Nested circulation modelling of inter-tidal zones: details of a nesting approach incorporating moving boundaries

Abstract

Nested circulation models developed to date either exclude the flooding and drying process or prohibit flooding and drying of nested boundaries; they are therefore ill-suited to the accurate modelling of inter-tidal areas. The authors have developed a nested model with moving boundaries which permits flooding and drying of both the interior domain and the nested boundaries. The model uses a novel approach to boundary formulation; ghost cells are incorporated adjacent to the nested boundary cells so that the nested boundaries operate as internal boundaries. When combined with a tailored adaptive interpolation technique, the approach facilitates a dynamic internal boundary. Details of model development are presented with particular emphasis on the treatment of the nested boundary. Results are presented for Cork Harbour, a natural coastal system with an extensive inter-tidal zone and a complex flow regime which provided a rigorous test of model performance. The nested model was found to achieve the accuracy of a high resolution single grid model for a much lower computational cost.

Appendix A: spatial interpolation formulae

1.1 Zeroth-order interpolation scheme

If ϕ _i represents the value of a parent grid variable at cell i, then ϕ _k for all the child grid cells within its confines can be written as:

$$ {\phi_k} = {\phi_i}\quad k = 1,...,m $$

(9)

where m is the number of child grid cells within the parent grid cell, i.e. $m = r_s^2,{r_s}$ being the spatial nesting ratio.

1.2 Linear interpolation scheme

The variable ϕ in a child grid cell k which lies within the parent grid cell i may be calculated as:

$$ {\phi_k} = {\phi_{i - 1}} + \omega \left( {{\phi_i} - {\phi_{i - 1}}} \right) $$

(10)

with ω, the proportional interpolation coefficient, further expressed as:

$$ \omega = \frac{{2k + {r_s} - 1}}{{2{r_s}}},\quad k = 1,...,{r_s} $$

(11)

1.3 Quadratic interpolation scheme

For a given nesting ratio r _s , the interpolation of the parent grid cell variable ϕ _i to any child grid cell k of any horizontal row of enclosed child grid cells may be specified as (adapted from Alapaty et al. (1998)):

$$ {\phi_k} = E_{i - 1}^k{\phi_{i - 1}} + E_i^k{\phi_i} + E_{i + 1}^k{\phi_{i + 1}} $$

(12)

with the function E further expressed as:

$$ E_{i - 1}^k = \frac{{{\lambda_k}\left( {{\lambda_k} - 1} \right)}}{2} + \alpha $$

(13)

$$ E_i^k = \left( {1 - \lambda_k^2} \right) - 2\alpha $$

(14)

$$ E_{i + 1}^k = \frac{{{\lambda_k}\left( {{\lambda_k} + 1} \right)}}{2} + \alpha $$

(15)

where λ represents a normalised local coordinate pointing in the same direction as the global coordinate and whose origin coincides with the centre of the parent grid cell i. The value of λ for the child grid cell k is defined as:

$$ {\lambda_k} = \frac{{\left( {2k - 1} \right)\Delta {x_c} - \Delta {x_p}}}{{2\Delta {x_p}}},\quad k = 1,...,{r_s} $$

(16)

and

$$ \alpha = \frac{1}{{24}}\left[ {{{\left( {\frac{{\Delta {x_c}}}{{\Delta {x_p}}}} \right)}^2} - 1} \right] $$

(17)

Δx _p and Δx _c are the parent and child grid spacings, respectively. The parameter α is introduced to ensure mass conservation following Clark and Farley (1984).

1.4 Inverse distance weighted interpolation scheme

The variable ϕ in a child grid cell k which lies within an enclosing parent grid cell is calculated as:

$$ {\phi_k} = \sum\limits_{i = 1}^n {{w_i}{\phi_i}} /\sum\limits_{i = 1}^n {{w_i}} \quad k = 1,...,m $$

(18)

where n is the number of parent grid cells used in the interpolation and w _i is the weighting function. For the nested model n = 9, i.e. the enclosing grid cell and the eight adjacent grid cells were used. The weighting function is written as:

$$ {w_i} = \frac{1}{{d_{\left( {k,i} \right)}^2}} $$

(19)

where d _(k,i) is the distance from the child grid point k to the parent grid point i. The weighting function varies from a value of unity at the child grid point of interest to a value approaching zero as the distance from the grid point increases.