TAU0

TAU0 is an input in the fort.15 file that influences the degree of numerical diffusion in ADCIRC’s governing equations. Specifically, it influences the weighting factor that determines the relative contribution of the primitive and wave portions of the the Generalized Wave-Continuity Equation (GWCE). The weighting factor, \(\tau_0\), is affected by values in both the fort.15 file and the fort.13 file, if the primitive weighting in continuity equation or min and max primitive weighting in continuity equation nodal attributes are specified. This page addresses both the TAU0 value in the fort.15 file and the \(\tau_0\) parameter more generally.

Parameter Summary

Because the TAU0 value specified in the fort.15 file can be either a flag (indicating how ADCIRC should operate) or the actual \(\tau_0\) value used in solving the GWCE, it is important to distinguish between the two. All negative TAU0 are flags, all positive TAU0 are \(\tau_0\). Furthermore, some values are overridden by the primitive weighting in continuity equation nodal attribute. The following table is a summary of possible TAU0 values and their meaning. Note that for TAU0 = -x.1 (where x is an integer), behavior is the same as -x, but the \(\tau_0\) values are written to the fort.90 file. More on this below in Outputting.

fort.15 `TAU0`	>= 0	-1	-2	-3	-5	-6	-7
Varies in space	no (mostly)	yes	yes	yes	yes	yes	yes
Varies in time	no	no	no	yes	yes	yes	yes
Space-averaged	no	no	no	yes	yes	yes	yes
Time-averaged	no	no	no	no	no	yes	yes
primitive weighting in continuity equation	yes	discouraged	discouraged	yes	discouraged	yes	yes
min and max primitive weighting in continuity equation	no	no	no	no	yes	no	no
Minimum	N/A	0.002	0.002	N/A	`Tau0FullDomainMin`	N/A	N/A
Maximum	N/A	0.005	1	0.2	`Tau0FullDomainMax`	0.2	0.2
Code Flags
`HighResTimeVaryingTau0`	.FALSE.	.FALSE.	.FALSE.	.TRUE.	.FALSE.	.TRUE.	.TRUE.
`FullDomainTimeVaryingTau0`	.FALSE.	.FALSE.	.FALSE.	.FALSE.	.TRUE.	.FALSE.	.FALSE.
`TimeAveragedTau0`	.FALSE.	.FALSE.	.FALSE.	.FALSE.	.FALSE.	.TRUE.	.FALSE.
`BackLoadedTimeAveragedTau0`	.FALSE.	.FALSE.	.FALSE.	.FALSE.	.FALSE.	.FALSE.	.TRUE.

Nodal Attributes

The primitive weighting in continuity equation nodal attribute permits a spatially variable \(\tau_0\) that is (at least initially) set equal to the nodal attribute values.

TAU0 >= 0, \(\tau_0\) is constant in time and equal to primitive weighting in continuity equation values
TAU0 = -3, -6, or -7, \(\tau_0\) varies in time and based on primitive weighting in continuity equation values

The min and max primitive weighting in continuity equation only works with TAU0 = -5, details on it are below.

Positive TAU0

A positive value is ignored if the primitive weighting in continuity equation nodal attribute is specified in the fort.15 file. For all positive values, the value specified in the fort.15 file is spatially and temporally constant and applied directly, i.e. TAU0 =\(\tau_0\). Practical guidance on setting a constant TAU0 is provided below in Selecting Values. Mathematically, the GWCE is a pure wave equation for \(\tau_0=0\), and approaches a pure primitive equation as \(\tau_0\rightarrow\infty\), however in practice, it behaves like a pure primitive equation once \(\tau_0\) reaches 1 or so.

Negative TAU0

Unexpected behavior may arise if a mismatching nodal attribute is specified
alongside one of these values, and this is discouraged.
Spatially varying, temporally constant
*TAU0 = -1, \(\tau_0\) is computed as follows:

- If depth >= 10; then\(\tau_0\)= 0.005
- If depth < 10; then\(\tau_0\)= 0.020
- This value is ignored if primitive weighting in continuity equation is specified in the fort.15 file.
TAU0 = -2, \(\tau_0\) is is computed according to depth as follows:
- If depth >= 200; then\(\tau_0\)= 0.005
- If 1 < depth < 200; then\(\tau_0\)= 1/depth
- If depth < 1; then\(\tau_0 = 1\)
- This value is ignored if primitive weighting in continuity equation is specified in the fort.15 file.

Spatially and temporally varying

In the unlikely event that one of the below options not paired with its corresponding nodal attribute (the code will exit if this is done for some TAU0), then default values are set using the TAU0 = -1 logic.

TAU0 = -3, \(\tau_0\) is computed from TAU0Base (read in from the nodal attribute) as follows:
- If TAU0Base < 0.025; then\(\tau_0\)= TAU0Base (constant in time)
- If TAU0Base >= 0.025; then\(\tau_0\)= TAU0Base+1.5*TK where TK=speed*Cd/depth
TAU0 = -5, \(\tau_0\) is computed similar to TAU0 = -3 as follows:
- \(\tau_0\)= Tau0Min+1.5*TK
- It is limited by `Tau0FullDomainMin <Tau0FullDomainMin>`__<=\(\tau_0\)<=`Tau0FullDomainMax <Tau0FullDomainMax>`__, which are specified on the line following TAU0 in the fort.15 file (only when TAU0 = -5)
- If the min and max primitive weighting in continuity equation nodal attribute is used, then the nodal minimum and maximum values replace the full-domain values in the above calculation.
TAU0 = -6, \(\tau_0\) is defined using the rules for TAU0 = -3, and then is set equal to the time-average of the current and previous (time-averaged) values.
TAU0 = -7, \(\tau_0\) is defined using the rules for TAU0 = -3, and then is set equal to the weighted time-average of the current and previous (time-averaged) values as follows:
- \(\tau_0\)= AlphaTau0*TAU0VAR + (1.0-AlphaTau0)*LastTau0, where TAU0VAR is Tau0Base after spatial averaging, and AlphaTau0 = 0.25 is hard-coded into the model. This means that \(\tau_0\) is weighted 75% toward older values.

Spatial and Temporal Updating

For TAU0 = -3, -5, -6, or -7, \(\tau_0\) is updated (via the CalculateTimeVaryingTau0 subroutine) in space and in time. An initial “update” is done when the model starts. After this, updates are done only after a time step in which there is a change in wet/dry state somewhere in the model domain. For use cases that contain large number of nodes near the wet/dry boundary, this can be the equivalent of updating every time step. However, for use cases that have little or no wet/dry changes, there may be little or no updating. The rules listed above in TAU0 Values are applied during the update. Each node’s \(\tau_0\) is then spatially averaged with all immediate neighbors. Time-averaging (for TAU0 = -6 or -7) is applied last.

Selecting Values

TAU0 = -3, paired with the primitive weighting in continuity equation nodal attribute is generally the most popular formulation. In this case, TAU0Base nodal attribute values can be generated with the ADCIRC utility program tau0_gen.f. The program bases generation on the following logic applied to each node individually:

If the avg. dist. to neighboring nodes < 1750 m; then TAU0Base = 0.03
Otherwise
- If depth < 10m; then TAU0Base = 0.02 (TAU0 is constant in time)
- If depth > 10m; then TAU0Base = 0.005 (TAU0 is constant in time)

\(\tau_0\) needs to be smaller in deeper water where there is little dissipation, and can also be sensitive to mesh resolution. The spatial variation, spatial smoothing, and physics-driven time-updating typically allow for a good balance between stability and conservation.

For manually-specified positive values (TAU0=\(\tau_0\)), a good rule of thumb for setting TAU0 is to set it equal to the largest value of an equivalent linear friction factor: for linear friction TAU0 =`TAU <TAU>`__; for quadratic friction TAU0 = max(speed*Cd/depth). Typical values for TAU0 are in the range of 0.005 – 0.1.

Outputting

For TAU0 formulations that vary spatially or temporally, ADCIRC can output the internally-calculated nodal \(\tau_0\) values. They are written to the fort.90 file, which has the same format and output frequency as the water surface elevation output file (fort.63). fort.90 output is activated by placing a 1 in the tenths place of TAU0 in the fort.15 file. For example, if TAU0 = -3.1, the calculation of \(\tau_0\) is still carried out according to the description of TAU0 = -3 above, and the fort.90 output file will also be produced.