JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 93, NO. C12, PAGES 15,749-15,750, DECEMBER 15, 1988 Comment on "Tracking Fronts in Solutions of the Shallow-Water Equations" by A. F. Bennett and P. F. Cummins JAMES O'DONNELL Department of Marine Sciences, University of Connecticut,Groton RICHARD W. GARVINE College of Marine Studies, University of Delaware, Newark We agree with Bennett and Cummins [1988] (hereinafter referred to as B & C) that it is expedient and practical to treat the interior singularities that develop in solutions to the non- linear shallow water wave equations as interior-free bound- aries. Our purposes here are to remark on the use of the Lax-Wendroff class of numerical schemes, since we believe that B & C have overstated the undesirableproperties in their example; to point out some additional advantages we have found in applying "shock patching" or "front tracking" to problems of oceanographic relevance; to describe a class of problems in which shock patching is essential; and finally, to compare the scheme described by B & C and attributed to Chern et al. [1986] to earlier work of ours [O'Donnell, 1981; O'Donnell and Gar•,ine, 1983] which is in many respectssimi- lar. Though probably unfamiliar to most meteorologists and oceanographers, variants of the original Lax and Wendroff [1960] scheme have proliferated in the field of compressible gas dynamics, and several have found applications in atmo- spheric and oceanic problems. B & C have employed the two- step version, often attributed to Richtmyer [1963], and have demonstrated its well-known tendency to develop large- amplitude oscillations in the neighborhood of discontinuities in flow properties that are referred to as fronts in oceano- graphic contexts. They then show that the inclusion of integral constraints at the discontinuity removes the spurious oscil- lations with little computational penalty. The requirement of nonoscillatory solutions alone is not sufficient justification for the use of the front-tracking ap- proach, since many alternative and equally effective methods exist. In a study of the effect of rotation on the formation of nonlinear lee waves by flow over a mountain ridge, Houghton [1969] employed the original, single-step Lax-Wendroff method successfully, even though large jumps in flow proper- ties occurredthat were very similar to those shown in Figure 6 of B & C. Sinions [1978] employed the same scheme suc- cessfully to investigate the evolutions of a lake thermocline during the relaxation from a wind-driven upwelling event, de- monstrating that internal surgesof large amplitude are pro- duced. It was also applied by O'Donnell and Garvine [1983] in a study of the dynamics of buoyant discharges to explore the consequences of fluctuations in the discharge transport on flow in the plume. Again, large-amplitude internal jumps characterized the response, yet in none of theseexamples were Copyright 1988 by the American Geophysical Union. Paper number 88JC03643. 0148-0227/88/88JC-03643502.00 post-shock oscillations significant. Both Pratt [1983] and O'Donnell [1986] have applied the two-dimensional version of the scheme employed by B & C to problems in which flow discontinuities develop. In these studies a pseudo-viscosity term added to the finite difference equations effectively con- trolled the magnitude of the spurious oscillations.Recently, Roe [1986] reviewed advances in the development of more sophisticated nonoscillatory schemes for compressible gas computations that should also be considered for application to nonlinear shallow water wave problems, though they do not employ front patching. It may be argued that the front patching approach allows more efficient computation of smooth solutions, since the need for high spatial resolution is reduced, but the additional pro- gramming effort required to implement the technique is sig- nificant for one-dimensional problems and formidable in two- dimensional problems, which is the primary reason that shock capturing, or smearing, has eclipsed shock patching in com- pressible gas dynamics in the last decade. We encourage the further development and application of front patching, not out of a desire for nonoscillatory solutions, since there are many satisfactory schemes available, but be- causein oceanography and meteorology there are problems in which the inclusion of integral conditions at discontinuities allows a better representation of the physical processesand others for which there are no alternative numerical schemes available. It is clear from the theoretical work of Houghton [1969], Simons [1978], O'Donnell and Garvine [1983], and Garvine [1984] that there are physically realistic cases in which jumps in layer depth evolve from smooth initial and boundary con- ditions, and it is reasonable to expect that there should be increased interlayer transport of buoyancy and momentum at these singularities.Thus additional physical processes become important, and the shallow-water wave equations are no longer enough to fully model the physicsat the jump. Gatvine [1981] derived a useful formulation of integral conditions, in- cluding interlayer friction and entrainment, which may be em- ployed instead of the simple conditions stated by B & C [equations (4) and (6)]. It would be impractical to include such a submodel without shock patching. O'Donnell [1981] demonstrated the application of these conditions in the Rie- mann problem described by B & C and compared the solution to that obtained by the Lax-Wendroff schemeand the analytic solution of Gatvine [1981]. The shock-patching method was shown to perform significantly better than the ordinary Lax- Wendroff method, particularly in the representation of the shock speed(seeFigure 1). 15,749