Geoscience Reference
In-Depth Information
Thus, the curl of the velocity is twice the angular velocity. A vector field for which V = 0
is called irrotational .
The Laplacian operator
In Cartesian coordinates, the Laplacian operator
2
is defined by
2
x 2
2
y 2
2
z 2
2
= ∇·∇ =
+
+
(A1.17)
which is the divergence of the gradient. It is a scalar operator:
2 T
x 2
2 T
y 2
2 T
z 2
+
+
2 T
(A1.18)
= ∇·∇ T
=
2
To define a Laplacian operator
for a vector, it is necessary to use the identity
∇·
(
V )
=
(
∇·
V )
(
V )
(A1.19)
In Cartesian coordinates, this is the same as applying the Laplacian operator to each
component of the vector in turn:
2 V = (
2 V x ,
2 V y ,
2 V z )
(A1.20)
However, in curvilinear coordinate systems this is not true because, unlike the unit vectors
(1, 0, 0), (0, 1, 0) and (0, 0, 1) in the Cartesian coordinate system, those in curvilinear
coordinate systems are not constants with respect to their coordinate system. The calculation
of the Laplacian operator applied to a vector in cylindrical and spherical polar coordinates is
long and is left to the reader as an extracurricular midnight activity. (Hint: use Eqs. (A1.19),
(A1.22)-(A1.24) and (A1.28)-(A1.30).)
Curvilinear coordinates
In geophysics it is frequently advantageous to work in curvilinear instead of Cartesian
coordinates. The curvilinear coordinates which exploit the symmetry of the Earth, and are
thus the most often used, are cylindrical polar coordinates and spherical polar coordinates .
Although not every gradient, divergence, curl and Laplacian operator is used in this topic in
each of these coordinate systems, all are included here for completeness.
Cylindrical polar coordinates ( r,
φ
,z )
In cylindrical polar coordinates (Fig. A1.3), a point P is located by specifying r , the radius of
the cylinder on which it lies, φ , the longitude or azimuth in the x - y plane, and z , the distance
from the x - y plane to the point P, where r 0, 0 φ 2 π and −∞ < z < .From Fig.
A1.3 it can be seen that
x = r cos φ
y = r sin φ
(A1.21)
Figure A1.3.
z = z
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