A basic shallow water system with variable topography is analyzed from the point of view of a Lagrangian derivation of momentum, energy, and pseudomomentum balances. A two-dimensional action and associated momentum equation are derived. The latter is further manipulated to derive additional equations for energy and pseudomomentum. This revealed structure emphasizes broken symmetries in space and a reference configuration and preserved symmetry in time.
References
1.
H.
Singh
and J. A.
Hanna
, “Pseudomomentum: Origins and consequences
,” Z. Angew. Math. Phys.
72
, 122
(2021
);H.
Singh
and J. A.
Hanna
“Correction to: Pseudomomentum: Origins and consequences
,” ibid.
73
, 189
(2022
).2.
D. G.
Andrews
and M. E.
McIntyre
, “On wave-action and its relatives
,” J. Fluid Mech.
89
, 647
–664
(1978
).3.
T. G.
Shepherd
, “Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics
,” Adv. Geophys.
32
, 287
–338
(1990
).4.
G. K.
Vallis
, Atmospheric and Oceanic Fluid Dynamics
(Cambridge University Press, Cambridge
, 2017
).5.
O.
Bühler
, Waves and Mean Flows
(Cambridge University Press
, New York
, 2014
).6.
G. B.
Whitham
, “A general approach to linear and non-linear dispersive waves using a Lagrangian
,” J. Fluid Mech.
22
, 273
–283
(1965
).7.
G. B.
Whitham
, “Variational methods and applications to water waves
,” Proc. R. Soc. London, Ser. A
299
, 6
–25
(1967
).8.
J. C.
Luke
, “A variational principle for a fluid with a free surface
,” J. Fluid Mech.
27
, 395
–397
(1967
).9.
J. W.
Miles
, “On Hamilton's principle for surface waves
,” J. Fluid Mech.
83
, 153
–158
(1977
).10.
J.
Miles
and R.
Salmon
, “Weakly dispersive nonlinear gravity waves
,” J. Fluid Mech.
157
, 519
–531
(1985
).11.
P. J.
Dellar
and R.
Salmon
, “Shallow water equations with a complete Coriolis force and topography
,” Phys. Fluids
17
, 106601
(2005
).12.
R.
Camassa
, D. D.
Holm
, and C. D.
Levermore
, “Long-time effects of bottom topography in shallow water
,” Physica D
98
, 258
–286
(1996
).13.
D.
Clamond
and D.
Dutykh
, “Practical use of variational principles for modeling water waves
,” Physica D
241
, 25
–36
(2012
).14.
This form would have been obtained directly by a slightly different choice of the vertical coordinate , instead of as used here. This changes the limits of integration and makes the lateral coordinate lines in the reference configuration share the shallow curvilinearity of the topography instead of being flat.).
15.
M. E.
Gurtin
, E.
Fried
, and L.
Anand
, The Mechanics and Thermodynamics of Continua
(Cambridge University Press
, Cambridge
, 2010
).16.
A.
Golebiewska Herrmann
, “On conservation laws of continuum mechanics
,” Int. J. Solids Struct.
17
, 1
–9
(1981
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
