李陶，方欣(中國(guó)科學(xué)技術(shù)大學(xué)地球和空間科學(xué)學(xué)院，安徽合肥 230052)

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鈉激光雷達(dá)對(duì)中間層頂大氣溫度和風(fēng)場(chǎng)的探測(cè)

李陶，方欣
(中國(guó)科學(xué)技術(shù)大學(xué)地球和空間科學(xué)學(xué)院，安徽合肥 230052)

摘 要:中高層大氣溫度和風(fēng)場(chǎng)是研究中高層大氣波動(dòng)的重要參數(shù)．鈉高光譜分辨率激光雷達(dá)能夠?qū)χ虚g層頂(80－105km)大氣溫度和風(fēng)場(chǎng)進(jìn)行高精度觀測(cè)．2011年中國(guó)科學(xué)技術(shù)大學(xué)成功研制了我國(guó)首臺(tái)高光譜分辨率鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)系統(tǒng)．文中對(duì)該激光雷達(dá)系統(tǒng)進(jìn)行了詳細(xì)介紹，其中包括探測(cè)的基本原理，發(fā)射機(jī)，接收機(jī)和采集控制部分的設(shè)計(jì)．給出了鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)于2011年12月9日晚同時(shí)探測(cè)的大氣溫度、緯向風(fēng)、經(jīng)向風(fēng)和鈉原子密度的結(jié)果．結(jié)果發(fā)現(xiàn)中間層頂區(qū)域大氣溫度、緯向風(fēng)、經(jīng)向風(fēng)變化范圍較大，分別是175K～235K，－70～60m/s，－100～110m/s，有明顯的半日或周日潮汐振蕩的成分．探測(cè)結(jié)果表明中國(guó)科學(xué)技術(shù)大學(xué)鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)可對(duì)中間層頂區(qū)域溫度和大氣風(fēng)場(chǎng)進(jìn)行高時(shí)空分辨率的探測(cè)，其探測(cè)數(shù)據(jù)對(duì)于研究中高層大氣動(dòng)力學(xué)具有重要的意義．

關(guān)鍵詞:大氣溫度;緯向風(fēng);經(jīng)向風(fēng);中高層大氣

引用格式:李陶，方欣．鈉激光雷達(dá)對(duì)中間層頂大氣溫度和風(fēng)場(chǎng)的探測(cè)［J］．安徽師范大學(xué)學(xué)報(bào):自然科學(xué)版，2015，38(2) :103－109．

引言

中間層頂區(qū)域是大氣波動(dòng)(重力波、潮汐波、行星波等)的活躍區(qū)域．這些波動(dòng)攜帶的能量和動(dòng)量會(huì)隨著波的破碎和耗散過(guò)程直接影響背景大氣，從而驅(qū)動(dòng)中間層大氣環(huán)流以及各大氣層間的禍合．因此對(duì)該區(qū)域的大氣溫度和風(fēng)場(chǎng)的探測(cè)對(duì)于研究這些中高層大氣物理過(guò)程具有非常重要的意義．中頻雷達(dá)和流星雷達(dá)可以實(shí)現(xiàn)對(duì)該區(qū)域風(fēng)場(chǎng)的垂直分辨探測(cè)，但很難獲得溫度信息．由于對(duì)鈉熒光光譜認(rèn)識(shí)的提升，推動(dòng)了激光雷達(dá)對(duì)中間層頂區(qū)域溫度和風(fēng)場(chǎng)探測(cè)的快速發(fā)展．Gibson和Thomas等于1979年首次實(shí)現(xiàn)了對(duì)中間層頂區(qū)域溫度的測(cè)量，在鈉層峰值附近溫度誤差±15K［1］．Fricke和von Zahn(1985)利用準(zhǔn)分子泵浦染料激光器實(shí)現(xiàn)了在10分鐘內(nèi)1km垂直分辨率和5K精度的溫度廓線測(cè)量．1990年美國(guó)科羅拉多州立大學(xué)(CSU) She帶領(lǐng)的研究組和伊利洛伊大學(xué)Gardner研究組合作，研制了利用雙頻技術(shù)的高光譜分辨率窄帶系統(tǒng)，并于1991年開始了對(duì)中間層頂(80－105 km)溫度的常規(guī)觀測(cè)(She et al．，1992)［2］，于1997年開始同時(shí)對(duì)溫度和水平風(fēng)場(chǎng)的常規(guī)觀測(cè)，并于2002年開始了對(duì)溫度和風(fēng)場(chǎng)的24hr連續(xù)常規(guī)觀測(cè)(She et al．，2003)．美國(guó)科羅拉多州立大學(xué)于2000年在ALOMAR建了另外一臺(tái)Na測(cè)溫測(cè)風(fēng)激光雷達(dá)(She et al．，2002)［3］．美國(guó)伊利洛伊大學(xué)也于1994年與科羅拉多州立大學(xué)合作在Urban建了一臺(tái)類似的Na溫測(cè)風(fēng)激光雷達(dá)，其技術(shù)指標(biāo)和性能與CSU激光雷達(dá)近似．在亞洲，日本的信州大學(xué)的N．激光雷達(dá)系統(tǒng)可以探測(cè)中間層頂區(qū)域的溫度(Kawahara et al．，2003)［4］，但目前未有風(fēng)場(chǎng)數(shù)據(jù)的報(bào)道．近幾十年來(lái)，國(guó)際上鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)為中高層大氣研究做出了突出貢獻(xiàn)．

針對(duì)國(guó)內(nèi)對(duì)中間層頂區(qū)域探測(cè)資料醫(yī)乏的現(xiàn)狀，中國(guó)科學(xué)技術(shù)大學(xué)(USTC)成功研制了一臺(tái)鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)系統(tǒng)，是國(guó)內(nèi)首次通過(guò)了國(guó)內(nèi)外專家鑒定的鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)系統(tǒng)(CLi et al．［5］)．本文將對(duì)該鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)系統(tǒng)設(shè)計(jì)進(jìn)行詳細(xì)介紹，并給出其鈉原子密度、溫度和風(fēng)場(chǎng)的探測(cè)結(jié)果，及其對(duì)重力波動(dòng)量通量的探測(cè)結(jié)果．

1　鈉高光譜分辨率激光雷達(dá)系統(tǒng)

鈉高光譜分辨率激光雷達(dá)系統(tǒng)由發(fā)射機(jī)、接收機(jī)、采集部分和控制部分組成．圖1給出了該激光雷達(dá)系統(tǒng)的原理示意圖．其探測(cè)的基本原理是:通過(guò)發(fā)射鈉原子熒光譜D2a線的三個(gè)頻率窄帶激光(υ0，υ0+ 630MHz，υ0－630MHz)，其中υ0對(duì)應(yīng)的波長(zhǎng)為589．158nm．系統(tǒng)分別接收這三個(gè)頻率的回波信號(hào)．由于溫度和風(fēng)速的變化，會(huì)引起鈉熒光光譜的多普勒增寬和多普勒頻移，這會(huì)體現(xiàn)在三個(gè)頻率回波信號(hào)強(qiáng)度的變化．根據(jù)實(shí)際探測(cè)的三個(gè)頻率回波信號(hào)光子數(shù)，可反演出大氣溫度和風(fēng)場(chǎng)．利用任一頻率回波信號(hào)光子數(shù)可反演出鈉原子密度．

下面將分別對(duì)系統(tǒng)各個(gè)部分進(jìn)行闡述．

1．1發(fā)射機(jī)

鈉高光譜分辨率激光雷達(dá)系統(tǒng)發(fā)射機(jī)主要由:半導(dǎo)體連續(xù)激光器(CW YAG)、環(huán)形染料激光器(Ring dye laser)、脈沖泵浦激光器(30Hz Pulsed YAG)和脈沖染料放大激光器(PDA)、激光頻率絕對(duì)鎖定跟蹤系統(tǒng)(Doppler free system)、三頻激光發(fā)生器(AOM)和自動(dòng)準(zhǔn)直系統(tǒng)(Beam steering)組成．

半導(dǎo)體連續(xù)激光器泵浦環(huán)形激光器用于產(chǎn)生整個(gè)激光雷達(dá)系統(tǒng)所需的窄帶激光光源．半導(dǎo)體連續(xù)激光器輸出波長(zhǎng)532nm的激光，其最大輸出率為6W左右．半導(dǎo)體連續(xù)激光器泵浦環(huán)形染料激光器Matisse DS，產(chǎn)生單模種子光源．環(huán)形染料激光器通過(guò)外部參考腔(reference cavity)采用側(cè)邊緣鎖定技術(shù)(side offringe)可使得激光器輸出的激光線寬均方根值在500KHz以內(nèi)．通過(guò)掃描外部參考腔的方法實(shí)現(xiàn)精細(xì)掃描激光頻率，其掃描范圍約為6GHz．最后輸出的波長(zhǎng)為589．158nm．

由于環(huán)形染料激光器本身的參考腔只能相對(duì)鎖定激光器的頻率，隨著環(huán)境的變化(溫度、震動(dòng)等)和自身壓電陶瓷原件的松弛，其輸出的激光頻率會(huì)產(chǎn)生漂移，可能在幾秒或幾分鐘內(nèi)產(chǎn)生幾百M(fèi)Hz的頻率漂移．而對(duì)于測(cè)量大氣風(fēng)場(chǎng)來(lái)說(shuō)，1MHz的頻移將會(huì)引起風(fēng)速測(cè)量誤差約為0．6m/s(Li，2005)［6］．因此，激光器輸出激光的頻移將引起很大的測(cè)量誤差．我們需絕對(duì)鎖定環(huán)形染料激光器輸出激光的頻率．激光頻率絕對(duì)鎖定跟蹤系統(tǒng)用于保證環(huán)形染料激光器的輸出波長(zhǎng)精確穩(wěn)定在589．158nm．我們選用充滿鈉原子的蒸汽池(鈉原子池，Sodium cell)中的鈉原子在受激輻射情況下輸出的熒光信號(hào)譜線作為頻率的絕對(duì)標(biāo)準(zhǔn)．激光頻率絕對(duì)鎖定和自動(dòng)跟蹤系統(tǒng)結(jié)合環(huán)形染料激光器外部參考腔的側(cè)邊緣(side offringe)鎖定技術(shù)(Fritschel，1989)［7］，利用相位鎖定技術(shù)(Arie et al．，1992; Gianlucaet al．，2003; Smith et al．，2008)，通過(guò)基于La bview的鎖定控制軟件，在外部參考腔的帶有壓電陶瓷的鏡片上加上頻率為的參考抖動(dòng)信號(hào)，將光電倍增管探測(cè)的熒光信號(hào)和參考抖動(dòng)信號(hào)進(jìn)行相位比較，得到偏差信號(hào)，通過(guò)PID(比例積分微分)控制算法將該偏差信號(hào)轉(zhuǎn)換為NI(美國(guó)國(guó)家儀器公司)的多功能數(shù)據(jù)采集卡輸出電壓控制壓電陶瓷執(zhí)行動(dòng)作．從而使得環(huán)形染料激光器始終鎖定在鈉無(wú)多普勒熒光光譜的D2a峰值凹陷處．圖2是激光頻率絕對(duì)鎖定跟蹤系統(tǒng)掃描得到的鈉原子飽和熒光光譜．鎖定系統(tǒng)精度為±2MHz．

三頻激光發(fā)生器用于產(chǎn)生鈉高光譜分辨率激光雷達(dá)系統(tǒng)所需的三個(gè)頻率(υ0，υ0+630MHz，υ0－630MHz)的種子光源．我們主要利用聲光相互作用的基本原理，實(shí)現(xiàn)激光頻率頻移的目的．環(huán)形激光器的輸出激光以布拉格入射角入射到聲光晶體器件(AOM)．其工作原理圖3如圖所示．通過(guò)外部施加丁TL電平信號(hào)給聲光晶體的控制信號(hào)源，使得聲光晶體是否處在調(diào)制頻率的狀態(tài)．在第一個(gè)脈沖周期(0～1/ 30s)內(nèi)，施加給AOM1和AOM2的控制信號(hào)源TTL電平為低電平，兩塊聲光器件都不調(diào)制激光頻率，聲光調(diào)制器系統(tǒng)輸出的激光頻率為D2apeak，圖3上圖所示．在第二個(gè)脈沖周期(1/30～2/30)，施加給AOM1控制信號(hào)源低電平，施加給AOM2控制信號(hào)源高電平AOM1不調(diào)制激光頻率，AOM2向上頻移激光頻率315MHz，由于兩次通過(guò)AOM2，聲光調(diào)制系統(tǒng)輸出的激光頻率為υ0+ 630MHz，圖3中間圖所示．在第三個(gè)脈沖周期(2/30～3/30)內(nèi)，施加給AOM1控制信號(hào)源高電平，施加給AOM2控制信號(hào)源低電平，AOM1向下頻移激光頻率315MHz，AOM2不調(diào)制激光頻率，同樣兩次通過(guò)AOM1，聲光調(diào)制系統(tǒng)輸出的激光頻率為υ0－630MHz，圖3下圖所示．整個(gè)聲光調(diào)制器系統(tǒng)最終輸出光的頻率是30Hz，交替輸出，其順序?yàn)棣?，υ0+630MHz和υ0－630MHz．

脈染料沖放大器用于放大三個(gè)頻率種子激光的功率，并將其轉(zhuǎn)換成脈沖激光．其由聲光調(diào)制器輸出的三種頻率激光在染料池DC1，DC2和DC3中被脈沖ND: YAG激光器輸出的532nm激光泵浦，從而實(shí)現(xiàn)激光的三級(jí)放大．圖4是我們自主研制的脈沖染料放大器的原理圖．從聲光調(diào)制器輸出的589nm激光由1．5～3mm小孔光闌入射，經(jīng)介質(zhì)反射鏡M1反射后由平凸透鏡L1匯聚通過(guò)第一級(jí)染料流動(dòng)池DC1，被染料介質(zhì)吸收后的激光由50微米小孔Pinhole 2改善光斑形狀，經(jīng)由雙凹透鏡L2和雙凸透鏡L3匯聚通過(guò)第二級(jí)染料流動(dòng)池DC2內(nèi)染料介質(zhì)后，由小孔Pinhole 3改善光斑形狀，經(jīng)雙凹透鏡L4和雙凸透鏡L5改善光束發(fā)散角后通過(guò)第三級(jí)染料流動(dòng)池DC3．脈沖ND: YAG激光器輸出的532nm從PDA左上部左側(cè)入射，經(jīng)第一片光束分束器BS1分束，部分反射光由棱柱形透鏡CL1匯聚到染料流動(dòng)池DC1內(nèi)染料上．透射光再由第二片分束器分束，部分反射光由棱柱形透鏡CL2匯聚到染料流動(dòng)池DC2內(nèi)染料上，透射光由反射晶體P1和P2反射進(jìn)第三個(gè)染料流動(dòng)池DC3和589nm光束成小角度通過(guò)染料介質(zhì)．激光染料介質(zhì)由若丹明640(Rodamine 640，也叫Rhodamine 101)和奇通紅620(Kiton Red 620，也叫Sulforhodamine B)溶解在1000m1高純度甲醇混合而成．第一級(jí)和第二級(jí)使用同一種濃度染料介質(zhì)，Rodamine 640含量為9．16mg，Kiton Red含量為44．3mg．第三級(jí)使用一種濃度染料介質(zhì)，Rodamine 640含量為1．9mg，Kiton Red含量為14．3mg．染料介質(zhì)通過(guò)染料循環(huán)器抽運(yùn)，以7L/min的流速循環(huán)．在17W激光泵浦情況下，當(dāng)輸入的589nm頻率為υ0激光為350mW時(shí)，PDA輸出功率約為1．65W，當(dāng)輸入的種子激光υ0+630MHz的功率為300mW時(shí)，PDA輸出功率約為1．5W，當(dāng)輸入的種子激光υ0－630MHz的功率為250mW時(shí)，PDA輸出功率約為1．45W．PDA輸出激光的脈沖寬度和上升沿時(shí)間通過(guò)快速光電二極管和示波器測(cè)量，測(cè)得其輸出脈沖寬度約10ns，上升沿時(shí)間約4ns．

自動(dòng)準(zhǔn)直系統(tǒng)用于準(zhǔn)直鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)系統(tǒng)收發(fā)光軸，保證系統(tǒng)工作在最佳效率狀態(tài)．我們采用的回波信號(hào)法進(jìn)行準(zhǔn)直，通過(guò)驅(qū)動(dòng)電機(jī)控制發(fā)射天線在傾斜方向和旋轉(zhuǎn)方向進(jìn)行掃描，電機(jī)每動(dòng)作一次后記錄回波信號(hào)強(qiáng)度．準(zhǔn)直過(guò)程可以分為三步:1)回波信號(hào)追蹤; 2)單軸掃描;3)矩陣掃描．圖5和圖6給出了自動(dòng)準(zhǔn)直系統(tǒng)其中一個(gè)調(diào)整鏡架在傾斜和旋轉(zhuǎn)方向(對(duì)應(yīng)于地理東西和南北方向)掃描的回波信號(hào)曲線．脈沖累加次數(shù)為24，積分距離為4km，激光發(fā)射平均功率約1．4W．圖中顯示了三個(gè)不同高度(22km，30km和90km)的回波信號(hào)，從圖中可以看出近似的梯形函數(shù)關(guān)系，根據(jù)梯形的半高寬算出傾斜方向視場(chǎng)角約1．05mrad，旋轉(zhuǎn)方向視場(chǎng)角為0．975mrad．圖7給出自動(dòng)準(zhǔn)直系統(tǒng)進(jìn)行矩陣掃描的二維強(qiáng)度圖結(jié)果．脈沖累加次數(shù)為48，圖中顯示的信號(hào)是21km－23km的2km范圍的積分，由中心計(jì)算公式(1)計(jì)算出傾斜方向位置為27μrad，旋轉(zhuǎn)方向中心位置為70μrad．(中心位置的弧度值表示的含義是最后定準(zhǔn)的中心位置偏離自動(dòng)準(zhǔn)直鏡架初始位置的角度)．最終發(fā)射光束位置定位在矩陣掃描的等值線圖頂部的中心位置．由誤差傳遞公式，在能量波動(dòng)為5%的情況下，根據(jù)實(shí)際掃描信號(hào)計(jì)算出自動(dòng)準(zhǔn)直系統(tǒng)的準(zhǔn)直誤差約為10μrad．

1．2接收機(jī)

鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)接收系統(tǒng)主要用于接收鈉原子共振熒光后向散射回波信號(hào)．其主要由望遠(yuǎn)鏡、斬光盤系統(tǒng)、后繼接收光路組成．圖8是鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)接收系統(tǒng)和采集系統(tǒng)結(jié)構(gòu)圖．鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)系統(tǒng)選用牛頓反射式望遠(yuǎn)鏡．望遠(yuǎn)鏡直徑為76cm，F(xiàn)/#為F/2．4．在觀測(cè)經(jīng)緯向風(fēng)場(chǎng)模式下，兩臺(tái)望遠(yuǎn)鏡其中一臺(tái)向正東傾斜和天頂?shù)慕?0°，另一臺(tái)向正北傾斜和天頂角傾斜30°．望遠(yuǎn)鏡接收的回波信號(hào)由光纖傳輸?shù)胶罄m(xù)接收通道，包括斬光盤系統(tǒng)和后繼禍合光路．光纖芯徑為2000μm，數(shù)值孔徑NA =0．37，長(zhǎng)度15m，內(nèi)部透過(guò)率＞90%．?dāng)毓獗P系統(tǒng)由一高速旋轉(zhuǎn)的盤片、驅(qū)動(dòng)電機(jī)和電機(jī)控制器組成．其工作轉(zhuǎn)速為5400轉(zhuǎn)/分鐘，盤片直徑為200mm．光纖出射光經(jīng)過(guò)斬光盤片后由焦距f =30mm，直徑為25．4mm的透鏡準(zhǔn)直后通過(guò)濾光片．濾光片的直徑為25．4mm，中心波長(zhǎng)為589．1nm，帶寬為1nm．經(jīng)濾光片后的光信號(hào)再經(jīng)過(guò)f =40mm、直徑為25．4mm的透鏡匯聚到光電倍增管陰極接收面上．光譜響應(yīng)范圍為300～720nm，在589nm處其量子效率可達(dá)40%．該倍增管集成了內(nèi)部放大轉(zhuǎn)換電路，直接輸出0～5V的光子電脈沖信號(hào)，脈沖寬度為8ns，單光子脈沖分辨率為20ns．

1．3采集和控制部分

光電倍增管輸出的脈沖信號(hào)由PCI插槽的MCA－3系列光子計(jì)數(shù)卡P7882進(jìn)行采集，采集卡時(shí)間間隔設(shè)置為1μs(對(duì)應(yīng)距離分辨率為150m)，采樣長(zhǎng)度為2048．采集軟件基于LabVIEW開發(fā)環(huán)境開發(fā)．利用光子計(jì)數(shù)卡提供的動(dòng)態(tài)鏈接庫(kù)接口函數(shù)，可方便對(duì)光子計(jì)數(shù)卡進(jìn)行控制．程序界面左側(cè)用于常規(guī)參數(shù)的輸入和關(guān)鍵參數(shù)的顯示，右側(cè)是回波信號(hào)顯示．程序內(nèi)部集成有環(huán)形染料激光器報(bào)警功能模塊，用于提醒觀測(cè)人員在跳模情況下對(duì)系統(tǒng)進(jìn)行適當(dāng)調(diào)整操作．

時(shí)序控制部分是保證鈉測(cè)溫測(cè)風(fēng)激光雷達(dá)系統(tǒng)按照一定秩序正常運(yùn)行的關(guān)鍵，只有在時(shí)序正確的情況下，才能獲取到正確有效的回波信號(hào)．需要協(xié)同工作的設(shè)備有:接收系統(tǒng)斬光盤，聲光晶體，AOM系統(tǒng)斬光盤，控制計(jì)算機(jī)，光子計(jì)數(shù)卡，脈沖泵浦Nd: YAG激光器．圖9給出了該激光雷達(dá)系統(tǒng)時(shí)序控制框圖．接收斬光盤輸出脈沖外觸發(fā)第一臺(tái)數(shù)字延遲脈沖發(fā)生器DG645－1．DG645－1一個(gè)通道輸出同步脈沖激光器(30Hz pulsed YAG)，一個(gè)通道輸出同步第二臺(tái)數(shù)字延遲脈沖發(fā)生器DG645－2．DG645－2的三個(gè)通道輸出分別同步三頻激光發(fā)生器中聲光晶體1，聲光晶體2和AOM斬光盤．脈沖激光器的Q開關(guān)輸出、控制計(jì)算機(jī)和采集計(jì)算機(jī)的同步信號(hào)及DG645－2的二個(gè)通道輸出信號(hào)輸入給自制的轉(zhuǎn)換電路，轉(zhuǎn)換后的信號(hào)作為光子計(jì)數(shù)卡的觸發(fā)信號(hào)和光譜標(biāo)記信號(hào)．

2　探測(cè)結(jié)果

系統(tǒng)在進(jìn)行探測(cè)時(shí)，其存儲(chǔ)原始回波信號(hào)文件是150m距離分辨率，1分鐘時(shí)間分辨率的數(shù)據(jù)文件．在數(shù)據(jù)反演過(guò)程中，我們先進(jìn)行壞數(shù)據(jù)的判斷，將無(wú)效的回波信號(hào)剔除．接著對(duì)回波信號(hào)進(jìn)行預(yù)處理，為了提高系統(tǒng)的信噪比，積分15分鐘范圍內(nèi)回波信號(hào)，扣除背景噪聲，并以2km漢寧窗進(jìn)行平滑處理．最后，預(yù)處理后的回波信號(hào)，計(jì)算得到大氣溫度、緯向風(fēng)、經(jīng)向風(fēng)和鈉原子密度信息．圖10給出2011年12月09日晚探測(cè)的大氣溫度、緯向風(fēng)、經(jīng)向風(fēng)和鈉原子密度結(jié)果．從圖中可以看出，整晚溫度在175K～235K范圍內(nèi)．隨著時(shí)間的變化，溫度變化非常劇烈，14:00UT在93km處為整晚最低值175K，溫度最大值出現(xiàn)81Km左右．在87～105km高度內(nèi)，可以看到較為明顯的相位向下移動(dòng)的波動(dòng)傳播結(jié)構(gòu)，有較為明顯的12小時(shí)半日潮汐波動(dòng)，100km高度處幅度約為15K．緯向風(fēng)結(jié)果顯示緯向風(fēng)整晚變化范圍是－70m/s～60m/s．也可看到較為明顯的12小時(shí)半日潮汐波動(dòng)，幅度值約在101km高度最大，約55m/s振幅．經(jīng)向風(fēng)整晚在－100～110m/s范圍內(nèi)變化，也可看到較為明顯的相位向下傳播波動(dòng)結(jié)構(gòu)．可以看到明顯的周期12小時(shí)半日潮汐，其振幅在105km高度達(dá)到最大值，約100m/s．90km處可以看到較為明顯的周期24小時(shí)的周日潮汐，其幅度值約為30m/s．鈉原子密度隨高度和時(shí)間變化也比較劇烈，其最大值出現(xiàn)在18:30UT左右93km高度處，約4400原子數(shù)/cm3．

3　小結(jié)

本文詳細(xì)地?cái)⑹隽蒜c高光譜分辨率測(cè)溫測(cè)風(fēng)激光雷達(dá)的基本原理，并對(duì)激光雷達(dá)系統(tǒng)的發(fā)射子系統(tǒng)，接收子系統(tǒng)和控制子系統(tǒng)進(jìn)行了詳細(xì)的敘述．最后給出了測(cè)量結(jié)果，結(jié)果顯示在中間層頂區(qū)域(80－105km)，大氣溫度和風(fēng)場(chǎng)的變化與大氣重力波和潮汐波有直接的關(guān)系．目前我們己經(jīng)積累了大量的高精度探測(cè)數(shù)據(jù)，這對(duì)于研究中高層大氣動(dòng)力學(xué)，大氣重力波和潮汐波的相互關(guān)系具有重要的科學(xué)意義．同時(shí)該激光雷達(dá)對(duì)于提高我國(guó)空間環(huán)境的保障能力具有非常重要的應(yīng)用價(jià)值．

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High-Spectra Resolution Sodium Lidar for Mesopause Region Temperature and Wind Measurements

LI Tao，F(xiàn)ANG Xin
(School of Earth and Space Sciences，University of Science and Technology of China，Hefei 230052，China)

Classification No: O175．8

Document code: A

Paper No:1001－2443(2015) 02－0110－07

Received date: 2014－04－10

Foundation item: Supported by the National Natural Science Foundation of China(11026213)．

Corresponding author: YAN Jie(1990－)，F(xiàn)emale，born in Jiangsu，master，major in differential equation．

引用格式:嚴(yán)潔，肖建中．帶積分邊界條件的奇異多點(diǎn)邊值問(wèn)題的正解［J］．安徽師范大學(xué)學(xué)報(bào):自然科學(xué)版，2015，38(2) :110－116．

In this paper，we consider the existence of positive solutions for the following singular integral boundary value problem (BVP for short)

where m≥3，0＜η1＜…＜ηm－2＜1，αi≥0(i = 1，2，…，m－2) withf∈C(［0，∞)，［0，+∞) ) and g∈L1［0，1］is nonnegative; the function a∈C((0，1)，［0，+∞) ) may be singular at t = 0，1．

The multi-point boundary value problems for ordinary differential equations arise in areas of applied mathematics and physics．For example，many problems in the theory of elastic stability，non Newtonian fluid and the turbulence theory of gases in porous media can be handled by the method of multi-point problems (see［1，2］)．In 1980s，Bitsadze and Samarkii firstly researched the nonlocal elliptic boundary value problems (see［3］)．Motivated by their work，the study of multi-point boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moiseev (see［4］)．Since then，multi-point boundary value problems for nonlinear ordinary differential equations has received much attention for many authors．To identify a few，we refer the reader to［5－7］and references therein．

In［6］，the following second order three-point boundary value problem was considered，where 0＜α＜1，0＜η＜1，f∈C(R，R)．The authors obtained the existence results of signchanging solutions for the above three-point boundary value problem by using the fixed-point index method．

A class of boundary value problems with integral boundary conditions for ordinary differential equations arise in the study of various physical，biological and chemical processes，such as heat conduction，undergroundwater flow，thermoelasticity and plasma physics．The existence of positive solutions for such class of problems has attracted much attention (see［8，9］)．

In［8］，F(xiàn)eng et al．investigated the existence and nonexistence of positive solutions of the following second order boundary value problem with integral boundary conditions in Banach space．

In［9］，Liu et al．investigated the existence of positive solutions for the singular second order integral boundary value problem where a，b∈C［0，1］，c∈C((0，1)，(0，+∞) )，f∈C((0，+∞)，［0，+∞) ) and g，h∈L1［0，1］are nonnegative; c may be singular at t = 0，1 and f may be singular at u = 0．The authors established the existence of positive solutions for the above problem by applying the fixed point index theorems．

Motivated by the works mentioned above，in this paper，we consider the BVP(1)．To the best of our knowledge，there are very few results for multi-point boundary value problems with integral boundary conditions．By applying Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type，we establish the existence of positive solutions．

1 Preliminaries

Let E = C［0，1］be a Banach space with the norm，and letbe a cone in E，where σ is a positive constant．For ΩP，the closure of Ω．The function u∈C［0，1］∩C2(0，1) is said to be a positive solution of the BVP (1)，if it satisfies (1) and u(t)≥0 for t∈(0，1)．The function e will be defined by e(s) = s(1－s)，s∈［0，1］．

We will frequently use the following constants:

For convenience in presentation，we here list two assumptions to be used throughout the paper．

(H1) f∈C(［0，+∞)，［0，+∞) )，g∈L1［0，1］is nonnegative such that k＞0．

(H2) a: (0，1)→［0，+∞) is continuous，and，and a(t)0 on any subset of (0，1)．

Our approach is based on the following Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type．

Lemma 1．1(see［10］) Let E a Banach space，and PE be a cone in E．Assume Ω1，Ω2are open subset ofE withand A:be a completely continuous operator such that，either

(1)‖Au‖≤‖u‖，u∈P∩Ω1，‖Au‖≥‖u‖，u∈P∩Ω2; or

(2)‖Au‖≥‖u‖，u∈P∩Ω1，‖Au‖≤‖u‖，u∈P∩Ω2，then A has a fixed point in

2 Lemmas

Lemma 2．1 Assume that (H1) (H2) hold，y(t)∈C［0，1］．Then the BVPhas a unique solution can be expressed in the form

Proof．Integrating both sides of (5) on［0，t］twice，we have By (5)，we get

Combining with (10)，(11) and (12)，we get

Thus，by (9) and (8)，we obtain

Multiplying g(t) on the both sides of (13)，and integrating from 0 to 1，and then solving∫10g(t) u(t) dt，we have

Combining with (13)，(14) and (7) we get

which shows the lemma．

Remark 2．1 For any t，s∈［0，1］，it is easy to check that

Lemma 2．2 If (H1) hold，then for any t，s∈［0，1］，it holds where

Proof．Firstly，from (8)，(9) and (15)，we have

Combining with (7) and (17)，we get ?

From (7) and (19)，we get

Thus，combining with (18) and (20)，Lemma 2．2 is proved．It follows from Lemma 2．1 that u∈E is a solution of the BVP (1) if and only if u is a fixed point of T．Lemma 2．3 For any t1，t2，s∈［0，1］，it holdsProof．For any t1，t2，s∈［0，1］，if t1，t2≤s，then

| l(t1，s)－l(t2，s) | = (1－s) | t1－t2|≤| t1－t2|．If t1，t2≥s，then | l(t1，s)－l(t2，s) | = s | t1－t2|≤| t1－t2|．Without loss of generality，we suppose t1＜t2and t1≤s≤t2．Since

－(t2－t1)≤(s－1) (t2－t1)≤l(t1，s)－l(t2，s)≤s(t2－t1)≤(t2－t1)，

which shows that (22) is also true in this case．

Lemma 2．4 If (H1)，(H2) are satisfied，then T: P→P is a completely continuous operator．

Proof．For each u∈P，by the definition of T and the assumptions (H1) and (H2)，we have (Tu) (t)≥0，t∈［0，1］．It follows from Lemma 2．2 that

Combining with (23) and (24)，we have

i．e．，Tu∈P．This shows that T(P)P．

Step 1．T is bounded．If D = { u∈P:‖u‖≤r} is a bounded subset of P，then for each u∈D，we have u≤r，let M = max{ f(u) : u≤r} be a constant．By (21) and (16)，

thus T(D) is bounded in P．

Step 2．T(D) is equicontinuous．For any ε＞0，there exists δ=ε＞0．where the constant

for any | t1－t2|＜δ，(Tu) (t1)，(Tu) (t2)∈T(D)，from Lemma 2．2 and Lemma 2．3，

So T(D) is equicontinuous．

Step 3．T is continuous．Take { um}∞m =0with‖um－u0‖→0(m→∞)．There exists R0such that { um}∈［0，

R］．Let M= max{ f(u) :0≤u≤R}，L=1mk．Since | H(t，s) a(s) f(u(t) ) |≤mke(s) a(s)≤

0000402m02

L0a(s)，then from (H2) and Lebesgue dominated convergence theorem，we have

Thus T is continuous．

Above knowable，T is a completely continuous operator．

3 Main results

In the following，we shall give the main results of this paper．

Let A，B be two positive constants defined by A =

Theorem 3．1 Suppose that (H1)，(H2) are satisfied，and there exist two positive constants R1and R2with R1≠R2such that

then the BVP (1) has at least one positive solution u*∈P，with min { R1，R2}≤‖u*‖≤max{ R1，R2}．

Proof．Without loss of generality，we suppose R1＜R2．Let Ω1= { u∈C［0，1］:‖u‖＜R1}，Ω2= { u∈C［0，1］:‖u‖＜R2}．Then from (H3) and Lemma 2．2，for any u∈P∩Ω1，we have‖u‖= R1，

thus‖Tu‖≤‖u‖，u∈P∩Ω1．(25)

On the other hand，from (H4) and Lemma 2．2，for any u∈P∩Ω2，we have‖u‖= R2，and σR2≤u≤R2，

thus‖Tu‖≥‖u‖，u∈P∩Ω2．(26)

Thus，from Lemma 1．1 and Lemma 2．4，T has a fixed point u*∈P，with min { R1，R2}≤‖u*‖≤max{ R1，R2}，which is one positive solution of the BVP (1)．

Theorem 3．2 Suppose that (H1)，(H2) are satisfied，and there exist two positive constants R1and R2with R1≠R2such that then the BVP (1) has at least one positive solution u*∈P，with min { R1，R2}≤‖u*‖≤max{ R1，R2}．

Proof．Without loss of generality，we suppose R1＜R2．Let Ω1= { u∈C［0，1］:‖u‖＜R1}，Ω2= { u∈

C［0，1］:‖u‖＜R2}．Then from (H5) and Lemma 2．2，for any u∈P∩Ω1，we have‖u‖= R1， thus‖Tu‖≤‖u‖，u∈P∩Ω1，(27)

On the other hand，from (H6) and Lemma 2．2，for any u∈P∩Ω2，wehave‖u‖= R2，and σR2≤u≤ R2， thus‖Tu‖≥‖u‖，u∈P∩Ω2．

Thus，from Lemma 1．1 and Lemma 2．4，T has a fixed point u*∈P，with min { R1，R2}≤‖u*‖≤max{ R1，R2}，which is one positive solution of the BVP (1)．

Corollary 3．3 Suppose that (H1)，(H2)，(H7) and (H8) are satisfied，

Then the BVP (1) has at least one positive solution u∈P．

To illustrate how our main results can be used in practice we present an example．

Example 3．4 Consider the singular integral boundary value problem

Proof．From the BVP (29)，we can get m = 3，．By calculating，．Thus，applying Corollary 3．3，we know that the BVP (29) has a positive solution．

References:

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［3］BITSADZE A V．On the theory of nonlocal boundary value problems［J］．Sov Math Dokl，1984，30:8－10．

［4］IL’IN V A，MOISEEV E I．Nonlocal boundary value of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects ［J］．Differ Equ，1987，23(7) :803－810．

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Abstract:Middle and upper atmosphere temperature and wind are important parameters to study atmospheric waves．A high-spectra resolution sodium lidar was developed at the University of Science and Technology of China (USTC) in 2011．It can be used for temperature and wind measurements in mesopause region(80－105km)．First，the lidar system is introduced in details，including principle of measurements，the transmitter，receiver，and acquisition and control subsystems．The results of mesopause region temperature，zonal wind，meridional wind and sodium density，observed simultaneously by this lidar on the night of December 9，2011，are then present．The results show that mesopause temperature，zonal wind，meridional wind vary in significantly with 175～235K，－70m～60m/s and－100～100m/s respectively，likely induced by the solar diurnal and/or semidiurnal tides．These results further indicate that the USTC high-spectra resolution sodium lidar is an effective tool for mesopause region temperature and wind measurements with high temporal and spatial resolutions，and the observational data enable us to study the middle and upper atmosphere dynamics in great high resolutions． In this paper，we are concerned with the existence of positive solutions for singular second order multi-point boundary value problems with integral boundary conditions．The arguments are based upon a specially constructed cone and the fixed point theorem of cone expansion-compression type．Meanwhile，an example is given to demonstrate the main results．

Key words:atmospheric temperature; zonal wind; meridional wind; middle and upper atmosphere boundary value problems; positive solutions; fixed point theorem; existence of positive solutions

Positive Solutions for Singular Multi-Point Boundary Value Problems with Integral Boundary Conditions

YAN Jie，XIAO Jian-zhong
(School of Mathematics and Statistics，Nanjing University of Information Science and Technology，Nanjing 210044，China)

作者簡(jiǎn)介:李陶(1974－)，男，安徽巢湖人，安徽師范大學(xué)物理系1992級(jí)校友．中國(guó)科學(xué)技術(shù)大學(xué)地球和空間科學(xué)學(xué)院教授，博士生導(dǎo)師，中國(guó)科學(xué)院“百人計(jì)劃”入選者(2009年)，國(guó)家杰出青年基金獲得者(2012年)．

基金項(xiàng)目:國(guó)家自然科學(xué)基金(41225017)．

收稿日期:2015－03－03

DOI:10．14182/J．cnki．1001－2443．2015．02．001 10．14182/J．cnki．1001－2443．2015．02．002

文章編號(hào):1001－2443(2015) 02－0103－07

文獻(xiàn)標(biāo)志碼:A

中圖分類號(hào):TN249