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

Mean density anomaly Mean principal-axis flow

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:

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.

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.

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

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

Along estuary: pressure gradient balanced by friction

S0S0

Mean density anomaly Mean principal-axis flow S0S0

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

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

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

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

u 1 = 0.10 m/s u 2 = 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 = 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 Observed = 7 m in 8 km = 8.8 x 10 -4

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

EFFECTS OF CHANNELS ON DENSITY-INDUCED FLOW

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

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

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

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

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 )

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

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

(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

out in BRiRi

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

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