PHYSICS OF FLUIDS 18, 036601 共2006兲
The effect of a background shear current on large amplitude internal
solitary waves
Wooyoung Choia兲
Department of Mathematical Sciences, Center for Applied Mathematics and Statistics,
New Jersey Institute of Technology, Newark, New Jersey 07102-1982
共Received 18 October 2005; accepted 24 January 2006; published online 14 March 2006兲
Large amplitude internal waves interacting with a linear shear current in a system of two layers of
different densities are studied using a set of nonlinear evolution equations derived under the long
wave approximation without the smallness assumption on the wave amplitude. For the case of
uniform vorticity, solitary wave solutions are obtained under the Boussinesq assumption for a small
density jump, and the explicit relationship between the wave speed and the wave amplitude is found.
It is shown that a linear shear current modifies not only the wave speed, but also the wave profile
drastically. For the case of negative vorticity, when compared with the irrotational case, a solitary
wave of depression traveling in the positive x direction is found to be smaller, wider, and slower,
while the opposite is true when traveling in the negative x direction. In particular, when the
amplitude of the solitary wave propagating in the negative x direction is greater than the critical
value, a stationary recirculating eddy appears at the wave crest. © 2006 American Institute of
Physics. 关DOI: 10.1063/1.2180291兴
I. INTRODUCTION
Large amplitude internal waves are no longer rare phenomena evidenced by an increasing number of field observations and the study of their physical properties has been an
active research area in recent years; for example, see Lynch
and Dahl,1 Ostrovsky and Stepayants,2 and Helfrich and
Melville.3 Classical weakly nonlinear theories including the
well-known Korteweg–de Vries 共KdV兲 equation have been
found to fail in this strongly nonlinear regime, and it has
been suggested that new theoretical approaches are necessary
to better describe large amplitude internal waves.
These strongly nonlinear waves have often been studied
numerically for both continuously stratified and multilayer
fluids 共Lamb;4 Grue et al.5兲. Alternatively, with the long
wave approximation but no assumption on the wave amplitude, the strongly nonlinear asymptotic models were proposed 共Miyata;6 Choi and Camassa;7 Choi and Camassa8兲,
and it was found that these simplified models are indeed
valid for large amplitude waves. In particular, for traveling
waves, the system of coupled nonlinear evolution equations
can be reduced to a single ordinary differential equation, and
its solitary wave solution shows excellent agreement with
laboratory experiments and numerical solutions of the Euler
equations 共Camassa et al.9兲. The success of this asymptotic
model even for large amplitude waves can be attributed to
the fact that the internal solitary wave becomes wider as the
wave amplitude increases and, therefore, the underlying long
wave assumption is valid even for large amplitude waves.
Many of these previous studies have neglected the background shear current effect, which is important in real ocean
environments but is not well understood. In this paper, the
strongly nonlinear model is further generalized to include the
a兲
Electronic mail: [email protected]
1070-6631/2006/18共3兲/036601/7/$23.00
shear current effect, and its solitary wave solutions and their
wave characteristics are studied.
For continuously stratified flows, Maslowe and
Redekopp10 considered weakly nonlinear internal waves in
shear flows and showed that the KdV and Benjamin-Ono
equations can be derived for long waves in the shallow and
deep configurations, respectively, both with and without
critical layers. Stastna and Lamb11 investigated numerically
the effect of shear currents on fully nonlinear internal waves.
For a linear shear current of negative constant vorticity, they
found the internal solitary wave of elevation traveling in the
positive x direction is taller and narrower, while the wave
propagating in the opposite direction becomes shorter and
wider. Their numerical solutions showed that the maximum
wave amplitude solution approaches the conjugate flow limit
共or a front solution兲, although the so-called wave breaking
and shear instability limits are also possible for some parameter ranges.
For two-layer system, Breyiannis et al.12 considered internal waves in a linear shear current numerically using a
boundary integral method. Since their focus was on surface
waves at the air-water interface, they added wind of constant
velocity in the upper layer, resulting in a velocity discontinuity at the interface. Therefore, along with the large density
jump that they considered, their results are not directly applicable to internal waves of our interest.
Here, we consider a system of two layers, each of which
has constant density and constant vorticity by assuming the
detailed vorticity distribution is not important for long internal waves. Solitary wave solutions are computed and compared with those of the irrotational model, and particular
attention is paid to wave profiles, wave speeds, and streamlines. This paper is organized as follows. With the governing
equations in Sec. II, the strongly nonlinear model is presented in Sec. III and is reduced to a single equation for
18, 036601-1
© 2006 American Institute of Physics
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036601-2
Phys. Fluids 18, 036601 共2006兲
Wooyoung Choi
rotational and, therefore, the perturbation velocities can be
written as ui = ␾x and wi = ␾z, where the velocity potential ␾i
satisfies the Laplace equation ⵜ2␾i = 0. By substituting into
the linearized system of 共2.2兲 and 共2.4兲,
␾1 = A1 cosh关k共z − h1兲兴ei共kx−␻t兲 ,
共2.7兲
␾2 = A2 cosh关k共z + h2兲兴ei共kx−␻t兲 ,
the linear dispersion relation between wave speed c and
wave number k can be obtained from
c2共␳1coth kh1 + ␳2coth kh2兲 − c共␳1⍀1 − ␳2⍀2兲/k
− g共␳2 − ␳1兲/k = 0.
FIG. 1. Internal waves in two-layer system interacting with linear shear
currents.
solitary waves. In Sec. IV, using the Boussinesq assumption
and assuming vorticity is uniform for simplicity, the properties of large amplitude internal solitary waves propagating in
both horizontal directions are described in detail with concluding remarks in Sec. V.
II. GOVERNING EQUATIONS
For an inviscid and incompressible fluid of density ␳i,
the velocity components in Cartesian coordinates 共u*i , w*i 兲
and the pressure p*i satisfy the continuity equation and the
Euler equations,
u*ix + wiz* = 0,
共2.1兲
u*it + u*i u*ix + w*i uiz* = − p*ix/␳i ,
共2.2兲
w*it + u*i w*ix + w*i wiz* = − piz* /␳i − g,
共2.3兲
where g is the gravitational acceleration and subscripts with
respect to space and time represent partial differentiation.
Here, i = 1 共i = 2兲 stands for the upper 共lower兲 fluid 共see Fig.
1兲 and ␳1 ⬍ ␳2 is assumed for a stable stratification.
The boundary conditions at the interface are the continuity of normal velocity and pressure,
␨t + u*1␨x = w*1,
␨t + u*2␨x = w*2,
p*1 = p*2
at z = ␨共x,t兲,
共2.4兲
where ␨ is the interface displacement. At the upper and lower
rigid surfaces, the kinematic boundary conditions are given
by
w*1共x,h1,t兲
= 0,
w*2共x,−
h2,t兲 = 0,
u*i 共x,y,z兲 = ⍀iz + ui共x,z,t兲,
u*i
w*i
w*i = wi共x,z,t兲,
For long waves 共khi → 0兲, the linear long wave speed is
found to be
c=
h 1h 2共 ␳ 1⍀ 1 − ␳ 2⍀ 2兲
2共␳1h2 + ␳2h1兲
±
冑
h21h22共␳1⍀1 − ␳2⍀2兲2 gh1h2共␳2 − ␳1兲
+
,
4共␳1h2 + ␳2h1兲2
␳ 1h 2 + ␳ 2h 1
共2.6兲
and
are the total horizontal and vertical velociwhere
ties, respectively, while ui and wi represent the perturbation
velocities. From conservation of vorticity, any twodimensional perturbation to uniform shear flows must be ir-
共2.9兲
which, for ⍀i = 0, can be reduced to
c= ±
冑
gh1h2共␳2 − ␳1兲
.
␳ 1h 2 + ␳ 2h 1
共2.10兲
III. A STRONGLY NONLINEAR MODEL
For long surface gravity waves in a single layer with
constant vorticity, a system of nonlinear evolution equations
was derived by Choi13 by adopting the similar asymptotic
expansion method used in Choi and Camassa.8 Since the
detailed derivation of the model can be found in Choi,13 it
will be omitted here.
Assuming the ratio of water depth to the characteristic
wavelength is small for long waves, the pressure in 共2.3兲 is
expanded to the second order in this small parameter to include the nonhydrostatic pressure effect, which, when combined with 共2.1兲 and 共2.2兲, results in the nonlinear evolution
equations for the lower layer,
␩2t − ⍀2h2␩2x + ⍀2␩2␩2x + 共␩2u2兲x = 0,
␩2 = h2 + ␨ ,
共3.1兲
共2.5兲
where h1 共h2兲 is the undisturbed thickness of the upper
共lower兲 fluid layer.
We assume that the total velocity can be decomposed
into a background uniform shear of constant vorticity ⍀i and
a perturbation so that
共2.8兲
u2t − ⍀2h2u2x + u2u2x + g␨x = − Px/␳2 +
冉
冊
1 2 3
␩G ,
␩2 3 2 2 x
共3.2兲
where P = p共x , ␨ , t兲 is the pressure at the interface providing
coupling with the upper layer, and the depth-averaged velocity u2 and the nonhydrostatic pressure effect G2 are defined
by
u2共x,t兲 =
1
␩2
冕
␨
u2共x,z,t兲dz,
共3.3兲
−h2
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036601-3
Phys. Fluids 18, 036601 共2006兲
The effect of a background shear current
G2共x,t兲 = u2xt − ⍀2h2u2xx + ⍀2␩2u2xx + u2u2xx − 共u2x兲2 .
共3.4兲
The kinematic equation 共3.1兲 is exact, while the dynamic
equation 共3.2兲 contains an error of O共⑀4兲, where ⑀ is defined
by the ratio of water depth to the characteristic wavelength.
The second term of the right-hand side of 共3.2兲 represents the
nonlinear dispersive effect of large amplitude internal waves.
For the upper layer, by simply replacing 共␨ , g , ⍀兲 by
共−␨ , −g , −⍀兲 and the subscript 2 by 1, we can obtain an
appropriate set of equations. Then, the complete set of equations for the four unknowns 共␨ , u1 , u2 , P兲 is, in dimensional
form, given by
␩it + Ui␩ix −
Ui
␩i␩ix + 共␩iui兲x = 0,
hi
共3.5兲
冋冉
1
1 ␩3i
uit + Uiuix + uiuix + g␨x = − Px +
u + Uiuixx
␳i
␩i 3 ixt
−
Ui
␩iuixx + uiuixx − u2ix
hi
冊册
,
x
共3.6兲
where Ui is the velocity at the rigid boundary given by
U 1 = ⍀ 1h 1,
U 2 = − ⍀ 2h 2 .
共3.7兲
For ⍀i = 0, the system given by 共3.5兲 and 共3.6兲 can be reduced to the system of equations of Choi and Camassa.8
When computing traveling wave solutions, it is useful to
write the horizontal momentum equation 共3.6兲 in the following conserved form:
冉
冊 冉
1
1
ui + ␩2i uixx + Uiui + u2i + g␨
6
2
t
1
= − Px +
␳i
冋冉
冊
冉 冊 册
␩3i
x
dt
d
dt
冕
␨dx = 0,
冕冉
␩ iu i −
冊
册
1 Ui 2
␩ dx = 0,
2 hi i
ui共x,t兲 = ui共X兲,
冉 冊
ui = 共c − Ui兲 1 −
共3.9兲
x
共3.10兲
共3.11兲
共3.12兲
X = x − ct.
共3.13兲
hi
␩i
+
冉 冊
h2
1 Ui
␩i 1 − i2 ,
2 hi
␩i
共3.14兲
where we have used the boundary conditions at infinity: ␩i
→ hi and ui → 0 as 兩X 兩 → ⬁. After substituting 共3.13兲 into
共3.8兲 and integrating with respect to X once, Eq. 共3.8兲 can be
written as
1 2
␳ i␩
3 i
冋冉
冋
− c + Ui + ui −
冊
3 2
Ui
␩i uiXX − uiX
2
hi
册
册
1
= ␳i 共− c + Ui兲ui + u2i + g␨ + P,
2
共3.15兲
where we have used uiX, uiXX → 0 as X → −⬁. When we subtract 共3.15兲 for i = 1 and 2 to eliminate P, we have
2
1
冋冉
− c + Ui + ui −
冋
冊
Ui
2
␩i uiXX − uiX
hi
册
册
共3.16兲
On the other hand, when we add 共3.9兲 for i = 1 , 2 and integrate once with respect to X, we have
1
␳i␩2i
兺
i=1 3
2
冋冉
− c + Ui + ui −
冋
冊
Ui
2
␩i uiXX − uiX
hi
册
U2
1
= 兺 ␳i 共− c + Ui + ui兲ui − 共− c + Ui兲 i ␩i
2
hi
i=1
+
From 共3.5兲, 共3.8兲, and 共3.9兲, it is easy to see that the following three conservation laws hold:
⬅
␳i
共i = 1,2兲,
Substituting 共3.13兲 into 共3.5兲 and integrating once with respect to X yields
2
␩3i
␩i
Ui
= − Px +
uixt + Uiuixx − ␩iuixx
3
␳i
hi
dM
i=1
␨共x,t兲 = ␨共X兲,
x
.
冋兺
2
2
1 U2i
+ uiuixx − u2ix
冊
1
ui + ␩2i uixx dx = 0
6
which physically correspond to conservation laws for mass,
irrotationality, and horizontal momentum of a perturbation to
the background linear shear current.
In order to find solitary wave solutions traveling with
constant speed c, we assume that
+ g ␩ i␨ x
冋冉
冊册
冕冉
1
= 兺 共− 1兲i−1␳i 共− c + Ui兲ui + u2i + g␨ .
2
i=1
共3.8兲
,
冊 冋
2
d
⬅
dt
dt
x
1 Ui 2
U
␩ + ␩iUiui + ␩iu2i −
␩ 2 − i ␩ 2u i
2 hi i t
2 hi i hi i
1 Ui
+
3 hi
dP
2
2 Ui
uixt + Uiuixx −
␩iuixx
2
3
3 hi
冊册
d
dt
␳i␩2i
兺
3
i=1
or, after multiplying 共3.6兲 by ␩i,
␩ iu i −
dt
⬅
␩2i
2
+ uiuixx − u2ix
3
冉
dI
册
1 U2i 3
␩ − g共− 1兲i−1共␩2i − h2i 兲 + hi P ,
3 h2i i
共3.17兲
where we have used ␩1 + ␩2 = h1 + h2 to obtain the last term in
the right-hand side. After using 共3.15兲 with i = 1 for P and
adding 共3.16兲 and 共3.17兲, the final equation for ␨X can be
found, after a lengthy manipulation, as
␨X2 =
3␨2共A1␨3 + A2␨2 + A3␨ + A4兲
,
B 1␨ 5 + B 2␨ 4 + B 3␨ 3 + B 4␨ 2 + B 5␨ + B 6
共3.18兲
where
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036601-4
Phys. Fluids 18, 036601 共2006兲
Wooyoung Choi
A1 = −
␳1 U21 ␳2 U22
+
,
12 h21 12 h22
B2 =
A2 = −
␳1 U21
␳2 U22
共h
−
4h
兲
−
共h1 − 4h2兲 + 共␳2 − ␳1兲g,
2
1
12 h21
12 h22
B3 = ␳1
冉
␳1 U21
␳2 U22
共h
−
4h
兲
+
共h1 − 4h2兲,
2
1
4 h21
4 h22
U21
h21
共h21 − h1h2兲 − ␳2
U22
h22
共h22 − h1h2兲
+ c共␳1U1 − ␳2U2兲,
冊
U2
U2
1
A3 = ␳1 21 − ␳2 22 h1h2 − c共␳1U1 − ␳2U2兲
3
h1
h2
B4 = ␳1
+ c 共␳1 − ␳2兲 + 共␳2 − ␳1兲g共h2 − h1兲,
2
U21
U22
h1
h22
2
2 h 1h 2 + ␳ 2
h1h22 + ␳1cU1共h2 − 2h1兲
+ ␳2cU2共h1 − 2h2兲,
A4 = − c共␳1U1h2 + ␳2U2h1兲 + c 共␳1h2 + ␳2h1兲
2
B5 = − 2ch1h2共␳1U1 − ␳2U2兲 + c2共␳1h21 − ␳2h22兲,
− 共␳2 − ␳1兲gh1h2 ,
B1 =
␨X2 =
␳1 U21
4 h21
−
B6 = c2h1h2共␳1h1 + ␳2h2兲.
␳2 U22
,
4 h22
When Ui = 0, 共3.18兲 becomes the equation for irrotational internal waves 共Miyata;14 Choi and Camassa8兲,
3␨2兵g共␳2 − ␳1兲␨2 + 共␳2 − ␳1兲关g共h2 − h1兲 − c2兴␨ + c2共␳1h2 + ␳2h1兲 − 共␳2 − ␳1兲gh1h2其
c2关共␳1h21 − ␳2h22兲␨ + 共␳1h1 + ␳2h2兲h1h2兴
.
共3.19兲
For the one-layer case 共␳1 = 0兲 共3.18兲 can be reduced to the model for surface waves in a uniform shear flow,13
␨X2 = −
␨2关␨2 + 共4 + 12gh22/U22兲␨ + 共12h22/U22兲共c2 − cU2 − gh2兲兴
,
关␨2 + 2h2␨ + 2共c/U2兲h22兴2
共3.20兲
where, from 共3.7兲, U2 = −⍀2h2.
IV. BOUSSINESQ ASSUMPTION WITH UNIFORM
VORTICITY
B5 = − 2␳0⍀ch1h2共h1 + h2兲 + ␳0c2共h21 − h22兲,
To further simplify Eq. 共3.18兲, we assume a small density change across the interface 共true for most oceanic applications兲 and use the Boussinesq assumption, so that we can
replace ␳1 = ␳0 − ⌬␳ / 2 and ␳2 = ␳0 + ⌬␳ / 2 by ␳0 except for
terms proportional to the gravitational acceleration g. Also,
for simplicity, uniform vorticity is assumed and, thus, ⍀1
= ⍀2 ⬅ ⍀. Then, the equation for ␨X given by 共3.18兲 can be
simplified to
B6 = ␳0c2h1h2共h1 + h2兲.
␨X2 =
3␨2共A2␨2 + A3␨ + A4兲
,
B 2␨ 4 + B 3␨ 3 + B 4␨ 2 + B 5␨ + B 6
where the coefficients are given by
共4.1兲
From the fact that the numerator has to vanish at the wave
crest where ␨ = a with a being the wave amplitude, the wave
speed c can be found, in terms of a, as
c=
⍀a
±
2
A4 = ␳0c2共h1 + h2兲 − ⌬␳gh1h2 ,
B2 = − 3␳0⍀2共h1 + h2兲/4,
B3 = ␳0⍀2共h21 − h22兲 + ␳0⍀c共h1 + h2兲,
B4 = ␳0⍀2h1h2共h1 + h2兲 − 2␳0⍀c共h21 − h22兲,
g共⌬␳/␳0兲共h1 − a兲共h2 + a兲
.
h1 + h2
共4.2兲
Notice that, compared with the irrotational 共⍀ = 0兲 case, the
nonlinear wave speed is increased by ⍀a / 2, while the linear
wave speed c0 given, from 共4.2兲 with a = 0, by
A2 = ␳0⍀2共h1 + h2兲/4 + ⌬␳g,
A3 = − ␳0⍀c共h1 + h2兲 + ⌬␳g共h2 − h1兲,
冑
c0 = ±
冑
g共⌬␳/␳0兲h1h2
,
h1 + h2
共4.3兲
is unaffected by constant vorticity under the Boussinesq
assumption.
As in the irrotational case, when A2a2 + A3a + A4 = 0 in the
numerator of 共4.1兲 has double roots, we have the maximum
wave amplitude, at which, instead of a solitary wave, a front
solution appears and beyond which no solitary wave solution
exists. This condition can be written as A23 = 4A2A4 to yield
the following expression for the maximum wave speed cm:
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036601-5
Phys. Fluids 18, 036601 共2006兲
The effect of a background shear current
FIG. 2. Maximum wave amplitude am given by 共4.5兲 for varying U1 / c0,
where U1 = ⍀h1 and c0 is the linear long internal wave speed defined by
共4.3兲: - - -, h2 / h1 = 2.5; —, h2 / h1 = 5. The density ratio is ␳2 / ␳1 = 1.001, and
the superscripts ⫹ and ⫺ indicate the waves propagating in the positive and
negative x directions, respectively.
±
cm
=
1
关⍀共h1 − h2兲
4
± 冑4g共⌬␳/␳0兲共h1 + h2兲 + ⍀2共h1 + h2兲2兴 .
共4.4兲
Then, the maximum wave amplitude am can be found, from
共4.2兲, as
±
am
=
±
2g共⌬␳/␳0兲共h1 − h2兲 + 2⍀共h1 + h2兲cm
.
2
4g共⌬␳/␳0兲 + ⍀ 共h1 + h2兲
共4.5兲
For ⍀ = 0, the maximum wave speed and the maximum wave
amplitude can be reduced to
±
= ±
cm
1
2
冑g共⌬␳/␳0兲共h1 + h2兲,
±
am
= 21 共h1 − h2兲,
8
共4.6兲
which are the results of Choi and Camassa under the Boussinesq assumption.
In this paper, without loss of generality, we assume ⍀
⬎ 0 and consider waves traveling in both positive and negative x directions. The density ratio ␳2 / ␳1 = 1.001 is chosen for
the results presented here, and h2 ⬎ h1 is assumed so that a
solitary wave of depression is expected, unless ⍀ is large.
As shown in Fig. 2, when propagating in the negative x
−
兲 increases condirection, the maximum wave amplitude 共am
FIG. 3. Wave speed c vs wave amplitude a given by 共4.2兲 for the case of
h2 / h1 = 5 and ␳2 / ␳1 = 1.001: —, U1 / c0 = 0; - - -, U1 / c0 = 0.3464 共or
⍀ / 冑g / h1 = 0.01兲, where U1 = ⍀h1 and c0 is the linear long internal wave
speed defined by 共4.3兲. Notice that 兩c+ 兩 = 兩c−兩 for ⍀ = 0. The density ratio is
␳2 / ␳1 = 1.001, and the superscripts ⫹ and ⫺ indicate the waves propagating
in the positive and negative x directions, respectively.
siderably as ⍀ 共or, equivalently, U1兲 increases, approaching
−h2. On the other hand, when propagating in the positive x
+
兲 decreases and
direction, the maximum wave amplitude 共am
its polarity changes as ⍀ increases. Even though the thickness of the lower layer is greater than that of the upper layer,
it is possible for a solitary wave of elevation to exist, for
example, when ⍀h1 / c0 ⬎ 0.8 for h2 / h1 = 5. Figure 2 also
shows that, for ⍀ ⫽ 0, the maximum wave amplitude can be
zero, implying that no solitary wave solution exists, even at a
depth ratio different from the critical ratio 共h2 / h1 = 1兲 for
⍀ = 0.
From Fig. 3, for a given depth ratio and ⍀h1 / c0, it can
be seen that the wave speed 共c−兲 is greater for the wave
traveling in the negative x direction compared with that of
the irrotational solitary wave of same amplitude, while the
opposite is true for the wave traveling in the positive x
direction.
As shown in Fig. 4共a兲, solitary wave profiles of a / h1
= −0.5 are obtained by solving 共4.1兲 numerically using MATHEMATICA, and it is found that the solitary wave traveling in
the positive x direction is wider than that of the solitary wave
FIG. 4. Solitary wave profiles for U1 / c0 = 0.3464 共⍀ / 冑g / h1 = 0.01, dashed line兲, where U1 = ⍀h1, compared with those for ⍀ = 0 共solid line兲 for ␳2 / ␳1
= 1.001 and h2 / h1 = 5. 共a兲 a / h1 = −0.5, 共b兲 a = 0.99am, where am / h1 ⬇ −0.9165 and − 3.0835 for waves traveling to the right and left, respectively.
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036601-6
Phys. Fluids 18, 036601 共2006兲
Wooyoung Choi
FIG. 5. Streamlines for a solitary wave propagating in the negative x direction, without and with a recirculating eddy, for U1 / c0 = 0.3464 共⍀ / 冑g / h1 = 0.01,
dashed line兲, ␳2 / ␳1 = 1.001, and h2 / h1 = 5: 共a兲 a = 0.7am and 共b兲 a = 0.99am, where am / h1 ⬇ −3.0835. Notice that the frame of reference moving with the solitary
wave is used and the vertical scale is exaggerated.
traveling in the negative x direction. Also, as can seen in Fig.
4共b兲, the wave profiles near the maximum wave amplitudes
are very different for two waves traveling in opposite directions. For example, the wave propagating in the negative x
direction is much taller and narrower, which is consistent
with the numerical solution of Stastna and Lamb11 for continuously stratified shear flows.
It is interesting to notice that, when a large amplitude
wave is traveling in the negative x direction, it is possible for
a recirculating eddy to appear at the wave crest. Figure 5
shows the streamlines for solitary waves of two different
wave amplitudes. A well-defined stationary recirculating
⌿共X,z兲 =
再
1
2
2 ⍀z
1
2
2 ⍀z
eddy can be observed for a / h1 = 0.99am, where am / h1
= −3.0835, as shown in Fig. 5共b兲, while it disappears for
a / h1 = 0.7am. In the absence of shear, the presence of a stationary recirculating eddy was observed previously in twolayer system with a constant Brunt-Väisälä frequency along
with a density jump across the interface by Voronovich.15
The criterion of existence of this recirculating eddy can
be found from the fact that a stagnation point where the
velocity vanishes should appear in the upper layer, more specifically, a 艋 z 艋 0. At the leading order, the total stream
function ⌿ can be approximated by
− cz + u1共X兲共z − h1兲
for
␨共X兲 艋 z 艋 h1 ,
− cz + u2共X兲共z + h2兲
for
− h2 艋 z 艋 ␨共X兲.
Then, the total horizontal velocity in the upper layer at the
origin 共X = 0兲 can be found, after substituting 共3.14兲 for u1, as
⳵⌿
= ⍀z − c− + u1共0兲
⳵z
1
= ⍀z − ⍀a + h1
2
冑
共4.7兲
z0 =
a h1
−
2 ⍀
冑
g共⌬␳/␳0兲共h2 + a兲
.
共h1 + h2兲共h1 − a兲
共4.9兲
For this stagnation point to lie in a 艋 z0 艋 0, the following
inequality between a and ⍀ must be satisfied:
g共⌬␳/␳0兲共h2 + a兲
,
共h1 + h2兲共h1 − a兲
共4.8兲
from which the vertical location of a stagnation point z0 can
be found as
⍀艌−
2h1
a
冑
g共⌬␳/␳0兲共h2 + a兲
,
共h1 + h2兲共h1 − a兲
共4.10兲
which is shown in Fig. 6.
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036601-7
Phys. Fluids 18, 036601 共2006兲
The effect of a background shear current
Here, a simple, uniform vorticity distribution is adopted
and a similar approach used here can be applied to a more
general vorticity distribution which might be approximated
by the stack of constant vorticity layers.
ACKNOWLEDGMENT
This work was supported in part by the U.S. Office of
Naval Research through Grant No. N00014-05-1-0273.
1
FIG. 6. Domain of existence of recirculating eddy for ␳2 / ␳1 = 1.001 and
h2 / h1 = 5. Solid line: the maximum wave amplitude given by 共4.5兲; dashed
line: Eq. 共4.10兲.
V. CONCLUDING REMARKS
We have studied large amplitude internal solitary wave
solutions in a linear shear current using the strongly nonlinear asymptotic model. It is found that, even for small background vorticity, the wave characteristics are significantly
different for waves traveling in opposite directions. The
background linear shear current changes not only the wave
speed, but also the wave shape drastically. Therefore, the
knowledge of background currents is essential for the
accurate interpretation of an increasing number of field
measurements.
Validity of the strongly nonlinear model in the absence
of shear has been examined by Ostrovsky and Grue16 and
Camassa et al.9 It was found that solitary wave solutions of
the model show excellent agreement with Euler solutions
and laboratory/field experiments in shallow configuration in
which the depth ratio h2 / h1 is less than roughly 10. The
model proposed here is therefore expected to be valid for the
same depth ratio range. As the thickness of one layer is much
greater than the other, a deep configuration model similar to
that in Choi and Camassa8 should be adopted, although its
validity might be limited to intermediate wave amplitudes
for a fixed depth ratio.9
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