1、2() (),1(),aaijVVp 为涡管的倾斜效应为涡管的伸长效应为力管项 因为斜压性 为涡度源为摩擦力项 为涡度汇(due to friction effect)(due to baroclinicity)(for vortex strectching)(for vortex tilting)()aaadVdtt 涡度可以通过对流传输(Convection effect)aaauvwxyz1上次课堂作业上次课堂作业21()()aaaijdpVVdt Chapter 5 Inviscid shallow water model and Geostropic flow无粘浅水模型 和地转流局
2、地直角坐标（or z coordinate): 随地球转动而运动的正交坐标系The variables x and y are distances eastward and northward on the globe. We will ignore the effects of sphericity except in the Coriolis term. Then (x, y, z) are equivalent to Cartesian coordinates.= 0 i+ cos f j + sin f kAEkman number 埃克曼数埃克曼数Ek1 viscosity can
3、be neglectedEk1 viscosity is important埃克曼层埃克曼层（Ekman layer）是流体中压力梯度力、科氏力和湍流粘性力三力平衡的一层, 粘性力不可忽视。由瑞典海洋学家埃克曼提出。埃克曼层理论适用于许多地区，包括大气层底部（接近地球表面和海洋），大洋底部（海床附近）和表层海水（海气界面附近）。22VkVkAEHAEHEkman depth/thickness (厚度) in atmosphere and sea flows5 For ocean 海洋, with an eddy viscosity AV as large as 102 m2/s ，= 7.3
4、 105 s1 d10m H = 100 m For atmosphere 大气, with an eddy viscosity AV as large as 5 m2/s ，= 7.3 105 s1 d103m=1km H = 10 km 5.1 Shallow water model 浅水模型无粘，正压均质 且不可压缩 （无层结结构）远离Ekman层处， 无粘流体方程: 不考虑粘性*00*00()()()1()()()1()()()1xyxxxzyxyyyzzyzxzzuuvuwupuf wfvtxyzxxyzuvvvwvpvfutxyzyxyzuwvwwwpwf ugtxyzzxyz I
5、t is generally not required to discriminate between the two horizontal directions, and we assign the same length scale L to both coordinates and the same velocity scale U to both velocity components. The same, however, cannot be said of the vertical direction. Geophysical flows are typically confine
6、d to domains that are much wider than they are thick, and the aspect ratio H/L is small. The atmospheric layer that determines our weather is only about 10 km thick, yet cyclones and anticyclones spread over thousands of kilometers. Similarly, ocean currents are generally confined to the upper hundr
7、ed meters of the water column but extend over tens of kilometers or more, up to the width of the ocean basin. It follows that for large-scale motionsHL; WU,VlH/L1 the fluid is shallow, 特征深度或运动的垂直特征尺度H远小于水平特征尺度L; lwu, v; 且垂直速度存在上届且垂直速度存在上届wuH/L; 即w/uH/L1, 换句话说，不可压缩流体的运动是准水平的x, y(x, y)fH0uvzz准水平10*00*
8、0111uuuupuvwf wfvtxyzxvvvvpuvwfutxyzywwwwpuvwf ugtxyzz0;uvwxyz不可压缩垂直速度 wu and v00;uvpgzzz 准水平准静压平衡We assume w u and v, and we can neglect the term. Scale analysis in geophysical flows (shallow water theory, Ro1 or Ro1) HL; WU, V; For large scale WL/(UH)1 or 1 11*01uuuupuvwf wfvtxyzxUT2UL2ULWUHW0PL1T
9、UL/W H UU LL0PLUUL1/()1ORUL大尺度, 惯性力不重要, 科氏力和压力梯度重要/()1ORUL中尺度, 惯性力, 科氏力和压力梯度 都重要UWU/()1ORUL 小尺度, 科氏力不重要01PLUScale analysis in shallow water model of geophysical flows (z) HL; WU, V12WTUWLUWLWWH0PH*001wwwwpuvwf ugtxyzz10pgz 001PPLUULUHH当科氏力重要时Ro1(中尺度)Hydrostatic balanceU/1UWWUULLL当科氏力重要时Ro1We assume
10、w u and v, and we can neglect the term. For horizontal components柯氏参数the Coriolis parameter, f = 2 sin , the latitude比湿方程能量方程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 for u and v, we note th
11、at 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 b b-plane approximation b b平面近似平面近似 当地流运动的径向范围(尺度)与地球半径(a6000km) 相比很小时 y/a1We neglect sphericity of the earth and treat the earth as a flat plane-f-planeapproximation 平面近似: f =f0= 2 sin
12、0=constant If y1000km, the sphericity is simplified.当方程中f不被微分时， f =f0 2 sin0=constant 当方程中f被微分时, f =f0+by=f0 2 sin0b b =df/dy=2 sin0/a=constantA性质 1x, y(x, y)fH()0hdHuvdHHHVdtxydt 水平辐散时，流体柱水平面积增大，厚度减小，体积不变11hdHdAVH dtA dt;BHhh0;uvwxyz不可压缩Continuity equation 证明过程证明过程()0hdHuvdHHHVdtxydt ()()BhhuvuvdzH
13、xyxy()( )()()BhBhwdHdzw hw hzdt边界条件();huvVVolAHxy ;BHhhHere水平辐散时，流体柱水平面积增大，厚度减小，体积不变从连续性方程得到0;uvwxyz不可压缩性质 2To Show证明证明: 垂直速度的动量方程消失垂直速度的动量方程消失,方程简化方程简化Vertical velocity w is a linear function of depth z0()0()()( )()0;( )()()()BzhhhhBBhBBwwVVdzzzVzhw zw hw zVzhw h 0zwyvxuThe horizontal divergence水平散
14、度)0 ,(vuVh)(yvxuVh垂直速度w是高度z的线性函数, 且与水平辐散有关性质 30hdHHVdt （2.24）5.2 浅水模型中的涡度方程24 无粘,不可压缩,均质(等密度), 准水平21()()(2cos )(2sin )0aaaijaxyzdpVVdtijkuvzz 准水平Vorticity equation in shallow water equation25(/)(/)(/); ,/1 xzyzzzxywvwOULyzyuwwOULzxxvuO ULxyD L 为浅水近似条件； 准水平W/DD, 无粘无粘,不可压缩不可压缩,匀质匀质, 准水平准水平, 正压，静压平衡正压，
15、静压平衡) Combining and playing with mathematicszvxwzuywpfdtfdz1kv101010uuupuvfvtxyxvvvpuvfutxyypgz 1zdfw uw vfVpdtyzxz k zz+=0zdffVdt zzvuxyz26()0;dHuvHdtxy+=0zdffVdt zzPotential-Vorticity conservation in shallow water theory (LD, 无粘无粘,不可压缩不可压缩,匀质匀质, 准水平准水平, 正压）正压）()dffdHdtHdtzz绝对涡度分量的相对变化反比于水平辐散 (colu
16、mn/tube height)Relative rate-of-change of absolute vorticity is equal to minus divergence0BfddtHHhhz浅水理论中势涡守恒性质 4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; 适用于不同尺度,可存在垂直运动,存在边界效应影响。290fdabsolutevorticitydtHfluiddepthz5.3 浅水模型的边界3031Exchange may take place at the air-sea interface, in b
17、ottom layers, along coasts and/or at any other boundary of the domainBoundary condition: Effect of outside on the domain浅水模型浅水模型-no penetration/impermeability condition理想不可压缩流动的固壁边界理想不可压缩流动的固壁边界 V n=U n V n=0 , ruu粘性流动采用的是固壁上的无滑移条件，由于理想流体动量方程中失掉了高阶粘性项，欧拉方程比N-S方程低了一阶，她就不需要象粘性流方程组那样多的边界条件。对理想流体采用法向无穿透
18、条件，理想流体采用法向无穿透条件，壁面上允许存在切向滑移速度壁面上允许存在切向滑移速度，固壁静止时，上述边界条件相当于要求固体壁面是流场中的一条流线。无穷远边界条件， Homogeneous flows over an solid irregular bottomwdxdyudbdyvdbdxbbwuvxyAt a shallow sea bottom (Homogeneous water) At the bottom:( , )0()0dzdbbbbwuvdtdttxxdzb x yzbdtFor a solid material surfaceNo penetration, no norm
19、al vector, the flow climb up/down the slope34For a free surface, the boundary is moving with the fluidAt the free surface The simplest mode:A flat bottom and a free surface of which the vertical displacement are neglected, the vertical velocity is zero. ()0dzdtdzdwuvdttxxdt35Difference between sea a
20、nd atmosphere (lateral boundary 侧边界问题)Open boundaries for a coast36Condition at sea-atmosphere interface and in sea1. Ignoring the surface tension (short water waves)Patm=psea at sea level 2.With the sea surface elevationPsea (interface)=Patm at sea level + 0 g ( , , )zx y t3. 大气中压强? 海水中压强?(均质流体, 水平
21、压强梯度可从上向下传)x, y(x, y)h(x, y) is the height of the free surface at (x, y); p0 = p(x, y, h) the pressure at the top of fluid layerp0(x, y, h) 0();()0;()0pgpg hzpzphphggxxyyppzxzy 以及5.4 地转流浅水模型用于各种尺度浅水模型在大尺度流动时得到地转流形式 (罗斯贝数 Ro1)Scale analysis in geophysical flows (x ,y) HL; WU, V; For large scale WL/(U
22、H)1 or 1 39222*22201()HVuuuupuuuuvwf wfvAAtxyzxxyzUT2UL2ULWUHU0PL2HA UL2VAUH1TULWL UUHL0PLU2HAL2VAHUL1/()1ORULfor large scale motion22;VHVHAAEEHLVertical/horizontal Ekman number大尺度运动, 惯性力不重要, 科氏力和压力梯度重要地转流 Geostropic flow, 当大尺度时Ro11hhdVpf k Vdt We saw that the acceleration term is relatively small.
23、Omitting it, we get a diagnostic relationship called geostrophic balance:101hhpf k VVkpf 0011pfvxpfuy矢量形式分量形式大尺度 地转平衡关系式 (二力平衡)00011pfvxpfuypgzGeostropic flow (地转流2) between Coriolis term and pressure gradient term1hVkpf Geostropic flow or current (地转流) 地转流：水平压强梯度力与科氏力取得平衡时的定常流动。中高纬度大气和海洋大尺度准水平运动，流体
24、平行于等压线流动， 无垂直运动，无辐散。在北半球垂直于压强梯度力指向右方，当观测者顺流而立时，右侧等压面高，左侧低 何时不成立？(尺度分析/边界/粘性）1) tropics where Coriolis parameter 0 and = 0 at equator; 2) small scale motion like atmospheric turbulence, where Earth rotation is negligible; 3) In small-scale motion and near wall boundary where friction/viscosity is sig
25、nificant.11hVkpkff 43 实际大气和海洋中存在各种破坏地转平衡,产生地转偏差的因素,例如粘性作用/斜压性/边界的存在/中小尺度等. 地转流是一种理想的流动. 地转平衡关系是大尺度地球流体运动的第一近似,准地转运动是该类流动的显著特征.44海水中压力梯度：密度倾斜度 海水中也存在海水中也存在 地转流地转流: 在忽略湍流摩擦力作用的较深的理想海洋中，由海水密度分布不均匀所产生的水平压强梯度力与水平地转偏向力平衡时的海流。虽然它和埃克曼漂流都是理想化的海流，但都能近似地反映海水的一些运动规律。 两种大洋的基本流动: 较厚的大洋下层水中的海流，近似于地转流；较薄的大洋上层水中，同时存
26、在着地转流和埃克曼漂流。这两种流动同为大洋的基本流动。45Geostropic flow or current (地转流)Large-scale motion (RoD, 准水平；准静力平衡; 势涡守恒，改变涡度的唯一机制是涡管的伸缩效应。适用于海洋和大气(2.)若浅水模型大尺度 (Ro1), 则为地转流, 两维化水平无辐散水平无辐散 (no horizontal divergence/convergence)530;,zxyuvzz 0fddtHz()0;/0zuvVxywzw 常数/ 0zdfdt地转流为浅水模型中的特例地转流的边界效应 If the fluid is limited in
27、 the vertical by a flat bottom (horizontal ground or sea for atmosphere) or by a flat lid (sea surface for the ocean), then vertical velocity is zero. The flow is prevented from climbing up or down the bottom slope, but have to go around the slope.54/00wzw 地转流的边界效应 Geostropic flow in a closed domain
28、 and over irregular topography. Solid lines are isobaths (Contours of equal depth 等深线), Flows are only constrained along closed isobaths. 对海洋和大气5556A meteorological example showing the high degree of parallelism between wind velocities and pressure contours (isobars), indicative of geostrophic balan
29、ce. Wind vectors are depicted by arrows with flags and barbs. The dashed lines are isotherms (PH). In flat-bottomed regions a geostrophic flow can assume arbitrary patterns, and the actual pattern reflects the initial conditions. But, over a bottom where the slope is non-zero almost everywhere, the
30、geostrophic flow has no choice but to follow the depth contours (called isobaths 等深线). Pressure contours are then aligned with topographic contours. These lines are sometimes also called geostrophic contours. Note that a relation between pressure and fluid thickness exists but cannot be determined w
31、ithout additional information on the flow. Open isobaths that start and end on a side boundary cannot support any flow, otherwise fluid would be required to enter or leave through lateral boundaries. The flow is simply blocked along the entire length of these lines. In other words, geostrophic flow
32、can occur only along closed isobaths.57涡度方程的地转平衡形式58Geostropic flow or current (地转流)Absolute vorticity in z direction for large-scale motion (Ro1) turns out that 地转流中涡度分量守恒 (Kelvin Theorem) 0;zdfdt()0;0zuvwVxyz 1;?zzVkpfgpgpuvfyfx 59Low p气旋气旋Cyclonic flow is Around low pressure; Anticycloniczvuxy+=0
33、zdffVdt zzTaylor-Proudman theorem 旋转以后,短柱体平移时,其上部会形成一相应的流体柱, 并且该流体柱具有旋转刚性,会跟着其下部的短柱体一起挺直的移动Taylor-Proudman ColumnIf relative motion is created in a rotating container by heating or stirring with an obstacle at the bottom of the tank, the moving fluid must flow around it, then the streamlines will fo
34、rm a column, going around the obstacle as if it extends to the top. Taylors column60Taylor-Proudman theorem 证明 在均质或正压旋转流体中,流体准定常和缓慢地运动(强旋转条件, Ro1),其速度将独立于旋转轴的方向,即运动将趋于两维化（从地转平衡关系证明）11()()(0)()0-()()()()zzzzzzzzf kVpf kVpVf kVf kVf VkfV kfk V 均质正压及准水平61()00()zzVf kVz 水 平 流 场 不 随 高 度 变 化水平无辐散62Equatio
35、n of divergence 散度方程散度方程10(1)10(2)uuupuvfvtxyxvvvpuvfutxyy Taking the x-derivative of (1) and adding it to the y-derivative of (2)，then we get the horizontal divergence equation:22()2()0zzzzzVVVuvftxyuvv uVugpxyxyb 02cosdfdyabHere地转平衡2zgpf浅水模型的特点（流动/涡度方程） 流体均质 （无层结），无粘不可压缩；水平尺度远大于垂直尺度；静力平衡 (p 高度 重力势
36、) 流体是旋转的，且旋转轴与z轴重合；准水平（u,v对z的偏导数为0）;具有自由面和刚性下边界；位涡守恒。方程得到简化。 水平散度与固定长度液柱的截面相对变化有关或与固定长度液柱的高度相对变化有关。垂直速度w是高度z的线性函数。每个流体元在运动的过程中，将保持其相对于下底面的高度不变。(w=dz/dt)0;( , , )hBHHVH x y thht ( , , , )()( , , )hBw x y z tVzw x y ht ()0;()0BzzhfdddtHdtH63作业 1. 地转流属什么尺度的运动，解释什么实际现象？ 2.地转平衡是什么力之间的平衡？ 3.与地转平衡对应的三维空间流场有什么特征？影响范围？ 4.浅水模型的优缺点？浅水模型的。 优点：简单，在保留流体主要的动力学特征的前提下，避免了地转退化，密度变化和复杂的热力学问题，使方程组大大简化。能描述大气和海洋的一些重要的大尺度特征。 缺点：。？65
