Methods for solving the transport equation

5.3.2.1 Horizontal transport solver

The horizontal transport terms are approximated with the Van Leer-2 scheme (Van Leer, 1974). The Van Leer-2 scheme is a combination of a first order upwind scheme and a second order upwind scheme. The first order upwind scheme is applied in the case of a local minimum or maximum. Fromm’s second order upwind scheme is used in case of a smooth numerical solution. The horizontal fluxes are calculated by (Deltares, 2014):

65 𝐹𝑚,𝑛,𝑘 = 𝑢𝑚,𝑛,𝑘ℎ𝑚,𝑛,𝑘∆𝑦 { 𝑐𝑚,𝑛,𝑘+ 𝛼(1 − 𝐶𝐹𝐿𝑎𝑑𝑣−𝑢)(𝑐𝑚,𝑛,𝑘− 𝑐𝑚−1,𝑛,𝑘) 𝑐𝑚+1,𝑛,𝑘− 𝑐𝑚,𝑛,𝑘 𝑐𝑚+1,𝑛,𝑘− 𝑐𝑚−1,𝑛,𝑘 when 𝑢_{𝑚,𝑛,𝑘} ≥ 0, 𝑐𝑚+1,𝑛,𝑘+ 𝛼(1 + 𝐶𝐹𝐿𝑎𝑑𝑣−𝑢)(𝑐𝑚,𝑛,𝑘− 𝑐𝑚−1,𝑛,𝑘) 𝑐_{𝑚+1,𝑛,𝑘}− 𝑐_{𝑚+2,𝑛,𝑘} 𝑐𝑚,𝑛,𝑘− 𝑐𝑚+2,𝑛,𝑘 when 𝑢_{𝑚,𝑛,𝑘} < 0, (5-7) with: 𝐶𝐹𝐿_{𝑎𝑑𝑣−𝑢} =∆𝑡|𝑢| ∆𝑥 (5-8) and: 𝛼 = { 0, |𝑐𝑚+1,𝑛,𝑘− 2𝑐𝑚,𝑛,𝑘+ 𝑐𝑚−1,𝑛,𝑘

𝑐𝑚+1,𝑛,𝑘 − 𝑐𝑚−1,𝑛,𝑘 | > 1, (local max.or min. ) ,

1, |𝑐𝑚+1,𝑛,𝑘− 2𝑐𝑚,𝑛,𝑘+ 𝑐𝑚−1,𝑛,𝑘

𝑐_{𝑚+1,𝑛,𝑘} − 𝑐_{𝑚−1,𝑛,𝑘} | ≤ 1, (monotone).

(5-9)

5.3.2.2 Vertical transport solver

Vertical fluxes in Delft3D are discretized with a central scheme where time integration in the vertical is fully implicit which leads to a tri-diagonal system in the vertical (Deltares, 2014): (𝑤𝑐)_{𝑚,𝑛,𝑘}− (𝑤𝑐)_{𝑚,𝑛,𝑘−1} = 𝑤_{𝑚,𝑛,𝑘}(𝑐𝑚,𝑛,𝑘+ 𝑐𝑚,𝑛,𝑘+1 2 ) − 𝑤𝑚,𝑛,𝑘−1( 𝑐𝑚,𝑛,𝑘+ 𝑐𝑚,𝑛,𝑘−1 2 ) (5-10) 5.3.2.3 Forester filter

Central differences in the horizontal and vertical directions may give rise to non- physical spurious oscillations in the solution resulting in negative concentrations. A filter

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may be applied to dampen these numerical wiggles. In Delft3D, if concentration 𝑐𝑚,𝑛,𝑘is

negative, then the iterative filtering process in the x-direction is given by (Deltares, 2014):

𝑐_{𝑚,𝑛,𝑘}𝑝+1 = 𝑐_{𝑚,𝑛,𝑘}𝑝 +𝑐𝑚+1,𝑛,𝑘

𝑝 _{+ 2𝑐}

𝑚,𝑛,𝑘𝑝 + 𝑐𝑚−1,𝑛,𝑘𝑝

4 (5-11)

where p denotes the iteration number. A maximum of 100 iterations are performed, and a warning is generated if there is still a grid cell with negative concentration after 100 iterations.

Similarly, a filter may be applied to smooth the vertical density profile to smooth out local maximums and minimums whereby a local maximum satisfies:

𝑐_{𝑚,𝑛,𝑘}> max(𝑐_{𝑚,𝑛,𝑘+1}, 𝑐_{𝑚,𝑛,𝑘−1}) + 0.001 (5-12)

Whereas a local minimum satisfies:

𝑐_{𝑚,𝑛,𝑘} < min(𝑐_{𝑚,𝑛,𝑘+1}, 𝑐_{𝑚,𝑛,𝑘−1}) + 0.001 (5-13)

For example if salinity 𝑠_{𝑚,𝑛,𝑘}> 𝑠_{𝑚,𝑛,𝑘−1}+ 0.001, then the vertical filter is applied such that:

𝑠_{𝑚,𝑛,𝑘} = 𝑠_{𝑚,𝑛,𝑘}− min (∆𝑧_𝑘, 𝑧_𝑘−1)(𝑠𝑚,𝑛,𝑘− 𝑠𝑚,𝑛,𝑘−1) 2∆𝑧𝑘

(5-14)

𝑠_{𝑚,𝑛,𝑘−1}= 𝑠_{𝑚,𝑛,𝑘−1}+ min (∆𝑧_𝑘, 𝑧_𝑘−1)(𝑠𝑚,𝑛,𝑘− 𝑠𝑚,𝑛,𝑘−1)

2∆𝑧_𝑘−1 (5-15)

This filtering process is only applied to the salinity and temperature constituents in Delft3D (Deltares, 2014).

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5.3.3 Turbulence closure

The Delft3D Z-level model code is limited to the use of the k-ε method for turbulence closure. The production, buoyancy, and dissipation terms are assumed to be the

dominating terms and therefore the conservation of the turbulent quantities is less important and the transport equation is implemented in a non-conservative form.

The eddy viscosity is based on information from the previous half time step and the eddy viscosity and turbulent transport quantities, k and ε are positioned at the layer interfaces in the center of the computational cell. In this way, the vertical gradients in the production term and buoyancy term are accurately discretized and the vertical boundary conditions at the bed and free surface may be implemented. Positive solutions are

provided by first order upwind differencing for advection (Uittenbogaard, van Kester, and Stelling, 1992).

5.3.3.1 The Ozmidov Length Scale

The k-ε model is the only turbulence closure model currently available for the Delft3D Z-level model, however, the k-ε turbulence model is incapable of reproducing the turbulence resulting from interfacial instabilities associated with strongly stratified flow. These instabilities are referred to as Kelvin-Helmholtz billows and Holmboe waves.

Therefore, in strongly stratified flows, the turbulent eddy viscosity at the interface reduces to zero and the vertical mixing reduces to molecular diffusion. In order to account for this shortcoming in the k-ε turbulence model, the minimal eddy diffusivity, DV, may be

based on the Ozmidov length scale, LOZ (Deltares, 2014):

𝐷_𝑉 = 𝑚𝑎𝑥 (𝐷_3𝐷, 0.2𝐿2_{𝑂𝑍√−}𝑔

𝜌 𝜕𝜌

𝜕𝑧) (5-16)

The Ozmidov scale represents the largest eddy size that can be supported by a given turbulent dissipation rate within a region of specified stratification:

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𝐿_𝑂𝑍 = (𝜀 𝑁_⁄ 3₎1₂ (5-17)

ε represents the rate of dissipation of turbulent kinetic energy. N is the Brunt–Väisälä buoyancy frequency or tendency of a fluid parcel to oscillate:

𝑁 = (−𝑔 𝜌₀ 𝜕𝜌 𝜕𝑧) 1 2 _(5-18)

The condition for stable stratification exists when 𝑁2 _{> 0 and unstable stratification exists}

when 𝑁2 _{< 0. The Ozmidov length scale parameter may be used as a calibration}

adjustment to control the position of the leading edge of the saltwater wedge in numerical simulations.