The Goddard Coastal Wave Model. Part II: Kinematics'' *
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JUNE 1998 NOTES AND CORRESPONDENCE 1305 Comments on ``The Goddard Coastal Wave Model. Part II: Kinematics'' * HENDRIK L. TOLMAN1 Ocean Modeling Branch, Environmental Modeling Center, NOAA/NCEP, Camp Springs, Maryland WAYNE L. NEU Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, Virginia LESLIE C. BENDER Deparment of Oceanography, Texas A&M University, College Station, Texas 2 December 1996 and 18 July 1997 1. Introduction etc. are based on linear wave kinematics. Lin and Huang (1996b) aim to improve upon this approach by consid- Lin and Huang (1996b, hereafter denoted as LH) dis- ering nonlinear kinematics in the derivation of the char- cuss kinematic aspects of the new Goddard Coastal acteristic velocities. As illustrated by Willebrand (1975), Wave Model (GCWAM). Their model is compared to such adaptations to Eq. (1) indeed might be important the well-known WAM model (WAMDI Group 1988; for shallow water, although they are generally irrelevant Komen et al. 1994) and to the WAVEWATCH model for deep water. We therefore applaud the attempt by LH. (Tolman 1991b, 1992) using theoretical considerations Unfortunately, we disagree with LH on three major and test cases. The basic equation of GCWAM is given points. First, the nonlinear group velocity [LH Eq. (6)] 1 as [LH, Eq. (11)] is based on a monochromatic instead of spectral wave concept, and therefore is critically dependent on sub- ]A ]cAgl 1 ]cAgf cosf ]cAuv]cA 11 115S, jective assumptions. Second, the characteristic veloci- ]t ]l cosf ]f ]u ]v ties c and c [LH Eqs. (17) and (18)] include new terms (1) u v that are attributed to nonlinear kinematics, but that ap- which is identical to the governing equation of WAVE- pear to be inconsistent with conventional nonlinear ex- WATCH2 and represents an extension of the governing pansion and with elementary physical properties of equation of WAM. The left side of this equation rep- waves. Third, previous wave models are seriously mis- resents the effects of wave propagation as dictated by represented, and appropriate comparisons with previous the dispersion relation and is inherently linear. The right models are not presented in the test cases. We will dis- side represents source and sink functions, including sev- cuss these three points in the following sections and eral effects of nonlinear wave propagation. Focusing on provide conclusions. kinematic aspects of wave propagation, LH assume S [ 0 throughout the paper. 2. Group velocity In WAM and WAVEWATCH purely linear propa- In their Eq. (6), LH present their nonlinear group gation is considered, and the characteristic velocities cg, velocity cg. This equation is based on a monochromatic nonlinear dispersion relation [LH Eq. (4), taken from Whitham (1974)]. Application to inherently random * Ocean Modeling Branch Contribution Number 144. wind waves requires subjective assumptions (as will be 1 UCAR visiting scientist. discussed below). It appears more appropriate to start 1 The notation of LH is adopted throughout this discussion. with a spectral model for the wave ®eld and allow for 2 The equation for WAVEWATCH is originally expressed for a nonlinear interactions between all spectral components. plane grid, but this longitude±latitude version is available in the mod- el. Willebrand (1975) shows that such an approach not only leads to conventional resonant interactions3 but also Corresponding author address: Dr. Hendrik L. Tolman, Ocean Modeling Branch, Environmental Modeling Center, NOAA/NCEP, 5200 Auth Road, Room 209, Camp Springs, MD 20746. 3 That is, the nonlinear source terms that are outside the scope of E-mail: Hendrik.Tolman@NOAA.gov LH. q 1998 American Meteorological Society Unauthenticated | Downloaded 10/01/21 08:18 AM UTC 1306 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 28 gives rise to nonlinear corrections to the characteristic (1990b),6 and where the additional terms are the ``new'' velocities in (1). We contend that the latter approach is terms of LH. superior to the former and that it therefore should have Equation (5) or LH Eq. (18) is correct, provided that formed the basis of GCWAM. Because LH have elected the spatial and temporal dependence of k [and a, if such otherwise, their approach requires further inspection. a dependence is assumed as in LH Eq. (9)] is treated The nonlinear shallow-water dispersion relation in explicitly. If the new term in Eq. (5) is expanded al- LH Eq. (4) is a nonlinear series expansion in terms of lowing for these dependencies, it vanishes with the use a small parameter4 of the proper form7,8 of LH Eq. (10). Alternatively, x and k may be considered as independent variables in e 5 (ka) 2 (2) the derivation of the characteristic velocities, in which representing the wave steepness. When applied to a case Eq. (5) can be written as spectral description of the wave ®eld, has to be re- e Dv ]v placed by an estimate of the steepness from the spec- cv 551xÇ ´=xkv 1 k˙ ´= v. (6) trum, typically Dt ]t The ®rst and second term on the right are identical to e 5 EE kN2 ( f, u) df du, (3) ´ ´ ´ and the ``new'' term in (5). Using the equations for du d f the conservation of waves and the irrotationality of the wavenumber vector,8 where k 2N is the steepness spectrum, and where du and df are integration bands for which estimates have to be Dk k˙ 552= v. (7) provided.5 Consistent with the spectral description of Dt x wind waves, an in®nite number of wave components Furthermore, xÇ , so that the second and third can be considered with du, df → 0. Consequently, e → 5 =kv 0, and this approach reproduces linear propagation (con- terms in Eq. (6) cancel. Thus, for steady media, cv 5 sistent with the inherently linear nature of the spectral 0, contrary to the assertion in LH. This will be true even description of a random wave ®eld). Finite nonlinear if it is assumed that v and cv explicitly depend upon expansion terms can only be obtained if ®nite band- the wave amplitude (i.e., in a nonlinear approach). widths du and df are retained. The magnitude of the We have not been able to trace the origin of the new nonlinear corrections then strongly depends on the ar- term in Eq. (4). If, however, x and k again are taken as bitrary choice of the bandwidth, or following LH, on independent variables, ]k/]n [ 0, and the new term the arbitrary spectral resolution.5 This is obviously vanishes. Thus, the new terms in Eqs. (4) and (5) appear aphysical, and LH Eq. (4) should therefore not be used to be due to mathematical errors. Lin and Huang appear to justify the new terms by as the basis of a nonlinear expansion for the group ve- 9 locity for irregular waves and cannot be considered an labeling them as effects of nonlinear kinematics. The improvement over the well-established linear approach magnitude of terms introduced by a nonlinear series as used in previous models. expansion are expected to depend on the wave energy or steepness and vanish for N → 0. The new term in Eq. (4), however, is always ®nite for shallow water with 3. Refraction and frequency shift varying depths, as k, (cg 2 c) and ]k/]n then are all Equations (17) and (18) present the characteristic ®nite. On p. 856 LH furthermore point out that the new term in Eq. (5) is similarly ®nite in shallow water for propagation velocities cu and cv of GCWAM. These equations are given as the linear test case 1. It therefore appears inconsistent to claim that both terms are expressions of nonlinear 1 ]k wave kinematics. c 5 ´´´1 (c 2 c) , (4) u k g ]n It should ®nally be noticed that the new frequency shift term results in aphysical model behavior. Consider, cv 5 ´´´1 (cg 1 V)´=(s 1 k´V), (5) for instance, an old swell ®eld approaching the coast in where ´ ´ ´ represent the conventional linear terms as presented by, for instance, LeBlond and Mysak (1978, §6), Christoffersen (1982), Mei (1983, p. 96), or Tolman 6 Assuming that n is a coordinate perpendicular instead of tangen- tial to k. 7 LH mistakenly use the intrinsic frequency s rather than the ap- parent frequency v in their Eq. (10). The proper form of this equation 4 Formally, e 5 ka, (2) is adopted for convenience of notation. follows from the conservation of waves. 5 LH Eq. (6) is dimensionally inconsistent because the integration 8 The conservation of waves and the irrotationality of k follow over the spectrum has been omitted. In replies to an earlier version directly from the de®nitions of k and v in the phase function of of this comment, Lin identi®ed this as a typographical error and stated monochromatic waves or spectral components, for example, Phillips à ∫ ∫ that the spectral density N should be replaced by N 5 du df N( f, u) (1977), LeBlond and Mysak (1978), and Mei (1983). df du throughout LH Eq. (6). The integration increments are de®ned 9 Whereas this was not clear to us from LH, this argument is re- by the spectral resolution and are given as du 5 158 and df 5 0.1 f. peatedly used in the reply to the original version of this discussion. Unauthenticated | Downloaded 10/01/21 08:18 AM UTC JUNE 1998 NOTES AND CORRESPONDENCE 1307 intermediate or shallow water. Such swell ®elds are es- effect (in spite of the diffusive numerics), as is illus- sentially monochromatic, and their low steepness im- trated in Fig. 4.48 of Komen et al. (1994, p. 346).13 plies linear dispersion. According to LH Eq.