DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL HALF SPACE

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n ＋ ： u tt u ＋ u t ＋ divf （u） = 0, t 〉 0, x = （x 1 , x ′ ） ∈ R n ＋ ：= R ＋ × R n 1 , u（0, x） = u 0 （x） → u ＋ , as x 1 → ＋ ∞ , u t （0, x） = u 1 （x）, u（t, 0, x ′ ） = u b , x ′ = （x 2 , x 3 , ··· , x n ） ∈ R n 1 . （I） For the non-degenerate case f ′ 1 （u ＋ ） 〈 0, it was shown in [10] that the above initialboundary value problem （I） admits a unique global solution u（t, x） which converges to the corresponding planar stationary wave φ（x 1 ） uniformly in x 1 ∈ R ＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u（t, x） for t → ＋ ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ （u） 〈 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 （u b ） ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.

Stationary Wave Activity Associated with the East Asian Winter Monsoon Pathway

The pathway of the East Asian winter monsoon(EAWM) that usually leads to the out-of-phase pattern of surface air temperature between northern and southern East Asia is an important feature in the variability of the EAWM besides its strength.Using the European Centre for Medium-Range Weather Forecasts 40-year(ERA40) reanalysis dataset,this study investigates the pathway-related stationary wave activity to explore the mechanism of the interannual variations in the EAWM pathway.It reveals that when the southern pathway of the EAWM is strong,the phase of the climatological stationary wave tends to be shifted westward significantly in both the horizontal and vertical directions by an anomalous wavenumber 2 pattern at mid-latitudes,whereas the changes are relatively small in the subtropics.The horizontal changes in the stationary wave phase facilitate a north-south-oriented East Asian trough in the middle troposphere that eventually produces the strong southern pathway of the EAWM.The vertical changes in the stationary wave,in contrast,feature a westward-tilted phase line with height over the North Pacific,indicating enhanced upward propagation of waves into the stratosphere.This result suggests that the phase of stationary waves at mid-latitudes dominate the interannual variations in the EAWM pathway.Moreover,it supports our previous interpretation of the possible role of the North Pacific sea surface temperature(SST) in the EAWM pathway variability.It also implies that the excitation of anomalous mid-latitude stationary waves may be the key in the response of the EAWM pathway to the North Pacific SST.

ANALYSES OF THE DYNAMIC EFFECTS ON WINTER CIRCULATION OF THE TWO MAIN MOUNTAINS IN THE NORTHERN HEMISPHERE——Ⅰ.RELATIONSHIP AMONG GENERAL CIRCULATION,TELECONNECTION AND STATIONARY WAVES

The distribution of troughs and ridges of geopotential height,the teleconnection patterns and the propagation pat- terns of stationary waves are the main features of the January mean geopotential height field at 500hPa.Data analyses and numerical experiments indicate that these three characteristics are associated to one another and closely related to the mechanical forcing of the Rocky Mountains and Tibetan Plateau.There exists a prominent negative correlation in the intensity variation between the American trough and the Asian trough at high and middle latitudes.Such negative correlation,in connection with the interannual variation of the intensity of the jets in front of the two troughs,leads to the existence of similar teleconnection patterns in North America and East Asia.On the other hand,the different propa- gation behaviour of quasi-stationary waves downstream of the two main mountains results in a fundamental difference in the distribution of correlation chains in North America and East Asia.

DECAY RATES TOWARD STATIONARY WAVES OF SOLUTIONS FOR DAMPED WAVE EQUATIONS

Abstract This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R＋{utt-txx＋ut＋f（u）x=0,t〉0,x∈R＋,u（0,x）=u0（x）→u＋,asx→＋∞,ut（0,x）=u1（x）,u（t,0）=ub.For the non-degenerate case f]（u＋） 〈 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u（t,x） which converges to the stationary wave φ（x） uniformly in x ∈ R＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates （including the algebraic convergence rate and the exponential convergence rate） of u（t, x） toward φ（x） are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f＇（ub） 〈 0. The main purpose of this paper is devoted to discussing the case of f＇（ub）= 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.

THE FORCING OF ANOMALOUS DIVERGENT WIND AND TRANSIENT VORTICITY FLUXES TO THE ANOMALY OF STATIONARY WAVES

The forcing of anomalous divergent wind and transient vorticity fluxes to the anomaly of stationary waves of DJF (December—January—February) 1982/1983 is studied.The results show that the anomaly of the transient vorticity flux convergence tends to cancel or dissipate the anomaly of the vorticity sources generated by divergent wind.The stationary wave anomalies are maintained by both of these forcings.It is also found that,for the DJF 1982/1983 case,both of the vorticity source anomaly over the tropical and equatorial eastern Pacific and that over the midlatitude North Pacific are important to maintain the atmospheric circulation anomaly over the Pacific/North America region.

SPATIAL AND TEMPORAL VARIATIONS OF THE KINEMATIC CHARACTERISTICS IN THE PROPAGATION OF ATMOSPHERIC STATIONARY WAVES

Time-mean global general circulation data are employed to analyze the temporal and spatial variations of the meridional gradient of zonal mean potential vorticity,the critical wavenumber n_s for horizontal wave- propagation,and the critical wavenumber K_c for vertical wave-propagation.Thereby the kinematic charac- teristics in the propagation of atmospheric stationary waves and their annual variations are studied.Results show that in the troposphere n_s and K_c usually decrease with the increase of either latitude or altitude. Synoptic and near-resonant Rossby waves could be trapped during their upward and meridional propagations. These characteristics possess prominent annual variations,especially in the Northern Hemisphere.It is found that the spatial and temporal variations of these kinematic characteristics are in good agreement with those of the atmospheric wave patterns.

THE ROLE OF THE MERIDIONAL CIRCULATION IN STATIONARY WAVE PROPAGATION

In a general baroclinic atmosphere,when the basic state includes meridional circulation,the sta- tionary waves might not only pass through the equatorial easterlies,but also strengthen significantly. The orographic forcing in the Northern Hemisphere mid-latitude might cause marked responses in the low latitude atmosphere.This suggests that the meridional circulation plays an important role in the connection of stationary responses in mid and low latitudes,and so does the heating forcing in the Northern Hemisphere mid-latitude.Forced by the heating forcing in the Northern Hemisphere mid- latitude,the features similar to the Northern Hemisphere summer monsoon circulation can be ob- tained.It appears that the meridional circulation plays certain role in the formation of summer mon- soon circulation.The heating anomaly forcing located at the eastern equatorial Pacific makes the sta- tionary waves present PNA(Pacific-North America)pattern in the winter hemisphere,but it does not in the summer hemisphere.It suggests that the meridional circulation has a marked influence on the route of stationary wave propagation both in the winter and summer hemispheres.

The East Pacific Wavetrain:Its Variability and Impact on the Atmospheric Circulation in the Boreal Winter

The East Pacific wavetrain （EPW） refers to here the intense stationary wave activity detected in the troposphere over the East Pacific and North America in 45 northern winters from 1958 to 2002. The EPW is generated in the lower troposphere over the East Pacific, propagating predominantly eastward into North America and slightly upward then eventually into the stratosphere. The intensity of the EPW varies from year to year and exhibits apparent decadal variability. For the period 1958-1964, the EPW was in its second maximum, and it was weakest for the period 1965-1975, then it was strongest for the period 1976-1987. After 1987, the EPW weakened again. The intensity and position of the members （i.e., the Aleutian low, the North American trough, and the North American ridge） of the EPW oscillate from time to time. For an active EPW versus a weak EPW, the Aleutian low deepens abnormally and shifts its center from the west to the east of the date line, in the middle and upper troposphere the East Asian trough extends eastward, and the Canadian ridge intensifies at the same time. The opposite is true for a weak EPW. Even in the lower stratosphere, significant changes in the stationary wave pattern are also observed. Interestingly the spatial variability of the EPW assumes a Pacific-North American （PNA）-like telecon- nection pattern. It is likely that the PNA low-frequency oscillation is a reflection of the oscillations of intensity and position of the members of the EPW in horizontal direction.

Long-term variation of Arctic Sudden Stratospheric Warmings(SSW)and potential causes

Utilizing the European Centre for Medium-Range Weather Forecast(ECMWF)Reanalysis v5(ERA5),for the first time,we have confirmed close links among Sudden Stratospheric Warmings(SSWs)in the Northern Hemisphere(NH),the polar vortices,and stratospheric Planetary Waves(PWs)by analyzing and comparing their trends.Interestingly,within overall increasing trends,the duration and strength of SSWs exhibit increasing and decreasing trends before and after the winter of 2002,respectively.To reveal possible physical mechanisms driving these trends,we analyzed the long-term trends of the winter(from December to February)polar vortices and of stratospheric PWs with zonal wave number 1.Notably,our results show that in all three time periods(the entire period of 41winters,1980 to 2020,and the two subperiods—1980-2002 and 2002-2020)enhancing SSWs were always accompanied by weakening winter polar vortices and strengthening polar PWs like Stationary Planetary Waves(SPWs)and 16-day waves,and vice versa.This is the first proof,based on ERA5 long-term trend data,that weakening polar vortices and enhancing stratospheric PWs(especially SPWs)could cause an increase in SSWs.

Generalized Wave Operators in Von Neumann Algebras

Let M⊆B(H)be a countable decomposable properly infinite von Neumann algebra with a faithful normal semifinite tracial weightτwhere B(H)is the set of all bounded linear operators on Hilbert space H.The main purpose of this article is to introduce generalized weak wave operators Wf_(±),generalized weak abelian wave operators ■ and generalized stationary wave operators U_(±) in M and then to explore the relation among W_(±),■,U_(±) and generalized wave operators W_(±).

Spatiotemporal Variations of Summer Rainfall over Eastern China during 1880-1999(1)

By applying rotated complex empirical orthogonal function (RCEOF) analysis on 1880-1999 summer rainfall at 28 selected stations over the east part of China, the spatio-temporal variations of China summer rainfall are investigated. Six divisions are identified, showing strong temporal variability, the middle and lower reaches of the Yangtze River, the Huaihe River, Southeast China, North China, Southwest China, and Northeast China. The locations of all divisions except Southwest China are in a good agreement with those of the rainband which moves northward from Southeast China to Northeast China from June-August. The phase relationship revealed by the RCEOF analysis suggests that rainfall anomalies in the middle and lower reaches of the Yangtze River, Southeast China, and Northeast China are all characterized by a stationary wave, while a traveling wave is more pronounced in the Huaihe River division, North China, and Southwest China. The fourth RCEOF mode indicates that rainfall anomalies can propagate from south of Northeast China across lower reaches of the Huanghe River and the Huaihe River to the lower reaches of the Yangtze River. A 20-25-year oscillation is found at the middle and lower reaches of the Yangtze River, the Huaihe River valley, North China, and Northeast China. The middle and lower reaches of the Yangtze River and Northeast China also show an approximately-60-year oscillation. Northeast China and the Huaihe River division are dominated by a 36-year and a 70-80-year oscillation, respectively. An 11-year oscillation is also evident in North China, with a periodicity similar to sunspot activity. The interdecadal variability in the middle and lower reaches of the Yangtze River, the Huaihe River valley, and North China shows a significant positive correlation with the solar activity.

Interannual Variability of the Winter Stratospheric Polar Vortex in the Northern Hemisphere and Their Relations to QBO and ENSO

We investigated the interannual variations of the winter stratospheric polar vortex in this paper. EOF analysis shows that two modes of variability dominate the stratospheric polar vortex on interannual timescales The leading mode （EOF1） reflects the intensity variation of the polar vortex and is characterized by a geopotential height seesaw between the polar region and the mid-latitudes. The second one （EOF2） exhibits variation in the zonal asymmetric part of the polar vortex, which mainly describes the stationary planetary wave activity. As the strongest interannual variation signal in the atmosphere, the QBO has been shown to influence mainly the strength of the polar vortex. On the other hand, the ENSO cycle, as the strongest interannual variation signal in the ocean, has been shown to be mainly associated with the variation of stationary planetary wave activity in the stratosphere. Possible influences of the stratospheric polar vortex on the tropospheric circulation are also discussed in this paper.

A Comparison of Polar Vortex Response to Pacific and Indian Ocean Warming

During recent decades, the tropical Indo-Pacific Ocean has become increasingly warmer. Meanwhile, both the northern and southern hemispheric polar vortices （NPV and SPV） have exhibited a deepening trend in boreal winter. Although previous studies have revealed that the tropical Indian Ocean warming （IOW） favors an intensifying NPV and a weakening SPV, how the tropical Pacific Ocean warming （POW） influences the NPV and SPV remains unclear. In this study, a comparative analysis has been conducted through ensemble atmospheric general circulation model （AGCM） experiments. The results show that, for the Northern Hemisphere, the two warmings exerted opposite impacts in boreal winter, in that the IOW intensified the NPV while the POW weakened the NPV. For the Southern Hemisphere, both the IOW and POW warmed the southern polar atmosphere and weakened the SPV. A diagnostic analysis based on the vorticity budget revealed that such an interhemispheric difference in influences from the IOW and POW in boreal winter was associated with different roles of transient eddy momentum flux convergence between the hemispheres. Furthermore, this difference may have been linked to different strengths of stationary wave activity between the hemispheres in boreal winter.

The Characteristics of the Shear-Vibrating Quartz Crystal Resonator

<正>According to the piezoelectric equation and the vibration theory of the quartz crystal,the relations between the vibrating frequency and structural parameters under the thickness-shear-vibration of AT-cut quartz crystal have been studied. The frequency conditions under which quartz crystal resonator formed stationary wave inside the electrode district and the transmission characteristics of wave outside the electrode district have also been discussed.A quartz crystal resonator was developed based on this analysis.The experiment showed that the force-sensing characteristics were independent of the fixation of the crystal edge.The detecting distinguish ability was up to 0.001°,and the short-term frequency stability was up to 1.38×10~ (-10)/min.

Role of Atlantic Multidecadal Oscillation(AMO) in winter intraseasonal variability over Ural

The modulation of winter intraseasonal variability(ISV)by the Atlantic Multidecadal Oscillation(AMO)is investigated through three sets of reanalysis data and numerical experiments with the NCEP’s atmospheric general circulation model(AGCM).Results show that the positive phase of the AMO tends to intensify ISV activity over the northern Atlantic and shift ISV activity over the Ural Mountains toward the south,causing weakened ISV activity at 200 hPa in the north to the Urals and intensified activity in the south.The modulation of ISV activity by the AMO over the Urals is then explored through comparison of the composite evolution of anomalous ISV cases for the different AMO phases.Fewer ISV cases are found in the AMO positive phase than the negative phase,but no substantial difference in the temporal evolution of anomalous ISV events between the two opposing phases of the AMO.Thus,the AMO exerts its modulation through influencing the occurrence frequency of ISV events,rather than their development or evolution processes.A similar result is seen in the AGCM sensitivity experiments.

On the Blocking Flow Patterns in the Euro–Atlantic Sector:A Simple Model Study

The flow patterns of Euro-Atlantic blocking events in winter are investigated by dividing the sector into three sub- regions： 60°-30°W （Greenland region）; 20°W-30°E [eastern Atlantic-Europe （EAE） region]; and 50°-90°E （Ural region）. It is shown that blocking events in winter are extremely frequent in the three sub-regions. Composite 500-mb geopotential height fields for intense and long-lived blocking events demonstrate that the blocking fields over Greenland and Ural regions exhibit southwest-northeast （SW-NE） and southeast-northwest （SE-NW） oriented dipole-type patterns, respectively, while the composite field over the EAE region exhibits an Ω-type pattern. The type of composite blocking pattern seems to be related to the position of the blocking region relative to the positive center of the climatological stationary wave （CSW） anomaly existing near 10°W. The physical cause of why there are different composite blocking types in the three sub-regions is identified using a nonlinear multiscale interaction model. It is found that when the blocking event is in almost the same position as the positive CSW anomaly, the planetary-scale field can exhibit an Ω-type pattern due to the enhanced positive CSW anomaly. Neverthe- less, a SW-NE （SE-NW） oriented dipole-type block can occur due to the reduced positive CSW anomaly as it is farther in the west （east） of the positive CSW anomaly. The total fields of blocking in the three regions may exhibit a meandering flow comprised of several isolated anticyclonic and cyclonic vortices, which resembles the Berggren-Bolin-Rossby meandering jet type.

Analysis of Stationary-Wave Nonstationarity in the Northern Hemisphere 500-hPa Height Field

In this paper,the concept of stationary-wave nonstationarity is presented and elucidated in the framework of the Lorenz circulation decomposition.This concept indicates the relative magnitude of the zonal nonuniform abnormity to the intensity of stationary waves on the monthly mean scale.Based on the Lorenz circulation decomposition,the nonstationarity degree I_（us）（I_（us）~1） of the global（local） stationary waves is defined,and then used to analyze the stationary-wave nonstationarity at 30°-60°N,where the intensity of stationary waves at 500 hPa in the Northern Hemisphere,as is well known,is very high.The following findings are obtained：（1） There exist seasonal southward and northward movements in the position of the nonstationarity zones of the global stationary waves.The steady stationary waves occur in midlatitudes （35°-55°N） in winter and in the subtropical region（south of 35°N） in summer,associated with the major troughs over East Asia and North America and the weak European trough in winter,and with the relatively steady subtropical high system in summer.A high value center of I_（us） is at 35°N in spring and 50°N in summer,which might be caused by the seasonal variation of stationary-wave intensity,particularly in association with the interannual variability of trough/ridge positions of stationary waves on the monthly mean maps.（2） There exists obvious asymmetry in I_（us）~1,with the steady zones always located in the areas controlled by strong troughs/ridges and the unsteady ones in the areas where the stationary-wave intensity is low.The I_（us）~1 in the subtropics（south of 35°N） is larger in winter than in summer,and vice versa in the midlatitude region（north of 35°N）.The summertime distribution of I_（us）~1 on the whole shows a rather complicated structure.However,North Europe is the most unsteady area for local stationary waves,as represented by high values of I_（us）~1 in both summer and winter,while over the North American continent （about 120°E-60°W）,the I_（us）~1 is slightly less than 1 in summer,indicating that the stationary waves in this region are more steady than those over other mid and high latitude regions.（3） From North China to Northwest Pacific,there is a high value zone of I_（us）~1 in summer,with its center（45°N,130°E） located in the east of Heilongjiang Province.This influences the summer climate of northern China,including Northeast, North,and Northwest China.It is obvious that the nonstationarity is an intrinsic attribute of stationary waves,and can be regarded as being of the same importance as the intensity and energy-spectrum structure of stationary waves in the studies of the general circulation system.

INFLUENCE OF THE BASIC FLOW IN THE TROPICS ON THE STATIONARY PLANETARY WAVES AT MIDDLE AND HIGH LATITUDES DURING THE NORTHERN HEMISPHERE WINTER

The relationship between the quasi-stationary planetary waves forced by topography and heat source during the Northern Hemisphere winter is investigated by means of a quasi-geostrophic,34-level,spherical coordinate model with the Rayleigh friction,the Newtonian cooling and the horizontal eddy thermal diffu- sion. The calculated results show that when the basic flow is the westerly in the tropical stratosphere,the amplitude of quasi-stationary planetary wave for zonal wavenumber 2 at middle and high latitudes is larger during the Northern Hemispheric winter;while when the basic flow is the easterly,it is smaller.This is in agreement with the observed results. The calculated results also show that influence of the basic flow in the tropical troposphere on the quasi- stationary planetary waves is larger than that of the basic flow in the tropical stratosphere on the quasi- stationary planetary waves.

Stationary phase and practical numerical evaluation of ship waves in shallow water

A simple and highly-efficient method for numerically evaluating the waves created by a ship that travels at a constant speed in calm water, of large depth or of uniform depth, is given. The method, inspired by Kelvin＇s classical stationary-phase analysis, is suited for evaluating far-field as well as near-field waves. More generally, the method can be applied to a broad class of integrals with integrands that contain a rapidly oscillatory trigonometric function with a phase function whose first derivative（and possibly also higher derivatives） vanishes at one or several points, commonly called points of stationary phase, with the range of integration.

The East Asian Upper-Tropospheric Jet Streams and Associated Transient Eddy Activities Simulated by a Climate System Model BCC_CSM1.1

In this paper, major features of the upper-tropospheric jet streams simulated by a coupled Climate System Model BCC_ CSMI.1 are evaluated through comparison with the NCEP/NCAR reanalysis. The jet streams consist of the East Asian subtropical jet （EASJ） and the East Asian polar-front jet （EAPJ）. Associated stationary wave and synoptic-scale transient eddy activities （STEA） are also examined. The results show that the climatological positions of the westerly jet streams are well captured by BCC_CSMI.1, but with slight intensity biases. Statistics from the 6-h model outputs reveal that the jet core number （JCN） of ESPJ is significantly underestimated. Examination of the simulated seasonal evolution of the westerly jet stream indicates that the model has produced a westward movement of the EASJ core in May, one month earlier than that in the reanalysis. Analysis of stationary wave activities shows that the overestimated meridional wind component may have caused considerable enhancement of meridional momentum and heat transport. The stationary Rossby wave represented by the wave activity flux at the southern flank of the Tibetan Plateau is favorable to the growth of asymmetric zonal wind and the multiple-center pattern of JCN. Unlike the stationary wave heat flux transport, the model tends to systematically generate weaker transient heat flux over East Asia. Further analysis of STEA exhibits a general consistent pattern between the simulation and the reanalysis, while the intensity of the northern STEA branch associated with the EAPJ is greatly reduced. The deficiencies of eddy momentum and heat flux transport and accompanied eddy forcing may contribute to the biases of the simulated upper-tropospheric jet streams, suggesting the potential importance of midlatitude internal atmospheric dynamics in shaping the tropospheric general circulation, which is not yet fully and accurately resolved in the current BCC-CSMI.1.