1. Typical Mean Dynamic Balances in Estuaries Along-Estuary Component 1. Barotropic pressure gradient vs. friction Steady state, linear motion, no rotation, homogeneous fluid, friction in the vertical only (A z is a constant) The balance can then be rewritten as: z x

2. Let’s solve this differential equation; integrating once: Integrating again: This is the solution, but we need two boundary conditions: This makes c 1 = 0 Substituting in the solution at z = -H: z x

3. General Solution from boundary conditions Particular solution, which can be re-arranged:

4. 2. 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:

5. 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.

6. 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.

7. 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

8. 0.0125 Pa

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

11. Momentum balance, steady with friction (tidally averaged) Geyer et al (2000) approximations: U T is the tidal current amplitude U E is the magnitude of the estuarine circulation. where a o is an O (1) constant related to the vertical variability of stress. Also considering that the surface slope scales in proportion to the baroclinic pressure gradient, the momentum eqn becomes

12. solving for U E where a o  0.3 based on Hudson data. (Its value depends on whether the layer average or the maximum is considered). -U E UEUE

13. a o  0.3 on Hudson (2000, JPO, 30, 2035)

14. What drives Gravitational Circulation? Pressure Gradient x z River Ocean isopycnals isobars?

15. 3. Advection vs. Pressure GradientTake: Upper layer homogeneous and mobile Lower layer immobile Consider inertial effects Ignore friction @ lower layer there is no horizontal pressure gradient x h1h1 u1u1 U 2 = 0 z interface surface At interface: At lower layer: The interface slope is of opposite sign to the surface slope

16. The basic balance of forces is Over a volume enclosing the upper layer: But x h1h1 u1u1 U 2 = 0 z interface surface Using Leibnitz rule for differentiation under an integral (RHS of last equation; see Officer ((1976), p. 103) we get: This is the momentum balance integrated over the entire upper layer (i.e., energy balance) The quantity inside the brackets (kinetic and potential energy) must remain constant

17. Defining transport per unit width: Total energy (Kinetic plus Potential energy) remains constant along the system If the density ρ 1 and the g’ do not change much along the system, we can estimate the changes in h 1 as a function of q 1 (i.e. how upper layer depth changes with flow) x h1h1 u1u1 U 2 = 0 z interface surface Differentiating with Respect to ‘q 1 ’

18. Subcritical flow causesand supercritical flow causes This results from a flow slowing down as it moves to deeper regions or accelerating as it moves to shallower waters or through constrictions x z h1h1 supercritical flow x z h1h1 subcritical flow

19. Apparent PARADOX!? Pressure gradient vs. friction: i.e., u proportional to H 2 This can also be seen from scaling the balance: If we include non-linear terms: Which may be scaled as: i.e., U proportional to H -2 !!! Ahaa! If L is very large, we go back to u proportional to H 2 Physically, this tells us that when L is small enough the non-linear terms are relevant to the dynamics and the strongest flow will develop over the shallowest areas (fjords). When frictional effects are more important than inertia, then the strongest flow appears over the deepest areas (coastal plain estuaries)!!!

20. This competition inertia vs. friction to balance the pressure gradient can be explored with a non-dimensional number: When this ratio > 1, inertia dominates When the ratio < 1, friction dominates Alternatively: When H / L > C b, inertia dominates When H / L < C b, friction dominates

21. 4. Surface Pressure + Advection vs. Interfacial Friction x h1h1 u1u1 u2u2 z interface surface h2h2 Similar situation as before (advection vs. presure gradient) but with interfacial friction (f i ). Flow in the lower layer but interface remains. There is frictional drag between the two layers. The drag slows down the upper layer and drives a weak flow in the lower layer. In the upper layer, over a volume enclosing the layer: The momentum balance becomes (Officer, 1976; pp. 106-107):

22. The solution has a parabolic shape Boundary conditions: u 1 = u 2 at the interface u 2 =0 at the bottom, z = -H because interface touches the bottom In the lower layer, the balance is: x h1h1 u1u1 u2u2 z interface surface h2h2 u z Salt-wedge

23. Typical Mean Momentum Balances Barotropic pressure gradient vs. FrictionRivers, Homogeneous Estuaries Total pressure gradient vs. FrictionPartially Mixed Estuaries Total pressure gradient vs. AdvectionFjords Total pressure gradient vs. Advection + Interfacial FrictionSalt Wedge