垂直速度（w）的关系如下：
The 2-D (vertically integrated) momentum and continuity equations are written as 二维（垂直积分）动量方程和连续方程如下：
where Gu and Gv are defined as
其中Gu和Gv定义如下：
并且
where the definitions of variables are the same as those described in the Cartesian coordinates. The spherical-coordinate version of FVCOM was developed based on the
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Cartesian coordinate version, in which all the boundary cond itions and forcing used in the spherical-coordinate system are the same. The only difference is in the discrete approach, which is described later in chapter 3.
其中变量的定义与笛卡尔坐标系相同。FVCOM的球坐标版本基于笛卡尔坐标版本，用于球坐标系中的所有边界条件和强迫与笛卡尔坐标系下的相同。唯一的不同是离散逼近，浙江在第三章中详细给出。
Chapter 3: The Finite-Volume Discrete Method
第三章 有限体积离散法
3.1. Design of the Unstructured Triangular Grids
3.1 不规则三角网格的设计
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured
triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:
与三角有限元方法相似，水平数学计算域可细分为许多不重复的无规则三角元。一个不规则三角形包含三个节点、一个质心和三条边（图3.1）。令N和M分别为计算域中的质心总数和节点总数，质心位置可表示为
图3.1 FVCOM不规则三角网格示意图。变量位置：节点?：H，?，?，D，s，?，q，
2q2l，Am，Kh；质心?：u，v。
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and the locations of nodes can be specified as:
节点位置可由下式给出：
Since none of the triangles in the grid overlap, N should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as Ni(j)where j is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as
NBEi(j) wherejis counted clockwise from 1 to 3. At open or coastal solid
^^^^boundaries, NBEi(j) is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as NT( j) , and they are counted using integral numbers NB i (m) where m is counted clockwise from 1 to NT( j) .
因为网格中三角形没有交叠，所以N也是三角形总数。在每个三角元中，三个节点由积分数Ni(j)确定其中j为顺时针方向从1到3积分。有一条共同边的相邻三角形可由积分数
^^^NBEi(j)计算，其中j为顺时针方向从1到3积分。在开边界或海岸固边界，NBEi(j)为
0。在每个节点处，相邻三角形的总数与这个节点有关，表示为NT（j），并可由积分数NBi(m)计算，其中m为顺时针方向从1到NT（j）积分。
^^^To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, u and v are placed at centroids and all scalar variables,
such as ?, H, D, w, S, T, ?, K m ,K h ,A m and A h are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the ―tracer control element‖ or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the ―momentum control element‖ or MCE).
为了给出海面上升、水流通量、盐度通量和温度通量的精确估计，将u，v放置在质心处，所有的标量变量放置在节点处，如?，H，D，?，S，T，?，Km，Kh，Am和Ah。在节点处的标量变量由穿过连接质心和相邻三角形邻边中点截面的净通量决定（称为“追踪控制元”或TCE）；在质心处的u和v由穿过三角形三条边的净通量计算（称为“动量控制元”或MCE）。
Similar to other finite-difference models such as POM and ROM, all the model variables except ??(vertical velocity on the sigma- layer surface) and turbulence
variables (such as q2and q2l ) are placed at the mid-level of each ?? layer (Fig. 3.2). There are no restrictions on the thickness of the ?- layer, which allows users to use
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either uniform or non-uniform ?-layers.
与其他的有限差分法（如POM和ROM）类似，除了?（?层表面的垂直速度）和湍流变量（如q2和q2l）外，所有的模型变量位于每个?层的中间（图3.2）。对?层的厚度没有约束，用户可使用统一的或不统一的?层。
图3.2 垂直?坐标中模型变量的位置
3.2. The Discrete Procedure in the Cartesian Coordinates
3.2 笛卡尔坐标下的离散方法
3.2.1. The 2-D External Mode.
3.2.1 二维外部模式
Let us consider the continuity equation first. Integrating Eq. (2.30) over a given triangle area yields:
首先考虑连续方程。结合方程（2.30）给出三角形面积流量：
wherevn is the velocity component normal to the sides of the triangle and s??is the closed trajectory comprised of the three sides. Eq. (3.3) is integrated numerically
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using the modified fourth-order Runge-Kutta time-stepping scheme. This is a
multi-stage timestepping approach with second-order temporal accuracy. The detailed procedure for this method is described as follows:
其中vn为三角形边的垂直法线分量；s?为包含三条边的闭合曲线。方程（3.3）是改进的四阶Runge-Kutta时间步长方案的积分。这是多阶时间步长趋近于二阶时间精确性。这种方法的具体步骤如下：
where k =1,2,3,4 and (?1，?2，?3，?4 ) = (1/4, 1/3,1/2, 1). Superscript n represents the nth time step. ??j?is the area enclosed by the lines through centroids and mid-points of the sides of surrounding triangles connected to the node where ?j is
nnlocated. um and vm are defined as:
其中k=1,2,3,4；（?，?，?，?）=（1/4，1/3，1/2，1）；上标n代表第n个时间步长；
nn和vm定义如下： ??j为质心、相邻三角形边中点与?j所在节点围成的面积。um1234
?t is the time step for the external mode, and
?t为外部模式的时间步长，并有：
Similarly, integrating Eqs. (2.31) and (2.32) over a given triangular area gives:
相似的，在给定的三角区域积分方程（2.31）和（2.32）
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