Quasi-geostrophic Equation Research Articles

The North China record-breaking rainfall in July 2021: The atmospheric influential factors and precursory signal

Abstract In July 2021, the southeastern part of North China (SENC) suffered a record-breaking extreme rainfall event that caused devastating flooding and enormous losses. In this study, the major atmospheric influential factors and the precursory signal of heavy rainfall in 2021 are investigated using the correlation, regression, power spectrum, and filtering methods, the quasi-geostrophic velocity equation, observational data and numerical simulation of a linear baroclinic model (LBM). The results show that the extremity of a quasi-barotropic high anomaly over Northeast Asia (NEA) contributes to the deep anomalous upward motion within SENC by inducing positive vorticity and temperature advections. On the other hand, the anomalous southeasterly flow at the southwestern flank of the NEA high anomaly transports sufficient moisture to SENC in the lower troposphere. The local deep upward motion combined with the lower-tropospheric moisture convergence directly leads to the occurrence of this extreme rainfall event. Further analysis shows that the intensification of the NEA high in July 2021 is closely tied to the westward migration of atmospheric disturbance originating from the vicinity of Northeast Pacific-North America, which could be supported by numerical simulation in LBM. The variation of the geopotential height anomaly over Northeast Pacific-North America precedes that of the NEA high by two weeks, which is likely to provide a potential source of predictability for the extreme rainfall in SENC.

Fourier Neural Operator Networks for Solving Reaction–Diffusion Equations

In this paper, we used Fourier Neural Operator (FNO) networks to solve reaction–diffusion equations. The FNO is a novel framework designed to solve partial differential equations by learning mappings between infinite-dimensional functional spaces. We applied the FNO to the Surface Quasi-Geostrophic (SQG) equation, and we tested the model with two significantly different initial conditions: Vortex Initial Conditions and Sinusoidal Initial Conditions. Furthermore, we explored the generalization ability of the model by evaluating its performance when trained on Vortex Initial Conditions and applied to Sinusoidal Initial Conditions. Additionally, we investigated the modes (frequency parameters) used during training, analyzing their impact on the experimental results, and we determined the most suitable modes for this study. Next, we conducted experiments on the number of convolutional layers. The results showed that the performance of the models did not differ significantly when using two, three, or four layers, with the performance of two or three layers even slightly surpassing that of four layers. However, as the number of layers increased to five, the performance improved significantly. Beyond 10 layers, overfitting became evident. Based on these observations, we selected the optimal number of layers to ensure the best model performance. Given the autoregressive nature of the FNO, we also applied it to solve the Gray–Scott (GS) model, analyzing the impact of different input time steps on the performance of the model during recursive solving. The results indicated that the FNO requires sufficient information to capture the long-term evolution of the equations. However, compared to traditional methods, the FNO offers a significant advantage by requiring almost no additional computation time when predicting with new initial conditions.

Self-similar spirals for the generalized surface quasi-geostrophic equations

We construct a large class of nontrivial (nonradial) self-similar solutions of the generalized surface quasi-geostrophic equations (gSQG). To the best of our knowledge, this is the first rigorous construction of any self-similar solution for these equations. The solutions are of spiral type, locally integrable, and may have mixed sign. Moreover, they bear some resemblance to the finite time singularity scenario numerically proposed by Scott–Dritschel [J. Fluid Mechanics 863, article no. R2 (2019)] in the SQG patch setting.

Global Well-Posedness and Refined Regularity Criterion for the Uni-Directional Euler-Alignment System

Abstract We investigate global solutions to the Euler-alignment system in $d$ dimensions with unidirectional flows and strongly singular communication protocols $\phi (x) = |x|^{-(d+\alpha )}$ for $\alpha \in (0,2)$. Our paper establishes global regularity results in both the subcritical regime $1<\alpha <2$ and the critical regime $\alpha =1$. Notably, when $\alpha =1$, the system exhibits a critical scaling similar to the critical quasi-geostrophic equation. To achieve global well-posedness, we employ a novel method based on propagating the modulus of continuity. Our approach introduces the concept of simultaneously propagating multiple moduli of continuity, which allows us to effectively handle the system of two equations with critical scaling. Additionally, we improve the regularity criteria for solutions to this system in the supercritical regime $0<\alpha <1$.

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.

A Unified Formulation of Quasi‐Geostrophic and Shallow Water Equations via Projection

AbstractThis paper introduces a unified model for layered rotating shallow‐water (RSW) and quasi‐geostrophic (QG) equations, based on the intrinsic relationship between these two sets of equations. We propose a novel formulation of the QG equations as a projection of the RSW equations. This formulation uses the same prognostic variables as RSW, namely velocity and layer thickness, thereby restoring the proximity of these two sets of equations. It provides direct access to the ageostrophic velocities embedded within the geostrophic velocities resolved by the QG equations. This approach facilitates the study of differences between QG and RSW using a consistent numerical discretization. We demonstrate the effectiveness of this formulation through examples including vortex shear instability, double‐gyre circulation, and a simplified North Atlantic configuration.

Global Unique Solutions with Instantaneous Loss of Regularity for SQG with Fractional Diffusion

In this work we construct global unique solutions of the dissipative Surface quasi-geostrophic equation (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-SQG) that lose regularity instantly when there is super-critical fractional diffusion.

Rossby waves with barotropic–baroclinic coherent structures and dynamics for the (2 + 1)-dimensional coupled cylindrical KP equations with variable coefficients

Starting from the classical quasi-geostrophic potential vorticity equation with equal depth two-layer fluid, the coupled cylindrical Kadomtsev–Petviashvili (KP) equations with variable coefficients for Rossby waves are studied. To be more general, the phase velocity is considered an indefinite integral about time and improves the analysis procedure. So the variable coefficients are obtained and some previous studies are reasonably explained. The cylindrical wave theory is therewith utilized to reduce the coupled cylindrical KP equations with variable coefficients, and based on the modified Hirota bilinear method, the lump solutions and interaction solutions are found. Through numerical simulations, the Rossby lump waves on both sides of the y axis move closer to the center, and their amplitude gradually decreases and tends to flatten with the generalized Rossby parameter growth. In the Rossby waves flow field, the dipole structures propagate to the east and lead to the appearance of the compress phenomenon during barotropic–baroclinic interaction. It is possibly useful for further theoretical research on atmospheric phenomena.

Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model

Ocean circulation plays an important role in the formation and occurrence of extreme climate events. The study shows that the periodic variation of ocean circulation under strong wind stress is closely related to climate oscillation. Ocean circulation is a nonlinear dynamic system, which shows complex nonlinear characteristics, so the essence behind ocean circulation has not been clearly explained. Therefore, the response and evolution of the circulation system to wind stress are studied based on the bifurcation and catastrophe theories in nonlinear dynamics. First, the quasi-geostrophic gyre equation and the normalized gravity model are introduced and developed to study ocean circulation driven by wind stress, and solved using the Galerkin method. Then, the bifurcation and catastrophe behaviors of the system governed by the low-order ocean circulation model during the change in wind stress intensity and the coexistence of multiple equilibria in the circulation system are studied in detail. The results show that saddle and unstable nodes appear in the system after a cusp catastrophe. With the change in model parameters, the unstable node becomes the unstable focus, and then the subcritical Hopf bifurcation occurs. The system forms a bistable interval when the system undergoes a catastrophe twice, and the system shows hysteresis. In addition, multiple equilibrium states are coexisting in the circulating system, and the unstable equilibrium state always changes into a stable equilibrium state through vortex movement. Therefore, there is a route for the system to induce short-term climate oscillation, that is, in the multi-stable equilibrium state of the system, the vortex oscillates after being subject to small disturbances, and then triggers climate oscillation.

Distinguishing Extreme Precipitation Mechanisms Associated with Atmospheric Rivers and Nonatmospheric Rivers in the Lower Yangtze River Basin

Abstract This study investigates the disparity in quantitative moisture contribution and synoptic-scale vertical motion in the lower reaches of the Yangtze River basin (LYRB) for different extreme precipitation (EP) types, which are categorized as EP associated with atmospheric river (AR&EP) or EP associated with nonatmospheric river (non-AR&EP). To analyze moisture contribution, backward tracking using the Water Accounting Model-2layers is performed. In general, the remote moisture contribution is 9.7 times greater than the local contribution, with the ocean contribution being 1.67 times stronger than the land contribution. However, terrestrial and oceanic contributions obviously increase in the EP types, especially for oceanic contribution being double in magnitude. Notably, the west Pacific (WP) contribution emerges as the dominant differentia between the EP types, playing a crucial role in the AR formation. By solving the quasigeostrophic omega equation, the upper-level jet (ULJ) stream acts as the primary dynamic forcing for transverse vertical motion in AR&EP, while the baroclinic trough exhibits a relatively weaker influence. However, both systems have a nearly equal impact on vertical velocity in non-AR&EP. The enhanced shearwise elevation in the non-AR&EP type is the response of the stronger upper-level ridge over the Tibetan Plateau (TP), which induces enhanced Q vector, the divergence pointing toward the LYRB. However, the main dynamic difference is the location of ULJ, which serves as the trigger role although weak. Diabatic forcing proves to be the decisive factor for vertical motion development, the difference attributed to the released excessive latent heating with excess moisture contribution from the WP in AR&EP with enhanced precipitation. Significance Statement The main objective of this study is to investigate quantitative moisture contribution by applying Water Accounting Model-2layers and vertical motion attribution using the quasigeostrophic omega equation for extreme precipitation types based on the presence or absence of atmospheric river. Our findings reveal excessive moisture from the west Pacific serving not only as key in atmospheric river formation but also as the primary trigger for intensified diabatic vertical motion, inducing enhanced precipitation. The direction of strong winds in the north of the Tibetan Plateau holds crucial forecasting implications, which determine the location of the upper-level jet stream downstream. The transverse vertical motion, induced by the upper-level jet stream, plays the dominant dynamic role in both extreme precipitation (EP) types.

Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation

In this paper, we systematically study the existence, asymptotic behaviors, uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds for the generalized surface quasi-geostrophic (gSQG) equation. Firstly we obtain the existence of a new family of global solutions via the variational method. Secondly we show the uniqueness of maximizers under our variational setting. Thirdly by using the variational framework, the uniqueness of maximizers and a concentration-compactness principle we establish some stability theorems. Moreover, after a suitable transformation, these solutions constitute the desingularization of traveling point vortex pairs.

Non-existence and Strong lll-posedness in Ck,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{k,\\beta }$$\\end{document} for the Generalized Surface Quasi-geostrophic Equation

We consider solutions to the generalized Surface Quasi-geostrophic equation (γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}-SQG) when the velocity is more singular than the active scalar function (i.e. γ∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\in (0,1)$$\\end{document}). In this paper we establish strong ill-posedness in Ck,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{k,\\beta }$$\\end{document} (k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 1$$\\end{document}, β∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in (0,1]$$\\end{document} and k+β>1+γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+\\beta >1+\\gamma $$\\end{document}) and we also construct solutions in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document} that initially are in Ck,β∩L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{k,\\beta }\\cap L^2$$\\end{document} but are not in Ck,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{k,\\beta }$$\\end{document} for t>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t>0$$\\end{document}. Furthermore these solutions stay in Hk+β+1-2δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{k+\\beta +1-2\\delta }$$\\end{document} for some small δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} and an arbitrarily long time.

Zeitlin Truncation of a Shallow Water Quasi‐Geostrophic Model for Planetary Flow

AbstractIn this work, we consider a Shallow‐Water Quasi Geostrophic equation on the sphere, as a model for global large‐scale atmospheric dynamics. This equation, previously studied by Verkley (2009, https://doi.org/10.1175/2008jas2837.1) and Schubert et al. (2009, https://doi.org/10.3894/james.2009.1.2), possesses a rich geometric structure, called Lie‐Poisson, and admits an infinite number of conserved quantities, called Casimirs. In this paper, we develop a Casimir preserving numerical method for long‐time simulations of this equation. The method develops in two steps: first, we construct an N‐dimensional Lie‐Poisson system that converges to the continuous one in the limit N → ∞; second, we integrate in time the finite‐dimensional system using an isospectral time integrator, developed by Modin and Viviani (2020, https://doi.org/10.1017/jfm.2019.944). We demonstrate the efficacy of this computational method by simulating a flow on the entire sphere for different values of the Lamb parameter. We particularly focus on rotation‐induced effects, such as the formation of jets. In agreement with shallow water models of the atmosphere, we observe the formation of robust latitudinal jets and a decrease in the zonal wind amplitude with latitude. Furthermore, spectra of the kinetic energy are computed as a point of reference for future studies.

Understanding the weakening patterns of inner Tibetan Plateau vortices

This study focuses on changes in the Tibetan Plateau vortices (TPVs) by using ERA5 reanalysis, covering the summers from 1979 to 2022 within the Tibetan Plateau (TP) region. These TPVs were identified using a geopotential height analysis. We discovered that the central-western TP had the most TPV activity and observed a clear decreasing trend in both the intensity and frequency of the TPVs in this region. This decrease was also accompanied by a decline in the strength of the associated vertical upward motion. To better understand this change, we employed the quasi-geostrophic omega equation. This allowed us to examine the dynamic, diabatic, and topographic factors contributing to the vertical motion during different phases of TPV activity in this region. Our results indicate that the main reason behind the weakened TPVs is the diminishing upper-level jet stream, which exerts dynamic forcing on the system. In the later stage, we observed that intensive moisture transport induces heightened diabatic vertical motion. However, this effect is not potent enough to counterbalance the diminishing dynamic influence. Therefore, our findings suggest a significant shift in TPV activity, transitioning from a dynamic-dominated regime to a latent heating-dominated diabatic regime. This new insight enhances our understanding of the complex mechanisms that influence TPV behavior.

The QG limit of magnetohydrodynamic rotating shallow water system

Magnetohydrodynamic rotating shallow water system (MRSW) is a proposed model for a thin layer of electrically conducting fluid, which plays an important role in astrophysical plasma studies. For the spatial periodic domain, a mathematically rigorous framework is developed for deriving reduced systems for MRSW equations with general unbalanced initial data. It is shown that the reduced slow dynamics are the magnetohydrodynamic quasi-geostrophic equations.

Wind stress curl as a driving force of annual waves in the upper ocean for interpreting energetics at all latitudes

The dynamics of waves and eddies in the upper ocean plays an important role in the climate variation of tropical and subtropical regions. Previous diagnoses for annual Rossby waves in oceanic model outputs manifested zonally alternating signals (ZASs) in the time-averaged distributions of wind input as well as pressure-flux divergence terms in the budget equation of wave energy. This is the case when the annual mean of the wind input is estimated as the inner product of simulated velocity vector and wind stress vector in previous studies. The present study proposes a new mathematical expression for estimating the wind input that is analogous to one derived from the quasi-geostrophic potential vorticity equation. Namely, the wind input is estimated as the negative of the product of pseudo-streamfunction and wind stress curl, the latter of which is associated with the horizontal divergence of Ekman velocity. This can be interpreted as replacement of kinetic energy input with gravitational potential energy input. Pseudo-streamfunction in the present study is inverted from Ertel’s potential vorticity anomaly and is seamlessly available at all latitudes. This contrasts with the quasi-geostrophic streamfunction which is singular at the equator. The new expression enables reducing ZASs in the horizontal distributions of both wind input and pressure-flux divergence terms, without harming the qualitative advantage of energy flux vectors to indicate the group velocity of waves at all latitudes.

Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise

Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise

A 3-D minimum-enstrophy vortex in stratified quasi-geostrophic flows

Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

A linear filter regularization for POD-based reduced-order models of the quasi-geostrophic equations

We propose a regularization for reduced-order models (ROMs) of the quasi-geostrophic equations (QGE) to increase accuracy when the proper orthogonal decomposition (POD) modes retained to construct the reduced basis are insufficient to describe the system dynamics. Our regularization is based on the so-called BV-α model, which modifies the nonlinear term in the QGE and adds a linear differential filter for the vorticity. To show the effectiveness of the BV-α model for ROM closure, we compare the results computed by a POD-Galerkin ROM with and without regularization for the classical double-gyre wind forcing benchmark. Our numerical results show that the solution computed by the regularized ROM is more accurate, even when the retained POD modes account for a small percentage of the eigenvalue energy. Additionally, we show that, although computationally more expensive than the ROM with no regularization, the regularized ROM is still a competitive alternative to full-order simulations of the QGE.

Geometric analysis of the generalized surface quasi-geostrophic equations

We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach