102
DYNAMICAL OCEANOGRAPHY
Transformation of the problem (5.21) in the coordinate ζ for the function ˜
y(ζ; ǫ)
gives,
y
′ (x)=
1
ǫ p ˜
y
′ (ζ),
(5.27)
and the problem near ζ = x =0becomes
ǫ
1−p ˜
y
′′ +˜ y
′ = ǫ
p a,
(5.28a)
˜
y(0) = 0,
(5.28b)
and the highest order derivative can only participate in a balance when p =1.
We now proceed with the inner expansion
˜
y(ζ; ǫ)=˜ y 0 (ζ)+ǫ˜ y 1 (ζ)+O(ǫ
2 ),
(5.29)
from which the O(1) system becomes
˜
y
′′
0 +˜ y
′
0 =0,
(5.30a)
˜
y 0 (0) = 0,
(5.30b)
with as a general solution (the inner solution)
˜
y 0 (ζ)=C 2 (e
−ζ − 1).
(5.31)
The constant C 2 follows now from the fact that the inner solution must continuously connect to the outer solution y 0 (x) near x =0. This results in the matching
condition
lim
ζ→∞
˜
y 0 (ζ)=−C 2 = lim
x→0
y 0 (x)=1− a,
(5.32)
with the result C 2 = a − 1. With the procedure, we find the O(1) asymptotic
solution
y(x; ǫ)=(1− a)+ax for ǫ → 0,x > 0 fixed
(5.33a)
y(ζ; ǫ)=(1− a)(1 − e
−ζ ) for ǫ → 0,ζ = x/ǫ > 0 fixed.
(5.33b)
These solutions are plotted (together with the ones in Fig. 5.21b) in Fig. 5.5 and
the boundary layer character of the solution is well represented by the asymptotic
solutions.
◭
Additional Material
D: Asymptotic methods and their application to problems in Fluid Mechanics are
well described in Kevorkian and Cole (1996). Another excellent reference on
these methods is chapter 7 of Bender and Orszag (1999).
DYNAMICAL OCEANOGRAPHY
Transformation of the problem (5.21) in the coordinate ζ for the function ˜
y(ζ; ǫ)
gives,
y
′ (x)=
1
ǫ p ˜
y
′ (ζ),
(5.27)
and the problem near ζ = x =0becomes
ǫ
1−p ˜
y
′′ +˜ y
′ = ǫ
p a,
(5.28a)
˜
y(0) = 0,
(5.28b)
and the highest order derivative can only participate in a balance when p =1.
We now proceed with the inner expansion
˜
y(ζ; ǫ)=˜ y 0 (ζ)+ǫ˜ y 1 (ζ)+O(ǫ
2 ),
(5.29)
from which the O(1) system becomes
˜
y
′′
0 +˜ y
′
0 =0,
(5.30a)
˜
y 0 (0) = 0,
(5.30b)
with as a general solution (the inner solution)
˜
y 0 (ζ)=C 2 (e
−ζ − 1).
(5.31)
The constant C 2 follows now from the fact that the inner solution must continuously connect to the outer solution y 0 (x) near x =0. This results in the matching
condition
lim
ζ→∞
˜
y 0 (ζ)=−C 2 = lim
x→0
y 0 (x)=1− a,
(5.32)
with the result C 2 = a − 1. With the procedure, we find the O(1) asymptotic
solution
y(x; ǫ)=(1− a)+ax for ǫ → 0,x > 0 fixed
(5.33a)
y(ζ; ǫ)=(1− a)(1 − e
−ζ ) for ǫ → 0,ζ = x/ǫ > 0 fixed.
(5.33b)
These solutions are plotted (together with the ones in Fig. 5.21b) in Fig. 5.5 and
the boundary layer character of the solution is well represented by the asymptotic
solutions.
◭
Additional Material
D: Asymptotic methods and their application to problems in Fluid Mechanics are
well described in Kevorkian and Cole (1996). Another excellent reference on
these methods is chapter 7 of Bender and Orszag (1999).