Up-Sliding Slantwise Vorticity Development and the Complete Vorticity Equation with Mass Forcing

The moist potential vorticity (MPV) equation is derived from complete atmospheric equations including the effect of mass forcing, with which the theory of Up-sliding Slantwise Vorticity Development, (USVD) is proposed based on the theory of Slantwise Vorticity Development (SVD). When an air parcel slides up along a slantwise isentropic surface, its vertical component of relative vorticity will develop, and the steeper the isentropic surface is, the more violent the development will he. From the definition of MPV and the MPV equation produced here in, a complete vorticity equation is then put forward with mass forcing, which explicitly includes the effects of both internal forcings, such as variations of stability, baroclinicity, and vertical shear of horizontal wind, arid external forcings, such as diabatic heating, friction, and mass forcing. When isentropic surfaces are flat, the complete vorticity equation matches its traditional counterpart. The physical interpretations of some of the items which are included in the complete- vorticity equation but not in the traditional one are studied with a simplified model of the Changjiang-Huaihe Meiyu front. A 60-h simulation is then performed to reproduce a torrential rain event in the Changjiang-Huaihe region and the output of the model is studied qualitatively based on the theory of USVD. The result shows that the conditions of the theory of USVD are easily satisfied immediately in front of mesoscale rainstorms in the downwind direction, that is, the theory of USVD is important to the development and movement of these kinds of systems.

Derivation of baroclinic Ertel–Rossby invariant-based thermally-coupled vorticity equation in moist flow

For the potential vorticity （PV） invariant, there is a PV-based complete-form vorticity equation, which we use heuris- tically in the present paper to answer the following question： for the Ertel-Rossby invariant （ERI）, is there a corresponding vorticity tendency equation？ Such an ERI-based thermally-coupled vorticity equation is derived and discussed in detail in this study. From the obtained new vorticity equation, the vertical vorticity change is constrained by the vertical velocity term, the term associated with the slope of the generalized momentum surface, the term related to the horizontal vorticity change, and the baroclinic or solenoid term. It explicitly includes both the dynamical and thermodynamic factors＇ influence on the vorticity change. For the ERI itself, besides the traditional PV term, the ERI also includes the moisture factor, which is excluded in dry ERI, and the term related to the gradients of pressure, kinetic energy, and potential energy that reflects the fast-manifold property. Therefore, it is more complete to describe the fast motions off the slow manifold for severe weather than the PV term. These advantages are naturally handed on and inherited by the ERI-based thermally-coupled vorticity equation. Then the ERI-based thermally-coupled vorticity equation is further transformed and compared with the traditional vorticity equation. The main difference between the two equations is the term which describes the contribution of the solenoid term to the vertical vorticity development. In a barotropic flow, the solenoid term disappears, then the ERI-based thermally-coupled vorticity equation can regress to the traditional vorticity equation.

APPLICATION OF ADVECTIVE VORTICITY EQUATION TO TYPHOON FUNG-WONG

The momentum advection vorticity equation in the form of cross multiplication is introduced, in which the divergence term in the classic vorticity equation does not appear explicitly. This equation includes the rotation effect of the horizontal wind advection, which are not explicitly included in the classic vortieity equation. The vorticity and its tendency of Typhoon Fung-Wong （0808） that occurred in July 2008 are analyzed. The computed results show that the rotation effect of the advection of the horizontal wind is a leading factor in determining the change of vertical vorticity for Fung-Wong during its life cycle, especially in the period leading up to landfall. The advection term represents the tendency variation of the vertical vortieity, and the positive-value region of the vertical vorticity tendency is almost in accord with the track of Fung-Wong, which may be taken as a factor to locate the key observational region of Fung-Wong. The equation provides a supplementary diagnostic tool for the systems related with strong advection of horizontal wind.

Symmetry analysis and explicit solutions of the (3+1)-dimensional baroclinic potential vorticity equation

This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.

Symmetry Analysis of Barotropic Potential Vorticity Equation

Recently F. Huang [Commun. Theor. Phys. 42 （2004） 903] and X. Tang and P.K. Shukla [Commun. Theor. Phys. 49 （2008） 229] investigated symmetry properties of the barotropic potential vorticity equation without forcing and dissipation on the beta-plane. This equation is governed by two dimensionless parameters, F and β, representing the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth＇ angular rotation, respectively. In the present paper it is shown that in the case F ≠ 0 there exists a well-defined point transformation to set β = 0. The classification of one- and two-dimensional Lie subalgebras of the Lie symmetry algebra of the potential vorticity equation is given for the parameter combination F ≠ 0 and β = 0. Based upon this classification, distinct classes of group-invariant solutions are obtained and extended to the case β ≠0.

A Note on Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

Applying the classical Lie symmetry approach to the barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a β-plane channel, we find a new symmetry, which is not presented in a previous work [F. Huang, Commun. Theor. Phys. （Beijing, China） 42 （2004） 903]. A general finite transformation group is obtained based on the full Lie point symmetry, Furthermore, two new types of similarity reduction solutions are obtained.

Three types of generalized Kadomtsev-Petviashvili equations arising from baroclinic potential vorticity equation

By means of the reductive perturbation method, three types of generalized （2＋l）-dimensional Kadomtsev- Petviashvili （KP） equations are derived from the baroclinic potential vorticity （BPV） equation, including the modified KP （mKP） equation, standard KP equation and cylindrical KP （cKP） equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.

Bcklund Transformations and Explicit Solutions of (2+1)-Dimensional Barotropic and Quasi-Geostrophic Potential Vorticity Equation

By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the （2＋1）-dimensional BQGPV equation. Some figures are also given out to describe these solutions.

Multiscale interaction analysis of Landfall Typhoon Lekima(2019)based on vorticity equation diagnosis

To investigate the multiscale interaction characteristics of Landfall Typhoon Lekima(2019),this study analyzed the characteristics of the different scale vortex structure and interactions among different scales based on vorticity equation diagnosis.The analysis is based on the simulation results of the WRF model which has been thoroughly verified.The main results are as follows:the original vorticity dominated by the meso-αscale vorticity increases with height and then decreases,with maximum vorticity distributed at 900 hPa.The meso-βscale vorticity varies significantly with altitude,while the meso-γscale vorticityfield exhibits obvious positive vorticity below 850 hPa.The meso-αscale vorticity tendency primarily maintains negative,contributing significantly to the overall reduction in the original vorticityfield over time.The increase in mid-to-upper-level(above 550 hPa)original vorticity is mainly related to the variations in the meso-βand meso-γscale vorticityfields.The original vorticity dominated by the meso-αscale vorticity increases with height and then decreases,and the whole layer vorticity decreases over time.The meso-βscale vorticity varies significantly with altitude and time,while the meso-γscale vorticityfield consistently exhibits significant positive vorticity below 850 hPa.The vorticity equation diagnosis revealed that the primary source terms of the vorticity tendencies are the twisting and stretching terms,and the main sink terms being horizontal and vertical vorticity transport terms below 900 hPa.The source terms and sink terms exchange above 850 hPa.Scale separation results show that the primary contributions of all impact factors originate from the meso-αand meso-γscalefields(accounting for over 80%of the total),with the contribution of the meso-αscale being less than that of the meso-γscale and a notable contribution over 35.5%of the interactions between different scales.

Discussion on the Complete-Form Vorticity Equation and Slantwise Vorticity Development

The complete form of the vertical vorticity tendency equation （the complete-form vorticity equation） is derived from the Ertel potential vorticity equation to contain thermodynamic factors. In this study, a new complete-form vorticity equation, which has the same form as the original complete-form vorticity equation, is deduced from the absolute vorticity vector equation combined with the continuity equation and the expression of three-dimensional （3D） entropy gradient. By comparing the complete-form vorticity equation with the classical vertical vorticity equation, it is found that regardless of whether or not the isentropic surface is tilting, the two vorticity equations are in essence the same. The ＂baroclinic term＂ of the complete-form vorticity equation is exactly equal to the solenoidal term of the classical one, and there is a significant amount of cancellation between the two baroclinic items （the ＂slantwise term＂ and the horizontal vorticity change term） in the complete-form vorticity equation. In operational weather analysis, the tilt of the isentropic surface can be diagnosed according to the density of the isotherm on the upper-level isobaric map. For synoptic-scale motion, the vertical vorticity produced by the tilt of the isentropic surface is due to the contribution of atmospheric baroclinicity, which is measured by the solenoid. The 3D solenoid is parallel to the isentropic surface, so the more tilted the isentropic surface, the bigger the projection of the 3D solenoid in the vertical direction. The baroclinic contribution can be interpreted based on the PV thinking theory, but the relationship between the vorticity field and the potential vorticity field is not immediate.

ANALYTICAL SOLUTION OF NONLINEAR BAROTROPIC VORTICITY EQUATION

The C^k （k ≥ 3） stability of nonlinear barotropic vorticity equation was proved. The necessary and sufficient conditions for the initial value problem to be well-posed were presented. Under the conditions of well-posedness, the corresponding analytical solution was also gained.

FOURIER-CHEBYSHEV PSEUDOSPECTRAL METHOD FOR THREE-DIMENSIONAL VORTICITY EQUATION

In this paper, a Fourier-Chebyshev pseudospectral scheme with mixed filtering is proposed for three-dimensional vorticity equation. The generalized stability and convergence are proved. The numerical results show the advantages of this method.

A PSEUDOSPECTRAL METHOD FOR VORTICITY EQUATIONS ON SPHERICAL SURFACE

The pseudospectral method for solving vorticity equations on spherical surface is discussed. An interpolation procedure, which is different from the usual onesl is proposed. Based on such an interpolation, the pseudospectral scheme is constructed. Its generalized stability and convergence are analyzed rigorously. The theoretical analysis and computational skills can also be applied to other nonlinear partial differential equations defined on spherical surface.

SPECTRAL-FINITE ELEMENT METHOD FOR SOLVING TWO-DIMENSIONAL VORTICITY EQUATIONS

In this paper, we construct a spectral-finite element scheme for solving semi-periodical two-dimensional vorticity equations. The error between the genuine solution and approximate solutionis estimated strictly. The numerical results show the advantages of such a method. The techniqueused in this paper can be easily generalized to three-dimensional problems.

PSEUDOSPECTRAL－FINITE DIFFERENCE METHOD FOR THREE－DIMENSIONAL VORTICITY EQUATION WITH UNILATERALLY PERIODIC BOUNDARY CONDITION

A Fourier pseudospectral-finue difference scheme is proposed for three-dimensional vorticity. equalion with unilalerally periodic boundary. condilion. Thegeneralized stability. and conrergence are analyzed. The nunierical results show. theadvantage of this method.

CHEBYSHEV SPECTRAL-FINITE ELEMENTMETHOD FOR THREE-DIMENSIONALVORTICITY EQUATION

Combined Chebyshev spectral-finite element schemes are constructed for three-dimensionalunsteady vorticity equation. The generalized stability and convergence are proved strictly.

THE RELATIONSHIP BETWEEN HORIZONTAL VORTICITY INDUCED BY VERTICAL SHEAR AND VERTICAL MOTION DURING A SQUALL LINE PROCESS

The horizontal vorticity equation used in this study was obtained using the equations of motion in the pressure coordinate system without considering friction, to reveal its relationship with vertical shear. By diagnostically analyzing each term in the horizontal vorticity equation during a squall line process that occurred on 19 June 2010, we found that the non-thermal wind term had a negative contribution to the local change of upward movement in the low-level atmosphere, and that its impact changed gradually from negative to positive with altitude, which could influence upward movement in the mid-and upper-level atmosphere greatly. The contribution of upward vertical transport to vertical movement was the largest in the low-level atmosphere, but had negative contribution to the upper-level atmosphere. These features were most evident in the development stage of the squall line. Based on analysis of convection cells along a squall line, we found that in the process of cell development diabatic heating caused the subsidence of constant potential temperature surface and non-geostrophic motion, which then triggered strong convergence of horizontal acceleration in the mid-level atmosphere and divergence of horizontal acceleration in the upper-level atmosphere. These changes of horizontal wind field could cause a counterclockwise increment of the horizontal vorticity around the warm cell, which then generated an increase of upward movement. This was the main reason why the non-thermal wind term had the largest contribution to the strengthening of upward movement in the mid-and upper-level atmosphere. The vertical transport of large value of horizontal vorticity was the key to trigger convection in this squall line process.

A NOVEL METHOD FOR CALCULATING VERTICAL VELOCITY:A RELATIONSHIP BETWEEN HORIZONTAL VORTICITY AND VERTICAL MOVEMENT

The present work provides a novel method for calculating vertical velocity based on continuity equations in a pressure coordinate system.The method overcomes the disadvantage of accumulation of calculating errors of horizontal divergence in current kinematics methods during the integration for calculating vertical velocity,and consequently avoids its subsequent correction.In addition,through modifications of the continuity equations,it shows that the vorticity of the vertical shear vector(VVSV) is proportional to-ω,the vertical velocity in p coordinates.Furthermore,if the change of ω in the horizontal direction is neglected,the vorticity of the horizontal vorticity vector is proportional to-ω.When ω is under a fluctuating state in the vertical direction,the updraft occurs when the vector of horizontal vorticity rotates counterclockwise;the downdraft occurs when rotating clockwise.The validation result indicates that the present method is generally better than the vertical velocity calculated by the ω equation using the wet Q-vector divergence as a forcing term,and the vertical velocity calculated by utilizing the kinematics method is followed by the O'Brien method for correction.The plus-minus sign of the vertical velocity obtained with this method is not correlated with the intensity of d BZ,but the absolute error increases when d BZ is >=40.This method demonstrates that it is a good reflection of the direction of the vertical velocity.

Effect of condensation latent heat release on the relative vorticity tendency in extratropical cyclones:a case study

Taking an extratropical cyclone that produced extreme precipitation as the research object,this paper calculates the contribution of condensation latent heat release(LHR)to relative vorticity tendency based on the complete-form vertical vorticity tendency equation.The results show that the heating rate of convectional condensation LHR can reach up to about 40 times that of stable condensation LHR.Both the stable and convectional heating centers are higher than 700 hPa,which would cause∂Q/∂z>0 and a positive vorticity source in the lower troposphere.The vertical gradient of stable condensation LHR contributes little to the growth of relative vorticity,while the relative vorticity tendency associated with the vertical gradient of convectional condensation LHR can be an order of magnitude higher than the former.The positive vorticity source is always located right below the latent heating center,and its maximum value can always be found in the lower troposphere.Convectional LHR is the primary factor for cyclone development from the perspective of diabatic heating.The horizontal gradient of total condensation LHR can contribute about 65%of the actual vorticity growth,but the effect of the vertical gradient of convectional condensation(LHR)can reach twice as much.The adiabatic heating from LHR can cause vorticity tendency directly.However,it can also change the vertical and horizontal gradient of potential temperature,which can further induce vorticity tendency.

Availability of the Boundary Condition for Vorticity

By applying a boundary condition for vorticity [1] in addition to that for velocity, a velocity distribution on a flat plate set in a parallel homogeneous flow has been numerically obtained through a one-way calculation from surface to infinity, without the “matching” procedure between an analysis from surface to infinity and that from infinity to surface. The numerical results obtained were in excellent agreement with those by Howarth [2]. The usage of the boundary condition for vorticity has raised the accuracy of velocity distribution near a plate’s surface and made it possible to realize the one-way calculation from surface to infinity.