-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Dispersive dam breaks and lock-exchanges ina two-layer fluid
Gavin Esler and Joe Pearce
Department of MathematicsUniversity College London
BIRS: Dispersive hydrodynamics
May 21, 2015
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
The Physical ProblemUndular bores in the atmosphere and
ocean
Two-layer problem: initial conditions
Consider a ‘lock release’ in a Boussinesq two-layer fluid.Waves
do not break: No turbulent internal bores.Waves can be strongly
nonlinear, but are long (kH . 1).
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
The Physical ProblemUndular bores in the atmosphere and
ocean
Undular bore over the Gulf of Mexico
(Loading ub.avi)
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
ub.aviMedia File (video/avi)
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IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
The Physical ProblemUndular bores in the atmosphere and
ocean
Single layer problem
Serre-Su-Gardner-Green-Naghdi equations.El, Grimshaw, Smyth
(Phys. Fluids, 2006).Generic behaviour: Rightwards-propagating
undularbore, leftwards rarefaction wave.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
The Physical ProblemUndular bores in the atmosphere and
ocean
Two-layer problem
Will the fluid behave as the single-layer system?How are the
details determined...?
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
Miyata-Choi-Camassa Equations - I
The following nondimensional extended shallow water set1 canbe
used to model internal undular bores:
u1t + u1u1x = −Πx + 13h1(
h31G[u1])
x+ τ2 h1xxx
u2t + u2u2x = −Πx − h2x + 13h2(
h32G[u2])
x+ τ2 h2xxx
h1t + (u1h1)x = 0,h2t + (u2h2)x = 0,
ui velocity, hi thickness in i th layer. Π pressure at rigid
lid.G[·] is a nonlinear differential operator, acting on any f (x ,
t),
G[f ] = fxt + ffxx − (fx )2.τ is a Bond number - a measure of
the surface tension.
1Miyata 1985; Choi & Camassa 1999.Gavin Esler and Joe Pearce
Internal dam breaks and lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
Miyata-Choi-Camassa Equations - II
In the absence of topography, the MCC equations are just
twocoupled PDEs in disguise.
1 Work in the zero-total momentum frame withu1h1 + u2h2 = 0.
2 Use h1 + h2 = 1 to define h = h2 with h1 = 1− h.3 Introduce a
baroclinic velocity v = u2 − u1.
Then the MCC equations can be written
ht + (vh(1− h))x = 0,vt +
(12v
2(1− 2h) + h)
x= τhxxx + 13h
(h3G[v(1− h)]
)x
− 13(1−h)(
(1− h)3G[−vh])
x.
Only two dependent variables (v ,h) remain.Gavin Esler and Joe
Pearce Internal dam breaks and lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
Dispersionless limit
For long waves (k � 1) neglect dispersion and surface
tension
ht + (vh(1− h))x = 0,vt +
(12v
2(1− 2h) + h)
x= 0.
The corresponding single layer shallow water equations are
σt + (uσ)x = 0,
ut +(
12u
2 + σ)
x= 0,
for depth σ and velocity u.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
Exact Correspondence...!?
Both systems can be mapped into the following form2:[∂
∂t+(3
4L +14R) ∂∂x
]L = 0,
[∂
∂t+(3
4R +14L) ∂∂x
]R = 0,
where L and R are the left and right Riemann invariants:
2-layer
L = v(1− 2h)− 2√
h(1− h)(1− v2)R = v(1− 2h) + 2
√h(1− h)(1− v2)
∣∣∣∣∣∣∣∣
1-layer
L = u − 2√σR = u + 2
√σ
The dynamics of the 2-layer and 1-layer SWE
appearidentical...!?
2E.g. Cavanie, 1969; Chumakova et al. 2009Gavin Esler and Joe
Pearce Internal dam breaks and lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
Two-layer SWE: Riemann invariants
Contours: L(v ,h) (dotted)Contours: R(v ,h) (solid).
The hyperbolic region of (v ,h) space is [−1,1]× [0,1].In
general four points in the region have the same value of(L,R).
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
Two-layer SWE: ‘St Andrew’s Cross’ mapping
1
1−layer
u
v
2−layer
h
0
1
−1 1 1−1
SD
BD
BD
SD
σ
Mapping 2-layer SWE→ 1-layer SWE is surjective.Four quadrants in
the 2-layer (v ,h) domain→ a triangularsubset of the 1-layer (u, σ)
domain.Physical reasoning =⇒ the existence of an up-downsymmetry h→
1− h, v → −v .Less-obvious symmetry: h→ 1− v/2, v → 1− 2h.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
MCC equations: Steadily propagating waves
Wave-like solutions of the MCC (steady propagation at speedc)
can be sought: use3
∂t → −c∂xThe mass equation can be integrated to eliminate v and
themomentum equation integrated twice to get the potential
form:
(hx )2 =3P(h)N(h)
,
P(h) = (h − h1)(h − h2)(h − h3)(h − h4)N(h) = h1h2h3h4(1 − h) +
(1 − h1)(1 − h2)(1 − h3)(1 − h4)h − 3τh(1 − h)
Steady wave solutions are entirely described by the roots hi
.
3following Choi and Camassa, 1999.Gavin Esler and Joe Pearce
Internal dam breaks and lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
MCC equations: Steadily propagating waves
Interface height h
Waves can be understood us-ing the potential function.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Equation set: The Miyata-Choi-Camassa equationsThe long wave
limit: Two-layer shallow waterWaves in the MCC equations
MCC equations: The Solibore
18 W. Choi and R. Camassa
h1
h2
h1
ʹ′
h2
ʹ′
u1
ʹ′
u2
ʹ′
q1
q2
u1
= c
u2
= c
Figure 9. Steady flow in a frame moving with a left-going
front.
where ζ0 ≡ ζ(X = 0) = am/2,
κm =
����3g(ρ2 − ρ1)
c2m(ρ1h21 − ρ2h
22)
����1/2
, (3.71)
and the function Xf is
Xf(x, y, z) = (y − z)1/2 log
(x − z)1/2 + (y − z)1/2
(x − z)1/2 − (y − z)1/2− (−z)1/2 log
(x − z)1/2 + (−z)1/2
(x − z)1/2 − (−z)1/2.
(3.72)
As figure 8 shows, the limiting wave form is a front slowly
varying from 0 to am.We remark that for the case of a homogeneous
layer, the analogue of the present
fully nonlinear model is offered by the GN system given by
(3.20) and (3.22) withP = 0. Its solitary wave solutions are
ζ(X) = a sech2(κ1X), κ21 =
3a
4(1 + a), c
2 = 1 + a , (3.73)
and it is interesting to notice that no limiting wave amplitude
exists in this case.Because we have not imposed any assumption on
the magnitude of the wave
amplitude in order to derive our long-wave model, and since the
solution (3.70)–(3.72) is consistent with the long-wave assumption
(3.1), it is natural to expect that asimilar front solution exists
for the full Euler system. In fact, this was demonstratedby
Funakoshi & Oikawa (1986), who also used their proof to
validate their numericalsolutions. For completeness, we now come
back to the existence proof of Euler frontsolutions (rederiving the
result of Funakoshi & Oikawa 1986) and compare these tothe
highest wave solutions of the present long-wave model.
3.4. Internal bore: exact theory
We assume the set-up illustrated in figure 9 for an internal
bore moving from rightto left at constant speed c into a two-layer
stratified fluid at rest at x = −∞. Thus,in a frame moving with
such a left-going front of speed c, the velocity at x = −∞is c in
both fluids, i.e. u1 = u2 = c. On the other hand, the velocity at x
= ∞ can beexpected to be different, say u�1 (u
�
2), in an upper (lower) fluid layer whose thicknessis h�1 (h
�
2). The question of existence of front-like solutions is then
equivalent to thatof finding c, u�
iand h�
ifor given ρi and hi, i = 1, 2, such that all three basic
physical
conservation laws of mass, momentum and energy hold.Mass
conservation in each fluid implies
ch1 = u�
1h�
1, ch2 = u�
2h�
2, (3.74)
Solibores connect states with equal (L,R)
Solibore speed : cb = 12 +14(L + R).
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Riemann problem
Consider general step-like initial conditions for the
MCCequations
v(x ,0) ={
v− x < 0v+ x > 0
h(x ,0) ={
h− x < 0h+ x > 0
Consider in particular states subject to a further
relationship
Φ(v−,h−, v+,h+) = 0,
so that only a rightwards propagating undular bore
(norarefaction wave) emerges at long times.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Whitham-El method
The MCC equations are not integrable.
Whitham modulation theory can be used to obtainWhitham equations
from conservation laws.
Whitham-El method∗ can be used to ‘fit’ dispersive shocksinto
solutions of non-dispersive system.
∗See El (2005), El et al. (2006, Phys. Fluids) and Esler and
Pearce (JFM,2011) for details.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Whitham Modulation Equations
Use Whitham averaging:
F̄ =k
2π
∫ 2π/k
0F (v ,h) dx =
k√3π
∫ h3h2
F (v(h),h)√
N(h)√P(h)
dh,
F̄ = F̄ (h1,h2,h3,h4) modulation variables. Apply toconservation
laws
(P̄i)t + (Q̄j)x = 0, (P1,P2,P3,P4) = (M, I, E , k)(Q1,Q2,Q3,Q4)
= (V,J ,F , ck)
Result: Four first-order PDEsyt + Byx = 0,
where yT = (h1,h2,h3,h4) and B = P−1Q, with P and Q 4× 4matrices
with components pij = ∂P̄i/∂hj , qij = ∂Q̄i/∂hj .
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Time-reversibility argument
The MCC system is timereversible.
Characteristics of theWhitham system arecontinuous across t =
0
=⇒ that a undular boreevolves into a rarefactionwave in -ve
time.
=⇒ rightward-propagating undular bore satisfies samejump
condition as a -ve time rarefaction wave: L− = L+.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Expansion fan solution
For our step-like initial conditions a similarity solution
y(x/t)can be sought. Obtain
(B−
( xt
)I)
y′ = 0.
Deduce that the similarity solution must be (schematically)
ofthe form
Fi(h1,h2,h3,h4) = 0, λj(h1,h2,h3,h4) =xt, i = 1,2,3.
for unknown functions Fi , and where λj is one of theeigenvalues
of B.
But what next...?
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Whitham-El method: Matching conditions
Linear (trailing) edge: At the linear edge (a→ 0) of theundular
bore the Whitham system reduces to:
a = 0 : h̄t +(v̄ h̄(1− h̄)
)x = 0,
v̄t +(
12 v̄
2(1− 2h̄) + h̄)
x= 0.
kt +(kc0(v̄ , h̄, k)
)x = 0
One of the eigenvalues of the reduced system must equals− = x/t
at the linear edge. Deduce that
s− =∂(kc0)∂k
(k−, v−,h−)
It remains to determine the trailing edge wavenumber k−.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Whitham-El method: Linear edge
How can the trailing wavenumber k− be determined?
Key Insight: The a = 0 reduction of the Whitham system canbe
integrated across the undular bore. (Intersects actualsolution at
x/t = s±). Integrate
Rt +(3
4R +14L±
)Rx = 0, kt + (c0(k ,L±,R)k)x = 0.
After some considerable working, get the first order
ODE:d(k2)
dR=
34 R +
54 L − 2c
r0 + 2k
2[HR(
13 (c
r 20 + 2H) − τ − 148 (R − L)
2)− 124 H(R − L)
]k−2(V r − cr0)(R + L − 2(1 +
13 k
2H)cr0) − 2H(13 (c
r 20 + H) − τ −
148 (R − L)2)
,
with the boundary condition k(R+) = 0.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Whitham-El method: Solitary wave edge
Solitary wave amplitude a+ and solitary wave speed s+obtained
analogously. Use conjugate wavenumber k̃ .
k̃ =√
3π
(∫ h2h1
√N(h)/
√−P(h) dh
)−1
to parametrise a.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Dam break: Resolution of initial data
Stationary initial conditions lead to a rightwards undularbore
and a leftwards rarefaction wave.The mid-state is defined by
(Lm,Rm) = (L+,R−).
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Dam break: Resolution of initial data - II
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Dam break: Resolution of initial data - III
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Dam break: numerical results- I
Distance x
Initial conditions: (L−,R−) = (−0.8660,0.9701),Initial
conditions: (L+,R+) = (−0.8660,0.8660).Linked with a tanh-function
(Half-width 2H).Periodic domain (400H). 2048 Fourier modes.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Dam break: numerical results- II
Distance x
Resolves into rightward propagating undular bore only.Edges of
the undular bore: predicted from theory (dottedlines).
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Dam break: numerical results- III
Distance x
Agreement improves at later times.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Dam break: Summary of edge speed predictions
Interface height h−
Theoretical predictions are good for small step-sizes, butbecome
poorer as R− → 1 (h− →≈ 0.35).
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Lock exchange: Resolution of initial data
Solutions can be constructed from ‘solibores’ and undularbores /
rarefaction waves.Correct causality: solibores connect to left and
right (notmid-)states.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Lock exchange: (x , t)-plane
Solibores propagate ahead of undular bore / rarefactioninto
undisturbed fluid.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Lock exchange: Resolution of initial data - II
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Lock exchange: Resolution of initial data - III
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Lock exchange: numerical results- I
Distance x
Initial conditions: (L−,R−) = (−0.97,0.99),Initial conditions:
(L+,R+) = (−0.97,0.98).Rightward propagating solibore followed by
undular bore.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Lock exchange: numerical results- II
Distance x
Predicted positions: dotted lines.Asymptotic solution takes
relatively long to emerge.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Lock exchange: numerical results- III
Distance x
A constant ‘image’ state emerges between solibore andundular
bore.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
The Gardner equation
16
−200 −100 0 100 200 300 400 5000
0.2
0.4
0.6
0.8
1Analytical Region 2
x
u
(a)
−200 −100 0 100 200 300 400 5000
0.2
0.4
0.6
0.8
1Numerical Region 2
x
u
(b)
FIG. 11: Decay of an initial discontinuity for the
Gardnerequation with α = 1. The step parameters are u− = 0.7,u+ =
0.1 (Region 2, {u− ← UB (u∗) SB → u+}). (a)analytical solution in
the form of a modulated periodic wave;(b) numerical solution; the
analytically found edges of theundular bore and solibore are shown
by dashed lines. Bothplots correspond to t = 300.
One can see from (114) that for u− = 1/2α the speed ofthe
solibore coincides with the speed of the leading soli-ton in the
undular bore so the “borderline” wave patternin Fig. 12b can be
interpreted in both ways: as a normal,“bright”, undular bore or,
equivalently, as a reversed,“dark”, undular bore with an attached
solibore. Whenwe increase u−, the solibore separates from the
undularbore, which in its turn acquires the reversed waveformwith a
distinct dark soliton structure near the leadingedge (see Fig.
12d).
Region 3, 1/α−u− < u+ < 12α < u−, u−+u+ > 1/α.{u− ←
RW (u∗) SB → u+}
This region is analogous to Region 2, since the valuesu− and u+
again lie in different domains of monotonic-ity of the function
w(u), thus a single-wave resolutionis not possible. However, now
the intermediate stateu∗ = 1/α − u+ < u+ (see Fig. 10b) so a
reversed rar-efaction wave is generated instead of reversed
undularbore. The solution for the rarefaction wave is given
byformula (80), where ul = u
− and ur = u∗. The soliboresolution connecting u∗ and u+ is the
same as in Region2. The numerical plot is presented in Fig. 13a
alongwith the analytically found positions of the solibore andthe
rarefaction wave boundaries. We note that, sincemaxu+(1 − αu+) =
14α , the speed of the solibore sk =
50 100 150 200 250 300 350 400 4500
0.2
0.4
0.6
0.8
1
x
u
(a)
50 100 150 200 250 300 350 400 4500
0.2
0.4
0.6
0.8
1
x
u
(b)
50 100 150 200 250 300 350 400 4500
0.2
0.4
0.6
0.8
1
x
u
(c)
50 100 150 200 250 300 350 400 4500
0.2
0.4
0.6
0.8
1
x
u
(d)
FIG. 12: The transformation of the undular bore structurefrom
the normal, “bright soliton”, pattern in Region 1 (plot1) to the
reversed, “dark soliton”, pattern in Region 2 (plot 4).Numerical
simulations of the Gardner equation with α = 1.The downstream
state, u+ = 0.1, is the same for all cases, theupstream state u− is
taken in the range u− = 0.49 < 1/2α(Region 1) to u− = 0.58 >
1/2α (Region 2)
1α +2u
+(1−αu+) is always greater than that of the rightedge of the
rarefaction wave, s+ = sr = 6u
+(1 − αu+)(see (81)). At the boundary between Regions 3 and
4,when u+ = 1/(2α), we have sk = sr and the solibore gets“attached”
to the right edge of the rarefaction wave.
Region 4, 1/(2α) ≤ u+ < u−. {u− ← RW u+}.A single reversed
rarefaction wave is produced. It is
Model equation:
ut +6uux−6αu2ux +uxxx = 0
Whitham equations areintegrable.
Solutions fully classified.
Kamchatnov et al.(Phys. Rev. E, 2012)
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges
-
IntroductionThe Boussinesq two-layer fluid
Dam breaks and lock exchanges
Whitham modulation theoryDam breaksLock exchanges
Conclusions
The MCC equations can be used to study dam-breaks
andlock-exchanges in a 2-layer fluid.Whitham-El method can be used
to ‘fit’ dispersive undularbores into 2-layer SWE solutions.A clear
distinction can be made between ‘dam-break’ and‘lock-exchange’
flows.All phenomenology seen in the Gardner equation
isobserved.
Gavin Esler and Joe Pearce Internal dam breaks and
lock-exchanges