

We show how mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite difference methods. There are a few major advantages: the mixed finite element methods do not require an orthogonal grid, and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation. These methods preserve several properties of the C-grid method when applied to linear barotropic wave propagation, namely: (a) energy conservation, (b) mass conservation, (c) no spurious pressure modes, and (d) steady geostrophic modes on the f-plane. We explain how these properties are preserved, and describe two examples that can be used on pseudo-uniform grids: the recently-developed modified RTk-Q(k-1) element pairs on quadrilaterals and the BDFM1-P1DG element pair on triangles. All of these mixed finite element methods have an exact 2:1 ratio of velocity degrees of freedom to pressure degrees of freedom. Finally we illustrate the properties with some numerical examples.

Numerical wave propagation for the triangular P1 DG– P2 finite element pair

The f-plane and β-plane wave propagation properties are examined for discretisations of the linearised rotating shallow-water equations using the P1 DG – P2 finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the f-plane (planar geometry, Coriolis force independent of space) case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the P1 DG – P2 finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency f, and which do not propagate. A restriction of the P1 DG velocity space is proposed in which these modes are not present, leading to a finite element discretisation which is completely free of spurious modes. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised P1 DG – P2 equations in the β-plane (planar geometry, Coriolis force linear in space) case resulting in a Rossby wave equation which is also third-order accurate. This means that the dispersion relation for the wave propagation is very accurate; an illustration of this is provided by a numerical dispersion analysis in the case of a triangulation consisting of equilateral triangles.

Energy cascade processes in rotating stratified turbulence with application to the atmospheric mesoscale

Numerical experiments of rotating stratified turbulence are conducted in order to investigate the mesoscale energy cascade process in the free atmosphere. We examine the downscale energy cascade process originated by energy injection at large scales. The energy spectrum created by this injection follows a power law, and its slope depends on planetary rotation rate and static stability. While the spectral slope tends to be steeper with increased static stability in cases without planetary rotation, it becomes insensitive to the planetary rotation rate and static stability and converges to the range from −1.9 to −2.1 in cases with an increased rotation rate. The roles of geostrophic and gravity modes in the cascade processes are discussed. It is shown that nonlinear interactions between geostrophic and gravity modes play a crucial role in the downscale energy cascades. In particular, the effect of the planetary rotation mainly influences the triad interaction composed of one geostrophic and two gravity modes in the energy cascade process. Energy flux to smaller scales is intensified by this type of triad interaction with increased planetary rotation rate.