Οι νόµοι διατήρησης στη Φυσική Ωκεανογραφία

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1 Οι όµοι διατήρησης στη Φσική Ωκεαογραφία 4. The conservation las in the ocean dnamics Sarantis Sofianos Dept. of Phsics, niversit of Athens Fndamental dnamics and the eqations for geophsical flids Oceanic approimations Rotation Dominant scales in ocean dnamics Redced eqations Scaling the ocean dnamics

2 Fndamental dnamics and the eqations for geophsical flids i. Momentm Eqation: D Material derivative D t local rate of change Non-linearit advection t 1 p Φ F ρ Φ g Force potential Pressre gradient 1 p gˆ ρ L Re L Non-conservative forces e.g. F see net transparenc L In the ocean, sall Re >>1

3 Miing and diffsion Moleclar random motions transport arond a propert FlForce/Resistance De Groot, 1963 Fl of q F q κ q q medim Resistance Force gradient If the concentration of q changes linearl along a path, there shold be a constant fl of q along this path and the concentration of q is constant in time along the path Or the opposite: Constant concentration in an point in a path, means constant fl along the path. q a at a Fl Concentration q q b at b b If there is a change of fl of q along the path F it ill prodce a change of q in time. q t q t F κ q q a Concentration q For velocit: resistance is µ moleclar viscosit and µ/ρ kinematic viscosit

4 Wh is viscosit a non-conservative force? Ignoring an other force and densit variations and orking onl in one dimension for simplicit: ρ t ρ t ρ Mltipling b In three-dimensions t ρ / ρ t t ρ / ρv ρ / t ρv ρv / ρ / ρv $ ρ ' & % $ & ρ / % * $ ' ρ,&,% $ ' ρ & % $ & ρ / % ' $ ρ ' & % $ ' & % $ ' & % - /. / ' $ ρ ' & % The heat added to the ater increases its entrop at the rate of heat generation divided b absolte temperatre entrop sorce term. Local rate of change of energ concentration Energ Fl Divergence of viscosit times the gradient of energ Loss of Mechanical Energ Transformation to heat alas negative

5 ii. Conservation of mass continit eqation: Mass fl to the volme direction ρδδ ρ Mass fl from the volme direction ρ δ δ δδ ρ ρ ρ Accmlation ρ δ δδδ O δ δδδ small terms ρ In 3-D: ρ ρ δδ δ This accmlation mst be accompanied b increase of mass in the volme densit volme ρ t t δδ δ ρ ρ ρ δδ δ ρ t ρ ρ ρ ρ t ρ ρ t ρ ρ Dρ ρ

6 iii. Conservation of dissolved material concentration: ρq t and sing the continit eqation Dq κ q ρq S q S iv. Internal energ conservation: De ρ p ρq Change of internal energ de to pressre variations v. Entrop conservation: Temperatre ΤΚ q cooling/heating De Dη T Q µ ks non conservative sorces and sinks of q e.g. p D κ q For the ocean and the atmosphere κ is of the order of, i.e. the Prandtl nmber / κ O 1 ths advection dominates diffsion as Re is ver large. and sing the continit eqation k cooling/heating 1 Q ρ 1 st La of Thermodnamics q k µ k chemical potential nd La of Thermodnamics

7 vi. Eqation of state: ρ ρ T, p, q ocean Sps " dρ ρ % $ ' # T & S,p α 1 " ρ % $ ' ρ # T & β 1 " ρ % $ ' ρ # S & γ 1 " ρ % $ ' ρ # p & S,p T,p S,T " dt ρ % $ ' # S & T,p " ds ρ % $ ' # p & S,T thermal epansion coefficient K -1 dp ρ αdt βds γdp haline contraction coefficient ps -1 compressibilit coefficient Pa -1 For small variations and assming α,β,γ independent of T,S, p ρρ 1α T T β S S γ p p *, In realit the thermal epansion coefficient depends on temperatre cabelling and the non-linearit indced has the reslt that to ater masses characteried b a different temperatre and salinit bt same densit hen mi together, the reslting mied ater can become denser and eventall sinks. Thermal epansion coefficient is also dependent on pressre thermobaricit. For this reason to ater masses ith different salinit and temperatre bt same densit at the srface, don't have the same densit belo the srface.

8 Oceanic approimations 1 Dρ ρ 1 Dρ ρ ~ 1 δρ T ρ δρ L ρ << L ~ Bossinesq Continit Eqation δρ ρ O 13 Incompressibilit Compressibilit and Dρ 1 β V DV DP 1 V DV DP 1 V DV If the volme does not change nder changes in pressre 1 DV β ; V sing 1 D ρ V D m 1 ρ m V V DV 1 Dρ ρ So, for incompressible flid DP Tpical vales ρ 18 kg/m 3, T 1 o C83 o K, S 35 α Linearied eqation of state nder incompressibilit 1 ρ 4 K 1 ρ T 1 Thermal epansion coefficient [ α T T S ] ρ ρ β 1 S β 1 ρ 1 ρ S ppt Haline contraction coefficient Bossinesq Eqation of State Neglecting possible thermobaric instabilit and cabelling instabilit

9 Oceanic approimations t 1 ρ p g ρ ρ ẑ 1 S t S κ S S S S 3 T t T κ T T Q c p ρ 4 ρ ρ " # 1α T T β S S $ % 5 Bossinesq Eqations Entrop and densit conservation eqations become redndant de to the thermodnamic approimations Bossinesq approimations cancel acostic and shock aves, it is a good approimation since Mach nmber : C s 15 m 1 s M 1, C s << 1 1m 1 s 1 c 4 1 p 3 J kg 1 K 1 When the right hand side of eqations 3 and 4 is ero, the processes are called adiabatic.

10 Bondar conditions η,, t h Sides/bottom: n ˆ at H, & H, Top: Dη at η,, t Dη & p at Rigid - lid p p atm,, t Interior: h δp / gρ atm Inverse Barometer Effect no oceanic acceleration p p p at lo h

11 The effect of rotation β f The β term Ωsinϕ R earth ϕ Ω f R earth ϕ Ωcosϕ R earth : : : Or epanding Ω arond ϕ or ith ϕ ϕ <<1, << R earth... Ω Ω # e $ sinϕ cosϕ ϕ ϕ D I I Dr I Dr Ω r I D R I I 1& D R R D R I Ω r R 1 Ω I DΩ Dr r Ω R Ω D R DΩ Ω R Ω R Coriolis acceleration small term Ωsinϕ v Ωcosϕ correction in g Ωsinϕ g effective g Ω Ω r Ωcosϕ fωsinφ... % & 1 # $ f β % & Ω r r Ω r R No Coriolis force on stationar bodies. Dominant horiontal deflection right/left. Coriolis force does no ork on a bod becase it is perpendiclar to the velocit.

12 t 1 ρ Ω p g ˆ ρ ρ S T S S S κ S S T κ T t t ρ ρ [ 1α T T β S S ] T Q c ρ p WORKING EQATIONS t 1 P ρ Ωsinϕ t 1 P ρ Ωsinϕ t 1 P ρ g S t T t S S S κs T T T κ S [ α T T S ] ρ ρ β 1 S T S κs κ S T S T S T κ κ T T S Q c ρ p

13 The mean state 1 f P t ρ 1 f P t ρ 1 f P P t ρ 1 v f P t ρ P P P ; ; ;

14 The continit eqation: Removing the mean state: 1 1 g P t f P t ρ ρ Similarl:

15 First order closre scheme: A H A H A V Moleclar diffsion and viscocit κ T.14 cm /sec temperatre κ S.13 cm /sec salinit.18 cm /sec at C.1 at C Trblence has the same coefficients for temepratre, salinit and momentm bt is anisotropic vertical/horiontal Edd diffsivit and viscosit A H horiontal 1 to 1 4 m /sec A V vertical 1-5 to 1-4 m /sec

16 SCALING The basic eqations inclde a ver large nmber of distinct tpes of phsical processes, ranging from moleclar to global scales. Depending on the processes nder investigation e can define appropriate scales that describe these processes., W, L t T L H W H Goal: Estimation of the relative importance of each term in the process nder investigation and the possibilit to simplif the basic eqation b defining a dominant balance of the dnamics/ thermodnamics involved. t 1 P ρ fv SCALING L L L W H ΔP ρl f L L H

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18 Scaling nmbers: Aspect ratio: H L Non -linear Term 1 R o Coriolis Term L f fl E k Viscosit Terms 1 Coriolis Term L f fl fh Non - linear Term R E AH Viscosit Term L L F R N H B NH fl L A H These nmber characterie the ocean dnamics Non-linear Geometricall similar Viscos Diffsive Bonded

19 Geostrophic hdrostatic approimation the large scale limit: R o << 1 fl E k fl << 1 H L <<1 t 1 P ρ f t 1 P ρ f t 1 P ρ g f 1 P ρ 1; 1 P 1 P f ; g 3 ρ ρ The geostrophic hdrostatic limit

20 The Talor Colmns 1 1 ρ f P P Non-divergent horiontal flo D Bossinesq continit or

21 f 1 P ρ 1 P f ρ 1 P g ρ Δ Δ g ρ f Δρ Δ 1 g ρ f 1 3 Δρ Δ Δ ρ f 1 ρ f sing 3 f ρ since ρ Δ Δ v g ρ f P P g ρ ; f ρ >> Δρ Δ P gρ g ρ ρ ; g ρ f P gρ e can rite ρ Thermal ind eqations Geostroph comptes the velocit shear and not the absolte velocit ρ 1 Level of no knon motion ρ