《地球物理流體力學(xué)》課件：Lecture 5_淺水模型和地轉(zhuǎn)流

1、2() (),1(),aaijVVp 為渦管的傾斜效應(yīng)為渦管的伸長效應(yīng)為力管項 因為斜壓性 為渦度源為摩擦力項 為渦度匯(due to friction effect)(due to baroclinicity)(for vortex strectching)(for vortex tilting)()aaadVdtt 渦度可以通過對流傳輸(Convection effect)aaauvwxyz1上次課堂作業(yè)上次課堂作業(yè)21()()aaaijdpVVdt Chapter 5 Inviscid shallow water model and Geostropic flow無粘淺水模型 和地轉(zhuǎn)流局

2、地直角坐標（or z coordinate): 隨地球轉(zhuǎn)動而運動的正交坐標系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 埃克曼數(shù)?？寺鼣?shù)Ek1 viscosity can

3、be neglectedEk1 viscosity is important?？寺鼘影？寺鼘樱‥kman layer）是流體中壓力梯度力、科氏力和湍流粘性力三力平衡的一層, 粘性力不可忽視。由瑞典海洋學(xué)家?？寺岢?。?？寺鼘永碚撨m用于許多地區(qū)，包括大氣層底部（接近地球表面和海洋），大洋底部（海床附近）和表層海水（海氣界面附近）。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 大氣, 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 淺水模型無粘，正壓均質(zhì) 且不可壓縮 （無層結(jié)結(jié)構(gòu)）遠離Ekman層處， 無粘流體方程: 不考慮粘性*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, 特征深度或運動的垂直特征尺度H遠小于水平特征尺度L; lwu, v; 且垂直速度存在上屆且垂直速度存在上屆wuH/L; 即w/uH/L1, 換句話說，不可壓縮流體的運動是準水平的x, y(x, y)fH0uvzz準水平10*00*

8、0111uuuupuvwf wfvtxyzxvvvvpuvwfutxyzywwwwpuvwf ugtxyzz0;uvwxyz不可壓縮垂直速度 wu and v00;uvpgzzz 準水平準靜壓平衡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, 慣性力不重要, 科氏力和壓力梯度重要/()1ORUL中尺度, 慣性力, 科氏力和壓力梯度 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當科氏力重要時Ro1(中尺度)Hydrostatic balanceU/1UWWUULLL當科氏力重要時Ro1We assume

10、w u and v, and we can neglect the term. For horizontal components柯氏參數(shù)the Coriolis parameter, f = 2 sin , the latitude比濕方程能量方程For equation of horizontal velocity (u, v, 0) 水平運動(u, v)與高度z無關(guān) 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準水平1/pdu dtfvx 1/pdv dtfuy b b-plane approximation b b平面近似平面近似 當?shù)亓鬟\動的徑向范圍(尺度)與地球半徑(a6000km) 相比很小時 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.當方程中f不被微分時， f =f0 2 sin0=constant 當方程中f被微分時, f =f0+by=f0 2 sin0b b =df/dy=2 sin0/a=constantA性質(zhì) 1x, y(x, y)fH()0hdHuvdHHHVdtxydt 水平輻散時，流體柱水平面積增大，厚度減小，體積不變11hdHdAVH dtA dt;BHhh0;uvwxyz不可壓縮Continuity equation 證明過程證明過程()0hdHuvdHHHVdtxydt ()()BhhuvuvdzH

13、xyxy()( )()()BhBhwdHdzw hw hzdt邊界條件();huvVVolAHxy ;BHhhHere水平輻散時，流體柱水平面積增大，厚度減小，體積不變從連續(xù)性方程得到0;uvwxyz不可壓縮性質(zhì) 2To Show證明證明: 垂直速度的動量方程消失垂直速度的動量方程消失,方程簡化方程簡化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的線性函數(shù), 且與水平輻散有關(guān)性質(zhì) 30hdHHVdt （2.24）5.2 淺水模型中的渦度方程24 無粘,不可壓縮,均質(zhì)(等密度), 準水平21()()(2cos )(2sin )0aaaijaxyzdpVVdtijkuvzz 準水平Vorticity equation in shallow water equation25(/)(/)(/); ,/1 xzyzzzxywvwOULyzyuwwOULzxxvuO ULxyD L 為淺水近似條件； 準水平W/DD, 無粘無粘,不可壓縮不可壓縮,勻質(zhì)勻質(zhì), 準水平準水平, 正壓，靜壓平衡正壓，

15、靜壓平衡) Combining and playing with mathematicszvxwzuywpfdtfdz1kv101010uuupuvfvtxyxvvvpuvfutxyypgz 1zdfw uw vfVpdtyzxz k zz+=0zdffVdt zzvuxyz26()0;dHuvHdtxy+=0zdffVdt zzPotential-Vorticity conservation in shallow water theory (LD, 無粘無粘,不可壓縮不可壓縮,勻質(zhì)勻質(zhì), 準水平準水平, 正壓）正壓）()dffdHdtHdtzz絕對渦度分量的相對變化反比于水平輻散 (colu

16、mn/tube height)Relative rate-of-change of absolute vorticity is equal to minus divergence0BfddtHHhhz淺水理論中勢渦守恒性質(zhì) 4淺水模型中 粘性，斜壓性和渦管的傾斜機制均不存在，所以改變渦度的唯一機制是渦管的伸縮效應(yīng)，或水平輻散/輻合效應(yīng); 適用于不同尺度,可存在垂直運動,存在邊界效應(yīng)影響。290fdabsolutevorticitydtHfluiddepthz5.3 淺水模型的邊界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淺水模型淺水模型-no penetration/impermeability condition理想不可壓縮流動的固壁邊界理想不可壓縮流動的固壁邊界 V n=U n V n=0 , ruu粘性流動采用的是固壁上的無滑移條件，由于理想流體動量方程中失掉了高階粘性項，歐拉方程比N-S方程低了一階，她就不需要象粘性流方程組那樣多的邊界條件。對理想流體采用法向無穿透

18、條件，理想流體采用法向無穿透條件，壁面上允許存在切向滑移速度壁面上允許存在切向滑移速度，固壁靜止時，上述邊界條件相當于要求固體壁面是流場中的一條流線。無窮遠邊界條件， 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 側(cè)邊界問題)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. 大氣中壓強? 海水中壓強?(均質(zhì)流體, 水平

21、壓強梯度可從上向下傳)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 地轉(zhuǎn)流淺水模型用于各種尺度淺水模型在大尺度流動時得到地轉(zhuǎn)流形式 (羅斯貝數(shù) 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大尺度運動, 慣性力不重要, 科氏力和壓力梯度重要地轉(zhuǎn)流 Geostropic flow, 當大尺度時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 地轉(zhuǎn)平衡關(guān)系式 (二力平衡)00011pfvxpfuypgzGeostropic flow (地轉(zhuǎn)流2) between Coriolis term and pressure gradient term1hVkpf Geostropic flow or current (地轉(zhuǎn)流) 地轉(zhuǎn)流：水平壓強梯度力與科氏力取得平衡時的定常流動。中高緯度大氣和海洋大尺度準水平運動，流體

24、平行于等壓線流動， 無垂直運動，無輻散。在北半球垂直于壓強梯度力指向右方，當觀測者順流而立時，右側(cè)等壓面高，左側(cè)低 何時不成立？(尺度分析/邊界/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 實際大氣和海洋中存在各種破壞地轉(zhuǎn)平衡,產(chǎn)生地轉(zhuǎn)偏差的因素,例如粘性作用/斜壓性/邊界的存在/中小尺度等. 地轉(zhuǎn)流是一種理想的流動. 地轉(zhuǎn)平衡關(guān)系是大尺度地球流體運動的第一近似,準地轉(zhuǎn)運動是該類流動的顯著特征.44海水中壓力梯度：密度傾斜度 海水中也存在海水中也存在 地轉(zhuǎn)流地轉(zhuǎn)流: 在忽略湍流摩擦力作用的較深的理想海洋中，由海水密度分布不均勻所產(chǎn)生的水平壓強梯度力與水平地轉(zhuǎn)偏向力平衡時的海流。雖然它和?？寺鞫际抢硐牖暮Ａ?，但都能近似地反映海水的一些運動規(guī)律。 兩種大洋的基本流動: 較厚的大洋下層水中的海流，近似于地轉(zhuǎn)流；較薄的大洋上層水中，同時存

26、在著地轉(zhuǎn)流和?？寺鳌＿@兩種流動同為大洋的基本流動。45Geostropic flow or current (地轉(zhuǎn)流)Large-scale motion (RoD, 準水平；準靜力平衡; 勢渦守恒，改變渦度的唯一機制是渦管的伸縮效應(yīng)。適用于海洋和大氣(2.)若淺水模型大尺度 (Ro1), 則為地轉(zhuǎn)流, 兩維化水平無輻散水平無輻散 (no horizontal divergence/convergence)530;,zxyuvzz 0fddtHz()0;/0zuvVxywzw 常數(shù)/ 0zdfdt地轉(zhuǎn)流為淺水模型中的特例地轉(zhuǎn)流的邊界效應(yīng) 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 地轉(zhuǎn)流的邊界效應(yīng) Geostropic flow in a closed domain

28、 and over irregular topography. Solid lines are isobaths (Contours of equal depth 等深線), Flows are only constrained along closed isobaths. 對海洋和大氣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 等深線). 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渦度方程的地轉(zhuǎn)平衡形式58Geostropic flow or current (地轉(zhuǎn)流)Absolute vorticity in z direction for large-scale motion (Ro1) turns out that 地轉(zhuǎn)流中渦度分量守恒 (Kelvin Theorem) 0;zdfdt()0;0zuvwVxyz 1;?zzVkpfgpgpuvfyfx 59Low p氣旋氣旋Cyclonic flow is Around low pressure; Anticycloniczvuxy+=0zdffVdt zzTaylor-Proudman theorem 旋轉(zhuǎn)以后,短柱體平移時,其上部會形成一相應(yīng)的流體柱, 并且該流體柱具有旋轉(zhuǎn)剛性,會跟著其下部的短柱體一起挺直的移動Taylor-Proudman ColumnIf relative motion is created in a rotating container by heating or stirring with an obstacle at the bottom o