Κύµατα παουσία βαύτητας 8. Grait as in th ocan Sarantis Sofianos Dpt. of hsics, Unirsit of thns Was in th ocan Srfac grait as Short and long limit in grait as Wa charactristics Intrnal as
Charactristic proprtis of as η λ Walngth (λ): Th distanc btn to conscti paks and Wanmbr (K) π K λ riod (Τ): Th tim it taks for to conscti paks to pass from a point in spac and Frqnc (ω) ω π T ω λ has Spd (C): Th spd of a monochromatic a C K T Grop Spd (C g ): Th spd of a a packt C g ω K Wa Enrg (Ε ): E g η Significant Wa ight (Η /3 ): Th arag hight (dobl amplitd) of th /3 largst as /3 4 η / π l Ftch: Th lngth of atr or hich a gin ind has blon λ π K K π k
t ( ) Ω g V dt d f t V f t V g t V Working qations Momntm Consration Mass Consration (Incomprssibl flo)
Th ocanic a spctrm ind as sll tsnamis long grait as tids h 4h plantar as f ks months capillar - 3 4 5 6 7 λ T s m
Linar grait as in barotropic flid Coriolis Trm Changs Tmporal >> ft fu T U R ot Coriolis Trm Viscosit Trms << f fl E V k F R U N >> Scaling paramtrs f p t V f p t V g p t V t () υ t () t (3) hr p g
() () (3) t (Continit) Εξίσωση LLCE Looking for a-lik soltions: Bondar Conditions: at gη η t at η η ( k l ω ) - D soltions cos t for simplicit ill instigat a -D (-) a Wa soltion for Laplac qation: ( k k B ) cos( k
Using th srfac bondar B gη condition: Using th ocan bottom bondar condition: gη k ( ) k( ) k k From qation (3): t cos k B k ( k gη cosh cosh kgη sinh[k( )] cos cosh( k ) Using th srfac bondar condition: kgη sinh cosh ( k ) ( k ) cos ω gη B gη k k [ k( )] ( k ) ( k Disprsion rlation Σχέση διασποάς η t t ( k ωt) ω η cos( k gk tanh( k ) k k cos k k η t ( k
Short as λ << Η (short alngth or dp ocan) > k>> Long as λ >> Η (long alngth or shallo ocan) > k<< tanh( k ) tanh( k ) k ω gk ω gk has Spd : C C g k Grop Vlocit : Disprsi ω k C g ω k C has Spd : C g Grop Vlocit : ω k Non-disprsi C g ω k C g C C g C
Watr parcl orbits ξ t ζ t ξ η cosh cosh( k ) smimajor ais [ k( )] ξ cosh kgη t ω cosh( k ) [ k( )] ζ sinh kgη t ω cosh( k ) cos( k sin( k [ k( )] cosh ξ η cosh( k ) [ k( )] sinh ζ η cosh( k ) [ k( )] η sinh[ k( )] ζ cosh( k ) smiminor ais sin( k cos( k ξ ζ (, ) ( ξ, ζ ) Both as dcras ith dpth Th smiminor ais is ro at bottom (-)
Short as λ << Η (short alngth or dp ocan) > k>> [ k( )] sinh[ k( )] k cosh cosh( k ) cosh( k ) ξ η k ζ η sin( k k cos( k Circls Both dcras ponntiall ith dpth Long as Constant λ >> Η (long alngth or shallo ocan) > k<< ξ η sin(k cosh[ k( )] ζ kη ( )cos(k sinh k k [ ( )] ( ) cosh( k ) Dcras linarl ith dpth and bcoms ro at
Stoks Drift If do not s th approimation (i.. th Lagrangian locit at tim t is qal ith th Elrian locit at, at tim t) (, Th Lagrangian locit L (,,t) can b dfind as th Talor sris pansion of th Elrian locit (,,t) arond, L For short grait as (dp atr limit): (,, t) (,, t) ( ) ( )..., t) η η η ω k k sin( k k cos( k cos( k raging or on priod s L η ω k k In gnral [ k( )] cos s ηωk sinh ( k ) Stoks locit
η η cos( k η cos( k ωt) η cos( k)cos( ωt) kgη cosh[ k( )] sin( k)sin( ωt) ω cosh( k ) Sichs Bondar condition: at και L n kl (n )π n,,,... ω λ L n πg( n ) ( n ) π tanh L L L n Wa Rfraction and th island ffct on as
Intrnal grait as: Lard stratification Εσωτεικά κύµατα βαύτητας: Απλή στωµάτωση Δύο στώµατα. Οι εξισώσεις Laplac για κάθε στώµα: k k i ( ) ( k ωt B ) ( k ) i( k ω t ) C k D i η a ζ b ( k ωt ) i ( k ωt ) η ζ - Οιακές συνθήκες: at at gη at η t ( η ζ ) η t g ζ ζ t Χησιµοποιώντας τις οιακές συνθήκες: ω gk ω gk [ sinh( k ) cosh( k )] ( ) sinh( k )
First soltion: ω gk Barotropic mod (Βαοτοπικός τόπος ταλάντωσης) Scond soltion: ω ( ) gk sinh( k ) cosh( k ) sinh( k ) In gnral: For an ocan of n lars, thr is on barotropic mod and n- baroclinic mods (a total of n mods of oscillation). Baroclinic mod (Βαοκλινικός τόπος ταλάντηωσης)
Intrnal as: Baroclinic mod Short as k coth( k ) ω ( ) gk sinh( k ) cosh( k ) sinh( k ) ( ) gk coth( k ) Long as k << sinh( k ) k cosh( k ) c gʹ, gk ω gʹ ( ) g ω gk η ζ k For a tpical ocanic stratificatiob: ΔΟ(kg/m 3 ), O(kg/m 3 ) g -3 g η ζ