The kinematic mechanism study of Hawaii-Emperor seamount chain: Evidence from paleomagnetic records
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摘要: 夏威夷-皇帝海山链位于北太平洋中部,是一条自西北至东南延伸的海底火山链,并在47 Ma存在一个方向弯折。厘定这一特征的成因机制和运动学过程,是西太平洋海区软流圈与岩石圈相互作用以及跨圈层物质能量交换的关键科学问题,对于解译东亚大陆动力学演化过程亦具有重要意义。目前对于夏威夷-皇帝海山链47 Ma弯折的机制有两种争论:太平洋板块运动方向改变和热点移动。古地磁学是研究大陆漂移和板块演化的最有效手段之一,其最大优势是可以定量化研究地质历史时期中岩石圈板块的运动学过程。本文首先通过回顾与总结前人对夏威夷-皇帝海山链成因及转向机制的研究,重点探讨古地磁学在该问题上所提供的约束证据,并对存在的关键科学问题进行了梳理和展望。Abstract: The Hawaiian-Emperor seamount chain is located in the middle of North Pacific Ocean extending in a direction from northwest to southeast. It consists of two segments, the older Emperor chian trending in N10°W and the Hawaiian chanin extending in N110°E. The research interests of the Hawaiian-Emperor seamount chain remain in the origin of seamount chain and the the sharp bend of the chain, which are the key to the investigation of the upwelling in the mantle, the movement of the lithosphere, and the exchange of material and energy between different layers. Paleomagnetism is the best tool for the kinematic studies on the seamount chain. In this paper, we summarized the previous studies on the formation mechanism of the Hawaiian-Emperor chain and the bend formed 47 Ma, with emphasis on the paleomagnetic evidence for the kinematics process of the Hawaiian-Emperor seamount chain. Key scientific topics and research directions were also discussed.
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海底浊流作为密度流或重力流的一种[1],是将陆地沉积物、有机碳和营养物质输送到深海的主要机制[2]。海底峡谷是浊流携带沉积物向深海输运的重要通道[3-4],浊流在峡谷中的沉积体能够延伸几百公里,厚度可达几十米,是良好的捕油、储油场所[5-7]。然而,高流速、强携沙能力的海底浊流会对海底通信电缆、海底管道等结构物造成严重危害。如2009年高屏峡谷曾发生线缆断裂事件,台风莫拉克的登陆引发暴雨导致高屏河水沉积物含量大大增加,在洪水期间形成了一个高密度流并沿着峡谷向下流动,导致两条电缆损坏,而三天后在河流水位恢复正常的情况下,却形成了第二次更具破坏力的沉积物密度流,导致了至少6根电缆断裂[8]。考虑到浊流的速度主要受其含沙量影响,如果两股浊流发生汇流,可能会使汇流后的浊流含沙量增大,进而导致其流速增大、破坏力增强。同时,高屏峡谷附近的枋寮峡谷是一个潜在的沉积物来源,因此推测更具破坏力的第二次沉积物流可能是由高屏峡谷和枋寮峡谷产生的两股浊流汇流后形成的。
海底峡谷根据地貌特征可分为单支型与树枝型海底峡谷体系[9],相较于单支型海底峡谷,树枝型海底峡谷体系多分布于构造活跃陆缘,具有活动性强和覆盖面积广等特征。中国南海海域就分布着多个分支型峡谷,在北部海底峡谷群中,台湾峡谷是典型的“Y”型峡谷,峡谷上段有两个主要分支[10];澎湖峡谷则是一个头部分支较多的峡谷,上段主要由3个分支组成[11];分布在东沙海域的峡谷也是一个典型的树枝型峡谷体系,该体系由一个主干峡谷和三个分支峡谷所构成[8];菲律宾北海岸的Malaylay峡谷是由多个海底峡谷向下游汇聚而成的一个主峡谷,曾出现强烈的浊流导致海底管道断裂事件[12]。可见树枝型海底峡谷分布广阔,因此浊流的汇流现象也可能是普遍存在的。现有研究多聚焦于单支峡谷中的浊流运动[13-14],但多股浊流发生汇流后可能会产生更大的破坏力,会对海底电缆、海底管道等结构物造成更严重的危害。
由于浊流具有突发性和强破坏力特征,对高速、偶发性的浊流进行实地观测的难度较大[7],因此研究浊流的主要方式是水槽试验与数值模拟。在地质学家Forel[15]最早提出了浊流概念后,Kuenen[16]通过固定坡水槽试验对Forel关于“浊流”的野外观测进行了验证;1959年浊流的深度平均特征首次被研究,并量化了浊流的垂向速度与含沙量分布。在此基础上Felix等[17]进行了3组不同含沙量的浊流试验,结果表明浊流流速和含沙量变化的相似性取决于其含沙量和流动位置。随后各国学者相继针对浊流流速、含沙量、粒径、动力学和沉积特性等方面开展了一系列水槽试验的研究,推动了浊流理论的发展[17-19]。随着科技的发展,在浊流理论知识的支撑下,浊流的数值模拟研究也进入蓬勃发展阶段,由简单的一维模型[20]到包含流速和含沙量垂直剖面的二维[21]再到多参数的三维数值模拟[22-24],浊流的数值模型日益成熟。Sun等[25]通过建立双层平均模型发现水库中支流的高浓度泥沙会加强主通道中的浊流,从而提高泥沙冲刷能力。FLOW-3D作为一款三维流体仿真软件,因其采用特殊的计算方法可以得到多尺度、多物理场耦合下的非稳态水动力模型,从而被广泛使用。Heimsund在利用Flow-3D模拟了水槽尺度下浊流的运动特征后[26],又以蒙特利峡谷为原型进行了实际尺度的模拟,发现浊流的运动特征对地形的复杂程度非常敏感,并且模拟的速度剖面与定点观测的速度剖面具有一定的可比性[27]。
本文聚焦海底分支峡谷的浊流汇流到主干峡谷中,与主干峡谷中浊流汇流后的含沙量与速度变化问题,通过自制汇流水槽进行室内试验,并开展数值模拟分析,研究海底浊流汇流后含沙量与速度的变化,对浊流现场监测方案的制定以及实际速度的推算等给出基本的规律认识,并对浊流汇流的形成演化过程给出试验参照。
1. 室内水槽试验
利用自制的汇流水槽,配置不同含沙量的浊流在主流情景(有主流无支流)和汇流情景(有主流有支流)2种情况下进行试验,利用自制取样器获取浑水样品,得到浊流头部的含沙量,并使用流速仪获得含沙量取样点附近的流速数据,为后续建立数值模型以获得含沙量取样点处的垂向流速变化提供验证依据。
1.1 试验装置及设备
为了模拟海底峡谷浊流的汇流过程,自制水槽并制作具有一个支流的海底峡谷模型,用以开展支流浊流与主流浊流汇流的试验。水槽长300 cm、宽200 cm、高40 cm(平面型式见图1a;后面文中涉及点位位置,均以图1a所给出的坐标,以cm为单位给出数据)。海底峡谷模型利用石膏制作完成,包括主流通道(主干峡谷)和支流通道(分支峡谷),其后在石膏表面涂刷油漆。本文是对浊流汇流后的含沙量和速度变化进行初步定性分析,因此在试验过程中固定通道的宽深比,地形坡度的选取则参考了南海北部陆坡海底峡谷的坡度数据,该区域峡谷地形起伏相对平缓,坡度通常在1.5°左右[28],因此本模型地形坡度也选为1.5°左右,模型地形等高线见图1b。模型一侧设置有浊流储流区,支流浊流储流区域尺寸为长50 cm×宽40 cm×高40 cm,主流浊流储流区域尺寸为长100 cm×宽50 cm×高40 cm(见图1a),储流区与外侧流动区用隔板分开,通过抽出插板可以实现浊流的释放。
浊流含沙量测量采用取样后泥沙沉淀烘干法测量。取样采用自制的插板封闭分层取样盒(图2),材质为有机玻璃,壁厚0.5 cm。取样盒自底向上每4 cm分为一层,宽度4 cm,长度10 cm,其长度方向顺浊流流动方向布置,上下游侧利用插板封闭水体,实现取样。试验时,在浊流释放前,将取样盒放置在取样点位置(图1a),取样点S1是浊流汇流处,位置为(130,100),取样点S2是浊流汇流后的继续流动点,位置为(120,90),浊流释放后,当浊流的头部刚好到达取样盒的末端时,快速插入取样盒的插板,封闭浊流水体,然后取出,通过根据层高逐渐提拉一侧插板以获得分层的浊流水体。预试验已知浊流在流动区主要为近底流动,设置的取样盒高度可覆盖浊流高度。
浊流速度测量使用挪威Nortek AS公司生产的Vector多普勒流速仪,布设位置见图1a,vA点位置为(150,90)、vB点位置为(190,90)。试验用土为黏土和粉土。黏土为高岭土,其中值粒径d50为3.7 μm。粉土为来源于黄河三角洲的原土,经晒干、碾碎、过筛去除杂质之后进行颗粒分析,黏粒含量为8.8%。配置沉积物流时,粉土与黏土的质量比为1∶1,混合后的土样经处理分析得到粒度级配曲线(图3)。
1.2 试验工况及流程
试验采用对照的方法,在主流情景和汇流情景两种情况下,配置不同含沙量的浊流进行试验,工况见表1。
表 1 试验浊流初始含沙量配置表Table 1. Configuration in initial sand content in the turbidity current experiment序号 试验情景 浊流含沙量/(g/L) 主流 支流 1 主流 100 0 2 汇流 100 100 3 主流 200 0 4 汇流 200 200 5 主流 300 0 6 汇流 300 300 7 主流 400 0 8 汇流 400 400 9 主流 500 0 10 汇流 500 500 试验采用抽板法制造浊流。每次试验开展前,在流动区注入清水的同时,向储流区注入如表1配置的浊流浑水体,保持储流区与流动区水面齐平。开展的10次试验,水位高度均设置在距底25 cm。释放浊流前,不断搅动储流区的浊流浑水体,以避免泥沙沉降。浊流体搅拌均匀后静置5 s左右,可使浊流体液面无明显晃动,然后迅速抽出储流区的插板,进行浊流的释放。当浊流到达取样点S1、S2处获取浑水样品,并保存在样品瓶中,经静置、烘干等处理后获得含沙量数据。多普勒流速仪在释放浊流前开机保持测量状态直至1次试验结束。
1.3 试验结果
在主流情景和汇流情景两种情况下,将取得的浊流样品进行处理后,得到浊流头部含沙量数据(图4、图5)。图4给出取样点S1、S2处的总含沙量(为浊流分层含沙量的总和)与配置的储流区浊流体含沙量之间的关系。在两种情景下,总含沙量均随着初始含沙量的增加而增大,汇流情景明显比主流情景的要大。试验条件范围内,汇流情景与主流情景相比,含沙量在S1点处增加53%~153%,在S2点处增加18%~53%。总含沙量在汇流与主流情景下在汇流点S1处浊流含沙量的差值,要明显大于浊流继续流动的S2处。
图5给出主流情景和汇流情景两种情况下,不同初始含沙量的浊流在取样点S1、S2处分层含沙量的变化情况(分层及样品编号见表2)。数据结果表明,泥沙主要分布在底层和中层,其中底层是高浓度含沙层。主流情景时,S1和S2位置的底层含沙量相近,S2的中层含沙量大于S1,上层含沙量接近于0。发生汇流情景时,在取样点S1、S2处的各层含沙量均大于只有主流时的各层含沙量,其中,中层含沙量的增幅最大,而且S1处的各层含沙量均大于S2处的各层含沙量。另外,当发生汇流时,S1处的上层含沙量较主流情景时有所增加,但S2处的上层含沙量小于S1处的上层含沙量。
表 2 含沙量样品编号与取样盒层位对应关系Table 2. Correspondence between sand content sample serial number and sampling box layer取样盒分
层名称距底高度
范围/cmS1处各层
样品编号S2处各层
样品编号上层 8~12 S1-3 S2-3 中层 4~8 S1-2 S2-2 底层 0~4 S1-1 S2-1 通过室内试验获得了含沙量和流速数据,其中流速数据主要用于验证模型,因此本节仅给出含沙量数据结果,试验测量的流速数据结果在2.2节给出。
2. 浊流流动数值模拟
通过室内水槽试验得到了浊流发生汇流后取样点S1、S2的含沙量变化情况,为了分析浊流头部流速与含沙量间的关系,利用数值模拟建立汇流模型,并以室内水槽试验获得的vA、vB点流速来验证模型,最终获得取样点S1、S2处的垂向流速数据,进而完整描述出浊流的汇流过程。
2.1 模型建立
使用三维流体仿真软件FLOW-3D建立汇流试验的数值模型,以室内汇流水槽为原型,等比例建立几何实体模型(图6a)。计算网格全部用结构化正交网格来划分(图6b),网格大小为0.016 m,网格总数为71.82万个。
2.2 模型验证
以在室内水槽试验中初始含沙量为200 g/L的浊流流速数据进行模型验证(表3),对比了浊流头部在流速测点vA、vB距底5 cm处的实测流速与模拟流速(图7)。
表 3 初始含沙量为200 g/L的浊流模拟流速最大值与试验流速最大值对比Table 3. Comparison between the maximum simulated currents velocity and the maximum experimental flow velocity for turbidity flow with initial sand content of 200 g/L项目 主流情景
vA点主流情景
vB点汇流情景
vA点汇流情景
vB点试验值/(m/s) 0.275 0.304 0.306 0.356 模拟值/(m/s) 0.270 0.291 0.305 0.351 绝对误差/(m/s) 0.005 0.013 0.001 0.006 相对误差/% 1.91 4.42 0.42 1.63 注:相对误差的计算方法为相对误差=$ \dfrac{|{\text{模拟值}}-{\text{试验值}}|}{\text{试验值}}$。 图 7 初始含沙量为200 g/L的浊流模拟流速与实测流速对比a:vA处主流情景,b:vB处主流情景,c:vA处汇流情景,d:vB处汇流情景。Figure 7. Comparison of simulated current velocity and measured current velocity for turbidity currents with initial sand content of 200 g/La: Main current scenario at vA, b: main current scenario at vB, c: confluence scenario at vA, d: confluence scenario at vB.在水槽试验中仅获得了初始含沙量为100 g/L和200 g/L的浊流流速,更高初始含沙量的浊流因含沙量过高导致流速仪无法获取流速数据。初始含沙量为100和200 g/L的浊流,汇流情景相比于主流情景,vA处最大流速分别增加17%和27%,vB处最大流速分别增加21%和30%。
2.3 模拟结果
利用已建立的模型模拟浊流在汇流水槽中的运动过程,对室内试验的10种工况进行模拟,得到主流情景和汇流情景下浊流速度的变化情况。为了便于对比分析浊流发生汇流后含沙量与速度结构的关系,提取浊流到达取样点S1、S2处时沿主要流动方向即图6b中x方向的流速vx,绘制垂向速度剖面图(图8)。
图 8 浊流在取样点S1、S2处的垂向速度剖面a、c、e、g、i:S1处的垂向速度剖面;b、d、f、h、j:S2处垂向速度剖面。速度负值代表反向流动。Figure 8. Vertical velocity profiles of turbidity currents at sampling points S1 and S2a, c, e, g, i: Vertical velocity profile at S1; b, d, f, h, j: vertical velocity profile at S2. The negative values of velocity represent reverse flow.模拟结果表明,浊流的运动速度与浊流初始含沙量呈正相关,随着初始含沙量的增加,浊流头部的流速形态变得更加尖锐。试验条件范围内,汇流情景相比仅有主流情景,流速在S1点处增加5%~27%,在S2点处增加12%~17%。在主流情景和汇流情景2种情况下,S1处的底层最大流速均小于S2处。在汇流情景下,各层速度均大于主流情景时各层的速度,二者的差值在S1处较大,并且在8 cm高度范围内,速度差值随着距底高度的升高而加大;二者的差值在S2处变小,只有底层最大速度的差值在S1和S2处变化不大。
3. 讨论
3.1 汇流过程浊流含沙量变化
浊流按入流状态分为持续入流型和突然释放型。持续入流型浊流的形态特点是头部较小,主体较大,行进机制是高含沙量主体推动着浊流头部前进;突然释放型浊流的特点是头部较大,主体较小,行进机制是高含沙量的头部驱散前方水体,进而牵引着含沙量较低的主体前进[29]。本试验中,在抽出挡板后浊流开始形成,此时储流区内的浊流浑水还较多,因此此时在产生浊流的边界处相当于有持续性的物料输入,其前进动力的来源是含沙量较高的主体,在这个过程中,浊流的头部高度较小(图9a)。
而当储流区内的浊流浑水完全流出后,浊流开始转变为突然释放型,此时浊流头部的高度开始增高,直至完全转变为突然释放型浊流的形态特征(图9b)。在主流情景下,当浊流经过S1处时,储流区内的尚未完全流出,因此此时的浊流头部高度较小,而当浊流经过S2处时,储流区内的浊流浑水已完全流出,因此浊流的头部高度有所提高,因此在含沙量数据中,S1处中层的含沙量小于S2处中层的含沙量。
当分支峡谷浊流与主干峡谷浊流发生汇流时,两股浊流头部在交叉口相遇交汇,水流和泥沙颗粒发生碰撞,水流紊动使得浊流头部高度增加。由此发生汇流情景下,取样点S1处的含沙量明显大于主流情景时的含沙量,并且由于碰撞产生的紊动作用,底层泥沙也被向上搅动,使得中层含沙量的增幅最大,而汇流时底层含沙量增加较小。在主流情景下,上层含沙量接近于0,是因为浊流高度较低,没有达到上层的高度,而在发生汇流时,浊流高度增加,因此上层含沙量也有了少量增加。
在汇流后形成的“新浊流”到达S2点时,各层的含沙量较主流情景时有明显增加,但均小于发生汇流时S1的分层含沙量(图8)。分析原因,应该是在汇流点S1,浊流由于有不同方向水流交汇产生强烈紊动,支持了泥沙在垂向上的向上悬浮,而随着汇流后向下游继续运动,在运动速度没有明显增加的情况下,水流结构不断调整趋向于层流结构,中上层泥沙失去紊动支持条件,出现过饱和而沉降。
3.2 汇流过程浊流速度变化
由于与周围水体的作用,浊流流动并非仅有水平流速,还有垂向流速,这里仅提取浊流主要流动方向的水平流速用于分析,并规定以浊流主要流动方向即图6b中的x方向为正方向。
由模拟结果可知(图10),浊流体内部速度为正方向,流体外部速度则是反方向,这是浊流在近底流动时对前方水体挤压导致的前方水体在上部返流,以达到水位高度均衡。在浊流上部与清水处,会出现水平流速等于0的界面,该界面高度可以视为浊流头部的高度。主流情景时,浊流在S2处的流速值和头部高度均大于S1处,这是由于浊流在斜坡上受到重力影响做加速运动,因而速度是逐渐增大的,同时由于浊流刚产生时属于持续输入型浊流,经过S1点后转变为突然释放型浊流,导致浊流头部形态发生变化,高度增加(图9b)。
发生汇流后,浊流在S1处的流速和头部高度都大于主流情景时的流速和高度,且中上层流速增量明显大于底层流速的增量,这是由于两股浊流相遇碰撞后导致浊流头部增高和整体含沙量增大,含沙量的增加会导致浊流与周围水体的密度差增大,进而导致流速增大。而底层含沙量的增量小,中层含沙量的增量最大,这就导致了中层流速增值最大而底层流速增值较小。与S1处相比,S2处的浊流头部高度有所降低且中层速度明显减小。其中浊流头部高度降低是由于在S1处碰撞增加的中上层泥沙在流动过程中不断沉降而导致的,而沉降的上层泥沙由于距离近底高含沙量沙层较远,受到的拖曳影响较小,同时其与周围水体密度差较小,在二者的共同影响下,导致沉降的上层泥沙速度较中层泥沙速度低,当这部分泥沙以相对较低的速度沉降到中层时,会使得中层泥沙的速度降低。同时,近底含沙层因距离上层泥沙较远,未受到上述情况的影响,因此在运动过程中与主流情景时一样,其受重力影响,做加速度运动,其加速度与主流情景时相同,因此汇流情境中底层最大速度与主流情景中底层最大速度的差值在S1处与S2处的相差不大。
4. 结论
(1)发生汇流处,浊流头部的高度和含沙量均有增加,其中含沙量较低的中层含沙量增加最多,含沙量较高的底层含沙量增加较少。
(2)发生汇流处,浊流头部的速度变化与含沙量变化一致,即中层流速增加最大,底层流速增加较小。
(3)发生汇流并流动一段距离后,与汇流点处相比,浊流头部的高度有所降低,各层含沙量也均有减少,其中中层含沙量减少幅度最大,中层速度也有所减小;整体来看,各层的含沙量和速度仍大于主流情景在该处的数据。
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图 4 太平洋板块的视极移曲线(APWP)(蓝色粗虚线)
其中红色星号代表地磁极位置[60, 61],星号附近的数字代表对应的年龄(Ma),实线椭圆代表每个极位置95%的置信区间。蓝色圆点和蓝色细虚线为根据热点轨迹得到的太平洋板块的极移曲线[15, 62]。插图表明了极移的不同阶段,引自Sager[56]
Figure 4. Apparent polar wander path in the Pacific (APWP)
The red stars denote pole positions defining the most likely APWP shown by the blue dashed line. Poles are surrounded by 95% confidence ellipses and labeled by age in Ma. Thin dashed lines show predicted polar wander path from plate/hotspots motion models of Duncan and Clague[15] and Wessel et al[62]. Inset sketch map shows interpreted phases of polar wander, revised from Sager[56]
图 5 ODP197航次站位(1206,光孝海山;1205,推古海山;1204和1203,底特律海山),ODP884站位(底特律海山)以及DSDP433站位(仁德海山))的古纬度值
蓝色三角形为热退磁结果,紫色三角形为交变退磁结果。433站位为热退和交变退磁结果(修改自Tarduno等[25])
Figure 5. Paleolatitude data from ODP Leg 197 (sites 1206, Koko seamount; 1206, Nintoku Seamount; 1204 and 1203, Detroit Seamount), ODP site (884 Detroit Seamount), and DSDP (Site 433, Suiko Seamount)
Blue triangle, results of thermal demagnetization; Purple, results of alternating field demagnetization; Results from 433 is based on AF and thermal data (Revised from Tarduno et al[25])
图 6 夏威夷-皇帝海山链地幔柱牵引和反弹机制
81 Ma时,地幔柱在1 200~1 500 km深度处,地幔上升流受太平洋-库拉洋中脊牵引,之后随着上升流减弱,地幔柱逐渐往回折返,并在47 Ma恢复到它本来位置(修改自Tarduno等[22])
Figure 6. Schematic diagram of plume capture and release for the Hawaiian-Emperor chain
The plume is bent between 1 200 and 1 500 km depth toward the mantle upwelling associated with the Pacific-Kula ridge system at 81 Ma; upwelling abates thereafter, allowing the plume to return to its original position relative to the deep mantle by 47 Ma (Revised based on Tarduno et al[22])
图 7 夏威夷热点的纬向运动
古地磁方法得到的皇帝海山链以及夏威夷岛链各海山或岛礁相对于夏威夷热点现今纬度位置(19.4°N)的纬度偏移量[22, 24- 25, 64, 77-79](黑色实心圆)以及经过真极移校正的数据(黄色六角星)(修改自Torsvik等[13])
Figure 7. Latitude motion of the Hawaiian hotspot
Paleomagnetically derived latitudes (blue ovals connected with the blue line) from seamounts along Emperor chain and islands/atolls along the Hawaiian chain plotted with 95% confidence bars. The data are shown as latitude offsets from the present latitude of Hawaii (observed latitude minus latitude of Hawaii, 19.4°N) (Revised from Torsvik et al[13])
图 8 皇帝海山链形成过程中夏威夷热点和太平洋板块运动向量模型图
hVp太平洋板块相对于热点的运动速度(红色向量),mVp太平洋板块相对于地幔的运动(假设地幔相对于自转轴固定)(黑色向量)。二者的矢量和mVh代表热点相对于地幔的运动(黄色向量),具有一个很大的西向分量。水平向量(紫色)代表了太平洋板块在没有北向速度分量的情况下表现出来的西向运动。虚线指示了太平洋板块没有北向运动的情况下,热点相对于地幔的运动(引自Sager[56])
Figure 8. Sketch of motion vectors indicating Hawaiian hotspot drift during the formation of the Emperor seamounts
Motion of plate relative to hotspot, hVp (red vector), given by trend of Emperor seamounts. Motions of plate relative to mantle (assumed fixed relative to spin axis), mVp (black vector), is assumed to be same as at present (Hawaiian chain). Sum is motion of hotspot relative to the mantle, mVh (yellow vector), which has a large westward component. Horizontal vector at bottom (purple) shows Pacific plate motion if the plate had no northward component of velocity. Dashed-line vectors show predicted motion of hotspot relative to mantle if Pacific plate motion has no northward component. Different dashed lines correspond to different westward velocities. Background is a shaded relief plot of Hawaiian-Emperor chain bathymetry (Referred from Sager[56])
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