The Governing Equations

Transcription

1 The Governng Equatons L. Goodman General Physcal Oceanography MAR 555 School for Marne Scences and Technology Umass-Dartmouth

2 Dynamcs of Oceanography The Governng Equatons- (IPO-7) Mass Conservaton and Momentum Equaton Conservaton of Mass Flow Water s approxmately ncompressble Volume flow of water gong n = Volume flow of water gong out.

3 Text: IPO- Introducton to Physcal Oceanography by Bob Stewart Download pdf fle verson for complete notes and/or use the on lne verson for summary. Introducton to Physcal Oceanography by Bob Stewart, Texas A&M

4 Mass Conservaton Q Q Q 1 2 = = v = v Q A A Volume Flow n Equals Volume flow Out Q 1 Q 2 3 m Q = volume flow rate, unts of sec m v = velocty, unts of sec A = Area perpendcular to velocty, unts of m 2

5 Example: Channel Flow of Varyng Depth L L v 1 H 1 H 2 v 2 Suppose the flow of velocty v 1 = 1 m/sec from a rver, L = 1 km wde and H 1 =3 m deep, emptes nto the contnental shelf at a depth of H 2 = 20 m, what s the ext velocty v 2 of ths flow? 3 Q 1 = nput volume flow rate = v1a 1 = 1 x(10 m)x(3m) = Q 2 = output volume flow rate v2a 2 = v2x((10 m)x(20m) 3 m m Usng Q 1 = Q2! v 2 = = sec sec m sec 3 m sec

6 Mass and Scalar Conservaton: Box Models Example Medterranean Outflow Salt Flow Conservaton V V S = V O O S VO S O (.79Sv)(38.3) = = S 36.2 m Sv = sec =! V + R + P = V + E Volume Flow Conservaton Input + Rverne + Precp = Output + Evaporaton o Estmate of fresh water nput R + P - E = V - V o = ? 10 m sec

7 Seawater s only slghtly compressble d! = dt 0 ρ Path of a flud parcel of densty ρ d! "! " u! " x u! " = + + y + u! z = 0 dt " t " x " y " z ds dt Homework Queston : Are = 0? = 0? dt dt T = temperature; S = salnty. Explan n words.

8 Generalzng Conservaton of Mass flow: Contnuty Equaton!u A = 0 sum of velocty tmes area over all 6 faces = 0 Contnuty Equaton!! u! v! w " # u = + + = 0! x! y! z

9 Example of Applcaton of Contnuty Equaton Estmate value of vertcal velocty n coastal upwellng 200m w =? u = cm 10 sec! 20km!! u! v! w " u " w # $ u = + + = 0 % + & 0! x! y! z " x " z Dmensonal Analyss! z 200 cm cm m! w " w "! u = 10 =.1 = 86! x sec sec day

10 Momentum (Euler) Equatons Total Force actng on a body = mass tmes ts acceleraton F = ma The forces on a cube of water (PD) Pressure dfference (PD) (Co) Corols Force (W) Weght (Fr) Frcton F du 1 a = " = ( PD + Co + Fr + W ) m dt! Note the subscrpt. "! = V densty of seawater

11 Fr = 0 No Frcton Case PD = "!p Why s there a mnus sgn?! x!! Co =(u? f)? s the "cross" product of vectors, f =2Ùsn(è), è s lattude; f =f =0 (an approxmaton) z x y W = -mg W =W =0 z x y f!! u! f! Corols force s always perpendcular to earth rotaton vector u!!!!! u " f! f " u Exercse Show that = f! u! and

12 Momentum Equaton Component Form (No Frcton) du 1 " p = # ( ) + f v dt! " x dv 1 " p = # ( ) # fu dt! " y dw 1 " p = # ( ) # g dt! " z where d " " " " = + u + v + w dt " t " x " y " z du 1 " p # fv = # ( ) dt! " x dv + fu = # dt 1 " p ( )! " y dw 1 " p = # ( ) # g dt! " z where d " " " " = + u + v + w dt " t " x " y " z

13 Summary of Fundamental Equatons du!! " p I. Momentum + ( f # u) = $ $ g + Fr dt " x! " u II. Contnuty % & u = = 0 " x d! III. Incompressblty = 0 dt Note that g g z x = = g g y =0 Homework: Wrte out equatons (1), (2) and (3) n component form (x,y,z); use the expanson of the total dervatve n tme.

14 The Hydrostatc Condton and Buoyancy Startng pont: vertcal momentum equaton 0 5 dw 1 " P = # ( ) # g dt! " z If the densty,!, s constant " p = #! g $ p = p0 #! gz " z where z = 0 surface, z = -H bottom, p s the pressure at the surface, atmospherc pressure, N p0 % 10 = 1 bar 2 m What s Pressure? Pressure s force (Unts of Newtons)per unt Area.)

15 p = 10 n m C z = 20m D z=4m A B Homework Problem kg! = 1000 m 3 A cylndrcal vessel of heght H = 20 m s flled wth water of densty to a heght of 4m. What s the pressure at: () pont A located on the bottom at the center, () at pont B located at the bottom of the vessel but at the rght sdewall; () at pont C at the surface of the water; (v) at pont D located at the vessel sdewall 5 2 m above the bottom.? Take the atmospherc pressure as p0 = 10 n 2. m

16 kg! = 1000 m 3 Homework Problems Contnued. II. Usng for an approxmaton of seawater densty at what depth would the pressure double from that of atmospherc pressure? A decbar s.1 bar, how much would you have to go beneath water for the pressure to ncrease 1 dbar pressure above atmospherc pressure. Can you see why some oceanographers use decbars nstead of meter to ndcate depth?