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Physical Vs. Numerical Dispersion in Nonhydrostatic Ocean Modeling ⇑ Sean Vitousek , Oliver B

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Physical Vs. Numerical Dispersion in Nonhydrostatic Ocean Modeling ⇑ Sean Vitousek , Oliver B Ocean Modelling 40 (2011) 72–86 Contents lists available at ScienceDirect Ocean Modelling journal homepage: www.elsevier.com/locate/ocemod Physical vs. numerical dispersion in nonhydrostatic ocean modeling ⇑ Sean Vitousek , Oliver B. Fringer Environmental Fluid Mechanics Laboratory, Stanford University, Stanford, CA 94305, United States article info abstract Article history: Many large-scale simulations of internal waves are computed with ocean models solving the primitive Received 11 September 2010 (hydrostatic) equations. Under certain circumstances, however, internal waves can represent a dynamical Received in revised form 29 May 2011 balance between nonlinearity and nonhydrostasy (dispersion), and thus may require computationally Accepted 1 July 2011 expensive nonhydrostatic simulations to be well-resolved. Most discretizations of the primitive equa- Available online 14 July 2011 tions are second-order accurate, inducing numerical dispersion generated from odd-order terms in the truncation error (3rd-order derivatives and higher). This numerical dispersion mimics physical dispersion Keywords: due to nonhydrostasy. In this paper, we determine the numerical dispersion coefficient associated with Dispersion common discretizations of the primitive equations. We compare this coefficient to the physical disper- Nonhydrostatic effects Internal waves sion coefficient from the Boussinesq equations or KdV equation. The results show that, to lowest order, 2 Ocean modeling the ratio of numerical to physical dispersion is C = Kk , where K is an O(1) constant dependent on the dis- Grid resolution requirements cretization of the governing equations and k is the grid leptic ratio, k Dx/h1, where Dx is the horizontal Leptic ratio grid spacing and h1 is the depth of the internal interface. In addition to deriving this relationship, we ver- Lepticity ify that it indeed holds in a nonhydrostatic ocean model (SUNTANS). To ensure relative dominance of physical over numerical effects, simulations require C 1. Based on this condition, the horizontal grid spacing required for proper resolution of nonhydrostatic effects is k < O(1) or Dx < h1. When this condi- tion is not satisfied, numerical dispersion overwhelms physical dispersion, and modeled internal waves exist with a dynamical balance between nonlinearity and numerical dispersion. Satisfaction of this con- dition may be a significant additional resolution requirement beyond the current state-of-the-art in ocean modeling. Published by Elsevier Ltd. 1. Introduction and de-Vries, 1895). These equations represent the depth-averaged conservation of mass and momentum and include dispersive Dispersion, a characteristic of nonlinear or nonhydrostatic effects due to nonhydrostasy by retaining the lowest-order effects waves, occurs when the wave speed is a non-constant function of of integrating the nonhydrostatic pressure over the depth. The KdV the wave amplitude or wave frequency (or similarly wavelength) equation reduces conservation of mass and momentum to a single (Whitham, 1974). Distinction is sometimes made between these equation which is valid only for unidirectional wave propagation. two types of dispersion, namely amplitude dispersion and In this paper, we rely heavily on the Boussinesq equations and frequency dispersion, respectively (LeBlond and Mysak, 1978). KdV equation as models that faithfully represent nonlinear and The two are synonymous with nonlinearity and nonhydrostasy in nonhydrostatic behavior in more complicated natural systems, in studies of gravity waves, and hence are typically called nonlinear- this case, oceanic internal gravity waves. ity and dispersion. Many researchers have relied on weakly nonlinear, weakly non- Interesting wave phenomena exist when both nonlinearity and hydrostatic KdV theory to understand both qualitative and quanti- dispersion occur together and with approximately equal magni- tative details of internal waves (Helfrich and Melville, 2006). tude. Under such circumstances, nonlinear steepening balances Internal wave dynamics have been examined using the KdV dispersive spreading, and propagating waves of constant form framework which shows good comparison of internal wave pro- known as ‘‘solitary’’ waves are possible (Korteweg and de-Vries, files with the analytical result from the KdV equation (Liu et al., 1895). The most common equations satisfying the balanced motion 1998; Stanton and Ostrovsky, 1998; Duda et al., 2004; Warn- include the Boussinesq equations (Boussinesq, 1871, 1872; Pere- Varnas et al., 2010). Developments extending KdV theory to fully grine, 1967) and the Korteweg–de Vries (KdV) equation (Korteweg nonlinear dynamics have made significant contributions to the literature, a complete list of which is given in Helfrich and Melville ⇑ Corresponding author. (2006). However, weakly nonlinear, weakly nonhydrostatic scaling E-mail addresses: [email protected] (S. Vitousek), [email protected] (O.B. of many internal wave problems justifies use of traditional KdV Fringer). theory. 1463-5003/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.ocemod.2011.07.002 S. Vitousek, O.B. Fringer / Ocean Modelling 40 (2011) 72–86 73 Many oceanic internal waves are long relative to the depth and ing equations. Often second-order central differences are used, hence are nearly hydrostatic. Therefore, modeling waves of such implying that the dominant numerical dispersion coefficient is pro- scale with a hydrostatic model may be appropriate (Kantha and portional to the horizontal grid spacing squared. In this paper, we Clayson, 2000). In many other cases, near continental shelves and determine the numerical dispersion coefficient from the modified ocean ridges, internal waves can be highly nonlinear and nonhy- equivalent partial differential equation given from a particular dis- drostatic, often appearing as rank-ordered solitary-like wave trains cretization (Hirt, 1968). Such analyses would be of limited use with- (Hunkins and Fliegel, 1973; Apel et al., 1985; Liu et al., 1998; Scotti out, as a point of comparison, the physical dispersion coefficient. For and Pineda, 2004). Thus, simulations of internal waves may require internal waves, as mentioned previously, this is provided by the fully nonhydrostatic models (such as MITgcm (Marshall et al., Boussinesq equations or the KdV equation. Realistic nonhydrostatic 1997), SUNTANS (Fringer et al., 2006), nonhydrostatic ROMS simulations require that the physical dispersion is much larger than (Kanarska et al., 2007)). The drawback of fully nonhydrostatic the numerical dispersion. Because the physical dispersion for a gi- models is the computational cost associated with solving a 3-D ven problem is typically fixed, the problem of minimizing numerical elliptic equation for the nonhydrostatic pressure, which may re- dispersion in nonhydrostatic ocean models reduces to the problem quire an order of magnitude increase in computational time (Frin- of providing adequate horizontal resolution. As we will show, this ger et al., 2006). The prohibitive computational cost requirement as adequate horizontal resolution is Dx < h1 where Dx is the horizontal well as the nearly hydrostatic scales make primitive (hydrostatic) grid spacing and h1 is the depth of the internal interface. ocean models the preferred method for simulating internal waves The remainder of this paper is divided into four sections. Sec- in the ocean. Although not as computationally expensive, the prim- tion 2 presents the governing equations for modeling dynamics itive equations do not resolve physical dispersion resulting from of internal waves used in our analysis and derives the ratio of nonhydrostatic effects. However, this may not be a problem when numerical to physical dispersion and associated resolution require- physical dispersion is negligible. An excellent discussion concern- ments of a model. Section 3 presents numerical simulations of the ing the application of hydrostatic models to the problem of simu- KdV equation and the fully nonhydrostatic ocean model SUNTANS lating internal waves is presented in Hodges et al. (2006). They and illustrates how both models are prone to numerical dispersion. propose that future use of primitive ocean models is justified due Finally, Sections 4 and 5 present the discussion and conclusions of to the clear hydrostatic scaling of most problems of interest. How- the methods in this paper, respectively. ever, they also point out issues that modelers must contend with when simulating processes which include internal waves with 2. Governing equations hydrostatic models. As we will show, it is not only the lack of phys- ical dispersion that adversely influences model results of internal We begin with the nondimensional two-layer Boussinesq equa- waves, but the presence of numerical dispersion. tions representing the weakly nonlinear and weakly nonhydrostat- The concept of numerical dispersion is not particularly new. ic motion of the two-layer system depicted in Fig. 1. In this paper, Long has it been known that even if the governing equations are we indicate nondimensional quantities with a ⁄. Following Lynett nondispersive, their discrete model may be (artificially) dispersive and Liu (2002), the governing nondimensional momentum and (Trefethen, 1982). In early work, some authors referred to this as continuity equations in one horizontal dimension which retain ‘‘spurious dispersion’’ (Vichnevetsky, 1980; Vichnevetsky and Bow- the first-order nonlinear and nonhydrostatic effects are given by les, 1982). Numerical
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