ESCI 485 Air/sea Interaction Lesson 4 Waves Dr. DeCaria

Transcription

1 GENERAL WAVE PROPERTIES ESCI 485 Air/ea Interaction Leon 4 Wave Dr DeCaria Each tye of wave ha it own dierion characteritic or dierion relation ο The dierion relation i an equation that ive the frequency () of the wave a a function of hyical arameter and/or the wave number The eed of an individual wave cret i called the hae eed and i iven a c k The enery of the wave roaate at the rou velocity The rou velocity i not necearily the ame a the hae eed The rou velocity i found by c k If the hae eed of a wave doen t deend on wave number then the wave i called non-dierive In thi cae it turn out that c = c If the hae eed i a function of wave number then the wave are dierive DISPERSION CHARACTERISTICS OF CAPILLARY WAVES The dierion relation for urface caillary/ravity wave i ( σ ρ ) = ± k + k tanh kh where σ i the urface tenion The urface tenion term i only imortant for the hortet wavelenth (on the order of a few centimeter or le) ο For wavelenth maller than around 17 cm only urface tenion i imortant and the wave are ure caillary wave The dierion relation for ure caillary wave i ( σ ρ) tanh kh = ± k Caillary wave are anomalouly dierive meanin that the rou velocity i larer than the hae eed and that horter wavelenth are fater than loner wavelenth Caillary wave attenuate very quickly due to molecular vicoity They are not found far from the ource reion

2 DISPERSION CHARACTERISTICS OF SRFACE GRAVITY WAVES For wavelenth loner than a few centimeter the urface tenion can be inored We therefore have the dierion relation for ure ravity wave = ± k tanh kh We can break thee urface ravity wave into three reime Dee water wave ο Thee are wave whoe wavelenth i much le than the deth of the water In thi cae kh i lare and o the dierion relation become = ± k ο Dee water wave are dierive ο The reure erturbation in dee water wave decreae exonentially with deth with an e-foldin cale on the order of the wavelenth ο Dee water wave are non-hydrotatic Shallow water wave ο Thee are wave whoe wavelenth i much reater than the deth of the water In thi cae kh i mall o that tan kh kh and the dierion relation become = ±k H ο Shallow water wave are non-dierive ο The hallower the water the lower the wave ο Shallow-water wave are hydrotatic ο The reure erturbation ha the ame manitude all the way to the bottom of the fluid In between the dee water and hallow water wave we mut ue the full dierion relation for ravity wave SEA AND SWELL The wave on the ocean are broken into two cateorie Sea tee irreular wave

3 ο Sea diiate quickly after the wind die Swell reular loner low and rounded wave that are left after the wind die down or that roaate away from the windy reion ο Swell can roaate for thouand of mile INITIAL GENERATION OF WIND WAVES The urface wave oberved on the ocean are enerated by the wind In a laboratory when air i blown over initially calm water ο a thin hear flow develo in the uer urface of the water ο turbulence develo in the air directly above the water ο Lon-creted reular wave quickly develo on the urface ο After econd or o the lon-creted reular wave bein chane into hort-creted intability wave ο The intability wave aear imultaneouly with turbulence in the hear flow on the water ide The intability wave have wavelenth near 17 cm which i alo the wavelenth of minimum eed for urface ravity/caillary wave In nature thi henomenon occur with wind ut eneratin the familiar cat aw on the urface of the water MEASRES OF WAVES Autocorrelation function R( t) = η( t ) η( t + t) 0 0 ο The time between eak in the autocorrelation function i an indicator of the dominant eriod of the wave ο The erie of eak in the autocorrelation function decay over about 5 eriod Thi i why wave often eem to come in et of five or o Power ectrum ο The Fourier coefficient for the wave are found by takin the Fourier tranform of the ocean heiht time erie data

4 F( ) = η( t)ex( it ) dt ο Plottin the manitude of F() a a function of ive u a lot of the ower ectrum of the wave and can tell u omethin about which frequencie of wave contain the mot enery ο A the wind become troner the ectral eak become taller and move to lower frequencie ο The ectral denity φ() i jut the ower ectrum divided by the frequency rane (called bandwidth) over which each Fourier coefficient i valid Jut remember that ectral denity i roortional to the ower ectrum ο Wave ectra uually exhibit a inle narrow eak which correond to the dominant frequency of the wave Root mean quare elevation H rm ( η ) 1 = Sinificant wave heiht H 1/ or H defined a the mean of the 1/ hihet wave ο In the abence of well the followin relation hold H = 4 0 H rm WAVES IN A FLLY DEVELOPED SEA A fully develoed ea i one in which the wave have rown to their maximum amlitude for the iven wind condition Thi imlie ο The wind ha been blowin for a lon enouh duration o that the wave ectrum ha become aturated (no more enery can be added) ο The ea i not fetch-limited meanin that the hore i far away In a fully develoed ea the factor that we would exect to be imortant for decribin the wave field are the ectral denity the wind eed at ten-meter above the urface wave frequency and ravity Thee can be formed into two non-dimenional rou 4

5 φ( ) and 5 (Note: u* can be ued intead of A tyical relation between thee two value i /u* = 75) ο We exect thee rou to be related and they have been found to relate quite well ο If thee non-dimenional rou are lotted aaint each other the rah are nearly identical reardle of the wind eed Obervation in fully develoed ea alo how relation between other nondimenional rou formed from other wave arameter The followin relation have been found to hold H H rm = 005 = 0 ο In a fully develoed ea exreion uch a thee can be ued to etimate wave heiht from the -meter wind eed The characteritic wave frequency ( ) i related to and via = 088 which how that a wind eed increae the characteritic wave frequency decreae ο If we aume the wave are dee-water wave then = k = C where C i the hae eed of the dominant wave o that = C Combined with the exreion in the recedin bullet we et a relation between the -meter wind and the eed of the characteritic wave C = 114 5

6 ο The characteritic move lihtly fater than the -meter wind a relationhi that can ometime be caually oberved if you ail downwind at the eed of the wind WAVE GROWTH Near the coatline or in reion where the wind ha not acted lon enouh for the ea to become fully develoed an additional arameter that mut be taken into account i the fetch denoted by X An additional non-dimenional rou that can be formed i X/u* o that there are now three non-dimenional rou φ( ) 5 We exect relationhi to hold between thee three non-dimenional rou and One uch emirical relation that ha been found i X = 71 which relate the characteritic wave frequency to the fetch and how that a fetch increae the wave frequency decreae 1 X The rm wave heiht ha been found to relate to fetch via 4 X ( 16 ) H rm 4 u = * howin wave heiht increae with the quare-root of the fetch Thee two relation can be combined eliminatin the fetch to et H rm = 0057 in the relation H = 4H rm and the definition of eriod T = π/ we et T 0061 * * = u H u Thi relation how that a the wave heiht increae o doe the eriod of the characteritic wave 6

7 Since the loner wave travel fater than the horter wave and a the wave row the characteritic eriod become loner than a time oe on the hae eed (C ) of the dominant wave increae Therefore the ratio of C /u* can be ued a a non-dimenional wave ae A meaure of wave teene i H /λ where λ i the wavelenth of the characteritic wave From the relationhi above a formula for wave teene a a function of wave ae can be found to be H λ C = 015 * u 1 and how that the wave become le tee with time WIND WAVE FORECASTING Theory and obervation ha allowed u to come u with everal formula relatin imortant wave arameter (heiht and eriod notably) in term of the -meter wind eed fetch and duration of the wind ο The formula are uually dilayed in rahical form The key to a ood wave-heiht forecat i to have a ood forecat of the wind field Fetch mut be accounted for even in the middle of the ocean If the ea i not fully develoed due to inadequate fetch the wave are fetchlimited If the ea i not fully develoed due to inadequate duration of wind the wave are duration-limited The effect of well mut alo be taken into account when forecatin wave ο The combined effect of ea and well are accounted for by definin a combined wave heiht accordin to the formula combined ea ( ) well H ( H ) H = + 7