Environmental Engineering Reference
InDepth Information
(
k
)
γ
f'
(
0
) = 0
φ
k
max
k
f'
(
0
) >
a
c
f'
(
0
) =
a
c
φ
φ
f'
(
0
) <
a
c
φ
Figure B.4. Dispersion relation
(
k
) as a function of the wave number
k
for various
values of the bifurcation parameter
f
(
γ
φ
0
).
a
c
is the critical value for
f
(
φ
0
) that
discriminates the situations of stability and instability.
with
k
is infinitely extended in both the
x
and
y
directions,
W
(
k
)is
the Fourier transform of the kernel function. The dispersion relation obtained with
kernel (
B.13
) is shown in Fig.
B.4
. Notice how the shape of the dispersion relation
is entirely determined by
W
(
k
), i.e., by the effect of the kernel function on the
spatial dynamics, whereas the local dynamics affect Eq. (
B.15
) only through the con
stant
f
(
=
k

.If
0
). In fact, changes in this constant determine a vertical shift of the curves
in Fig.
B.4
without modifying their shape. This vertical shift affects the sign of
γ
φ
(
k
), thereby determining the stability or instability of the system and the range
of unstable modes. All modes with
γ<
0 are linearly stable because they vanish
as time
t
passes. Conversely, all modes with
0 are linearly unstable and tend
to grow with time. However, even in this case, when the amplitude of the unstable
modes becomes finite, the assumptions underlying this linearstability analysis (i.e.,
that perturbations are “small” or infinitesimal) are no longer valid. Thus the linear
stability analysis does not shed light on the state approached by the system as an
effect of the unstable modes. However, as noted for Turing's instability, the dominant
wavelength of patterns emerging from this instability is dictated by the most unstable
mode,
k
max
, (which grows faster than the other unstable modes, thereby determining
some key aspects of pattern geometry). This wavelength depends on only the shape
of the kernel function and is not affected by the term
f
(
γ>
φ
0
)[seeEq.(
B.15
)], even
though
f
(
φ
0
) determines the stability of the system and the emergence of spatial
patterns: For relatively low values of
f
(
φ
0
),
γ<
0 for all wave numbers
k
(see
Fig.
B.4
), whereas as
f
(
φ
0
) increases above a critical value,
γ
(
k
max
) becomes positive
and the mode
k
max
is unstable. Larger values of
f
(
φ
0
) correspond to broader ranges
of unstable wave numbers.
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