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# Horizontal eddy viscosity

**Horizontal eddy viscosity** is a term in the momentum equations to cover the well-known turbulence closure problem. The term is also useful numerically for increased stability. For background on the turbulence closure problem, see, e.g. the AMS glossary definition, the Wikipedia entry on turbulence modeling, or a fluid mechanics textbook such as Kundu's^{[1]}. Readers unfamiliar with these concepts should note that, contrary to the name, eddy viscosity is unrelated to true viscosity, even though the units (length squared over time) are the same. Typically, one should expect eddy viscosity to be many orders of magnitude larger than the viscosity of actual water (around 1x10^{-6} meters^{2}/second for seawater). In ADCIRC, the user can either specify a constant in time horizontal eddy viscosity or employ the Smagorinsky model to update the horizontal eddy viscosity based on local flow conditions with each time step. In both cases the horizontal eddy viscosity or Smagorinsky coefficient can be either spatially constant or spatially varying.

## Contents

## User-Specified Constant in Time Eddy Viscosity

A spatially constant eddy viscosity can be supplied via the `ESLM`

parameter in the fort.15 file. Alternatively, the `average_horizontal_eddy_viscosity_in_sea_water_wrt_depth`

nodal attribute permits a spatially variable eddy viscosity.

### Usage Notes

Commonly used values range from 1 to 50 meters^{2}/second, with 10 being a good starting point for many coastal ocean modeling scenarios. By the nature of turbulence closure, one might expect the value to be smaller as mesh resolution increases. However, this may not always be the case since areas with high resolution may also be more turbulent. In particular, some modelers use a larger value in overland areas, which can also improve stability. However, specifying a value that is too large can induce instabilities, as well.

## Smagorinsky Eddy Viscosity

ADCIRC also allows a Smagorinsky-type turbulence closure model^{[2]}. There are currently two slightly different ways of enabling this in the model. One is by specifying a negative value for `ESLM`

in the fort.15 file. If this is done, the Smagorinsky turbulence closure is enabled and the absolute value of `ESLM`

is used as the coefficient. Alternatively, the `Smag_Control`

namelist can be used, e.g.

`&Smag_Control SMAG_LOWER_LIM=1.0d-8, SMAG_UPPER_LIM=100 /`

When this is done, the value of `ESLM`

is taken as the coefficient. This also allows the user to optionally specify lower and upper bounds on the actual eddy viscosity determined via the Smagorinsky formulation. The values given in the above example of 1E-8 m^{2}/s and 100 m^{2}/s are the default values. There is no way to supply upper and lower bounds with the other method.

If the `average_horizontal_eddy_viscosity_in_sea_water_wrt_depth`

nodal attribute is used when the Smagorinsky eddy viscosity is employed, then the nodal attribute values are taken to be the Smagorinsky coefficient values.

### Usage Notes

The two methods currently result in different flags being set in the code, which have slightly inconsistent behavior. Specifically, the "negative `ESLM`

method" results in a check being run at model initialization to verify that the default (Kolar-Gray) formulation of the lateral stresses in the GWCE is not in use, and the model errors out if it is. That check is not done in the namelist method. The lateral stress formulation is specified by the first digit in a six-digit IM code being set to 1. It is also set for any regular IM value for 2D ADCIRC. Mathematically, the Kolar-Gray formulation of the lateral stress does not take into account a time derivative of the eddy viscosity terms, meaning it is technically inconsistent with a Smagorinsky turbulence closure model. However, it is not clear whether the time derivative term is ever large enough to be consequential, and therefore merit departure from ADCIRC's default GWCE formulation.

## References

- ↑ Pijush K. Kundu, Ira M. Cohen, David R. Dowling (2012). Fluid Mechanics.
- ↑ Smagorinsky, J. “General Circulation Experiments with the Primitive Equations.” Monthly Weather Review 91, no. 3 (March 1, 1963): 99–164. https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.