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EVAT 554 OCEAN-ATMOSPHERE DYNAMICS

EVAT 554 OCEAN-ATMOSPHERE DYNAMICS. OCEAN BOUNDARY CONDITIONS. LECTURE 17. (Reference: Peixoto & Oort, Chapter 8,10). Z= h. f = f N. Z=0. l = l W. l = l E. L. Z=-H. f = f S. Consider an idealized ocean basin. Z= h. Z=0. l = l W. l = l E. L. Z=-H.

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EVAT 554 OCEAN-ATMOSPHERE DYNAMICS

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  1. EVAT 554OCEAN-ATMOSPHERE DYNAMICS OCEAN BOUNDARY CONDITIONS LECTURE 17 (Reference: Peixoto & Oort, Chapter 8,10)

  2. Z= h f= fN Z=0 l= lW l= lE L Z=-H f= fS Consider an idealized ocean basin

  3. Z= h Z=0 l= lW l= lE L Z=-H Consider an idealized ocean basin f= fN f= fS An idealized global basin can be envisioned as a system of coupled such basins

  4. Z= h Z=0 l= lW l= lE L Z=-H Consider an idealized ocean basin First, consider the boundary conditions on mass No mass flux through the side boundaries, f= fN (rv)/f=0 at f= fN, f=fS f= fS (ru)/l=0 at l=lE, l=lW No mass flux through the vertical boundaries, (rw)/z=0 at z= h,-H

  5. Z= h Z=0 l= lW l= lE L Z=-H Consider an idealized ocean basin Now, consider the boundary conditions on momentum No normal flow at side boundaries, f= fN v=0 at f= fN, f=fS f= fS u=0 at l=lE, l=lW No normal flow at lower boundary w=0 at z= -H ‘Rigid lid’ approximation at upper boundary w=0 at z= h

  6. Z= h Z=0 l= lW l= lE L Z=-H Consider an idealized ocean basin Now, consider the boundary conditions on momentum No lateral stress at side boundaries, f= fN u/f, w/f =0 at f= fN, f=fS f= fS v/l, w/l =0 at l=lE, l= lW No vertical stress at lower boundary u/z=0, v/z=0, w/z=0 at z= -H Atmospheric windstress forcing of upper boundary (rnV/f) u/z=tl (rnV/f) v/z=tf w/z=0 at z= h

  7. Z= h Z=0 l= lW l= lE L Z=-H at z= h Consider an idealized ocean basin Now, consider the boundary conditions on temperature No heat flux through side boundaries, f= fN T/f =0 at f= fN, f=fS f= fS T/l =0 at l=lE, l=lW No heat flux at lower boundary T/z =0at z= -H Heat flux at top boundary

  8. Z= h Z=0 l= lW l= lE L This last term represents brine formation due to the freezing of ocean water Z=-H at z= h Consider an idealized ocean basin Now, consider the boundary conditions on salinity No salt flux through side boundaries, f= fN S/f =0 at f= fN, f=fS f= fS S/l =0 at l=lE, l=lW No salt flux at lower boundary S/z =0at z= -H Salt flux at top boundary

  9. Z= h Z=0 l= lW l= lE L Z=-H at z= h Consider an idealized ocean basin Return to the boundary conditions on temperature No heat flux through side boundaries, f= fN T/f =0 at f= fN, f=fS f= fS T/l =0 at l=lE, l=lW No heat flux at lower boundary T/z =0at z= -H Heat flux at top boundary

  10. Z=-h Z= h Z=0 l= lW l= lE L Z=-H at z= h Recall the temperature equation Integrate from the base of the thermocline f= fN f= fS Heat flux at top boundary

  11. at z= h Recall the temperature equation T Z=-h Integrate from the base of the thermocline Z to the surface Heat flux at top boundary

  12. = Qrad - Qrad - Qconv - Qlat  =S(f,l)+easTA4 Recall the temperature equation T Z=-hT Integrate from the base of the thermocline Z to the surface (steady state) -essTS4 -a(TS-TA)-b

  13. =S(f,l)+easTA4 -essTS4 -a(TS-TA)-b = Qrad - Qrad - Qconv - Qlat  =S(f,l)+easTA4 T Z=-hT Z -essTS4 -a(TS-TA)-b

  14. =S(f,l)+easTA4 -essTS4 -a(TS-TA)-b T Z=-hT Z Consider departures from steady state, This is sometimes approximated by a ‘restoring’ boundary condition, Why ‘restoring’?

  15. Reconsider temperature equation T Integrate over the mixed layer (depth = h) Z=-h Z Consider departures from steady state,

  16. Reconsider temperature equation T Integrate over the mixed layer (depth = h) Z=-h Z Consider departures from steady state, t= t/t0 TS’/t=-l TS’+TA’

  17. Mixed Layer Temperatures T We will assume that TA’ represents random atmospheric surface temperature variations due to e.g. mid-latitude storm systems. Z=-h Z For simplicity, this can be approximated as Gaussian white noise [Hasselmann, K. (1976) Stochastic climate models. Part I: Theory. Tellus 28:473-485.] This model exhibits a “Red Noise” spectrum: t= t/t0 TS’/t=-l TS’+TA’

  18. Power Spectral Density frequency Mixed Layer Temperatures T Z=-h Z This model exhibits a “Red Noise” spectrum: This simple model can be generalized upon t= t/t0 TS’/t=-l TS’+TA’

  19. Z=-hT This simple model can be generalized upon t= t/t0 TS’/t=-l TS’+TA’ Mixed Layer Temperatures T “Upwelling-diffusion” EBM Z=-h Z Generalizes upon the simple red noise model by allowing for both a mixed layer and a “deep ocean” with exchange of heat by advection and diffusion between the two layers

  20. Mixed Layer Temperatures T “Upwelling-diffusion” EBM Z=-h Z=-hT Z Wigley and Raper (1990) Natural Variability of the Climate System and Detection of the Greenhouse Effect. Nature 344:324-327.

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