1. Salty water inflow (near the bottom) Freshwater input Along the Estuary: Pressure Gradient balanced by Friction (Pritchard, 1956) 0

4. Mean density anomaly Mean principal-axis flow

5. Pressure gradient vs. vertical mixing expanding the pressure gradient: We can write: The momentum balance then becomes: O.D.E. with general solution obtained from integrating twice:

6. General solution: c 1 and c 2 are determined with boundary conditions: This gives the solution: Third degree polynomial proportional to depth and inversely proportional to friction. Requires knowledge of I, G, and wind stress.

7. We can express I in terms of River Discharge R, G,and wind stress if we restrict the solution to: i.e., the river transport per unit width provides the water added to the system. Integrating u(z) and making it equal to R, we obtain: Which makes: Note that the effects of G and R are in the same direction, i.e., increase I. The wind stress tends to oppose I.

8. Substituting into: We get: Density-induced: sensitive to H and A z ; third degree polynomial - two inflection points River induced: sensitive to H; parabolic profile Wind-induced: sensitive to H (dubious) and A z ; parabolic profile

10. If we take no bottom stress at z = -H (instead of u(-H) = 0):

14. Along estuary: pressure gradient balanced by friction

15. S0S0

16. Mean density anomaly Mean principal-axis flow S0S0

17. C Burchard & Hetland (2010, JPO, 40, 1243)

18. Jay (2010, in Contemporary Issues in Estuarine Physics, Chap. 4)

19. Across Channel Momentum Balance Geostrophic balance (frictionless, steady and linear motion) in the lower layer y z LNM h1h1 h2h2

20. Geostrophic Balance in the upper layer Geostrophic Balance in the lower layer: y z LNM h1h1 h2h2  Margules’ Relation

22. u 1 = 0.10 m/s u 2 = -0.05 m/s ρ 1 = 1017 kg/m 3 ρ 2 = 1022 kg/m 3 f = 8.8 e-5 s -1 u 1 = 0.08 m/s u 2 = -0.06 m/s ρ 1 = 1021 kg/m 3 ρ 2 = 1023 kg/m 3 f = 8.8 e-5 s -1 Observed = 2 m in 8 km = 2.5 x 10 -4 Observed = 7 m in 8 km = 8.8 x 10 -4

24. Generally, the outflow modified by rotation will be restricted by the internal radius of deformation R, derived from geostrophy: Scaling: Internal Radius of Deformation or Internal Rossby Radius L

25. EFFECTS OF CHANNELS ON DENSITY-INDUCED FLOW

26. looking into the bay Further seaward: surface outflow; bottom inflow

27. Looking into the bay IN OUT Outflow over shoals; inflow in channels WHY THE DIFFERENCE? E = A z / [ f H 2 ]

28. Examples in the lower Chesapeake Bay with different E. Which has larger E = A z / [ f H 2 ] ? Larger E Smaller E

29. Along-basin: Across-basin: Pressure Gradient + Friction + Coriolis Friction/Coriolis E = A z / (f H 2 )

30. Ekman # -- proxy for dynamical DEPTH y z y z Small E Deep Basin frictionalinfluence Large E Shallow Basin Friction/Coriolis E E = A z / (f H 2 )

31. Along-basin: Across-basin: Pressure Gradient + Friction + Coriolis Friction/Coriolis E = A z / (f H 2 )

32. z – vertical distance from the surface H – total water column depth D – density gradient N – sea level slope α = (1 + i )/D E, where D E = [2A z / f ] ½ w = u + iv

33. (red is inflow; white is outflow) (contours are normalized with the maximum flow) Valle-Levinson et al, 2003, JPO deep shallow Deep It matters how Deep ! depth distance Friction/Coriolis E = A z / (f H 2 ) Friction increases Coriolis increases

34. out in BRiRi

35. low E med E high E (wide) (narrow) (looking into the estuary; orange is inflow; white is outflow) Ke = B/R; E = A z / (f H 2 ) deep shallow Width Width matters! Valle-Levinson, 2008, JGR

36. low E med E high E (wide) (narrow) Ke = B/R E = A z / (f H 2 ) deep shallow Valle-Levinson, 2008, JGR Adriatic & Med Rías Gibraltar Estuaries