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Downloaded 09/24/21 11:26 AM UTC 966 JOURNAL of PHYSICAL OCEANOGRAPHY VOLUME 46 MARCH 2016 G R I M S H A W E T A L . 965 Modelling of Polarity Change in a Nonlinear Internal Wave Train in Laoshan Bay ROGER GRIMSHAW Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom CAIXIA WANG AND LAN LI Physical Oceanography Laboratory, Ocean University of China, Qingdao, China (Manuscript received 23 July 2015, in final form 29 December 2015) ABSTRACT There are now several observations of internal solitary waves passing through a critical point where the coefficient of the quadratic nonlinear term in the variable coefficient Korteweg–de Vries equation changes sign, typically from negative to positive as the wave propagates shoreward. This causes a solitary wave of depression to transform into a train of solitary waves of elevation riding on a negative pedestal. However, recently a polarity change of a different kind was observed in Laoshan Bay, China, where a periodic wave train of elevation waves converted to a periodic wave train of depression waves as the thermocline rose on a rising tide. This paper describes the application of a newly developed theory for this phenomenon. The theory is based on the variable coefficient Korteweg–de Vries equation for the case when the coefficient of the quadratic nonlinear term undergoes a change of sign and predicts that a periodic wave train will pass through this critical point as a linear wave, where a phase change occurs that induces a change in the polarity of the wave, as observed. A two-layer model of the density stratification and background current shear is developed to make the theoretical predictions specific and quantitative. Some numerical simulations of the variable coefficient Korteweg–de Vries equation, and also the extended variable coefficient Korteweg–de Vries equation, are reported that confirm the theoretical predictions and are in good agreement with the observations. 1. Introduction riding on a negative pedestal [see the aforementioned references and Grimshaw et al. (1998, 1999, 2004)]. Internal solitary waves in the ocean are modeled by There are now many observations of this kind of phe- nonlinear evolution equations of the Korteweg–de Vries nomenon in the ocean [see the review by Grimshaw type [see the reviews by Grimshaw (2001) and Helfrich et al. (2010)]. and Melville (2006) for instance]. When the topography However, recently an observation of a polarity change and hydrology vary in the horizontal direction, then a of a different kind was observed in 2012 in Laoshan Bay variable coefficient Korteweg–de Vries (vKdV) equa- off the Qingdao coast, China. A temperature time series tion is used [see the recent review by Grimshaw et al. at a single mooring in shallow water showed an internal (2010)]. Here, we are concerned with the situation when quasi-periodic wave train of elevation waves being the coefficient of the quadratic term in the vKdV replaced by a wave train of depression waves as the equation changes sign, typically from negative to posi- thermocline rises on a rising tide. The observations have tive vis-a-vis the positive coefficient of the linear dis- been reported in detail in our companion paper Li et al. persive term as the waves propagate shoreward. This (2015), where we also argue that it is the cooling effect of change of polarity causes a solitary wave of depression the rising tide that causes the thermocline to rise. A to transform into a train of solitary waves of elevation sample of the data obtained in Laoshan Bay is shown again here in Fig. 1. The top panel shows a temperature time series for two periods of the semidiurnal barotropic Corresponding author address: Caixia Wang, Physical Ocean- ography Laboratory, Ocean University of China, 238 Songling tide, and the bottom panel shows the interval of interest Road, Qingdao, 266100, China. of 1300 to 1400 local time (LT) (all times referred to E-mail: [email protected] herein are local time). The total water depth was 10 m, DOI: 10.1175/JPO-D-15-0136.1 Ó 2016 American Meteorological Society Unauthenticated | Downloaded 09/24/21 11:26 AM UTC 966 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46 FIG. 1. Time series of temperature recorded at different depths at the mooring in Laoshan Bay. (top) A 24-h period; (bottom) a 1-h period from 1300 to 1400. and the bottom panel in Fig. 1 shows that initially at 1300 internal waves, but we suggest that this might be caused the thermocline was below middepth, implying that the by the well-known mechanism of steepening of an in- quadratic coefficient of the vKdV equation is positive ternal tide generated offshore by barotropic tidal in- and hence there are waves of elevation, as observed. But teraction with a topographic feature. However, whether the thermocline rises and after the time 1330 it is above it is due to the rising barotropic tide or due to an internal the middepth, implying that the quadratic coefficient has tide, or some combination, the observations clearly become negative and hence waves of depression form, show that over the period 1300 to 1400 UTC there is a as observed. The top panel in Fig. 1 does show some rising thermocline and consequent change in polarity of indication of a semidiurnal internal tide, although here the observed quasi-periodic wave train. It is this feature the water depth is very shallow (only 10 m), and since the that we are modeling in this paper. observations were taken in the summer (30 July) there Essentially this observation suggests that a periodic should also be an indication of diurnal heating and wave train will behave quite differently to a single soli- cooling. Our interest is specifically in the daytime period tary wave when passing a critical point where there is a 0900 to 1500 when one would normally expect to see polarity change. Consequently, recently a theory was some heating of the water column. Instead there was developed for this case by Grimshaw (2015); see also significant cooling especially over the period 1300 to Pelinovsky and Shavratsky (1977, 1981) for an earlier 1600, which we conjectured in Li et al. (2015) was due to theory with a different approach but with the same the rising tide bringing in cooler water. Nevertheless the general conclusion. This asymptotic theory is based on depression of the thermocline may well be at least partly the vKdV equation and describes the polarity change because of a semidiurnal internal tide, and we especially induced in a periodic wave train passing through a crit- note the quite large elevation of the thermocline around ical point where the coefficient of the quadratic term in 1200 immediately preceding the period of interest of the vKdV equation changes sign. The outcome is totally 1300 to 1400 when a quasi-periodic wave train undergoes different from the case when a solitary wave approaches a polarity change. It is likely that these are related in such a critical point, described above, and instead pre- some way, although a detailed analysis of that is beyond dicts that a periodic wave train will pass through a crit- the scope of this paper. Further, with only a single-site ical point as a linear wave train but of finite amplitude, measurement, we cannot determine the origin of these while a phase change is induced causing a change of Unauthenticated | Downloaded 09/24/21 11:26 AM UTC MARCH 2016 G R I M S H A W E T A L . 967 polarity. In this paper, we apply that general theory to 2 2 1 2 5 2 , , [r0(c u0) fz]z r0N f 0, for h z z0, and the Laoshan Bay observation, using a two-layer model (7) of the density stratification. In our companion paper Li 5 52 2 2 5 5 et al. (2015), the theory is examined using the observed f 0atz h,(c u0) fz gf at z z0 . density stratification and simulated using a variable co- (8) efficient extended KdV (veKdV) equation that contains both quadratic and cubic nonlinearity. These determine f(z; T) and the speed c(T), where the In section 2, we describe the vKdV equation relevant T dependence is parametric. for these observations and present a summary of the The theory developed in Grimshaw (2015) describes a asymptotic theory, developed here in section 3 for a two- slowly varying cnoidal wave solution of (2). At leading layer model. In section 4, we present some numerical order, the solution is the cnoidal wave: ð simulations of the vKdV equation and also of the veKdV s equation that has an extra cubic nonlinear term. We A; a[b(m)1cn2(gc; m)] 1d, c 5 k u2 Vds , conclude in section 4. 1 2 m E(m) where b 5 2 , aa 5 12mbk2g2, m mK(m) 2. Variable coefficient KdV equation aa 2 2 m 3E(m) and V 5 ad 1 2 . (9) In general, the vKdV equation can be developed for a 3 m mK(m) background stratification and current that vary both spatially and temporally and for spatially variable to- Here, cn(x; m) is the Jacobian elliptic function of mod- pography (see Grimshaw 1981). Here, we use the more ulus m,0, m , 1; K(m) and E(m) are the elliptic in- convenient form for unidirectional propagation de- tegrals of the first and second kind. The (trough to crest) scribed in Zhou and Grimshaw (1989) and Grimshaw amplitude is a, the mean value over one period is d, (2015), where the background depends only on a g 5 K(m)/p, and the spatial period is 2p/k.
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