An analytical solution of the shallow water equations

Volume 123, number 6

PHYSICS LETTERS A

17 August 1987

AN ANALYTICAL SOLUTION OF THE SHALLOW WATER EQUATIONS E.A. NOVIKOV Institute fo r Nonlinear Science (INLS), University of California, San Diego, La Jolla, CA 92093, USA Received 14 January 1987; revised manuscript received 29 April 1987; accepted for publication 8 June 1987 Communicated by D.D. Holm

The general analytical solution of one-dimensional shallow water equations is presented. The solution corresponds to the propagation of spherical waves in four-dimensional space. The problem of two-dimensional rotating shallow water is discussed.

1. The classical system of one-dimensional shallow water equations has the form [ 1]:

ur + uux +ghx=O,

(la)

h, +uh.,.+hux=O,

(lb)

where u(t, x) is the velocity, h(t, x) is the depth of the layer over tile horizontal boundary, g is the acceleration of gravity; the suffixes in (1) correspond to the differentiat!on over time t and position x. We will use a m o r e symmetric form of the system

(1): u,+uux+½wwx=O,

wt+uw~+½wu~=O,

(2)

where w= 2(gh ) t/z. Let us introduce the "hodograph" transformation [ 1 ]. Consider (t, x) as functions of (u, w). Thejacobian should be non-zero:

O-:--u,w,.-wtux= ½w(u ~ - w~)-O, consequently, w-=0,

and system (2) takes the form of linear equations

x,-utu+½wtw=O,

xw-utw+½wtu=O.

(3)

By cross-differentiation we eliminate x from (3) and get

tw~ + 3w -~ t~-tuu=O.

(4)

The first two terms in (4) correspond to the spherically-symmetric four-dimensional Laplace operator, and eq. (4) describes the propagation of spherical waves in four-dimensional space. Having the solution of (4), we can get x(u, w) from (3). If instead of coefficient 3 in (4) we have an even integer, then the solution can be represented as a finite series [ 1,2 ]. In the case of an odd coefficient, such representation gives only the polynomial solution which, as well as other solutions presented in refs. [ 1-5], correspond to particular initial conditions. In the special case of eq. (4), we shall use a different procedure. The Fourier-transformation

u+ w~const

[from (2) it follows that u+ w cannot be a function of only t]. The case u+ w=const corresponds to the well-known solution, based on the Riemann invariants [ I ]. We have the relations

ut =Dx,,, u~ = -Dtw,

wt = -Dxu,

t(u, w)= ~ eiguz(k, w) dk,

(5)

--oo

being applied to (4), gives Z~,~+ 3 W - l Z w + k 2 z = O ,

w2¢w,~+wOw+O(k2w2-1)=O,

wx =Dt,

0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

¢=wr.

(6)

The solutions of eq. (6) are the Bessel functions 287

Volume 123, number 6

PHYSICS LETTERSA

of the first order [ 6]. Thus, we have the general solution

O(k, w)=A(k)Zl')(kw) +B(k)Zl2)(kw),

(7)

where A and B are the arbitrary functions, and Z/~> and Z/2~ are the two independent Bessel functions of the first order (for example, we can choose ZI~) =J~, Z~ 2) = Y~). Finally, from ( 5 ) - ( 7 ) we have

t(u, w)=w -) ~f eiX"[A(k)Z~))(kw) +B(k)ZlZ)(kw)] dk.

(8)

The detailed analysis of the particular cases of the general solution (8), which may contain shocks, will be presented elsewhere. We plan to investigate the shallow water "turbulence" with random A(k) and B(k), taking into account the inviscid dissipation of energy in shocks. 2. The analytical investigation of two-dimensional shallow water equations is much more complicated, and we will discuss here only the first simple steps. Let us consider the quasi-one-dimensional case, when the scale L in the direction of the second coordinate y is much larger than the x-scale 1. In the first approximation, we can drop the y-derivatives. The system (1) does not change, and additionally we have for the second component of velocity v, + uv,. =0,

v(t,x)=vo(a).

(,(t, a)=u(t, ~), ~(0, a) =a, where u(t, x) is the solution of the system (1). In the presence of rotation [7] with Coriolis parameter f, we get on the right side of eq. (9) the additional term -fu, and the solution takes the form

288

For the evolution of vorticity we have

t2(t, x) =vx(t, x) = [t20(a) + f l a x - f where t2o(x) is the initial vorticity. The additional term fv on the right-hand side of eq. ( l a ) is small, if ot ~ f ~ - 2 << 1, where u, v are the characteristic values of velocities. Having two small parameters o~ and fl =--vu-ilL-~, we can play with approximations around the general solution of the system (1). I f these parameters are not small, the system is generally not integrable and exhibits chaos (simply because it contains, as a particular case, the quasigeostrophic two-dimensional hydrodynamics). The topography of the bottom also may cause chaos even in the one-dimensional case. I wish to thank H. Abarbanel and J. McWilliams for interesting discussions and the anonymous referee for useful remarks. I am grateful to M. Farnsworth, H. Hamilton and P. Velasquez for help in preparing the manuscript. A portion of this work was done during a stay at the National Center for Atmospheric Research, which is sponsored by the National Science Foundation. My visit to the Institute for Nonlinear Science is supported through the Office of Naval Research DARPA University Research Initiative program under grant number N00014-86-K0758.

(9)

Here Vo(X) is the initial velocity, and a(t, x) is the initial position of the fluid particle, which at moment t has the position x. For the trajectory of the particle we have

v(t, x) = vo(a) - (x-a)f.

17 August 1987

References [ 1] G.B. Whitham, Linear and nonlinear waves (Wiley, New York, 1974). [2 ] R. von Mises. Mathematicaltheory of compressiblefluid flow (Academic Press, New York, 1958). [ 3] R. Courant and K.O. Fridrichs, Supersonic flow and shock waves (Interscience,New York, 1948). [4] E.T. Capson, Proc. R. Soc. A 216 (1953) 539. [5] A.G. Mackie, J. Rational Mech. Anal. 4 (1955) 733. [6 ] M. Abramowitzand J.A. Stegun, Handbook of mathematical functions (Dover, New York, 1972). [ 7] J. Pedlosky, Geophysicalfluid dynamics (Springer, Berlin, 1979).