《地球物理流体力学》课件：Lecture 6 Ekman flows 和海洋风生环流.pptx

1、 Objectives: to understand the common character of large scale atmospheric dynamics and oceanic dynamics.大尺度大气和海洋动力学的共性特征. Major Contents:(1.) Dynamic behavior of geophysical fluid under conditions of gravitation, rotation, stratification层结, viscosity and heating粘性和加热; (2.) Theory for solutions and

2、mechanisms: inviscid shallow-water theory无粘旋转流体的浅水理论 friction and viscous flow,摩擦和粘性流 wind-driven oceanic circulation,海洋风生环流Shallow water model 浅水模型For equation of horizontal velocity (u, v, 0) 水平运动(u, v)与高度z无关 We assumed that, the velocity (u, v) is independent of depth, z. Examining the equations

3、for u and v, we note that the accelerations do not vary with depth z. (du/dz=dv/dz=0)0 ,(vuVh1/2hhhdVdtpV 0uvzz准水平1/pdu dtfvx 1/pdv dtfuy 浅水模型中 粘性，斜压性和涡管的倾斜机制均不存在，所以改变涡度的唯一机制是涡管的伸缩效应，或水平辐散/辐合效应40fdabsolutevorticitydtHfluiddepthz21()()aaaijdpVVdt 地转流 Geostropic flow, 当大尺度时Ro0) 和南半球和南半球 （f0)， 高压中心附近沿等

4、压线高压中心附近沿等压线呈顺和逆时针运动；在低压中心附近则相反呈顺和逆时针运动；在低压中心附近则相反EqHLHLChapter 6 Ekman flows Near wall surfaces Near free surfacesEkman模型 (考虑粘性和湍流的情况 viscosity and turbulence effect)(Ekman flows/layers)6.1 考虑粘性效应的边界层89Figure : Formation of a Boundary Layer Boundary Layer Let us now follow the effects as a flow app

5、roaches a flat plate, Consider a uniform (inviscid) flow in front of a flat plate at a speed . As soon as the flow hits the plate No Slip Conditions gets into action. As a result, the velocity on the body becomes zero. Since the effect of viscosity is to resist fluid motion, the velocity close to th

6、e solid surface continuously decreases towards downstream. But away from the flat plate the speed is equal to the free stream value of . Consequently a velocity gradient is set up in the fluid in a direction normal to flow. Thus a layer establishes itself close to the wall with a velocity gradient.

7、This is what we call the Boundary Layer. Planetary boundary layer 行星边界层大气边界层（百度）近地面层或近地面层或表面层（亦称常通量层constant fluxes layer ）：是大气边界层最接近地表面的部分。厚度为100米左右，该层内湍流黏性力为主导力。风速与高度同增。湍流动量通量(湍流切应力)、热通量和水汽通量近似不随高度变化的气层。按照稳定度性质区分为不稳定近地面层、中性近地面层和稳定近地面层。厚度在十米至百米左右，不稳定或地面粗糙度大的情形下厚度较大，稳定或地面粗糙度小的时候较浅薄。近地面层中温度、湿度、风速等气象。

8、要素随高度的变化很大，湍流运动对该层的性质起着决定性的作用，进而又决定了整个大气边界层的特征。近地面层是人类和生物直接接触的气层和大气污染影响最主要的表现场所。艾克曼层：地转形成的科里奥氏力在该层当中相当重要。风向在这里随高度改变，艾克曼层的厚度随着强烈太阳辐射和夜晚低风速影响而从100米高度到20003000米不等。12 Ludwig Prandtl (1875-1953) German physicist who is considered to be the father of aerodynamics. Prandtl made decisive advances in boundar

9、y-layer and wing theories, and his work became the fundamental material of aerodynamics.Perhaps Prandtls greatest achievement was his ability to produce so many great scientists. It is mind boggling to look at the long list of those who were his students and colleagues. There is no one who educated

10、as many great scientists as Prandtl. Prandtl changed the field of fluid mechanics and is called the father of modern fluid mechanics because of his introduction of boundary layer, turbulence mixing theories etc.13Ludwig Prandtl was born in Freising, Bavaria 巴伐利亚巴伐利亚, in 1874. His father was a profes

11、sor of engineering and his mother suffered from a lengthy illness. As a result, the young Ludwig spent more time with his father which made him interested in his fathers physics and machinery books. This upbringing fostered the young Prandtls interest in science and experimentation. Prandtl started

12、his studies at the age of 20 in Munich, Germany and he graduated at the age of 26 with a Ph.D. Interestingly, his Ph.D. was focused on solid mechanics. His interest changed when, in his first job, he was required to design factory equipment that involved problems related to the field of fluid mechan

13、ics (a suction device). Later he sought and found a job as a professor of mechanics at a technical school in Hannover汉诺威汉诺威, Germany (1901). During this time Prandtl developed his boundary layer theory and studied supersonic fluid flows through nozzles. In 1904, he presented the revolutionary paper

14、Flussigkeitsbewegung Bei Sehr Kleiner Reibung (Fluid Flow in Very Little Friction), the paper which describes his boundary layer theory. His 1904 paper raised Prandtls prestige. He became the director of the Institute for Technical Physics at the University of Gttingen. He developed the Prandtl-Glau

15、ert rule for subsonic airflow. Prandtl, with his student Theodor Meyer, developed the first theory for calculating the properties of shock and expansion waves in supersonic flow in 1908 (two chapters in this book). As a byproduct they produced the theory for oblique shock. In 1925 Prandtl became the

16、 director of the Kaiser Wilhelm Institute for Flow Investigation at Gttingen. By the 1930s, he was known worldwide as the leader in the science of fluid dynamics. Prandtl also contributed to research in many areas, such as meteorology and structural mechanics. Ludwig Prandtl (1) 14Ludwig Prandtl wor

17、ked at Gttingen until his death on August 15, 1953. His work and achievements in fluid dynamics resulted in equations that simplified understanding, and many are still used today. Therefore many referred to him as the father of modern fluid mechanics. Ludwig Prandtl died in Gttingen, Germany on Augu

18、st 15th 1953. Prandtls other contributions include: the introduction of the Prandtl number in fluid mechanics, airfoils and wing theory (including theories of aerodynamic interference, wing-fuselage机身机身, wing-propeller, biplane, etc); fundamental studies in the wind tunnel, high speed flow (correcti

19、on formula for subsonic compressible flows), theory of turbulence. His name is linked to the following: Prandtl number (heat transfer problems) Prandtl-Glauert compressibility correction Prandtl-格劳厄特格劳厄特Prandtls boundary layer equation Prandtls lifting line theory Prandtls law of friction for smooth

20、 pipes Prandtl-Meyer expansion fans (supersonic flow) Prandtls Mixing Length Concept (theory of turbulence) Ludwig Prandtl (2) 15The failure of inviscid an irrotational flow theory to predict drag on objects when a fluid flows past them provided the impetus for Prandtl to put forward a theory of the

21、 boundary layer adjacent to a rigid surface. Prandtls principal assumptions are listed below.Assumptions: 1. When a fluid flows past an object at large values of the Reynolds number, the flow region can be divided into two parts.(i) Away from the surface of the object, viscous effects can be conside

22、rednegligible, and inviscid flow can be assumed.(ii)In a thin region near the surface of the object, called the boundary layer, viscous effects cannot be neglected, and are as important as inertia. Laminar layer The laminar boundary layer of air adhering to all surfaces In the laminar sub-layer ther

23、e is no convection due to the absence of vertical velocities Insulating barrier between the surface and the atmosphere Steep climatic response16层流层流湍流湍流过渡区过渡区Buffer regionof 5617 *yuyuw Different regions are defined on y+ y+ 5 viscous sublayer (Reynolds shear stress negligible compared with viscous

24、stress) y+ 50 outer layer (viscosity negligible)18The Character of flow past an object is dependent on the value of the Reynolds number. Why the boundary layer theory is limited to large values of the Reynolds number ? Re=0.1 Re=10 Re=107 Dark area is the area where viscous effects are important19Cl

25、early the actual fluid viscosity is the same throughout(viscosity is same in the boundary layer and outside the boundary layer ) ; only the relatively importance of the viscous effects (due to the velocity gradients ) is different within or outside of the boundary layer. What is the essential differ

26、ence between the boundary layer and outside the boundary layer ? Fluid particles within the boundary layer experience viscous effects ( because of velocity shear there ) . Prandtl assumed that the lump of fluid is transported freely with conserved momentum by turbulence over a certain mixing length.

27、20Prandtl mixing length theory其基本思想是把湍流输送用平均量来表示。设开始位于(z-l)高度的湍流微团与周围流体具有相同的属性量,当微团向上移动时, w0, 当其运行不超过l时,属性不变,当达l时,立刻与周围空气发生混合. l 为混合长，为混合长， l以内物理以内物理属性不变属性不变000000()()()( )()()( )zu zlu zu zlu zu zlu zlzu zulz Similarly, we assume the same for the w2222( )( )( )Mu zwK uKlzu zu zu wK u wKllzz 22( )(

28、)( )xzMVMu zu zlzzu zAlz Experiment shows the mixing length is of the order of the flow dimension. When velocity reaches maximum du/dy = 0, Reynolds stresses vanish.模仿分子平均自由程而引入了混合长的理论，即湍流系数与平均风速梯度的绝对值和混合长的平方两者的乘积成正比的理论。Eddy viscosity (AV) modeling*;VmmAzul ulz0 xzVduAdz*0( )lnuuzduu zzdzz2*0 xzVuuA

29、zPrandtl mixing length lm2/VmAluz为中性稳定层结的情况混合长正比于高度z模仿分子平均自由程而引入了混合长的理论，即湍流系数与平均风速梯度的绝对值和混合长的平方两者的乘积成正比的理论。23Six components of Reynolds stresses摩擦速度摩擦速度In fluid dynamics, the Von Krmn constant (or Krmn constant), named for Theodore von Krmn, is a dimensionless constant describing the logarithmic vel

30、ocity profile of a turbulent fluid flow near a boundary with a no-slip condition. The equation for such boundary layer flow profiles is:The Krmn constant is often used in turbulence modeling, for instance in boundary-layer meteorology to calculate fluxes of momentum, heat and moisture from the atmos

31、phere to the land surface.粗糙度 z0对数律对数律Theodore von Krmn (original Hungarian name: Szllskislaki Krmn Tdor) (May 11, 1881 May 7, 1963) was a Hungarian-American aerospace engineer and physicist who was active primarily in the fields of aeronautics and astronautics. He is responsible for many key advanc

32、es in aerodynamics, notably his work on supersonic and hypersonic airflow characterization.粗糙度 z0湍流区速度满足对数律Logarithmic law of wind 对数律0log*zzuzUVon Karmans constant 0.4Surface velocity Roughness length Surfacez0Sand0.01 0.1Snow0.1 0.5Low grass, steppe0.1 5Forest80 100Suburbs20 40Towns80 120Metropoli

33、s150 250Water surface23. 0103 .4410Uefor U10 粘性输运 (AM is turbulent kinematic viscosity);iiijM jjjjpEjuuu uAxxTu c TKx35Turbulent momentum transportTurbulent energy transportWater vapor or pollutant transport湍流交换系数湍流动量交换系数湍流热量输运通量密度湍流水汽通量密度湍流动量输运通量密度wjwqjqu qKxVagn Walfrid Ekman (3 May 1874 9 March 1

34、954) was a Swedish oceanographer.The Ekman Layer is the layer in a fluid where there is a force balance between pressure gradient force, Coriolis force and turbulent drag. It was first described by Vagn Walfrid Ekman.Ekman developed the theory of the Ekman layer after Fridtjof Nansen observed that i

35、ce drifts at an angle of 20-40 to the right of the prevailing wind direction while on an Arctic expedition aboard the Fram. Nansen asked his colleague, Vilhelm Bjerknes to set one of his students upon study of the problem. Bjerknes tapped Ekman, who presented his results in 1902 as his doctoral thes

36、is.(Fridtjof Nansen, 1861-1930, 挪威北极探险家、博物学家及外交家挪威北极探险家、博物学家及外交家, 曾获曾获1922年诺贝尔和平年诺贝尔和平奖奖)大气层底部（接近地球表面和海洋, 1-1.5km 以下），大洋底部（海床附近）表层海水（海气界面附近）考虑湍流输送。3800()()(.)xzVat surfaceyzVat surfaceEat surfaceuAzvAzTKFz0uvwAt wall surface, no slip wall boundary conditionEkman depth/thickness d in atmosphere and s

37、ea flows39 For ocean, with an eddy viscosity AV as large as 102 m2/s ，= 7.3 105 s1 d10m H = 100 m For atmosphere, with an eddy viscosity AV as large as 5 m2/s ，= 7.3 105 s1 d103m=1km d ) toward interior考虑均质均质正压大气,边界层内气压梯度力不随高度变化452202200111VVpufvAxzpvfuAyzpgz001;010;0ggpuUwfypvVwfx 内区内区()()0ppzyzxzd

38、f Ug2222(0)0;()0VVguAf vzvAf u Uz42420ggVdu Ufu UdzAf Vg=0In Ekman 公式, d is called Ekman depth46boundary conditions at wall (A=-U, B=0)2/VdAfThe distance over which it approaches the interior solution is on the order of d. Thus d give the order of boundary layer thickness./( cossin)( cossin)z dgz dz

39、zuUeABddzzveBAdd/(1cos)sinz dgz dgzuUedzvU ed472/VdAfThe distance over which it approaches the interior solution is on the order of d. Thus d give the order of boundary layer thickness./(1cos)sinz dgz dgzuUedzvU edu/Ugv/Ugz/d地转流(Ug) xEM The velocity spiral in the bottom Ekman layer for northern Heis

40、phere (f0)And the deflection is to the left of the current above the layer.风矢量随着高度是变化的,把各高度上的风矢量投影到同一平面内,其矢量端的连线称为Ekman 螺线. 经过原点,在地面上风向与等压线(地转风)方向夹角45o,风速随高度增加而增加,逐渐趋于地转风. 在北半球实际风向偏向于地转风的左方.大气中的埃克曼螺线 (Ekman spiral) , 海水中? General cases: 地转流 (Ug, Vg)4950The balance between the Coriolis effect and the

41、 pressure gradient force. u=v=w=0 at wall surfaces远离地面/surface(Ug, Vg)近地 near surface512202200111VVpufvAxzpvfuAyzpgz下边界条件下边界条件No slip condition u=v=w=011;0ggppuUvVwfyfx 上边界条件上边界条件 (zd )Satisfy geostrophic balance 三力的平衡In Ekman models, only boundaries are different Homogeneous fluid (均质正压,不可压缩) Geost

42、rophic balance in interior region(内区地转平衡) Ekman layer (AV is constant, 水平压力梯度不随高度变化)522202200111VVpufvAxzpvfuAyzpgz三力的平衡1;1;ggpUfypVfx 0uvwxyz53摩擦产生地转偏差的量化/(1cos)sinsin(1cos)z dz dggz dz dggzzuUeV eddzzvU eVedd随高度增加,摩擦力影响则愈来愈小; 当z 增大时, 速度呈指数律迅速趋向于内区无粘地转速度.直接受摩擦影响的区域具有Ekman层厚度的量级, 该厚度随着Av的增大而增加, 但随着旋

43、转强度的加强而减小, 旋转越强, 直接受摩擦力影响的区域越小. 2/VdAf/cossinsincosz dz dgggz dz dgggzzu UU eV eddzzv VU eV edd6.3 Ekman transport on flat terrianEkman通量00( ( );( ( )xgygMu zUdzMv zV dz56Ekman transport (穿越等压线的总体积通量-Ekman通量)0000( ( )( ( )( ( )()( ( )()2(0)(0)()EggxyxgggygggVEMiu zU dzjv zV dzM iM jdMu zU dzUVzdMv z

44、V dzUVuvA ijzzkMf 每一层流体都受到位于其下的流体层的摩擦阻滞作用, 但Ekman层作为一个整体, 所受到的力只有压力梯度力和下边界的摩擦力. Ekman通量和下边界摩擦应力垂直,而且在北半球, Ekman通量沿着该应力方向指向右侧.2/VdAf地转流(Ug, Vg)EM 摩擦力57020()()222yxggzgMMuvdzxyxyVUdxyddpfThe flow in the Ekman layer converges or divergences if the interior flow has a relative vorticity。内区相对涡度大，Ekman通量的

45、散度大2/VdAfEkman transport (Ekman通量)的散度 与内区涡度的关系00( ( );( ( )xgygMu zU dz Mv zV dzWhere the vertical velocity in the interior is from?(为一种补偿运动为一种补偿运动) How large? 高压58Vertical velocity in the interior (Ekman pumping,埃克曼抽吸埃克曼抽吸)020()()(0)()()()221222gggVzgzgwuvzxyuvw zw zdzxyVUdw zwxyAddpff The greater

46、for relative vorticity, the stronger upwelling and downwelling motion (larger vertical velocity in the interior). Also the Ekman pumping effect increases toward to the equator (decreasing f) and increasing d.内区相对涡度越大，Ekman厚度越大，则抽吸变强。2/VdAf固壁无滑移u=v=w=00(0)wzw z在内区在固壁 =059在边界层受在边界层受Ekman流影响流影响 （压强梯度力、

47、科氏力、摩擦力平衡）（压强梯度力、科氏力、摩擦力平衡） （1）在刚壁附近，摩擦力起抵抗运动的作用，所以）在刚壁附近，摩擦力起抵抗运动的作用，所以Ekman流具有流具有从高压流向低压的分量从高压流向低压的分量低压高压Effect of Ekman Pumping (抽吸) on weather in atmosphere 抽吸把质量/热量/动量和水汽垂直输送. 例如台风登陆后的迅速填塞, 为地面摩擦加大所致. 近地面低压系统周围，辐合发生，湿度等会被向上输运，容易引起降雨等坏天气；近地面高压系统周围，辐散发生，湿度等会被向下输运，易为晴朗天气。 Around the low pressure s

48、ystem near the ground, convergences occur and moistures are transported upward. It tends to rain and cause bad weather conditions. Around the high pressure system near the ground, divergences occur and the downward motions are dominated. Sunny and fine weather conditions occur. 61Bottom Ekman layer

49、with flat terrain从方程解释622201VpvfuAyz从方程解释, 摩擦(no slip)产生穿越等压线的运动, 方向为从高压到底压 (摩擦破坏地转平衡): u small at z=0 (no slip) larger pressure gradient induces v-Vgor u-Ug, 产生Ekman抽吸, 垂直性补偿运动22()gVvf u UAz固壁无滑移内区为地转平衡固壁无滑移u=v=w=0u=v=w=0从高压到低压穿过等压线的运动中, 压力梯度做功, 补偿了摩擦消耗的动能, 使流场定常稳定. Ekman层中压力场做功的能量来源是内区地转流的动能向下输送.

50、单位面积下:63从能量角度22220022()()222VVpuuWvdzA uvdzyzzUUWfdA f 深度为D的地转流体层, 单位面积具有的动能为:22UKD 消耗完地转流体柱动能的时间为旋转减弱时间 (the spin-down time):221;2VVVVAKDEfDA ffEWEV is Ekman numberFor sea, three months; for atmosphere, 1 day64能量角度因水平速度切变造成的垂直涡度的倾斜，从而导致水平涡度的产生; Ekman层的建立作为边界的涡度扩散与行星涡度(f) 的补偿倾斜之间的平衡:从涡度角度理解Ekman lay