Vorticity Equation Research Articles

Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations

In this paper, we prove the global existence and uniqueness of the weak solutions to the inviscid velocity-vorticity model of the g-Navier-Stokes equations. The system is performed by entegrating the velocity-pressure system which is involved by using the rotational formulation of the nonlinearity and the vorticity equation for the g-Navier-Stokes equations without viscosity term. In this study we particularly interest the inviscid velocity-vorticity system of the g-Navier-Stokes equations over the two dimensional periodic box Ω=(0,1)^2⊂R^2.

A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach

A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to find this quantity based on the known magnitudes computed at the previous instant of time.

A 15-m resolution numerical simulation of a tornado in South China: Structure and vorticity budget of the suction vortices

An EF3 tornado occurred in the spiral cloud belt surrounding Typhoon Rainbow on October 4, 2015. We performed six nested numerical simulations with horizontal resolutions of 4050, 1350, 450, 150, 50 and 15 m to analyze the structure and mechanisms of formation of the tornado. The typical structure of the tornadic supercell was simulated at a resolution of 444 m. The tornado-like vortex system developed until the grid spacing was ≤148 m. The suction vortex inside the tornado were captured at a resolution of 15 m. These suction vortices developed and rotated rapidly before being organized into the main vortex of the tornado. We used the vorticity equation to analyze the vorticity budget of the suction vortex. The main contribution to the vorticity was the solenoidal term. The horizontal convergence term mainly influenced the high-level outflow, whereas the tilting term affected the mid-level updraft. The frictional term mainly affected the lower levels where the suction vortex developed. We propose a conceptual model based on our results. Influenced by the frictional force, the suction vortex generated, developed and merged into the tornado vortex near the surface. The vertical circulation of the strong spiral region was influenced by convergence at higher levels and inflow at the lower rear of the tornado. Tilting of the vorticity tube and merging of the suction vortex caused the hydrometeors to aggregate and fall at the location where the tornado was generated.

Steady Normal-Mode Vortices in Circular Cavities

Summary This article establishes elementary families of steady two-dimensional (2D) vortices in circular enclosures filled with inviscid fluid. A normal-mode vortex is a continuous vortex that satisfies a Helmholtz equation for the streamfunction. An individual normal-mode vortex satisfies the steady 2D vorticity equation. A way to satisfy the vorticity equation with two superposed normal-mode vortices is to let one of them have circular streamlines, trivially satisfying the kinematic boundary condition at the circle boundary. The requirement for steady flow is that the wavenumber eigenvalues for the Helmholtz equations are identical. This eigenvalue has to be dictated by a normal-mode vortex that is fully 2D. In a circle sector with a designed angle, it is possible to add two fully 2D normal-mode vortices so that their superposed flow is steady. One such example is demonstrated, where the circle sector has an angle of $63.77^\circ$. While conventional stability analysis does not apply to these families of steady normal-mode vortices, there is a latent algebraic instability in terms of linearly growing offspring vortices evolving from an initial state where a small initial perturbation is added to the steady normal-mode vortex.

The Basic Equations under Weak Temperature Gradient Balance: Formulation, Scaling, and Types of Convectively Coupled Motions

Abstract The weak temperature gradient (WTG) approximation is extended to the basic equations on a rotating plane. The circulation is decomposed into a diabatic component that satisfies WTG balance exactly and a deviation from this balance. Scale analysis of the decomposed basic equations reveals a spectrum of motions, including unbalanced inertio-gravity waves and several systems that are in approximate WTG balance. The balanced systems include equatorial moisture modes with features reminiscent of the MJO, off-equatorial moisture modes that resemble tropical depression disturbances, “mixed systems” in which temperature and moisture play comparable roles in their thermodynamics, and moist quasigeostrophic motions. In the balanced systems the deviation from WTG balance is quasi nondivergent, in nonlinear balance, and evolves in accordance to the vorticity equation. The evolution of the strictly balanced WTG circulation is in turn described by the divergence equation. WTG balance restricts the flow to evolve in the horizontal plane by making the isobars impermeable to vorticity and divergence, even in the presence of diabatically driven vertical motions. The vorticity and divergence equations form a closed system of equations when the irrotational circulation is in WTG balance and the nondivergent circulation is in nonlinear balance. The resulting “WTG equations” may elucidate how interactions between diabatic processes and the horizontal circulation shape slowly evolving tropical motions. Significance Statement Many gaps in our understanding of tropical weather systems still exist and there are still many opportunities to improve their forecasting. We seek to further our understanding of the tropics by extending a framework known as the “weak temperature gradient approximation” to all of the equations for atmospheric flow. Doing this reveals a variety of motions whose scales are similar to observed tropical weather systems. We also show that two equations describe the evolution of slow systems: one that describes tropical thunderstorms and one for the rotating horizontal winds. The two equations may help us understand the dynamics of slowly evolving tropical systems.

Deep Learning Augmented Data Assimilation: Reconstructing Missing Information with Convolutional Autoencoders

Abstract Remote sensing data play a critical role in improving numerical weather prediction (NWP). However, the physical principles of radiation dictate that data voids frequently exist in physical space (e.g., subcloud area for satellite infrared radiance or no-precipitation region for radar reflectivity). Such data gaps impair the accuracy of initial conditions derived from data assimilation (DA), which has a negative impact on NWP. We use the barotropic vorticity equation to demonstrate the potential of deep learning augmented data assimilation (DDA), which involves reconstructing spatially complete pseudo-observation fields from incomplete observations and using them for DA. By training a convolutional autoencoder (CAE) with a long simulation at a coarse “forecast” resolution (T63), we obtained a deep learning approximation of the “reconstruction operator,” which maps spatially incomplete observations to a model state with full spatial coverage and resolution. The CAE was applied to an incomplete streamfunction observation (∼30% missing) from a high-resolution benchmark simulation and demonstrated satisfactory reconstruction performance, even when only very sparse (1/16 of T63 grid density) observations were used as input. When only spatially incomplete observations are used, the analysis fields obtained from ensemble square root filter (EnSRF) assimilation exhibit significant error. However, in DDA, when EnSRF takes in the combined data from the incomplete observations and CAE reconstruction, analysis error reduces significantly. Such gains are more pronounced with sparse observation and small ensemble size because the DDA analysis is much less sensitive to observation density and ensemble size than the conventional DA analysis, which is based solely on incomplete observations. Significance Statement Data assimilation plays a critical role in improving the skills of modern numerical weather prediction by establishing accurate initial conditions. However, unobservable regions are common in observation data, particularly those derived from remote sensing. The nonlinear relationship between data from observable regions and the physical state of unobservable regions may impede DA efficiency. As a result, we propose that deep learning be used to improve data assimilation in such cases by reconstructing a spatially complete first guess of the physical state with deep learning and then applying data assimilation to the reconstructed field. Such deep learning augmentation is found effective in improving the accuracy of data assimilation, especially for sparse observation and small ensemble size.

The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System

In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique.

Study on numerical calculation of flow structures in a curved open channel with advanced depth-integrated models

A better understanding of flow structures distribution in rivers is crucial to determine the safety degree of rivers. A practical and reliable model is required to overcome the issue of the long computational time of a three-dimensional calculation model and the lack of computation detail of a two-dimensional calculation model for flow structures distribution simulation in rivers. This paper presented an advanced depth-integrated numerical calculation method called the bottom velocity calculation (BVC) to reproduce the flow structures in a curved open channel. BVC method is an integrated multiscale simulation of flows in rivers that can evaluate vertical distributions of velocities and bottom velocity distributions by introducing depth-averaged horizontal vorticity and horizontal momentum equations on a water surface. It has several models, such as simplified bottom velocity calculation (SBVC) with shallow water assumption and general bottom velocity calculation (GBVC) method without the assumptions. The advantages of the BVC method, including SBVC and GBVC models, are validated in this paper using experimental datasets of a curved open channel and compared to two-dimensional and three-dimensional models. The results show that the BVC method has good reproducibility to simulate the flow structures distribution in the channel.

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.

A numerical method based on the Nambu bracket for the 3D vorticity equation

Abstract In this study, numerical methods for the 3D vorticity equation and Euler equation, which preserve the structure of Nambu mechanics for fluid dynamics, are investigated. Discrete vector fields of the stream function (or vector potential), velocity, and vorticity, as well as discrete counterparts of the gradient, curl, and divergence operators acting on them, are defined such that the structure of the de Rham complex in 3D Euclidean space is preserved. The inner products of the discrete vector fields are defined such that discrete counterparts of integration-by-parts formulae for the gradient, curl, and divergence operators hold. In addition, cross products of the discrete vector fields are introduced to define a skew-symmetric trilinear form. A discrete Nambu bracket, as well as the (kinetic) energy, helicity, and enstrophy of the discrete flow field, are defined straightforwardly. They are employed to derive a discrete vorticity equation in the same way as in the continuum setting. A discrete Euler equation is derived from the discrete vorticity equation based on the discrete counterpart of the Poincaré lemma, which holds under some typical conditions. It is proved that any solution to these discretized equations satisfies discrete analogues of the balances of energy, helicity, and enstrophy. Numerical experiments on a periodic array of rolls are conducted to examine the effectiveness of the method.

Helicity budget in turbulent channel flows with streamwise rotation

The streamwise rotation effects in turbulent channel flows are reflected not only in the appearance of the secondary flows but also in the weakened streamwise velocity and spanwise vorticity. In this paper, we investigate the secondary flows from three perspectives: the mean spanwise velocity, the mean streamwise vorticity, and combined mean and fluctuating helicity. We found that the combined helicity is also an alternative perspective to characterize the streamwise rotation effect, especially for the secondary flows. The budget equations of the mean and fluctuating helicity in physical space are derived theoretically and analyzed numerically. The streamwise rotation effects on the secondary flows are directly reflected on the pressure and Coriolis terms, which provides an essential source for helicity within the near-wall regions. The production term could be decomposed into two terms, which originate from the momentum and vorticity equations, respectively. The helical stress (velocity–vorticity correlation) originating from the vorticity equation shows a simple profile distribution and is dominant for the production for the helicity within the near-wall regions. The high helical structures in the core regions can be explained as an intense wall-normal transportation, which transfers produced helicity within the near-wall regions into the core regions.

Optimal Precursor Triggering Kuroshio Large Meander Decay Obtained in a Regional Ocean Model

AbstractThis study investigates the optimal precursor (OPR) triggering Kuroshio large meander (LM) decay using a conditional nonlinear optimal perturbation (CNOP) approach and a regional ocean modeling system (ROMS). The results show that the large amplitudes of the OPR are mainly concentrated at 1,000–2,500 m in the LM upstream region (west of 137°E), and their propagation downstream along the Kuroshio axis contributes to triggering the transition from the LM path to the nonlarge meander (NLM) path. Using the growth of the OPR, it is found that baroclinic instability contributes to the fast vertical development of the perturbations in the early stage of the LM decay, while barotropic processes play a leading role during the cyclonic ring shedding of the LM decay. In particular, the LM decay process is evaluated by the depth‐integrated vorticity equation, implying that the advection term dominates over the beta effect in the LM decay. Furthermore, the vorticity balance associated with the cyclonic eddy indicates that the advection and divergence terms represent a main balance in the spin‐down process of the cyclonic eddy. Under the effects of both, high potential vorticity water flows out from the LM meander tip and then the amplitude of the LM experiences a remarkable decrease. With the detachment of a cyclonic ring, a transition from the LM path to the NLM path occurs. This study further deepens our understanding of the LM decay and provides us some useful hints about designing the observation system of the LM.

Damped perturbations in inviscid shear flows: van Kampen modes and Landau damping

We compare initial value and eigenvalue problems for two-dimensional perturbations of the inviscid shear flow in a channel. Singular solutions, known in plasma physics as van Kampen (vK) modes, are constructed. They form a complete set of eigenfunctions for decomposition of any initial perturbation for stable wavy perturbations. A pair of discrete modes appears to ensure completeness in the unstable case. Expansion coefficients for eigenmodes are found, and equivalence of temporal evolution obtained with the help of the evolutionary equation for vorticity and expansion over eigenmodes is presented. This alternative description of the evolution using vK modes is analogous to ones found earlier in plasma and in stellar dynamics. In particular, for stable wavy perturbations, an initial state decays first exponentially due to Landau damping, then algebraically. It has been established (numerically and analytically) that the final decay law is t−1. Also, we numerically demonstrate that Landau-damped perturbations are not true eigenmodes, but rather a superposition of vK-modes with a real frequency, which does not retain its shape over time. However, solution on contours in the complex plane may exhibit properties of a true eigenmode, that is, decay without changing its spatial form. Energy redistribution between perturbation and the flow, in stable and unstable regimes, is analyzed.

A novel dynamical diagnosis of relative vorticity equation based on three-pattern decomposition of global atmospheric circulation: a case study of the western Pacific subtropical high in 2020

A novel dynamical diagnosis of relative vorticity equation based on three-pattern decomposition of global atmospheric circulation: a case study of the western Pacific subtropical high in 2020

VIScous Vorticity Equation (VISVE) model applied to 2-D turbulent flow over hydrofoils

The k−ω SST turbulence model is implemented in the VIScous and Vorticity Equation (VISVE) method, enabling VISVE to solve turbulent flows independently. The discretization of the vorticity equation and of the turbulence model is based on the Finite Volume Method (FVM). The major advantages of the VISVE method are: it needs a small computational domain, the continuity equation is automatically satisfied, and the pressure can be calculated after solving for the flow field. The formulation and the computational scheme for the turbulent VISVE method are introduced. Then the current method is applied on a 2-D hydrofoil in turbulent flow and the results are shown to be in very good agreement with those predicted by a Reynolds-Averaged Navier–Stokes (RANS) solver. The effect of Reynolds number (Re) on the predicted vorticity, velocity, and turbulent viscosity is presented for a range of Re from 5∗105 to 4∗106. The turbulent VISVE model is also applied on a hydrofoil with cupped trailing edge, which leads to separation. The results show that the turbulent VISVE solver is capable of predicting turbulent flow at high Reynolds numbers, as well as turbulent separated flow.

A new nonlinear integral-differential equation describing Rossby waves and its related properties

This paper focuses on algebraic Rossby solitary waves models in stratified fluids. Starting with the quasi-geostrophic potential vorticity equation, we present the Boussinesq-intermediate long-wave (Boussinesq-ILW) model for the first time using the multi-scale analysis and the perturbation expansion method. Compared with previous models describing algebraic Rossby solitary waves, the Boussinesq-ILW model is more general: the equation transforms into the Boussinesq-BO equation when h1→∞, and when [Ξ(B)]XXX changes to BXXXX, the model reduces to the Boussinesq equation. The conservation law is important for exploring the properties of the model, therefore we propose several common conservation laws: mass, momentum, and energy conservation. Finally, based on the trial function method, we obtain the solution of the Boussinesq-ILW equation and investigate the wave-wave interaction. The results show that the peak value of the Mach stem increases as the parameter γ increases, but the length becomes shorter and changes more rapidly.

Theory of Parcel Vorticity Evolution in Supercell-Like Flows

Abstract In a prior paper, insights into tornadogenesis in supercell storms were gained by discovering analytical formulas for vorticity variations along streamlines in idealized, steady, frictionless, isentropic inflows of dry air imported from a horizontally uniform environment. This work is simplified and extended to the evolution of parcel vorticity in unsteady, nonisentropic flows by integrating the vorticity equation using nonorthogonal Lagrangian coordinates. The covariant basis vectors , , and are material line elements attached to each parcel. Initially they form an orthonormal set with in the direction of and left normal to the storm-relative wind at each level in the environment, and upward. The surface containing all parcels with the same initial height constitutes a material surface, within which initially streamwise and transverse material lines are reoriented and stretched or shrunk. The basis vectors propagate a parcel’s barotropic vorticity through time by factoring in the “frozen-field” effect. With a horizontally uniform environment, the barotropic vorticity of a parcel depends on its initial streamwise vorticity times its current plus its initial crosswise vorticity times its current . For baroclinic and frictional vorticity, each contravariant component is the integral from initial to current time of the corresponding contravariant component of the generation vector. The “river-bend” effect acting on all parts (baroclinic, frictional, and barotropic) of transverse vorticity produces streamwise vorticity (parallel to 3D wind). In left-turning steady flow, it arises from rotating toward . For steady, frictionless, dry isentropic flow, previous vorticity formulas are recovered. Significance Statement Air parcels rising in a tornado spin rapidly about their direction of motion. Theory herein describes the processes that can produce this streamwise spin in supercells. Cyclonic updraft rotation originates from strong low-level environmental storm-relative winds that turn clockwise with height. Parcels flowing into the updraft have initially large streamwise spins that are amplified by streamwise stretching. Rain curtains falling through the cyclonic updraft cause other parcels to descend and turn leftward. Buoyancy and frictional torques give them horizontal spin. Even if these spins are transverse to the flow initially, they are turned streamwise by secondary flow that develops in left-hand bends. As the parcels reach the ground and converge into the tornado, streamwise stretching greatly magnifies their streamwise spins.

Unified theory of tearing mode growth from linear to Rutherford regime in the presence of local equilibrium current gradients

Using the two-field reduced magneto-hydrodynamic (MHD) model, a new theory of tearing mode transition from linear to Rutherford regime in the presence of local equilibrium current gradient (LCG) effects is proposed. By containing the LCG term in the vorticity equation, I reproduce the modified Rutherford equation with saturation terms via a quasi-linear approach. Therefore, the LCG term in the vorticity equation is responsible for the saturation mechanism of the classical tearing mode. The unified formula of the tearing mode evolution from linear exponential growth to quasi-linear saturation is obtained if one contains the LCG and inertial terms in the vorticity equation simultaneously. Since LCG is essential for the linear growth rate of tearing mode in the marginal stable regime, the new theory is also valid when the mode close to marginality.

Vorticity Budget and Formation Mechanisms of a Mesoscale Convective Vortex in a Heavy-Rainstorm Episode

Mesoscale convective vortices (MCVs) often cause rainstorms. To deepen our understanding of MCV formation mechanisms, reanalysis data from the National Centers for Environmental Prediction and the Weather Research and Forecasting model were used to simulate MCV activity in East China in August 2009. The simulations could reproduce the MCV and associated convective activities well. The vorticity budget and MCV formation mechanisms were then analyzed. The results show that the planetary vorticity advection is much smaller than other terms of the vorticity equation. The MCV initiates in the convective precipitation region and below 800 hPa. When the MCV initiates, there are vorticity-variation couplets within the vortex, and the MCV moves towards the positive vorticity-variation direction. In positive vorticity-variation areas, the divergence term and the tilting term are the vorticity source. The equilibrium response to diabatic heating is one of the forming mechanisms of this MCV. The latent-heating level is relatively low in this MCV case, and the MCV-forming level is also relatively low. Another forming mechanism of this MCV is the tilting of the horizontal vortex tube caused by the upward motion. At the MCV initiation, the perturbation scale of the vortex is found to be larger than the Rossby deformation radius, and thus the MCV could have a long duration.

Pressure-exerted steady laminar flow of an incompressible fluid along a porous parallel-walled channel with an impermeable wall

Abstract This paper is devoted to the pressure-exerted steady laminar flow of an incompressible Newtonian fluid along a parallel-walled horizontal channel with a porous upper wall and an impermeable lower wall. The fluid is sucked or blown through the porous wall, at constant and uniform velocity, orthogonally to the wall. At the same time, an external pressure gradient constant in time is applied between the two ends of the channel. The aim of this work is to determine and analyze the effects of the external pressure gradient on the flow, the suction/blowing velocity being kept constant. The two-dimensional configuration of the flow with zero-divergence velocity field allows the existence of the stream function given by a single nonlinear partial differential equation which replaces the Navier–Stokes equations and is called the vorticity equation. This latter equation is demonstrated by applying an unusual approach which uses the vector momentum equation in its general form. From the similarity-solutions assumption, it is shown that the vorticity equation leads to a two-point boundary value problem whose solutions are computed by means of a numerical shooting technique including the Newton–Raphson optimization algorithm. Physical understandings of the flow under consideration are derived from the results obtained.