Csanady (1998) presents solutions for time-dependent wind-driven flow in a barotropic coastal ocean. We disagree with two of his three boundary condition options and wish to clarify the origin of the non-wavelike aspect of the flow

Allen, J. S.

Brink, K. H.

English

Yale University

      P.O. BOX 208118 | NEW HAVEN CT 06520-8118 USA | PEABODY.YALE. EDU   JOURNAL OF MARINE RESEARCH The Journal of Marine Research, one of the oldest journals in American marine science, published important peer-reviewed original research on a broad array of topics in physical, biological, and chemical oceanography vital to the academic oceanographic community in the long and rich tradition of the Sears Foundation for Marine Research at Yale University.  An archive of all issues from 1937 to 2021 (Volume 1–79) are available through EliScholar,  a digital platform for scholarly publishing provided by Yale University Library at  https://elischolar.library.yale.edu/.  Requests for permission to clear rights for use of this content should be directed to the authors, their estates, or other representatives. The Journal of Marine Research has no contact information beyond the affiliations listed in the published articles. We ask that you provide attribution to the Journal of Marine Research.  Yale University provides access to these materials for educational and research purposes only. Copyright or other proprietary rights to content contained in this document may be held by individuals or entities other than, or in addition to, Yale University. You are solely responsible for determining the ownership of the copyright, and for obtaining permission for your intended use. Yale University makes no warranty that your distribution, reproduction, or other use of these materials will not infringe the rights of third parties.  This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. https://creativecommons.org/licenses/by-nc-sa/4.0/   Comments on ‘‘The non-wavelike response of acontinental shelf to wind’’ by G. T. Csanadyby K. H. Brink1 and J. S. Allen21. IntroductionCsanady (1998) presents solutions for time-dependent wind-driven  ow in a barotropiccoastal ocean. We disagree with two of his three boundary condition options and wish toclarify the origin of the non-wavelike aspect of the  ow.2. Offshore boundary conditionThere is nothing arbitrary about the appropriate choice of boundary condition at x 5 l(where the topography meets the  at-bottom deep ocean). This point was dealt with clearlyby Buchwald and Adams (1968), Gill and Schumann (1974), and Allen (1976b), forexample. The cross-shelf transport and the pressure must be continuous at x 5 l. Thisimplies that cross-shelf velocity must be continuous if depth h and wind stress G arecontinuous. Hence, from Csanady’s equation (1), both components of the pressuregradient, z x and z y must be continuous at x 5 l. This in turn implies that pressure and itsderivative normal to the boundary, z and z x must be continuous at x 5 l. Csanady’scondition 1 ( z de ned as zero offshore) allows z to be continuous, but not z x. His second(channel) condition causes pressure to be discontinuous at x 5 l, another unsatisfactoryproperty. His third condition is the traditional z x 5 0 at x 5 l, which allows both z and z x tobe continuous.Csanady (1998) advocates that pressure approach zero far from shore in the ‘‘boundarylayer’’ or ‘‘long wave’’ approximation.This is equivalent to taking limits—far offshore vs.long wave—in the wrong order, as we now demonstrate. Once the long-wave approxima-tion is made, Csanady’s (4) reduces toz xx 5 0 (1)when the bottom is  at, far offshore. Requiring that z be bounded far from shore yieldsz 5 z (y, t)z x 5 0 6 for x . l. (2)1. Woods Hole Oceanographic Institution,Woods Hole, Massachusetts, 02543, U.S.A.2. College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon, 97331-5503,U.S.A.Journal of Marine Research, 56, 789–792, 1998789Choosing z 5 0 for all x . l at this point makes it impossible, in general, for z x to becontinuous at x 5 l.On the other hand, if the long-wave approximation is not made  rst, then the equivalentof Csanady’s (4), for hx 5 0, becomes (in the absence of a wind stress curl)z xx 1 z yy 5 0 (3)or, for harmonic alongshore behavior with wavenumber k,z xx 2 k2 z 5 0. (4)In this case, the bounded solution indeed decays as x ® ` , but at a rate k 2 1. Matching to ashelf solution allows both z and z x to be continuous. The long-wave limit (1) is equivalentto letting k become in nitesimally small, so the offshore decay scale for pressure becomesin nitely large. This, then, requires that z offshore be independent of x in the long-wavelimit, so z x is zero at x 5 l, and z is generally nonzero at this location. Thus, a carefulconsideration of behavior far from shore leads to a nonzero solution offshore in the longwave limit.The outer shelf boundary condition for pressure can be derived quite systematically forthe wind-forced problem (by a derivation analogous to that in Allen (1976b)). Curiously,Csanady (1998) does not present a mathematical argument for de ning pressure to be zerooffshore. Instead, he argues on intuitivegrounds that, in a realistic ocean, coastal in uencesshould not be felt far from the shelf-slope topography. Thus, he comes to choose solutionsthat do not conserve interior mass transport because of imposed discontinuities in z or z x.Of course, in the real ocean, coastal-trappedwaves are not appreciably felt far offshore, butthe reason is that the inclusion of a realistic, continuous strati cation leads to decay of thedominant baroclinic component (offshore of the shelf-slope topography) on the relativelyshort scale of the internal Rossby radius of deformation.Stated succinctly,Csanady (1998) advocates the use of a non-mass-conserving boundarycondition as a means to compensate for incomplete model physics.As Csanady (1998) states, ‘‘the best boundary condition [at x 5 l] is no boundarycondition.’’ We agree. It is straightforward to solve Csanady’s shelf wave problemnumerically, without using a conditionat x 5 l, or assuming a priori that z x 5 0 offshore. Ifthe long-wave approximation is not made, then the inviscid governing equation (equivalentto Csanady’s (4)) becomes(h z xt)x 1 h z yyt 1 fhx z y 5 0. (5)Wave solutions are sought in the formz 5 z (x) exp [i( v t 1 ky)] (6)sov (hz x)x 2 v hk2 z 1 fhx z k 5 0. (7)790 Journal of Marine Research [56, 4This is solved subject toh(v z x 1 kz ) 5 0 at x 5 0, (8)andz x 1 * k * z 5 0 at x 5 L, (9)where L . l. This offshore condition derives from (3), and in fact allows z ® 0 as x ® ` .This problem (7–9) can readily be solved using resonance iteration on a  nite-differencegrid, using the code of Brink and Chapman (1987). We use the geometry:h 5 h01 a x x # l (10a)h 5 h0 1 a l x $ l (10b)where h0 ½ a l since the numerical code does not allow h 5 0 anywhere. We chose h0 50.5 m, a l 5 4000 m, f 5 10 2 4 s 2 1, l 5 100 km and L 5 2l. We use 300 grid points.The longwave limit corresponds to letting kl ® 0, so we solve the problem with k 5 10 2 6, 10 2 7, and10 2 8 m 2 1. The results, respectively, for the  rst mode are c1 5 2.65, 2.72, and 2.72 m/s.Results do not change, to this accuracy, when h0 is halved (0.25 m). Clearly, in the ‘‘longwave’’ or ‘‘boundary layer’’ limit, kl ® 0, c1 ® 2.72 m/s or c1 5 0.272 , thus vindicatingCsanady’s third (correct) x 5 l boundary condition and invalidating the other two.Given that there is only one correct choice of x 5 l boundary condition, Csanady’s(1998) Eq. (6) also needs correction toge0`h z xdx 5 e y`Gdy 2 ey`gh z ydy * x 5 ` (11a)although part of his (7),c ` 5 e y`(G/f )dy (11b)is correct.3. InterpretationThe ‘‘non-wavelike’’ aspect of Csanady (1998) was found in the solutions obtained byAllen (1976a) by expanding the long-wave wind-forced problem in terms of its free modes.This is, of course, a valid procedure since Huthnance (1975) demonstrated that thelong-wave modes represent a complete set.The governing wind-forced, undamped  rst order wave equations arecn2 1Fnt 2 Fny 5 bnG, (12)where Fn( y, t) is a modal amplitude function and bn is a coupling coefficient computedknowing across-shelf modes f n(x) and wind structure. If there is no alongshore variation in1998] 791Brink & Allen: Comments on Csanady (1998)the wind, thenFnt 5 cnbnG (13)for all modes. Following Allen (1976a) or by analogy with Csanady (1998), we nowchooseG 5 G0H( 2 y)H(t). (14)The total solution for pressure is thenp 5 G0 on5 0`bnf n(x)[H(t)H( 2 y 2 cnt)cnt 2 H( 2 y)H(y 1 cnt)y ]. (15)At y 5 2 Y, the response for a given mode is governed by (13), a two-dimensionalnon-wavelike behavior, until time t 5 cn2 1Y, when the information that an edge to theforcing exists reaches the site. More complicated geometries lead to the same result: that‘‘knowledge’’ of alongshore variations reach a site after a wave propagation time. Beforethat time, there is no way to tell that the system is not two dimensional. Further, in the rigidlid limit that is relevant here, the mode equivalent to the barotropic Kelvin wave has c0 5 `and f 0 5 constant. Thus, a spatially uniform, steady alongshore pressure gradient is set upinstantly. In both senses (short time or fast waves), the wave Eq. (12) yields solutions withnon-wavelike aspects.This physical behavior is difficult to observe in Csanady’s (1998) unbounded wedgeproblem because all coastal-trapped waves propagate with in nite celerity in that case: theknowledge of the edge is immediate and the adjustment is continuous.REFERENCESAllen, J. S. 1976a. Some aspects of the forced wave response of strati ed coastal regions. J. Phys.Oceanogr., 6, 113–119.—— 1976b. On forced, long continental shelf waves on an f-plane. J. Phys. Oceanogr., 6, 426–431.Brink, K. H. and D. C. Chapman. 1987. Programs for computing properties of coastal-trappedwavesand wind-driven motions over the continental shelf and slope—2nd ed., Woods Hole Oceano-graphic InstitutionTechnical Report WHOI-87-24, 119 1 iii pp.Buchwald, V. T. and J. K. Adams. 1968. The propagation of continental shelf waves. Proc. Roy. Soc.London,A 305, 235–250.Csanady, G. T. 1998. The non-wavelike response of a continental shelf to wind. J. Mar. Res., 56,773–788, (this issue).Gill, A. E. and E. H. Schumann. 1974. The generation of long shelf waves by the wind. J. Phys.Oceanogr., 4, 83–90.Huthnance, J. M. 1975. On trapped waves over a continental shelf. J. Fluid Mech., 69, 689–704.Received: 21 April, 1998.792 Journal of Marine Research [56, 4

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