地球形成和演化過程中的分異能計(jì)算方法研究

耿煜， 王君恒,2,3*

1 中國(guó)地質(zhì)大學(xué)(北京)地球物理與信息技術(shù)學(xué)院， 北京 100083 2 中國(guó)地質(zhì)大學(xué) 地質(zhì)過程與礦產(chǎn)資源國(guó)家重點(diǎn)實(shí)驗(yàn)室， 北京 100083 3 地下信息探測(cè)技術(shù)與儀器教育部重點(diǎn)實(shí)驗(yàn)室(中國(guó)地質(zhì)大學(xué)，北京)， 北京 100083

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地球形成和演化過程中的分異能計(jì)算方法研究

耿煜1， 王君恒1,2,3*

1 中國(guó)地質(zhì)大學(xué)(北京)地球物理與信息技術(shù)學(xué)院， 北京 100083 2 中國(guó)地質(zhì)大學(xué) 地質(zhì)過程與礦產(chǎn)資源國(guó)家重點(diǎn)實(shí)驗(yàn)室， 北京 100083 3 地下信息探測(cè)技術(shù)與儀器教育部重點(diǎn)實(shí)驗(yàn)室(中國(guó)地質(zhì)大學(xué)，北京)， 北京 100083

地球形成初期，構(gòu)成地球的物質(zhì)在組成上是大致均一的.目前地球的地核-地幔-地殼圈層結(jié)構(gòu)，是由分異作用形成的.分異過程釋放的能量稱為分異能.Sorokhtin和Chilingarian等人從行星吸積的定義出發(fā)，導(dǎo)出了基于地球內(nèi)部密度分布的勢(shì)能計(jì)算公式，計(jì)算出的分異能大小為1.698×1031J.本文采用計(jì)算球體勢(shì)能的思路，導(dǎo)出分異能計(jì)算的解析公式和數(shù)值計(jì)算公式，通過求取原始地球模型與均勻分層模型、PREM模型的勢(shì)能差計(jì)算分異能.兩種方法的計(jì)算結(jié)果分別為1.535×1031J和1.698×1031J.前者與Sorokhtin等的結(jié)果相近，后者與之相同.本文初步分析了方法間的異同以及造成結(jié)果偏差的主要原因.

重力分異； 勢(shì)能； 分異能； 吸積能； PREM

1 引言

為說明地球的成因，國(guó)內(nèi)外已有四十多種假說(王君恒等, 2010, 2012, 2013)，其中較為普遍認(rèn)同的有我國(guó)天文學(xué)家戴文賽首次提出的新星云假說(戴文賽和胡中為, 1980).該假說認(rèn)為：地球的形成與太陽(yáng)系形成密不可分，要經(jīng)過“原始星云→星云盤→塵層→星子→行星”共5個(gè)階段(戴文賽和胡中為, 1979).

吸積是形成行星地球的最終階段.關(guān)于吸積有兩種不同的觀點(diǎn)，即均一吸積說和非均一吸積說(朱志祥, 1982).通常認(rèn)為，均一吸積說可能性較大，即原始地球是一個(gè)接近均質(zhì)的球體，并沒有明顯的分層現(xiàn)象(戴文賽和陳道漢, 1976).根據(jù)對(duì)地球外核成分的認(rèn)識(shí)不同，均一吸積說又可分為金屬化核說和鐵核說(Schmidt, 1957).沖擊波實(shí)驗(yàn)表明(朱志祥, 1980)：外核物質(zhì)的密度比鐵在外核條件下的密度小15%左右.所以外核物質(zhì)除了鐵外，還應(yīng)有少量的輕元素.較可能的輕元素是硫和氧.

但是，與原始地球不同，目前的地球內(nèi)部分為地殼、上下地幔和內(nèi)外地核等幾個(gè)大的圈層.這種圈層結(jié)構(gòu)是由分異作用形成的(騰吉文, 2003).在地球自身引力和內(nèi)部溫度的共同作用下，流動(dòng)的輕物質(zhì)上涌形成外層，流動(dòng)的重物質(zhì)下降形成內(nèi)層，故構(gòu)成了地球圈層物質(zhì)的分異過程(Rubie et al., 2007).在圈層分異、調(diào)整過程中，地球內(nèi)部能量的產(chǎn)生、遷移、轉(zhuǎn)化和消耗，是制約整體作用過程的決定要素.因此，分異能的計(jì)算是一個(gè)關(guān)鍵性問題.

Sorokhtin等(2010)從行星吸積的定義出發(fā)，導(dǎo)出了基于地球內(nèi)部密度分布的勢(shì)能計(jì)算公式，計(jì)算出的分異能大小為1.698×1031J.Flasar和Birch (1973)計(jì)算了目前地球和原始地球兩種不同情形下地球吸積過程中重力所做的功，兩者的差1.66×1031J即為地球分異過程中損失的勢(shì)能.另有不同學(xué)者的估算(Lyubimov,1968; Vityazev,1973; Keondjian and Monin,1977)表明，地球分異過程中釋放的重力勢(shì)能在1.46×1031J到2×1031J之間.

本文采用計(jì)算球體勢(shì)能的思路，通過求取原始地球和目前地球的勢(shì)能差計(jì)算分異能.首先在均勻分層模型下推導(dǎo)出地球勢(shì)能的解析表達(dá)式，計(jì)算所得分異能大小為1.535×1031J，與Sorokhtin等的結(jié)果相近.該方法能夠以解析形式表達(dá)出地球的勢(shì)能，可以避免數(shù)值求和的繁瑣步驟，較前人方法相比計(jì)算更加簡(jiǎn)潔.本文進(jìn)一步在分層更加精細(xì)的PREM全球參考模型下，應(yīng)用地球勢(shì)能的數(shù)值計(jì)算公式，得出的分異能大小為1.698×1031J，在所給精度范圍內(nèi)與Sorokhtin等結(jié)果一致.該公式使用壓強(qiáng)表示地球的勢(shì)能，降低了由模型間差異所帶來的誤差，較前人方法具有更高的準(zhǔn)確性.本文初步分析了方法間的異同以及造成結(jié)果偏差的主要原因.

2 前人計(jì)算所用吸積做功法

Sorokhtin等(2010)認(rèn)為：在數(shù)值上，地球的吸積能Ea等于其重力勢(shì)能的相反數(shù)(根據(jù)定義勢(shì)能總是負(fù)的).任何系統(tǒng)的勢(shì)能取決于該系統(tǒng)的構(gòu)造格局，在此處討論的情形中則是地球內(nèi)部的密度分布，表達(dá)式為

(1)

(2)

圖1 普遍接受的地球內(nèi)部密度分布(1為目前地球，2為原始地球)Fig.1 Accepted density distribution within Earth(1 is present-day Earth；2 is primordial Earth)

其中U為地球的勢(shì)能；m(r)是半徑為r的球體內(nèi)部所包含的地球質(zhì)量；ρ(r)為地球在半徑r處的物質(zhì)密度；γ=6.673×10-11m3·kg-1·s-2為引力常數(shù)；R=6.371×106m為地球的平均半徑.目前和原始地球內(nèi)部密度分布見圖1(Naimark and Sorokhtin, 1977a,b).

為了確定原始地球的吸積能，明確其內(nèi)部密度分布是必需的.該分布是建立在地球物質(zhì)的平均組分(表1)及硅酸鹽和金屬?zèng)_擊壓縮數(shù)據(jù)(Naimark and Sorokhtin, 1977a,b)之上的.高壓下基于沖擊壓縮數(shù)據(jù)的造巖氧化物密度測(cè)定的目前技術(shù)具有2%～4%的精度(Sorokhtin et al., 2010).用這種方法測(cè)定出的原始地球內(nèi)部密度分布見圖1(Naimark and Sorokhtin, 1977a,b).

表1 目前地球和原始地球的物質(zhì)組成(Sorokhtin et al., 2010)Table 1 Composition of present-day Earth and primordial Earth matter (Sorokhtin et al., 2010)

地球質(zhì)量：M=5.9772×1027g；地核質(zhì)量：Mcore=1.9404×1027g；內(nèi)核質(zhì)量：Mcore1=0.1083×1027g；過渡帶質(zhì)量：Mcore2=0.1299×1027g；外核質(zhì)量：Mcore3=1.8321×1027g；地幔質(zhì)量：Mm=4.0143×1027g；大陸地殼質(zhì)量：Mcc=2.25×1025=0.0225×1027g.aRonov and Yaroshevsky, 1978;bRingwood, 1966; Dmitriyev, 1973;cUrey and Craig, 1953;dBarsukov, 1981.

使用式(1)及(2)來計(jì)算46億年前地球形成過程中釋放的吸積能.該能量(約等于其初始勢(shì)能)是巨大的：U(4.6)≈ -23.255×1031J.在數(shù)值上，重力分異能等于分異過程剛好開始(即約40億年前)之前均勻地球的勢(shì)能與目前分層地球的勢(shì)能差為(Sorokhtin et al., 2010)

Eg=U4.0-U0.0,

(3)

目前地球的勢(shì)能為-24.952×1031J(Sorokhtinetal., 2010).因此根據(jù)定義，重力分異的總能量為[-23.255-(-24.952)]×1031J=1.698×1031J.

除該方法外，F(xiàn)lasar和Birch(1973)計(jì)算了目前地球和原始地球兩種不同模型下地球形成過程中重力所做的功.分別基于Dziewonski和Gilbert(1972)的目前地球模型與Birch(1965)的原始地球模型，他們得出目前地球的吸積能為2.490×1032J，原始地球的吸積能為2.324×1032J.根據(jù)分異能的定義，兩者的差1.66×1031J即為地球分異過程中損失的勢(shì)能.

此外，Monteux等(2009)一起研究了行星分異的相關(guān)數(shù)值模型，給出了分異過程中的勢(shì)能損失計(jì)算公式為

(4)

其中Ω為行星的體積.雖然作者沒有給出具體的推導(dǎo)思路和地球分異能的計(jì)算結(jié)果，但是其推導(dǎo)可能應(yīng)用了與Flasar和Birch (1973)相同的思路.

3 均勻分層模型下的解析積分法

本文嘗試在均勻分層地球模型下通過球坐標(biāo)積分推導(dǎo)出原始地球和目前地球的勢(shì)能表達(dá)式，分別計(jì)算原始地球和目前地球的勢(shì)能.再根據(jù)分異能的定義，用原始地球的勢(shì)能減去目前地球的勢(shì)能，得出分異能的大小.

圖2 密度均勻的原始地球模型Fig.2 A primordial Earth model which consists of a homogeneous mixture of the materials of the present core and mantle

作為近似，將原始地球看作密度均勻的標(biāo)準(zhǔn)球體，并以球心為原點(diǎn)建立球坐標(biāo)系(圖2).設(shè)無限遠(yuǎn)處的勢(shì)能U∞=0，原始、均勻地球的密度為ρ0，原始地球的半徑為R0，以原點(diǎn)為球心選取一個(gè)半徑為r、厚度為dr的薄球殼(0

dm=ρ0×4πr2dr,

(5)

勢(shì)能為

dU4.0=-r×g(r)dm=-4πr3ρ0g(r)dr.

(6)

從0到R0積分，得原始地球的總勢(shì)能為(Solomon, 1979)

(7)

其中

(8)

式(8)是半徑為r處的重力加速度(Monteux et al., 2009).代入地球勢(shì)能的表達(dá)式，得：

(9)

圖3 以地核的平均密度和地殼加地幔的平均密度代替實(shí)際密度分布的目前地球模型Fig.3 A present-day earth model in which its density distribution is substituted by the mean density of core and the mean density of crust and mantle

為了給出目前地球勢(shì)能的解析表達(dá)式，設(shè)理想的目前地球及其地核均為標(biāo)準(zhǔn)球體，并將地殼并入地幔之中.以地核的平均密度代替地核的實(shí)際密度分布，以地殼加地幔的平均密度代替地殼和地幔的實(shí)際密度分布.這樣，本文提出的目前地球模型內(nèi)部是一個(gè)勻質(zhì)的地核，外部則是地幔與地殼合在一起的殼幔層，構(gòu)成“地核-殼幔層”的雙層結(jié)構(gòu).

以球心為原點(diǎn)建立球坐標(biāo)系(圖3)，設(shè)地核的平均密度為ρc，地核的平均半徑為Rc.類比原始地球勢(shì)能的推導(dǎo)方法，并參考式(7)的形式，寫出地核勢(shì)能的積分表達(dá)式為

(10)

其中

(11)

式(11)是半徑為r處的重力加速度(0

(12)

對(duì)于地殼和地幔，依舊參考式(7)的形式，寫出其勢(shì)能表達(dá)式為(Rc

(13)

此時(shí)，r處的重力加速度由兩部分質(zhì)量提供：一是地核質(zhì)量，二是所取薄球殼包圍的地殼和地幔質(zhì)量，即：

(14)

代入整理得

(15)

將地核與地殼和地幔的勢(shì)能相加，便得到目前地球的勢(shì)能表達(dá)式為

U0.0=Uc+Um.

(16)

假設(shè)原始地球和目前地球表面重力加速度相同，原始地球和目前地球的平均密度相同.計(jì)算所需數(shù)據(jù)(Anderson, 1989;JeffreysandSinger, 2009)如下：地球表面重力加速度g0=9.8156 m·s-2；原始、均勻地球的密度ρ0=5.514×103kg·m-3；原始地球的半徑R0=6.355×106m；目前地球的半徑R=6.371×106m；地核的平均密度ρc=10.76×103kg·m-3；地核的平均半徑Rc=3.485×106m；地殼加地幔的平均密度ρm=4.400×103kg·m-3.計(jì)算所得原始地球的勢(shì)能為U4.0≈ -22.131×1031J，目前地球的勢(shì)能為U0.0≈ -23.665×1031J.于是根據(jù)定義，分異能為Eg=U4.0-U0.0≈[(-22.131)-(-23.665)]×1031J=1.535×1031J.

作為對(duì)地核-殼幔層雙層結(jié)構(gòu)的改進(jìn)，可以在內(nèi)核—外核—下地幔—上地幔—地殼的五層結(jié)構(gòu)下推導(dǎo)并計(jì)算目前地球的勢(shì)能.但是，在模型被進(jìn)一步細(xì)化之后，以解析形式給出的地球勢(shì)能表達(dá)式非常復(fù)雜，從而給均勻分層解析法的推導(dǎo)和計(jì)算帶來不便.因此，本文僅給出在地核-殼幔層模型下目前地球勢(shì)能的推導(dǎo)與計(jì)算過程.

4 PREM模型下的數(shù)值求和法

總結(jié)前人的計(jì)算公式(1)、(2)、(3)及(4)發(fā)現(xiàn)，他們均采用地球內(nèi)部密度分布表示出地球的吸積能.經(jīng)比較，不同的地球模型(Bolt, 1957; Bullen, 1965; Dziewonski et al., 1975; Martinec et al., 1986)在壓強(qiáng)分布上的差異要低于密度分布上的差異，特別是在地心附近.根據(jù)計(jì)算，在深度為6371 km處，Bullen地球密度模型(Bullen, 1938)所給出的壓強(qiáng)數(shù)值相對(duì)于PREM模型(Dziewonski and Anderson, 1981)偏小約3.53%，密度偏小約7.02%.可見，若能以壓強(qiáng)表示出地球的勢(shì)能，則能夠降低因模型間差異而帶來的誤差.

對(duì)于一個(gè)密度僅為半徑函數(shù)的處于流體靜力平衡狀態(tài)下的球體，勢(shì)能可以表達(dá)為多種不同的形式(王君恒等, 2010, 2012, 2013)：

φdmr

(17)

其中

(18)

(19)

總質(zhì)量為M，球體的半徑為R，壓強(qiáng)為P，密度為ρ.當(dāng)壓強(qiáng)P為關(guān)于r的已知函數(shù)時(shí)，最后一種形式對(duì)于勢(shì)能的計(jì)算是較為方便的.并且從不同密度模型中壓強(qiáng)的差異較小這一事實(shí)可以推知：若使用地球內(nèi)部半徑和壓強(qiáng)分布計(jì)算目前地球的勢(shì)能，在不同密度模型下計(jì)算出的勢(shì)能差異不會(huì)過大.對(duì)于原始地球的勢(shì)能，本文將采用Birch的原始地球模型(表2)進(jìn)行計(jì)算；對(duì)于目前地球的勢(shì)能，本文將采用Dziewonski和Anderson的初步地球參考模型(表3)進(jìn)行計(jì)算.

將式(17)改寫為求和形式為

(20)

其中i為由內(nèi)向外逐層所做的編號(hào)，N為所用模型的數(shù)據(jù)長(zhǎng)度，即可用程序逐步完成該計(jì)算.在編寫了程序后，本文計(jì)算出原始地球的勢(shì)能為U4.0≈ -23.338×1031J(詳見表4)，目前地球的勢(shì)能為U0.0≈ -25.036×1031J(詳見表5).于是按照定義，分異能Eg=U4.0-U0.0≈ [(-23.338)-(-25.036)]×1031J=1.698×1031J.

在本文所選取的原始和地球模型下，PREM數(shù)值求和法得出了與Sorokhtin等相同的結(jié)果.由于選用壓強(qiáng)可以減低不同模型間差異帶來的誤差，當(dāng)不同模型間密度分布差異較大時(shí)，該方法比Sorokhtin等的方法具有更高的準(zhǔn)確性.

表2 Birch原始地球模型下半徑、密度和壓強(qiáng)分布(Birch，1965)Table 2 Radius，density and pressure distribution under Birch primordial earth model (Birch，1965)

表3 PREM模型下半徑、密度和壓強(qiáng)分布(Dziewonski and Anderson，1981)Table 3 Radius，density and pressure distribution under preliminary reference earth model (Dziewonski and Anderson，1981)

續(xù)表3

表4 Birch原始地球模型下勢(shì)能的計(jì)算步驟Table 4 Calculation procedures of potential energy under Birch primordial earth model

續(xù)表4

表5 PREM模型下勢(shì)能的計(jì)算步驟Table 5 Calculation procedures of potential energy under preliminary reference earth model

續(xù)表5

5 結(jié)論

(1) 不同于前人的吸積做功法，均勻分層解析法能夠以解析形式表達(dá)出原始地球和目前地球的勢(shì)能，計(jì)算過程簡(jiǎn)潔，避免了數(shù)值求和的繁瑣步驟.

(2) 在實(shí)際情況下，地核、地幔和地殼的密度隨半徑增大而逐漸減小.這使得均勻分層模型下計(jì)算出的目前地球勢(shì)能比實(shí)際情況偏大，該偏差進(jìn)一步導(dǎo)致了分異能的計(jì)算結(jié)果偏小.

(3) 由于分層更多的地球模型會(huì)給均勻分層解析法的推導(dǎo)和計(jì)算帶來困難，本文只給出地核-殼幔層雙層結(jié)構(gòu)下目前地球勢(shì)能的推導(dǎo)與計(jì)算過程.

(4) 考慮到不同的地球模型在壓強(qiáng)分布上的差異小于密度分布上的差異，PREM數(shù)值求和法采用了壓強(qiáng)表達(dá)地球的勢(shì)能，可降低由模型間差異所帶來的誤差.

(5) 在本文所選取的原始和目前地球模型下，PREM數(shù)值求和法得出了與Sorokhtin等的方法相同的結(jié)果.并且當(dāng)不同模型間密度分布差異較大時(shí)，該方法比Sorokhtin等的方法具有更高的準(zhǔn)確性.

(6) 目前地球的分異活動(dòng)仍沒有停止，只是分異不再是形成核-幔-殼結(jié)構(gòu)的全面大規(guī)模活動(dòng).在此過程中，一部分分異能被地球的彈性壓縮所消耗，絕大部分分異能轉(zhuǎn)化為地球內(nèi)部的熱量.

(7) 后續(xù)研究中深入探討的主要內(nèi)容有：地球演化過程中的其他物理機(jī)制(如放射性元素的衰變等)能夠?yàn)榈厍蛱峁┑臒崮埽碚撋线@些熱能總共能使地球升高的溫度；分異能的釋放速率與釋放總量隨時(shí)間的變化規(guī)律等重要問題.

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(本文編輯 張正峰)

Research on calculation methods of differentiation energy during the formation and evolution of the earth

GENG Yu1， WANG Jun-Heng1,2,3*

1SchoolofGeophysicsandInformationTechnology,ChinaUniversityofGeosciences,Beijing100083,China2ChinaUniversityofGeosciences,StateKeyLaboratoryofGeologicalProcessesandMineralResources,Beijing100083,China3KeyLaboratoryofGeo-detection(ChinaUniversityofGeosciences,Beijing),MinistryofEducation,Beijing100083,China

According to Dai Wensai′s nebular hypothesis, the formation of the Earth was closely related to the formation of the solar system, which can be described as “primordial nebular-protoplanetary disc-konisphere-planetesimal-planet”. Accretion was the last stage during the formation of the Earth. Generally considered, homogeneous accretion has a greater possibility, which means that the primordial Earth was a nearly homogeneous body without significant stratification. However, different from the primordial Earth, the present-day Earth is divided into crust, upper mantle, lower mantle, outer core and inner core. This layering structure was formed by the differentiation process. During the differentiation and adjustment of the layers, the generation, migration, conversion and consumption of the Earth′s internal energy was the decisive factor that restricted the whole process. Therefore, the calculation of differentiation energy is a pivotal issue.Starting from the definition of planet accretion, Sorokhtin et al. derived a potential calculation formula which is based on the density distribution within the Earth, and the calculated differentiation energy is 1.698×1031J. Flasar and Birch calculated the work done by gravity in the process of the Earth′s accretion in the light of primordial Earth and present-day Earth. The difference between these two quantity, 1.66×1031J is the potential energy loss in the process of the Earth′s differentiation. Estimates given by other authors suggest that the gravitational potential energy released during the process of the Earth′s differentiation is between 1.46×1031J and 2×1031J.The idea of calculating the potential energy of a sphere was adopted in this paper, and differentiation energy was calculated by evaluating the potential energy difference between primordial Earth and present-day Earth. Firstly, the analytic formula of the Earth′s potential energy was derived based on a uniformly layered Earth model. The calculated differentiation energy is 1.535×1031J which is close to the result given by Sorokhtin et al. Further, using a more sophisticated model, the preliminary reference Earth model (PREM), and by applying the numerical formula of the Earth′s potential energy, the differentiation energy was calculated to be 1.698×1031J, which is the same as the result of Sorokhtin et al. within the given precision.Different from the “accretion work method” in previous studies, the“uniform layered analytic method” gives the analytic formula for the potential energy of primordial Earth and present-day Earth, from which the tedious steps of numerical summation were avoided. In the actual case, the density of core, mantle and crust decreases with radius increasing. This will make the potential energy of present-day Earth under uniform layered Earth model larger than that in the actual case, which can further make the calculated differentiation energy small. Since an Earth model with more layers can bring inconvenience to the derivation and calculation of the “uniform layered analytic method”, only the derivation and calculation on the “core-mantle two-layer structure” was given in this paper.Considering that the difference of pressure is smaller than the difference of density between different Earth models, the “PREM numerical summation method” uses pressure instead of density to describe the Earth′s potential energy, which can reduce the error brought by the differences between models. Using the Earth models adopted in this paper, the “PREM numerical summation method” gives the same result as the method of Sorokhtin et al. Moreover, when density distributions given by different Earth models vary significantly, this method can lead to more reliable results than the method of Sorokhtin et al.At present, the Earth′s differentiation has not yet stopped, but it is no longer comprehensive and large-scale activity which forms the core-mantle-crust structure. In this process, a portion of the differentiation energy was consumed by the Earth′s elastic compression, while most of it was converted into the Earth′s internal heat. Subsequent research should focus on the heat sources provided by other physical processes during the evolution of the Earth and other relevant issues, such as the decay of radioactive elements, the total temperature the Earth raised by absorbing this heat, and the releasing rate and releasing amount of differentiation energy with time.

Gravitational differentiation； Potential energy； Differentiation energy； Accretion energy； PREM

10.6038/cjg20151009.

Geng Y, Wang J H. 2015. Research on calculation methods of differentiation energy during the formation and evolution of the earth.ChineseJ.Geophys. (in Chinese),58(10):3530-3539,doi:10.6038/cjg20151009.

耿煜, 男, 1989年生, 天津人, 中國(guó)地質(zhì)大學(xué)(北京)地球物理與信息技術(shù)學(xué)院,碩士，目前在美國(guó)孟菲斯大學(xué)地震研究中心，主要從事天然地震研究. E-mail: ygeng1@memphis.edu

*通訊作者 王君恒, 男, 1962年生, 山東青島人, 博士, 中國(guó)地質(zhì)大學(xué)(北京)地球物理與信息技術(shù)學(xué)院副教授, 主要從事應(yīng)用地球物理和理論地球物理研究. E-mail: w1128@cugb.edu.cn

10.6038/cjg20151009

P311

2015-01-06，2015-10-08收修定稿

耿煜，王君恒. 2015. 地球形成和演化過程中的分異能計(jì)算方法研究.地球物理學(xué)報(bào)，58(10):3530-3539,