View
217
Download
0
Category
Tags:
Preview:
Citation preview
ANALYTIC SOLUTIONSANALYTIC SOLUTIONS FOR LONG INTERNAL WAVE MODELS FOR LONG INTERNAL WAVE MODELS
WITH IMPROVED NONLINEARITYWITH IMPROVED NONLINEARITY
Alexey Slunyaev Alexey Slunyaev Insitute of Applied Physics RASInsitute of Applied Physics RAS
Nizhny Novgorod, RussiaNizhny Novgorod, Russia
z
x
1
U1
g
0
H
U2
2
2-layer fluid rigid-lid boundary conditionBoussinesq approximation
0
X
p
X
UU
T
U jjj
jj
0
jjj UH
XT
H
HHH 21
02211 HUHU
z
x
1
U1
g
0
H
U2
2 012
wxt
012
wxt
w
H
HWU 2
1 2H
HWU 1
2 2
21
HHH
22
HHH
HgC
2
121
CwW HH
tC
HT HxX
12
0
x
SV
t
S
)1)(1(22 22 wwS
)1)(1(2 22 wwV
Representation inRepresentation inRiemann invariantsRiemann invariants
[Baines, 1995;Lyapidevsky & Teshukov 2000;
Slunyaev et al, 2003]
z
x
1
U1
g
0
H
U2
2
2-layer fluid rigid-lid boundary conditionBoussinesq approximation
0
xV
t
212121
22121
22
231
hhhhhh
hhhhV
z
x
1
U1
g
0
H
U2
2
The fully nonlinear The fully nonlinear (but dispersiveless) (but dispersiveless)
modelmodel
The full nonlinear velocityThe full nonlinear velocity
[Slunyaev et al, 2003; Grue & Ostrovsky, 2003]
z
x
1
U1
g
0
H
U2
2
0
xV
t
212121
22121
22
231
hhhhhh
hhhhV
The full nonlinear velocityThe full nonlinear velocity
00 cos4
12cos
4
3 V
)sin(12
1 h )sin(12
101 h
1
1
2
2
u1
u1
u2u2
clinclin
V+
V+
Velocity profilesVelocity profiles
hh = 0.1 = 0.1 hh = 0.5 = 0.5
z
x
1
U1
g
0
H
U2
2
0
xV
t
212121
22121
22
231
hhhhhh
hhhhV
The full nonlinear velocityThe full nonlinear velocity
asymptotic expansions for asymptotic expansions for any-order nonlinear coefficientsany-order nonlinear coefficients
0
xV
t
543
32
211 OVV lin
21
21
2
3
HH
HH 2
21
212
1
4
8
3
HH
HHH
321
212
2 16
3
HH
HHH
421
221
2
3 128
15
HH
HHH
etc…etc…
The full nonlinear velocityThe full nonlinear velocity
0
xV
t
543
32
211 OVV lin
Exact Exact relation relation
for for HH11 = = HH22
The full nonlinear velocityThe full nonlinear velocity
2
2
121H
VV lin
Corresponds to the Corresponds to the Gardner eqGardner eq
2-layer fluid rigid-lid boundary conditionBoussinesq approximation
Exact fully nonlinear velocity for asymp eqsExact fully nonlinear velocity for asymp eqs
Exact velocity fields (hydraulic approx)Exact velocity fields (hydraulic approx)
Strongly nonlinear wave steepening (dispersionless approx)Strongly nonlinear wave steepening (dispersionless approx)
The GE is exact when the layers have equal depthsThe GE is exact when the layers have equal depths
z
x
(z)
U (z)
g
0
H
Rigorous way for Rigorous way for obtaining asymptotic eqsobtaining asymptotic eqs
stratified fluid free surface condition
z
x
(z)
U (z)
g
0
H
Rigorous way for Rigorous way for obtaining asymptotic eqsobtaining asymptotic eqs
stratified fluid free surface condition
02
512
33323
31
214
33
2
21
Ouuuuuuu
uuuuuuuu
uuuuuu
xxxxxxxx
xxxxxxxx
xxxxxt extGE
0 Ouuuu xxxxt
02121
21
Ouuuuuuu
uuuu
xxxxxxxxxxxx
xxxxt
xxxx
x
x
xxx vvvxvdxvvvvu 4322
1
0
02 Ovvvv xxxxt
Asymptotical integrability Asymptotical integrability (Marchant&Smyth, Fokas&Liu 1996)
2nd order KdV
KdV
021 Ouuuuuu xxxxxt
02
512
33323
31
214
33
2
21
Ouuuuuuu
uuuuuuuu
uuuuuu
xxxxxxxx
xxxxxxxx
xxxxxt
xxxxx
x
x
x
x
x
xxx vvvvvxdxvvvdxvvvvvu 216
2543
32
21
00
0243
32
21 Ovvvvvvv xxxxt
Almost asymptotical integrabilityAlmost asymptotical integrability
GE
extGE
021 Ouuuuuu xxxxxt
02
512
33323
31
214
33
2
21
Ouuuuuuu
uuuuuuuu
uuuuuu
xxxxxxxx
xxxxxxxx
xxxxxt
xxxxx
x
x
x
x
x
xxx vvvvvxdxvvvdxvvvvvu 216
2543
32
21
00
0243
32
21 Ovvvvvvv xxxxt
Almost asymptotical integrabilityAlmost asymptotical integrability
GE
extGE
021 Ouuuuuu xxxxxt
02
512
33323
31
214
33
2
21
Ouuuuuuu
uuuuuuuu
uuuuuu
xxxxxxxx
xxxxxxxx
xxxxxt
xxxxx
x
x
x
x
x
xxx vvvvvxdxvvvdxvvvvvu 216
2543
32
21
00
0243
32
21 Ovvvvvvv xxxxt
Almost asymptotical integrabilityAlmost asymptotical integrability
GE
extGE
0243
32
21 Ovvvvvvv xxxxt
Sv
v
dVxx0
2/163524132
0 151063β
Solitary waves
2-order GE theory as perturbations of the GE solutions2-order GE theory as perturbations of the GE solutions
Qualitative closeness of the GE and its extensionsQualitative closeness of the GE and its extensions
066 2 xxxxxt uuuuuuGE
-20 0 20
0.0
0.2
0.4
0.6
0.8
1.0
-4 0 4
-6
-4
-2
0
2
4
066 2 xxxxxt uuuuuuGE
066 2 xxxxxt uuuuuuGE
2
1
x
x
u
u
1
Initial Problem
AKNS approach
066 2 xxxxxt uuuuuuGE
x
x
u
u
1
AKNS approach
mKdV
)(qQ AKNS approach
06 2 xxxxt qqqq
x
x
q
066 2 xxxxxt uuuuuuGE
mKdV 06 2 xxxxt qqqq
066 2 xxxxxxt uuuuucuu
26ac a2
atxutxq ),(),(
GE
mKdV 06 2 xxxxt qqqq
066 2 xxxxxxt uuuuucuu
)(uU
x
x
au
uU
2
AKNS approach
22)(
222 aaUQ u
22)(
2)( aqu
GE
mKdV 06 2 xxxxt qqqq
066 2 xxxxxxt uuuuucuu
26ac a2
a – is an arbitrary number
GE 066 2 xxxxxxt uuuuucuu
26ac a2
Passing through a turning point?
t
Tasks:
GE 066 2 xxxxxxt uuuuucuu
Passing through a turning point?
t
Tasks:
A solitary-like wave over a long-scale wave
22)(
2)( aqu
GE 066 2 xxxxxxt uuuuucuu
A solitary-like wave over a long-scale wave
22)(
2)( aqu
222)(
2)( aauu
GE+
mKdV+ 06 2 xxxxt qqqq
066 2 xxxxxxt uuuuucuu
22)(
2)( aqu
a soliton cannot pass a soliton cannot pass through a too high through a too high
wave being a solitonwave being a soliton
discrete eigenvalues discrete eigenvalues may become may become continuouscontinuous
a
GE+
mKdV+ 06 2 xxxxt qqqq
066 2 xxxxxxt uuuuucuu
soliton amplitude soliton amplitude ((ss denotes polarity) denotes polarity)
asA qsolu 22 )(
)()(
22)(
)()( 24 aC q
solu
soliton velocitysoliton velocity )(2
2)(
2)()(
cosh411
4
uu
usol
s
u
tctx uuu2
)()()( 42
Solitons
GE-
mKdV- 06 2 xxxxt qqqq
066 2 xxxxxxt uuuuucuu
22)(
2)( aqu 02
)( q
22)( au
at the turning point at the turning point all spectrum becomes all spectrum becomes
continuouscontinuous
GE-
mKdV- 06 2 xxxxt qqqq
066 2 xxxxxxt uuuuucuu
soliton amplitudesoliton amplitude
aA qsolu 22 2
)()(
)(
22)(
)()( 24 aC q
solu
soliton velocitysoliton velocity
This approach was applied to the NLS eq
peri
odic
al b
ound
ary
cond
itio
ns
peri
odic
al b
ound
ary
cond
itio
ns
an e
nve
lope
sol
iton
plane wave
plane wave
The initial conditions: an envelope soliton and a plane wave background
Spatio-temporal evolution NLS “breather”
envelope soliton
This approach was applied to the NLS eq
Solitary wave dynamics on pedestals may be interpretedSolitary wave dynamics on pedestals may be interpreted
Strong change of waves may be predicted (“turning” points)Strong change of waves may be predicted (“turning” points)
Thank you for attention!Thank you for attention!
Gavrilyuk S.
Grimshaw R.
Pelinovsky E.
Pelinovsky D.
Polukhina O.
Talipova T.
Co-authorsCo-authors
Recommended