高等大气动力学Rossbywave.ppt

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Physics of Rossby Wave,Yihua,Lin,AOPP,Clarendon Laboratory,University of Oxford,UK,4,th,June,2008,Wave,Rossby,wave in atmosphere&ocean,Mechanism of,R,ossby,wave,Propagation&dispersion of,R,ossby,wave,Outline,2,What is wave,？,In,physics,a,wave,is a disturbance that travels through,space,and,time,usually accompanied by the transfer of,energy,.,Waves travel and the,wave motion,transfers,energy,from one point to another,often with no permanent displacement of the particles of the mediumthat is,with little or no associated mass transport.They consist,instead,of,oscillations,or vibrations around almost fixed locations,.,Wave,3,Mechanical,wave,which propagates through a medium in which the substance of this medium is deformed.The deformation reverses itself owing,to restoring forces resulting,from its deformation.For example,sound,waves.,Electromagnetic radiation,(,including visible light,ultraviolet radiation,infrared radiation,gamma rays,X-rays,and radio waves).,This type wave,can travel through a,vacuum and,consists of periodic oscillations in electrical and magnetic fields.,4,Transverse,and,longitudinal,waves.Transverse,waves occur when a disturbance sends waves perpendicular(at right angles)to the original wave.Longitudinal waves occur when a disturbance sends waves in the same direction as the original wave.,5,Waves in uids result from the action of restoring forces on uid parcels that have been displaced from their equilibrium positions.The restoring forces maybe due to compressibility,gravity,rotation,or electromagnetic effects,.,Acoustic,wave,shallow water gravity wave,buoyancy,wave,or internal,gravity,wave,inertial,wave,Rossby,wave,(,planetary waves,),mixed,wave,.,6,Waves in Atmosphere&Ocean,7,usually,are 4-6 of them around the globe at a given time,wavelength,varies between 4000-8000 km,are,generally stationary-or move very slowly east or west,Atmosphere,8,Height of the 500hPa isobaric surface(dam),03 March,2005 06Z.,Surface,pressure,analysis;,03 March 2005 12Z.,9,10,The 2 small black circles off the coast of the Americas correspond to sea level highs on the trailing edge of Kelvin waves.The leading edge of this wave group has bounced off the coastline,creating,Rossby,waves whose rising and falling sea levels are marked by solid and dashed lines,respectively.,In the July image,the circles are moving west with the,Rossby,waves.The X marks a relative sea level low caused by a Kelvin wave moving east.,In December,the,Rossby,waves continue to move westward.The shapes of the solid and dashed lines indicate that the,Rossby,waves are moving away from the Americas faster at the equator than at higher latitudes.,White and red indicated higher than average levels;purple and magenta indicate lower than average levels.These scenes show eastward-moving Kelvin waves,and westward,-moving,Rossby,waves.Superimposed black circles show how the elevated sea surface moves east in April,then west as,Rossby,waves during July and,December.,Ocean,Rossby,Waves,The,most important class of large-scale atmospheric,waves,are called planetary waves,or,Rossby,waves,.These,waves are characterized by oscillations in the,rotational,part of the horizontal wind that are parallel,to the,horizontal gradient in the potential,vorticity,.,11,Mechanism of,R,ossby,Wave,Restoring force is the north-south gradient of background,potential,vorticity,(f/H).That gradient can be due to either the variation in,f with,latitude,or to a variation in H(“topographic,Rossby,waves,”and,when trapped near the coast“continental shelf waves”),.,Note that,Rossby,waves are,tranverse,waves,that is the,particles move,perpendicular to the direction of propagation,.,Conserves PV=(,f,+,),/H.,12,13,Sketch:northern hemisphere,Imagine that something pushes a water column perpendicular to,the planetary,vorticity,.What will happen?,The,parcel will,conserve total,potential,vorticity,(,f,+,/,H).As it moves into a region,of higher,f,it will tend to decrease relative,vorticity,a,tendency,to clockwise,motion.Likewise if it were displaced towards,lower relative,vorticity,it will increase relative,vorticity,a,tendency towards,counterclockwise motion.,14,15,Now lets look at the,Coriolis,forces on the induced,circulation,pattern,.,Size of restoring force(for a given displacement)is proportional to,=,df,/dy.PV equation:,D,/,Dt,+v=0,The,same thing will hold true on a slope,pushed,towards shallower,water will require decreased relative,vorticity,and,vice versa,so shallower water is the analog of northward over a,flat bottom,and there are analogous waves generated on,strong topographic,slopes called,topographic,Rossby,waves,.,Hf,.,16,In,last section,we derived the properties of,Rossby,waves.,Our key findings,was that,Rossby,waves rely on the,-effect,for their,existence.,We also discussed the mechanism for,Rossby,wave propagation,through potential,vorticity,conservation.,17,Rossby,wave propagation,Potential,vorticity,equation,Consider small-amplitude disturbances to a,uniform zonal background flow,(,U,0,0)for which,the,streamfunction,is,and,the,quasigeostrophic,potential,vorticity,is,The,streamfunction,for the total flow(background,plus small,disturbance,)is thus,and the potential,vorticity,18,Where,We,now substitute the above in the,quasigeostrophic,potential,vorticity,equation and,neglect terms that are quadratic in the,disturbance quantities:,The first term represents the rate of change of the perturbed,potential,vorticity,following the mean flow,and the second term represents,the advection,of mean potential,vorticity,by the perturbed flow.,19,Dispersion relation,We now look for plane-wave solutions,We shall also assume constant stratification,i.e.,N,2,=,constant,.This,gives the dispersion relation for,quasigeostrophic,Rossby,waves,:,As,in,last section,the,-,effect is crucial for these,Rossby,waves.,Setting,=0 gives,=,Uk,which implies c,(x,),p,=c,(x,),gU,i.e.,the waves,are merely,carried along by the background flow,U,.,20,For the barotropic motion,we,get the,simpler dispersion,relation for Rossby,waves,(,on a motionless,state,U,=0),This function is plotted,as below(Note,that,l,/=,x,/(,x,2,+1),where,x,=,k,/,l,.),21,The function,x,/(,x,2,+1).,1./,k,=,/(,k,2,+,l,2,)0 for,k,/,l,1,and,d,/,d,k,0 for,k/l,0,i.e.,where the critical background,velocity,depends on the horizontal wavelengths of the wave.,Thus for vertical propagation we must,have,26,In particular,for stationary waves,whose crests and troughs do not,move relative,to the ground such that c,(x,),p,=0,we obtain the,Charney-,Drazin,criterion,Stationary waves propagate vertically only in eastward background,flows(,U 0,)that are not too strong(,U,/,(U c,(x,),p,),then,m,is imaginary,.If,we set,m=,i,where,is real and positive,then the wave solutions,are of,the,form,i.e.,the waves propagate horizontally but not vertically,and decay,with height.,Equivalently,this happens,if,For stationary waves(c,(x,),p,=0),vertical trapping occurs if the,backgroundwinds,are westward(,U,U,c,).,29,Vertical modes,In the ocean,Rossby,waves are strongly constrained by the surface,and bottom,boundariesWe can seek,Rossby,wave solutions with modal structures in the vertical.,Thus we seek solutions of the,form,Substituting the above into the potential,vorticity,equation,gives,This separable solution only works if,both,&,Where,is a separation constant.,30,Here,we shall consider only waves in which the surface and bottom,density anomalies,vanish,giving the boundary conditions,:,where,z=0,is the sea surface and,z=H,is the sea floor,.The,vertical modes,satisfy,where,The corresponding dispersion relation for the waves,is,31,This,is,the dispersion relation for,Rossby,waves,and we set,known as the,Rossby,deformation radius,for the,n,th mode,.,The deformation radius is infinite for the,barotropic,mode(,n,=0),and increasingly,smaller for each,baroclinic,mode(,n,1,).,The first,baroclinic,deformation radius is approximately 1000 km in,the atmosphere,and 30 km in the ocean.,32,33,Illustration,of the different deformation radii in the atmosphere and,ocean as,revealed through(nonlinear,breaking),Rossby,waves,.,In,fact,for,barotropic,motions,the theory does not discriminate between atmosphere and ocean,.,Rossby,waves in the ocean,The ocean supports,Rossby,waves,just as the atmosphere does,obeying,the same,dispersion relation,and for the same reasons.,The,coastal boundaries of the ocean prevent a sustained east-west,circulation(,except in the Southern Ocean)and sustained east-west propagation of,the waves,themselves,so in practice there are many,differences,.The presence,of coasts,means that ocean basins can support trapped modes,for one thing,.However,much of the large-scale variability of the ocean can be,described as,Rossby,waves,albeit in a less organized way than for the atmosphere,.However,there is one central aspect of ocean dynamics that may not,appear to,involve,Rossby,wavesbut in fact does:the existence of western,boundary currents,.,34,Western,intensification,It is,obvious that,Rossby wave,behavior is,zonally asymmetric.In particular,we saw that the group propagation,of long,Rossby waves is fast and westward,while that of short waves is,slow and,eastward.,This,has dramatic,consequences,for ocean,dynamics.,35,Eastward,slow,Westward,fast,Any large-scale disturbance in mid-ocean will,generate,Rossby,waves;the larger scale of these will propagate rapidly westward,.Before,long,they will reach the western boundary of the ocean where,they will,be,reflected,.Unlike gravity(or light)waves,the,reflected,waves will,not simply,be a mirror image of the incident waves:the,reflected,waves,must,have an eastward component of group velocity and so must be of short,zonal wavelength,.,36,Moreover,they will propagate relatively slowly,more so than the incoming waves.Thus,there will be a kind of traffic jam at the western boundary information can get in more readily that it gets out.The information that accumulates there will involve motions of small zonal scale.This is the underlying dynamical reason for the existence of strong boundary currents on the western,rather than eastern,sides of the ocean.The underlying reason for the east-west asymmetry is,the northward gradient of planetary,vorticity,.,37,An Introduction to,Dynamic Meteorology,J,.R,.,Holton,Academic Press,2004,(4th,edition,).,Atmosphere,-Ocean Dynamics,A.E.,Gill,Academic,Press,1982.,38,Further reading,