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Wave hindcast sensitivity to wind forcing 1986

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WAVE HINDCAST SENSITIVITY TO WIND FORCING by SANDRA LEELLA MARGARET HODGINS B.A.Sc, University of Waterloo, 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1986 © Sandra Lee11a Margaret Hodgins, 1986 In presenting this thesis in partial fulfillment of the requirements of an advanced degree at THE UNIVERSITY OF BRITISH COLUMBIA, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It Is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l Engineering THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: October 1986 Abstract Efficient planning and safe operation of marine vessels and coastal structures require good understanding of wave properties, u s u a l l y expressed i n s t a t i s t i c a l terras. In the absence of very long direct measurement records, spectral wave hindcasting is routinely used to derive design and operating c r i t e r i a . The method i n v o l v e s s o l u t i o n of a time-dependent energy balance equation including spatial propagation, growth due to local wind, non-linear transfers between frequency components and dissipation processes. The most important source of errors i n a c a r e f u l hindcast i s the input wind which i s normally derived from historical surface pressure data. Since spectral wave models generate a sea-state that is almost in balance with local winds, wind sea is largely independent of the preceding storm history. Swell energy, on the other hand, propagates freely at off-wind angles perpetuating any errors that occurred in i t s generation. Shallow water near-shore zones are usually modelled with a small-area, nested subgrid that receives boundary data from the surrounding large-area model. Since the wave energy solution within the submodel can be dominated by the boundary conditions, errors in the large-area model must be minimized. The purpose of this study was to investigate the sensitivity of hindcast wave f i e l d s to prescribed errors i n the e v o l u t i o n of the wind patterns. Using a r a d i a l l y symmetric model of surface pressure, f i v e parameters were used to define storm evolution: storm trajectory, minimum central low pressure, rate of storm intensification, advection rate and storm size. Sensitivity analyses of each parameter were used to obtain the l i k e l y maximum att r i b u t a b l e v a r i a t i o n i n the s i g n i f i c a n t wave height f i e l d . On this basis, and considering wind sea only, the three most important parameters are (1) trajectory, (2) minimum central pressure and (3) advection rate. With careful wind f i e l d re-construction, wave hindcasts of specific events can be performed to acceptable engineering standards for extreme value analysis. However, for the present, climatological (continuous multi-year) wave databases constructed by s p e c t r a l hindcasting w i l l be too inaccurate to warrant their cost. The most promising advance for wave hindcasting (and forecasting) i s the p o s s i b i l i t y of acquiring high-resolution wind and wave data with s a t e l l i t e sensors to eliminate the need for inherently inaccurate surface pressure data. - i i i - Table of Contents Page A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f T a b l e s v L i s t o f F i g u r e s v i Acknowledgement x i i 1.0 I n t r o d u c t i o n 1 2 . 0 S e a - S t a t e D e s c r i p t i o n : The B a s i s f o r Wave M o d e l l i n g 10 2 .1 The E n e r g y B a l a n c e E q u a t i o n 10 2 . 2 Wave M o d e l P a r a m e t e r i z a t i o n s 11 2 . 2 . 1 Wave D i r e c t i o n a l i t y 11 2 . 2 . 2 The E n e r g y S o u r c e F u n c t i o n 13 2 . 2 . 3 The E n e r g y S p e c t r u m 15 P a r a m e t r i c Forms 15 D i s c r e t e Forms 16 2 . 3 O t h e r C o n s i d e r a t i o n s 16 2 . 3 . 1 W i n d - S e a and S w e l l 16 2 . 3 . 2 S h a l l o w Wa te r E f f e c t s 18 2 . 3 . 3 Wind I n p u t 18 3 .0 S p e c t r a l Wave M o d e l s 23 3.1 P a r a m e t r i c S p e c t r a l Wave M o d e l s 24 3 . 2 D i s c r e t e S p e c t r a l Wave M o d e l s 26 3 . 2 . 1 The ODGP M o d e l 27 3 . 2 . 1 The ADWAVE M o d e l 30 P r o p a g a t i o n . 30 S o u r c e Terms 33 4 . 0 Wind F i e l d S e n s i t i v i t y : An A p p l i c a t i o n o f ADWAVE 39 4 .1 S t r u c t u r e o f t h e S e n s i t i v i t y A n a l y s i s 39 4 . 2 Wind F i e l d S p e c i f i c a t i o n 40 4 . 2 . 1 The S u r f a c e P r e s s u r e F i e l d M o d e l 42 4 . 2 . 2 G r a d i e n t and S u r f a c e Wind F i e l d s 47 4 . 3 Wave M o d e l and G r i d S e t up 47 4 . 4 P r e s s u r e P a r a m e t e r S p e c i f i c a t i o n 51 4 . 4 . 1 T r a j e c t o r y 53 4 . 4 . 2 C e n t r a l P r e s s u r e 53 4 . 4 . 3 S t o r m Speed 53 4 . 4 . 4 R a d i a l S c a l i n g P a r a m e t e r 55 - i v - Table of Contents ( C o n t i n u e d ) Page 4 . 5 Summary o f M o d e l T e s t C a s e s 56 4 . 5 . 1 S to rm 1: The M e d i a n B a s e S c e n a r i o 56 4 . 5 . 2 S to rm 2 : E x p l o s i v e D e e p e n i n g 57 4 . 5 . 3 S to rm 3 : E a s t e r l y S h i f t e d N o r t h w a r d T r a j e c t o r y 58 4 . 5 . 4 S to rm 4 : I n c r e a s e d R a d i a l E x t e n t 59 4 . 5 . 5 S to rm 5 : A d v e c t i o n R a t e ( S t a l l e d Wea the r Sys tem i n t h e G u l f o f A l a s k a ) 60 4 . 5 . 6 S to rm 6 : D e e p e s t C e n t r a l Low 61 4 . 5 . 7 S to rm 7 : H i g h e s t Min imum C e n t r a l P r e s s u r e 62 4 . 5 . 8 S to rm 8 : I d e a l i z a t i o n of t h e F e b r u a r y 5 - 7 , 1960 S t o r m 63 5 . 0 D i s c u s s i o n o f S e n s i t i v i t y A n a l y s i s R e s u l t s 66 5.1 The Base S c e n a r i o 66 5 . 2 S to rm I n t e n s i t y V a r i a t i o n s 73 5 . 3 S to rm T r a j e c t o r y 91 5 . 4 A d v e c t i o n R a t e o f t h e C e n t r a l Low 95 5 . 5 The I d e a l i z a t i o n o f t h e F e b r u a r y 5 - 7 , 1960 S t o r m 99 6 . 0 Summary o f R e s u l t s and C o n c l u s i o n s 104 6 .1 Summary of S e n s i t i v i t y T e s t R e s u l t s 104 6 . 1 . 1 S to rm I n t e n s i t y 105 R a d i a l E x t e n t 105 Lowes t Min imum C e n t r a l Low P r e s s u r e 106 H i g h e s t Minimum C e n t r a l Low P r e s s u r e 107 R a t e o f I n t e n s i f i c a t i o n 108 6 . 1 . 2 S to rm T r a j e c t o r y 109 6 . 1 . 3 S to rm A d v e c t i o n R a t e 110 6 . 1 . 4 C o m p a r i s o n o f Maximum E r r o r s 111 6 . 2 C o n c l u s i o n s 113 6 . 2 . 1 The M e t e o r o l o g i c a l P e r s p e c t i v e 113 6 . 2 . 2 Wave H i n d c a s t i n g P e r s p e c t i v e 113 6 . 2 . 3 E n g i n e e r i n g A p p l i c a t i o n s P e r s p e c t i v e 114 7 . 0 R e f e r e n c e s 116 - v - L i s t o f T a b l e s Page Table 3.1 Comparison of Predicted and Observed Wave Heights as a Function of Modelled Energy Source Terms 38 Table 4.1 Storm 1 Scenario Parameters 56 Table 4.2 Storm 2 Scenario Parameters 57 Table 4.3 Storm 3 Scenario Parameters 58 Table 4.4 Storm 4 Scenario Parameters 59 Table 4.5 Storm 5 Scenario Parameters 60 Table 4.6 Storm 6 Scenario Parameters 61 Table 4.7 Storm 7 Scenario Parameters 62 Table 4.8 Storm 8 Scenario Parameters 63 Table 6.1 Summary of S e n s i t i v i t y to Radial Extent 105 Table 6.2 Summary of S e n s i t i v i t y to Lowest Minimum C e n t r a l Pressure 106 Table 6.3 Summary of S e n s i t i v i t y to Highest Minimum C e n t r a l Pressure 107 Table 6.4 Summary of S e n s i t i v i t y to I n t e n s i f i c a t i o n 108 Table 6.5 Summary of S e n s i t i v i t y to Storm Trajectory 109 Table 6.6 Summary of S e n s i t i v i t y to Storm Advection. Rate 109 Table 6.7 Maximum Sea-State V a r i a b i l i t y A t t r i b u t a b l e to Pressure Parameter S e n s i t i v i t y Tests 112 - v i - L i s t of Figures Page F i g . 1.1 The west coast of Canada showing the major landform features as w e l l as wave monitoring stations. 2 F i g . 1.2 L e v e l l e d contour a n a l y s i s of the ocean surface. Contour i n t e r v a l 0.2 mm, where 0.1016 mm = 1 foot. Shaded areas are below mean sea l e v e l and c l e a r areas are above sea l e v e l . (From Cote et a l . , 1960). 4 F i g . 1.3 A d i r e c t i o n a l wave spectrum c a l c u l a t e d from data measured off the west coast of Canada. The presentation i s F(f ,9) i n the lower p a n e l and E(f) d i r e c t l y above on the same frequency s c a l e . (From Hodgins et a l . , 1985). 6 F i g . 1.4 A contour map of s i g n i f i c a n t wave height as hindcast f o r the west coast of B r i t i s h Columbia. (From Hodgins and Nikleva, 1986). 7 F i g . 2.1 The cosine-power s p r e a d i n g f u n c t i o n . The l e f t p a n el i l l u s t r a t e s the shape v a r i a t i o n with s (from Sarpkaya and I s a a c s o n , 1981); t h e r i g h t p a n e l shows the d i s t r i b u t i o n of s as a f u n c t i o n of frequency (from Hasselmann et a l . , 1980). 12 F i g . 2.2 A schematic i l l u s t r a t i o n of the three energy source t e r m s t h a t t o g e t h e r a r e r e s p o n s i b l e f o r the r e d i s t r i b u t i o n of wave e n e r g y as a f u n c t i o n of frequency. (From Hasselmann et a l . , 1973) 14 F i g . 2.3 A comparison of s p e c t r a l frequency d i s c r e t i z a t i o n i n equal frequency and equal p e r i o d increments f o r N=15, fmin= 0' 0 5 H z a n d fmax= 0- 2 H z « 1 7 F i g . 2.4 A coarse deep water hindcast g r i d with a nested shallow water model g r i d as used for a hindcast of waves on the west coast of B r i t i s h Columbia. (From Hodgins and N i k l e v a , 1986). 19 F i g . 2.5 V a r i a t i o n s i n the drag c o e f f i c i e n t as a function of the 10-m e l e v a t i o n wind speed. (From Hsu, 1986). 21 - v i i - L i s t of Figures ( C o n t i n u e d ) Page F i g . 3 . 1 M e a s u r e d a n d h i n d c a s t s i g n i f i c a n t w a v e h e i g h t t i m e - s e r i e s a t t h r e e s t a t i o n s i n t h e G u l f o f M e x i c o d u r i n g H u r r i c a n e C a m i l l e . The h i n d c a s t s i t e numbers d e s i g n a t e g r i d p o i n t s c l o s e t o t h e m e a s u r e m e n t s i t e s . ( F r o m Cardone e t a l . , 1 9 7 5 ) . 2 9 F i g . 3 . 2 R e p r e s e n t a t i o n o f a t w o - d i m e n s i o n a l s p e c t r u m . The u p p e r p a n e l ( f r om S a r p k a y a and I s a a c s o n , 1 9 8 1 ) shows a p o r t i o n o f F ( f , 6 ) i n t h r e e - d i m e n s i o n a l r e l i e f a n d t h e l o w e r p a n e l s h o w s t h e d i s c r e t i z a t i o n o f t h a t s p e c t r u m i n i n c r e m e n t s o f f r e q u e n c y a n d d i r e c t i o n a s a p p l i e d i n ADWAVE. 3 1 F i g . 3 . 3 The b i l i n e a r i n t e r p o l a t i o n scheme i n ADWAVE (a) i n deep w a t e r and (b) i n s h a l l o w w a t e r . 3 2 F i g . 3 . 4 E n e r g y r e g i m e s w i t h i n a s p e c t r u m d u r i n g a c t i v e w a v e g r o w t h . (From R e s i o , 1 9 8 2 ) . 3 4 F i g . 4 . 1 The i n t e r - r e l a t i o n s h i p o f m e t e o r o l o g i c a l and s e a - s t a t e p a r a m e t e r s . 4 1 F i g . 4 . 2 S u r f a c e p r e s s u r e a s a f u n c t i o n o f d i s t a n c e c a l c u l a t e d f r o m measurements made d u r i n g t h e p a s s a g e o f H u r r i c a n e C a m i l l e . (From C a r d o n e e t a l . , 1 9 7 5 ) . 4 3 F i g . 4 . 3 An i d e a l i z e d p r e s s u r e f i e l d f o r a m o d e r a t e l y i n t e n s e s t o r m (uppe r p a n e l ) and an a c t u a l s u r f a c e p r e s s u r e c h a r t ( bo t tom p a n e l ) , b o t h a t a p p r o x i m a t e l y t h e same s c a l e f o r t h e same l o w p r e s s u r e s y s t e m . 4 4 F i g . 4 . 4 P a r a m e t e r s d e f i n i n g d i s t a n c e be tween a r b i t r a r y p o i n t s on a s p h e r e . (From P e a r s o n , 1 9 8 4 ) . 4 6 F i g . 4 . 5 A t y p i c a l i d e a l i z e d p r e s s u r e f i e l d ( uppe r p a n e l ) and i t s c o r r e s p o n d i n g s u r f a c e w i n d (U^Q) f i e l d ( l o w e r p a n e l ) . 4 8 - v i i i - L i s t o f F i g u r e s ( C o n t i n u e d ) Page F i g . 4 . 6 The w ind and wave m o d e l g r i d s h o w i n g t h e l ° x l ° l a t i t u d e - l o n g i t u d e g r i d , t h e c o a s t l i n e r e p r e s e n t a t i o n , and t he s p e c i a l o u t p u t p o i n t s . 50 F i g . 4 . 7 The s t o r m t r a j e c t o r i e s o f (a) s o u t h w e s t f r o n t a l l o w s and (b) s o u t h w e s t c o l d l o w s . (Adap ted f r o m L e w i s and M o r a n , 1 9 8 5 ) . 52 F i g . 4 . 8 T h e t h r e e s e g m e n t s o f a n i d e a l i z e d s t o r m t r a j e c t o r y b a s e d on t h e s o u t h w e s t f r o n t a l l o w s a n d t h e s o u t h w e s t c o l d l o w s . 54 F i g . 4 . 9 The s u r f a c e p r e s s u r e c h a r t f o r F e b r u a r y 6 , 1960 a t 12Z a t t h e p e a k o f t h e s t o r m ( u p p e r p a n e l ) and. t h e s t o r m t r a j e c t o r y ( l o w e r p a n e l ) w i t h 1 2 - h o u r l y c e n t r a l p r e s s u r e s ( k P a ) . (From L e w i s and M o r a n , 1985) . 64 F i g . 4 . 1 0 I d e a l i z a t i o n of t h e s u r f a c e p r e s s u r e c h a r t c o r r e s p o n d i n g t o F e b r u a r y 6 , 1 9 6 0 a t 12Z ( u p p e r p a n e l ) a n d t h e s t o r m t r a j e c t o r y ( l o w e r p a n e l ) a t r o u g h l y t h e same s c a l e s as t h e a c t u a l s t o r m d a t a i n F i g . 4 .9 . 65 F i g . 5 .1 Peak w i n d f i e l d f o r s t o r m 1 s h o w i n g maximum w inds o f 60 k n o t s . 67 F i g . 5 . 2 S i x - h o u r l y s i g n i f i c a n t w a v e h e i g h t f i e l d s p r o d u c e d by s t o r m 1 a l o n g l e g s B -C and C-D o f t h e s t o r m t r a j e c t o r y . T h e p e a k o f t h e s t o r m w i n d s o c c u r s a t 8 6 0 1 0 2 1 8 ( d a y 02 h o u r 18) i n t h e f o u r t h p a n e l ; t h e maximum waves a r e s i x h o u r s l a t e r a t 86010300 (day 03 h o u r 0 0 ) . 68 F i g . 5 . 3 T i m e - s e r i e s o f s i g n i f i c a n t wave h e i g h t ( H g ) , peak p e r i o d (Tp) and mean wave d i r e c t i o n a t t he s p e c i a l o u t p u t p o i n t a t t he e n t r a n c e t o Queen C h a r l o t t e Sound (mode l g r i d c o - o r d i n a t e s (31 ,13 ) ) d u r i n g s t o r m 1. 72 F i g . 5 . 4 The e v o l u t i o n o f P Q i n s t o r m s 1 and 2 . 74 - i x - L i s t o f F i g u r e s ( C o n t i n u e d ) Page F i g . 5 . 5 C o n t o u r e d f i e l d s o f A H G c a l c u l a t e d as s t o r m 2 l e s s s t o r m 1 f o r t h e p e r i o d d a y 02 h o u r 06 t o d a y 03 h o u r 00 i n 6 - h o u r l y t ime s t e p s . 75 F i g . 5 . 6 T i m e - s e r i e s o f s i g n i f i c a n t wave h e i g h t ( H ) , peak p e r i o d s (Tp) and mean wave d i r e c t i o n a t t h e s p e c i a l o u t p u t p o i n t a t t h e e n t r a n c e t o Queen C h a r l o t t e Sound (mode l g r i d c o - o r d i n a t e s (31 ,13 ) ) d u r i n g s t o r m 2 . 77 F i g . 5 .7 T i m e - s e r i e s o f s i g n i f i c a n t wave h e i g h t ( H „ ) , peak p e r i o d s (Tp) and mean wave d i r e c t i o n a t t h e s p e c i a l o u t p u t p o i n t n o r t h o f t he Queen C h a r l o t t e I s l a n d s ( g r i d c o o r d i n a t e s (26 ,17) ) d u r i n g s t o r m s 1 and 2 . 78 F i g . 5 . 8 The e v o l u t i o n o f P Q i n s t o r m s 1, 6 a n d 7 . I n a l l c a s e s P Q ( m i n ) i s imposed a t 53°N 145°W. 80 F i g . 5 . 9 C o m p a r i s o n o f s t o r m 1 a n d s t o r m 6 r e s p o n s e a s e x e m p l i f i e d by s i g n i f i c a n t w a v e h e i g h t a t t h e s p e c i a l o u t p u t p o i n t s n o r t h o f t h e Q u e e n C h a r l o t t e I s l a n d s ( 2 6 , 1 7 ) and a t t h e e n t r a n c e t o Q u e e n C h a r l o t t e S o u n d ( 3 1 , 1 3 ) . 81 F i g . 5 . 1 0 The f i e l d o f A H G a t t h e t i m e o f maximum wave h e i g h t s - i n s t o r m 1 ( d a y 03 h o u r 0 0 ) d i f f e r e n c e d a s s t o r m 6 m i n u s s t o r m 1 (upper p a n e l ) and t h e c o r r e s p o n d i n g f i e l d o f H G f r o m s t o r m 1. 82 F i g . 5 .11 The f i e l d o f AH a t t h e t i m e o f maximum wave h e i g h t s i n s t o r m 1 ( d a y 03 h o u r 0 0 ) d i f f e r e n c e d a s s t o r m 7 m i n u s s t o r m 1 (upper p a n e l ) and t h e c o r r e s p o n d i n g f i e l d o f H G f r o m s t o r m 1. 84 F i g . 5 . 12 T i m e - s e r i e s o f H a n d T i n t h e e n t r a n c e t o Q u e e n s p C h a r l o t t e Sound and a t t he c o a s t a l s i t e t h a t i s n o r t h o f t h e Queen C h a r l o t t e I s l a n d s . 85 - x - L i s t o f F i g u r e s ( C o n t i n u e d ) F i g . 5 . 1 3 C o m p a r i s o n o f i s o b a r r a d i i a s a f u n c t i o n o f t h e r a d i a l Page s c a l e rggo f ° r t he c a s e o f P Q =958 mb. 86 F i g . 5 . 14 T h e f i e l d o f A H g a t t h e t i m e o f max imum w i n d s ( d a y 02 h o u r 1 8 ; u p p e r p a n e l ) a n d a t t h e e n d o f t h e m o d e l l i n g s e q u e n c e ( d a y 03 h o u r 1 2 ; l o w e r p a n e l ) d i f f e r e n c e d a s s t o r m 4 minus s t o r m 1. 88 F i g . 5 . 1 5 T i m e - s e r i e s c o m p a r i s o n o f H and T f r o m s t o r m s 4 and 1 s p a t t h e e n t r a n c e t o Queen C h a r l o t t e Sound . 89 F i g . 5 . 1 6 T i m e - s e r i e s c o m p a r i s o n o f H and T f r o m s t o r m s 4 and 1 s p a t t h e w e s t e r n m o s t o u t p u t s i t e . 90 F i g . 5 . 17 S i g n i f i c a n t w a v e h e i g h t f i e l d u n d e r max imum w i n d s i n s t o r m 1 (uppe r p a n e l ) and i n s t o r m 3 ( l o w e r p a n e l ) . The s t o r m t r a j e c t o r y i n s t o r m 3 i s 5 ° c l o s e r t o t h e c o a s t . 92 F i g . 5 . 1 8 C o m p a r i s o n o f t h e t i m e - s e r i e s o f H „ , T_ a n d mean s p d i r e c t i o n i n s t o r m s 3 and 1 a t t h e s p e c i a l o u t p u t p o i n t due wes t o f t he e n t r a n c e t o Queen C h a r l o t t e Sound. 93 F i g . 5 . 1 9 C o m p a r i s o n o f t h e t i m e - s e r i e s o f H g , Tp a n d mean d i r e c t i o n i n s to rms 3 and 1 a t t h e s p e c i a l o u t p u t p o i n t i n t h e e n t r a n c e t o Queen C h a r l o t t e Sound . 94 F i g . 5 . 2 0 T h e s i g n i f i c a n t w a v e h e i g h t f i e l d 12 h o u r s a f t e r t h e s t o r m s t a l l (upper p a n e l ) and 24 h o u r s a f t e r t h e s t a l l ( l o w e r p a n e l ) . The max imum H g i s 12 .5 m on d a y 03 h o u r 12 and 13.5 m on day 04 h o u r 00 . 96 F i g . 5 .21 T h e AH f i e l d c a l c u l a t e d a s H a t d a y 03 h o u r 00 ( t h e s s b e g i n n i n g o f t h e s t a l l ) m i n u s H g 12 h o u r s l a t e r ( u p p e r p a n e l ) a n d m i n u s H g 24 h o u r s l a t e r ( l o w e r p a n e l ) . Maximum d i f f e r e n c e s a r e on t h e o r d e r o f 6 t o n e a r l y 8 m n e a r t h e s t o r m c e n t r e and 1 t o 3 m a l o n g t h e B.C. c o a s t . 97 - x i - L i s t o f F i g u r e s ( C o n t i n u e d ) P a § e F i g . 5 . 22 T h e AHg f i e l d c a l c u l a t e d a s H g a t d a y 04 h o u r 12 i n s t o r m 5 m i n u s H a t d a y 03 h o u r 12 i n s t o r m 1. A t t h e s e t i m e s the s t o r m s y s t e m s a r e e q u i v a l e n t and h a v e been f o r t he p r e c e d i n g 12 h o u r s . 98 F i g . 5 . 2 3 T i m e - s e r i e s r e s p o n s e o f H , T and mean wave d i r e c t i o n & p i n t he e n t r a n c e to Queen C h a r l o t t e Sound f o r s t o r m s 1, 6 a n d 8 . A s i d e f r o m t h e v e r y e a r l y s p i n u p p e r i o d , t h e mean d i r e c t i o n i n a l l s t o r m s I s e s s e n t i a l l y t h e same. 101 F i g . 5 .24 The s e a - s t a t e p a t t e r n i n t e r r a s o f H g i n s t o r m 8 a t 1 2 - h o u r l y i n t e r v a l s b e g i n n i n g on day 02 h o u r 18. 102 F i g . 5 . 2 5 The s e a - s t a t e p a t t e r n i n t e r m s o f H g i n s t o r m 6 a t 1 2 - h o u r l y i n t e r v a l s b e g i n n i n g on day 02 h o u r 12. 103 - x i i - Acknowledgement A l l compute r and s u p p o r t i n g s o f t w a r e r e s o u r c e s , i n c l u d i n g the ADWAVE m o d e l , were p r o v i d e d by S e a c o n s u l t M a r i n e R e s e a r c h L t d . o f V a n c o u v e r , B . C . - 1 - 1.0 INTRODUCTION E v e r y y e a r be tween O c t o b e r and M a r c h t he B r i t i s h C o l u m b i a c o a s t l i n e ( F i g . 1.1) comes under t he i n f l u e n c e o f one o r more s e v e r e n o r t h e a s t P a c i f i c Ocean s t o r m s ( L e w i s and M o r a n , 1 9 8 5 ) . Some o f t h e s e l o w p r e s s u r e s y s t e m s move d i r e c t l y i n l a n d , t y p i c a l l y w i t h i n a b r o a d band be tween n o r t h e r n V a n c o u v e r I s l a n d and t h e Q u e e n C h a r l o t t e I s l a n d s . M o r e c o m m o n l y , t h e t r a j e c t o r y s h i f t s t o t h e n o r t h , w e l l o f f s h o r e , and e n t e r s t h e G u l f o f A l a s k a . E a c h o f t h e s e e v e n t s has t h e p o t e n t i a l t o s e r i o u s l y i m p a c t on some a s p e c t o f m a r i n e a c t i v i t y a l o n g t h e c o a s t : i n t e r n a t i o n a l s h i p p i n g , f i s h i n g , b a r g e t r a f f i c , l o g booming o r o f f s h o r e r e s o u r c e d e v e l o p m e n t schemes. Be tween 1957 and 1983 , L e w i s and M o r a n r e p o r t t h a t e i g h t s h i p s a r e k n o w n t o h a v e s u n k w i t h a s many a s 60 c r e w members a b o a r d , and abou t t he same number of v e s s e l s h a v e s u f f e r e d e x t e n s i v e damage i n t h e s e s t o r m s . The r a r e r r e p o r t s o f c o a s t a l p r o p e r t y damage a r e u s u a l l y a t t r i b u t e d d i r e c t l y t o t h e w i n d s , a l t h o u g h o c c a s i o n a l l y w a v e damage h a s o c c u r r e d as i n F e b r u a r y 1979 when a s e c t i o n o f a f l o a t i n g b r i d g e on Hood C a n a l (Wash ing ton ) was washed away. The s e v e r i t y o f the l o c a l s e a - s t a t e was h i g h l i g h t e d a g a i n i n O c t o b e r o f 1984 when s i x f i s h i n g v e s s e l s a n d f o u r c r e w m e m b e r s w e r e l o s t o f f t h e n o r t h w e s t c o a s t o f V a n c o u v e r I s l a n d d u r i n g a s e v e r e s t o r m . T h i s e v e n t l e d t o a C o m m i s s i o n o f E n q u i r y ( L e B l o n d , 1984) recommending i m p r o v e m e n t s i n w e a t h e r and s e a - s t a t e f o r e c a s t i n g f o r B.C. c o a s t a l w a t e r s . F o r e n g i n e e r i n g p u r p o s e s , t he d a t a r e s o u r c e s f o r s e a - s t a t e s p e c i f i c a t i o n on t h e w e s t c o a s t o f C a n a d a a r e v e r y m e a g r e . T h e r e i s a s e m i - p e r m a n e n t n o n - d i r e c t i o n a l wave m e a s u r i n g s t a t i o n a t T o f i n o on t h e wes t c o a s t o f V a n c o u v e r I s l a n d ( F i g . 1 .1 ) . A W a v e r i d e r b u o y h a s b e e n p o s i t i o n e d a n d m a i n t a i n e d h e r e s i n c e J u n e 1 9 7 0 by t h e M a r i n e E n v i r o n m e n t a l D a t a S e r v i c e (MEDS) o f t h e D e p a r t m e n t o f F i s h e r i e s a n d O c e a n s . The u t i l i t y o f t h e s e d a t a i s q u i t e l i m i t e d s i n c e t he buoy i s l o c a t e d c l o s e t o t he c o a s t i n r e l a t i v e l y s h a l l o w w a t e r ( 4 0 m) , a n d p r i o r t o J a n u a r y 1981 t h e s e d a t a w e r e g e n e r a l l y o f p o o r q u a l i t y (MEDS, 1984) . Between O c t o b e r 1982 and May 1984 MEDS s p o n s o r e d a m a j o r wave c l i m a t e s t u d y o f t h e n o r t h e r n B r i t i s h C o l u m b i a c o a s t (Seakem, 1985 ; H o d g i n s e t a l . , 1985) . S i x w a v e b u o y s w e r e d e p l o y e d i n Q u e e n C h a r l o t t e S o u n d , H e c a t e S t r a i t a n d D i x o n E n t r a n c e a t t h e l o c a t i o n s s h o w n i n F i g 1 .1 . R e l i a b l e d i r e c t i o n a l d a t a w e r e o b t a i n e d o n l y a t t he M c l n n e s I s l a n d s i t e . T h i s p rog ram o f d a t a c o l l e c t i o n was c o n t i n u e d t h r o u g h May 1985 u n d e r t h e s u p p o r t o f t h e E n v i r o n m e n t a l S t u d i e s R e v o l v i n g Fund (ESRF) and w i l l be r e p o r t e d by D o b r o c k y S e a t e c h (1986) . F i g . 1.1 The w e s t c o a s t o f C a n a d a s h o w i n g t h e m a j o r l a n d f o r m f e a t u r e s as w e l l as wave m o n i t o r i n g s t a t i o n s . - 3 - The w a v e c l i m a t e a t m o s t o f t h e s e s i t e s i s s t r o n g l y i n f l u e n c e d by a d j a c e n t l a n d f o r m s and by b a t h y m e t r i c r e l i e f . As a r e s u l t , e v e n t he few months o f d a t a t h a t a r e a v a i l a b l e h a v e l i m i t e d r e g i o n a l a p p l i c a b i l i t y t o t he d e s i g n e r o f a p a r t i c u l a r c o a s t a l s t r u c t u r e o r h a r b o u r f a c i l i t y . T h i s l i m i t a t i o n i s a common p r o b l e m i n m o s t m a r i t i m e l o c a t i o n s t h a t i s t r a d i t i o n a l l y o v e r c o m e by w a v e h i n d c a s t i n g . T h i s a p p r o a c h i s p o s s i b l e b e c a u s e t h e d a t a a r c h i v e s o f w inds ( o r s u r f a c e p r e s s u r e f i e l d s f r o m w h i c h w i n d maps c a n be d e r i v e d ) e x t e n d f o r many y e a r s o v e r l a r g e r e g i o n s o f t h e g l o b e a n d b e c a u s e t h e p h y s i c a l m e c h a n i s m s g o v e r n i n g s e a - s t a t e e v o l u t i o n by w i n d f o r c i n g a r e s u f f i c i e n t l y w e l l - k n o w n . To g a i n an i m p r e s s i o n o f how waves u s u a l l y a p p e a r i n t h e o c e a n , a c o n t o u r p l o t o f one s t e r e o p a i r o f a e r i a l images i s shown i n F i g . 1.2. T h i s p l o t , i n w h i c h t h e w h i t e a r e a s a r e w a v e c r e s t s a n d t h e d a r k a r e a s t r o u g h s , was d r a w n f r o m a r e g u l a r 90 by 60 p o i n t g r i d o f s p o t w a v e h e i g h t s d e r i v e d f r o m t h e s t e r e o i m a g e s . The w a v e s s h o w n h e r e w e r e i n t h e g e n e r a t i o n a r e a w i t h a mean o v e r - w a t e r w ind o f 18 k n o t s f r o m 330°T a t t he t i m e o f t h e p h o t o g r a p h s . T h e r e i s a d i s c e r n i b l e a l i g n m e n t o f t he c r e s t s i m p l y i n g a mean d i r e c t i o n o f p r o p a g a t i o n a t a b o u t 3 0 ° t o t h e r i g h t o f t h e w i n d v e c t o r . S u p e r i m p o s e d on t h i s a v e r a g e p a t t e r n i s a g r e a t d e a l o f f i n e s t r u c t u r e (down t o abou t 20 m i n l e n g t h ) w h i c h c a n be i n t e r p r e t e d as t he s u p e r p o s i t i o n o f a l a r g e number o f wave p a t t e r n s o f d i f f e r e n t wave l e n g t h s t r a v e l l i n g a t d i f f e r e n t a n g l e s to t h e w i n d . I f t h e range of a n g l e s a s s o c i a t e d w i t h t h e i n d i v i d u a l wave p a t t e r n s i s w i d e , t h e n t h e o v e r a l l w a v e p a t t e r n a p p e a r s d i s o r d e r e d ( a s i n F i g . 1 . 2 ) , a n d t h e w i n d w a v e s a r e s h o r t - c r e s t e d w i t h t h e l e n g t h a l o n g a c r e s t o n l y a f e w t i m e s t h e dominan t wave l e n g t h . I t i s o n l y a f t e r waves h a v e p r o p a g a t e d away f r o m t h e a c t i v e g e n e r a t i o n a r e a t h a t t h e y become s o r t e d ou t i n t o l o n g - c r e s t e d s w e l l w a v e s . The v a r i a b i l i t y i l l u s t r a t e d i n F i g . 1.2 d o e s n o t h a v e a n y p r a c t i c a l s i g n i f i c a n c e f o r most c o a s t a l e n g i n e e r i n g a p p l i c a t i o n s . The t y p i c a l a p p r o a c h i s t o d e s c r i b e s u c h a n e n s e m b l e o f w a v e s by a s i n g l e c h a r a c t e r i s t i c w a v e h e i g h t ( s u c h as s i g n i f i c a n t w a v e h e i g h t , H g ) , w a v e p e r i o d a n d ( i f p o s s i b l e ) d i r e c t i o n ; i n t h i s manner a r e l i a b l e s t a t i s t i c a l r e p r e s e n t a t i o n o f t he s e a - s t a t e i s d e v e l o p e d o v e r t i m e f r o m w h i c h n o r m a l and ex t reme d e s i g n c o n d i t i o n s may be d e r i v e d . The v a s t m a j o r i t y o f wave d a t a c o l l e c t e d a r e no t s p a t i a l s n a p s h o t s l i k e F i g . 1 .2 , b u t a r e t i m e - s e r i e s o f s e a s u r f a c e e l e v a t i o n ( a n d p e r h a p s t i l t ) a t a f i x e d p o i n t . S u c h d a t a a r e r o u t i n e l y a n d c o n v e n i e n t l y a n a l y z e d i n t h e - 4 - F i g . 1.2 L e v e l l e d contour a n a l y s i s of the ocean surface. Contour i n t e r v a l 0.2 mm, where 0.1016 mm = 1 foot. Shaded areas are below mean sea l e v e l and c l e a r areas are above sea l e v e l . (From Cotg et a l . , 1960) - 5 - f r e q u e n c y domain by F o u r i e r t r a n s f o r m i n g t he t i m e doma in s i g n a l . T h i s r e s u l t s i n a d i s t r i b u t i o n o f w a v e e n e r g y E a s a f u n c t i o n o f f r e q u e n c y f w h i c h i s c a l l e d a v a r i a n c e (o r e n e r g y o r power) s p e c t r u m . G e n e r a l l y one s p e c t r u m c a n be c a l c u l a t e d w i t h c o n f i d e n c e f r om abou t 20 m in o f d a t a s a m p l e d a t abou t 2 t o 4 H z . I f t i l t i s a l s o m e a s u r e d , a d i r e c t i o n a l e n e r g y s p e c t r u m F ( f ,0) c a n be c a l c u l a t e d . A s a m p l e m e a s u r e d d i r e c t i o n a l s p e c t r u m i s s h o w n i n F i g . 1.3 a s F ( f , 0 ) a n d , by i n t e g r a t i n g o v e r d i r e c t i o n , as E ( f ) . T h i s i l l u s t r a t i o n shows a s e a s t a t e d o m i n a t e d by w a v e s w i t h p e r i o d s o f 9 t o 11 s . The w a v e f r e q u e n c y c o m p o n e n t w i t h t h e p e a k e n e r g y c o n t e n t ( f p ) i s t r a v e l l i n g t o t h e n o r t h , a n d t he mean d i r e c t i o n a t a l l f r e q u e n c i e s i s a l s o t o the n o r t h . The p r i m a r y p r o d u c t o f w a v e h i n d c a s t i n g i s a s p a t i a l a n d t e m p o r a l a r r a y o f e n e r g y s p e c t r a F ( f , 0 ) ( o f w h i c h F i g . 1.3 i s a t y p i c a l s i n g l e e l e m e n t ) . T h e m o s t u s e f u l s e c o n d a r y p r o d u c t i s t h e e q u i v a l e n t a r r a y o f s i g n i f i c a n t w a v e h e i g h t w h i c h i s r e a d i l y o b t a i n e d by i n t e g r a t i o n of F ( f , 0 ) o v e r f r e q u e n c y and d i r e c t i o n . F i g . 1.4 f r o m a r e c e n t B . C . c o a s t a l h i n d c a s t s t u d y ( H o d g i n s a n d N i k l e v a , 1 9 8 6 ) shows a c o n t o u r map o f H g w h i c h i s a s p a c e p l a n e o f t h i s s e c o n d a r y d a t a a r r a y a t a p a r t i c u l a r t i m e . A t i m e - s e r i e s o f s u c h p l o t s d e p i c t s t h e b e s t e s t i m a t e o f r e g i o n a l s e a - s t a t e e v o l u t i o n t h a t c a n n o r m a l l y be o b t a i n e d . T r a d i t i o n a l t i m e - s e r i e s o f H s a t p a r t i c u l a r p o i n t s a r e r e a d i l y e x t r a c t e d by s p a t i a l i n t e r p o l a t i o n . Such wave h e i g h t f i e l d s c o n t a i n e r r o r s , some o f w h i c h may be i n h e r e n t i n t h e wave m o d e l : i n a c c u r a t e c a l i b r a t i o n o f w a v e g r o w t h c o e f f i c i e n t s , i n a d e q u a t e r e s o l u t i o n o f b a t h y m e t r i c and l a n d f o r m f e a t u r e s , i n a c c u r a t e wave p r o p a g a t i o n s c h e m e s , o m i s s i o n o f i m p o r t a n t w a v e f r e q u e n c i e s , o m i s s i o n o f p h y s i c a l p r o c e s s e s a s s o c i a t e d w i t h w a v e - w a v e i n t e r a c t i o n s and w h i t e c a p i n g . The e f f e c t s o f e a c h o f t h e s e p o t e n t i a l p r o b l e m s c a n be m i n i m i z e d by c a r e f u l p r o b l e m d e f i n i t i o n and a t h o r o u g h m o d e l c a l i b r a t i o n and v e r i f i c a t i o n s t a g e p r i o r t o p r o d u c t i o n h i n d c a s t i n g . N e v e r t h e l e s s , t he most s e r i o u s s o u r c e o f e r r o r i n h i n d c a s t wave f i e l d s w i l l u s u a l l y be i n t he f o r c i n g w i n d d a t a , and s e v e r a l r e c e n t s t u d i e s h a v e begun t o a d d r e s s t h e c h a r a c t e r i s t i c s o f t h e s e e r r o r s . S t a t i s t i c a l a s s e s s m e n t s o f w i n d f i e l d s p r o d u c e d by t h e o p e r a t i o n a l n u m e r i c a l w e a t h e r p r e d i c t i o n (NWP) m o d e l s o f the C a n a d i a n M e t e o r o l o g i c a l C e n t r e (CMC) h a v e been r e p o r t e d by H o d g i n s and N i k l e v a ( 1 9 8 6 ) , H o d g i n s and H o d g i n s ( 1 9 8 6 ) and M a c L a r e n P l a n s e a r c h ( 1 9 8 5 ) . C a r d o n e e t a l . ( 1 9 8 0 ) h a v e c o n d u c t e d a s i m i l a r s t u d y o f U . S . m e t e o r o l o g i c a l d a t a . - 6 - 8214 Mclnnes Island at 03:00:00 10/19/83 28 20 16 13 1110 9 P . T 0 V s e C , 6 ''' 5 4 _i fsi X a in c c u cu 3 O 0_ Deg. of Freedo»» 64. Banduidth«0.0200 H2 Hs« 4.33 • . Tp« 3.59 sec a c z • a o Max. Contour-10.97 (95x of Peakl Contour interval »0.744. Theta « 9. i i i ' i i ' ' ' ' ' I ' 1 ' I - i — i — ] — i — r — r — r — r - t 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 Frequency (Hz) F i g . 1.3 A d i r e c t i o n a l wave spectrum c a l c u l a t e d from data measured off the west coast of Canada. The presentation i s F(f ,9) i n the lower p a n e l and E ( f ) d i r e c t l y above on the same frequency s c a l e . (From Hodgins et a l . , 1985) - 7 - F i g . 1.4 A c o n t o u r map o f s i g n i f i c a n t wave h e i g h t as h i n d c a s t f o r t h e w e s t c o a s t o f B r i t i s h C o l u m b i a . ( F r o m H o d g i n s a n d N i k l e v a , 1986). - 8 - Hodg ins and N i k l e v a (1986) i n v e s t i g a t e d t h e e f f e c t s on e r r o r s t a t i s t i c s i n t h e n o r t h e a s t P a c i f i c o f t h e CMC a n a l y s i s t i m e ( " n o w - c a s t " ) w i n d s , o f t h e w e s t c o a s t r e g i o n a l s u r f a c e p r e s s u r e r e - a n a l y s i s p r e p a r e d by t he P a c i f i c Wea the r C e n t r e (PWC) f r om w h i c h t he a u t h o r s d e r i v e d a s u r f a c e w i n d , and o f w ind f i e l d k i n e m a t i c a n a l y s i s i n w h i c h n u m e r i c a l mode l w ind f i e l d s a r e c a l i b r a t e d by hand t o m a r i n e o b s e r v a t i o n s r e p o r t e d b y s h i p s a t s e a . On a v e r a g e , t h e NWP m o d e l w i n d f i e l d s a r e i m p r o v e d by human i n t e r v e n t i o n , m o s t o f a l l by i n p u t o f c a l i b r a t i o n d a t a i n t h e k i n e m a t i c a n a l y s i s a n d t o a l e s s e r e x t e n t by t h e a n a l y t i c a l e x p e r i e n c e o f t he r e g i o n a l w e a t h e r f o r e c a s t e r . F o r t he k i n e m a t i c w inds e x c e e d i n g 10 k n o t s , t he r e p o r t e d b i a s i s 2 t o 3 k n o t s , r o o t - m e a n - s q u a r e (rms) speed e r r o r i s 5 t o 7 k n o t s and rms d i r e c t i o n e r r o r i s 20° t o 2 8 ° . Cardone e t a l . (1980) a l s o i n v e s t i g a t e d e r r o r c h a r a c t e r i s t i c s o f NE P a c i f i c w inds g e n e r a t e d by a NWP m o d e l and k i n e m a t i c w i n d f i e l d s , bu t a l s o i n c l u d e d " o b j e c t i v e " w ind f i e l d s i n w h i c h s h i p - r e p o r t e d w i n d s were b l e n d e d by compute r w i t h t h e NWP m o d e l w i n d s . T h e s e r e s e a r c h e r s f o u n d t h a t t h e k i n e m a t i c w i n d f i e l d s had c o n s i s t e n t l y l o w e r b i a s and rms e r r o r s t h a n t he o t h e r w ind t y p e s . B i a s was l e s s t h a n 1 k n o t , rms w i n d s p e e d e r r o r was a b o u t 6 k n o t s and rms d i r e c t i o n e r r o r was 3 0 ° , i n c l u d i n g a l l o b s e r v a t i o n s i r r e s p e c t i v e o f w i n d s p e e d . A l l o f t h e s e s t u d i e s were l i m i t e d t o s t a t i s t i c a l i n t e r c o m p a r i s o n o f a h i n d c a s t o r f o r e c a s t w ind w i t h a p o i n t w i n d measuremen t , and none of them has been a b l e t o r e l a t e t h e e r r o r p a r a m e t e r s t o c o n s e q u e n c e s f o r t h e m o d e l l e d s e a - s t a t e . O t h e r i n v e s t i g a t o r s s u c h a s C l a n c e y e t a l . ( 1 9 8 6 ) , J a n s s e n e t a l . ( 1 9 8 4 ) , G o l d i n g ( 1 9 8 3 ) , a n d m o s t o t h e r w a v e f o r e c a s t i n g and h i n d c a s t i n g g r o u p s i n Eu rope and N o r t h A m e r i c a h a v e p u b l i s h e d s i m i l a r t y p e s o f s t a t i s t i c a l a n a l y s e s b a s e d on w a v e m o d e l l i n g r e s u l t s i n w h i c h w a v e f i e l d e r r o r i s l a r g e l y a t t r i b u t e d t o known o r s u s p e c t e d w i n d e r r o r s . B e c a u s e t h e s e c o m p a r i s o n s a r e made a t i s o l a t e d p o i n t s w i t h i n c o m p l e x w e a t h e r s y s t e m s , t h e o p p o r t u n i t y t o i d e n t i f y more s p e c i f i c c a u s e - a n d - e f f e c t c o n n e c t i o n s i s u n l i k e l y . Wave m o d e l s w h i c h a r e c u r r e n t l y s t a t e - o f - t h e - a r t c a n g e n e r a l l y h i n d c a s t s e a - s t a t e s t o w i t h i n +1 m i n h e i g h t and +2 s i n p e r i o d g i v e n s u f f i c i e n t l y a c c u r a t e i n p u t w i n d s . Of t h e s e v e r a l s o u r c e s o f w i n d f i e l d e r r o r s , some a r e s y s t e m a t i c and o t h e r s a r e random i n s p a c e o r t i m e . The p u r p o s e o f t h i s i n v e s t i g a t i o n i s t o s t u d y t h e e f f e c t s o n w a v e m o d e l r e s u l t s o f a s e t o f common s y s t e m a t i c e r r o r s i n t h e f o r c i n g w i n d i n p u t , w i t h t h e end o b j e c t i v e o f i d e n t i f y i n g w h i c h o f t h e s e e r r o r s h a v e s e r i o u s e f f e c t s on wave mode l p e r f o r m a n c e . By c a r e f u l l y c o n t r o l l i n g t he s t r u c t u r e and e v o l u t i o n o f t h e f o r c i n g w ind f i e l d s , v a r i a t i o n s - 9 - i n t h e t i m e - s e r i e s o f t h e i n t e g r a t e d e n e r g y f i e l d ( H g ) h a v e b e e n d i r e c t l y l i n k e d t o t h e c a u s a t i v e f a c t o r s i n t he w i n d h i s t o r y . R e p o r t i n g o f t h i s r e s e a r c h has been d i v i d e d i n t o two e s s e n t i a l e l e m e n t s : t h e t h e o r y o f wave m o d e l l i n g and t h e e x p e r i m e n t a l a p p l i c a t i o n o f one p a r t i c u l a r m o d e l . C h a p t e r 2 d e s c r i b e s t he e l e m e n t s o f t he p h y s i c s o f s e a - s t a t e e v o l u t i o n t h a t may be p a r a m e t e r i z e d i n w a v e h i n d c a s t m o d e l s . The h e a r t o f t h i s d i s c u s s i o n i s t h e e n e r g y b a l a n c e e q u a t i o n i n c l u d i n g e n e r g y s o u r c e a n d s i n k t e r m s , p r o p a g a t i o n a n d t h e f o r m o f e n e r g y s p e c t r a . I m p o r t a n t p e r i p h e r a l c o n s i d e r a t i o n s t h a t a r e i n c l u d e d a r e s w e l l e n e r g y p r o p a g a t i o n , s h a l l o w w a t e r e f f e c t s and w i n d p a r a m e t e r i z a t i o n . C h a p t e r 3 i s d e v o t e d t o a more s p e c i f i c d e s c r i p t i o n o f t h r e e wave m o d e l s t h a t i l l u s t r a t e t h r e e f a i r l y d i s t i n c t a p p r o a c h e s t o s o l u t i o n o f t h e t o t a l p r o b l e m , t h a t i s , w a v e g e n e r a t i o n , p r o p a g a t i o n a n d d i s s i p a t i o n i n g e n e r a l i z e d g e o g r a p h i c a l and b a t h y m e t r i c c o n f i g u r a t i o n s . E m p h a s i s i s p l a c e d on t h e ADWAVE mode l w h i c h was u s e d e x c l u s i v e l y i n t he e x p e r i m e n t a l p a r t o f t h i s r e s e a r c h . The e x p e r i m e n t was c o m p r i s e d o f a s e n s i t i v i t y a n a l y s i s i n w h i c h t h e e f f e c t s o f s y s t e m a t i c v a r i a t i o n s i n t he i n p u t w i n d f o r c i n g f u n c t i o n were q u a n t i f i e d i n t e r m s o f w a v e e n e r g y a t d i s c r e t e l o c a t i o n s i n t h e s o l u t i o n d o m a i n . T h e s e e f f e c t s h a v e b e e n i n t e r p r e t e d a s i n p u t a n d o u t p u t e r r o r c h a r a c t e r i s t i c s t o p r o v i d e a p r e l i m i n a r y i n d i c a t i o n o f t h e r a n g e o f a c c e p t a b l e w i n d p a r a m e t e r e r r o r s f o r t o l e r a b l e u n c e r t a i n t y i n s e a - s t a t e s t a t i s t i c s . The i n p u t w i n d s and t h e w a v e m o d e l s e t - u p c o n f i g u r a t i o n a r e d e s c r i b e d i n d e t a i l i n C h a p t e r 4, f o l l o w e d by d i s c u s s i o n o f t h e o u t p u t r e s u l t s i n C h a p t e r 5. The c o n c l u s i o n s drawn f r o m t h i s a n a l y s i s a r e r e p o r t e d i n C h a p t e r 6. - 10 - 2.0 SEA-STATE DESCRIPTION: THE BASIS FOR WAVE MODELLING V i r t u a l l y a l l wave p r e d i c t i o n m o d e l s a r e b a s e d on e q u a t i o n s d e s c r i b i n g t he c o n s e r v a t i o n o f w a v e e n e r g y a t s p e c i f i e d l o c a t i o n s . The s o l u t i o n o f t h i s s y s t e m d e s c r i b e s t he t i m e - s p a c e e v o l u t i o n o f wave s p e c t r a F ( f , 0 ) , s u c h as t h a t i l l u s t r a t e d i n F i g . 1 .3 , a t s e l e c t e d p o i n t s . The s p e c t r a a r e s t a t i s t i c a l l y a v e r a g e d r e p r e s e n t a t i o n s t h a t a r e assumed t o be v a l i d f o r some f i n i t e p e r i o d o f t i m e (on t h e o r d e r o f 1 t o 3 h) and c h a r a c t e r i s t i c o f s e a - s t a t e c o n d i t i o n s o v e r a d i s c r e t e a r e a (up t o 2 .5° o f l a t i t u d e and l o n g i t u d e ) . 2.1 The Energy Balance Equation The p r o c e s s e s t h a t must be c o n s i d e r e d a r e p r o p a g a t i o n o f wave e n e r g y t oward a n d away f r o m t h e p o i n t , g r o w t h d u e t o l o c a l w i n d s , n o n - l i n e a r t r a n s f e r p r o c e s s e s t h a t r e d i s t r i b u t e e n e r g y b e t w e e n f r e q u e n c y c o m p o n e n t s , a n d d i s s i p a t i o n due t o w h i t e c a p p i n g a n d b o t t o m f r i c t i o n . The e n e r g y b a l a n c e d e s c r i b i n g t h e s e p r o c e s s e s i s w r i t t e n as ± F ( f , 9 ) - c ^ ( f ) . V F ( f , 9 ) = S ( f , 6) (2.1) 3 t where F i s wave e n e r g y as a f u n c t i o n o f f r e q u e n c y ( f ) and d i r e c t i o n (9) Cg i s t h e wave g roup v e l o c i t y as a f u n c t i o n o f f r e q u e n c y V i s t he g r a d i e n t o p e r a t o r / _ 3 _ , _3_ \ and \9x dy) S i s t h e ne t e n e r g y s o u r c e f u n c t i o n . U n d e r s t a n d i n g o f t h e p h y s i c s g o v e r n i n g t h e p r o c e s s embodied i n S ( f , 9 ) and t h e p a r a m e t e r i z a t i o n o f i t i n wave m o d e l s has e v o l v e d as a d i r e c t r e s u l t o f wave m e a s u r e m e n t e x p e r i m e n t s . T h e two m o s t i m p o r t a n t s t u d i e s , b o t h s t a r t e d i n 1968, a r e t h e Ocean D a t a G a t h e r i n g P r o g r a m (ODGP) and t h e J o i n t ] Jo r th Sea Wave P r o j e c t ( J O N S W A P ) . ODGP was a j o i n t o i l i n d u s t r y p r o g r a m c o n d u c t e d b e t w e e n 1968 and 1971 a t s i x l o c a t i o n s i n t h e G u l f o f M e x i c o t o c o l l e c t o c e a n o g r a p h i c and m e t e o r o l o g i c a l d a t a w i t h e m p h a s i s o n s e v e r e c o n d i t i o n s i n h u r r i c a n e s . JONSWAP was a n i n t e r n a t i o n a l u n d e r t a k i n g b e t w e e n c o o p e r a t i n g r e s e a r c h i n s t i t u t e s t o d e t e r m i n e t h e s t r u c t u r e o f t h e s o u r c e t e r m s i n t h e e n e r g y b a l a n c e e q u a t i o n . The l a t t e r e x p e r i m e n t l a s t e d 10 weeks i n 1968 and 1969 w i t h c o m p r e h e n s i v e d a t a c o l l e c t i o n a t up t o 13 l o c a t i o n s i n t h e N o r t h S e a . T h e o b j e c t i v e was t o r e c o r d d u r i n g s t e a d y o f f s h o r e w i n d s to f a c i l i t a t e s t u d y i n g the e v o l u t i o n o f t h e wave e n e r g y s p e c t r u m and t h e r e b y e l u c i d a t e t h e p h y s i c a l mechanisms r e s p o n s i b l e f o r t h a t e v o l u t i o n . A l t h o u g h ODGP and JONSWAP were so d i f f e r e n t i n p u r p o s e , t he e v e n t u a l outcome o f e a c h was a n i m p o r t a n t , b u t a g a i n v e r y d i f f e r e n t , a d v a n c e i n w i n d - w a v e - 11 - m o d e l l i n g . The ODGP d a t a s e t a l l o w e d , f o r t h e f i r s t t i m e , t h e d e v e l o p m e n t o f a v e r i f i e d w a v e m o d e l — t h e ODGP m o d e l by C a r d o n e e t a l . ( 1 9 7 5 ) w h i c h h a s become t he b a s i s o f v i r t u a l l y a l l c o m m e r c i a l wave f o r e c a s t i n g i n t h e U n i t e d S t a t e s . The JONSWAP e x p e r i m e n t h i g h l i g h t e d t h e i m p o r t a n c e o f n o n - l i n e a r e n e r g y t r a n s f e r p r o c e s s e s be tween wave f r e q u e n c i e s ( w a v e - w a v e i n t e r a c t i o n ) , w h i c h a r e o m i t t e d f r o m t h e ODGP m o d e l a n d i t s s u c c e s s o r s . Many o f t h e E u r o p e a n o p e r a t i o n a l f o r e c a s t i n g w a v e m o d e l s , i n a d d i t i o n t o H a s s e l m a n n ' s r e s e a r c h m o d e l s ( H a s s e l m a n n e t a l . , 1976; G u n t h e r e t a l . , 1979a,b) and R e s i o ' s ( 1981 , 1982) h i n d c a s t m o d e l , u t i l i z e t h e JONSWAP r e s u l t s . 2.2 Wave Model Parameterizations 2.2.1 Wave Directionality I n many m o d e l s e q u a t i o n ( 2 . 1 ) i s i n t e g r a t e d w i t h r e s p e c t t o 6 t o y i e l d t h e e n e r g y e q u a t i o n , w r i t t e n h e r e i n o n e s p a t i a l d i m e n s i o n ( H a s s e l m a n n e t a l . , 1976) as _3_ E ( f ) + c g ( f ) . 3 E ( f ) = S ( f ) (2.2) o-t 9x F ( f , 8 ) c a n t h e n be o b t a i n e d by a p p l y i n g a s p r e a d i n g f u n c t i o n G ( f , 0 ) t o t h e s o l u t i o n o f (2.2) s u c h t h a t F ( f , 0) = E ( f ) . G ( f , 6 ) (2.3) where G must s a t i s f y 0)d = 1 (2.4) fa, —TT The most w i d e l y u s e d f u n c t i o n a l fo rms o f G a r e t h e c o s i n e - s q u a r e d ( S t . D e n i s and P i e r s o n , 1953) : / 2 c o s 2 0 , | 0| ITT/2 G(6) = \ * (2.5) ( 0 , o t h e r w i s e w h e r e G i s i n d e p e n d e n t o f w a v e f r e q u e n c y ; a n d t h e c o s i n e - p o w e r ( L o n g u e t - H i g g i n s e t a l . , 1961) G(0) = C ( s ) c o s 2 s / 6 - 6 \ ( 2 . 6 ) where C(s ) i s t h e n o r m a l i z i n g f a c t o r needed t o e n s u r e t h a t (2.4) i s s a t i s f i e d , s i s a f u n c t i o n o f f r e q u e n c y , a n d 0 i s t h e mean w a v e d i r e c t i o n . The c o s i n e - power s p r e a d i n g f u n c t i o n i s shown i n F i g . 2.1 f o r v a l u e s o f s be tween 1 and 10 and as a f u n c t i o n o f f r e q u e n c y . T h i s f i g u r e shows t h a t t h e l a r g e s t v a l u e s o f - 12 - F i g . 2 .1 The c o s i n e - p o w e r s p r e a d i n g f u n c t i o n . The l e f t p a n e l i l l u s t r a t e s t h e s h a p e v a r i a t i o n w i t h s ( f r o m S a r p k a y a a n d I s a a c s o n , 1 9 8 1 ) ; t h e r i g h t p a n e l s h o w s t h e d i s t r i b u t i o n o f s a s a f u n c t i o n o f f r e q u e n c y ( f r o m H a s s e l m a n n e t a l . , 1 9 8 0 ) . - 13 - s, w h i c h a r e abou t 10, o c c u r a t t he peak f r e q u e n c y and t h a t t he wave e n e r g y i s w e l l - f o c u s s e d a l o n g t h e mean w a v e d i r e c t i o n ( w i t h i n a b o u t + 4 5 ° ) . A t o t h e r w a v e f r e q u e n c i e s t h e e n e r g y i s mo re s p r e a d o u t a r o u n d t h e mean d i r e c t i o n o f wave p r o p a g a t i o n . 2.2.2 The E n e r g y S o u r c e F u n c t i o n The ne t e n e r g y s o u r c e f u n c t i o n S w h i c h g o v e r n s t he i n p u t o f ene rgy t o t he wave f i e l d was d e s c r i b e d by H a s s e l m a n n e t a l . (1973) i n t e rms o f t h r e e p r o c e s s e s : S " S i n + S n l + S d s < 2 ' 7> w i t h t he f o l l o w i n g m e a n i n g s : S ^ n = t h e i n p u t o f e n e r g y f r o m the w i n d S n ^ = t h e n o n - l i n e a r e n e r g y t r a n s f e r p r o c e s s S ^ s = t h e e n e r g y d i s s i p a t i o n p r o c e s s ( w h i t e c a p p i n g ) T h e s e p r o c e s s e s a r e s h o w n s c h e m a t i c a l l y i n F i g . 2 . 2 , t o g e t h e r w i t h t h e r e s u l t a n t n e t t r a n s f e r c u r v e as a f u n c t i o n o f f r e q u e n c y . T h i s f i g u r e e m p h a s i z e s t h e i m p o r t a n c e o f S n ^ w h i c h i s t h e dom inan t e n e r g y i n p u t mechan ism a t f r e q u e n c i e s l o w e r t h a n f p . N e a r a n d j u s t b e l o w f p , S n ^ and S i n a r e a b o u t b a l a n c e d , b u t a t h i g h e r f r e q u e n c i e s S n ^ r e m o v e s mo re e n e r g y t h a n S ^ n c o n t r i b u t e s . O n l y a t v e r y h i g h f r e q u e n c i e s does S^s c o n t r i b u t e t o t h e e n e r g y b a l a n c e . The ne t r e d i s t r i b u t i o n o f e n e r g y i s t h e r e f o r e f r om f r e q u e n c i e s a b o v e f p to t h o s e b e l o w on t h e f o r w a r d f a c e o f t h e wave s p e c t r u m . Of t h e s e e n e r g y s o u r c e t e r m s , o n l y S n ^ a t p r e s e n t h a s an a n a l y t i c f o r m u l a t i o n (known as t he B o l t z m a n n i n t e g r a l ) . A l t h o u g h t h i s f u n c t i o n a l f o rm o f S ^ c a n be e v a l u a t e d a n a l y t i c a l l y , t h e c o m p u t a t i o n s a r e t o o c o m p l e x to' be r o u t i n e l y i n c l u d e d i n wave m o d e l s . I n JONSWAP, t h e S n ^ t e r m was e v a l u a t e d f o r i n d i v i d u a l t e s t c a s e s and combined w i t h t h e d a t a f o r S t o examine t he r e m a i n i n g two t e rms S ^ n + S j ^ . The l a t t e r was assumed t o a c c o u n t f o r a l l d i s s i p a t i o n m e c h a n i s m s , p r i m a r i l y w h i t e c a p p i n g , b u t a l s o f o r e n e r g y t r a n s f e r s b e t w e e n s p e c t r a l c o m p o n e n t s s u c h a s t h e a t t e n u a t i o n o f l o n g w a v e s by damped s h o r t w a v e s ( H a s s e l m a n n e t a l . , 1 9 7 3 ) . A s s u m i n g t h a t S . , n h a s a s i m i l a r d i s t r i b u t i o n t o t h e s p e c t r u m i t s e l f , i n a c c o r d a n c e w i t h l i n e a r wave g r o w t h t h e o r i e s , t he p r o c e s s mode l shown i n F i g . 2 .2 c a n be i n f e r r e d . The p a r t i t i o n o f e n e r g y b e t w e e n S ^ n a n d S ^ g c o u l d n o t , h o w e v e r , be d e r i v e d d i r e c t l y f r om t h e JONSWAP d a t a . - 14 - F i g . 2 . 2 A s c h e m a t i c i l l u s t r a t i o n o f t h e t h r e e e n e r g y s o u r c e t e r m s t h a t t o g e t h e r a r e r e s p o n s i b l e f o r t h e r e d i s t r i b u t i o n o f w a v e e n e r g y a s a f u n c t i o n o f f r e q u e n c y . (F rom H a s s e l m a n n e t a l . , 1973) - 15 - 2 . 2 . 3 The E n e r g y S p e c t r u m P a r a m e t r i c Forms T h e r e a r e two g e n e r a l a p p r o a c h e s t o t h e s o l u t i o n o f t h e e n e r g y b a l a n c e e q u a t i o n ( 2 . 1 ) . The f i r s t m e t h o d a s s u m e s t h a t e v e r y s p e c t r u m E ( f ) c a n be a p p r o x i m a t e d by a u n i v e r s a l p a r a m e t r i c s p e c t r u m E ( f , a ^ ) . The g o v e r n i n g p a r a m e t e r s a ^ a r e u s u a l l y r e l a t e d t o w i n d s p e e d , peak s p e c t r a l f r e q u e n c y , and s p e c t r a l shape f u n c t i o n s , a l l o f w h i c h must be r e a d i l y d e t e r m i n e d f r om d a t a o r f r om e m p i r i c a l l a w s . The b e s t - k n o w n fo rms o f E a r e t h e P i e r s o n - M o s k o w i t z (PM) s p e c t r u m ( P i e r s o n and M o s k o w i t z , 1 9 6 4 ) f o r f u l l y - d e v e l o p e d w i n d s e a s a n d t h e JONSWAP s p e c t r u m ( H a s s e l m a n n e t a l . , 1973 ) w h i c h was d e v e l o p e d by a n a l y s i s o f f e t c h - l i m i t e d g r o w i n g w i n d - s e a s p e c t r a . The PM s p e c t r u m i s g i v e n by E ( f ) = a g 2 ( 2 7 T ) - 4 f - 5 e x p [ - 0 . 7 4 / _ g _ \ 4 / f 4 ] (2 .8) \27TU / i n w h i c h a i s P h i l l i p s ' c o n s t a n t (=0.0081) and U i s w i n d s p e e d . The JONSWAP p a r a m e t r i c s p e c t r a l e q u a t i o n i s E(f) = a g 2 ( 2 T r)~ 4f - 5 e x p / - 5 Tf " T 4 \ . Y c ( f ) (2.9) my where a = t h e P h i l l i p s ' s c a l i n g p a r a m e t e r , now a f u n c t i o n o f f r a t h e r t h a n a c o n s t a n t f = f r e q u e n c y a t t h e p o i n t o f maximum e n e r g y c ( f ) Y = t h e peak enhancement f a c t o r c ( f ) = e x p ( [ - ( f - f p ) 2 ] / [ 2 a 2 f 2 ] ) o = t h e s p e c t r a l shape p a r a m e t e r d e f i n e d by o& ( i f f < fp ) a n d o b ( i f f > f p ) The peak enhancement f a c t o r , w h i c h l a r g e l y a c c o u n t s f o r t he d i f f e r e n c e be tween PM a n d JONSWAP f o r m s , h e i g h t e n s a n d s h a r p e n s t h e s p e c t r a l p e a k i n f e t c h - l i m i t e d s e a s t o m o d e l t h e " o v e r s h o o t " e f f e c t m e a s u r e d i n JONSWAP. The t e r m o v e r s h o o t d e s c r i b e s t h e o b s e r v a t i o n t h a t u n d e r s t e a d y w i n d c o n d i t i o n s , t h e ene rgy a t and n e a r f i n d e v e l o p i n g s e a s c a n e x c e e d t h e e v e n t u a l e q u i l i b r i u m e n e r g y l e v e l a t t a i n e d a t t h o s e f r e q u e n c i e s . F o r Y= l , t h e JONSWAP e q u a t i o n r e d u c e s t o t h e PM s p e c t r u m w h i c h i m p o s e s f p = 0 . 1 4 ( g / U ) and a=0 .0081 . F o r Y>1. the maximum e f f e c t i s a t f = f p ( s i n c e c ( f ) = l ) and d i m i n i s h e s e x p o n e n t i a l l y w i t h i n c r e a s i n g | f - f | . H a s s e l m a n n e t a l . (1976) r e p o r t t h a t most w i n d - s e a s p e c t r a - 16 - adhe re t o t h e JONSWAP f o r m w i t h a v e r a g e p a r a m e t e r v a l u e s o f Y=3.3, 03=0.07 and a f e = 0 . 0 9 , i n d e p e n d e n t o f f e t c h . T h i s f o r m o f E i s c a l l e d t h e mean JONSWAP s p e c t r u m . Discrete Forms The second g e n e r a l s o l u t i o n t e c h n i q u e i n v o l v e s t h e d i s c r e t i z a t i o n o f F ( f , 0 ) . The d i r e c t i o n a l dependence i s a l w a y s d i v i d e d i n t o a number o f e q u a l s e c t o r s s u c h t h a t MA8 = 2-rrand t y p i c a l v a l u e s o f M a r e 1 2 , 1 6 , o r 24 t o g i v e c o r r e s p o n d i n g r e s o l u t i o n s o f 3 0 ° , 2 2 . 5 ° , and 1 5 ° . D i s c r e t i z a t i o n of f r e q u e n c y i s o f t e n d o n e i n a s i m i l a r w a y : NAf = ^ f m a x ~ f m i n ^ » i n w n i c n c a s e £ m ± n i s t y p i c a l l y 0 .05 Hz (T=20 s ) , f m o v i s 0 .2 Hz (T=5 s ) , a n d A f=0 .01 H z . The TO ci x d i s a d v a n t a g e o f t h i s method i s t h a t t he r e s o l u t i o n o f t h e s p e c t r u m i s c o a r s e s t i n t h e l o w e r f r e q u e n c i e s . I n m o d e r a t e a n d s e v e r e s e a - s t a t e s , m o s t o f t h e s p e c t r a l e n e r g y w i l l be i n t h i s p o o r l y d e f i n e d p a r t o f t h e s p e c t r u m . A l o g i c a l a l t e r n a t i v e i s t o i m p o s e e q u a l p e r i o d e l e m e n t s : NAT = ( T m a x ~ T m ^ n ) . F o r t h e same f r e q u e n c y ( p e r i o d ) l i m i t s a n d t h e same number o f d i s c r e t e e l e m e n t s , the l o w f r e q u e n c y r e s o l u t i o n c a n be g r e a t l y i m p r o v e d as i l l u s t r a t e d i n F i g . 2 .3 . T h i s means t h a t s p e c t r a l r e s o l u t i o n c a n , and s h o u l d , be t a i l o r e d t o more p r e c i s e l y s p e c i f y t h e s h a p e a n d p e a k o f t h e s p e c t r u m i n t h o s e s e a - s t a t e s o f most i m p o r t a n c e f o r t h e m o d e l l i n g a p p l i c a t i o n . 2.3 Other Considerations 2.3.1 Wind-Sea and Swell One i m p o r t a n t c o n s i d e r a t i o n i n b u i l d i n g a m o d e l o f t h e e n e r g y b a l a n c e e q u a t i o n ( 2 . 1 ) i s t h e p a r a m e t e r i z a t i o n o f l o c a l l y g e n e r a t e d w i n d - s e a a n d o f s w e l l ene rgy t h a t p r o p a g a t e s i n t o t he a r e a o f i n t e r e s t f r o m a f a r . I n t he m o d e l l i n g c o n t e x t , a s w e l l wave may be d e f i n e d as any s e a component whose phase v e l o c i t y exceeds t h a t o f t h e l o c a l w i n d , s i n c e i t i s assumed t h a t no a t m o s p h e r i c i n p u t can be a b s o r b e d u n d e r t h e s e c o n d i t i o n s . B e c a u s e t h e s e a r e f r e e l y p r o p a g a t i n g w a v e s , t h e e n e r g y c o n s e r v a t i o n e q u a t i o n i s r e d u c e d t o o n l y a n a d v e c t i o n e q u a t i o n w i t h e n e r g y s o u r c e te rms s e t t o z e r o . S i n c e t h e r e i s no p a r a m e t r i c m o d e l o f s w e l l , t h e s o l u t i o n t e c h n i q u e c h o s e n i s e i t h e r a f i n i t e d i f f e r e n c i n g a p p r o a c h ( t o h i g h - o r d e r a c c u r a c y t o c o n t r o l n u m e r i c a l d i s p e r s i o n ) o r t he method o f c h a r a c t e r i s t i c s . F r e q u e n t l y t h e l a t t e r scheme, w h i c h i s a r a y t r a c i n g t e c h n i q u e , i s e m p l o y e d b e c a u s e s h a l l o w w a t e r r e f r a c t i v e e f f e c t s a r e r e a d i l y i n c o r p o r a t e d . - 17 - Period (s) 28 20 16 13 1110 9 8 7 6 5 I y-1• i • i -1 • i • itt>11• 1111. • . . t . . . . . . . . . 1 1 1 1 1 1 1 1 1 i i * ' 1 * * * ' • 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 Frequency (Hz) Period (s) 28 20 1 6 1 3 1 1 1 0 9 8 7 6 5 A • m i l i u m 11 I . I . I . i . I . I . . i . • . . i i i i in c c u O o _ O - 1 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 Frequency (Hz) F i g . 2.3 A c o m p a r i s o n o f s p e c t r a l f r e q u e n c y d i s c r e t i z a t i o n i n e q u a l f r e q u e n c y and e q u a l p e r i o d i n c r e m e n t s f o r N=15, f m i n = 0 - 0 5 H z a n d f m a x = 0 - 2 H z - - 18 - I n p r i n c i p l e , s w e l l e n e r g y c o u l d be i n p u t as a f u n c t i o n o f t i m e a l o n g t he wave mode l b o u n d a r i e s , bu t t h e i n f o r m a t i o n t o do s o , F s w e n ( f > 9 ) » i s n o t g e n e r a l l y a v a i l a b l e . T h i s r e p r e s e n t s a r e s t r i c t i o n on l i m i t e d - a r e a wave m o d e l l i n g (as c o n t r a s t e d w i t h h e m i s p h e r i c o r g l o b a l o c e a n b a s i n a p p l i c a t i o n s ) s i n c e s w e l l t h a t w o u l d be g e n e r a t e d by o t h e r s t o r m s y s t e m s o u t s i d e t he m o d e l ' s p h y s i c a l d o m a i n c a n n o t be a c c o u n t e d f o r . I n some c a s e s t h i s o m i s s i o n may c a u s e s i g n i f i c a n t e r r o r s i n wave h i n d c a s t r e s u l t s . The f r e q u e n c y o f s w e l l w a v e s i s g e n e r a l l y l o w . On b o t h t h e e a s t a n d w e s t c o a s t s of Canada wave measurements c a n c o n t a i n s i g n i f i c a n t e n e r g y i n t h e T=20 t o 30 s r a n g e t h a t i s a s s o c i a t e d w i t h t h e p a s s a g e o f i n t e n s e s t o r m s y s t e m s . I t i s i m p o r t a n t , t h e r e f o r e , t o r e s o l v e t h i s f r e q u e n c y r a n g e i f s u c h m e t e o r o l o g i c a l e v e n t s a r e t o be a c c u r a t e l y m o d e l l e d . The a d v a n t a g e o f t h e e q u a l p e r i o d s p e c t r a l d i s c r e t i z a t i o n ( F i g . 2.3) i s e v i d e n t i n t h i s c a s e . 2 . 3 . 2 S h a l l o w W a t e r E f f e c t s I n t h e i n t e r e s t s o f c o m p u t a t i o n a l e f f i c i e n c y , most wave h i n d c a s t s a r e r u n as deep w a t e r a p p r o x i m a t i o n s u n l e s s a c c u r a t e n e a r - s h o r e s e a - s t a t e c o n d i t i o n s a r e r e q u i r e d . I n t h a t e v e n t , t h e u s u a l a p p r o a c h i s t o s o l v e t h e d e e p w a t e r e q u a t i o n s o n a c o a r s e g r i d w h o s e p r i m a r y d o m a i n i s t h e d e e p o c e a n and t o u s e t h e s e r e s u l t s a s b o u n d a r y c o n d i t i o n s f o r a n e s t e d , f i n e - g r i d s h a l l o w w a t e r s o l u t i o n d o m a i n . An e x a m p l e o f a n e s t e d g r i d a r r a n g e m e n t i s shown i n F i g . 2.4 f o r the wes t c o a s t h i n d c a s t by S e a c o n s u l t (Hodg ins and N i k l e v a , 1986) . P h y s i c a l m e c h a n i s m s t h a t may be a c c o u n t e d f o r i n s h a l l o w w a t e r m o d e l s a r e b a t h y m e t r i c r e f r a c t i o n , wave s h o a l i n g , e n e r g y d i s s i p a t i o n by bo t t om f r i c t i o n and s h a l l o w w a t e r w a v e b r e a k i n g . S p e c t r a l r e f r a c t i o n a n d s h o a l i n g m o d e l s d e s c r i b e t he e v o l u t i o n o f wave e n e r g y a l o n g c h a r a c t e r i s t i c r a y s " o r t h o g o n a l t o w a v e c r e s t s . Wave b r e a k i n g i s m o d e l l e d by i m p o s i t i o n o f a d e p t h - l i m i t e d s a t u r a t e d s p e c t r a l f o r m . A s e c o n d a r y b e n e f i c i a l e f f e c t o f a f i n e - g r i d n e s t e d mode l i s i m p r o v e d l a n d f o r m r e s o l u t i o n t o g i v e b e t t e r m o d e l l i n g o f s h e l t e r i n g c o n d i t i o n s . T h i s c a n be a c r i t i c a l c o n s i d e r a t i o n i n h i n d c a s t i n g n e a r - s h o r e s e a - s t a t e s i n a r e a s l i k e Queen C h a r l o t t e Sound o r H e c a t e S t r a i t . 2 . 3 . 3 Wind I n p u t A l l w a v e g r o w t h m o d e l s e x p e c t w i n d i n p u t as a t i m e - s e r i e s o f n e a r - s u r f a c e speed and d i r e c t i o n (o r v e c t o r componen ts ) . These w inds may be d e r i v e d f r o m s u r f a c e p r e s s u r e d a t a , o r t h e y may be g e n e r a t e d by a n u m e r i c a l w e a t h e r p r e d i c t i o n m o d e l , a n d e i t h e r o f t h e s e may i n c o r p o r a t e d i r e c t m e a s u r e m e n t s . - 19 - F i g . 2 . 4 A c o a r s e deep w a t e r h i n d c a s t g r i d w i t h a n e s t e d s h a l l o w w a t e r mode l g r i d as used f o r a h i n d c a s t o f waves on t he w e s t c o a s t o f B r i t i s h C o l u m b i a . ( F r o m H o d g i n s a n d N i k l e v a , 1 9 8 6 ) . - 20 - There a r e two key a s p e c t s t o s p e c i f i c a t i o n o f t he w i n d f o r c i n g t e r m : s c a l e s o f r e s o l u t i o n ( b o t h t e m p o r a l and s p a t i a l ) and t he a c t u a l p a r a m e t e r i z a t i o n o f w i n d f o r c e . G e n e r a l l y h i n d c a s t w inds a r e d e r i v e d f rom 6 - h o u r l y s u r f a c e p r e s s u r e c h a r t s as a g r a d i e n t w i n d t h a t i s t h e n r e d u c e d t o a n e a r - s u r f a c e v e c t o r . T h i s t i m e i n t e r v a l i m p l i e s a n i n t e r p o l a t i o n down t o t h e m o d e l i n t e g r a t i o n t i m e s t e p w h i c h i s t y p i c a l l y o f t h e o r d e r o f one h o u r . I n t h e t i m e d o m a i n , v e c t o r i a l i n t e r p o l a t i o n o f t h e w i n d v e c t o r U i s common, b u t may no t be v e r y a c c u r a t e I n r a p i d l y t u r n i n g w i n d f i e l d s . I f p r e s s u r e c h a r t s a r e d i g i t i z e d i n l e n g t h i n c r e m e n t s t h a t a r e much l e s s t h a n t he mode l g r i d s c a l e and w i t h f a i r l y f i n e i s o b a r s p a c i n g (1 t o 4 mb), t h e n s p a t i a l i n t e r p o l a t i o n o f t h e w i n d f i e l d s w i l l be r e a s o n a b l y a c c u r a t e . A t t e m p t s t o u s e h i s t o r i c a l g r i d d e d w i n d f i e l d s a s a r c h i v e d f r o m a NWP mode l a r e u s u a l l y v e r y u n s u c c e s s f u l b e c a u s e t he t y p i c a l l y c o a r s e s p a t i a l r e s o l u t i o n (o f t he o r d e r o f 400 km) i s i n a d e q u a t e t o embody t he a c t u a l g r a d i e n t s i n w i n d speed (and p o s s i b l y d i r e c t i o n ) . S i n c e t h e e n e r g y i n p u t t e r m S i n i s n o r m a l l y f o r m u l a t e d i n t e r m s o f t h e f r i c t i o n v e l o c i t y a t t h e s e a s u r f a c e u * , the i n p u t w i n d U must be c o n v e r t e d by t h e wave m o d e l t o u * as u£ = C D U 2 ( 2 . 1 0 ) w h e r e t h e d r a g c o e f f i c i e n t C Q i s a f u n c t i o n o f U. G a r r a t t ( 1 9 7 7 ) f o u n d a g e n e r a l dependence o f t h e d r a g c o e f f i c i e n t on w i n d speed o f t h e f o r m C D = 0 . 5 1 x l O ~ 3 U 0 ' 4 6 ( 2 . 1 1 ) Thus u * w i t h t h i s f o r m u l a t i o n i s p r o p o r t i o n a l t o a p p r o x i m a t e l y U ^ 4 . T h e r e a r e s e v e r a l o t h e r f o r m u l a t i o n s w i t h d i f f e r e n t d e p e n d e n c i e s on U s u c h as t h e one p u b l i s h e d by L a r g e and Pond ( 1 9 8 1 ) : C D = ( 0 . 4 9 + 0 . 0 6 5 U ) * 1 0 ~ 3 ( 2 . 1 2 ) w h e r e U i s t h e w i n d s p e e d a t 10 m e l e v a t i o n a n d i s d e f i n e d i n t h e r a n g e 111U£25 m / s . R e c e n t l y a n u m b e r o f t h e s e f o r m u l a e h a v e b e e n c o m p a r e d by H s u (1986) and h i s f i n d i n g s a r e i l l u s t r a t e d i n F i g . 2 . 5 . P r i o r t o a b o u t 1 9 7 0 , w a v e m o d e l f o r m u l a t i o n s may h a v e a s s u m e d t h a t Cp was a c o n s t a n t . I f s u c h a m o d e l w e r e w e l l - c a l i b r a t e d f o r l o w a n d m o d e r a t e w i n d s p e e d s , i t w o u l d i n c r e a s i n g l y u n d e r - e s t i m a t e wave g r o w t h as w i n d s s t r e n g t h e n . - 21 - 1 Kondo.1975 2 Qarratt,1977 3 Smith, 1980 4 Amorocho and 0aVrtas,19S1 5 Large and Pond, 1981 6 Don*lan,1982 7 Wu.1982 8 Graf at al.,1984 9 Thl« atudy >8 10 15 20 U10,(nM-') 25 30 F i g . 2.5 Variations in the drag c o e f f i c i e n t as a function of the 10-m elevation wind speed. (From Hsu, 1986). - 22 - B e c a u s e t he w ind v e c t o r v a r i e s w i t h e l e v a t i o n a b o v e t h e s e a s u r f a c e , a g i v e n m o d e l w i l l be c a l i b r a t e d t o a c c e p t i n p u t a t a p a r t i c u l a r r e f e r e n c e e l e v a t i o n — 10 m and 19.5 m a r e common s t a n d a r d s . I f w i n d i n p u t i s o n l y a v a i l a b l e a t o t h e r t h a n t he r e f e r e n c e l e v e l , i t must f i r s t be c o n v e r t e d by a p p l i c a t i o n o f an a t m o s p h e r i c boundary l a y e r m o d e l . The s i m p l e s t o f t h e s e i s an e m p i r i c a l r e l a t i o n s h i p s u c h as where U"M i s t h e measured w i n d speed a t e l e v a t i o n h , U R i s t he w ind speed a t t h e r e f e r e n c e l e v e l h r , and A i s an e m p i r i c a l c o n s t a n t a p p r o x i m a t e d by Det n o r s k e V e r i t a s (1982) as A c c u r a t e h i n d c a s t i n g r e q u i r e s more p r e c i s e a t m o s p h e r i c b o u n d a r y l a y e r m o d e l s s u c h a s t h e m e t h o d u s e d i n t h e C M C f o r e c a s t s p e c t r a l w i n d m o d e l ( D e l a g e , 1985) . The same p r o c e d u r e i s a l s o r e q u i r e d t o d e r i v e s u r f a c e w inds f r o m t h e s u r f a c e p r e s s u r e d a t a v i a t he g r a d i e n t w i n d . W i t h t h i s me thod , t he f r i c t i o n v e l o c i t y u * i s d e t e r m i n e d as a f u n c t i o n o f t he f r e e - a t m o s p h e r e ( o r g r a d i e n t ) w i n d u Q , t he a i r c o l u m n s t a b i l i t y w h i c h depends on t h e u p p e r a i r t e m p e r a t u r e a n d t h e s e a s u r f a c e t e m p e r a t u r e , a n d a n e m p i r i c a l f o r m f o r t h e s u r f a c e r o u g h n e s s l e n g t h ( c o n c e p t u a l l y e q u i v a l e n t t o CQ). A l o g a r i t h m i c p r o f i l e o f U i s a s s u m e d , a s a f u n c t i o n o f a i r c o l u m n s t a b i l i t y a n d s u r f a c e r o u g h n e s s l e n g t h , t o e x t r a c t w i n d speed U(z) a t t he r e q u i r e d r e f e r e n c e e l e v a t i o n z . F o r t h e n e u t r a l s t a b i l i t y c a s e , t h e s o l u t i o n i s d e t e r m i n e d f r o m where < i s von K a r m a n ' s c o n s t a n t ( 0 . 4 ) zQ i s t h e s u r f a c e r o u g h n e s s l e n g t h ( 0 . 3 5 u * / g ) g i s t h e g r a v i t a t i o n a l c o n s t a n t T h e a n g u l a r r o t a t i o n o f t h e w i n d v e c t o r a t t h e s e a s u r f a c e d u e t o t h e f r i c t i o n a l boundary l a y e r i s a l s o c a l c u l a t e d and depends on t h e same s t a b i l i t y a n d s u r f a c e r o u g h n e s s l e n g t h p a r a m e t e r s as w e l l a s t h e b o u n d a r y l a y e r t h i c k n e s s . ( 2 . 1 3 ) 0 . 1 5 . u * = K u 0 { l n ( l n ( z / z 0 ) ) } 1 U (z ) = (U*/K) l n ( l n ( z / z 0 ) ) ( 2 . 1 4 ) - 2 3 - 3 . 0 SPECTRAL WAVE MODELS S p e c t r a l m o d e l s h a v e e v o l v e d w i t h imp rovemen ts i n u n d e r s t a n d i n g o f t he p h y s i c s o f w a v e g e n e r a t i o n . The " f i r s t g e n e r a t i o n " m o d e l s , d a t i n g f r o m t h e 1 9 6 0 » s , a r e d i s c r e t e s p e c t r a l f o r m u l a t i o n s t h a t c o n s i d e r t h e f r e q u e n c y b i n s t o be i n d e p e n d e n t o f e a c h o t h e r . H a s s e l m a n n e t a l . (1973) h a v e shown no t o n l y t h a t i n t e r a c t i o n s b e t w e e n t h e f r e q u e n c i e s ( w a v e - w a v e i n t e r a c t i o n s ) a r e i m p o r t a n t , bu t t h a t d u r i n g the g r o w t h s t a g e o f w i n d s e a s , t h e y a r e t he p r i n c i p a l s o u r c e o f e n e r g y on t h e l o w f r e q u e n c y f o r w a r d f a c e o f t h e s p e c t r u m . The " s e c o n d g e n e r a t i o n " o f d i s c r e t e s p e c t r a l m o d e l s m i m i c k t h e s e i n t e r a c t i o n s by r e d i s t r i b u t i o n o f t he e n e r g y o v e r t h e f r e q u e n c y b i n s a f t e r e a c h m o d e l l i n g t i m e s t e p . I t was t h e d i f f i c u l t y o f g e n e r a l i z i n g t h i s e n e r g y r e d i s t r i b u t i o n mechan ism f o r a r b i t r a r y s p e c t r a t h a t g a v e i m p e t u s t o t h e p a r a m e t r i c s p e c t r a l m o d e l l i n g method . These m o d e l s t oo a r e " s e c o n d g e n e r a t i o n " , h o w e v e r , i n t h a t t h e y h a v e had to i n c l u d e j i p r i o r i r e s t r i c t i o n s on s p e c t r a l shape b e c a u s e an a c c u r a t e r e p r e s e n t a t i o n o f t h e n o n - l i n e a r w a v e - w a v e i n t e r a c t i o n p r o c e s s t h a t c a n be e v a l u a t e d e c o n o m i c a l l y i s no t a v a i l a b l e . The " t h i r d g e n e r a t i o n " mode l p r o p o s e d by t h e WAM Group (Komen, 1984) i s b a s e d on a c o m p l e t e r e p r e s e n t a t i o n of a l l s o u r c e te rms i n t h e e n e r g y c o n s e r v a t i o n e q u a t i o n t h a t r e m o v e s a l l a p r i o r i r e s t r i c t i o n s o n s p e c t r a l s h a p e . Some p r e l i m i n a r y r e s u l t s o f t h i s t h i r d g e n e r a t i o n m o d e l h a v e b e e n p r e s e n t e d by Komen, bu t f u l l i m p l e m e n t a t i o n i s s t i l l a few y e a r s away . I n t h i s c h a p t e r , t h r e e mode l c l a s s e s a r e d e s c r i b e d w i t h s p e c i f i c d e t a i l s drawn a p a r t i c u l a r wave m o d e l . The m o d e l s r e p r e s e n t e d a r e as f o l l o w s : M o d e l C l a s s M o d e l Name R e f e r e n c e ( s ) s e c o n d g e n e r a t i o n p a r a m e t r i c s p e c t r a l The HYPA M o d e l H a s s e l m a n n e t a l . ( 1976 ) G u n t h e r e t a l . ( 1 9 7 9 a , b ) f i r s t g e n e r a t i o n d i s c r e t e s p e c t r a l The ODGP M o d e l C a r d o n e e t a l . ( 1975 ) s e c o n d g e n e r a t i o n d i s c r e t e s p e c t r a l The ADWAVE M o d e l R e s i o ( 1 9 8 1 , 1982 , 1985) - 24 - 3 .1 P a r a m e t r i c S p e c t r a l Wave M o d e l s The re a r e many p a r a m e t r i c s p e c t r a l wave m o d e l s t h a t h a v e been d e v e l o p e d f o r r e s e a r c h , o p e r a t i o n a l wave f o r e c a s t i n g , and wave h i n d c a s t i n g . Of t h e s e , t h e b e s t k n o w n a r e b a s e d on t h e w o r k o f K. H a s s e l m a n n and h i s c o - w o r k e r s ( H a s s e l m a n n e t a l . , 1 9 7 6 ; G u n t h e r e t a l . , 1 9 7 9 a , b ) and t h e f o l l o w i n g d i s c u s s i o n i s based on t h e i r t e c h n i q u e s . The b a s i c c o n c e p t i n p a r a m e t r i c s p e c t r a l wave m o d e l l i n g i s t o s o l v e t h e e n e r g y b a l a n c e e q u a t i o n by a s s u m i n g a p a r a m e t r i c f o rm f o r the ene rgy s p e c t r u m E ( f ) . The p a r a m e t r i c s p e c t r u m E i s e x p r e s s e d i n t e r m s o f a s e t o f s p e c t r a l p a r a m e t e r s a.^ t h a t a r e r e a d i l y s p e c i f i e d b a s e d on t h e r e s u l t s o f w a v e measurement e x p e r i m e n t s . To o b t a i n s o l u t i o n s t o ( 2 . 1 ) n a t u r a l l y i n v o l v e s s e v e r a l s i m p l i f y i n g a s s u m p t i o n s . H a s s e l m a n n e t a l . ( 1 9 7 6 ) i n t e g r a t e d ( 2 . 1 ) o v e r d i r e c t i o n t o o b t a i n m o d e l e q u a t i o n s i n t h e f o r m o f ( 2 . 2 ) a n d u s e d t h e JONSWAP p a r a m e t r i c s p e c t r a l f o r m f o r E ( f ) t o s u b s t i t u t e f o r E ( f ) . T h i s s u b s t i t u t i o n y i e l d s e q u a t i o n s o f t h e f o r m 9 a i + D i j k - ^ i = T i ^ 3 - 1 > 3t 3x where a ^ = t h e s e t o f f i v e JONSWAP p a r a m e t e r s {f p , a , y , 0"a, o^} D j j k = t h e wave p r o p a g a t i o n v e l o c i t i e s ( f u n c t i o n s o f a^) T^ = t h e s o u r c e te rms ( f u n c t i o n s o f a^) H a s s e l m a n n e t a l . (1976) a r g u e d t h a t o f t h e f i v e s p e c t r a l p a r a m e t e r s o n l y t h e f i r s t two , f and cx, a r e n e c e s s a r y as m o d e l l i n g v a r i a b l e s because o f t h e shape i n v a r i a n c e o f t h e s p e c t r u m . T h i s s i m p l i f i c a t i o n r e s u l t s i n a p r o g n o s t i c m o d e l f o r wave e n e r g y i n terras of v= f p U / g ( n o n - d i m e n s i o n a l peak f r e q u e n c y ) and ot w i t h e q u a t i o n s s i m i l a r i n f o r m t o ( 3 . 1 ) . G u n t h e r e t a l . ( 1 9 7 9 ) i n c l u d e d a l l f i v e JONSWAP p a r a m e t e r s p a c e s . I n o r d e r t o s o l v e t h e s e e q u a t i o n s , e m p i r i c a l f o r m u l a e a r e s u b s t i t u t e d f o r t h e w i n d i n p u t and d i s s i p a t i o n p r o c e s s e s , and S ^ i s p a r a m e t e r i z e d by a l o o k - u p t a b l e o f e x a c t s o l u t i o n s t o t h e B o l t z m a n i n t e g r a l f o r a r a n g e o f s p e c t r a E ( f ; a ^ ) . I n p r a c t i c e , t h i s a p p r o a c h i s s i m p l i f i e d by u s i n g t h e s o l u t i o n o f o n l y t h e m e a n J O N S W A P s p e c t r u m . T h e r e s u l t i s q u i t e s i m p l e S n ^ p a r a m e t e r i z a t i o n s : - 25 - S v - 0 . 5 4 a 2 f 2 S a = - 5 . 0 a 3 f p S y = - 1 6 . 0 ( Y - 3 . 3 ) a 2 f p S o a = - [ 2 5 . 5 ( a a - 0 . 0 7 ) - 0 . 5 ( a b - 0 . 0 9 ) ] a 2 f p S a b - - [ 2 5 . 5 ( a b - 0 . 0 9 ) - 0 . 5 ( a a - 0 . 0 7 ) ] a 2 f The c o n s t a n t c o e f f i c i e n t o f S v i s c a l i b r a t e d by e m p i r i c a l r e l a t i o n s o f f p and a as f u n c t i o n s o f n o n - d i m e n s i o n a l f e t c h . G u n t h e r e t a l . (1979b) r e p o r t u s i n g a v a l u e of - 0 . 5 8 6 f o r t he N o r t h S e a . The e x p r e s s i o n f o r S i n i s g i v e n by G u n t h e r e t a l . (1979b) as S . = 5 . 0 2 2 x l 0 " 3 v 4 / 3 a f i n p and o c c u r s i n o n l y t h e a e q u a t i o n o f t h e s e t s p e c i f i e d by (3 .1 ) . The c o n s t a n t c o e f f i c i e n t i s a l s o d e t e r m i n e d f r om t h e e m p i r i c a l f e t c h dependence o f f p and a . I n t h e m o d e l s d e s c r i b e d by H a s s e l m a n n e t a l . ( 1 9 7 6 ) a n d G u n t h e r e t a l . ( 1979b) , i t has been assumed t h a t t h e s p e c t r a l doma in o f i n t e r e s t l i e s b e l o w the r e g i o n o f S ^ g i n f l u e n c e so t h a t t h i s d i s s i p a t i o n t e r m was o m i t t e d . The e q u a t i o n s a r e s o l v e d o n a s p a t i a l g r i d a s a f u n c t i o n o f t i m e by a p r e d i c t o r - c o r r e c t o r m e t h o d ( G u n t h e r e t a l . , 1 9 7 9 a ) . The s o l u t i o n s c h e m e i s r e p o r t e d t o be s e c o n d o r d e r a c c u r a t e i n s p a c e a n d t i m e a n d t h e s t a b i l i t y c r i t e r i o n r e q u i r e s t h a t t he r a t i o o f g r i d s p a c i n g t o t i m e s t e p A x / A t be l a r g e r t h a n t h e l a r g e s t g r o u p v e l o c i t y o f w a v e s i n t h e s o l u t i o n d o m a i n . I n t h e G u n t h e r e t a l . ( 1 9 7 9 a ) a p p l i c a t i o n , a 2 km g r i d a n d a 5 m i n i n t e g r a t i o n s t e p w e r e u s e d t o m o d e l s e l e c t e d p e r i o d s o f t h e JONSWAP e x p e r i m e n t . The c o s i n e - s q u a r e d d i r e c t i o n a l s p r e a d i n g f u n c t i o n , i n d e p e n d e n t o f f , was assumed t o a p p l y and t he mean wave d i r e c t i o n was s p e c i f i e d by t he a v e r a g e l o c a l w i n d d i r e c t i o n . The e n t i r e p a r a m e t r i c m o d e l i s t h u s a s t r a i g h t f o r w a r d s e t o f f i v e e q u a t i o n s i n t he f i v e p a r a m e t r i c s p e c t r a l unknowns a . , . Once t h e s e a ^ a r e d e t e r m i n e d , t h e s p e c t r u m E ( f ) i s s p e c i f i e d , a n d a p p l i c a t i o n o f t h e d i r e c t i o n a l s p r e a d i n g f u n c t i o n p r o v i d e s F ( f , 9). One s e v e r e l i m i t a t i o n o f p a r a m e t r i c s p e c t r a l m o d e l s i s t h a t t h e y e x c l u d e any s w e l l e n e r g y c o n t r i b u t i o n s t o t h e wave s p e c t r u m . The c o u p l i n g o f a p a r a m e t r i c s p e c t r a l w i n d - s e a m o d e l a n d a s w e l l m o d e l i s r e f e r r e d t o as a h y b r i d p a r a m e t r i c s p e c t r a l m o d e l . The d i s a d v a n t a g e s w i t h t h e h y b r i d m o d e l s a r e w e l l - known (Gun the r e t a l . , 1979b; J a n s s e n e t a l . , 1984) . The f i r s t p r o b l e m i s the l a r g e c o m p u t a t i o n a l h o u s e k e e p i n g r e q u i r e m e n t t o c o n v e r t b e t w e e n r a y c h a r a c t e r i s t i c s a n d t h e w i n d - s e a f i x e d g r i d mesh t o i n c o r p o r a t e a p p r o p r i a t e - 26 - c o n t r i b u t i o n s o f s w e l l e n e r g y t o t h e c a l c u l a t e d E ( f ) s o l u t i o n — b u t t h i s i s c h i e f l y an i n c o n v e n i e n c e . The o t h e r d i s a d v a n t a g e a r i s e s f r o m t h e l a r g e number o f somewhat a r b i t r a r y a s s u m p t i o n s r e q u i r e d t o d i s t i n g u i s h w i n d - s e a f r o m s w e l l a n d t o d e a l w i t h t h e i n t e r a c t i o n o f t h e t w o . T h i s d i f f i c u l t y i s much more f u n d a m e n t a l and r a t h e r poo r p e r f o r m a n c e o f t h e D u t c h o p e r a t i o n a l f o r e c a s t i n g mode l has been a t t r i b u t e d t o i t ( J a n s s e n e t a l . , 1984) . F o r e x a m p l e , G u n t h e r e t a l . (1979b) made t h e f o l l o w i n g a s s u m p t i o n s . (1) W i n d - S e a Becoming S w e l l : I f f p < 0 . 1 3 g / U ( i . e . b e l o w t he PM v a l u e o f f f o r a 10 m e l e v a t i o n w i n d ) t h e n f p = f p ( P M ) ; a i s a d j u s t e d t o c o n s e r v e e n e r g y i n t h e w i n d - s e a p o r t i o n o f E ( f ) a b o v e f p ( P M ) ; e n e r g y a t f < f Q ( a c u t - o f f f r e q u e n c y ) becomes s w e l l where f i s d e t e r m i n e d n u m e r i c a l l y t o m a i n t a i n t h e same t o t a l e n e r g y i n E ( f ) b e f o r e and a f t e r s w e l l s e p a r a t i o n ; c o s i n e - s q u a r e d d i r e c t i o n a l s p r e a d i n g o f s w e l l i s i n c l u d e d . (2) S w e l l Becoming W i n d - S e a : I f s w e l l f r e q u e n c y e x c e e d s 0 . 9 f p t h e n s w e l l i s a b s o r b e d i n s t a n t a n e o u s l y i n t o t h e w i n d - s e a s p e c t r u m w i t h o u t r e g a r d to d i r e c t i o n ; e n e r g y i s c o n s e r v e d by a d j u s t i n g f w i t h a and Y f i x e d . (3 ) S w e l l I n t e r a c t i n g W i t h W i n d - S e a : S w e l l may e x i s t a t f r e q u e n c i e s s u b j e c t t o wave g r o w t h , bu t o u t s i d e t h e range o f n o n - l i n e a r i n t e r a c t i o n , i . e . f Q < f < 0 . 9 f p ( d e f i n i n g f Q a s g/2TTUcos^ w h e r e i s t h e a n g l e s e p a r a t i o n be tween w i n d and s w e l l d i r e c t i o n s ) ; t h e M i l e s - P h i l l i p s g r o w t h mechan ism i s assumed s u c h t h a t S ^ n = 2irf [ ( f - f Q ) / f Q ] [c p a / p ^ ] f o r f > f Q and o t h e r w i s e S i n = 0 ( w h e r e p a = d e n s i t y o f a i r , p w = d e n s i t y o f w a t e r and c = 0 . 0 5 ) . T h e s e a s s u m p t i o n s by G u n t h e r e t a l . ( 1 9 7 9 b ) a r e d e s c r i b e d as i n t u i t i v e , a n d h a v e no doub t been r e f i n e d , bu t t h e y do i l l u s t r a t e t h e f u n d a m e n t a l o b j e c t i o n t o p a r a m e t r i c s p e c t r a l wave m o d e l l i n g . 3 . 2 D i s c r e t e S p e c t r a l Wave M o d e l s D i s c r e t e s p e c t r a l w a v e m o d e l s a r e s o l u t i o n s o f (2 . 1 ) a c h i e v e d by f i n i t e d i f f e r e n c e t e c h n i q u e s i n w h i c h t h e s p e c t r u m F ( f ,9) i s d i v i d e d i n t o d i s c r e t e f r e q u e n c y and d i r e c t i o n doma ins . These m o d e l s h a v e t h r e e o b v i o u s a d v a n t a g e s : ( 1 ) t h e y do n o t r e l y on e m p i r i c a l s p r e a d i n g f u n c t i o n s t o r e d i s t r i b u t e w a v e e n e r g y by d i r e c t i o n , (2) t h e y do n o t i m p o s e a u n i v e r s a l p a r a m e t r i c s p e c t r a l shape e x c e p t when e n e r g y s a t u r a t i o n i s a c h i e v e d , and (3) t h e y t r e a t l o c a l w ind s e a and s w e l l as a combined phenomenon r a t h e r t h a n as a h y b r i d c o m b i n a t i o n of d i f f e r e n t wave t y p e s . - 27 - The v a r i o u s o p e r a t i o n a l and r e s e a r c h m o d e l s i n t h i s c l a s s a r e d i s t i n g u i s h e d f rom one a n o t h e r p r i m a r i l y by t h e i r p a r a m e t e r i z a t i o n o f t h e e n e r g y s o u r c e terra S ( f , 8 ) , and two f u n d a m e n t a l l y d i f f e r e n t f o r m u l a t i o n s h a v e d e v e l o p e d . One o f t h e s e , e x e m p l i f i e d by t he ODGP m o d e l (Cardone e t a l . , 1975) , bases S i n + S d g on e m p i r i c a l f o r m u l a t i o n s to d e s c r i b e wave i n i t i a t i o n by t u r b u l e n t a t m o s p h e r i c p r e s s u r e f l u c t u a t i o n s a t the s e a s u r f a c e and wave g r o w t h by t he M i l e s - P h i l l i p s i n s t a b i l i t y mechan ism. F u r t h e r m o r e , t he Cardone m o d e l assumes t h a t S -i i s a ' n l n e g l i g i b l e c o n t r i b u t o r t o t h e e n e r g y b a l a n c e . The o t h e r a p p r o a c h t o p a r a m e t e r i z a t i o n o f S ( f , 9 ) i s much more c l o s e l y b a s e d on t h e JONSWAP r e s u l t s i n c l u d i n g each o f the S±n, S d g and S n l t e r m s , and R e s i o ' s ( 1 9 8 1 , 1982) ADWAVE mode l i s t he most w e l l - d e v e l o p e d , w e l l - p u b l i s h e d e x a m p l e . 3 .2 . 1 The ODGP M o d e l The ODGP m o d e l was d e v e l o p e d by Ca rdone e t a l . (1975) u n d e r t he o i l i n d u s t r y s p o n s o r s h i p o f t h e O c e a n D a t a G a t h e r i n g P r o g r a m i n t h e G u l f o f M e x i c o . The i m p e t u s f o r t h e s t u d y was t o m o n i t o r w i n d a n d w a v e c o n d i t i o n s d u r i n g h u r r i c a n e s w i t h a v i e w t o b u i l d i n g a - c a l i b r a t e d w a v e h i n d c a s t i n g m o d e l t o d e s c r i b e t h e e v o l u t i o n o f h u r r i c a n e waves t h r o u g h o u t t h e G u l f . The ODGP wave mode l i s d e s c r i b e d by i t s a u t h o r s as " a r a t h e r s t r a i g h t f o r w a r d a p p l i c a t i o n and c a l i b r a t i o n of t he P i e r s o n - T i c k - B a e r (PTB) m o d e l " , p u b l i s h e d by P i e r s o n e t a l . (1966) . B e c a u s e t he wave m o d e l came t o be c o m m e r c i a l l y e x p l o i t e d by C a r d o n e , t he p u b l i s h e d d e t a i l s of i t s code and c a l i b r a t i o n p r o c e d u r e a r e s k e t c h y . Wave g r o w t h and d i s s i p a t i o n a r e a p p l i e d t o d o w n - w i n d s p e c t r a l c o m p o n e n t s a c c o r d i n g t o the e m p i r i c a l f o r m u l a e 2- S i n A ( f ,0 ,U) + B ( f , 6 , U ) . F ( f ,9) 1 - / F ( f , 9 ) \ wf,e,u); ( 3 . 2 ) where A i s t h e l i n e a r g r o w t h c o e f f i c i e n t w h i c h i n i t i a t e s t he e x c i t a t i o n o f g r a v i t y waves on a c a l m s e a s u r f a c e , B .F i s t h e e x p o n e n t i a l g r o w t h t e r m i n c o r p o r a t i n g t h e M i l e s - P h i l l i p s i n s t a b i l i t y mechan i sm, U i s t h e w i n d s p e e d , F ^ i s t h e s a t u r a t e d PM s p e c t r u m f o r a g i v e n w i n d speed m u l t i p l i e d by a d i r e c t i o n a l s p r e a d i n g f u n c t i o n G ( f , 9 ) . The f o r m u l a t i o n s o f t h e A and B c o e f f i c i e n t s a r e u n p u b l i s h e d . The A terra i s r e l a t i v e l y u n i m p o r t a n t , s e r v i n g o n l y t o i n i t i a t e s p e c t r a l ene rgy g r o w t h . - 28 - D i s s i p a t i o n o f upw ind s p e c t r a l components i s no t d e s c r i b e d by Ca rdone e t a l . (1975) , bu t i n the o r i g i n a l PTB m o d e l i t . h a d t h e f o r m : F d i s s = [exp(-78v^ 0fJ)]N (3.3) F o T h a t i s , d i s s i p a t i o n was a p p l i e d a s a d a m p i n g f u n c t i o n o n t h e u p w i n d components o f t he i n i t i a l s p e c t r u m F Q ( f , 6 ) . E q u a t i o n ( 3 . 3 ) i s c a l i b r a t e d t o be a p p l i e d a t 2 - h o u r l y i n t e r v a l s where ra0 i s t h e z e r o t h moment ( f t ) o f t h e w i n d s e a s p e c t r u m ( i . e . t h e h a l f - p l a n e down-w ind s p e c t r u m ) , N i s a f u n c t i o n of t h e d e v i a t i o n o f t h e wave component d i r e c t i o n f r om t h e l o c a l w i n d d i r e c t i o n and c a l c u l a t e d as N = 4 - [ | Q , - 6 W | - 1 8 0 ] / 1 5 , 6^ i s t he upw ind wave component d i r e c t i o n , 6 i s t h e l o c a l w i n d d i r e c t i o n , w D e t a i l s o f t he G u l f o f M e x i c o h i n d c a s t i l l u s t r a t e a n o t h e r d i f f e r e n c e be tween d i s c r e t e a n d p a r a m e t r i c s p e c t r a l m o d e l s , t h e l a r g e c o m p u t e r m e m o r y r e q u i r e m e n t s f o r t h e f i n i t e d i f f e r e n c e s o l u t i o n s t o ( 2 .1 ) . The G u l f o f M e x i c o s t u d y a r e a was r e p r e s e n t e d by 1265 d i s c r e t e p o i n t s a r r a n g e d i n a t r i a n g u l a r g r i d p a t t e r n on an i c o s a h e d r a l - g n o m o n i c map p r o j e c t i o n i n w h i c h s t r a i g h t l i n e s a r e g r e a t c i r c l e s and hence a r e d e e p w a t e r wave t r a j e c t o r i e s . The n o m i n a l g r i d s p a c i n g was 20 n a u t i c a l m i l e s . A t each of t h e s e g r i d p o i n t s a d i s c r e t e d i r e c t i o n a l s p e c t r u m was c a l c u l a t e d and s t o r e d t h a t was c o m p r i s e d o f 13 f r e q u e n c y b a n d s a n d 24 d i r e c t i o n b a n d s . I n a l l t h e n , a l m o s t 4 x 1 0 ^ c o m p o n e n t s o f s p e c t r a l e n e r g y w e r e d e t e r m i n e d a t e a c h h o u r l y t i m e s t e p . T h i s demand f o r c o m p u t e r memory a n d i n p u t / o u t p u t t o mass s t o r a g e d e v i c e s i s t he p r i m a r y l i m i t a t i o n t o i m p l e m e n t a t i o n o f s u c h wave mode l s on s m a l l c o m p u t e r s . F i g . 3.1 i s f r o m p u b l i s h e d r e s u l t s o f t he ODGP h u r r i c a n e C a m i l l e h i n d c a s t and i l l u s t r a t e s t h e p e r f o r m a n c e o f t h i s m o d e l . B a s e d on t h e h u r r i c a n e C a m i l l e r e s u l t s , t h r e e o t h e r h u r r i c a n e h i n d c a s t s a n d o n e t r o p i c a l s t o r m h i n d c a s t , Cardone e t a l . (1975) r e p o r t t h a t t he r o o t - m e a n - s q u a r e e r r o r i n the p r e d i c t e d maximum w a v e h e i g h t i s 4 .9 f e e t ( 1 . 5 m) a n d t h e b i a s i s +1.5 f e e t ( 0 . 5 m) i n s e a s t a t e s o f maximum wave h e i g h t r a n g i n g be tween 20 and 80 f e e t . Such v a l u e s r e p r e s e n t g e n e r a l l y s a t i s f a c t o r y a c c u r a c i e s , bu t t h e m e a s u r e m e n t - p r e d i c t i o n c o m p a r i s o n p a i r s a r e a l m o s t c e r t a i n l y t h e max imum m e a s u r e d a n d t h e maximum p r e d i c t e d wave h e i g h t s , w i t h o u t r e g a r d t o t h e t i m e o f o c c u r r e n c e of e i t h e r . - 2 9 - 32 24 u. 16 x 2 8 UJ X > •I < o E 32 z 19 STATION 2 • 002 0"2" • 801 Hindcoit 801 — Hindcoit 802 24 IC STATION 3 f * t t T I 1 1 1 * " 7 5 Hindcoit 874 * 3 Hindcoit 875 • 8M 1 i i I 0900 1000 1100 1200 1300 1400 1500 1600 l?0O TIME (COT) / t 1 . 1 / DlSSHHfH 1 1 1 ALABAMA I G/O'S.'.t 1 . 1 1 T LOUISIANA I \ l / ; i A noMtDA \ - _ _ V . T » - T R A N S W O R L D ^ " \ 7 9URW0OD-1T».B R I G 50 \ 5 a ? 2 \ A \ OOGf> S T A T I O N S » \ \ 0 so 100 \ v - S C A . t N . M l . \ CAMIUP S T O K - M T R A C K F i g . 3 .1 M e a s u r e d a n d h i n d c a s t s i g n i f i c a n t w a v e h e i g h t t i m e - s e r i e s a t t h r e e s t a t i o n s i n t h e G u l f o f M e x i c o d u r i n g H u r r i c a n e C a m i l l e . The h i n d c a s t s i t e numbers d e s i g n a t e g r i d p o i n t s c l o s e t o t h e m e a s u r e m e n t s i t e s . ( F r o m Cardone e t a l . , 1 9 7 5 ) . - 30 - 3 . 2 . 1 ADWAVE The ADWAVE h i n d c a s t m o d e l h a s e v o l v e d f r o m p r e c u r s o r s d e s c r i b e d by R e s i o ( 1 9 8 1 , 1 9 8 2 ) . I t s p r e s e n t f o r m u l a t i o n ( R e s i o , 1 9 8 5 ) i s a f i n i t e d i f f e r e n c e s o l u t i o n t o t h e c o m p l e t e e n e r g y b a l a n c e e q u a t i o n ( 2 . 1 ) i n w h i c h t h e s o u r c e terras i n c l u d e w i n d i n p u t , n o n - l i n e a r e n e r g y t r a n s f e r be tween wave f r e q u e n c i e s (wave-wave i n t e r a c t i o n ) and d i s s i p a t i o n . P r o p a g a t i o n i s done by t he method o f c h a r a c t e r i s t i c s w h i c h i s g e n e r a l i z e d f o r s h a l l o w w a t e r c a l c u l a t i o n s . The e n e r g y s p e c t r u m F ( f , 8 ) i s d i s c r e t i z e d i n f r e q u e n c y a n d d i r e c t i o n ( a s i l l u s t r a t e d s c h e m a t i c a l l y i n F i g . 3 .2 ) a t e a c h p o i n t on a r e g u l a r x - y g r i d a p p l i e d o v e r t he m o d e l l i n g doma in . P r o p a g a t i o n and t h e s o u r c e / s i n k mechanisms a r e c a l c u l a t e d as changes i n ene rgy i n e a c h o f t h e s e f and 6 e l e m e n t s . P r o p a g a t i o n The p r o p a g a t i o n o f wave ene rgy i s most r e a d i l y u n d e r s t o o d i n terras of a s i n g l e f r e q u e n c y - d i r e c t i o n e l e m e n t P ^ k * ^ ) i n a d e e p w a t e r l o c a t i o n . A t a f i x e d s o l u t i o n g r i d l o c a t i o n ( i , j ) t h e e n e r g y c o n t e n t a t t i m e n+At i n t h e d i s c r e t i z e d e l e m e n t F ( f 9 m ; i , j ) c a n o n l y h a v e o r i g i n a t e d f r o m t h e p o s i t i o n ( i + 6 x , j + 6 y ) a t t i m e n , a n d w i l l h a v e t r a v e l l e d a d i s t a n c e c A t a l o n g t h e c h a r a c t e r i s t i c r a y . A l o n g t h i s r a y t h e q u a n t i t y F . c . c „ i s c o n s e r v e d , f i s i n v a r i a n t a n d 0 o n l y c h a n g e s as t h e r a y m o v e s f r o m d e e p i n t o s h a l l o w w a t e r . The d e e p w a t e r ( c o n s t a n t 6 ) c a s e i s i l l u s t r a t e d i n F i g . 3 . 3 a . The e n e r g y c o n t e n t o f F ( f j c , 9 m ) a t ( i + 6 x , j + 6y) i s d e t e r m i n e d by b i l i n e a r i n t e r p o l a t i o n of F a t t i m e l e v e l n , f i r s t a t t h e g r i d i n t e r s e c t i o n ( i . e . a t p o i n t B i n F i g . 3 .3a ) a n d t h e n a t t h e r a y o r i g i n a t i o n p o i n t ( i + 6 x , j + 6 y ) ( i . e . a t p o i n t 0 i n F i g . 3 . 3 a ) . The same p r i n c i p l e a p p l i e s f o r t h e g e n e r a l i z e d d e p t h c a s e , b u t t h e c h a r a c t e r i s t i c r a y s a r e c u r v i l i n e a r a s d e p i c t e d i n F i g . 3 . 3 b . The b a c k r a y t r a c i n g t o l o c a t i o n p o i n t s 0 and B i s done i n s m a l l i n c r e m e n t a l s t e p s i n o r d e r t o v a r y c and c as f u n c t i o n s o f d e p t h . & S i n c e t h e m o d e l i s w r i t t e n i n t e r m s o f f i x e d d i s c r e t e f r e q u e n c i e s and d i r e c t i o n s w i t h d e p t h s p e c i f i e d i n d e p e n d e n t l y o f t i m e , a l l t h e i n t e r p o l a t i o n c o n s t a n t s ( l o c a t i o n of p o i n t s B and 0 ) , phase and g r o u p v e l o c i t i e s may be p r e - c a l c u l a t e d f o r e a c h g r i d p o i n t ( i , j ) a t e a c h f r e q u e n c y a n d d i r e c t i o n . T h i s p r e - p r o c e s s i n g a d d s v e r y s i g n i f i c a n t l y t o the compu te r memory r e q u i r e m e n t s , but speeds t he t i m e s t e p p i n g c a l c u l a t i o n phase o f t h e m o d e l . - 31 - F ( f,9) centre of frequency-direction element F(f,9) F i g . 3 . 2 R e p r e s e n t a t i o n o f a t w o - d i m e n s i o n a l s p e c t r u m . The uppe r p a n e l ( f r o m S a r p k a y a and I s a a c s o n , 1981) shows a p o r t i o n o f F ( f , 0 ) i n t h r e e - d i m e n s i o n a l r e l i e f a n d t h e l o w e r p a n e l s h o w s t h e d i s c r e t i z a t i o n o f t h a t s p e c t r u m i n i n c r e m e n t s o f f r e q u e n c y a n d d i r e c t i o n a s a p p l i e d i n ADWAVE. - 32 - I I I B position of wave | component at j beginning of time step position of wave component at end of time step interpolation along ray i F i g . 3 . 3 The b i l i n e a r i n t e r p o l a t i o n scheme i n ADWAVE (a) i n deep w a t e r and (b ) i n s h a l l o w w a t e r . - 33 - S o u r c e T e r m s : S i n + S n l + S d s H a s s e l m a n n e t a l . (1973) and R e s i o ( 1 9 8 1 , 1982) h a v e shown t h e o r e t i c a l l y and e m p i r i c a l l y t h a t i n d e e p w a t e r t h e b a l a n c e among t h e s o u r c e / s i n k t e r m s i n v o l v e s p r i m a r i l y t h e w i n d i n p u t S ^ n a n d t h e w a v e - w a v e i n t e r a c t i o n s S n ^ . The re i s a l s o a g r o w i n g r e s o u r c e o f measurements t o g i v e e m p i r i c a l e v i d e n c e t h a t t h i s i s a l s o so i n s h a l l o w w a t e r (Bouws e t a l . , 1985) . F u r t h e r e v i d e n c e o f t h e s u i t a b i l i t y o f t h e R e s i o ( 1 9 8 5 ) f o r m u l a t i o n f o r t h e w a v e - w a v e i n t e r a c t i o n mechan ism comes f r om a r e c e n t s t u d y s p o n s o r e d by t he ESRF Waves Commi t tee ( H o d g i n s e t a l . , 1986) . The t r a n s f o r m a t i o n o f deep w a t e r waves i n t o s h a l l o w w a t e r as p r e d i c t e d by ADWAVE was compared w i t h measurements i n 12 m of w a t e r on S a b l e I s l a n d Bank w i t h a d i r e c t i o n a l wave buoy d u r i n g w i n t e r s to rms i n 1 9 8 4 - 8 5 . P r e l i m i n a r y r e s u l t s i n d i c a t e t h a t t h e a g r e e m e n t i s q u i t e a c c e p t a b l e , w i t h i n abou t 10% i n t o t a l e n e r g y and s i g n i f i c a n t wave h e i g h t bu t somewhat l o w i n peak p e r i o d . I n ADWAVE t h e e n e r g y s p e c t r u m i s c o n s i d e r e d to be composed of t h r e e segments a s s h o w n i n F i g . 3 . 4 : t h e f o r w a r d f a c e a t f r e q u e n c i e s b e l o w f p , a n i n t e r m e d i a t e r a n g e where S ^ n i s o p e r a t i o n a l and t he h i g h f r e q u e n c y ' t a i l o f t he s p e c t r u m . The h i g h f r e q u e n c y t a i l ( r e g i o n I I I ) i s r e g a r d e d as f u l l y s a t u r a t e d a t a l l t i m e s a n d g o v e r n e d by t h e E ( f ) = f - ^ p o w e r l a w . T h i s p a r t o f t h e ene rgy s p e c t r u m i s a l s o r e f e r r e d t o as t h e P h i l l i p s ' e q u i l i b r i u m range where e n e r g y i n p u t i s b a l a n c e d by d i s s i p a t i o n t h r o u g h w h i t e c a p p i n g and w a v e b r e a k i n g . Modern h y p o t h e s e s ( R e s i o , 1985 ; K i t a i g o r o d s k i i , 1983) a r g u e t h a t the s o u r c e o f ene rgy t o r e g i o n I I I i s no t f r o m t h e l o c a l w i n d , bu t f r o m r e s o n a n t n o n l i n e a r w a v e - w a v e i n t e r a c t i o n s o r i g i n a t i n g f r o m r e g i o n I I . The t r a n s i t i o n be tween reg imes I I and I I I i n deep w a t e r i s a p p r o x i m a t e l y ( K i t a i g o r o d s k i i , 1983) * g " 27TfgUa = JL ( 3 . 4 ) a a 6 u where ot i s a p p r o x i m a t e l y 4 . 4 x 1 0 , B i s a p p r o x i m a t e l y 1 .5x10 , and U„ i s t he a v e r a g e w i n d s p e e d . Thus f „ i s abou t 5 . 3 2 / U „ . F o r a modera te w i n d o f 20 k n o t s (U =10 m/s) t h e n f_ g a a g i s a b o u t 0 .5 Hz o r T =2 s . F o r t h e t r a n s i t i o n t o be a t t h e n o r m a l h i g h f r e q u e n c y c u t o f f i n wave m o d e l l i n g (T =5 s ) , U must be 26.6 m/s o r j u s t o v e r -5 50 k n o t s . I t w o u l d t h e r e f o r e be r a r e t o h a v e a r e g i o n I I I ( i . e . a n f s l o p e reg ime) i n a h i n d c a s t wave s p e c t r u m . - 34 - ENERGY GAINED ON FORWARD FACE OF SPECTRUM DUE TO WAVEWAVE INTER- ACTION SOURCE z UJ a >• a E WAVE SPECTRUM lEtf)] WA VE-WAVE- INTER ACTION SOURCE/SINK DYNAMIC EQUILIBRIUM MAINTAINED BETWEEN ATMOSPHERIC INPUT AND WAVEWAVE INTERACTION SINK ATMOSPHERIC SOURCE OF ENERGY m ENERGY INPUT FROM WAVEWAVE INTER- ACTIONS LOST TO VISCOUS AND TURBU- LENT DISSIPATION $ ° ° > o £ < c c F i g . 3 . 4 E n e r g y r e g i m e s w i t h i n a s p e c t r u m d u r i n g a c t i v e w a v e g r o w t h . (From R e s i o , 1 9 8 2 ) . - 35 - R e s i o (1985) and K i t a i g o r o d s k i i (1983) h y p o t h e s i z e t h a t r e g i o n I I a l s o has an e q u i l i b r i u m s h a p e c h a r a c t e r i z e d by a p o w e r l a w , s u p p o r t f o r w h i c h h a s b e e n found by K i t a i g o r o d s k i i i n e x p e r i m e n t a l d a t a f r o m s e v e r a l s o u r c e s : E(UJ) = a u g U a " 4 ( 3 . 5 ) where i s a n o n - d i m e n s i o n a l c o n s t a n t ( - 4 . 4 x 1 0 ) , g i s t h e g r a v i t a t i o n a l c o n s t a n t , U A i s t h e mean w i n d speed ( u A / U " a - 1 / 3 0 ) , and w i s a n g u l a r f r e q u e n c y (2T r f ) . I n o t h e r w o r d s , t h e s a t u r a t e d shape o f t h i s p a r t o f t h e s p e c t r u m i s i n v a r i a n t , but the e n e r g y c o n t e n t depends d i r e c t l y on mean l o c a l w i n d s p e e d . The l i m i t s o f t h e r a n g e a r e f a n d f , a n d a s s u c h t h e y h a v e d y n a m i c d e f i n i t i o n s t h a t d e p e n d , f o r t h e m o s t p a r t , o n l o c a l w i n d s p e e d . T h i s c e n t r a l r e g i o n o f t h e ene rgy s p e c t r u m has been d e s i g n a t e d by K i t a i g o r o d s k i i (1983) as K o l m o g o r o f f ' s e q u i l i b r i u m range i n r e c o g n i t i o n o f t h e p a r a l l e l s b e t w e e n K i t a i g o r o d s k i i ' s t h e o r i e s o f e n e r g y t r a n s f e r f r o m l o w t o h i g h f r e q u e n c i e s i n wave s p e c t r a and K o l m o g o r o f f ' s t h e o r y o f a t u r b u l e n t e n e r g y c a s c a d e f r o m l o w t o h i g h f r e q u e n c i e s . I n ADWAVE, t h e w i n d e n e r g y i n p u t t o r e g i o n I I i s d e f i n e d as S i n = X f a C D ^ 1 0 ( 3 . 6 ) Pw « . where A i s a p a r t i t i o n i n g c o e f f i c i e n t e x p r e s s i n g t h e f r a c t i o n o f w ind momentum t h a t e n t e r s t h e wave f i e l d , P a i s a i r d e n s i t y , P w i s w a t e r d e n s i t y , i s t h e s u r f a c e d r a g c o e f f i c i e n t , U J Q i s t h e 10-m e l e v a t i o n w i n d s p e e d , and g i s t h e g r a v i t a t i o n a l c o n s t a n t . H a s s e l m a n n e t a l . (1973) c o n c l u d e d t h a t t he minimum a t m o s p h e r i c momentum f l u x i n t o t h e w a v e f i e l d i s o f t h e o r d e r o f 10 t o 40% o f t h e t o t a l momentum t r a n s f e r a c r o s s t h e a i r - s e a i n t e r f a c e , bu t I t may be up t o 100% i f d i s s i p a t i o n i s i m p o r t a n t . R e s i o (1985) a r g u e s t h a t t he p a r t i t i o n i n g c o e f f i c i e n t i s more r e a s o n a b l y abou t 60%. - 36 - The d r a g c o e f f i c i e n t has t he f o r m p r o p o s e d by L a r g e and Pond (1981) w i t h s l i g h t l y d i f f e r e n t c o e f f i c i e n t s : C D = ( 0 . 7 5 + 0 .054 U 1 0 ) x l 0 - 3 ( 3 . 7 ) R e s i o ( 1 9 8 5 ) h a s a s s u m e d t h a t w h e r e b o t t o m e f f e c t s a r e n e g l i g i b l e S ^ n i s e x a c t l y b a l a n c e d by S ^ when t h e e n e r g y s p e c t r u m i s s a t u r a t e d i n r e g i o n I I . B a s e d on t h i s e q u i l i b r i u m b a l a n c e a n d a n a p p r o x i m a t e i n t e g r a t i o n o f t h e c o l l i s i o n ( B o l t z m a n n ) i n t e g r a l , t h e e x p r e s s i o n f o r t h e t o t a l ene rgy f l u x r a t e i n te rms o f s p e c t r a l p a r a m e t e r s i s g i v e n as S n l = ^ E 3 * P - 5 ( 3 . 8 ) t a n h 3 / 4 ( k p d ) where £ i s a n o n - d i m e n s i o n a l c o n s t a n t ( e v a l u a t e d n u m e r i c a l l y t o be o f t h e o r d e r o f 1 0 0 ) , g i s t he g r a v i t a t i o n a l c o n s t a n t , E Q i s t h e t o t a l s p e c t r a l e n e r g y i n t h e doma in f < f<°° , kp i s t h e peak wave number (hii f^/g i n deep w a t e r ) , and d i s t he l o c a l w a t e r d e p t h . When t h e w i n d - w a v e s y s t e m i s no t i n b a l a n c e , S ^ n - S n ^ p r o v i d e s a ne t s o u r c e o r s i n k o f e n e r g y t o o r f r o m t h e K o l m o g o r o f f r a n g e a l t h o u g h t h e m e t h o d o f d i s t r i b u t i n g t h e n e t e n e r g y among t h e w a v e n u m b e r s o r f r e q u e n c i e s ( a n d d i r e c t i o n s ) i s no t d i s c u s s e d by R e s i o i n any o f t he r e f e r e n c e d works ( R e s i o , 1 9 8 1 ; 1 9 8 2 ; 1 9 8 5 ) . E n e r g y g r o w t h on t h e f o r w a r d f a c e o f t h e s p e c t r u m ( r e g i o n I i n F i g . 3 .4 ) i s d e f i n e d as a f i x e d p r o p o r t i o n o f S n ^ , t h e r e m a i n d e r o f w h i c h i s d i s s i p a t e d by i m p l i c i t t r a n s f e r t o t h e p e r m a n e n t l y s a t u r a t e d s p e c t r a l r e g i o n I I I . The shape o f t h e f o r w a r d f a c e i s p r e s u m e d t o h a v e a n e v o l u t i o n a r y r a t h e r t h a n a n e q u i l i b r i u m f o r m ( R e s i o , 1 9 8 5 ) . T h e r a t e o f e n e r g y t r a n s f e r among t h e f r e q u e n c i e s i s v i e w e d as an a n a l o g u e o f h e a t t r a n s f e r i n a medium o f c o n s t a n t c o n d u c t i v i t y . I n deep w a t e r , t h i s i d e a y i e l d s E ( f ) = E ( f p ) e x p { - 6 ( f / f p ) ~ 4 } ( 3 . 9 ) where 6 i s a d i m e n s i o n l e s s c o n s t a n t . O t h e r e n e r g y s o u r c e - s i n k terras i n c l u d e d i n ADWAVE a r e s h o a l i n g , r e f r a c t i o n and bo t t om i n t e r a c t i o n e f f e c t s , a l t h o u g h none o f t h e s e f a c t o r s a r e i n c l u d e d i n t h e deep w a t e r t e s t s p e r f o r m e d as p a r t o f t h i s r e s e a r c h . - 37 - U s i n g h i s t o r i c a l wave measurements f r o m f o u r d i f f e r e n t l o c a t i o n s (Duck, N o r t h C a r o l i n a ; H i e z u C o a s t , J a p a n ; N i s h i k i n o h a m a C o a s t , J a p a n ; and M e l k b o s s t r a n d , S o u t h A f r i c a ) , R e s i o ( 1 9 8 5 ; a l s o i n c l u d e d i n H o d g i n s e t a l . , 1986 a n d S e a c o n s u l t , 1 9 8 6 a ) h a s b e e n a b l e t o v e r i f y t h e p e r f o r m a n c e o f t h e s h a l l o w w a t e r i m p l e m e n t a t i o n o f ADWAVE f o r a w i d e v a r i e t y o f e n e r g y s o u r c e - s i n k c o n d i t i o n s . T a b l e 3.1 p r e s e n t s R e s i o ' s summary of t h e s e t e s t c a s e s . The f o u r r i g h t h a n d m o s t c o l u m n s g i v e t he r e l a t i v e i m p o r t a n c e of s h o a l i n g , r e f r a c t i o n , w ind i n p u t and n o n l i n e a r i n t e r a c t i o n s i n each s p e c t r a l t e s t c a s e . I n c a s e 1, f o r e x a m p l e , a s t h e w i n d i n p u t d i m i n i s h e s w i t h t i m e , t h e S n ^ t e r m q u i c k l y b e c o m e s n e g l i g i b l e a n d s h o a l i n g b e g i n s t o be t h e d o m i n a n t s o u r c e o f w a v e g r o w t h . C o l u m n s 2 a n d 3 g i v e t h e s i m u l t a n e o u s H g v a l u e s a t t h e d e e p e r H^ l o c a t i o n a n d t h e s h o r e w a r d s h a l l o w H2 s i t e . I t i s t h e d i f f e r e n c e b e t w e e n t h e s e H s v a l u e s t h a t i s b e i n g a t t r i b u t e d t o t h e v a r i o u s S ( f ) te rms i n t h e l a s t f o u r c o l u m n s . The ADWAVE p r e d i c t i o n o f H g a t s i t e H2 i s g i v e n i n c o l u m n 4 a n d i s a l m o s t a l w a y s w i t h i n 10% o f t he measu red v a l u e w i t h o u t any a p p a r e n t b i a s . I n t he two e x a m p l e s t h a t e x c e e d t h e 10% d e v i a t i o n , t h e r e l a t i v e i m p o r t a n c e o f t h e S ( f ) terras i s abou t t he same bu t c e r t a i n l y no t u n i q u e l y s o . These r e s u l t s i n d i c a t e v e r y a c c e p t a b l e p e r f o r m a n c e by ADWAVE o v e r a b r o a d r a n g e o f p h y s i c a l p r o t o t y p e c o n d i t i o n s . - 38 - T a b l e 3 .1 C o m p a r i s o n o f P r e d i c t e d and O b s e r v e d Wave H e i g h t s as a F u n c t i o n o f M o d e l l e d E n e r g y S o u r c e Terms Date-Time «1 H 2 H p r e d Dev % Dev *1 % 2 • *3 »4 8210100100 1.8 1.8 1.8 .0 -.1 -4.5 .0 49.7 -45.8" 8210101300 2.4 2.6 2.8 .2 7.8 54.1 .0 30.6 -15.3 8210110100 2.1 2.7 2.6 -.1 -3.7 59.0 .0 31.7 -9.3 8210111300 2.0 2.6 2.4 -.2 -6.1 83.7 .0 10.5 -5.8 -8210120100 2.0 2.4 2.5 .1 5. 3 91.9 .0 1.9 -6.2 8210121300 2.4 3.1 3.0 -.1 -3.3 93.1 .0 .6 -6.3 8210121900 2.2 3.1 2.8 -. 3 -9.9 94.9 .0 .7 -4.4. 8210241300 4.1 3.4 3.5 .0 1.1 .9 -.9 42.5 -55.7 " 8210251300 3.6 3.2 3.6 .4 13.5 50. 4 -11.7 .1 -37.9 — 8410060700 1.1 1.0 1.0 .0 -3.0 -.9 .0 48.0 -51.1 " 8410061900 1.4 1.2 1.3 .1 5.7 -12.2 -.4 41.8 -45.5 -8410070700 1.5 1.4 1.5 .0 2.1 -18.3 -.9 50.0 -30.9 6412021405 1.9 1.8 2.0 .1 6.1 59.1 -22.2 .0 -18.7" 6412021434 2.4 2.1 2.4 .3 12.0 61.4 -16.4 . 1 -22.1 6412021527 2.0 2.0 2.1 .1 4.3 69.6 -14.6 . 2 -15.6 6410021546 1.9 1.9 1.9 -.1 -2.9 -63.5 -18.8 .3 -17.4. 6401311620 .8 .7 .6 .0 -2.2 -.9 -.2 37. 6 -61.4" 6401311710 .8 .8 .8 .0 6.6 -.5 -.6 52.6 -46.4 6901160000 1.7 1.5 1.5 .1 4.0 -44.6 -31.3 15. 3 -8.9" 6905130000 3.4 3.1 3.3 .2 6.4 27. 4 -13.6 . 3 -58. 8 6905290000 2.7 2.4 2.6 . 2 8.4 17. 4 -49.3 .1 -33.2 6908190000 3.4 3.0 3.0 .0 -.5 -59.3 -2.2 1.4 -37.0 -6910070000 2.1 2.2 2.3 .1 3.4 81.9 -5.4 7.1 -5.6 6911180000 4.4 3.5 3.6 . 2 4.5 17.4 -15.7 6.3 -60.6 7001270000 2.7 2.6 2.5 -.1 -3.6 -7.7 -69.8 5.9 -16.6. KEY H l h e i g h t h e i g h t H p r e d . . p r e d i c t e d wave h e i g h t • • H p r e d " H 2 <m> % Dev . . 1 0 0 x < H P r e d " H 2 ) / H 2 %1 change %2 change %3 change 7.4 change (m) a t s i t e 1 ( d e e p e r ) (m) a t s i t e 2 ( s h a l l o w e r ) (m) a t s i t e 2 due t o s h o a l i n g due t o r e f r a c t i o n due t o w ind i n p u t due t o n o n l i n e a r i n t e r a c t i o n s - 39 - 4 . 0 WIND F I E L D S E N S I T I V I T Y : AN APPL ICATION OF ADWAVE To i n v e s t i g a t e t h e e f f e c t s t h a t w i n d f i e l d e r r o r s may c a u s e i n w a v e h i n d c a s t i n g , a s y s t e m a t i c s e n s i t i v i t y a n a l y s i s h a s b e e n c o n s t r u c t e d . T h e c h o i c e o f ADWAVE f o r t h e w a v e h i n d c a s t m o d e l was d i c t a t e d l a r g e l y by a v a i l a b i l i t y and f a m i l i a r i t y . On t he o t h e r h a n d , i t i s a good c h o i c e b e c a u s e i t r e p r e s e n t s a v e r y u p - t o - d a t e s o l u t i o n method f o r t he c o n s e r v a t i o n o f e n e r g y e q u a t i o n i n a r b i t r a r y w a t e r d e p t h s . S i n c e i t c o n t a i n s p a r a m e t e r i z a t i o n s o f more p h y s i c s t h a n i s f o u n d i n a f i r s t g e n e r a t i o n m o d e l l i k e ODGP, i t i s e x p e c t e d t o more a c c u r a t e l y mode l d i v e r s e i n p u t c o n d i t i o n s w i t h o u t c a l i b r a t i o n f o r e a c h new s i t u a t i o n . ADWAVE h a s b e e n s u c c e s s f u l l y a p p l i e d o n t h e w e s t c o a s t o f Canada (Hodg ins and N i k l e v a , 1986 ) , i n t h e B e a u f o r t Sea ( S e a c o n s u l t , 1986b) and on S a b l e I s l a n d Bank on C a n a d a ' s e a s t c o a s t (Hodg ins e t a l . , 1986; S e a c o n s u l t , 1986a) . B e c a u s e i t i s a c o m p l e x c o m p u t e r p r o g r a m , i t i s v e r y d e m a n d i n g o f c o m p u t e r r e s o u r c e s : CPU t i m e , v i r t u a l memory and p e r i p h e r a l mass s t o r a g e . H o w e v e r , f o r i m p o r t a n t h i n d c a s t i n g a p p l i c a t i o n s i t s u s e i s j u s t i f i e d on t h e g r o u n d s o f m i n i m i z i n g m o d e l l i n g e r r o r s , e s p e c i a l l y i f w i n d f i e l d e r r o r s may be l a r g e ( R e s i o a n d V i n c e n t , 1 9 7 9 ) . T h e s e a u t h o r s n o t e t h a t i n p u t a n d w a v e m o d e l e r r o r s a r e g e n e r a l l y a d d i t i v e , s o t h a t i n m o s t c a s e s r e d u c i n g e i t h e r o n e a u t o m a t i c a l l y r e d u c e s t h e t o t a l rms e r r o r . The c h a r a c t e r i s t i c w a v e m o d e l e r r o r i s u s u a l l y a b i a s i n t o t a l e n e r g y ( s i g n i f i c a n t w a v e h e i g h t ) a n d / o r f r e q u e n c y d i s t r i b u t i o n ( p e a k p e r i o d ) . F o r e n g i n e e r i n g a p p l i c a t i o n s , i t i s i m p o r t a n t t o remove as much b i a s as p o s s i b l e f r o m a wave h i n d c a s t and t h e n t o u n d e r s t a n d t h e p r o b a b l e n a t u r e o f a n y r e m a i n i n g r a n d o m e r r o r s . I n t h i s way s a f e , e c o n o m i c a l d e s i g n s a r e a c h i e v e d w i t h o p t i m a l s a f e t y f a c t o r s . 4 . 1 S t r u c t u r e o f t h e S e n s i t i v i t y A n a l y s i s The p u r p o s e o f t h i s i n v e s t i g a t i o n i s t o q u a n t i f y t h e v a r i a t i o n s i n h i n d c a s t s e a - s t a t e p a r a m e t e r s i n t h e B.C. c o a s t a l w a t e r s t h a t a r e a t t r i b u t a b l e t o known v a r i a t i o n s i n t h e m e t e o r o l o g i c a l f o r c i n g p a r a m e t e r s . The s e a - s t a t e p a r a m e t e r s o f i n t e r e s t a r e s i g n i f i c a n t w a v e h e i g h t ( H g ) , p e a k s p e c t r a l p e r i o d ( T p ) , mean w a v e d i r e c t i o n ( 0 ) a n d s p e c t r a l s h a p e ( E ( f ) o r F ( f , 9 ) ) . The p h y s i c a l d e t e r m i n a n t s o f t h e s e p a r a m e t e r s ( i n d e e p w a t e r ) a r e w i n d s p e e d , f e t c h a n d d u r a t i o n . I n t u r n t h e s e f a c t o r s a r e g o v e r n e d by m e t e o r o l o g i c a l c o n d i t i o n s o f t he s u r f a c e p r e s s u r e f i e l d : i n t e n s i t y o f the l o w p r e s s u r e s y s t e m ( r a d i a l e x t e n t o f t h e s y s t e m a n d d e p t h o f t h e c e n t r a l l o w ) , p a t h o f t h e l o w i n r e l a t i o n t o t he c o a s t l i n e o r t o s i t e s o f i n t e r e s t , r a t e of s t o r m p a s s a g e a l o n g i t s t r a j e c t o r y a n d r a t e o f s t o r m i n t e n s i f i c a t i o n . F i g . - 40 - 4.1 i l l u s t r a t e s how t h e s e f a c t o r s a r e r e l a t e d . I n p a r t i c u l a r i t s h o w s t h a t m o s t s u r f a c e p r e s s u r e p a r a m e t e r s i n f l u e n c e t h e f e t c h , p r i m a r i l y t h r o u g h c o n t r o l o f t h e w i n d f i e l d c u r v a t u r e . To s y s t e m a t i c a l l y a l t e r the s u r f a c e p r e s s u r e p a t t e r n s , an i d e a l i z e d m o d e l was u t i l i z e d i n w h i c h a r a d i a l l y s y m m e t r i c p r e s s u r e f i e l d P i s d e f i n e d a s a f u n c t i o n o f r a d i u s r i n t e r m s o f t h e c e n t r a l p r e s s u r e P , t h e mean p r e s s u r e f i e l d , a n d a r a d i a l s c a l e R. I n a c c o r d a n c e w i t h t h e u s u a l w i n d - w a v e h i n d c a s t i n g c o n v e n t i o n s , one s u r f a c e p r e s s u r e map was c o n s t r u c t e d f o r e a c h 6 - h o u r l y h i n d c a s t i n t e r v a l o f a 2 .5 t o 3 .5 d a y s t o r m d u r a t i o n . A n a n a l y s i s o f 27 y e a r s o f s e v e r e s t o r m s i n t h e n o r t h e a s t P a c i f i c by L e w i s and M o r a n (1985) was u s e d t o s e l e c t r e a s o n a b l e v a l u e s f o r P Q a n d R a s w e l l a s f o r t h e t r a j e c t o r y o f t h e c e n t r a l l o w X Q , t h e v e l o c i t y o f t h e l o w a l o n g t h a t t r a j e c t o r y A X Q / A t , t he p o s i t i o n o f P Q ( m i n ) , and t h e r a t e s o f i n t e n s i f i c a t i o n and f i l l i n g o f t h e l o w A P Q / A t . G r a d i e n t and n e a r - s u r f a c e w i n d f i e l d s w e r e c a l c u l a t e d f r o m the s u r f a c e p r e s s u r e maps and t h e s e l a t t e r v e c t o r w inds were i n t e r p o l a t e d i n t i m e t o t he wave mode l t i m e s t e p . The h i n d c a s t m o d e l c o n f i g u r a t i o n was d e s i g n e d to p r o v i d e a v e r y d e t a i l e d l ° x l ° l a t i t u d e - l o n g i t u d e s p e c i f i c a t i o n o f t h e d e e p w a t e r s e a - s t a t e . The p r e s e n t a p p l i c a t i o n o u t p u t w o u l d p r o v i d e t h e i n p u t s p e c t r a l boundary c o n d i t i o n s f o r s h a l l o w w a t e r c o a s t a l h i n d c a s t i n g . H i n d c a s t r e s u l t s c o n s i s t o f t i m e - s e r i e s o f o n e - and t w o - d i m e n s i o n a l s p e c t r a ( E ( f ) and F ( f , 6 ) ) a n d d e r i v e d s e a - s t a t e p a r a m e t e r s H „ , T^ a n d 8 a t s e l e c t e d - l o c a t i o n s w i t h i n t h e m o d e l l i n g d o m a i n a s w e l l a s c o m p l e t e f i e l d s o f t h e d e r i v e d p a r a m e t e r s a t e v e r y p o i n t on t h e s o l u t i o n g r i d . 4 . 2 Wind F i e l d S p e c i f i c a t i o n The i d e a l i z e d w i n d f i e l d s a r e d e r i v e d f r om s u r f a c e p r e s s u r e p a t t e r n s t h a t a r e f u l l y s p e c i f i e d i n t i m e and s p a c e by a t h r e e p a r a m e t e r mode l based on - t h e t i m e h i s t o r y o f t he c e n t r a l low p r e s s u r e p o s i t i o n , - c e n t r a l l o w p r e s s u r e as a f u n c t i o n o f t i m e , and - a r a d i a l s c a l i n g p a r a m e t e r a s s u m i n g a mean b a c k g r o u n d p r e s s u r e e q u a l t o 1015 mb. A g r a d i e n t w i n d c a l c u l a t i o n was a p p l i e d t o t h e s u r f a c e p r e s s u r e maps t o d e r i v e the f r e e a tmosphe re w i n d . The bounda ry l a y e r m o d e l d e s c r i b e d i n S e c t i o n 2.3.3 was used t o c a l c u l a t e s u r f a c e w inds s u i t a b l e f o r d i r e c t i n p u t t o t he ADWAVE h i n d c a s t wave m o d e l . - 41 - d e r i v e d H s = 4EL Fkm.Af.A9 s e a - s t a t e k m p a r a m e t e r s T P = l / f ( m a x ( F k m ) ) 6 = 0 ( m a x ( F k m ) ) m o d e l l e d s e a - s t a t e F ( f,0) Wf^fm a x O°<0<36O° g o v e r n i n g p h y s i c a l p a r a m e t e r s d u r a t i o n c o n t r o l l i n g m e t e o r o l o g i c a l c o n d i t i o n s s t o r m i n t e n s i t y t r a j e c t o r y o f s t o r m v e l o c i t y o f s t o r m d e e p e n i n g & f i l l i n g s u r f a c e P p r e s s u r e A P / A r X Q A X Q / A t A P Q / A t p a r a m e t e r s R F i g . 4 .1 T h e i n t e r - r e l a t i o n s h i p o f m e t e o r o l o g i c a l a n d s e a - s t a t e p a r a m e t e r s . - 42 - 4 . 2 . 1 The S u r f a c e P r e s s u r e F i e l d M o d e l T h e m o d e l u s e d h a s b e e n a d a p t e d f r o m one by C a r d o n e e t a l . ( 1 9 7 5 ) t h a t was a p p l i e d t o h u r r i c a n e s i n t h e G u l f o f M e x i c o . I n t h i s m o d e l a n a x i a l l y s y m m e t r i c s t o r m p r e s s u r e f i e l d (P) i s s u p e r i m p o s e d on a l a r g e s c a l e p r e s s u r e f i e l d (P) o f c o n s t a n t g r a d i e n t . The c o m b i n e d s u r f a c e p r e s s u r e d i s t r i b u t i o n g o v e r n s t h e f r e e a t m o s p h e r e w i n d f i e l d f r o m w h i c h t h e s u r f a c e w i n d i s d e t e r m i n e d . The a x i a l l y s y m m e t r i c s t o r m p r e s s u r e f i e l d i s d e f i n e d by t he s e t o f i s o b a r s P as a f u n c t i o n o f r a d i u s r f r om the s t o r m c e n t r e as P = P Q + AP e ~ R / r (4.1) where P Q i s t he c e n t r a l p r e s s u r e Ap i s t h e p r e s s u r e g r a d i e n t be tween r-*=° and r = 0 ( i . e . P - P Q ) R i s a r a d i a l s c a l e p a r a m e t e r s u c h t h a t a t r = R P = P Q + 0 .37AP F i g . 4 .2 f o r H u r r i c a n e C a m i l l e i l l u s t r a t e s how P Q , Ap a n d R a r e d e t e r m i n e d f r o m s u r f a c e p r e s s u r e m e a s u r e m e n t s d u r i n g t h e p a s s a g e o f a s t o r m . I n t h i s a p p l i c a t i o n , P was assumed e q u a l t o 1015 mb, and P Q and R were s p e c i f i e d f o r a g i v e n g e o g r a p h i c a l l o c a t i o n , X Q , o f P Q . I f AP i s r e p r e s e n t e d by t h e A E S s t a n d a r d 4 mb i s o b a r i n c r e m e n t , t h e n t h e r a d i u s o f t h o s e i s o b a r s i s e x p r e s s e d by r e a r r a n g e m e n t o f (4.1) as v± = - R / l n ( 4 i / A P ) f o r i = 1,1 (4 .2) where I = I N T ( ( A P - l ) / 4 ) , and t h e p r e s s u r e v a l u e s o f t h o s e i s o b a r s a r e P ± = 4 i + P Q f o r i = 1,1 (4.3) T h i s mapp ing o f t he p r e s s u r e f i e l d i s e q u i v a l e n t t o t h e f o r m a t o f a s t a n d a r d s u r f a c e p r e s s u r e c h a r t p r e p a r e d by A E S . F i g . 4 . 3 a i l l u s t r a t e s a s y n t h e t i c p r e s s u r e f i e l d i n 5 mb i n c r e m e n t s f o r Ap=60 mb, P Q = 9 5 5 mb, a n d R=479 km. S i n c e w i n d s p e e d i s i n v e r s e l y p r o p o r t i o n a l t o d P / d r , e v a l u a t i n g d P / d r =0 p r o v i d e s t h a t t h e maximum v e l o c i t y w i l l be f o u n d a t R / 2 . F o r c o m p a r i s o n , a n a c t u a l h a n d - d r a w n p r e s s u r e c h a r t i s shown i n F i g . 4 . 3 b , b a s e d on t h e f e w a v a i l a b l e o b s e r v a t i o n s o f P . The i d e a l i z e d p r e s s u r e c o n t o u r s a r e mo re c o n c e n t r a t e d n e a r t h e c e n t r e o f t h e s y s t e m a n d much mo re d i s p e r s e d n e a r i t s p e r i p h e r y . The r e s u l t w i l l be somewhat s t r o n g e r w i n d s i n a s m a l l r e g i o n n e a r t h e eye o f t he s t o r m and a b r o a d s u r r o u n d i n g a r e a o f v e r y weak w i n d s . N e i t h e r - 43 - I O I C 1000 930 980 970 .o 960 6 u i 3 950 cc o. 9'10 930 920 910 S r . - * * ^ - A V - 3 I— OA X AO / o o / / / / . / / / r i soo x BILOXI O GULF PORT • NEW ORLEANS (NAS) A NEW ORLEANS ( WSO) O BAY ST. LOUIS • MINIMA ALONG DELTA Ap« 105 R = 10 C A M I L L . E P R E S S U R E P R O F I L E H I S T O R I C A L D A T A / * usisstrri lOVlSIA MA • 7 I . - — TRANS WORLD " _3URWOt5ETr£\«j R i o JO 2 OOG? STATIONS \ V S C i . t N.MI. CAMIUF STO«M TRACK O 20 4 0 60 80 100 120 D I S T A N C E ( n- m i . ) 140 160 ISO F i g . 4 . 2 S u r f a c e p r e s s u r e a s a f u n c t i o n o f d i s t a n c e c a l c u l a t e d f r o m measurements made d u r i n g t h e p a s s a g e o f H u r r i c a n e C a m i l l e . (F rom C a r d o n e e t a l . , 1 9 7 5 ) . - 44 - A t 86010:1 00 h o u r s lit arm c e n t r e d a t 5 0 . 0 0 d e g r e e s N 1 4 5 . 0 0 d e g r e e s W C e n t r a l P r e s s u r e = 9 5 5 . mb Mean G e o s t r o p h i c P r e s s u r e - 1 0 1 5 . mb P r e s s u r e D i f f e r e n c e -• 6 0 . mb R a d i a l S c a l e = 4 7 9 . 4 6 km P<i> = 9 6 0 . mb R ( i ) = 1 9 3 . km P < i ) = 9 6 5 . mb R ( i ) = 2 6 8 . km P ( i ) = 9 7 0 . mb R C i ) = 3 4 6 . km P(:L) = 9 7 5 . mb R ( . i ) = 4 3 6 . km P ( i ) ~ 9 8 0 . mb R ( i ) = 5 4 8 . km P < i ) 9 8 5 . mb R(:L) = 6 9 2 . km P < i ) 9 9 0 . mb R ( i ) = 8 9 0 . km P<:i) = 9 9 5 . mb R ( i ) = 1 1 8 2 . km P < i ) 1 0 0 0 . mb R ( i ) = 1 6 6 7 . km F i g . 4.3 An i d e a l i z e d p r e s s u r e f i e l d f o r a m o d e r a t e l y i n t e n s e s t o r m (upper p a n e l ) and an a c t u a l s u r f a c e p r e s s u r e c h a r t (bottom p a n e l ) , both at a p p r o x i m a t e l y the same s c a l e f o r the same low p r e s s u r e system. - 45 - o f t h e s e d i f f e r e n c e s s h o u l d r e s t r i c t t h e u s e f u l n e s s o f t h e p r e s s u r e f i e l d mode l f o r i d e a l i z e d w i n d f i e l d g e n e r a t i o n . The s t o r m p r e s s u r e map i n F i g . 4 .3a i s l o c a t e d on t he e a r t h ' s s u r f a c e w i t h t h e c e n t r a l p r e s s u r e a t X Q C ^ Q . ^ Q ) where <f> i s t h e l a t i t u d e r e l a t i v e t o t he e q u a t o r a n d A Q i s t h e l o n g i t u d e r e l a t i v e t o t h e G r e e n w i c h m e r i d i a n ( h e n c e w e s t l o n g i t u d e s a r e n e g a t i v e ) . A s s u m i n g t h e e a r t h i s a p e r f e c t s p h e r e , t h e n t h e o r t hod rome ( o r g r e a t c i r c l e a r c ) f r o m X Q t o an a r b i t r a r y p o i n t X.2^2'^2^ n a s a l e n g t h d d e f i n e d i n F i g . 4.4 as d = a cos * [sincf>osin4>2 + cos<)>0cos<}>2cos AX] (4.4) w h e r e a i s t h e e a r t h ' s r a d i u s ( 6 3 7 1 km) a n d AX= ^ 2 ~ \ ) * The p r o b l e m , g i v e n X Q a n d d = r ^ , i s t o d e t e r m i n e X 2 . F o r a u n i q u e s o l u t i o n , t h e a z i m u t h ( t h e a n g u l a r d e f l e c t i o n a o f t h e o r t h o d r o m e m e a s u r e d c l o c k w i s e f r o m the m e r i d i a n A ) must a l s o be i m p o s e d . W i t h r e f e r e n c e t o F i g . 4 . 4 , t h e a n g l e S s u b t e n d e d by t h e a r c d i s 6 = d (4.5) a The l a w of c o s i n e s f o r a s p h e r i c a l t r i a n g l e may t h e n be i n v o k e d to s p e c i f y t h e l a t i t u d e o f X 2 as <j>2 = s i n - ' ' ' [cos6sin<{> 0 + s in0cos<J> o cosa] (4.6) A p p l y i n g t h e l a w o f s i n e s t o t h e a z i m u t h a n g l e a y i e l d s t he s p e c i f i c a t i o n o f t h e l o n g i t u d e o f X2 as .-1 r. A 2 = A Q + s i n s i n a s i n G COS<f>n ( 4 . 7 ) I n o t h e r w o r d s , an i s o b a r a t r a d i a l d i s t a n c e r^ f r om X Q may be d i s c r e t i z e d i n l a t i t u d e - l o n g i t u d e c o o r d i n a t e s a s a f u n c t i o n o f a z i m u t h a n g l e ct. I n t h i s a p p l i c a t i o n , t he c i r c u m f e r e n c e C o f t he i s o b a r i c c o n t o u r was d i v i d e d i n t o 2M e l e m e n t s s u c h t h a t t he a r c l e n g t h be tween d i s c r e t i z a t i o n p o i n t s c o u l d no t e x c e e d a minimum l e n g t h . T a k i n g a d v a n t a g e o f symmetry abou t t he m e r i d i a n A 0, a was v a r i e d f r o m Act t o MAa r a d i a n s w h e r e Aa = Tr/M. I n a l l c a s e s two a d d i t i o n a l p o i n t s were added a t a =0 and cx =TT. A t t h e same t i m e , a t l e a s t 12 p o i n t s w e r e r e q u i r e d t o d e f i n e e a c h i s o b a r i c c o n t o u r . T h e s e n u m b e r s w e r e c h o s e n t o c o r r e s p o n d r o u g h l y t o t h e r a t e o f d i g i t i z a t i o n o f a s t a n d a r d AES - 46 - F i g . 4 .4 P a r a m e t e r s d e f i n i n g d i s t a n c e be tween a r b i t r a r y p o i n t s on a s p h e r e . (From P e a r s o n , 1 984 ) . - 47 - s u r f a c e p r e s s u r e c h a r t . The t r a j e c t o r y o f X Q was d e f i n e d as a f u n c t i o n o f t i m e a t d i s c r e t e l a t i t u d e - l o n g i t u d e p o s i t i o n s . The speed o f t r a n s l a t i o n i s r e a d i l y c a l c u l a t e d f o r any segment of t he p a t h as v=d /A t u s i n g ( 4 . 4 ) t o c a l c u l a t e d . 4.2.2 Gradient and Surface Wind Fields The g r a d i e n t w i n d f i e l d , a l s o k n o w n as t h e f r e e a t m o s p h e r e w i n d f i e l d , i s c a l c u l a t e d f r o m the g r a d i e n t o f t h e s u r f a c e p r e s s u r e d i s t r i b u t i o n . The b a s i s o f t he c o m p u t a t i o n i s a f o r c e b a l a n c e among p r e s s u r e g r a d i e n t , C o r i o l i s and a c e n t r i f u g a l f o r c e t e r m due t o c u r v a t u r e o f t he i s o b a r s . T h i s g r a d i e n t w ind i s a s s u m e d t o be a t 500 m e l e v a t i o n , a b o v e t h e i n f l u e n c e o f t h e a t m o s p h e r i c boundary l a y e r . The g r a d i e n t w ind i s r e d u c e d t o an a r b i t r a r y n e a r - s u r f a c e e l e v a t i o n z u s i n g a b o u n d a r y l a y e r m o d e l b a s e d o n t h e w o r k o f D e l a g e ( 1 9 8 5 ) a t . t h e C a n a d i a n M e t e o r o l o g i c a l C e n t r e and Yamada (1976 ) . As d i s c u s s e d i n S e c t i o n 2 . 3 . 3 , t he m e t h o d d e t e r m i n e s u * a s a f u n c t i o n o f t h e g r a d i e n t w i n d , u p p e r a i r a n d s e a t e m p e r a t u r e s , and an e m p i r i c a l s u r f a c e r o u g h n e s s l e n g t h . Wind a t a s p e c i f i c e l e v a t i o n z i s t h e n c a l c u l a t e d f r o m u * u s i n g e m p i r i c a l p r o f i l e s , the f o rm o f w h i c h depend on a i r c o l u m n s t a b i l i t y . I n t h i s a p p l i c a t i o n t h e n e a r - s u r f a c e r e f e r e n c e l e v e l i s z=10 m and n e u t r a l s t a b i l i t y i s i m p o s e d . T h i s w i n d f i e l d w h i c h i s d e n o t e d by U^Q i s c a l c u l a t e d on a l ° x l ° l a t i t u d e - l o n g i t u d e g r i d t h a t e n c o m p a s s e s t h e d o m a i n o f t h e w a v e m o d e l . F i g . 4 . 5 s h o w s a n e x a m p l e o f a s u r f a c e p r e s s u r e map and t h e c o r r e s p o n d i n g U^Q f i e l d . I s o b a r s a r e drawn a t 1 mb s p a c i n g and t h e w i n d v e c t o r s a r e d r a w n i n t h e s t a n d a r d m e t e o r o l o g i c a l c o n v e n t i o n i n w h i c h e a c h f u l l f e a t h e r r e p r e s e n t s 10 k n o t s , each h a l f f e a t h e r i s 5 k n o t s a n d t h e " v " f e a t h e r i s a s h o r t h a n d f o r 50 k n o t s . D i r e c t i o n o f t h e w i n d i s f r om the f e a t h e r s t o t he h e a d of t h e v e c t o r . T h u s , f o r e x a m p l e , means 25 k n o t s f r o m t h e wes t and means 60 k n o t s f rom the e a s t . 4.3 Wave Model and Grid Setup The s t u d y a r e a f o r m o d e l l i n g p u r p o s e s i s bounded by 39°N on t h e s o u t h , by 60°N o n t h e n o r t h , by 160°W on t h e w e s t a n d by 124°W on t h e e a s t . The s p a t i a l d i s c r e t i z a t i o n i s i n 1 ° l a t i t u d e a n d 1° l o n g i t u d e i n c r e m e n t s g i v i n g a g r i d w h i c h i s 37 by 22 e l e m e n t s . A t e a c h o f t h e s e p o i n t s t h e e n e r g y s p e c t r u m F ( f , 0 ) i s d e t e r m i n e d a t e a c h m o d e l t i m e s t e p . The mapp ing o f the mode l doma in on a p o l a r s t e r e o g r a p h i c p r o j e c t i o n o f t h e n o r t h e a s t P a c i f i c Ocean i s shown i n F i g . 4.5 A t y p i c a l i d e a l i z e d pressure f i e l d (upper panel) and i t s corresponding surface wind (U^Q) f i e l d (lower panel). - 49 - F i g . 4 .6 . The r e s o l u t i o n o f the c o a s t l i n e i s n e c e s s a r i l y c o a r s e and w o u l d no t be s u i t a b l e f o r h i n d c a s t i n g of wave c l i m a t e n e a r t he c o a s t . B e c a u s e t h e m a j o r i t y o f t h e c a l c u l a t i o n p o i n t s a r e i n d e e p w a t e r , o p t i o n a l c a l c u l a t i o n o f s h o a l i n g a n d r e f r a c t i v e e f f e c t s i n s h a l l o w w a t e r h a v e b e e n o m i t t e d , a l t h o u g h t h e s e e f f e c t s w o u l d h a v e t o be i n c l u d e d f o r a c c u r a t e h i n d c a s t m o d e l l i n g of n e a r s h o r e s e a - s t a t e s . The deep w a t e r c o n f i g u r a t i o n on a c o a r s e g r i d i s t h e t y p i c a l f i r s t s t e p i n c o a s t a l w a v e m o d e l l i n g ; s p e c t r a l v a l u e s c a l c u l a t e d a t t h e e d g e o f t h e s h e l f b r e a k b e c o m e t h e i n p u t b o u n d a r y c o n d i t i o n s f o r a f i n e r m e s h , s h a l l o w w a t e r c a l c u l a t i o n a l o n g t h e c o a s t . M o d e l l i n g i n v e r y s h a l l o w w a t e r o r i n r e g i o n s o f c o m p l e x b a t h y m e t r y may r e q u i r e a t h i r d , y e t f i n e r g r i d . D i r e c t i o n a l r e s o l u t i o n was f i x e d a t 16 a n g l e s (A8=2TT/ 16=22 .5° ) . F r e q u e n c y was p a r a m e t e r i z e d i n 16 e q u a l f r e q u e n c y c l a s s e s s u c h t h a t f m i n = 0 . 0 3 0 Hz (T=32 .8 s ) , A f = 0 . 0 1 1 3 Hz a n d f m o =0.2 Hz (T=5 .0 s ) . U s i n g l i n e a r w a v e t h e o r y a n d d e e p w a t e r a p p r o x i m a t i o n s , t h e l o n g e s t p e r i o d w a v e o f 33 s w i l l h a v e a g r o u p v e l o c i t y c o f gT/47T o r 26 m / s . S i n c e t h e s m a l l e s t g r i d e l e m e n t l e n g t h i s a b o u t 57 km, t h e maximum t i m e s t e p f o r c o m p u t a t i o n a l s t a b i l i t y o f t h e p r o p a g a t i o n scheme i s 5 7 x 1 0 / 2 6 = 2 1 9 2 s . As a r e s u l t , a c o n v e n i e n t t i m e s t e p o f 1800 s (0.5 h) was s e l e c t e d . T i m e - s e r i e s o u t p u t o f H , T , 6, E ( f ) a n d F ( f , 6 ) was o b t a i n e d a t 11 s p e c i a l s p p o i n t s d e s i g n a t e d i n F i g . 4 . 6 : f o u r a l o n g t h e c o a s t ( i n c l u d i n g one wave buoy l o c a t i o n i n Queen C h a r l o t t e Sound marked by a s t a r s y m b o l ) , two p a r a l l e l rows o f t h r e e a d d i t i o n a l p o i n t s e a c h r u n n i n g s o u t h w e s t f r o m t h e c e n t r a l c o a s t a l l o c a t i o n s ( i n c l u d i n g a U . S . A . d e e p o c e a n m e t e o r o l o g i c a l d a t a buoy s t a t i o n ) , and a s e c o n d deep o c e a n d a t a buoy s i t e o f f t h e W a s h i n g t o n - O r e g o n c o a s t . These t i m e - s e r i e s , o b t a i n e d a t h o u r l y i n t e r v a l s , p r o v i d e t h e f i r s t l e v e l o f c o m p a r i s o n s b e t w e e n t h e v a r i o u s c a s e s t u d i e s . H o u r l y o u t p u t o f t h e e n t i r e f i e l d o f H T a n d 0 was a l s o made f r o m w h i c h c o n t o u r e d maps o f H„ w e r e s p s c o n s t r u c t e d t o i l l u s t r a t e t he o v e r a l l d e v e l o p m e n t and e v o l u t i o n o f t he s e a - s t a t e . To c o m p a r e b e t w e e n c e r t a i n i n p u t s c e n a r i o s , f i e l d s o f A H g w e r e c o n t o u r e d t o h i g h l i g h t r e g i o n s w h e r e t h e s e a - s t a t e d e v e l o p m e n t w a s s i g n i f i c a n t l y d i f f e r e n t . s p e c i a l - 51 - 4.4 Pressure Parameter Specification R e c e n t l y , L e w i s and M o r a n (1985) p r e p a r e d a c a t a l o g u e of s e v e r e s to rms i n t h e n o r t h e a s t P a c i f i c O c e a n . F r o m t h i s d o c u m e n t , i t was p o s s i b l e t o d e r i v e r e a l i s t i c mean o r m e d i a n and e x t r e m e v a l u e s f o r t h e p r e s s u r e p a r a m e t e r s g o v e r n i n g t h e s u r f a c e p r e s s u r e f i e l d e v o l u t i o n . Two m e t e o r o l o g i c a l s t o r m c l a s s e s were c o n s i d e r e d : t h e s o u t h w e s t f r o n t a l l o w and t he sou thwes t c o l d l ow as d e f i n e d by L e w i s and M o r a n . B o t h u s u a l l y o c c u r r e d b e t w e e n O c t o b e r a n d M a r c h , b u t t h e f r o n t a l l o w i s somewhat more common w i t h 21 e v e n t s i n t h e 27 y e a r d a t a b a s e f r om 1957 t o 1983. W a v e s g e n e r a t e d by t h e s e s t o r m s a r e r e p o r t e d l y as g r e a t a s 25 m w i t h c o n c u r r e n t 16 .5 ra s w e l l , a l t h o u g h m o r e t y p i c a l r e p o r t s a r e 8 t o 10 m. S h i p s i n k i n g o r s e v e r e damage i s k n o w n t o h a v e h a p p e n e d d u r i n g f i v e o f t h e s e 21 s t o r m s . M a n y o f t h e s t o r m c e n t r e s t r a c k f a i r l y c o n s i s t e n t l y f r o m t h e w e s t be tween t h e 4 0 ° and 45°N p a r a l l e l s and t u r n s h a r p l y to t h e n o r t h a l o n g 145°W, a p p r o x i m a t e l y , as s k e t c h e d i n F i g . 4 . 7 a . The g r e a t e s t s t o r m i n t e n s i f i c a t i o n , a s i n d i c a t e d by t h e l o w e s t c e n t r a l p r e s s u r e , i s o f f s h o r e o f t h e Q u e e n C h a r l o t t e I s l a n d s b e t w e e n 5 0 ° a n d 5 5 ° N . As t h e s t o r m s e n t e r t h e G u l f o f A l a s k a t h e y u s u a l l y b e g i n t o d i s s i p a t e q u i t e r a p i d l y . The re were 14 s o u t h w e s t c o l d l o w s t o r m s i n t h e 27 y e a r s s t u d i e d by L e w i s and M o r a n ( 1 9 8 5 ) . A b o u t h a l f o f t h e s e h a v e a s i m i l a r t r a j e c t o r y t o t h e f r o n t a l l o w s as shown i n F i g . 4 .7b . These s t o r m s a r e g e n e r a l l y l e s s s e v e r e i n t e rms o f w a v e g e n e r a t i o n , a l t h o u g h one d i d r e p o r t e d l y c a u s e 20 m w a v e s . One s h i p s i n k i n g has been a t t r i b u t e d t o t h i s s t o r m t y p e , bu t s h i p - r e p o r t e d wave h e i g h t s a t t h e t i m e were no t e s p e c i a l l y l a r g e . C l e a r l y t h e s e a r e s e v e r e s t o r m s w h i c h must be a c o n c e r n t o C a n a d a ' s wes t c o a s t s h i p p i n g and f i s h i n g i n d u s t r i e s . F o r t u n a t e l y , t h e y a r e g e n e r a l l y s u f f i c i e n t l y f a r o f f s h o r e t h a t s e r i o u s damage a l o n g t h e c o a s t d o e s n o t o c c u r . N e v e r t h e - l e s s , t he n e a r s h o r e wave c l i m a t e r e s u l t i n g f r o m s u c h s t r o n g m e t e o r o l o g i c a l f o r c i n g w i l l be v e r y i m p o r t a n t i n a c o m p r e h e n s i v e s e a - s t a t e d e s c r i p t i o n f o r d e s i g n p u r p o s e s , e s p e c i a l l y f o r t h e e x p o s e d o u t e r c o a s t . B e c a u s e t h e y a r e r e c o r d e d q u i t e f r e q u e n t l y , r e a s o n a b l e s t a t i s t i c s o f t h e i r ma jo r c h a r a c t e r - i s t i c s c a n be d e r i v e d . - 52 - 4 . 7 The s t o r m t r a j e c t o r i e s o f (a) s o u t h w e s t f r o n t a l l o w s and (b) s o u t h w e s t c o l d l o w s . ( A d a p t e d f r om L e w i s and M o r a n , 1 9 8 5 ) . - 53 - 4.4.1 Trajectory T h e s e s t o r m s w e r e i d e a l i z e d i n t e r ras o f t h r e e s e g m e n t s i l l u s t r a t e d i n F i g . 4 .8 : a s p i n - u p l e g be tween t h e w e s t e r n m o d e l bounda ry ( p o i n t A) and t he p o i n t where t h e t r a j e c t o r y t u r n s n o r t h ( B ) , a s t o r m i n t e n s i f i c a t i o n l e g p r o c e e d i n g n o r t h to t h e l o c a t i o n o f minimum c e n t r a l l o w p r e s s u r e ( C ) , and a s t o r m f i l l i n g l e g o v e r w h i c h t h e c e n t r a l p r e s s u r e i n c r e a s e s a s t h e s t o r m a p p r o a c h e s t h e n o r t h e r n edge o f t h e mode l doma in (D). The mean t r a j e c t o r y p a t h was imposed f r o m t h e w e s t a l o n g 4 2 ° N (A t o B) a n d t h e n c e n o r t h a l o n g 145°W (B t o C t o D ) . V a r i a t i o n s w i t h i n + 5 ° o f t h e 145°W l o n g i t u d e a r e a l s o r e a l i s t i c . 4.4.2 Central Pressure The a v e r a g e minimum c e n t r a l l o w p r e s s u r e b a s e d on t h e 21 s o u t h w e s t f r o n t a l and c o l d l o w s s k e t c h e d i n F i g . 4.7 i s 958 mb f o r d a t a r a n g i n g be tween t he minimum r e c o r d o f 944 mb and t he l a r g e s t v a l u e of 970 mb. The med ian p o s i t i o n (C) o f t h i s l o w p r e s s u r e was imposed a t 53°N 145°W, due wes t o f t he Queen C h a r l o t t e I s l a n d s . A t t h e w e s t e r n e d g e ( A ) , t h e mean p r e s s u r e o f t h e 21 s t o r m s was 984 mb f r o m d a t a r a n g i n g b e t w e e n 954 a n d 1010 mb. A t t h e n o r t h e r n e d g e ( D ) , t h e mean c e n t r a l p r e s s u r e was 970 mb w i t h a m i n i m u m o f 952 mb and a maximum o f 982 mb. The v a l u e s a t p o i n t B a r e more d i f f i c u l t t o s p e c i f y s i n c e t h e t r a j e c t o r i e s ( F i g . 4 . 7 ) v a r y g r e a t l y i n t h i s a r e a . I n s t e a d , t h e mean r a t e o f c h a n g e o f P Q b e t w e e n A a n d C was u s e d t o e s t i m a t e a r e a s o n a b l e v a l u e a t B. T h e a v e r a g e r a t e o f d e e p e n i n g o f t h e c e n t r a l p r e s s u r e A p Q / A t was d e t e r m i n e d t o be a p p r o x i m a t e l y - 0 . 6 m b / h w h i c h i m p l i e s P Q = 9 6 9 mb a t B f o r t y p i c a l r a t e s o f s t o r m t r a n s l a t i o n . 4.4.3 Storm Speed T h e r e a r e two a s p e c t s t o s t o r m s p e e d : t h e r a t e o f t r a n s l a t i o n o f t h e s t o r m c e n t r e A X Q / A t t h r o u g h t h e s e g m e n t s o f t h e s t u d y a r e a and t h e r a t e o f d e v e l o p m e n t ( o r d e e p e n i n g ) o f the l o w p r e s s u r e s y s t e m A p Q / A t . The speed w i t h w h i c h the s t o r m moves depends on i t s s t a g e o f d e v e l o p m e n t ( l e g A - B , B -C o r C - D). The f i n a l s t a g e u s u a l l y has t he s l o w e s t movement and o c c a s i o n a l l y a s t o r m s y s t e m w i l l become s t a t i o n a r y i n t h i s a r e a f o r up t o 24 h o u r s . A v e r a g e s p e e d s a r e on t he o r d e r o f 35 t o 40 km/h w h i c h i s the e q u i v a l e n t o f 18 h o u r s to move f r o m t h e l a t i t u d e o f t h e Q u e e n C h a r l o t t e I s l a n d s t o t h e s o u t h e r n A l a s k a n c o a s t . D u r i n g t h e e a r l y s t o r m f o r m a t i o n i n t h e s o u t h w e s t e r n r e g i o n o f t h e s t u d y a r e a , t y p i c a l s p e e d s a r e 45 t o 55 k m / h b e t w e e n 160°W a n d 145°W a l o n g 4 2 ° N . T h e h i g h e s t v e l o c i t i e s o c c u r d u r i n g t h e i n t e n s i f i c a t i o n s t a g e , a v e r a g i n g abou t 70 k m / h . These s p e e d s c a n n o t be c a l c u l a t e d v e r y a c c u r a t e l y - 54 - 4.8 The t h r e e s e g m e n t s o f a n i d e a l i z e d s t o r m t r a j e c t o r y b a s e d on t h e s o u t h w e s t f r o n t a l l o w s a n d t h e s o u t h w e s t c o l d l o w s . - 55 - f r o m t h e 1 2 - h o u r l y c e n t r a l p r e s s u r e p o s i t i o n s , b u t t h e y do s u f f i c e f o r t h e d e v e l o p m e n t o f t h e i d e a l i z e d w i n d f i e l d t i m e - s e r i e s . The a v e r a g e r a t e o f c e n t r a l p r e s s u r e d e e p e n i n g A P Q / A t i s - 0 . 6 rab/h. O c c a s i o n a l l y , i n 8 o f t h e 21 s t o r m s , t h i s r a t e e x c e e d s . - 1 mb/h o v e r a 12 h o u r p e r i o d , and t h i s s i t u a t i o n i s k n o w n a s " e x p l o s i v e d e e p e n i n g " o f t h e l o w p r e s s u r e s y s t e m . S i n c e t he w i n d s s t r e n g t h e n so q u i c k l y , a v e r y h i g h s e a - s t a t e c a n d e v e l o p w i t h o u t much w a r n i n g a n d many o f t h e s e s t o r m s w i l l be among t h e most s e v e r e a f f e c t i n g an o f f s h o r e o r o u t e r c o a s t a l l o c a t i o n . F o r t h e e i g h t e v e n t s c o n t a i n i n g a p e r i o d o f e x p l o s i v e d e e p e n i n g , t h e a v e r a g e r a t e was - 1 . 8 rab/h. D u r i n g one o f t h e s e s t o r m s - 3 . 3 mb/h i n t e n s i f i c a t i o n o c c u r r e d f o l l o w e d by r e p o r t s o f 10 m l o c a l w i n d waves and 12.5 m s w e l l w a v e s . 4.4.4 Radial Scaling Parameter Of a l l t h e s t o r m c h a r a c t e r i s t i c s , t h e r a d i a l s c a l i n g p a r a m e t e r R was t he most s u b j e c t i v e l y d e t e r m i n e d due t o l i m i t a t i o n s i n t he a v a i l a b l e d a t a f o r t h e 21 s t o r m s . From i n s p e c t i o n of t h e f u l l y d e v e l o p e d s t o r m maps i n L e w i s and M o r a n ( 1 9 8 5 ) a n d t h e e v o l u t i o n i n t i m e a n d s p a c e o f a f e w o t h e r l o w p r e s s u r e s y s t e m s , i t was d e t e r m i n e d t h a t t h e p h y s i c a l d i m e n s i o n o f t h e s e p r e s s u r e p a t t e r n s i s more o r l e s s c o n s t a n t f r o m t h e c e n t r e t o t h e 990 mb i s o b a r o v e r t he t i m e h i s t o r y o f a g i v e n s y s t e m . F o r the 21 f u l l y d e v e l o p e d p r e s s u r e maps, t h e a v e r a g e o f t h i s r a d i a l d i m e n s i o n r ^ g g was a b o u t 8 ° o f l a t i t u d e ( o r 8 9 0 k m ) , w h i l e t h e o t h e r d a t a maps w h i c h w e r e o f v e r y i n t e n s e s t o r m s s u g g e s t e d r<jgg c o u l d be as s m a l l as 3 o r 4 ° o f l a t i t u d e . A compromise o f 6 ° o f l a t i t u d e was s e l e c t e d f o r s u b s t i t u t i o n i n ( 4 . 1 ) t o d e t e r m i n e R f o r e a c h s p e c i f i c P Q s i n c e by r e a r r a n g e m e n t ( 4 . 8 ) I n e s s e n c e t h e n , b e c a u s e t h e r a d i a l d i m e n s i o n r ^ g g w a s s p e c i f i e d , R v a r i e s t h r o u g h o u t a s t o r m as a f u n c t i o n o f P Q . As a r e s u l t , t he s e n s i t i v i t y t o R c a n o n l y be t e s t e d i n d i r e c t l y t h r o u g h a d j u s t m e n t o f r g Q Q . R = - r 9 9 Q I n 990 - P c A P ~ - 56 - 4.5 Summary of Model Test Cases 4.5.1 Storm 1: The Median Base Scenario A b a s e l i n e s t o r m was c o n s t r u c t e d a g a i n s t w h i c h t o c o m p a r e mos t o t h e r t e s t c a s e s . I t was c o m p o s e d o f t h e m e d i a n t r a j e c t o r y f r o m t h e w e s t a l o n g 4 2 ° N t u r n i n g n o r t h a l o n g 145°W w i t h t h e a v e r a g e P Q v a l u e s o f 984 mb i m p o s e d a t p o i n t A , 969 mb a t p o i n t B , P Q ( m i n ) o f 958 mb a t p o i n t C ( 5 3 ° N 145°W) a n d 9 7 0 mb a t p o i n t D. T h e a v e r a g e r a t e o f d e e p e n i n g A P Q / A t = - 0 . 6 m b / h was s e l e c t e d f o l l o w e d by a t y p i c a l r a t e o f f i l l i n g w i t h A P Q / A t = + 0 . 7 m b / h . The r a t e o f s t o r m a d v a n c e a l o n g t h e t r a j e c t o r y p a t h i s n o t a n i n d e p e n d e n t p a r a m e t e r a f t e r t h e o t h e r v a r i a b l e s a r e i m p o s e d . The r e s u l t i n g speeds A X Q / A t a r e r e p r e s e n t a t i v e o f t h e 21 s t o r m s t h a t were a n a l y z e d f r o m L e w i s and M o r a n ( 1 9 8 5 ) . T a b l e 4 .1 summar i zes t he s t o r m p a r a m e t e r s . Table 4.1 Storm 1 Scenario Parameters P o i n t L e g L a t i t u d e L o n g i t u d e L e n g t h (km) Time YYMMDDHH D u r a t i o n (h ) A X Q / A t ( km/h ) P o (mb) A P Q / A t (mb/h) A A - B 42°N 160°W 1238 86010100 24 52 984 - 0 . 6 B B-C 42°N 145°W 1223 86010200 18 68 969 - 0 . 6 C C-D 53°N 145°W 667 86010218 18 37 958 +0.7 D 59°N 145°W 86010312 970 - 57 - 4.5.2 Storm 2; Explosive Deepening The c e n t r a l p r e s s u r e s o f s t o r m 1 a t p o i n t s A , C and D were unchanged i n s t o r m 2 , b u t A P Q / A t was a d j u s t e d a l o n g l e g s A - B a n d B - C s o t h a t i n t h e 12 h o u r s p r e c e d i n g t h e a c h i e v i n g o f P Q ( m i n ) a t p o i n t C t h e r a t e o f d e e p e n i n g was - 1 . 8 m b / h . T h i s was t h e a v e r a g e e x p l o s i v e d e e p e n i n g r a t e i n t h e 8 c a s e s t o r m s d i s c u s s e d i n S e c t i o n 4 .4 .3 . T a b l e 4.2 c o n t a i n s t he c o m p l e t e s p e c i f i c a t i o n o f t h e p r e s s u r e p a r a m e t e r s . Table 4.2 Storm 2 Scenario Parameters L a t i t u d e Length Time Dura t i on A X Q / A t P o i n t Leg Long i tude (km) YYMMDDHH (h) (km/h) P o (mb) A P Q / A t (mb/h) A-B B-C C-D 42°N 160°W 42°N 145°W 53°N 145°W 59°N 145°W 1238 1223 667 86020100 86020200 86020218 86020312 24 18 18 52 68 37 984 980 958 970 - 0 . 2 - 1 . 2 (avg . ) - 1 . 8 (max.) +0.7 - 58 - 4.5.3 S t o r m 3: E a s t e r l y S h i f t e d N o r t h w a r d T r a j e c t o r y As i l l u s t r a t e d i n F i g . 4 . 7 , t h e B - C - D p o r t i o n o f t h e s t o r m t r a j e c t o r y c a n e a s i l y be 5 ° c l o s e r t o t h e B . C . c o a s t l i n e . S t o r m 3 was u s e d t o i n v e s t i g a t e t h e e f f e c t o f t h i s e a s t w a r d t r a n s l a t i o n by s i m p l y m o v i n g p o i n t B a n d t h e s u b s e q u e n t s t o r m p a t h f r o m 145°W i n s t o r m 1 t o 140°W i n s t o r m 3 . T a b l e 4 .3 summar i zes t h e r e s u l t i n g p r e s s u r e p a r a m e t e r s and c o n f i r m s t h a t t h e o n l y o t h e r c o n s e q u e n t i a l v a r i a t i o n f r o m s t o r m 1 i s a s m a l l i n c r e a s e i n Ax Q /A t i n l e g A - B . T a b l e 4.3 S t o r m 3 S c e n a r i o P a r a m e t e r s L a t i t u d e L e n g t h Time D u r a t i o n A X Q / A t P Q A P Q / A t P o i n t Leg L o n g i t u d e (km) YYMMDDHH (h ) ( km/h ) (mb) (mb/h) A 42°N 86030100 984 160°W A - B 1649 24 69 - 0 . 6 B 42°N 86030200 969 140°W B-C 1223 18 68 - 0 . 6 C 53°N 86030218 958 140°W C-D 667 18 37 +0.7 D 59°N 86030312 970 140°W - 59 - 4 . 5 . 4 Storm 4 : Increased Radial Extent Storm 4 has the same t r a j e c t o r y , c e n t r a l p r e s s u r e h i s t o r y and speed of t r a n s l a t i o n as storm 1, but the r a d i a l extent was increased from r QgQ=6° to 8° of l a t i t u d e . The e f f e c t of t h i s change i s , on average, to reduce Ap/Ar and hence reduce the wind speeds. As d i s c u s s e d i n S e c t i o n 4.4.4, 8° was the average radius of the 990 mb isobar i n the 21 f u l l y developed pressure maps from Lewis and Moran (1985) that were studied. The storm scenario parameters are repeated from Table 4.1 i n Table 4.4. Table 4 . 4 Storm 4 Scenario Parameters Latitude Length Time Duration ^ 0 ^ 1 p 0 AP Q/At Point Leg Longitude (km) YYMMDDHH (h) (km/h) (mb) (mb/h) A 42°N 86040100 984 160°W A-B 1238 24 52 -0.6 B 42°N 86040200 969 145°W B-C 1223 18 68 -0.6 C 53°N 86040218 958 145°W C-D 667 18 37 +0.7 D 59°N 86040312 970 145°W - 60 - 4.5.5 Storm 5: Advection Rate (Stalled Weather System In the Gulf of Alaska) Storm 5 i s i d e n t i c a l to storm 1 u n t i l day 03 hour 00 which i s 6 hours past p o i n t C i n the t r a j e c t o r y . At t h i s p o i n t (55°N 145°W) the system remains constant i n character and stationary f o r 24 hours. It then proceeds north i n the same manner as storm 1. Table 4.5 describes the c h a r a c t e r i s t i c s of t h i s storm. Table 4.5 Storm 5 Scenario Parameters Latitude Length Time Duration p 0 AP Q/At Point Leg Longitude (km) YYMMDDHH (h) (km/h) (mb) (mb/h) A 42°N 86050100 984 160°W A-B 1238 24 52 -0.6 B 42°N 86050200 969 145°W B-C 1223 18 68 -0.6 C 53°N 86050218 958 145°W C - s t a l l 222 6 37 +0.7 24 0 962 0.0 s t a l l - D 445 12 37 +0.7 s t a l l 55°N 86050300 145°W 86050400 D 59°N 86050412 970 145°W - 61 - 4.5.6 S t o r m 6: D e e p e s t C e n t r a l Low I n t h i s s c e n a r i o t h e m i n i m u m c e n t r a l p r e s s u r e i m p o s e d a t t h e same p o i n t C ( 5 3 ° N 145°W) was s e t t o t h e m i n i m u m r e c o r d e d P Q ( m i n ) among t h e s t o r m s i l l u s t r a t e d i n F i g . 4.7 w h i c h was 944 mb. The r a t e o f d e e p e n i n g a l o n g l e g B-C and t he r a t e o f f i l l i n g a l o n g l e g C-D were a d j u s t e d t o keep the v a l u e s o f P Q a t p o i n t s B a n d D n e a r l y e q u a l t o t h o s e i n s t o r m 1. T h e r e s u l t i n g r a t e o f d e e p e n i n g was - 1 . 3 m b / h b e t w e e n B a n d C a n d t h e r a t e o f f i l l i n g d o u b l e d t o +1.4 mb/h . The c h a r a c t e r of t h i s s t o r m i s , t h e r e f o r e , a r a p i d d e v e l o p m e n t o f a v e r y i n t e n s e l o w p r e s s u r e s y s t e m f o l l o w e d by v e r y r a p i d d i s s i p a t i o n a s d e s c r i b e d by T a b l e 4 .6 . T a b l e 4.6 S t o r m 6 S c e n a r i o P a r a m e t e r s L a t i t u d e L e n g t h Time D u r a t i o n P Q A P Q / A t P o i n t L e g L o n g i t u d e (km) YYMMDDHH (h ) ( km/h ) (mb) (mb/h) A 42°N 86060100 984 160°W A-B 1238 24 52 - 0 . 7 B 42°N 86060200 968 145°W B-C 1223 18 68 - 1 . 3 C 53°N 86060218 944 145°W C-D 667 18 37 +1.4 D 59°N 86060312 970 145°W - 62 - 4.5.7 Storm 7: Highest Minimum Central Pressure I n s t o r m 7 t h e m i n i m u m c e n t r a l p r e s s u r e d e e p e n s o n l y a s f a r a s 974 mb. T h i s was t h e h i g h e s t v a l u e among t h e 21 s t o r m s i l l u s t r a t e d i n F i g . 4 . 7 . H o w e v e r , t h e p r o t o t y p e s t o r m s y s t e m was s t i l l r e s p o n s i b l e f o r c r e a t i n g s t o r m f o r c e w inds w h i c h a r e d e f i n e d by L e w i s and M o r a n (1985) as a w i n d speed e x c e e d i n g 48 k n o t s ( i . e . B e a u f o r t s c a l e 10 o r g r e a t e r ) . I n t h e i d e a l i z e d s c h e m e , t h e s y s t e m d i d n o t d e e p e n a t a l l i n l e g A - B a n d t h e n P Q d e c r e a s e d by a b o u t - 0 . 6 m b / h t o t h e 974 mb m in imum a t p o i n t C a n d d i d n o t f i l l i n l e g C - D . T h e d e t a i l s o f t h i s s c e n a r i o a r e l i s t e d i n T a b l e 4 .7 . Table 4.7 Storm 7 Scenario Parameters L a t i t u d e L e n g t h Time D u r a t i o n A X 0 / A t P o A P Q / A t P o i n t L e g L o n g i t u d e (km) YYMMDDHH (h ) ( km/h ) (mb) (mb/h) A 42°N 86070100 984 160°W A - B 1238 24 52 0 . 0 B 42°N 86070200 984 145°W B-C 1223 18 68 - 0 . 6 C 53°N 86070218 974 145°W C-D 667 18 37 0 . 0 D 59°N 86070312 974 145°W - 63 - 4.5.8 Storm 8: Idealization of the February 5-7, 1960 Storm The s t o r m o f F e b r u a r y 5 - 7 , 1960 i s t h e one I n w h i c h P Q ( m i n ) r e a c h e d t h e l o w e s t r e c o r d e d v a l u e o f 944 mb among t h e 21 s t o r m s e x t r a c t e d f r o m L e w i s a n d M o r a n ( 1 9 8 5 ) . I t c o n s t i t u t e s q u i t e a s e v e r e s t o r m , e s p e c i a l l y s i n c e t h e m i n i m u m p r e s s u r e was r e a c h e d v e r y e a r l y i n t he t r a j e c t o r y . F i g . 4.9 shows t he w e a t h e r s y s t e m on F e b r u a r y 6 a t 12Z and t h e h i s t o r y of i t s c e n t r a l p r e s s u r e i n terras o f X Q a n d P Q a t 12 h o u r I n t e r v a l s . The i d e a l i z e d s t o r m i s p r e s e n t e d i n t h e same terras i n F i g . 4 .10 . The s y n t h e t i c v e r s i o n r e a c h e s i t s P 0 ( r a i n ) a t a p o i n t 6 h o u r s wes t o f p o i n t B and r e m a i n s c o n s t a n t a t t h i s l o w v a l u e f o r 24 h as t h e s t o r m p r o g r e s s e s a l o n g t he t r a j e c t o r y o f s t o r m 1 (and 6 ) . The r a t e o f a d v a n c e o f t h e i d e a l i z e d l o w p r e s s u r e s y s t e m i s f a i r l y s t e a d y a t a b o u t 50 k m / h t h r o u g h o u t i t s h i s t o r y a s s e e n i n T a b l e 4 . 8 . One o t h e r d i f f e r e n c e b e t w e e n s t o r m 8 a n d t h e o t h e r t e s t c a s e s i s t h a t t h e maximum p r e s s u r e c h a n g e i s o n l y 20 mb o f d e e p e n i n g i n s t o r m 8 s i n c e i t i s a l r e a d y s i g n i f i c a n t l y d e e p e n e d a t t h e w e s t e r n e d g e o f t h e s t u d y d o m a i n . On the o t h e r h a n d , P Q ( m i n ) i s f o l l o w e d by 20 mb o f f i l l i n g . By c o m p a r i s o n , s t o r m 1 deepens by 26 mb bu t f i l l s by o n l y 12 mb. The d u r a t i o n o f s t o r m 8 i s 6 h o u r s l o n g e r t h a n s t o r m 1 and p o i n t C i s d e f i n e d t o be a t t he same t i m e i n t he two s t o r m s (day 02 h o u r 18) bu t i t i s p o s i t i o n e d 3 ° of l a t i t u d e f u r t h e r s o u t h i n s t o r m 8 . Table 4.8 Storm 8 Scenario Parameters L a t i t u d e Length Time Dura t i on A x 0 / A t P o A p 0 / A t P o i n t Leg Long i tude (km) YYMMDDHH (h) (km/h) (mb) (rab/h) A - P 0 ( m i n ) P 0 (m in ) 42°N 160°W 42°N 149°W P 0 ( m i n ) - B B-C C-D 42°N 145°W 50°N 145°W 59°N 145°W 86080100 908 18 86080118 331 6 86080200 890 18 86080218 1001 24 86080318 50 55 49 42 964 944 944 944 964 - l . H a v g . ) - 1 . / ( m a x . ) 0 .0 0 .0 +0.8 - 64 - 12Z 6th February 1960 (+6) N IM ' 1»t' H i * U»* . 4 . 9 T h e s u r f a c e p r e s s u r e c h a r t f o r F e b r u a r y 6 , 1960 a t 12Z a t t h e p e a k o f t h e s t o r m ( u p p e r p a n e l ) a n d t h e s t o r m t r a j e c t o r y ( l o w e r p a n e l ) w i t h 1 2 - h o u r l y c e n t r a l p r e s s u r e s ( k P a ) . (From L e w i s and M o r a n , 1 9 8 5 ) . - 65 - S to rm 8 F i g . 4 . 1 0 I d e a l i z a t i o n of t h e s u r f a c e p r e s s u r e c h a r t c o r r e s p o n d i n g t o F e b r u a r y 6 , 1 9 6 0 a t 12Z ( u p p e r p a n e l ) a n d t h e s t o r m t r a j e c t o r y ( l o w e r p a n e l ) a t r o u g h l y the same s c a l e s as t he a c t u a l s t o r m d a t a i n F i g . 4 . 9 . - 66 - 5 . 0 DISCUSSION OF SENSIT IV ITY ANALYSIS RESULTS 5 .1 The B a s e S c e n a r i o S to rm 1 i s t he base s c e n a r i o a g a i n s t w h i c h most o t h e r c a s e s a r e compared . The c h a r a c t e r i s t i c s o f i t s p r e s s u r e f i e l d w e r e s e l e c t e d t o r e p r e s e n t a v e r a g e c o n d i t i o n s f o r a s e v e r e w i n t e r s t o r m . Maximum w i n d speeds a t t he peak o f t h e e v e n t a r e 60 k n o t s a s s h o w n i n F i g . 5 . 1 . The 6 - h o u r l y t i m e - s e r i e s o f s i g n i f i c a n t w a v e h e i g h t f i e l d s i s p r e s e n t e d i n F i g . 5.2 f r o m t h e e n d o f t h e s p i n - u p p e r i o d (P c e n t r e d a t p o i n t B o f t h e s t o r m t r a j e c t o r y ) t o t h e e n d o f the s i m u l a t i o n 36 h o u r s l a t e r (P c e n t r e d a t p o i n t D) . S e a - s t a t e s e x c e e d i n g 9 m a r e p r e s e n t f r o m d a y 02 h o u r 12 t o t h e e n d o f t h e s e q u e n c e on d a y 03 h o u r 12 . The max imum H g i s a b o u t 10 m o n d a y 03 h o u r 00 w h i c h i s 6 h a f t e r t h e s t o r m w inds a r e maximum. A l o n g t he wes t c o a s t o f B r i t i s h C o l u m b i a , waves a r r i v e f r o m the s o u t h w e s t . As a r e s u l t , t h e s e a - s t a t e e v o l v e s f r o m a f a i r l y u n i f o r m H g o f 0.5 t o 1.0 m t o a g r a d i e n t i n c r e a s i n g f r o m a b o u t 1.8 m n e a r T o f i n o t o 4 .5 m a t t h e n o r t h e n d o f t he Queen C h a r l o t t e I s l a n d s o v e r a 24 hou r p e r i o d b e g i n n i n g on day 01 h o u r 12. F i g . 5 .3 s h o w s t h e (H >T ,"§) h i s t o r y a t t h e s i t e i n t h e e n t r a n c e t o Q u e e n s p C h a r l o t t e Sound . A t t h i s l o c a t i o n , t he maximum w i n d speed i s abou t 15 k n o t s f r o m t h e s o u t h f r o m d a y 02 h o u r 12 t o d a y 03 h o u r 0 0 ; t h e r e a f t e r w i n d s p e e d d i m i n i s h e s a n d s h i f t s p r o g r e s s i v e l y mo re s o u t h w e s t e r l y . U n d e r s u c h l i g h t f o r c i n g , H g w i l l n o t e x c e e d a b o u t 1.5 m a s s u m i n g a s t e a d y w i n d f o r 12 h o u r s w i t h u n r e s t r i c t e d f e t c h ( B r e t s c h n e i d e r nomogram i n U . S . A r m y , 1 9 7 7 ) . T h e a b r u p t i n c r e a s e i n T o n d a y 03 a t a b o u t 03 h o u r s i n F i g . 5.3 a n d t h e c o r r e s p o n d i n g i n c r e a s e i n H g i s due t o s w e l l p r o p a g a t i n g s h o r e w a r d f r o m t h e g e n e r a t i o n r e g i o n on t h e s o u t h e a s t e r n s i d e o f t h e s t o r m s y s t e m a s i t was p o s i t i o n e d abou t 15 h o u r s p r e v i o u s l y . T h i s f i n d i n g h i g h l i g h t s t h e i m p o r t a n c e o f s w e l l e n e r g y t o t h e c o r r e c t h i n d c a s t i n g o f the c o a s t a l wave r e g i m e . 73 - 67 - I ( f f f r r f r f r 1111 ( ( f f f r f t f t»f * ( J < ' ? i g ' 5 \s.tiot8. sVl0«itvg A d t o t s t o r m - 68 - F i g . 5 . 2 S i x - h o u r l y s i g n i f i c a n t w a v e h e i g h t f i e l d s p r o d u c e d by s t o r m 1 a l o n g l e g s B-C and C-D o f t h e s t o r m t r a j e c t o r y . T h e p e a k o f t h e s t o r m w i n d s o c c u r s a t 8 6 0 1 0 2 1 8 ( d a y 02 h o u r 18) i n t h e f o u r t h p a n e l ; t h e maximum waves a r e s i x h o u r s l a t e r a t 86010300 (day 03 h o u r 0 0 ) . F i g . 5 . 2 C o n t i n u e d . F i g . 5.2 C o n t i n u e d . F i g . 5.2 C o n t i n u e d . - 72 - 5 . 3 T i m e - s e r i e s o f s i g n i f i c a n t wave h e i g h t ( H g ) , peak p e r i o d (Tp) and mean wave d i r e c t i o n a t t he s p e c i a l o u t p u t p o i n t a t the e n t r a n c e t o Queen C h a r l o t t e Sound (mode l g r i d c o - o r d i n a t e s ( 3 1 , 1 3 ) ) d u r i n g s t o r m 1. - 73 - 5 . 2 S t o r m I n t e n s i t y V a r i a t i o n s S to rm 2 t e s t s t he e f f e c t o f more r a p i d d e v e l o p m e n t o f an e q u a l l y i n t e n s e s t o r m a s m e a s u r e d by P Q ( m i n ) . I n e s s e n c e , t h i s s c e n a r i o h a s a s p i n - u p o f a d i m i n i s h e d i n t e n s i t y f o l l o w e d by a s h o r t p e r i o d o f e x p l o s i v e d e e p e n i n g a s i l l u s t r a t e d i n F i g . 5.4. The e x p e c t a t i o n i s , t h e r e f o r e , t h a t t he e a r l y s e a - s t a t e d e v e l o p m e n t w i l l be r e t a r d e d and maximum s e a - s t a t e s w i l l be l e s s h i g h t h a n i n s t o r m I. I f t h e l o c a l s e a s a r e r o u g h l y i n e q u i l i b r i u m w i t h l o c a l w i n d s , t h e d i f f e r e n c e i n t he f i e l d o f H g by t he end o f t he s l o w m o v i n g C-D l e g s h o u l d no t be v e r y g r e a t . Because t he e a r l y d e v e l o p m e n t i s w e a k e r , c o a s t a l s w e l l e n e r g y s h o u l d no t be as g r e a t as i n s t o r m 1. F i g . 5.5 s h o w s t h e d i f f e r e n c e i n t h e H g f i e l d e v e r y 6 h o u r s f r o m d a y 02 h o u r 06 a t t he b e g i n n i n g o f t he e x p l o s i v e d e e p e n i n g u n t i l day 03 h o u r 00 w h i c h i s 6 h o u r s a f t e r t h e end o f t h e d e e p e n i n g t o P Q ( m i n ) a n d w h i c h c o r r e s p o n d s t o t he t i m e o f maximum H g v a l u e s i n s t o r m 1. The d i f f e r e n c e A H g i s c a l c u l a t e d as H g ( s t o r m 2) m inus H g ( s t o r m 1) so t h a t n e g a t i v e d i f f e r e n c e s i n d i c a t e a r e g i o n w h e r e i n t he s e a - s t a t e o f s t o r m 2 i s l e s s t h a n t h a t o f s t o r m 1. I n t h i s c a s e , the l a r g e s t d i f f e r e n c e s a r e somewhat g r e a t e r t h a n 5 m. C o m p a r i s o n w i t h F i g . 5.2 i n d i c a t e s t h a t t h e s e d i f f e r e n c e s a r e c o i n c i d e n t w i t h t h e h i g h e s t s e a - s t a t e a r e a s where H g ( s t o r m 1) i s on the o r d e r o f 7.5 t o 8.0 m. As s t o r m 2 d e v e l o p s , t h e d i f f e r e n c e s d i m i n i s h t o a b o u t 3 .5 m by t h e s t o r m p e a k on d a y 02 a t h o u r 18. F u t h e r m o r e , t h e l a r g e s t d i f f e r e n c e s a r e no l o n g e r c o i n c i d e n t w i t h peak H g r e g i o n s . By d a y 03 h o u r 0 0 , t h e r e i s v i r t u a l l y no d i f f e r e n c e i n t h e a r e a o f l o c a l w i n d g e n e r a t i o n n o r t h o f abou t - ^ ° N . S o u t h o f t h i s l a t i t u d e i s a b r o a d r e g i o n i n w h i c h H g i s 1 t o 2 m l e s s . T h i s r e s u l t s i n s w e l l ene rgy o f abou t 2 m l o w e r s i g n i f i c a n t wave h e i g h t 12 h o u r s l a t e r a l o n g t h e c o a s t . T h i s f i n d i n g i s i l l u s t r a t e d by t he Queen C h a r l o t t e Sound s i t e i n F i g . 5.6. At t he c o a s t a l s i t e n o r t h o f t he Queen C h a r l o t t e I s l a n d s t h e r e i s v i r t u a l l y no d i f f e r e n c e i n t he t i m e - s e r i e s r e s p o n s e be tween s t o r m s c e n a r i o s 1 and 2 ( F i g . 5.7) . T h i s i s b e c a u s e t he s i t e i s w e l l - s h e l t e r e d d u r i n g t he e a r l y s t a g e s o f t h e s t o r m d e v e l o p m e n t ( i . e . o f f s h o r e w i n d s ) a n d t h e r e c a n be l i t t l e s w e l l e n e r g y a r r i v i n g a t t h i s s i t e u n t i l t h e s t o r m h a s p a s s e d t h i s n o r t h e r l y l a t i t u d e . 79 - 74 - 9 4 0 -i ; : — i i 00 12 0 0 12 0 0 12 D a y 01 D a y 02 D a y 0 3 F i g . 5 .4 The e v o l u t i o n o f P Q i n s t o r m s 1 and 2 . - 75 - F i g . 5.5 C o n t o u r e d f i e l d s o f AHG c a l c u l a t e d as s t o r m 2 l e s s s t o r m 1 f o r t h e p e r i o d d a y 02 h o u r 06 t o d a y 03 h o u r 00 i n 6 - h o u r l y t i m e s t e p s . - 76 - - 77 - Storm 2 Q u e e n C h a r l o t t e S o u n d (31,13) F i g . 5 . 6 T i m e - s e r i e s o f s i g n i f i c a n t wave h e i g h t ( H g ) , peak p e r i o d (Tp) and mean wave d i r e c t i o n a t t h e s p e c i a l o u t p u t p o i n t a t t he e n t r a n c e t o Queen C h a r l o t t e Sound (mode l g r i d c o - o r d i n a t e s ( 3 1 , 1 3 ) ) d u r i n g s t o r m 2 . Storm 1 N o r t h o f Q u e e n C h a r l o t t e I s l a n d s S t o r m 2 N o r t h o f Q u e e n C h a r l o t t e I s l a n d s (26,17) (26,17) F i g . 5.7 T i m e - s e r i e s o f s i g n i f i c a n t w a v e h e i g h t ( H g ) , p e a k p e r i o d (T ) a n d mean w a v e d i r e c t i o n a t the s p e c i a l o u t p u t p o i n t n o r t h o f t he Queen C h a r l o t t e I s l a n d s ( g r i d c o o r d i n a t e s (26 ,17) ) d u r i n g s to rms 1 and 2. - 79 - S torms 6 and 7 t e s t t h e s e n s i t i v i t y o f t he wave f i e l d t o v a r i a t i o n s i n s t o r m i n t e n s i t y c r e a t e d by t h e d i f f e r e n c e s i n c e n t r a l p r e s s u r e t h a t a r e shown i n F i g . 5.8. I n s t o r m 6 t h e 14 mb d i f f e r e n c e i n P Q ( m i n ) i s q u i t e i m p o r t a n t s i n c e t h e maximum w i n d s a r e s t r e n g t h e n e d f r o m 60 t o 75 k n o t s ( o r 25%). T h e c o n s e q u e n c e f o r t h e wave f i e l d i s an i n c r e a s e i n t h e l a r g e s t H g f r o m 10 m t o 14 m ( o r 40%) . A l o n g t h e c o a s t t h e d i f f e r e n c e s a r e e x e m p l i f i e d by t h e n o r t h e r n Q u e e n C h a r l o t t e I s l a n d s a n d Q u e e n C h a r l o t t e S o u n d s i t e s i n F i g . 5 . 9 . The f i r s t d e v i a t i o n a p p e a r s a t a b o u t t h e t i m e o f p e a k w i n d s ( d a y 02 h o u r 18) a n d p r o g r e s s i v e l y b u i l d s t o a n i n c r e a s e o f a b o u t 1.0 m o v e r 3 m f o r s t o r m 1 i n Q u e e n C h a r l o t t e S o u n d and a b o u t 1.5 m i n 5 m n o r t h o f t h e Q u e e n C h a r l o t t e I s l a n d s . I n t e r m s o f s p e c i f y i n g a b o u n d a r y c o n d i t i o n t o a f i n e g r i d n e s t e d c o a s t a l m o d e l , e r r o r s o f a t l e a s t 30% c o u l d e a s i l y r e s u l t f r o m a s e r i o u s unde r e s t i m a t e o f P Q ( m i n ) f o r t h i s t y p e of s t o r m t r a j e c t o r y . W i t h i n t h e a r e a d o m i n a t e d by l o c a l w i n d s e a , F i g . 5 . 1 0 s h o w s max imum d i f f e r e n c e s o f 5 m i n abou t 9 m as p r e d i c t e d by s t o r m 1, o r a p o t e n t i a l e r r o r o f n e a r l y 60% r e s u l t i n g f r o m a 25% c h a n g e i n P Q ( m i n ) w h i c h i s c a l c u l a t e d r e l a t i v e t o P = 1015 mb. T h e r e a r e , o f c o u r s e , a l s o d i f f e r e n c e s i n t h e s t o r m h i s t o r y i n t e r m s o f A P Q / A t w h i c h m i t i g a t e a g a i n s t l a r g e s e a - s t a t e c h a n g e s . S i n c e t h e r a t e o f d e v e l o p m e n t i n s t o r m 6 i s m o r e r a p i d t h a n i n s t o r m 1 ( i n f a c t , i t i s s i m i l a r t o s t o r m 2 i n t h i s r e s p e c t ) , t h e r e w i l l be l e s s t i m e f o r the l o c a l s e a to r e a c h i t s maximum p o t e n t i a l f o r t h e g i v e n w ind f o r c i n g . . . . 83 - 80 - D a y 01 D a y 02 D a y 0 3 F i g . 5.8 The e v o l u t i o n o f P Q i n s t o r m s 1, 6 a n d 7. I n a l l c a s e s P Q ( m i n ) i s imposed a t 53°N 145°W. - 81 - S t o r m 6 n o r t h o f t h e Q u e e n C h a r l o t t e I s l a n d s (26,17) S t o r m 6 Q u e e n C h a r l o t t e S o u n d (31,13) . 5 . 9 C o m p a r i s o n o f s t o r m 1 a n d s t o r m 6 r e s p o n s e a s e x e m p l i f i e d by s i g n i f i c a n t w a v e h e i g h t a t t h e s p e c i a l o u t p u t p o i n t s n o r t h o f t h e Q u e e n C h a r l o t t e I s l a n d s ( 2 6 , 1 7 ) a n d a t t h e e n t r a n c e t o Q u e e n C h a r l o t t e S o u n d ( 3 1 , 1 3 ) . - 82 - Dashed l i n e s i n d i c a t e n e g a t i v e d i f f e r e n c e s Heavy s o l i d l i n e s i n d i c a t e a d i f f e r e n c e of z e r o L i g h t s o l i d l i n e s i n d i c a t e p o s i t i v e d i f f e r e n c e s F i g . 5 . 1 0 The f i e l d o f A H g a t t h e t i m e o f maximum wave h e i g h t s i n s t o r m 1 ( d a y 03 h o u r 0 0 ) d i f f e r e n c e d a s s t o r m 6 m i n u s s t o r m 1 (uppe r p a n e l ) and the c o r r e s p o n d i n g f i e l d o f H g f r o m s t o r m 1. - 83 - S t o r m 7 h a s c o n s i d e r a b l y w e a k e r w i n d s t h a n s t o r m 1 s i n c e i t s c e n t r a l l o w o f 974 mb I s 16 mb h i g h e r r e p r e s e n t i n g a change o f 28% i n P Q r e l a t i v e to P. Peak w inds r e a c h o n l y 35 k n o t s , j u s t a l i t t l e more t h a n h a l f o f t h e 60 k n o t maximum i n s t o r m 1. The l a r g e s t d i f f e r e n c e i n t h e s i g n i f i c a n t w a v e h e i g h t f i e l d i s b e t w e e n 5 a n d 6 m i n 10 m f o r s t o r m 1 as s h o w n I n F i g . 5 . 1 1 . C o m p a r i s o n w i t h t h e s t o r m 1 H g f i e l d i n d i c a t e s a g e n e r a l r e d u c t i o n , o n t h e o r d e r o f 50%, t h r o u g h o u t the r e g i o n d o m i n a t e d by t h e l o c a l w i n d - s e a . The c o a s t a l s w e l l c l i m a t e i s a l s o g r e a t l y r e d u c e d i n b o t h h e i g h t and p e r i o d due t o t h e w e a k e r s e a - s t a t e d e v e l o p m e n t i n t h e e a r l y s t a g e s o f t h e s t o r m . F i g . 5.12 i l l u s t r a t e s t h i s t r e n d a t two s t a t i o n s : one n o r t h w e s t o f t he Queen C h a r l o t t e I s l a n d s and t he o t h e r s o u t h e a s t o f t h e s e i s l a n d s i n t h e e n t r a n c e t o Queen C h a r l o t t e Sound. A d i f f e r e n c e o f 2 m i n H comb ined w i t h a 6 s l o w e r T s p i s p r e d i c t e d i n s t o r m 7 f o r t h e mo re n o r t h e r l y s i t e by t h e end o f t h e t e s t w h i c h i s 18 h a f t e r t h e s t o r m p e a k . A t t h e Q u e e n C h a r l o t t e S o u n d s i t e , t h e H g d i f f e r e n c e i s a l m o s t as g r e a t , bu t some l o n g p e r i o d s w e l l b e g i n s to a r r i v e j u s t b e f o r e t he end o f t he mode l r u n . S to rm 4 i s a l s o a t e s t o f s t o r m i n t e n s i t y i n t h a t t h e r a d i a l e x t e n t d e f i n e d by r 9 9 0 ^ s i n c r e a s e d t o 8 ° o f l a t i t u d e f r o m 6 ° i n s t o r m 1 (and a l l o t h e r s t o r m s ) . T h i s r e p r e s e n t s a n i n c r e a s e i n s t o r m s i z e o f 222 km (33%) f r o m t h e c e n t r e t o t h e 990 mb i s o b a r . S i n c e t he d e r i v a t i o n o f w ind speed depends on d P / d r , l o w e r w i n d s p e e d s a r e e x p e c t e d . H o w e v e r , a s s h o w n i n F i g . 5 . 1 3 , t h e l o c a l r a t e o f c h a n g e i n P may a c t u a l l y i n c r e a s e f o r t h e l a r g e r s t o r m s y s t e m d u e t o t h e l o g a r i t h m i c s c a l i n g o f p r e s s u r e . I n F i g . 5 . 13 t h i s s o m e w h a t a n o m a l o u s b e h a v i o u r i s s e e n a t r a d i a l d i s t a n c e s be tween abou t 800 and 1000 km. I n t h i s e x a m p l e , t h e r e i s e s s e n t i a l l y no d i f f e r e n c e i n t he p r e s s u r e g r a d i e n t be tween abou t 300 and 800 km and a s l i g h t d e c r e a s e i n t h e g r a d i e n t f rom t h e c e n t r e ou t t o abou t 300 km f o r t h e l a r g e r s t o r m s y s t e m . . . 87 - 84 - Heavy s o l i d l i n e s i n d i c a t e a d i f f e r e n c e of z e r o L i g h t s o l i d l i n e s i n d i c a t e p o s i t i v e d i f f e r e n c e s F i g . 5 .11 The f i e l d o f A H g a t t he t i m e o f maximum wave h e i g h t s i n s t o r m 1 ( d a y 03 h o u r 00) d i f f e r e n c e d a s s t o r m 7 m i n u s s t o r m 1 ( u p p e r p a n e l ) and t he c o r r e s p o n d i n g f i e l d o f H g f r om s t o r m 1. S t o r m 7 Q u e e n C h a r l o t t e S o u n d (31.13) S t o r m 7 N o r t h o f Q u e e n C h a r l o t t e I s l a n d s (26,17) F i g . 5 .12 T i m e - s e r i e s o f H g and T p i n t h e e n t r a n c e t o Q u e e n C h a r l o t t e Sound and a t the c o a s t a l s i t e t h a t i s n o r t h o f the Queen C h a r l o t t e I s l a n d s . - 86 - F i g . 5 . 13 C o m p a r i s o n o f i s o b a r r a d i i a s a f u n c t i o n o f t h e r a d i a l s c a l e r^gQ f o r the c a s e o f P Q =958 mb. - 87 - F i g . 5 .14 s h o w s t h e A H g f i e l d ( s t o r m 4 m i n u s s t o r m 1) a t t h e t i m e o f peak, w i n d s i n t h e u p p e r p a n e l and a t t h e e n d o f t h e m o d e l r u n i n t h e l o w e r p a n e l . These r e s u l t s i n d i c a t e t h a t d u r i n g s t o r m d e v e l o p m e n t t h e wave f i e l d a l o n g t h e c o a s t i s s l i g h t l y g r e a t e r i n s t o r m 4 t h a n i n s t o r m 1 a l t h o u g h i n t h e a r e a d o m i n a t e d by l o c a l w i n d - s e a t h e l a r g e r s t o r m s y s t e m H g v a l u e s a r e up t o 2 m l e s s t h a n i n t he s m a l l e r base s t o r m . By t h e end of: t he s t o r m t h e r e i s a l m o s t no d i f f e r e n c e i n H a t mos t o f t h e c o a s t a l s t a t i o n s . T h i s c o n c l u s i o n i s s u p p o r t e d by t he t i m e - s e r i e s of H g and T p c o m p a r i s o n s i n t h e e n t r a n c e t o Queen C h a r l o t t e S o u n d ( F i g . 5 . 1 5 ) . A t t h e p e a k o f t h e s t o r m e a r l y on d a y 03 t h e s t o r m 1 p e r i o d s a r e l o n g e r by 2 t o 10 s i n s t o r m 1. The w a v e h e i g h t s a r e e s s e n t i a l l y e q u a l t h r o u g h o u t t h e s t o r m , a l t h o u g h t h e s t o r m 4 r e s u l t s a r e n u m e r i c a l l y l a r g e r . T h i s i s due t o t h e s l i g h t w i n d i n t e n s i f i c a t i o n a t t h e s t o r m p e r i p h e r y t h a t i s e x p l a i n e d by F i g . 5 . 1 3 . A t t he w e s t e r n m o s t s p e c i a l o u t p u t p o i n t , w h i c h i s w i t h i n 380 km o f t he s t o r m c e n t r e e a r l y on d a y 0 2 , t h e f i r s t 42 h o u r s o f r e s u l t s a r e n e a r l y e q u a l i n s t o r m s 1 and 4. H o w e v e r , as t h e o v e r a l l s t o r m wave f i e l d peaks (day 03 h o u r 0 0 ) a n d d e c a y s , t h e m o d e l r e s u l t s a t t h i s s i t e ( F i g . 5 .16 ) s h o w t h a t t h e s e a - s t a t e i s 0 .5 t o 1.0 m l e s s when f o r c e d by t h e l a r g e r s t o r m 4 . Of t h e t h r e e i n t e n s i t y p a r a m e t e r s t e s t e d ( s p e e d o f d e v e l o p m e n t , m i n i m u m c e n t r a l p r e s s u r e , a n d r a d i a l e x t e n t ) , t h e w a v e r e s u l t s a r e c e r t a i n l y l e a s t s e n s i t i v e t o changes i n r a d i a l e x t e n t , a l t h o u g h t h e r a n g e o f v a l u e s o f t h i s p a r a m e t e r t h a t were t e s t e d i s q u i t e s m a l l . Of t h e r e m a i n i n g two p a r a m e t e r s , t he v a l u e o f P Q ( m i n ) a p p e a r s t o be t h e more c r i t i c a l and i t i s an e x c e e d i n g l y d i f f i c u l t p a r a m e t e r t o d e t e r m i n e a c c u r a t e l y f r o m t h e s c a t t e r e d measurements made by s h i p s a t s e a and t he few m e t e o r o l o g i c a l b u o y s . F u r t h e r o b s e r v a t i o n s o n t h e s p e e d o f s t o r m d e v e l o p m e n t w i l l be d r a w n i n S e c t i o n 5 .5 i n t h e d i s c u s s i o n o f t he i d e a l i z a t i o n o f t h e F e b r u a r y 5 - 7 , 1960 s t o r m . 91 - 88 - . 5 .14 The f i e l d o f A H g a t t h e t i m e o f maximum w i n d s ( d a y 02 h o u r 18 ; u p p e r p a n e l ) a n d a t t h e end o f t h e m o d e l l i n g s e q u e n c e ( d a y 03 h o u r 1 2 ; l o w e r p a n e l ) d i f f e r e n c e d a s s t o r m 4 m i n u s s t o r m 1. - 89 - F i g . 5 . 1 5 T i m e - s e r i e s c o m p a r i s o n o f H s and T p f r o m s t o r m s 4 and 1 a t t he e n t r a n c e t o Queen C h a r l o t t e S o u n d . - 90 - F i g . 5 . 1 6 T i m e - s e r i e s c o m p a r i s o n o f H s and Tp f r om s to rms 4 and 1 a t t he w e s t e r n m o s t o u t p u t s i t e . - 91 - 5.3 S t o r m T r a j e c t o r y S to rm 3 was d e s i g n e d to t e s t the e f f e c t s o f t r a n s l a t i o n o f the n o r t h w a r d p a r t o f the s t o r m t r a j e c t o r y by 5° o f l o n g i t u d e t o 140°W. F i g . 4.7 shows t h a t t h i s i s a r e a s o n a b l e a l t e r n a t e t r a j e c t o r y f o r b o t h t h e s o u t h w e s t f r o n t a l l o w s and t h e s o u t h w e s t c o l d l o w s c o n s i d e r e d i n t h i s s t u d y . B e c a u s e t h e n o r t h w a r d p o r t i o n s o f the s t o r m 1 and s t o r m 3 p a t h s a r e c o n v e r g e n t , t h e a b s o l u t e s p a t i a l d i f f e r e n c e d i m i n i s h e s w i t h i n c r e a s i n g l a t i t u d e . A t 4 2 ° N , 5 ° of l o n g i t u d e i s a d i s t a n c e o f 413 km, b u t a t 5 9 ° N i t i s o n l y 286 k m , a n d t h i s c o n v e r g e n c e w i l l c o n t r i b u t e t o a s i m i l a r i t y i n t h e s e a - s t a t e s t o w a r d t h e end o f t h e t e s t . N e v e r t h e l e s s , a s i d e f r om s m a l l d i s t o r t i o n s i n t h e wave f i e l d nea r the c o a s t , t he e f f e c t i s a l i t e r a l t r a n s l a t i o n o f t he wave f i e l d by 5 ° towards t he c o a s t w h i c h i s c l e a r l y a p p a r e n t i n F i g . 5 .17. P e r c e n t a g e d i f f e r e n c e s i n H g a r e g r e a t e s t a l o n g t h e c o a s t . A t t h e Q u e e n C h a r l o t t e Sound s i t e , t h e i n c r e a s e i s f r o m 1 m t o 3 m on day 02 hou r 18 ( F i g . 5 . 1 7 ) . The m o s t d r a m a t i c d i f f e r e n c e i n s e a - s t a t e n e a r t h e s t o r m p e a k i s a t the s p e c i a l o u t p u t s i t e due west o f t h e Queen C h a r l o t t e Sound s i t e . Here H g c h a n g e s f r o m 3.5 m i n s t o r m 1 t o a l m o s t 10 m i n s t o r m 3 . H o w e v e r , F i g . 5 .18 shows t h a t t owards t h e end o f t h e s t o r m s e q u e n c e on day 03 h o u r 12, a l l t h r e e s e a - s t a t e p a r a m e t e r s a t t h i s s i t e a r e v e r y n e a r l y t h e same i n t h e two s t o r m c a s e s . T h i s i s c o n s i s t e n t w i t h the r e s u l t s o f s t o r m 1: c o n s i d e r i n g t h e f i n a l p a n e l o f F i g . 5 . 2 , i t i s c l e a r t h a t i n t h e v i c i n i t y o f t h e s p e c i a l o f f s h o r e o u t p u t p o i n t s t he s p a t i a l g r a d i e n t i n t h e s e a - s t a t e i s no t v e r y l a r g e so t h a t a s h i f t i n g o f t h e H f i e l d by 5 ° o f l a t i t u d e c a n o n l y make a b o u t 0.5 t o 1.5 m d i f f e r e n c e . F o r the c o a s t a l s i t e s , c o n t i n u i t y d i c t a t e s t h a t t h e g r a d i e n t i n wave h e i g h t m u s t s t e e p e n when t h e s t o r m t r a c k i s c l o s e r t o t h e c o a s t . T h i s i s s e e n i n F i g . 5 .17 a l o n g V a n c o u v e r I s l a n d . F r o m t h e t i m e - s e r i e s i n Q u e e n C h a r l o t t e Sound i n F i g . 5 .19, i t a p p e a r s t h a t t h e s e a - s t a t e b e g i n s i t s s h a r p r i s e a t t h e c o a s t a b o u t 18 h o u r s e a r l i e r i n s t o r m 3 t h a n i n s t o r m 1 ( d a y 02 h o u r 06 i n s t e a d o f d a y 03 h o u r 0 0 ) . The d e v e l o p m e n t o f T p i n s t o r m 3 i n d i c a t e s t h a t t h e r e i s a n i m p o r t a n t l o c a l w i n d - s e a c o m p o n e n t d u r i n g d a y 0 2 , w h i c h was n o t w e l l - d e v e l o p e d i n s t o r m 1, b e c a u s e t he c o a s t a l w inds a r e somewhat h i g h e r when t h e s t o r m c e n t r e i s c l o s e r t o t he c o a s t . A t t h e c o a s t , t h i s s c e n a r i o p r o d u c e s g r e a t e r c h a n g e s t h a n a n y o f t h e s t o r m i n t e n s i t y t e s t s , i n c l u d i n g i m p o s i t i o n o f t h e d e e p e s t c e n t r a l l o w . 95 F i g . 5 .17 S i g n i f i c a n t w a v e h e i g h t f i e l d u n d e r maximum w i n d s i n s t o r m 1 (uppe r p a n e l ) and i n s t o r m 3 ( l o w e r p a n e l ) . The s t o r m t r a j e c t o r y i n s t o r m 3 i s 5 ° c l o s e r to the c o a s t . Storm 3 - 93 - (25,13) 5.18 C o m p a r i s o n o f Che t i m e - s e r i e s o f H g, T and mean d i r e c t i o n i n storms 3 and 1 at the s p e c i a l o u t p u t p o i n t due west of the e n t r a n c e t o Queen C h a r l o t t e Sound. Storm 3 - 94 - Q u e e n C h a r l o t t e S o u n d (31,13) . 5 . 1 9 C o m p a r i s o n o f t h e t i m e - s e r i e s o f H , T and mean d i r e c t i o n i n s t o r m s 3 and 1 a t the s p e c i a l o u t p u t p o i n t i n the e n t r a n c e t o Queen C h a r l o t t e S o u n d . - 95 - 5.4 A d v e c t i o n R a t e o f t h e C e n t r a l Low I n t h i s s e c t i o n o n l y o n e i m p o r t a n t c a s e i s c o n s i d e r e d : t h e s t a l l e d l o w . I n L e w i s and M o r a n (1985) t h e r e a r e f o u r e x a m p l e s among t h e 21 s to rms i n F i g . 4.7 i n w h i c h a l o w p r e s s u r e s y s t e m became s t a t i o n a r y f o r up to 24 h o u r s , u s u a l l y n o r t h o f t h e Q u e e n C h a r l o t t e I s l a n d s a f t e r t h e s t o r m peak.. R e p o r t e d s e a - s t a t e s i n t h e s e s t o r m s a r e i n v a r i a b l y h i g h (8 t o 10 m l o c a l w i n d sea ) and s h i p s i n k i n g was r e c o r d e d i n two o f t he f o u r s t o r m s . Two o f t h e s e e v e n t s a l s o had e x p l o s i v e d e e p e n i n g i n t he d e v e l o p m e n t s t a g e s . I n t h e i d e a l i z e d s t o r m 5 e x a m p l e , t h e s t o r m 1 w i n d f i e l d s a r e u s e d w i t h a 24 h o u r i n s e r t i o n of c o n s t a n t w ind p a t t e r n b e g i n n i n g on day 03 hou r 00 c e n t r e d a t 55°N 145°W. A t t h i s t i m e t h e maximum w inds a r e 50 t o 55 k n o t s w i t h a s o u t h t o n o r t h g r a d i e n t o f 10 t o 20 k n o t s o u t h e r l y w inds a l o n g t h e B.C. c o a s t . A t day 03 h o u r 00 t h e s e a - s t a t e has d e v e l o p e d as shown f o r s t o r m 1 i n F i g . 5.2 a t t h e same t i m e ( f i f t h p a n e l ) . T h e c o n d i t i o n s 12 a n d 24 h o u r s l a t e r a r e p r e s e n t e d i n F i g . 5.20. The maximum H g has i n c r e a s e d f r o m 10 m on day 03 h o u r 00 t o j u s t o v e r 13 .5 m 24 h o u r s l a t e r . T h e c h a n g e i n t h e s e a - s t a t e p a t t e r n d u r i n g t h e s t a l l p e r i o d i s s h o w n i n F i g . 5 . 2 1 . The l a r g e s t d i f f e r e n c e i s a n i n c r e a s e o f m o r e t h a n 7 m due s o u t h o f t h e c e n t r a l l o w i n a r e g i o n t h a t was w e a k l y d e v e l o p e d (4 t o 5 m) unde r t h e m o v i n g s t o r m s y s t e m . A l o n g t he c o a s t t h e s e a - s t a t e i n c r e a s e s by more t h a n 2 m by the end of t he 24 h o u r s t a l l ( F i g . 5 .21) . A t t h e o f f s h o r e s p e c i a l o u t p u t p o i n t s , t he change i s 1 m o r l e s s f o r t h o s e s i t e s wes t of 132°W and 2 m f o r the two s o u t h e r l y s i t e s t h a t a r e e a s t o f t h i s l o n g i t u d e . F i g . 5 .22 s h o w s t h e n e t d i f f e r e n c e i n H g a t t he end o f s t o r m 5 and t h e base s t o r m ( s t o r m 5 day 04 h o u r 12 l e s s s t o r m 1 day 03 h o u r 1 2 ) . A t t h i s p o i n t , t h e two s t o r m s a r e t h e same a n d h a v e b e e n f o r 12 h o u r s : t h e o n l y d i f f e r e n c e i s t he p r e c e d i n g s t o r m h i s t o r y . I n t h e n o r t h w e s t h a l f o f t h e G u l f o f A l a s k a , the s t o r m 5 s e a - s t a t e i s g e n e r a l l y e l e v a t e d by 1 m o v e r t h a t o f s t o r m 1. B e c a u s e t h e w ind and wave c i r c u l a t i o n i s a n t i c l o c k w i s e , t h i s i n c r e a s e i n H , c a n be a t t r i b u t e d l a r g e l y t o t h e s t a l l . I n t h e s o u t h w e s t e r n q u a r t e r o f t he s t u d y d o m a i n , t h e two s c e n a r i o s e a - s t a t e s a r e t h e same w i t h i n l e s s t h a n +1 ra. I n t he s o u t h e a s t , h o w e v e r , H g i s e l e v a t e d by 1 t o 2 m o f f s h o r e and somewhat more c l o s e r to the c o a s t . B a s e d on the f i n a l p a n e l i n F i g . 5.2 f o r d a y 03 h o u r 12 i n s t o r m 1 a n d t h e f a c t t h a t w a v e s a r r i v i n g a t t h e c o a s t a r e f r o m t h e w e s t s o u t h w e s t , m o s t o f t h i s p r e d i c t e d i n c r e a s e i n H g a r i s e s f r o m p r o p a g a t i n g s w e l l e n e r g y t h a t w o u l d a l s o a r r i v e f o r s t o r m 1 w i t h i n t h e e x t r a 24 h o u r s of m o d e l l i n g t i m e t h a t was a v a i l a b l e t n s t o r m 5. . . . 99 - 96 n F i g . 5 . 2 0 The s i g n i f i c a n t wave h e i g h t f i e l d 1 2 h o u r s a f t e r t h e storm s t a l l (upper p a n e l ) and 2 4 hours a f t e r the s t a l l ( l o w e r p a n e l ) . The maximum H g i s 1 2 . 5 m on day 0 3 h o u r 1 2 and 1 3 . 5 m on day L ) 4 hour 0 U . - 97 - F i g . 5 .21 The A H g f i e l d c a l c u l a t e d a s H a t d a y 03 h o u r 00 ( t h e b e g i n n i n g o f t h e s t a l l ) m i n u s H G 12 h o u r s l a t e r ( u p p e r p a n e l ) and m i n u s H g 24 h o u r s l a t e r ( l o w e r p a n e l ) . Maximum d i f f e r e n c e s a r e on the o r d e r o f 6 t o n e a r l y b m n e a r t h e s t o r m c e n t r e and 1 t o 3 m a l o n g t h e B . C . c o a s t . - 98 - F i g . 5 . 22 The AH f i e l d c a l c u l a t e d a s H g a t d a y 04 h o u r 12 i n s t o r m 5 m i n u s H g a t d a y 0 3 h o u r 12 i n s t o r m 1. A t t h e s e t i m e s t he s t o r m s y s t e m s a r e e q u i v a l e n t and h a v e been f o r t he p r e c e d i n g 12 h o u r s . - 99 - The l o c a t i o n o f t h e s t a l l w i l l d e t e r m i n e w h i c h s i t e s r e c e i v e t h e m o s t a d d i t i o n a l w a v e e n e r g y . T h e m o r e n o r t h e r l y t h e s t a l l p o s i t i o n , t h e l e s s consequence i t w i l l h a v e f o r t h e s o u t h e r n c o a s t , and t h e more i m p a c t i t w i l l h a v e n o r t h of the Queen C h a r l o t t e I s l a n d s f r om g r e a t e r d e v e l o p m e n t o f l o c a l w i n d - s e a i n t h a t r e g i o n and t h e p o t e n t i a l f o r much more s w e l l e n e r g y f r om the s o u t h e a s t s i d e o f a s t a t i o n a r y w e a t h e r s y s t e m . 5.5 The Idealized Storm of February 5 to 7, 1960 T h e a c t u a l and i d e a l i z e d v e r s i o n s o f t h i s s t o r m a r e i l l u s t r a t e d i n F i g . 4 .9 a n d 4 . 1 0 . I t h a s b e e n i n c l u d e d t o e m p h a s i z e a n i m p o r t a n t p o i n t i n w a v e h i n d c a s t i n g . E v e n f o r i d e a l i z e d c a s e s , i t c a n n o t b e s a i d , w i t h o u t q u a l i f i c a t i o n , t h a t any one o f t he s t o r m p a r a m e t e r s i s more i m p o r t a n t t h a n t h e o t h e r s . T h e r e i s a m p l e scope w i t h i n the n a t u r a l v a r i a b i l i t y o f t h e s e ( f a c t o r s , i f c o m b i n e d i n u n f a v o u r a b l e w a y s , t o g e n e r a t e a m o r e s e v e r e s e a - s t a t e t h a n t h a t p r o d u c e d by any o f t he e x t r e m e c a s e s a l r e a d y i n v e s t i g a t e d . S to rm 8 has a number o f c h a r a c t e r i s t i c s i n common w i t h o t h e r t e s t s c e n a r i o s , bu t t h e y o c c u r i n u n i q u e c o m b i n a t i o n s . F o r e x a m p l e , i t has t he same minimum c e n t r a l l o w p r e s s u r e o f 944 mb a s s t o r m 6; L t h a s a 12 h o u r e x p l o s i v e d e e p e n i n g segment p r e c e d i n g P Q ( m i n ) t h a t , a t - 1 . 7 m b / h , i s abou t t he same r a t e a s i n s t o r m 2 ; i t h a s t h e same t r a j e c t o r y a s s t o r m 1 ( a n d a l l o t h e r s e x c e p t number 3). The i m p o r t a n t c h a r a c t e r i s t i c s w h i c h make s t o r m 8 more s e v e r e t h a n any o f t he o t h e r s a r e t h r e e f o l d : ( 1 ) P Q ( m i n ) i s r e a c h e d i n t h e s o u t h w e s t q u a d r a n t o f the mode l g r i d . As a r e s u l t , t h e s o u t h w e s t e r l y s w e l l r e a c h i n g t h e s o u t h e r n B.C. c o a s t w i l l be more e n e r g e t i c . ( 2 ) P Q ( m i n ) p e r s i s t s f o r 24 h o u r s as the s t o r m a d v e c t s e a s t w a r d and t h e n n o r t h w a r d ( s e e F i g . 4 . 1 0 ) . T h i s means t h a t t h e d u r a t i o n o f maximum w inds i s much l o n g e r t h a n i n t h e a v e r a g e ( s t o r m 1) c a s e . ( 3 ) The c e n t r a l p r e s s u r e i s l o w e r , and hence t h e w inds a r e s t r o n g e r , as t h e s t o r m e n t e r s and e x i t s f rom the s t u d y a r e a . I t i s more u n c e r t a i n what t he e x a c t e f f e c t o f d i f f e r e n c e s i n t he s t o r m speed a r e . I t i s s l o w e r i n t he B-C l e g o f the t r a j e c t o r y w h i c h w i l l t e n d t o b u i l d t h e maximum wave f i e l d a b i t h i g h e r , bu t i t i s a l s o a l i t t l e f a s t e r i n t h e C-D s e c t i o n w h i c h w i l l t e n d t o d i m i n i s h t h e s e a - s t a t e s o m e w h a t o n t h e n o r t h e r n c o a s t . A n o t h e r f a c t o r w h i c h o f f s e t s t h e p r o m o t i o n o f h i g h s e a - s t a t e s , e s p e c i a l l y f o r t he n o r t h e r n c o a s t , i s t h a t the f i l l i n g p r o c e s s b e g i n s 3 ° o f - 100 - l a t i t u d e f u r t h e r s o u t h i n s t o r m 8 t h a n i n any o t h e r t e s t c a s e . B e c a u s e t h i s s t o r m i s much s t r o n g e r w h i l e i t i s c e n t r e d i n t h e s o u t h e r l y l a t i t u d e s o f t he mode l g r i d , the s o u t h w e s t e r l y s w e l l e n e r g y t h a t p r o p a g a t e s t owards V a n c o u v e r I s l a n d and Queen C h a r l o t t e Sound ought t o be c o n s i d e r a b l y g r e a t e r t h a n i n o t h e r s c e n a r i o s : h i g h e r H and l o n g e r T . T h i s i s c o n f i r m e d s p i n F i g . 5 . 2 3 i n Queen C h a r l o t t e S o u n d . The o r i g i n and e v o l u t i o n o f t he c o a s t a l s w e l l e n e r g y i s i l l u s t r a t e d i n F i g . 5 . 24 w i t h 1 2 - h o u r l y H f i e l d s f rom s t o r m 8, u s i n g t h e h i g h l i g h t e d 7 m c o n t o u r as a r e f e r e n c e . T h e w a v e p a t t e r n n e a r t h e s o u t h e r n g r i d e d g e a t d a y 02 h o u r 18 g r a d u a l l y b e c o m e s mo re i n d e p e n d e n t o f t h e a c t i v e s t o r m c e n t r e a n d p r o p a g a t e s t o t h e e a s t f r o m t h e r e g i o n c e n t r e d a r o u n d 145°W a n d t o t h e n o r t h e a s t f r o m a r e a s on t h e s o u t h e a s t e r n s i d e o f t h e s t o r m c e n t r e . A t t h e same t i m e e n e r g y i s d i s p e r s e d w i t h t h e h i g h e r waves t r a v e l l i n g f a s t e r t h a n t he s m a l l e r o n e s , a n d e n e r g y i s d i s s i p a t e d t h r o u g h t h e w a v e - w a v e i n t e r a c t i o n mechan ism t h a t c a s c a d e s e n e r g y to h i g h e r f r e q u e n c i e s where t h e s p e c t r u m may a l r e a d y be s a t u r a t e d . Based on t h e H g p a t t e r n on day 03 h o u r 18 and t he w e s t s o u t h w e s t e r l y mean wave d i r e c t i o n ( F i g . 5 . 2 3 ) , t h e s e a - s t a t e c o u l d be e x p e c t e d t o p e a k a t a b o u t 8 .5 m i n Q u e e n C h a r l o t t e S o u n d . By c o n t r a s t , H g i n s t o r m 6 ( w i t h t h e same t r a j e c t o r y and t he same minimum c e n t r a l l o w p r e s s u r e ) w i l l p r o b a b l y no t e x c e e d abou t 6.5 ra a t t h i s l o c a t i o n . F i g . 5.25 shows t h e e v o l u t i o n a r y p a t t e r n o f H g f o r t h i s l e s s s e v e r e s t o r m u s i n g t h e 5 m c o n t o u r a s a r e f e r e n c e t h a t i s r o u g h l y e q u i v a l e n t t o t h e 7 ra c o n t o u r f r o ra s t o r m 8 ( F i g . 5 . 2 4 ) . T h e s i m i l a r i t y o f t h e s w e l l f i e l d p a t t e r n s i n t h e s e two s t o r m s e m p h a s i z e s , f o r t h i s p a r t i c u l a r t r a j e c t o r y , t h a t (1 ) t h e e n e r g y c o n t e n t i n t h e s w e l l t h a t w i l l a r r i v e a t t h e c o a s t Ls a l m o s t t o t a l l y d e t e r m i n e d by t he e a r l y s t o r m h i s t o r y , and ( 2 ) t h e p r o p a g a t i o n o n t o t h e c o a s t , p a r t i c u l a r l y s o u t h o f t h e Q u e e n C h a r l o t t e I s l a n d s , i s e s s e n t i a l l y i n d e p e n d e n t o f t h e l a t e r l o w p r e s s u r e s y s t e m e v o l u t i o n . 104 - 101 - S t o r m 8 Q u e e n C h a r l o t t e S o u n d (31,13) x 31 , 1 , 2 , 3 , 4 , 5 STORM 8 / S T O R M 6 / / / STORM 1 1 1 1 l 1 1 1 1; . — STORM 8 (" ^ STORM 6 1 l! 1 S T O R M 1 J 1 ! 1 1 1 JULY 19BG AUGUST 1986 F i g . 5 . 2 3 T i m e - s e r i e s r e s p o n s e o f H , T and mean wave d i r e c t i o n i n t h e e n t r a n c e t o Queen C n a r l o t t e Sound f o r s t o r m s 1, 6 a n d 8 . A s i d e f r o m t h e v e r y e a r l y s p i n u p p e r i o d , t h e mean d i r e c t i o n i n a l l s t o r m s I s e s s e n t i a l l y t he same. F i g . 5 . 24 T h e s e a - s t a t e p a t t e r n i n t e r m s o f H g i n s t o r m 8 a t 1 2 - h o u r l y i n t e r v a l s b e g i n n i n g on d a y 02 h o u r 18 . F i g . 5.25 The s e a - s t a t e p a t t e r n i n t e r m s o f H g i n s t o r m 6 a t 1 2 - h o u r l y i n t e r v a l s b e g i n n i n g on d a y 02 h o u r 12. - 104 - 6 . 0 SUMMARY OF RESULTS AND CONCLUSIONS T h i s a n a l y s i s has i n v e s t i g a t e d the s e n s i t i v i t y o f h i n d c a s t s e a - s t a t e s on t he d e e p o c e a n w e s t o f t h e c o a s t o f B r i t i s h C o l u m b i a t o v a r i a t i o n s i n m e t e o r o l o g i c a l p a r a m e t e r s f o r a p a r t i c u l a r c l a s s o f s t o r m t r a j e c t o r y . I n t h i s c h a p t e r t h e s t u d y f i n d i n g s a r e s u m m a r i z e d a n d q u a n t i f i e d f o r e a c h s t o r m p a r a m e t e r i n t u r n . T h e r e a f t e r , g e n e r a l c o n c l u s i o n s on the r e l a t i v e e f f e c t o f e a c h p a r a m e t e r i s d i s c u s s e d i n t e r m s o f d e e p o c e a n a n d c o a s t a l s e a - s t a t e h i n d c a s t i n g . 6 . 1 Summary o f S e n s i t i v i t y T e s t R e s u l t s To c h a r a c t e r i z e t h e maximum l o c a l l y g e n e r a t e d s e a - s t a t e d i f f e r e n c e s , t h r e e c o m p a r a t i v e s i t u a t i o n s were c o n s i d e r e d : (1 ) t h e maximum v a l u e o f H g i n t h e b a s e ( s t o r m 1) c a s e a n d t h e c o r r e s p o n d i n g v a l u e i n t h e s e n s i t i v i t y c a s e ; ( 2 ) t h e maximum H g i n t h e s e n s i t i v i t y t e s t a n d i t s c o r r e s p o n d i n g b a s e h e i g h t ; and ( 3 ) t h e g r e a t e s t A H g a t o r f o l l o w i n g t h e s t o r m peak and the minimum base H g w i t h i n t h i s r e g i o n . W h i l e t h e s e c o m p a r i s o n s do n o t n e c e s s a r i l y i s o l a t e t h e l a r g e s t p e r c e n t c h a n g e s , t h e y do c h a r a c t e r i z e t h e s e a - s t a t e v a r i a t i o n s s u f f i c i e n t l y t o r a n k t h e i m p o r t a n c e o f the v a r i o u s p r e s s u r e p a r a m e t e r s f o r wave h i n d c a s t i n g . A l l d i f f e r e n c e s a r e r e p o r t e d a s s e n s i t i v i t y s t o r m m i n u s t h e b a s e s t o r m ; h e n c e n e g a t i v e A H g v a l u e s mean t he s e n s i t i v i t y t e s t r e s u l t s a r e l o w e r t h a n the s t o r m 1 p r e d i c t i o n s . The c o a s t a l s w e l l v a r i a t i o n s were c o n s i d e r e d s e p a r a t e l y i n s l i g h t l y d i f f e r e n t t e r m s . The c h a n g e s b e t w e e n t h e b a s e H g a n d t h e s e n s i t i v i t y c a s e s w e r e examined a t t h r e e s p e c i f i c l o c a t i o n s : t h e s p e c i a l o u t p u t p o i n t s n e a r T o f i n o , i n t he e n t r a n c e t o Queen C h a r l o t t e Sound and j u s t n o r t h o f t h e Queen C h a r l o t t e I s l a n d s . A t t h e s e s i t e s , t h e d i f f e r e n c e b e t w e e n h i n d c a s t m a x i m a w e r e d e t e r m i n e d , i n d e p e n d e n t o f t h e t i m e o f o c c u r r e n c e o f e i t h e r o n e , i n t h e way t h a t most mode l p e r f o r m a n c e c h a r a c t e r i s t i c s a r e r e p o r t e d . - 105 - 6.1.1 Storm Intensity R a d i a l e x t e n t o f t he l o w p r e s s u r e s y s t e m , t h e minimum c e n t r a l p r e s s u r e and t he r a t e o f i n t e n s i f i c a t i o n a r e t h e key f a c t o r s c o n s i d e r e d h e r e . Radial Extent The s t o r m s i z e was d e f i n e d i n te rms o f a c o n s t a n t r a d i u s o f t he 990 mb i s o b a r , and as s u c h i t may v a r y w i d e l y f r o m 3° o f l a t i t u d e (334 km) t o abou t 13° (1445 km) w i t h a mean o f 8° (890 k m ) . I n t h i s s t u d y t h e b a s e v a l u e was s e t a t 6° (667 km) and t he s e n s i t i v i t y to a change t o 8° was i n v e s t i g a t e d . The r e s u l t s a r e summar ized i n T a b l e 6.1 Table 6.1 Summary of S e n s i t i v i t y to Radial Extent Base Va lue : S e n s i t i v i t y V a l u e : Per Cent Change: Wind Field Variations 6° o f l a t i t u d e (667 km) 8° of l a t i t u d e (890 km) 33% i n c r e a s e i n r ad ius 78% i n c r e a s e i n a rea near s torm c e n t r e , decreases ~ 5 to 10 kno ts remote from storm cen t re ( i n the v i c i n i t y of the 990 mb i s o b a r ) , i n c r e a s e s ~ 5 knots no change i n between Maximum Local Wind-Sea Difference Maximum Base V a l u e : 10 ra on day 03 hour 00 AH = -1 m (-10%) Maximum S e n s i t i v i t y T e s t : 9 m on day 03 hour 00 A H = -1 m (-1J Maximum D i f f e r e n c e : Coastal Swell To f i no Queen C h a r l o t t e Sd N Queen C h a r l o t t e I -2 m on day 02 hour 18 m i n . base H g = 8.5 m (-24%) ( ve ry l o c a l i z e d under the maximum winds) s torm maxima to end of t e s t run Base H g S e n s i t i v i t y H g (m) (m) (m) D i f f e r e n c e 1.3 3.1 4.7 1.6 3.3 4.7 0.3 0.2 0.0 23% 6% 0% - 106 - Lowest Minimum Central Low Pressure The minimum c e n t r a l p r e s s u r e was imposed a t 53°N 145°W. I n t he b a s e c a s e i t h a d t h e mean P Q ( m i n ) v a l u e o f 958 mb a n d i n t h e s e n s i t i v i t y t e s t i t was r e d u c e d t o 944 mb. T h i s l a t t e r v a l u e r e p r e s e n t s a v e r y i n t e n s e l o w p r e s s u r e s y s t e m . The s e n s i t i v i t y t e s t r e s u l t s a r e summar i zed i n T a b l e 6 . 2 . Table 6.2 Summary of S e n s i t i v i t y to Lowest Minimum Central Pressure 958 mb a t 53°N 145°W 944 mb a t 53°N 145°W 25% deeper r e l a t i v e to P = 1015 mb Base V a l u e : S e n s i t i v i t y Va lue Per Cent Change: Wind Field Variations near s torm c e n t r e , i n c r e a s e s ~ 10 to 15 knots 5 knot i n c r e a s e out to about 600 km from storm cen t re Maximum Local Wind-Sea Difference Maximum Base V a l u e : 10 m on day 03 hour 00 AH = +1.5 m (15%) Maximum S e n s i t i v i t y Tes t :14 m on day 03 hour 00 AH S = +5 m (36%) Maximum D i f f e r e n c e : Coastal Swell To f i no Queen C h a r l o t t e Sd N Queen C h a r l o t t e I +5 m on day 02 hour 18 m i n . base H g = 6 .5 m (77%) (very l o c a l i z e d under the maximum winds) s torm maxima to end of t e s t run Base H g S e n s i t i v i t y H g (m) (m) A « s (m) D i f f e r e n c e 1.3 3.1 4 .7 1.7 4 .0 5 .9 0.4 0 .9 1.2 31% 29% 26% - 107 - Highest Minimum Central Low Pressure The h i g h e s t v a l u e of P 0 ( m i n ) imposed a t 53°N 145°W was 974 mb. I n t h i s c a s e , t h e s t o r m d i d not i n t e n s i f y v e r y g r e a t l y d u r i n g i t s p a s s a g e t h r o u g h t h e m o d e l d o m a i n . I t s c h a r a c t e r i s t i c s e a - s t a t e d i f f e r e n c e s a r e p r e s e n t e d i n T a b l e 6 . 3 . Table 6.3 Summary of S e n s i t i v i t y to Highest Minimum Central Pressure Base V a l u e : S e n s i t i v i t y V a l u e : Per Cent Change: 958 mb at 53°N 145°W 974 mb a t 53°N 145°W 28% l e s s deep r e l a t i v e to P = 1015 mb Wind Field Variations near s torm c e n t r e , decreases - 15 to 25 knots 5 to 10 knot i n c r e a s e out to about 600 km from storm cen t re Maximum Local Wind-Sea Difference Maximum Base V a l u e : Maximum S e n s i t i v i t y T e s t : 5 m on day 03 hour 00 Maximum D i f f e r e n c e : Coastal Swell T o f i n o Queen C h a r l o t t e Sd N Queen C h a r l o t t e I 10 m on day 03 hour 00 A H g = - 5 m (-50%) AH S = - 5 m (-100%) 5 m on day 03 hour 00 m i n . base H g = 8 m (-38%) (ve ry l o c a l i z e d under the maximum winds) storm maxima to end of t e s t run Base H S e n s i t i v i t y H (m) (m) A * s (m) D i f f e r e n c e 1.3 3.1 4 .7 0 .5 1.5 2.7 - 0 . 8 - 1 . 6 - 2 . 0 -62% -52% -43% - 108 - Rate of I n t e n s i f i c a t i o n O c c a s i o n a l l y s t o rms t h a t t r a v e l a t t y p i c a l r a t e s a l o n g t h e i r t r a j e c t o r y w i l l i n c l u d e a 12 t o 24 h o u r segment d u r i n g w h i c h the c e n t r a l l o w p r e s s u r e f a l l s by 1 rab/h o r m o r e . S u c h a h i g h r a t e o f i n t e n s i f i c a t i o n i s c a l l e d e x p l o s i v e d e e p e n i n g . I n t h e l a s t two y e a r s ( s i n c e t he f i s h i n g boa t l o s s e s i n November 1984) , wes t c o a s t s t o r m s w i t h t h i s c h a r a c t e r h a v e r e c e i v e d more a t t e n t i o n f r o m w e a t h e r f o r e c a s t e r s b e c a u s e i t i s d i f f i c u l t t o p r e d i c t w h i c h s t o r m s w i l l deepen a t t h e s e a c c e l e r a t e d r a t e s . I n many c a s e s , t he e x p l o s i v e c y c l o g e n e s i s r a p i d l y t r a n s f o r m s a m i l d w e a t h e r s y s t e m i n t o a s e v e r e o n e . C o r r e c t m o d e l l i n g o f t h e r a t e o f d e e p e n i n g i s a l s o i m p o r t a n t f o r s t o r m h i n d c a s t i n g i n t h a t i t a f f e c t s b o t h t he shape (hence f e t c h ) o f t he w ind f i e l d a n d t h e d u r a t i o n o f e a c h w i n d p a t t e r n f o r up t o a d a y p r e c e d i n g t h e s t o r m peak . The r e s u l t s o f one i n t e n s i f i c a t i o n r a t e s e n s i t i v i t y t e s t a r e p r e s e n t e d i n T a b l e 6 . 4 . Table 6 . 4 Summary of S e n s i t i v i t y to Rate of I n t e n s i f i c a t i o n Base V a l u e : S e n s i t i v i t y V a l u e : Per Cent Change: Wind Field Variations - 0 . 6 mb/h f rom day 02 hour 06 to day 02 hour 18 - 1 . 8 mb/h from day 02 hour 06 to day 02 hour 18 200% i n c r e a s e i n r a t e of deepening f o r the 12 hours p reced ing the s torm peak none a t the storm peak up t o 25 knots weaker near the storm cen t re 12 hours be fo re the storm peak Maximum Local Wind-Sea Difference Maximum Base V a l u e : 10 m on day 03 hour 00 A H G = 0 m (0%) Maximum S e n s i t i v i t y T e s t : 1 0 m on day 03 hour 00 A H G = 0 m (0%) Maximum D i f f e r e n c e : Coastal Swell - 5 m on day 02 hour 06 m i n . base H G = 7 m (-71%) - 5 ra on day 02 hour 12 m in . base H G = 7 m (-71%) (very l o c a l i z e d under the maximum winds) s torm maxima to end of t e s t run Base H S e n s i t i v i t y H AH D i f f e r e n c e (m) (m) (m) To f i no 1.3 0.7 - 0 . 6 -46% Queen C h a r l o t t e Sd 3.1 2.2 - 0 . 9 -29% N Queen C h a r l o t t e I 4 .7 4 . 0 - 0 . 7 -15% - 109 - 6.1.2 Storm Trajectory As c l e a r l y i l l u s t r a t e d i n F i g . 4 . 7 , w i t h i n t h e c l a s s o f s to rms c o n s i d e r e d i n t h i s a n a l y s i s , t he n o r t h w a r d l e g o f t he s t o r m t r a j e c t o r y c a n v a r y w i d e l y . The a v e r a g e p a t h a l o n g 145°W c a n e a s i l y be d e f l e c t e d 5 ° o f l o n g i t u d e c l o s e r to t he c o a s t , a n d t h i s was t h e s e n s i t i v i t y c a s e t h a t was t e s t e d . The r e s u l t s a r e summar i zed i n T a b l e 6.5. I n t h e more g e n e r a l h i n d c a s t i n g c o n t e x t , t h i s i s a v e r y i m p o r t a n t p a r a m e t e r . The r e s u l t s a r e s e n s i t i v e t o i t s s p e c i f i c a t i o n and i t i s q u i t e d i f f i c u l t t o d e t e r m i n e w i t h c e r t a i n t y , e s p e c i a l l y f o r s t o r m s t h a t a r e no t w e l l - m a p p e d by s a t e l l i t e i m a g e r y . Those w e a t h e r s y s t e m s t h a t c r o s s t h e c o a s t a r e l i k e l y t o be more a c c u r a t e l y t r a c k e d as t h e y n e a r t h e c o a s t by t h e more d e n s e a n d s y s t e m a t i c l a n d - b a s e d m e t e o r o l o g i c a l r e p o r t i n g n e t w o r k . Base V a l u e : S e n s i t i v i t y V a l u e : Per Cent Change: Table 6.5 Summary of Sensitivity to Storm Trajectory no r the rn t r a j e c t o r y a long 145°W nor the rn t r a j e c t o r y a long 140°W 26% l e s s d i s t a n c e to coas t a t 49°N 42% l e s s d i s t a n c e to coast at 55°N Wind Field Variations none i n r e l a t i o n to the s torm cen t re a long the c o a s t , peak winds s t reng then by ~5 knots on the sou th coast and -10 knots on the no r th coas t a t day 02 hour 18 change i n d i r e c t i o n i s n e g l i g i b l e Maximum Local Wind-Sea Difference Maximum Base V a l u e : 10 m on day 03 hour 00 AHg = -4 m (-40%) Maximum S e n s i t i v i t y T e s t : 1 0 m on day 03 hour 00 AHg = +6 m (60%) Maximum D i f f e r e n c e : +6 m on day 03 hour 00 m in . base H g = 3 .5 m (171%) - 5 m on day 02 hour 18 m in . base H g = 8 m (-63%) (very l o c a l i z e d under the maximum winds) Coastal Swell To f i no Queen C h a r l o t t e Sd N Queen C h a r l o t t e I s torm maxima to end of t e s t run Base H g S e n s i t i v i t y H g (m) 1.3 3.1 4.7 4 .7 (m) A H s (m) D i f f e r e n c e 3 .3 4 .8 8 . 2 * 5.8 2 .0 154% 1.7 55% 3 . 5 * 74%* 1.1 23% *due to l o c a l wind sea - 110 - 6.1.3 Storm Advectlon Rate The most s i g n i f i c a n t change that can occur i n the storm a d v e c t i o n r a t e i s a s s o c i a t e d with low pr e s s u r e systems that s t a l l ( i . e . become s t a t i o n a r y i n space) without change i n any other pressure c h a r a c t e r i s t i c s . T y p i c a l l y s t a l l s l a s t 24 hours and occur somewhere i n the n o r t h e r n G u l f of A l a s k a a f t e r the storm has peaked and begun to diminish i n i n t e n s i t y . Table 6.6 summarizes the res u l t s of t h i s s e n s i t i v i t y t e s t . Table 6.6 Summary of S e n s i t i v i t y to Storm Advection Rate Base Value: Sensitivity Value: Per Cent Change: Wind Field Variations Coastal winds •Deep ocean 37 km/h from 53°N to 59°N along 145°W 16 km/h from 53°N to 59°N along 145°W 57% slower on average none i n space except as a function of time after day 03 hour 12 up to 5 knots stronger from Queen Charlotte Sound to northern Queen Charlotte Islands 25 to 30 knots stronger due W of Queen Charlotte I except 10 to 35 knots weaker near storm centre 15 to 25 knots stronger due W of Queen Charlotte Sd 5 to 15 knots stronger due W of Vancouver Island more rapid direction change with longitude Maximum Local Wind-Sea Difference Maximum Base Value: 10 m on day 03 hour 00 A H g = 0 m (0%) Maximum Sensitivity Test:12.5 m on day 03 hour 12 AHG = 6.2 m (50%) (up to end of base) Maximum Difference: 7 m on day 03 hour 12 min. base HG = 3 m (233%) (very localized under the maximum winds; this difference w i l l increase as the s t a l l continues) Coastal Swell Tofino Queen Charlotte Sd N Queen Charlotte I storm maxima to end of test run at 0312 (and 0412) Difference (m) " (m) A H s (m) 1.3 (3.8) 1.4 (3.9) 0.1 (0.1) 8% ( 3%) 3.1 (4.8) 3.0 (3.7) -0.1 (-1.1) -3% (-23%) 4.7 (5.5) 4.4 (5.8) -0.3 (0.3) -6% ( 5%) - I l l - 6 . 1 . 4 C o m p a r i s o n o f Maximum E r r o r s B a s e d on t h e s e a - s t a t e d i f f e r e n c e s n o t e d i n T a b l e s 6.1 t h r o u g h 6.6 a n d e n c a p s u l a t e d i n T a b l e 6 . 7 , t h e r e a r e c l e a r d i s t i n c t i o n s t o be made on t h e r e l a t i v e i m p o r t a n c e o f each s e n s i t i v i t y p a r a m e t e r . The one w h i c h c a u s e s t he m o s t s e a - s t a t e c h a n g e o v e r t h e w i d e s t a r e a i s t h e s t o r m t r a j e c t o r y . D i f f e r e n c e s i n e x c e s s o f 150% i n b o t h l o c a l w i n d - s e a and s w e l l r e s u l t e d f r o m a l a r g e ( b u t n o t t h e maximum r e a l i z a b l e ) s h i f t i n t h e p a t h o f t h e c e n t r a l l o w p r e s s u r e . A p p l y i n g a l i n e a r s c a l i n g t o p e r c e n t c h a n g e as a f u n c t i o n o f d i s t a n c e , t he t r a j e c t o r y must be c o r r e c t t o w i t h i n 1° o f l o n g i t u d e t o r e d u c e H g e r r o r s t o abou t 10%. T h i s i s a p h y s i c a l d i s t a n c e of abou t 70 km n e a r 50°N , b u t c o r r e s p o n d s t o l e s s t h a n 1 cm a t t h e s c a l e o f a m e t e o r o l o g i c a l s u r f a c e a n a l y s i s c h a r t . I t seems u n r e a l i s t i c , t h e r e f o r e , t o e x p e c t t h i s d e g r e e o f a c c u r a c y , e s p e c i a l l y i n h i s t o r i c a l s u r f a c e p r e s s u r e d a t a . G r e a t e r changes i n H g may be c a u s e d by e x t r e m e changes i n t he s t o r m a d v e c t i o n r a t e ( i . e . by a s t a l l e d s t o r m s y s t e m ) , bu t t h e s e a r e r e s t r i c t e d t o l o c a l w i n d - s e a unde r t he s t a l l e d s y s t e m and t o any i n c r e a s e d s w e l l t he e v o l v e s f r o m t h i s a r e a . The m i n i m u m c e n t r a l p r e s s u r e was v a r i e d w i t h i n v i r t u a l l y i t s h i s t o r i c a l l i m i t s . I n t e n s i f y i n g P Q ( m i n ) by 25% ( -14 mb) c r e a t e d H g i n c r e a s e s o f r o u g h l y t h e same p e r c e n t a g e e x c e p t i n v e r y l o c a l i z e d p o c k e t s unde r t he maximum w i n d s . R e d u c i n g t h e i n t e n s i f i c a t i o n by 28% ( + 16 mb) g e n e r a t e d s e a - s t a t e s t h a t a r e a b o u t 50% l o w e r . I t i s t h e n r e a s o n a b l e t o e x p e c t t h a t a 5 mb o v e r - i n t e n s i f i c a t i o n w i l l no t change H g by more t h a n 10%; h o w e v e r , t he e q u i v a l e n t u n d e r - e s t i m a t e of t he d e p t h o f P Q ( m i n ) c a n l e a d t o u n d e r e s t i m a t i n g H g by abou t 20%. E r r o r s o f +5 mb a r e v e r y p o s s i b l e o v e r t h e deep o c e a n s i n c e P Q ( m i n ) must be i n f e r r e d f r o m measurements f r o m a r e p o r t i n g n e t w o r k c o n s i s t i n g o f a v e r y few m e t e o r o l o g i c a l b u o y s , s h i p s - o f - o p p o r t u n i t y , l i g h t h o u s e s and o t h e r l a n d s t a t i o n s . I t i s most u n l i k e l y t h a t any o f t h e s e d a t a s o u r c e s w i l l be c l o s e to a s t o r m c e n t r e s i n c e s h i p s ' c a p t a i n s w i l l t r y t o a v o i d s e v e r e w i n d a n d w a v e c o n d i t i o n s a n d t h e d i s t r i b u t i o n o f t h e r e m a i n i n g p a r t o f t h e n e t w o r k i s s o s p a r s e . Assum ing t h a t the minimum c e n t r a l p r e s s u r e v a l u e i s a g i v e n , t he changes i n t he r a t e o f i n t e n s i f i c a t i o n t o t h a t v a l u e c a n h a v e a t most two c o n s e q u e n c e s . S i n c e t h e w i n d - s e a w i l l r e s p o n d t o t h e l o c a l w i n d , t h e d u r a t i o n o f a g i v e n w i n d speed and d i r e c t i o n w i l l be s h o r t e r f o r more r a p i d l y d e e p e n i n g s t o r m s , a n d t h e r e s u l t i n g s w e l l e n e r g y w i l l be d i m i n i s h e d . S i g n i f i c a n t p r e s s u r e e r r o r s o f t h i s t y p e w o u l d no t n o r m a l l y a r i s e u n l e s s s u r f a c e p r e s s u r e c h a r t s - 112 - Table 6.7 Maximum Sea-State Variability Attributable to Pressure Parameter Sensitivity Tests S e n s i t i v i t y P a r a m e t e r L o c a l W i n d - S e a D i f f e r e n c e S w e l l M a x . From Max . From M a x . AH Max . AH * Base S e n s i t i v i t y s & R a d i a l E x t e n t (R) -10% -11% -24% 0 t o 23% Minimum C e n t r a l P r e s s u r e minimum P Q ( m i n ) 15% 36% 77% 26 t o 31% maximum P Q ( m i n ) -50% -100% -38% - 4 3 t o -62% R a t e o f I n t e n s i f i c a t i o n n / a n / a -71% - 1 5 t o -46% T r a j e c t o r y -40% 60% - 6 3 t o 171% 55 t o 154% S to rm A d v e c t i o n R a t e n / a 50% 233% - 2 3 t o 8% n / a : no a p p l i c a b l e c o m p a r i s o n * i n d e p e n d e n t of t i m e - 113 - a r e v e r y i n a c c u r a t e o r u n l e s s t h e y a r e o n l y a v a i l a b l e a t w i d e l y s p a c e d i n t e r v a l s d u r i n g s t o r m d e v e l o p m e n t . The s t o r m s i z e , a c c o r d i n g t o t he d e f i n i t i o n u s e d i n t h i s s t u d y , has the l e a s t e f f e c t on t he h i n d c a s t r e s u l t s . S m a l l a d j u s t m e n t s i n t h e r a d i u s (±25%) w i l l h a v e a n i m p a c t o f n o t more t h a n +10% i n t h e w a v e f i e l d m a x i m a , a l t h o u g h g r e a t e r p e r c e n t changes a r e p o s s i b l e i n l o w l e v e l c o a s t a l s w e l l e n e r g y . 6.2 Conclusions 6.2.1 The Meteorological Perspective Two c o n c l u s i o n s a r e a p p a r e n t f r o m t h e r e s u l t s o f t h i s s t u d y : ( 1 ) W i t h t h e a d v e n t of r o u t i n e s a t e l l i t e images o f c l o u d f o r m a t i o n s , i t i s e x p e c t e d t h a t s i g n i f i c a n t i m p r o v e m e n t s i n t h e a c c u r a c y o f s t o r m t r a j e c t o r y p o s i t i o n i n g w i l l h a v e o c c u r r e d . S i n c e t h i s i s a m o s t s e n s i t i v e p a r a m e t e r f o r w a v e m o d e l l i n g , b e t t e r w a v e h i n d c a s t i n g r e s u l t s , on a v e r a g e , s h o u l d be f o u n d when s a t e l l i t e d a t a a r e u s e d i n p r e p a r i n g the s u r f a c e p r e s s u r e c h a r t s . ( 2 ) A n y d e g r a d a t i o n i n t h e s u r f a c e p r e s s u r e m o n i t o r i n g n e t w o r k a t s e a ( e s p e c i a l l y d e c o m m i s s i o n i n g o f w e a t h e r s h i p s , p o o r m a i n t e n a n c e o r r e m o v a l o f m e t e o r o l o g i c a l b u o y s w h i c h n o r m a l l y p r o v i d e t h e m o s t r e l i a b l e g u i d a n c e t o m e t e o r o l o g i s t s ) w i l l t e n d t o r e s u l t i n l e s s a c c u r a t e p r e s s u r e f i e l d s p e c i f i c a t i o n . 6.2.2 Wave Hindcasting Perspective T h i s s t u d y d e m o n s t r a t e s t h a t ADWAVE (and p r e s u m e a b l y o t h e r m o d e l s t h a t embody t h e same e l e m e n t s o f t h e p h y s i c s of wave g e n e r a t i o n ) p r o d u c e s s e a - s t a t e s t h a t a r e l o c a l l y i n b a l a n c e w i t h t he w i n d i n t h e s e n s e t h a t the H g f i e l d u n d e r t he a c t i v e s t o r m c e n t r e a n d t h e s t r o n g e s t w i n d s i s l a r g e l y i n d e p e n d e n t o f t h e p r e c e d i n g s t o r m h i s t o r y . As a r e s u l t , peak s e a - s t a t e s i n t he deep open o c e a n w i l l no t depend s t r o n g l y on any a r b i t r a r y w i n d f i e l d e r r o r s t h a t p r e c e d e o r f o l l o w t h e s t o r m peak . The wave f i e l d ' s "memory" o f a w i n d f i e l d e r r o r w i l l have two f o r m s : ( 1 ) T h r o u g h t h e t e m p o r a l i n t e r p o l a t i o n f r o m ( s a y ) 6 - h o u r l y w i n d maps ( c o r r e s p o n d i n g t o a v a i l a b l e s u r f a c e p r e s s u r e c h a r t s ) t o t h e w a v e mode l t i m e s t e p , an e r r o r i n a s i n g l e w i n d f i e l d i n f l u e n c e s a l m o s t a f u l l 12 h o u r s o f wave h i n d c a s t i n g . ( 2 ) S w e l l e n e r g y t h a t p r o p a g a t e s f r e e l y a t o f f - w i n d a n g l e s w i l l p e r p e t u a t e any e r r o r t h a t o c c u r r e d i n i t s g e n e r a t i o n . - 114 - N e s t e d w a v e m o d e l g r i d s p r e s e n t a n a d d e d d i m e n s i o n o f c o m p l e x i t y t o a n a s s e s s m e n t o f t he e f f e c t s o f w i n d e r r o r s . I n s u c h a c a s e , t he n e s t e d g r i d i s l i n k e d t o t he c o a r s e r s u r r o u n d i n g one t h r o u g h t h e s u p p l y o f boundary s p e c t r a f r om the c o a r s e r u n to the f i n e r n e s t e d one . I n d e e p w a t e r c o a r s e g r i d h i n d c a s t m o d e l l i n g , t h e b o u n d a r y c o n d i t i o n s a r e i n v a r i a b l y s e t t o z e r o e n e r g y f l u x . U s u a l l y t h e o p e n w a t e r b o u n d a r i e s a r e s u f f i c i e n t l y w e l l - r e m o v e d f r o m p o i n t s o f i n t e r e s t t h a t t h e a p p r o x i m a t i o n i s q u i t e a d e q u a t e u n l e s s s i g n i f i c a n t s w e l l e n e r g y c a n p r o p a g a t e i n t o t he mode l d o m a i n f r o m r e m o t e s t o r m s y s t e m s t h a t a r e n o t b e i n g m o d e l l e d . On t h e o t h e r h a n d , n e s t e d g r i d s a r e u s u a l l y o f s m a l l o v e r a l l p h y s i c a l d i m e n s i o n s and i n t h e s e c a s e s bounda ry c o n d i t i o n s w i l l o f t e n be as i m p o r t a n t , and a t t i m e s more i m p o r t a n t t h a n the l o c a l w i n d f o r c i n g . I n s h a l l o w w a t e r , where n e s t e d g r i d s a r e u s u a l l y a p p l i e d , r e f r a c t i o n , s h o a l i n g and bo t t om f r i c t i o n a n d / o r p e r c o l a t i o n a r e a d d i t i o n a l s o u r c e / s i n k terms t h a t a r e e x c l u d e d f r o m deep w a t e r a p p l i c a t i o n s . R e f r a c t i o n , i n p a r t i c u l a r , depends on c o r r e c t bounda ry s p e c i f i c a t i o n o f t h e p a r t i t i o n o f t h e t o t a l e n e r g y ( H g ) as a f u n c t i o n o f p e r i o d ( f r e q u e n c y ) and d i r e c t i o n . 6 . 2 .3 E n g i n e e r i n g A p p l i c a t i o n s P e r s p e c t i v e I n terms o f e x t r e m e v a l u e a n a l y s i s o f t he l o n g - t e r r a wave c l i m a t e f o r c o a s t a l d e s i g n a p p l i c a t i o n s , t h e i m p o r t a n c e o f s w e l l c a n be l a r g e l y d i s c o u n t e d s i n c e the a n n u a l e x t r e m e e v e n t ( f o r e x a m p l e ) w i l l a l m o s t c e r t a i n l y be a l o c a l w i n d - s e a dom ina ted s e a - s t a t e a t i t s peak . I n t h a t c a s e , t h e p r e s s u r e p a r a m e t e r s o f r a n k e d i m p o r t a n c e w i l l be (1 ) t h e t r a j e c t o r y o f P Q and hence t he l o c a t i o n o f P Q ( r a i n ) , (2 ) t he v a l u e o f P Q ( m i n ) , and (3 ) t h e a d v e c t i o n r a t e o f P Q . The r a t e o f d e e p e n i n g i s e x p e c t e d t o be much l e s s i m p o r t a n t so l o n g as P Q ( m i n ) i s as c o r r e c t as p o s s i b l e . T h u s , w i t h due c o n s i d e r a t i o n o f t he p i t f a l l s , w i t h r e a n a l y s i s o f • t h e key s u r f a c e p r e s s u r e f i e l d s , and w i t h i n c o r p o r a t i o n of as much m e a s u r e d w i n d d a t a a s p o s s i b l e i n a k i n e m a t i c a n a l y s i s , i t i s q u i t e f e a s i b l e to h i n d c a s t a s e t o f e x t r e m e wave e v e n t s u s i n g a s p e c t r a l mode l s u c h as ADWAVE t o w i t h i n e n g i n e e r i n g s t a n d a r d s . B e c a u s e t he d a t a r e q u i r e m e n t s a r e r e l a t i v e l y s m a l l , t h e demand f o r man- t ime and compu te r r e s o u r c e s can be met w i t h i n t he e c o n o m i c c o n s t r a i n t s o f most m a j o r p r o j e c t s . - 115 - I n many c a s e s c l i m a t o l o g i c a l s t a t i s t i c s o f w i n d - s e a and o f s w e l l r a t h e r t h a n , o r i n a d d i t i o n t o , l o n g r e t u r n p e r i o d e x t r e m e v a l u e s a r e r e q u i r e d . T h e s e s i t u a t i o n s i n c l u d e p l a n n i n g , c o n s t r u c t i o n a n d o p e r a t i o n o f f i x e d c o a s t a l i n s t a l l a t i o n s s u c h as h a r b o u r s , e x p o s e d b e r t h i n g f a c i l i t i e s and b r e a k w a t e r s ; and many a s p e c t s o f o f f s h o r e h y d r o c a r b o n e x p l o i t a t i o n s u c h a s p i p e l i n e i n s t a l l a t i o n , t a n k e r and o t h e r t e n d e r v e s s e l r o u t i n g and o p e r a t i o n , c e r t a i n phases o f c o n c r e t e g r a v i t y - b a s e p r o d u c t i o n p l a t f o r m c o n s t r u c t i o n , r i g t o w i n g , and d r e d g i n g f o r c o n s t r u c t i o n and m a i n t e n a n c e o f a r t i f i c i a l i s l a n d s . I n each c a s e t h e e f f i c i e n c y o f p l a n n i n g a n d e x e c u t i n g e a c h o f t h e s e a c t i v i t i e s i s e n h a n c e d by i n f o r m e d u n d e r s t a n d i n g o f t h e n o r m a l w a v e c l i m a t e : j o i n t d i s t r i b u t i o n o f h e i g h t , p e r i o d a n d d i r e c t i o n , p e r s i s t e n c e s t a t i s t i c s a n d s h o r t - t e r m e x t r e m e v a l u e e x p e c t a t i o n s . U n l e s s many s i t e s a r e i n v o l v e d o r d e c i s i o n - m a k i n g t i m e - f r a m e s a r e v e r y s h o r t , c o l l e c t i o n o f wave measurements w i l l be p r e f e r r a b l e t o wave h i n d c a s t i n g f r o m b o t h c o s t and a c c u r a c y p o i n t s o f v i e w . F o r e x a m p l e , one y e a r o f w ind and wave h i n d c a s t i n g w o u l d r e q u i r e many m a n - m o n t h s t o p r e p a r e w i n d i n p u t a n d a b o u t 1500 h o u r s o f c o m p u t e r CPU t i m e ( u s i n g a v e r y f a s t m i c r o - c o m p u t e r as was done i n t h i s s t u d y ) t o g e n e r a t e t h e d i g i t a l w i n d f i e l d s , t o p r e p a r e a deep w a t e r wave h i n d c a s t and t o examine some o f t he r e s u l t s . A s u b s e q u e n t s h a l l o w w a t e r n e s t e d a p p l i c a t i o n w o u l d r e q u i r e a t l e a s t as much compute r t i m e as w e l l . T h e r e f o r e , c o n s i d e r i n g o n l y t h e number o f s o u r c e s o f s i g n i f i c a n t e r r o r s t h a t may be i n t r o d u c e d i n t o d i g i t a l w i n d f i e l d s , i t d o e s n o t seem p r u d e n t t o a t t e m p t h i n d c a s t i n g s i t e - s p e c i f i c n e a r s h o r e wave c l i m a t o l o g i c a l d a t a u n t i l w i n d d a t a c a n be g e n e r a t e d w i t h much g r e a t e r r e l i a b i l i t y . I n t h i s r e s p e c t , t h e m o s t p r o m i s i n g d e v e l o p m e n t , t h e r e a l i z a t i o n o f w h i c h i s s t i l l i n t h e f u t u r e , i s t he a b i l i t y t o o b j e c t i v e l y i n c o r p o r a t e r e a l - t i m e r e m o t e l y - s e n s e d ( s a t e l l i t e ) w i n d d a t a i n t o t h e n u m e r i c a l a n a l y s i s and p r o g n o s i s p r o c e d u r e s o f t he n a t i o n a l w e a t h e r s e r v i c e . I n t h e n e x t decade o r s o , when t h e r e a r e enough s a t e l l i t e s w i t h t h e a p p r o p r i a t e s c a t t e r o m e t e r o r p a s s i v e m i c r o w a v e s e n s o r s , i t s h o u l d be p o s s i b l e t o a r c h i v e w i n d d a t a i n s t e a d o f s u r f a c e p r e s s u r e s , t h e r e b y g r a d u a l l y t o remove t h e r e l i a n c e on p r e s s u r e d a t a f o r wave h i n d c a s t i n g and to b e g i n t h e m o d e l l i n g p r o c e s s w i t h a c t u a l d a t a - b a s e d d i g i t a l s u r f a c e w i n d s . - 116 - 7.0 REFERENCES Bouws, E . , H. G u n t h e r , W. R o s e n t h a l and C.L . V i n c e n t , 1985. S i m i l a r i t y o f t he Wind S p e c t r u m i n F i n i t e D e p t h W a t e r . J . G e o p h y s . R e s . , 90 (C1 ) , 9 7 5 - 9 8 6 . C a r d o n e , V . J . , W . J . P i e r s o n a n d E . G . W a r d , 1 9 7 5 . H i n d c a s t i n g t h e D i r e c t i o n a l S p e c t r a o f H u r r i c a n e G e n e r a t e d W a v e s . O f f s h o r e T e c h n o l o g y C o n f e r e n c e , P a p e r OTC 2332 . C a r d o n e , V . J . , A . J . B r o c c o l i , C V . G r e e n w o o d a n d J . A . G r e e n w o o d , 1 9 8 0 . E r r o r C h a r a c t e r i s t i c s o f E x t r a t r o p i c a l - S t o r m W i n d F i e l d s S p e c i f i e d F r o m H i s t o r i c a l D a t a . J . P e t r o l . T e c h . , 8 , 8 7 2 - 8 8 0 . C l a n c y , R . M . , J . E . K a i t a l a , a n d L . F . Z a m b r e s k y , 1 9 8 6 . The F l e e t N u m e r i c a l O c e a n o g r a p h y C e n t e r G l o b a l S p e c t r a l O c e a n W a v e M o d e l . B u l l . Am. M e t e o r . S o c , 6 7 ( 5 ) , 4 9 8 - 5 1 2 . C o t e , L . J . , J . O . D a v i s , W. M a r k s , R . J . M c G o u g h , E . M e h r , W . J . P i e r s o n , J . F . R o p e k , G. S t e p h e n s o n a n d R . C . V e t t e r , 1 9 6 0 . T h e D i r e c t i o n a l S p e c t r u m o f a Wind G e n e r a t e d Sea as D e t e r m i n e d From D a t a O b t a i n e d by t h e S t e r e o Wave O b s e r v a t i o n P r o j e c t . M e t e o r o l o g i c a l P a p e r s , New Y o r k U n i v e r s i t y , C o l l e g e o f E n g i n e e r i n g , ^ ( 6 ) . D e l a g e , Y . , 1985. S u r f a c e T u r b u l e n t F l u x F o r m u l a t i o n i n S t a b l e C o n d i t i o n s f o r A t m o s p h e r i c M o d e l s . M o n . Wea. R e v . , 1 1 3 . Det n o r s k e V e r i t a s , 1982. R u l e s f o r t h e D e s i g n a n d I n s p e c t i o n o f O f f s h o r e S t r u c t u r e s , A p p e n d i x A : E n v i r o n m e n t a l C o n d i t i o n s . O s l o , R e p r i n t 1982 , A l l . D o b r o c k y S e a t e c h , 1 9 8 6 . T e c h n i c a l r e p o r t p r e p a r e d f o r t h e E n v i r o n m e n t a l S t u d i e s R e v o l v i n g F u n d , O t t a w a . G o l d i n g , B . , 1 9 8 3 . A Wave P r e d i c t i o n S y s t e m f o r R e a l - T i m e S e a S t a t e F o r e c a s t i n g . Q u a r t . J . R. M e t . S o c , 1 0 9 , 3 9 3 - 4 1 6 . G a r r a t t , J . R . , 1977. R e v i e w o f D r a g C o e f f i c i e n t s O v e r Oceans and C o n t i n e n t s . Mon . Wea. R e v . , 1 0 5 , 9 1 5 - 9 2 9 . G u n t h e r , H . , W. R o s e n t h a l a n d K. R i c h t e r , 1 9 7 9 a . A p p l i c a t i o n o f t h e P a r a r a e t r i c a l S u r f a c e Wave P r e d i c t i o n M o d e l t o R a p i d l y V a r y i n g Wind F i e l d s D u r i n g JONSWAP 1973. J . Geophys . R e s . , 84 (C8 ) , 4 8 5 5 - 4 8 6 4 . G u n t h e r , H . , W. R o s e n t h a l , T . J . W e a r e , B .A . W o r t h i n g t o n , K. H a s s e l m a n n a n d J . A . E w i n g , 1979b. A H y b r i d P a r a m e t r i c a l Wave P r e d i c t i o n M o d e l . J . G e o p h y s . R e s . , 84_(C9), 5 7 2 7 - 5 7 3 7 . H a s s e l m a n n , K . , T . P . B a r n e t t , E . B o u w s , H. C a r l s o n , D . E . C a r t w r i g h t , K. E n k e , J . A . E w i n g , H. G i e n a p p , D . E . H a s s e l m a n n , P . K r u s e m a n , A . M e e r b u r g , P . M u e l l e r , D . J . O l b e r s , K. R i c h t e r , W. S e l l a n d H. W a l d e n , 1 9 7 3 . M e a s u r e m e n t s o f W ind -Wave Growth and S w e l l Decay D u r i n g t h e J o i n t N o r t h S e a W a v e P r o j e c t ( J O N S W A P ) . D e u t s c h e H y d r o g r a p h i s c h e Z e i t s c h r i f t , R e i h e A . , N r . 1 2 . H a s s e l m a n n , K . , D . B . R o s s , P . M u e l l e r a n d W. S e l l , 1 9 7 6 . A P a r a m e t r i c W a v e P r e d i c t i o n M o d e l . J . P h y s . O c e a n o g r . , 6 ^ , 2 0 0 - 2 2 8 . - 117 - H a s s e l m a n n , D . E . , M. D u n c k e l a n d J . A . E w i n g , 1 9 8 0 . D i r e c t i o n a l Wave S p e c t r a O b s e r v e d D u r i n g JONSWAP 1973. J . P h y s . O c e a n o g r . , ^ 0 ( 8 ) , 1264 -1280 . H o d g i n s , D .O. , P . H . L e B l o n d , D . S . D u n b a r a n d C . T . N i w i n s k i , 1 9 8 5 . A Wave C l i m a t e S t u d y o f T h e N o r t h e r n B r i t i s h C o l u m b i a C o a s t , V o l u m e I I . T e c h n i c a l r e p o r t p r e p a r e d f o r F i s h e r i e s a n d O c e a n s , C a n a d a by S e a c o n s u l t M a r i n e R e s e a r c h L t d . , V a n c o u v e r . H o d g i n s , D.O. a n d S . N i k l e v a , 1 9 8 6 . On t h e I m p a c t o f New O b s e r v i n g S i t e s o n S e v e r e Sea S t a t e Warn ings f o r the B.C. C o a s t . U n p u b l i s h e d t e c h n i c a l r e p o r t p r e p a r e d f o r F i s h e r i e s & Oceans Canada by S e a c o n s u l t M a r i n e R e s e a r c h L t d . , V a n c o u v e r . H o d g i n s , D.O. and S. H o d g i n s , 1986. An E v a l u a t i o n o f Wave F o r e c a s t i n g M o d e l s a n d F o r e c a s t W i n d F i e l d s i n t h e C a n a d i a n C o n t e x t . D r a f t r e p o r t p r e p a r e d f o r the E n v i r o n m e n t a l S t u d i e s R e v o l v i n g F u n d , O t t a w a . H o d g i n s , D .O. , C . T . N i w i n s k i a n d D.T. R e s i o , 1 9 8 6 . C o m p a r i s o n a n d V a l i d a t i o n o f Two S h a l l o w - W a t e r S p e c t r a l Wave M o d e l s . D r a f t r e p o r t p r e p a r e d f o r The E n v i r o n m e n t a l S t u d i e s R e v o l v i n g F u n d s , O t t awa by S e a c o n s u l t M a r i n e R e s e a r c h L t d . H s u , S . A . , 1 9 8 6 . A M e c h a n i s m f o r t h e I n c r e a s e o f W i n d S t r e s s ( D r a g ) C o e f f i c i e n t W i t h Wind Speed O v e r Wate r S u r f a c e s : A P a r a m e t r i c M o d e l . J . P h y s . O c e a n o g r . , 16(1), 1 4 4 - 1 5 0 . J a n s s e n , P . A . E . M . , G . J . Komen a n d W . J . P . de V o o g t , 1 9 8 4 . An O p e r a t i o n a l C o u p l e d H y b r i d Wave P r e d i c t i o n M o d e l . J . G e o p h y s . R e s . , 8 9 ( C 3 ) , 3 6 3 5 - 3 6 5 4 . K i t a i g o r o d s k i i , S . A . , 1 9 8 3 . On t h e T h e o r y o f t h e E q u i l i b r i u m R a n g e i n t h e S p e c t r u m o f W i n d - G e n e r a t e d G r a v i t y W a v e s . J . P h y s . O c e a n o g r . , 1 3 ( 5 ) , 8 1 6 - 8 2 7 . Komen, G . J . , 1 9 8 4 . T h e Wave M o d e l l i n g (WAM) P r o j e c t : P r o p o s a l f o r t h e D e v e l o p m e n t and O p e r a t i o n a l I m p l e m e n t a t i o n o f a T h i r d G e n e r a t i o n O c e a n W a v e M o d e l . F i r s t d r a f t i s s u e d by C h a i r m a n WAM G r o u p , De B i l t , The N e t h e r l a n d s . L a r g e , W.G. a n d S . P o n d , 1 9 8 1 . O p e n O c e a n Momentum F l u x M e a s u r e m e n t s i n M o d e r a t e t o S t r o n g W i n d s . J . P h y s . O c e a n o g r . , 11 (3 ) , 3 2 4 - 3 3 6 . L e B l o n d , P . H . , 1 9 8 4 . F i n a l R e p o r t o f t h e I n v e s t i g a t i o n on t h e S t o r m o f O c t o b e r 1 1 - 1 2 , 1984 on t h e West C o a s t o f V a n c o u v e r I s l a n d . M i n i s t r y o f t h e E n v i r o n m e n t , V i c t o r i a , B.C. L e w i s , C . J . a n d M.D. M o r a n , 1 9 8 5 . S e v e r e S t o r m s O f f C a n a d a ' s Wes t C o a s t : A C a t a l o g u e Summary f o r t h e P e r i o d 1957 t o 1 9 8 3 . C a n a d i a n C l i m a t e C e n t r e U n p u b l i s h e d R e p o r t No. 8 5 - 7 . L o n g u e t - H i g g i n s , M . S . , D . E . C a r t w r i g h t a n d N.D. S m i t h , 1 9 6 1 . O b s e r v a t i o n s o f t h e D i r e c t i o n a l S p e c t r u m o f S e a W a v e s U s i n g t h e M o t i o n s o f a F l o a t i n g B u o y , i n O c e a n Wave S p e c t r a . P r e n t i c e - H a l l , E n g l e w o o d C l i f f s , 1 1 1 - 1 3 2 . - 118 - M a c L a r e n P l a n s e a r c h , 1985. E v a l u a t i o n o f t he S p e c t r a l Ocean Wave M o d e l (SOWM) f o r S u p p o r t i n g R e a l - T i m e F o r e c a s t i n g i n t h e C a n a d i a n E a s t C o a s t O f f s h o r e . U n p u b l i s h e d t e c h n i c a l r e p o r t p r e p a r e d f o r t h e M e t e o r o l o g i c a l S e r v i c e s R e s e a r c h B r a n c h , A t m o s p h e r i c E n v i r o n m e n t S e r v i c e , D o w n s v i e w , On t . MEDS, 1 9 8 4 . H i s t o r i c a l Wave M e a s u r i n g S t a t i o n s , S t a t i o n L o c a t i o n s Summary. D e p t . o f F i s h e r i e s a n d O c e a n s , M a r i n e E n v i r o n m e n t a l D a t a S e r v i c e . Upda ted 2 8 - 0 9 - 8 4 . P e a r s o n , F . , 1984. Map P r o j e c t i o n M e t h o d s . S igma S c i e n t i f i c , I n c . , B l a c k s b u r g , V i r g i n i a . P i e r s o n , W . J . a n d L . M o s k o w i t z , 1 9 6 4 . A P r o p o s e d S p e c t r a l F o r m f o r F u l l y - D e v e l o p e d W i n d S e a B a s e d o n t h e S i m i l a r i t y T h e o r y o f S . A . K i t a i g o r o d s k i i . J . G e o p h y s . R e s . , 6 9 , 5 1 8 1 - 5 1 9 0 . P i e r s o n , W . J . , L . J . T i c k a n d L . B a e r , 1 9 6 6 . C o m p u t e r B a s e d P r o c e d u r e s f o r P r e p a r i n g G l o b a l Wave F o r e c a s t s and Wind F i e l d A n a l y s e s C a p a b l e o f U s i n g Wave D a t a O b t a i n e d by a S p a c e c r a f t . S i x t h N a v a l H y d r o d y n a m i c s Sympos ium, O f f i c e o f N a v a l R e s e a r c h , W a s h i n g t o n , D.C. , 4 9 9 - 5 3 2 . R e s i o , D .T . , 1 9 8 1 . The E s t i m a t i o n o f W i n d - W a v e G e n e r a t i o n i n a D i s c r e t e S p e c t r a M o d e l . J . P h y s . O c e a n o g r . , 1 1 , 5 1 0 - 5 2 5 . R e s i o , D .T . , 1 9 8 2 . Wave P r e d i c t i o n i n S h a l l o w W a t e r . P r o c . 1 4 t h A n n u a l O f f s h o r e T e c h n o l o g y C o n f e r e n c e , OTC 4 2 4 2 , V o l . 2 , 1 4 7 - 1 5 2 . R e s i o , D .T . , 1 9 8 5 . Wave T r a n s f o r m a t i o n s R e l a t e d t o N o n l i n e a r F l u x e s . M a n u s c r i p t s u b m i t t e d t o ASCE J . o f Wa te rway , P o r t , C o a s t a l and Ocean E n g i n e e r i n g . R e s i o , D.T. a n d C . L . V i n c e n t , 1 9 7 9 . A C o m p a r i s o n o f V a r i o u s N u m e r i c a l W a v e P r e d i c t i o n T e c h n i q u e s . P r o c . 1 1 t h A n n u a l O f f s h o r e T e c h n o l o g y C o n f e r e n c e , OTC 3 6 4 2 , 2 4 7 1 - 2 4 7 8 . S t . D e n i s , M. a n d W . J . P i e r s o n , 1 9 5 3 . On t h e M o t i o n o f S h i p s i n C o n f u s e d S e a s . T r a n s . SNAME, 6 1 , 2 8 0 - 3 5 7 . S a r p k a y a , T . a n d M. I s a a c s o n , 1 9 8 1 . M e c h a n i c s o f W a v e F o r c e s on O f f s h o r e S t r u c t u r e s . V a n N o s t r a n d R e i n h o l d C o . , New Y o r k . S e a c o n s u l t , 1 9 8 6 a . U n p u b l i s h e d r e p o r t p r e p a r e d f o r M o b i l O i l C a n a d a , L t d . S e a c o n s u l t , 1986b . U n p u b l i s h e d r e p o r t p r e p a r e d f o r E s s o R e s o u r c e s Canada L t d . Seakem, 1 9 8 5 . A Wave C l i m a t e S t u d y o f t h e N o r t h e r n B r i t i s h C o l u m b i a C o a s t . T e c h n i c a l r e p o r t p r e p a r e d f o r M a r i n e E n v i r o n m e n t a l D a t a S e r v i c e s B r a n c h , Depa r tmen t o f F i s h e r i e s and O c e a n s , O t t awa by Seakem O c e a n o - g r a p h y L t d . , S i d n e y , B . C . ' U . S . Army, 1977. S h o r e P r o t e c t i o n M a n u a l , V o l . I. U.S. Government P r i n t i n g O f f i c e , W a s h i n g t o n . Yamada, T . 1 9 7 6 . On t h e S i m i l a r i t y F u n c t i o n s A , B , a n d C o f t h e P l a n e t a r y B o u n d a r y L a y e r . J . A t m o s . S c i . , 2 3 , 7 8 1 - 7 9 3 .

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