BASIC RELATIONS BETWEEN TSUNAMIS CALCULATION AND THEIR PHYSICS II

Transcription

1 BASIC RELATIONS BETWEEN TSUNAMIS CALCULATION AND THEIR PHYSICS II Zygmnt Kowalik Institte of Marine Science, University of Alaska Fairbanks, AK 99775, USA ABSTRACT Basic tsnami physics of propagation and rn-p is discssed for the simple geometry of a channel. Modifications of a nmerical techniqe are sggested for the long-distance propagation and for the nonlinear processes in tsnami waves. The principal modification is application of the higher order of approximations for the first derivative in space. Presently, tsnami calclations employ the high resoltion 2D and 3D models for generation and rnp processes, while propagation is resolved by the reglar 2D models. Sch approach reqires bondary conditions which will seamlessly connect the high resoltion calclations to the propagation models. These conditions are described with the help of the method of characteristics. Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 154

2 1. Introdction This is the second part of the paper on relations between tsnami calclations and their physics (Kowalik, 21, hereinafter, K1). The prpose is to consider modifications of nmerical techniqes sed for the long-distance open ocean propagation and for the pslope propagation. While compting an open-ocean propagation at the resoltion of 1 natical mile (approximately 2km) the nmerical dispersion slowly changes tsnami wave parameters by changing amplitde and generating the short dissipative waves. This is especially important for the long-distance deep-ocean propagation of the short-period waves (5min-15min period). To alleviate these nmerical effects we sggest sing of the higher order of approximations especially for the spatial derivative. While tsnami wave starts to climb pslope towards the shore the nonlinear advective terms in the eqations of motion and the nonlinear terms in the continity eqations start to play important role. Tsnami wave steepens p and starts to break into shorter waves. This dissipative process cannot be flly reprodced throgh the nmerical means becase it is connected to the short wave domain, which is not resolved by the applied nmerical grid. In this domain the short waves of nmerical origin occr along with the physical processes. Neither higher space resoltion nor higher time resoltion is completely rectifyingthis problem which starts in the sbgrid domain. Application of the simple filter reslts in deletingthe short nmerical waves ths allowingto observe how tsnami wave changes while part of its energy is otflowing into the short wave dring the tsnami wave breaking. Is this remnant tsnami wave is actally observed in natre or only in the compter models? Investigations in this paper and reslts given by Lynett et al. (21) show that the physics of the nonlinear processes can be described by nmerical soltion bt the nmerical soltion needs modifications to take into accont the wide spectra of processes. Applications of the different nmerical approaches for the different ocean domains reqire bondary conditions which will seamlessly connect these domains. A problem to be considered is constrction of semi-transparent bondaries. The bondary between oceanic and coastal domains shold allow the signal arriving from the ocean to enter the coastal domain withot any reflection or dissipation and afterwards when the signal is reflected by shoreline back towards the ocean it mst cross the same bondary withot any reflections as well. If, for example, sch bondary is not completely transparent for tsnami which has entered coastal domain and a portion of the signal is reflected from the bondary back, tsnami will pmp energy towards the shore casing permanent increase of the amplitde at the shore. The varios conditions at the open bondaries are described with the help of the method of characteristics. 2. Nmerical approximations for the spatial derivatives. Consider the nmerical soltion of the eqations of motion and continity t = g ζ x (1) ζ t = (H) (2) x Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 155

3 Soltion of this system is sally searched by the two-time-level or the three-time-level nmerical schemes (Kowalik and Mrty, 1993a, Imamra, 1996). For constrction of the space derivatives in the eqs. (1) and (2), a space staggered grid (Figre 1) is sally sed (Arakawa C grid). The two-time-level nmerical scheme ( m+1 j m j ) T = g ζm j ζ m j 1 h (3) ζ m+1 j T ζ m j = (m+1 j+1 H j+1 m+1 j H j ) h is of the second order of approximation in space and only the first order in time. All notations are standard: is velocity, ζ denotes sea level changes, t is time, x denotes horizontal coordinate, g is the Earth s gravity acceleration (g=981 cm s 2 ), and H is depth. h h m+1 (4) T j=-2 j=-1 ζ j=1 j=2 j=3 Figre 1 Space-time grid for the tsnami propagation. j is space index, m is time index. The space-staggered grid given in Figre 1 is sed to constrct space derivatives in the above eqations. Variables (dashes) and ζ (crosses) are located in sch a way that the second order of approximation in space is achieved. The depth is taken in the sea level points. The space step alongthe x direction is h. Index m stands for the time stepping and the time step is T. Let s consider a simple problem of a sinsoidal wave propagating over the longdistance in the channel of the constant depth. At the left end of the channel a sinsoidal wave is given as ζ = ζ sin( 2πt ) (5) T p Here the amplitde is ζ =1 cm, and the period T p will be taken from 5 min to.5 hor range. m Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 156

4 AMPLITUDE (CM) Wavelength L=12 KM SEA LEVEL DX=1 KM, T=5 S Period=1 min 1 AMPLITUDE (CM) 5-5 DX=1 KM, T=.5 S -1 1 AMPLITUDE (CM) 5-5 DX=2 KM, T=5 S DISTANCE IN KM Figre 2 Propagation of the monochromatic wave of 1 min period along the channel of constant 477 m depth. DX denotes spatial step and T is time step. Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 157

5 Propagation of this monochromatic wave toward the right end of the channel will be stdied throgh eqs. (3) and (4). The right end of channel is open, and a radiating condition will be sed so that the wave can propagate beyond the channel withot reflection (see eq.(29) Sec. 5, and Reid and Bodine 1968). At the left end of the channel, eq (5) is applied for one period only; after that, the radiatingcondition is sed as well. The channel is 6 km longand 477 m deep. The wave period nder consideration is 1 min, which reslts in a 12-km wavelength. The time step of nmerical integration will be taken eqal to5sor.5s. Theinitialspacestepischoseneqalto1km(closeto5 represented by gridded topography). The 1-km space grid sets 12 steps per wavelength (SPW). Sch a resoltion will slowly introdce nmerical errors into reprodced waves. In Figre 2, reslts of comptation are given for the space step of 1 km (SPW=12) and the time step 5 s (pper panel), for the space step of 1 km and the time step.5 s (middle panel), and for the space step of 2 km (SPW= 6) and the time step 5 s (bottom panel). The wave propagation from the left end has been depicted at distances of 2 km, 4 km, and 6 km. The relatively poor space resoltion in the pper and middle panels reslts in the wave dampingalongthe channel. At approximately 15 km, the amplitde of the first wave became smaller than the amplitde of the second wave. Travelingwave train has a tail of secondary waves trailing behind the main wave. The shorter time step does not correct dispersive behavior (see the middle panel), only the shorter space step which increases the nmber of SPW, allows the nondispersive propagation. Dispersive nmerical error is cmlative, i.e. for the longer travel distances it will become large enogh to generate dispersive waves again. Therefore, the choice of the SPW index will depend on the propagation distance as well. The time step is aimed at resolvingthe tsnami wave period and the space step at resolvingwavelength. The above discssion shows that the encontered problems are related to the space resoltion. This is becase we are able to control time resoltion bt the spatial resoltion depends on the resoltion of the available bathymetric data. To improve soltions obtained by the nmerical methods we shall apply higher order of approximations for the first derivatives in space. We start by constrctingthe space derivative for the sea level in the eqation of motion. The central point ( point) located in the j grid point is srronded by the two sea level grid points at the distance h/2 from the velocity point (see Fig. 1). Consider Taylor series in these points, ζ j+1/2 = ζ j = ζ j + ζ j x h ζj 2 x 2 (h 2 ) ζj 3! x 3 (h 2 ) ζj 4! x 4 (h 2 )4 + O(h 5 ) (6) ζ j 1/2 = ζ j 1 = ζj ζ j h x ζj 2 x (h 2 2 )2 1 3 ζj 3! x (h 3 2 ) ! Alongwith Taylor series at the distance h3/2, ζ j+3/2 = ζ j+1 = ζ j + ζ j x ζ j 3/2 = ζ j 2 = ζ j ζ j x 4 ζ j x 4 (h 2 )4 O(h 5 ) (7) 3h ζj 2 x (3h 2 2 ) ζj 3! x (3h 3 2 ) ζj 4! x (3h 4 2 )4 + O(h 5 ) (8) 3h ζj 2 x 2 (3h 2 )2 1 3 ζj 3! x 3 (3h 2 ) ζj 4! x 4 (3h 2 )4 O(h 5 ) (9) Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 158

6 AMPLITUDE (CM) AMPLITUDE (CM) AMPLITUDE (CM) SEA LEVEL DX=1 KM, T=5 S DX=1 KM, T=5 S, FOURTH ORDER OF APPROXIMATION DX=2 KM, T=5 S DISTANCE IN KM Figre 3 Propagation of the monochromatic wave of 1 min period along the channel of constant 477 m depth. DX denotes spatial step and T is time step. Middle panel shows application of the higher order derivatives Here ζ j denotes the sea level in the point. Space derivative for the sea level in (3) is Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 159

7 obtained by sbtracting(7) from (6). Similar formla follows from (8) and (9), bt with the longer space step of 3h. The errors (the order of approximation) in both formlas, for the first derivative are proportional to the third derivatives. Ths by combiningthe two formlas the higher order formla can be constrcted. The new formla for the first derivative of the sea level in the point reads, ζ x = [27(ζ j ζ j 1 ) (ζ j+1 ζ j 2 )]/24h + O(h 4 ) (1) Space derivative for the velocity in the continity eqation (2) can be constrcted in the similar way by noticingthat the central point for sch derivative is the sea level and the space index shold be moved to the right so that j oght to be sbstitted by j +1. Ths the derivative for the velocity in the ζ point reads x (H)={27[ j+1(h j + h j+1 )/2 j (h j + h j 1 )/2] [ j+2 (h j+2 + h j+1 )/2 j 1 (h j 2 + h j 1 )/2]}/24h + O(h 4 ) (11) The propagation of the monochromatic wave described with the new derivatives is given in Fig. 3. This is repetition of the Fig. 2 with the middle panel reslting from application the new formlas. It shows essential improvement when compared against the reslts obtained with the second order derivatives (pper panel). 3. Propagation in sloping channel We shall proceed to constrct a simple algorithm for the propagation along the pslopingchannel as we did in the previos paper (K1). This nmerical scheme allows s to investigate processes occrring across the shelf. We will be able to pinpoint the inflence of friction and nonlinear terms on the process of propagation and dissipation and hopeflly nderstand how nmerical schemes change tsnami physics. Consider eqation of motion and continity alongx direction: t + x ζ = g x r D (12) ζ t = (D) (13) x Soltion of this system will be searched throgh the two-time-level nmerical scheme. The nonlinear (advective) term will be approximated by the pwind/downwind scheme. D in the above eqations denotes the total depth D = H + ζ. The followingnmerical scheme is sed to march in time: ( m+1 j m j ) + p (m j m j 1 ) + n (m j+1 m j ) T h h = g (ζm j ζ m j 1 ) h + r m j m j.5(d m j + D m j 1 ) (14) Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 16

8 Here: p =.5( m j + m j ), and n =.5(m j m j ) ζ m+1 j T ζ m j = [( m+1 j+1.5(dm j + D m j+1) m+1 j.5(d m j 1 + D m j )]/h (15) 2 TSUNAMI, PERIOD=5MIN AMPLITUDE (CM) VELOCITY (CM/S) DISTANCE IN KM Figre 4 Amplitde (pper panel) and velocity (lower panel) of 5 min period wave. Advective term and bottom friction are inclded. In this experiment the pslopingchannel of 1 km length is considered. Depth is changing from 5 m at the entrance to 5 m at the end of the channel. We start by comptingpropagation of a 5 min period tsnami wave from the deep water towards the shallow water. Reslts we had obtained previosly are depicted again in Fig.4. Again, it is important to notice the large disparity in the sea level and velocity. While the sea Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 161

9 level amplitde changes over a small range along the channel, velocity, on the other hand, displays mch greater variations and is less prone to the dissipation. The wider range of changes in the velocity field offers better opportnity to compare models and observations. The comparison of the models against the sea level amplitde only, often show that the lack of the bottom friction reslts in an increase of the amplitde, bt the reslts are not very different from the frictional models (Titov and Synolakis, 1998). VELOCITY (CM/S) BASIC PERIOD, T = 5MIN NONLINEAR INTERACTION VELOCITY (CM/S) 1-1 ADVECTIVE TERMS TA=T/2-2 2 VELOCITY (CM/S) 1-1 BOTTOM FRICTION TB=T/ TIME IN MIN Figre 5 Velocity wave of the nit amplitde (pper panel). The nonlinear wave prodced by the advective term (middle panel) and by the bottom friction (lower panel) Here we investigate the wave breakingprocess in the pslopingchannel. Is this tre physical process of the longwave breakingor this is a nmerical artifact? The period is 6 s and the time step.1 s, ths the temporal resoltion is 6SPP (step per period). Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 162

10 The wavelength is changing from approximately 7 km at the 5 m depth to 2 km at the 5 m depth. The latter is resolved with 25 m grid reslting in 8SPW, which is still an excellent resoltion. Unfortnately, a simple notion of the spatial and temporal resoltion needs to be reexamined since in the shallow water a strongnonlinear interaction occrs. This phenomenon shold change or approach to analyzing the nmerical stability of the basic set of eqations, becase previosly we relied on the linear stability analysis only. Strong nonlinearities are often sorce of comptational instabilities (Lewis and Adams, 1983). It follows from Fig.4 that even if the entire spectra of incident waves is limited to only one period the nonlinear interactions shold reslt in the new periods and in the rectified crrent. The average velocity calclated over an incident wave period is not eqal to zero, resltingtherefore in the rectified crrents. Let s consider a wave of the nit amplitde in velocity and of the 5 min period (Fig. 5, pper panel) and calclate the inflence of the advective terms (middle panel) or the bottom friction terms (lower panel). The new period in the middle panel is 2.5 min, and in the bottom panel is 1.67 min. Generally, the wave of the period T generates throgh the advective terms the new oscillations whose periods are T A = T/2i, wherei =1, 2, 3,... The new periods de to the bottom friction are T B = t/(2i +1),wherei =1, 2, 3,... (Parker, 1991). It is of interest to notice that the amplitde of the secondary oscillations in the lower panel is significantly smaller that the amplitde of oscillations in the middle panel. Conclsion is that the advective mechanism more effectively transfers linear motion into the nonlinear motion. The process of breaking longer waves into shorter waves proceeds continosly, ths the waves shown in Fig. 5 will break into the shorter period waves as well. To test whether the short period waves are the part of physical phenomenon we carry ot a simple experiment with an improved spatial resoltion by taking step h=5m. The reslt of calclation is given in Fig. 6. In the pper panel for comparison the reslt with h=25 m is also shown. The improvement in resoltion (lower panel) leads to decreasingof the short wave oscillations. Therefore, we may conclde that the short wave oscillations is an comptational artifact cased by the poor space resoltion. The main sorce of nonlinear effects is advective term. To the advective term in eq. (14) the pwind method of the first order approximation in space is applied. The pstream approach is sed for the stability reason. To deal with stability problems even with the higher order of approximation for the first derivatives (see Kowalik and Bang, 1987, Kowalik and Mrty, 1993a) the pstream approach is needed. Constrction of the first derivative can be carried ot on the three-point or for-point stencil. For the three-point stencil the followingconstrction can be sed for the advective term, x p(3m j 4m j 1 + m j 2 ) + n ( m j+2 +4m j+1 3m j ) + O(h 3 ) (16) 2h 2h Application of this approach to the advective term together with the higher order derivatives (1) and (11) for the remainingspace derivatives leads again to the improved reslts shown in the lower panel of Fig. 7. Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 163

11 AMPLITUDE (CM) AMPLITUDE(CM) TSUNAMI, PERIOD=5MIN DX=25M, T=.1 SEC; SECOND ORDER APPROXIMATION DX=5M, T=.1 SEC; SECOND ORDER APPROXIMATION DISTANCE IN KM Figre 6 Wave traveling in psloping channel. Soltion obtained by eqs. (14) and (15). Upper panel: space step 25 m, lower panel: space step 5 m. AMPLITUDE (CM) AMPLITUDE(CM) TSUNAMI, PERIOD=5MIN DX=25M, T=.1 SEC; SECOND ORDER APPROXIMATION DX=25M, T=.1 SEC; THIRD ORDER APPROXIMATION DISTANCE IN KM Figre 7 Wave traveling in psloping channel. Upper panel: soltion by eqs. (14) and (15), lower panel: soltion by the third order approximation Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 164

12 AMPLITUDE (CM) TSUNAMI, PERIOD=5MIN DX=25M, T=.1 SEC; SECOND ORDER APPROXIMATION DX=25M, T=.1 SEC; THIRD ORDER APPROXIMATION AMPLITUDE(CM) AND FILTER DISTANCE IN KM Figre 8 Wave traveling in psloping channel. Upper panel: soltion by eqs. (14) and (15), lower panel: soltion by the third order approximation and space filter Conclsion from the above experiments is that the parasitic short wave oscillations can be deleted throgh an application of the high spatial and temporal resoltions. A somewhat different and easier soltion is application of a simple space filter. It is applied only to the compted velocity. The new velocity m+1 j is filtered in the followingmanner, UN(J) = m+1 j (1 ALP )+.25 ( m+1 j m+1 j + m+1 j+1 ) ALP (17) The filter parameter ALP =.5. The comptation carried ot with the high order derivatives and the above filter are shown in the lower panel of Fig. 8 Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 165

13 4. Rn-p in channel We se here the algorithm previosly given in K1, see also Kowalik and Mrty, (1993b). The eqation of motion is solved by (14), while the continity eqation is approximated by the pwind/downwind approach as well. This approach introdced by Mader (1986) makes eqation of continity qite stable at the bondary between wet and dry domains. The followingnmerical scheme is sed to march in time for the eqation of continity: ζ m+1 j T ζ m j = (pj1 D m j + nj1 D m j+1 pj D m j 1 nj D m j ) (18) In the above eqation: pj1 =.5( m+1 j+1 + m+1 j+1 ) and nj1 =.5(m+1 j+1 m+1 j+1 ) pj =.5( m+1 j + m+1 j ) and nj =.5( m+1 j m+1 j ) To simlate the rn-p and rn-down, the variable domain of integration is established after every time step by checkingwhether the total depth is positive. This was done throgh a simple algorithm proposed by Flather and Heaps (1975) for the storm srge comptations. To answer whether j is a dry or wet point, the sea level is tested at this point; { j is wet point, if.5(d j 1 + D j ) ; (19) j is dry point, if.5(d j 1 + D j ) < Figre 9, middle and lower panels, describes an experiment in which 15 min and 3 min period waves of 1 m amplitde are continosly generated at the open end of the channel. After this signal is reflected from the sloping bondary, a standing wave is settled in the constant bottom domain. One can glean from this figre that the rnp for the 15 min period is mch bigger than the rnp for the 3 min. Sch growth sally show conditions close to the resonance. The sea level distribtion in the channel depends strongly on the open bondary condition sed for the comptation. Settingonly velocity or sea level at the open bondary may generate an additional error. The bondary condition mst be semitransparent. With the help of sch bondary condition we shold be able to set reqired sea level (or velocity) at the bondary and when the incident wave reflects from a shore and arrives to the open bondary it shold cross the bondary withot any reflections. These bondary conditions are discssed in the next sections. Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 166

14 -5 DEPTH (M) AMPLITUDE (CM) PERIOD=15 MIN AMPLITUDE (CM) PERIOD=3 MIN DISTANCE (KM) Figre 9 Upper panel: depth distribtion. Propagation of 1 m amplitde wave: Middle panel: 15 min period, lower panel: 3 min period. Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 167

15 5. Bondary conditions for the tsnami problems Presently, to calclate tsnami generation and rnp the high resoltion 2D or 3D models are sed, while the open ocean physics is well resolved by 2D models. To constrct a bondary condition for connecting the propagation and generation domains let s consider first a simple flow in a channel described by eqs. (1) and (2). Introdcingsoltion in the form (, ζ) =(,ζ )Φ(x ct) into these eqations gives risetoasimpleset c + gζ = (2) H cζ =, (21) whose soltion defines the well known dispersion relation c = ± (gh). Soltions to eqs.(1) and (2) can be written now as sperposition of two waves traveling into positive and negative directions along the x axis, ζ = ζ + Φ(x ct)+ζ Φ(x + ct) (22) = + Φ(x ct)+ Φ(x + ct) (23) Throgh eq.(2) and (21) the velocity amplitdes are related in the following way to the sea level amplitdes With this sbstittion eq.(23) reads, + = g H ζ+ and = g H ζ (24) = g H ζ+ Φ(x ct) g H ζ Φ(x + ct) (25) Combiningeqs.(22) and (23) the two dependent variables and ζ are expressed by two distrbances ζ + Φ(x ct) andζ Φ(x + ct) which we denote as Φ + and Φ. Throgh eqs.(22) and (25) these fnctions are expressed as Φ + = ζ + H/g 2 (26a) Φ = ζ H/g (26b) 2 The Φ + is a fnction of x ct, therefore it mst be constant alongany line x ct=constant. Sch line is called characteristic and speed c is the slope of the characteristic (Abbot and Minns, 1998, Drran, 1999). In the finite difference domain a characteristic located between two spatial grid points at the old time step can be followed to predict the vale of Φ + at the new time step. Similar conclsion can be dedced with respect to Φ. Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 168

16 The variables Φ + and Φ canbealsosedtoconstrcttwoeqationsinsteadofeqs.(1) and (2), Φ + t Φ t + c Φ + x c Φ x = (27) = (28) These will better serve for the bondary condition constrction since the vales of Φ + and Φ are preserved alongcharacteristics. Consider, the wave propagation in a channel with the left end located at x =and the right end at the distance x = L. From the left end enters a wave denoted as Φ +,it propagates towards the right end. If the right bondary oght to be transparent to this wave the reqirement is that there will be no reflection, or Φ =. From eq.(26b) it follows that the sea level at the right end of the channel is ζ = H/g (29) Similarly, if the left end of channel oght to be transparent for an incoming wave, eq. (26a) will prescribe the sea level nder condition that Φ + = ζ = H/g (3) Problem to solve is a constrction of semi-transparent bondaries, when e.g., at the left-hand bondary a permanent signal is generated and the right-hand bondary is the reflective one. A signal, reflected from the right bondary when arriving to the left bondary mst find the way ot becase if this bondary is not transparent the reflected signal will pmp energy casing permanent increase of the amplitde in the channel. To this prpose, can serve eq. (26a), assmingthat incomingwave from the left bondary is constant Φ + = Const, the relations between amplitde and velocity follows. There are a few variations of this approach e.g., setting sea level at the left bondary as a constant and calclatingφ + and Φ alongcharacteristics in proximity to the bondary and afterwards insertingthis vales for the bondary conditions to calclate velocity from eq.(23). In many applications, while going from the larger-scale domain to the smaller-scale, the open bondary for the smaller domain are taken from the larger scale comptations or observations. Sppose at the left-hand bondary the sea level (ζ b )andvelocity( b )are given either from measrements or comptations. The incoming vale of the Φ + is defined as Φ + = ζ b + b H/g (31) 2 and for the smooth propagation into domain this vale oght to be eqal to the invariant specified inside comptational domain in the close proximity to the bondary Φ + = ζ + H/g 2 (32) Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 169

17 or, ζ = ζ b +( b ) H/g (33) Some nderstandingof the above condition can be gleaned by comparison to the radiation conditions given by eqs. (29) and (3). Generally eq.(33) reqires that at the bondary, calclated variables in the smaller domain be eqal to the measred (or inpt variables). One cannot expect this condition to be flfilled at the initial stage of a comptation, especially when the comptation start from zero velocity. The second term at the right-hand-side of eq.(33) is actally a radiation condition, which radiates difference between prescribed and compted velocity. When stationary conditions will be achieved the difference of velocity will be close to zero. Condition (33) is often sed to establish open bondary condition for the tidal comptations (Flather, 1976). The seflness of eq.(33) for the transient tsnami processes reqires frther testing. 6. Nmerical implementation of the bondary condition Fig. 1 depicts grid distribtion at the left-hand bondary. Two-time level nmerical scheme is considered. Both sea level and velocity are havingthe same space index. Let s start by consideringradiation bondary condition defined for the velocity. Here a simple implementation of eq.(3) reads, m+1 = ζ m+1 1 g/h (34) The qestion to be answered is that eq.(34) is actally valid alongthe characteristic and not at the grid points. First, it is sefl to notice that this radiating condition is also flfilled by the eqation for the sea level ζ t c ζ x = (35) and ths the vale of the sea level is constant alongthe characteristic which propagates from the old time (m) into the new time (m + 1) domains as shown in Fig. 1 by dashed line. The distance dx = ct,wherec denote phase velocity and T is time step. The sea level at the old time step is defined at the point p on the characteristic, ζ p = ζm 1 (h dx)+ζ2 m dx (36) h This sea level is eqal to the sea level at the new time step (ζ1 m+1 = ζ p ), and since dx = ct, and time step and space step are given; denoting ct/h as γ, the above eqation can be written as, ζ1 m+1 = ζ1 m (1 γ)+ζm 2 γ (37) Eq.(37) can be introdced into (34) to calclate velocity. Notice that this velocity is defined in the sea level point (j = 1). Calclations show that actally (34) works qite well even if the variables are defined in the grid points and not along the characteristics. Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 17

18 Figre 1 The bondary conditions defined throgh the modeling or observations reqire also smooth transition between conditions and compted variables. To this prpose serves well eq.(33); at the left bondary it can be written as (assmingsea level is prescribed at the j=1 grid point and velocity at the j=2 grid point), ζ m+1 1 = ζ m+1 b +( m+1 b m+1 2 ) H/g (38) The major problem arises when only one variable is given at the open bondary: sea level or velocity. With sch condition an incomingcharacteristic is not flly defined, therefore the above relation cannot be applied. It is easy to start comptation with the prescribed sea level bt when the reflected wave arrives to the open bondary and its magnitde differs from the open bondary vale, the energy bild p ense reslting in an instability. Several soltions are feasible bt none is resolvingthe problem completely. Generally, a difference between the prescribed sea level magnitde at the bondary and the sea level generated by the reflected wave at the same bondary is de to an initial adjstment problem or de to transient character of the signal. The prescribed bondary condition shold actally inclde both incomingand otgoingsignal. One possible soltion for arrivingat the stationary signal, is to introdce the radiatingmechanism into a bondary condition and slowly remove this mechanism in time. Assme, at the left bondary the amplitde is prescribed as ζ1 m+1 = a cos ω(m +1)T, now introdcingaccordingly to eq.(3) a radiatingsignal, the bondary condition at the left bondary reads, ζ1 m+1 = a cos ω(m +1)T m+1 2 H/g (39) The second term at the right-hand-side of the above eqation oght to be slowly removed in time after the initial adjstment process is over. If only the sea level (ζ m+1 b )isgiven Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 171

19 at the bondary a different approach can be worked ot startingfrom eq.(38). The new bondary vale (ζ1 m+1 ) at the same grid point can be calclated by the radiation condition, sing, e.g., eq.(37). This sea level will differ from prescribed bondary vale ζ m+1 b. The difference may be sed to calclate a correction for the bondary vale of velocity at the point j = 2. We rewrite eq.(38) as m+1 2,c = m g/h(ζ m+1 b ζ m+1 1 ) (4) Velocity m+1 2 has been compted in the reglar way, i.e., by application of ζ m+1 b bondary vale. Acknowledgments as the This stdy was spported by the Arctic Region Spercompting Center at the University of Alaska Fairbanks. I am indebted to my stdents Jan Horrillo and Elena Sleimani for their help and discssions throghot the work. References Abbot, M.B. and A.W. Minns Comptational Hydralics. Ashgate, Aldershot, 557pp. Drran, D.R Nmerical Methods for Wave Eqations in Geophysical Flid Dynamics. Springer, 465pp. Flather, R.A A tidal model of the north-west Eropean continental shelf. Mem. Soc. R. Sci. Liege, 6, Flather, R.A. and Heaps, N.S Tidal comptations for Morecambe Bay. Geophys. J. Royal Astr. Soc., 42, Imamra F Review of tsnami simlation with a finite difference method. In Long- Wave Rnp Models, H.Yeah, P. Li and C. Synolakis, Eds, World Scientific, Kowalik, Z. 21. Basic relations between tsnami calclation and their physics, Science of Tsnami Hazards, 19, 2, Kowalik, Z. and I. Bang Nmerical comptation of tsnami rn-p by the pstream derivative method.science of Tsnami Hazards, 5, 2, Kowalik, Z. and Mrty, T.S. 1993a. Nmerical Modeling of Ocean Dynamics, World Scientific, 481 pp. Kowalik, Z. and Mrty, T.S. 1993b. Nmerical simlation of two-dimensional tsnami rnp. Marine Geodesy, 16, Lewis, C.H. III and Adams, W.M Development of a tsnami-floodingmodel havingversatile formlation of movingbondary conditions. The Tsnami Society MONOGRAPH SERIES, No.1, 128 pp. Lynett, P.J.,Tso-Ren W and P. L.-F. Li. 21. Modelingwave rnp with Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 172

20 depth-integrated eqations. Sbmitted to Coastal Engineering. Mader, C. L Nmerical Modeling of Water Waves, Univ. Calif. Press, Berkeley, Los Angeles, 26 pp. Parker, B. B The relative importance of the varios nonlinear mechanisms in a wide range of tidal interactions (Review). In: Tidal Hydrodynamics, B.B.Parker,Ed., Wiley and Sons, Reid, R.O. and B.R. Bodine Nmerical model for storm srges in Galveston Bay. J. Waterway Harbor Div., 94(WWI), Titov, V.V. and Synolakis, C.S Nmerical modelingof tidal wave rnp. J. Waterway, Port, Coastal and Ocean Eng., 124, 4, Science of Tsnami Hazards, Vol 21, Nmber 3 (23) page 173