## Scale considerations in grids: A start from Bob Beardsley

Greenberg et al. (2007) reviewed the need for adequate horizontal resolution regarding both tidal and subtidal current simulations.

The key considerations follow.

### Tidal motion

Δx ≤ (T√gH)/n

where Δx is the grid size; T is the tidal period; g is the gravity; H is the mean water depth; and n is the number of nodes n is the number of nodes per wavelength. Based on Le Provost et al. (1995)'s analysis, n = 30.

### Subtidal flow

For subtidal flow, the dynamics over the slope are controlled by a cross-shelf motion scale defined as ∇H/H (Hannah and Wright, 1995; Greenberg et al., 2007). On an assumption of steady state and no along-isobath variation, Greenberg et al. (2007) derived this scale through the continuity equation, which was given as

∂U/∂x ~ -1/H dH/dx U = -αU

For non-dimensional scaling, it yielded

U/Lx ~ -αU ⟹ Lx ~ 1/α=H/∇H

where Lx is the cross-shelf motion scale and α=1/HdH/dx. Using a numerical model to produce an analytically-derived barotropic slope current, Hannah and Wright (1995) concluded that in order to reasonably reproduce this flow, the model resolution (Δx) must satisfy the criterion given as

Δx ≤ 0.33 H/∇H = 0.33Lx

which is one-third of the cross-shelf motion scale.

Others to add are external and internal Rossby radius of deformation, probably more.