Boundary Conditions for Momentum and Vorticity at an Interface between Two Fluids

Boundary conditions for momentum and vorticity have been precisely derived, paying attention to the physical meaning of each mathematical expression of terms rigorously obtained from the basic equations: Navier-Stokes equation and the equation of vorticity transport. It has been shown first that a contribution of fluid molecules crossing over a conceptual surface moving with fluid velocity due to their fluctuating motion is essentially important to understanding transport phenomena of momentum and vorticity. A notion of surface layers, which are thin layers at both sides of an interface, has been introduced next to elucidate the transporting mechanism of momentum and vorticity from one phase to the other at an interface through which no fluid molecules are crossing over. A fact that a size of δV, in which reliable values of density, momentum, and velocity of fluid are respectively defined as a volume-averaged mass of fluid molecules, a volume-averaged momentum of fluid molecules and a mass-averaged velocity of fluid molecules, is not infinitesimal but finite has been one of the key factors leading to the boundary conditions for vorticity at an interface between two fluids. The most distinguished characteristics of the boundary conditions derived here are the zero-value conditions for a normal component of momentum flux and tangential components of vorticity flux, at an interface.

The Early Mesozoic NE-SW Extensional Model and Exhumation Processes at the Southeastern Margin of the Central Asian Orogenic Belt:Insights from the Strain and Kinematic Vorticity Analysis of the Sonid Zuoqi Ductile Detachment Zone

The Sonid Zuoqi ductile detachment zone is located at the southeastern margin of the Central Asian orogenic belt(CAOB),striking EW and dipping to the S.The major rock type of the Sonid Zuoqi ductile detachment zone is mylonite derived from granite.The sequence of mylonite features is:(1)S and C foliations of mylonite,and(2)extensional crenulation cleavage(ecc)or C′and the kinematic vorticity(Wk)value changed from 0.70 to 0.95 and from 0.37 to 0.69,respectively;the strain type of the mylonites within the Sonid Zuoqi ductile detachment zone is compressional to planar strain.The strong deformation mylonite and Halatu plutons yielded a zircon U-Pb age of 244 Ma and a zircon(U-Th)/He age of 214 Ma,respectively.Based on the strain and kinematic vorticity analysis,together with the zircon U-Pb and zircon(U-Th)/He ages and the regional tectonic background,the study area experienced three stage evolution:tangential simpleshear(244 Ma),simple-shear-dominated general shear represented by upper crustal extension(224 Ma)and pure-shear-dominated general shear represented by the Halatu pluton doming(214 Ma),which constrained the early Mesozoic NE-SW crustal extension at the southeastern margin of the CAOB.This NE-SW extension probably originated from the postorogenic extensional collapse of the CAOB,subsequent exhumation being controlled by the far afield effects of the closure of the Mongol-Okhotsk belt.

Extreme Cold Events in North America and Eurasia in November-December 2022: A Potential Vorticity Gradient Perspective

From 17 November to 27 December 2022, extremely cold snowstorms frequently swept across North America and Eurasia. Diagnostic analysis reveals that these extreme cold events were closely related to the establishment of blocking circulations. Alaska Blocking(AB) and subsequent Ural Blocking(UB) episodes are linked to the phase transition of the North Atlantic Oscillation(NAO) and represent the main atmospheric regimes in the Northern Hemisphere. The downstream dispersion and propagation of Rossby wave packets from Alaska to East Asia provide a large-scale connection between AB and UB episodes. Based on the nonlinear multi-scale interaction(NMI) model, we found that the meridional potential vorticity gradient(PVy) in November and December of 2022 was anomalously weak in the mid-high latitudes from North America to Eurasia and provided a favorable background for the prolonged maintenance of UB and AB events and the generation of associated severe extreme snowstorms. However, the difference in the UB in terms of its persistence,location, and strength between November and December is related to the positive(negative) NAO in November(December). During the La Ni?a winter of 2022, the UB and AB events are related to the downward propagation of stratospheric anomalies, in addition to contributions by La Ni?a and low Arctic sea ice concentrations as they pertain to reducing PVyin mid-latitudes.

LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY

In this paper we investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed.We assume that the free surface is almost periodic in the horizontal direction.Using conformal mappings,one can change the free boundary problem into a fixed boundary problem for some unknown functions with the boundary condition.By virtue of the Hilbert transform,the problem is equivalent to a quasilinear pseudodifferential equation for an almost periodic function of one variable.The bifurcation theory ensures that we can obtain an existence result.Our existence result generalizes and covers the recent result in[15].Moreover,our result implies a non-uniqueness result at the same bifurcation point.

Variation of Dynamical Parameters with Upper Tropospheric Potential Vorticity in Tropical Cyclone over the North Indian Ocean Using WRF Model

Meteorologists are experiencing many challenges in the reliable forecasting of the track and intensity of tropical cyclones(TC).Uses of the potential vorticity(PV)technique will enrich the current forecasting system.The use of PV analysis of TC intensification over the North Indian Ocean(NIO)is rare.In this study,the authors analyze the behaviour of upper-level PV with dynamic parameters of TCs over NIO.The authors used NCEP FNL reanalysis 1×1 degree data as input in WRF model version 4.0.3 with one-way nesting between the parent and child domains.The authors used a coupling of the Kain-Fritsch(new Eta)scheme and the WSM 6-class graupel scheme as cumulus and microphysics options to run the model.The authors found that at least one potential vorticity unit(PVU)(1 PVU=10^(-6) m^(2)s^(-1)KKg^(-1))upper PV is required to maintain the intensification of TC.Larger upper PV accelerates the fall of central pressure.The high value of upper PV yields the intensification of TC.The wind shear and upper PV exhibited almost identical temporal evolution.Upper PV cannot intensify the TCs at negative wind shear and shear above the threshold value of 12 ms^(-1).The upper PV and geopotential heights of 500 hPa change mutually in opposite trends.The upper PV calculated by the model is comparable to that of ECMWF results.Therefore,the findings of this study are admissible.

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.

DNS Study on Volume Vorticity Increase in Boundary Layer Transition

The issue whether transition from laminar flow to turbulent flow on a flat plate should be characterized as a vorticity redistribution process or a vorticity increasing process is investigated by a high-order direct numerical simulation on a flat plate boundary layer.The local vorticity can either increase or decrease due to tilting and stretching of vortex filaments according to the vorticity transport equation while the total vorticity cannot be changed in a boundary layer flow in conforming to the Fppl theorem of total vorticity conservation.This seemingly contradictory problem can be well resolved by the introduction of a new term:volume vorticity of a vorticity tube,defined as vorticity flux timed by the vorticity tube length.It has been shown that,although vorticity flux must keep conserved,the total volume vorticity is significantly increased during boundary layer transition according to our direct numerical simulation(DNS)computation,which directly results from the lengthening(stretching and tilting)of vortex filaments.Therefore,the flow transition is a process with appreciable increase of volume vorticity,and cannot be only viewed as a vorticity redistribution process.

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.

Potential Vorticity Structure and Inversion of the Cyclogenesis Over the Yangtze River and Huaihe River Valleys

In this paper, the potential vorticity structure and inversion of the cyclogenesis over the Yangtze River and Huaihe River valleys during 21 23 June 2003 are investigated with a potential vorticity （PV） framework. The cyclogenesis is manifested by a lower-tropospheric PV anomaly over the Yangtze River and Huaihe River valleys at early stages mainly due to latent heat release, which greatly affects the evolution of the associated lower-tropospheric geopotential height and wind fields as demonstrated by piecewise PV inversion. At later stages, an upper-tropospheric PV anomaly develops, resulting in the growth of ridges over the cyclone in both the upstream and downstream, which provide a favorable background field for the low-level cyclone development. But the effect of a surface thermal anomaly always impedes the development of the cyclone to different extents during this cyclogenesis. It is further demonstrated that the position and the strength of the PV anomaly are closely related to the low-level cyclone development, and the lower-tropospheric PV anomaly seems to constitute the most significant feature, for instance, contributing about 60% to the low-level jet （LLJ）.

THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.

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.

ANALYSIS OF EFFECT OF ENVIRONMENTAL POTENTIAL VORTICITY LOCATED AT SOUTH EDGE OF SOUTH ASIA HIGH ON INTENSIFICATION OF TROPICAL CYCLONES

The NCEP 1°×1°reanalysis of June-to-September dataset between 2002 to 2009 is used in this study to conduct statistical analysis of the relationship between the environmental potential vorticity(PV)on 150 hPa located at the south edge of South Asia High(SAH)and TCs making landfall.The results show that 23 of the TCs are affected by the PV on 150 hPa located at the south edge of SAH between 2002 to2009,and three TCs'center pressure decline after the high-value environmental PV moves to the center of the TCs.These three TCs are Senlaku(0216),Bilis(0604)and Linfa(0903).Through diagnostic analysis from the viewpoint of isolines,we determined the relationship between the intensification of these TCs and the PV anomaly at high levels;the isentropic surface is close to the high level’s PV anomaly under the influence of the 150-hPa PV anomaly,leading to the decline of isentropic surfaces on both sides of the PV anomaly.Then the warm core of the middle and high levels of the TC strengthens and PV increases at the middle level,and both of them are beneficial to the reinforcement of the cyclonic vorticity in the low level.As a result,the center pressure of the TC declines.According to Wu’s theory of Slantwise Vorticity Development(SVD),the incline of the isentropic surfaces leads to the development of vertical vorticity,contributing to the vertical motion and the release of the latent heat.Then the warm core of the TC strengthens and the TC strengthens,too.Otherwise,piecewise PV inversion also shows that the high-level PV influences the mid-level more than the low level.

On the Significance and Use of the Generalized Moist Potential Vorticity Equation

In this paper,the continuity and thermodynamic equations including moisture forcings were derived.Using these two equations and the basic momentum equation of local Cartesian coordinates,the budget equation of generalized moist potential vorticity(GMPV) was derived.The GMPV equation is a good generalization of the Ertel potential vorticity(PV) and moist potential vorticity(MPV) equations.The GMPV equation is conserved under adiabatic,frictionless,barotropic,or saturated atmospheric conditions,and it is closely associated with the horizontal frontogenesis and stability of the real atmosphere.A real case study indicates that term diabatic heating could be a useful diagnostic tool for heavy rainfall events.

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.

Impacts of Cloud-Induced Mass Forcing on the Development of Moist Potential Vorticity Anomaly During Torrential Rains

The impacts of cloud-induced mass forcing on the development of the moist potential vorticity (MPV) anomaly associated with torrential rains are investigated by using NCEP/NCAR 1? × 1? data. The MPV tendency equation with the cloud-induced mass forcing is derived, and applied to the torrential rain event over the Changjiang River-Huaihe River Valleys during 26–30 June 1999. The result shows that positive anomalies are located mainly between 850 hPa and 500 hPa, while the maximum MPV, maximum positive tendency of the MPV, and maximum surface rainfall are nearly collocated. The cloud-induced mass forcing contributes to the positive tendency of the moist potential vorticity anomaly. The results indicate that the MPV may be used to track the propagation of rain systems for operational applications.

CFD Simulation of Flow Features and Vorticity Structures in Tuna-Like Swimming

The theoretical research on the propulsive principle of aquatic animal becomes more important and attracted more researchers to make efforts on it. In the present study, a computational fluid dynamic （CFD） simulation of a three-dimensional traveling-wave undulations body of tuna has been developed to investigate the fluid flow features and vorticity structures around this body when moving in a straight line. The undulation only takes place in the posterior half of the fish, and the tuna-tail is considered as a lunate fin oscillating with the mode combined swaying with yawing. A Reynolds-averaged Navier-Stokes （RANS） equation is developed, employing a control-volume method and a k-omega SST turbulent model; meanwhile an unstructured tetrahedral grid, which is generated for the three-dimensional geometry, is used based on the deformation of the hind parts of the body and corresponding movement of the tail. We calculated the hydrodynamic performance of tuna-like body when a tuna swims in a uniform velocity, and compared the input power coefficient, output power coefficient and propulsive efficiency of the oscillating tuna-tail with or without body vortex shedding. Additionally, the load distribution on the body, flow features and vorticity structures around the body were demonstrated. The effect of interaction between the body-generated vortices and the tail-generated vorticity on the hydrodynamic performance can be obtained.

Study on Moist Potential Vorticity and Symmetric Instabilityduring a Heavy Rain Event Occurred inthe Jiang-Huai Valleys

In the light of the theory on moist potential vorticity (MPV) investigation was undertaken of the 700 hPa vertical (horizontal) component MP1 (MPV2) for the heavy rain event occurring in July 5–6, 1991. Results show that the distribution features of the two components were closely related to the development of a mesoscale cyclone as a rainstorm-causing weather system in the lower troposphere in such a way that the ambient atmosphere of which MPV1 > 0 and MPV2 < 0 with |MPV1| ≥ |MPV2| favored the genesis of conditional symmetric instability (CSI) and that, as indicated by calculations, a CSI sector was really existent in the lower troposphere during the heavy rain happening and contributed greatly to its development.

Vorticity Budget Investigation of a Simulated Long-Lived Mesoscale Vortex in South China

A vorticity budget investigation is performed using the output data from a numerical simulation of a typical MCV (mesoscale convectively generated votex) case in South China. Results suggest that the divergence caused by convection in the low troposphere is the main producer of positive vorticity, while vertical vorticity transferred by the tilting term from the horizontal vorticity compensates the upward output of cyclonic vorticity. Scale analyses of the vorticity equation suggest that the advection of planetary vorticity can be neglected owing to the low latitude, which is di?erent from the larger scale systems in high latitude areas. In addition, the distribution of relative vorticity tendency on pressure level is not uniform. A vortex will move along the vector from the negative to the positive vorticity tendency region. The mechanism of the phenomenon—that nearly all of the convectively ascending region is located southward/southeastward of the vortex center—is also discussed. Convergence with regard to latent heat release would be in favor of the spin-up of meso-vortex, however, the horizontal vorticity caused by wind shear is tilted by vertical motion due to convection. Consequently, the negative and positive vorticity tendencies are located symmetrically about the convective center, which suggests that the vortex southward movement is dynamically driven by convection.

Diagnostic Analyses of the Modified Convective Vorticity Vector in Non-uniformly Saturated Moist Flow

Because the real atmosphere is non-uniformly saturated, the generalized potential temperature is introduced. The convective vorticity vector, which can depict the occurrence and development of mesoscale deep convective systems, is modified and re-derived in a nonuniformly saturated moist atmosphere (C*). Then, a case study is performed for a frontal rainfall event which occurred near the middle and lower reaches of the Yangtze River in China. The diagnostic results of C* show that, in the lower troposphere, the vertical component of C* (Cz*) can diagnose developments and movements of precipitation and convection better than those of Cm (Cmz, in saturated moist flow) and C (Cz, in dry flow). Cz* is a good predictor for precipitation analyses as well.

DNS in evolution of vorticity and sign relationship in wake transition of a circular cylinder:(pure)mode A

In the present paper,the spatio-temporal evolution of vorticity in the first wake instability,i.e.,(pure)mode A,is investigated in order to understand the wake vortex dynamics and sign relationships among vorticity components.Direct numerical simulation(DNS)for the flow past a circular cylinder is performed,typically at a Reynolds number of 200,in the three-dimensional(3-D)wake transition.According to characteristics of time histories of fluid forces,three different stages are identified as the computational transition,the initial stage and fully developed wake.In the second initial stage,the original two-dimensional spanwise vortices become obviously three-dimensional associated with the streamwise or vertical vorticity intensified up to about 0.1.As a matter of fact,these additional vorticities,caused by the intrinsic 3-D instability,are already generated firstly on cylinder surfaces early in the computational transition,indicating that the three-dimensionality appeared early near the cylinder.The evolution of additional components of vorticity with features the same as mode A shows that(pure)mode A can be already formed in the late computational transition.Through careful analysis of the vorticity field on the front surface,in the shear layers and near wake at typical times,two sign laws are obtained.They illustrate intrinsic relationships among three vorticity components,irrelevant to the wavelength or Fourier mode and Reynolds number in(pure)mode A.Most importantly,the origin of streamwise vortices is found and explained by a new physical mechanism based on the theory of vortex-induced vortex.As a result,the whole process of formation and shedding vortices with these vorticities is firstly and completely illustrated.Other characteristics are presented in detail.