28
D. Parra-Guevara and Y.N. Skiba
Fig. 2.1 View of domain D
from above
Fig. 2.2 Cross-sectional
area of domain D
vector to the boundary ∂ D = S T ∪ S + ∪ S − ∪ S B of domain D, ∂/∂n is the derivative
in the normal direction, and k = (0, 0, 1) t is the unit vector directed upward in the
Cartesian coordinate system (Fig. 2.2). We observe that
k · n = 0 on S
+
∪ S
− and U · n = 0 on S T ∪ S B .
(2.12)
Also note that the boundary conditions (2.6)–(2.9) are general (i.e., not only for
horizontal free and bottom surfaces S T and S B ), and hence, the dispersion model can
take into account free surface wave motion and marine topography.
First of all we show that the solution of dispersion model (2.4)–(2.11) satisfies
the mass balance equation. Indeed, integrating Eq. (2.4) over domain D we get
∂
∂t
D
φ dr +
D
U · φ dr −
D
∇ · μ∇φ dr +
D
σ φdr +
D
∇ · φ s dr
=
N
i=1
D
Q i (t)δ(r − r i ) dr.
D. Parra-Guevara and Y.N. Skiba
Fig. 2.1 View of domain D
from above
Fig. 2.2 Cross-sectional
area of domain D
vector to the boundary ∂ D = S T ∪ S + ∪ S − ∪ S B of domain D, ∂/∂n is the derivative
in the normal direction, and k = (0, 0, 1) t is the unit vector directed upward in the
Cartesian coordinate system (Fig. 2.2). We observe that
k · n = 0 on S
+
∪ S
− and U · n = 0 on S T ∪ S B .
(2.12)
Also note that the boundary conditions (2.6)–(2.9) are general (i.e., not only for
horizontal free and bottom surfaces S T and S B ), and hence, the dispersion model can
take into account free surface wave motion and marine topography.
First of all we show that the solution of dispersion model (2.4)–(2.11) satisfies
the mass balance equation. Indeed, integrating Eq. (2.4) over domain D we get
∂
∂t
D
φ dr +
D
U · φ dr −
D
∇ · μ∇φ dr +
D
σ φdr +
D
∇ · φ s dr
=
N
i=1
D
Q i (t)δ(r − r i ) dr.