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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
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(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)
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(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,