**Analytic Solutions for Computer Flow Model Testing**,” Lynch and Gray present solutions for water levels and depth-averaged velocities, for tidal and/or wind forcing, and for Cartesian and polar domains. These solutions have been useful for model validation, especially for tides, and especially within the ADCIRC community — the first example problem in the ADCIRC documentation is based on one of their solutions. That problem, for tidal flows in a polar domain, has been used to validate several model advancements over the years.

However, we found an error in their solution for wind-driven setup on a polar domain. It appears to be a typographical error — the variables are not updated correctly at the last step, when the solution is generalized for a wind with arbitrary direction. This solution is not used frequently, and we did not find a correction to this error in the literature (although we were unable to access every subsequent manuscript that cited the Lynch and Gray solution). So we are documenting it here.

**Basic Equations**

To derive an analytical solution, Lynch and Gray start with the linearized shallow water equations:

and:

in which is the water level relative to mean sea level, is the depth averaged velocity, is the bathymetric depth, is gravity, is a linear bottom friction, and is the constant wind stress (with units of ).

These equations are combined into a single wave equation:

which is then solved for the water level, . For the no-tides solution , they assume steady flow and thus the friction can be ignored.

**Wind Forcing on Polar Domain**

We are interested in their solution for wind-driven flows on a polar domain, as shown below. Flow is required to be tangential to the solid boundaries at , , and , where is arbitrary. For the wind-only case, the water levels are pinned at the outer boundary, . A constant wind stress is applied in an arbitrary direction. Bathymetry is described by , in which is constant and may assume any real value.

**Original Solution**

A boundary-value problem is defined on this polar domain, and then a solution is derived by using separation of variables, an assumed form, and a Fourier series. The solution is drived fully for a wind stress acting in the direction (i.e. toward the right in the figure), but then the solution is generalized for a wind stress with arbitrary direction:

in which and are the wind stress components in the and directions. (For example, if , then would act toward the top in the figure.) With this setup, they derive the analytic solution:

in which the coefficients are defined in their paper (linked above). For example, the first coefficient is:

However, this solution is incorrect because it does not fully account for the arbitrary direction in these coefficients, which were derived only for the direction . To account correctly for the additional component in the direction , the coefficients need to be re-defined, and the solution needs to be re-written.

**Corrected Solution**

That first coefficient should be:

in which the numerator is now a function of direction , and can be either (for ) or (for ). This correction must be carried through the other coefficients and .

The correct analytical solution should be:

which accounts correctly for the contributions to the water levels from the two wind stress components. With this corrected solution, the water levels should respond correctly inside this polar domain.

**Example**

We considered a simple test case with the following parameters:

so the domain is a quarter annular harbor with a constant bathymetry, and the wind stress is acting toward the bottom left. The water levels show a setup in the lower-left of the domain, with peaks of about 12 cm:

**References**