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Conservation laws associated with long surface waves
Journal of Computational and Applied Mathematics 190 (2006) 136 – 141 www.elsevier.com/locate/cam
Conservation laws associated with long surface waves D.J. Benney Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 24 September 2004 This paper is dedicated to Professor Roderick Wong on the occasion of his 60th birthday
Abstract Long waves at the free surface of an inviscid fluid are known to have many interesting properties. In this review the existence or nonexistence of conservation laws is explored. While it is known that there are an infinite number of moment type conservation laws in two spatial dimensions, this does not appear to be true in three and higher dimensions. © 2005 Elsevier B.V. All rights reserved.
1. Two-dimensional analysis Consider the classical problem of time-dependent motion on an inviscid fluid of constant density under the action of gravity g. In two dimensions let y = 0 be the rigid bottom and y = h(x, t) the free surface. The two velocity components satisfy the Euler equations, namely, ju jx ju jt jv jt
+
jv jy
+u +u
= 0,
ju jx jv jx
+v +v
(1.1) ju jy jv jy
=− =−
1 jp
,
(1.2)
1 jp − g. jy
(1.3)
jx
E-mail address: [email protected]. 0377-0427/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2005.04.011
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
137
The appropriate boundary conditions are that v = 0,
y = 0,
p = p0 , jh jt
+u
(1.4)
y = h,
jh jx
(1.5)
− v = 0,
y = h,
(1.6)
where p0 is the constant atmospheric pressure. At this stage several possible theoretical developments arise depending on the relative magnitude of the wave amplitude, wavelength and vertical extent. In this study interest is focused on nonlinear long wave theory (sometimes referred to as nonlinear shallow water theory), and it is assumed that the ratio of vertical to horizontal scales is a small parameter, but the wave amplitude is arbitrary. It is well known that the consequence of these assumptions being applied to the preceding problem is the following simplified system: ju jx ju jt
+
jv jy
+u
= 0,
ju jx
(1.7) ju
+v
jy
= −g
jh
(1.8)
jx
subject to the boundary conditions v = 0, jh jt
+u
y = 0, jh jx
(1.9)
− v = 0,
y = h(x, t).
(1.10)
This new initial value problem remains difficult, save in the very special (classical) case where u = u(x, t),
v = −y
ju
(1.11)
jx
in which case the following hyperbolic system for u and h is obtained: ju jt jh jt
+u +u
ju jx jh jx
+g +h
jh jx ju jx
= 0,
(1.12)
= 0.
(1.13)
These equations predict wave breaking so that the theory eventually fails, unless some dispersion or dissipation is included. For the more general case where u(x, y, 0) and h(x, 0) are prescribed, the flow may be locally unstable and possible motion becomes much more complex. However, the general two-dimensional system does have some interesting properties. These are seen if Eq. (1.8) is multiplied by un−1 and integrated with respect to y from y = 0 to y = h. The resulting equation is jAn jt
+
jAn+1 jx
+ gnAn−1
jA0 jx
= 0,
n 1,
(1.14)
138
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
where moments Ar are defined by
h
Ar (x, t) =
ur (x, y, t) dy,
r 0.
(1.15)
0
Laws corresponding to conservation of mass, momentum and energy are readily found to be jA0 jt
+
jA1
= 0,
jx
(1.16)
1 2 + A2 + gA0 = 0, jt jx 2
jA1
j jt
j
(A2 + gA20 ) +
j jx
(1.17)
(A3 + 2gA0 A1 ) = 0
(1.18)
and as has been shown by Benney [1] there are an infinity of such moment conservation laws. Additionally Miura [2] has proved that there exist an infinity of local conservation laws. In dealing with the moment conservation laws it is convenient to use either of the generalizing functions F or G where F (x, t; ) =
∞
n
0
n=0
G(x, t; ) =
∞ An n n=0
h
An =
h
=
n!
dy , 1 − u
(1.19)
eu dy.
(1.20)
0
The relevant equations for F and G are found to be
jF jt
jG jt
+
+
jF jx j2 G
jjx
2 j
= 1 − g
+ g G
jG jx
j
(F )
jF jx
(x, t; 0),
(x, t; 0) = 0.
(1.21)
(1.22)
2. An analogous mathematical system A slight mathematical extension for the sequential moments considered earlier is the infinite system jCn jt
+ an
jCn+1 jx
+ gbn Cn−1
jC0 jx
= 0,
n 0,
(2.1)
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
139
where an and bn are prescribed functions of n. The question of concern is whether this more general system has an infinity of conservation laws. With this purpose in mind, it is instructive to attempt to derive some of the low order laws. Not unexpectedly the first three laws are jC0 jt
+ a0
jC1 jx
= 0,
gb1 2 + a1 C2 + C = 0, jt jx 2 0 j j gb2 2 C2 + C0 + (a2 C3 + gb2 C0 C1 ) = 0. jt 2a0 jx jC1
j
(2.2)
Less obvious is the fact that two further conservation laws exist. These are gb3 ga 0 b3 2 g 2 b1 b3 3 j j C0 C2 + C + C0 = 0, C3 + a3 C4 + gb3 C0 C2 + jt a1 jx 2a1 1 3a1 2 j gb gb a0 g 2 b4 g b2 b4 C4 + 4 C0 C2 + 4 C12 + C03 + jt a2 2a1 a2 6a0 a2 2a1 a2 2 j gb4 a0 g b2 b4 g 2 a0 b4 2 + a4 C5 + gb4 C0 C3 + C1 C0 = 0. C1 C2 + + jx a2 2a2 2a1 a2
(2.3) (2.4)
(2.5)
(2.6)
However, without constraints on the constants an and bn , no further laws of moment type appear to be possible. For example, an attempt for a sixth law yields j gb ga b5 g 2 b2 b5 2 C0 C1 C5 + 5 C0 C3 + 0 C1 C2 − jt a a2 a3 a1 a3 3 j ga b5 ga a1 b5 2 + a5 C6 + gb5 C0 C4 + 0 C1 C3 + 0 C jx a3 2a2 a3 2 g 3 a0 b12 b5 4 jC1 g 2 a0 b2 g 2 b5 a0 b1 2 2 = 0, (2.7) b3 + C0 C2 + + C0 C 1 + C + K6 C02 a2 a3 2a3 a2 jt 8a1 a2 a3 0 where
g 2 b5 a0 b1 b2 b3 . − + K6 = a3 2a1 a2 a2 2a1
(2.8)
Unless K6 = 0 the system has only five conservation laws. Even if K6 = 0, additional constraints are to be anticipated at higher moment levels in order to preserve conservation properties. Note that for the An defined by Eq. (1.19), an = 1 and bn = n while for those defined by (1.20), an = n + 1 and bn = 1. In each case K6 = 0. Indeed, it is a simple matter to show that the condition an bn+1 = constant n+1 is sufficient to ensure an infinity of conservation laws.
(2.9)
140
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
3. Three-dimensional analysis Analogous mathematical issues arise in higher dimensions. More complicated physical mechanisms might be expected to limit conservation properties. Indeed, this seems to be the case. In three dimensions, with z as the second horizontal coordinate, the nonlinear long wave equations are ju
+
jx ju
jv
+u
jt jw
ju jx
+u
jt
jw
+
jy
= 0,
jz
+v
jw
ju
+w
jy
+v
jx
(3.1)
jw
ju jz
+w
jy
= −g
jw jz
jh jx
= −g
,
(3.2)
jh
(3.3)
jz
with boundary conditions v = 0, jh jt
y = 0,
+u
jh jx
+w
(3.4) jh jz
− v = 0,
y = h(x, z, t).
(3.5)
Using the moments Ar,s defined by h Ar,s = ur ws dy
(3.6)
0
leads to the evolution equations jAr,s jt
jAr+1,s
+
jx
jAr,s+1
+
jz
+ grAr−1,s
jA0,0 jx
+ gsAr,s−1
jA0,0 jz
= 0.
(3.7)
Once again mass, momentum (z) and energy conservation give the four laws jA0,0 jt jA1,0 jt jA0,1 jt j jt
+ +
+
jA1,0 jx j
+
jx
jz
A2,0 +
jA1,1 jx
jA0,1
+
j jz
= 0,
gA20,0
j jz
+
2 A0,2 +
(A2,0 + A0,2 + gA20,0 ) + +
(3.8)
j jx
jA1,1 jz
gA20,0 2
= 0,
(3.9)
= 0,
(3.10)
(A3,0 + A1,2 + 2gA1,0 A0,0 )
(A0,3 + A2,1 + 2gA0,1 A0,0 ) = 0.
(3.11)
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
141
However, attempts by the author to find other moment conservation laws lead him to believe that there are no more. References [1] D.J. Benney, Stud. Appl. Math. 52 (1973) 45. [2] R.M. Miura, Stud. Appl. Math. 53 (1974) 45.
Conservation laws associated with long surface waves D.J. Benney Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 24 September 2004 This paper is dedicated to Professor Roderick Wong on the occasion of his 60th birthday
Abstract Long waves at the free surface of an inviscid fluid are known to have many interesting properties. In this review the existence or nonexistence of conservation laws is explored. While it is known that there are an infinite number of moment type conservation laws in two spatial dimensions, this does not appear to be true in three and higher dimensions. © 2005 Elsevier B.V. All rights reserved.
1. Two-dimensional analysis Consider the classical problem of time-dependent motion on an inviscid fluid of constant density under the action of gravity g. In two dimensions let y = 0 be the rigid bottom and y = h(x, t) the free surface. The two velocity components satisfy the Euler equations, namely, ju jx ju jt jv jt
+
jv jy
+u +u
= 0,
ju jx jv jx
+v +v
(1.1) ju jy jv jy
=− =−
1 jp
,
(1.2)
1 jp − g. jy
(1.3)
jx
E-mail address: [email protected]. 0377-0427/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2005.04.011
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
137
The appropriate boundary conditions are that v = 0,
y = 0,
p = p0 , jh jt
+u
(1.4)
y = h,
jh jx
(1.5)
− v = 0,
y = h,
(1.6)
where p0 is the constant atmospheric pressure. At this stage several possible theoretical developments arise depending on the relative magnitude of the wave amplitude, wavelength and vertical extent. In this study interest is focused on nonlinear long wave theory (sometimes referred to as nonlinear shallow water theory), and it is assumed that the ratio of vertical to horizontal scales is a small parameter, but the wave amplitude is arbitrary. It is well known that the consequence of these assumptions being applied to the preceding problem is the following simplified system: ju jx ju jt
+
jv jy
+u
= 0,
ju jx
(1.7) ju
+v
jy
= −g
jh
(1.8)
jx
subject to the boundary conditions v = 0, jh jt
+u
y = 0, jh jx
(1.9)
− v = 0,
y = h(x, t).
(1.10)
This new initial value problem remains difficult, save in the very special (classical) case where u = u(x, t),
v = −y
ju
(1.11)
jx
in which case the following hyperbolic system for u and h is obtained: ju jt jh jt
+u +u
ju jx jh jx
+g +h
jh jx ju jx
= 0,
(1.12)
= 0.
(1.13)
These equations predict wave breaking so that the theory eventually fails, unless some dispersion or dissipation is included. For the more general case where u(x, y, 0) and h(x, 0) are prescribed, the flow may be locally unstable and possible motion becomes much more complex. However, the general two-dimensional system does have some interesting properties. These are seen if Eq. (1.8) is multiplied by un−1 and integrated with respect to y from y = 0 to y = h. The resulting equation is jAn jt
+
jAn+1 jx
+ gnAn−1
jA0 jx
= 0,
n 1,
(1.14)
138
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
where moments Ar are defined by
h
Ar (x, t) =
ur (x, y, t) dy,
r 0.
(1.15)
0
Laws corresponding to conservation of mass, momentum and energy are readily found to be jA0 jt
+
jA1
= 0,
jx
(1.16)
1 2 + A2 + gA0 = 0, jt jx 2
jA1
j jt
j
(A2 + gA20 ) +
j jx
(1.17)
(A3 + 2gA0 A1 ) = 0
(1.18)
and as has been shown by Benney [1] there are an infinity of such moment conservation laws. Additionally Miura [2] has proved that there exist an infinity of local conservation laws. In dealing with the moment conservation laws it is convenient to use either of the generalizing functions F or G where F (x, t; ) =
∞
n
0
n=0
G(x, t; ) =
∞ An n n=0
h
An =
h
=
n!
dy , 1 − u
(1.19)
eu dy.
(1.20)
0
The relevant equations for F and G are found to be
jF jt
jG jt
+
+
jF jx j2 G
jjx
2 j
= 1 − g
+ g G
jG jx
j
(F )
jF jx
(x, t; 0),
(x, t; 0) = 0.
(1.21)
(1.22)
2. An analogous mathematical system A slight mathematical extension for the sequential moments considered earlier is the infinite system jCn jt
+ an
jCn+1 jx
+ gbn Cn−1
jC0 jx
= 0,
n 0,
(2.1)
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
139
where an and bn are prescribed functions of n. The question of concern is whether this more general system has an infinity of conservation laws. With this purpose in mind, it is instructive to attempt to derive some of the low order laws. Not unexpectedly the first three laws are jC0 jt
+ a0
jC1 jx
= 0,
gb1 2 + a1 C2 + C = 0, jt jx 2 0 j j gb2 2 C2 + C0 + (a2 C3 + gb2 C0 C1 ) = 0. jt 2a0 jx jC1
j
(2.2)
Less obvious is the fact that two further conservation laws exist. These are gb3 ga 0 b3 2 g 2 b1 b3 3 j j C0 C2 + C + C0 = 0, C3 + a3 C4 + gb3 C0 C2 + jt a1 jx 2a1 1 3a1 2 j gb gb a0 g 2 b4 g b2 b4 C4 + 4 C0 C2 + 4 C12 + C03 + jt a2 2a1 a2 6a0 a2 2a1 a2 2 j gb4 a0 g b2 b4 g 2 a0 b4 2 + a4 C5 + gb4 C0 C3 + C1 C0 = 0. C1 C2 + + jx a2 2a2 2a1 a2
(2.3) (2.4)
(2.5)
(2.6)
However, without constraints on the constants an and bn , no further laws of moment type appear to be possible. For example, an attempt for a sixth law yields j gb ga b5 g 2 b2 b5 2 C0 C1 C5 + 5 C0 C3 + 0 C1 C2 − jt a a2 a3 a1 a3 3 j ga b5 ga a1 b5 2 + a5 C6 + gb5 C0 C4 + 0 C1 C3 + 0 C jx a3 2a2 a3 2 g 3 a0 b12 b5 4 jC1 g 2 a0 b2 g 2 b5 a0 b1 2 2 = 0, (2.7) b3 + C0 C2 + + C0 C 1 + C + K6 C02 a2 a3 2a3 a2 jt 8a1 a2 a3 0 where
g 2 b5 a0 b1 b2 b3 . − + K6 = a3 2a1 a2 a2 2a1
(2.8)
Unless K6 = 0 the system has only five conservation laws. Even if K6 = 0, additional constraints are to be anticipated at higher moment levels in order to preserve conservation properties. Note that for the An defined by Eq. (1.19), an = 1 and bn = n while for those defined by (1.20), an = n + 1 and bn = 1. In each case K6 = 0. Indeed, it is a simple matter to show that the condition an bn+1 = constant n+1 is sufficient to ensure an infinity of conservation laws.
(2.9)
140
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
3. Three-dimensional analysis Analogous mathematical issues arise in higher dimensions. More complicated physical mechanisms might be expected to limit conservation properties. Indeed, this seems to be the case. In three dimensions, with z as the second horizontal coordinate, the nonlinear long wave equations are ju
+
jx ju
jv
+u
jt jw
ju jx
+u
jt
jw
+
jy
= 0,
jz
+v
jw
ju
+w
jy
+v
jx
(3.1)
jw
ju jz
+w
jy
= −g
jw jz
jh jx
= −g
,
(3.2)
jh
(3.3)
jz
with boundary conditions v = 0, jh jt
y = 0,
+u
jh jx
+w
(3.4) jh jz
− v = 0,
y = h(x, z, t).
(3.5)
Using the moments Ar,s defined by h Ar,s = ur ws dy
(3.6)
0
leads to the evolution equations jAr,s jt
jAr+1,s
+
jx
jAr,s+1
+
jz
+ grAr−1,s
jA0,0 jx
+ gsAr,s−1
jA0,0 jz
= 0.
(3.7)
Once again mass, momentum (z) and energy conservation give the four laws jA0,0 jt jA1,0 jt jA0,1 jt j jt
+ +
+
jA1,0 jx j
+
jx
jz
A2,0 +
jA1,1 jx
jA0,1
+
j jz
= 0,
gA20,0
j jz
+
2 A0,2 +
(A2,0 + A0,2 + gA20,0 ) + +
(3.8)
j jx
jA1,1 jz
gA20,0 2
= 0,
(3.9)
= 0,
(3.10)
(A3,0 + A1,2 + 2gA1,0 A0,0 )
(A0,3 + A2,1 + 2gA0,1 A0,0 ) = 0.
(3.11)
D.J. Benney / Journal of Computational and Applied Mathematics 190 (2006) 136 – 141
141
However, attempts by the author to find other moment conservation laws lead him to believe that there are no more. References [1] D.J. Benney, Stud. Appl. Math. 52 (1973) 45. [2] R.M. Miura, Stud. Appl. Math. 53 (1974) 45.