Potential Enstrophy Research Articles

Analysis of a secondary 16-day planetary wave generation through nonlinear interactions in the atmosphere

Using a nonlinear model of the general circulation of the middle and upper atmosphere (MUAM), spatio-temporal structures of planetary waves (PWs) during boreal winter were studied. Modeling of global atmospheric circulation was performed for January–February. Despite the tropospheric PW sources shaped in the model, the phenomenon of 16-day PW excitation arise out of internal atmospheric sources in the southern lower thermosphere was discovered. To explain the observed phenomenon, a number of numerical experiments were carried out according to different scenarios with a selective turning (on/off) tropospheric sources of PW individual modes (having periods of 4–16 days) in the model. Also, the evolution of perturbed potential enstrophy for a 16-day PW, as well as the contribution of nonlinear interactions between individual PW to it, was studied. This made it possible for the first time to demonstrate explicitly the process of generation a secondary 16-day PW as a result of the nonlinear interconnection of 4-day and 5-day PWs.Graphical

A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations

Methods for stabilising the turbulent cascade of potential enstrophy are analysed and compared for a compatible finite element discretisation of the rotating shallow water equations. These different approaches to upwinding the potential vorticity include the well-known anticipated potential vorticity method (APVM), streamwise upwind Petrov-Galerkin (SUPG) method, and a more recent method where the trial functions are evaluated downstream within the reference element. In all cases the upwinding scheme conserves both potential vorticity and energy, since the antisymmetric structure of the equations is preserved. The APVM leads to a symmetric definite correction to the potential enstrophy that is dissipative and inconsistent. The SUPG scheme introduces a consistent correction to the APVM scheme that acts as a backscatter term ensuring a richer depiction of turbulent dynamics. The downwinded trial function formulation results in the advection of downwind corrections, which analysis and numerical experiments show to be quantitatively similar to the SUPG scheme for a turbulent shear flow. The main difference between the SUPG and downwinded trial function schemes is in the energy conservation and residual errors. If just two nonlinear iterations are applied then the energy conservation errors are improved for the downwinded trial function formulation, reflecting a smaller residual error than for the SUPG scheme.We also present new temporal formulations by which potential enstrophy is exactly integrated across each time level. Results using these formulations are observed to be stable in the absence of any dissipation, despite the aliasing of grid scale turbulence. Using such a formulation and the APVM with a coefficient O(100) times smaller that its regular value leads to turbulent spectra that are greatly improved at the grid scale over the SUPG and downwinded trial function formulations with unstable potential enstrophy errors.

Backscatter in energetically-constrained Leith parameterizations

This paper combines two strains in the literature on subgrid-scale parameterizations for eddying ocean models to develop six new parameterizations and test them in an idealized quasigeostrophic model. The first strain develops nonlinear, Smagorinsky-like Leith scalings for the viscous coefficient based on reasoning about the turbulent forward enstrophy cascade of geophysical turbulence. The second introduces backscatter whose amplitude is scaled to re-inject a portion of the energy dissipated by other parameterizations. In the new parameterizations developed here the backscatter and viscous coefficients depend on each other and are set to simultaneously absorb the forward enstrophy (or potential enstrophy) cascade and backscatter a portion of the dissipated energy. The addition of backscatter to Leith-scaled nonlinear viscosity improves the simulations at resolutions from 4 km to 24 km by increasing the total kinetic energy and by reducing the fraction of the total energy dissipation rate associated with the net effect of viscosity and backscatter. Versions that use biharmonic viscosity to absorb the enstrophy cascade perform better than versions using a harmonic viscosity, and purely viscous closures perform better than closures that dissipate both kinetic and potential energy. Stochastic and deterministic backscatter schemes are developed, and though similar, the deterministic schemes perform slightly better.

A Multifaceted Isoneutral Eddy Transport Diagnostic Framework and Its Application in the Southern Ocean

AbstractWe propose an isoneutral eddy transport diagnostic framework and apply it in the Southern Ocean to analyze eddy potential vorticity (PV) transport dynamics. We leverage a thickness‐weighted spatiotemporal scale separation to define the generalized eddy flux. A local sampling method is employed and validated to estimate the transport tensor. We introduce Leonard's decomposition in Large Eddy Simulations (LES) to split the process into the Reynolds and non‐Reynolds parts. We also use the stationary‐transient decomposition. The non‐Reynolds term and the stationary effect contribute significantly to the transport properties. The Reynolds and transient parts have distinctions, that show oceanic mesoscale dynamics are not synergistic in time and spatial scales. We investigate the spatial scale dependence of each part and find that the transient and stationary components of the Reynolds (non‐Reynolds) term have consistent (dissimilar) scale dependence. The scale dependence of the generalized eddy flux and the transport eigenvalues is controlled by the non‐Reynolds term. We further investigate the eigenvalues in terms of the potential enstrophy and anisotropy. The potential enstrophy in the Southern Ocean holistically has a forward cascade on the researched scales. The Reynolds and transient effects efficiently promote the forward enstrophy cascade. The non‐Reynolds and stationary effects stimulate drastically anisotropic eddy transport. The PV gradient barrier mechanism dominates the drastic anisotropic case. When the anisotropy is weak, a special anisotropic case with a locally balanced potential enstrophy budget overwhelms the case of real isotropy. We also introduce several LES concepts, hoping to enlighten any potential ocean modeling and parameterization.

Potential Vorticity Generation in Breaking Gravity Waves

Potential vorticity (PV) is an important quantity in stratified flows because it is conserved following the flow in the absence of forcing and viscous and diffusive effects. However, as shown by previous work for unstratified turbulence, viscosity and diffusion, when present, are not purely dissipative and can create potential vorticity even when none is present initially. In this work, we use direct numerical simulations to investigate the viscous and diffusive generation of potential vorticity and potential enstrophy (integrated square PV) in stratified turbulence. Simulations are initialized with a two-dimensional standing internal gravity wave, which has no potential vorticity apart from some low-level random noise; as a result, all potential vorticity and enstrophy comes from viscous and diffusive effects. Significant potential enstrophy is found when the standing wave breaks, and the maximum potential enstrophy increases with increasing Reynolds number. The mechanism for the initial PV generation is spanwise diffusion of buoyancy perturbations, which grow as the standing wave three-dimensionalizes, into the direction of spanwise vorticity. The viscous and diffusive terms responsible are small-scale and are sensitive to under-resolution, so high resolution is required to obtain robust results.

Stirring of interior potential vorticity gradients as a formation mechanism for large subsurface-intensified eddies in the Beaufort Gyre

Abstract The Beaufort Gyre (BG) is hypothesized to be partially equilibrated by those mesoscale eddies that form via baroclinic instabilities of its currents. However, our understanding of the eddy field’s dependence on the mean BG currents and the role of sea ice remains incomplete. This theoretical study explores the scales and vertical structures of eddies forming specifically due to baroclinic instabilities of interior BG flows. An idealized quasi-geostrophic model is used to show that flows driven only by the Ekman pumping contain no interior potential vorticity (PV) gradients and generate weak and large eddies, ℴ(200km) in size, with predominantly barotropic and first baroclinic mode energy. However, flows containing realistic interior PV gradients in the Pacific halocline layer generate significantly smaller eddies of about 50 km in size, with a distinct second baroclinic mode structure and a subsurface kinetic energy maximum. The dramatic change in eddy characteristics is shown to be caused by the stirring of interior PV gradients by large-scale barotropic eddies. The sea ice-ocean drag is identified as the dominant eddy dissipation mechanism, leading to realistic sub-surface maxima of eddy kinetic energy for drag coefficients higher than about 2×10−3. A scaling law is developed for the eddy potential enstrophy, demonstrating that it is directly proportional to the interior PV gradient and the square root of the barotropic eddy kinetic energy. This study proposes a possible formation mechanism of large BG eddies and points to the importance of accurate representation of the interior PV gradients and eddy dissipation by ice-ocean drag in BG simulations and theory.

Conservative Numerical Schemes with Optimal Dispersive Wave Relations: Part II. Numerical Evaluations

A new energy and enstrophy conserving scheme (EEC) for the shallow water equations is proposed and evaluated using a suite of test cases over the global spherical or bounded domain. The evaluation is organized around a set of pre-defined properties: accuracy of individual operators, accuracy of the whole scheme, conservation of key quantities, control of the divergence variable, representation of the energy and enstrophy spectra, and simulation of nonlinear dynamics. The results confirm that the scheme is between the first and second order accurate, and conserves the total energy and potential enstrophy up to the time truncation errors. The scheme is capable of producing more physically realistic energy and enstrophy spectra, indicating that it can help prevent the unphysical energy cascade towards the finest resolvable scales. With an optimal representation of the dispersive wave relations, the scheme is able to keep the flow close to being non-divergent, and maintain the geostrophically balanced structures with large-scale geophysical flows over long-term simulations.

Phase transition to blob-hole coherent structure in the Hasegawa–Mima model for plasmas

An equilibrium statistical mechanics theory for the Hasegawa–Mima equations of toroidal plasmas, with canonical constraint on energy and microcanonical constraint on potential enstrophy, is solved exactly as a spherical model. The use of a canonical energy constraint instead of a fixed-energy microcanonical approach is justified by the preference for viewing real plasmas as an open system. A significant consequence of the results obtained from the partition function, free energy and critical temperature, is the condensation into a ground state exhibiting a blob-hole-like structure observed in real plasmas.

Potential-enstrophy lengthscale for the turbulent/nonturbulent interface in stratified flow

Turbulent/nonturbulent interfaces typically have a thickness of the order of the Kolmogorov scale. However, when we add a density or temperature stratification, we observe that the interface thickness scales differently. We analyze two different turbulent flows at moderate Reynolds numbers using direct numerical simulation and demonstrate that the results collapse on a single curve using a lengthscale based on the potential enstrophy.

Selective decay for the rotating shallow-water equations with a structure-preserving discretization

Numerical models of weather and climate critically depend on the long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2), 495 (2013)], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [Pavlov et al., “Structure-preserving discretization of incompressible fluids,” Physica D 240(6), 443–458 (2011)] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

Modified View of Energy Budget Diagram and Its Application to the Kuroshio Extension Region

AbstractThe nonlocality of eddy–mean flow interactions, which appears explicitly in the modified Lorentz diagram as a form of the interaction energy, and its link to other estimation methods are revisited, and a new formulation for the potential enstrophy is proposed. The application of these methods to the Kuroshio Extension region suggests that the combined use of energy analysis with other methods, including the potential enstrophy diagram, provides more comprehensive understandings for the eddy–mean flow interactions in the limited region. It is shown that the interaction energy is transported from the nearshore and upstream regions to the downstream region in the form of the interaction energy flux, causing acceleration of the Kuroshio Extension jet in the downstream region. The potential enstrophy diagram indicates that the eddy field decelerates (accelerates) the jet in the nearshore (downstream) region, which is a consistent result with the energy analysis. It turns out that the interaction potential enstrophy flux is radiated from a region of the eddy kinetic energy maximum toward the upstream region, which is the opposite direction from the interaction energy flux. The interaction potential enstrophy flux that originated from this eddy kinetic energy maximum region also convergences near the center of the northern recirculation gyre of the Kuroshio Extension region and tends to stabilize the structures of the recirculation gyre. Together with the energy analysis that indicates the eddy field accelerates the northeastern part of the recirculation gyre through the local interactions, the present analyses support the arguments on the eddy-driven northern recirculation gyre.

Downscale transfer of quasigeostrophic energy catalyzed by near-inertial waves

Abstract

Comparing Numerical Accuracy of Icosahedral A-Grid and C-Grid Schemes in Solving the Shallow-Water Model

AbstractA single software framework is introduced to evaluate numerical accuracy of the A-grid (NICAM) versus C-grid (MPAS) shallow-water model solvers on icosahedral grids. The C-grid staggering scheme excels in numerical noise control and total energy conservation, which results in exceptional stability for long time integration. Its weakness lies in the lack of model error reduction with increasing resolution in specific test cases (especially the root-mean-square error). The A-grid method conserves well potential enstrophy and shows a linear reduction of error with increasing resolution. The gridpoint noise manifests itself clearly on A-grid, but much less on C-grid. We show that the Coriolis force term on C-grid has a larger error than on A-grid. To treat the Coriolis term and kinetic energy gradient on an equal footing on C-grid, we propose combining these two quantities into a single tendency term and computing its value by a linear combination operation. This modification alone reduces numerical errors but still fails to converge the maximum error with resolution. The method of Peixoto can solve the maximum-error nonconvergence problem on C-grid but degrades the numerical stability. For the steady-state thin-layer test (0.01 m in depth), the A-grid method is less susceptible than C-grid methods, which are presumably disrupted by the Hollingsworth instability. The effect of horizontal diffusion on model accuracy and energy conservation is shown in detail. Programming experience shows that software implementation and optimization can strongly influence computational performance for models, although memory requirement and computational load of the two schemes are comparable.

The role of the asymmetric Ekman dissipation term on the energetics of the two-layer quasi-geostrophic model

In the two-layer quasi-geostrophic model, the friction between the flow at the lower layer and the surface boundary layer, placed beneath the lower layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the lower layer. The Ekman term appears in the governing equations asymmetrically; it is placed at the lower layer, but does not appear at the upper layer. A variation, proposed by Phillips and Salmon, uses extrapolation to place the Ekman term between the lower layer and the surface boundary layer, or at the surface boundary layer. We present theoretical results that show that in either the standard or the extrapolated configurations, the Ekman term dissipates energy at large scales, but does not dissipate potential enstrophy. It also creates an approximately symmetric stable distribution of potential enstrophy between the two layers. The behavior of the Ekman term changes fundamentally at small scales. Under the standard formulation, the Ekman term will unconditionally dissipate energy and also dissipate, under very minor conditions, potential enstrophy at small scales. However, under the extrapolated formulation, there exist small “negative regions”, which are defined over a two-dimensional phase space, capturing the distribution of energy per wavenumber between baroclinic energy and barotropic energy, and the distribution of potential enstrophy per wavenumber between the upper layer and the lower layer, where the Ekman term may inject energy or potential enstrophy.

Nonlinear interactions of stationary planetary waves during February 2016 sudden stratospheric warming

An analysis of nonlinear wave-wave and wave-mean flow interactions has been carried out during a sudden stratospheric warming (SSW) event in February 2016. This approach is based on a study of the perturbed potential enstrophy (Ertel’s potential vorticity squared) balance equation. The results obtained by using UK Met Office reanalysis data are presented. It has been shown that an increase in the nonlinear interactions occurs at higher-middle latitudes of the boreal stratosphere during the SSW. It is noted that it is necessary to include the stationary planetary wave with zonal wave number m=3 in the analysis of major warmings.

Numerical investigation of the Danilov inequality for two-layer quasi-geostrophic systems

In the statistical steady state of forced-dissipative turbulence governed by the 2D incompressible Navier–Stokes (NS) equation, the difference between the energy flux ΠE(k) and the enstrophy flux ΠZ(k), k2ΠE(k) − ΠZ(k), where k is the wavenumber, has been proved to be negative outside the forcing wavenumber range. This is often referred to as the Danilov inequality and is important for determining the directions of the energy and enstrophy fluxes in the inertial ranges for 2D NS turbulence. We investigate numerically whether or not this inequality holds for two-layer quasi-geostrophic (QG) potential vorticity equations, which are a generalization of the vorticity equation for the 2D NS system. In two-layer QG systems forced by a horizontally homogeneous, baroclinically unstable basic flow, the corresponding difference between the total energy and total potential enstrophy fluxes is mathematically sign-indefinite due to the presence of the internal forcing term. Moreover, even if we concentrate on outside of the internal forcing range, the flux difference is still sign-indefinite when the dissipation mechanisms are asymmetric between the upper and lower layers. However, in our numerical experiments adopting both the internal forcing and asymmetric dissipations, the flux difference is negative across the whole wavenumber range. That is, the Danilov inequality continues to hold for two-layer QG systems, even though this cannot be proved mathematically.

Scale selection and feedback loops for patterns in drift wave-zonal flow turbulence

The scale selection and feedback loops for the formation and sustainment of a mesoscopic staircase profile structure are investigated for drift wave-zonal flow turbulence. A mean field model derived from the Hasegawa–Wakatani system and including the evolution of mean density, mean vorticity and perturbed potential enstrophy (PE) is used. It is found that a quasi-periodic zonal staircase forms from self-sharpening of modulation. The principle feedback loop is through the nonlinear dependence of mixing length on electron density gradient, which enters by way of the potential vorticity gradient. Counterintuitively, shearing is not effective. Moreover, the number of steps in the staircase is sensitive to both the drive (production rate of PE and initial density gradient) and damping (flow viscosity and collisional diffusivity) factors. The minimal step scale is selected by competition between the initial density gradient and diffusive dissipation. Finite turbulence spreading is necessary to form the staircase, but moderate enhancement of turbulence spreading tends to wash out the pattern. The staircase retains a memory of its initial state. Both the mean shear and zonal shear affect the staircase evolution. A strong mean shear quenches the pattern by suppressing the drift wave turbulence. The implications of these findings are also discussed.

Diffusion-rotational parameterization of eddy fluxes of potential vorticity: barotropic flow in the zonal channel

The problem of parametrization of the eddy fluxes of a potential vorticity is discussed. Traditional diffusion parameterization is complemented by the inclusion of a rotational component. For the analysis of the new scheme, a quasi-geostrophic model of the dynamics of the barotropic flow in a zonal channel with a non-uniform bottom is used. An analytical solution of the problem is found and the influence of topography on the flow disturbances is discussed. It is shown that the equation for the eddy potential enstrophy allows to relate diffusion and «rotational» coefficients.

Generalized Z-Grid Model for Numerical Weather Prediction

Z-grid finite volume models conserve all-scalar quantities as well as energy and potential enstrophy and yield better dispersion relations for shallow water equations than other finite volume models, such as C-grid and C-D grid models; however, they are more expensive to implement. During each time integration, a Z-grid model must solve Poisson equations to convert its vorticity and divergence to a stream function and velocity potential, respectively. To optimally utilize these conversions, we propose a model in which the stability and possibly accuracy on the sphere are improved by introducing more stencils, such that a generalized Z-grid model can utilize longer time-integration steps and reduce computing time. Further, we analyzed the proposed model’s dispersion relation and compared it to that of the original Z-grid model for a linearly rotating shallow water equation, an important property for numerical models solving primitive equations. The analysis results suggest a means of balancing stability and dispersion. Our numerical results also show that the proposed Z-grid model for a shallow water equation is more stable and efficient than the original Z-grid model, increasing the time steps by more than 1.4 times.

Diffusion-Rotational Parameterization of Eddy Fluxes of Potential Vorticity: Barotropic Flow in the Zonal Channel

The problem of parametrization of the eddy fluxes of a potential vorticity is discussed. Traditional diffusion parameterization is complemented by the inclusion of a rotational component. For the analysis of the new scheme, a quasi-geostrophic model of the dynamics of the barotropic flow in a zonal channel with a non-uniform bottom is used. An analytical solution of the problem is found and the influence of topography on the flow disturbances is discussed. It is shown that the equation for the eddy potential enstrophy allows to relate diffusion and «rotational» coefficients.