Inundation or Wet and Dry (WAD) Processes

[Oey, 2005, 2006; Oey, Ezer, Hu & Muller-Kager, 2007]

The land-sea boundaries in coastal and/or regional modeling are in general variable. Grid cells near a land-sea boundary are sometimes wet (i.e. inundated) and sometimes dry (muddy!). The Princeton Ocean Model used in PROFS is the first general-purpose ocean model (accounting for large or semi-basin scales, mesoscale eddies, and also fine-scale near-shore processes) that is equipped with WAD physics.

Modeling the land-sea interface is tricky. There exist analytical solutions in idealized settings - homogeneous bottom properties, no vegetation etc. Carrier and Greenspan [1956] gave elegant special solutions of water running up (and down) a beach. The difficulty arises not only because ocean models in general do not accommodate zero water depth (D = 0), but also because in general the real beach has complicated bottom shapes and properties which can themselves interact with the flow.

For coastal processes where bottom friction often dominates, WAD may be modeled by relatively simple equations that offer physical insights; the following equation may be obtained (assuming incompressible and homogeneous fluid, hydrostatic etc);:

This equation is similar to the Burger’s equation except that the diffusion term is nonlinear; it governs how the water depth D varies with time t and position x subject to some initial and boundary conditions, and D = 0 indicates dry points included as part of the solution. The H(x) measures the shape of the ocean’s bottom, and is conveniently defined as the distance from some fixed datum above the highest water level to the bottom, r is the bottom friction coefficient (unit is m/s) and g is the acceleration due to gravity. Imagine an initially dry channel closed at left, x = 0, and water rushes in from the right so that D increases with time at x = L. If the bottom is flat, the channel’s water level rises much like the temperature rises in a metal rod heated at one end, the conductivity of the rod being a quadratic function of the temperature; this analogy allows estimate of the temporal and flood-intrusion scales [Oey, 2005]. For a general bottom shape (H_x and H_xx ¹ 0), two other effects arise. First, there is a down-gravity advective effect represented by the second term on the left-hand-side of the above equation that, for positive (negative) H_x, opposes (accelerates) the leftward diffusive inundation. For sufficiently steep negative slope, this advective effect can give rise to shock-formation. The second effect (of comes from the last term on the right-hand-side of the above equation. This represents a source (sink) if the bottom is a bowl or concave (hump or convex) shape, i.e. if H_xx < (>) 0. In the source (sink) case, water represented by “D” tends to be accumulated (repelled) by the bowl (hump).

The figure below illustrates the solution for a linear slope in a channel with a rather large value of r = 0.01 m/s during a 12-hour tide. To mimic tides, sea-level at the right end of the channel is raised during the first six hours (left column); it is then lowered during the next six hours. Color indicates the velocity and shaded region is water (i.e. inundated). Because of friction, sea-level tilts for water to flood. During ebb there is a time lag so that water near the right-end ebbs before the previously-inundated water near the left-end also ebbs (panels G & H), creating a sea-level hump.

Analytical or semi-analytical solutions (Carrier & Greenspan’s and the above friction-dominated one) are useful for verifying WAD algorithm in ocean models. POM is coded to work for r = 0 (or very small) also. The following illustration (r = 0) shows a case with slightly more complicated topography. Please also consult the above references for more examples such as Tsunami and other interesting cases.