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^2 = ^

^1 = '

e^ e ^3 = ^

er (10.47)

;

and that the velocity in this rotating coordinate system is given by

v =u^1 + v^2 + w^3 :

eee (10.48)

Problem m: We will assume that the axes of the spherical coordinate system are chosen

in such a way that the direction = 0 is aligned with the rotation axis. This is a

di erent way of saying the rotation vector is parallel to the z-axis: = ^. Usez

the rst two expressions of equation (3.13) of section (3.1) to show that the rotation

vector has the following expansion in the unit vectors ^ and ^:

r

= cos ^ sin ^ :

r (10.49)

;

Problem n: In the rotating coordinate system, the Coriolis force is given by Fcor =

v. Use the expressions (10.47)-(10.49) and the relations (3.11) of section

;2m

(3.1) for the cross product of the unit vectors to show that the Coriolis force is given

by

Fcor = 2m sin u ^+2m cos u ^ + 2m (v cos w sin ) ' :

r ^ (10.50)

;

Problem o: Both the ocean or atmosphere are shallow in the sense that the vertical

length scale (a few kilometers for the ocean and around 10 kilometers for the atmo-

sphere) is much less than the horizontal length scale. This causes the vertical velocity

CHAPTER 10. LINEAR ALGEBRA

116

to be much smaller than the horizontal velocity. For this reason the vertical velocity

w will be neglected in expression (10.50). Use this approximation and the de nition

(10.47) to show that the horizontal component aH of the Coriolis acceleration is in

cor

this approach given by:

aH = ^3 v e (10.51)

cor ;f

with

f = 2 cos (10.52)

This result is widely used in meteorology and oceanography, because equation (10.51)

states that in the Cartesian coordinate system aligned with the earth's surface, the Coriolis

force generated by the rotation around the true earth's axis of rotation is identical to the

Coriolis force generated by the rotation around a vertical axis with a rotation rate given

by cos . This rotation rate is largest at the poles where cos = 1, and this rotation

rate vanishes at the equator where cos = 0. The parameter f in equation (10.51) acts as

a coupling parameter, it is called the Coriolis parameter. (In the literature on geophysical

uid dynamics one often uses latitude rather than the co-latitude that is used here, for

this reason one often sees a sin-term rather than a cos-term in the de nition of the Coriolis

parameter.) In many applications one disregards the dependence of f on the co-latitude

in that approach f is a constant and one speaks of the f-plane approximation. However, the

dependence of the Coriolis parameter on is crucial in explaining a number of atmospheric

and oceanographic phenomena such as the propagation of Rossby waves and the formation

of the Gulfstream. In a further re nement one linearizes the dependence of the Coriolis

parameter with co-latitude. This leads to the -plane approximation. Details can be found

in the books of Holton 30] and Pedlosky 46].

10.5 The eigenvalue decomposition of a square matrix

In this section we consider the way in which a square N N matrix A operates on a

vector. Since a matrix describes a linear transformation from a vector to a new vector,

the action of the matrix A can be quite complex. However, suppose the matrix has a set

of eigenvectors v(n) . We assume these eigenvectors are normalized, hence a caret is used

^

in the notation v(n) . These eigenvectors are extremely useful because the action of A on

^

an eigenvector v(n) is very simple:

^

A^(n) = nv(n)

v ^ (10.53)

where n is the eigenvalue of the eigenvector v(n) . When A acts on an eigenvector, the

^

resulting vector is parallel to the original vector, the only e ect of A on this vector is

to either elongate the vector (when n 1), compress the vector (when 0 n < 1) or

reverse the vector (when n < 0). We will restrict ourselves to matrices that are real and

symmetric.

Problem a: Show that for such a matrix the eigenvalues are real and the eigenvectors

are orthogonal.

10.5. THE EIGENVALUE DECOMPOSITION OF A SQUARE MATRIX 117

The fact that the eigenvectors v(n) are normalized and mutually orthogonal can be ex-

^

pressed as

v(n) v(m) = nm

^^ (10.54)

where nm is the Kronecker delta, this quantity is equal to 1 when n = m and is equal to

zero when n = m. The eigenvectors v(n) can be used to de ne the columns of a matrix V:

^

6

0 .. .. .. 1

. . .

B (1) (2) (N) C

V =B v v vC

@ ^. ^. ^A (10.55)

..

.. .. .

this de nition implies that

Vij vi(j) : (10.56)

Problem b: Use the orthogonality of the eigenvectors v(n) (expression (10.54)) to show

^

that the matrix V is unitary, i.e. to show that

VT V = I (10.57)

where I is the identity matrix with elements Ikl = kl . The superscript T denotes

the transpose.

Since there are N eigenvectors that the orthonormal in an N-dimensional space, these

eigenvectors form a complete set and analogously to (10.13) the completeness relation can

be expressed as

X (n) (n)T

N

I= v v :

^^ (10.58)

n=1

When the terms in this expression operate on a arbitrary vector p, an expansion of p in

the eigenvectors is obtained that is completely analogous to equation (10.11):

X (n) v(n)T p = X v(n)

N N

p= v ^^ ^ v(n) p :

^ (10.59)

n=1 n=1

This is a useful expression, because is can be used to simplify the e ect of the matrix A

on an arbitrary vector p.

Problem c: Let A act on expression (10.59) and show that:

X (n) (n)

N

Ap = nv v p : ^^ (10.60)

n=1

This expression has an interesting geometric interpretation. When A acts on p, the vector

p is projected on each of the eigenvectors, this is described by the term v(n) p . The

^

corresponding eigenvector v(n) is multiplied with the eigenvalue v(n)

^ ^ n v(n) and the

^

!

result is summed over all the eigenvectors. The action of A can thus be reduced to

a projection on eigenvectors, a multiplication with the corresponding eigenvalue and a

summation over all eigenvectors. The eigenvalue n can be seen as the sensitivity of the

eigenvector v(n) to the matrix A.

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