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further . In order to transform the system Eq. (1. 5. 2 4 ) , introducing a transformation of dependent variables w = w ( u ) , multiplying each equation by the components of a cer- tain vector a respectively, adding the three resultant equations, and comparing them with those of the Eq. (1. 5. 35), we get the following conditions that a should satis- fy : (1 .5 .37 ) where a, and ay are elements of A, and A,, while a,, u, and w, are elements of u, u and w , respectively. Eliminating u, from Eq. (1. 5. 3 7 ) , in which each equalitity corresponds to three equations, yielding 45 (1. 5. 38) The number of the components of w , G and H amount to 3m, while the number of e- quations is 2mZ. The problem is overdetermined, so there may be no solution at all. Of course, in some special cases one or more solutions may exist. The 2-D SSWE originates from the conservation laws of mass and momentum, can certainly be written in conservative form, but it depends on the choice of depen- dent variables. The Eq. (1. 5. 35) is just one of possible forms. Alternatively, if we take w= ( u , v, h ) T as unknown vector, the 2-D shallow- water equations can also be written in conservative form, in which the fluxes G = ( u 2 / 2 f gh, v , uh )'and H = ( u , v 2 / 2 + g h , v h ) T . Especially in the I-D case, a lot of conservative forms can easily be found. For example, taking energy ( u 2 h + g h 2 ) / 2 as a dependent variable, the associated flux is u3h/2+ugh2. Higher-order forms also hold mathematically but without an explicit physical meaning. However, if we take pressure or gravity wave celerity as a dependent variable, the shallow-water equa- tions cannot be rewritten in conservative form. The main advantages of the conservative form are as follows: ( i ) There is a close relationship between conservation law, symmetric system and hyperbolic sys- tem, which will be useful in theoretical studies (cf. Section 2. 3 ) . (ii)It is conve- nient for constructing a conservative finite difference scheme, so that conservation of mass and momentum can be guaranteed in the numerical solution (cf. Section 5. 2 ) . For the normal form, such a scheme can only be established in some special cases. (iii) It is the only appropriate form for defining and calculating a discontinuous solu- tion (cf. Sections 3. 4 , 9. 1). X . FORM IN AN ORTHOGONAL CURVILINEAR CCK)RDINATE iWiYTEM (OCCS) Construct a fixed OCCS in the x-y plane and weave a mesh by two sets of isolines {(z, y)=const and ~ ( z , y)=const (cf. Section 8. 1). At any point P ( x , y ) with curvilinear coordinates (<, 71, unit vectors e, and e2 in the <-and ?-coordinate direc- tions constitute a local orthogonal coordinate system. The orthogonality condition be- tween e , and ez is The length of a differential arc in an OCCS is d s = h, d<el + hz dqel where (1. 5. 39) (1. 5. 40) where hl and hz , called measuring coefficients or Lami coefficients, constitute an or- 46 der-1 tensor, the measuring tensor. The sums under the square root symbol are de- noted by gts and gon respectively. OCCS formulas in common use are listed below: d A = h l h z d t d ~ Gradient of scalar function Area of a differential element spanned by d< and dv (1. 5. 42) Divergence of vector function Vorticity of vector function Gradient of vector function i a i a V = e l - - h, at + e2 h, & Laplacian operator of scalar (Hamiltonian operator) function (1.5. 43) (1. 5. 4 4 ) (1. 5. 45) (1. 5. 4 6 ) (1. 5. 47) Laplacian operator of vector function 02v = ( V v >v = v ( V V ) - v x (0 x v> (1. 5. 48) Product of a vector and a gradient of another vector (1. 5. 49) Using the above formulas, we can derive the equation of continuity in an OCCS (1. 5. 50) and also the equation of motion (cf. Eq. ( 1. 5. 19) ) in the <-direction (expressed by 47 unit vector a ) ( I . 5. 51) where { t , > } denotes an order-2 tensor (1. 5. 52) 1 1 9h2 P P 2 T hVV - -U = hVV f -(PI - Z) = hVV + --I - 2vthE The components of symmetric deformation-rate tensor E can be expressed as ( 1 . 5. 53) (1. 5. 54) (1. 5. 55) Denote depth-averaged velocities in the x-and y-directions by u, and v,, and those in the <-and 7-directions by uc and v,. The relations between them are ug = u, - + VY - - (: [ 5 :I& v , = u z - + v y - - ( I . 5. 56) ( 1 . 5. 57) Then, the 2-D SSWE in an OCCS on the x-y plane (or in a rectangular coordinate system on the 5-71 plane) can be written as (1. 5. 58) v, aA v, aB - F , + ~ - + - - - & a7 ( I . 5. 59) ( I . 5. 60) where 6 and & are local transformation coefficients of differential arcs in the <-and 7-directions ; g = gcs grs is the square Jacobian of the orthogonal coordinate 48 aq ay a< 3 a7l a< transformation, i . e. , = - - - - - ; ( Fc , F,)T denotes external forces, e- qua1 to averaged horizontal turbulent kinetic viscosity. / W ( u , , v , , ) ~ when considering bottom friction only; and v,=depth- C2 h (1. 5 . 61) (1. 5. 61a) In the equation of motion in the <-direction, &,/a< represents a physical convective term, while a &/a< is a convective term due to correction for coordinate curva- ture. In numerical solutions, special care should be taken of these. The situation in the q-direction is similar. In oceanography, a commonly-used OCCS is the spherical coordinate system, in which the governing differential equations are a( hvcosp) ]= 0 at+- Rcosp[aYi) - + a? ah 1 av u al) v all + - - f u2tang, at+~cosp% ~ a p (1. 5. 62) (1. 5. 62a) (1. 5. 62b) where $ = east longitude, p = latitude, R = radius of the globe, and p = density of sea-water. X I . FORM IN A SQECIAL TYPE OF LINEARLY HOMOTOPIC CIIRVILINEAR COORDINATE SYSTEM (LHCCS) Due to the difficulty of fulfilling the orthogonality condition for an OCCS, it is possible to utilize the concept of linear homotopy , taken from topology, for the con- struction of a special type of LHCCS. We shall describe such a mesh in detail in Chapters 6 and 8. The coordinate mesh is composed of two families of curves: one is a family of straight lines 4 (2, y ) = CI , parallel to the y-axis, and the second is a family of curves, ~ ( z , y > = C2, which intersect the former at equidistant points (or yielding proportional intervals), so that in a neighborhood of any point the two curves consti- tute an oblique coordinate system (Fig. 1. 2). 49 I X Fig. 1. 2 A linearly hornotopic mesh Derivation of the 2-D SSWE in such a coordinate system can be done by two approaches: ( 1 ) Establish the equations of mass and momentum conservation for a fluid element d < X d q (direct approach). (2 ) Convert the 2-D SSWE into a rectangu- lar coordinate system by using the following two transformations ( indirect approach ) . ( i ) Coordinate transformation. From the total differential formula (chain rule) we get equations for ( a f / a x , a f / a y > in terms of ( a f / a < , a f / a q ) , and then substitute them into the original system. (ii) Dependent variable transformation. The two unknowns u and v, velocities in the .r-and y-directions, are obviously inconvenient for a curvilinear coordinate sys- tem. So they are transformed into ug and ZI,, in the <-and q-directions by projection, in which transformation coefficients are functions of I and y. The transformation is pointwise linear locally, but, in general, nonlinear globally. Adopting the indirect approach, under the condition that each cell of the mesh can be approximated by a rhombus, we obtain the 2-D SSWE on the 5-q plane where (1. 5. 63) (1. 5. 6 4 ) (1. 5. 65) (1. 5 . 66) (1. 5. 67) (1. 5. 68) 50 (1. 5. 69) a = cosB, c = sine (1. 5. 70) In the above equations only gravity, bottom friction and geostrophic force are considered. As compared with the related formulas in a rectangular coordinate sys- tem,